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LOWENERGY LUNAR
TRAJECTORY DESIGN
Jeffrey S. Parker and Rodney L. Anderson
Jet Propulsion Laboratory
Pasadena, California
July 2013
ii
DEEP SPACE COMMUNICATIONS AND NAVIGATION SERIES
Issued by the Deep Space Communications and Navigation Systems
Center of Excellence
Jet Propulsion Laboratory
California Institute of Technology
Joseph H. Yuen, EditorinChief
Published Titles in this Series
Radiometric Tracking Techniques for DeepSpace Navigation
Catherine L. Thornton and James S. Border
Formulation for Observed and Computed Values of Deep Space Network Data
Types for Navigation
Theodore D. Moyer
BandwidthEfﬁcient Digital Modulation with Application to DeepSpace
Communication
Marvin K. Simon
Large Antennas of the Deep Space Network
William A. Imbriale
Antenna Arraying Techniques in the Deep Space Network
David H. Rogstad, Alexander Mileant, and Timothy T. Pham
Radio Occultations Using Earth Satellites: A Wave Theory Treatment
William G. Melbourne
Deep Space Optical Communications
Hamid Hemmati, Editor
Spaceborne Antennas for Planetary Exploration
William A. Imbriale, Editor
Autonomous SoftwareDeﬁned Radio Receivers for Deep Space Applications
Jon Hamkins and Marvin K. Simon, Editors
LowNoise Systems in the Deep Space Network
Macgregor S. Reid, Editor
CoupledOscillator Based ActiveArray Antennas
Ronald J. Pogorzelski and Apostolos Georgiadis
LowEnergy Lunar Trajectory Design
Jeffrey S. Parker and Rodney L. Anderson
LOWENERGY LUNAR
TRAJECTORY DESIGN
Jeffrey S. Parker and Rodney L. Anderson
Jet Propulsion Laboratory
Pasadena, California
July 2013
iv
LowEnergy Lunar Trajectory Design
July 2013
Jeffrey Parker:
I dedicate the majority
of this book to my wife
Jen, my best friend and
greatest support
throughout the
development of this book
and always. I dedicate
the appendix to my son
Cameron, who showed up
right at the end.
Rodney Anderson:
I dedicate this book to
my wife Brooke for her
endless support and
encouragement.
We both thank our
families and friends for
their support throughout
the process.
CONTENTS
Foreword
Preface
Acknowledgments
Authors
xi
xiii
xv
xxi
1
Introduction and Executive Summary
1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Direct, Conventional Transfers . . . . . . . . . . . . . . .
1.3.2 LowEnergy Transfers . . . . . . . . . . . . . . . . . . . .
1.3.3 Summary: LowEnergy Transfers to Lunar Libration Orbits
1.3.4 Summary: LowEnergy Transfers to Low Lunar Orbits . .
1.3.5 Summary: LowEnergy Transfers to the Lunar Surface . . .
1.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The Lunar Transfer Problem . . . . . . . . . . . . . . . . . . . . .
1.6 Historical Missions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Missions Implementing Direct Lunar Transfers . . . . . . .
1.6.2 LowEnergy Missions to the Sun–Earth Lagrange Points . .
1.6.3 Missions Implementing LowEnergy Lunar Transfers . . .
1.7 LowEnergy Lunar Transfers . . . . . . . . . . . . . . . . . . . . .
1
1
1
2
5
6
7
8
10
11
12
14
15
15
20
23
2
Methodology
2.1 Methodology Introduction . . . . . . . .
2.2 Physical Data . . . . . . . . . . . . . . .
2.3 Time Systems . . . . . . . . . . . . . . .
2.3.1 Dynamical Time, ET . . . . . . .
2.3.2 International Atomic Time, TAI .
2.3.3 Universal Time, UT . . . . . . .
2.3.4 Coordinated Universal Time, UTC
2.3.5 Lunar Time . . . . . . . . . . . .
2.3.6 Local True Solar Time, LTST . .
2.3.7 Orbit Local Solar Time, OLST .
2.4 Coordinate Frames . . . . . . . . . . . .
2.4.1 EME2000 . . . . . . . . . . . .
2.4.2 EMO2000 . . . . . . . . . . . .
27
27
28
29
29
29
30
30
30
31
31
32
32
33
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33
33
34
35
36
39
40
41
42
43
49
50
52
67
74
77
81
86
95
106
113
114
114
114
115
Transfers to Lunar Libration Orbits
3.1 Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Direct Transfers Between Earth and Lunar Libration Orbits . . . .
3.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 The PerigeePoint Scenario . . . . . . . . . . . . . . . . .
3.3.3 The OpenPoint Scenario . . . . . . . . . . . . . . . . . .
3.3.4 Surveying Direct Lunar Halo Orbit Transfers . . . . . . . .
3.3.5 Discussion of Results . . . . . . . . . . . . . . . . . . . .
3.3.6 Reducing the ΔV Cost . . . . . . . . . . . . . . . . . . . .
3.3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 LowEnergy Transfers Between Earth and Lunar Libration Orbits .
3.4.1 Modeling a LowEnergy Transfer using Dynamical Systems
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Energy Analysis of a LowEnergy Transfer . . . . . . . . .
3.4.3 Constructing a LowEnergy Transfer in the Patched
ThreeBody Model . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Constructing a LowEnergy Transfer in the Ephemeris
Model of the Solar System . . . . . . . . . . . . . . . . . .
117
117
120
122
122
125
127
130
152
157
158
161
2.5
2.6
2.7
3
viii
2.4.3
Principal Axis Frame . . . . . . . . . . . . .
2.4.4
IAU Frames . . . . . . . . . . . . . . . . . .
2.4.5
Synodic Frames . . . . . . . . . . . . . . . .
Models
. . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
CRTBP . . . . . . . . . . . . . . . . . . . . .
2.5.2
Patched ThreeBody Model . . . . . . . . . .
2.5.3
JPL Ephemeris . . . . . . . . . . . . . . . . .
LowEnergy Mission Design . . . . . . . . . . . . . .
2.6.1 Dynamical Systems Theory . . . . . . . . . .
2.6.2 Solutions to the CRTBP . . . . . . . . . . . .
2.6.3 Poincare´ Maps . . . . . . . . . . . . . . . . .
2.6.4 The State Transition and Monodromy Matrices
2.6.5 Differential Correction . . . . . . . . . . . . .
2.6.6 Constructing Periodic Orbits . . . . . . . . .
2.6.7 The Continuation Method . . . . . . . . . . .
2.6.8 Orbit Stability . . . . . . . . . . . . . . . . .
2.6.9 Examples of Practical ThreeBody Orbits . . .
2.6.10 Invariant Manifolds . . . . . . . . . . . . . .
2.6.11 Orbit Transfers . . . . . . . . . . . . . . . . .
2.6.12 Building Complex Orbit Chains . . . . . . . .
2.6.13 Discussion . . . . . . . . . . . . . . . . . . .
Tools . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Numerical Integrators . . . . . . . . . . . . .
2.7.2 Optimizers . . . . . . . . . . . . . . . . . . .
2.7.3 Software . . . . . . . . . . . . . . . . . . . .
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163
169
177
183
CONTENTS
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187
190
208
221
224
Transfers to Low Lunar Orbits
4.1 Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Direct Transfers Between Earth and Low Lunar Orbit . . . . . . . .
4.4 LowEnergy Transfers Between Earth and Low Lunar Orbit . . . .
4.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Example Survey . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Arriving at a FirstQuarter Moon . . . . . . . . . . . . . .
4.4.4 Arriving at a ThirdQuarter Moon . . . . . . . . . . . . . .
4.4.5 Arriving at a Full Moon . . . . . . . . . . . . . . . . . . .
4.4.6 Monthly Trends . . . . . . . . . . . . . . . . . . . . . . .
4.4.7 Practical Considerations . . . . . . . . . . . . . . . . . . .
4.4.8 Conclusions for LowEnergy Transfers Between Earth and
Low Lunar Orbit . . . . . . . . . . . . . . . . . . . . . . .
4.5 Transfers Between Lunar Libration Orbits and Low Lunar Orbits .
4.6 Transfers Between Low Lunar Orbits and the Lunar Surface . . . .
227
227
229
231
233
233
235
239
246
250
253
257
3.5
4
5
6
3.4.5 Families of LowEnergy Transfers . . . . . . . . . . .
3.4.6 Monthly Variations in LowEnergy Transfers . . . . . .
3.4.7 Transfers to Other ThreeBody Orbits . . . . . . . . . .
ThreeBody Orbit Transfers . . . . . . . . . . . . . . . . . . .
3.5.1 Transfers from an LL2 Halo Orbit to a Low Lunar Orbit
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.
ix
258
258
258
Transfers to the Lunar Surface
5.1 Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Introduction for Transfers to the Lunar Surface . . . . . . . . . . .
5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Analysis of Planar transfers between the Earth and the Lunar Surface
5.5 LowEnergy Spatial Transfers Between the Earth and the Lunar
Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Trajectories Normal to the Surface . . . . . . . . . . . . .
5.5.2 Trajectories Arriving at Various Angles to the Lunar Surface
5.6 Transfers Between Lunar Libration Orbits and the Lunar Surface . .
5.7 Transfers Between Low Lunar Orbits and the Lunar Surface . . . .
5.8 Conclusions Regarding Transfers to the Lunar Surface . . . . . . .
277
277
287
294
298
298
Operations
6.1 Operations Executive Summary . . .
6.2 Operations Introduction . . . . . . .
6.3 Launch Sites . . . . . . . . . . . . .
6.4 Launch Vehicles . . . . . . . . . . .
6.5 Designing a Launch Period . . . . . .
6.5.1 LowEnergy Launch Periods
6.5.2 An Example Mission Scenario
6.5.3 Targeting Algorithm . . . . .
299
299
300
301
301
304
305
307
311
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263
263
265
267
268
x
CONTENTS
6.6
6.7
6.5.4 Building a Launch Period . . . . . . . . . . . . . . . . .
6.5.5 Reference Transfers . . . . . . . . . . . . . . . . . . . .
6.5.6 Statistical Costs of Desirable Missions to Low Lunar Orbit
6.5.7 Varying the LEO Inclination . . . . . . . . . . . . . . . .
6.5.8 Targeting a Realistic Mission to Other Destinations . . .
6.5.9 Launch Period Design Summary . . . . . . . . . . . . .
Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 Launch Targets . . . . . . . . . . . . . . . . . . . . . . .
6.6.2 StationKeeping . . . . . . . . . . . . . . . . . . . . . .
Spacecraft Systems Design . . . . . . . . . . . . . . . . . . . . .
Appendix A: Locating the Lagrange Points
A.1 Introduction . . . . . . . . . . . . . . . .
A.2 Setting Up the System . . . . . . . . . .
A.3 Triangular Points . . . . . . . . . . . . .
A.4 Collinear Points . . . . . . . . . . . . .
A.4.1 Case 132: Identifying the L1 point
A.4.2 Case 123: Identifying the L2 point
A.4.3 Case 312: Identifying the L3 point
A.5 Algorithms . . . . . . . . . . . . . . . .
A.5.1 Numerical Determination of L1 .
A.5.2 Numerical Determination of L2 .
A.5.3 Numerical Determination of L3 .
References
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Terms
377
Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
FOREWORD
The Deep Space Communications and Navigation Systems Center of Excellence
(DESCANSO) was established in 1998 by the National Aeronautics and Space
Administration (NASA) at the California Institute of Technology’s Jet Propulsion
Laboratory (JPL). DESCANSO is chartered to harness and promote excellence and
innovation to meet the communications and navigation needs of future deepspace
exploration.
DESCANSO’s vision is to achieve continuous communications and precise navi
gation—any time, anywhere. In support of that vision, DESCANSO aims to seek out
and advocate new concepts, systems, and technologies; foster key technical talents;
and sponsor seminars, workshops, and symposia to facilitate interaction and idea
exchange.
The Deep Space Communications and Navigation Series, authored by scientists
and engineers with many years of experience in their respective ﬁelds, lays a foun
dation for innovation by communicating stateoftheart knowledge in key technolo
gies. The series also captures fundamental principles and practices developed during
decades of deepspace exploration at JPL. In addition, it celebrates successes and
imparts lessons learned. Finally, the series will serve to guide a new generation of
scientists and engineers.
Joseph H. Yuen, DESCANSO Leader
xi
PREFACE
The purpose of this book is to provide highlevel information to mission managers
and detailed information to mission designers about lowenergy transfers between
the Earth and the Moon. This book surveys thousands of trajectories that one can
use to transfer spacecraft between the Earth and various locations near the Moon,
including lunar libration orbits, low lunar orbits, and the lunar surface. These surveys
include conventional, direct transfers that require 3–6 days as well as more efﬁcient,
lowenergy transfers that require more transfer time but which require less fuel.
Lowenergy transfers have been shown to be very useful in many circumstances and
have recently been used to send satellites to the Moon, including the two ARTEMIS
spacecraft and the two GRAIL spacecraft. This book illuminates the trade space of
lowenergy transfers and illustrates the techniques that may be used to build them.
xiii
ACKNOWLEDGMENTS
We would like to thank many people for their support writing this book, including
people who have written or reviewed portions of the text, as well as people who have
provided insight from years of experience ﬂying spacecraft missions to the Moon
and elsewhere. It is with sincere gratitude that we thank Ted Sweetser for his selﬂess
efforts throughout this process, providing the opportunity for us to perform this work,
and reviewing each section of this manuscript as it has come together. We would like
to thank Al Cangahuala, Joe Guinn, Roby Wilson, and Amy Attiyah for their valuable
feedback and thorough review of this work in each of its stages. We would also like
to thank Tim McElrath for his feedback, insight, and excitement as we considered
different aspects of this research.
We would like to give special thanks to several people who provided particular
contributions to sections of the book. We thank Ralph Roncoli for his assistance
with Sections 2.3 and 2.4, as well as his feedback throughout the book. Kate
Davis assisted with Sections 2.6.3 and 2.6.11.3, most notably with the discussions of
Poincar´e sections. Roby Wilson provided particular assistance with Section 2.6.5 on
the subject of the multiple shooting differential corrector. We would like to sincerely
thank Andrew Peterson for his contribution to the development of Chapter 4. Finally,
George Born and Martin Lo provided guidance for this research as it developed in
xv
xvi
its early stages, leading to the authors’ dissertations at the University of Colorado at
Boulder.
Jeffrey Parker’s Ph.D. dissertation (J. S. Parker, LowEnergy Ballistic Lunar Trans
fers, Ph.D. Thesis, University of Colorado, Boulder, 2007) provides the backbone
to this manuscript and much of the dissertation has been repeated and ampliﬁed in
this book. Much of the additional material that appears in this manuscript has been
presented by the authors at conferences and published in journals. Such material
has been reprinted here, with some signiﬁcant alterations and additions. Finally, a
number of additional journal articles and conference proceedings directly contributed
to each chapter in the following list. In addition to their listing here, they are cited in
text where the related material appears.
Chapter 2:
• J. S. Parker, K. E. Davis, and G. H. Born, “Chaining Periodic ThreeBody
Orbits in the Earth–Moon System,” ACTA Astronautica, vol. 67, pp. 623–638,
2010.
• M. W. Lo, and J. S. Parker, “Chaining Simple Periodic ThreeBody Orbits,”
AAS/AIAA Astrodynamics Specialist Conference (Lake Tahoe, California), Pa
per No. AAS 2005380, August 7–11, 2005, vol. 123, Advances in Astro
nautical Sciences (B. G. Williams, L. A. D’Amario, K. C. Howell, and F. R.
Hoots, editors), AAS/AIAA, Univelt Inc., San Diego, CA, 2006.
• R. B. Roncoli, Lunar Constants and Models Document, JPL D32296 (inter
nal document), Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, California, September 23, 2005.
• R. L. Anderson and J. S. Parker, “Survey of Ballistic Transfers to the Lunar
Surface,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 4,
pp. 1256–1267, July–August 2012.
Chapter 3:
• J. S. Parker, “Monthly Variations of LowEnergy Ballistic Transfers to Lu
nar Halo Orbits,” AIAA/AAS Astrodynamics Specialist Conference, (Toronto,
Ontario, Canada), Paper No. AIAA 20107963, August 2–5, 2010.
• J. S. Parker, “Targeting LowEnergy Ballistic Lunar Transfers,” AAS George H.
Born Special Symposium, (Boulder, Colorado), May 13–14, 2010, American
Astronautical Society, 2010.
• J. S. Parker, “Targeting LowEnergy Ballistic Lunar Transfers,” Journal of
Astronautical Sciences, vol. 58, no. 3, pp. 311–334, July–September, 2011.
xvii
• J. S. Parker, “LowEnergy Ballistic Transfers to Lunar Halo Orbits,” AAS/AIAA
Astrodynamics Specialist Conference, (Pittsburgh, Pennsylvania, Paper No.
AAS 09443, August 9–13, 2009, Advances in Astronautical Sciences, Astrodynamics
2009 (A. V. Rao, A. Lovell, F. K. Chan, and L. A. Cangahuala, editors), vol.
135, pp. 2339–2358, 2010.
• J. S. Parker, and G. H. Born, “Modeling a LowEnergy Ballistic Lunar Transfer
Using Dynamical Systems Theory,” AIAA Journal of Spacecraft and Rockets,
vol. 45, no. 6, pp.1269–1281, November–December 2008.
• J. S. Parker and G. H. Born, “Direct Lunar Halo Orbit Transfers,” Journal of
the Astronautical Sciences, vol. 56, issue 4, pp. 441–476, October–December
2008.
• J. S. Parker and G. H. Born, “Direct Lunar Halo Orbit Transfers,” AAS/AIAA
Spaceﬂight Mechanics Conference (Sedona, Arizona, January 28–February 1,
2007), Paper No. AAS 07229, Advances in Astronautical Science, vol. 127,
pp. 1923–1945, 2007.
• J. S. Parker, “Families of LowEnergy Lunar Halo Transfers,” AAS/AIAA Space
ﬂight Dynamics Conference, (Tampa, Florida, January 22–26, 2006) Paper No.
AAS 06132, (S. R. Vadali, L. A. Cangahuala, J. P. W. Schumacher, and J. J.
Guzman, editors), vol. 124 of Advances in Astronautical Sciences, San Diego,
CA, AAS/AIAA, Univelt Inc., 2006.
• J. S. Parker and M. W. Lo, “Shoot the Moon 3D,” Paper AAS 05383, AAS/AIAA
Astrodynamics Conference held August 7–10, 2005, South Lake Tahoe, Cali
fornia, (originally published in) AAS publication, Astrodynamics 2005 (edited
by B. G. Williams, L. A. D’Amario, K. C. Howell, and F. R. Hoots) American
Astronautical Society (AAS) Advances in the Astronautical Sciences, vol. 123,
pp. 2067–2086, 2006, American Astronautical Society Publications Ofﬁce,
San Diego, California (Web Site: http://www.univelt.com), pp. 2067–2086.
Chapter 4:
• J. S. Parker and R. L. Anderson, “Targeting LowEnergy Transfers to Low Lu
nar Orbit,” Astrodynamics: Proceedings of the 2011 AAS/AIAA Astrodynamics
Specialist Conference, (Girdwood, Alaska, July 31–August 4), Paper AAS
11459, edited by H. Schaub, B. C. Gunter, R. P. Russell, and W. T. Cerven,
Vol. 142, Advances in the Astronautical Sciences, American Astronautical
Society, Univelt Inc., San Diego, California, pp. 847–866, 2012.
• J. S. Parker, R. L. Anderson, and A. Peterson, “A Survey of Ballistic Transfers to
Low Lunar Orbit,” 21st AAS/AIAA Space Flight Mechanics Meeting, (February
13–17, 2011, New Orleans, Louisiana), Paper AAS 11277, Vol. 140, Advances
in the Astronautical Sciences (edited by M. K. Jah, Y. Guo, A. L. Bowes, and
xviii
P. C. Lai), American Astronautical Society, Univelt Inc., San Diego, California,
pp. 2461–2480, 2011.
Chapter 5:
• R. L. Anderson, and J. S. Parker, “Survey of Ballistic Transfers to the Lunar
Surface,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 4,
pp. 1256–1267, July–August 2012.
• R. L. Anderson and J. S. Parker, “Comparison of LowEnergy Lunar Transfer
Trajectories to Invariant Manifolds,” Celestial Mechanics and Dynamical As
tronomy, vol. 115, DOI 10.10075 1056901294663, pp. 311–331, published
online February 16, 2013.
• R. L. Anderson, and J. S. Parker, “Comparison of LowEnergy Lunar Trans
fer Trajectories to Invariant Manifolds,” AAS/AIAA Astrodynamics Specialist
Conference (Girdwood, Alaska, July 31–August 4, 2011), Paper AAS 11423,
edited by H. Schaub, B. C. Gunter, R. P. Russell, and W. T. Cerven, Vol.
142, Advances in the Astronautical Sciences, American Astronautical Society,
Univelt Inc., San Diego, California, pp. 333–352, 2012.
• R. L. Anderson, and J. S. Parker, “A Survey of Ballistic Transfers to the Lunar
Surface,” Proceedings of the 21st AAS/AIAA Space Flight Mechanics Meeting
(New Orleans, Louisiana, February 13–17, 2011), Paper AAS 11278, edited
by M. K. Jah, Y. Guo, A. L. Bowes, and P. C. Lai, Vol. 140, Advances in
the Astronautical Sciences, vol. 140, American Astronautical Society, Univelt
Inc., San Diego, California, pp. 2481–2500, 2011.
Chapter 6:
• J. S. Parker, “Targeting LowEnergy Ballistic Lunar Transfers,” Journal of
Astronautical Sciences, vol. 58, no. 3, pp. 311–334, July–September, 2011.
• J. S. Parker and R. L. Anderson, “Targeting LowEnergy Transfers to Low
Lunar Orbit,” Astrodynamics 2011: Proceedings of the AAS/AIAA Astrody
namics Specialist Conference (Girdwood, Alaska, July 31–August 4, 2011),
Paper AAS 11459, edited by H. Schaub, B. C. Gunter, R. P. Russell, and
W. T. Cerven, Vol. 142, Advances in the Astronautical Sciences, American
Astronautical Society, Univelt Inc., San Diego, California, pp. 847–866, 2012.
• J. S. Parker, “Targeting LowEnergy Ballistic Lunar Transfers,” AAS 09443,
AAS George H. Born Special Symposium (Boulder, Colorado, May 13–14),
American Astronautical Society, 2010.
A large portion of the research in this book, and all of the compiling of related
research documentation from other sources, were carried out at the Jet Propulsion
xix
Laboratory, California Institute of Technology, under a contract with the National
Aeronautics and Space Administration. This work has been supported through
funding by the Multimission Ground System and Services Ofﬁce (MGSS) in support
of the development of the Advanced MultiMission Operations System (AMMOS).
Reference herein to any speciﬁc commercial product, process, or service by trade
name, trademark, manufacturer, or otherwise, does not constitute or imply its endorse
ment by the United States Government or the Jet Propulsion Laboratory, California
Institute of Technology.
Jeffrey S. Parker & Rodney L. Anderson
AUTHORS
Jeffrey S. Parker received his B.A. in 2001 in physics and astronomy from Whitman
College (Walla Walla, Washington) and his M.S. and Ph.D. in aerospace engineering
sciences from the University of Colorado at Boulder in 2003 and 2007, respectively.
Dr. Parker was a member of the technical staff at the Jet Propulsion Laboratory
(JPL) from January 2008 to June 2012. While at JPL he supported spacecraft
exploration as a mission design and navigation specialist. He worked both as a
spacecraft mission designer and as a navigator on the GRAIL mission, which sent two
spacecraft to the Moon via lowenergy ballistic lunar transfers. He supported India’s
Chandrayaan1 mission to the Moon, also as a mission designer and spacecraft
navigator. Dr. Parker led the mission design development for numerous design
studies and mission proposals, including missions to the Moon, nearEarth objects,
the nearby Lagrange points, and most of the planets in the Solar System. At present,
Dr. Parker is an assistant professor of astrodynamics at the University of Colorado
at Boulder, teaching graduate and undergraduate courses in many subjects related
to space exploration. His research interests are focused on astrodynamics and the
exploration of space, including the design of lowenergy trajectories in the Solar
System, the optimization of lowthrust trajectories in the Solar System, autonomous
spacecraft operations, and use of these engineering tools to provide new ways to
achieve scientiﬁc objectives.
xxi
xxii
Rodney L. Anderson received his B.S. in 1997 in aerospace engineering from
North Carolina State University at Raleigh and his M.S. and Ph.D. in aerospace
engineering sciences from the University of Colorado at Boulder in 2001 and 2005,
respectively. Upon the completion of his Ph.D., he worked as a research associate
at the University of Colorado at Boulder conducting a study for the U.S. Air Force
that focused on understanding the effects of atmospheric density variations on orbit
predictions. Dr. Anderson has been a member of the JPL technical staff since 2010
where he has participated in mission design and navigation for multiple missions and
continues to work on the development of new methods for trajectory design. His
research interests are concentrated on the application of dynamical systems theory to
astrodynamics and mission design. Some speciﬁc applications that he has focused
on are the design of lunar trajectories, tour and endgame design in the Jovian system
using heteroclinic connections, missions to nearEarth asteroids, and lowenergy
trajectories in multibody systems. He has worked closely with multiple universities
and has taught at both the University of Colorado at Boulder and the University
of Southern California with an emphasis on the intersection of dynamical systems
theory with astrodynamics.
CHAPTER 1
INTRODUCTION AND EXECUTIVE
SUMMARY
1.1
PURPOSE
This book provides sufﬁcient information to answer highlevel questions about the
availability and performance of lowenergy transfers between the Earth and Moon
in any given month and year. Details are provided to assist in the construction of
desirable lowenergy transfers to various destinations on the Moon, including low
lunar orbits, halo and other threebody orbits, and the lunar surface. Much of the
book is devoted to surveys that characterize many examples of transfers to each of
these destinations.
1.2
ORGANIZATION
This document is organized in the following manner. The remainder of this chapter
ﬁrst provides an executive summary of this book, presenting an overview of lowenergy lunar transfers and comparing them with various other modes of transportation
from near the Earth to lunar orbit or the lunar surface. It then provides background
information, placing lowenergy lunar transfers within the context of historical lunar
1
2
INTRODUCTION AND EXECUTIVE SUMMARY
missions. The chapter describes very highlevel costs and beneﬁts of lowenergy
transfers compared with conventional transfers.
Chapter 2 provides information about the methods, coordinate frames, models,
and tools used to design lowenergy lunar transfers. This information should be
sufﬁcient for designers to reconstruct any transfer presented in this book, as well as
similar transfers with particular design parameters.
Chapter 3 presents information about transfers from the Earth to highaltitude
threebody orbits, focusing on halo orbits about the ﬁrst and second Earth–Moon
Lagrange points. The chapter includes surveys of the transfer types that exist and
discussions about how to construct a particular, desirable transfer.
Chapter 4 presents information about transfers from the Earth to lowaltitude
lunar orbits, focusing on polar mapping orbits. The techniques presented may be
used to survey and construct conventional direct lunar transfers as well as lowenergy
transfers.
Chapter 5 presents information about transfers from the Earth to the lunar surface,
including discussions and surveys of transfers that intersect the lunar surface at a
steep 90 degree (deg) angle, as well as transfers that target a shallow ﬂight path angle.
The techniques illustrated in Chapter 5 may be used to generate conventional direct
transfers as well as lowenergy transfers.
Chapters 3–5 also include discussions about the variations of these transfers from
one month to the next. The discussions are useful for mission designers and managers
to predict what sorts of transfers exist in nearly any month and what sorts of transfers
are particular to speciﬁc months.
Chapter 6 discusses several important operational aspects of implementing a lowenergy lunar transfer. The section begins with a discussion of the capabilities of
current launch vehicles to inject spacecraft onto lowenergy trajectories. The section
then describes how to design a robust launch period for a lowenergy lunar trans
fer. Additional discussions are provided to address navigation, stationkeeping, and
spacecraft systems issues.
1.3
EXECUTIVE SUMMARY
This book characterizes lowenergy transfers between the Earth and the Moon as
a resource to mission managers and trajectory designers. This book surveys and
illustrates transfers between the Earth and lunar libration orbits, low lunar mapping
orbits, and the lunar surface, including transfers to the Moon and from the Moon to
the Earth.
There are many ways of transporting a spacecraft between the Earth and the
Moon, including fast conventional transfers, spiraling lowthrust transfers, and lowenergy transfers. Table 11 summarizes several of these methods and a sample of the
missions that have ﬂown these transfers.
The vast majority of lunar missions to date have taken quick, 3–6 day direct
transfers from the Earth to the Moon. The Apollo missions took advantage of
3–3.5 day transfers: transfers that were as quick as possible without dramatically
EXECUTIVE SUMMARY
3
Table 11 A summary of several different methods used to transfer between the Earth
and the Moon.
Transfer Type
Typical Duration
Beneﬁts
Example Missionsa
Direct, conventional
3–6 days
Well known, quick
Apollo, LRO, others
Direct, staging
2–10 weeks
Quick, many launch days
Clementine, CH1
Direct to lunar L1
1–5 weeks
Staging at L1
None to date
Lowthrust
Many months
Low fuel, many launch days
SMART1
Lowenergy
2.5–4 months
Low fuel, many launch days
Hiten, GRAIL, ARTEMIS
a
Missions referred to include Lunar Reconnaissance Orbiter (LRO), Chandrayaan1 (CH1),
Small Missions for Research in Technology 1 (SMART1), and Mu Space Engineering Space
craft (MUSES 1, Hiten)
increasing the transfers’ fuel requirements. The Lunar Reconnaissance Orbiter
(LRO) followed a slightly more efﬁcient 4.5day transfer. The additional transfer
duration saved fuel and relaxed the operational timeline of the mission. The Apollo
missions and LRO had very limited launch opportunities: they had to launch within
a short window each month. Clementine and Chandrayaan1 implemented phasing
orbits about the Earth to alleviate this design constraint and expand their launch
periods. SMART1 was also able to establish a wider launch period using lowthrust propulsion. The lowthrust system requires less fuel mass than conventional
propulsion systems, but the transfer required signiﬁcantly more transfer time than
any typical ballistic transfer.
The Gravity Recovery and Interior Laboratory (GRAIL) mission was the ﬁrst
mission launched to the Moon directly on a lowenergy transfer. GRAIL’s lowenergy
transfer required much less fuel than a conventional transfer, though it required a
longer cruise that traveled farther from the Earth. The longer cruise (∼90–114 days)
made it possible to establish a wide, 3plus week long launch period and signiﬁcantly
relaxed the operational timeline. Furthermore, GRAIL launched two satellites on
board a single launch vehicle and leveraged the longer cruise to separate their orbit
insertion dates by more than a day. Finally, GRAIL’s lowenergy transfer reduced the
orbit insertion change in velocity (ΔV) for each vehicle, permitting each spacecraft
to perform its lunar orbit insertion with a smaller engine and less fuel.
In general, a lowenergy transfer is a nearly ballistic transfer between the Earth
and the Moon that takes advantage of the Sun’s gravity to reduce the spacecraft’s
fuel requirements. The only maneuvers required are typical statistical maneuvers
needed to clean up launch vehicle injection errors and small deterministic maneuvers
to target speciﬁc mission features. A spacecraft launched on a lowenergy lunar
transfer travels beyond the orbit of the Moon, far enough from the Earth and Moon
to permit the gravity of the Sun to signiﬁcantly raise the spacecraft’s energy. The
spacecraft remains beyond the Moon’s orbit for 2–4 months while its perigee radius
rises. The spacecraft’s perigee radius typically rises as high as the Moon’s orbit,
permitting the spacecraft to encounter the Moon on a nearly tangential trajectory.
This trajectory has a very low velocity relative to the Moon: in some cases the
4
INTRODUCTION AND EXECUTIVE SUMMARY
spacecraft’s twobody energy will even be negative as it approaches the Moon,
without having performed any maneuver whatsoever. As the spacecraft approaches
the Moon, it may target a trajectory to land on the Moon, to enter a low lunar orbit,
or to enter any number of threebody orbit types, such as halo or Lissajous orbits. No
matter what its destination, the spacecraft requires less fuel to reach it than it would
following a conventional transfer.
Lowenergy transfers provide many beneﬁts to missions when compared with
conventional transfers. Six example beneﬁts include the following:
1. They require less fuel. A lowenergy transfer to a lunarlibration orbit saves
400 meters per second (m/s) of ΔV and often more. This is a signiﬁcant
savings, which is fully demonstrated in Chapter 3. A lowenergy transfer to a
100kilometer (km) lunar orbit saves more than 120 m/s of ΔV for cases when
a mission can use an optimized conventional transfer. The savings are far more
dramatic for missions that cannot use an optimized conventional transfer.
2. Lowenergy transfers are more ﬂexible than conventional transfers and may be
used to transfer spacecraft to many more orbits on a given date. It is shown
in Chapter 4 that lowenergy transfers may be used to reach polar orbits with
any node at any arrival date—conventional transfers may only target speciﬁc
nodes at any given date.
3. Lowenergy transfers have extended launch periods. It requires very little fuel
to establish a launch period of 21 days or more for a mission to the Moon
that implements a lowenergy transfer. Conventional transfers may be able to
accomplish similar launch periods, but they require multiple passes through
the Van Allen Belts, necessitating improved radiation protection. The lowΔV
costs of establishing a launch period for a lowenergy transfer are discussed in
Chapter 6.
4. Lowenergy transfers have a relaxed operational timeline. Modern launch
vehicles, such as the Atlas V family with their Centaur upper stages, place
spacecraft on their trajectories with small errors. Missions such as GRAIL,
which launched aboard a Delta II launch vehicle, may be able to wait 6 days
or more before performing a maneuver. In fact, GRAIL was able to cancel
the ﬁrst trajectory correction maneuver (TCM) for both spacecraft; the ﬁrst
TCM performed was executed 20 days after launch. In this way, a spacecraft
operations team has a great deal more time to prepare the spacecraft before
requiring a maneuver, when compared to conventional transfers that typically
require a maneuver within a day or less.
5. Lowenergy transfers may place several vehicles into very different orbits at
the Moon using a single launch vehicle. The GRAIL mission separated two
lunarorbit insertions by over a day using very little fuel. Chapter 3 illustrates
how to place multiple spacecraft in many different orbit types using a single
launch vehicle. This typically requires a large amount of fuel when using
conventional transfers.
EXECUTIVE SUMMARY
5
6. Lowenergy transfers may be used to transfer a spacecraft from the Moon
directly to any location on the surface of the Earth. Typical conventional
transfers, for example, those used by the Apollo missions, return spacecraft
to a nearequatorial landing site. Lowenergy transfers may be used to target
any location (such as the different hemispheres of the Utah Test and Training
Range in North America and the Woomera Weapons Testing Range in South
Australia) using relatively small quantities of fuel.
The typical drawbacks of lowenergy transfers between the Earth and the Moon
are the longer transfer durations for missions that are very timecritical and the longer
linkdistances, as the spacecraft travels as far as 1.5–2 million kilometers away from
the Earth.
The next sections deﬁne direct and lowenergy transfers to provide a clear under
standing of what trajectories are presented in this book.
1.3.1
Direct, Conventional Transfers
A direct lunar transfer is a trajectory between the Earth and the Moon that requires
only the gravitational attraction of the Earth and Moon. A spacecraft typically begins
from a low altitude above the surface of the Earth as a result of an injection by a
launch vehicle, as a result of a maneuver performed by the spacecraft, or as a result
of some intermediate orbit. The spacecraft then cruises to the Moon on a trajectory
that typically remains within the orbit of the Moon about the Earth. It is a trajectory
whose dynamics are dominated by the gravitational attraction of the Earth and Moon,
and all other forces (such as the Sun or any spacecraft events) may be considered
to be perturbations. The spacecraft then enters some orbit about the Moon via a
maneuver. Direct transfers may be constructed from the Moon to the Earth in much
the same way as they are constructed to the Moon.
Figure 11 illustrates a 3day transfer nearly identical to the one the Apollo 11
astronauts used to go from the Earth to the Moon in 1969 [1]. The mission imple
mented a lowEarth parking orbit with an inclination of approximately 31.38 deg.
From there, the launch vehicle was required to attain a translunar injection energy
(C3 ) of approximately −1.38 km2 /s2 to reach the Moon in approximately 3.05 days.
Upon arrival at the Moon, the vehicle injected into an elliptical orbit with a peri
apse altitude of approximately 110 km and an apoapse altitude of approximately
310 km, followed soon after by a circularization maneuver [1]. In order to compare
the Apollo 11 transfer with the transfers in the surveys presented here, the Apollo 11
transfer would have a velocity of approximately 2.57 kilometers per second (km/s)
at an altitude of 100 km above the mean lunar surface, requiring a hypothetical,
impulsive ΔV of approximately 0.94 km/s to insert into a circular 100km orbit.
Direct transfers may be constructed between the Earth and the Moon with durations
as short as hours or as long as a few weeks. In general, the most fuelefﬁcient direct
transfers require about 4.5 days of transfer duration. Any longer duration typically
sends the spacecraft beyond the orbit of the Moon before it falls back and encounters
the Moon.
6
INTRODUCTION AND EXECUTIVE SUMMARY
Figure 11 A modiﬁed version of the Apollo 11 Earth–Moon transfer, as if it had performed
an impulsive lunarorbit insertion (LOI) maneuver directly into a circular 100km lunar orbit
c 2011 by American Astronautical Society Publications Ofﬁce, all rights
[2]. (Copyright ©
reserved, reprinted with permission of the AAS.)
Direct transfers may also be constructed between the Earth and lunar libration
orbits for similar amounts of fuel as required to transfer directly to low lunar orbits.
The launch energy requirement is very similar for missions to the Moon, to Lagrange 1
(L1 ), and to Lagrange 2 (L2 ), and to a ﬁrst order may be treated as equal. A direct
transfer requires 400–600 m/s of ΔV to insert into a lunar libration orbit about either
L1 or L2 , though a powered lunar ﬂyby en route to a libration orbit about L2 may be
used to reduce the total transfer cost by 100–200 m/s. These transfers are examined
in Chapter 3.
Several missions have added Earth phasing orbits to their mission itineraries, such
that they launch into a highaltitude, temporary Earth orbit and remain in that orbit
for several orbits before arriving at the Moon. A mission designer may add these
orbits to a ﬂight plan for several reasons. First, they may be used to establish an
extended launch period, since the mission planners can adjust the size of the phasing
orbits to compensate for varying launch dates. Second, they may be used to reduce
the operational risk of the mission by increasing the amount of time between each
maneuver en route to the Moon. They may also be used if the launch vehicle is not
powerful enough or accurate enough to send the spacecraft directly to the Moon, such
as Chandrayaan1 [3]. Drawbacks of Earth phasing orbits include additional passes
through the Van Allen Belts and an extended transfer duration.
1.3.2
LowEnergy Transfers
Lowenergy transfers take advantage of the Sun’s gravity to reduce the transfer fuel
costs. They involve trajectories that take the spacecraft beyond the orbit of the Moon,
where the Sun’s gravity becomes more inﬂuential. The Sun’s gravity works slowly
EXECUTIVE SUMMARY
7
and steadily, gradually raising the spacecraft’s periapse altitude until it has risen to the
altitude of the Moon’s orbit about the Earth. When the spacecraft falls back toward
the Earth, it arrives at the Moon with a velocity that closely matches the Moon’s
orbital velocity. The result is that the spacecraft’s lunar orbit insertion requires much
less fuel than required by a conventional, direct lunar transfer. Figure 12 illustrates
an example 84day lowenergy transfer that arrives at the Moon when the Moon is at
its ﬁrst quarter. More explanation of these transfers is provided in Section 1.7 and in
later chapters.
Lowenergy transfers typically travel far beyond the orbit of the Moon; hence, they
may be designed to take advantage of one or more lunar ﬂybys on their outbound
segment. The lunar ﬂybys may be used to reduce the injection energy requirements,
or to change the spacecraft’s orbital plane, similar to the ﬂight of each of the two
Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interac
tion with the Sun (ARTEMIS) spacecraft [4]. If a mission takes advantage of a lunar
ﬂyby immediately after launch, it may be useful to add one or more Earth phasing
orbits into the design, as described above.
1.3.3
Summary: LowEnergy Transfers to Lunar Libration Orbits
Lowenergy transfers may be used to save a great deal of fuel when a mission’s
destination is a lunar libration orbit, such as a halo orbit, a Lissajous orbit, or
Figure 12 An example 84day lowenergy lunar transfer to a low, polar lunar orbit [2].
c 2011 by American Astronautical Society Publications Ofﬁce, all rights reserved,
(Copyright ©
reprinted with permission of the AAS.)
8
INTRODUCTION AND EXECUTIVE SUMMARY
some other threebody orbit. Many studies have demonstrated practical applications
of lunar libration orbits, including locations for communication satellites [5–7],
navigation satellites [8–13], staging orbits [14–18], and science orbits [4, 19]. The
ARTEMIS mission took advantage of the geometries of several orbits about both the
lunar L1 and L2 points, and it used two different lowenergy transfers to arrive at
those orbits.
Chapter 3 presents a full study of the characteristics and performance of lowenergy transfers to lunar libration orbits. The results demonstrate that a typical
transfer requires 70–120 days to travel from Earth departure to an arrival state that is
within 100 km of the target libration orbit. The transfers arrive asymptotically, such
that they do not require any insertion maneuver. This is an extraordinary beneﬁt: it
saves a mission upwards of 500 m/s of ΔV when compared to conventional, direct
transfers to lunar libration orbits. The typical transfers studied in Chapter 3 depart the
Earth with a C3 of −0.7 to −0.3 km2 /s2 , which is higher than the conventional transfer
that has a C3 of approximately −2.0 km2 /s2 , but the lowenergy transfer requires
only small TCMs after the Earthdeparture maneuver. Studies show (Section 6.5)
that two or three deterministic maneuvers with a total of only ∼70 m/s of ΔV may be
used to depart the Earth from a speciﬁc inclination (such as 28.5 deg), and from any
day within a 21day launch period, and arrive at a particular location in a speciﬁed
libration orbit.
Figures 13 and 14 illustrate two example direct transfers and two example lowenergy transfers to lunar libration orbits, respectively. One can see that these transfers
are ballistic in nature: they require a standard translunar injection maneuver, a
few TCMs, and an orbit insertion maneuver (which is essentially zero for the lowenergy transfers). One may also add Earth phasing orbits and/or lunar ﬂybys to the
trajectories, which change their performance characteristics. Figure 15 illustrates
two transfers that a spacecraft may take to depart the libration orbit using minimal
fuel and transfer to a low lunar orbit or to the lunar surface.
1.3.4
Summary: LowEnergy Transfers to Low Lunar Orbits
Robotic spacecraft may take advantage of the beneﬁts of a lowenergy transfer when
transferring to a low lunar orbit, such as GRAIL’s target lunar orbit. The transfer
duration is about the same as a lowenergy transfer to a lunar libration orbit, namely,
70–120 days. This duration is typically far too long for human occupants, unless
the purpose of the mission is to demonstrate a long deepspace transfer. There
are many beneﬁts for robotic missions, including smaller orbit insertion maneuver
requirements, the capability to establish an extended launch period, and a relaxed
operational schedule. The GRAIL mission took advantage of these beneﬁts, as well
as the characteristic that it requires very little ΔV to separate the two spacecraft
from their joint launch. GRAIL’s two spacecraft ﬂew independently to the Moon and
arrived 25 hours apart: a feat that requires a great deal more ΔV and/or operational
complexity when implementing direct lunar transfers. Lowenergy transfers may
also access a much broader range of lunar orbits for a particular arrival date than
direct transfers.
EXECUTIVE SUMMARY
9
Figure 13 The proﬁle for a simple, direct transfer from the Earth to a lunar libration orbit
about either the Earth–Moon L1 or L2 point, viewed from above in the Earth–Moon rotating
coordinate frame.
Figure 14 The proﬁle for a simple, lowenergy transfer from the Earth to a lunar libration
orbit about either the Earth–Moon L1 or L2 point, viewed from above in the Earth–Moon
rotating coordinate frame.
Chapter 4 presents a full study on the characteristics and performance of lowenergy transfers to low lunar, polar orbits. The examination uses 100km circular,
polar orbits as the target orbits to simplify the trade space. It remains relevant
to practical mission design since many spacecraft missions have inserted into very
similar orbits, including Lunar Prospector, Kaguya/ Selenological and Engineering
Explorer (SELENE), Chang’e 1, LRO, and GRAIL, among others. The results of the
study indicate that lowenergy transfers typically depart the Earth with an injection
C3 of –0.7 to –0.3 km2 /s2 , much like lowenergy transfers to lunar libration orbits,
10
INTRODUCTION AND EXECUTIVE SUMMARY
Figure 15 The proﬁle for a simple, lowenergy transfer from a libration orbit to either a
low lunar orbit or the surface of the Moon, viewed from above in the Earth–Moon rotating
coordinate frame.
and require 70–120 days to reach the Moon. A spacecraft may implement a lunar
ﬂyby on the outbound segment to reduce the launch energy requirement, but such an
event would increase the complexity and operational risk of the mission. When the
spacecraft arrives at the Moon, it arrives traveling at a slower relative speed than if it
had used a direct lunar transfer. The examination shows that the lunarorbit insertion
maneuver is at least 120 m/s smaller for any lowenergy mission; the ΔV savings are
often much greater.
Lowenergy transfers may also be used in such a way that a spacecraft transfers
to a lunar libration orbit, or some other threebody orbit, before transferring to the
target orbit. This strategy was used in the ARTEMIS mission and has been used in a
number of spacecraft proposals.
Figure 16 illustrates an example direct transfer and an example lowenergy trans
fer to two low lunar orbits. The transfers are very similar to those presented in the
previous section, except of course that these target low lunar orbits instead of lunar
libration orbits.
1.3.5 Summary: LowEnergy Transfers to the Lunar Surface
Lowenergy transfers from the Earth to the lunar surface may be constructed in much
the same way as transfers to low lunar orbit. They have the same sorts of beneﬁts
and drawbacks as other lowenergy transfers.
Chapter 5 presents a full study on the characteristics and performance of lowenergy transfers to the lunar surface. There are two main classes of missions studied:
those that arrive at the surface with a high impact angle and those that arrive at the
surface with a shallow ﬂight path angle. The shallow angles are useful for missions
that aim to land on the surface, and then it is useful that the lowenergy transfers
BACKGROUND
Figure 16
lunar orbit.
11
The proﬁles for both a direct and a lowenergy transfer from the Earth to a low
yield trajectories that arrive at the surface with lower velocities. The steeper arrival
conditions are useful for lunar impactors, such as the Lunar Crater Observatory and
Sensing Satellite (LCROSS). In this case, higher velocities are typically preferred.
Lowenergy transfers may not result in the highest impact velocities achievable, but
they do offer the capability of targeting any location on the surface of the Moon with
ease.
As with the lowenergy transfers studied in Chapters 3 and 4, the typical transfers
to the lunar surface require 70–120 days. They typically depart the Earth with C3
values between –0.7 and –0.3 kilometers squared per square second (km2 /s2 ) and
only require small trajectory correction maneuvers after launch. The same sort of
two or threeburn strategies may be used to target a particular lowenergy transfer
from a speciﬁed low Earth parking orbit, and from any day within a 21day launch
period.
The lunar surface may also be accessed from a lunar libration orbit or from a low
lunar orbit. Hence, a mission may implement a lowenergy transfer to either type of
orbit studied in Chapters 3 or 4 and then follow a transfer to the lunar surface. This
sort of trajectory design is also studied in Chapter 5.
Figure 17 illustrates an example direct transfer and an example lowenergy trans
fer to the lunar surface. Again, the transfers are very similar to those presented in the
previous two sections, except (of course) that these target the lunar surface.
1.4 BACKGROUND
This section reviews historical lunar missions as a reference for the discussions about
designing future lunar missions, including future missions that use direct transfers as
well as lowenergy transfers. Nearly one hundred spacecraft have ﬂown conventional,
12
INTRODUCTION AND EXECUTIVE SUMMARY
Figure 17 The proﬁles for both a direct and a lowenergy transfer from the Earth to the
lunar surface. Transfers may be constructed to arrive with a shallow or steep ﬂight path angle.
direct transfers between the Earth and the Moon, including the Union of Soviet
Socialist Republics’ (USSR’s) Luna spacecraft, the USA’s Apollo spacecraft, and
the most recent international missions. Only ﬁve spacecraft have ﬂown lowenergy
lunar transfers, though several others have followed lowenergy transfers to other
destinations near the Earth. The complexity of lunar missions has gradually grown,
as has the need to save money and collect a greater scientiﬁc return using less fuel.
Modern ﬂight operations, spacecraft hardware, and infrastructure have opened the
door to lowenergy techniques as a method to reduce costs.
The ﬁrst two missions to implement lowenergy transfers—Hiten and ARTEMIS—
demonstrated the technique as a method to extend their missions to the Moon, despite
not having the fuel to reach lunar orbit using conventional techniques. The GRAIL
mission, launched on September 10, 2011, was the ﬁrst mission to implement a lowenergy lunar transfer as part of its primary mission. The GRAIL mission beneﬁted
from its lowenergy route to the Moon in more ways than just saving fuel. It is
fully expected that more missions will follow this lead, and lowenergy transfers will
become common among lunar missions.
1.5
THE LUNAR TRANSFER PROBLEM
Soon after the dawn of the Space Age, people were designing trajectories for space
craft to travel to the Moon [20, 21]. In fact, not even a full year had elapsed since
the launch of Sputnik (October 4, 1957) before the United States attempted to launch
the Pioneer 0 probe to the Moon (August 17, 1958). The ﬁrst probes designed to
explore the Moon were plagued with launch vehicle failures, including four Pioneer
failures by the United States and three Luna failures by the Soviet Union. It was not
THE LUNAR TRANSFER PROBLEM
13
until 1959 that Luna 1 ﬁnally ﬂew by the Moon. Later in 1959, Luna 2 became the
ﬁrst probe to impact the Moon.
As technology improved, spacecraft were able to ﬂy to the Moon using less
fuel. Several general bounds exist that limit the movement of a spacecraft in the
Earth–Moon system when other perturbations, such as the Sun’s gravity, are ignored.
Research in the circular restricted threebody problem (examined in Section 2.6.2)
reveal that a spacecraft with enough energy to reach the Earth–Moon L1 point has
the minimum energy required to transfer to the Moon, without considering other
perturbations. Sweetser computed that the theoretical minimum ΔV that a space
craft would require to travel from a 167km altitude circular orbit at the Earth to a
100km altitude circular orbit at the Moon, just passing through L1 , is approximately
3.721 km/s [22]. Actual trajectories have since been computed that approach this
theoretical minimum [23].
Early investigations concluded that it is impossible to launch from the Earth and
arrive at the Moon such that the spacecraft becomes captured without performing
a maneuver [21]; however, these analyses did not include the effects of the Sun’s
gravity. As early as 1968, Charles Conley began using dynamical systems methods
to explore the construction of a theoretical trajectory that could become temporarily
captured by the Moon without performing a capture maneuver [24]. A spacecraft with
the proper energy could target the neck region near one of the collinear libration points
in the Earth–Moon system (see Section 2.6.2). A planar periodic orbit exists in each
of those regions that acts as a separatrix, separating the interior of the Moon’s region
from the rest of the Earth–Moon region. Conley’s method implemented dynamical
systems techniques to construct the transfer by targeting the gateway periodic orbit.
His transfers were restricted to the Moon’s orbital plane.
In the late 1980s and early 1990s, Belbruno and Miller began developing a method
to construct lunar transfers using a new technique, which they have referred to as
the weak stability boundary (WSB) theory [25–27]. The method involves targeting
the region of space that is in gravitational balance between the Sun, Earth, and
Moon, without involving any threebody periodic orbits or other dynamical structures.
Ballistic capture occurs when the spacecraft’s twobody energy becomes negative,
as described by Yamakawa [28, 29]. In 1991, the Japanese mission Hiten/MUSESA
used the effects of the Earth, Moon, and Sun for its transfer to the Moon [30].
In the early 2000s, Ivashkin also developed a method to construct transfers between
the Earth and Moon using the Sun’s gravitational inﬂuence [31–34]. His methods
involve beginning from a low lunar orbit, or from the surface of the Moon, and
numerically targeting trajectories that depart from the Moon in the direction of the
Earth’s L1 or L2 points. A spacecraft on such a trajectory departs from the Moon
with a negative twobody energy with respect to the Moon, but as it climbs away
from the Moon, it gains energy from the effect of the Earth’s and Sun’s gravity.
Eventually, it gains enough energy to escape the Moon’s vicinity. The trajectory is
then targeted such that it lingers near the chosen Lagrange point long enough to allow
the Sun to lower the perigee radius of the next perigee passage down to an altitude of
approximately 50 km.
14
INTRODUCTION AND EXECUTIVE SUMMARY
In the mid 1990s, other methods were developed to construct a lunar transfer
that takes advantage of the chaos in the Earth–Moon threebody system. Bollt and
Meiss constructed a trajectory that sent a spacecraft into an orbit without sufﬁcient
energy to immediately reach the Moon, but with enough to get close enough to
become substantially perturbed by the Moon [35]. Using a series of four very small
maneuvers, the spacecraft could then hop between nearby trajectories in the chaotic
sea of possible trajectories to become captured by the Moon using far less energy
than standard direct transfers. In 1997, Schroer and Ott reduced the time of transfer
for the chaotic lunar transfer by targeting speciﬁc threebody orbits near the Earth
[36]. The total cost remained approximately the same as the transfer constructed
by Bollt and Meiss, but the transfer duration was reduced from approximately 2.05
years to 0.8 years.
In 2000, Koon et al. [37, 38] constructed a planar lunar transfer that was almost
entirely ballistic using the techniques involved in Conley’s method [38]. Following
Conley, Koon et al. [37] observed that the planar libration orbits act as a gateway
between the interior and exterior regions of space about the Moon. Koon et al. [37, 38]
constructed a trajectory that targets the interior of the stable invariant manifold of
a planar libration orbit about the Earth–Moon L2 point. Once inside the interior
of the stable manifold, the spacecraft ballistically arrives at a temporarily captured
orbit about the Moon. Many authors have debated what it means to be temporarily
captured at the Moon; Koon et al., deﬁne a similar term, “ballistically captured” to
be a trajectory that comes within the sphere of inﬂuence of the Moon and revolves
about the Moon at least once [38].
Further advances have been made since 2004 to apply dynamical systems theory
to the generation of threedimensional lowenergy lunar transfers [39–44]. Parker
mapped out numerous families of lowenergy transfers, illuminating different ge
ometries that are available for spacecraft to travel to the Moon and arrive in lunar
libration orbits without requiring any capture maneuver [2, 45–47]. Several authors
have begun applying lowthrust techniques to further improve lowenergy transfers,
including transfers from the Earth to the Moon and transfers from one libration or
bit to another [48–55]. In 60 years, research has advanced the knowledge of lunar
transfers from the early spacecraft missions that implemented direct lunar transfers
to modern analyses that reveal maps of entire families of lowenergy transfers to the
Moon.
1.6
HISTORICAL MISSIONS
Many historical missions have taken direct transfers from the Earth to the Moon,
including a large number of spacecraft in the Luna, Zond, Ranger, Surveyor, Lunar
Orbiter, and Apollo programs. A few of these missions implemented direct transfers
back to the Earth again: most notably Luna16 and the nine Apollo missions that
ventured to the Moon and returned. Several other missions have also ﬂown direct
transfers since the 1960s, and they are summarized below.
HISTORICAL MISSIONS
15
Lowenergy lunar transfers are closely related to lowenergy transfers in the Sun–
Earth system, as is described later in this book. Since the 1970s, several spacecraft
have been placed on threebody trajectories in the Sun–Earth system to conduct their
scientiﬁc and technological missions, including International Sun–Earth Explorer3
(ISEE3), Solar and Heliospheric Observatory (SOHO), Advanced Composition Ex
plorer (ACE), Wind, Wilkinson Microwave Anisotropy Probe (WMAP), and Genesis,
among others. Three spacecraft are known to have followed threebody trajectories
in the Earth–Moon system, including SMART1 and the two ARTEMIS spacecraft.
Between 1971 and 2011, ﬁve spacecraft have traversed lowenergy transfers from the
Earth to the Moon, including Hiten/MUSESA in 1971, the two ARTEMIS spacecraft
in 2010 and the two GRAIL spacecraft in 2011. A brief summary of each of these
missions will be presented here.
1.6.1
Missions Implementing Direct Lunar Transfers
Table 12 summarizes many historical missions that have taken direct lunar transfers,
noting their launch date and transfer duration, among other things. One notices that
early missions implemented very quick transfers that required fewer than 1.5 days
to reach the Moon. These involved lunar ﬂybys or impacts, with no intention of
inserting into orbit or landing softly. Indeed, their velocities at the Moon would be
quite high. The ﬁrst soft landing was performed by the Soviet Union’s Luna 9, which
took a 79hour transfer to the Moon. The ﬁrst robotic sample return attempt was
performed by the Soviet Union’s Luna 15, which took a 101.6hour transfer to the
Moon: longer to save fuel mass so that it would be capable of returning to the Earth.
Luna 16 was the ﬁrst successful robotic sample return, taking a 105.1hour lunar
transfer. The ﬁrst human landing, and ﬁrst successful sample return was performed
earlier, by Apollo 11. The direct transfer that Apollo 11 took required about 73 hours,
which was shorter in time and required more fuel, but required less total consumable
mass than a longer transfer since the mission involved human occupants.
1.6.2
LowEnergy Missions to the Sun–Earth Lagrange Points
ISEE3. On August 12, 1978, the International Sun–Earth Explorer 3 (ISEE3)
spacecraft was launched and placed in a halo orbit about the Sun–Earth L1 point. It
was the ﬁrst spacecraft to be inserted into an orbit about a Lagrange point. On June
10, 1982, the spacecraft began performing 15 very small maneuvers to guide it on
a series of lunar ﬂybys. Its ﬁfth and ﬁnal lunar ﬂyby was performed on December
22, 1983, coming within 120 km of the lunar surface. The lunar ﬂyby ejected the
spacecraft from the Earth–Moon system and it entered a heliocentric orbit. The
spacecraft was renamed the International Cometary Explorer (ICE) as it readied for
its encounter with the comet GiacobiniZinner. On June 5, 1985, ICE entered the
comet’s tail and collected scientiﬁc information about the tail. ICE is expected to
return to the vicinity of the Earth in 2014, when it may be captured and brought back
to Earth, or repurposed for another comet observation mission. Figure 18 shows a
plot of the trajectory of ISEE3/ICE [60, 61].
16
INTRODUCTION AND EXECUTIVE SUMMARY
Table 12 The transfer durations, among other information, of several historical
missions that have implemented direct lunar transfers [56–59].
Launch Date
Spacecraft
Nationa Transfer Duration Notes
There were 24 successful Soviet Luna missions; examples include:
2 Jan. 1959
12 Sept. 1959
4 Oct. 1959
2 Apr. 1963
9 May 1965
Luna 1
Luna 2
Luna 3
Luna 4
Luna 5
USSR
USSR
USSR
USSR
USSR
31 Jan. 1966
31 Mar. 1966
13 July 1969
12 Sep. 1970
9 Aug. 1976
Luna 9
Luna 10
Luna 15
Luna 16
Luna 24
USSR
USSR
USSR
USSR
USSR
34 hr (1.42 days) First lunar ﬂyby (5995 km)
33.5 hr (1.40 days) First lunar impact (29.10 N, 0.00 E)
60 hr (2.50 days) Flyby (6200 km)
77.3 hr (3.22 days) Flyby (8336.2 km)
∼83 hr (3.4 days) First softlanding attempt;
impact (31 S, 8 W)
79 hr (3.29 days) First soft landing (7.08 N, 64.37 W)
78.8 hr (3.29 days) First orbiter
101.6 hr (4.23 days) First sample return attempt
105.1 hr (4.38 days) First sample return (101 grams)
103.0 hr (4.29 days) Sample return, landing within 1 km
of Luna 23 (170 grams returned)
There were eight Soviet Zond missions; little accurate information is available.
18 July 1965
14 Sept. 1968
Zond 3
Zond 5
USSR
USSR
33 hr (1.38 days)
∼3.4 days
Flyby (9200 km)
First circumlunar return
There were nine American Ranger missions; examples include:
26 Jan. 1962
23 Apr. 1962
18 Oct. 1962
30 Jan. 1964
28 July 1964
17 Feb. 1965
21 Mar. 1965
Ranger 3
Ranger 4
Ranger 5
Ranger 6
Ranger 7
Ranger 8
Ranger 9
USA
USA
USA
USA
USA
USA
USA
2–3 days
64 hr (2.67 days)
2–3 days
65.5 hr (2.73 days)
68.6 hr (2.86 days)
64.9 hr (2.70 days)
64.5 hr (2.69 days)
Flyby (∼36,800 km)
Impact (15.5 S, 130.7 W)
Flyby (725 km)
Impact
Impact (10.70 S, 20.67 W)
Impact (2.71 N, 24.81 E)
Impact (12.91 S, 2.38 W)
There were seven American Surveyor missions, including:
30 May 1966
20 Sept. 1966
17 Apr. 1967
14 July 1967
8 Sept. 1967
7 Nov. 1967
Surveyor 1
Surveyor 2
Surveyor 3
Surveyor 4
Surveyor 5
Surveyor 6
USA
USA
USA
USA
USA
USA
7 Jan. 1968
Surveyor 7
USA
63 hr (2.63 days) Landed (2.45 S, 43.21 W)
∼1.9 days
Impact (5.5 N, 12 W)
64.5 hr (2.69 days) Landed (3.01 S, 23.34 W)
∼2.6 days
Impact (0.4 N, 1.33 W)
64.8 hr (2.70 days) Landed (1.41 N, 23.18 E)
65.0 hr (2.71 days) Landed (0.49 N, 1.4 W);
First powered takeoff
66.0 hr (2.75 days) Landed (40.86 S, 11.47 W)
There were ﬁve American Lunar Orbiter missions; examples include:
10 Aug. 1966 Lunar Orbiter 1
6 Nov. 1966 Lunar Orbiter 2
5 Feb. 1967 Lunar Orbiter 3
USA
USA
USA
91.6 hr (3.82 days)
92.5 hr (3.85 days)
92.6 hr (3.86 days)
Orbiter
Orbiter
Orbiter
There were 9 American Apollo missions that orbited or orbited and
landed on the Moon; examples include:
21 Dec. 1968
18 May 1969
16 July 1969
7 Dec. 1972
a
Apollo 8
Apollo 10
Apollo 11
Apollo 17
USA
USA
USA
USA
66.3 hr (2.76 days)
73.3 hr (3.05 days)
73.1 hr (3.04 days)
83.0 hr (3.46 days)
First manned lunar orbiter
Orbit and return
First manned landing
Final manned landing
35km traverse,
110.5 kg returned
Union of Soviet Socialist Republic (USSR) and United States of America (USA)
HISTORICAL MISSIONS
Launch Date
Table 12 Continued.
Spacecraft
Nation Transfer Duration Notes
Additional missions that have implemented direct transfers include:
3 Mar. 1959
Pioneer 4
USA
19 July 1967
10 June 1973
25 Jan. 1994
Explorer 35
Explorer 49
Clementine
USA
USA
USA
24 Dec. 1997 Asiasat 3 / HGS1 China
7 Jan. 1998 Lunar Prospector
26 Oct. 2006 STEREO Ahead
USA
USA
26 Oct. 2006 STEREO Behind
USA
14 Sept. 2007 Kaguya/Selene
24 Oct. 2007
Chang’e 1
Japan
China
22 Oct. 2008
Chandrayaan1
India
18 June 2009
1 Oct. 2010
LRO/LCROSS
Chang’e 2
USA
China
29.3 hr (1.22 days) Flyby, ﬁrst USA
spacecraft to reach
escape velocity
∼2 days
Orbiter
113.1 hr (4.71 days) Orbiter
∼4 days
Orbiter
+ 12 days phasing
∼4.5 days
2 lunar ﬂybys en
route to GEO
105 hr (4.38 days) Orbiter
85 hr (3.54 days) 1 lunar ﬂyby
+ 47 days phasing
83 hr (3.46 days) 2 lunar ﬂybys
+ 47 days phasing
127 hr (5.29 days) Orbiter
∼120 hr (∼5 days) Orbiter
+ 7 days phasing
107.9 hr (4.50 days) Orbiter/impactor
+ 13 days phasing
108 hr (4.5 days) Orbiter/impactor
112.1 hr (4.7 days) Orbiter
Figure 18 The trajectory of ISEE3 / ICE [62]. (See color insert.)
17
18
INTRODUCTION AND EXECUTIVE SUMMARY
Wind. The Wind mission was launched on November 1, 1994, and placed in a halo
orbit about the Sun–Earth L1 point. Its scientiﬁc objectives were to characterize
the solar wind using a variety of particle and ﬁeld measurements, all of which
complemented several other spacecraft in a variety of other orbits, including the
Polar and Geotail satellites, as part of the International SolarTerrestrial Physics
(ISTP) Science Initiative. After several years of measurements from the Sun–Earth
L1 environment, Wind’s orbit was altered to give it access to new areas in the nearEarth environment, including a campaign of “petal orbits” to send it out of the ecliptic
plane (1998–1999), a lunar backﬂip (April, 1999), several revolutions about a distant
prograde orbit (2001–2003), and a complex orbit that involved repeated lunar ﬂybys
and excursions out beyond the Sun–Earth L1 and L2 points (2003–2006). The ﬁrst
part of Wind’s trajectory resembles the ﬁrst part of ISEE3’s trajectory shown in
Fig. 18. Figure 19 illustrates Wind’s orbits in the Sun–Earth system from 2003
through 2006 [63], illustrating a unique aspect of its lowenergy mission design.
Figure 19 The trajectory of Wind from 2003 through 2006, viewed from above in the
Sun–Earth rotating frame [63].
HISTORICAL MISSIONS
19
SOHO. The Solar and Heliospheric Observatory (SOHO) was launched on Decem
ber 2, 1995, on a path taking it directly toward a libration orbit about the Sun–Earth
L1 point. On March 17, 1996, SOHO performed a small orbit insertion maneuver to
formally enter the quasihalo L1 orbit 1.5 million kilometers away from the Earth.
The L1 halo orbit is ideal for the observatory because it provides an unobstructed
view of the Sun on one side and a nearconstant view of the Earth on the other side.
Hence, it can collect scientiﬁc data about the Sun continuously, while being able to
communicate with the Earth at any time. Figure 110 shows a plot of the trajectory
that SOHO used to transfer to its halo orbit [64–67].
ACE. In 1997, the Advanced Composition Explorer (ACE) was launched and placed
in a Lissajous orbit about the Sun–Earth L1 point. Its mission, much like SOHO’s,
is dedicated to collecting energetic particles to study the solar corona, interplanetary
medium, solar wind, and cosmic rays. Its transfer appears very similar to SOHO’s
transfer, shown in Fig. 110 [68, 69].
WMAP. Launched on June 30, 2001, the Wilkinson Microwave Anisotropy Probe
(WMAP) is currently residing in a smallamplitude Lissajous orbit about the Sun–
Earth L2 point. From this orbit, WMAP continues to measure cosmic background
radiation, unobstructed by the radiation originating from the Sun, Earth, or Moon.
Figure 111 shows a plot of the trajectory that WMAP used to reach its libration orbit
about L2 [70].
Figure 110
of ESA.
The transfer trajectories and mission phases of SOHO [68], used with permission
20
INTRODUCTION AND EXECUTIVE SUMMARY
Figure 111 The transfer trajectory of WMAP [76].
Genesis. On August 8, 2001, Genesis launched and was quickly injected into a
halo orbit about the Sun–Earth L1 point. It traversed the halo orbit approximately ﬁve
times, spending more than 2 years in the libration orbit collecting solar wind samples
before turning back toward the Earth. Before returning to the Earth, however, it made
a 3millionmile (4.8 × 106 km) detour to visit the Sun–Earth L2 point. The detour
allowed it to deposit its science payload on the sunlitside of the Earth. Figure 112
shows a plot of the trajectory that Genesis followed during its primary mission
[71, 72].
Herschel and Planck. The Herschel and Planck space observatories were launched
together on May 14, 2009 [73–75]. The two spacecraft separated soon after launch
and traveled separately to Lissajous orbits about the Sun–Earth L2 point. Their orbit
transfers were heuristically similar to WMAP’s transfer to L2 , illustrated in Fig. 111.
Future Missions. There are plans to place the proposed James Webb Space Tele
scope [78] and the proposed Terrestrial Planet Finder [79] missions, among others, at
the Sun–Earth L2 point. Lowenergy trajectories to the Sun–Earth Lagrange points
have been shown to be very useful for solar observatories (L1 ) and astrophysics
observatories (L2 ), and they frequently appear in spacecraft proposals.
1.6.3
Missions Implementing LowEnergy Lunar Transfers
Hiten/MUSESA. In 1991, the Japanese spacecraft Hiten was the ﬁrst spacecraft
to transfer to the Moon using a lowenergy lunar transfer. The spacecraft was not
designed to go to the Moon, but rather to send a probe to the Moon. After the probe’s
communication system failed, mission designers scrambled to ﬁnd a new mission
for Hiten. Edward Belbruno and James Miller constructed a new trajectory—a
“WSB transfer”—that required less fuel than traditional lunar transfers [80, 81]. The
HISTORICAL MISSIONS
21
Figure 112 The lowenergy trajectory that the Genesis spacecraft followed [77], viewed
from above in the Sun–Earth rotating frame.
spacecraft Hiten did not have the fuel required for a conventional lunar transfer, but
had the fuel to use this new lunar transfer to reach the Moon. Hiten became Japan’s
ﬁrst lunar mission.
SMART1. On September 27, 2003, the European Space Agency’s SMART1 space
craft followed a lowthrust 2year trajectory to reach the Moon, becoming the ﬁrst
lowthrust spacecraft to transfer to the Moon [82].
ARTEMIS. The Time History of Events and Macroscale Interactions during Substorms (THEMIS) constellation was launched on February 17, 2007, to monitor the
Earth’s magnetic ﬁeld from ﬁve different vantage points in highaltitude orbits, track
ing the largescale evolution of substorms. In 2009, two of those spacecraft were
maneuvered to begin an extended mission called ARTEMIS [4]. The two spacecraft
performed numerous maneuvers near their orbital perigees to gradually raise their
orbits until they could take advantage of several lunar ﬂybys to propel them onto
two lowenergy transfers. Both ARTEMIS spacecraft arrived at the Moon near the
Earth–Moon L2 point; one of them remained there and one immediately transferred
to a libration orbit about the Earth–Moon L1 point. After several months, the second
spacecraft made the transfer and both orbited the L1 point. After several more months,
the two spacecraft departed their respective L1 orbits, descended to the Moon, and
entered smaller Keplerian orbits about the Moon. The two ARTEMIS spacecraft are
the ﬁrst two spacecraft to orbit either LL1 or LL2 , and they each orbited both points.
22
INTRODUCTION AND EXECUTIVE SUMMARY
GRAIL. The GRAIL mission (Fig. 113) [83–85] was launched on September 10,
2011, aboard a Delta II Heavy launch vehicle. Two vehicles, GRAILA (Ebb) and
GRAILB (Flow), were separated soon after launch and ﬂew independently to the
Moon via two similar lowenergy transfers. The two spacecraft arrived at the Moon
approximately 25 hours apart, on December 31, 2011 and January 1, 2012. After a
few months of orbit reductions and adjustments, the two spacecraft inserted into a
formation, such that one spacecraft trailed the other in almost identical orbits about
the Moon. By tracking each other, the two spacecraft were able to recover the Moon’s
gravity ﬁeld to unprecedented precision and map the interior structure of the Moon.
The two GRAIL spacecraft were the ﬁrst ever to ﬂy lowenergy lunar transfers as
part of their primary mission, and they were the ﬁrst ever to arrive at the Moon and
perform lunar orbit insertions directly from lowenergy transfers.
GRAIL’s trajectory design is illustrated in Fig. 113, including the ﬁrst and last
launch opportunity in a 26day launch period. This is the launch period published in
Ref. [83], however, it was actually extended by many days as the mission developed.
As one can see in Fig. 113, GRAIL’s mission design includes two signiﬁcant deter
ministic maneuvers executed per spacecraft during the cruise, performed primarily
to separate their lunar orbit insertion dates.
Figure 113 GRAIL’s mission design, including a 26day launch period and two deterministic
maneuvers for both GRAILA and GRAILB, designed to separate their lunar orbit insertion
times by 25 hours (Ref. [83], originally published by AAS).
LOWENERGY LUNAR TRANSFERS
1.7
23
LOWENERGY LUNAR TRANSFERS
Lowenergy transfers between the Earth and the Moon are the focus of this book; this
section heuristically describes these transfers and how they are used.
A lowenergy lunar transfer includes several segments and a wide variety of
possible itineraries. The transfer may begin from a direct launch, a parking orbit,
or some previous mission orbit. From the initial state, the spacecraft may depart
immediately toward the lowenergy transfer, or it may target an outbound lunar ﬂyby.
If the trajectory employs a lunar ﬂyby, the mission may beneﬁt by incorporating one
or more Earth phasing orbits to target that ﬂyby. The lunar transfer then spends
3–4 months before returning to the Moon. Upon arriving at the Moon, the spacecraft
may immediately inject into a libration orbit or some other threebody orbit, a low
lunar orbit, or it may immediately descend to the surface for a soft landing or a targeted
impact. If the mission inserts into an orbit, it may later transfer to a different orbit
and/or transfer to the surface. These itinerary choices and approximate performance
parameters are illustrated in the ﬂowchart shown in Fig. 114. This section describes
each of these options in more detail.
Figure 114 A ﬂowchart illustrating different lowenergy lunar transfer itineraries, with
approximate C3 values, transfer times, and ΔV values shown. For instance, a mission could
use this ﬂowchart to determine the approximate C3 of taking a direct injection to a lowenergy
transfer (upper half), followed by the transfer duration and ΔV cost needed to transfer to a low
lunar orbit (lower half). From there, one could transfer to the lunar surface, if desired (lower
right).
24
INTRODUCTION AND EXECUTIVE SUMMARY
Earth Parking Orbit. Lowenergy lunar transfers may begin in any Earth parking
orbit, including those compatible with a launch from any launch site around the
world; they may also begin from nearly any preexisting mission orbit, which was the
case for the Hiten and ARTEMIS missions [4, 81]. It is typically easier to tailor a
mission to launch into a parking orbit and then depart that orbit onto a lowenergy
transfer than it is to adjust the orbit of a preexisting spacecraft to achieve a particular
lowenergy transfer. The surveys in this book assume that the mission begins in a
185km circular lowEarth parking orbit, unless otherwise noted.
It will be shown that a given lowenergy transfer has a natural Earth departure
geometry—one that does not necessarily align with a desirable Earth parking orbit.
Section 6.5 provides targeting procedures to connect a desirable Earth parking orbit,
for example, one with an inclination of 28.5 deg, with a given lowenergy transfer
using 1–3 maneuvers and a minimal amount of fuel.
TransLunar Injection. The translunar injection (TLI) is modeled in this book as
an impulsive ΔV tangent to the parking orbit. This maneuver is typically performed
by the launch vehicle’s upper stage. The launch vehicle’s target C3 value is typically
in the range of –0.7 to –0.4 km2 /s2 , where C3 is a parameter equal to twice the target
speciﬁc energy. Since this target is negative, the resulting orbit is still captured by the
Earth. If the trajectory is designed to implement a lunar gravity assist on the way out
to the long cruise, then the launch target may be reduced to a C3 of approximately
–2 km2 /s2 . Launch vehicles typically target the right ascension and declination of
the outbound asymptote for interplanetary missions to other planets. Since a lowenergy lunar transfer is still captured by the Earth there is no outbound asymptote.
The GRAIL targets included the right ascension and declination of the instantaneous
apogee vector at the target interface time, referred to as RAV and DAV [83].
TransLunar Cruise. A spacecraft’s translunar cruise on its lowenergy lunar
transfer takes it beyond the orbit of the Moon and typically in a direction toward
either the second or fourth quadrant in the Sun–Earth synodic coordinate system [86].
The spacecraft typically ventures 1–2 million kilometers away from the Earth, where
the Sun’s gravity becomes very inﬂuential. As the spacecraft traverses its apogee the
Sun’s gravity constantly pulls on it, raising the spacecraft’s perigee altitude. By the
time the spacecraft begins to return to the Earth its perigee has risen high enough that
it encounters the Moon. Further, the trajectory is designed to place the spacecraft on
a lunar encounter trajectory. The GRAIL mission design involves two deterministic
maneuvers and three statistical maneuvers for each spacecraft to navigate its translunar cruise [84]. The transfers in this book may include up to two deterministic
maneuvers performed during the translunar cruise, and it is reasonable to assume that
two or three statistical maneuvers are sufﬁcient to implement a lowenergy transfer
unless a spacecraft has particularly challenging characteristics.
During this transfer, the spacecraft requires stationkeeping to remain on its proper
trajectory. The stationkeeping cost is minimal and may be accounted for by trajectory
correction maneuvers; the Genesis spacecraft followed a similar lowenergy transfer
and required only approximately 8.87 m/s of ΔV per year [71, 72, 87].
LOWENERGY LUNAR TRANSFERS
25
The lowenergy transfer may include one or more Earth phasing orbits and/or one
or more lunar ﬂybys. These add complexity to the mission and may increase the
number of maneuvers required to perform the mission, but may reduce the injection
energy requirements or orbit insertion requirements upon arriving at the Moon.
Lunar Arrival. As the spacecraft approaches the Moon, it arrives on a trajectory
that leads it to its initial lunar destination, be it a highaltitude threebody orbit, a low
lunar orbit, or the surface of the Moon. If the spacecraft’s destination is a threebody
orbit, then the spacecraft often does not require any signiﬁcant maneuver to enter the
orbit (studied in Chapter 3); if the spacecraft’s destination is a low lunar orbit, then the
trajectory guides the spacecraft to its lunar orbit insertion state (studied in Chapter 4);
ﬁnally, if the spacecraft’s destination is the lunar surface, then the trajectory guides
the spacecraft there at the designed ﬂight path angle (studied in Chapter 5).
CHAPTER 2
METHODOLOGY
2.1
METHODOLOGY INTRODUCTION
This chapter introduces all of the models, coordinate frames, and methodology used
in the analysis and construction of lunar transfers. The chapter begins by simply
deﬁning the physical constants used in these analyses, including the masses and
radii of the Sun, the Moon, and the planets. It then deﬁnes the time systems used,
coordinate frames, and models, including the circular restricted threebody problem
and the Jet Propulsion Laboratory (JPL) developmental ephemerides used to model
the motion of the planets and the Moon. A large portion of this chapter is then devoted
to describing the dynamical systems methods employed in this work for the analysis
and design of lowenergy transfers in the Solar System. These methods may be used
to design lowenergy transfers from one orbit to another and/or one celestial body to
another, such as lowenergy transfers between the Earth and the Moon. Finally, this
chapter discusses the tools used to generate the trajectories in this work.
27
28
2.2
METHODOLOGY
PHYSICAL DATA
The trajectories generated in this work have been propagated using point masses for
the Sun, the Moon, and the planets. Early analyses include just the Sun, Earth, and
Moon, often in circular orbits that approximate the real orbits. Once early analyses are
complete, highﬁdelity trajectories are generated that include all of the planets, such
that their positions are determined at each moment in time using accurate planetary
ephemerides. Table 21 presents the masses, gravitational parameters, and average
radii used to generate each trajectory, where it is assumed that the gravitational
constant, G is equal to 6.673 × 10−20 cubic kilometers per second squared per
kilogram (km3 /s2 /kg).
The values of gravitational constant times mass (GM) shown in cubic kilometers
per second squared (km3 /s2 ) in Table 21 are the best estimates of those values
when modeling the entire Solar System as point masses. However, other GM values
represent the best estimate for different cases, such as when one is modeling the
gravity of the Moon using the spherical harmonic expansion. For instance, the
LP150Q gravity ﬁeld estimates the GM of the Moon to be approximately 4902.801076
km3 /s2 : slightly different than the value in the table [88].
Table 21 The masses, gravitational parameters, and average radii of the Sun, Moon,
and planets used in this work [89, 90]. If the planet has natural satellites, the mass and
gravitational parameter of the barycenter of the planetary system have been used.
Mass (kg)
GM (km3 /s2 )
Sun
Earth
Moon
Earth Barycenter
1.98879724 × 1030
5.97333183 × 1024
7.34722101 × 1022
6.04680404 × 1024
1.32712440 × 1011
3.98600433 × 105
4.90280058 × 103
4.03503233 × 105
Mercury
Venus
Mars
Mars Barycenter
3.30167548 × 1023
4.86825414 × 1024
6.41814926 × 1023
6.41814990 × 1023
2.20320805 × 104
3.24858599 × 105
4.28283100 × 104
4.28283143 × 104
Jupiter
Jupiter Barycenter
Saturn
Saturn Barycenter
Uranus
Uranus Barycenter
Neptune
Neptune Barycenter
Pluto
Pluto Barycenter
1.89849445 × 1027
1.89888757 × 1027
5.68552375 × 1026
5.68569250 × 1026
8.68269993 × 1025
8.68357412 × 1025
1.02429180 × 1026
1.02450683 × 1026
1.32300764 × 1022
1.47100388 × 1022
1.26686534 × 108
1.26712768 × 108
3.79395000 × 107
3.79406261 × 107
5.79396566 × 106
5.79454901 × 106
6.83509920 × 106
6.83653406 × 106
8.82843000 × 102
9.81600888 × 102
Body
Radius (km)
696000.
6378.14
1737.4
–
2439.7
6051.8
3396.19
–
71492.
–
60268.
–
25559.
–
24764.
–
1195.
–
TIME SYSTEMS
29
The radius of the Earth at the Equator is equal to approximately 6378.14 km, ac
cording to the International Astronomical Union/International Association of Geodesy
(IAU/IAG) 2000 Report [89]. The distance from the Earth’s center to either pole is ap
proximately 6356.75 km, shorter than at the Equator since the Earth has a signiﬁcant
oblateness about the Equator [89]. The radius that deﬁnes the atmospheric bound
ary at the Earth for sample return missions is equal to approximately 6503.14 km,
approximately 125 km above the Earth’s Equator [91].
2.3
TIME SYSTEMS
The passage of time may be represented in countless ways. One may deﬁne broad
deﬁnitions of four types of time systems that are in common use in physics and astron
omy. To varying degrees, each of these types of time systems, and the relationships
between them, is important to the mission analyst [91, 92].
1. Dynamical time, in which the unit of duration is based on the orbital motion
of the Earth, Moon, and planets.
2. Atomic time, in which the unit of duration corresponds to a deﬁned number of
wavelengths of radiation of a speciﬁed atomic transition of a chosen isotope.
3. Universal time, in which the unit of duration represents the solar day, deﬁned
to be as uniform as possible, despite variations in the rotation of the Earth.
4. Sidereal time, in which the unit of duration is the period of the Earth’s rotation
with respect to a point nearly ﬁxed with respect to the stars.
It is very difﬁcult to be both succinct and technically correct when deﬁning the
different types of time systems that exist. See Seidelmann, 1992, for more details
[92].
2.3.1
Dynamical Time, ET
To a mission analyst, “ephemeris time” or “ET” refers to the independent variable in
the equations of motion governing the motion of bodies in the Solar System. The time
scale represents a smoothﬂowing time coordinate that is used in the development of
the numerically integrated Solar System ephemerides produced at JPL and distributed
worldwide [91], as well as barycentric dynamical time (TDB). This time scale has
also been referred to as Teph in other studies [93]. Unfortunately, the label “ET” has
a history of referring to a variety of slightly different time scales in previous studies.
2.3.2
International Atomic Time, TAI
As of 2012, the fundamental time period of a second is deﬁned in the Syste` me
International (SI) system to be a speciﬁc number of oscillations of an undisturbed
30
METHODOLOGY
cesium atom. Speciﬁcally, the second is deﬁned as the duration of time required for
9,192,631,770 periods of the radiation corresponding to the transition between the
two hyperﬁne levels of the ground state of the cesium 133 atom. The Temps Atomique
International (TAI), or international atomic time, is deﬁned as a continuous time scale
resulting from the statistical analysis of a large number of atomic clocks operating
around the world, performed by the Bureau International des Poids et Mesures
(BIPM). The difference between Terrestrial Time (TT) and TAI is approximately
32.184 seconds (s); that is, TT − TAI = 32.184 s. The difference between TAI
and ET is: ET − TAI = 32.184 s + relativistic terms, where the relativistic terms
contribute less than 2 milliseconds (ms) of variation [91].
2.3.3
Universal Time, UT
Universal Time (UT) is a time scale that is based upon the mean solar day. The
time scale “UT1” represents the daily rotation of the Earth and is independent of the
observing location, that is, it is independent of corrections for polar motion on the
longitude of the observing site. The Earth’s rotation rate changes continuously as its
shape and mass distribution shifts; hence, this time scale is unpredictable. UT1 is
computed using a combination of a variety of different types of observations, includ
ing very long baseline interferometry (VLBI) measurements of extragalactic radio
sources (quasars), lunar laser ranging, satellite laser ranging, and Global Positioning
System (GPS) measurements, to name a few.
2.3.4 Coordinated Universal Time, UTC
The Coordinated Universal Time (UTC) is the time scale that is used as the basis
for the worldwide system of civil timekeeping and is available from radio broadcast
signals. It is the time system used by ﬂight operations teams and tracking stations.
UTC was set equal to TAI in 1958; it was reset in 1972 such that the TAI time scale
was 10 s ahead of UTC, corresponding to the approximate accumulation of drift by
1972. From then on it has been adjusted using leap seconds so that it remains within
0.9 s of UT1. As of early 2012, a total of 24 leap seconds had been added, such that
the TAI time scale was 34 s ahead of UTC, that is, TAI − UTC = 34 leap seconds.
The “ET” time scale was 66.184 s (excluding periodic relativistic terms) ahead of
UTC, as it had been since January 1, 2009 [91].
2.3.5
Lunar Time
A “day” on the Moon is typically associated with a mean solar day, namely, the
duration of time between sunrises and sunsets at a particular location on the surface.
Put another way, a day on the Moon is equal to the mean interval of time between
successive crossings of the Sun on a particular lunar longitude, that is, the lunar
prime meridian. As a result, the period of one mean lunar day is equal to the period
of a mean synodic lunar month, namely, approximately 29.53059 Earth days. The
actual lunar month may vary from this mean value by nearly ± 2 hours due to the
TIME SYSTEMS
31
eccentricity of the Earth’s orbit and small periodic variations in the Moon’s rotation
rate.
A lunar month may be deﬁned in a variety of ways. Table 22 summarizes ﬁve
ways that one may deﬁne a lunar month and their corresponding durations of time
[92].
2.3.6
Local True Solar Time, LTST
The Local True Solar Time (LTST) represents the instantaneous time of day of an
observer at a site on the Moon. It is a time system that does not ﬂow constantly, but
it is useful to the mission planner when measuring time for a lunar lander. The LTST
on the Moon is deﬁned as follows
LTST = (λp − λT S deg)
24 hr
+ 12 hr
360 deg
where λp is the east longitude of a point on the surface of the Moon and λT S is the
east longitude of the true Sun. Using this relationship, 12 lunar hours corresponds
to the time when the Sun is crossing the local meridian of the reference site, for
example, local noon, and the lunar day includes 24 lunar hours.
2.3.7
Orbit Local Solar Time, OLST
During the development and operations of nearly all planetary and satellite orbiting
missions, understanding how the geometry of the orbit plane changes relative to
the Sun over time is extremely important, both from an engineering and a science
perspective. A useful way to characterize the orbit geometry, particularly for highinclination orbiters, is to report the local solar time of the ascending or descending
node of the orbit, namely, the Orbit Local Solar Time (OLST). To be clear, this
measurement describes the orientation of the orbit relative to the Moon’s surface.
The convention generally adopted is to report the local time of the orbit node relative
to the true Sun. The Moon’s gravity ﬁeld will have an effect on the orbit’s OLST
Table 22 Five ways to deﬁne a lunar month and their corresponding durations of
Earth time [92].
Month
Synodic (new moon to new moon)
Anomalistic (perigee to perigee)
Sidereal (ﬁxed star to ﬁxed star)
Tropical (equinox to equinox)
Nodical / Draconic (node to node)
Duration
(Earth days)
days
29.53059
27.55455
27.32166
27.32158
27.21222
29
27
27
27
27
Duration
hr
min
12
13
07
07
05
44
18
43
43
05
s
03
33
12
05
36
32
METHODOLOGY
over time, but to ﬁrst order the orbit remains essentially ﬁxed in inertial space. The
main reason that the orbit’s OLST will change over time is due to the motion of the
Earth–Moon system about the Sun. Thus, the following relationship describes the
gross change in the OLST over time, derived from the mean synodic and sidereal
periods of the Moon’s orbit.
OLSTin LTST changes by −3.94
minutes
Earth day
= −27.60
minutes
week
= −1.94
hours
mean lunar day
Since the change in OLST over time is primarily a function of the rate at which
the Earth–Moon system moves about the Sun, the partial will change slightly as a
function of time due to the eccentricity of the Earth’s orbit. For example, the partial
will vary roughly within the following range each year during the 3year period from
2009–2012
OLSTin LTST changes by −4.1
2.4
minutes
Earth day
to
−3.8
minutes
Earth day
within 2009–2012
COORDINATE FRAMES
This section describes several coordinate frames that are frequently used in lunar
mission analysis. Each coordinate frame has its use: some are useful to describe
states on the surface of the Earth, Moon, or other body; others are useful to describe
the relative geometry between the Sun, Earth, and/or Moon.
Coordinate systems include a reference frame and an origin, and are often rotating
or translating relative to other bodies. A coordinate system is inertial only when
it is not accelerating. When referencing motion in the Solar System, the only truly
“inertial” coordinate system is one that is not rotating and centered at the Solar System
barycenter. Strictly speaking, no Earthcentered coordinate system can be inertial,
even one that is not rotating, since the Earth is accelerating in its orbit as it revolves
about the Sun. Although it is inaccurate, coordinate systems may be referred to in
this book as “inertial” when they are merely nonrotating.
2.4.1
EME2000
The Earth Mean Equator and Equinox of J2000 (EME2000) coordinate frame is a
nonrotating coordinate frame that is approximately aligned with the Earth’s Equator.
It is almost identical to the International Celestial Reference Frame (ICRF) [94]. The
ICRF is deﬁned by the IAU and is tied to the observations of a selection of quasars
and other distant bright radio objects. It is a reference frame that is ﬁxed as well as
possible to the observable universe. Each of the quasars moves relative to the others,
but very slowly; the motion of each of the quasars is averaged out in order to best
approximate inertial space relative to the Earth’s position in the universe. According
to Feissel and Mignard [95], the pole of the EME2000 frame differs from the ICRF
pole by ∼18 milliarcseconds and the right ascension of the EME2000 xaxis differs
from the right ascension of the ICRF xaxis by 78 milliarcseconds.
COORDINATE FRAMES
33
The coordinate axes are deﬁned as follows:
• The zaxis of the EME2000 is deﬁned as the pole vector of the Earth Mean
Equator at the J2000 epoch, namely, at 1 January 2000 12:00:00 ET, or at
Julian date 2451545.0 ET.
• The xaxis of the EME2000 is deﬁned as the cross product of the zaxis and
the Earth Mean Orbit pole of J2000, that is, the ecliptic pole of J2000. This
axis deﬁnes the vernal equinox of J2000.
• The yaxis completes the righthanded coordinate frame.
This coordinate frame provides the fundamental reference for the deﬁnitions of other
coordinate frames.
2.4.2 EMO2000
The Earth Mean Orbit of J2000 (EMO2000) coordinate frame is a nonrotating coordi
nate frame that is approximately aligned with the ecliptic. The frame shares the same
xaxis as the EME2000 frame, but is rotated about that axis such that the EMO2000
zaxis is aligned with the mean ecliptic pole of J2000. This involves a rotation
of approximately 23.4393 degrees (deg). The yaxis completes the righthanded
coordinate frame.
2.4.3
Principal Axis Frame
The principal axis frame of a body is a bodyﬁxed coordinate frame, that is, rotating
frame, aligned with the principal axes of the body.
The coordinate axes are deﬁned as follows:
• The zaxis points in the direction of the maximum moment of inertia; for the
Earth and the Moon, this is the North Pole principal axis.
• The xaxis points in the direction of the minimum moment of inertial, that is,
the prime meridian principal axis.
• The yaxis completes the righthanded coordinate frame.
It is common practice to deﬁne lunar gravity ﬁelds in the lunar principalaxis bodyﬁxed frame (LPABF).
2.4.4 IAU Frames
The International Astronomical Union has developed deﬁnitions for coordinate
frames that are tied to the surface of each planet, many satellites, and some other
bodies in the Solar System. There are typically two variations of each coordinate
frame: a ﬁxed frame that rotates with the motion of the body about its primary spin
axis and an “inertial” frame that shares the same zaxis but which does not rotate.
34
METHODOLOGY
Detailed deﬁnitions of the IAU frames are described by Archinal et al. [96], and an
overview of these frames is given here. The zaxis of the IAU bodyﬁxed frame for
a given body is aligned with the direction of the spinaxis of that body. The positive
direction of the North Pole is deﬁned to be on the north side of the invariable plane
of the Solar System (deﬁned by the angular momentum of the Solar System), and
the pole’s orientation is deﬁned using measured values for the right ascension and
declination [96]. Relatively simple lowdegree polynomial approximations are used
to compute the direction of this pole vector for most of the planets. Longitude is
typically deﬁned relative to a ﬁxed surface feature for rigid bodies. In each case these
quantities are speciﬁed relative to the ICRF, which varies slightly from the EME2000
coordinate frame as described above.
To give some idea for the variations between the Earth’s IAU frame and EME2000,
Fig. 21 illustrates the mapping of Greenwich, England, from the inertial IAU Earth
frame to EME2000, where Greenwich has been deﬁned in the IAU Earth frame to be
at a latitude of 51.48 deg North and a longitude of 0.0 deg at the J2000 epoch.
2.4.5
Synodic Frames
It is often useful to describe a synodic frame that rotates with the motion of two mas
sive bodies about their barycenter. Two synodic reference frames that are frequently
used in this work are the Earth–Moon synodic frame, which rotates with the motion
of the Earth and the Moon about their barycenter, and the Sun–Earth synodic frame,
which rotates with the motion of the Earth–Moon barycenter about the Sun. The
Figure 21
The latitude and longitude of Greenwich, England, in EME2000, where
Greenwich has been deﬁned in the inertial IAU Earth frame to be at a latitude of 51.48 deg
North and a longitude of 0.0 deg at the J2000 epoch.
MODELS
35
synodic frame may be constructed to rotate at a constant rate or at a rate that varies
with the instantaneous motion of the bodies. In this book, the frames are always
constructed such that the xaxis points from the larger body to the smaller body at
each instant in time, the zaxis points in the direction of the angular momentum of the
system, and the yaxis completes the righthanded coordinate system. This deﬁnition
deﬁnes a frame that rotates at a rate that varies with the motion of the bodies in their
orbits. Of course, if the bodies orbit their barycenter in circular orbits, then this frame
rotates at a constant rate.
2.5
MODELS
This section describes the different models that have been used in this work to
approximate the motion of spacecraft in the Solar System. Each model has a use in
the analyses provided here.
The most basic model is the twobody model, which is used to approximate the
motion of a spacecraft about a simple massive body without any other perturbations.
This model is useful because one can use conic sections to approximate the space
craft’s motion, which are predictable and very quick to generate. This model is very
wellknown [97] and will not be further described here.
The next step up in complexity is a model that includes the gravitational attraction
of two large bodies, namely, the model formulated by the circular restricted threebody problem (CRTBP). The CRTBP more closely approximates the motion of a
spacecraft in the Earth–Moon and Sun–Earth threebody systems than the twobody
model. Working within the CRTBP allows a mission designer to bring a wealth of
techniques that have been developed over hundreds of years to a design problem.
These techniques provide many qualitative insights that assist in the design of useful
lowenergy orbit transfers.
The patched threebody model gracefully introduces a fourth body into the design
problem. The patched threebody model approximates the motion of a spacecraft
using the Sun–Earth threebody model for all times, except when the spacecraft is
within close proximity to the Moon, at which point the model approximates the
motion of the spacecraft using the Earth–Moon threebody model. These features
permit the design of fourbody trajectories, such as lowenergy lunar transfers, while
retaining much of the useful structure found in the CRTBP.
Finally, the fourth and most complex model frequently used in this work is the JPL
developmental ephemerides model, which approximates the motion of a spacecraft
under the inﬂuence of the gravitational attraction of any or all of the planets and the
Moon, using accurate ephemerides to model the motion of the planets and the Moon
relative to the Sun.
Each of these models is described in detail in this section.
36
2.5.1
METHODOLOGY
CRTBP
The CRTBP describes a dynamical model that is used to characterize the motion of
a massless particle, for example, a spacecraft, in the presence of two massive bodies,
such as the Earth and the Moon [86]. The model assumes the two massive bodies
orbit their barycenter in circular orbits.
2.5.1.1 Equations of Motion It is convenient to characterize the motion of the
third body, that is, the spacecraft, in a synodic reference frame that rotates at the same
rate as the orbital motion of the two primary masses. The coordinate frame is centered
at the barycenter of the system and oriented such that the xaxis extends from the
barycenter toward the smaller primary, the zaxis extends toward the primary bodies’
orbit normal, and the yaxis completes the righthanded coordinate frame. In that
synodic frame, the two massive bodies are stationary, and the spacecraft moves about
the system in nonKeplerian motion [46, 86]. It is convenient to normalize the units
in the system such that the following measurements are equal to one: the distance
between the two primaries, the sum of the mass of the two primaries, the rotation rate
of the system, and the gravitational parameter. The threebody constant, µ, relates
all of these normalized measurements and is easily computed by dividing the mass
of the smaller primary by the total mass in the system. The equations of motion for
the third body in the normalized rotating frame are equal to [86]
x
¨
=
y¨ =
z¨ =
x+µ
x−1+µ
−µ
r13
r23
y
y
−2x˙ + y − (1 − µ) 3 − µ 3
r1
r2
z
z
− (1 − µ) 3 − µ 3
r1
r2
2y˙ + x − (1 − µ)
(2.1)
(2.2)
(2.3)
where r1 and r2 are equal to the distance from the third body to the larger and smaller
primary, respectively
r12
r22
=
=
2
(x + µ) + y 2 + z 2
2
2
(x − 1 + µ) + y + z
(2.4)
2
(2.5)
The dynamics in the circular restricted threebody system depend only on the threebody constant, µ. Furthermore, as µ goes to zero, the dynamics approach twobody
dynamics, although represented in a rotating frame.
2.5.1.2 Lagrange Points There are ﬁve wellknown equilibrium solutions to
the CRTBP, known as the ﬁve Lagrange points [86], or the ﬁve libration points. These
points are referred to as L1 –L5 ; this book adopts the nomenclature that L1 lies between
the two primary masses and L2 lies on the far side of the smaller primary, relative
to the barycenter of the system. The Lagrange points in the Earth–Moon system
are abbreviated using the nomenclature LL1 –LL5 ; the Sun–Earth Lagrange points
are abbreviated EL1 –EL5 . The seven Lagrange points near the Earth are depicted
in Fig. 22. More discussion about their locations and dynamics are provided in
Sections 2.6.2 and 2.6.10.
MODELS
37
Figure 22 A plot depicting the relative proximity of the ﬁve Earth–Moon Lagrange points
and the two nearby Sun–Earth points (ﬁrst published in Ref. [97]; reproduced with kind
permission from Springer Science+Business Media B.V.).
2.5.1.3 Jacobi Constant The dynamics of the CRTBP permit an integral of
motion to exist in the synodic reference frame, known as the Jacobi integral or Jacobi
constant [46, 97, 98]. The Jacobi constant of a spacecraft in the threebody system
may be written simply as
C
U
V2
where
2U − V 2 ,
1 2
1−µ
µ
=
x + y2 +
+
r2
2
r1
= x˙ 2 + y˙ 2 + z˙ 2
=
(2.6)
(2.7)
(2.8)
The spacecraft’s position and velocity coordinates in Equations 2.1–2.8 are given
in nondimensional normalized synodic coordinates, relative to the barycenter of the
threebody system. The Jacobi constant of a spacecraft moving in the CRTBP may not
change unless the spacecraft is perturbed in some way other than by the gravitational
attraction of the two primary bodies.
It is useful to consider the Jacobi constant of spacecraft in different practical orbits
in order to place the value of the Jacobi constant in context. From twobody analyses,
we know that spacecraft in orbits about the Earth below the geosynchronous Earth
orbit (GEO) belt are only slightly perturbed by the gravity of the Moon. A spacecraft
in a 185km nearcircular orbit about the Earth has a Jacobi constant of approximately
58.0, though there is some variation depending on the location of the Moon, the time
of year, and the inclination of the orbit. If the spacecraft’s orbital altitude is increased,
its Jacobi constant decreases. A spacecraft in a 1000km nearcircular orbit has a
Jacobi constant near 51.5, a GPS satellite has a Jacobi constant near 14.6, a GEO
satellite has a Jacobi constant near 9.6, and so forth. The same trend exists for orbits
about the Moon. A spacecraft in a nearcircular lunar orbit at an altitude near 100 km
has a Jacobi constant near 5.5, and a satellite in a lunar orbit at an altitude near
1000 km has a Jacobi constant near 4.7, and so forth. A spacecraft on a direct transfer
to the Moon has a Jacobi constant in the vicinity of 2.3, depending on the particulars
38
METHODOLOGY
of the transfer. Likewise, a spacecraft on a lowenergy transfer to the Moon departs
with a Jacobi constant of about 0.8, though it changes signiﬁcantly before it arrives
at the Moon due to the gravity of the Sun.
One observes that a spacecraft with a smaller Jacobi constant can traverse further
from either central body. A useful analysis is to identify the boundary of possible
motion for a spacecraft with a particular Jacobi constant. These boundaries are
computed by setting the velocity of the spacecraft equal to zero in Eq. (2.6); they are
hence known as zerovelocity curves. Figure 23 illustrates the zerovelocity curves
for several Jacobi constants for motion in the x–y plane in the Earth–Moon system.
2.5.1.4 Forbidden Regions A spacecraft traversing the Earth–Moon system
with a Jacobi constant less than 2.988 (the approximate Jacobi constant of the L4 and
L5 points) can theoretically reach any point in the entire system. Its velocity in the
rotating frame will decrease to a minimum if it traverses through the L4 or L5 points,
but no region is inaccessible. Any spacecraft that has a Jacobi constant greater than
about 2.988 cannot physically reach all regions, but is bounded by the zerovelocity
curves to regions of space that permit its Jacobi constant value. Those regions in
space that the spacecraft cannot reach are known as forbidden regions.
Figure 23 An illustration of zerovelocity curves for several Jacobi constant values in the
planar Earth–Moon system.
MODELS
39
Consider a spacecraft with a Jacobi constant of 3.18 in the Earth–Moon system
(see Fig. 23). Its forbidden region encircles the Earth–Moon system, including the
L2 –L5 points. If the spacecraft begins at a point near the Earth or Moon, it can traverse
anywhere between the Earth and Moon within the corresponding zerovelocity curve,
including transferring through the gap at the L1 point. If the spacecraft begins well
outside of the system, then it must remain beyond the zerovelocity curve. It cannot
match the angular velocity of the rotating frame any nearer than its zerovelocity
curve.
2.5.1.5 Symmetries The existence of symmetries in the CRTBP is of particular
interest for some of the analyses encountered in this book. One symmetry that is
quite useful was demonstrated by Miele in his examination of image trajectories
in the Earth–Moon space [99]. He showed that if (x, y, z, x,
˙ y,
˙ z,
˙ t) is a solution
in the CRTBP, then (x, −y, z, −x,
˙ y,
˙ −z,
˙ −t) is also a solution. In other words,
if a trajectory is reﬂected about the xz plane, a valid trajectory is obtained by
traveling along the reﬂected trajectory in reverse. This property eliminates the need
to compute approach and departure trajectories separately in the CRTBP. Another
useful symmetry that exists in the CRTBP is that if (x, y, z, x,
˙ y,
˙ z,
˙ t) is a solution in
the CRTBP, then (x, y, −z, x,
˙ y,
˙ −z,
˙ t) is also a solution. This permits trajectories
to have a Northern and a Southern variety. Since the CRTBP approximates many
aspects of the real Solar System, one may also frequently use CRTBP reﬂections as
approximations for trajectories in the real Solar System.
2.5.2 Patched ThreeBody Model
The patched threebody model [38–40, 45, 46] uses purely threebody dynamics
to model the motion of a spacecraft in the presence of the Sun, Earth, and Moon.
It retains many of the desirable characteristics of the CRTBP, while permitting a
spacecraft in the nearEarth environment to be affected by all three massive bodies,
albeit only two massive bodies at any given moment. When the spacecraft is near the
Moon, the spacecraft’s motion is modeled by the Earth–Moon threebody system.
Otherwise, the spacecraft’s motion is modeled by the Sun–Earth threebody system,
where the secondary body is the barycenter of the Earth and Moon. For simplicity it
is assumed that the Earth–Moon system is coplanar with the Sun–Earth system. The
boundary of these two systems is referred to as the threebody sphere of inﬂuence
(3BSOI); it is analogous to the twobody sphere of inﬂuence used in the patched
conic method of interplanetary mission design.
Parker describes the 3BSOI as the boundary of a sphere centered at the Moon with
a radius rSOI computed using the following relationship [46]
rSOI = a
mMoon
mSun
2/5
(2.9)
where mMoon and mSun are the masses of the Moon and Sun, respectively, and a is the
average distance between the Sun and Moon, equal to approximately 1 astronomical
40
METHODOLOGY
unit (AU). Thus, the 3BSOI has a radius of approximately 159, 200 km, which is
large enough to include LL1 and LL2 .
2.5.3 JPL Ephemeris
The Jet Propulsion Laboratory and the California Institute of Technology have de
veloped the DE421 Planetary and Lunar Ephemerides, which is the most accurate
model of the Solar System used in this work. The model includes ephemerides of
the positions and velocities of the Sun, the four terrestrial planets, the four gasgiant
planets, the Pluto/Charon system, and the Moon [100]. The lunar orbit is accurate to
within a meter; the orbits of Earth, Mars, and Venus are accurate to within a kilometer
[100].
Lowenergy lunar transfers modeled in the patched threebody model repeat per
fectly from one synodic month to the next, since the dynamics and the Sun–Earth–
Moon geometry are perfectly symmetric. It is often possible to build a very similar
lowenergy lunar transfer from one month to the next in the DE421 model of the
Solar System, but its characteristics will vary in each month. This will be further
discussed in later chapters.
2.5.3.1 Earth Orbit The Earth–Moon system orbits the Sun in a nearly circular
orbit, but its nonzero eccentricity has an impact on the performance of a particular
lowenergy lunar transfer from one month to the next. Furthermore, its orbit changes
over time due to the inﬂuence of Jupiter and the other planets. Figure 24 illustrates
the Earth’s osculating eccentricity, semimajor axis, and inclination over time in the
DE421 model of the Solar System, relative to the Sun. One notices a nearly annual
signal in the eccentricity and a biannual signal in the semimajor axis. This is
predominantly due to the inﬂuence of Jupiter’s gravity, which has a synodic period
of about 399 days.
2.5.3.2 Lunar Orbit For the purposes of mission planning, the Moon’s orbit
about the Earth may be modeled as circular and coplanar with Earth’s orbit about the
Sun. In reality, the Moon’s orbit about the Earth is inclined by about 5.15 deg relative
to the ecliptic, and it has an average eccentricity of about 0.05490—quite a bit higher
than the Earth’s orbital eccentricity. Figure 25 illustrates the Moon’s osculating
eccentricity, semimajor axis, and inclination over time in the DE421 model of the
Solar System, relative to the Earth. The Moon’s orbit is strongly perturbed by the
gravity of the Sun on several time scales. First, one can see a very clear signal in the
time series of the Moon’s orbital parameters that has a frequency of about 29.53 days,
corresponding to the length of an average synodic month. Another very strong signal
in the time series of the Moon’s orbital parameters has a frequency of about 6 months,
corresponding to the biannual impact of the Earth’s orbit about the Sun. The relative
orientation of the Moon’s orbit to the Sun cycles over the course of a year, as well as
the distance to the Sun. Both the orientation and the distance have a direct effect on
the orbit. In addition to the solar perturbation, the planets Venus and Jupiter impact
the lunar orbit, as does the Earth’s asymmetric gravity ﬁeld.
LOWENERGY MISSION DESIGN
41
Figure 24 The instantaneous eccentricity (top), semimajor axis (middle), and inclination
(bottom) of the Earth–Moon barycenter over time relative to the Sun in the EMO2000
coordinate frame.
In addition to the three orbital parameters illustrated in Fig. 25, the orientation
of the Moon’s orbit about the Earth undergoes both secular and periodic variations.
Most notably, the Moon’s orbit precesses about the ecliptic North Pole. The period
of regression of the longitude of the lunar orbit’s ascending node (Ω) is equal to about
18.6 years. The period of precession of the lunar orbit’s argument of periapse (ω) is
equal to about 6.0 years. Finally, the period of precession of the longitude of periapse
(Ω + ω) is equal to about 8.85 years.
2.6
LOWENERGY MISSION DESIGN
The ﬁeld of lowenergy mission design relates to the study of trajectories that traverse
unstable threebody orbits and take advantage of the dynamics to perform orbit trans
fers using very little fuel. This section will describe threebody orbits, their unstable
manifolds, and how to construct lowenergy transfers between them. Indeed, an ex
42
METHODOLOGY
Figure 25 The instantaneous eccentricity (top), semimajor axis (middle), and inclination
(bottom) of the Moon over time relative to the Earth in the EMO2000 coordinate frame.
ample lowenergy lunar transfer is described later in dynamical systems terminology
as a trajectory that ﬁrst departs the Earth on the stable manifold of a Sun–Earth orbit,
transfers from the stable manifold to an unstable manifold, and traverses that until
it intersects the stable manifold of an orbit in the Earth–Moon system. This section
describes dynamical systems analyses and how those methods may be applied to
practical spacecraft mission design.
2.6.1
Dynamical Systems Theory
A dynamical system may be described as a state space with a set of rules, where the
rules govern the evolution of objects’ states through time within the system. The
rules are deterministic; that is, the evolution of a state through a particular amount of
time yields only one future state.
There are different types of dynamical systems depending on the mathematics
involved and the allowable values of time. If time is continuous, capable of taking
LOWENERGY MISSION DESIGN
43
any value in the set of real numbers, then the dynamical system is smooth and is
called a ﬂow. If time may only take discrete values, then the dynamical system is a
map. Models of the Solar System are generally described by ﬂows. A spacecraft’s
trajectory in such dynamical systems is the set of states that the spacecraft will take
as it moves through time, given its initial state. When integrating the equations of
motion for a spacecraft through time using a machine, time cannot truly take on any
value in the set of real numbers. The process of integration is a mapping of the
spacecraft’s state from one point in the state space to another point. A spacecraft’s
mapped trajectory is therefore only an approximation of the true trajectory.
There are many techniques that are commonly used to analyze dynamical systems.
In this work, we begin our analysis of the CRTBP by identifying ﬁxed points and
periodic orbits that exist in the system. We continue by studying the stability of those
solutions. These techniques provide an understanding of the motion of trajectories
near those solutions. Further analysis gradually provides more information about the
motion of trajectories throughout the dynamical system.
2.6.2
Solutions to the CRTBP
The CRTBP is a good example of a system in which dynamical systems methods of
analysis work well. The CRTBP contains ﬁve ﬁxedpoint solutions and an inﬁnite
number of periodic orbit solutions. The characterization of these solutions helps to
understand the ﬂow of particles and spacecraft in the system. Useful trajectories may
then be constructed that take advantage of the ﬂow in the system, rather than forcing
their way through the system. The following sections describe some of the simplest
solutions to the CRTBP.
2.6.2.1 FixedPoint Solutions: Five Lagrange Points The most basic so
lutions to the CRTBP are ﬁxedpoint solutions, that is, the trajectories in the CRTBP
that particles may follow such that they stay at rest in the system indeﬁnitely. There
are ﬁve such ﬁxedpoint solutions in the CRTBP, namely, the ﬁve Lagrange points.
These points were introduced in Section 2.5.1 and are displayed again in Fig. 26 for
the case of the Earth–Moon CRTBP.
The locations of the ﬁve Lagrange points in the Sun–Earth and Earth–Moon
circular threebody systems are given in Table 23, using the planetary masses and
distances provided in the Constants, page 382. Appendix A provides an analytical
derivation for the locations of the ﬁve Lagrange points for any threebody system,
as well as algorithms to determine their locations. Table 24 summarizes the Jacobi
constant of each of the ﬁve Lagrange points for both threebody systems.
2.6.2.2 Periodic and Quasiperiodic Orbit Solutions The CRTBP permits
the existence of numerous families of periodic and quasiperiodic orbits. Authors have
been studying such orbits since the late 1800s, though the introduction of modern
computing capability dramatically improved the quantity and complexity of orbits
that could be generated.
A periodic orbit in the threebody system may just be a twobody orbit about one
of the bodies that is slightly perturbed by the other massive body and is in resonance
44
METHODOLOGY
Figure 26 The locations of the ﬁve Lagrange points in the Earth–Moon CRTBP.
Table 23 The locations of the ﬁve Lagrange points in the Sun–Earth and
Earth–Moon circular threebody systems. The positions are given in nondimensional
normalized units and kilometers with respect to the barycenter of the system, assuming
the masses and distances given in the Methodology Introduction in Section 2.1.
Lagrange
Point
Position in normalized units
x
y
Position in kilometers
z
x
y
z
L1
L2
L3
L4
L5
0.9899859823
0
1.0100752000
0
−1.0000012670
0
0.4999969596
08660254038
0.4999969596 −08660254038
0
0
0
0
0
148, 099, 795.0
151, 105, 099.2
−149, 598, 060.2
74, 798, 480.5
74, 798, 480.5
0
0
0
129, 555, 556.4
−129, 555, 556.4
0
0
0
0
0
L1
Earth– L2
Moon L3
L4
L5
0.8369151324
0
1.1556821603
0
−1.0050626453
0
0.4878494157
08660254038
0.4878494157 −08660254038
0
0
0
0
0
321, 710.177
444, 244.222
−386, 346.081
187, 529.315
187, 529.315
0
0
0
332, 900.165
−332, 900.165
0
0
0
0
0
Sun–
Earth
with the motion of the primaries, that is, a low Earth orbit with a period that is
perfectly resonant with a sidereal month. Such an orbit has characteristics not unlike
any other low Earth orbit, except that its orbital elements were carefully selected to
be periodic with the Moon in the presence of the Moon’s perturbations. Further, such
a low Earth orbit is not quite periodic from one revolution to the next about the Earth,
LOWENERGY MISSION DESIGN
45
Table 24 The Jacobi constant of each Lagrange point in the Earth–Moon and the
Sun–Earth threebody systems, given in normalized coordinates.
Lagrange Point
Earth–Moon C
Sun–Earth C
L1
L2
L3
L4
L5
3.18834129
3.17216060
3.01214717
2.98799703
2.98799703
3.00089794
3.00089388
3.00000304
2.99999696
2.99999696
due to the Moon’s perturbations; it is only perfectly periodic over the course of its
resonant cycle with the Moon.
Alternatively, one can construct a trajectory that carefully balances the threebody
dynamics and can only exist in any form under the gravitational attraction of both
bodies. Examples of three families of such periodic orbits are illustrated in Fig. 27.
These orbits include libration orbits about the Earth–Moon L1 and L2 points and
distant prograde orbits about the Moon. It should be noted that the smallest distant
prograde orbits are very similar to twobody orbits about the Moon in resonance
with the orbital motion of the Earth and Moon. Clearly the libration orbits about the
Lagrange points only exist within a threebody system.
Between the 1890s and the 1930s, George Darwin [102, 103], George Hill [104],
Henry Plummer [105], Forest Moulton [106], Elis Stro¨ mgren [107], and their col
Figure 27 Several example orbits from three families of unstable periodic Earth–Moon
threebody orbits, viewed from above in the Earth–Moon synodic reference frame. The
orbits shown are from the family of Lyapunov orbits about L1 (left), the family of distant
prograde orbits about the Moon (center), and the family of Lyapunov orbits about L2 (right).
The arrows indicate the motion of objects traversing these orbits; the Moon’s orbital radius
about the Earth–Moon barycenter is shown in gray for reference [101] (Acta Astronautica by
International Academy of Astronautics, reproduced with permission of Pergamon in the format
reuse in a book/textbook via Copyright Clearance Center).
46
METHODOLOGY
leagues contributed to the discovery of the ﬁrst known periodic orbits in the circular
restricted threebody problem. Over the course of 40 years, it is unlikely that more
than 150 orbits were ever computed [108]; however, the general aspects of orbits in
the threebody problem became wellunderstood.
In the 1960s, modern computers became accessible, and numerical techniques
could be used to swiftly identify and compute periodic orbits. In 1968, Roger
Broucke published a large catalog of families of planar periodic orbits that exist in
the CRTBP with Earth–Moon masses [108]. Also in the 1960s, researchers computed
and cataloged a large number of threedimensional periodic orbits; signiﬁcant con
tributors include Michel He´ non [109–113], Arenstorf [114], Goudas [115], Bray and
Goudas [116, 117], and Kolenkiewicz and Carpenter [118], among others. Halo and
quasihalo orbits were discovered and analyzed beginning in the late 1960s (see, for
example, Farquhar [119], Farquhar and Kamel [120], Breakwell and Brown [121],
and Howell [122]). In 1980, David Richardson used the Lindstedt–Poincar´e method
to analytically produce periodic orbits, such as halo orbits, about the collinear libra
tion points [123]. Additional work was accomplished toward the end of the 20th
Century studying Lissajous and other quasihalo orbits (see, for example, Farquhar
and Kamel [120], Howell and Pernicka [124], and Go´ mez et al. [67, 125]). Many
authors have studied how to take advantage of libration orbits for practical spacecraft
missions, including scientiﬁc missions such as WMAP and SOHO, communication
relays [5–7, 11], transportation nodes [14, 126], and navigation services [8, 10–
13, 127, 128].
In this section, we demonstrate how to analytically construct one set of periodic
and quasiperiodic orbits that exist about each of the collinear Lagrange points. This
demonstration sheds light on why many periodic orbits exist [106, 123, 124].
We begin by translating the origin of the synodic frame to one of the collinear
libration points, Li . The parameter γ is deﬁned to be equal to the distance from Li
to the smaller primary. The value of γ is positive when referring to L2 and negative
when referring to L1 and L3 . The new position coordinates x' , y ' , and z ' are thus
deﬁned by the following
x'
=
x − (1 − µ + γ)
y
'
=
y
z
'
=
z
If we now linearize the equations of motion of the CRTBP given in Eqs. (2.1)–(2.3)
under this transformation, we ﬁnd the following
x
¨' − 2y˙ ' − (1 + 2c)x'
'
'
y¨ + 2x˙ + (c − 1)y
'
z¨ + cz
=
0
'
=
0
'
=
0
(2.10)
where c is a constant coefﬁcient. The analytical solution to the outofplane z motion
describes simple harmonic motion. The solution for the inplane x–y motion involves
a characteristic equation that has two real roots and two imaginary roots. The roots
LOWENERGY MISSION DESIGN
47
represent modes of motion, one divergent and one nondivergent. If the nondivergent
mode is excited, then the solution is bounded and may be written as
x'
=
−kAy cos (λt + φ)
y
'
=
Ay sin (λt + φ)
z
'
=
Az sin (ν t + ψ)
(2.11)
This motion is described by six variables: the amplitudes of the inplane and outofplane motion (Ay and Az ), the frequency of oscillation in the inplane and outofplane motion (λ and ν), and the phase angles for the inplane and outofplane
motion (φ and ψ). The linearized approximation to the analytical solution for periodic
motion about a Lagrange point may thus be characterized by oscillatory motion. If
the two frequencies λ and ν are equal or otherwise commensurate, the resulting
motion will be periodic; if the frequencies are incommensurate, the resulting motion
will be quasiperiodic. The periodic orbits whose frequencies are equal are known as
halo orbits, the more general quasiperiodic trajectories are known as Lissajous orbits
or quasihalo orbits. A portion of the family of halo orbits about the Earth–Moon L2
is shown in Fig. 28, and characteristic views of several types of Lissajous orbits are
shown in Fig. 29.
It should also be noted that there is a symmetry that exists in the CRTBP, as
described in Section 2.5.1. If the CRTBP permits an orbit to exist, then it also
permits a symmetric orbit to exist that is a reﬂection across the y = 0 plane. Hence,
there are two families of halo orbits, a northern and a southern. By convention, if a
spacecraft spends more than half of its time above the y = 0 plane in a halo orbit,
then the spacecraft is following a northern halo orbit.
If Az is set to zero in Eq. (2.11) the resulting orbits are planar and are known as
Lyapunov orbits. Figure 27 shows a portion of the families of Lyapunov orbits about
L1 and L2 in the Earth–Moon system.
These orbits may be constructed analytically since the linearization process near
one of the Lagrange points produces a good approximation of the true dynamics
found in the system. Other orbits do not have welldescribed linear approximations
and must be constructed numerically. The process of numerically constructing simple
periodic orbits is discussed in Section 2.6.6.
Periodic orbits in the threebody system exist that revolve about all ﬁve Lagrange
points, the primary, the secondary, and also about the entire system. Periodic orbits
exist that revolve about either body in a prograde sense and a retrograde sense.
One may construct simple symmetric periodic orbits, such as those illustrated in
this section, and one may construct asymmetric, complex orbits. A wide variety of
periodic orbits exist that may be useful to the mission planner.
2.6.2.3 Orbit Parameters An orbit and a position in that orbit may be uniquely
speciﬁed in the twobody problem using six parameters. Typical sets of twobody
parameters include the Cartesian and Keplerian sets. Parameterization of orbits in
the threebody problem has proven to be much more difﬁcult, since there are no
general analytical solutions to the system. Dynamical systems theory is very useful
48
METHODOLOGY
Figure 28 A portion of the family of halo orbits about L2 in the Earth–Moon system, shown
from four perspectives.
in this regard because the methodology lends itself to many useful parameters. One
such parameter, τ , is useful when describing periodic orbit solutions to the CRTBP.
This parameter is described here; others are introduced in later chapters as their uses
become apparent.
The parameter τ mimics the twobody mean anomaly. For the case of halo orbits,
and other symmetric periodic orbits in the CRTBP, τ advances at a steady rate over
time, beginning at one landmark (typically where the orbit pierces the y = 0 plane)
and resetting when it completes an entire period. In some studies, τ takes on values
in the range of 0–360 deg, much like the mean anomaly [11]. In other studies, τ
is deﬁned to take on values in the range of 0–1, indicating the periodic revolution
number [46]. Most libration orbits, for example, halo and Lyapunov orbits, have a
shape that resembles a conic section; in those cases it is intuitive to use an angular
unit of measurement for τ . However, there are many classes of periodic orbits that
LOWENERGY MISSION DESIGN
49
Figure 29 A sample of Lissajous curves representing the view of Lissajous orbits in
the Earth–Moon system as viewed from an observer at the Earth looking toward the Moon;
ωinplane and ωout are multiples of some base frequency ωbase . The curves on the left are
perfectly periodic; the curves on the right have incommensurate frequencies and have only
been propagated for a short amount of time (ﬁrst published in Ref. [97]; reproduced with kind
permission from Springer Science+Business Media B.V.).
do not resemble any sort of conic section, and it may be confusing to refer to τ in
angular units. Figure 210 shows two orbits, demonstrating how τ advances along
each orbit, where τ has been represented as a revolution number for a complex orbit
and as an angle for an L2 libration orbit.
2.6.3
Poincare´ Maps
A Poincare´ map is a useful tool for analyzing dynamical systems and is often used to
identify orbits and orbit transfers in a complex system. A Poincare´ map is created by
intersecting a trajectory in the ndimensional ﬂow x˙ = f (x) by an (n−1)dimensional
surface of section Σ. Thus, the Poincar´e mapping replaces the ﬂow of an nth order
system with a discrete system of order (n − 1) [129]. A Poincar´e mapping, P , may
be described as a function that maps the state of a trajectory at the kth intersection
with the surface of section, xk , to the next intersection, xk+1
xk+1 = P (xk )
(2.12)
If a trajectory pierces Σ at the state x∗ at time t and then returns to x∗ at time t + T ,
then one may conclude that the trajectory is a periodic orbit with a period T [130].
There are three different types of Poincare´ maps considered in this research,
deﬁned as follows [130]:
50
METHODOLOGY
Figure 210 The two orbits shown demonstrate how the parameter τ advances from 0 to 1
about a complex orbit (left) or from 0 deg to 360 deg about a libration orbit (right). Both orbits
are viewed from above in the Earth–Moon CRTBP synodic frame.
• P+ : The Poincare´ map created from only the positive intersections of the
trajectory with the surface of section. For instance, in the CRTBP, Σ may
be deﬁned as a y–z plane set to some xvalue and P+ includes only those
intersections that have positive values of x˙ .
• P− : The Poincare´ map created from only the negative intersections of the
trajectory with the surface of section.
• P± : The Poincare´ map created from all intersections of the trajectory with the
surface of section.
The maps P+ and P− are called onesided maps, while P± is called a twosided map
[130]. Figure 211 provides a simple illustration of a onesided Poincar´e mapping of
two orbits, where one is periodic and one is not immediately periodic.
2.6.4
The State Transition and Monodromy Matrices
The state transition matrix Φ(t, t0 ) monitors the divergent dynamics along a trajectory.
Essentially, it approximates how a slight deviation in any of the state variables
propagates along the trajectory. Its practical uses are twofold in this study:
1. to provide a means of adjusting the initial conditions of a trajectory to correct
for unwanted motion, and
2. to provide information about the stability of an orbit, including the orientation
of the eigenvectors along the orbit.
LOWENERGY MISSION DESIGN
51
Figure 211 An illustration of a onesided Poincare´ mapping of two trajectories. The
point x∗ indicates a ﬁxed point on the surface, corresponding to a periodic trajectory [101]
(Acta Astronautica by International Academy of Astronautics, reproduced with permission of
Pergamon in the format reuse in a book/textbook via Copyright Clearance Center).
The second use involves the monodromy matrix, a special case of the state transition
matrix. We explore (1) in Sections 2.6.5 and 2.6.6 and (2) in Sections 2.6.8 and 2.6.10.
This section discusses how to construct the state transition matrix and the monodromy
matrix.
Let us deﬁne the state vector X to be a column vector that contains all of the state
variables of interest in the system. In most cases, one is usually only interested in
computing the six state variables of a particle or spacecraft in a system. Hence, X is
deﬁned as
T
X = [x y z x˙ y˙ z˙]
Then the state transition matrix is a 6 × 6 matrix composed of the partial derivatives
of the state
∂X(t)
(2.13)
Φ(t, t0 ) =
∂X(t0 )
with initial conditions Φ(t0 , t0 ) = I. The state transition matrix is propagated using
the following relationship
˙ t0 ) = A(t)Φ(t, t0 )
Φ(t,
(2.14)
where the matrix A(t) is equal to
A(t) =
˙ (t)
∂X
∂X(t)
(2.15)
52
METHODOLOGY
In the CRTBP, A(t) is equal to
⎡
A(t) =
0
UX X
I
2Ω
,
where
⎤
0 1 0
Ω = ⎣ −1 0 0 ⎦
0 0 0
(2.16)
and UX X is the symmetric matrix composed of second partial derivatives of U with
respect to the third body’s position evaluated along the orbit
⎡
⎤
∂x
¨ ∂x
¨ ∂x
¨
⎢ ∂x ∂y ∂z ⎥
⎢
⎥
⎢ ∂y¨ ∂y¨ ∂y¨ ⎥
⎢
⎥
(2.17)
UX X = ⎢
⎥
⎢ ∂x ∂y ∂z ⎥
⎢
⎥
⎣ ∂ z¨ ∂ z¨ ∂ z¨ ⎦
∂x ∂y ∂z
The monodromy matrix, M , exists for periodic orbits and is computed by propa
gating the state transition matrix one entire orbit: M = Φ(t0 + P, t0 ) [131]. After
being propagated for a full orbit, the matrix contains information about every region
that a spacecraft would pass through along that orbit. This matrix’s use is further
explored in Sections 2.6.8 and 2.6.10.
2.6.5 Differential Correction
Differential correction, as it is implemented here, is a process by which the state
transition matrix is used to change a set of initial conditions in order to better satisfy
a set of criteria. It is a targeting scheme that converges on its constraints very swiftly
within the basin of convergence. Two types of differential correction routines are
used in this work: singleshooting and multipleshooting correction routines.
2.6.5.1 SingleShooting Differential Correction In the standard singleshooting differential correction routine used in this work, a spacecraft begins at
some state X0 , following a nominal trajectory T (t), where X0 is composed of a po
sition vector R0 and a velocity vector V0 . It is desired that the spacecraft’s trajectory
ˆf
be shifted such that at a later time, tf , the trajectory encounters a desired state X
ˆ
ˆ
(including a desired position vector Rf and a velocity vector Vf ). There are usually
two constraints to the problem: (1) the spacecraft’s initial position may not change;
and (2) the spacecraft’s new trajectory leads it to a ﬁnal speciﬁed position vector
ˆ f . The routine is allowed to vary the initial velocity of the spacecraft (simulating
R
a change in velocity (ΔV) in the mission design), and is oftentimes allowed to vary
the time at which the spacecraft arrives at its ﬁnal desired position. The velocity of
the spacecraft at the ﬁnal position is usually a free variable, and mission designers
typically plan to perform an additional ΔV at that time. If the routine converges, a
new ballistic trajectory is constructed, Tˆ(t), that satisﬁes the two conditions
ˆ0
Tˆ(tˆ0 ) = X
ˆf
Tˆ(tˆf ) = X
ˆ 0 = R0 , and V
ˆ 0 free
with tˆ0 = t0 , R
ˆ f constrained, and V
ˆ f free
with tˆf constrained or ﬁxed, R
LOWENERGY MISSION DESIGN
53
This routine is diagrammed in Fig. 212.
The singleshooting method uses the state transition matrix Φ(tf , t0 ) to estimate
what change to make in the initial velocity of the state, ΔV0 , in order to eliminate
the deviation in position at the end of the trajectory δRf . The state transition matrix
maps perturbations in the state over time using the following linearized equations
δXf
δRf
δVf
=
Φ(tf , t0 )ΔX0
=
ΦRR (tf , t0 )
ΦV R (tf , t0 )
or
ΦRV (tf , t0 )
ΦV V (tf , t0 )
ΔR0
ΔV0
(2.18)
Since ΔR0 = 0 and δVf is unconstrained, we may simplify Eq. (2.18) and solve for
ΔV0 to ﬁnd
−1
ΔV0 = [ΦRV (tf , t0 )] δRf
(2.19)
Since the state transition matrix is propagated with linearized equations, the al
gorithm must be iterated until convergence. When the algorithm is converging, each
iteration typically improves the solution by a factor of 10, although factors anywhere
between 2 and 100 have been observed [46].
2.6.5.2 MultipleShooting Differential Correction Multipleshooting dif
ferential correction takes a series of states and adjusts them all simultaneously to
construct a complicated trajectory that satisﬁes a set of constraints. It is very useful
when mission designers wish to construct a long trajectory in an unstable environment
in the presence of machine precision. For example, the Genesis spacecraft departed
the Earth, traversed a quasihalo orbit about the Sun–Earth L1 point several times,
transferred to the vicinity of the Sun–Earth L2 point, and then returned to Earth. This
entire trajectory may in theory be constructed without a single maneuver. However,
in this unstable environment, deviations even as small as roundoff errors due to ma
chine precision grow exponentially. A computer using ﬁniteprecision mathematics
Figure 212 The singleshooting differentialcorrection routine. The solidline trajectory,
T (t), is the initial trajectory; the dashedline trajectory, Tˆ(t), is the corrected trajectory that
encounters the target position, indicated by a bull’s eye, at the target time.
54
METHODOLOGY
does not normally have the precision required to propagate the spacecraft through its
entire trajectory before the roundoff errors grow large enough to create a largescale
deviation in the spacecraft’s trajectory. To get around this numerical problem, mission
designers break the trajectory into many segments and patch the segments together
with very small maneuvers. The maneuvers counteract the buildup of propagated
roundoff errors. They may be as small as desired, depending on the length of each
patched segment, and are typically much smaller than any expected stationkeeping
maneuver. Hence, they are not normally considered to be deterministic maneuvers
in the mission.
The multipleshooting differential correction method is described in detail by
Pernicka [132, 133] and by Wilson [134, 135], among other authors [136–138].
This section provides a summary of the process. Section 2.6.5.3 then returns to the
details and derives the tools needed to implement the multipleshooting differential
correction technique. The derivation follows that presented by Wilson [134].
The ﬁrst step in the process of differentially correcting a complex trajectory is to
deﬁne a series of patchpoints. Several things must be considered when setting up the
patchpoints; a discussion of these considerations is given below. From then on, a twolevel process is iterated until either the differential corrector fails or each constraint in
the system is satisﬁed. The ﬁrst level of the process adjusts the patchpoints’ velocities;
the second level of the process adjusts the patchpoints’ positions and epochs. If the
patchpoints fall into some basin of convergence for the differential corrector, then
the process converges on a continuous trajectory swiftly. The following paragraphs
provide more details about the twolevel process:
Level 1. The velocities of every patchpoint along the trajectory except the last one
are adjusted using the singleshooting differential corrector given in Sec
tion 2.6.5.1. The velocities are set such that the position of each segment
ends at the following patchpoint. When this step has been completed, the
trajectory is continuous, although a spacecraft must perform a ΔV at each
patchpoint in order to follow the trajectory.
Level 2. The positions and epochs of every patchpoint, including the last patchpoint,
are adjusted using a leastsquares method that is designed to reduce the total
ΔV cost of the trajectory. The result is a discontinuous trajectory that should
require less total ΔV after the following iteration of Level 1.
This iteration process is repeated until the discontinuity at each patchpoint in position
and velocity is below some tolerance.
The choice of patchpoints has a strong effect on the differential corrector’s perfor
mance. First of all, since the singleshooting method is invoked between every adja
cent patchpoint, the patchpoints must be close enough to permit the singleshooting
method to converge within the desirable tolerance given the numerical precision of
the machine. That is, if the patchpoints are too far apart, Level 1 will not converge.
Secondly, Level 2 of the twolevel process is generally designed with the assumption
that each patchpoint is evenly spaced in time. If the patchpoints are not evenly spaced
LOWENERGY MISSION DESIGN
55
in time, then the time system should be normalized in some way. This improves the
convergence characteristics of the algorithm, but it certainly does not guarantee that
the system will converge on a desirable solution. Finally, it has been observed that
the differential corrector converges more readily if there are more patchpoints where
the dynamics become more unstable, for example, near massive bodies, though the
time system often must be normalized for this to beneﬁt the stability of the algorithm.
Section 2.6.5.1 provides the details of Level 1. In order to shed some light onto
the functionality of Level 2, let us assume that we have a trajectory built from three
patchpoints. The two segments meet in the middle at the second patchpoint, but are
not continuous in velocity, that is, there is some velocity deviation ΔV2 . Let us
assume that it is desirable to remove that discontinuity, and to do so we allow the
positions and epochs of all three patchpoints to be adjusted. Section 2.6.5.3 provides
details about how to numerically compute the variations of the target (ΔV2 ) to the
controls (R1 , t1 , R2 , t2 , R3 , t3 ). In this way, one can construct the following partial
derivatives
∂ΔV2
∂ΔV2
∂ΔV2
,
,
,
∂R1
∂R2
∂R3
∂ΔV2
∂ΔV2
∂ΔV2
,
,
∂ t1
∂ t2
∂t3
With these partial derivatives, one can build an approximation of the change in ΔV2
when each of the control parameters are perturbed
⎤
δR1
δt1 ⎥
∂ΔV2 ∂ΔV2 ∂ΔV2 ∂ΔV2 ∂ΔV2 ∂ΔV2 ⎢
⎢ δR2 ⎥
[δΔV2 ] =
⎢ δt ⎥
2 ⎦
∂ t1
∂R2
∂ t2
∂R3
∂t3
∂R1
⎣
�
�
δR3
δt3
[M ]
(2.20)
In general, we wish to determine the appropriate changes to make to each of the
control variables in order to reduce the value of ΔV2 to zero. The linear system given
in Eq. (2.20) is underdetermined; it is common practice to use the smallest Euclidean
norm to produce a good solution [134]
⎡
⎤
δR1
⎢ δt1 ⎥
⎢
⎥
⎢ δR2 ⎥
T
T −1
⎢
⎥
[δΔV2 ]
(2.21)
⎢ δt2 ⎥ = M M M
⎢
⎥
⎣ δR3 ⎦
δt3
⎡
These deviations in position and epoch are then added to the patchpoints’ states
to complete the Level 2 iteration. This example includes only three patchpoints;
additional patchpoints may be added on indeﬁnitely. With many patchpoints in
the system, the majority of the matrix M is ﬁlled with zeros, since each velocity
56
METHODOLOGY
discontinuity is only dependent on the positions and epochs of the three nearest
patchpoints.
In its simplest form, the Level 2 corrections are only constrained by the velocity
discontinuities at each patchpoint. Wilson describes how to add many other types
of constraints to the differential corrector [134]. Some examples of constraints that
may be added include:
• Desirable Position Vector. One may target a particular position vector or
position magnitude for any patchpoint in the trajectory. This may be with
respect to a point in the coordinate axes or with respect to another body.
• Desirable Inclination. One may target many different orbital parameters, such
as the inclination of one or more speciﬁed patchpoints.
• Maximum Change in Position. One may limit the differential corrector’s
capability to change one or more patchpoints’ positions during each iteration
of Level 2. This helps to keep a trajectory near some initial guess.
Many other types of constraints may be placed on the system. The inclusion of
additional constraints is very useful for practical spacecraft missions, where the
trajectory must be designed to begin from a particular state or to end at a particular
state; however, it does often make it more difﬁcult for the differential corrector to
converge.
There are many practical applications of the multipleshooting differential cor
rector. To demonstrate its use, we will examine its performance as it is used to
differentially correct a periodic halo orbit from the CRTBP into a quasihalo orbit in
the DE421 model of the Solar System. Figure 213 provides several representative
plots of the differential corrector in action. The plots are exaggerated to demonstrate
the procedure clearly. The plot shown in (a) depicts the initial periodic halo orbit in
the CRTBP. The trajectory is broken into four segments, separated by ﬁve patchpoints
as shown in (b), where the ﬁfth patchpoint is coincident with the ﬁrst in the synodic
frame. The ﬁrst iteration of Level 1, shown in (c), forces the new trajectory to be
continuous in position and time in the DE421 model, but permits velocity discontinu
ities at each interior patchpoint. The ﬁve patchpoints’ positions and epochs are then
adjusted in the ﬁrst iteration of Level 2 as shown in (d). The plots shown in (e) and
(f) give an exaggerated representation of the second iteration of Levels 1 and 2. The
plot shown in (g) depicts the trajectory after the third iteration of Level 1; one can
see that the trajectory is approaching a continuous trajectory. The plot shown in (h)
depicts the ﬁnal, converged trajectory that is continuous in the DE421 model within
some tolerance limits.
Studying Figure 213, one can see that the differential corrector permits the ends
of the trajectory to be altered substantially since there are no boundary conditions. If
more revolutions of the halo orbit were originally sent into the differential corrector,
then the ﬁnal trajectory would resemble the original halo orbit more closely. This
process is shown in more detail in Section 2.6.6.3.
The multipleshooting differential corrector typically operates on a set of patchpoints that deﬁne a single trajectory, presumably to be followed by a single spacecraft.
LOWENERGY MISSION DESIGN
57
Figure 213 An exaggerated demonstration of the implementation of the multipleshooting
differential corrector used to convert a halo orbit from the CRTBP into the DE421 model.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
The initial CRTBP halo orbit.
The initial placement of patchpoints in the DE421 model.
Level 1: Differential correction to determine the ΔVs necessary to make the trajectory continuous
in the DE421 model.
Level 2: The adjustments of the patchpoints’ positions and epochs to reduce the total ΔV.
Level 1: The second adjustments of the patchpoints’ velocities to make the trajectory continuous.
Level 2: The second adjustments of the patchpoints’ positions and epochs to reduce the total ΔV.
Level 1: The third adjustments of the patchpoints’ velocities to make the trajectory continuous.
The ﬁnal converged trajectory in the DE421 model after several additional iterations.
58
METHODOLOGY
However, the differential corrector may certainly be designed to operate on segments
that represent more than one spacecraft, including segments that branch, segments
that rendezvous, and/or segments that deﬁne a formation.
2.6.5.3 MultipleShooting Implementation The multipleshooting differen
tial corrector is such a useful tool in the design of lowenergy trajectories that further
attention is given here to derive the algorithms needed to implement it. As described
earlier, the multipleshooting differential corrector involves a process that repeats
two steps until a trajectory is generated that satisﬁes all given constraints. Level 1 is
fully described in Section 2.6.5.1, including everything needed to generate software
to implement it. Level 2 is introduced in Section 2.6.5.2, but the details have been
omitted in order to demonstrate its operation. Those details are provided here.
The engine of the most basic implementation of Level 2 is given by Eq. (2.21),
which computes a linear approximation of the changes that must be made to the
positions and/or times of the three patchpoints in the scenario in order to reduce
the ΔV at the interior patchpoint. The multipleshooting differential corrector may
certainly be extended to include many trajectory segments and a wide variety of
constraints. Further, the trajectory segments are not restricted to a single trajectory,
but may deﬁne multiple trajectories that are simultaneously optimized.
This section begins by describing a basic formulation of Level 2 that involves a
single trajectory deﬁned by at least three patchpoints such that the only goal is to
reduce the total ΔV required to traverse that trajectory. Next, the section describes the
algorithms required to add constraints to the patchpoints in the trajectory. Finally, the
section includes a discussion about how to implement the multipleshooting technique
such that it operates on several codependent trajectories simultaneously. In each case,
it is always assumed that a ΔV or constraint applied to a particular patchpoint is only
affected by the position and/or time of that patchpoint and its neighbors, which is an
important feature in the formulation of Level 2.
Basic Level 2. The basic Level 2 formulation is one that operates on a single
trajectory and works only to reduce the ΔV of each interior patchpoint. It is assumed
that the position and/or time of each patchpoint may be changed to accomplish this
goal. Hence, the ΔV at the second patchpoint, ΔV2 , may be reduced by changing
the position, the time, or both of the ﬁrst, second, and third patchpoints. Any other
patchpoints do not directly inﬂuence ΔV2 , though their inﬂuences are transmitted
through the connecting patchpoints.
Equation (2.21) captures the linear estimate of the change in the positions and
times of three patchpoints needed to reduce ΔV2 , the velocity discontinuity at the
interior patchpoint. This expression may be extended to include multiple patchpoints
as follows
LOWENERGY MISSION DESIGN
⎡
δR1
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎥
⎥
δt1 ⎥
⎥
⎥
δR2 ⎥
⎥
⎥
δt2 ⎥
⎥
⎥
δR3 ⎥ = M T M M T
⎥
⎥
δt3 ⎥
⎥
.. ⎥
. ⎥
⎥
⎥
δRn ⎥
⎦
59
⎤
−1
⎡
δΔV2
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
δΔV3
..
.
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(2.22)
δΔVn−1
δtn
where the matrix M is constructed using the relationship
⎡
⎡
⎢
⎢
⎢
⎣
δΔV2
δΔV3
..
.
δΔVn−1
Thus, M is equal to
δR1
δt1
δR2
δt2
δR3
δt3
..
.
⎢
⎢
⎢
⎢
⎢
⎥
⎢
⎥
⎢
⎥=M⎢
⎢
⎦
⎢
⎢
⎢
⎢
⎣ δRn
δtn
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(2.23)
60
METHODOLOGY
⎡
∂ΔV2 ∂ΔV2 ∂ΔV2 ∂ΔV2 ∂ΔV2
⎢ ∂R1 ∂ t1 ∂R2 ∂ t2 ∂R3
∂ΔV2
∂ t3
0
∂ΔV3 ∂ΔV3 ∂ΔV3
∂R2 ∂ t2 ∂R3
∂ΔV3
∂ t3
∂ΔV3
∂R4
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
.
..
0
0
0
.
..
0
.
..
0
0
∂ΔV3
∂ t4
...
...
...
...
0
0
0
0
.
..
∂ΔVn−1 ∂ΔVn−1 ∂ΔVn−1 ∂ΔVn−1 ∂ΔVn−1 ∂ΔVn−1
∂Rn−2 ∂ tn−2 ∂Rn−1 ∂ tn−1
∂Rn
∂ tn
�
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
�
[M ]
(2.24)
In order to generate M , one requires knowledge of each of the partials given in
Eq. (2.24). Fortunately, each of these partials may be constructed using the state
transition matrix, provided that the linear approximations are acceptable. In order to
derive the formulae needed to represent each of these partials, we must examine the
problem deﬁnition more closely.
We again consider the ﬁrst two segments, deﬁned by the ﬁrst three patchpoints:
P1 , P2 , and P3 . Each of these patchpoints is characterized by its position R, velocity
V, and time t. After the application of the Level 1 correction, Segment 1 traverses
from P1 to P2 and Segment 2 traverses from P2 to P3 . The resulting trajectory is
continuous in position over time (within some small tolerance at P2 ) and continuous
in velocity over time except at P2 , where ΔV2 deﬁnes the difference between V2+
(the velocity at the start of Segment 2) and V2− (the velocity at the end of Segment 1)
ΔV2 = V2+ − V2−
The superscripts “−” and “+” differentiate between the incoming and outgoing
parameters, respectively, at a particular patchpoint. The position, velocity, and time
−
−
of the end of Segment 1 are indicated as R−
2 , V2 , and t2 , respectively. Likewise,
the position, velocity, and time of the initial state of Segment 2 are indicated as R+
2,
+
−
+
V2+ , and t2+ , respectively. After applying Level 1 to P1 , R−
2 = R2 and t2 = t2 .
These are ﬁxed constraints and assumed in the formulation of Level 2.
The state transition matrix, Φ, may be mapped from P1 to P2 and from P3 to P2 to
approximate the response of V2− and V2+ , respectively, given a change in the states
of P1 and P3 . The basic Level 2 formulation deﬁnes the state transition matrix to be
a 6 × 6 matrix as given in Eq. (2.18)
δRf
δVf
=Φ
ΔR0
ΔV0
where we indicate a change in parameters performed by the user by a “Δ” and the
response by a “δ”. The 6 × 6 state transition matrix may be broken up into four 3 × 3
submatrices as illustrated previously in Eq. (2.18) and repeated here
δRf
δVf
=
ΦRR (tf , t0 )
ΦV R (tf , t0 )
ΦRV (tf , t0 )
ΦV V (tf , t0 )
ΔR0
ΔV0
(2.25)
LOWENERGY MISSION DESIGN
61
In order to simplify the nomenclature, we abbreviate the pieces of Eq. (2.25) as
follows, where the state transition matrix is now demonstrating a mapping of the
deviations from patchpoint P1 to patchpoint P2
δR−
2
δV2−
=
A21
C21
B21
D21
ΔR+
1
ΔV1+
(2.26)
This simpliﬁed notation is commonly found in literature [134]. Using this simpliﬁ
cation, the matrix C23 would describe the change in V2+ caused by a perturbation in
the position of P3 , namely, R−
3.
The Level 1 differential corrector process given in Eq. (2.19) may be written using
this simpliﬁed notation as follows, where we have again applied it to describe the
linear approximation of the change in velocity of P1 needed to achieve a position
difference at the time t2
−1
ΔV1+ = B21
δR−
2
The following two linear systems represent approximations of the changes in P2
that are caused by deviations in the patchpoints P1 and P3 [134], where it is assumed
that the deviations are all small enough to be in the linear regime of the dynamics
along each trajectory
− −
δR−
2 − V2 δt2
−
− −
δV2 − a2 δt2
=
A21
C21
B21
D21
+
+
ΔR+
1 − V1 Δt1
+
+
+
ΔV1 − a1 Δt1
(2.27)
δR2+ − V2+ δt2+
+
δV2+ − a+
2 δt2
=
A23
C23
B23
D23
−
−
ΔR−
3 − V3 Δt3
−
−
−
ΔV3 − a3 Δt3
(2.28)
The formulation for this particular Level 2 differential corrector includes the
ﬁxed constraints that the trajectory be continuous in position and time across each
+
−
+
patchpoint. Hence, R−
2 = R2 = R2 and t2 = t2 = t2 . For most applications, this
−
+
also implies that a2 = a2 = a2 , though that may not be the case in the presence
of dynamics that are velocitydependent, such as atmospheric drag. These ﬁxed
constraints will be applied to each and every patchpoint in turn as the matrix M is
constructed.
The targets for this Level 2 are that V2− = V2+ in order that the trajectory require
no ΔV. The formulation is nearly identical for the case when a mission designer
wishes to specify that a particular ΔV be performed at a patchpoint. Hence, the more
general target is given by
ˆ 2 − V+ − V− = 0
ΔV
2
2
(2.29)
ˆ 2 is speciﬁed by the designer.
where the vector ΔV
ˆ 2 include the
As described earlier, the controls available to achieve the target ΔV
position vectors and times of P1 , P2 , and P3 . The controls and constraints applied to
ˆ 2 permit V+ and V− to be free variables, though those may
achieve the target ΔV
1
3
be targeted by neighboring constraints as the matrix M is constructed.
62
METHODOLOGY
In summary, there are 28 parameters involved with the goal of achieving a desirable
ΔV across P2 , including the position, velocity, and time at the beginning and end of
each trajectory segment, organized as follows
+
R−
t2− = t2+ = t2
2 = R2 = R2 ,
ΔR1 , Δt1 , ΔR2 , Δt2 , ΔR3 , Δt3
ΔV1+ , ΔV3−
ˆ 2 − V+ − V−
δV+ − δV− = ΔV
Fixed Constraints:
Controls:
Free Variables:
Targets:
2
2
2
2
A similar set of parameters is deﬁned for each patchpoint that is included in the
differential correction process.
The ﬁrst row of the M matrix requires six partial derivatives. These in turn require
other partial derivatives, as follows
∂ΔV2
∂V2−
=−
∂R1
∂R+
1
∂ΔV2
∂V2+
∂V2−
=
− −
∂R2
∂R2
∂R+
2
∂ΔV2
∂V2+
=
∂R3
∂R−
3
∂ΔV2
∂V−
= − +2
∂ t1
∂ t1
∂ΔV2
∂V2+
∂V2−
=
−
∂t2
∂ t−
∂ t+
2
2
∂ΔV2
∂V2+
=
∂ t3
∂ t−
3
Wilson provides details to construct each of these partials [134]; we will demon
∂V−
strate the process and illustrate the construction of ∂R2+ ; the process may be applied
1
in the same manner to construct each of these partials.
In order to construct
control to zero, namely
∂V2−
,
∂R+
1
we ﬁrst set the perturbation of every other independent
δt1
=
δR2
δt2
=
=
δR3
δt3
=
=
0
0
These values may then be inserted into Eqs. (2.27) and (2.28) or their inverses,
whichever generates the most practical result. There are often many ways to describe
the partials, and we are interested in the simplest relationships. For this particular
case, the simplest relationship comes from substituting these values into the inverse
of Eq. (2.27)
δR1+ − V1+ δt1+
+
δV1+ − a+
1 δt1
=
A12
C12
B12
D12
− −
δR−
2 − V2 δt2
−
− −
δV2 − a2 δt2
(2.30)
δR+
1
δV1+
=
A12
C12
B12
D12
0
δV2−
(2.31)
This yields a system of two equations
δR+
1
δV1+
=
B12 δV2−
=
D12 δV2−
and
(2.32)
(2.33)
LOWENERGY MISSION DESIGN
63
The ﬁrst equation provides the relationship we are interested in, namely
δV2−
−1
= B12
δR+
1
(2.34)
Be aware that although Φ21 = Φ−1
12 , the submatrices do not typically follow such
−1
inverse relationships; that is, B12
= B21 .
This procedure may be followed to generate relationships for each partial required
for the production of the matrix M . The result is the following
∂ΔV2
∂R1
=
−
∂V2−
−1
= −B12
∂R+
1
∂ΔV2
∂ t1
=
−
∂V2−
−1 +
= B12
V1
∂t+
1
∂ΔV2
∂R2
=
∂V2+
∂V2−
−1
−1
−
= −B32
A32 + B12
A12
∂R2−
∂R+
2
∂ΔV2
∂ t2
=
∂V2+
∂V2−
−
=
∂ t2−
∂t+
2
∂ΔV2
∂R3
=
∂V2+
−1
= B32
∂R−
3
∂ΔV2
∂ t3
=
∂V2+
−1 −
= −B32
V3
∂ t−
3
−1
−1
a2+ − a2− + B32
A32 V2+ − B12
A12 V2−
Finally, we have all of the pieces to use Eq. (2.22) to determine an approximation
of the adjustments that must be made in the positions and times of each patchpoint
as a function of the unwanted velocity changes in each patchpoint.
Level 2 with Constraints. The Level 2 differential corrector can be modiﬁed to
place a wide variety of constraints on the patchpoints in the system. For instance,
we already observed in the derivation of the partials above that it is quite arbitrary
to enforce the ΔV at each patchpoint to zero; rather, one can specify a list of ΔV
values to perform at particular times and drive the trajectory to that solution instead.
Before doing that, we must have a way of preventing the Level 2 corrector from
adjusting a patchpoint’s time. This is one example of a constraint that may be
placed on the system. Other examples include constraining a patchpoint to have a
particular inclination relative to some body, or to be located at a particular position
or distance relative to a body. These constraints are very important when designing
a practical trajectory for a spacecraft mission. For instance, the trajectory being
designed may be an extension to a spacecraft’s mission that is already in orbit, such
that the trajectory must originate from the spacecraft’s current trajectory. Or perhaps
the trajectory being designed must land on the Moon at a particular landing site. The
multipleshooting differential corrector can accommodate any of these scenarios.
64
METHODOLOGY
Any constraint may be added to the Level 2 architecture as long as it may be
described in the form
(2.35)
αij = f (Ri , Vi , ti )
where the subscript i represents the patchpoint that the constraint is placed upon
and the subscript j indicates the constraint number applied to that patchpoint. This
nomenclature is consistent with that used by previous authors [134]. In this form,
a constraint may be treated precisely the same as the ΔV targets described in the
previous section. The constraint will be added to the list of targets for the differential
corrector. It will be assumed, once again, that the only controls that may inﬂuence
the constraint are the position and time of the patchpoint that the constraint is applied
to as well as the positions and times of the two neighboring patchpoints (or the
single neighboring patchpoint in the case of a constraint placed on the ﬁrst or last
patchpoint of a trajectory). The differential corrector may certainly be rederived to
operate with constraints that act upon many patchpoints, but this discussion is limited
to constraints that act upon a single patchpoint.
It is straightforward to add a constraint to the list of targets in the differential
corrector. The relationship given in Eq. (2.23) is augmented as follows
⎡
⎤
∂ΔVi ∂ΔVi
⎥
⎢
δΔVi
∂tk ⎥ δRk
⎢ ∂Rk
=⎢
(2.36)
⎥
δαij
⎣ ∂αij
∂ αij ⎦ δtk
∂Rk
∂tk
�
�
[P ]
where the matrix P is known as the augmented state relationship matrix (SRM).
Equation (2.36) is highly compressed: P is typically sparsely populated roughly
along the diagonal, such that each constraint and each ΔV may only be inﬂuenced
by the patchpoint it is assigned to and that patchpoint’s nearest neighbors. Much
like the ΔV constraints described in the previous section, each constraint requires the
deﬁnition of the following six partials
∂ αij
∂Ri−1
∂ αij
∂ti−1
∂ αij
∂Ri
∂ αij
∂ti
∂ αij
∂Ri+1
∂ αij
∂ ti+1
(2.37)
A quick observation shows that the ΔV constraints described in the previous section
are a speciﬁc case of a constraint, where αij = ΔVi .
Any constraint that is a function of the position and/or time of one of the control
patchpoints, and not a direct function of the velocity of any patchpoint, may be easily
deﬁned. For instance, if one wishes to constrain the time of patchpoint Pi , one simply
characterizes that constraint as
αij = ti − tˆi
LOWENERGY MISSION DESIGN
65
where tˆi is the desired time. One then computes the partials given in Eq. (2.37) and
ﬁnds that the only nonzero partial is
∂αij
=1
∂ti
Similarly, if one wishes to constrain the position vector of patchpoint Pi , one
characterizes that constraint as
ˆi
αij = Ri − R
ˆ i is the desired position vector. One then ﬁnds that the only nonzero partial
where R
is
∂ αij
= I3×3
∂Ri
Constraints that depend on velocity are more complex, as demonstrated by the ΔV
constraints given above. In order to compute the partials given in Eq. (2.37), one must
perform the chain rule and compute additional partial derivatives. Fortunately, many
of these were computed in the previous section, and many go to zero for numerous
constraint formulations. The relationships are
∂ αij
∂Ri−1
=
∂αij
∂αij ∂Vi−
+
∂Ri−1
∂Vi− ∂Ri−1
∂ αij
∂ ti−1
=
∂αij
∂ αij ∂Vi−
+
∂ ti−1
∂Vi− ∂ ti−1
∂ αij
∂Ri
=
∂ αij
∂ αij ∂Vi−
∂ αij ∂Vi+
+
+
−
∂Ri
∂Vi ∂Ri
∂Vi+ ∂Ri
∂ αij
∂ ti
=
∂ αij
∂ αij ∂Vi−
∂ αij ∂Vi+
+
+
∂ ti
∂Vi− ∂ ti
∂Vi+ ∂ ti
∂αij
∂Ri+1
=
∂ αij
∂ αij ∂Vi+
+
∂Ri+1
∂Vi+ ∂Ri+1
∂ αij
∂ ti+1
=
∂ αij
∂αij ∂Vi+
+
∂ ti+1
∂Vi+ ∂ ti+1
Wilson derives the formulae that may be used to constrain a patchpoint’s velocity,
velocity magnitude, inclination, apse location, ﬂight path angle, declination, right
ascension, and conic energy [134]. For example, the conic energy relative to a
massive body may be described as
αij
=
=
Vi 2
µ
−
2
Ri 
V i · Vi
µ
−
2
(Ri · Ri )1/2
66
METHODOLOGY
where µ is the gravitational parameter for the central body. The majority of the
partials given above are either zero or already known. The remaining partials may
be computed as follows
∂ αij
∂Ri
∂ αij
∂Vi±
∂ αij
∂ ti
=
µRTi
Ri 3
= Vi±T
=
0
The implementation of additional constraints is left to the designer.
A practical constraint that is not formulated in the same way is to restrict the size
of the steps that the Level 2 differential corrector may take between iterations. The
differential corrector estimates the change in each patchpoint’s position, time, or both
in order to achieve the given targets, and it does so using a large system of linearized
equations. It is often the case that small perturbations drive the realized deviations in
the trajectory into highly nonlinear regimes. In practice, it is often the case that the
application of a full adjustment in the controls will push the trajectory further from
the desired solution than it started. If a designer observes the trajectory diverging
from the desired target, one common solution is to limit the maximum deviation that
the patchpoints may shift in position or time per iteration of the differential corrector.
If implemented properly, the smaller steps should keep the trajectory within the basin
of convergence of the solution.
Level 2 with Multiple Trajectories. The Level 2 differential corrector formulated
here operates on a large system of controls, targets, and constraints, where ultimately
each patchpoint in the system contributes to the satisfaction of all goals, though each
patchpoint is only directly inﬂuenced by its neighboring patchpoints at any given
iteration. This system may be applied to multiple trajectories simultaneously in
much the same way as it is applied to a single trajectory. This has clear practical
applications for many spacecraft missions that involve deployments, separations,
and/or formationﬂying activities.
One may formulate the Level 2 differential corrector with multiple trajectories by
augmenting the SRM, P , once again, such that it includes the patchpoints, targets,
and constraints of every trajectory. One must be sure to permit the system some ΔV
leverage to allow any given pair of trajectories to separate (either forward in time or
backward in time, as appropriate).
For example, let us assume we have a scenario that involves one spacecraft de
ploying a secondary payload via a spring mechanism, which imparts a speciﬁed ΔV
between the two spacecraft. Let us also assume that the differential corrector is
permitted to vary the trajectory of the joined system prior to the deployment, as well
as both trajectories after the deployment, and the deployment ΔV may occur in any
direction. One way to model this scenario is to set up two series of patchpoints that
deﬁne each spacecraft and then carefully lock the two spacecraft together. A practical
LOWENERGY MISSION DESIGN
67
way to lock the two trajectories together is to deﬁne the ﬁrst patchpoint of the de
ployed payload to be prior to the deployment, such that its position, velocity, and time
are all constrained to be equal to the relevant parameters of the host spacecraft. That
is, it is entirely constrained to match the corresponding patchpoint on the host space
craft’s trajectory. The second patchpoint in the deployed payload’s trajectory is then
deﬁned to be the deployment event, such that its position and time are constrained
to be equal to the position and time of the host spacecraft at the deployment, and
its outgoing velocity is constrained to have the appropriate ΔV magnitude applied.
From there the trajectory departs in the same way as any other trajectory. In this
scenario, one would also have to take care to model the appropriate reaction to the
host spacecraft’s trajectory via constraints.
The augmented SRM for the case of multiple trajectories is very sparse, and it
may be beneﬁcial to implement numerical algorithms that take advantage of this
feature. The simple Step 2 SRM includes nonzero elements only on a diagonal
swath six elements wide. The SRM shifts further away from diagonal each time it is
augmented by an additional constraint or an additional trajectory, though it remains
approximately diagonal.
2.6.6
Constructing Periodic Orbits
Periodic orbits are important when analyzing and constructing trajectories using
dynamical systems methods, since they help to characterize the ﬂow in the system.
There are many methods that are frequently used to identify and construct periodic
orbits. Three categories of methods are described here:
1. Analytical Expansion Techniques. The discussion given in Section 2.6.2.2
demonstrates how to use basic analytical techniques to identify planar and
threedimensional periodic orbits in the CRTBP. Many authors have constructed
analytical expansions that may be used to approximate periodic orbits in the
CRTBP or in more complex systems [67, 123, 139].
2. Shooting Techniques. One may numerically construct a periodic orbit by
targeting a single state as both the initial and ﬁnal states in a trajectory using
either a single or multipleshooting technique. This technique is difﬁcult
without any constraints, but it has proven to be very useful when numerically
constructing certain types of periodic orbits, such as simple symmetric periodic
orbits [46, 107, 108, 122].
3. The Poincare´ Method. The Poincare´ Method is a notable method that has
proven to be very successful at identifying periodic and quasiperiodic orbits,
especially stable orbits. Poincar´e’s technique involves numerically integrating
many trajectories for a large amount of time. Trajectories that are close to
periodic tend to linger near the same regions of the state space. One can
readily identify stable periodic orbits or trajectories near such orbits if one
places a plane in the state space, that is, a Poincare´ Surface of Section. Then
one records the state of each trajectory as the trajectory pierces the plane.
68
METHODOLOGY
A periodic orbit appears as a ﬁxedpoint in the plane; a quasiperiodic orbit
appears as a closed loop in the plane. Regions that are unstable in the state
space appear as a chaotic sea of points, since unstable trajectories are very
sensitive to their initial conditions.
Many other types of methods certainly exist, but these three categories provide a
good overview of the variety of methods that are frequently used.
2.6.6.1 Periodic Orbits in the CRTBP If the Lagrange points represent the
ﬁve simplest solutions to the CRTBP, it may be argued that the next set of solutions
to introduce is the set of simple periodic symmetric orbits in the CRTBP. Simple
periodic symmetric orbits are orbits that are symmetric about the y = 0 plane, pierce
the y = 0 plane exactly twice per orbit, and pierce the plane orthogonally each time.
Libration orbits, such as halo and Lyapunov orbits, are good examples of such orbits.
A simple singleshooting differential correction scheme may be used to construct
these orbits by taking advantage of their welldeﬁned structure. Section 2.6.6.2
provides more information about this differential correction scheme. It should be
noted that although this class of orbits does include what might be argued to be the
simplest periodic orbits in the CRTBP, this class of orbits also includes families of
very complex orbits.
Many other types of periodic orbits exist in the CRTBP, including orbits that
pierce the y = 0 plane multiple times per orbit and orbits that are not symmetric,
such as orbits about the triangular Lagrange points. One may also construct arbitrarily
complex periodic orbits by chaining simple unstable orbits together, as is discussed
in Section 2.6.11.
2.6.6.2 SingleShooting Method for Constructing Simple Periodic Sym
metric Orbits in the CRTBP One may formulate many types of shooting tech
niques to identify periodic orbits using the techniques introduced in Section 2.6.5.
Howell identiﬁed a simple procedure that has been used by many researchers in the
ﬁeld [122]. The technique is easily applied to the families of halo orbits, Lyapunov
orbits, distant prograde orbits, distant retrograde orbits, symmetric resonant orbits,
and a variety of other classes of symmetric periodic orbits [46]. Since it is a very
common and straightforward procedure, and since it has been used repeatedly in
relevant research to construct halo orbits and other similar orbits, it is reviewed here.
As mentioned earlier, simple periodic symmetric orbits are orbits that are symmet
ric about the y = 0 plane, pierce the y = 0 plane exactly twice per orbit, and pierce
the plane orthogonally each time. Let us deﬁne X(t0 ) to be the state of a simple
periodic symmetric orbit at the y = 0 planecrossing with a positive y˙ and X(tT /2 )
to be the state of the orbit half of its orbital period later at the y = 0 planecrossing
with a negative y˙. For this orbit to be periodic and symmetric, these states must have
the following form
X(t0 )
X(tT /2 )
=
=
[ x0
0
xT /2
z0
0
0
zT /2
y˙ 0
0
T
0]
y˙ T /2
(2.38)
0
T
LOWENERGY MISSION DESIGN
69
ˆ (t0 ), that is near the initial state of a
Let us assume that we have an initial guess, X
desirable orbit. When we integrate this state forward in time until the next y = 0
ˆ (t ˆ )
plane, we obtain the state X
T /2
ˆ ˆ )=
X(t
T /2
xTˆ/2
0
zTˆ/2
x˙ Tˆ/2
y˙ Tˆ/2
z˙Tˆ/2
rT
We now wish to adjust the initial state of the trajectory in such a way as to drive the
values of x˙ Tˆ/2 and z˙Tˆ/2 to zero. One notices that by adjusting the initial state, not
only do the values of x˙ and z˙ change, but the propagation time, Tˆ/2, needed to pierce
the y = 0 plane also changes. In order to target a proper state X(tT /2 ), one may vary
the initial values of x, z, and/or y˙.
The linearized system of equations relating the ﬁnal state to the initial state may
be written as
∂X
δX(tT /2 ) ≈ Φ tT /2 , t0 δX(t0 ) +
δ(T /2)
(2.39)
∂t
where δX(tT /2 ) is the deviation in the ﬁnal state due to a deviation in the initial
state, δX(t0 ), and a corresponding deviation in the orbit’s period, δ(T /2). The
timederivative of the state, ∂X/∂t, may be computed at the second planecrossing,
namely, at time t = T /2. Equation (2.39) may be used as the driver for a differential
corrector by setting δX(tT /2 ) to be the desired change in the ﬁnal state’s components
and solving for δX(t0 ), the approximate correction to the initial state needed to
produce such a change.
We now consider what the desired change in the ﬁnal state’s components should
be. For our purposes, the only desired change in the ﬁnal state is a change in the
values of x˙ and z˙, but it is not important if the other components of the ﬁnal state
change. However, we know that the deviation in the ﬁnal value of y will always be
equal to zero since the trajectory is always propagated to that point. Thus we set
δX(tT /2 ) to
δX(tT /2 ) =
δxT /2
0
δzT /2
−x˙ T /2
−z˙T /2
δy˙ T /2
T
Furthermore, in order to restrict our search to simple periodic symmetrical orbits, we
restrict the allowed correction in the initial conditions to
δx0
δX(t0 ) =
Now Eq. (2.39) simpliﬁes to
⎡
⎤ ⎡
δxT /2
φ11 φ12 φ13
⎥ ⎢φ21 φ22 φ23
⎢ 0
⎢ δzT /2 ⎥ ⎢φ φ φ
31
32
33
⎥
⎢
⎢−x˙ T /2 ⎥≈⎢
φ41
φ42 φ43
⎣
⎦ φ φ φ
⎣ δ y˙
51
52
53
T /2
φ61 φ62 φ63
−z˙T /2
φ14
φ24
φ34
φ44
φ54
φ64
φ15
φ25
φ35
φ45
φ55
φ65
0
δz0
0
δy˙ 0
0
T
⎤
⎡
⎤ ⎡ ⎤
φ16
x˙
δx0
φ26 ⎥
⎢ 0 ⎥ ⎢y˙ ⎥
φ36 ⎥ t , t ⎢ δz0 ⎥ + ⎢ z˙ ⎥δ(T /2) (2.40)
⎢ ¨⎥
T /2 0 ⎢ 0 ⎥
φ46 ⎥
⎦
⎦
⎣
⎦ ⎣x
y¨
δy
˙
φ56
0
z¨
0
φ66
The value of δ(T /2) may be determined from the second line of Eq. (2.40) to be
δ(T /2) =
−φ21 δx0 − φ23 δz0 − φ25 δy˙ 0
y˙
(2.41)
70
METHODOLOGY
Substituting this value into the fourth and sixth lines of Eq. (2.40) yields
x
¨
x
¨
x
¨
δx0 + φ43 − φ23 δz0 + φ45 − φ25 δy˙ 0
y˙
y˙
y˙
z¨
z¨
z¨
−z˙T /2 ≈ φ61 − φ21 δx0 + φ63 − φ23 δz0 + φ65 − φ25 δy˙ 0
y˙
y˙
y˙
−x˙ T /2 ≈ φ41 − φ21
(2.42)
(2.43)
Equations (2.42) and (2.43) give expressions for the approximate deviation in the
ﬁnal x and zvelocities as functions of the deviation in all three initial conditions
x0 , z0 , and y˙ 0 . It is sufﬁcient to change only two of the initial conditions, if that is
desirable, or a combination of all three. For the purpose of this description, the value
of x0 will be kept constant, and the values of z0 and y˙ 0 will be permitted to vary. The
following expression summarizes the approximate changes that must be made to z0
and y˙ 0 to produce a desirable change in the ﬁnal state (while keeping the other initial
conditions constant)
δz0
δy˙ 0
�
≈
φ43 − φ23 xy¨˙
φ63 − φ23 yz¨˙
φ45 − φ25 xy¨˙
φ65 − φ25 yz¨˙
�−1
−x˙ T /2
−z˙T /2
(2.44)
Since the system was linearized in order to produce this procedure, the adjustments
will not correct the unwanted motion perfectly; this procedure must be iterated until
it converges on an orbit.
When all is said and done, a simple, symmetric periodic orbit has three nonzero
states at its orthogonal y = 0 plane crossing: x0 , z0 , and y˙ (see Eq. (2.38)). The
procedure outlined here is used to generate the periodic orbit given one of those
parameters and estimates of the other two. Because of this, a family of periodic
orbits may be well represented by plotting its initial y˙ values or its Jacobi constant
values as a function of its initial x values. Figure 214 illustrates these curves using
the family of Lyapunov orbits about LL1 as an example.
2.6.6.3 Differentially Correcting Orbits into the DE421 Model An orbit
that is perfectly periodic in the CRTBP is not perfectly periodic in the real Solar
System since the planets and moons in the real Solar System do not move in circular,
coplanar orbits. Various perturbations lead the orbit to diverge from being periodic;
the most notable of which is the nonzero eccentricity of the orbits of the primary
bodies in the system [100].
To produce a quasiperiodic orbit in the real Solar System, one can use a multipleshooting differential corrector with the periodic CRTBP orbit as the initial guess of the
real trajectory. This technique was demonstrated in Section 2.6.5.2. The differential
corrector takes the CRTBP orbit and perturbs it to keep it near its initial guess while
eliminating the need to perform large maneuvers. In the case of generating a quasiperiodic halo or Lissajous orbit in the DE421 model of the Solar System, one may
use an analytical approximation of the orbit as the initial guess to the differential
corrector [123, 139]. This has been demonstrated on many occasions and has been
shown to work well [47].
LOWENERGY MISSION DESIGN
71
Figure 214 Plots of x0 vs. y˙0 (top) and x0 vs. C (bottom) for the family of Lyapunov
orbits about the Earth–Moon L1 point. The initial values of the other Cartesian coordinates
in the Earth–Moon synodic frame are all equal to zero for each orbit in this family [101]
(Acta Astronautica by International Academy of Astronautics, reproduced with permission of
Pergamon in the format reuse in a book/textbook via Copyright Clearance Center).
Figure 215 shows the difference between a halo orbit about the lunar L2 point
produced in the CRTBP compared with the same halo orbit differentially corrected
into the DE421 model of the real Solar System. One can see that the real halo
orbit is quasiperiodic, tracing out the same vicinity of space on each orbit, but never
truly retracing itself. For this illustration, the realistic quasihalo orbit is plotted in
a coordinate frame that is normalized over time based on the instantaneous distance
between the Earth and the Moon, and then rescaled to the average distance between
the Earth and the Moon.
72
METHODOLOGY
Figure 215 A comparison between a halo orbit produced in the CRTBP and a quasihalo
orbit produced in the DE421 model of the real Solar System. The orbits are shown in the
Earth–Moon synodic reference frame [44, 46].
The perfectly periodic CRTBP orbit is typically a very close approximation of the
real quasiperiodic orbit, enough so that the early mission design may be developed
in the CRTBP. This is convenient because the motion of a spacecraft in a perfectly
periodic orbit is more predictable than the motion of a spacecraft in a quasiperiodic
orbit.
On several occasions, it has been observed that some of the structure of a periodic
orbit in the CRTBP becomes lost or signiﬁcantly altered as the orbit is differentially
corrected into the DE421 model. This is often seen when a single revolution of a
periodic orbit is sent into the differential corrector. Ordinarily, a differential corrector
converges on a continuous trajectory more readily if the trajectory’s endpoints are
not constrained. Without the boundary values constrained, it is often the case that the
differential corrector signiﬁcantly alters the states of the trajectory’s endpoints. The
resulting trajectory, although continuous, may not resemble the original orbit much
at all. This effect may be observed in Fig. 216.
One way to combat this effect is to differentially correct several orbits of the
periodic orbit together. For the purpose of this discussion, let us say that four periodic
orbits are differentially corrected together. Then, two of the orbits are “outer” orbits
(the ﬁrst and last orbits) that are vulnerable to substantial changes in the differential
correction process, and two of the orbits are “inner” orbits (the second and third
orbits) that are more protected from signiﬁcant alteration in the process. Normally,
the differential corrector converges on a continuous trajectory before the inner orbits
are substantially altered. Once the differential corrector has converged on the ﬁnal
trajectory, then the outer orbits may be pruned off in order to observe the structure of
LOWENERGY MISSION DESIGN
73
Figure 216 A single L2 halo orbit in the Earth–Moon CRTBP (left) is differentiallycorrected into the DE421 model (right) [46].
the resulting quasiperiodic orbit. Ordinarily, this procedure results in quasiperiodic
orbits that exist in the DE421 model that retain the same structure as the periodic
orbits that exist in the CRTBP. Figure 217 shows an example of this process.
Since halo orbits are used frequently in later chapters of this book, some discus
sion is given here regarding the largest observable deviations between the perfectly
periodic halo orbit in the CRTBP and the quasihalo orbit in the real Solar System.
Arguably the most substantial deviation between the CRTBP and the real Solar Sys
Figure 217 An example of the process of differentially correcting and pruning a halo orbit
from the Earth–Moon CRTBP into the DE421 model. Left: the nominal periodic halo orbit in
the Earth–Moon CRTBP; center: the differentially corrected trajectory in the DE421 model;
right: the pruned quasiperiodic halo trajectory in the DE421 model [44, 46].
74
METHODOLOGY
tem, at least in the case of the Sun–Earth and Earth–Moon threebody systems, is
the nonzero eccentricity of the orbits of the primary bodies in the system. The real,
eccentric orbit of the primaries imparts a deviation in the quasihalo orbits that has
a period equal to the orbital period of the primaries. Since most halo orbits have a
period equal to approximately half of the orbital period of the primaries, this dynamic
perturbation tends to appear as a resonant pulsation. One quasihalo revolution tends
to deviate from the perfectly periodic halo orbit in one direction, and the next rev
olution tends to deviate in the opposite direction. The result is that a spacecraft on
a quasihalo orbit tends to retrace its path very closely every other revolution. This
effect is less visible when the reference frame is centered on a Lagrange point rather
than the barycenter, since the Lagrange point pulses in and out as the primary bodies
follow their noncircular orbits.
Figure 218 illustrates the pulsation that exists in the Earth–Moon system by
showing a plot of the distance between the Moon and a spacecraft traversing an orbit
much like that one shown in Figs. 216 and 217. One can see that every other
revolution retraces a similar path. The moments in time when the Moon reaches
its perigee and apogee are indicated for reference. Figure 219 illustrates how this
same quasihalo orbit appears in the DE421 ephemeris when viewed in different
synodic coordinate systems, including an Earthcentered synodic frame, out to an
LL2 centered synodic frame.
2.6.7
The Continuation Method
Periodic orbits in the CRTBP may be grouped into families, where a family consists
of an inﬁnite number of periodic orbits whose properties vary continuously from
one end of the family to the other. All orbits in the same family may be uniquely
identiﬁed by a single parameter of that family, for example, their position on a
perpendicular y = 0 plane crossing, their velocity at that crossing, or some other
speciﬁed parameter. This property of the CRTBP is due to the existence of the Jacobi
Figure 218 The distance between the orbit and the Moon over time for a realistic quasihalo
orbit. The moments in time when the Moon reaches its perigee and apogee are indicated by
the symbols “p” and “a”, respectively.
LOWENERGY MISSION DESIGN
75
Figure 219 An illustration of how the same quasihalo orbit appears in different synodic
coordinate systems. From left to right, the systems include an Earthcentered synodic frame,
an L1 centered frame, a Mooncentered frame, and an L2 centered frame.
constant, the CRTBP’s unique integral of motion. He´ non provides more discussion
about the existence of families of solutions in the CRTBP [113].
Once a single periodic orbit is known in the CRTBP, then the continuation method
may be used to traverse that orbit’s family. The method starts by perturbing some
parameter of the known periodic orbit and then differentially correcting the new con
ditions to ﬁnd that periodic orbit’s neighbor in its family. The differential corrector
presented in Section 2.6.6 is wellsuited to this method for simple periodic symmetri
cal orbits because one may vary the initial position and correct for the initial velocity
that corresponds to the next periodic orbit in the family (or vice versa, if desired).
To demonstrate this method, the continuation method has been applied to the
family of Lyapunov orbits that exist about the Earth–Moon L2 point. First, a single
Lyapunov orbit is identiﬁed, for example, the gray orbit in Figs. 220 and 221. The
orbit’s initial position, x0 , is then systematically varied while a differential corrector
ﬁlls out the curve shown in Fig. 220. The initial conditions in the curve correspond
to the family of orbits shown in Fig. 221.
The continuation method works well when the perturbations are small; in practice
it is beneﬁcial to predict the differential corrector’s adjustment to the perturbation
because this allows larger jumps in the varying parameter. Furthermore, if the
perturbations are too large, the differential corrector may converge on a solution of a
different family. Thus smaller steps or better prediction methods may be required to
make the continuation method more reliable. The work for this study has implemented
a quadratic prediction method that uses the three previous data points of the family to
predict the next data point. This has been sufﬁcient to allow the differential corrector
to converge quickly while allowing the curve of the family to evolve naturally over
the state space. Twodimensional curve tracking algorithms may also work well since
76
METHODOLOGY
Figure 220 A plot of the initial conditions of the family of Lyapunov orbits about the
c
Earth–Moon L2 point (LL2 ) [140] (Copyright ©2006
by American Astronautical Society
Publications Ofﬁce, San Diego, California [website http://www.univelt.com], all rights
reserved; reprinted with permission of the AAS).
Figure 221 Plots of the orbits in the family of LL2 Lyapunov orbits corresponding to those
c
initial conditions shown in Fig. 220 [140] (Copyright ©2006
by American Astronautical
Society Publications Ofﬁce, San Diego, California [website http://www.univelt.com], all rights
reserved; reprinted with permission of the AAS).
LOWENERGY MISSION DESIGN
77
state space curves are not necessarily wellmodeled by polynomials. For instance,
one may extrapolate curves using a constant arclength of two parameters [141].
2.6.8
Orbit Stability
The stability of a periodic orbit may be determined by analyzing the eigenvalues of
the orbit’s monodromy matrix. A random perturbation in the state of a spacecraft on
an unstable orbit will cause the spacecraft’s state to exponentially diverge from that
of the original orbit over time; hence, the monodromy matrix of an unstable orbit
includes at least one eigenvalue for which the real component is outside of the unit
circle. This section explores the stability characteristics of periodic orbits via the
eigenvalues of their monodromy matrices.
2.6.8.1 Eigenvalues of an Orbit’s Monodromy Matrix The monodromy
matrices of orbits in the CRTBP have six eigenvalues, λi for i = 1, 2, . . . , 6, corre
sponding to the eigenvectors vi . The eigenvalues of the monodromy matrix occur in
reciprocal pairs [142], which is a direct consequence of the symplectic nature of the
monodromy matrix, and of the state transition matrix in the CRTBP in general [143].
Additionally, a pair of eigenvalues of the monodromy matrix will be equal to unity
because of the Jacobi integral of motion in the CRTBP [131, 142]. The eigenvalues
are thus related in the following way
λ2 =
1
λ1
λ4 =
1
λ3
λ5 = λ6 = 1
(2.45)
The monodromy matrices of periodic orbits in the planar CRTBP only have four
eigenvalues: (λ1 , 1/λ1 , 1, 1). Since those orbits may be computed in the spatial
CRTBP by setting their z and z˙components to zero, the remainder of this section
only considers orbits in the full threedimensional system.
The eigenvalues of the monodromy matrix of a periodic orbit in the CRTBP are the
roots of a characteristic equation; furthermore, each has a characteristic exponent,
α, where λ = eαT and T is the period of the orbit. Then, the reciprocal of that
eigenvalue is equal to: 1/λ = e−αT . The characteristic exponents are sometimes
referred to as Lyapunov characteristic exponents [144].
The monodromy matrices of Keplerian orbits, such as low Earth orbits (LEOs),
have three pairs of eigenvalues that are all equal to 1, indicating that after a full orbit
any given perturbation neither grows nor decays exponentially. The monodromy
matrices of periodic orbits in the CRTBP may have other eigenvalue pairs, including
real values not equal to 1 and pairs of complex numbers. Table 25 provides a
summary of the resulting motion of a spacecraft in a periodic orbit, whose state is
perturbed along the eigenvector corresponding to any type of given eigenvalue.
The stability of a periodic orbit may be identiﬁed by observing the resulting motion
of a perturbed particle in that orbit or by computing the eigenvalues of the orbit’s
monodromy matrix and comparing those eigenvalues to the results given in Table 25.
The following classiﬁcation scheme for an orbit’s stability is used in this work:
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METHODOLOGY
Table 25 A summary of the resulting motion of a spacecraft in a periodic orbit,
whose state is perturbed along the eigenvector corresponding to a given eigenvalue. The
result of a perturbation along the eigenvector corresponding to a complex eigenvalue
includes a combination of the imaginary result and one of the real results listed.
Eigenvalue
Result of the perturbation
Real, within the range [−1,1]
Real, equal to 1 or −1
The perturbation exponentially decays.
The perturbation neither exponentially
decays nor grows.
The perturbation exponentially grows.
Real, outside of the range [−1,1]
Imaginary
After each orbital period, the perturbation
oscillates about the spacecraft’s original
state.
• If an eigenvalue exists whose real component is outside of the range [−1,1],
then the periodic orbit is asymptotically unstable, referred to here as unstable,
along the corresponding eigenvector.
• If the real component of each and every eigenvalue of the monodromy matrix is
between −1 and 1, then the orbit is stable. Given the relationships in Eq. (2.45),
an orbit may only be stable in the CRTBP if each and every eigenvalue pair is
complex with real components in the range [−1,1].
• If the orbit is not unstable and there is at least one eigenvalue pair whose real
component is equal to 1, then the periodic orbit is neutrally stable, or a center
[130].
Since every periodic orbit in the CRTBP has at least one pair of eigenvalues with
values are equal to unity, then it is customary to ignore that pair of eigenvalues when
classifying the stability of the orbit [108, 145].
To determine the eigenvalues of the monodromy matrix, it is useful to consider the
characteristic equation, since many of the roots of this equation are already known
det (M − λI) = (λ − λ1 )(λ − λ2 )(λ − λ3 )(λ − λ4 )(λ − λ5 )(λ − λ6 ) = 0
= (λ − 1)2 (λ − λ1 )(λ − 1/λ1 )(λ − λ3 )(λ − 1/λ3 ) = 0 (2.46)
The relationship given in Eq. (2.46) may be rewritten in terms of the new parameters
p and q, keeping consistent with the nomenclature found in the literature [116]
(λ − 1)2 λ2 + pλ + 1
λ2 + qλ + 1 = 0
(2.47)
Thus, p = −(λ1 + 1/λ1 ) and q = −(λ3 + 1/λ3 ). Equation (2.47) may also be
factored in the following manner
(λ − 1)2 λ4 + (p + q)λ3 + (pq + 2)λ2 + (p + q)λ + 1 = 0
(2.48)
79
LOWENERGY MISSION DESIGN
Equation (2.48) may be rewritten using the new parameters α, β, and γ, once again
to keep consistent with the nomenclature found in the literature [116] (where α should
not be confused with the characteristic exponent that corresponds to each eigenvalue)
(λ − 1)2 λ4 + αλ3 + β λ2 + αλ + γ = 0
(2.49)
In this form it is clear that α = p + q, β = pq + 2, and γ = 1. The beneﬁts of
factoring the characteristic equation into the parameters α, β, and γ arises at this
point. Bray and Goudas derive a fast and simple method to compute α and β using
the monodromy matrix [116, 145]
α
β
2 − trace(M )
α2 − trace(M 2 )
=
+1
2
=
(2.50)
(2.51)
It is then simple to determine the parameters p and q using knowledge of α and β
p
q
=
α±
α2 − 4β + 8
2
(2.52)
It then follows that with knowledge of p and q one may determine the corresponding
eigenvalues
λ1
1/λ1
=
λ3
1/λ3
=
p2 − 4
−p ±
2
q2 − 4
−q ±
2
(2.53)
(2.54)
The ﬁnal two eigenvalues have already been predetermined and are given in Eq. (2.45)
as λ5 = λ6 = 1. Thus, Eqs. (2.50)–(2.54) provide a fast and simple method to com
pute the six eigenvalues of the monodromy matrix. The corresponding eigenvectors
may be computed in any standard way using the equation M vi = λi vi . It should be
noted that the stable and unstable eigenvalues, λS and λU , of an orbit’s monodromy
matrix, are equal to the pair of real eigenvalues with the smallest and largest values,
respectively, if they exist.
2.6.8.2 The Stability Index An orbit’s stability index is deﬁned in various ways
in the literature depending on the author. Several authors, for example, Broucke [108],
deﬁne the stability of a periodic orbit on the value of k, where k is equal to the sum
of the real eigenvalues of the orbit. If k > n, where n is equal to the number of real
eigenvalues in the orbit’s monodromy matrix, then the orbit is unstable; if k < n,
the orbit is stable; otherwise k = n and the orbit is neutrally stable. One problem
with such a deﬁnition is that the value of n may change depending on the orbit.
Another deﬁnition of the stability index is deﬁned by Howell, among others, as
follows [122]. If one considers the deﬁnition ki = λi + 1/λi , one notices several
things. First, the values of ki may be easily computed using the parameters p and
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METHODOLOGY
q that were introduced above, namely: k1 = −p, k2 = −q, and k3 = 2. Next,
the value of ki is always real and in the range −2 ≤ k ≤ 2 for stable orbits since
the sum of a real pair is real and the sum of a complex conjugate pair is also real.
Furthermore, if ki  > 2, then the real component of at least one of the eigenvalues
summed is greater than 1 and the orbit is unstable. Since two of the eigenvalues
of the orbit’s monodromy matrix are equal to unity and it is conventional to ignore
them, the stability index, k, may then be given by
k = sup{Re(k1 ), Re(k2 )}
(2.55)
where the operator Re() only observes the real component of the operand. We have
the ﬁnal test: if k > 2 the orbit is unstable, if k = 2 the orbit is neutrally stable, and
if k < 2 the orbit is stable.
2.6.8.3 The Perturbation Doubling Time The stability index deﬁned in
Eq. (2.55) certainly provides information about the stability of the orbit in ques
tion. However, it only provides limited information about the relative stability of
different orbits. A highly unstable orbit may appear to be more stable than a weakly
unstable orbit if the weakly unstable orbit’s period is much greater than the highly
unstable orbit’s period. It is now of interest to ﬁnd a parameter that may be used to
directly compare the stability of two orbits regardless of their relative orbital periods.
The eigenvalues of the monodromy matrix of a periodic orbit are a function of the
orbit’s period, T , and a characteristic exponent, α, as follows
λ = eαT
(2.56)
To compare the stability of several orbits directly, one may either normalize the
eigenvalues of the monodromy matrices or, equivalently, compare the characteristic
exponents in some way.
An intuitive measure for comparison is the orbit’s perturbation doubling time (for
unstable orbits) or the orbit’s perturbation halflife (for stable orbits). In this work,
we refer to this time measurement as τˆ for two reasons: ﬁrst, to indicate that it is
a normalized measurement and second, to distinguish it from the parameter τ that
is used to identify points along an orbit (see Section 2.6.2.3). Given a spacecraft
in an unstable orbit, the perturbation doubling time characterizes the length of time
that is required for a perturbation in the spacecraft’s state to double in magnitude.
Similarly, given a spacecraft in a stable orbit, the perturbation halflife characterizes
the length of time that is required for a perturbation in the spacecraft’s state to be
reduced by one half. For simplicity, we refer to this time measurement only as the
perturbation doubling time, since it is generally more useful when designing real
missions to compare this time measurement for unstable orbits.
After determining the eigenvalues of the orbit’s monodromy matrix, one may use
Eq. (2.56) to determine the corresponding characteristic exponents. If a spacecraft’s
state is perturbed at time t = t0 from its nominal state by a perturbation with
magnitude δ(t0 ) along the eigendirection corresponding to the characteristic exponent
α, the perturbation magnitude grows over time by the following expression
δ(t) = δ(t0 )eα(t−t0 )
(2.57)
LOWENERGY MISSION DESIGN
81
Given a random perturbation in the spacecraft’s state, the spacecraft’s deviation over
time is dominated by the component of that deviation that exists in the most unstable
eigendirection, namely, by the direction indicated by the unstable eigenvalue λU . The
perturbation doubling time may be computed by identifying the time, t = t0 +τˆ, when
the spacecraft’s perturbed state is twice as far from its nominal position compared to
its perturbed state at time t = t0 . One can ﬁnd the perturbation doubling time by
solving for τˆ in Eq. (2.58), derived as follows
δ(t) = δ(t0 )eα(t−t0 )
2δ(t0 ) = δ(t0 )eαˆτ
2
=
eαˆτ
(2.58)
where α is the characteristic exponent that corresponds to the unstable eigenvalue,
λU , of the orbit’s monodromy matrix. The value of α may be computed using the
simple relationship αT = ln λU , derived from Eq. (2.56). Hence, the time duration
τˆ may be computed using the expression
τˆ =
2.6.9
ln 2
T
ln λU
(2.59)
Examples of Practical ThreeBody Orbits
The threebody problem contains a wide variety of interesting and potentially useful
periodic and quasiperiodic orbits. Numerous authors have catalogued families of
orbits and a brief history of these efforts is given in Section 2.6.2.2. This section il
lustrates several example families of threebody orbits, all of which appear frequently
in the literature, and often in spacecraft mission proposals.
2.6.9.1 Lyapunov Orbits Lyapunov orbits were introduced in Section 2.6.2.2;
they are twodimensional periodic solutions to the circular restricted threebody
problem. Lyapunov orbits exist about all three of the collinear Lagrange points,
as illustrated in Fig. 222. The LL1 and LL2 families include orbits with orbital
periods between two and four weeks—closer to two weeks for orbits closer to the
Lagrange point; the LL3 family includes orbits with orbital periods of approximately
four weeks [140]. These orbits are all unstable.
2.6.9.2 Distant Prograde Orbits Periodic threebody orbits certainly exist
about the Earth and the Moon as well as the Lagrange points. Figure 223 illustrates
the family of planar distant prograde orbits and shows how that family of orbits ﬁts
in between the family of L1 and L2 Lyapunov orbits. A spacecraft only needs to
adjust its state slightly to transfer from a Lyapunov orbit to a distant prograde orbit
and vice versa. This is explored in Section 2.6.11.3. Most distant prograde orbits
are unstable; their orbital periods vary from two weeks to four weeks, much like the
Lyapunov orbits.
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METHODOLOGY
Figure 222 Example orbits in the families of Lyapunov orbits about the Earth–Moon
L3 point (left), L1 point (middle), and L2 point (right), viewed in the Earth–Moon rotating
frame from above [46].
Figure 223 Example orbits in the families of Lyapunov orbits about the Earth–Moon L1
point (left), distant prograde orbits about the Moon (middle), and Lyapunov orbits about the
Earth–Moon L2 point (right), viewed in the Earth–Moon rotating frame from above [46].
2.6.9.3 Distant Retrograde Orbits Distant retrograde orbits (DROs) are pe
riodic threebody orbits that exist about the smaller primary, for example, the Moon
in the Earth–Moon system, such that a spacecraft revolves about the body in a retro
grade fashion. They are commonly found in the literature and in proposed spacecraft
missions because they are frequently stable. They behave just like a normal twobody
orbit, but occur in resonance with the motion of the threebody system. Figure 224
illustrates several examples of Earth–Moon DROs of varying radii from the Moon.
2.6.9.4 Halo Orbits Halo orbits are very wellknown threedimensional peri
odic solutions [119, 121, 122] to the circular restricted threebody problem. Fig
LOWENERGY MISSION DESIGN
83
Figure 224 Example orbits in the family of distant retrograde orbits about the Moon, viewed
in the Earth–Moon rotating frame from above.
ure 225 shows a plot of several example halo orbits about the lunar L1 and L2 points.
Many authors have studied how to take advantage of halo orbits for practical missions
to the Moon [5–7]. Halo orbits are of particular use for lunar communication and
navigation satellites [11]: a satellite in a halo orbit has an unimpeded view of both
the Earth and either the nearside of the Moon or the farside of the Moon, for lunar
L1 and L2 halo orbits, respectively. Furthermore, a satellite may be placed in a halo
orbit such that its view of the Sun is also never impeded, simplifying the satellite’s
power and thermal systems.
Since the force ﬁeld in the CRTBP is symmetric about the xy plane (see Sec
tion 2.5.1), and since halo orbits are assymetric about this plane, each halo orbit
solution to the CRTBP comes in a symmetric pair with a northern and a southern
variety [121]. As one can see in Fig. 225, a satellite in a southern orbit spends more
than half of its time below the Moon’s orbital plane, which gives that satellite beneﬁts
for communicating with objects in the southern hemisphere of the Moon.
It is convenient to specify a halo orbit by its zaxis amplitude, Az , since one may
formulate an analytical approximation to a halo orbit using that parameter as an input
[106, 123, 124]. Other studies have speciﬁed a halo orbit using its Jacobi constant or
its x0 value, namely, the xvalue of the location where the orbit has a yposition of
0 km and a positive yvelocity in the synodic reference frame [46, 122]. Figure 226
shows several northern LL1 and LL2 halo orbits from the side in the synodic frame
to illustrate the relationship between a halo orbit’s shape and its zaxis amplitude.
Figure 227 shows the relationship between a halo orbit’s zaxis amplitude and its
period for reference.
84
METHODOLOGY
Figure 225 An illustration of four example halo orbits about the lunar L1 and L2 points.
The halo orbits are viewed from above (left) and from the side (right) in the Earth–Moon
synodic reference frame [47] (ﬁrst published by the American Astronautical Society).
Figure 226 Northern Earth–Moon halo
c 2008 by
orbits [146] (Copyright ©
American Astronautical Society Publications
Ofﬁce, San Diego, California [Web Site:
http://www.univelt.com], all rights reserved;
reprinted with permission of the AAS).
Figure 227
Earth–Moon halo orbit
c 2009 by American
periods [44] (Copyright ©
Astronautical Society Publications Ofﬁce, all
rights reserved, reprinted with permission of
the AAS).
2.6.9.5 Vertical Lyapunov Orbits Another family of libration orbits that exist
about each of the collinear Lagrange points is the family of vertical Lyapunov orbits,
also known as vertical orbits for short. Vertical orbits oscillate out of the xy plane,
piercing the plane at the Lagrange point itself. They are symmetric orbits, traversing
the same route above the plane as below it. Figure 228 provides several views of
example orbits in the family of LL1 vertical Lyapunov orbits.
2.6.9.6 Resonant Orbits Although there are numerous other interesting fam
ilies of periodic orbits in the threebody system, the last type of orbit that will be
LOWENERGY MISSION DESIGN
85
Figure 228 Four perspectives of example orbits within the family of L1 vertical Lyapunov
orbits computed in the Earth–Moon CRTBP.
described here is the resonant orbit. Resonant orbits in the Earth–Moon threebody
problem are essentially twobody orbits about the Earth that are in resonance with the
Moon, and which have been signiﬁcantly perturbed by the Moon. As one may expect,
there are different families of resonant orbits for each resonant period, namely, 3:1,
3:2, 5:1, 5:2, and so forth, where an m:n resonant orbit is one where the spacecraft
traverses the resonant orbit n times while the primaries orbit their barycenter m times.
Figure 229 illustrates four families of resonant orbits in the Earth–Moon system,
shown in the synodic reference frame.
The resonant orbits shown in Fig. 229 are particularly unstable as they pass by
the Moon, but they are generally stable elsewhere. It is possible to transition a
spacecraft off of one threebody orbit, such as a Lyapunov orbit, and onto a resonant
orbit for very little fuel, if the transition is performed near the Moon. A spacecraft
that arrives onto a resonant orbit may then sit in it without requiring any signiﬁcant
stationkeeping fuel, until the spacecraft returns to the Moon. In that way, resonant
orbits may play a useful role as a staging orbit, quarantine orbit, or a destination for a
spacecraft to remain to avoid performing stationkeeping maneuvers. One may also
select how much time should pass between lunar swingbys, based on the resonance;
for instance, a spacecraft traversing a 7:3 resonant orbit will spend far longer between
lunar swingbys than a spacecraft traversing a 3:2 resonant orbit.
86
METHODOLOGY
Figure 229 Four example families of resonant orbits in the Earth–Moon system, viewed
from above in the Earth–Moon rotating frame.
2.6.10
Invariant Manifolds
The dynamics in the circular restricted threebody system permit the existence of ﬁve
ﬁxed points (Section 2.6.2.1) and numerous periodic orbits (Section 2.6.6.1). The
three collinear libration points and many of the periodic orbit solutions in the Earth–
Moon threebody system are unstable (Section 2.6.8). An unstable orbit has at least
one stable and one unstable eigenvalue with corresponding eigenvectors. A spacecraft
traveling along an unstable orbit that experiences a perturbation even slightly in the
unstable direction will exponentially fall away from its nominal position on that orbit,
tracing out a smooth trajectory away from the orbit. In a similar sense, a spacecraft
that has the right initial conditions will follow a smooth trajectory that exponentially
approaches an unstable orbit and eventually arrives on that orbit from the orbit’s
stable direction. These two trajectories describe what is known as an orbit’s stable
and unstable invariant manifolds.
An orbit’s unstable invariant manifold (W U ) contains the set of all trajectories
that a spacecraft may take if it was perturbed anywhere on that orbit in the direction
of the orbit’s unstable eigenvector. Similarly, an orbit’s stable invariant manifold
(W S ) contains the set of all trajectories that a spacecraft may take to asymptotically
LOWENERGY MISSION DESIGN
87
arrive onto that orbit along the orbit’s local stable eigenvector. Put another way, the
orbit’s stable invariant manifold is the set of all trajectories that a spacecraft may
take backward through time after a perturbation in the direction of the orbit’s stable
eigenvector.
Mathematically, the invariant manifolds are deﬁned as follows. First, the CRTBP
may be deﬁned as a vector ﬁeld bound in R6 . One and only one vector is bound to
every point in the vector ﬁeld. Thus, the integration of any point p in the vector ﬁeld
with respect to time generates only one trajectory. Let us deﬁne Tp as the trajectory
generated by the point p. The α and ωlimits are deﬁned to be the set of points in R6
as Tp tends toward −∞ and +∞, respectively. The α and ωlimits may include a
single point, a periodic orbit, or, if Tp has no asymptotic behavior, they may include
a large portion of the state space. The set of all points deﬁning trajectories that have
the same αlimit set is called the unstable manifold of that limit set. Similarly, the
set of all points deﬁning trajectories that have the same ωlimit set is called the stable
manifold of that limit set.
2.6.10.1 Invariant Manifolds of the Unstable Lagrange Points The three
collinear Lagrange points are unstable in both the Sun–Earth and Earth–Moon threebody systems; hence, they have associated invariant manifolds. Since the Lagrange
points are single points in space, their invariant manifolds are onedimensional struc
tures. To produce them, one ﬁrst computes the eigenvalues of the Jacobian of
their states. If X is the state of one of the collinear Lagrange points, equal to
[x y z x˙ y˙ z˙]T , then its Jacobian is equal to
⎤
∂x˙ ∂x˙
∂x˙
···
⎢ ∂x ∂y
∂z˙ ⎥
⎢ ∂y˙ ∂y˙
∂y˙ ⎥
⎥
⎢
···
˙
⎢
⎥
∂X
⎢
∂
x
∂
y
∂z
˙
⎥
J=
=⎢
(2.60)
..
.. ⎥
∂X ⎢ ..
..
⎥
.
.
. ⎥
⎢ .
⎣ ∂z¨ ∂z¨
∂z¨ ⎦
···
∂x ∂y
∂z˙
After plugging in the equations of motion of the CRTBP given in Eqs. (2.1)–(2.3) in
Section 2.5.1, Eq. (2.60) simpliﬁes to
⎡
⎤
0
0
0
1 0 0
⎢ 0
0
0
0 1 0 ⎥
⎥
⎢
⎢ 0
0
0
0 0 1 ⎥
⎥
⎢
⎥
⎢ ∂x
¨ ∂x
¨ ∂x
¨
⎢
0 2 0 ⎥
(2.61)
J = ⎢ ∂ x ∂ y ∂z
⎥
⎥
⎢ ∂y¨ ∂y¨ ∂y¨
⎥
⎢
−2 0 0 ⎥
⎢
⎥
⎢ ∂ x ∂ y ∂z
⎦
⎣ ∂z¨ ∂z¨ ∂z¨
0 0 0
∂ x ∂ y ∂z
⎡
It is apparent that the Jacobian is the same as the Amatrix given in Eq. (2.16).
88
METHODOLOGY
The eigenvalues of the Jacobian for each of the three collinear Lagrange points
include two pairs of imaginary numbers and one pair of real numbers. Tables 26
and 27 summarize the six eigenvalues for the Jacobian of each of the ﬁve Lagrange
points for the Earth–Moon system and for the Sun–Earth system, respectively. The
eigenvector corresponding to the larger real eigenvalue indicates the unstable direc
tion: vU ; the eigenvector corresponding to the other real eigenvalue indicates the
stable direction: vS . The unstable manifold of the Lagrange point, W U , may be
mapped by propagating the state XU forward in time, where XU = X ± EvU and E
is some small perturbation. Similarly, the stable manifold, W S , may be mapped by
propagating the state XS backward in time, where XS = X ± EvS .
The perturbation Ev may be applied to the state X in either a positive or a negative
sense, corresponding to two halves of each manifold. One perturbation will result in
motion that departs the Lagrange point toward the smaller body (for example, toward
the Moon in the Earth–Moon system), and one will result in motion that departs the
Lagrange point away from the smaller body. It is conventional to refer to the half of
the manifold that moves toward the smaller body as the interior manifold, since it
remains in the interior of the smaller body’s inﬂuence, at least for a short while, and
Table 26 A summary of the eigenvalues of the Jacobian of each Lagrange point in
the Earth–Moon CRTBP.
Component
λ1
λ2
λ3
λ4
λ5
λ6
LL1
2.932056
2.932056
2.334386i
2.334386i
2.268831i
2.268831i
LL2
2.158674
2.158674
1.862646i
1.862646i
1.786176i
1.786176i
LL3
0.177875
0.177875
1.01041991i
1.01041991i
1.00533144i
1.00533144i
LL4
1i
1i
0.95450078i
0.95450078i
0.29820842i
0.29820842i
LL5
1i
1i
0.95450078i
0.95450078i
0.29820842i
0.29820842i
Table 27 A summary of the eigenvalues of the Jacobian of each Lagrange point in
the Sun–Earth CRTBP.
Component
λ1
λ2
λ3
λ4
λ5
λ6
EL1
2.532659
2.532659
2.0864535i
2.0864535i
2.0152106i
2.0152106i
EL2
2.484317
2.484317
2.057014i
2.057014i
1.985075i
1.985075i
EL3
0.002825
0.002825
1.00000266i
1.00000266i
1.00000133i
1.00000133i
EL4
1i
1i
0.99998974i
0.99998974i
0.00453024i
0.00453024i
EL5
1i
1i
0.99998974i
0.99998974i
0.00453024i
0.00453024i
LOWENERGY MISSION DESIGN
89
to refer to the half that moves away from the smaller body as the exterior manifold
[37].
The process of analyzing and constructing the invariant manifolds of the unstable
Lagrange points may be visualized by considering that each unstable Lagrange point
is a dynamical saddle point, as illustrated by the plot shown in Fig. 230. One can
see that a spacecraft’s motion will follow the unstable manifold when propagated
forward in time after a perturbation, and it will follow the point’s stable manifold
when propagated backward in time. Figure 230 also demonstrates how there are two
halves of each manifold.
Figures 231–233 show plots of the stable and unstable manifolds of the ﬁrst
three Lagrange points in the Earth–Moon threebody system. The forbidden region
is shown shaded in gray in each plot.
The eigenvalues of the Jacobian of the triangular Lagrange points include three
imaginary pairs for the Sun–Earth and Earth–Moon threebody systems; hence, they
do not have interesting associated invariant manifolds. A spacecraft following a
trajectory near one of these Lagrange points will oscillate about the point. If the
spacecraft is perturbed, its motion will change but it will not exponentially deviate
from its nominal path.
2.6.10.2 Invariant Manifolds of Unstable Periodic Orbits Every unstable
periodic orbit in the CRTBP has a set of invariant manifolds, much like the Lagrange
points. The only substantial difference between the invariant manifolds of periodic
Figure 230 A dynamical saddle point, such as that of the unstable Lagrange points in the
CRTBP, with a vector ﬁeld shown that indicates the motion of a spacecraft near the point.
There are two lines of stable (W S ) and unstable (W U ) manifolds of the saddle point (ﬁrst
published in Ref. [97]; reproduced with kind permission from Springer Science+Business
Media B. V.).
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Figure 231 The stable and unstable invariant manifolds of the ﬁrst Lagrange point of the
Earth–Moon threebody system. See inset at right for expanded view of the lunar vicinity (First
published in Ref. [97]; reproduced with kind permission from Springer Science+Business
Media B. V.).
Figure 232 The stable and unstable invariant manifolds of the second Lagrange point of the
Earth–Moon threebody system. See inset at right for expanded view of the lunar vicinity (First
published in Ref. [97]; reproduced with kind permission from Springer Science+Business
Media B. V.).
orbits and of the Lagrange points is that an additional dimension is added when
considering periodic orbits: periodic orbits are onedimensional structures where
the Lagrange points are zerodimensional structures. Consequently, the invariant
manifolds of unstable periodic orbits are twodimensional structures. They are
constructed of a set of trajectories, where each trajectory corresponds to a point along
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91
Figure 233 The stable and unstable invariant manifolds of the third Lagrange point of the
Earth–Moon threebody system. See inset at right for an expanded view in the vicinity
of Earth (First published in Ref. [97]; reproduced with kind permission from Springer
Science+Business Media B. V.).
the periodic orbit. The set of trajectories wraps about itself, forming a topological
tube. This is further explained below.
To produce the invariant manifolds of an unstable periodic orbit, one requires
information about the local stability characteristics of each point along the orbit.
In theory, one may evaluate the eigenvalues and eigenvectors of the Jacobian at
each and every state along the orbit, and use that information to produce the orbit’s
invariant manifold. However, evaluating so many eigenvalues requires a great deal
of computation. A more efﬁcient manner of producing the invariant manifolds uses
the eigenvalues and eigenvectors of the monodromy matrix [147, 148].
Since the monodromy matrix is produced by propagating the state transition
matrix all the way around the orbit, from time t = t0 to time t = t0 + T , it contains
information about the stability of the entire orbit. To determine the stable and
unstable directions at each point along the orbit, one only has to propagate the stable
and unstable eigenvectors of the monodromy matrix about the orbit using the state
transition matrix. That is, the stable and unstable vectors at time ti about the orbit, viS
and viU , respectively, may be determined using the stable and unstable eigenvectors
of the monodromy matrix, vS and vU , respectively, using the following equations
viS
viU
=
Φ(ti , t0 )vS
(2.62)
=
U
(2.63)
Φ(ti , t0 )v
A small perturbation, E, is then applied to the state of the orbit at that time, Xi , and
the result is propagated in time. Since the state transition matrix grows exponentially
along an unstable orbit, the magnitudes of the vectors viS and viU grow along the orbit.
It is therefore important to normalize the vectors so that a consistent perturbation is
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applied to each orbit state. The ﬁnal equations to produce the initial conditions for
the stable and unstable manifolds at time ti about the orbit, XSi and XU
i , respectively,
are then equal to
XSi
=
Xi ± E
viS
viS 
(2.64)
XU
i
= Xi ± E
viU
viU 
(2.65)
The sign of the perturbation differentiates between interior and exterior manifolds,
as discussed in Section 2.6.10.1.
Some discussion should be provided regarding the magnitude of the perturbations
applied to the state to produce the manifolds. The theoretical invariant manifolds of
the orbit include the set of all trajectories that asymptotically approach the orbit as time
goes either forward or backward. In fact, they never truly arrive on the orbit in ﬁnite
time, but just come arbitrarily close to the orbit. To map them, one approximates the
manifolds by perturbing a state slightly off of the orbit and then propagating that state
in time. The smaller the perturbation, the closer the approximation comes to mapping
the true manifolds; however, small perturbations require more time to depart from
the orbit than larger perturbations. When designing practical missions, one is less
interested in precisely mapping the invariant manifolds of the orbits, and generally
more interested in computationallyswift algorithms. Additionally, the dynamics of
the trajectories depend the greatest on the largest eigenvalues since motion in those
directions grows exponentially faster than motion in any other direction. Hence,
somewhat large perturbations may be used to map out the motion of the trajectories
in the manifolds, for example, on the order of 100 km in the Earth–Moon system
and 1000 km in the Sun–Earth system. In practice, the perturbation magnitudes
are given in either units of position or units of velocity, but the perturbation is
applied proportionally to all six components. A 100km perturbation means that the
magnitude of the perturbation applied to the position coordinates is equal to 100 km,
and the resulting proportionality is used to apply the perturbation to the velocity
components, that is
100 km
E= (2.66)
2
vx + vy2 + vz2
The structure of the manifolds of an orbit greatly depends on the stability charac
teristics of each portion of the orbit. Orbits such as libration orbits are fairly uniformly
unstable; that is, the local Lyapunov exponent does not vary much along the orbit
(Anderson, among others, provides a detailed exploration about the local Lyapunov
exponent of libration orbits [149]). Consequently, their manifolds are fairly smooth
as they extend from the orbit. Various other orbits are unstable due to a close ﬂyby
of one of the primary bodies. The local stability of these orbits changes drastically,
becoming very unstable as the orbit approaches one of the massive bodies. Hence,
their manifolds spread out quickly near the body and remain fairly close together
elsewhere.
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93
Figure 234 shows the stable and unstable manifolds of a twodimensional Lya
punov orbit about the Earth–Moon L2 point. One can see that the manifolds are
smooth and form a tubelike structure. They remain welldeﬁned until they encounter
the Moon, at which time they spread out very rapidly, and the tubelike structure be
comes less obvious. One can also see that the stable and unstable manifolds are
Figure 234 The stable (left) and unstable (right) manifolds of a Lyapunov orbit about the
Earth–Moon L2 point.
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symmetric about the xaxis, due to the symmetry in the CRTBP. For comparison,
Fig. 235 shows the stable manifold of a resonant ﬂyby orbit, which shows how the
structure of the manifolds depends on the local stability of the orbit. One notices that
the trajectories in the manifold diverge quickly near the Moon but remain near the
orbit elsewhere. The unstable manifold is not shown but is symmetric to the stable
manifold.
2.6.10.3 Invariant Manifolds of Unstable Quasiperiodic Orbits Unstable
quasiperiodic orbits have associated stable and unstable invariant manifolds, much
like unstable periodic orbits; however, the structure and the procedures required to
produce them are slightly different. Quasiperiodic Lissajous and quasihalo orbits in
the CRTBP are twodimensional structures [125]. Hence, their invariant manifolds
are threedimensional structures. The additional dimension adds beneﬁts as well as
complexity when using them in practical mission designs.
Since quasiperiodic orbits never retrace their path, one cannot produce them
entirely, although one can use a variety of numerical tools to represent them and to
produce desirable segments of them [150]. Since these orbits are not periodic, they do
not have associated monodromy matrices. Hence, one cannot use the same simpliﬁed
procedures to produce their invariant manifolds as those procedures discussed in
Section 2.6.10.2.
To produce a quasiperiodic orbit’s invariant manifolds, one can always compute
the eigenvectors of the Jacobian of the states at each point along sample segments
of the orbit, and follow the same procedures as given in the previous sections.
However, that procedure is numerically intensive and slow. Alternatively, to reduce
the computational load, one may approximate the manifolds by producing an analog
to the monodromy matrix. One may propagate the state transition matrix from one
y = 0 plane crossing in the synodic frame to the next (or to any later crossing) and
Figure 235 The stable manifold of a resonant lunar ﬂyby orbit. See inset at right for
expanded view of the lunar vicinity.
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95
use the resulting matrix as a pseudomonodromy matrix. When one propagates this
matrix’s stable and unstable eigenvectors about the orbit segment, and then follows
the process outlined in Section 2.6.10.2, one produces approximations of the invariant
manifolds of the quasiperiodic orbit. These approximations are often good enough to
be used for preliminary spacecraft mission design, such as that used for the Genesis
spacecraft mission [137].
2.6.11
Orbit Transfers
Dynamical systems theory provides the tools needed to systematically produce trans
fers to/from unstable orbits in the CRTBP. This section discusses several example
orbit transfers as demonstrations of the application of dynamical systems theory.
Section 2.6.11.1 discusses the construction of a transfer from the Earth to a halo
orbit about the Sun–Earth L2 point, and that transfer is very similar to that used by
the WMAP mission [70]. Section 2.6.11.3 discusses the construction of a chain of
periodic orbits in the CRTBP, which is relevant to missions like Genesis [71, 72] and
Wind [63]. That is, trajectories are constructed that transfer a spacecraft back and
forth between several periodic orbits in the CRTBP. These examples demonstrate the
procedures that may be followed to construct any type of orbit transfer in the CRTBP
using dynamical systems theory.
2.6.11.1 Surface to Orbit Transfers Several missions (including WMAP,
Herschel, and Planck) have demonstrated the beneﬁts of operating in a libration
orbit about the Sun–Earth L2 point; many other missions have been proposed to op
erate in similar orbits, including the James Webb Space Telescope and the Terrestrial
Planet Finder. In this section, we demonstrate how to construct a ballistic transfer
from the Earth to a halo orbit about EL2 , a transfer that might prove to be very useful
for missions such as these proposed missions. The transfers produced here do not
require any orbit insertion maneuvers; after their LEO departures, each transfer is
thereafter entirely free of any deterministic maneuvers. The process used here may
be generalized to compute a transfer from the surface of the secondary body in most
threebody systems into many unstable threebody periodic orbits, or viceversa.
We ﬁrst consider the family of halo orbits about the EL2 point, illustrated in
Section 2.6.9.4. The family begins as a bifurcation of the family of planar Lyapunov
orbits about EL2 . The orbits in the family gradually move farther out of the plane
until they eventually make close approaches with the Earth. Example orbits in the
family of northern EL2 halo orbits are shown in Fig. 236.
We next consider a single unstable halo orbit and produce its stable invariant
manifold. This manifold includes all the trajectories that a spacecraft may take to
arrive onto the orbit. A plot of the example halo orbit and its stable manifold is shown
in Fig. 237. The trajectories shown in blue have a perigee altitude below 500 km.
The halo orbit chosen here has a Jacobi constant equal to approximately 3.00077207.
The CRTBP is a good model of the real Solar System for trajectories propagated for
a reasonably short amount of time, namely, for one orbital period of the two primary
masses about their barycenter, or about a year in the Sun–Earth system. Beyond
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Figure 236 Example halo orbits in the family of northern halo orbits about the Sun–Earth
L2 point. The orbits are shown from four perspectives.
that, the accumulation of errors due to perturbations in the real Solar System causes
the CRTBP approximation to break down. The trajectories shown in Fig. 237 have
been propagated for at most 365 days—they are only plotted in the ﬁgure until they
cross the plane of the Earth for clarity. As the propagation time is increased, the
trajectories may make additional close approaches to the Earth. In some cases the
second or third perigee passes closer to the Earth than the ﬁrst. These features will
be explored below.
Each trajectory shown in Fig. 237 may be characterized using several parameters.
The parameter τ , deﬁned in Section 2.6.2.3, indicates the point where the trajectory
arrives at the halo orbit. The closest approach of each trajectory with the Earth
is identiﬁed to compute the perigee altitude and ecliptic inclination with respect to
the Earth. These two parameters are useful because they indicate the altitude and
inclination of a low Earth orbit that may be used as a staging orbit to transfer a
spacecraft to the halo orbit. Figure 238 shows two plots: one of the perigee altitude
and one of the corresponding ecliptic inclination as functions of τ , where the vertical
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97
Figure 237 An example unstable halo orbit (green) about the Sun–Earth L2 point and its
stable invariant manifold (blue). (See insert for color representation of this ﬁgure.)
bars indicate the locations in the manifold that have perigee altitudes below 500 km.
For example, one can see that the trajectory with a τ value of 0.751 encounters a
closest approach with the Earth with a perigee altitude of approximately 185 km and
an ecliptic inclination of approximately 34.8 deg. Hence, a spacecraft in a circular
low Earth parking orbit with an altitude of 185 km and an ecliptic inclination of
34.8 deg may perform a tangential ΔV to transfer onto the manifold; once on the
manifold, the spacecraft ballistically follows it and asymptotically arrives on the halo
orbit.
There are two statements in these results that need to be addressed. The ﬁrst is
that the inclination values displayed in Fig. 238 are the inclination values computed
in the axes of the CRTBP: namely, in a plane that is very similar to the ecliptic.
The equatorial inclination values of these perigee points depend on which date a
spacecraft launches. Since the Earth’s rotational axis is tilted by approximately
23.45 deg with respect to the ecliptic [97], many equatorial inclinations may be used
to inject onto a desired trajectory, depending on the date. The second statement
that should be addressed is that the results shown in Fig. 238 depend greatly on
the perturbation magnitude, E, described in Section 2.6.10. Implementing a different
perturbation magnitude results in a change in the τ values required to obtain a certain
trajectory. For example, if E were reduced, the trajectories modeling the orbit’s
manifold would spend more time asymptotically approaching/departing the orbit.
Once the trajectories are sufﬁciently far from the orbit, their characteristics are nearly
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Figure 238 The perigee altitude (top) and corresponding ecliptic inclination (bottom) of
the trajectories in the stable manifold shown in Fig. 237 as functions of τ .
unchanged. The result is that the τ value for a desired trajectory is strongly related
to the value of E. This has no signiﬁcant effect for practical spacecraft mission
designs; a spacecraft following a trajectory in the halo orbit’s stable manifold will
asymptotically approach the halo orbit—the value of E is only used for modeling the
stable manifold.
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99
It is now of interest to identify how the manifolds change and how the plots shown
in Fig. 238 change as the manifold’s propagation time is varied. Figure 239 shows
plots of the stable manifold of the same halo orbit propagated for successively longer
amounts of time. One can see that the trajectories on the manifold spend some
amount of time near the halo orbit (where, once again, the amount of time depends
on the value of E), and then depart. It may be seen that many of the trajectories in the
manifold make closer approaches with the Earth after their ﬁrst perigee. Figure 240
shows many plots of the closest approach each manifold makes with the Earth for
varying amounts of propagation time. It is clear that the longer propagation times
yield closer perigee passages.
The procedures given in this section may be repeated for each halo orbit in the
entire family of halo orbits, and maps may be produced showing the range of perigee
altitudes and the range of inclination values obtainable for each halo orbit. These
are useful for identifying the approximate location of desirable trajectories in the real
Solar System.
2.6.11.2 Homoclinic and Heteroclinic Connections Many unstable peri
odic orbits in the CRTBP contain homoclinic connections with themselves and/or
heteroclinic connections with other unstable periodic orbits [71, 98, 151]. If a tra
jectory in an orbit’s unstable manifold departs that orbit, traverses the threebody
system for some time, and then later arrives back onto the same orbit, it makes what
is known as a homoclinic connection with the host orbit [151]. This trajectory is
contained within both the orbit’s unstable and stable manifolds. McGehee proved the
existence of homoclinic connections in both the interior and exterior regions of the
threebody system [152]. In a similar sense, a different trajectory within the unstable
manifold of one orbit may depart that orbit and eventually arrive onto a second orbit.
The trajectory is thus contained within both the unstable manifold of the ﬁrst orbit
and the stable manifold of the second orbit. Such a trajectory forms what is known
as a heteroclinic connection between the two unstable orbits [151].
In theory, heteroclinic connections asymptotically depart one orbit and asymp
totically approach another orbit. In practice, the spacecraft is never truly on any
host periodic orbit, but is instead within some small distance from the orbit. For the
purpose of the discussions provided here, it is assumed that a spacecraft departs an
orbit when its state is perturbed off of that orbit, and it arrives on the new orbit when
it arrives at the state that corresponds to the perturbation that generated the stable
manifold. For the case of orbit transfers in the Earth–Moon system, this means that
the duration of an orbit transfer includes all time that the spacecraft is further than
100 km from a host orbit.
Many authors have explored homoclinic and heteroclinic transfers between threebody orbits as transport mechanisms for spacecraft and comets [98, 147, 149, 151,
153–160]. Using dynamical systems theory, Lo and Ross noted that the orbit of
the comet Oterma appeared to shadow the invariant manifolds of libration orbits
about the L1 and L2 points in the Sun–Jupiter threebody system [161]. Koon et al.
later showed that the comet closely followed a homoclinicheteroclinic chain [151].
G´omez et al. began exploring the numerical construction of orbits with prescribed
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Figure 239 The stable manifold of a Sun–Earth L2 halo orbit propagated for successively
longer amounts of time. The duration of each propagation is indicated in each plot by the
value Δt.
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101
Figure 240 The altitude of closest approach of each trajectory in the stable manifolds
shown in Fig. 239 with respect to the Earth. The propagation times of each manifold are
shown in the legend. Longer propagation times yield closer perigee passages. (See insert for
color representation of this ﬁgure.)
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itineraries to describe the resonant transitions exhibited by the comet Oterma [98].
The material presented in Section 2.6.11.3 extends their work, applying a method for
the construction of prescribed orbit transfers in the Earth–Moon system [101].
2.6.11.3 Orbit Transfers and Chains Once a spacecraft is on an unstable
periodic orbit in the threebody system, then it may theoretically stay there for an
arbitrarily long time, or it may depart that orbit by following any trajectory on that
orbit’s unstable manifold. The practical ΔV cost for a spacecraft to depart an orbit
is the same as the cost of stationkeeping to remain on that orbit: both are arbitrarily
small given good navigational support. These considerations are further explored in
Chapter 6.
Section 2.6.10.2 shows several examples of stable and unstable invariant manifolds
of unstable periodic orbits. One may notice by studying these manifolds that by
controlling exactly when the spacecraft departs from its periodic orbit, it may be able
to transfer to numerous other locations in the state space, including, but not limited
to, the surface of the Moon, any of the ﬁve lunar Lagrange points, another unstable
periodic orbit in the system, or an escape trajectory away from the vicinity of the
Moon or Earth. If the spacecraft were carefully navigated onto the correct trajectory
within the unstable manifold of one orbit, it would then encounter the stable manifold
of a different unstable threebody orbit.
After considering a spacecraft’s options, several categories of orbit transfers may
be identiﬁed. Table 28 summarizes a few characteristic categories of orbit transfers.
In the table, a “stable orbit” includes conventional twobody orbits about either of the
two primaries in the system, as well as stable threebody orbits, and even transfers
to/from the surface of one of the primary bodies. The minimum number of ΔVs
indicates the fewest number of maneuvers that may typically be used to perform
the given transfer. There are many cases when a particular transfer might require
more maneuvers, such as a transfer from the surface of a body to a particular orbit
in space with a time constraint. There are also certain special cases when a transfer
might require fewer maneuvers, such as a transfer between two stable orbits where
the two orbits intersect in space. Nonetheless, Table 28 gives a good idea about the
minimum number of required maneuvers for orbit transfers in several circumstances.
Table 28
A summary of several categories of orbit transfers in the CRTBP.
Minimum
Number of ΔVs
Orbit 1
Orbit 2
Constraints
Stable Orbit
Stable Orbit
Unstable Orbit
Unstable Orbit
Stable Orbit
Unstable Orbit
Stable Orbit
Unstable Orbit
None
None
None
None
2
1
1
1
Any Orbit
Unstable Orbit
Any Orbit
Unstable Orbit
Transfer Time
Same Jacobi Constant
2
0
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103
In this section, lowenergy orbit transfers are introduced that may be useful for
practical mission design, and which are useful background for the discussions of
lowenergy lunar transfers in later chapters. These orbit transfers are in the category
represented by the last row of Table 28, transfers that are free of deterministic
maneuvers.
Lowenergy transfers between unstable orbits may be located in the CRTBP by
analyzing Poincare´ maps (Section 2.6.3) [156]. Suppose there are two unstable
Lyapunov orbits in the Earth–Moon threebody system: one about LL1 and the
other about LL2 . Both of these orbits have a set of stable and unstable invariant
manifolds. In the planar CRTBP, each point along a manifold may be characterized
by a fourdimensional state [ x, y, x,
˙ y˙ ]. If a surface of section is placed in R4 at
some xposition, the resulting intersection is a surface in R3 . If it is further speciﬁed
that the two Lyapunov orbits have the same Jacobi constant, then each point along
any trajectory within both orbits’ manifolds will have the same Jacobi constant and
the phase space of the problem is reduced to R2 . The state at any intersection in
the surface may only be reconstructed if the Poincar´e map is onesided, since the
Jacobi constant has a sign ambiguity. The stable and unstable manifolds of both
orbits appear as curves in the twodimensional Poincare´ map. Any intersection of
these curves corresponds to a free transfer between the two orbits.
Figure 241 illustrates the process of identifying free transfers from a Lyapunov
orbit about LL1 to a Lyapunov orbit about LL2 . In this case, the value of the Jacobi
constant of both orbits has been selected to be 3.13443929. A P+ Poincar´e map
has been constructed, where the surface, Σ, has been placed at the xcoordinate
of the Moon, namely, at a value of approximately 379,730 km with respect to the
barycenter of the Earth–Moon system. The top of Fig. 241 illustrates the unstable
and stable manifolds integrated to the ﬁrst intersection with the surface of section.
The intersection of both manifolds with the surface of section is shown on the bottom
of Fig. 241. One can see that there are two intersections that correspond to the two
free transfers indicated in the ﬁgure.
The simple illustration shown in Fig. 241 may be extended by propagating the
manifolds longer and identifying intersections in the manifolds that correspond to
longer, more complicated heteroclinic connections. The Poincar´e map shown in
Fig. 242 is produced by propagating the unstable manifold of the LL1 Lyapunov
orbit and the stable manifold of the LL2 Lyapunov orbit for 60 days each. In
addition, the map shown in Fig. 242 is a P± map, displaying all intersections of
both manifolds with the surface of section. In this particular mapping, the majority
of the points shown below the y = 0 line are members of the P+ map (including the
points shown in Fig. 241), the majority of the points shown above it are members of
the P− map, and all observed intersections of the two manifolds do indeed intersect,
even accounting for the sign ambiguity of x˙ .
Figure 242 includes eight example orbit transfers to illustrate what sort of hetero
clinic connections exist between these two libration orbits. Certain types of motion
appear in more than one heteroclinic connection. For example, the trajectories la
beled (1), (2), and (7) appear to graze a distant prograde orbit, whereas the trajectories
labeled (1), (3), and (4) appear to traverse a ﬁgureeight type orbit. The appearance of
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Figure 241 An illustration of the process of using a Poincar´e map to identify free transfers
between two Lyapunov orbits. Both orbits have a Jacobi constant of 3.13443929. Top: the
unstable manifold of an LL1 Lyapunov orbit and the stable manifold of an LL2 Lyapunov
integrated to the surface of section. Bottom: the corresponding P+ Poincar´e map and two
free transfers [101] (Acta Astronautica by International Academy of Astronautics, reproduced
with permission of Pergamon in the format reuse in a book/textbook via Copyright Clearance
Center).
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105
Figure 242 The P± Poincar´e map produced from the same system and surface of section
shown in Fig. 241, now with an extended manifold propagation duration of 60 days. The
plots shown surrounding the Poincar´e map illustrate several example free transfers that have
been identiﬁed in the central map [101] (Acta Astronautica by International Academy of
Astronautics, reproduced with permission of Pergamon in the format reuse in a book/textbook
via Copyright Clearance Center).
such orbits in the Poincare´ maps reinforces the idea that one may construct a speciﬁc
chain of simple orbits to construct a complicated itinerary of orbit transfers.
The Poincare´ map is a useful tool to identify what sorts of orbit transfers exist, but
it does not immediately reveal the shape or geometry of the transfers. For instance,
the transfer labeled (8) in Fig. 242 includes a lunar ﬂyby, which may or may not
be desirable. Section 2.6.12 introduces a method that may be used to construct a
desirable sequence of orbit transfers after identifying that such orbit transfers exist.
Free transfers only exist in the CRTBP between two unstable orbits that have the
same Jacobi constant. Figure 243 shows a plot of several families of threebody
orbits in the Earth–Moon CRTBP, where the orbits’ Jacobi constant values are plotted
as functions of their x0 values. The curves shaded in black correspond to unstable
threebody orbits; the curves shaded in gray correspond to orbits that are neutrally
stable [130]. The horizontal line indicates the Jacobi constant value used to produce
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METHODOLOGY
Figure 243 A plot of several families of threebody orbits in the Earth–Moon CRTBP,
where the orbits’ Jacobi constant values are plotted as functions of their x0 values. The curves
shaded in black correspond to unstable threebody orbits; the curves shaded in gray correspond
c
to orbits that are neutrally stable [162] (Copyright ©2006
by American Astronautical Society
Publications Ofﬁce (web site: http://www.univelt.com), all rights reserved, reprinted with
permission of the AAS).
the heteroclinic connections observed in Figs. 241 and 242. The ﬁgure veriﬁes that
the families of Lyapunov orbits about LL1 and LL2 both include unstable orbits at
the same indicated Jacobi constant value, along with the family of distant prograde
orbits, which helps to explain the appearance of such an orbit in the transfers labeled
(1), (2), and (7) in Fig. 242.
2.6.12
Building Complex Orbit Chains
In the previous section, a technique was presented that may be used to identify
the heteroclinic connections between two unstable periodic orbits. Previous papers
have theorized using symbolic dynamics that if a heteroclinic connection exists, it
is possible to ﬁnd a trajectory that transfers back and forth arbitrarily between those
orbits. Robinson provides a thorough review of the background of symbolic dynamics
[163]. Canalias et al. [155], provide a methodology to search for a combination of
homoclinic transfers that may be used to change the phase of a spacecraft traversing
an unstable periodic orbit. In this section we study a practical method to construct
a complex orbit chain given a desired sequence of homoclinic and/or heteroclinic
transfers.
LOWENERGY MISSION DESIGN
107
2.6.12.1 Constructing a Complex Orbit Chain One may describe a space
craft’s itinerary between simple periodic orbits in the CRTBP by considering its state
at each xaxis crossing. A spacecraft traversing any simple periodic orbit pierces the
xaxis twice: once with positive and once with negative values of y˙. One may model
a simpliﬁed orbit transfer by considering that the spacecraft departs the initial orbit at
one xaxis crossing, is midway through the transfer at the next xaxis crossing, and
completes the transfer at a later xaxis crossing. Using this conceptualization, one
may construct a set of xaxis states to describe a given itinerary between two orbits.
A set of eight states are summarized in Fig. 244 and Table 29 for transfers between
an example LL1 Lyapunov orbit and an example distant prograde orbit (DPO) that
have the same Jacobi constant.
The states given in Fig. 244 and Table 29 have been collected from two sources.
The states corresponding to the simple periodic orbits (A, B, E, and F) have been
taken directly from those orbits; the algorithm described in Section 2.6.6.2 is wellsuited to generate the states of a periodic orbit at their orthogonal xaxis crossings.
The states that correspond to the orbit transfers (C, D, G, and H) have been taken
from their heteroclinic connections identiﬁed using the Poincare´ analysis described
in Section 2.6.3. A theoretical heteroclinic connection between these orbits asymp
Figure 244 A summary of the states needed to produce complex itineraries between two
orbits. In this case, the two orbits are a Lyapunov orbit about L1 and a DPO about the Moon.
The states “D” and “H” are on the xaxis, although the labels are offset [101] (Acta Astronautica
by International Academy of Astronautics, reproduced with permission of Pergamon in the
format reuse in a book/textbook via Copyright Clearance Center).
108
METHODOLOGY
Table 29 The eight states shown in Fig. 244. The state coordinates are given in the
Earth–Moon synodic reference frame, relative to the Earth–Moon barycenter, in both
nondimensional normalized units and SI units [101] (Acta Astronautica by International
Academy of Astronautics, reproduced with permission of Pergamon in the format reuse
in a book/textbook via Copyright Clearance Center).
State
x
Units
y
x˙
y˙
A:
X+
LL1 −LL1
Normalized
SI (km, m/s)
0.812255
312,230
0.0
0.0
0.0
0.0
0.248312
254.418
B:
X−
LL1 −LL1
Normalized
SI (km, m/s)
0.878585
337,728
0.0
0.0
0.0
0.0
−v0.281719
−288.647
C:
X+
LL1 −DP O
Normalized
SI (km, m/s)
0.813049
312,536
0.0
0.0
0.0
0.0
0.247532
253.618
D:
X−
LL1 −DP O
Normalized
SI (km, m/s)
0.890940
342,477
0.0
0.0
0.049050
50.256
−0.311179
−318.830
E:
X+
DP O−DP O
Normalized
SI (km, m/s)
1.061692
408,115
0.0
0.0
0.0
0.0
0.403877
413.809
F:
X−
DP O−DP O
Normalized
SI (km, m/s)
0.909845
349,745
0.0
0.0
0.0
0.0
−0.386264
−395.762
G:
X+
DP O−LL1
Normalized
SI (km, m/s)
1.056340
406,057
0.0
0.0
0.0
0.0
0.432104
442.729
H:
X−
DP O−LL1
Normalized
SI (km, m/s)
0.890940
342,477
0.0
0.0
−0.049050
−50.256
−0.311179
−318.830
totically wraps off one orbit and onto the next as E in Eq. (2.65) approaches 0; an
inﬁnite number of xaxis crossings precede the theoretical heteroclinic connection.
The states D and H correspond to the xaxis crossings that are furthest from either
host orbit. The states C and G correspond to the previous respective xaxis crossing.
As one can see in Table 29, state C is approximately 306 km and 0.8 m/s away from
state A, and state G is approximately 2058 km and 28.9 m/s away from state E. These
state differences are small enough to proceed without difﬁculty.
The states summarized in Fig. 244 and Table 29 may be used to construct a
sequence of states that represent any itinerary between the two given orbits. This
sequence may then be converted into a series of patchpoints that may be inputted into
a differential corrector in order to produce a continuous trajectory. For instance, the
trajectory of a spacecraft in orbit about the LL1 Lyapunov orbit may be represented
by the sequence
{. . . , A, B, A, B, . . . }
LOWENERGY MISSION DESIGN
109
A differential corrector may be used to convert this sequence into a continuous
trajectory. If a mission designer wishes to transfer the spacecraft from the LL1 orbit
to the distant prograde orbit, the designer would construct the sequence
{. . . , A, B, A, B, C, D, E , F, E , F, . . . }
and input that sequence into the differential corrector. The differential correction
process adjusts every state in the sequence to accommodate the slight differences
between the states A and C to make the transfer continuous.
Table 210 provides two example sequences that may be used as inputs to a differ
ential corrector in order to produce continuous trajectories with different itineraries.
To demonstrate this process, the ﬁrst sequence in Table 210 has been converted into
patchpoints and processed by the multipleshooting differential corrector described in
Section 2.6.5.2. Table 211 displays the results of the differential correction process,
comparing the states of the patchpoints before and after the process. One can see
that the differential corrector adjusted each patchpoint away from the xaxis in order
to produce a continuous trajectory, however, none of the patchpoints moved far. In
this example, the differential corrector achieved a trajectory that met the requested
continuity tolerances: the largest position and velocity discontinuities that were ob
served in any of the patchpoints along the ﬁnal trajectory were less than 0.4 mm and
3.1 × 10−9 m/s, respectively.
2.6.12.2 Complex Periodic Orbits A complex periodic orbit may be con
structed by repeating a given sequence of states ad inﬁnitum and inputting that
theoretical sequence into the differential corrector. For instance, the following se
quence may be used to represent a periodic orbit that consists of two revolutions
Table 210 Two sequences that may be used as inputs to a differential corrector in
order to produce continuous trajectories with different example itineraries. The letters
correspond to the states summarized in Fig. 244 [101] (Acta Astronautica by
International Academy of Astronautics, reproduced with permission of Pergamon in the
format reuse in a book/textbook via Copyright Clearance Center).
Example 1
Sequence
�
A
B �
C
D �
E
F �
E
F �
G
H
Objective
Traverse LL1
Transfer to DPO
Traverse DPO (1)
Traverse DPO (2)
Transfer to LL1
Example 2
Sequence
�
A
B �
C
D �
G
H �
C
D �
E
F
Objective
Traverse LL1
Transfer to DPO
Transfer to LL1
Transfer to DPO
Traverse DPO
A:
H:
G:
F:
E:
F:
E:
D:
C:
B:
312,230
342,477
406,057
349,745
408,115
349,745
408,115
342,477
312,536
337,728
312,230
A: X+
LL1 −LL1
X−
LL1 −LL1
X+
LL1 −DP O
X−
LL1 −DP O
X+
DP O−DP O
X−
DP O−DP O
X+
DP O−DP O
X−
DP O−DP O
X+
DP O−LL1
X−
DP O−LL1
X+
LL1 −LL1
x (km)
Patchpoint
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
y (km)
x (km)
16.916
2.644
−395.762 349,595
442.729 407,079
0.000
−0.175
−395.762 349,595
413.809 408,081
19.004
413.809 407,079
254.092
y˙ (m/s)
253.675
427.761
414.071
−7.171
427.755
−1.094 −393.772
0.000
1.079 −393.771
6.862
51.463 −319.032
1.180
0.034 −288.236
0.000
x˙ (m/s)
0.0
254.418 312,555
0.000
0.000
254.418
−50.256 −318.830 342,557 −12.750 −51.431 −319.076
0.0
0.0
0.0
0.0
0.0
12.782
50.256 −318.830 342,553
−0.003
0.000
y (km)
0.640
−288.647 337,704
254.418 312,242
y˙ (m/s)
Final State
253.618 312,399
0.0
0.0
0.0
x˙ (m/s)
Initial State
Table 211 The results of a differential correction process that converted the ﬁrst sequence presented in Table 210 into a continuous trajectory.
The coordinates shown here are in the Earth–Moon synodic reference frame, relative to the Earth–Moon barycenter [101] (Acta Astronautica by
International Academy of Astronautics, reproduced with permission of Pergamon in the format reuse in a book/textbook via Copyright Clearance
Center).
110
METHODOLOGY
LOWENERGY MISSION DESIGN
111
about the LL1 Lyapunov orbit, followed by one revolution about the distant prograde
orbit, repeating itself indeﬁnitely, where the orbits and letters are deﬁned in Fig. 244
{. . . , A, B, A, B, C, D, G, H , A, B, A, B, C, D, G, H , . . . }
�
� �
�
Figure 245 shows a plot of such a periodic orbit. One can see that a trajectory
following a complex itinerary gets very close to one of its generating threebody
orbits even with as few as two revolutions about the orbit.
Since each unstable threebody orbit exists in a family, where the characteristics
of each orbit in the family vary continuously from one end of the family to the other,
it is hypothesized that a complex periodic orbit also exists in a family. The family of
any given periodic orbit is limited in extent to some range of parameters [46]. The
extent of the family of complex orbits is also limited in extent, and it is hypothesized
that the family may only extend through a range where each of its fundamental orbits
and orbit transfers exists. Figure 246 shows several example complex periodic orbits
that exist in the same family as the orbit shown in Fig. 245. Each of these orbits has
a different Jacobi constant, but the same morphology.
2.6.12.3 Generalization The method demonstrated here has been illustrated by
a very straightforward example, namely, the construction of orbit transfers between
an LL1 Lyapunov orbit, and a distant prograde orbit, two simple periodic threebody
orbits. These orbits have been used because they are easily visualized and may
be characterized using only a handful of states. Each state is placed at an xaxis
crossing, although one can see in Table 211 that the states may be displaced during
the differential correction process.
Figure 245 A complex periodic orbit that consists of two revolutions about the LL1
Lyapunov orbit, followed by one revolution about the distant prograde orbit, repeating itself
indeﬁnitely. This orbit is viewed from above in the Earth–Moon synodic reference frame.
112
METHODOLOGY
Figure 246 Several example complex periodic orbits that exist in the same family as the
orbit shown in Fig. 245.
This method may certainly be applied to orbit transfers between other unstable
threebody orbits, including nonsymmetric orbits. In addition, a chain of orbits may
certainly contain more than two different threebody orbits. Longer orbits and orbit
transfers will likely require more states per segment for the differential corrector to
converge. In that case, it is easier to visualize the problem by deﬁning a sequence of
states per segment and using symbols that represent sequences rather than individual
LOWENERGY MISSION DESIGN
113
states. Table 212 provides an example where the states A–H given above have been
mapped to four such sequences.
If one refers to Fig. 241, one notices that there are two lowenergy transfers
between the example Lyapunov orbits about LL1 and LL2 . One may construct a
different sequence of states for each of those transfers, for example, S1LL1−LL2 and
S2LL1−LL2 , which may be constructed from three or more states, including an initial
state and two intermediate states in order to keep the trajectory segment lengths short
enough to permit the differential corrector to converge.
Figure 242 shows several lowenergy transfers that exist from an orbit about LL1
to an orbit about LL2 that were generated using a Poincare´ map. The transfer labeled
(7) may be described as a complex chain that starts in an orbit about LL1 , transfers
to a DPO, remains in that orbit for a revolution, and then transfers from there to an
orbit about LL2 . This complex chain was identiﬁed using a Poincare´ map, but it may
be quickly generated by differentially correcting the series of states represented by
the following sequence of
{. . . , SLL1 , SLL1 , SLL1−DP O , SDP O , SDP O−LL2 , SLL2 , SLL2 , . . . }
2.6.13
Discussion
This section introduced the tools that may be used to construct interplanetary transfers
in the CRTBP using dynamical systems theory. It introduced the basic solutions of
the CRTBP, including the Lagrange points and simple periodic orbits. It discussed
several methods that may be used to build periodic orbits in the CRTBP. The sta
bility of a trajectory or an orbit may be evaluated using the eigenvalues of the state
transition or monodromy matrices. The state transition matrix is also very useful
when implementing targeting tools such as the differential corrector. The unstable
nature of many trajectories in the CRTBP leads to divergent behavior and chaos, but
it also permits a mission designer to build lowenergy transfers from one orbit to
another. Mission designers trace structure in an orbit’s stable and unstable manifolds
and use that information to target a transfer to/from that orbit. Such transfers require
Table 212 The mapping of the states A–H to sequences [101] (Acta Astronautica by
International Academy of Astronautics, reproduced with permission of Pergamon in the
format reuse in a book/textbook via Copyright Clearance Center).
Sequence
SLL1
SLL1−DP O
SDP O
SDP O−LL1
States
{ A, B }
{ C, D }
{ E, F }
{ G, H }
Purpose
Traverse the LL1 Orbit
Transfer from LL1 to DPO
Traverse the DPO
Transfer from DPO to LL1
114
METHODOLOGY
very little energy and may be used to move a spacecraft a great distance around the
threebody system without expending much fuel. These transfers are the basis for
building ballistic transfers between the Earth and the Moon, which is the topic of the
next few chapters.
2.7
TOOLS
Many tools are used in the design of lowenergy lunar transfers. Dynamical sys
tems methods and the corresponding tools, such as the differential corrector and
Poincar´e sections, are described earlier in this chapter. Other tools include numerical
integrators and optimizers. These will be brieﬂy described here.
2.7.1
Numerical Integrators
The two primary integrators used in the analyses contained in this work are the DIVA
integrator [164–166] and a RungeKuttaFehlberg seventhorder (RKF78) integrator
with stepsize control [167]. The DIVA integrator is currently implemented in both
the Missionanalysis, Operations, and Navigation Toolkit Environment (MONTE)
and libration point mission design tool (LTool) software (see Section 2.7.3) and has
a rich heritage spanning more than three decades as an integrator for interplanetary
missions at the Jet Propulsion Laboratory. It uses a variableorder Adams method
for solving ordinary differential equations that has been written speciﬁcally for inte
grating trajectories. The RKF78 integrator is implemented in JPL’s LTool. It allows
for a variable step size as described by Fehlberg [167], and it is also widely used for
astrodynamics and mission design.
2.7.2
Optimizers
Many mission designs presented in this book take advantage of the software package
SNOPT (Sparse Nonlinear OPTimizer) [168, 169] to adjust the various parameters in
order to identify solutions that require minimal amounts of fuel. SNOPT is written
to use a particular sequential quadratic programming (SQP) method, one that takes
advantage of the sparsity of the Jacobian matrix of the constraints of the system while
maintaining a quasiNewtonian approximation of the Hessian of the Lagrangian of
the system. The details of the algorithms are beyond the scope of this discussion,
except to say that they are written to be highly effective when applied to a system
that has smooth nonlinear objective functions [169].
The objective functions and constraints studied here are indeed nearly smooth
functions. There are two reasons why the functions studied in this paper are not
perfectly smooth. First, the unstable nature of lowenergy lunar transfers combined
with ﬁniteprecision computers yields functions that have discontinuities. In general,
these discontinuities are several orders of magnitude smaller than the trends being
studied in this paper and are therefore ignored. Second, several objective functions
studied in this paper involve iterative algorithms; there are discontinuities between a
TOOLS
115
set of parameters that require n iterations to generate a solution and a neighboring set
of parameters that require n + 1 iterations to converge. The discontinuities observed
are small relative to the topography in the state space; thus, SNOPT tends to work
well in these studies.
2.7.3
Software
JPL’s MONTE software [170] has been used for the majority of the analyses con
tained in this book. It provides an interface with JPL’s DE421 Planetary and Lunar
Ephemerides as well as integration using the DIVA propagator. JPL’s LTool has
been used for many of the computations involving libration orbits and their invariant
manifolds. Targeting and optimization algorithms have been implemented in both
sets of software for analyses in the CRTBP and in the ephemeris model. All of the
coordinate frames described in Section 2.4 are accessed through the SPICE Toolkit
in both software suites [171].
CHAPTER 3
TRANSFERS TO LUNAR LIBRATION
ORBITS
3.1
EXECUTIVE SUMMARY
This chapter focuses on the performance of lowenergy transfers to lunar libration
orbits and other threebody orbits in the Earth–Moon system. This chapter presents
surveys of direct transfers as well as lowenergy transfers to libration orbits, and
provides details about how to construct a desirable transfer, be it a shortduration
direct transfer or a longerduration lowenergy transfer. The work presented here
uses lunar halo orbits as destinations, but any unstable threebody orbit may certainly
be used in place of those example destinations.
For illustration, Figs. 31 and 32 show some example direct and lowenergy
transfers to lunar halo orbits, respectively. One can see that these transfers are
ballistic in nature: they require a standard translunar injection maneuver, a few
trajectory correction maneuvers, and a halo orbit insertion maneuver. One may also
add Earth phasing orbits and/or lunar ﬂybys to the trajectories, which change their
performance characteristics.
Many thousands of direct and lowenergy trajectories are surveyed in this chapter.
Table 31 provides a quick guide for several types of transfers that are presented here,
comparing their launch energy costs, the breadth of their launch period, that is, the
117
118
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 31 The proﬁle for a simple direct transfer from the Earth to a lunar libration orbit
about either the Earth–Moon L1 or L2 point.
Figure 32 The proﬁle for a simple, lowenergy transfer from the Earth to a lunar libration
orbit about either the Earth–Moon L1 or L2 point.
119
EXECUTIVE SUMMARY
Table 31 A summary of several parameters that are typical for different mission
scenarios to libration orbits about either the Earth–Moon L1 or L2 points. EPOs = Earth
phasing orbits, BLT = lowenergy ballistic lunar transfer.
Mission
Element
Launch C3
(km2 /s2 )
Launch Period
Direct
Transfer
Direct
w/EPOs
Simple
BLT
−2.2 to −1.5 < −1.5 −0.7 to −0.4
BLT w/Outbound BLT
Lunar Flyby
w/EPOs
−2.1 to −0.7
< −1.5
Short
Extended
Extended
Short
Extended
Transfer Duration
(days)
3–6
13+
70–120+
70–120+
80–130+
Outbound Lunar
Flyby
No
No
No
Yes
Yes
∼500
∼500
∼0
∼0
∼0
Libration Orbit
Insertion ΔV (m/s)
number of consecutive days they may be launched, their transfer duration, and the
relative magnitude of the orbit insertion change in velocity (ΔV) upon arriving at the
lunar libration orbit. These are representative and may be used for highlevel mission
design judgements, though the details will likely vary from mission to mission.
Direct transfers to lunar libration orbits are presented in Section 3.3. That section
surveys thousands of transfers to libration orbits about both the Earth–Moon L1 and
L2 points and presents methods to construct them. The trajectories minimize the
halo orbit insertion ΔV cost while keeping the total transfer duration low, between
5 days and 2 months. The trajectories include no maneuvers other than the translunar injection maneuver and the halo orbit insertion maneuver. Hence, there are
no highrisk maneuvers, such as powered lunar ﬂybys, though such maneuvers may
indeed reduce the total transfer ΔV cost [172].
The surveys show that one may depart the Earth from any parking orbit, certainly
including lowaltitude parking orbits with an inclination of 28.5 degrees (deg). The
transfers involve translunar injections with launch injection energy (C3 ) requirements
as low as −2.6 kilometers squared per second squared (km2 /s2 ) and as high as
−2.0 km2 /s2 for transfers to LL1 or as high as −1.0 km2 /s2 for transfers to LL2 .
The halo orbit insertion maneuver may be as low as 430 meters per second (m/s)
or as high as 950 m/s, depending on the mission’s requirements, though most are
in the range of 500–600 m/s. The quickest transfers arrive at their libration orbit
destinations within 5 or 6 days. Some missions can reduce the total transfer ΔV by
∼50 m/s by implementing a longer, 30day transfer. In some cases it is beneﬁcial to
extend the duration to 40 or 50 days. Finally, direct lunar transfers exist in families,
such that very similar transfers exist to neighboring libration orbits. That is, if a
120
TRANSFERS TO LUNAR LIBRATION ORBITS
mission’s requirements change slightly and a new libration orbit is required, one can
usually build a very similar transfer to that orbit as to the original orbit.
Lowenergy transfers to lunar libration orbits are presented in Section 3.4. Much
like the analyses of direct transfers, Section 3.4 surveys thousands of transfers to
libration orbits about both the Earth–Moon L1 and L2 points and presents methods
to construct them. The trajectories are always entirely ballistic, except for the translunar injection maneuver. None of the transfers studied requires an orbit insertion
maneuver; every trajectory asymptotically arrives at the target orbit and inserts au
tomatically. Trajectories are studied with a wide variety of geometry characteristics,
but all require less ΔV than direct transfers.
Much like the analyses of direct transfers, the surveys in Section 3.4 show that
one may depart the Earth from any given low Earth parking orbit, or any higher
orbit as needed. The transfers involve translunar injections with C3 requirements
as low as −0.75 km2 /s2 and as high as −0.35 km2 /s2 . This C3 requirement may be
reduced to about −2.1 km2 /s2 if a lunar ﬂyby is implemented at an altitude of about
2000 km. The quickest transfers identiﬁed require about 83 days between the translunar injection and the point when the trajectory has arrived within 100 km of the
lunar libration orbit. Many transfer options exist that require 90–140 days between
the injection point and the orbit arrival point. Since the transfers asymptotically
approach the target libration orbit, they are essentially at the target orbit as many
as 10 days prior to the “arrival” time. Finally, much like direct transfers to lunar
libration orbits, lowenergy transfers exist in families, such that very similar transfers
exist to neighboring libration orbits. Very similar transfers also exist to the same
orbit when the arrival time or arrival location is adjusted.
This chapter summarizes nearly ballistic transfers between the Earth and lunar
libration orbits. Techniques to use these transfers in practical spacecraft mission
design (for example, building launch periods, and budgeting station keeping ΔV) are
studied in Chapter 6.
3.2
INTRODUCTION
This chapter describes methods to construct both direct and lowenergy transfers
between the Earth and libration orbits near the Moon. The focus of this book is on
the analysis and construction of lowenergy transfers, but it is helpful to have a good
understanding of the costs and beneﬁts of direct transfers as well. In addition, this
chapter provides some transfers that one may take after arriving at a lunar libration
orbit; transfers are presented from those libration orbits to other libration orbits, to
low lunar orbits, and to the lunar surface.
Direct transfers include any sort of highenergy conventional trajectories using
chemical propulsion systems. Lowenergy transfers use the same propulsion systems
but travel well beyond the orbit of the Moon, taking advantage of the Sun’s gravity to
reduce the ΔV cost of the transfer. Direct transfers to lunar libration orbits (and other
INTRODUCTION
121
threebody orbits) typically require 3–6 days, though there are beneﬁts to increasing
the transfer duration as long as 1 or 2 months. Lowenergy transfers typically require
3–4 months of transfer time or more in some circumstances.
Figure 31 illustrates two example direct transfers between the Earth and libration
orbits about the Earth–Moon L1 and L2 points. Figure 32 illustrates two lowenergy
transfers to the same two libration orbits, viewed in the same reference frame. One
can see that the trajectories traverse beyond the orbit of the Moon and return after
2–3 months to arrive at the Moon in such a way that they insert into the target
orbits without requiring any insertion maneuver. The lack of a large orbit insertion
maneuver is the primary reason why these transfers save so much fuel (the direct
transfers require an orbit insertion maneuver near 500 m/s).
Figure 33 illustrates two different lowenergy transfers viewed in the Sun–Earth
rotating frame to show that spacecraft may ﬂy either toward the Sun or away from it
during their transfers.
This chapter describes techniques to build direct and lowenergy transfers to lunar
libration orbits and surveys the performance of both types of transfers. Section 3.3
describes the techniques and provides performance data for direct transfers to lunar
libration orbits. Section 3.4 does the same for lowenergy transfers to the same
orbits. Section 3.5 provides information about orbit transfers from the libration
orbits to other libration orbits, to low lunar orbits, and to the lunar surface. Finally,
Section 3.5 discusses transfers that a spacecraft could take to depart its lunar libration
orbit and travel to another threebody orbit, a low lunar orbit, the lunar surface, or
back to the Earth.
Figure 33 Two example lowenergy transfers between the Earth and an LL2 libration
orbit. The transfers are viewed from above in the Sun–Earth rotating coordinate frame [44]
c 2009 by American Astronautical Society Publications Ofﬁce, all rights reseved,
(Copyright ©
reprinted with permission of the AAS.).
122
TRANSFERS TO LUNAR LIBRATION ORBITS
3.3 DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION
ORBITS
As of 2012 no missions have ﬂown a direct transfer from the Earth to a lunar libration
orbit. Many researchers have considered the problem, dating back to 1970 when
Edelbaum studied the case of transferring a spacecraft from the Earth to the L1 point
itself via a direct transfer [173]. Certainly NASA has considered the problem as it
considers destinations for future missions [126]. The work presented here is based
upon the work of Parker and Born [174, 175], who performed a robust survey of
direct transfers to lunar halo orbits about both L1 and L2 . Several other authors have
also studied this problem, including Rausch [176], Gordon [177], and Alessi et al.
[178].
The trajectories generated here are constructed by intersecting a low Earth orbit
(LEO) parking orbit with a trajectory within the stable invariant manifold of the target
libration orbit. Hence, the trajectories include two maneuvers: a maneuver to depart
the Earth and a maneuver to inject onto the target orbit’s stable manifold. Once on
the stable manifold, the spacecraft asymptotically arrives at the target orbit.
3.3.1
Methodology
Direct transfers are constructed here by targeting states within the stable manifold of
a desirable halo orbit or other libration orbit. This strategy has been implemented
before for transfers to many types of Sun–Earth libration orbits, yielding trajectories
for missions such as Genesis [72], Wilkinson Microwave Anisotropy Probe (WMAP)
[70], and Solar and Heliospheric Observatory (SOHO) [66]. The technique has been
highly successful for missions in the Sun–Earth system because the stable manifolds
of many Sun–Earth halo orbits intersect the Earth. Unfortunately, as one can begin
to see in Fig. 34, the stable manifolds of libration orbits near the Earth–Moon L1
and L2 points do not intersect the Earth within as much as two months of time.
Consequently, at least two maneuvers must be performed to directly transfer onto the
lunar halo orbit’s stable manifold from an initial LEO parking orbit, rather than the
single maneuver required to inject onto the stable manifold of a Sun–Earth halo orbit.
In theory, a direct transfer to a lunar halo orbit could involve many burns, each
performed in some arbitrary direction. We have chosen to survey the simplest type
of direct lunar halo orbit transfers, namely, transfers with only two burns that are
each performed in a direction tangential to the spacecraft’s velocity vector. These
transfers are not guaranteed to have the lowest ΔV cost of any type of direct lunar
halo transfer, but they should provide a good estimate for the ΔV requirement of such
transfers. Even with this simpliﬁcation, this design problem yields a very rich design
space and is a useful foundation for future studies.
Figure 35 shows two perspectives of a scenario that illustrates the strategy used
here to transfer a spacecraft from a 185km LEO parking orbit to a lunar L1 halo
orbit. The scenario requires a large maneuver at the LEO injection point (ΔVLEO ;
also known as the translunar injection maneuver) and a second large maneuver at the
manifold injection point (ΔVMI ). The two ballistic mission segments are referred to
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
123
Figure 34 Plots of the stable manifolds of example L1 and L2 halo orbits, viewed from
above in the Earth–Moon synodic reference frame. A spacecraft that travels along any one of
these trajectories will asymptotically arrive onto the corresponding halo orbit [174] (Copyright
c 2008 by American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted
©
with permission of the AAS).
as the bridge segment and the manifold segment. Once the spacecraft arrives onto the
manifold segment, after performing the ΔVMI maneuver, it asymptotically transfers
onto the lunar halo orbit.
It is assumed that each transfer constructed here begins in a 185km circular
prograde Earth parking orbit. In this way, the performance of each transfer may be
directly compared. In reality, the same sorts of transfers that are constructed here
may begin from a LEO parking orbit at any altitude and with any eccentricity, or even
from the surface of the Earth, provided that the vehicle is at the correct position at
the correct time to perform the ΔVLEO maneuver successfully.
The following strategy has been followed to construct direct transfers to lunar halo
orbits:
Step 1. Construct the desired halo orbit.
Step 2. Construct the manifold segment:
1. Choose a τ value, that is, a point along the halo orbit as illustrated
in Fig. 210 (page 50); choose a direction, that is, either “interior” or
“exterior” as shown in Fig. 34; and choose a manifold propagation
duration, Δtm .
2. The manifold segment is constructed by propagating the speciﬁc tra
jectory in the halo orbit’s stable manifold that corresponds to the given
τ value. The trajectory departs the halo orbit either in the interior or exte
rior direction, as indicated. It is propagated in the Earth–Moon threebody
system backward in time for the given duration.
124
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 35 Two perspectives of an example scenario that may be used to transfer a spacecraft
from a 185km LEO parking orbit to a lunar L1 halo orbit. The transfer is shown in the Earth–
Moon rotating frame (top) and the corresponding inertial frame of reference (bottom). The
halo orbit is shown in the inertial frame only for reference.
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
125
Step 3. Deﬁne XMI to be the ﬁnal state of the manifold segment. This is the state
that a spacecraft would need to obtain in order to inject onto the manifold
segment.
Step 4. Construct ΔVMI and the bridge segment:
1. Deﬁne ΔVMI to be the tangential ΔV that may be applied to XMI in order
to construct the bridge segment.
2. When propagated further backward in time, the bridge segment will
encounter the prograde 185km LEO orbit at the bridge’s ﬁrst perigee
point. The bridge segment is propagated in the Earth–Moon threebody
system.
Step 5. Construct ΔVLEO , the tangential ΔV that may be applied to transfer the
spacecraft from its LEO orbit onto the bridge segment.
This procedure is used here to produce a direct, twoburn transfer to a lunar halo
orbit given an arbitrary lunar halo orbit and any given value for those parameters
speciﬁed in Step 2 above. A signiﬁcant beneﬁt of this procedure is that it requires no
knowledge of what a transfer should look like, except that the bridge segment is only
propagated backward in time to its ﬁrst perigee passage.
This process generates threedimensional transfers in the idealized Earth–Moon
circular restricted threebody problem (CRTBP). The inclination of the Earth de
parture is a free variable; it is computed and reported, but not targeted in any way.
Furthermore, since no date is speciﬁed, the inclination is presented relative to the
orbital plane of the Moon. The performance of actual transfers to real halo orbits
will vary based on the date and orientations of each body and its orbit in the Solar
System. Nevertheless, this exploration sheds light on what sorts of transfers exist
and their approximate performance.
Several scenarios have been explored to identify optimal transfers, given the
conﬁnes of this survey. The ﬁrst suspicion is that the optimal transfer may be
constructed by building a bridge segment that connects the LEO departure with the
manifold segment’s perigee point. Since energychange maneuvers are more efﬁcient
when a spacecraft is traveling faster [97], the perigee of the manifold segment seems
like a good location to perform the ΔVMI maneuver. The best transfer for a speciﬁc
halo orbit would then be the one that requires the least total ΔV over all τ values.
This perigeepoint scenario is presented ﬁrst. It turns out that this strategy does not
produce the most efﬁcient transfers—the next strategy generates better transfers—but
the perigeepoint scheme will still be presented because it illuminates the problem
very well.
3.3.2
The PerigeePoint Scenario
Figure 36 shows two perspectives of several example trajectories that may be used
to transfer a spacecraft onto a single lunar L1 halo orbit using the perigeepoint
scheme. Each transfer implements a different τ value about the same halo orbit. For
126
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 36 Example trajectories that implement the perigeepoint scheme to directly transfer
from LEO to a lunar L1 halo orbit. The transfers are shown in the Earth–Moon rotating
frame (top) and the corresponding inertial frame of reference (bottom). (See insert for color
representation of this ﬁgure.)
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
127
reference, the halo orbit is a northern L1 halo orbit with an x0 value of approximately
319,052 kilometers (km). The manifold segment in each case has been propagated
to its perigee point, and the corresponding bridge segment has been constructed to
transfer from a 185km prograde LEO orbit to that perigee point. The trajectories are
shown in both the Earth–Moon rotating frame and the corresponding inertial frame
of reference.
Several of the trajectories shown in the left plot of Fig. 36 appear to have nontangential ΔVMI maneuvers; this is only a visual effect caused by the rotating frame
of reference. As the spacecraft departs the Earth on the bridge segment, it quickly
crosses a point where the frame of reference rotates about the Earth faster than the
spacecraft. After that point, the spacecraft appears to travel in a retrograde fash
ion about the Earth, seemingly in conﬂict with its inertially prograde orbit. If the
spacecraft then performs a large enough ΔVMI maneuver, the spacecraft’s rotational
velocity will once again exceed the rotational velocity of the frame of reference. The
spacecraft will appear to have switched directions when it actually just increased its
inertial velocity.
Figure 37 shows plots of the magnitudes of the two required maneuvers, ΔVLEO
and ΔVMI , as well as the total maneuver cost as functions of the parameter τ . One
can see that the minimum ΔV cost to transfer from the 185km LEO orbit to this halo
orbit using the perigeepoint scheme is approximately 4.14 kilometers per second
(km/s). One can also see that this minimum occurs at the point where ΔVLEO is at its
maximum. Figure 38 shows plots of the minimum and maximumΔV transfers and
veriﬁes that the minimumΔV transfer involves the largest bridge segment observed
in Fig. 36. The total transfer duration from the point where the spacecraft performs
its ΔVLEO maneuver to the point where it is within 100 km of the given halo orbit
ranges between approximately 17.7 days (τ ≈ 0.30) and 22.9 days (τ ≈ 0.83).
3.3.3
The OpenPoint Scenario
Although it may be intuitive to perform ΔVMI at the manifold segment’s perigee
point because of the energy considerations, it is actually better to perform a larger
ΔVLEO and a smaller, although lessefﬁcient, ΔVMI . This is because the maneuver at
LEO can take advantage of its close proximity to the Earth to make the total energy
change required in the transfer as efﬁcient as possible. That is, it is most efﬁcient
to change as much of the spacecraft’s energy at LEO as possible, since that is the
location where the spacecraft will be traveling the fastest during the lunar transfer.
This result is evident by studying the results of the perigeepoint scheme.
An alternate scheme is presented here where the second maneuver, ΔVMI , may
be placed anywhere along the stable manifold of the halo orbit. The manifold
segment may be propagated well beyond its perigee point, although it has an imposed
maximum propagation time of 1 or 2 months: 1 month for exterior manifolds since
they depart the Moon’s vicinity quickly and 2 months for interior manifolds since they
linger near the Moon for longer amounts of time. The transfers have an additional
degree of freedom compared with the perigeepoint scheme, but they are otherwise
constructed in exactly the same manner as listed above. This new scheme will be
128
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 37 Plots of the maneuver requirements to transfer onto a lunar L1 halo orbit using
the perigeepoint scheme. Top: the magnitudes of the two maneuvers ΔVLEO and ΔVMI as
functions of τ ; bottom: the total ΔV cost as a function of τ .
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
129
Figure 38 The minimum and maximumΔV transfers produced using the perigeepoint
scheme. One can see that the minimumΔV transfer contains the largest bridge segment
c 2008 by American Astronautical Society Publications
observed in Fig. 36 [174] (Copyright ©
Ofﬁce, all rights reserved, reprinted with permission of the AAS).
130
TRANSFERS TO LUNAR LIBRATION ORBITS
referred to as the openpoint scheme, since the manifold insertion point has had its
position constraint opened.
To demonstrate the openpoint transfer strategy, Fig. 39 shows several transfers
that may be constructed from LEO to an arbitrary trajectory along the stable manifold
of a particular halo orbit. The halo orbit shown in Fig. 39 is the same northern L1
halo orbit presented in Section 3.3.2, and the manifold shown has a τ value of 0.3.
Figure 310 shows the maneuver cost associated with transferring to various points
along the manifold, where the location of ΔVMI is speciﬁed by the manifold propa
gation duration, Δtm . One can see that there are two local minima that correspond
to lowenergy lunar transfers: one at a Δtm of approximately 10.0 days and the next
at a Δtm of approximately 22.7 days, neither of which corresponds to a mission that
transfers to the manifold segment’s perigee point, which has a Δtm of approximately
16.86 days. In fact, these transfers correspond to missions where the bridge segment
connects the spacecraft to a point very near the apogee of the manifold segment.
Figure 311 shows plots of the extreme cases, namely, the four transfers indicated by
the labels (1)–(4) in Fig. 310. One can see that the two local minima observed in
Fig. 310, that is, the trajectories marked with a (2) and a (4), coincide very near to
the manifold segment’s apogee locations.
Figures 39 to 311 have demonstrated the openpoint scheme applied to a single
trajectory (where τ = 0.3) on the stable manifold of a single halo orbit (the lunar
L1 halo orbit with an x0 value of approximately 319,052 km). The openpoint
scheme is easily extended to cover many trajectories along the halo orbit’s stable
manifold. Figure 312 summarizes the required maneuvers and the total maneuver
cost associated with the least expensive lunar transfer for each trajectory on the stable
manifold of the same halo orbit. One can see that the lowestenergy openpoint
transfer constructed to this particular halo orbit requires a total ΔV of approximately
3.62 km/s. This lowenergy transfer implements the trajectory in the orbit’s stable
manifold with a τ value of approximately 0.48. For veriﬁcation, Fig. 312 shows
that the trajectory with a τ value of 0.3 requires a minimum ΔV of approximately
3.67 km/s: the same result as that shown in Fig. 310.
Note that in Fig. 312 the leastexpensive transfers to this halo orbit use the ﬁrst
maneuver, ΔVLEO , to perform the vast majority of the spacecraft’s energy change.
This is consistent with the notion that the most efﬁcient transfer performs as much
ΔV as possible deep within the Earth’s gravity well where the spacecraft is traveling
fastest.
3.3.4
Surveying Direct Lunar Halo Orbit Transfers
The previous section illustrates the openpoint scheme applied to a single halo orbit
about the Earth–Moon L1 point using the halo orbit’s exterior stable manifold. The
process results in a lowenergy, twomaneuver, direct lunar transfer to that halo orbit,
following the exterior stable manifold. This section surveys lowenergy direct lunar
transfers to a large number of orbits within the families of halo orbits about both
the Earth–Moon L1 and L2 points, taking advantage of both the exterior and interior
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
131
Figure 39 Example trajectories that implement the openpoint scheme to directly transfer
from LEO to a speciﬁc manifold of a particular lunar L1 halo orbit.
132
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 310 Plots of the maneuver requirements to transfer onto a speciﬁc manifold of a
speciﬁc lunar L1 halo orbit using the openpoint scheme. Top: the magnitudes of the two
maneuvers ΔVLEO and ΔVMI as functions of Δtm ; bottom: the total ΔV cost as a function
of Δtm .
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
133
Figure 311 The four transfers with locally extreme ΔV requirements as indicated by the
labels (1)–(4) in Fig. 310. The transfers are shown in the Earth–Moon rotating frame (top) and
c 2008 by American
the corresponding inertial frame of reference (bottom) [174] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission of the
AAS).
134
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 312 Plots of the magnitudes of the two required maneuvers ΔVLEO and ΔVMI
(top) and the total ΔV cost (bottom) associated with the leastexpensive lunar transfer for each
trajectory on the stable manifold of a single lunar L1 halo orbit.
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
135
stable manifolds. These results should be representative of other threebody orbits
as well, such as Lissajous and vertical Lyapunov orbits.
The following sections summarize the results of four surveys performed here:
Section
Section 3.3.4.1
Section 3.3.4.2
Section 3.3.4.3
Section 3.3.4.4
Halo Family
Interior/Exterior
Stable Manifold
Page Number
L1
L1
L2
L2
Exterior
Interior
Exterior
Interior
136
140
142
146
In each of these four cases, it would be ideal to perform an exhaustive search
for the very best transfer to each halo orbit implementing the given stable manifold.
However, it is very timeconsuming to construct a transfer to each point along each
trajectory in each halo orbit’s stable manifold. The corresponding phase space is
threedimensional, and every combination of parameters takes a signiﬁcant amount
of computation time. To reduce the computation load, while still performing a survey
of a large portion of the phase space, several numerical optimization routines have
been implemented.
It has been found that a combination of hillclimbing and genetic algorithms per
forms very well at identifying the leastexpensive transfers to a given halo orbit very
swiftly [46]. The numerical algorithms use the state X = [x0 , τ , Δtm ]T to deﬁne
a direct twomaneuver lunar transfer, given the procedure outlined in Section 3.3.1.
The numerical optimization process begins by implementing a genetic algorithm to
identify a local ΔVminimum in the phase space. The implementation of the genetic
algorithm will not be discussed here for brevity, but may be found in many sources in
literature [179]. After several iterations of the genetic algorithm, the state that corre
sponds to the leastexpensive lunar transfer is reﬁned using a dynamic hillclimbing
algorithm, also known as the steepestdescent algorithm [180]. In this way, the local
minima of the threedimensional phase space are quickly explored. In order to survey
speciﬁc orbits within a family of halo orbits, the parameter x0 is held constant and
the remaining two parameters are varied.
The majority of the locallyoptimal transfers found in this work were identiﬁed
by specifying a value for x0 and varying the values of τ and Δtm using ten iterations
of a genetic algorithm with a population of twenty states. The leastexpensive state
resulting from the genetic algorithm was then iterated in the dynamic hillclimbing
algorithm until a solution was found whose ΔV cost could not be improved by varying
τ by more than 1 × 10−5 or by varying Δtm by more than 4 seconds.
The numerical optimization routine is not guaranteed to converge on the most
efﬁcient transfer, but it easily converges on relatively efﬁcient transfers. The results
given in the following sections include the most efﬁcient transfers identiﬁed, as well
as somewhat less efﬁcient transfers. The results then trace out a Pareto front of
optimal solutions [181]. Other nonoptimal points have been added to the results to
give an impression of the range of costs of transfers that exist. Each result is discussed
in more detail in the following sections.
136
TRANSFERS TO LUNAR LIBRATION ORBITS
3.3.4.1 Survey of Exterior Transfers to L1 Halo Orbits This section presents
the results of openpoint transfers constructed between 185km LEO parking orbits
and the exterior stable manifold of halo orbits in the family of lunar L1 halo orbits.
Figure 313 shows the cost of many such example transfers to halo orbits in the fam
ily. One can see that there are several types of efﬁcient transfers. To help identify the
trends and differences between each type of transfer, Fig. 314 shows plots of several
example transfers. Finally, Tables 32 through 35 provide additional information
about sample transfers of several varieties observed in the ﬁgures. Table 32 sum
marizes the characteristics of the numbered transfers shown in Fig. 314; Table 33
provides details about the shortestduration transfers identiﬁed; Tables 34 and 35
summarize the transfers labeled “efﬁcient” and “complex” in Fig. 313, respectively.
Figures 313 and 314 show many interesting patterns. After studying the transfers
presented in these ﬁgures, as well as the corresponding data presented in Tables 32
through 35, the following observations have been made:
• The majority of the leastexpensive transfers of this type are very fast transfers,
requiring only ﬁve days to transfer to a close proximity of each corresponding
halo orbit. Table 33 provides details about examples of such fast transfers.
Their bridge segments take the spacecraft nearly directly to the halo orbit.
These transfers compose the majority of the Pareto front observed in the ﬁgures.
Figure 313 The total ΔV cost of many surveyed transfers to the exterior stable manifold of
orbits in the family of lunar L1 halo orbits.
137
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Figure 314 Several example transfers between 185km LEO parking orbits and the exterior
stable manifold of lunar L1 halo orbits. The parameters of the numbered transfers are
summarized in Table 32.
Table 32 Characteristics of the numbered transfers identiﬁed in Fig. 314 [174]
c 2008 by American Astronautical Society Publications Ofﬁce, all rights
(Copyright ©
reserved, reprinted with permission of the AAS).
#
x0
(km)
1
2
3
4
5
6
7
8
320265
348963
357643
357177
342539
334016
322568
317035
∗ The
ΔVLEO ΔVMI Total ΔV Inc∗ Transfer
Bridge Manifold
(m/s) (m/s)
(m/s) (deg) Δt (days) Δt (days) Δt (days)
3128.0
3134.1
3132.9
3129.0
3136.2
3135.9
3119.2
3111.3
539.0
934.1
923.1
579.0
453.5
493.6
503.1
531.7
3667.0
4068.2
4056.0
3708.0
3589.7
3629.5
3622.2
3643.0
26.3
8.2
16.1
25.7
48.1
46.9
9.8
5.0
13.1
8.8
7.5
30.6
22.1
28.8
31.3
23.5
inclination of the LEO parking orbit in the CRTBP.
4.5
4.7
4.6
4.6
4.9
4.9
4.0
3.4
8.5
4.1
2.9
26.0
17.2
23.9
27.3
20.1
τ
0.179
0.888
0.500
0.501
0.461
0.911
0.800
0.281
138
TRANSFERS TO LUNAR LIBRATION ORBITS
Table 33 Characteristics of example fast transfers identiﬁed in Figs. 313 and 314
c 2008 by American Astronautical Society Publications Ofﬁce, all
[174] (Copyright ©
rights reserved, reprinted with permission of the AAS).
x0
(km)
317406
318240
320569
324912
328382
332715
335440
339191
341814
345948
347333
350325
353906
357643
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3123.6
3125.3
3129.4
3133.4
3134.8
3135.7
3135.9
3136.1
3136.1
3135.8
3135.2
3133.4
3132.5
3132.9
572.1
558.3
540.9
522.2
511.3
497.9
488.6
471.9
457.9
876.0
915.6
940.5
940.7
923.1
3695.7
3683.6
3670.3
3655.6
3646.2
3633.5
3624.5
3608.0
3594.0
4011.8
4050.8
4073.9
4073.2
4056.0
17.0
21.1
30.0
38.3
42.2
45.9
47.7
49.3
49.5
26.0
13.4
8.6
12.7
16.1
7.6
9.0
5.4
6.6
6.8
5.7
5.7
6.5
8.2
8.3
7.3
5.9
6.7
7.5
4.4
4.4
4.6
4.8
4.8
4.9
4.9
4.8
4.8
4.9
4.8
4.7
4.6
4.6
3.1
4.5
0.7
1.8
2.0
0.7
0.8
1.7
3.5
3.4
2.5
1.3
2.1
2.9
τ
0.757
0.863
0.543
0.638
0.664
0.562
0.566
0.673
0.878
0.874
0.745
0.500
0.500
0.500
Table 34 Characteristics of example transfers from the family labeled “Efﬁcient
c 2008 by American Astronautical Society
Transfers” in Fig. 313 [174] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
x0
(km)
316507
317035
317353
317721
318219
318745
319497
320179
320899
321932
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3108.9
3111.3
3112.1
3112.9
3113.8
3114.7
3115.8
3116.7
3117.6
3118.7
561.7
531.7
524.8
519.0
513.2
508.7
504.6
502.3
501.2
501.1
3670.6
3643.0
3636.9
3631.9
3627.0
3623.4
3620.3
3619.0
3618.8
3619.8
1.4
5.0
5.7
6.3
6.9
7.5
8.1
8.6
9.1
9.6
22.3
23.5
24.0
24.6
25.4
26.2
27.4
28.3
29.2
30.7
3.4
3.4
3.5
3.5
3.6
3.6
3.7
3.7
3.8
3.8
18.9
20.1
20.5
21.1
21.8
22.6
23.7
24.6
25.5
26.8
τ
0.204
0.281
0.312
0.348
0.398
0.453
0.525
0.583
0.645
0.743
• The bridge segments that do connect the spacecraft nearly directly with the halo
orbit appear to do so in an organized manner. For halo orbits with x0 values
below a value of approximately 345,000 km, the bridge segments connect the
spacecraft with the far side of the halo orbit. Beyond x0 values of 345,000 km,
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
139
Table 35 Characteristics of example transfers from those labeled “Complex
c 2008 by American Astronautical Society
Transfers” in Fig. 313 [174] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
x0
(km)
348011
352619
354615
358106
358150
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3137.0
3135.9
3136.4
3134.2
3131.1
430.7
587.1
669.3
501.1
519.1
3567.6
3723.0
3805.7
3635.3
3650.2
31.9
32.7
19.7
24.8
15.7
28.4
31.7
29.9
31.2
31.8
4.2
4.5
4.3
4.9
4.6
24.2
27.3
25.6
26.3
27.2
τ
0.476
0.904
0.703
0.499
0.499
that is, for very large zamplitude halo orbits, the optimal direct transfers
tend to connect closer to the nearside of the halo orbit. This pattern may be
observed in the plots shown around the perimeter of Fig. 314.
• A family of very efﬁcient direct transfers of this kind appears for transfers
to halo orbits with x0 values between approximately 316,000 km and ap
proximately 323,000 km. The bridge segments of these transfers connect the
spacecraft with the ﬁrst apogee of the manifold segments after the manifold
segments traverse to the opposite side of the Earth–Moon system. This family
of transfers may be seen on the left side of the ﬁgures and corresponds to halo
orbits that have small zamplitudes. Table 34 summarizes additional details
about these transfers.
• A few transfers have been found that require less total ΔV than the vast
majority of locally optimal transfers. These transfers appear toward the lower
right portion of the plot shown in Fig. 313 and are labeled as complex transfers.
These transfers tend to involve several close ﬂybys of the Moon. This study
has not fully explored these transfers, since they are much more complicated
by nature, but Table 35 provides details about several example transfers of this
type.
• The transfers shown in Figs. 313 and 314 implement LEO parking orbits
with ecliptic inclinations anywhere between 0 deg and 50 deg. The equatorial
inclination, by comparison, depends on the speciﬁc launch date and varies from
the ecliptic inclination by as much as ±23.45 deg.
• The duration of time required to transfer within 100 km of the halo orbit may
be anywhere between 5–30 days. Transfers may certainly be constructed that
require more time; however, these transfers are not considered in this study
since they may be more inﬂuenced by the Sun’s gravity.
• The leastexpensive transfers to lunar L1 halo orbits following their exterior
stable manifolds generally require a total ΔV no smaller than approximately
140
TRANSFERS TO LUNAR LIBRATION ORBITS
3.60 km/s, depending on the halo orbit of choice. Halo orbits with x0 values
greater than approximately 345,000 km tend to require more total ΔV: in the
range of 4.05 km/s ≤ ΔV ≤ 4.08 km/s.
In many practical missions, the launch vehicle provides a set amount of ΔV,
given a payload mass, and mission designers must optimize their transfer trajectories
around that performance. Hence, many times it is useful to consider the two transfer
maneuvers separately as well as the total cost of the transfer. Figure 315 shows the
magnitudes of the two maneuvers separately, which combine to produce the total ΔV
cost of the transfers shown in Figs. 313 and 314. One can see that nearly all of
the transfers require the magnitude of the translunar injection maneuver (ΔVLEO )
to be between 3.120 and 3.136 km/s. This suggests that the same launch vehicle can
perform the translunar injection maneuver for nearly all of these transfers given the
same payload mass. Although it is difﬁcult to see in these plots, the leastexpensive
transfers require the mostexpensive ΔVLEO magnitudes. The second maneuver,
ΔVMI , contributes most of the variations seen in the total cost of the lunar transfer.
3.3.4.2 Survey of Interior Transfers to L1 Halo Orbits This section presents
the survey of transfers constructed between 185km LEO parking orbits and the inte
rior stable manifold of halo orbits in the family of lunar L1 halo orbits. Figure 316
shows the cost of many such example transfers, where several families of locally
optimal transfers have been plotted in a more prominent shade. Other nonoptimal
transfers have been scattered about the plot to demonstrate that an entire ﬁeld of
options are available. To help identify the trends and differences between each type
of transfer, Fig. 317 shows plots of several example transfers and Tables 36 through
39 summarize the characteristics of many of these transfer types.
The following observations may be made after studying the plots shown in
Figs. 316 and 317 and the data displayed in Tables 36 through 39:
• The same types of fast transfers exist to L1 halo orbits via their interior stable
manifolds as via their exterior stable manifolds, because the manifold segments
of those transfers do not extend far beyond the halo orbits. Hence, the cost and
performance of such fast transfers closely resemble the cost of the fast transfers
explored in Section 3.3.4.1. This is apparent when comparing the data shown
in Tables 33 and 37.
• Many families of longerduration transfers exist that often require less total
ΔV than the faster transfers. Examples of these transfers may be seen in
the lower left and lower right regions of Figs. 316 and 317, as well as in
Tables 38 and 39. In general, each of these transfers involves at least one close
lunar encounter, and many are constructed by intersecting the transfer’s bridge
segment with a point very near apogee of the transfer’s manifold segment.
• The transfers shown in Figs. 316 and 317 implement LEO parking orbits with
ecliptic inclinations anywhere between 0 deg and 60 deg. Again, the equatorial
inclinations of the LEO parking orbits depend on the launch date.
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
141
Figure 315 The two transfer maneuver magnitudes that combine to produce the total ΔV
cost of the transfers shown in Figs. 313 and 314. Top: The magnitudes of the translunar
injection maneuvers (ΔVLEO ) in each transfer; bottom: The magnitudes of the manifoldinsertion maneuvers (ΔVMI ) in each transfer.
• The leastexpensive transfers to lunar L1 halo orbits following their interior
stable manifolds generally require a total ΔV no smaller than approximately
3.60 km/s, depending on the halo orbit of choice. The trend is very similar to
that presented in Section 3.3.4.1 for shortduration lunar halo transfers.
142
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 316 The total ΔV cost of many surveyed transfers to the interior stable manifold
of orbits in the family of lunar L1 halo orbits. Dark points correspond to locally optimal
trajectories; faint points represent additional nonoptimal solutions.
To continue this analysis, Fig. 318 shows the magnitudes of the two determin
istic maneuvers separately. One can see that the total ΔV cost of each transfer is
divided between the two maneuvers in a very similar way as the exterior transfers
shown in Section 3.3.4.1. Many of the transfers require a translunar injection ma
neuver magnitude (ΔVLEO ) between 3.120 and 3.136 km/s. Some of the families of
moreefﬁcient transfers require smaller ΔVLEO magnitudes. Even with these slight
reductions, the second maneuver, ΔVMI , still contributes most of the variations seen
in the total cost of the lunar transfer.
3.3.4.3 Survey of Exterior Transfers to L2 Halo Orbits This section presents
the survey of transfers constructed between 185km LEO parking orbits and the ex
terior stable manifold of halo orbits in the family of lunar L2 halo orbits. Figure 319
shows the cost of many such example transfers to halo orbits in the family, including a
Pareto front of optimal transfers. To help identify the trends and differences between
each type of transfer, Fig. 320 shows plots of several example transfers, and Tables
310 through 313 summarize the characteristics of many sample transfers of this
type.
Figures 319 and 320 show many interesting patterns. After studying the transfers
presented in these ﬁgures, and the data summarized in Tables 310 through 313, the
following observations have been made:
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
143
Figure 317 Several example transfers between 185km LEO parking orbits and the interior
stable manifold of lunar L1 halo orbits. The parameters of the numbered transfers are
summarized in Table 36. Dark points correspond to locally optimal trajectories; faint points
represent additional nonoptimal solutions.
• Two dominant types of efﬁcient transfers exist that transfer to the halo orbits’
exterior stable manifold. The ﬁrst one, indicated by the upper prominent curve
in Fig. 319, includes transfers whose bridge segments connect the spacecraft
directly with the far side of the L2 halo orbit. These are shortduration trans
fers, characterized by data shown in Table 311, and they are similar to the
shortduration transfers explored in Sections 3.3.4.1 and 3.3.4.2. The second
dominant type of transfer, indicated by the lower prominent curve in Fig. 319,
includes trajectories whose bridge segments send the spacecraft well beyond
the Moon, where they intersect the corresponding manifold segments near the
segments’ apogee points. The ﬁrst type of transfer requires only 5–6 days to
accomplish, whereas the second type requires as many as 35–50 days before
the spacecraft is within 100 km of the lunar halo orbit.
• Additional beneﬁt may be obtained for transfers to L2 halos with x0 values
greater than approximately 425,000 km by ﬂying near the Moon en route to
the ΔVMI maneuver. The lunar ﬂyby reduces the total required ΔV, albeit at
the expense of more sensitive navigation requirements near that lunar ﬂyby.
144
TRANSFERS TO LUNAR LIBRATION ORBITS
Table 36 Characteristics of the numbered transfers identiﬁed in Fig. 317 [174]
c 2008 by American Astronautical Society Publications Ofﬁce, all rights
(Copyright ©
reserved, reprinted with permission of the AAS).
#
x0
(km)
1
2
3
4
5
6
7
8
9
10
11
12
326808
348529
353325
358234
358745
357400
353001
341601
326786
321441
325594
317083
ΔVLEO ΔVMI Total ΔV Inc∗ Transfer
Bridge Manifold
(m/s) (m/s)
(m/s) (deg) Δt (days) Δt (days) Δt (days)
3111.3
3118.2
3132.6
3129.9
3116.7
3127.1
3133.0
3136.2
3121.0
3117.8
3133.8
3111.4
902.6
837.7
941.9
920.8
796.4
729.5
498.9
462.4
477.5
469.6
520.1
593.1
4013.9
3955.9
4074.5
4050.8
3913.1
3856.6
3631.9
3598.6
3598.5
3587.4
3653.9
3704.5
12.2
12.1
12.0
18.3
17.5
10.6
27.0
51.9
9.3
7.9
39.5
6.7
48.8
58.4
8.7
24.1
39.4
57.3
37.6
25.7
48.5
39.7
6.2
31.5
3.5
3.9
4.6
4.5
3.9
4.3
4.7
4.9
3.9
3.8
4.8
3.5
45.4
54.6
4.1
19.6
35.5
53.0
32.9
20.8
44.6
35.9
1.4
28.0
τ
0.447
0.193
0.772
0.499
0.660
0.008
0.456
0.848
0.756
0.121
0.609
0.079
*The inclination of the LEO parking orbit in the CRTBP.
Table 37 Characteristics of example fast transfers identiﬁed in Figs. 316 and 317
c 2008 by American Astronautical Society Publications Ofﬁce, all
[174] (Copyright ©
rights reserved, reprinted with permission of the AAS).
x0
(km)
316536
318562
320977
324263
328038
331309
335684
339602
341979
345722
347918
349968
351974
354725
358661
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3121.0
3126.3
3130.0
3133.1
3134.8
3135.5
3135.9
3136.1
3136.1
3135.9
3134.8
3133.6
3132.9
3132.5
3133.3
607.9
556.5
538.2
523.2
513.0
500.8
485.5
469.3
456.2
859.6
924.3
939.0
942.7
939.9
910.4
3728.9
3682.9
3668.2
3656.4
3647.8
3636.2
3621.5
3605.4
3592.3
3995.5
4059.2
4072.6
4075.6
4072.5
4043.7
7.0
23.3
31.3
37.5
41.7
44.2
47.6
50.0
49.9
30.4
10.1
8.4
10.4
13.7
16.7
8.2
7.5
6.5
9.5
4.9
9.2
9.2
7.7
9.5
6.6
7.1
8.5
8.3
5.0
7.8
4.3
4.5
4.7
4.7
4.8
4.8
4.8
4.8
4.8
4.9
4.8
4.6
4.6
4.7
4.6
3.9
3.0
1.8
4.7
0.1
4.3
4.4
2.9
4.7
1.7
2.4
3.8
3.7
0.3
3.2
τ
0.834
0.738
0.635
0.892
0.487
0.905
0.946
0.805
0.025
0.671
0.707
0.828
0.764
0.256
0.500
145
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Table 38 Characteristics of example transfers from the region labeled “Efﬁcient
Transfers” in Fig. 316. The rows of the table are organized in groups, where each group
c 2008 by American
describes example transfers in a different family [174] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission
of the AAS).
x0
(km)
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s)
(m/s)
(deg) Δt (days) Δt (days) Δt (days)
τ
320688
321925
323219
324345
326087
327737
3117.0
3118.2
3119.4
3120.2
3120.9
3121.3
473.3
468.3
470.2
476.2
477.5
480.6
3590.3
3586.6
3589.6
3596.4
3598.4
3601.8
7.9
7.9
8.0
8.3
8.8
9.5
38.3
40.6
44.6
46.2
47.8
49.6
3.8
3.8
3.8
3.9
3.9
3.9
34.6
36.8
40.8
42.4
43.9
45.7
0.057
0.173
0.456
0.556
0.686
0.859
327189
328326
329353
330278
3127.3
3129.4
3131.5
3133.7
497.7
491.6
486.5
480.9
3625.0
3621.0
3618.0
3614.6
15.3
20.1
25.9
33.9
45.4
46.3
47.1
47.9
4.3
4.5
4.7
4.9
41.1
41.8
42.5
43.1
0.658
0.745
0.830
0.933
322265
325061
326012
328613
3126.8
3128.2
3129.2
3130.8
495.6
498.4
496.2
496.3
3622.4
3626.6
3625.4
3627.1
18.1
20.1
22.5
26.6
31.2
36.2
36.9
39.5
4.4
4.4
4.5
4.6
26.8
31.8
32.5
34.9
0.212
0.599
0.669
0.947
329737
329778
330195
330545
3136.3
3136.5
3136.2
3136.2
486.6
486.8
484.3
487.4
3622.8
3623.3
3620.6
3623.6
52.5
52.9
51.8
51.5
29.0
30.0
33.7
30.4
4.9
5.0
4.8
4.9
24.1
25.0
28.8
25.5
0.861
0.972
0.326
0.018
These transfers may be seen in the lower right portions of the plots shown
in Figs. 319 and 320; Tables 312 and 313 compare the characteristics of
transfers with and without the lunar ﬂyby.
• The transfers shown in Figs. 319 and 320 implement LEO parking orbits
with different ranges of ecliptic inclinations. The transfers indicated by the
upper prominent curve in Fig. 319 may be launched from LEO parking orbits
with ecliptic inclination values anywhere in the range of 0 deg–25 deg. Those
transfers indicated by the lower prominent curve have a narrower range of
0 deg–19 deg. Finally, the lowest ΔV transfers shown in the lower right portion
of the ﬁgures may implement LEO parking orbits with a much more broad
range of ecliptic inclinations: anywhere in the range of 20 deg–120 deg and
possibly beyond.
146
TRANSFERS TO LUNAR LIBRATION ORBITS
Table 39 Characteristics of example transfers in the region labeled “Complex
Transfers” in Fig. 316. The examples summarized here belong to many different
c 2008 by
families, demonstrating the variety of transfers that exist [174] (Copyright ©
American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with
permission of the AAS).
x0
(km)
351166
351444
352138
353639
355251
355550
355848
358221
358332
358677
358837
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3134.5
3121.5
3135.3
3120.0
3135.1
3131.2
3121.8
3135.3
3130.8
3122.8
3135.6
501.3
506.1
528.4
522.0
529.1
504.5
508.6
532.3
501.3
502.7
523.4
3635.8
3627.5
3663.7
3642.0
3664.2
3635.7
3630.4
3667.6
3632.1
3625.5
3659.0
34.8
11.6
40.2
12.7
35.3
19.5
12.1
31.6
15.9
13.0
28.1
37.4
49.9
35.2
50.0
35.3
37.5
49.2
35.1
37.9
50.2
36.6
4.8
3.9
4.7
3.8
4.7
4.6
4.0
4.7
4.5
4.0
4.7
32.6
46.0
30.5
46.3
30.5
32.9
45.3
30.4
33.4
46.2
31.8
τ
0.448
0.515
0.260
0.508
0.250
0.405
0.454
0.108
0.317
0.374
0.142
• The total ΔV cost of the leastexpensive transfers to lunar L2 halo orbits
following their exterior stable manifolds greatly depend on which halo orbit is
being targeted. Halo orbits with x0 values less than 385,000 km, that is, very
large zamplitude halo orbits, require no less than approximately 3.95 km/s to
reach in this way. The cost steadily decreases for halo orbits with x0 values
between 385,000 km and 415,000 km. Halo orbits with x0 values greater than
approximately 415,000 km, that is, very low zamplitude halo orbits, require
no less than approximately 3.77 km/s to reach in this way. Finally, those halo
orbits that may be reached using an additional lunar ﬂyby en route have a total
ΔV requirement that may be reduced to as low as approximately 3.69 km/s.
Once again, to continue this analysis, Fig. 321 shows the magnitudes of the two
transfer maneuvers separately. One can see that the total ΔV cost of each transfer
is divided between the two maneuvers in a similar way as the transfers shown in
Sections 3.3.4.1 and 3.3.4.2. However, in these exterior transfers to the L2 halo
orbits, the ﬁrst maneuver, ΔVLEO , must perform somewhat larger ΔVs than it did for
transfers to L1 halo orbits: between 3.145 and 3.185 km/s. The second maneuver,
ΔVMI , still contributes most of the variations seen in the total cost of the lunar transfer.
3.3.4.4 Survey of Interior Transfers to L2 Halo Orbits This section presents
the survey of transfers constructed between 185km LEO parking orbits and the inte
rior stable manifold of halo orbits in the family of lunar L2 halo orbits. Figure 322
shows the cost of many such example transfers to halo orbits in the family. Sev
eral families of locally optimal transfers have been highlighted in a more prominent
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
147
Figure 318 The two transfer maneuver magnitudes that combine to produce the total ΔV
cost of the transfers shown in Figs. 316 and 317. Dark points correspond to locally optimal
trajectories; faint points represent additional nonoptimal solutions. Top: The magnitudes of
the translunar injection maneuvers (ΔVLEO ) in each transfer; Bottom: The magnitudes of the
manifoldinsertion maneuvers (ΔVMI ) in each transfer.
shade to be distinguished from the scattered nonoptimal transfers. To help identify
the trends and differences between each type of transfer, Fig. 323 shows plots of
several example transfers and Tables 314 through 316 summarize the characteristics
of many sample transfers of this type.
148
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 319 The total ΔV cost of many surveyed transfers to the exterior stable manifold of
orbits in the family of lunar L2 halo orbits, including a Pareto front of optimal solutions. Dark
points correspond to locally optimal trajectories; faint points represent additional nonoptimal
solutions.
The following observations may be made after studying the plots shown in
Figs. 322 and 323 and the data presented in Tables 314 through 316:
• The most prominent upper curve in Fig. 322 is nearly identical to the most
prominent curve in Fig. 319 from Section 3.3.4.3. This is because the manifold
segments of the transfers along both of those curves do not depart far from the
corresponding halo orbits. Both of these curves correspond to the shortestduration transfers to lunar L2 halo orbits, although they are certainly not the
leastexpensive in most cases.
• Many transfers exist that may be modeled as a transfer from LEO to an orbit
about the Moon’s L1 point, followed by a transfer from L1 to L2 . It makes
sense, then, that many transfers to L2 require no more ΔV than transfers to L1 .
These transfers require more transfer time than the shortestduration transfers
previously described.
• The transfers shown in the lower left plots in Fig. 323 include manifold
segments that extend well beyond the lunar vicinity. The bridge segments
in those transfers connect with a point near one of the apogee points of the
corresponding manifold segments. Several such families exist; in fact, a
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
149
Figure 320 Several example transfers between 185km low Earth orbits and the exterior
stable manifold of lunar L2 halo orbits. Dark points correspond to locally optimal trajectories;
faint points represent additional nonoptimal solutions. Parameters of the transfers shown are
summarized in Table 310.
different family may be produced for transfers that connect with any given
apogee of the corresponding manifold segments. Figure 323 shows two
plots of transfers that connect with the manifold segment’s ﬁrst apogee point
opposite of the Moon, as well as one plot of a transfer that connects with the
manifold segment’s second apogee point. Families of transfers that intersect
with later apogee points have not been produced here because they require
longer transfer durations. The characteristics of example transfers from several
of these families are shown in Table 316.
• There exist many types of transfers that make at least one close lunar passage
en route to the L2 halo orbit. It is apparent when studying the ﬁgures that the
total required ΔV of a transfer is very dependent on the distance between the
Moon and the manifoldinsertion maneuver. That is, as the proximity of ΔVMI
with the Moon is reduced the total required ΔV in the transfer is reduced. This
makes sense because more of the energy change in the transfer is performed
150
TRANSFERS TO LUNAR LIBRATION ORBITS
Table 310 Characteristics of the numbered transfers identiﬁed in Fig. 320 [174]
c 2008 by American Astronautical Society Publications Ofﬁce, all rights
(Copyright ©
reserved, reprinted with permission of the AAS).
ΔVLEO ΔVMI Total ΔV Inc∗ Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
#
x0
(km)
1
2
3
4
5
6
7
379441
406016
430307
427287
430167
399548
391748
∗ The
3142.6
3150.5
3152.4
3185.5
3162.2
3173.9
3169.2
820.4
860.0
957.0
588.5
536.4
659.5
696.2
3963.0
4010.4
4109.4
3774.1
3698.6
3833.4
3865.3
22.9
13.4
1.7
3.2
85.4
9.1
9.8
25.6
9.4
13.3
44.2
51.5
39.2
37.4
5.3
5.9
6.0
16.9
24.2
11.9
10.1
20.3
3.5
7.3
27.3
27.3
27.3
27.3
τ
0.046
0.740
0.970
0.465
0.593
0.004
0.203
inclination of the LEO parking orbit in the CRTBP.
Table 311 Characteristics of example fast transfers observed in Figs. 319 and 320
c 2008 by American Astronautical Society Publications Ofﬁce, all
[174] (Copyright ©
rights reserved, reprinted with permission of the AAS).
x0
(km)
382074
388716
400469
407319
412311
418589
423782
430202
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3146.3
3148.2
3149.9
3150.6
3151.0
3151.4
3151.8
3152.1
801.5
811.5
842.4
864.3
881.8
905.9
927.8
957.8
3947.7
3959.7
3992.3
4014.8
4032.8
4057.3
4079.6
4109.9
19.0
17.1
14.8
13.4
12.1
10.0
7.7
2.0
6.2
7.5
7.3
6.2
6.2
6.2
6.3
6.3
5.6
5.7
5.9
6.0
6.0
6.0
6.0
6.0
0.6
1.8
1.5
0.2
0.3
0.3
0.3
0.2
τ
0.547
0.636
0.594
0.506
0.506
0.506
0.506
0.506
deeper in a gravity well, where the spacecraft is traveling faster. The transfer
shown in the lower right plot of Fig. 323 is a good example of this effect: its
ΔVMI is performed very close to the Moon; hence, its total ΔV cost is lower.
• Several of the nonoptimal transfers (plotted in a lighter shade in Fig. 322)
appear to require less total ΔV than other locally optimal transfers. It is likely
that those nonoptimal transfers are in a different class of transfer, that is, they
require a different combination of lunar ﬂybys en route to the L2 halo orbit,
such that the optimized transfers of that class require a longer transfer time.
Only transfers requiring fewer than 60 days are plotted in the ﬁgures; the
locally optimal transfers that require more than 60 days, and perhaps less total
ΔV, are not displayed.
151
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Table 312 Characteristics of example longduration transfers observed in Figs. 319
c 2008 by American
and 320 that do not include a lunar ﬂyby [174] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission
of the AAS).
x0
(km)
384950
392617
399992
407598
415027
422804
430370
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3152.7
3169.9
3173.8
3178.5
3181.8
3184.3
3186.4
795.4
689.9
657.8
607.7
593.0
588.5
589.4
3948.1
3859.8
3831.5
3786.1
3774.7
3772.7
3775.8
12.3
9.7
9.1
8.6
7.2
5.0
0.9
33.2
37.6
39.1
41.0
42.4
43.6
44.7
6.4
10.4
11.8
13.7
15.1
16.3
17.4
26.8
27.3
27.3
27.3
27.3
27.3
27.3
τ
0.650
0.172
0.988
0.725
0.600
0.508
0.440
Table 313 Characteristics of example longduration transfers seen in Figs. 319
and 320 that do include a lunar ﬂyby in their corresponding bridge segments.
x0
(km)
424719
426208
427590
428819
430167
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3183.8
3182.5
3179.4
3174.2
3162.2
592.0
578.6
562.6
549.2
536.4
3775.9
3761.1
3742.0
3723.4
3698.6
22.1
23.5
30.9
48.4
85.4
50.9
50.8
51.0
51.2
51.5
23.6
23.6
23.7
23.9
24.2
27.3
27.3
27.3
27.3
27.3
τ
0.684
0.655
0.631
0.612
0.593
• The transfers shown in Figs. 322 and 323 implement LEO parking orbits with
ecliptic inclinations generally in the range of 0–55 deg.
• The duration of time required to transfer within 100 km of the halo orbit may
be anywhere between 5 and 60 days.
• The leastexpensive transfers to lunar L2 halo orbits following their exterior
stable manifolds generally require a total ΔV no smaller than approximately
3.60 or 3.65 km/s, depending on the halo orbit of choice.
The ﬁnal analysis in this section is to study the performance of the two maneuvers
separately for each interior lunar L2 halo transfer. Figure 324 shows the magnitudes
of the two transfer maneuvers. One can see that the majority of each transfer’s ΔV
cost is performed in the ﬁrst maneuver, ΔVLEO , but the variations in the magnitude
of ΔVLEO between transfers is very small, ranging between approximately 3.11 and
3.15 km/s. The second maneuver, ΔVMI , although much smaller, has a great deal
more variability and therefore determines the total cost of the transfer.
152
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 321 The two transfer maneuver magnitudes that combine to produce the total ΔV
cost of the transfers shown in Figs. 319 and 320. Dark points correspond to locally optimal
trajectories; faint points represent additional nonoptimal solutions. Top: The magnitudes of
the translunar injection maneuvers (ΔVLEO ) in each transfer. Bottom: The magnitudes of
the manifoldinsertion maneuvers (ΔVMI ) in each transfer.
3.3.5
Discussion of Results
The previous four sections surveyed four different types of direct lunar halo transfers;
this section studies them together to draw several overall conclusions.
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
153
Figure 322 The total ΔV cost of many surveyed transfers to the interior stable manifold
of orbits in the family of lunar L2 halo orbits. Dark points correspond to locally optimal
trajectories; faint points represent additional nonoptimal solutions.
Each of the results presented above implemented direct lunar transfers found by
searching through only one half of the stable manifold of the targeted halo orbits.
In reality, it most likely doesn’t matter whether a particular trajectory implements
an interior or an exterior transfer—just that the spacecraft arrives at the halo orbit in
some way. Figure 325 shows a summary of the ΔV requirements for both interior and
exterior transfers to lunar L1 halo orbits, plotted in the same axes. Hence, Fig. 325
may be used to identify the leastexpensive transfers to any lunar L1 halo orbit no
matter which type of manifold is taken. Figure 326 shows the same ΔV summary
for transfers to lunar L2 halo orbits.
Theoretically, it is possible to transfer to any given lunar L2 halo orbit from a lunar
L1 halo orbit with the same Jacobi constant, and vice versa. The dynamical systems
methodology presented in this work has been used in previous studies to construct
lowenergy orbit transfers and orbit chains [162]. To explore this concept further,
Fig. 327 shows a plot of the Jacobi constant, C, of the lunar halo orbits surveyed in
this work as a function of the halo orbits’ x0 values. One can see that there is a lunar
L1 halo orbit with the same Jacobi constant as each and every lunar L2 halo orbit in
this study. The family of lunar L2 halo orbits includes orbits with Jacobi constants in
the approximate range 3.015 < C < 3.152; the family of lunar L1 halo orbits spans
that entire range and then extends a bit further in each direction. In theory, it is thus
154
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 323 Several example transfers between 185km LEO parking orbits and the interior
stable manifold of lunar L2 halo orbits. Dark points correspond to locally optimal trajectories;
faint points represent additional nonoptimal solutions. The parameters of the numbered
transfers are summarized in Table 314.
possible to transfer to any lunar L2 halo orbit from the corresponding lunar L1 halo
orbit for very little energy.
Figure 328 shows the same results shown in Figs. 325 and 326, but now plotted
as a function of the halo orbits’ Jacobi constant values (Cvalues) rather than their
x0 values. In this way, one can observe the minimum total ΔV required to reach
any halo orbit of a particular Jacobi constant. Then, once in that orbit, one can
theoretically transfer to a different desired orbit, provided the desired orbit has the
same Jacobi constant. The left part of Fig. 328 shows transfers that may be used to
reach only lunar L1 halo orbits, since there are no lunar L2 halo orbits with Jacobi
constant values below 3.015. Figure 328 also shows that if a lowenergy transfer
can be found between halo orbits about L1 and L2 of a given Jacobi constant, it is
almost always more efﬁcient to transfer directly to the lunar L1 halo orbit ﬁrst, and
then take the lowenergy transfer over to the lunar L2 halo orbit.
Halo orbits exist in two families: a northern and a southern family as illustrated in
Fig. 225. Every lunar L1 halo orbit explored in this work has been a member of the
155
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Table 314 Characteristics of the numbered transfers identiﬁed in Fig. 323 [174]
c 2008 by American Astronautical Society Publications Ofﬁce, all rights
(Copyright ©
reserved, reprinted with permission of the AAS).
#
x0
(km)
1
2
3
4
5
6
7
8
9
10
11
12
394370
394096
411239
429222
415075
425204
430641
420255
406534
403368
396769
393789
∗ The
ΔVLEO ΔVMI Total ΔV Inc∗ Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3132.7
3149.0
3150.8
3152.0
3129.0
3126.7
3133.4
3112.2
3112.8
3113.6
3122.9
3135.7
817.6
824.6
878.7
953.8
776.3
705.0
464.8
605.2
599.5
676.7
728.2
792.6
3950.3
3973.6
4029.4
4105.8
3905.3
3831.7
3598.2
3717.4
3712.3
3790.3
3851.1
3928.3
31.9
16.2
12.4
3.6
19.9
13.0
5.6
9.7
20.9
16.3
17.9
26.9
38.6
6.4
12.4
9.2
25.4
23.5
18.3
31.5
49.2
43.6
31.3
25.9
4.7
5.8
5.9
5.9
4.5
4.5
5.2
3.6
3.8
3.6
4.1
4.9
τ
34.0
0.6
6.5
3.3
20.9
19.0
13.1
27.9
45.4
40.0
27.2
21.0
0.951
0.537
0.948
0.708
0.288
0.177
0.034
0.868
0.112
0.284
0.356
0.485
inclination of the LEO parking orbit in the CRTBP.
Table 315 Characteristics of example fast transfers observed in Figs. 322 and 323
c 2008 by American Astronautical Society Publications Ofﬁce, all
[174] (Copyright ©
rights reserved, reprinted with permission of the AAS).
x0
(km)
383881
392708
401069
408865
416071
429548
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
3147.0
3148.8
3149.9
3150.7
3151.2
3152.0
802.9
821.0
844.3
869.6
896.0
954.7
3949.9
3969.8
3994.2
4020.3
4047.3
4106.7
18.4
16.5
14.9
13.1
11.0
3.2
5.7
5.9
6.0
6.0
6.1
6.1
5.7
5.8
5.9
5.9
6.0
6.1
0.1
0.1
0.1
0.1
0.1
0.0
τ
0.497
0.495
0.494
0.494
0.494
0.493
northern family; every lunar L2 halo orbit has been a member of the southern family.
To access the symmetric family of halo orbits, in either case, the transfer must be
reﬂected about the z = 0 plane. The only difference that would be noticeable in such
a symmetric transfer would be that the LEO parking orbit’s inclination relative to the
Moon’s orbital plane would have the opposite sign.
156
TRANSFERS TO LUNAR LIBRATION ORBITS
Table 316 Characteristics of example transfers within a collection of seven different
sample families observed in Figs. 322 and 323. The families are identiﬁed by the
number of the corresponding example plot shown around the perimeter of Fig. 323
c 2008 by American Astronautical Society Publications Ofﬁce, all
[174] (Copyright ©
rights reserved, reprinted with permission of the AAS).
x0
(km)
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
τ
Family (12)
390299 3138.3
395448 3134.5
400189 3130.9
405365 3128.0
410586 3126.4
415399 3125.9
801.0
787.5
769.1
744.0
715.0
685.8
3939.3
3922.0
3900.0
3872.0
3841.4
3811.8
26.9
27.0
25.6
22.9
19.9
17.7
27.6
25.1
23.7
22.3
21.3
20.5
5.0
4.8
4.6
4.4
4.4
4.4
22.5
20.4
19.1
17.9
16.9
16.1
0.607
0.429
0.302
0.200
0.126
0.070
Family (1)
394170 3132.7
395343 3131.6
395820 3130.3
396907 3129.3
818.8
814.3
812.9
808.0
3951.5
3945.9
3943.3
3937.4
31.4
35.1
34.7
31.4
38.7
38.2
37.5
35.7
4.6
4.6
4.6
4.5
34.1
33.6
33.0
31.2
0.961
0.873
0.821
0.737
Family (11)
396738 3122.9
402272 3119.7
407548 3117.3
413568 3116.3
419902 3121.3
425400 3126.1
430618 3131.1
721.1
717.4
712.2
695.0
651.3
600.1
510.9
3844.0
3837.1
3829.5
3811.4
3772.6
3726.1
3642.0
14.9
11.4
8.4
6.4
10.0
8.2
0.2
30.2
27.8
25.8
23.2
21.0
19.9
19.2
4.1
3.9
3.8
3.8
4.2
4.7
5.2
26.1
23.9
22.0
19.4
16.7
15.2
14.0
0.240
0.070
0.973
0.873
0.808
0.736
0.648
Family (5, 6)
401972 3121.6
406699 3130.4
411226 3130.0
415688 3128.7
420324 3127.5
425204 3126.7
429490 3126.4
782.7
803.3
793.3
772.4
742.3
705.0
668.3
3904.4
3933.7
3923.2
3901.0
3869.8
3831.7
3794.7
24.7
22.0
21.2
19.3
16.7
13.0
6.7
33.3
28.3
26.6
25.3
24.3
23.5
23.0
4.1
4.6
4.6
4.5
4.5
4.5
4.6
29.2
23.7
22.0
20.8
19.8
19.0
18.4
0.587
0.440
0.350
0.281
0.225
0.177
0.144
Family (8, 10)
399413 3114.9
403744 3112.9
408970 3112.5
413531 3112.7
418227 3112.5
687.4
679.1
654.2
626.8
607.8
3802.3
3792.0
3766.7
3739.5
3720.3
15.7
16.7
15.7
13.3
10.7
41.4
38.7
35.1
33.4
31.9
3.7
3.6
3.5
3.6
3.5
37.7
35.0
31.5
29.8
28.4
0.145
0.053
0.999
0.936
0.887
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
157
Table 316 Continued.
x0
(km)
ΔVLEO ΔVMI Total ΔV Inc Transfer
Bridge Manifold
(m/s) (m/s) (m/s) (deg) Δt (days) Δt (days) Δt (days)
τ
Family (8, 10) (cont’d)
422292 3112.0 608.8
425682 3111.4 640.5
427668 3110.3 690.6
3720.7
3751.9
3801.0
8.5
6.1
4.2
31.3
31.5
32.4
3.5
3.5
3.5
27.8
28.0
28.9
0.853
0.837
0.830
Family (9)
416120 3112.2
418419 3111.7
420564 3111.4
423526 3110.8
425847 3110.2
603.2
600.1
599.0
605.7
619.5
3715.5
3711.8
3710.4
3716.5
3729.7
10.4
10.1
9.2
7.6
6.0
43.7
42.9
42.4
42.3
43.0
3.5
3.5
3.5
3.5
3.5
40.2
39.4
38.9
38.8
39.5
0.950
0.898
0.873
0.849
0.836
Nearby (7)
417319 3111.8
420771 3110.7
422285 3110.5
425306 3110.8
426565 3110.6
617.5
586.4
569.7
536.4
529.4
3729.3
3697.1
3680.1
3647.2
3640.0
13.0
13.0
12.4
9.0
7.5
43.4
42.4
42.2
40.7
40.5
3.5
3.4
3.4
3.5
3.5
39.9
39.0
38.8
37.3
37.0
0.194
0.086
0.055
0.004
0.981
3.3.6
Reducing the ΔV Cost
One notices that the transfers that require the least ΔV presented in the previous
sections involve missions that perform the majority of the energychanging maneuvers
deep within either the Earth’s or the Moon’s gravity wells where the spacecraft is
moving the fastest. The most convincing example of this is the trajectory labeled (7)
in Fig. 323: the Earthdeparture maneuver is large enough to send the spacecraft out
to the radius of the Moon, and the manifoldinsertion maneuver is performed quite
close to the Moon.
The trajectories designed here do not purposefully place the manifoldinsertion
maneuver near the Moon, and in fact, may not converge well if the maneuver occurs
nearby. However, the total transfer ΔV may be reduced if the manifoldinsertion
maneuver were indeed performed near the Moon, and recent research supports this
[172].
Performing a maneuver near the Moon may have energy beneﬁts, but it does
increase the operational complexity of the mission. The manifoldinsertion maneuver
becomes very timecritical when performed close to the Moon, and any execution
errors tend to exponentially increase afterward. Other operational considerations are
discussed in Chapter 6.
158
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 324 The two transfer maneuver magnitudes that combine to produce the total ΔV
cost of the transfers shown in Figs. 322 and 323. Dark points correspond to locally optimal
trajectories; faint points represent additional nonoptimal solutions. Top: The magnitudes of
the translunar injection maneuvers (ΔVLEO ) in each transfer; bottom: The magnitudes of the
manifoldinsertion maneuvers (ΔVMI ) in each transfer.
3.3.7
Conclusions
This section has explored direct transfers to lunar halo orbits. It has been found
that shortduration transfers exist to both lunar L1 and L2 halo orbits, requiring
approximately 5 days of transfer time. Such shortduration transfers require between
DIRECT TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
159
Figure 325 The total ΔV cost of many transfers to lunar L1 halo orbits using either interior
or exterior transfers.
Figure 326 The total ΔV cost of many transfers to lunar L2 halo orbits using either interior
or exterior transfers.
160
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 327 The Jacobi constant, C, of the lunar halo orbits surveyed in this work as a
function of the halo orbits’ x0 values.
Figure 328 The total ΔV cost of direct lunar halo orbit transfers as a function of the halo
orbits’ Jacobi constant values.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
161
3.6 and 4.1 km/s, depending on the halo orbit, when launched from a 185km circular
parking orbit. It has also been found that transfers exist between LEO and every
halo orbit surveyed here that require as little as 3.59–3.65 km/s, although many of
these transfers require 3 weeks or more of transfer time. Figure 329 summarizes the
results, showing the least amount of total ΔV required to reach any halo orbit using the
fastest optimized transfers, that is, transfers with a duration of approximately 5 days,
as well as an envelope of longer lowΔV transfers that require at most 2 months of
transfer time. The curve representing the longer transfers is very approximate—it
was produced by tracing out points that were produced successfully and interpolating
between those points. Some of these transfers may be difﬁcult to construct; other
lowercost transfers may also exist. Figure 330 summarizes the same results as a
function of the halo orbits’ Cvalues rather than their x0 values.
3.4 LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR
LIBRATION ORBITS
Transfers between the Earth and lunar libration orbits may be constructed that re
quire less fuel than direct transfers by taking advantage of the gravity of the Sun.
The scenario involves propelling a spacecraft beyond the orbit of the Moon, about
1–2 million kilometers away from the Earth, and letting the Sun’s gravity raise the
spacecraft’s energy. When the spacecraft returns toward its perigee after 2–4 months,
it encounters the Moon. The spacecraft encounters the Moon at a much lower relative
velocity than that of a direct transfer. The trajectory is crafted such that the spacecraft
approaches the Moon on the stable manifold of the target lunar libration orbit.
This section illustrates lowenergy transfers that arrive at a variety of lunar li
bration orbits, such that they require no orbit insertion maneuver whatsoever. The
performance of many lowenergy transfers is surveyed. First, Section 3.4.1 demon
strates how to model a lowenergy transfer using dynamical systems theory. Then
Section 3.4.2 provides an energy analysis of an example transfer, which illuminates
how energy shifts and how one may use both twobody and threebody tools to design
and analyze a lowenergy transfer. Sections 3.4.3 and 3.4.4 describe the process of
constructing desirable lowenergy transfers in the patched threebody and DE421
ephemeris models, respectively. The dynamical systems methods used to construct
lowenergy transfers may be extended to construct entire families of transfers. Sec
tion 3.4.5 surveys many families of transfers that have different geometries and
performance characteristics. Section 3.4.6 discusses how these transfers vary from
one month to the next. Finally, Section 3.4.7 presents several additional example
analyses to design lowenergy transfers to different threebody orbits, including an
LL1 halo orbit and a distant prograde orbit.
162
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 329 A summary of the minimum amount of total ΔV required to reach any lunar
L1 halo orbit (top) and any lunar L2 halo orbit (bottom) surveyed here using the fastest
optimized transfers (approximately 5 days) as well as an envelope of longer lowΔV transfers
(1–2 months).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
163
Figure 330 A summary of the minimum amount of total ΔV required to reach halo orbits
with a given Jacobi constant.
3.4.1 Modeling a LowEnergy Transfer using Dynamical Systems
Theory
Many types of lowenergy transfers exist in any given month, and their characteristics
tend to repeat from one month to the next. The most complex lowenergy transfers
typically do not appear in many consecutive months due to the asymmetries in the
real Solar System; however, simple lowenergy transfers reappear in a predictable
fashion from one month to the next.
This section studies how to model lowenergy transfers using dynamical systems
theory and the Patched ThreeBody Model (introduced in Section 2.5.2). It turns
out that simple lowenergy transfers are represented well in this simpliﬁed model
of the Solar System, and that one may use the modeled trajectory as a guide to
construct a realistic transfer in a more accurate model of the solar system. Because
lowenergy transfers may be represented in the Patched ThreeBody Model, one may
take advantage of tools within dynamical systems theory to analyze these transfers.
The goal is to be able to build a useful lowenergy transfer quickly to meet a mission’s
needs; dynamical systems tools provides an avenue to do this.
A lowenergy ballistic transfer may be modeled as a series of heteroclinic transfers
between unstable threebody orbits in the Sun–Earth system and the Earth–Moon
system [39, 40, 45, 46]. Figure 331 illustrates these orbit transfers in the Patched
ThreeBody Model. One can see that a spacecraft departs the Earth on a trajectory
that shadows the stable invariant manifold of an unstable threebody orbit in the
164
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 331 Modeling a ballistic lunar transfer as a series of heteroclinic transfers between
unstable threebody orbits in the Patched ThreeBody Model [97] (ﬁrst published in Ref. [97];
reproduced with kind permission from Springer Science+Business Media B.V.). (See insert
for color representation of this ﬁgure.)
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
165
Sun–Earth/Moon threebody system. The spacecraft does not arrive on that orbit,
however, before it ballistically diverts and then shadows the unstable manifold of that
orbit. The trajectory is designed to arrive in the stable manifold of a target threebody
orbit in the Earth–Moon threebody system, for example, an LL2 halo orbit. This
process will be described in detail in this section.
A lowenergy, ballistic lunar transfer may be modeled as a series of transfers from
one threebody orbit to another. After the spacecraft launches from its LEO parking
orbit, the spacecraft transfers to the vicinity of a threebody orbit in the Sun–Earth
system, referred to in this section as the Earth staging orbit. The spacecraft’s LEO
departure trajectory follows the ﬂow of the Earth staging orbit’s stable manifold.
Once in the vicinity of the Earth staging orbit, the spacecraft falls away from the
staging orbit, following the ﬂow of that orbit’s unstable manifold. The trajectory is
chosen so that it encounters the stable manifold of a threebody orbit in the Earth–
Moon system, referred to in this section as the lunar staging orbit. The spacecraft
may use the lunar staging orbit as a ﬁnal destination or as a transitory orbit, as
discussed later in Section 3.5. To generalize the modeling process even further, a
ballistic lunar transfer may be modeled as a transfer from Earth to one or more Earth
staging orbits to one or more lunar staging orbits and then to some ﬁnal destination.
Earth Staging Orbits. Many types of threebody orbits may be used as Earth
staging orbits in the process of modeling or constructing a lowenergy transfer. A
proper staging orbit must meet the following requirements:
1. The orbit must be unstable;
2. If the orbit is the ﬁrst Earth staging orbit, then the orbit’s stable manifold must
intersect LEO or the launch asymptote; otherwise, the orbit’s stable manifold
must intersect the preceding staging orbit’s unstable manifold;
3. The orbit’s unstable manifold must intersect the following staging orbit’s stable
manifold, be it another Earth staging orbit or a lunar staging orbit.
A quasiperiodic Lissajous orbit has been selected to build the example transfer
shown in this section, because it meets each of these requirements. Unfortunately,
quasiperiodic orbits and their invariant manifolds are difﬁcult to visualize since they
never retrace their paths. This section illustrates the validity of a Lissajous orbit by
showing that halo orbits are viable candidates to be used as Earth staging orbits.
Figure 332 shows four perspectives of the family of northern halo orbits centered
about the Sun–Earth L2 point. Lissajous orbits span a very similar region of space,
but often do not extend as far in the zaxis.
Most libration orbits in the Sun–Earth system are unstable and hence meet Re
quirement 1 given above. This discussion will assume that a halo orbit from the
family shown in Fig. 332 will be used as the only Earth staging orbit en route to
a lunar staging orbit. Figure 333 shows two plots of an example halo orbit about
the Sun–Earth L2 point and the interior half of its stable manifold. One can see that
this stable manifold intersects the Earth. Thus, a spacecraft may make a single ma
neuver to transfer from a LEO parking orbit to a trajectory on this halo orbit’s stable
166
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 332
point.
Four perspectives of the family of northern halo orbits about the Sun–Earth L2
manifold; this satisﬁes Requirement 2 for this itinerary. Similarly, Fig. 334 shows
two plots of the same halo orbit’s unstable manifold, showing that trajectories exist
that intersect the Moon’s orbit about the Earth. Thus, a spacecraft on, or sufﬁciently
near, the halo orbit may use the orbit’s unstable manifold to guide it to intersect the
Moon (satisfying Requirement 3). The invariant manifolds of Lissajous orbits with
similar Jacobi constants also demonstrate the same properties, making them viable
candidates for lowenergy staging orbits.
Lunar Staging Orbits. Many different Earth–Moon threebody orbits may be used
as lunar staging orbits; the example lowenergy transfer modeled in this section uses
a halo orbit about the Earth–Moon L2 point as its lunar staging orbit because it meets
all of the requirements.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
167
Figure 333 Two perspectives of an example Northern halo orbit about the Sun–Earth L2
point, shown with the interior half of its stable manifold. One can see that the stable manifold
intersects the Earth.
Figure 334 Two perspectives of the same northern EL2 halo orbit shown in Fig. 333,
this time shown with the interior half of its unstable manifold. One can see that the unstable
manifold intersects the Moon’s orbit.
The requirements for a lunar staging orbit typically come from the requirements
of the mission itself. The following list summarizes the additional requirements
imposed on the lunar staging orbit:
1. The orbit must be unstable;
2. The orbit’s stable manifold must intersect the unstable manifold of the preced
ing staging orbit, be it the previous lunar or the previous Earth staging orbit;
168
TRANSFERS TO LUNAR LIBRATION ORBITS
3. If the orbit is the ﬁnal lunar staging orbit, then it must meet any require
ments derived from the mission; otherwise, the orbit’s unstable manifold must
intersect the following lunar staging orbit’s stable manifold.
There are many families of Earth–Moon threebody orbits that satisfy Requirement 1,
including the family of lunar L2 halo orbits. The family of halo orbits about the Earth–
Moon L2 point closely resembles the family of halo orbits about the Sun–Earth L2
point shown in Fig. 332 and won’t be shown here for brevity.
Figure 335 shows two perspectives of an example LL2 halo orbit along with its
exterior stable manifold, propagated in the Patched ThreeBody Model. If a spacecraft
were to target a trajectory on this manifold it would asymptotically approach and
eventually arrive onto the staging orbit. Thus, if a spacecraft were able to transfer
from the Earth staging orbit’s unstable manifold onto this LL2 halo orbit’s stable
manifold, then the spacecraft would have achieved a ballistic transfer to this lunar
orbit from LEO.
An Example Modeled Ballistic Lunar Transfer. An example ballistic lunar trans
fer has been modeled using dynamical systems theory and is presented here. It is a
fairly simple example of a transfer: it consists of a single Earth staging orbit and a
single lunar staging orbit. A Lissajous orbit about the Sun–Earth L2 point has been
selected to be the Earth staging orbit, although it is visualized here by a halo orbit
with the same Jacobi constant. A lunar L2 halo orbit has been selected to be the
only lunar staging orbit. The transfer has been produced in the Patched ThreeBody
Model (see Section 2.5.2).
Figure 336 shows the ﬁrst portion of the threedimensional transfer in two per
spectives. The spacecraft is launched from a 185km low Earth orbit, travels outward
toward the Sun–Earth L2 point along a trajectory that shadows the stable manifold
of an EL2 libration orbit, skims the periodic orbit, and then travels toward the Moon.
S
; the sta
Figure 336 shows the representative halo orbit and its stable manifold, WEL
2
ble manifold of the actual Lissajous staging orbit does an even better job of mapping
out the ﬂow of the spacecraft’s motion in space.
Figure 337 shows two perspectives of the same transfer trajectory, but this time
U
. One can see that
plotted with the Earth staging orbit’s unstable manifold, WEL
2
as the spacecraft departs the vicinity of the Earth staging orbit and approaches the
Moon, its trajectory shadows the unstable manifold of the Earth staging orbit.
Figure 338 shows the same two perspectives of the threedimensional lowenergy
S
transfer plotted alongside the lunar staging orbit’s stable manifold, WLL
. One can
2
see that the lowenergy transfer intersects the manifold in full phase space, indicating
that the spacecraft has injected into the LL2 halo orbit. Once in the ﬁnal Earth–Moon
halo orbit, the spacecraft has all of the options presented in Section 3.5 available to
it.
Figure 339 shows a topdown perspective of the entire threedimensional lowenergy transfer with all three manifolds displayed.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
169
Figure 335 Two perspectives of an example southern halo orbit about the Earth–Moon L2
point, shown with the exterior half of its stable manifold. One can see that the stable manifold
quickly departs the Moon’s vicinity and may then intersect the unstable manifold of the Earth
staging orbit.
3.4.2
Energy Analysis of a LowEnergy Transfer
Lowenergy lunar transfers harness the Sun’s gravity to reduce the ΔV requirements
of a lunar transfer. It is useful to observe how the twobody energy of the spacecraft
with respect to each of the massive bodies changes throughout the transfer. It is also
useful to observe how the Moon affects the spacecraft’s Sun–Earth Jacobi constant
and especially how the Sun affects the spacecraft’s Earth–Moon Jacobi constant.
These energy changes are explored in this section, applied to the example transfer
170
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 336 Two perspectives of the ﬁrst portion of the example lowenergy transfer,
modeled using the stable manifold of a halo orbit about the Sun–Earth L2 point. One can see
that the spacecraft’s outbound motion shadows the halo orbit’s stable manifold.
produced in the previous section. Other lowenergy transfers have been found to
behave in a very similar fashion.
To begin this analysis, Fig. 340 shows plots of the distance between the spacecraft
and both the Earth and Moon as the spacecraft traverses the lowenergy ballistic lunar
transfer. This is a useful illustration since both the spacecraft’s twobody energy and
its Jacobi constant vary as functions of distance to these bodies. By observing
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
171
Figure 337 Two perspectives of the second portion of the example lowenergy transfer,
modeled using the unstable manifold of a halo orbit about the Sun–Earth L2 point. One can
see that as the spacecraft departs the vicinity of the Earth staging orbit and approaches the
Moon, its trajectory shadows the unstable manifold of the Earth staging orbit.
Fig. 340, one can determine the time at which the spacecraft arrives at its lunar halo
orbit destination.
It is expected that the twobody energy of a spacecraft with respect to the Earth
increases over time due to the Sun’s gravity, since the spacecraft’s perigee radius
gradually rises throughout the transfer. Figure 341 shows the twobody speciﬁc
172
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 338 Two perspectives of the third portion of the example lowenergy transfer,
modeled using the stable manifold of a halo orbit about the Earth–Moon L2 point. Every
fourth trajectory has been darkened for visualization purposes. One can see that the transfer
intersects the manifold in full phase space, indicating that the spacecraft has injected into the
LL2 halo orbit.
energy of the spacecraft with respect to the Earth throughout the transfer. One can
see that the spacecraft’s energy does indeed rise while it is in the vicinity of the
Earth staging orbit. The energy then begins to vary wildly once it enters the lunar
halo orbit, which makes sense because the halo orbit only exists in the presence of
both the Earth and the Moon, balancing the gravity of both bodies. Figure 342
shows four other twobody orbital elements of the spacecraft with respect to the
Earth as the spacecraft traverses the ballistic transfer, including the spacecraft’s semimajor axis, perigee radius, eccentricity, and ecliptic inclination. One can see that
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
173
Figure 339 A topdown perspective of the example lowenergy transfer, shown with all
three manifolds that were used to model it (blue = the lowenergy transfer, green = stable
manifold of a threebody orbit, brown = unstable manifold of a threebody orbit). (See color
insert.)
the Sun’s gravity increases the spacecraft’s semimajor axis and perigee radius as the
spacecraft traverses the Earth staging orbit. The Sun’s gravity reduces the spacecraft’s
eccentricity and inclination with respect to the Earth. The spacecraft enters the lunar
halo orbit at approximately 110 days after launch, beyond which the Moon’s gravity
is the dominant source causing each of the spacecraft’s orbital elements to vary over
time.
It is interesting to notice that the spacecraft’s inclination changes dramatically
during the ﬁrst half of the transfer, while the perigee radius remains near zero; then
during the second half of the transfer the perigee radius rises dramatically while
the spacecraft’s inclination settles down. These effects may be correlated with the
location of the spacecraft relative to the four quadrants of the Sun–Earth state space.
In this particular transfer, the spacecraft spends several weeks near the boundary of
the ﬁrst and fourth quadrants before moving deﬁnitively into the fourth quadrant,
where the spacecraft’s perigee radius rises rapidly. Other lowenergy transfers have
varying geometries and their twobody orbital elements change in correspondingly
different fashions.
It is also expected that the spacecraft’s twobody energy with respect to the Moon
decreases as the spacecraft approaches and ballistically inserts into the lunar halo
orbit. Figure 343 shows the twobody speciﬁc energy of the spacecraft with respect
to the Moon throughout the lowenergy lunar transfer. One can clearly see that the
174
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 340 The magnitude of the radius vector of the spacecraft with respect to the Earth
and the Moon as the spacecraft traverses the example lowenergy lunar transfer.
Figure 341 The twobody speciﬁc energy of a spacecraft with respect to the Earth over
time as it traverses an example lowenergy lunar transfer.
spacecraft’s speciﬁc energy drops as it approaches the lunar halo orbit. Furthermore,
its energy drops below zero, satisfying some authors’ requirements to be temporarily
captured by the Moon [29, 46, 182].
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
175
Figure 342 Four twobody orbital elements of the spacecraft with respect to the Earth
as the spacecraft traverses the example lunar transfer: (a) the spacecraft’s semimajor axis,
(b) perigee radius, (c) eccentricity, and (d) ecliptic inclination.
Figures 344 and 345 show the evolution of the spacecraft’s Jacobi constant
with respect to the Sun–Earth and Earth–Moon threebody systems, respectively, as
the spacecraft traverses the example lunar transfer. The spacecraft’s trajectory has
been constructed in the Patched ThreeBody Model; hence, the spacecraft’s Jacobi
constant will be constant in one or the other threebody system at any given time,
depending on which threebody system is responsible for the given segment of the
spacecraft’s trajectory. The spacecraft’s motion has been modeled by the Sun–Earth
threebody system during the ﬁrst 105 days of the transfer. After the spacecraft has
crossed the Earth–Moon threebody sphere of inﬂuence (3BSOI), its motion is then
modeled by the Earth–Moon threebody system.
176
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 343 The twobody speciﬁc energy of a spacecraft with respect to the Moon over
time as it traverses an example lowenergy lunar transfer.
Figure 344 The evolution of the spacecraft’s Jacobi constant with respect to the Sun–Earth
threebody system as the spacecraft traverses the example lunar transfer.
Figure 345 presents a compelling case that it is possible to build lowenergy trans
fer to lunar halo orbits, or other unstable Earth–Moon threebody orbits, with a wide
variety of different Jacobi constants. If the spacecraft traversing the example transfer
had arrived at the Moon slightly earlier or slightly later, it could have transferred to a
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
177
Figure 345 The evolution of the spacecraft’s Jacobi constant with respect to the Earth–Moon
threebody system as the spacecraft traverses the example lunar transfer.
lunar halo orbit with a different Jacobi constant. Furthermore, it may be possible for a
spacecraft to depart one lunar halo orbit, traverse through the Sun–Earth environment
for some time, and return to the Moon on the stable manifold of a different lunar
halo orbit. Section 3.4.5 demonstrates that it is indeed possible to build lowenergy
transfers to lunar halo orbits within a wide range of Jacobi constants [46], but more
work needs to be accomplished to determine how to take advantage of the time series
shown in Fig. 345 to target a lunar halo orbit with a speciﬁed Jacobi constant.
3.4.3 Constructing a LowEnergy Transfer in the Patched ThreeBody
Model
Modeling a lowenergy transfer using dynamical systems theory involves the use of
several staging orbits and their corresponding invariant manifolds in the Earth–Moon
and Sun–Earth systems. If a mission designer wishes to construct a transfer that
intentionally visits certain staging orbits, then the transfer may be constructed in the
same manner that it is modeled. More often, a mission designer only wishes for the
spacecraft to reach the ﬁnal lunar orbit, no matter its route through the Sun–Earth
system. In that case, the methods used to construct a lowenergy transfer may be
simpliﬁed.
Ballistic lunar transfers are constructed here by propagating the stable manifold
of the ﬁnal lunar halo orbit backward in time for a set amount of time. After each
trajectory has been propagated, the perigee point of the trajectory is identiﬁed. A
proper transfer may be identiﬁed as one whose perigee point corresponds to some
178
TRANSFERS TO LUNAR LIBRATION ORBITS
desired value, for example, an altitude of 185 km. In this manner, a practical
transfer may be constructed between the Earth and the lunar threebody orbit without
identifying any required staging orbit.
3.4.3.1 Parameters The dynamical systems method of constructing ballistic
lunar transfers provides a natural set of six parameters that may be used to deﬁne
each transfer. In the Patched ThreeBody Model, this set may be described by the
parameters: [F, C, θ, τ , p, Δtm ]. Each of these parameters is described in this
section.
Orbit Family Parameter: F . Depending on the mission requirements, one may
wish to target any type of Earth–Moon threebody orbit. The parameter F is a
discrete variable that describes the orbit family that contains the desired target orbit.
The example transfer presented previously has had the parameter F set to describe the
family of southern LL2 halo orbits. There are certainly symbolic ways to represent
each family of threebody orbits, but using text to do so provides a clear description
of which family is being used.
Orbit Parameter: C. The Jacobi constant, C, of the targeted orbit is used in this
work to specify which orbit is being targeted within the family. There are numerous
ways to identify a particular threebody orbit within its family [108, 113]. The Jacobi
constant is used here because it also provides information about the corresponding
forbidden regions and allowable motion of spacecraft with that Jacobi constant [46].
Sun–Earth–Moon Angle: θ. The parameter θ is deﬁned to be the angle between
the Sun–Earth line and the Earth–Moon line. It is a required parameter needed
to convert between the two threebody systems in the Patched ThreeBody Model.
Figure 346 shows an example of the geometry and the deﬁnition of θ.
Arrival Location: τ . Each point on a periodic orbit may be uniquely described by
the parameter τ , a parameter analogous to a conic orbit’s true anomaly. This parameter
was introduced in Section 2.6.2.3, but is described again here. The parameter τ may
range from 0 to 1, representing a revolution number, or from 0 deg to 360 deg,
Figure 346 An illustration of θ, the Sun–Earth–Moon angle.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
179
representing an angle like the true anomaly [46]. Figure 347 shows a plot of the
deﬁnition of τ when applied to two halo orbits. For halo orbits, it is intuitive to use
an angle and model τ off of a conic orbit’s true anomaly; for other threebody orbits
it is confusing using an angle. In any case, the only use of τ here is to identify each
point about a threebody orbit, and either representation may be used.
Perturbation Direction: p. To construct a trajectory in the stable invariant manifold
of a given unstable orbit, one takes the state of the orbit at a given τ value and perturbs
that state along the direction of the stable eigenvector [46, 147]. The perturbation
may occur in two directions: an interior or an exterior direction, as illustrated in
Fig. 34. The parameter p is a discrete variable that may be set to interior or exterior,
indicating the direction of the perturbation.
Manifold Propagation Duration: Δtm . The trajectory in the given threebody
orbit’s stable manifold is propagated backward in time for an amount of time equal
to Δtm . Typically when propagated backward in time, the trajectories that lead to
desirable lowenergy transfers depart the vicinity of the Moon, traverse their apogee,
fall toward the Earth, and then intersect a desirable altitude above the surface of the
Earth. However, transfers may also be constructed that pass near the Earth once or
several times before intersecting the desirable altitude above the surface of the Earth.
Such trajectories must be propagated long enough to allow the desirable perigee
passage to occur. Thus, the parameter Δtm is important in order to ensure that the
proper perigee passage is being implemented by the lowenergy transfer.
Figure 347 The two halo orbits shown demonstrate how the parameter τ moves from 0 to 1
c 2008 by American Astronautical Society Publications
about an orbit [174] (Copyright ©
Ofﬁce, all rights reserved, reprinted with permission of the AAS).
180
TRANSFERS TO LUNAR LIBRATION ORBITS
Discussion Regarding Parameters. The set of parameters used here does not
contain all continuous variables as other sets of orbital elements do, such as the
Keplerian orbital element set of a twobody orbit. The present parameter set also
requires knowledge about how to use it, for example, how to build the target lunar
orbit given the parameters F and C. Nonetheless, this set may be used to uniquely
describe any lowenergy ballistic transfer between the Earth and an unstable lunar
threebody orbit. Table 317 summarizes the parameter set.
3.4.3.2 Producing the lowenergy transfer The process of producing a lowenergy transfer given the parameter set [F, C, θ, τ , p, Δtm ] is very simple and is
described henceforth.
Step 1. First, one must build the target Earth–Moon orbit. The desired orbit must
be unstable and may be identiﬁed using the parameters F and C, as deﬁned above.
The example lowenergy transfer presented in this section has been produced using
an orbit in the family, F , of southern halo orbits about the Earth–Moon L2 point. The
speciﬁc orbit has been identiﬁed in its family by the value of C, equal to 3.05.
Step 2.
The parameter θ speciﬁes the location of the Moon, and hence the target
orbit, with respect to the Earth and Sun in the Patched ThreeBody Model. The
example transfer has used an initial θvalue of approximately 293.75 deg. This may
be veriﬁed by inspecting the ﬁnal location of the Moon in Figs. 336–339. Since
the transfer is generated backward in time, the value of θ speciﬁes the ﬁnal position
of the Moon.
Step 3.
The parameter τ speciﬁes a particular state in the unstable threebody
orbit. The example transfer has implemented a τ value of approximately 0.74,
corresponding to a point roughly three quarters around the orbit from the orbit’s
reference point (the point where the orbit crosses the y = 0 plane with positive y˙)
[46, 108].
Step 4.
The particular state in the target orbit is then perturbed in order to
construct a single trajectory in the stable manifold of the orbit. The magnitude of this
Table 317 A summary of the six parameters used to produce lowenergy transfers in
the Patched ThreeBody Model.
Parameter
F
C
θ
τ
p
Δtm
Domain
Description
Discrete
Continuous
Continuous [ 0 deg, 360 deg ]
Continuous [ 0,1 ]
Discrete
Continuous
Target threebody orbit family
Jacobi constant of target orbit
Sun–Earth–Moon angle
Arrival location on the target orbit
Perturbation direction
Propagation duration
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
181
perturbation is given by E; the direction is given by the orbit’s monodromy matrix
[131] and the parameter p. The orbit’s monodromy matrix is used to compute the
orbit’s stable and unstable eigenvectors; the stable eigenvector is then mapped to the
given τ value using the orbit’s state transition matrix [46, 147]. The example lunar
transfer has implemented a trajectory in the halo orbit’s exterior manifold with the
value of E set proportional to a 100km perturbation.
Step 5.
The resulting state is then used as the initial condition to construct a
trajectory in the stable manifold of the threebody orbit. This trajectory is propagated
backward in time for a duration of time equal to Δtm . The trajectory that has
produced the example transfer has been propagated for approximately 28.53 nondimensional Earth–Moon time units (approximately 123.9 days) before encountering
the desired perigee point, that is, the desired LEO injection point.
Step 6. The ﬁnal step in the construction of a lowenergy transfer is to connect this
trajectory with a prescribed LEO parking orbit or with the surface of the Earth. It is
unlikely that an arbitrary set of parameters will yield a lunar transfer that connects
with its prescribed LEO starting conditions. In such a case, either the parameters
should be adjusted [46], or a bridge must be constructed to connect the spacecraft’s
origin with the lunar transfer, as discussed in Section 3.3 [174].
3.4.3.3 Discussion The parameter set derived here is very useful if a mission
designer needs to build a transfer to a speciﬁc lunar orbit that cannot exceed some
maximum transfer time. In that case, the parameters F , C, and Δtm are ﬁxed. By
setting Δtm to the maximum transfer duration, one ensures that no transfers are
constructed that require excessive transfer time, but one still permits transfers that
require less transfer time. The three remaining parameters are conveniently well
deﬁned. The parameter p is binary and the parameters θ and τ are cyclic. Thus,
mission designers can explore all possible lowenergy transfers to a target orbit by
producing two maps: one map of θ vs. τ with p set to “Exterior,” and another identical
map with p set to “Interior.” Examples of these two maps that survey all possible
lowenergy transfers to an example halo orbit about the LL2 point, along with several
representative transfers, are illustrated in Figs. 348 and 349. The exploration of
these maps will be the purpose of Section 3.4.5, and further description of these
ﬁgures will appear there.
Other methods have been described in the literature that also describe parameter
sets to target lowenergy lunar transfers. The majority of these methods start with a
spacecraft in orbit about the Earth and target a maneuver for that spacecraft to perform
in order to reach the Moon’s vicinity via a lowenergy transfer. For instance, Belbruno
and Carrico have developed a set of parameters that describe the sixdimensional state
that a spacecraft would need to obtain to reach the Moon’s vicinity via a lowenergy
transfer [27]. Five parameters are speciﬁed, including an epoch (t), the spacecraft’s
radial distance from Earth (rE ), its longitude (αE ), its latitude (δE ), and its ﬂight
path azimuth (σE ). Then, the spacecraft’s speed (VE ) and ﬂight path angle (γE ) are
varied to target a prescribed radial distance from the Earth (rM ) and a prescribed
inclination (iM ), which would ultimately send the spacecraft in the general direction
182
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 348 An example state space map, capturing a wide variety of lowenergy transfers
that exist between the Earth and an example LL2 halo orbit. Each trajectory arrives at the halo
orbit from the exterior direction, and arrives at the orbit in a geometry according to the given
(θ, τ ) combination. The color of the map indicates how close to the Earth the trajectory gets
when propagated from the LL2 halo orbit backward in time. All black points represent viable
lowenergy transfers. (See insert for color representation of this ﬁgure.)
of a lowenergy transfer. The advantage of this method is that the spacecraft’s initial
orbit at the Earth is welldeﬁned, which is useful when a transfer must be designed
for a spacecraft that is already in orbit about the Earth. However, the technique
requires a great deal of predetermined knowledge of the problem, including a priori
estimates for the values of rM , iM , VE , γE , and t (t is speciﬁed to obtain a proper
Sun–Earth–Moon angle). The procedure is therefore constrained to build a transfer
with a predeﬁned geometry that may not be ideal.
Operationally, it is likely that a combination of these two approaches will work the
best to produce practical lowenergy transfers. A transfer may then be constructed
that starts from a prescribed orbit, ends at a speciﬁed lunar orbit, and probably
includes one or two small trajectory correction maneuvers to connect the segments.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
183
Figure 349 An example state space map that is identical to the map illustrated in Fig. 348,
except that the trajectories arrive at the LL2 halo orbit from the interior direction. (See insert
for color representation of this ﬁgure.)
Chapter 6 presents such an algorithm, and the results demonstrate that it generates
very successful trajectories.
3.4.4 Constructing a LowEnergy Transfer in the Ephemeris Model of
the Solar System
The previous sections demonstrated how to analyze and construct a lowenergy
lunar transfer to a libration orbit using the Patched ThreeBody Model; this section
describes how to do so in the more accurate DE421 ephemeris model of the Solar
System.
There are two main strategies that have been shown to work to generate a lowenergy transfer in a realistic model of the Solar System, such as a model that uses
the JPL Ephemerides to approximate the motion of the planets and the Moon in the
184
TRANSFERS TO LUNAR LIBRATION ORBITS
Solar System. The ﬁrst strategy is to generate the transfer in a simpliﬁed model,
such as the Patched ThreeBody Model, and then convert the transfer into the more
realistic model of the Solar System. The conversion process typically involves some
combination of multiple shooting differential correction and continuation [40, 46].
The second strategy is to construct the lowenergy transfer directly in the realistic
model, using experience gained from the simpliﬁed models. This strategy is described
in this section.
The dynamical systems methods that enabled the clear analysis and construction
of lowenergy ballistic lunar transfers in the Patched ThreeBody Model apply to the
DE421 model of the Solar System as well. The Sun, Earth, and Moon orbit their
respective barycenters in orbits that are nearly circular and coplanar. Thus, many
trajectories that exist in the Patched ThreeBody Model are good approximations of
trajectories that exist in the real Solar System.
Lowenergy ballistic lunar transfers are constructed in the DE421 model of the
Solar System in the same way that they have been constructed in the Patched ThreeBody Model. An unstable threebody orbit is selected as a target orbit near the
Moon. The orbit’s stable manifold is propagated and intersected with the Earth.
Those trajectories that intersect the Earth may be used as ballistic transfers from
the Earth to the target orbit via the orbit’s stable manifold. The most signiﬁcant
adjustment to this procedure involves the construction of the target threebody orbit
in the DE421 model. This process is described in detail in Section 2.6.6.3.
Ballistic lunar transfers to realistic halo orbits may be uniquely speciﬁed in the
DE421 model using a set of six parameters that is similar to the set used to describe
transfers constructed in the Patched ThreeBody Model. This set includes the param
eters: { F, Az , Tref , p, τ , Δtm }, where Az replaces the Jacobi constant and Tref
replaces the parameter θ from the previous set of parameters. It is very straightfor
ward to generate a halo orbit in the DE421 model using an analytical approximation
as an initial guess to the multiple shooting differential corrector (Section 2.6.5.2).
The parameter Az speciﬁes the zaxis amplitude of the halo orbit in the analytical
approximation speciﬁed by Richardson [123]. The parameter Tref speciﬁes the ref
erence epoch that ties the initial guess of the states of the halo orbit to the DE421
model.
Table 318 summarizes the set of parameters that generates an example transfer
in the DE421 model, shown in Fig. 350. The parameters F , Az , and Tref deﬁne the
southern LL2 halo orbit that is shown in Fig. 351. One can see that the multiple
shooting differential corrector adjusted the state of the analytical approximation of
the halo orbit such that the reference epoch is no longer at the τ = 0 deg point, but
at the τ ≈ 3.84 deg point. A particular trajectory in the halo orbit’s stable manifold
is then generated that corresponds to the parameters τ and p in Table 318, which
propagates backward in time to a perigee with an altitude of 185 km. The distance
between this trajectory and the Moon is shown in Fig. 352. One can see that this
trajectory asymptotically arrives at the orbit from the exterior direction.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Table 318
Fig. 350.
185
The parameters used to produce the lowenergy transfer shown in
Parameter
F
Az
Tref
τ
p
Δtm
Value
The family of southern Earth–Moon L2 halo orbits
30,752 km (0.08 normalized distance units)
15 January 2017 12:57:36 Ephemeris Time
280.2 deg
Exterior
115.9 days
Figure 350 An example lowenergy transfer produced in the DE421 model using the
c 2009 by American Astronautical Society
parameters speciﬁed in Table 318 [44] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
186
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 351 The halo orbit speciﬁed by F , Az , and Tref in Table 318 [44] (Copyright
c 2009 by American Astronautical Society Publications Ofﬁce, San Diego, California (Web
©
Site: http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
Figure 352 The distance between the transfer and the Moon as the trajectory approaches
c 2009 by American Astronautical Society
and arrives at the LL2 halo orbit [44] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
3.4.5
187
Families of LowEnergy Transfers
A set of lowenergy parameters may be used to generate the initial conditions of a
trajectory that is propagated backward in time to construct a ballistic lunar transfer.
'
If one set of parameters {F ' , A'z , Tref
, p' , τ ' , Δt'm } generates a trajectory that
originates from a LEO with an altitude of 185 km, then it is typically the case that
'
a small deviation in either Tref
or τ ' will generate a trajectory that originates from
a LEO with a slightly different altitude. However, small deviations in both of those
parameters may often be designed to generate a new trajectory that originates from a
LEO with the same 185 km altitude. In that case, the two sets of parameters deﬁne
two different ballistic lunar transfers that are in the same family of transfers.
Figure 353 illustrates how transfers may be organized into families. In this
example, the lunar transfer shown in Fig. 350 with the parameters given in Table 318
is used as a reference trajectory. The transfer’s parameters are all held constant,
except for the parameters Tref and τ , which are systematically varied through all
combinations of values shown in Fig. 353. At each combination, a new trajectory
is propagated and analyzed to determine its new perigee altitude. One can see that
by reducing both Tref and τ , one builds trajectories that come closer to the Earth at
Figure 353 A map of the perigee altitude that each lowenergy trajectory encounters as a
function of Tref and τ . The 185km contour is highlighted, which includes the nominal ballistic
c 2009 by American Astronautical
lunar transfer presented in Table 318 [44] (Copyright ©
Society Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all
rights reserved; reprinted with permission of the AAS). (See insert for color representation of
this ﬁgure.)
188
TRANSFERS TO LUNAR LIBRATION ORBITS
their perigee point, and vice versa. By reducing Tref and increasing τ an appropriate
amount, one can produce new trajectories that also have a perigee altitude of 185 km.
The exercise given above may be extended to allow Tref to vary across an entire
month and τ to vary across 360 deg to observe full families of lowenergy lunar
transfers. Figure 354 shows such a Ballistic Lunar Transfer (BLT) state space map
given the parameter set summarized in Table 319. The ﬁgure shows a plot that maps
the perigee altitude of each trajectory generated using each combination of Tref and
τ . The darkest regions contain the parameters that produce useful transfers; the white
ﬁelds contain parameters that generate trajectories that do not approach the Earth.
Figure 355 shows the same map with several trajectories plotted to illustrate the
trajectories that may be generated using these parameters.
Families of transfers may be identiﬁed in the BLT state space map shown in
Fig. 354 by tracing those combinations of Tref and τ that have a perigee altitude
of some desirable value, for example, 185 km. Figure 356 shows samples of the
combinations of Tref and τ that generate ballistic transfers with injection altitudes of
185 km. The points displayed in black correspond to trajectories that traverse closer
to EL2 than EL1 and vice versa. Table 320 presents a summary of the characteristics
of a sample of the transfers identiﬁed in Fig. 356. Each of these transfers is a member
of a family of similar trajectories for which the characteristics vary smoothly away
from those presented in the table. There are certainly many families of ballistic
Figure 354 A BLT state space map that shows the perigee altitude of each generated
trajectory as a function of Tref and τ . The darkest regions include the combinations of
c 2009
Tref and τ that yield transfers that begin from low Earth orbits [44] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
Table 319
189
The parameters used to produce the results shown in Figs. 354–356.
Parameter
F
Az
Tref
τ
p
Δtm
Value
The family of southern Earth–Moon L2 halo orbits
30,752 km (0.08 normalized distance units)
1 Jan 2017 00:00:00 ET ≤ Tref ≤ 31 Jan 2017 00:00:00 ET
0 deg ≤ τ ≤ 360 deg
Exterior
180 days
Figure 355 The same BLT state space map shown in Fig. 354 with example transfers
c 2009 by American Astronautical Society
shown around the perimeter [44] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
transfers unrepresented in the table. Figure 357 illustrates six example families of
lowenergy transfers. One can see that the general characteristics of each family
varies in a smooth fashion from one transfer to the next in the family.
The quickest transfer identiﬁed in Fig. 356 requires fewer than 83 days between
the injection and the point when the trajectory has arrived within 100 km of the lunar
halo orbit. The vast majority of the transfers shown require a launch energy in the
range of −0.75 km2 /s2 ≤ C3 ≤ −0.35 km2 /s2 . The transfers that include a lunar
ﬂyby often require less launch energy, particularly those that involve a lunar ﬂyby on
the outbound trajectory soon after injection. Several transfers have been identiﬁed
that require a C3 as low as −2.1 km2 /s2 , implementing a lunar ﬂyby at an altitude
of approximately 2000 km. Figure 358 shows the relationship between the required
injection C3 and the transfer duration; Fig. 359 compares the required injection C3
190
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 356 Combinations of Tref and τ that generate ballistic transfers with perigee
injections at an altitude of 185 km. The points displayed in black correspond to trajectories
that traverse closer to EL2 than EL1 ; points shown in gray travel closer to EL1 than EL2 .
with the lowest lunar periapse altitude. One can see a clear correlation in Fig. 359
that the closer a trajectory gets to the Moon during the transfer, the lower the injection
C3 may be. Additional lunar ﬂybys or Earth phasing orbits may help provide the
geometry needed for a particular mission.
3.4.6
Monthly Variations in LowEnergy Transfers
The BLT state space map shown in Figs. 354–356 will repeat perfectly from one
synodic month to the next in the Patched ThreeBody Model, since the model is
symmetric. The characteristics of the BLT state space map generated in the DE421
model of the solar system will not repeat perfectly each month, although similar
features will be present in each month. Figure 360 shows a map of the perigee altitude
of trajectories generated from the same set of parameters presented in Table 318.
But for a wider range of Tref and τ , Tref is varied over 3 months, and τ is varied over
two halo orbit revolutions. One can see the same features from cycle to cycle, but
the details of the state space map vary. Signiﬁcant variations are observed between
the ﬁrst halo orbit revolution (0 deg ≤ τ ≤ 360 deg) and the second halo orbit
(360 deg ≤ τ ≤ 720 deg), mostly as a consequence of the nonzero eccentricity of the
Moon’s orbit about the Earth–Moon barycenter.
3.4.6.1 12Month Survey The state space map has been further extended to
12 months to study the variations that exist throughout an entire year. It has been
observed that the most prominent features continue to persist, and repeat regularly,
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
191
Table 320 Summary characteristics for a sample of the ballistic transfers identiﬁed in
c 2009 by American Astronautical Society Publications
Fig. 356 [44] (Copyright ©
Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights reserved;
reprinted with permission of the AAS).
Δ Reference
# Epoch∗ (days)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
∗
12.060302
12.211055
14.170854
15.226131
15.829146
20.351759
20.351759
22.311558
23.819095
20.050251
25.025126
27.286432
28.190955
28.040201
28.040201
0.000000
0.150754
0.452261
1.507538
8.592965
8.592965
6.030151
27.889447
28.040201
28.190955
28.341709
2.110553
2.412060
2.110553
2.261307
6.934673
6.783920
11.457286
14.170854
14.170854
14.170854
16.733668
16.733668
17.035176
22.160804
23.819095
28.190955
28.492462
3.165829
τ
(deg)
334.519
333.736
283.655
271.069
279.419
238.347
239.232
221.171
206.901
180.970
164.113
137.373
168.405
185.608
185.630
55.325
63.382
54.781
66.990
59.539
59.962
144.580
53.118
15.470
34.787
43.756
245.420
247.372
251.704
255.586
122.568
138.709
38.141
65.695
70.107
73.417
222.850
223.945
192.365
108.406
87.587
313.713
285.732
227.614
EL1 /
C3
Transfer # Earth # Lunar Injection Inclination (deg)
EL2 (km2 /s2 ) Δt (days) Flybys Flybys Equatorial
Ecliptic
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
−0.2902
−0.3457
−0.3444
−0.4944
−0.4296
−0.6556
−0.5856
−0.6904
−0.7153
−1.8533
−1.9222
−2.0307
−2.0880
−1.0318
−1.6144
−0.9032
−0.6429
−0.6608
−1.1266
−0.8393
−0.6791
−0.6940
−0.9637
−0.4261
−0.5891
−0.5740
−0.5465
−0.6290
−0.6311
−0.5150
−0.7340
−0.5098
−1.1299
−0.5599
−0.6869
−0.6246
−0.7658
−0.6178
−1.5154
−2.0107
−0.6915
−0.4043
−0.4568
−1.9572
133.76
132.91
118.36
108.76
171.62
130.11
145.20
137.51
129.17
171.79
146.35
176.72
122.46
145.08
145.75
179.35
97.90
132.55
113.39
178.32
165.37
170.11
140.22
172.37
105.30
96.55
91.66
172.42
178.46
154.75
165.38
164.58
167.55
143.25
123.22
115.20
179.64
171.17
156.53
129.17
167.13
177.60
109.17
169.47
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
2
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
7
0
0
0
1
0
0
0
1
0
1
1
2
2
1
2
1
0
0
1
1
0
3
2
1
0
0
0
0
2
0
0
2
2
0
0
0
1
0
1
1
0
0
1
2
23.225
131.701
51.319
32.329
85.326
115.737
21.877
35.973
22.180
97.684
20.490
38.302
19.325
34.251
103.995
143.590
23.372
145.538
166.454
99.214
14.732
23.140
11.452
27.743
148.336
20.962
20.003
54.249
59.547
65.164
20.624
124.809
39.917
19.771
106.493
87.048
137.534
11.994
28.596
18.754
50.748
140.309
10.097
153.358
28.192
151.274
69.045
51.431
103.860
137.694
22.738
13.527
10.275
92.972
4.271
36.809
30.359
11.315
126.244
121.792
0.836
168.969
144.152
87.676
20.434
17.669
28.632
40.712
171.495
3.797
4.747
30.825
36.213
44.035
28.138
129.384
26.275
14.374
129.791
110.261
126.323
14.627
51.902
5.377
32.372
130.765
14.214
172.197
The reference epoch is given as a duration of time, in days, away from 1 Jan 2017 00:00:00 Ephemeris Time.
192
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 357 Example trajectories within six families of lowenergy transfers that each may
be used to transfer a spacecraft from a 185km altitude state above the Earth to the same LL2
halo orbit, though at different arrival times.
while subtle features appear and disappear from month to month. Figure 361 shows
a plot of samples of the combinations of Tref and τ that yield lowenergy transfers
between 185km LEO parking orbits and the lunar halo orbit.
The reference epoch of each transfer shown in Fig. 361 may be wrapped into
one synodic month to observe the changes that occur in the state space map from
one synodic month to the next. Figure 362 shows the resulting plot, revealing the
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
193
Figure 358 The relationship between the injection C3 value and the duration of the transfer
for each transfer identiﬁed in Fig. 356. The points displayed in black correspond to trajectories
that traverse closer to EL2 than EL1 .
Figure 359 The relationship between the injection C3 value and the lowest lunar periapse
altitude during each lunar transfer identiﬁed in Fig. 356. The points displayed in black
correspond to trajectories that travel closer to EL2 than EL1 .
194
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 360 The same state space map shown in Fig. 354 extended to cover 90 days of
reference epochs and two revolutions of the halo orbit.
Figure 361 Sample combinations of Tref and τ that yield lowenergy transfers between
185km LEO parking orbits and the lunar halo orbit for reference dates that span the year
2017. From lightest to darkest, the shading corresponds to reference dates from 1/1/2017 to
1/1/2018 [47] (ﬁrst published by the American Astronautical Society).
variations in the locations of the curves as they shift throughout the 12 months. The
transfers are shaded in Fig. 362 in the same manner as they are in Fig. 361, that is,
the transfers that exist in the ﬁrst month, which starts at a reference epoch of January
1, 2017, are shown in the lightest shade and the transfers in each consecutive synodic
month thereafter are plotted in a darker shade. One can see that certain features
repeat very closely from one synodic month to the next. Other features only appear
in a subset of synodic months.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
195
Figure 362 The combinations of Tref and τ that yield transfers during 12 synodic months,
relative to the beginning of each synodic month. The ﬁrst month, which starts at a reference
epoch of 1 Jan 2017 00:00:00 Ephemeris Time, is shown in the lightest shade and each
consecutive synodic month thereafter is plotted in a darker shade. From lightest to darkest, the
shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47] (ﬁrst published by the
American Astronautical Society).
Quite a few patterns exist in the families of transfers that are observed. First of all,
the most pronounced curves observed in Figs. 361 and 362 correspond to transfers
that do not include any lunar ﬂybys or Earth phasing orbits. Most of them require
between 90 and 110 days to transfer between the Earth and 100 km from their target
orbit. Examples of these sorts of transfers may be seen in Fig. 355.
Several relationships exist between the launch energy of a lowenergy lunar trans
fer and how close it gets to the Moon on its Earthdeparture leg. If the transfer
does not encounter the Moon, it typically requires a launch energy in the range of
−0.75 km2 /s2 ≤ C3 ≤ −0.35 km2 /s2 . If a spacecraft traversing a lowenergy transfer
does encounter the Moon as it departs the Earth’s vicinity, one ﬁnds that the Moon
may either boost or reduce the spacecraft’s energy, depending on how the spacecraft
passes by the Moon. If it boosts the spacecraft’s energy, then the lunar transfer’s
required launch energy drops to as low as −2.1 km2 /s2 . Figure 363 shows a plot of
the relationship between the launch energy of each lowenergy transfer observed in
Fig. 362 and how close the transfer passes by the Moon.
One can also glean a great deal of understanding about the characteristics of
these transfers by observing the relationship between each transfer’s injection energy
and the transfer’s duration. Figure 364 shows this relationship for each transfer
in the 12month survey. One can see that the trends in this relationship are nearly
196
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 363 The relationship between injection C3 and the lowest perilune altitude for each
transfer in the 12month survey. The trajectories near the top of the plot do not include any
lunar ﬂyby; trajectories toward the bottom do, where those toward the bottomleft receive an
energy boost from the Moon and those toward the bottom right have energy removed by the
Moon. From lightest to darkest, the shading corresponds to reference dates from 1/1/2017
c 2009 by American Astronautical Society Publications Ofﬁce,
to 1/1/2018 [44] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS).
independent of the month of the transfer. Typical mission designs prefer the transfer
duration to be as short as possible. One can see that there are two types of transfers
that require fewer than 100 days to perform: those that require an injection C3 on
the order of −2.1 to −1.5 km2 /s2 and those that require an injection C3 on the order
of −0.7 to −0.5 km2 /s2 . Clearly, those that require less injection C3 pass near the
Moon on the way out of the Earth’s vicinity.
The inertial orientation of each lowenergy transfer observed in this 12month
survey clearly depends on which month the transfer departs the Earth. However, the
orientation of each similar lowenergy transfer is fairly constant throughout the year
when observed in the Sun–Earth rotating frame. One way to observe that is to track
each transfer’s departure from Earth in the Sun–Earth rotating frame. Figure 365
shows a plot that compares the departure state of each transfer in the 12month survey
by plotting the relationship of each transfer’s right ascension of apogee vector (RAV)
and declination of apogee vector (DAV) parameters of the transfer’s initial apogee
vector. The RAV and DAV values have been computed at the instant of the translunar
injection, before any perturbations change the orbit. Each transfer departs the Earth
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
197
Figure 364 The relationship between injection C3 and duration for each transfer in the
12month survey. From lightest to darkest, the shading corresponds to reference dates from
1/1/2017 to 1/1/2018 [47] (ﬁrst published by the American Astronautical Society).
Figure 365 The relationship between the right ascension and declination of the apogee
vector, RAV and DAV, respectively, for each transfer in the 12month survey. From lightest
to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47] (ﬁrst
published by the American Astronautical Society).
198
TRANSFERS TO LUNAR LIBRATION ORBITS
on an orbit that is highly eccentric, but still captured by the Earth. From Fig. 365,
one can see that this initial orbit is usually oriented near the ecliptic plane and usually
oriented either toward or away from the Sun. A RAV value of 0 deg corresponds
to an orbit that has its apogee vector pointing away from the Sun, in the direction
of positive x in the Sun–Earth rotating coordinate frame. The outlying points in the
ﬁgure correspond to transfers that include some combination of Earth phasing loops
and lunar ﬂybys and typically do not reappear in the same region of this ﬁgure from
one month to the next.
The largest variations observed from one synodic month to the next correspond to
differences in the lowenergy transfer’s injection inclination, in both equatorial and
ecliptic reference frames, as illustrated in Fig. 366. It is apparent when studying
the plots shown in Fig. 366 that transfers depart the Earth from orbital planes at
nearly any inclination during each synodic month. It is expected that the equatorial
inclination of the transfers’ injection points will vary from one synodic month to the
next due to the Earth’s obliquity angle; however, signiﬁcant variations also exist from
month to month when observing the transfers’ injection points’ ecliptic inclination
values. The variations in the geometry during the year have a more pronounced effect
when the trajectories ﬂy near the Earth or Moon.
3.4.6.2 Tracking One Family Through 12 Months The ﬁgures shown in
the previous sections, as well as analyses in the literature [46] show that one can trace
hundreds of different families of lowenergy lunar transfers in any given reference
month. The characteristics of these families often stack on top of each other in
each relationship presented in Figs. 362–365, making it difﬁcult to discern how
the characteristics of one family evolve from month to month. This section studies
a subset of transfers of the 12month survey, ﬁltered to isolate a particular set of
practical lowenergy transfers. It is often the case that a practical spacecraft mission
beneﬁts by shorter transfer durations; it is also usually beneﬁcial to avoid outbound
lunar ﬂybys because they add geometrical constraints to the system that make it more
difﬁcult to establish a wide launch period. Hence, the ﬁlters that have been applied
to the transfer selection include:
• Maximum duration: 105 days
• Minimum perilune altitude: 20,000 km
In addition, the set of all transfers that meets these criteria has been divided into two
subsets, split such that one subset includes those transfers that travel closer to EL1
than EL2 and vice versa. In this way, one can compare practical EL1 transfers and
practical EL2 transfers from one month to the next.
Figure 367 identiﬁes the transfers that meet the ﬁlter criteria in the state space
map. A visual comparison will conﬁrm that these transfers exist in the most prominent
features of the state space maps shown in Figs. 354, 355, 361, and 362. One can
see that the location of the curves of each family on these plots varies from month to
month; the variations appear to be contained within approximately 50 deg in τ and
at most 5 days in the orbit’s reference date, Tref .
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
199
Figure 366 The equatorial (top) and ecliptic (bottom) inclination of the transfers’ injection
point for each lowenergy lunar transfer identiﬁed in Fig. 362. The ﬁrst month, which starts
at a reference epoch of 1 Jan 2017 00:00:00 Ephemeris Time, is shown in the lightest shade
and each consecutive synodic month thereafter is plotted in a darker shade. From lightest to
darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [44] (Copyright
c 2009 by American Astronautical Society Publications Ofﬁce, San Diego, California (Web
©
Site: http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
Figure 368 shows the relationship of each transfer’s injection C3 and its duration
for every transfer that satisﬁes the ﬁlter criteria. One can clearly see that the transfers’
performance parameters vary along a curve for each month, and the performance
curve does not vary signiﬁcantly from one month to the next. The transfer duration
200
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 367 The relationship between the reference epoch and τ for each EL1 (top) and
EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria. From lightest
to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47] (ﬁrst
published by the American Astronautical Society).
may vary by several days between months, but the curves span very similar ranges
of injection C3 .
It is very interesting to plot the relationship between each transfer’s injection date
and its injection energy, C3 . Figure 369 shows this comparison for the EL1 and
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
201
Figure 368 The relationship between the injection C3 and transfer duration for each EL1
(top) and EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria. From
lightest to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47]
(ﬁrst published by the American Astronautical Society).
EL2 transfers. One can see that the families of transfers shift on this plot from
month to month. The comparison also shows that most families of transfers span an
injection date of 10 to 15 days. This suggests that there are 10 to 15 days in a launch
202
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 369 The relationship between the injection date and the injection C3 for each EL1
(top) and EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria. From
lightest to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47]
(ﬁrst published by the American Astronautical Society).
period to this lunar libration orbit via this type of transfer before the deep space ΔV
cost increases. This relationship, however, does not take into account differences in
the injection inclination throughout the family. Figure 369 also veriﬁes that EL1
transfers and EL2 transfers depart approximately two weeks apart from each other.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
203
The departure geometry of the ﬁltered transfers is very consistent and predictable
from month to month, given the proper analysis. Figures 370 and 371 show the
RAV and DAV parameters for the EL1 and EL2 transfers, respectively, computed
in the Sun–Earth rotating coordinate frame at the instant of translunar injection.
One can immediately observe that the ranges of RAV and DAV values are very
limited for each set of transfers: the EL1 transfers are conﬁned to the approximate
range of ∼140 deg ≤ RAV ≤ ∼170 deg, the EL2 transfers are conﬁned to the range of
∼320 deg ≤ RAV ≤ ∼355 deg, and both sets are conﬁned in DAV to the approximate
range ∼−10 deg ≤ DAV ≤ ∼10 deg. The RAV values appear to cover a very similar
span of values for each month, but there appears to be an annual signal in the DAV
values. This systematic variation may be isolated by observing the relationship
between a transfer’s DAV value and the orientation of the Moon’s orbital pole vector
at the arrival time. The Moon’s orbit has an inclination of approximately 5.1 deg
relative to the ecliptic. The Moon’s orbital plane is approximately ﬁxed in inertial
space, but rotates in the Sun–Earth rotating frame. Figure 372 shows the relationship
between the transfer’s injection DAV value and the right ascension of the lunar orbit
pole vector in the Sun–Earth rotating coordinate frame at the time of arrival. One
sees a clear annual signal in the data. A mission designer may be able to use this
information to improve an initial estimate of the translunar injection geometry. The
injection DAV value still varies by approximately 10 deg throughout a family after
accounting for the annual variation. This remaining variation may be explained by
the zaxis motion of the target orbit at the time of arrival, though that relationship has
not been studied sufﬁciently yet.
A relationship has also been observed between the injection RAV value and the
injection C3 . Figure 373 shows this relationship for both the EL1 and EL2 transfers.
One can see that higher RAV values require less injection energy and there is very
little monthly variation in the observed data.
Another parameter that depends closely on the relative orientation of the Moon’s
orbit about the Earth at the time of the transfer is the inclination of the LEO parking
orbit that is used to transfer onto these lowenergy transfers. The transfers are
constructed by building an initial state at the Moon and propagating backward in
time until they intersect a 185km parking orbit above the Earth’s surface. The
inclination of that parking orbit is driven by the geometry of the transfer. A real
mission launched from Cape Canaveral, Florida, would likely launch from an orbit
with an equatorial inclination near 28.5 deg and perform maneuvers to target the
desirable lowenergy transfer [183, 184]. This is the subject of Section 6.5. That
section shows that the closer the natural transfer is to having a parking orbit with a
particular, desired inclination, the less ΔV is required to target that transfer from the
desired parking orbit, though extended launch periods reduce the ΔV signiﬁcance.
204
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 370 The relationship between RAV and DAV (the right ascension and declination
of the apogee vector) at the time of translunar injection for the ﬁltered EL1 transfers. From
lightest to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018. Top:
one can see that RAV and DAV are conﬁned in a narrow box for these transfers; bottom: a
closer look at the parameter space [47] (ﬁrst published by the American Astronautical Society).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
205
Figure 371 The same relationship between RAV and DAV as Fig. 370, but for the ﬁltered
EL2 transfers. From lightest to darkest, the shading corresponds to reference dates from
1/1/2017 to 1/1/2018 [47] (ﬁrst published by the American Astronautical Society).
206
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 372 The relationship between the right ascension of the lunar orbit’s pole vector
at the time of arrival and the value of DAV at the time of injection, both computed in the
Sun–Earth rotating coordinate frame. From lightest to darkest, the shading corresponds to
reference dates from 1/1/2017 to 1/1/2018.This relationship is shown for each EL1 (top) and
EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria [47] (ﬁrst published
by the American Astronautical Society).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
207
Figure 373 The relationship between the right ascension of the apogee vector, RAV, at the
time of translunar injection and the injection energy, C3 , for each EL1 (top) and EL2 (bottom)
transfer in the 12month survey that satisﬁes the ﬁlter criteria. From lightest to darkest, the
shading corresponds to reference dates from 1/1/2017 to 1/1/2018 [47] (ﬁrst published by the
American Astronautical Society).
208
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 374 shows the relationship between the reference date of the lunar halo
orbit and the equatorial inclination of the natural LEO parking orbit needed to perform
the transfer. One can see that the inclination varies signiﬁcantly from one month to
the next. Figure 375 shows the relationship between the right ascension of the lunar
orbit pole vector and the ecliptic inclination of the LEO parking orbit. One can
clearly see that there is an evolution of the inclination from one month to the next.
Figure 376 shows the same plot, but this time presenting the relationship between
the lunar orbit pole vector and the equatorial inclination of the LEO parking orbit.
3.4.6.3 Annual Variations Much of the monthly variation observed in families
of lowenergy lunar transfers is caused by the Moon’s noncircular, inclined orbit
relative to the Earth. Other variations in the Solar System change over the course of
several years, evident in the analysis in Section 2.5.3. It is therefore of interest to
ensure that the relationships observed here hold over the course of several years. The
same analyses performed in the previous section have been performed again on a set
of transfers constructed with reference dates spanning the year 2021, four years after
the previous study. The results of this new examination coincide very well with the
previous study. Not all of the results will be shown here for brevity.
Figure 377 shows the relationship between Tref and τ , where the lighter shaded
points are lowenergy transfers that exist in 2017 and the darker points are lowenergy
transfers that exist in 2021. One can see that the combinations of the two parameters
are very similar for both years. Figure 378 shows a similar comparison between the
injection C3 and duration of the transfers in both 2017 and 2021. One can see that
there is very little noticeable difference between the points in 2017 and 2021.
The transfers that exist in 2021 have been ﬁltered in the same way as the transfers
presented in Section 3.4.6.2 in order to observe how the family might change during
the course of four years. Figures 379 and 380 show the same relationships as shown
in Figs. 372 and 375, except now for ﬁltered transfers in 2017 and 2021. One can
see that the 2021 parameters overlap the 2017 data very well, including the dramatic
monthly variations observed in the data.
The evidence suggests that the yearly variations are much more subtle than the
monthly variations that exist.
3.4.7
Transfers to Other ThreeBody Orbits
All of the analyses performed in Sections 3.4.3 through 3.4.6 have used the family of
halo orbits about the LL2 point as the example destination, but these analyses work
for any unstable threebody orbit in the Earth–Moon system.
Section 3.4.7.1 explores lowenergy lunar transfers that target an example lunar
L1 halo orbit. Since this orbit is on the interior side of the Moon, the trajectories that
target it must transfer from the lunar L2 region past the Moon before encountering
the target orbit.
Section 3.4.7.2 explores lowenergy lunar transfers that target an example distant
prograde orbit about the Moon. Orbits in this family traverse both the near and
far sides of the Moon. Hence, transfers that target these orbits may demonstrate
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
209
Figure 374 The relationship between the reference date of the lunar halo orbit and the
equatorial inclination of the LEO parking orbit needed to perform the transfer. From lightest
to darkest, the shading corresponds to reference dates from 1/1/2017 to 1/1/2018. This
relationship is shown for each EL1 (top) and EL2 (bottom) transfer in the 12month survey
that satisﬁes the ﬁlter criteria [47] (ﬁrst published by the American Astronautical Society).
210
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 375 The relationship between the right ascension of the lunar orbit pole vector
and the ecliptic inclination of the LEO parking orbit. From lightest to darkest, the shading
corresponds to reference dates from 1/1/2017 to 1/1/2018. This relationship is shown for each
EL1 (top) and EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria [47]
(ﬁrst published by the American Astronautical Society).
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
211
Figure 376 The relationship between the right ascension of the lunar orbit pole vector
and the equatorial inclination of the LEO parking orbit. From lightest to darkest, the shading
corresponds to reference dates from 1/1/2017 to 1/1/2018. This relationship is shown for each
EL1 (top) and EL2 (bottom) transfer in the 12month survey that satisﬁes the ﬁlter criteria [47]
(ﬁrst published by the American Astronautical Society).
212
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 377 The combinations of Tref and τ that yield lowenergy transfers between 185km
LEO parking orbits and the target lunar libration orbit during 2017 (gray points) and 2021
(black points) [47] (ﬁrst published by the American Astronautical Society).
characteristics similar to lowenergy lunar transfers that target either L1 or L2 halo
orbits.
These analyses are merely additional examples to demonstrate these analysis
techniques. All analyses will likely need to be repeated given speciﬁc mission
design requirements. That is, a given mission may require a spacecraft to transfer
to a particular unstable threebody orbit, perhaps for communication, staging, or
rendezvous reasons, and a new BLT map will need to be generated to study the
trajectory options that exist.
3.4.7.1 LowEnergy Transfers to a Lunar L1 Halo Orbit This section ex
plores lowenergy ballistic transfers to an example lunar L1 halo orbit. For simplicity
in this example analysis, the Patched ThreeBody Model is used; hence, the L1 halo
orbit is perfectly periodic.
In order to reach a halo orbit about the L1 point via a typical lowenergy transfer, a
spacecraft must depart the Earth and arrive in the lunar L2 vicinity in much the same
way as a spacecraft following a lowenergy transfer to a lunar L2 halo orbit. Then
from the vicinity of L2 , the spacecraft must transfer past the Moon before arriving
at its target L1 halo orbit. As usual, there are two types of transfers: transfers that
implement either the exterior or the interior stable manifold of the L1 halo orbit.
Interior transfers may arrive on the L1 halo orbit immediately after passing by the
Moon since the interior stable manifold is propagated in that direction. Exterior
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
213
Figure 378 The combinations of injection C3 and transfer duration that yield viable lowc 2009
energy lunar transfers in 2017 (gray points) and 2021 (black points) [44] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
transfers to most L1 halo orbits must ﬁrst traverse some sort of Earth staging orbit
prior to arriving on the L1 halo orbit.
Figure 381 shows an example interior lowenergy transfer to a lunar L1 halo orbit
in the Sun–Earth synodic reference frame. Figure 382 shows the same transfer in the
Earth–Moon synodic reference frame. The characteristics of this example transfer
are very similar to many of the lowenergy transfers previously studied in this work
that have transferred to L2 halo orbits. The only major difference is that this example
lowenergy transfer passes through the L2 region en route to the L1 region, where it
encounters its target L1 halo orbit.
Figures 383 and 384 show an example exterior lowenergy transfer to a lunar L1
halo orbit in the Sun–Earth and Earth–Moon synodic reference frames, respectively.
One can see that the transfer involves an Earth staging orbit, which permits it to
encounter the L1 halo orbit along the orbit’s exterior stable manifold. Every exterior
lowenergy transfer that has been constructed in this work between the Earth and this
L1 halo orbit requires the use of at least one Earth staging orbit. When propagated
backward in time, the exterior lunar transfers depart the L1 halo orbit away from the
Moon; hence, they must return to the Moon via an Earth staging orbit in order to
transfer out of the Earth–Moon system and into the Sun–Earth system.
Figures 385 and 386 show the interior and exterior BLT maps, respectively,
for lowenergy transfers to this halo orbit, making it possible to characterize many
214
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 379 The relationship between the right ascension of the lunar orbit pole vector and
the declination of the apogee vector at the time of injection. This relationship is shown for
each EL1 (top) and EL2 (bottom) transfer in both the 2017 (light) and 2021 (dark) surveys that
satisﬁes the ﬁlter criteria [47] (ﬁrst published by the American Astronautical Society).
transfers to this orbit simultaneously. Each ﬁgure also shows eight example transfers
to display some of the available transfer options that exist to this halo orbit. The BLT
maps are colored according to the altitude of closest approach that each trajectory
makes, given the values of θ and τ , when propagated backward in time at most
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
215
Figure 380 The relationship between the right ascension of the lunar orbit pole vector and
the ecliptic inclination of the LEO parking orbit. This relationship is shown for each EL1 (top)
and EL2 (bottom) transfer in both the 2017 (light) and 2021 (dark) surveys that satisﬁes the
ﬁlter criteria [47] (ﬁrst published by the American Astronautical Society).
216
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 381 An example interior lowenergy transfer to a lunar L1 halo orbit, shown in the
Sun–Earth synodic reference frame from above the ecliptic.
Figure 382 The same lowenergy transfer presented in Fig. 381, but now shown in the
Earth–Moon synodic reference frame from above the ecliptic.
195 days. Points colored black in each BLT map correspond to transfers that may be
used to depart the Earth from a lowaltitude orbit, or from the surface directly. The
lightest colors correspond to transfers that do not approach any closer to the Earth
than the L1 orbit itself when propagated backward in time. As usual, we are only
interested in the darkest regions of the BLT maps because those regions correspond
with trajectories that depart from practical low Earth orbits.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
217
Figure 383 An example exterior lowenergy transfer to a lunar L1 halo orbit, shown in the
Sun–Earth synodic reference frame from above the ecliptic.
Figure 384 The same lowenergy transfer presented in Fig. 383, but now shown in the
Earth–Moon synodic reference frame from above the ecliptic.
One can see that the two BLT maps shown in Figs. 385 and 386 are very complex.
This makes sense because the only ways to construct ballistic transfers between the
Earth and this lunar L1 halo orbit require some combination of lunar passages and
Earth staging orbits.
218
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 385 The interior BLT map for lowenergy transfers to the example lunar L1 halo
orbit. Eight example lowenergy transfers are shown around the BLT map to demonstrate
some of the types of transfers that may be constructed between 185km LEO orbits and this
halo orbit. (See insert for color representation of this ﬁgure.)
When studying Fig. 385, one notices many things. First, the BLT map is rather
simple in the range of τ values between 0.4 and 0.7. This region of τ values includes
ballistic lunar transfers that make only a single lunar passage en route to the L1
halo orbit. These transfers resemble the simplest lowenergy transfers to lunar L2
halo orbits and have very similar performance parameters. Somewhat more complex
transfers are shown in the BLT map for τ values between 0.7 and 0.96: most of these
involve several close lunar passages en route to the L1 halo orbit. Every transfer
constructed with a τ value between 0 and 0.35 involves at least one Earth staging
orbit, as may be seen in the two example transfers shown on the lowerleft edge of
the ﬁgure.
The exterior BLT map shown in Fig. 386 is more complex than the interior BLT
map. This is because each transfer must implement at least one Earth staging orbit in
addition to whatever lunar passages are required to complete the lowenergy transfer.
LOWENERGY TRANSFERS BETWEEN EARTH AND LUNAR LIBRATION ORBITS
219
Figure 386 The exterior BLT map for lowenergy transfers to the example lunar L1 halo
orbit. Eight example lowenergy transfers are shown around the BLT map to demonstrate
some of the types of transfers that may be constructed between 185km LEO orbits and this
halo orbit. (See insert for color representation of this ﬁgure.)
One may verify this by observing that every example trajectory shown around the
edge of Fig. 386 includes at least one Earth staging orbit. Otherwise, these transfers
are very similar to other lunar transfers previously studied.
3.4.7.2 LowEnergy Transfers to a Distant Prograde Orbit This section
explores lowenergy ballistic transfers to an example distant prograde orbit (DPO)
about the Moon. Like the previous section, this analysis is performed using the
Patched ThreeBody Model, making the DPO perfectly periodic. Distant prograde
orbits are interesting because they traverse both the near and far sides of the Moon.
One might suspect that the qualitative nature of a lowenergy transfer to such an orbit
might take on characteristics of transfers to either L1 or L2 halo orbits, depending on
how the speciﬁc transfer arrives at the orbit.
An example DPO has been generated here that has fairly large lobes and is easy
to view in the example transfers presented here. Figure 387 shows an example
220
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 387 An example lowenergy transfer to a distant prograde orbit, shown in the
Sun–Earth synodic reference frame from above the ecliptic.
lowenergy transfer to this distant prograde orbit in the Sun–Earth synodic reference
frame. Figure 388 shows the same example transfer in the Earth–Moon synodic
reference frame. One can see that this transfer does not enter any staging orbits, nor
make any lunar ﬂybys, but rather injects immediately into the distant prograde orbit.
Other ballistic transfers may be produced that do use staging orbits or other complex
lunar ﬂybys en route to the orbit.
Because of the symmetry in the distant prograde orbit’s shape, the two halves
of the orbit’s stable manifold are not clearly identiﬁable based on their immediate
motion. That is, both halves of the stable manifold include both interior and exterior
trajectories. However, the majority of one half of the distant prograde orbit’s stable
manifold propagates toward the Earth, and the majority of the other half propagates
away from the Earth. This discussion refers to the half that propagates toward the
Earth as the interior stable manifold and the other half as the exterior manifold.
Using this nomenclature, Figs. 389 and 390 show the exterior and interior BLT
maps, respectively, for lowenergy transfers to this distant prograde orbit.
Along with the exterior BLT map, Fig. 389 also shows eight example exterior
transfers that exist to this distant prograde orbit. One can see that these transfers
are very simple—they don’t require any lunar ﬂybys or staging orbits to reach the
target orbit. Because such simple transfers are prevalent in this exterior BLT map,
the map is consequently not nearly as chaotic as some of the previous BLT maps
studied in this chapter. The interior BLT map shown in Fig. 390, however, presents
more complex transfers to this distant prograde orbit, including several examples of
lowenergy transfers that require Earth staging orbits.
THREEBODY ORBIT TRANSFERS
221
Figure 388 The same transfer presented in Fig. 387, but now shown in the Earth–Moon
synodic reference frame from above the ecliptic.
The characteristics of the exterior transfers shown in Fig. 389 resemble the
characteristics of the exterior transfers to the lunar L2 halo orbit. The only real
complexity that may be introduced into the majority of such transfers is the addition of
a lunar ﬂyby en route to the transfers’ apogee passages. Conversely, the characteristics
of many of the interior transfers shown in Fig. 390 resemble the characteristics of
the exterior transfers to the lunar L1 halo orbit shown in Fig. 386. This makes sense
because the majority of both types of transfers involve Earth staging orbits, among
other features.
3.4.7.3 Discussion This section has demonstrated that the methodology pre
sented in this examination may be applied to many different families of unstable
threebody orbits. The same techniques may be applied to quasiperiodic and aperi
odic orbits as well, such as Lissajous orbits, though the parameters that generate the
BLT maps will not be perfectly cyclical. The lowenergy transfers and BLT maps
constructed using different target orbits may appear very different. Nonetheless,
families of lowenergy transfers may still be identiﬁed and systematically evaluated
in order to identify good candidates for practical lunar missions.
3.5 THREEBODY ORBIT TRANSFERS
Once a spacecraft has arrived at a lunar threebody orbit, the spacecraft has several
options. First, it may remain there for as long as desired, or at least until its stationkeeping fuel budget is exhausted (which may be years). Lunar halo orbits may be a
desirable location for communication and/or navigation satellites; they may also be
a desirable location for space stations or servicing satellites.
222
TRANSFERS TO LUNAR LIBRATION ORBITS
Figure 389 The exterior BLT map for lowenergy transfers to the example distant prograde
orbit about the Moon. Eight example lowenergy transfers are shown around the BLT map
to demonstrate some of the types of transfers that may be constructed between 185km LEO
orbits and this lunar orbit. (See insert for color representation of this ﬁgure.)
The spacecraft may transfer from the threebody orbit to a different threebody
orbit in the Earth–Moon system for very little energy, provided that both orbits
are unstable and have the same Jacobi constant [162, 185, 186]. For instance, the
spacecraft might arrive at a lunar L2 halo orbit and then later transfer to a lunar L1
halo orbit. Section 2.6.11 presents several methods that one may use to identify and
construct such transfers.
The spacecraft may also transfer from the nominal threebody orbit onto its unsta
ble manifold and follow that trajectory to a desirable stable lunar orbit. It has been
found that nearly any low lunar orbit is accessible in this way, and every transfer
studied has required a smaller orbitinsertion maneuver than any conventional, direct
transfer to the same low lunar orbit [46]. An example of such a transfer will be
described in more detail below.
Similarly, the spacecraft may follow the unstable manifold of the threebody orbit
down to the surface of the Moon. It has been found that any point on the surface
THREEBODY ORBIT TRANSFERS
223
Figure 390 The interior BLT map for lowenergy transfers to the example distant prograde
orbit about the Moon. Eight example lowenergy transfers are shown around the BLT map
to demonstrate some of the types of transfers that may be constructed between 185km LEO
orbits and this lunar orbit. (See insert for color representation of this ﬁgure.)
of the Moon may be reached, although some points require several orbits about the
Moon prior to touchdown [11, 46]. Again, the required ΔV to land from the lunar
threebody orbit is smaller than the required ΔV to land following a conventional,
direct transfer from the Earth.
Finally, the spacecraft has the option to return to the Earth following a lowenergy
Earthreturn trajectory. Every lowenergy lunar transfer has a symmetric Earthreturn counterpart; the Earthreturn trajectory does not need to be a mirror image of
the trajectory used to arrive at the lunar orbit.
If the spacecraft’s ﬁnal destination is not the lunar threebody orbit, then the
spacecraft does not need to inject into that orbit. Instead, the orbit’s stable manifold
may be used to guide the spacecraft to its ﬁnal destination rather than to inject the
spacecraft onto the threebody orbit. The stable manifold may be used as an initial
guess into a trajectory optimization routine, such as a multipleshooting differential
corrector (Section 2.6.5.2).
224
TRANSFERS TO LUNAR LIBRATION ORBITS
3.5.1
Transfers from an LL2 Halo Orbit to a Low Lunar Orbit
The discussion henceforth graphically illustrates some example options that a space
craft has upon arriving at a lunar halo orbit. Figure 391 shows one such lunar halo
staging orbit and its unstable manifold. A spacecraft on this halo orbit may depart
along any one of these trajectories. These trajectories ﬂy by the Moon at different
radii and inclinations, indicating that many different ﬁnal lunar orbits are accessible
from this staging orbit. When one considers all halo orbits in the family of L2 halo
orbits, one ﬁnds that nearly any low lunar orbit may be accessed by a lowenergy
lunar transfer. Figure 392 shows the available options that have been identiﬁed for
Figure 391 An example lunar halo staging orbit and its unstable manifold, viewed in the
Earth–Moon rotating frame from above (top) and from the side (bottom). A spacecraft on this
halo orbit may depart along any one of the trajectories shown.
THREEBODY ORBIT TRANSFERS
225
Figure 392 Available options identiﬁed for the radius and inclination of lunar orbits accessed
by southern lunar L2 halo orbits. Top: The radii and inclination combinations that may be
obtained at perilune of the unstable manifolds of six different lunar L2 halo orbits, where
each orbit’s available options are labeled with that orbit’s Jacobi constant. Bottom: The radii
and inclination combinations that may be obtained at perilune of the unstable manifolds of
many orbits in the family of southern halo orbits. The highlighted options in the plot at right
correspond to the available options for the halo orbit shown in Fig. 391.
226
TRANSFERS TO LUNAR LIBRATION ORBITS
the radius and inclination of lunar orbits that may be accessed by southern lunar L2
halo orbits. The shaded ﬁeld in the right plot has been constructed by sampling the
unstable manifolds of hundreds of halo orbits and interpolating between the results.
The highlighted points in the plot on the right are those points that are accessible from
the example southern halo staging orbit shown in Fig. 391. Northern halo orbits can
access the same set of lunar orbits except with a negative inclination. In each case,
it is assumed that the orbitinsertion maneuver is performed at the perilune of the
unstable manifold, but this is not required.
CHAPTER 4
TRANSFERS TO LOW LUNAR ORBITS
4.1
EXECUTIVE SUMMARY
This chapter examines lowenergy transfers that target low, 100kilometer (km), polar
lunar orbits. The analyses presented here may be applied to any lunar orbit insertion;
polar orbits are used as examples since mapping missions have historically been
frequently sent to nearpolar orbits about the Moon. This chapter presents surveys
of direct transfers as well as lowenergy transfers to low lunar orbits, and provides
details about how to construct a desirable transfer, be it a shortduration direct transfer
or a longer duration lowenergy transfer.
Figure 41 shows an example direct transfer, compared with an example lowenergy transfer to low lunar orbits. Much like the transfers presented in Chapter 3,
these trajectories are ballistic in nature; they require a standard translunar injection
(TLI) maneuver, a few trajectory correction maneuvers, and an orbit insertion maneu
ver. One may again add Earth phasing orbits and/or lunar ﬂybys to the trajectories,
if needed, which change their performance characteristics.
Many thousands of direct and lowenergy trajectories are surveyed in this chapter.
Table 41 provides a quick guide for several types of transfers that are presented here,
much like Table 31 from Chapter 3, comparing their launch energy costs, the breadth
227
228
TRANSFERS TO LOW LUNAR ORBITS
Figure 41
lunar orbit.
The proﬁles for both a direct and a lowenergy transfer from the Earth to a low
Table 41 A summary of several parameters that are typical for different mission
scenarios to low lunar orbits. EPOs = Earth Phasing Orbits, BLT = LowEnergy
Ballistic Lunar Transfer.
Mission
Element
Launch C3
(km2 /s2 )
Launch Period
Transfer Duration
(days)
Outbound Lunar
Flyby
Lunar Orbit
Insertion ΔV (m/s)
Direct
Transfer
Direct
w/EPOs
Simple
BLT
BLT w/Outbound
Lunar Flyby
BLT
w/EPOs
−2.2 to −1.5
< −1.5
−0.7 to −0.4
−2.1 to −0.7
< −1.5
Short
2–6
Extended
13+
Extended
70–120+
Short
70–120+
Extended
80–130+
No
No
No
Yes
Yes
∼820+
∼820+
∼640+
∼640+
∼640+
of their launch period, that is, the number of consecutive days they may be launched,
their transfer duration, and the relative magnitude of the orbit insertion change in
velocity (ΔV) upon arriving at the lunar orbit. The performance parameters are very
similar to lowenergy transfers to lunar libration orbits, except for the orbit insertion
INTRODUCTION
229
ΔV. These parameters are representative and may be used for highlevel mission
design judgements, though the details will likely vary from mission to mission.
Conventional lunar mission design is presented in Section 4.3 as a reference for the
analyses of lowenergy lunar transfers. The trajectories shown in that section require
translunar injection parameter (C3 ) values of at least −2.06 kilometers squared per
second squared (km2 /s2 ), realistic transfer durations between 2 and 6 days, and lunar
orbit insertion ΔV values of at least 813 m/s. One can certainly construct quicker
or longer transfers, but the injection C3 and lunar orbit insertion ΔV values increase
rapidly.
Direct transfers and lowenergy transfers to low lunar orbits are directly compared
and analyzed in Section 4.4. The surveys include many thousands of lunar transfers,
arriving at the Moon in any orientation and arriving at different times. The surveys
demonstrate that direct transfers must arrive at the Moon in a geometry such that the
orbital plane is roughly normal to the Earth–Moon line at the time of arrival. Whereas
lowenergy transfers may be constructed that arrive at any orbital plane. If a mission
must enter a lunar orbit with a particular node, then only certain values of the orbit’s
argument of periapse may be targeted, depending on the lunar arrival date; further,
those values are different for lowenergy transfers than they are for direct lunar
transfers. It has been found that lowenergy transfers require translunar injection
C3 values of about −0.6 km2 /s2 , compared with typical direct transfers that require
C3 values of about −2.0 km2 /s2 . Lowenergy transfers require about 70–120 days
of transfer duration, compared with direct transfers that require 2–6 days, though
either type of transfer may be designed to take more time. The lunar orbit insertion
ΔV is at least 640 m/s for lowenergy transfers, assuming an impulsive maneuver
to immediately target a 100km circular lunar orbit. Direct lunar transfers require
at least 120 m/s more ΔV, and often signiﬁcantly more ΔV than that to target the
same arrival conditions. Finally, lowenergy lunar transfers exist in families, such
that very similar transfers exist to neighboring libration orbits. Very similar transfers
also exist to the same orbit when the arrival time or arrival geometry is adjusted.
4.2 INTRODUCTION
This chapter is devoted to the analysis and construction of lowenergy transfers
to low lunar orbit. This is a rich problem; it is far too complex to present all
possible examinations of such transfers in a concise form. To simplify the problem,
while retaining a connection to practical spacecraft mission design, this book limits
the scope of this study and only examines lowenergy transfers to lowaltitude,
100km circular, polar orbits about the Moon. These orbits are very similar to
many mapping orbits ﬂown by historical lunar missions, including Lunar Prospector
[56], Kaguya/SELENE [187], Chang’e 1 [58], Chandrayaan1 (CH1) [3], the Lunar
Reconnaissance Orbiter (LRO) [188], and Gravity Recovery and Interior Laboratory
(GRAIL) [83]. The procedures presented in this chapter may easily be applied to
transfers that implement an eccentric capture orbit about the Moon: in that case the
argument of periapse of the target orbit becomes a design constraint and the orbit
230
TRANSFERS TO LOW LUNAR ORBITS
insertion ΔV is reduced appropriately. This chapter contains all of the information to
design such orbit insertions, assuming that the mission performs lunar orbit insertion
(LOI) at an altitude of 100 km and an inclination of 90 degrees (deg). Even so, the
procedures presented here may be applied to orbit insertions at other altitudes and in
other inclinations, though in those cases the design space will have to be reconstructed
by the mission designer. The surveys presented here provide a good representation
of the trade space of any direct and lowenergy transfer to any low orbit about the
Moon.
Although the general characteristics of lowenergy transfers to low lunar orbits
are similar to the characteristics of lowenergy transfers to lunar libration orbits, such
as those presented in Chapter 3, the geometry of transfers that arrive at polar orbits
is still signiﬁcantly different. Therefore, the analysis in this chapter is independent
of Chapter 3 and speciﬁcally tailored to study missions to low lunar orbit.
The GRAIL mission is the only mission in history, prior to 2012, to implement
a lowenergy transfer to a lowaltitude orbit about the Moon as part of its primary
mission. Its design features will be used as a reference in many of the discussions in
this chapter [83–85]. GRAIL’s trajectory design is illustrated in Fig. 42, including
the ﬁrst and last launch opportunity in a 21day launch period. This is the launch
period published in Ref. 83; however, it was actually extended by many days as
the mission developed. The GRAIL mission launched on September 10, 2011, on
the third day of its launch period. GRAIL’s mission design includes two signiﬁcant
Figure 42 An illustration of GRAIL’s mission design, including a 26day launch period and
two deterministic maneuvers for both GRAILA and GRAILB, designed to separate their lunar
orbit insertion times by 25 hours [83] (Originally published by the American Astronautical
Society).
DIRECT TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
231
deterministic maneuvers performed per spacecraft during the cruise, performed pri
marily to separate their lunar orbit insertion dates. The trajectories generated in this
chapter do not include these sorts of maneuvers. Chapter 6 explores the addition of
maneuvers like those in GRAIL’s design.
4.3
DIRECT TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
The purpose of this book is to illustrate the costs, beneﬁts, and characteristics of
lowenergy lunar transfers; the primary referent is the direct lunar transfer, which
has been used so frequently in lunar missions that it is known as the conventional
method. The ﬁrst spacecraft launched toward the Moon, Luna 1, followed a direct
transfer: a trajectory that required only 34 hours to reach the Moon, passing by within
6000 km of the surface. Since then, dozens of missions have implemented direct
lunar transfers with durations ranging from 1.4 to 5.5 days, not including any staging
orbits. Table 12 on page 16 summarize many example missions that implemented
such direct transfers. Many resources exist that describe these direct lunar transfers in
great detail [189]. This section only considers the ΔV of basic transfers as a function
of the transfer duration to be used as a reference when describing lowenergy lunar
transfers.
Direct lunar transfers are trajectories that depend only on the gravity of the Earth
and Moon. The Sun’s gravity is accounted for, but only as a perturbation to the
transfer. A very shortduration direct transfer departs the Earth on a hyperbola that
encounters the Moon. The most efﬁcient direct transfers typically require 4–5 days,
depending on the location of the Moon in its elliptical orbit, and resemble Hohmann
transfers. Figure 43 illustrates several direct lunar transfers that have varying transfer
durations.
Figure 43 illustrates how the ΔV cost of a direct transfer increases away from
the optimal transfer duration. But the cost doesn’t rise very rapidly until the transfer
duration has changed by several days. Recent spacecraft have taken advantage
of the optimal transfer durations to maximize the amount of payload sent to the
Moon. Conversely, it is apparent why the Apollo mission planners opted for a shorter
transfer: the ΔV cost does not rise very much by decreasing the transfer duration
from 4.5 days to 3.0 days, but the other consumables (including items such as food,
water, and electrical power) required 1.5 days less support time on both the outbound
and return transfer segments.
Since a spacecraft following a direct transfer only requires a few days to reach
the Moon, it must be prepared to perform a maneuver within hours, or perhaps at
most a day, to perform a trajectory correction maneuver. If this is an undesirably
short amount of time, the mission may implement an Earth phasing orbit to extend
the transfer duration. The spacecraft would be launched into an orbit that does not
encounter the Moon, and only after one or more perigee passages would the trajectory
ﬁnally arrive at the Moon.
The launch periods for many historical direct transfers were very short: only
a handful of opportunities to launch per month, when the geometry was aligned
232
TRANSFERS TO LOW LUNAR ORBITS
Figure 43 Five example direct transfers from 185km circular Earth orbits to 100km
prograde lunar orbits, shown in the rotating frame (top) and inertial frame (bottom). These
trajectories have been generated in the planar circular restricted threebody system. The
following information applies to the labeled trajectories:
Traj.
Duration
(days)
C3
(km2 /s2 )
ΔVTLI
(km/s)
ΔVLOI
(km/s)
Total ΔV
(km/s)
(a)
(b)
(c)
(d)
(e)
6.0
4.5
3.0
2.0
1.0
−1.976
−2.064
−1.670
0.264
13.654
3.138
3.134
3.152
3.240
3.831
0.829
0.813
0.893
1.248
3.024
3.966
3.948
4.045
4.488
6.854
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
233
properly. The Clementine and Chandrayaan1 missions implemented Earth phasing
orbits, which extended the launch periods. Chandrayaan1’s nominal mission proﬁle
included half a dozen Earth orbits prior to the lunar encounter. If the mission launched
a day late, then the orbital period of one or more of these orbits would be adjusted
to compensate for the change in transfer duration. The drawbacks of Earth phasing
orbits include an extended operational timeline, which may add to the costs of the
mission, and an increased dose of radiation as the spacecraft passes through the Van
Allen Belts multiple times.
4.4 LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR
ORBIT
This section discusses how to build a lowenergy ballistic transfer between the Earth
and a low lunar orbit. The algorithms and methodology used to build a lowenergy
transfer are ﬁrst described. Then, several example surveys are conducted, examining
lowenergy transfers that arrive at the Moon in some particular geometry at some
given arrival time. The surveys become more general as this analysis continues. It
then shows how to construct a map that tracks the minimum transfer ΔV cost required
for a spacecraft to target any lunar orbit at a particular arrival time. Finally, the arrival
time is opened up and transfers are examined that arrive at the Moon at many different
times. The goal is to capture the transfer ΔV cost for transfers to any polar orbit
about the Moon at any given arrival time in order to guide mission planners as they
deﬁne the orbits and timeline for a given mission.
4.4.1
Methodology
Each transfer in the surveys presented here departs the Earth, coasts to the Moon, and
injects directly into a low lunar orbit. To reduce the scope of the problem while still
yielding practical data, the surveys presented here assume that the mission targets a
circular 100km polar orbit about the Moon. This lunar orbit is akin to the mapping
orbits of several spacecraft, including Lunar Prospector [56], Kaguya/SELENE [187],
Chang’e 1 [58], Chandrayaan1 [3], the LRO [188], and GRAIL [83].
The LOI is modeled as a single impulsive maneuver that is performed at the
periapse point and places the vehicle directly into a circular orbit. This is not a
realistic maneuver, but it is useful to directly compare the total insertion cost of one
transfer to another. The orbit insertion cost needed to place a satellite into an elliptical
orbit, rather than a circular orbit, may be determined via the VisViva equation [97].
The surveys presented here have been generated using a method that does not
make many assumptions about what the lunar transfers look like. This permits each
survey to reveal trajectories that may not have been expected. Each trajectory in each
survey is constructed using the following procedure:
1. Construct the target lunar orbit. The following parameters are used in this
study, speciﬁed in the International Astronomical Union (IAU) Moon Pole
coordinate frame (see Section 2.4.4).
234
TRANSFERS TO LOW LUNAR ORBITS
Periapse radius,
Eccentricity,
Equatorial inclination,
Argument of periapse,
Longitude of the ascending node,
True anomaly,
rp :
e :
i :
ω :
Ω:
ν :
1837.4 km (∼100km altitude)
E
90 deg
Speciﬁed value
Speciﬁed value
0 deg
The argument of periapse is undeﬁned for a circular orbit. However, since all
practical missions to date have inserted into elliptical orbits, and some missions
remain in a highly elliptical orbit, the target orbit’s argument of periapse, ω, is
presented here rather than the true anomaly, which is kept at 0 deg to indicate
that LOI is performed at periapse. The orbit’s eccentricity is given as E: it is
approximately zero (1 × 10−9 ) while permitting ω to be deﬁned. One may
also use the argument of latitude, which is deﬁned for a circular orbit.
2. Construct the LOI state.
(a) Specify the date of the LOI, tLOI . Dates are given here in Ephemeris
Time (ET).
(b) Specify the magnitude of the impulsive orbit insertion maneuver, ΔVLOI .
Apply the ΔV in a tangential fashion to the LOI state.
3. Propagate the state backward in time for 160 days.
4. Identify the perigee and perilune passages that exist in the trajectory.
(a) If the trajectory ﬂies by the Moon within 500 km, label the trajectory as
undesirable.
(b) The latest perigee passage that approaches within 500 km of the Earth is
considered the earliest opportunity to inject into that trajectory.
(c) If no low perigees are observed, then the lowest perigee is identiﬁed as
the translunar injection (TLI) location.
5. Characterize the performance of the trajectory, making note of the following
values:
• TLI altitude, inclination, and C3 ;
• Duration of the transfer;
• Periapse altitude of any/all Earth and Moon ﬂybys; and
• LOI ΔV magnitude.
This procedure requires four inputs: the longitude of the ascending node of the target
orbit (Ω), the argument of periapse of the target orbit (ω), the ΔV of the impulsive LOI
(ΔVLOI ), and the date of the LOI (tLOI ). Figures 44 and 45 show two examples
of lunar transfers generated with this procedure using the inputs summarized in
Table 42. Figure 44 illustrates a direct 4day transfer and Fig. 45 illustrates an
84day lowenergy transfer.
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
235
c 2011 by American
Figure 44 An example 4day direct lunar transfer [2] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission of the
AAS).
All integrations performed here have been performed using a DIVA integrator
(Section 2.7.1) with tolerance set to 1 × 10−10 ; the force model includes the Sun,
Earth, Moon, and each of the planets, all conﬁgured as pointmass gravitating bodies
whose positions are estimated from JPL’s DE421 Planetary and Lunar Ephemeris
(Section 2.5.3).
Many surveys have been conducted, searching for practical lunar transfers. In
general, a survey ﬁxes the parameters Ω and tLOI and systematically varies the
other two parameters. This process generates a twodimensional map displaying a
parameter—typically the TLI altitude—which changes smoothly as either Ω or tLOI
shift. These surveys are described in more detail in the next sections.
4.4.2
Example Survey
Figure 46 shows the results of an example survey of lunar transfers. In this example,
Ω is set to 120 deg, the LOI date is set to 18 July 2010 09:50:08 ET, the value
236
TRANSFERS TO LOW LUNAR ORBITS
c 2011 by
Figure 45 An example 84day lowenergy lunar transfer [2] (Copyright ©
American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with
permission of the AAS).
Table 42 The inputs and performance parameters of the two example lunar transfers
shown in Figs. 44 and 45. Both transfers begin in a 185km circular low Earth orbit
(LEO) parking orbit before their injections, and both transfers arrive at the Moon at a
time tLOI of 18 July 2010 9:50:08 ET.
Figure
Ω
ω
ΔVLOI
Duration
#
(deg)
(deg)
(m/s)
(days)
Equatorial
Ecliptic
(km2 /s2 )
44
45
120.0
120.0
310.0
160.0
839.878
669.543
4.036
83.706
62.114
28.093
39.761
5.921
−2.064
−0.725
LEO Inclination (deg)
C3
of ω is systematically varied from 0–360 deg, and ΔVLOI is systematically varied
from 650–1050 meters per second (m/s), a range empirically determined to generate
practical transfers. Figure 46 shows the altitude of the translunar injection point
for each combination of ω and ΔVLOI , assuming a spherical Earth with radius of
6378.136 km. The points shaded white correspond to trajectories that arrive at the
Moon such that when propagated backward in time they never come any closer to
the Earth than the orbit of the Moon itself. The points shaded black correspond to
trajectories that arrive at the Moon such that when propagated backward in time they
approach within 10,000 km of the Earth: trajectories that may be used to generate
real missions [183, 184, 190, 191], assuming the departure time and geometry are
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
237
Figure 46 The altitude of the TLI location for each combination of ν and ΔVLOI , given a
lunar orbit insertion on July 18, 2010 into a lunar orbit with Ω equal to 120 deg [2] (Copyright
c 2011 by American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted
©
with permission of the AAS).
acceptable (see Section 6.5 for more information about generating a real mission
using a ballistic guess).
The plot shown in Fig. 46 contains many interesting features. First, roughly
half of the state space is white, corresponding to trajectories that arrive at the Moon
from heliocentric orbits. With a quick investigation, one ﬁnds that the large black
ﬁeld toward the top of the plot corresponds to direct transfers to the Moon, that is,
trajectories that take 2–12 days to reach the Moon, in family with the transfers that
were implemented by the Apollo program and LRO; though most of the trajectories
include Earth phasing orbits that extend the transfer’s duration. The black curve that
outlines the large white ﬁeld corresponds to lowenergy lunar transfers that require
80–120 days. There are many other curves throughout the plot that correspond to
trajectories that enter some sort of large Earth orbit, or perform a combination of one
or more ﬂybys.
The direct transfers that are observed in the upper part of the plot shown in Fig. 46
require ΔVLOI values from 760 m/s to 1000 m/s or more. The direct transfers that
don’t involve any Earth phasing orbits or any sort of lunar ﬂyby require at least
818 m/s, though nearly all require 845 m/s or more. Figure 47 explores the structure
of the direct transfer state space, presenting two additional maps that only show those
Figure 47 The transfers shown in Fig. 46 that approach within 1000 km of the Earth, shaded according to the number of Earth phasing orbits (top)
c 2011 by American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted
and lunar ﬂybys (bottom) that they make [2] (Copyright ©
with permission of the AAS). The upper portion of each map, corresponding with direct lunar transfers, is magniﬁed in the plots on the right. (See
insert for color representation of this ﬁgure.)
238
TRANSFERS TO LOW LUNAR ORBITS
239
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
trajectories that approach within 1000 km of the Earth; the two maps are shaded
according to the number of Earth perigee passages (top) and lunar ﬂybys (bottom)
that they make before arriving at their target orbit. One notices that direct transfers
with more phasing orbits and/or lunar ﬂybys may require less orbit insertion ΔV
than the most basic lunar transfers. In any case, simple lowenergy trajectories exist
that require as little as 669 m/s, ∼100 m/s less than most multirev direct transfers
observed and ∼170 m/s less than most simple direct transfers.
Tables 43 and 44 summarize the performance parameters of several example
direct lunar transfers and lowenergy lunar transfers, respectively. Several examples
of these trajectories are shown in Figs. 48 and 49, respectively. One can see that
the value of ΔVLOI is generally over 100 m/s lower for lowenergy transfers in
nearly all examples, though the TLI injection energy, C3 , is higher. The injection
energy of direct lunar transfers is very close to −2.0 km2 /s2 , compared to a value of
approximately −0.7 km2 /s2 for lowenergy transfers. Both types of transfers include
missions with a wide range of TLI inclinations, both relative to the Earth’s Equator
and to the ecliptic. This suggests that transfers can begin from any inclination about
the Earth. Section 6.5 demonstrates that one can add one to three maneuvers and
adjust a trajectory to depart from a speciﬁed TLI inclination rather than the ballistic
inclination value shown in the tables for a very modest ΔV cost. The total ΔV required
to make this adjustment is on the order of 1 m/s per degree of inclination change.
4.4.3
Arriving at a FirstQuarter Moon
All of the transfers presented in the previous section arrive at the Moon at a particular
time into a particular orbit, namely, a circular, polar orbit with a longitude of the
ascending node, Ω, of 120 deg and a time of arrival, tLOI , of 18 July 2010 at
9:50:08 ET. This time of arrival corresponds to a moment in time when the Sun–
Table 43 A summary of the performance parameters of several direct lunar transfers
c 2011 by American
shown in Fig. 46 and illustrated in Fig. 48 [2] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission
of the AAS).
Traj
Ω
ω
ΔVLOI
Duration
C3
# Earth
# Moon
#
(deg)
(deg)
(m/s)
(days)
Equatorial
Ecliptic
(km2 /s2 )
Flybys
Flybys
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
321.3
326.4
304.8
301.5
311.7
321.0
326.4
279.0
325.5
327.0
354.9
268.2
818.0
860.4
867.5
947.7
971.8
813.3
868.0
870.0
758.0
810.1
828.8
861.4
4.111
4.155
4.004
3.942
4.009
13.941
14.005
32.759
67.175
84.747
85.441
141.341
22.147
43.459
85.516
142.173
131.320
24.717
52.683
19.407
37.135
62.694
75.489
46.894
8.551
62.667
63.963
123.280
154.340
6.435
72.504
31.944
13.784
39.723
54.465
63.333
−2.078
−2.058
−2.045
−2.006
−2.002
−2.095
−2.071
−2.046
−2.292
−2.055
−2.061
−2.054
0
0
0
0
0
1
1
2
6
7
7
8
0
0
0
0
0
0
0
1
1
3
1
1
LEO Inclination (deg)
240
TRANSFERS TO LOW LUNAR ORBITS
Table 44 A summary of the performance parameters of several lowenergy lunar
c 2011 by
transfers shown in Fig. 46 and illustrated in Fig. 49 [2] (Copyright ©
American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with
permission of the AAS).
Traj
Ω
ω
ΔVLOI
Duration
C3
# Earth
# Moon
#
(deg)
(deg)
(m/s)
(days)
Equatorial
Ecliptic
(km2 /s2 )
Flybys
Flybys
L1
L2
L3
L4
L5
L6
L7
L8
L9
L10
L11
L12
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
120.0
169.2
103.8
70.2
225.3
99.9
186.9
61.5
59.7
36.3
348.6
262.2
244.2
669.3
692.1
743.9
716.0
697.5
673.2
660.4
651.3
661.5
675.1
656.1
657.8
83.483
85.287
93.598
93.621
110.060
122.715
143.360
129.422
144.417
155.107
141.982
136.687
29.441
25.688
57.654
134.322
83.127
23.941
18.624
73.143
146.592
36.598
153.641
179.084
6.129
34.778
74.955
112.840
61.624
3.088
35.412
96.544
138.491
16.583
176.867
156.890
−0.723
−0.723
−0.667
−0.657
−0.697
−0.712
−0.572
−0.612
−0.658
−0.645
−0.608
−0.640
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
1
3
1
1
3
6
LEO Inclination (deg)
Figure 48 Example plots of several of the transfers summarized in Table 43. The
trajectories are shown in the Sun–Earth rotating frame, such that the Sun is ﬁxed on the
c 2011 by American Astronautical Society Publications
xaxis toward the left [2] (Copyright ©
Ofﬁce, all rights reserved, reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
241
Figure 49 Example plots of several of the transfers summarized in Table 44. The
trajectories are shown in the Sun–Earth rotating frame, such that the Sun is ﬁxed on the
c 2011 by American Astronautical Society Publications
xaxis toward the left [2] (Copyright ©
Ofﬁce, all rights reserved, reprinted with permission of the AAS).
Earth–Moon angle is approximately equal to 90 deg at the Moon’s ﬁrst quarter. This
is very similar to the arrival geometry of the two GRAIL spacecraft, though in a
different month. In addition, the plane of the target orbit is nearly orthogonal to the
Earth–Moon line. A polar orbit with an Ωvalue of 111.9 deg (also 291.9 deg) is in
a plane that is as close to orthogonal to the Earth–Moon line as a polar orbit can get
on this date. The surveys presented in this section keep the time of arrival the same
and explore the changes to the lunar transfers that occur as the target orbit’s Ωvalue
is varied.
Figures 410 and 411 show surveys of the lunar transfer state space as Ω varies
from 0–80 deg and 160–270 deg, respectively. There is a clear progression of the
state space as Ω varies. Locations where direct and lowenergy transfers exist are
indicated. The state space varies much less discernibly when Ω is within ∼30 deg
of 111.9 deg or 291.9 deg, namely, when the orbit is close to being orthogonal to the
Earth–Moon line.
Many features are quickly apparent when studying the maps shown in Figs. 410
and 411. First, a large portion of each map is white, corresponding to combinations
of ΔVLOI and ω that result in trajectories that depart the Moon backward in time and
242
TRANSFERS TO LOW LUNAR ORBITS
Figure 410 Nine surveys of trajectories that arrive at the ﬁrstquarter Moon, where the
target orbit’s Ω varies from 0–80 deg. Points in black originate from the Earth; other points
are shaded according to how close they come to the Earth when propagated backward, using
c 2011 by American
the light–dark shading scheme presented in Fig. 46 [2] (Copyright ©
Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission of the
AAS).
traverse away from the Earth–Moon system. At lower ΔVLOI values, the trajectories
depart the Moon backward in time and later impact the Moon or remain very near
the Moon. One can see curves of black in each map, corresponding to trajectories
that depart the Moon backward in time and eventually come very near the Earth;
hence, making viable Earth–Moon transfers. The features are observed to shift in a
continuous fashion across the range of Ωvalues.
If one surveys these maps, one ﬁnds that lowenergy transfers exist to any lunar
orbit plane, but simple direct transfers only exist for certain ranges of Ωvalues.
Direct transfers can only reach orbits with Ωvalues between approximately 50 deg
and 170 deg and between approximately 230 deg and 350 deg for this particular
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
243
Figure 411 Twelve surveys of missions that arrive at the ﬁrstquarter Moon, where the
target orbit’s Ω varies from 160–270 deg. The maps are shaded according to the closest
approach distance that the trajectories make with the Earth, as illustrated in Figs. 46 and
c 2011 by American Astronautical Society Publications Ofﬁce, all rights
410 [2] (Copyright ©
reserved, reprinted with permission of the AAS).
arrival date. These orbit planes are within about 60 deg of being orthogonal to the
Earth–Moon line; furthermore, direct lunar transfers require less ΔV for their orbit
insertions the closer they are to being orthogonal to the Earth–Moon line.
Figure 412 captures the leastexpensive ΔVLOI for simple direct lunar transfers,
as well as simple lowenergy lunar transfers (that is, transfers that do not involve
244
TRANSFERS TO LOW LUNAR ORBITS
Figure 412 The minimum lunar orbit insertion ΔV for direct and lowenergy (L.E.) lunar
transfers, requiring no Earth phasing orbits nor lunar ﬂybys for transfers to a ﬁrstquarter
Moon. Polar orbits with Ωvalues of 111.9 deg and 291.9 deg are very close to orthogonal to
c 2011 by American Astronautical
the Earth–Moon line on this arrival date [2] (Copyright ©
Society Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
lunar ﬂybys or Earth phasing orbits) for any target orbit plane studied. Three curves
are presented: direct transfers involve transfers that require less than 40 days to
achieve (most require less than 10 days), fast lowenergy transfers require less than
95 days, and long lowenergy transfers require more than 95 days to achieve. The
transfer durations are not permitted to exceed 160 days in this study. There are many
trajectories that require more ΔV than what is shown in Fig. 412; the illustration
tracks the least expensive transfer in each case. Trajectories with Earth phasing orbits
and/or lunar ﬂybys may require even less ΔV, but those are not tracked here since there
are so many paths that a spacecraft can take through the system. One observes that
lowenergy transfers do indeed reach any target orbit, though the insertion ΔV costs
vary as the orbit plane changes. Direct lunar transfers are indeed limited to certain
orbital planes, and they require at least 120 m/s more LOI ΔV than a lowenergy
transfer to the same orbit. Further, the cost of longer lowenergy transfers remains
very constant—within 50 m/s of ΔV—for any target lunar orbit plane.
The lunar transfers with the least LOI ΔV and no low Earth or lunar periapse
passages have been identiﬁed for each combination of Ω and ω; their performance
parameters are plotted in Fig. 413. The left plot shows a map of the LOI ΔV cost
of these transfers; the plot on the right shows the corresponding transfer duration for
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
245
Figure 413 The combinations of Ω and ω that yield simple lunar transfers, that is, those
without low Earth or lunar periapse passages. If multiple transfers exist for the same
combination, then the one with the least LOI ΔV is shown. All of these transfers arrive
at a ﬁrstquarter Moon. The lowΔV transfers shown in Fig. 412 are indicated by dots in each
c 2011 by American Astronautical Society Publications Ofﬁce, all rights
map [2] (Copyright ©
reserved, reprinted with permission of the AAS). (See insert for color representation of this
ﬁgure.)
each trajectory. The lowΔV solutions identiﬁed in Fig. 412 are plotted in these
maps for reference, and to identify their ωvalues and durations. Direct transfers are
easily discerned by observing the dark ﬁelds in the plot on the right, corresponding
to shortduration transfers. One can see that there are large ﬁelds of combinations
of Ω and ω that yield lowenergy transfers, though the costs increase as one moves
246
TRANSFERS TO LOW LUNAR ORBITS
away from the lowΔV curves. One can see that the combinations of Ω and ω that
yield practical direct transfers are much more limited.
The maps shown in Fig. 413 are very useful: they illustrate what sorts of transfers
may be used to reach any given polar orbit at the Moon, given that the transfers must
arrive at the Moon at this particular arrival time. Missions that target an elliptical
orbit must consider which argument of periapsis value to target; missions that aim to
enter a circular orbit may likely use any ω for the initial orbit insertion, simplifying
the trade space. Similar maps may be generated for any lunar arrival time: two
different arrival times will be considered in the next sections.
4.4.4
Arriving at a ThirdQuarter Moon
All of the transfers studied so far have arrived at the Moon at the same time, when the
Moon is at its ﬁrst quarter. Yet spacecraft missions may need to arrive at the Moon
at any time of the month. As a second step in this survey, lunar transfers are studied
that arrive at the Moon on 3 August 2010 at 04:38:29 ET: a time when the Moon has
reached its third quarter. Figure 414 shows two example transfers that arrive at the
thirdquarter Moon, where the trajectory on the left is a direct lunar transfer and the
trajectory on the right is a lowenergy transfer. Neither transfer requires any extra
Earth phasing orbits or lunar ﬂybys. One notices that the lowenergy transfer extends
away from the Sun rather than toward it as seen in Figs. 45 and 49. Otherwise the
transfers appear very similar to those studied previously. The symmetry observed
here is expected according to the nearly symmetrical dynamics in the Sun–Earth
system [86]. The Sun–Earth L1 and L2 points are located nearly the same distance
from the Earth, and threebody libration orbits about those Lagrange points behave
in a very similar fashion [46].
Figure 414 Two example lunar transfers that arrive at a thirdquarter Moon. The transfers
are simple, direct (left) and lowenergy (right) lunar transfers with no Earth phasing orbits
nor lunar ﬂybys. The transfers are viewed from above in the Sun–Earth rotating frame of
c 2011 by American Astronautical Society Publications Ofﬁce, all
reference [2] (Copyright ©
rights reserved, reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
247
One may construct state space maps for transfers to a thirdquarter Moon in
the same way that maps have been constructed previously to a ﬁrstquarter Moon.
Figures 415 and 416 plot state space maps for transfers to target orbits with Ωvalues
of 0–80 deg and 180–260 deg, respectively. These ranges of Ωvalues track the
interesting features as the orbit plane changes; the maps of the Ωvalues between
those plotted in the ﬁgures vary little across the range. An orbit with an Ωvalue
of 126.9 deg (also 306.9 deg) is as close to orthogonal to the Earth–Moon axis as a
polar orbit can be at this time. Transfers within about 60 deg of this angle are all very
similar, though the cost of those transfers rises as the orbital plane moves away from
this optimal Ωvalue. When one compares the maps shown in Figs. 415 and 416
to those constructed earlier in Figs. 410 and 411, one sees that the maps are very
Figure 415 Nine surveys of missions that arrive at the thirdquarter Moon, where the target
orbit’s Ω varies from 0–80 deg. The points are again shaded according to how close they
approach to the Earth when propagated backward in time, using the same light–dark shading
c 2011 by American Astronautical Society
scheme applied in previous maps [2] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
248
TRANSFERS TO LOW LUNAR ORBITS
Figure 416 Nine surveys of missions that arrive at the thirdquarter Moon, where the target
c 2011 by American Astronautical Society
orbit’s Ω varies from 180–260 deg [2] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
similar with a 195 deg plane change. The transfers are arriving at the Moon when
it is 180 deg further along in its orbit in the Sun–Earth synodic frame and 195 deg
further in its orbit inertially, while the inertial coordinate axes that deﬁne Ω and ω
have not changed.
Figure 417 shows the same two plots as shown in Fig. 413 for these thirdquarter
lunar arrival transfers. The maps show the LOI ΔV cost and transfer duration for
simple lunar transfers that target different lunar orbits. As before, if there are multiple
lunar transfers that may be used to arrive at the same lunar orbit, then the maps present
the parameters for the transfer with the least LOI ΔV. The maps illustrate that the
same trends exist to thirdquarter lunar arrivals as do to ﬁrstquarter lunar arrivals,
but with a 195deg shift in Ω.
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
249
Figure 417 The combinations of Ω and ω that yield simple lunar transfers, that is,
those without low Earth or lunar periapse passages. If multiple transfers exist for the same
combination, then the one with the least LOI ΔV is shown. All of these transfers arrive at a
c 2011 by American Astronautical Society Publications
thirdquarter Moon [2] (Copyright ©
Ofﬁce, all rights reserved, reprinted with permission of the AAS).
250
TRANSFERS TO LOW LUNAR ORBITS
4.4.5
Arriving at a Full Moon
Trajectories have been studied that arrive at the Moon when the Sun–Earth–Moon
angle is near 90 deg; this section brieﬂy considers trajectories that arrive at a full
Moon, when the Sun–Earth–Moon angle is approximately 180 deg. Lunar transfers
that arrive at a new Moon have much the same characteristics as those that arrive at a
full Moon, but with a familiar 180 deg ± 15 deg shift in Ω; for brevity they will not
be shown here.
Figures 418 and 419 present state space maps for trajectories that arrive at the
full Moon in polar orbits with Ωvalues in the ranges 90–170 deg and 270–350 deg,
respectively. The maps not shown vary only gradually between these maps. One
observes that direct lunar transfers arrive at the full Moon with lowΔV insertions at
Ωvalues approximately 90 deg apart from those that arrive at the ﬁrstquarter and
Figure 418 Nine surveys of missions that arrive at a full Moon, where the target orbit’s
c 2011 by American Astronautical Society
Ω varies from 90–170 deg [2] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
251
Figure 419 Nine surveys of missions that arrive at a full Moon, where the target orbit’s
c 2011 by American Astronautical Society
Ω varies from 270–350 deg [2] (Copyright ©
Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
thirdquarter Moons. This demonstrates additional evidence that the minimum orbit
insertion ΔV requirements for direct lunar transfers occurs when the orbit’s plane is
nearly orthogonal to the Earth–Moon line.
The lowenergy lunar transfers’ locations in the fullMoon state space maps evolve
somewhat differently as Ω varies compared with their evolutions in the state space
maps for ﬁrst and thirdquarter Moons. Lowenergy transfers still arrive at the Moon
for any Ωvalue, but the range of ωvalues that may be used are bifurcated along the
range of Ωvalues. Many of the lowenergy transfers that require the least LOI ΔV
arrive at the full Moon at ωvalues near 75 deg and 255 deg. These transfers ﬂy
further out of the plane of the Moon’s orbit than others; those transfers that remain
closer to the Moon’s orbital plane require more ΔV and target ωvalues near 165 deg
and 345 deg. These characteristics are also apparent in Fig. 420, which shows the
LOI ΔV and transfer duration state space maps for lunar transfers to this arrival time.
252
TRANSFERS TO LOW LUNAR ORBITS
Figure 420 The combinations of Ω and ω that yield simple lunar transfers, that is, those
without low Earth or lunar periapse passages. If multiple transfers exist for the same
combination, then the one with the least LOI ΔV is shown. All of these transfers arrive
c 2011 by American Astronautical Society Publications Ofﬁce,
at a full Moon [2] (Copyright ©
all rights reserved, reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
4.4.6
253
Monthly Trends
The work presented here describing transfers between the Earth and low lunar polar
orbits has been extended, surveying transfers that arrive at the Moon at eight points
in its orbit for several consecutive months. This section ﬁrst presents results from
surveys throughout one month and then considers similarities and variations that
exist in lunar transfers across multiple months. The goal is to be able to predict the
performance of lunar transfers for any given month.
Figure 421 shows eight state space maps, including those previously studied in
Figs. 413, 417, and 420. These maps include simple transfers that arrive at the
Moon at eight different points in a synodic month. Each map only tracks lunar
transfers with no close lunar ﬂybys or Earth phasing orbits, though each map does
include both direct and lowenergy transfers.
These maps are very useful to identify the combinations of Ω and ω that may
be accessed via direct or lowenergy transfers for a particular lunar arrival time.
Similarly, the collection of these maps may be used to identify when to perform the
lunar orbit insertion for a transfer to a particular combination of Ω and ω. One can see
that lowenergy transfers with LOI ΔV values below 700 m/s may be constructed that
arrive at the Moon at any time. One also observes strong symmetry in the state space
maps. First, each map shows a strong symmetrical mapping by shifting both Ω and ω
by ±180 deg. This shift corresponds to the difference between arriving at the Moon
over the North Pole and arriving at the Moon over the South Pole. Second, a strong
symmetry appears between two maps that correspond to arrivals ±180 deg apart
in the Moon’s orbit: the maps show very similar characteristics when their arrival
position and their Ωvalues are both shifted by ±180 deg. This shift corresponds
to the symmetry that exists in the Sun–Earth threebody system: the dynamics are
very similar, with a 180 deg rotation about the Earth, for the case where a spacecraft
traverses from the Earth toward the Sun and for the case where a spacecraft traverses
away from the Sun.
Figure 422 shows eight scatter plots, corresponding to the same arrival times
presented in Fig. 421. The plots illustrate the relationships between each transfer’s
duration and its lunar orbit insertion ΔV. One can clearly see that direct transfers
and lowenergy transfers exist at every arrival time: direct transfers are shown on the
far left of each plot, corresponding with short transfer durations and raised LOI ΔV
requirements; lowenergy transfers are similarly shown toward the bottomright of
each plot, corresponding with longer transfers and lower LOI ΔV requirements.
Intermediate transfers exist for some arrival times, with transfer durations on the
order of 60 days. One can see the same symmetry described above, between a given
plot and the one that corresponds to a lunar arrival ±180 deg apart. These plots are
useful to quickly identify the limits of transfer duration and LOI ΔV for each type of
transfer at any given lunar arrival time.
Most characteristics of ballistic twoburn lunar transfers repeat from one month
to the next. The Moon’s orbital plane is nearly coplanar to the Earth’s, and the orbits
of the bodies involved are nearly circular. However, since these conditions are not
perfectly met, the characteristics of these lunar transfers do vary from one month
TRANSFERS TO LOW LUNAR ORBITS
254
Figure 421 State space maps, illustrating the LOI ΔV required to transfer from the Earth to low polar lunar orbit at different arrival times in a month
c 2011 by American Astronautical Society Publications Ofﬁce, all rights reserved, reprinted with permission of the AAS).
[2] (Copyright ©
Figure 422 Scatter plots showing the relationship between transfer duration and a transfer’s LOI ΔV cost for simple lunar transfers arriving at the
Moon at different points about its orbit.
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
255
256
TRANSFERS TO LOW LUNAR ORBITS
Figure 423 State space maps that illustrate the LOI ΔV for transfers to each combination
of Ω and ω that arrive at the Moon at its ﬁrst quarter in each of six consecutive months [2]
c 2011 by American Astronautical Society Publications Ofﬁce, all rights reserved,
(Copyright ©
reprinted with permission of the AAS).
LOWENERGY TRANSFERS BETWEEN EARTH AND LOW LUNAR ORBIT
257
to the next. The Moon’s orbital plane and equatorial plane are tilted approximately
5.1 deg and 1.5 deg, respectively, relative to the ecliptic. One may therefore assume
that the characteristics observed in the state space maps presented here vary by
several degrees in their ωvalues in a given month. The inclination of the translunar
departure state for a given type of lunar transfer may vary by many degrees from one
month to the next, particularly on account of the obliquity of the Earth’s spin axis.
It has been found that most types of simple lunar transfers appear in any given
month and their characteristics remain relatively constant relative to the ecliptic.
Figure 423 illustrates how little the state space maps vary from one month to the
next, when evaluating simple lunar transfers. The six state space maps shown capture
the LOI ΔV for transfers to each combination of Ω and ω that arrive at the Moon at
its ﬁrst quarter in each of six consecutive months. The only major apparent variation
is that the features in each map shift approximately 30 deg in Ω from one month to
the next. This is because Ω is deﬁned inertially and the Earth moves approximately
30 deg in its orbit from one month to the next, rotating the Sun–Earth geometry. The
more complex lunar transfers, such as those with multiple lunar ﬂybys, vary much
more on a monthly basis and may not even appear at all in a given month.
4.4.7
Practical Considerations
The surveys presented here study trajectories that are entirely ballistic—they do
not contain any correction maneuvers or targeting maneuvers of any sort. When
propagated backward in time from the Moon, if a trajectory arrives at the Earth
without impacting the Moon, then it is considered a viable Earth–Moon transfer.
However, the trajectory may have arrived at the Earth with an inclination that is
unsuitable for a mission that launches from a particular launch site. Ideally, a
mission would start in a lowEarth parking orbit with an inclination very close to
that of the latitude of the launch site, for example, near 28.5 deg for missions that
launch from Cape Canaveral, Florida. It is undesirable to perform a large plane
change during launch and translunar injection. Section 6.5 shows that one can add
1–3 small trajectory correction maneuvers to depart the Earth from a particular LEO
parking orbit and transfer onto a desirable lowenergy transfer to the Moon; and
doing so requires only about 1 m/s per degree of inclination change. This works for
lowenergy transfers particularly well since lowenergy transfers travel far from the
Earth and spend many weeks doing so. This method does not work well for direct
lunar transfers, which require far more ΔV to change planes.
Midcourse maneuvers may also be implemented to establish a launch period
for a lowenergy transfer to the Moon, extending or shrinking its transfer duration.
Missions that implement direct lunar transfers may establish a launch period using
Earth phasing orbits, making those sorts of transfers more desirable in the surveys
presented here.
258
TRANSFERS TO LOW LUNAR ORBITS
4.4.8 Conclusions for LowEnergy Transfers Between Earth and Low
Lunar Orbit
The surveys presented in this section characterize twoburn lunar transfers that arrive
at the Moon, targeting 100km polar orbits with any orientation. Transfers are studied
that arrive at an example ﬁrstquarter Moon, an example full Moon, and an example
thirdquarter Moon. Additional results are also presented for transfers that arrive
at eight different times during a month and for several consecutive months. Many
types of transfers are observed, including lowenergy transfers, shortduration direct
transfers, and variations that involve any number of lunar ﬂybys and Earth phasing
orbits, provided that they do not involve any deterministic maneuvers. The only two
burns considered are the translunar injection maneuver and orbit insertion maneuver.
It has been found that lunar transfers consistently require translunar injection C3
values on the order of −2.0 km2 /s2 for direct transfers and −0.6 km2 /s2 for lowenergy transfers. Simple transfers typically require 2–12 days for direct transfers
and 70–120 days for lowenergy transfers, though both types can require more time.
The lowenergy transfers that require the least LOI ΔV require 640 m/s, or more
depending on the target orbit and the arrival time; direct lunar transfers require at
least 120 m/s more ΔV than lowenergy transfers to the same arrival conditions.
Further, lowenergy transfers can reach many arrival conditions that direct transfers
cannot reach without additional maneuvers. Practical simple direct transfers only
exist that target a lunar orbit that is within 60 deg of being orthogonal to the Earth–
Moon line, though the ΔV cost rises signiﬁcantly when the orbit is beyond 30 deg
of orthogonal. Lowenergy transfers can target polar orbits with any argument of
periapse, ω, or with any longitude of ascending node, Ω; targeting one such parameter
restricts the other for a particular arrival date as illustrated in the state space maps
presented here.
4.5 TRANSFERS BETWEEN LUNAR LIBRATION ORBITS AND LOW
LUNAR ORBITS
Many mission designs may beneﬁt by transferring a spacecraft from the Earth to a
lunar libration orbit prior to descending to a low lunar orbit. For instance, Hill et
al. [11], designed a mission where two satellites transferred to a halo orbit about
the lunar L2 point. One satellite remained there as a navigation and communication
relay and the other satellite transferred to a low lunar orbit. Information about such
transfers is summarized in Section 3.5.1 on page 224.
4.6 TRANSFERS BETWEEN LOW LUNAR ORBITS AND THE LUNAR
SURFACE
Many historical missions have performed maneuvers to transfer a spacecraft from a
low lunar orbit to the lunar surface, for example, the Apollo missions [1]. A few
spacecraft, including Apollo missions, have then risen from the lunar surface and
TRANSFERS BETWEEN LOW LUNAR ORBITS AND THE LUNAR SURFACE
259
returned to lunar orbit. These maneuvers are very straightforward and may even be
well approximated by conic sections; nevertheless, it is useful to brieﬂy describe their
designs here.
Let’s assume we have a spacecraft in a 100km circular lunar orbit. That spacecraft
is traveling approximately 1633.5 m/s in its orbit and revolves about the Moon once
every 117.8 minutes. The minimum ΔV required to place the spacecraft on a collision
course with the Moon would reduce the spacecraft’s orbital periapse to an altitude of
0 km, at which point it would just graze the surface, that is, a Hohmann transfer. This
transfer requires a ΔV of approximately 23 m/s, sending the spacecraft on a 180deg
transfer in about 56.5 minutes. The spacecraft’s grazing velocity upon arriving at
its orbital periapse is approximately 1703.2 m/s. If the spacecraft performs a larger
braking burn from its 100 km orbit, then its transfer orbit will strike the surface of
the Moon at a steeper ﬂight path angle in less time.
Figure 424 illustrates the ﬂight path angles that may be achieved at the mean
radius of the Moon as a function of the deorbit burn ΔV for trajectories starting from
an altitude of 100 km. One can see that a ΔV of 23 m/s is indeed required to obtain a
ﬂight path angle of 0 deg, which is the limit of trajectories that have a passive abort
option, not including local geometry variations. Of course, by performing a braking
burn ΔV of 1633.5 m/s, the spacecraft completely removes its orbital velocity and
falls straight down to the surface, achieving a vertical impact.
Figure 425 illustrates the velocities that the spacecraft will have at the impact
point, assuming the impact point occurs at a radius of 1737.4 km, for example, the
mean radius of the Moon. Figure 426 shows the duration of time required to reach
the impact point.
260
TRANSFERS TO LOW LUNAR ORBITS
Figure 424 The ﬂight path angles that may be achieved at the mean surface of the Moon as
a function of the deorbit burn ΔV for trajectories starting from a circular orbit at an altitude
of 100 km.
TRANSFERS BETWEEN LOW LUNAR ORBITS AND THE LUNAR SURFACE
261
Figure 425 The impact velocity values that may be achieved at the mean surface of the
Moon as a function of the deorbit burn ΔV for trajectories starting from a circular orbit at an
altitude of 100 km.
262
TRANSFERS TO LOW LUNAR ORBITS
Figure 426 The duration of time required to reach the mean surface of the Moon as a
function of the deorbit burn ΔV for trajectories starting from a circular orbit at an altitude of
100 km.
CHAPTER 5
TRANSFERS TO THE LUNAR SURFACE
5.1
EXECUTIVE SUMMARY
In this chapter techniques are developed that allow an analysis of a range of different
types of transfer trajectories from the Earth to the lunar surface. Trajectories ranging
from those obtained using the invariant manifolds of unstable orbits to those derived
from collision orbits are analyzed. These techniques allow the computation of
trajectories encompassing lowenergy trajectories as well as more direct transfers. A
conceptual illustration of the types of trajectories discussed in this chapter is given in
Fig. 51. The range of possible trajectory options is summarized, and a broad range
of trajectories that exist as a result of the Sun’s inﬂuence are computed and analyzed.
The results are classiﬁed by type, and trades between different measures of cost are
discussed. The information in this chapter is largely derived from papers presented
by Anderson and Parker [192–195], and the results as presented here are oriented as
a guide for mission design.
The problem of designing transfers to the lunar surface is approached here as fol
lows. First, an analysis is given showing the types of trajectories that exist as a result
of the Sun’s inﬂuence in both the planar and spatial problems. A signiﬁcant set of
trajectories at high Jacobi constants, or low energies, is found to exist when the Sun’s
263
264
TRANSFERS TO THE LUNAR SURFACE
Figure 51 The proﬁles for both a direct and a lowenergy transfer from the Earth to the
lunar surface.
inﬂuence is taken into account. This result indicates that for trajectory design in this
energy regime, trajectories traveling to the Sun–Earth Lagrange points and following
the invariant manifolds of orbits around these points deserve careful consideration.
Monthly variations are examined, and it is determined that the monthly variations
capture the majority of the variations seen in the studied transfer trajectories. The
greatest variation over a month occurs between the cases traveling to the Moon when
it is at its apoapse and periapse. These trajectories are described for the spatial case
initially with trajectories normal to the surface, which illustrates in a succinct man
ner the types of options available. A more detailed analysis of trajectories arriving
at various angles to the surface is also presented, and these trajectory options are
summarized using several different plots of various parameters. These trajectories
may serve as initial guesses for future mission design, and they provide a general
overview of the range of trajectory options. Invariant manifold trajectories traveling
to the lunar surface are also described, and some sample trajectory options traveling
from libration orbits to the lunar surface are given.
The numerical results presented in this chapter are given primarily relative to the
Jacobi constant (C) of trajectories encountering the lunar surface. The velocity of the
trajectories varies little over the surface of the Moon for each Jacobi constant, and in
each case an approximate value of the velocity corresponding to each Jacobi constant
may be obtained by referring to Fig. 52. From this plot, a Jacobi constant of 2.5 gives
INTRODUCTION FOR TRANSFERS TO THE LUNAR SURFACE
265
Figure 52 Mean inertial velocities relative to the Moon at each Jacobi constant for the cases
c 2011 by American Astronautical
with velocities normal to the surface [192] (Copyright ©
Society Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all
rights reserved; reprinted with permission of the AAS).
a mean velocity at the Moon of approximately 2473.3 meters per second (m/s), while
a Jacobi constant of 3.0 gives a mean velocity of 2365.0 m/s. In the planar analysis it
is shown that trajectories from the Earth to the Moon exist without the Sun’s inﬂuence
up to a Jacobi constant of 2.78, while they exist up to a Jacobi constant of 3.16 when
the Sun’s inﬂuence is included in the computations. As expected, the time of ﬂight
of the trajectories generally increased and the launch injection energy parameter
(C3 ) generally decreased as the Jacobi constant increased. For the spatial case with
trajectories encountering the Moon perpendicular to the surface, the minimum time
of ﬂight varied from approximately 3.4 days at C = 2.2 to around 101.0 days and
78.7 days at C = 3.0 and 3.1, respectively. The range of possible elevation angles
that generate missions to the surface is very dependent on the Jacobi constant and
the target location on the surface. For the selected grid size, the maximum elevation
angle range for points on the surface changed from 0 deg to 90 deg at C = 2.6 to
between 57 deg and 90 deg at C = 3.1. Likewise the minimum elevation angle
range decreased from 0 deg to 72 deg at C = 2.6 to between 0 deg and 15 deg at
C = 3.1. These numbers are given to present a rough idea of the kinds of results that
are discussed in this chapter. The details of these cases along with a wide variety of
launch and approach parameters are contained in the plots presented throughout the
chapter.
5.2 INTRODUCTION FOR TRANSFERS TO THE LUNAR SURFACE
A wide variety of Earth–Moon trajectories have been employed for past missions,
ranging from the more direct transfers used for the Apollo missions [196] to more
recent missions such as ARTEMIS [4] that make use of the multibody dynamics
266
TRANSFERS TO THE LUNAR SURFACE
of the Earth–Moon and Sun–Earth–Moon systems. The design of trajectories in
multibody systems is a particularly rich problem because the twobody model is
often insufﬁcient to compute accurate trajectories, and the gravity of the Sun, Earth,
and Moon combine to form a highly nonlinear dynamical environment. These facts
limit the applicability of traditional patchedconic techniques commonly used for
interplanetary missions, and the threedimensional aspects of the problem further
complicate realworld missions. Mission designers must take into account the orien
tation of each body in addition to the relative orientations of the orbits of the Earth
and the Moon over time. Parker et al. [44, 47, 183] have studied trajectories that
include many of these complicating factors for insertion into a variety of orbit types
near the Moon. This chapter focuses on an analysis including these types of effects
with a focus on trajectories traveling to the lunar surface.
Lunar landing trajectories often have a different set of constraints from those of
orbiters, and the nature of this problem makes it possible to approach it with a different
set of techniques. Indeed, a theoretical basis for analyzing lunar landing trajectories
may be found in the computation of collision orbits. Collision orbits have been
studied extensively in the mathematical community by Easton [197] and McGehee
[198]. Anderson and Lo [199], Villac and Scheeres [200], and Von Kirchbach et
al. [201] have previously analyzed collision orbit trajectories for the Jupiter–Europa
system and categorized the different regions and trajectories that exist for orbits that
terminate or originate at Europa’s surface. While the theoretical basis for collision
orbits is focused on trajectories that intersect the surface of the selected body normal
to the surface, this type of analysis can be extended to trajectories coming in at
the various ﬂight path angles and declinations of interest to mission designers. A
study of these trajectories is almost directly applicable to impactor missions such as
the Lunar Crater Observation and Sensing Satellite (LCROSS) [202]. This mission
used an 86 deg impact angle relative to the lunar surface for the impact trajectory.
The techniques developed here are also easily applied to systems including the full
ephemeris and multiple bodies. Much of the work to design lowenergy trajectories
from the Earth to the Moon has focused on the use of libration point orbits along with
their stable and unstable manifolds [39, 45, 51, 203]. These techniques have proven
to be quite successful, and they are increasingly used for the design of Earth–Moon
trajectories. The invariant manifolds of libration orbit trajectories are also studied
here with an emphasis on their applicability to landing trajectories.
A wide range of trajectory types for lunar landing trajectories were computed for
the results given here, and presenting a complete picture of the possible trajectory
categories while remaining easily accessible was a goal of this research. In keeping
with this goal, the problem is approached and presented using several different levels
of analysis with increasing complexity. Presenting all different combinations of
velocities encountering the surface of the Moon with all different magnitudes and
orientations makes it difﬁcult to see the relevant structures in the solution space, so two
divisions were made in the approach to the analysis. The problem is ﬁrst approached
by analyzing planar cases covering selected velocities or energies with the trajectories
encountering the Moon at various angles relative to the surface. The characteristics
of these trajectories are observed to lay the groundwork for understanding the spatial
METHODOLOGY
267
trajectory cases. The spatial case is then attacked using the results from the planar
analysis to understand the dynamics in this more complex model. These trajectories
are categorized with the goal of providing a broad survey of the trajectory types
that may be available for transfers to the lunar surface from the Earth. The speciﬁc
trades between launch costs and time of ﬂight (TOF) are quantiﬁed and summarized
in addition to the topological characteristics of the trajectories. Other parameters
relevant to mission design such as the launch orientation are computed. The regions
of the Moon attainable using different types of trajectories are also characterized.
These results are summarized with the goal of providing a tool for mission designers
to quickly understand the trades between various measures of cost and time when a
particular mission is being designed to land on the Moon.
The results in this chapter are also presented using two key concepts. The ﬁrst
is to view the problem in terms of the limiting bounds that a mission designer
could use to reﬁne the search space. This practically takes the form of computing
parameters such as the velocities for which trajectories exist that travel from the
Earth to the Moon or the launch energies required to reach such trajectories. The
second, which is an overall theme of this work and one of the primary results, is
related to computationally examining in a more comprehensive sense the trajectories
available when the Sun’s perturbations are taken into account. To achieve this
objective, comparisons involving several different models are made. Many traditional
trajectories were computed using the Earth–Moon model or the circular restricted
threebody problem (CRTBP), and in general, similar types of trajectories exist in
the full ephemeris model. If simpler models are used, however, some solutions in
the full problem may be ignored. Some particular solutions employing the effects of
the Sun for transfers in the Earth–Moon system have been examined more recently.
The 1991 Japanese mission MUSESA (Hiten) used the effects of the Earth, Moon,
and Sun for its transfer to the Moon [30]. Koon et al. provided techniques for
systematically reproducing missions similar to Hiten using the invariant manifolds
of libration orbits [38]. In each of these techniques the Sun’s effects were included in
the mission design. Parker and Lo examined trajectories within the Sun–Earth–Moon
spatial problem and looked at multiple trajectories for transfer to lunar halo orbits
[39, 46]. The work here seeks to broaden the search space for landing trajectories
traveling to the Moon and characterize the effects of the fourthbody perturbations
of the Sun on the potential trajectories that may be used. A direct approach isolating
the effects of the Sun is taken here by comparing trajectories in the CRTBP, the
Earth–Moon system, and the Sun–Earth–Moon system.
5.3
METHODOLOGY
Two primary models are used for the analyses contained in this chapter. The ﬁrst
model, the CRTBP, closely approximates real world systems, and a signiﬁcant set of
tools exists within this model to bring to bear on the problem. The qualitative insights
gained in this model are very helpful in providing an overview of the categories of
trajectories that are available. The trajectories developed within the CRTBP are
268
TRANSFERS TO THE LUNAR SURFACE
also generally transferable to the full ephemeris although trajectories developed with
the effects of other bodies may not be transferable to the CRTBP. Refer back to
Section 2.5.1 for a more complete description of the CRTBP. The ephemeris model,
implemented using point masses, is used to capture additional types of trajectories
that are not found using the CRTBP model. Although the variations in the orbits of
the Earth and Moon are important, this model is primarily used to search for members
of the broad category of trajectories utilizing the Sun’s perturbation for Earth–Moon
transfers. See Section 2.5.3 for more details on the use of the ephemeris.
5.4 ANALYSIS OF PLANAR TRANSFERS BETWEEN THE EARTH AND
THE LUNAR SURFACE
The procedure described next involves varying the location of the landing site on
the Moon, the orientation of the incoming trajectory, and the energy/velocity of
the trajectory. Each trajectory must also be characterized or evaluated using some
ﬁgure of merit. While this can provide a relatively complete picture of the potential
trajectory options, it is helpful to ﬁrst gain insight into the dynamics by limiting the
scope of the problem to allow the results to be easily visualized.
Several different techniques have been used to achieve this goal in the Jupiter–
Europa system, and it is useful to consider their application here. One technique used
by Anderson and Lo [199] varied the Jacobi constant for trajectories intersecting
Europa on a sphere for several different trajectory orientations and characterized the
origin of the trajectories. Von Kirchbach et al. [201] examined the planar case for the
Jupiter–Europa system for additional velocity orientations leaving the surface. Both
of these techniques are applied here to the Earth–Moon system, and it is interesting
to start with the planar problem in order to gain some initial insight. First, the planar
results are computed in the Earth–Moon CRTBP system to allow for a comparison
with the results from Von Kirchbach et al. [201] in the Jupiter–Europa system. This
technique is then used to extend the analysis to the ephemeris case with the Earth and
Moon and then to the case where the Sun is included. The effect of adding the Sun
is examined in detail over a range of Jacobi constants.
For this planar analysis, a set of trajectories was integrated backward in time from
the surface of the Moon. Specifying the Jacobi constant gives the velocity magnitude
for each trajectory, while the location of the trajectory and the orientation of the
velocity are speciﬁed using α and θ as shown in Fig. 53. Multiple simulations have
been performed using these techniques, and the results for several selected Jacobi
constant values are given in Fig. 54. The resulting points are colored according to
the original location of each trajectory. Note that if a trajectory integrated backward
in time were to intersect the Moon and then encounter the Earth at an earlier time,
the trajectory would be gray. The (α, θ) point corresponding to the intermediate
intersection of the Moon would then be blue. The fact that points with only slightly
different initial conditions in the plot can travel to either the Earth or the Moon
conﬁrms the known existence of chaos in this problem. Comparison with the results
from the Jupiter–Europa system in Von Kirchbach et al. [201] reveals that the divid
ANALYSIS OF PLANAR TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
269
Figure 53 Diagram showing location and orientation of the velocity vector as it intersects
the lunar surface. The xy axes shown here are centered on the Moon in the same orientation
c 2011 by American Astronautical Society
as the axes in the rotating frame [192] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
ing lines between different regimes of motion are less distinct at equivalent Jacobi
constants for the Earth–Moon system. This existence of chaos indicates that it may
be possible to design trajectories that cover a relatively wide range of the surface
by carefully selecting landing sites. It is useful to note here again that Moon–Earth
transfers may be derived from Earth–Moon transfers, and the plot corresponding to
these trajectories may be obtained from Fig. 54 for the planar problem using x → x,
y → −y, x˙ → −x˙ , and y˙ → y˙. The transformation in position gives α → 2π − α
and then from examination of the transformed velocity vector, θ → −θ.
As can also be seen from the results, a signiﬁcant percentage of the trajectories do
not encounter either the primary or the secondary over the given time span. However,
it is useful to note that for low Jacobi constants (higher energies), a signiﬁcant
percentage of the trajectories do originate at the Earth. Determining the Jacobi
constant where Earth–Moon transfer trajectories no longer exist in the planar problem
can help provide a rough limit on energies or velocities for these trajectories and
provide a method for determining the potential beneﬁts of perturbations from other
bodies in trajectory design. To determine the approximate Jacobi constant above
which Earth–Moon trajectories computed in this simulation no longer exist, a series
of runs were made in parallel to step through the Jacobi constant. The grid resolution
used for this step was one degree in both α and θ. The percent of the total number
of trajectories that encountered the Earth for each Jacobi constant was computed and
270
TRANSFERS TO THE LUNAR SURFACE
Figure 54 Plots showing the origin of each trajectory as a function of the position and
orientation of the velocity vectors as the trajectories encounter the Moon’s surface. Blue
points indicate that the trajectory originated at the Earth and gray that it originated at the
Moon. If no point is plotted the integrated trajectory did not encounter the surface of either
c 2011
body over the given time span of 200 days (Earth–Moon CRTBP) [192] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS). (See
insert for color representation of this ﬁgure.)
plotted in Fig. 55. As expected from the previous plots, the number of trajectories
originating at the Earth generally decreases with increasing Jacobi constant, but it is
interesting that the slope of the curve varies signiﬁcantly over the plotted range. It is
also interesting that although the curve approaches zero percent near a Jacobi constant
of 2.7, for Jacobi constants as high as 2.78, the percent of trajectories originating at
the Earth remains at approximately 0.03 percent or approximately 19 out of 64800
trajectories. So even for this relatively low energy, some trajectories manage to travel
from the Earth to the Moon.
For mission design, it is helpful to be aware of the velocities of the trajectories as
they intersect the surface. They will actually vary somewhat as the constant for the
computations so far has been the Jacobi constant rather than velocity. In general the
inertial velocities relative to the Moon only vary at the m/s level. Figure 52 shows
the average velocities for the case with velocities normal to the surface as a reference
for each Jacobi constant.
ANALYSIS OF PLANAR TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
271
Figure 55 Percent of trajectories originating at the Earth for each Jacobi constant (CRTBP)
c 2011 by American Astronautical Society Publications Ofﬁce, San Diego,
[192] (Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
Although the CRTBP is known to provide an accurate approximation to realworld
trajectories, an obvious question for mission designers is related to how much the
inclusion of realworld effects would affect selected trajectories. This question can
be addressed by using planetary ephemerides and replicating the analysis for the
CRTBP in this model. This analysis was ﬁrst performed in the ephemeris model
initially including only the gravity of the Earth and Moon. The initial velocities were
computed for a given Jacobi constant in the CRTBP in the rotating frame, and then
the states were initialized in the integrator relative to the Moon in an instantaneous
rotating frame aligned with the Earth–Moon frame on an epoch of January 1, 2015.
As the distance between the Earth and the Moon varies over the course of the orbit, it is
difﬁcult to obtain a direct comparison to the results from the CRTBP, but this method
was selected because it was found to provide a good approximate comparison using
the important mission design parameter of velocity at the lunar surface. Although
the Jacobi constant will vary along the trajectory in this model, the ﬁnal impact
velocities at the Moon will be the same in each system. So the Jacobi constant
labels in the ephemeris model plots in this study serve to indicate the velocities that
were used at the lunar surface as they were computed in the CRTBP. The initial
conditions were originally planar for this case, but the trajectory was free to vary in
three dimensions for this problem. Using this method for a system including the Earth
and Moon ephemerides, the trajectories were integrated, and the results are plotted
in Fig. 56. Comparing the results for this system with the results in Fig. 54 reveals
few obvious differences. The Earth impacting cases for C = 2.6 have some slight
differences, but in general the trajectories match the expectation that the CRTBP is
a good approximation to the threebody problem including the ephemerides. If the
272
TRANSFERS TO THE LUNAR SURFACE
Figure 56 Plots showing the origin of each trajectory as a function of the position and
orientation of the velocity vectors as the trajectories encounter the Moon. Blue points
indicate that the trajectory originated at the Earth and gray that it originated at the Moon.
If no point is plotted the integrated trajectory did not encounter either body over the given
c 2011
time span of 200 days (Earth–Moon only Ephemeris system) [192] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS). (See
insert for color representation of this ﬁgure.)
percent of trajectories originating at the Earth are compared, some differences in the
shape of the curve can be found, but the overall trends are very similar. In this case,
the percent of trajectories in Fig. 57 originating at the Earth decreases down to 0.006
percent at C = 2.76, approximately the same Jacobi constant cutoff as the CRTBP.
Next, the same procedure was performed including the Sun in the integration,
and the results are plotted in Fig. 58. Now, comparison with the results in both
the CRTBP and the Earth–Moon systems reveal some obvious differences. Several
new bands of trajectories originating at the Earth spring into existence. The overall
structure remains generally similar, but the points appear chaotic. A new band of
solutions remains for C = 2.8 and a signiﬁcant number of Earth origin trajectories
still exist at C = 3.0. Remember that the ﬁnal velocities at the Moon are the same as
the other models, but the Jacobi constant will vary as a result of the Sun’s inﬂuence.
In this sense, the Sun may be thought of as changing the trajectory’s energy or Jacobi
constant to provide the transfer. If the percent of trajectories originating at the Earth
ANALYSIS OF PLANAR TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
273
Figure 57 Percent of trajectories originating at the Earth for each Jacobi constant (Earth–
c 2011 by American Astronautical Society
Moon only Ephemeris system) [192] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
is examined in Fig. 59(a), it can be seen that at C = 3.0, 0.3 percent of the trajectories
still originate at the Earth. Indeed, as high as C = 3.16, 0.15 percent of the trajectories
still originate at Earth.
One immediate question that arises is whether the selected epoch for lunar arrival
would signiﬁcantly affect these results, so three additional cases seven days apart were
computed and plotted in Fig. 59(b). Some variation is observed as the Moon travels
through its orbit with one case starting with a lower percent of trajectories for low
Jacobi constants and two of them possessing peaks just before C = 3.0. However, all
of them have approximately the same upper Jacobi constant cutoff of approximately
C = 3.16 where the percent of trajectories drops to near zero. The existence of
the additional bands of trajectories and the increase in the Jacobi constant where
trajectories connecting the Earth and Moon exist in this system raises the question
as to where these trajectories come from. These trajectories were plotted in both the
Earth–Moon system and the Sun–Earth–Moon system to examine the differences,
and a sample of one of these trajectories plotted in both rotating frames is given in
Fig. 510. As can be seen from the plots, the trajectory ends up in very different
places depending on whether the Sun is included in the integration or not. In the
Earth–Moon rotating frame with the Sun included, the trajectory travels far away
from the system with no close periapses until it approaches the Earth, while the case
without the Sun has two relatively close periapses at approximately the lunar distance
and ends up far from the Earth. The most telling plots, however, are in the Sun–
Earth rotating frame. Here, the characteristic shape of a trajectory using the libration
point dynamics of the Sun–Earth system is apparent when the Sun is included. The
trajectory travels out toward the L1 point, lingers there, and then ﬁnally falls back
274
TRANSFERS TO THE LUNAR SURFACE
Figure 58 Plots showing the origin of each trajectory as a function of the position and
orientation of the velocity vectors as the trajectories encounter the Moon. Blue points
indicate that the trajectory originated at the Earth and gray that it originated at the Moon.
If no point is plotted the integrated trajectory did not encounter either body over the given
c 2011
time span of 200 days (Sun–Earth–Moon Ephemeris system) [192] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS). (See
insert for color representation of this ﬁgure.)
toward the Earth. Without the Sun, the trajectory stays out near the Moon until it
eventually wanders further away from the system, unless there is a lunar ﬂyby.
ANALYSIS OF PLANAR TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
275
Figure 59 Percent of trajectories originating at the Earth for each Jacobi constant (Sun–
c 2011 by American Astronautical Society
Earth–Moon Ephemeris system) [192] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
276
TRANSFERS TO THE LUNAR SURFACE
Figure 510 Comparison of a single trajectory at C = 2.8 (α ≈ 197.5 deg, θ ≈ 9.5 deg)
integrated with and without the Sun’s gravity in different rotating frames [192] (Copyright
c 2011 by American Astronautical Society Publications Ofﬁce, San Diego, California (Web
©
Site: http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
277
5.5 LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND
THE LUNAR SURFACE
5.5.1
Trajectories Normal to the Surface
While the planar cases discussed up to this point are quite complicated, it is still
possible to plot many of the salient features of the design space given the relatively
small dimension of the problem. With the increase in dimension that occurs for the
spatial problem, the visualization of the resulting trajectories and their characteristics
becomes an even more difﬁcult issue. One of the stated objectives of this analysis is
to capture the characteristics of the major trajectory categories while also providing
adequate information to evaluate the usefulness of each trajectory. With this objective
in mind it is worth noting that if the plots in Figs. 54, 56, and 58 are examined, it can
be seen that the majority of the dominant types of trajectories seen in the ﬁgures may
be captured by making a particular cut at θ = 0. The trajectories obtained with θ = 0
correspond to those trajectories impacting the Moon normal to the lunar surface.
As previously mentioned, these types of trajectories are particularly applicable to
impactor missions similar to LCROSS. Given the results from the planar case, they
can also provide a good initial overview of the different categories of Earth–Moon
landing trajectories, including those with different ﬂight path angles. The results
presented next are restricted to those computed using impacts normal to the surface
for an epoch of January 1, 2015. They provide accurate results for impactortype
trajectories, and they also give a good indication of the types of trajectories that may
exist for trajectories coming in at other ﬂight path angles.
To allow for easy visualization of the trajectory characteristics, the results are
presented for each energy level, which corresponds to a slightly varying velocity
magnitude relative to the Moon that depends on the location of the ﬁnal point on the
trajectory at the Moon’s surface. The velocity can be used to provide an indication
of the change in velocity (ΔV) required for landing, although the speciﬁc ΔV
will depend on the particular landing trajectory. The regions of the Moon that
are accessible for each energy level can be evaluated for particular mission design
requirements by using the desired parameters plotted over the surface of the Moon
in α and β. α and β are measured in the rotating frame with α positive in the same
direction as shown in Fig. 53. β is measured like latitude and is positive above the
xy plane. Understanding how to connect the trajectory to the Earth becomes more of
a challenge in the spatial problem because a large number of possible Earthrelative
orientations and methods of injection onto the translunar trajectory are possible.
For this reason, a speciﬁc set of trajectory characteristics was selected for plotting.
The procedure in each case was to begin with the ﬁnal point on the trajectory with
a velocity normal to the lunar surface at the given α and β. The trajectory was then
propagated backward in time until it either encountered the Earth or the Moon, or
the trajectory duration reached 200 days. For those trajectories not encountering the
Earth or the Moon in this time period, a search was then made for the periapse closest
to the Earth. Several quantities were then computed using the point at encounter
or periapse. They included the periapse radius relative to the Earth, the TOF, the
278
TRANSFERS TO THE LUNAR SURFACE
launch energy (C3 ), and inclination in the Earth Mean Equator and Equinox of J2000
(EME2000) frame.
Results showing the origin of each trajectory encountering the Moon are given
in Fig. 511. For these cases threedimensional effects are included, and it is now
possible for the trajectory to miss encounters with the Earth and Moon by traveling
above or below them. A signiﬁcant number of encounters are still observed though,
and the features seen for the θ = 0 cases in the planar model may still be observed
here where β is 0 deg. Although a signiﬁcant number of Earthorigin trajectories
are observed for low Jacobi constants, as the Jacobi constant increases (energy
decreases), the number of Earth–Moon transfers decreases. Once a Jacobi constant
of 2.8 is reached, there are no more of these types of trajectories in the Earth–Moon
ephemeris model. However, there are still a signiﬁcant number of Earth–Moon
trajectories in the Sun–Earth–Moon ephemeris model. Indeed, a signiﬁcant number
still exist as the Jacobi constant is increased, even above 3.1. As in the planar case,
the Sun may be thought of as changing the energy or Jacobi constant of the trajectory
while the velocities at the Moon remain the same in each model. This observation
emphasizes the need to include the Sun’s inﬂuence in the trajectory design process,
but it raises the question as to what types of Earth–Moon trajectories exist at these
energies and how long are their times of ﬂight? It is difﬁcult to answer these questions
completely since trajectories are constantly changing with energy, but it is interesting
to observe some of the trajectories that exist in the Sun–Earth–Moon system with no
corollary in the Earth–Moon system. Two sample trajectories from the line of Earth
origin trajectories at C = 2.6 that do not exist in the Earth–Moon system are given
in Fig. 512. The majority of cases found in this line are similar to the trajectory in
Fig. 512(a), and they exhibit the characteristics of known trajectories designed to
utilize the dynamics of the invariant manifolds of libration orbits. They approach the
L1 Lagrange point from the Earth in the Sun–Earth system and then fall away toward
the Moon. Although almost all of the trajectories follow this type of orbit, some do
have characteristics similar to the trajectory in Fig. 512(b). In this case, the Sun’s
gravity is still inﬂuential, but an intermediate ﬂyby is inserted.
It is also interesting to observe the types of trajectories that exist for higher Jacobi
constants, or lower velocities, at the lunar surface in the Sun–Earth–Moon system
where no analogues in the Earth–Moon system have been found. Several samples
are shown along with the trajectory origin plots in Fig. 513 to provide an overview
of these types of trajectories. Here, an interesting phenomenon occurs. As the Jacobi
constant increases to 3.0, the trajectories originating at the Earth are scattered across
the map. The majority of the Earthorigin trajectories seem to require multiple ﬂybys
of the Earth or the Moon. The sample trajectories shown in Fig. 513(a) are intended
to be representative of the types of trajectories found across the map. Although a few
trajectories, such as those found in the lower left corner of the map, utilize the libration
dynamics more directly, the majority seem to require variations on different phasing
ﬂybys as shown by the various trajectories. As the Jacobi constant is increased further
to a value of 3.1 in Fig. 513(b), a line of trajectories appear. These trajectories, as
shown in the ﬁgure, once again utilize the libration orbit dynamics more directly,
sometimes making use of a single ﬂyby along the trajectory. The remaining scattered
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
279
Figure 511 Plots showing the origin of the spatial collision trajectories. Black indicates
the trajectory originated at the Earth, and gray indicates it originated at the Moon. If it is
white, no encounter occurred within 200 days (for epoch of January 1, 2015) [192] (Copyright
c 2011 by American Astronautical Society Publications Ofﬁce, San Diego, California (Web
©
Site: http://www.univelt.com), all rights reserved; reprinted with permission of the AAS, 2006)
280
TRANSFERS TO THE LUNAR SURFACE
Figure 512 Sample trajectories at C = 2.6 for the Sun–Earth–Moon system. The
trajectories correspond to the line of trajectories not found in the Earth–Moon plots [192]
c 2011 by American Astronautical Society Publications Ofﬁce, San Diego,
(Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
281
Figure 513 Plots showing the origin of the spatial collision trajectories. Black indicates
the trajectory originated at the Earth, and gray indicates it originated at the Moon. If it is
white, no impact occurred within 200 days. Trajectories are shown for select points in the
Earthcentered Sun–Earth rotating frame. The gray circular orbit is the Moon’s orbit while
the Sun is in the indicated direction. The scale is the same for all trajectories shown, and
c 2011 by American Astronautical
the trajectories all originate at the Earth [192] (Copyright ©
Society Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all
rights reserved; reprinted with permission of the AAS).
trajectories found near the center of the plot continue to use multiple gravity ﬂybys
to connect the Earth and the Moon.
Another interesting characteristic to include in the analysis is the TOF required
for each trajectory originating at the Earth. More speciﬁcally, what are the minimum
TOF values that may be achieved at each energy? The TOF values provide an
indication of whether the trajectories at each Jacobi constant fall more in the category
of direct transfers, lowenergy transfers, or somewhere in between. The existence
282
TRANSFERS TO THE LUNAR SURFACE
of trajectories in the Sun–Earth–Moon system that do not exist in the Earth–Moon
system already indicates the presence of trajectories utilizing multibody effects that
would be expected to fall more in the lowenergy category. The minimum TOF values
for selected Jacobi constants are listed in Table 51. These values were computed
using a grid with the points spaced at onedegree intervals in each variable. As
expected, the TOFs start near the 3day values seen for the Apollo program’s direct
transfers for a Jacobi constant of 2.2, and climb to over 100 days for a Jacobi constant
of 3.0. It is surprising though that the minimum TOF at a Jacobi constant of 3.1 drops
to 78.7 days. Although this point is lower than most of the others at this energy,
a number of trajectories still exist in the 90day time range. The reasons behind
this drop in the TOF may be more clearly understood by reexamining the typical
trajectories plotted for the Jacobi constants of 3.0 and 3.1 in Fig. 513. As mentioned
previously, the majority of the trajectories computed for the Jacobi constant of 3.0
required multiple phasing ﬂybys, while the C = 3.1 trajectories typically utilize the
libration dynamics without these phasing loops. This phenomenon would explain the
lower minimum TOF value at C = 3.1, since many of the trajectories at this energy
actually use a more direct approach.
The analysis so far has focused on categories of trajectories originating at the
Earth, with the expectation that trajectories from a given category may often be
modiﬁed to meet the particular requirements of a mission when they are supplied.
Often, however, trajectories that originate within some distance of the Earth may be
used by targeting them from low Earth orbit. It is also important to quantify the
orbital parameters of the initial conditions of the analyzed trajectories relative to the
Earth in order to determine the suitability of the trajectories for particular missions.
For example, if a launch from Cape Canaveral is selected, an inclination relative to
the Earth’s pole of 28.5 deg would be desirable. Particular quantities relevant to
mission design are presented next with the objective of presenting an overview of the
possible trajectories so that initial estimates may be made for future mission design.
Table 51 Minimum TOF values from the computed trajectories originating at the
Earth for selected Jacobi constants.
C
2.2
2.4
2.6
2.7
2.8
2.9
3.0
3.1
TOF (days)
3.4
29.8
58.3
57.8
74.0
94.9
101.0
78.7
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
283
The analysis here focuses on the Sun–Earth–Moon system so as to encompass the
complete range of trajectories.
The closest periapse values obtained over 200 days for selected Jacobi constants
in the Sun–Earth–Moon system are plotted in Fig. 514. Note that some of the gaps in
(a) are Earth intersection trajectories as can be seen be reexamining Fig. 513. It can
Figure 514 Periapse radius values for the computed trajectories plotted over the surface
c 2011 by American Astronautical
for a range of Jacobi constant values [192] (Copyright ©
Society Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all
rights reserved; reprinted with permission of the AAS).
284
TRANSFERS TO THE LUNAR SURFACE
immediately be seen from the plots that the majority of the trajectories never come
near the Earth. In general, the trajectories originating near α = 90 deg produce the
most trajectories with periapses closer to Earth. This does shift with Jacobi constant,
as was seen in the earlier Europa study [199]. As the Jacobi constant increases and
energy decreases, it appears that fewer trajectories come as close to the Earth, but
the majority stay near the system. The chaos present in the system can especially be
observed for C = 3.0, where trajectories very close to each other alternate with low
and high periapses.
From the analysis so far, it appears that a large portion of the lunar surface may
be physically accessible to trajectories coming from the Earth or near the Earth, but
the feasibility of ﬂying these trajectories will depend on mission design parameters
such as TOF, launch energy (C3 ) at Earth, and inclination. It is uncertain what twobody orbital element parameters (such as C3 and inclination) mean when they are
computed where multibody perturbations are signiﬁcant, but this problem may be
alleviated by computing these parameters where multibody effects are minimized.
With this objective in mind, only parameters for trajectories with periapses lower
than geosynchronous radius are plotted in the following ﬁgures. For these plots,
the parameters are now included for those trajectories originating at the Earth, and
in those cases their values are computed using the initial conditions at the Earth’s
surface.
The TOFs and C3 values are plotted in Fig. 515 for those trajectories with periapsis
relative to the Earth of less than geosynchronous radius. The immediate feature that
can be noticed is the sparsity of points compared to the previous plots, conﬁrming that
a large number of trajectories ending at the lunar surface never come near the Earth.
Indeed, for lower Jacobi constants, the locations between approximately 180 deg and
360 deg have almost no trajectories originating near the Earth. Curiously, around
a Jacobi constant of 3.0, the trajectories are more randomly distributed across the
surface with a combination of C3 values. This feature may be partly explained by
returning to the TOF values. From the plots, it can be conﬁrmed that the minimum
TOFs generally increase with Jacobi constant. The minimum TOF values at C = 3.0
are signiﬁcantly larger, indicating that lowenergy trajectories under the inﬂuence of
chaos are beginning to be more common. Given the variety of trajectory types and
the TOFs involved, it is not surprising that more of the lunar surface is potentially
covered. Examining the trends in the TOF plots, it may also be observed that
longer TOF trajectories appear to exist at each energy level. The lines of long TOF
trajectories correspond to lowenergy trajectories using the Sun’s perturbations and
approaching the libration points of the Sun–Earth system. It is also worth noting that a
variety of C3 options are available at each energy level for transfers to the Moon. Even
for low Jacobi constants, there still exist some relatively low C3 options, although
the minimum is higher than that found for the higher Jacobi constant cases. It is
important to realize that a small change in the landing location can result in a drastic
change in the required C3 even with similar TOFs and the same velocity at the Moon.
This fact is important for mission designers, as it may sometimes be possible to
move the landing site slightly to improve ΔV, or a similar effect may be obtained by
targeting with maneuvers along the trajectory. Trajectory correction maneuvers may
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
285
Figure 515 TOF and C3 for each trajectory plotted over the surface for a range of Jacobi
c 2011 by American Astronautical Society Publications Ofﬁce,
constants [192] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS).
286
TRANSFERS TO THE LUNAR SURFACE
also help aid in reducing the ΔV. In general, it is useful to be aware of the chaotic
nature of the design space as seen from these plots.
Finally, it is important for most mission designs to consider the inclination. The
inclination results in the EME2000 coordinate frame are given in Fig. 516. One of
the important features to notice here is that a variety of inclinations are possible. A
choice of trajectories exist with the lower inclinations suitable for launch from Cape
Figure 516 Inclination computed relative to the Earth in the EME2000 coordinate frame
c 2011 by American Astronautical Society Publications Ofﬁce, San Diego,
[192] (Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
287
Canaveral. The particular inclinations needed for a mission will depend on the target
location on the Moon and the particular constraints of the mission. They are provided
here as a sample of the range of the values that are possible.
5.5.2
Trajectories Arriving at Various Angles to the Lunar Surface
The planar CRTBP provides a convenient framework in which to understand and
visualize the relationship between invariant manifolds and lunar approach trajectories,
but the design of realworld equivalent trajectories often requires a landing at either
higher or lower latitudes. Indeed, many of the recently proposed landing sites at
the Moon are at northern or southern latitudes [204], and one of the locations that is
currently a focus for a lunar lander is more southern latitudes in the Aitken Basin.
In this analysis, lunar landing trajectories are analyzed over the threedimensional
surface of the Moon, and the approach geometry of the trajectories in three dimensions
is also analyzed.
The landing geometry of trajectories traveling from the Earth to the Moon is of
particular importance for mission design. In the previous section and in Anderson and
Parker [192, 195] we analyzed trajectories encountering the Moon normal to the sur
face to determine whether these trajectories originated at the Earth within the previous
200 days. Given this elevation angle constraint, only some locations of the Moon’s
surface were found to be accessible from the Earth. For this analysis, trajectories
were allowed to approach each point on the lunar surface from all directions. These
directions were speciﬁed relative to the surface at each point. The azimuth angle (Ω)
is measured clockwise from north where north is the lunar orbit’s North Pole, rather
than the Moon’s North Pole, to be consistent with the results from the CRTBP. The
elevation angle (φ) is measured positive above the Moon’s surface, with a trajectory
encountering the Moon’s surface normal to the surface having an elevation angle of
90 deg. (Note that this is different from θ used for the planar case, but it was chosen
to be more consistent with typical mission design parameters.) While the previous
analysis was ideal for impactors, the trajectories computed here are applicable for
a wide range of mission types traveling to the lunar surface. Additional parameters
for each trajectory related to the original characteristics relative to the Earth may be
computed, but the focus here is on characterizing the approach geometry. For the
following analysis the trajectories were computed over the surface of the Moon using
1deg increments in α and β. The same deﬁnition is used for α that is used in the
planar problem in Fig. 53. As described earlier, β is measured like latitude and
is positive above the xy plane. Two different grids were used for the azimuth and
elevation angles. In each case, the elevation angle was varied in even increments,
and the steps taken in azimuth angle were speciﬁed initially for an elevation angle of
0 deg. The number of azimuth points were then decreased with cos(φ) so that the
number of points decreased with elevation angle. Both a ﬁne grid and a coarser grid
were used in this analysis. For the ﬁne grid case, 1deg increments were taken at
0deg elevation for Ω, and 1deg increments were used for elevation. For the coarser
grid, 10deg increments were used for Ω at 0deg elevation, and 3deg increments
were used for elevation. This coarser grid was found to provide a good approximation
288
TRANSFERS TO THE LUNAR SURFACE
that conveyed the overall trends of the ﬁne grid, while allowing for a more reasonable
computation time. Even with this coarser grid, computing trajectories over the entire
surface in the ephemeris problem for each Jacobi constant required approximately
seven days running in parallel on 40 processors. Unless otherwise stated, this coarser
grid is the one used throughout the analysis.
As an initial step in the analysis, the set of trajectories was computed in the
CRTBP for a Jacobi constant of 2.6. The trajectories were computed for both the
ﬁne grid and the coarser grid. Comparing the maximum and minimum elevation
angles resulted in trajectories that originate at the Earth as shown in Fig. 517.
Using the symmetry about the xy plane mentioned earlier, it can be seen that the
northern and southern latitudes will be reﬂected for the elevation plots in the CRTBP.
Note that the azimuth angles would need to account for the reﬂection if they are
Figure 517 Maximum and minimum elevation angles for trajectories originating at the
Earth and encountering the Moon at each point on the surface. These cases are computed in
the CRTBP for C = 2.6. Results from two different grids (in elevation and azimuth angle)
c 2011 by American Astronautical Society Publications Ofﬁce,
are shown [193] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS). (See insert for color representation of this ﬁgure.)
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
289
plotted, and although similar results would be expected in the ephemeris problem,
the variations in the ephemeris require that the northern and southern hemispheres be
computed independently. Using this symmetry the values computed for the northern
and southern hemispheres were reﬂected in Fig. 517 to save computation time. By
comparing the plots, it can be seen that, as might be expected, the ﬁner grid captures
more trajectories at higher and lower elevation angles that originate at the Earth,
however, the overall trends in the data remain the same for both grids. In each
case the range of elevation angles from minimum to maximum is shifted higher near
α = 90 deg and lower near α = 270 deg. Referring back to Fig. 53, the 90deg
direction corresponds to the leading edge of the Moon, and the 270deg direction to
the trailing edge. The coarser grid is used in the remainder of this analysis, so it
should be remembered that details in the plots may change with a ﬁner grid, but the
overall trends can still be observed.
An analysis of trajectories for a Jacobi constant of C = 2.8 conﬁrmed our earlier
result for trajectories encountering the Moon normal to the surface of the Moon
that no Earthreturn trajectories were found for this Jacobi constant or higher ones
in the CRTBP. However, it is expected that Earthorigin trajectories with velocities
consistent with higher Jacobi constants in the CRTBP will exist in the ephemeris
problem because these trajectories may use the Sun’s perturbations to travel from
the Earth to the Moon. Those higher Jacobi constants, especially those approaching
the values near CL1 and CL2 are especially relevant for the computation of the
invariant manifolds of libration point orbits, which is useful for the comparison later
in this study. The elevation angle range results are shown in Fig. 518 for Jacobi
constants ranging from C = 2.6 to 3.1 in the ephemeris problem. Note that, as in
Anderson and Parker [192, 195], the Jacobi constant for the ephemeris plots is used
as a shorthand for a particular set of velocities computed around the Moon in the
CRTBP. These same velocities are attached to the Moon in the ephemeris problem
referenced to the instantaneous orbital plane of the Moon’s orbit around the Earth.
The symmetry used to simplify the computations in the CRTBP is no longer present
for the ephemeris problem, and trajectories were directly computed for the entire
plot. Once the trajectory is integrated backward from the Moon, the Jacobi constant
of the trajectory will vary in both the Earth–Moon and Sun–Earth systems.
Comparing the results from Fig. 517 for the Jacobi constant case of 2.6 in
the CRTBP and the ephemeris problem results reveals that they are quite similar.
The maximum and minimum elevation values still occur at approximately the same
locations on the surface for each case. However, several new bands of highelevation
angle cases occur for the ephemeris case near α = 180 deg for the maximum elevation
angle case and from approximately α = 290 deg to 360 deg. Additional bands also
seem to exist for the minimum elevation angle case, especially for high and low
latitudes. It is natural to expect from past work that these bands may represent
trajectory options that exist as a result of the Sun’s inﬂuence, and it is interesting that
these types of bands remain up through C = 2.8 (Figs. 518(a) through 518(d)). An
interesting topic planned for future study is to determine how these characteristics
vary with a ﬁner grid. However, the comparison performed here is with the same
grid in each case, indicating that these additional trajectories exist.
290
TRANSFERS TO THE LUNAR SURFACE
Figure 518 The minimum and maximum elevation angles of trajectories originating at
the Earth for each point on the lunar surface. These trajectories are computed in the
c 2011
Earth–Moon ephemeris system including the Sun’s perturbations [193] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS). (See
insert for color representation of this ﬁgure.)
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
291
As the Jacobi constant increases even more as seen in Figs. 518(e) through
518(h), the range of elevation angles for lunar landing at each point seems to increase
even more. The location of the peaks also seems to shift, and for the maximum
elevation angle plots, the peaks move to the right or eastward with increasing Jacobi
constant. When a Jacobi constant of 3.1 is reached, the maximum elevation angle
for points containing a trajectory originating at the Earth never drops below 57 deg,
and the minimum elevation angle for the same points never goes over 15 deg. It
is important to mention that although the points look dense across the surface in
the plots, this is because of the size of the plot and the points used for plotting.
There are individual points on the surface where no Earthorigin trajectory exists
for this grid, but there are always nearby points where such a trajectory exists. For
realworld mission design, a small ΔV can be used to target slightly different points,
and the surface of the Moon is covered in practice for mission design purposes. It
has also been found for particular points that if a much ﬁner grid is used, typically
some Earthorigin trajectories are found, and these points will be included in future
studies. The points with no Earthorigin trajectories for this grid are not included in
the elevation angle ranges listed in the plots. These results for higher Jacobi constants
agree generally with the normal trajectory cases seen in our previous work [192, 195].
The additional range of geometries available for landing at these energies appears
to be a result of the increasingly chaotic nature of the system as the Jacobi constant
approaches the values at the L1 and L2 libration points. In other words, the trajectories
are more able to take advantage of chaos to arrive at different elevation and azimuth
angles. This also indicates that these Jacobi constants are of particular interest for
comparison with the invariant manifolds of libration orbits. One interesting statistic
to examine with a ﬁxed grid is the maximum number of trajectories at a particular
point that originate at the Earth. Although this number is generally quite low, there
are some points where it peaks. The maximum number of trajectories at a particular
point is listed in Table 52 for different Jacobi constants. The higher values are
found for a Jacobi constant of 2.6 and 3.1. The C = 2.6 results include more direct
trajectories that still exist in the CRTBP and do not require the Sun’s inﬂuence, and
the C = 3.1 results include those trajectories that are heavily inﬂuenced by the Sun.
The total number of Earthorigin trajectories follows the same trend. These numbers
Table 52 Maximum number of Earthorigin trajectories at a single point on the lunar
surface for a ﬁxed grid including the corresponding location and the total number of
Earthorigin trajectories for various values of Jacobi constant.
Jacobi Constant
2.6
2.8
3.0
3.1
Maximum at a Point
27
16
14
36
Location (α, β)
Total Number
(193 deg, −36 deg)
(213 deg, −18 deg)
(225 deg, −11 deg)
(192 deg, 7 deg)
290,672
114,684
162,061
298,621
292
TRANSFERS TO THE LUNAR SURFACE
are a function of the grid that is being used and can be reﬁned by using a denser grid;
however, they do align with the results from the trajectories computed normal to the
lunar surface seen in our earlier work.
Because the trajectories are computed in the ephemeris problem for the cases just
discussed, the results will naturally vary with the initial epoch of the integration.
A sample of the results was computed for four different epochs around the Moon’s
orbit (with the time intervals each at approximately onequarter of the Moon’s orbit)
to determine how they might vary with the initial epoch. Representative results for
a Jacobi constant of 2.8 are shown in Fig. 519. The salient features of the plots
remain generally the same for each epoch in that the maximum values still occur near
α = 90 deg and the minimum values occur near α = 270 deg. The January 7 and 21
cases have more locations with higher elevation angles, especially near α = 270 deg,
mixed in with lower elevation angle points. These two cases appear better positioned
to take advantage of the Sun–Earth libration point dynamics, which could increase
Figure 519 Comparison of maximum elevation angle results around the lunar orbit at
c 2011 by American Astronautical Society
seven day intervals for C = 2.8 [193] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS). (See insert for color representation of this
ﬁgure.)
LOWENERGY SPATIAL TRANSFERS BETWEEN THE EARTH AND THE LUNAR SURFACE
293
the range of elevation angles that may be obtained for approaching the Moon. Overall
though, given this comparison, it is expected that the results from this study may be
extrapolated to other epochs without drastically changing the outcome.
Another important aspect of the approach for mission design is, of course, the
azimuth angle of the trajectory. Plotting this information in a global sense is difﬁcult,
but a sample of the types of results obtained for each Jacobi constant may be visualized
in Fig. 520 for a subset of the points. The azimuth angles are plotted for each point on
a grid computed at 30deg intervals in both α and β. For these plots, the ﬁne grid was
used at each point on the surface (which of course produced more trajectory options),
and the trajectories were limited to those with C3 < 0.0 km2 /s2 at the Earth. The
orientation of the lines centered on each point indicates the azimuth angle, and the
color is used to designate the corresponding elevation angle of each trajectory. Note
that the ± 90deg cases used azimuths that were rotated differently at each elevation
as a result of the transformation used to compute them. So the speciﬁc results differ,
but they generally show similar trends. It is interesting that there are deﬁnite regions
Figure 520 Azimuth angles at points on a 30deg grid on the lunar surface. The plotted lines
at each gridpoint are oriented in the proper azimuth direction for each individual trajectory.
The color corresponds to the elevation angle of that trajectory. The trajectories shown here
are limited to those with C3 < 0.0 km2 /s2 at the Earth [194] (ﬁrst published in Ref. [194];
reproduced with kind permission from Springer Science+Business Media B.V.). (See insert
for color representation of this ﬁgure.)
294
TRANSFERS TO THE LUNAR SURFACE
where the majority of Earthorigin trajectories appear to have similar elevation angles.
In each case though, there are often just a few high or low trajectories that result in
the extremes seen in the elevation angles plots. This fact is worth keeping in mind
for mission design since a particular elevation angle may be available in combination
with only a few azimuth angles. In general it appears that higherelevation options
are more available as the Jacobi constant increases, although there are typically at
least a few low elevation angle options at each point. The combinations of available
elevation and azimuth angles are evaluated in more detail for C = 3.1 in the following
comparison with invariant manifolds, which helps explain the features seen in these
plots a little more directly. In general, these plots can provide a broad overview of
the available trajectory options.
5.6 TRANSFERS BETWEEN LUNAR LIBRATION ORBITS AND THE
LUNAR SURFACE
A general framework and understanding does exist in regard to the relationship
between invariant manifolds of unstable orbits and the Moon. (Refer to Section
2.6.10 for more background on invariant manifolds.) Much of the work to design
lowenergy trajectories from the Earth to the Moon has focused on the use of libration
point orbits along with their stable and unstable manifolds [39, 45, 51, 203]. Koon, Lo,
Marsden, and Ross examined this problem for the planar case [37], and Parker studied
approach cases to lunar libration orbits using invariant manifolds in his dissertation
[46]. Baoyin and McInnes analyzed some speciﬁc cases of transfers from libration
points and planar Lyapunov orbits to the lunar surface [205]. In particular, they
searched for the Jacobi constant that would provide complete coverage of the lunar
surface by the invariant manifolds of the selected Lyapunov orbit. Von Kirchbach
et al. [201] looked at the characteristics of the invariant manifolds of a Lyapunov
orbit as they intersected the surface of Europa in the context of the escape problem.
Alessi, Go´ mez, and Masdemont [206] examined the locations of the Moon reachable
by the stable manifolds of a range of halo orbits and square Lissajous orbits. They
computed the intersections of these invariant manifolds with the surface of the Moon
with the expectation that they could be used for astronauts to escape to a libration
point orbit if necessary. Anderson [207] examined the approach problem within the
context of the invariant manifolds of unstable resonant and Lyapunov orbits as the
trajectory ties into the resonances of the Jupiter–Europa endgame problem following
invariant manifolds [158, 208–210].
One focus of the transfer to the lunar surface using invariant manifolds is on
the ﬁnal approach from a desired libration orbit to the lunar surface. The problem
may be most easily approached using the planar CRTBP and Lyapunov orbits. Two
sample Lyapunov orbits found in Anderson and Parker [192, 195] are replotted here
in Fig. 521. The Jacobi constants for these orbits were chosen so that the invariant
manifolds of the Lyapunov orbits just graze the surface of the Moon. The Jacobi
constants where the Lyapunov orbits cover the surface of the Moon were computed
TRANSFERS BETWEEN LUNAR LIBRATION ORBITS AND THE LUNAR SURFACE
295
Figure 521 Invariant manifolds of libration orbits computed for Jacobi constants where
c 2011 by
the manifolds are tangent to the surface of the Moon [193] (Copyright ©
American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS). (See
insert for color representation of this ﬁgure.)
by Baoyin and McInnes [205] as approximately C = 3.12185282430647 for an L1
Lyapunov orbit and C = 3.09762627497867 for an L2 Lyapunov orbit.
As a ﬁrst step in the comparison, the lunar landing geometry of the invariant
manifolds of various halo orbits is analyzed. Alessi, Go´ mez, and Masdemont [206]
examined similar trajectories for escaping the surface of the Moon to various halo
orbits and summarized the areas on the Moon from which such escape trajectories
are possible. We are concerned here with a combination of the landing location along
with the landing geometry, therefore, a similar technique to that used in Fig. 520 is
employed here. In subsequent ﬁgures, the intersections of the unstable manifolds of
the L1 halo orbits are indicated by a red point, and the intersections for the L2 halo
orbits are orange points. The azimuth angle and the elevation angle are indicated by
the direction and the color of the line segments, respectively.
The results for a halo orbit at C = 3.1 are shown in Fig. 522. It can be immediately
seen that for this energy, the L1 halo orbit invariant manifolds generally fall on the
leading edge of the Moon in its orbit, and the L2 halo orbit invariant manifolds fall
on the trailing edge of the Moon. As expected, the intersections of the northern
and southern halo orbits are reﬂected about β = 0. The elevation angles are some
what lower for the L1 halo orbits than the L2 halo orbits. All together, the unstable
manifolds provide relatively broad coverage of much of the lunar surface, although
signiﬁcant regions are still not intersected by the invariant manifolds. This may be
remedied by examining the invariant manifolds at additional energies. The unstable
manifold intersections with the Moon can change signiﬁcantly with the Jacobi con
stant as can be seen for the intersections plotted with a Jacobi constant of 3.08 in
Fig. 523. The intersections for the L1 case have divided into two different regions,
and the L2 intersection case has grown tighter together. It should be reiterated that
the unstable manifold intersections can increase if larger time intervals are used for
296
TRANSFERS TO THE LUNAR SURFACE
Figure 522 Unstable manifold intersections of the speciﬁed orbits with the Moon for
c 2011 by American Astronautical Society Publications Ofﬁce,
C = 3.1 [193] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS). (See insert for color representation of this ﬁgure.)
TRANSFERS BETWEEN LUNAR LIBRATION ORBITS AND THE LUNAR SURFACE
297
Figure 523 Unstable manifold intersections of the speciﬁed orbits with the Moon for
c 2011 by American Astronautical Society Publications Ofﬁce,
C = 3.08 [193] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS). (See insert for color representation of this ﬁgure.)
the integration, and these plots focus on shortduration trajectories. The unstable
manifold intersections also change even more as energy continues to change, but
these energies appear to provide some of the most direct transfers.
This analysis shows that the unstable manifolds of halo orbits can provide broad
coverage for landing at various points on the Moon, although not with the nearly
complete coverage found from the previous results. It is also interesting to explore
the relationship between the unstable manifolds and these Earthorigin trajectories
from the general analysis. A similar examination to the one made for the planar
problem would be desired, but the nature of the threedimensional problem makes
this drastically more complex. One possible method for performing this comparison
is to examine the origin of the trajectories coming from all azimuth and elevation
angles at each point that the unstable manifolds intersect the surface of the Moon. In
this case, only one unstable manifold intersection is plotted for each location on the
Moon relative to the trajectories coming in from all angles, but it still allows this point
to be placed in context of the dynamics indicated by the source of each trajectory.
298
TRANSFERS TO THE LUNAR SURFACE
5.7 TRANSFERS BETWEEN LOW LUNAR ORBITS AND THE LUNAR
SURFACE
Information about transfers from low lunar orbits to the lunar surface is provided in
Section 4.6 on page 258.
5.8 CONCLUSIONS REGARDING TRANSFERS TO THE LUNAR
SURFACE
A wide variety of trajectory options exist for transfer to the lunar surface. These range
from more direct trajectories that may be primarily computed using just the inﬂuence
of the Earth and Moon, to those at lower energies that require the inﬂuence of the Sun
to compute. The invariant manifolds of libration orbits may be used for transfers to
the lunar surface, and in combination with transfers to these libration orbits from the
Earth, can be used as a complete transfer from the Earth. The trajectories computed
for various Jacobi constants shown in the selected plots in this chapter may be used
to obtain an initial idea of the types of trajectories available for different energy
regimes. These energy regimes correspond to the constraints dictated by a particular
mission, such as the available launch vehicle. Once the general type of trajectory that
may be of interest is selected, more detailed initial guesses for particular trajectories
may be obtained from the various plots showing trajectories coming in at various
angles to the surface or from the invariant manifolds results. A mission designer may
then modify and constrain these trajectories, while incorporating the mission design
constraints of interest, to compute the ﬁnal desired trajectory.
CHAPTER 6
OPERATIONS
6.1
OPERATIONS EXECUTIVE SUMMARY
The purpose of this chapter is to address different ways that a lowenergy transfer
may impact the operations of a spacecraft, compared to conventional lunar transfers.
Most conclusions are very straightforward consequences of the fact that lowenergy
transfers require less change in velocity (ΔV), more time, and have longer link
distances during the transfer than direct lunar transfers. For instance, there are fewer
demands on the spacecraft’s propulsion system and operational schedule, but more
demands on the spacecraft’s communication capabilities due to the longer distances.
The operations team must be able to perform several trajectory correction maneuvers
(TCMs) during the translunar cruise, but these maneuvers are typically much more
separated in time from launch, lunar arrival, and other maneuvers than they are on
conventional lunar transfers.
The majority of discussion in this chapter is devoted to studying the availability
and ΔV cost of establishing an extended 21day launch period for a lunar mis
sion. Conventional lunar missions typically have very constrained launch periods,
reﬂecting the geometry in the Sun–Earth–Moon system. However, lowenergy lunar
299
300
OPERATIONS
transfers are very ﬂexible and may be adjusted in many ways to accommodate an
extended launch period. Several conclusions are drawn from these examinations.
First, the cost of a launch period is dependent on the number of launch days
in the period. The examination performed in Section 6.5 estimates that it costs
on average 2.5 meters per second (m/s) of ΔV per day added to a launch period;
hence, the average 21day launch period requires about 50 m/s more deterministic
ΔV than a 1day launch period for a given transfer. The cost of the 1day launch
period is dependent on the inclination change that must be performed to inject onto
the desirable lowenergy transfer from a constrained low Earth orbit (LEO) parking
orbit. Section 6.5.7 estimates that it costs approximately 0.97 m/s more transfer ΔV
per degree of inclination change that must be performed. The total cost of establishing
a 21day launch period from a 28.5degree (deg) LEO parking orbit to a given lunar
orbit is approximately 71.7 ± 29.7 m/s (1σ). Thus, to be very conservative when
estimating a preliminary ΔV budget for a mission, one may estimate that the ΔV cost
to transfer from a 28.5deg LEO parking orbit to a particular lunar orbit, including
a 21day launch period, will cost approximately 161 m/s, not including statistical
costs and/or other deterministic costs. Of course, the 161 m/s accounts for a 3sigma
high value, evaluated from a large set of random mission designs; it is likely that a
practical mission may be constructed for signiﬁcantly less ΔV.
A 21day launch period does not necessarily have to include 21 consecutive days;
in fact, most launch periods constructed in this examination include one or two gaps
when the launch operations would have to stand down. The average launch period
for the sample set used here requires a total of 27 days; the vast majority of the launch
periods may be contained within 40 days.
Finally, it has been found that there is no signiﬁcant trend between the total
launch period ΔV for the sample missions studied here and their reference departure
inclination values or their reference transfer durations, except that missions with short
durations (< 90 days) require more ΔV to establish an extended launch period, on
account of the reduced ﬂexibility of a shorter transfer.
6.2
OPERATIONS INTRODUCTION
This chapter discusses several aspects of a spacecraft mission that must be considered
for the lowenergy transfers presented in this book to be used in a real mission.
Numerous discussions throughout Chapters 3–5 have considered the latitude of the
mission’s launch site, since that strongly inﬂuences the inclination of the parking
orbit that may be used in a mission. But other aspects have not been fully discussed,
such as which launch vehicle may be used, how to establish a launch period, and what
considerations must be made to a spacecraft’s design to ﬂy a lowenergy transfer.
Sections 6.3 and 6.4 provide information and discussion about which launch sites
and launch vehicles are typically used and/or available for lunar missions. Section 6.5
provides a lengthy discussion, analysis, and several algorithms that may be used to
generate a 21day launch period for a given lowenergy transfer. The results indicate
that simple lowenergy transfers may be targeted from nearly any LEO parking
LAUNCH SITES
301
orbit with a 21day launch period for a modest fuel cost on the order of 72 m/s.
Section 6.6 discusses issues relevant to navigating a spacecraft while on a lowenergy
transfer, including the costs of stationkeeping and the beneﬁts of having 3–4 months
to perform the transfer instead of the conventional 3–6 days. Finally, Section 6.7
presents several considerations that must be made to the spacecraft systems and
operations design to accommodate a lowenergy lunar transfer.
6.3
LAUNCH SITES
Chapters 3–5 illustrated that lowenergy ballistic transfers may be constructed that
depart the Earth from parking orbits or direct departures with any orbital inclination.
By carefully selecting a particular transfer, one may build a mission that launches from
any given launch site and efﬁciently injects into the ballistic lunar transfer. While this
is very important for conventional mission design, Section 6.5 later demonstrates that
a mission can actually depart the Earth from virtually any inclination and transfer
to a particular lunar arrival for a modest ΔV cost. Still, it is of interest to build
a lowenergy transfer that is designed to depart the Earth with an inclination that
is very similar to the latitude of the mission’s launch site so that no sizable orbital
plane changes are needed. This is particularly useful for missions with a brief launch
period.
Table 61 provides a summary of the launch sites that have demonstrated the
capability of placing large payloads into orbit. This is not a complete list, but
provides a good review of the latitude and longitude of several sites for reference.
6.4 LAUNCH VEHICLES
Many launch vehicles are available to place spacecraft on lowenergy lunar transfers.
The NASA Launch Services Program (LSP) at Kennedy Space Center coordinates
contracts with several launch vehicle providers using NASA Launch Services (NLS)
contracts [211]. On September 16, 2010, NASA released the details about the
NLS II contacts that were awarded to four launch vehicle providers: Lockheed
Martin Space Systems Company of Denver, Colorado; Orbital Sciences Corporation
of Dulles, Virginia; Space Exploration Technologies of Hawthorne, California; and
United Launch Services, LLC of Littleton, Colorado. This contract includes several
families of launch vehicles, including Atlas V, Falcon 9, Pegasus XL, Taurus XL,
Athena I, and Athena II. The NLS II contract provides the minimum performance that
is contractually obligated by the launch vehicle; a mission may be able to negotiate
with the launch vehicle provider to increase the performance of the launch vehicle
depending on the mission’s requirements [211].
Table 62 summarizes the maximum payload capabilities of several launch vehicles
injected from Cape Canaveral, Florida, onto lowenergy lunar transfers with and
without an outbound lunar ﬂyby. The table captures two extreme cases: ﬁrst, the
case where the transfer includes an outbound lunar ﬂyby and requires an injection
C3 of −2.1 kilometers squared over seconds squared (km2 /s2 ), which is near the
302
OPERATIONS
Table 61 The locations of several launch sites that have been used to launch large
payloads into orbit, toward the Moon, and/or into Interplanetary space. This is not a
complete list, the locations are approximate, and some are representative of several
particular launch sites.
Latitude Longitude
Comments
(deg)
(deg)
Country
Location
USA
Cape Canaveral Air Force Station,
Florida
Kennedy Space Center, Florida
Vandenberg Air Force Base,
California
Kodiak Launch Complex, Alaska
MidAtlantic Regional Spaceport
(MARS), Delmarva Peninsula,
Virginia
Kwajalein Atoll
28.47 N
80.56 W
28.61 N
34.77 N
80.60 W
120.60 W
9.00 N
167.65 E
Alcˆantara Launch Center,
Maranh˜ao
Jiuquan Satellite Launch Center
Xichang Satellite Launch Center
Guiana Space Centre, Kourou
2.32 S
USA
USA
USA
USA
USA
Brazil
China
China
French
Guiana
India
Satish Dhawan Space Centre
(Sriharikota), Andhra Pradesh
Israel
Palmachim Air Force Base
Japan
Uchinoura Space Center
Japan
Tanegashima Space Center,
Tanegashima Island
Kazakhstan Baikonur Cosmodrome, Tyuratam
Marshall
Omelek
Island
Russia
Svobodny Cosmodrome, Amur
Oblast
Russia
Yasny Cosmodrome, Orenburn
Oblast
Russia
Kapustin Yar Cosmodrome,
Astrakhan Oblast
Sweden
Esrange, Kiruna
Several
Sea Launch / Ocean Odyssey
complex
Interplanetary
Lunar
High
inclinations
57.44 N 152.34 W Orbital
37.83 N
75.48 W Orbital
44.37 W
Orbital
Orbital
41.12 N 100.46 E
28.25 N 102.03 E
5.24 N
52.77 W
Orbital
Lunar
Interplanetary
13.74 N
Lunar
80.24 E
31.88 N
34.68 E
31.25 N 131.08 E
30.39 N 130.97 E
Orbital
Orbital
Orbital
45.96 N
63.35 E
9.05 N 167.74 E
Interplanetary
Orbital
51.83 N
128.28 E
Orbital
51.21 N
59.85 E
Orbital
48.58 N
46.25 E
Orbital
67.89 N
0.0 N
21.10 E
Varies
Orbital
Orbital
LAUNCH VEHICLES
303
Table 62 The payload capabilities of several launch vehicles injected from Cape
Canaveral, Florida, onto lowenergy lunar transfers with and without an outbound lunar
ﬂyby. This information has been captured from the NASA Launch Services (NLS)
Program’s Launch Vehicle Performance site under the NLS II contract [211].
Launch Vehicle
Maximum Payload Performance (kg)
C3 = −2.1 km2 /s2
C3 = −0.3 km2 /s2
Athena II
395.0
375.0
Falcon 9 Block 1
Falcon 9 Block 2
2125.0
2645.0
1995.0
2515.0
Atlas V 401
Atlas V 411
Atlas V 421
Atlas V 431
3170.0
4095.0
4845.0
5445.0
3050.0
3955.0
4680.0
5265.0
Atlas V 501
Atlas V 511
Atlas V 521
Atlas V 531
Atlas V 541
Atlas V 551
2215.0
3410.0
4365.0
5135.0
5815.0
6340.0
2110.0
3285.0
4215.0
4965.0
5625.0
6140.0
minimum injection energy typically required. The second case presented requires a
C3 of −0.3 km2 /s2 , which is near the maximum injection energy typically required
without a lunar ﬂyby. Most missions will fall between these two values: closer to one
depending on whether or not the mission aims to ﬂy past the Moon on the outbound
segment.
As of September 2011, Orbital Sciences estimates that the Taurus XL may be used
to inject as much as 425 kilograms (kg) to a C3 of 0 km2 /s2 , and presumably more to a
lowenergy lunar transfer. Further, although it is not currently in the NLS II contract,
Orbital Sciences estimates that the Taurus II launch vehicle may be able to inject
between 920 kg and 1120 kg to a C3 of −2.1 km2 /s2 depending on its conﬁguration.
The Taurus II’s performance drops about 40 kg when injecting payloads to a C3 of
−0.3 km2 /s2 .
In addition, the Pegasus XL launch vehicle may be used to place up to about
470 kg of payload into a 200km circular parking orbit [212]. A spacecraft could
then perform its own translunar injection to transfer to the Moon, much like the
proposed Dust Near Earth (DUNE) mission [146, 213], or similar to the Interstellar
Boundary Explorer (IBEX) mission [214–216].
Other launch vehicles may also be used to inject a spacecraft onto a lowenergy
lunar transfer, though they do not have a contract with NASA, including the Delta IV
family of vehicles. Certainly several international vehicles may be used, assuming the
304
OPERATIONS
vehicles’ guidance algorithms have the capability of targeting such orbital parameters,
including the Russian Soyuz and Proton vehicles, Arianespace’s Ariane V, China’s
Long March and CZ vehicles, Japan’s HIIA and HIIB, and Ukraine’s Zenit3SL,
among others. The Indian Space Research Organization’s (ISRO’s) Polar Satellite
Launch Vehicle (PSLVC11) was used to launch the Chandrayaan1 mission to the
Moon, though the launch vehicle only injected the spacecraft into a 6hour orbit about
the Earth and the spacecraft performed the remainder of the lunar injection.
6.5
DESIGNING A LAUNCH PERIOD
This section considers how to construct an extended launch period for a lowenergy
transfer to the Moon. The discussion begins by reviewing several interesting features
that exist in the Earth–Moon system and how historical launch periods have been
constructed around those features. This provides context for future discussions about
designing launch periods for lowenergy transfers.
First, the Moon’s orbit is nearly circular about the Earth. This means that one
may theoretically launch a spacecraft on a conventional, direct transfer with very
similar characteristics on any given day. The Moon’s elliptical orbit means that the
launch energy and transfer duration will vary across the month to some degree, but
this is a secondorder effect. The largest variation from one day to the next when
injecting into a direct direct transfer arises from the obliquity of the Earth relative to
the Moon’s orbit. The Earth’s spin axis is tilted approximately 23.5 deg relative to the
ecliptic, and the Moon’s orbit has an inclination of about 5.15 deg with respect to the
ecliptic. Together, this means that the relative orientation of the Earth’s spin axis and
the orbit of the Moon may be anywhere from 18.35 deg to 28.65 deg; the orientation
of the parking orbit must be adjusted to accommodate this shift. Ultimately this
means that the time of day that one must launch shifts from one day to the next, as
does the duration of time that the spacecraft coasts in a low Earth parking orbit prior
to injecting toward the Moon.
Next, a lunar day is approximately 29.5 Earth days long, which means that the
lighting conditions on the Moon roughly repeat every 29.5 days. There are variations
on top of this cycle that correspond with where the Moon is in its orbit about the
Earth relative to its perigee, and where the Earth is in its orbit about the Sun. The
net effect is that if one is interested in viewing a particular lighting condition as one
ﬂies by the Moon or impacts the Moon, then one may only be able to launch on a
direct transfer one or two days every month. This is very important for missions that
aim to land on the surface, including the Apollo missions. The Apollo missions were
designed to land on the surface soon after sunrise at the landing site to maximize
the amount of sunlit time they had on the surface before needing to ascend. These
considerations have a direct effect on the time of arrival at the Moon for any mission,
though missions that go into orbit prior to landing/impact can arrive early. The time
of arrival is highly correlated with the launch time for direct transfers, since direct
transfers have a short transfer duration that cannot be varied much. The time of arrival
is loosely correlated with the launch time for lowenergy transfers, since lowenergy
DESIGNING A LAUNCH PERIOD
305
transfers can vary their transfer durations by many days without a large penalty in
transfer ΔV.
Another consideration for a mission planner is that many lunar spacecraft are not
designed to survive a long eclipse. Lunar eclipses occur roughly every 6 months when
the Earth comes directly between the Sun and Moon. The Moon’s nonzero orbital
inclination relative to the ecliptic means that a lunar eclipse does not occur each and
every month, but only occurs when the Moon is near its ascending or descending
node when it traverses behind the Earth. Since the Moon’s orbit is ﬁxed in inertial
space, though subject to perturbations, one of the nodes traverses directly behind the
Earth twice per year. If the Moon is near that point in its orbit at that time, then the
eclipse will be a full lunar eclipse and any spacecraft on the surface or in a low orbit
will traverse through the umbra of the Earth. If the Moon is not near that point in
its orbit, then the spacecraft may be able to avoid the shadow, or at least avoid the
umbra of the Earth. The Gravity Recovery and Interior Laboratory (GRAIL) mission
was designed with lunar eclipses in mind, since the two GRAIL spacecraft were not
originally designed to survive an extended passage through shadow. GRAIL’s entire
science phase was designed to occur between two lunar eclipses in case one of the
spacecraft did not survive the following eclipse. This means that GRAIL’s launch
opportunities do not repeat every month, but only repeat once every six months.
6.5.1 LowEnergy Launch Periods
Lowenergy lunar transfers are more ﬂexible than direct lunar transfers since their
transfer durations are longer; hence, it is possible to build an extended, 21day launch
period such that every launch opportunity yields a trajectory that a spacecraft can
follow that arrives at the Moon at the same time. This is very useful for missions
such as GRAIL that depend on arriving at the Moon at a particular time of the year or
month.
There are often many ways to adjust a trajectory’s design so that it may depart
the Earth on multiple days, in order to establish a launch period. For this discussion
we assume that the trajectory begins with a launch from a particular site into a low,
nearcircular parking orbit; coasts in the parking orbit for some duration; performs
a translunar injection; and then follows a ballistic transfer to the Moon using one
or two trajectory correction maneuvers en route to the Moon. Given this trajectory
design, several examples of controls include the following:
• Adjust the launch time. By launching at a different time of day, one can change
the longitude of the ascending node of the parking orbit that the spacecraft uses
prior to its translunar injection.
• Adjust the launch and parking orbit geometry. One may be able to reduce the
total transfer ΔV cost and ultimately transfer more payload mass to the Moon
by changing the parking orbit’s inclination. This reduces the launch vehicle’s
performance, but it may be worthwhile.
• Adjust the location of the translunar injection maneuver in the parking orbit.
306
OPERATIONS
• Adjust the translunar injection maneuver. The maneuver magnitude and/or
direction may be adjusted, depending on the control algorithm that operates
the maneuver. In the studies presented in this chapter, only the maneuver
magnitude is adjusted.
• Add one or more trajectory correction maneuvers in the translunar cruise.
These maneuvers may be performed in any direction, though some missions
may place constraints on the magnitude or direction of these maneuvers. In the
studies presented here, two maneuvers are introduced that may be performed
in any direction with any maneuver magnitude, though no two maneuvers may
be placed within four days of each other to reduce operations complexity.
• Adjust the lunar arrival conditions as described below.
The available controls upon arriving at the Moon depend on the arrival orbit/geometry
and the mission design. Some examples of different missions and their controls
include:
Arriving at a lunar libration orbit. Arriving at a lunar libration orbit typically in
volves a ballistic, asymptotic arrival with a ﬁnal correction maneuver to ensure
that the spacecraft is placed in the target orbit. Controls include:
• Adjust the date/time of arrival. This may vary by mere seconds or by days,
depending on the mission’s requirements.
• Vary the target libration orbit. It is typically more desirable to maintain a single
target libration orbit throughout the launch period, though that depends on the
mission.
• Add a libration orbit insertion maneuver, which may vary in magnitude/direction.
This is typically much more useful if the target libration orbit is held ﬁxed across
a launch period.
Arriving at a low lunar orbit. Arriving at a low lunar orbit typically involves a
timecritical lunarorbit insertion (LOI) maneuver that places the spacecraft
into a capture orbit. Controls include:
• Adjust the date/time of the LOI. This may vary by mere seconds or by days,
depending on the mission’s requirements.
• Adjust the LOI’s magnitude and/or direction. Some spacecraft designs require
that the maneuver be a ﬁxedattitude maneuver, a pitchover maneuver, or a
maneuver that rotates about a speciﬁed axis at a constant rate. The studies
presented here model the maneuver using an impulsive burn and frequently
permit the burn to vary in both magnitude and direction.
• Adjust the location of the LOI within the target orbit. This is typically held
constant, or varied only a small amount, since the maneuver is much more
efﬁcient when performed at the orbit’s periapse.
DESIGNING A LAUNCH PERIOD
307
• Adjust the geometry of the capture orbit. The spacecraft’s mission design
may permit the orbit’s argument of periapse to vary, particularly if the goal
is to eventually enter a circular orbit. It may also be permissible to vary
the inclination or longitude of ascending node of the orbit, though those are
typically not varied more than a small amount.
Arriving at the lunar surface. A mission to the lunar surface may be targeting a
shallow ﬂight path angle with the goal to land softly, or it may be targeting a
steep ﬂight path angle for a targeted impact, similar to the design of the Lunar
Crater Observatory and Sensing Satellite (LCROSS) mission. Some examples
of trajectory controls include:
• Adjust the date/time of arrival. This may vary by mere seconds or by days,
depending on the mission’s requirements.
• Adjust the arrival velocity.
• Adjust the arrival geometry. It may be permissible to vary the ﬂight path angle
and/or azimuth of the arrival.
• Adjust the arrival location on the lunar surface.
In addition, it may be possible to incorporate a dramatic shift in a mission’s
trajectory. For instance, it may be preferable to break a 21day launch period into
two halves, where the early portion of the launch period sends the spacecraft toward
the Sun–Earth L1 vicinity and the second portion implements trajectories that travel
near the Sun–Earth L2 vicinity.
One can see that there are many ways to adjust a trajectory from one launch
opportunity to the next in order to establish a launch period. This section presents
several scenarios and their corresponding algorithms that may be used to establish a
launch period. The algorithms presented here may need to be adjusted for a particular
mission, though the results presented here are certainly useful for guiding the early
trades for a mission.
6.5.2 An Example Mission Scenario
There are many ways to construct an extended launch period for a lowenergy lunar
transfer, some of which are outlined above. This section studies one mission design
architecture and applies that to a large number of practical cases, in order to generate
some useful statistics about that architecture. The design studied here is similar to
GRAIL’s mission: a spacecraft is launched from a parking orbit that has an inclination
of 28.5 deg, for example, one that effectively supports launches from Cape Canaveral,
and uses a nearballistic lowenergy transfer to target a low, 100km, polar orbit about
the Moon. The trajectory includes as many as two deterministic trajectory correction
maneuvers (TCMs) to assist the construction of a 21day launch period. The results
of the studies presented here for this architecture are (of course) only relevant to
308
OPERATIONS
very similar missions, but hopefully they shed some light on other lowenergy lunar
architectures.
Figure 61 illustrates one example trajectory taken from the surveys presented in
Section 4.4. This trajectory departs the Earth on April 1, 2010, at 05:27 Coordinated
Universal Time (UTC) from a 185km parking orbit with an inclination of approx
imately 38.3 deg and transfers to the Moon using no maneuvers at all. It arrives at
a polar orbit 100 km above the mean radius of the Moon. A launch vehicle may
certainly target an outbound inclination of 38.3 deg on that date to inject a spacecraft
onto this transfer, but it would suffer a large penalty to its lift capability if it did
so from Cape Canaveral, compared to the vehicle’s capability to lift payloads to an
inclination of 28.5 deg. Further, the launch may slip. This section studies how to
adjust the transfer to permit it to depart the Earth from an inclination of 28.5 deg on
multiple days. As an example, a new trajectory has been generated using the ballistic
transfer shown in Fig. 61 as a reference. The new trajectory departs the Earth a full
day after the reference, on April 2, 2010, and departs from a 28.5 deg parking orbit.
Two maneuvers are required to correct this new outbound trajectory so that it arrives
at the same lunar orbit as the reference. Figure 62 illustrates the difference between
Figure 61 An illustration of the example reference lowenergy lunar transfer, shown in
the Sun–Earth rotating frame from above the ecliptic, where the Sun is ﬁxed to the left
c 2012 by American Astronautical Society Publications Ofﬁce, San Diego,
[190] (Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
DESIGNING A LAUNCH PERIOD
309
Figure 62 The targeted Earth departure compared with the reference Earth departure [191]
(Acta Astronautica by International Academy of Astronautics, reproduced with permission of
Pergamon in the format reuse in a book/textbook via Copyright Clearance Center). (See insert
for color representation of this ﬁgure.)
the Earth departures of the reference trajectory and the newly adjusted trajectory.
Figure 63 shows the difference between these transfers, as viewed from above in the
Sun–Earth rotating reference frame. Finally, Fig. 64 shows a toplevel view of 21
such trajectories that depart the Earth on 21 different days and all arrive at the Moon
at the same time at the same orbit. The details of these trajectories, and whether or
not they should vary in any given way, is described later.
The performance of the launch period for this example mission depends on which
controls are available. For instance, if one is only permitted to vary the launch time
and the translunar injection, while keeping the dates of the trajectory correction
maneuvers constant and keeping the geometry of the lunar orbit insertion constant,
then the spacecraft must be capable of performing at least 730 m/s of ΔV to reach
a 100km circular polar orbit about the Moon. But if the dates of the TCMs are
permitted to vary as well as the magnitude and direction of the lunar orbit insertion,
then the spacecraft’s fuel budget may be reduced such that it must perform only
706 m/s of ΔV on the most challenging launch day of a 21day launch period.
However, these controls may not be available to the mission. Figure 65 illustrates
the total Δ V that must be performed for a spacecraft in each of ﬁve different
launch period conﬁgurations. One can see that the launch period ΔV cost may be
reduced even by adjusting a single parameter; for instance, Launch Period 3 requires
approximately 10.7 m/s less ΔV than Launch Period 2 and the only thing different is
that the date of the second TCM is performed 10 days later in each trajectory.
The illustrations shown here are representative of one example lunar mission.
This section explores several hundred such missions and characterizes any statistical
310
OPERATIONS
Figure 63 The ﬁnal targeted lunar transfer compared to the reference transfer, viewed in
the Sun–Earth rotating frame from above the ecliptic [191] (Acta Astronautica by International
Academy of Astronautics, reproduced with permission of Pergamon in the format reuse in a
book/textbook via Copyright Clearance Center).
Figure 64 An example of 21 trajectories that depart the Earth from 21 different days and
all arrive at the Moon at the same time, inserting into the same lunar orbit. The trajectories are
viewed in the Sun–Earth rotating frame from above the ecliptic.
DESIGNING A LAUNCH PERIOD
311
Figure 65 Several example launch periods for the example lunar mission, depending on
which controls are ﬁxed, their ﬁxed values, and which controls are permitted to vary. (See
insert for color representation of this ﬁgure.)
ﬁndings that provide mission managers rules of thumb for estimating the costs of
establishing a launch period for a given lowenergy transfer.
6.5.3
Targeting Algorithm
Each lunar mission and its corresponding launch period is constructed here using a
straightforward procedure that is described as follows. Once again, this algorithm
is formulated for missions to low lunar orbits, though it may be easily modiﬁed for
other destinations.
Step 1. First, a target lunar orbit is selected and a reference lowenergy lunar transfer
is constructed. The transfers used here have been taken from the surveys pre
sented in Section 4.4, which provides many more details about these transfers,
but to summarize, each transfer targets a low lunar orbit that is constructed
by setting its semimajor axis to 1837.4 km, its eccentricity to zero, and its
inclination to 90 deg in the International Astronomical Union (IAU) Moon
Pole coordinate frame. This deﬁnes a circular, polar orbit with an altitude of
approximately 100 km. Its longitude of ascending node, Ω, and argument of
312
OPERATIONS
periapse, ω, are selected from the surveys and can take on a wide variety of
combinations.
An impulsive, tangential LOI is applied at the orbit’s periapse point on a
speciﬁed date. The LOI ΔV magnitude is taken from the surveys. It is set
to generate a trajectory that originates at the Earth via a simple lowenergy
transfer: one that contains no close lunar encounters or Earthphasing orbits.
The ΔV value is at least 640 m/s and is the least ΔV needed to construct a
transfer that requires fewer than 160 days to reach an altitude of 1000 km or
less above the Earth when propagated backward in time. Table 63 summarizes
several example transfers that target low lunar orbits that each have an Ω of
120 deg; these may be seen in the surveys illustrated in Figs. 46, 47, and 48
and in Table 44 in Section 4.4.
Each reference trajectory generated in this study has no maneuvers and does
not target any particular Earth orbit when propagated backward in time.
Step 2. Second, the mission’s LEO parking orbit and translunar injection time are
speciﬁed. The LEO parking orbits used in this study are all 185km circular
orbits with inclinations of 28.5 deg, as previously described. The orbit’s node,
ΩLEO , and the location of the translunar injection (TLI) maneuver about the
orbit, νLEO , are permitted to vary; the TLI is performed impulsively and tangent
to the orbit. The values of Ω and νLEO may initially be set to any arbitrary
angle, for example, to 0 deg.
Step 3. If the LOI maneuver is permitted to vary, which it is in the majority of the
missions studied here, then the third step is to adjust the lowenergy transfer
such that its perigee passage occurs at the time of the TLI. This is performed
−→
by searching for the smallest change in the LOI ΔV that results in a new
lowenergy transfer that originates at the Earth on the date of the TLI, or at
Table 63 A summary of the performance parameters of several example simple
lowenergy lunar transfers. None of these transfers includes any Earth phasing orbits or
c 2012 by American Astronautical Society Publications
lunar ﬂybys [190] (Copyright ©
Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights reserved;
reprinted with permission of the AAS).
Traj
#
Ω
(deg)
ω
(deg)
ΔVLOI
(m/s)
Duration
(days)
LEO Inclination (deg)
Equatorial Ecliptic
1
2
3
4
5
6
120.0
120.0
120.0
120.0
120.0
120.0
169.2
103.8
70.2
225.3
99.9
186.9
669.3
692.1
743.9
716.0
697.5
673.2
83.483
85.287
93.598
93.621
110.060
122.715
29.441
25.688
57.654
134.322
83.127
23.941
6.129
34.778
74.955
112.840
61.624
3.088
C3
(km2 /s2 )
−0.723
−0.723
−0.667
−0.657
−0.697
−0.712
DESIGNING A LAUNCH PERIOD
313
least one that has a perigee on that date even if the perigee altitude is higher
than 1000 km. The optimization package sparse nonlinear optimizer (SNOPT)
was used for the missions presented here, but other algorithms may certainly
be used.
Step 4. The radius of the lowenergy transfer with respect to the Earth at a time
20 days after the TLI is noted. The TLI ΔV magnitude, ΔVTLI , is set to a
value that takes the Earthdeparture trajectory out to that distance at that time.
The spacecraft is beyond the orbit of the Moon by that time, assuming no
Earthphasing orbits, and not yet at its apogee.
Step 5. The values of ΩLEO and νLEO are adjusted to minimize the difference in
position between the Earthdeparture and the target lowenergy transfer at a
time 20 days after TLI. After convergence, the algorithm is repeated, this time
permitting Δ VTLI to vary as well. It is typically the case that the Earthdeparture trajectory will intersect the target lowenergy transfer at that time
when all three variables are permitted to vary, though it is not necessary. Once
again this study implemented the SNOPT package to perform the optimization.
Step 6. Two deterministic maneuvers are added to the trajectory: TCM1 at a time
21 days after TLI, and TCM2 at a time halfway between TCM1 and LOI. It is
intentional that the ﬁrst maneuver be placed near 20 days but not at a value of
20 days in order to improve the performance of the optimization algorithm in
the next step [183]. A singleshooting differential corrector (Section 2.6.5.1)
may be used to target the values of Δ VTCM1 and Δ VTCM2 to generate a
continuous endtoend trajectory.
Step 7. Finally, all control parameters are varied using an optimizer to minimize the
total transfer ΔV of the trajectory. This study again used the SNOPT package
to perform the optimization. The missions generated here permitted eight
control variables to vary: the three Earthdeparture parameters ΩLEO , νLEO ,
and ΔVTLI ; the dates of the two translunar maneuvers tTCM1 and tTCM2 ; and
y
, and ΔVzLOI . When
the three components of the LOI ΔV, namely, ΔVxLOI , ΔVLOI
the eight parameters are adjusted, an Earthdeparture trajectory is generated
out to the time of TCM1, a lunararrival trajectory is generated backward in
time from LOI to the time of TCM2, and a bridge trajectory is generated
connecting TCM1 and TCM2 using a singleshooting differential corrector
(Section 2.6.5.1). The total transfer ΔV that is minimized includes the sum of
the maneuvers that are typically required by the spacecraft, namely, the sum of
ΔVTCM1 , ΔVTCM2 , and ΔVLOI , but not the TLI ΔV. The dates of the TLI and
LOI are ﬁxed, and the dates of TCM1 and TCM2 are constrained to be at least
four days from any other maneuver to facilitate relaxed spaceﬂight operations.
When the optimizer has converged, the performance of the trajectory compared
with the reference lowenergy transfer is recorded. It is often the case that the
differential corrector will converge on a local minimum and not the global minimum;
314
OPERATIONS
hence, this process is repeated with adjustments in the eight parameters to identify
the lowest local minimum possible. This will be discussed more later.
To summarize, this procedure constructs a practical, twoburn, lowenergy lunar
transfer between a speciﬁed Earth departure and a speciﬁed lunar arrival. The altitude,
eccentricity, and inclination of the Earth parking orbit are speciﬁed and ﬁxed, as is the
date of the translunar injection maneuver. The target lunar orbit, the LOI position,
and the LOI date are all speciﬁed and ﬁxed. The TLI maneuver is constrained to
be tangential to the parking orbit, though the orientation of the parking orbit may
vary; the LOI maneuver is not constrained to be tangential. Finally, the dates of
two translunar maneuvers are permitted to vary, which therefore changes their ΔV
values.
To illustrate this entire targeting process, Table 64 tracks the eight control vari
ables that have been used to generate the adjusted trajectory shown in Figs. 62
and 63. That is, Table 64 shows the steps that were taken to adjust the trajec
tory from the reference ballistic transfer, which has an Earth departure inclination
of 38.3 deg on April 1, 2010, to the desired transfer, which has an Earth departure
inclination of 28.5 deg on April 2, 2010. The reference trajectory is summarized in
Step 1: the only control variables set are the components of the LOI ΔV. Step 2 does
not change any control variables and is hence not shown. Step 3 illustrates the small
change in the components of the LOI ΔV vector that are required to shift the timing of
the trajectory’s perigee from April 1, 2010 to April 2, 2010, coinciding with the TLI
maneuver, though the perigee altitude is now 5200 km. The adjustment amounts to a
difference of only 3.3 centimeter per second (cm/s) in the LOI ΔV magnitude. Steps
4–6 construct initial guesses for the departure parameters and place two deterministic
maneuvers en route to construct a complete endtoend trajectory. Finally, Step 7
includes the full optimization, where all eight parameters are permitted to vary and
the transfer ΔV is minimized.
During Step 4, initial guesses for ΩTLI and νTLI are needed. In this example they
are both set to 0 deg; however, it has been observed that the entire procedure may
converge to different local minima using different combinations of initial guesses for
these parameters. There are often two local minima that correspond to the typical
short and long coasts for the Earth departure. In addition, the process often converges
on different local minima depending on the propagation duration of the initial Earth
departure. Research indicates that it is typically computationally efﬁcient to perform
Steps 4–6 numerous times with different initial guesses and then send only the best
one or two trajectories into Step 7 [190, 191]. This process ensures that the majority
of local minima are explored without spending too much time in Step 7, which is
by far the most computationally demanding step. It is likely that additional small
improvements may be made, but this procedure generates a reliable estimate of the
minimum transfer ΔV given a reference lunar transfer.
Taking the preceding into account, this targeting algorithm yields a trajectory that
requires only 24.1 m/s of deterministic ΔV to compensate for the change in departure
inclination and departure date. This deterministic ΔV will vary throughout a full
launch period, but this is a small ΔV penalty compared to the cost of launching into
parking orbits at widely varying inclinations.
0.00
27.18
27.18
27.32
0.00
−25.00
−25.00
−25.08
1
3
4
5
6
7

deg
deg
#
ν
Ω
Step

m/s
ΔV
3196.79
3197.44
3196.77
3196.77
TLI Parameters
20.63
21.00

days
Δt
24.09
26.10

m/s
ΔV
TCM1
34.86
34.84

days
Δt
0.00
6.37

m/s
ΔV
TCM2
681.500
−87.732, −271.103, −583.138
−87.732, −271.103, −583.138
−87.732, −271.103, −583.138
−87.732, −271.103, −583.138
673.155
−87.728, −271.090, −583.108
−87.736, −271.118, −583.167
Transfer
ΔV, m/s
Total
ΔVx , ΔVy , and ΔVz
m/s, EME2000
LOI
Table 64 The history of the example lunar transfer’s control variables as the mission is constructed, where ΔtTCM1 is the duration of time
c 2012 by American Astronautical
between TLI and TCM1 and ΔtTCM2 is the duration of time between TCM1 and TCM2 [190] (Copyright ©
Society Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission of the
AAS).
DESIGNING A LAUNCH PERIOD
315
316
OPERATIONS
6.5.4
Building a Launch Period
The process described in 6.5.1 may be repeated for each day in a wide range of dates
to identify a practical launch period. The total transfer ΔV typically rises as the TLI
date is adjusted further from a reference trajectory’s TLI date. For the purpose of
these studies, a search is conducted that extends 30 days on either side of the reference
trajectory’s TLI date and the best, practical, 21day launch period is identiﬁed within
those 61 days. The 21 days of opportunities do not have to be consecutive, though
they are typically collected in either one or two segments. Since lowenergy transfers
travel beyond the orbit of the Moon, they may interact with the Moon as they pass
by, even if they pass by at a great distance. The Moon may boost or reduce the
spacecraft’s energy as it passes by, depending on the geometry; typically there is a
point in a launch period where the geometry switches and it is often beneﬁcial to
avoid launching on one or several days when the geometry is not ideal.
Figure 66 illustrates the transfer ΔV cost required to target the reference lunar
transfer studied in the previous section as a function of TLI date. Each transfer has
been generated using the procedure outlined previously, but with a different TLI date.
The trajectories that launch 5–6 days prior to the reference transfer are signiﬁcantly
perturbed by the Moon, though not perturbed enough to break the launch period into
two segments. This perturbation is also visible in Fig. 64, where a sudden change
in the geometry of the transfers appears. One can see that the least expensive 21day
launch period requires a transfer ΔV of approximately 706.2 m/s.
Figure 66 An example 21day launch period, constructed using the reference lunar transfer
c 2012 by American Astronautical Society Publications
presented in Fig. 61 [190] (Copyright ©
Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted
with permission of the AAS).
DESIGNING A LAUNCH PERIOD
6.5.5
317
Reference Transfers
A total of 288 reference transfers have been used to generate lunar missions with
realistic, 21day launch periods, each starting from a 28.5deg LEO parking orbit.
These reference trajectories have been randomly sampled from lowΔV, simple,
lowenergy transfers presented in the surveys found in Section 4.4. The trajectories
target low lunar orbits with any longitude of ascending node and with any argument
of periapsis, though the combination of those parameters must yield a satisfactory
reference transfer. The transfers arrive at the Moon at any of eight arrival times
evenly distributed across a synodic month between July 11, 2010 at 19:41 UTC and
August 6, 2010 at 20:37 UTC. The majority of the reference transfers sampled here
implement lunar orbit insertion maneuvers with magnitudes between 640 m/s and
750 m/s, though reference transfers have been sampled with LOI ΔV values as high
as 1080 m/s. These ΔV values correspond with the full cost of capturing and reducing
the orbit to a 100km circular orbit; although that process typically involves many
maneuvers, in this study it will be performed by one maneuver. Finally, reference
transfers have been sampled with transfer durations between 65 and 160 days. This
collection of reference transfers makes no assumptions about what sort of mission a
designer may be interested in, except that each transfer is simple, that is, it includes
no Earthphasing orbits nor lunar ﬂybys, and each transfer targets a polar lunar orbit.
6.5.6
Statistical Costs of Desirable Missions to Low Lunar Orbit
In general, the algorithms described in this section generate successful launch periods
with similar characteristics. Figure 67 illustrates the total transfer ΔV of several
example launch periods that have been generated from these reference transfers. One
notices that many of these launch periods include a single main convex ΔV minimum,
from which a 21day launch period is easily identiﬁed. Other ΔV curves include two
or more local minima. The launch periods are designed to have at most two gaps,
where each gap must be less than 14 days in extent. A particular lunar mission may
have different requirements dictating the breadth of each segment and/or gap, which
will likely change the launch period’s cost; the requirements used here are simply
representative of a real mission.
It has been found that most 21day launch periods among the 288 missions studied
include the reference launch date, though there are many examples that do not,
including two of those shown in Fig. 67. In some cases a practical launch period
may have extended further than 30 days from the reference launch date and required
less total ΔV. A particular lunar mission may certainly relax this constraint, but
these extended launch periods are not explored here in order to keep the constraints
consistent across every mission studied.
Once again, one also sees frequent lunar perturbations in the 288 launch periods
studied, much like the example launch period shown in Section 6.5.4. Since each
transfer in a particular launch period departs the Earth in approximately the same
direction, the Moon passes near the transfer’s outbound leg about once every synodic
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Figure 67 Several example curves that illustrate the postTLI ΔV cost of transferring
from a 28.5deg LEO parking orbit at different TLI dates to a given reference lowenergy
c 2012
transfer, including a highlighted 21day launch period in each case [190] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
month. This causes a brief jump in the launch period. Some transfers do not
experience any signiﬁcant perturbations due to their outofplane motion.
Figures 68 and 69 illustrate two additional views of the six example launch
periods shown in Fig. 67. Figure 68 shows the view of each trajectory in each of
the six launch periods as if viewed from above the ecliptic in the Sun–Earth rotating
frame, such that the Sun is toward the left in each plot. One notices that some of
these launch periods traverse toward the Sun and others traverse away from the Sun.
The transfers arrive at the Moon at the exact same point in each mission, but each
mission arrives at the Moon at different points in its orbit. The reference transfers are
sampled randomly to include a wide variety of different target lunar orbit geometries,
arrival times, and transfer durations. Figure 69 illustrates the proﬁle of a spacecraft’s
distance from the Earth over time while traversing each trajectory in each of the six
launch periods. One can see that each design involves trajectories with different
transfer durations, and trajectories that traverse to different maximum distances. The
optimization processes often shift the epochs of the trajectory correction maneuvers,
though one can see that the TCM epochs are sometimes shifted more on one trajectory
than on its neighbors, depending on the sensitivity of that variable on the trajectory’s
ΔV costs. Similarly, some TLI parameters are shifted more in one trajectory than its
DESIGNING A LAUNCH PERIOD
319
Figure 68 Each trajectory in each of the six launch periods illustrated in Fig. 67, viewed
from above the ecliptic in the Sun–Earth rotating frame. The Moon’s orbit is shown for
reference.
neighbors. It is likely that a mission designer would use these results to guide further
reﬁnements in the optimization of a real mission.
The examples shown in Figs. 67–69 illustrate six missions; the remainder of this
discussion focuses on the random sample of 288 similar missions. Figure 610 shows
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Figure 69 The distance from Earth over time for each transfer in the six launch periods
shown in Figs. 67 and 68. The distance to the Moon over time is shown for reference.
the range of the transfer ΔV values that are contained in each 21day launch period
in these 288 missions as a function of their reference transfer ΔV. As an example,
the launch period illustrated in Fig. 66 was generated using a reference transfer with
DESIGNING A LAUNCH PERIOD
321
Figure 610 The range of transfer ΔV values contained in each 21day launch period as a
function of the reference transfer ΔV shown in normal view (top) and exploded view (bottom)
c 2012 by American Astronautical Society Publications Ofﬁce, San Diego,
[190] (Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
a ΔV of 649 m/s (the ordinate of the plots in Fig. 610), and the resulting launch
period included missions that had transfer ΔV values between 670.6 and 706.2 m/s.
One can see that the majority of transfers studied here have reference transfer ΔV
values less than 750 m/s, though the transfers sampled include those with reference
ΔV values as great as 1080 m/s. The launch period ΔV range often starts above the
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mission’s reference ΔV, since each mission starts from a 28.5deg LEO parking orbit
and the reference transfer typically departs from some other inclination. In a few
cases, and one extreme case, the launch period ΔV range starts below the reference
ΔV. This is often possible when the reference transfer has a natural Earth departure
far from 28.5 deg and a change in the transfer duration reduces the total ΔV. The plots
in Fig. 610 clearly illustrate that the ΔV cost of establishing a 21day launch period
is highly dependent on the reference transfer’s total ΔV. The launch period ΔV cost
of these 288 example transfers requires approximately 71.67 ± 29.71 m/s (1σ) more
deterministic ΔV than the transfer’s reference ΔV.
The launch periods studied here include missions that depart the Earth on 21
different days, and the launch period ΔV cost is the ΔV of the most expensive
transfer in that set. The departure days do not need to be consecutive, as described
earlier. In general, increasing the number of launch days included in a launch period
increases the ΔV cost of the mission. Figure 611 shows a plot of the change in
the launch period ΔV cost of the 288 missions studied here as one adds more days
to each mission’s launch period, relative to the case where each mission has only a
single launch day. The line of best ﬁt through these data indicates that on average it
requires approximately 2.480 m/s per launch day to add days to a mission’s launch
Figure 611 The change in the launch period ΔV cost of the 288 missions studied here as a
function of the number of days in the launch period. The linear trend has a slope of 2.480 m/s per
c 2012 by American Astronautical Society Publications Ofﬁce,
launch day [190] (Copyright ©
San Diego, California (Web Site: http://www.univelt.com), all rights reserved; reprinted with
permission of the AAS).
DESIGNING A LAUNCH PERIOD
323
period. There is a signiﬁcant jump in the launch period ΔV when one moves from
a 1day launch period to a 2day launch period. This is due to the fact that the
Moon’s perturbations often produce a single launch day with remarkably low ΔV
requirements. The change in a launch period’s required ΔV would be more smooth
if the effects of lunar perturbations on the Earthdeparture leg were ignored.
It has been noted, when studying Fig. 67, that a launch period does not necessarily
include the reference launch date. However, it is expected that the transfer duration
of a reference trajectory may be used to predict a mission’s actual transfer duration.
Figure 612 tracks the range of transfer durations within each 21day launch period
studied here as a function of the mission’s reference transfer duration. One can see
that the range of transfer durations is indeed correlated with the reference transfer
duration. Furthermore, it has been found that the maximum transfer duration of the
288 launch periods is approximately 15.95 ± 8.66 days longer than the mission’s
reference duration, the minimum transfer duration is approximately 10.91 ± 7.75
days shorter than the reference duration, and the total number of days between the
ﬁrst and ﬁnal launch date of a given launch period may be estimated at approximately
26.86 ± 6.95 days. Hence, one may predict that a mission’s launch period will
include 21 of about 27 days, centered on a date several days earlier than the reference
launch date, if one constructs a 21day launch period using the same rules invoked
here.
Figure 613 tracks the range of ΔV costs associated with each launch period as a
function of the duration of the mission’s reference transfer. One can see that there is
a wide spread of transfer ΔV across the range of durations. As the reference transfer
duration drops below 90 days, the launch period ΔV cost climbs, which makes sense
because there is less time to perform maneuvers during the shorter transfers. Beyond
90 days, there are launch periods with low ΔV requirements for any transfer duration.
It is expected that the launch period’s ΔV cost is dependent upon the reference
transfer’s natural Earth departure inclination. It is hypothesized that a reference
transfer that departs the Earth with an inclination near 28.5 deg will generate a launch
period that requires less total ΔV than a reference transfer that departs the Earth
with a far different inclination. Figure 614 tracks the launch period ΔV cost of the
288 missions constructed here as a function of their reference departure inclination
values. The bottom plot in Fig. 614 observes the range of transfer ΔV values as a
function of the difference between the reference departure inclination value and the
target 28.5deg value. A line has been ﬁt to the maximum ΔV for each launch period
using a leastsquares approach, which yields the relationship:
Launch Period ΔV ∼ (0.470 m/s/deg) × x + 756.5 m/s
where x is equal to the absolute value of the difference between the reference departure
inclination and 28.5 deg. The sample set of lunar transfers includes lowΔV and high
ΔV missions, which may swamp any signiﬁcant relationship between the launch
period’s ΔV cost and the reference departure inclination. Nevertheless, it is very
interesting to observe that the launch period’s ΔV cost does not present a strong
correlation with the reference departure inclination.
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Figure 612 The range of transfer durations contained in each 21day launch period as
a function of the reference transfer duration. The plot at the bottom shows an exploded
c 2012
view, focused on transfer durations between 75 and 115 days [190] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
To further test the relationship of a launch period to the reference LEO inclination,
each launch period’s ΔV has been reduced by its reference ΔV so that each launch
period may be more closely compared. Figure 615 shows the same two plots as
DESIGNING A LAUNCH PERIOD
325
Figure 613
The range of transfer Δ V costs contained in each 21day launch
c 2012 by
period as a function of the reference transfer’s duration [190] (Copyright ©
American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
shown in Fig. 614, but with each mission’s reference ΔV subtracted from its launch
period ΔV range. One can see that the launch period ΔV is not well correlated with the
reference departure inclination. The linear ﬁt has a slope of only 0.206 m/s per degree
of inclination away from 28.5 deg. It appears that a 21day launch period absorbs
most of the ΔV penalty associated with inclination variations. The natural Earth
departure inclination of a transfer certainly varies with transfer duration, and it has
already been noticed that the launch period is often not centered about the reference
transfer’s TLI date. This result is useful, because it indicates that the natural Earth
departure inclination is not a good predictor of the launch period ΔV requirement
of a reference transfer. The relationship of the lowenergy transfer ΔV and the TLI
inclination is further explored in the next section.
6.5.7
Varying the LEO Inclination
The results presented previously in this section have only considered missions that
begin in a LEO parking orbit at an inclination of 28.5 deg relative to the Equator,
corresponding to launch sites such as Cape Canaveral, Florida. Spacecraft missions
certainly depart the Earth from other launch sites; launch vehicles from those sites
typically deliver the most mass to low Earth orbit if they launch into a parking
orbit at an inclination approximately equal to their launch site’s latitude. Hence, it
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OPERATIONS
Figure 614 The range of transfer ΔV costs contained in each 21day launch period as a
function of the reference transfer’s Earth departure inclination (top) and the absolute value
of the difference between the reference inclination and 28.5 deg (bottom) [190] (Copyright
c 2012 by American Astronautical Society Publications Ofﬁce, San Diego, California (Web
©
Site: http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
is of interest to determine the ΔV cost required to depart the Earth from any LEO
inclination and transfer to the same lunar orbit using a particular lowenergy reference
transfer.
DESIGNING A LAUNCH PERIOD
327
Figure 615 The same two plots as shown in Fig. 614, but with each mission’s reference
c 2012 by
ΔV subtracted from its 21day launch period ΔV range [190] (Copyright ©
American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
The algorithms described here have been used to generate missions that depart
the Earth from LEO parking orbits at a wide range of inclinations and then target
the same reference lowenergy transfer discussed earlier (described in Section 6.5.2
and illustrated in Fig. 61). The reference trajectory naturally departs the Earth on
April 1, 2010, from an orbital inclination of approximately 38.3 deg; hence, a mission
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that departs the Earth at that time from that orbit requires no deterministic maneuvers
en route to the Moon. Upon arrival at the Moon, the reference trajectory requires a
649.0m/s orbit insertion maneuver to impulsively enter the desired 100km circular
lunar orbit. Any mission that departs the Earth from a different inclination will
require deterministic TCMs and/or a different orbit insertion maneuver.
Figure 616 illustrates how the deterministic ΔV varies for missions that depart
the Earth at different LEO inclination values and target the same lunar orbit. The
dates and times of the translunar injection and lunar orbit insertion are ﬁxed. The
total transfer ΔV is shown on the top, and the difference between each mission’s
total ΔV compared to the reference transfer’s total ΔV is shown on the bottom. One
can see that the ΔV cost of the mission rises as a function of the difference between
the mission’s departure inclination and the reference transfer’s departure inclination.
The cost is approximately 0.97 m/s per degree of inclination change for missions with
LEO inclinations greater than 20 deg. The transfer cost increases much more rapidly
as a mission’s departure approaches equatorial. As the departure inclination drops,
the system gradually loses a degree of freedom: the LEO parking orbit’s ascending
node becomes less inﬂuential on the geometry of the departure. The ascending node
is no longer deﬁned for equatorial departures, and the lunar transfer requires greater
than 120 m/s more deterministic ΔV than the reference transfer.
As Fig. 616 illustrates, the total ΔV of a mission to the reference lunar orbit is
minimized if the LEO parking orbit has an inclination of 38.3 deg, provided that the
translunar injection is performed on April 1, 2010. If the TLI date is shifted, then the
optimal LEO inclination is likely to shift as well. Hence, the ΔV cost of a full 21day
launch period cannot be strictly predicted by observing the difference in inclination
between a desired LEO parking orbit and the reference departure.
Figure 617 illustrates three launch periods, corresponding to missions that depart
from LEO parking orbits with inclinations of 20, 50, and 80 deg. One can see that the
launch period shifts in time, illustrating that the transfer duration may signiﬁcantly
alter the reference trajectory’s natural departure inclination. Figure 618 illustrates
the total transfer ΔV for each launch opportunity of a 21day launch period departing
from a wide range of departure inclinations. One can see that the launch period ΔV is
dramatically higher for low inclinations and that the ΔV changes very little from one
inclination to another for higher inclination values. It is interesting that the missions
with higher inclinations require less ΔV than missions near the reference transfer’s
departure inclination. The lowΔV points in the lower left part of the plot correspond
to brief opportunities in those launch periods when the Moon passes through an ideal
location in its orbit to reduce the transfer ΔV.
6.5.8
Targeting a Realistic Mission to Other Destinations
The algorithms presented in Section 6.5.3 have been applied to the problem of
constructing realistic missions to low lunar orbit. The algorithms require little mod
iﬁcation for missions to other destinations, such as lunar libration orbits or the lunar
surface.
DESIGNING A LAUNCH PERIOD
329
Figure 616 How deterministic ΔV varies for different LEO inclination values. Top: The
total transfer ΔV for missions that depart the Earth on April 1, 2010, at different inclinations
and arrive at the same reference lunar orbit. Bottom: The difference in the total transfer
ΔV for these missions compared with the reference lowenergy transfer, which departs at
c 2012 by American Astronautical Society
an inclination of ∼38.3 deg [190] (Copyright ©
Publications Ofﬁce, San Diego, California (Web Site: http://www.univelt.com), all rights
reserved; reprinted with permission of the AAS).
Missions to the Lunar Surface. Certainly a mission to the lunar surface may ﬁrst
target an intermediate lunar orbit, such as a low lunar orbit or a lunar libration orbit.
Intermediate orbits provide some riskreduction in the case of a contingency, because
one may postpone the landing until the system is fully prepared to land. Alternatively,
one may construct a mission that is designed to land immediately upon arrival at the
Moon, with the option to divert into a parking orbit of some kind in the event of a
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OPERATIONS
Figure 617 Three launch periods for missions to the reference lunar orbit, where each
launch period is designed to accommodate a speciﬁc LEO inclination; namely, 20 deg (top),
50 deg (middle), and 80 deg (bottom). The Moon perturbs the outbound trajectories for those
c 2012
missions that launch about 5 days before the reference transfer [190] (Copyright ©
by American Astronautical Society Publications Ofﬁce, San Diego, California (Web Site:
http://www.univelt.com), all rights reserved; reprinted with permission of the AAS).
contingency. In this scenario, or in the scenario where the mission design has no
option but to land immediately, the targeting algorithms described in Section 6.5.3
may be easily modiﬁed to accommodate a lander instead of an orbiter.
A lander may be able to adjust its time of arrival or its incoming velocity magnitude,
ﬂight path angle, or ﬂight path azimuth. If these parameters must be held ﬁxed, for
example, to reduce the complexity, risk, or cost of the design, then one may instead
introduce a third trajectory correction maneuver, performed some signiﬁcant amount
of time prior to landing, in order to minimize the total launch period ΔV.
Missions to Lunar Libration Orbits. There are many reasons why a mission to a
lunar libration orbit, or other threebody orbit, would beneﬁt by designing a single
libration orbit and constructing a launch period that inserted the spacecraft into that
same libration orbit. For instance, a mission design team building a lunar lander
and/or sample return mission may be interested in focusing their efforts to validate
one speciﬁc landing sequence, and would have to spend a great deal more effort to
support 21 different landing sequences, with varying geometry and timing. It may
therefore be less expensive and more reliable to implement a mission that targets
DESIGNING A LAUNCH PERIOD
331
Figure 618 The total transfer ΔV for each opportunity of a 21day launch period for missions
to the reference lunar orbit, departing from LEO parking orbits with varying inclination values
c 2012 by American Astronautical Society Publications Ofﬁce, San Diego,
[190] (Copyright ©
California (Web Site: http://www.univelt.com), all rights reserved; reprinted with permission
of the AAS).
a particular lunar libration orbit, no matter which day it launches on, even if that
mission design required slightly higher ΔV budget.
Studies have demonstrated that the algorithms presented here may be used very
successfully in conjunction with a libration orbit insertion maneuver [183, 184].
6.5.9
Launch Period Design Summary
The goal of this section is to characterize the ΔV costs associated with building a
21day launch period for a practical mission to the Moon via a lowenergy transfer.
We have sampled 288 different lowenergy transfers between the Earth and polar
orbits about the Moon and have constructed practical 21day launch periods for each
of them, using a 28.5deg LEO parking orbit and no more than two deterministic
maneuvers. The lunar orbits have a wide range of geometries, though they are all
polar and have an altitude of approximately 100 km. The reference lowenergy
transfers include no Earthphasing orbits nor close lunar ﬂybys, and they require
between 65 and 160 days of transfer duration. Each mission has been constructed
by using a sequence of steps, varying eight parameters to minimize the transfer ΔV
cost. The eight variable parameters include the parking orbit’s ascending node, the
translunar injection’s location in the parking orbit, the translunar injection’s ΔV, the
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times of two deterministic maneuvers en route to the Moon, and three components
of the lunar orbit insertion maneuver. All other aspects of the transfer are ﬁxed when
building a particular mission.
Several conclusions may be easily drawn from the results presented here. First of
all, the cost of a launch period is obviously dependent on the number of launch days
in the period. The transfers constructed here demonstrate that it costs on average
approximately 2.5 m/s per day added to a launch period; hence, the average 21day
launch period requires about 50 m/s more deterministic ΔV than a 1day launch period
for a given transfer. The cost of a particular launch period may rise nonlinearly as
one adds days to the launch period, such that it may be the case that additional days
cost exponentially more ΔV or perhaps that additional days do not cost any additional
ΔV. The statistical cost of establishing a 21day launch period to the 288 reference
transfers studied in this section is approximately 71.7 ± 29.7 m/s (1σ), where the
additional ΔV of more than the 50 m/s is required to accommodate a departure from
a 28.5deg LEO parking orbit. The 21 opportunities in the launch period may be
on 21 consecutive days, and frequently are, but typically include one or two gaps.
The average launch period for these 288 missions requires a total of 27 days; the
vast majority of the launch periods may be contained within 40 days. Finally, we
have shown that there is no signiﬁcant trend between the total launch period ΔV for
these 288 missions and their reference departure inclination values or their reference
transfer durations, except for short transfers with durations below 90 days.
An additional study has been performed to observe how a mission’s ΔV changes
as a function of the particular LEO inclination selected. A mission that departs
at a particular time requires approximately 0.97 m/s more transfer ΔV per degree
of inclination change performed, assuming that the departure inclination is greater
than 20 deg. The total transfer ΔV cost increases dramatically as the departure
inclination approaches 0 deg. These trends change when considering a full 21day
launch period. The required launch period ΔV is still high for missions that depart
from nearly equatorial LEO parking orbits, but the variation in the launch period ΔV
is reduced for missions that depart at higher inclinations.
6.6
NAVIGATION
Spacecraft traversing lowenergy lunar transfers may be navigated in very similar
fashions to those following interplanetary transfers. Indeed, there are many sim
ilarities: the trajectories require many weeks, they traverse well beyond the orbit
of the Moon, they require trajectory correction maneuvers, etc. There are several
differences, including the fact that lowenergy lunar transfers remain captured by
the Earth, they are not wellmodeled by conic sections, and they are unstable. This
section discusses how these similarities and differences impact the navigation and
operation of the spacecraft during such transfers.
NAVIGATION
6.6.1
333
Launch Targets
Launch vehicle operators typically target three parameters when injecting spacecraft
onto interplanetary trajectories: one describing the target energy, namely, C3 , and
two angular measurements describing the orientation of the departure asymptote,
namely, the right ascension and declination of the launch asymptote, RLA and DLA,
respectively. Lowenergy lunar transfers, conversely, remain captured by the Earth
and do not have launch asymptotes. The GRAIL project used two similar target
parameters to describe the orientation of the departure ellipse—the right ascension
and declination of the instantaneous apogee vector (RAV and DAV, respectively)
at the time of the launch vehicle’s target interface point (TIP). Combined with the
target C3 parameter, these three targets describe a departure that keeps the expected
correction ΔV after the TIP to a minimum.
6.6.2
StationKeeping
As illustrated in Chapter 2, lowenergy lunar transfers are unstable; they depend
on a careful balance of the gravitational attraction of the Sun, Earth, and Moon.
Any random deviation from the designed trajectories will grow exponentially over
time. Therefore, a spacecraft traversing a lowenergy lunar transfer in the presence of
realistic uncertainties will require TCMs to remain on a desirable course. Fortunately,
lowenergy lunar transfers are stable enough that maneuvers are typically only needed
every 4–8 weeks, though more are needed to support any lunar approach and/or lunar
ﬂybys.
The cost of performing statistical corrections on a lowenergy lunar transfer may
be estimated by considering the stability of trajectories in each region of space that
the transfer passes through. First, typical spacecraft missions plan to perform a
maneuver soon after the translunar injection in order to clean up any injection errors.
For instance, the GRAIL mission planned to have both spacecraft perform a maneuver
within a week after injection. Next, the spacecraft spend 1–3 months traversing a
region of space far from the Earth, typically near the Sun–Earth L1 or L2 points.
The stability of this portion of the trajectory may be approximated by measuring the
stability of typical Sun–Earth libration orbits—as illustrated in Section 3.4.1. As the
spacecraft approach the Moon either for a lunar ﬂyby or for their ﬁnal lunar approach,
the trajectories become more unstable. The stability of the trajectories near the Moon
may be approximated by measuring the stability of typical Earth–Moon libration
orbits.
There are many ways to measure the stability of a trajectory, but a rather intuitive
way is to consider the trajectory’s perturbation doubling time, that is, the amount of
time it takes for a spacecraft to double its distance away from a reference trajectory
(see Section 2.6.8.3 on page 80). If at time t0 a spacecraft is 100 km away from its
reference trajectory, then at time t0 + τˆ the spacecraft will be approximately 200 km
away from its reference, and at time t0 + 2ˆ
τ the spacecraft will be approximately
400 km away from its reference, and so on, where τˆ is the perturbation doubling
time.
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Typical halo orbits about the Sun–Earth L1 and L2 points have τˆ values of about
17 days. Hence, one may assume that the position error of a spacecraft traversing
a lowenergy lunar transfer near those orbits will double roughly every ∼17 days.
Fortunately, this is a rather long time for most spacecraft operations unless the space
craft has particularly strict ﬂight path requirements. The GRAIL mission scheduled
two trajectory correction maneuvers per spacecraft while they traversed the region of
space near EL1 , though there were deterministic needs for those maneuvers as well.
The Genesis mission performed maneuvers every couple of months while traversing
its EL1 orbit, requiring only about 10 m/s per year of stationkeeping [87]. The Solar
and Heliospheric Observer’s (SOHO) spacecraft has demonstrated the ability to re
main in orbit about the EL1 point for even less ΔV. SOHO’s ﬁrst eight stationkeeping
maneuvers (SKMs) were executed between May 1996 and April 1998, imparting a
total ΔV of approximately 4.77 m/s: an average of one maneuver per 99 days with
an average maneuver ΔV of only 0.596 m/s [217].
As a spacecraft approaches the Moon, either for the targeted arrival or for a
lunar ﬂyby, its trajectory becomes more unstable and the perturbation doubling time
shrinks. If the spacecraft arrives at a lunar libration orbit via a transfer such as
those presented in Chapter 3 then its stability may be measured by the perturbation
doubling time of typical halo orbits about the Earth–Moon libration points. If the
spacecraft’s destination is a low lunar orbit, the lunar surface, or a ﬂyby, then this
measurement is only an approximation and more analysis is needed. Nevertheless,
typical halo orbits about the Earth–Moon L1 and L2 points have τˆ values of about 1.4
days. Not surprisingly, a spacecraft’s position error doubles about twelve times faster
in the Earth–Moon system than it does in the Sun–Earth system. Depending on the
mission, it may be necessary to perform maneuvers as often as 1–2 times per week to
traverse Earth–Moon libration orbits. Even so, the two Acceleration, Reconnection,
Turbulence, and Electrodynamics of the Moon’s Interaction with the Sun (ARTEMIS)
spacecraft successfully navigated several months of libration orbits about both the
Earth–Moon L1 and L2 points, demonstrating that such operations are viable.
There are two fundamentally different strategies that have been implemented
when designing the SKMs of historical missions, namely, tight control and loose
control. The term tight control describes a strategy where each SKM is designed
to bring the spacecraft’s trajectory back to a designed reference trajectory. The
International Sun–Earth Explorer3 (ISEE3) and Genesis spacecraft maneuvers are
good examples of missions that implemented tight stationkeeping control [218, 219].
This strategy is used when the spacecraft’s trajectory has particular requirements; for
the case of Genesis, the trajectory ultimately placed the spacecraft on a course to
enter the atmosphere for a landing in Utah. Each SKM is designed such that the
resulting trajectory intersects the reference trajectory at the time of the next planned
SKM. Conversely, a loose stationkeeping strategy describes one where a spacecraft
may travel anywhere within some wide corridor and the particular route taken is
not important. For instance, the SOHO spacecraft must remain in orbit about the
Sun–Earth L1 point, but the particular path about the L1 point is not important. Thus,
SOHO’s trajectory is reoptimized each time an SKM is designed [217]. SOHO’s
NAVIGATION
335
loose control has resulted in stationkeeping ΔV costs just over 2 m/s per year, nearly
four times lower than ISEE3’s tight stationkeeping control costs.
In summary, one may expect to perform a TCM soon after launch in order to
clean up injection errors, followed by TCMs every 4–8 weeks when the spacecraft is
traversing the cruise phase far from the Earth or Moon, followed by one or two TCMs
per lunar approach. If the spacecraft’s itinerary includes lunar libration orbits, or other
unstable threebody orbits, then one may expect to perform TCMs every 3–7 days
during those phases. The total navigation ΔV cost depends on the spacecraft and
its propulsion system’s performance, but it is certainly possible to navigate such
trajectories for a modest ΔV – on the order of 1–10 m/s per year.
6.6.2.1 StationKeeping Strategies Numerous stationkeeping strategies have
been formulated since investigators began applying libration orbits to practical space
craft mission designs [6, 119]. Most developments have been in support of ﬂight
projects and proposals that involved trajectories in the Sun–Earth system [218, 220–
230]. More recent investigations have examined stationkeeping strategies within the
Earth–Moon system, especially with the development and success of the ARTEMIS
mission [17, 186, 231–234]. Folta et al., surveyed a wide variety of stationkeeping
strategies with the purpose of applying a desirable strategy to the ARTEMIS mission
[231]. Ultimately, each of the ARTEMIS stationkeeping maneuvers was designed
using a gradientbased optimizer that ensured the spacecraft would remain on the li
bration orbit for the next few revolutions. This method kept the total stationkeeping
fuel cost low without requiring the generation of a reference trajectory. After the ma
neuvers were designed, a later study found that each of the maneuvers closely aligned
with the local stable eigenvector at that point of the spacecraft’s orbit [235]. This
conclusion has certainly prompted researchers to investigate if the stable eigenvector
is a good initial guess for a nearoptimal stationkeeping strategy.
While ARTEMIS employed a loose stationkeeping strategy very successfully, its
strategy was focused on the short term: ensuring that the spacecraft remained on
a desirable trajectory for the next few revolutions about the Lagrange point. There
is concern that any shortterm strategy may fail over the long term, resulting in a
trajectory that diverges from the desired orbit. Recent work has applied tools such as
the multipleshooting differential corrector to the goal of achieving a minimumΔV
longterm stationkeeping strategy [234]. This goal is a rich, challenging problem
with a wide variety of possible constraints and degrees of freedom available. The
solution may differ for each spacecraft mission, with its own operational constraints
and desirable mission characteristics.
In the following sections, we study several aspects of the stationkeeping problem
in order to provide a background for the general problem of stationkeeping on a
libration orbit. The reader is encouraged to explore different strategies, particularly
those surveyed by Folta et al. [231]. We present the results of several analyses, in
cluding tight and loose stationkeeping strategies. Typical lowenergy lunar transfers
are highly constrained, such that there are often not enough degrees of freedom avail
able to the mission designer to employ a loose stationkeeping strategy. However, if
336
OPERATIONS
the goal of the lowenergy lunar transfer is to enter a libration orbit and to remain
there, then a loose strategy may be beneﬁcial.
6.6.2.2 StationKeeping Simulations Each of the simulations studied here
uses the same set of assumptions, varying only one or two aspects of the stationkeeping problem in order to keep the results as comparable as possible. The sim
ulations employ models that are representative of the real solar system, with some
simpliﬁcations to speed up the computations. The DE421 ephemerides are used to
approximate the motion of the Sun, Moon, and planets (Section 2.5.3) and each of the
bodies is approximated as a pointmass using the masses presented in Section 2.2. So
lar radiation pressure is modeled using a constant solar ﬂux of 1.019794376 × 1017 N
and a ﬂat plate model where the mass of the spacecraft is 1000 kg and the surface area
is 10 m2 . The trajectories are integrated using JPL’s MissionAnalysis, Operations,
and Navigation Toolkit Environment (MONTE) software (Section 2.7.1).
Each simulation includes a truth set of dynamics and an estimated set of dynamics,
which differ enough to introduce dynamical errors into the navigation problem. The
truth set includes the gravitational forces of the Earth, Sun, and Moon and uses a
value of 1.00 for the coefﬁcient of radiation of the solar pressure. The estimated set of
dynamics also includes the gravitational forces of all of the other planets in the Solar
System and uses a value of 1.03 for the coefﬁcient of radiation. These perturbations
are somewhat arbitrary and have been selected to approximate the level of accuracy
of ﬂight operations.
The reference trajectory for the simulations is a southern halo orbit about the
Earth–Moon L2 point with a zaxis amplitude of approximately 10,000 km (see
Section 2.6.6.3 and Section 2.6.9.4). The reference epoch is January 1, 2017. The
perturbations depend on the reference epoch, though they will not likely impact the
results very much. It is more likely that the choice of orbit will change the results of
the simulations.
Each SKM in each of these simulations is generated using a similar process.
First, the state of the spacecraft is propagated from one time to the next using the
truth dynamics. At the time of a stationkeeping maneuver, the estimated state of
the spacecraft at that maneuver is computed by taking the truth state at that time
and perturbing it with orbit determination errors. The resulting state is used as
the initial state for the stationkeeping strategy, whatever it may be. Each stationkeeping strategy studied here involves propagating estimated trajectories into the
future. These trajectories are propagated using the estimated dynamics, which again
differ from the truth dynamics. Once a stationkeeping ΔV is determined, that ΔV is
applied to the true spacecraft state. Finally, a maneuver execution error is added to
the state as well, and the resulting state is propagated using the truth dynamics. This
process is repeated for each stationkeeping maneuver in the simulation.
The orbit determination errors are modeled as spherically symmetric distributions,
such that each of the three Cartesian position coordinates and each of the three
Cartesian velocity components is perturbed using independent Gaussian errors, with
zero mean and standard deviations of 100 meters (m) for position and 1 millimeter
per second (mm/s) for velocity. Hence, the orbit determination errors may be in
NAVIGATION
337
any direction, with net 1σ position uncertainties of approximately 173 m and net
1σ velocity uncertainties of approximately 1.73 mm/s. These errors are similar
in magnitude to those observed by the ARTEMIS mission navigators [235]. The
maneuver execution error model applies a similar spherically distributed error, such
that a Gaussian perturbation of zero mean and 2 mm/s is applied to each of the three
components, no matter what size of maneuver it is. Hence, the net 1σ uncertainty
is approximately 3.46 mm/s. The maneuver execution error could be a realization
of a burn duration error, an efﬁciency error, a pressure regulation error, etc. Since it
is not clear what is causing the error, the execution error component of the net ΔV
is not included in any computation of the average or total stationkeeping maneuver
ΔV cost presented below.
Finally, each simulation is repeated at least 30 times to generate statistical results.
6.6.2.3 Tight StationKeeping A very common tight stationkeeping strategy
is to correct a spacecraft’s trajectory in the presence of errors by building each stationkeeping maneuver to target the position of the spacecraft’s reference trajectory at the
time of the following stationkeeping maneuver. If all goes well, the stationkeeping
maneuver will execute perfectly, and the modeled dynamics will perfectly match the
true dynamics. In that case, the spacecraft would arrive at the reference trajectory at
the time of the next maneuver and perform that maneuver to match its velocity with
the reference trajectory. Of course, in reality the spacecraft never arrives precisely
on the reference, but must perform another maneuver to correct for additional errors.
Figures 619 and 620 illustrate this strategy. Figure 619 shows a topdown view
of the reference halo orbit with a very exaggerated trajectory attempting to follow it.
In this case, the SKM are performed at 1day intervals and the errors are huge, just
for visualization purposes. The illustration in Fig. 620 shows the difference between
the estimated and reference trajectory for a simulation that uses the proper error
distributions. The black curve is the truth trajectory, the “x”s indicate the estimated
state of the spacecraft at the time of each SKM, and the gray curves illustrate the
target trajectories built with the intention to return the spacecraft to the reference.
This tight stationkeeping strategy has been applied to a wide range of SKM
periods, including periods as short as 0.5 days and as long as 12 days. Figures 621
and 622 present the resulting range of maneuver ΔV costs. One can see many
interesting features in the results. First, Fig. 621 presents a clear trend such that the
average SKM magnitude grows as the duration of time between maneuvers grows.
One exception to this is that if the maneuvers are performed too frequently, the average
maneuver magnitude rises as the frequent maneuvers ﬁght their collective execution
errors. Second, Fig. 622 illustrates that there is a minimum in the total expected
stationkeeping ΔV cost that occurs at a period of approximately 3 days, requiring
slightly less than 2 m/s per year. If maneuvers are performed more frequently, fuel is
wasted combating frequent maneuver execution errors. If maneuvers are performed
less frequently, then the spacecraft has more time to drift exponentially away from the
reference. Third, the relationships between stationkeeping ΔV cost and maneuver
execution period are very smooth until the maneuvers are executed approximately
7–10 days apart. This duration is slightly longer than half of a revolution period
338
OPERATIONS
Figure 619 A topdown view of a spacecraft following a reference halo orbit using a tight
stationkeeping strategy in the presence of very large, exaggerated errors. Stationkeeping
maneuvers are executed once per day.
Figure 620 The distance between a spacecraft’s trajectory and its reference trajectory for
an example tight stationkeeping scenario, with maneuvers performed once per day.
NAVIGATION
339
Figure 621 The average stationkeeping ΔV cost as a function of the duration of time
between maneuvers.
Figure 622 The total annual stationkeeping ΔV cost as a function of the duration of time
between maneuvers.
about the halo orbit. It is hypothesized that the stationkeeping sensitivity grows
signiﬁcantly when the target is on the opposite side of the orbit.
Figure 622 clearly indicates that if a navigation team intends to reduce the stationkeeping cost of a spacecraft on this halo orbit then it is best to perform maneuvers
every 2–6 days. From an operational perspective, it is convenient to work on a
schedule where a maneuver design cycle is performed every seven days. If the team
can support the operational pace, the best strategy may be to design maneuvers every
3.5 days, knowing that if a maneuver is missed then the cost will not grow too high
340
OPERATIONS
after 7 days. If this is the case, then it is desirable to estimate the total stationkeeping
cost of a mission performing approximately two maneuvers per orbit.
The next question is to decide where in the orbit to perform those two maneuvers.
Recall from Section 2.6.2.3 that the parameter τ may be used to specify a location
about a halo orbit, much like the mean anomaly of a conic orbit. We will refer to
τ = 0 deg to be at the y = 0 plane crossing with positive yvelocity (in the synodic
reference frame), and τ increases at a constant rate as the spacecraft traverses the orbit.
We have simulated scenarios where we have placed one stationkeeping maneuver at
a τ value anywhere from 0 deg to 180 deg and the other stationkeeping maneuver
at a τ value of 180 deg greater than the ﬁrst. Figures 623 and 624 illustrate the
resulting stationkeeping ΔV cost of each of these scenarios.
One can draw several conclusions after observing the relationships presented in
Figs. 623 and 624. First, the overall stationkeeping cost is roughly the same
order of magnitude anywhere around the orbit, except for the spikes observed near
τ = 10 deg and τ = 170 deg. These spikes are rather unexpected features of these
curves. The SKMs become very sensitive to variations at those points in the orbit. In
contrast, the best places to perform SKMs on this particular halo orbit are at τ values
near 30 deg and 150 deg, where the total cost is below 6 m/s per year. Apart from
the spikes, the worst locations to perform maneuvers are at τ values of 90 deg and
270 deg, namely, where the orbits extend the furthest from the y = 0 plane. It is of
interest to note that the stationkeeping cost is relatively low at τ values of 0 deg and
180 deg, namely, where the orbits cross the y = 0 plane, where they approach the
closest and furthest from the Moon, and also where they have their greatest zaxis
excursions.
If the mission operations plan calls for frequent small maneuvers, such that it is
okay—and perhaps even expected—to skip a maneuver from time to time, then it is
Figure 623 The average stationkeeping ΔV cost for two maneuvers performed per
revolution 180 deg apart in τ , as a function of the τ value of the ﬁrst maneuver.
NAVIGATION
341
Figure 624 The total annual stationkeeping ΔV cost for two maneuvers performed per
revolution 180 deg apart in τ , as a function of the τ value of the ﬁrst maneuver.
of interest to measure the stationkeeping cost of a slightly different scenario. In this
variation, stationkeeping maneuvers are planned every 7 days, but each maneuver
is targeted to generate a trajectory that would bring the spacecraft to the reference
trajectory in only 3.5 days. In a perfect situation, the trajectory would ﬂy past the
reference trajectory halfway between each stationkeeping maneuver. In reality it
will likely ﬂy past the reference trajectory, though at some distance. Figure 625
illustrates the distance between the trajectory and reference of one example instance
of this scenario. One can see that the position differences pass very close to zero
after most of the maneuvers. Further, the maximum excursions from the reference
trajectory rise over time. The ﬁgure includes a linear ﬁt and a quadratic ﬁt of the
maximum excursions over time, and it is clear that both trends are growing.
This strategy may be generalized in order to understand how the cost of stationkeeping depends on the stationkeeping period and the duration of time between
each SKM and the target state. Numerous simulations are studied here, varying the
stationkeeping period and the target duration in order to study these relationships.
Figure 626 illustrates a few example scenarios where the SKMs are performed every
day, while their targets are 1, 2, 4, and 5 days into the future. This stationkeeping
period is likely to be far too rapid for any realistic ﬂight operations, but it is easier to
see the features of the plots. One can see that the strategy converges for the cases of
1, 2, and 4, but it does not converge if the target is 5 days into the future. In addition,
there is a trend that the spacecraft remains further from the reference trajectory if the
SKM targets a point further into the future.
Figures 627–629 illustrate the results of a wide range of scenarios, where the
stationkeeping period varies from 1 day to 13 days and the target duration varies from
0.5 days to 24 days. Figure 627 presents the total annual stationkeeping ΔV cost for
each combination. Figure 628 illustrates the average stationkeeping magnitude for
342
OPERATIONS
Figure 625 The position difference between the simulated trajectory and the reference
trajectory for a scenario where SKMs are performed every 7 days, targeting to the reference
trajectory at a point 3.5 days later.
each scenario. Figure 629 summarizes the average distance between the resulting
trajectory and the reference trajectory for each scenario. In each case, the scenarios
shaded white exceed the data range and are not viable stationkeeping strategies.
One can draw many conclusions studying these charts. First, if one studies the
line of solutions that corresponds to the scenarios where the stationkeeping period
is equal to the target time, one recovers the results shown in Figs. 621 and 622.
These ﬁgures also provide further evidence that the stationkeeping performance
degrades when the stationkeeping period and target time are both around 9 days.
It is interesting that there are periodic bands of target durations that converge to
successful stationkeeping strategies for a given stationkeeping period, that is, the
three nearvertical dark stripes in each ﬁgure. When looking back at Fig. 626, it
is apparent that some target durations yield scenarios where the trajectories must
travel farther from the reference trajectory before returning to the reference. If the
stationkeeping period is too rapid, or set at an undesirable resonant period, then the
distance from the reference trajectory at one SKM is greater than the distance at the
previous maneuver, and the strategy diverges.
Nevertheless, the performance of the stationkeeping strategy does not signiﬁ
cantly improve by targeting a point 10 or more days beyond the given SKM. Statis
tically there is some beneﬁt derived by permitting the target time to be different than
the stationkeeping period, though it is typically not far from being equal. Figure 630
illustrates this by plotting three curves, tracking the stationkeeping performance for
1, 3, and 7day stationkeeping periods. One can see that the global minimum of
NAVIGATION
343
Figure 626 The progression in the position difference between the simulated trajectory and
the reference trajectory for scenarios where SKMs are performed every day, but their targets
are 1, 2, 4, and 5 days into the future.
each curve shown indeed exists toward the right, where the target duration is around
27 days. But the beneﬁts are slight compared to targeting a few days downstream,
which is also a more stable and computationallyefﬁcient stationkeeping strategy.
6.6.2.4 Loose StationKeeping A large number of different strategies have
been investigated by researchers in order to attempt to reduce the stationkeeping ΔV
cost. We present one such loose strategy, namely, a strategy that keeps the spacecraft
in the desired region of space without targeting any sort of reference trajectory. For
additional strategies, see for example, Folta et al. [231].
The strategy studied here is designed to work for libration orbits and other tra
jectories that pierce the y = 0 plane with an xvelocity of approximately zero in
the synodic reference frame. Halo orbits pierce the y = 0 plane orthogonally in
the circular restricted threebody problem (CRTBP) and nearly orthogonally in a
highﬁdelity model of the Solar System. Lissajous orbits are permitted to have some
nonzero velocity in the zaxis at those crossings. The loose stationkeeping strategy
is designed to take advantage of these orbital features.
344
OPERATIONS
Figure 627 The total annual stationkeeping cost for a wide range of scenarios, where the
xaxis sets the amount of time between each SKM and its target point along the reference
trajectory, and the yaxis sets the amount of time between each maneuver and the next.
Figure 628
The average SKM magnitude for the same trade space given in Fig. 627.
The idea is that a given SKM is designed to target a trajectory that pierces the
y = 0 plane orthogonally at either the next crossing or a subsequent crossing. Doing
this ensures that the spacecraft remains in the vicinity of its libration orbit for at
NAVIGATION
345
Figure 629 The average distance between the trajectory and the reference trajectory for the
same trade space given in Fig. 627.
Figure 630 The annual stationkeeping ΔV cost for three stationkeeping periods as
functions of the duration between each SKM and the target point along the reference trajectory
for that maneuver.
346
OPERATIONS
least some time, given the size of the orbit determination and maneuver execution
errors. Mission designers typically start the design by targeting the next y = 0
plane crossing to have zero velocity in the xaxis; once that design is complete it
is used to seed a search for a maneuver that pierces the following y = 0 plane
crossing with zero velocity in the xaxis. When targeting the second y = 0 plane
crossing, all constraints on the ﬁrst y = 0 plane crossing are removed. This may
be repeated a few times, but modern integrators cannot typically integrate more than
two revolutions about a libration orbit (four y = 0 plane crossings) into the future
accurately enough to achieve further targets. The further this process extends into the
future, the more likely it is that the spacecraft will remain on the particular libration
orbit of interest. This algorithm permits the spacecraft’s Jacobi constant to change;
hence, the spacecraft may wander from one orbit to a neighboring orbit in the state
space.
This algorithm has been implemented and tested on scenarios that target the ﬁrst
through fourth y = 0 plane crossing. In each case, each SKM is performed at a y = 0
plane crossing and targets a future y = 0 plane crossing. There may be beneﬁt to
placing the SKMs at different τ values, or even permitting each maneuver’s τ value
to vary. But these strategies have not been explored here for brevity.
It has been found that a modiﬁed singleshooting differential corrector (Sec
tion 2.6.5.1 and Section 2.6.6.2) works very well to generate each SKM rapidly. One
formulates the problem by permitting the SKM to be in any direction, targeting a
state on the subsequent y = 0 plane crossing such that its xvelocity is zero. The
following equation is very similar to Eq. (2.40), modiﬁed for this application
δxT /2
⎢ 0
⎢ δzT /2
⎢
⎢−x˙ T /2
⎣ δy˙
T /2
δz˙T /2
⎡
⎤ ⎡
φ11
⎥ ⎢φ21
⎥ ⎢φ
⎥≈⎢ 31
⎥ ⎣
φ41
⎦
φ
51
φ61
φ12
φ22
φ32
φ42
φ52
φ62
φ13
φ23
φ33
φ43
φ53
φ63
φ14
φ24
φ34
φ44
φ54
φ64
φ15
φ25
φ35
φ45
φ55
φ65
⎡ 0 ⎤ ⎡ ⎤
⎤
φ16
x˙
⎢ 00 ⎥ ⎢y˙ ⎥
φ26 ⎥
⎥
φ36 ⎥ t , t ⎢
z˙ ⎥δ(T /2) (6.1)
⎢ ˙ 0⎥ + ⎢
⎢x
T /2 0 ⎢δx
⎥
¨⎥
φ46 ⎥
⎦
⎣ δy˙ 0 ⎦ ⎣y¨⎦
φ56
δz˙0
z¨
φ66
0
In this application, the value of δ(T /2) may be determined from the second line
of Eq. (6.1) to be
δ(T /2) =
−φ24 δx˙ 0 − φ25 δy˙ 0 − φ26 δz˙0
y˙
(6.2)
Substituting this value into the fourth line of Eq. (6.1) yields
−x˙ T /2 ≈
φ44 − φ24
x
¨
y˙
δx˙ 0 + φ45 − φ25
x
¨
y˙
δy˙ 0 + φ46 − φ26
x
¨
y˙
δz˙0 (6.3)
One now has a choice about how to construct the SKM. Since there are three
degrees of freedom and one control, this algorithm works very well for a mission
whose maneuvers are constrained. If there are no further constraints, it is typically
347
NAVIGATION
best to build the maneuver that minimizes the ΔV. We construct the least squares
solution as follows
M
�
δ x˙ 0
δ y˙ 0
δ z˙0
=
x
¨
φ44 − φ24 ,
y˙
�
=
MT MMT
x
¨
φ45 − φ25 ,
y˙
−1
φ46 − φ26
−x˙ T /2
x
¨
y˙
(6.4)
(6.5)
Table 65 summarizes the performance of this loose stationkeeping strategy for
different combinations of maneuver parameters, including the least squares solution
−→
ΔV0 , and each case where the maneuver is constrained to be in one Cartesian
direction (in the Earth–Moon rotating coordinate frame). Further, Table 65 includes
information for scenarios that target different target y = 0 plane crossings. One can
see that the least squares solution performs better than any singlecomponent solution.
The zaxis burns did not converge often enough to characterize their performance
for the case when the target was the ﬁrst y = 0 plane crossing. The table illustrates
very clearly that it is signiﬁcantly better to target the second or third y = 0 plane
crossing rather than the ﬁrst. This makes sense given the amount of oscillation that
exists in the system on account of the Moon’s noncircular orbit about the Earth–Moon
barycenter. These results suggest that targeting the second y = 0 plane crossing is
the most optimal of these loose stationkeeping strategies, applied to these particular
constraints, errors, and dynamics.
Table 65 A summary of the results of the loose stationkeeping strategy
explored here.
y = 0 ΔV
Target Type
1
1
1
1
2
2
2
2
3
3
3
3
−→
ΔV0
ΔVx0
ΔVy0
ΔVz0
−→
ΔV0
ΔVx0
ΔVy0
ΔVz0
−→
ΔV0
ΔVx0
ΔVy0
ΔVz0
Avg SKM ΔV (m/s)
Annual SKM ΔV (m/s)
Mean
1σ
Avg Slope from
Ref (km/day)
0.5317
0.3547
0.6116
0.4880
1.2691
0.7243
Failed to converge
25.6863
29.3714
61.2268
0.4614
0.5992
1.2253
7.9279
4.3580
22.4134
0.0643
0.0793
0.1512
1.2563
0.0525
0.0613
0.1455
1.0133
3.1067
3.8287
7.2116
60.2222
0.1557
0.1828
0.3583
3.7733
1.9986
1.7720
2.7830
12.8971
0.0667
0.0846
0.1536
1.1862
0.0522
0.0600
0.1567
0.9755
3.2276
4.0560
7.3782
56.6230
0.1837
0.2046
0.4482
2.6863
1.9252
1.4319
3.2306
14.4563
Mean
1σ
348
OPERATIONS
If we compare the loose stationkeeping strategy studied here with the tight stationkeeping strategy considered earlier, we see that the loose strategy performs better for
similar stationkeeping periods. However, the tight strategy performs better if a
mission can perform maneuvers more frequently, on the order of 3–4 days between
maneuvers.
6.6.2.5 Maneuver Execution Errors All of the results presented previously
have kept the spacecraft maneuver execution error model the same, namely, set such
that each coordinate of a maneuver’s execution is perturbed by an error taken from
a normal distribution with mean zero and standard deviation of 2 mm/s. This error
model is consistent with the errors observed from the ARTEMIS mission. Naturally,
the stationkeeping ΔV budget is dependent on this execution error model. Figure 631
presents the annual stationkeeping ΔV budget as a function of maneuver execution
error for a scenario where SKMs are performed at each y = 0 plane crossing, and
each maneuver targets the subsequent plane crossing of the reference trajectory. One
can see a very linear relationship between the annual ΔV cost and maneuver execution
error. The line of best ﬁt of this data is equal to
ΔV = 1.8705x + 1.9267 m/s
The curve’s linearity is promising in the sense that the stationkeeping strategy
has kept the trajectory within the vicinity of the reference trajectory enough that
linear approximations are valid. One notices also that the curve does ﬂatten out
as the maneuver execution error gets very small. It is in this regime that the orbit
determination errors begin to dominate the stationkeeping performance.
Figure 631
The annual stationkeeping ΔV cost as a function of maneuver execution errors.
SPACECRAFT SYSTEMS DESIGN
6.7
349
SPACECRAFT SYSTEMS DESIGN
Several considerations must be made to a spacecraft’s design when evaluating lowenergy lunar transfers compared to conventional lunar transfers. This discussion is
meant to guide further analysis and not to reveal a full list of potential issues that one
may have with a lowenergy transfer, compared with a conventional transfer.
First, lowenergy transfers require much more time than conventional transfers
between the Earth and the Moon. This impacts the operations schedule, its risk,
and its cost. A lowenergy transfer’s schedule is typically much more relaxed than
a conventional transfer’s schedule, which must perform a maneuver within a day or
even within hours of injection. A spacecraft operations team has much more time
to recover from anomalies and safemode events when ﬂying a lowenergy transfer.
The spacecraft team also has more time to characterize the spacecraft, check out
the instruments, outgas, and so forth. The mission may even delay maneuvers as
needed. In addition, there is much more time to ensure that a spacecraft is on a
proper approach vector when arriving at the Moon via a lowenergy transfer than a
conventional transfer.
The communications systems for spacecraft traversing lowenergy transfers must
be capable of reaching out to 1–2 million kilometers, depending on the transfer.
This is 3–5 times further than a conventional transfer. This long link distance
may require larger ground station antennas, larger spacecraft antennas, and/or more
communications power. However, a spacecraft intending to perform its mission
objectives at the Moon may not have much data to transmit at its apogee passage,
alleviating some of the pressures caused by the long link distance.
A lowenergy transfer requires a smaller maneuver when arriving at the Moon,
compared with a conventional transfer to the same destination. This fact may beneﬁt
a lunar mission in many ways. First, the spacecraft does not require as much fuel and
can put more of the ΔV requirements on the launch vehicle rather than the spacecraft.
Second, the spacecraft may reduce the amount of gravity losses when performing an
insertion into a low lunar orbit using small engines. This was the case for the two
GRAIL spacecraft, and it could be the case for any lunar landers. Finally, a mission to
a lunar libration orbit does not even require a large orbit insertion maneuver, which
may open up many design options.
Lowenergy transfers commonly traverse through regions of space where the
Sun–Earth–spacecraft angle and/or the Sun–spacecraft–Earth angle drops near zero
degrees. This characteristic may be detrimental to the communications system on
board the spacecraft, though it may only impact the mission for a day or two.
The ﬁnal consideration presented here is that lowenergy transfers typically do not
pass through the Van Allen Belts more than once, which may reduce the radiation risks
for a lunar spacecraft, compared with a conventional transfer that may implement
Earthphasing orbits.
APPENDIX A
LOCATING THE LAGRANGE POINTS
A.1
INTRODUCTION
The discussion given here, previously authored by Parker [46], is devoted to deriving
analytical expressions for the Lagrange points in the circular restricted threebody
problem (CRTBP). Szebehely provides more details and a clear description of this
derivation [86]. Other authors have provided similar derivations, including Moulton
[106] and Broucke et al. [236].
A.2
SETTING UP THE SYSTEM
Let us begin with a system of two masses, m1 and m2 , such that m1 ≥ m2 .
Furthermore, each of these masses is orbiting the center of mass of the system in
a circle. Then there exist cases where a third body, m3 , of negligible mass can be
placed in the system in such a way that the force of gravity from both bodies and
the rotational motion in the system balance to produce a conﬁguration that does not
change in time with respect to the rotating system. That is, each body rotates about
the center of mass at exactly the same rate and is seemingly ﬁxed in the rotating frame
351
352
LOCATING THE LAGRANGE POINTS
of reference. Euler and Lagrange located ﬁve of these cases, and those locations have
henceforth been known as the ﬁve Lagrange points in a threebody system.
To locate the Lagrange points, we begin with the three bodies stationary in the
corotating frame of reference. That is
θ˙1 = θ˙2 = θ˙3 = θ˙(t)
(A.1)
where θ˙i is the angular velocity of the body of mass mi about the center of mass.
Furthermore, if the shape of the conﬁguration does not alter over time, the relative
distances r12 (t), r23 (t), and r31 (t) are given by
r23 (t)
r31 (t)
r12 (t)
=
=
= f (t)
r23 (t0 )
r31 (t0 )
r12 (t0 )
(A.2)
So far, there are no constraints on the relative size of the conﬁguration, only on the
angular velocity and the shape of the conﬁguration.
Ri
Next, we move the origin to the center of mass of the conﬁguration. Then R
th
describes the vector position of the i mass, satisfying the constraint
3
3
Ri = 0
mi R
(A.3)
i=1
Equation (A.3) may be written
R 1 + m2 ( R
R2 − R
R 1 ) + m3 (R
R3 − R
R 1 ) = 0,
(m1 + m2 + m3 )R
or
R 1 = −m2Rr12 − m3Rr13
MR
(A.4)
where M is equal to the sum of the masses in the system. Squaring this relationship
produces
2
2
M 2 R12 = m22 r12
+ m23 r13
+ 2m2 m3Rr12 • Rr13
(A.5)
R i and Rri , respectively. Since
where Ri and ri denote the magnitudes of the vectors R
we know that the relative shape of the conﬁguration does not change, as seen above,
we may substitute in the relationships for the relative angles and distances (Eqs. (A.1)
and (A.2)) into Eq. (A.5) to ﬁnd that, in general
Ri (t) = Ri (t0 )f (t)
(A.6)
If Fi is the magnitude of the force per unit mass acting on the mass mi , then the
total force acting on mi is mi Fi and the equation of motion of that mass along the
direction of the force satisﬁes
¨ i − Ri θ˙2
mi Fi = mi R
i
(A.7)
Since all of the particles are rotating at the same rate, we can reduce this relationship
to the following
r
mi Fi = mi Ri (t0 )f¨(t) − Ri θ˙2
TRIANGULAR POINTS
353
or equivalently
mi Fi = Ri mi f¨(t)/f (t) − θ˙2
r
(A.8)
Hence, we have the proportionality relationship
F1 : F2 : F3 = R1 : R2 : R3
(A.9)
There are two cases that will satisfy the conditions given in Eqs. (A.8) and (A.9).
The two cases are
Ri × R
R¨ i = 0
R i × FRi = 0
or
R
(A.10)
R
When we set i = 1 and look at the ﬁrst particle, we have the following force function
R¨ 1 = G
m1 R
m1 m2
m1 m3
Rr12 + 3 Rr13
3
r12
r13
(A.11)
R 1 with each side of Eq. (A.11), we obtain the
When we take the cross product of R
following expression
R3 = 0
R 1 × m2 R
R 2 + m3 R
R
(A.12)
3
3
r12
r13
Using the center of mass relationship given in Eq. (A.3), this can be simpliﬁed to
R1 × R
R2
m2 R
1
1
− 3
3
r12
r13
=0
(A.13)
Once, again, there are two similar equations for the other two particles. For Eq. (A.13)
to hold, either of the following expressions must be true
r12 = r23 = r31 = r
(A.14)
(the equilateral triangle solution), or
R2 = R
R2 × R
R3 = R
R3 × R
R1 = 0
R1 × R
R
(A.15)
(the collinear solution).
The triangular and collinear cases are addressed separately in Sections A.3 and
A.4.
A.3 TRIANGULAR POINTS
In the equilateral triangle case given in Eq. (A.14), we arrive at the following rela
tionship for the ﬁrst particle
R
R¨ 1 + GM1 R1 = 0
R
R13
(A.16)
354
LOCATING THE LAGRANGE POINTS
where
m22 + m23 + m2 m3
M1 =
(m1 + m2 + m3 )
3/2
(A.17)
2
This result is the familiar twobody equation of motion. In this case, the ﬁrst particle
moves about the center of mass of the system in any conic orbit as if it had unit mass
and a mass of M1 were placed at the center of mass of the system. Each particle
moves in a corresponding trajectory, and the ﬁgure remains in an equilateral triangle
conﬁguration (although its size may oscillate or grow indeﬁnitely).
A.4
COLLINEAR POINTS
In the collinear case given in Eq. (A.15), we can also ﬁrst show that each particle’s
orbit is a conic section. Beginning with the ﬁrst particle, we can take the collinear
axis to be the x axis; the force acting on m1 is then
F1 = m2
(x3 − x1 )
(x2 − x1 )
+ m3
3
x12
x313
(A.18)
But we also know from Eq. (A.6) that
xi (t) = xi (t0 )f (t)
so that
F1 =
1
(x2 − x1 )
(x3 − x1 )
m2
+ m3
3
2
x12
x313
f
=
0
constant
f2
(A.19)
Since f is proportional to distance, m1 is acted upon by an inversesquarelaw central
force. Hence, the particle’s orbit is a conic section.
Now we will impose the condition from Eq. (A.9) that
F1 : F2 : F3 = x1 : x2 : x3 .
This condition introduces the proportionality constant A, such that
F1
F2
F3
=
Ax1
=
Ax2
=
Ax3
(A.20)
or equally
Ax1
=
Ax2
=
Ax3
=
x2 − x 1
x3 − x1
+ m3
x312
x313
x3 − x 2
x1 − x2
m3
+ m1
3
x23
x321
x1 − x 3
x2 − x3
m1
+ m2
x331
x332
m2
(A.21)
COLLINEAR POINTS
355
We are looking for the placement of the particle of mass m3 with respect to the other
two particles such that the relative positions are constant in the rotating frame. The
equilibrium positions possible for m3 are in the arrangements m1 − m3 − m2 (case
132), m1 − m2 − m3 (case 123), and m3 − m1 − m2 (case 312). Each case will be
observed separately.
A.4.1
Case 132: Identifying the L1 point
For case 132, we are looking for a positive value of X such that
x2 − x3
x32 ⎫
⎪
X=
=
⎬
x13
x3 − x1
x2 − x1
x12 ⎪
⎭
X +1=
=
x13
x3 − x1
(A.22)
We identify X using a series of steps. We ﬁrst subtract Ax1 from Ax3 and Ax3 from
Ax2 from Eq. (A.21) to arrive at Ax13 and Ax32
m1 + m3
+ m2
x213
m2 + m3
=−
+ m1
x232
Ax13 = −
Ax32
1
1
− 2
x232
x12
1
1
− 2
x213
x12
(A.23)
Using Eq. (A.22), we know that x32 = X x13 and x12 = (X + 1)x13 . When we
substitute these relationships into Eq. (A.23), we ﬁnd two different relationships for
the quantity Ax313 . When we set them equal and arrange in powers of X, we arrive
at Lagrange’s quintic equation
(m1 + m3 )X 5 + (3m1 + 2m3 )X 4 + (3m1 + m3 )X 3
− (3m2 + m3 )X 2 − (3m2 + 2m3 )X − (m2 + m3 ) = 0
(A.24)
We can use a quintic solver to solve for X (see Section A.5). Since the coefﬁcients
of Eq. (A.24) change sign only once, there can be only one positive real root. We can
then use that value for X to determine the relative location of the massless particle,
that is, the location of L1 , with respect to the other two particles by solving for x3 in
Eq. (A.22)
x2 − x3
x2 − x1
X=
⇒
x3 = x1 +
(A.25)
X +1
x 3 − x1
A.4.2
Case 123: Identifying the L2 point
For case 123, we are looking for a positive value of X such that
x3 − x2
x23 ⎫
⎪
X=
=
⎬
x12
x2 − x1
x3 − x1
x13 ⎭
⎪
X +1=
=
x12
x2 − x1
(A.26)
356
LOCATING THE LAGRANGE POINTS
In order to identify X, we follow a similar derivation as in case 132. We ﬁrst subtract
Ax2 from Ax3 and Ax1 from Ax2 from Eq. (A.21) to arrive at Ax23 and Ax12
m2 + m3
+ m1
x223
m1 + m2
=−
+ m3
x212
Ax23 = −
Ax12
1
1
− 2
2
x12
x13
1
1
− 2
2
x23
x13
(A.27)
We then substitute in X and (X + 1) from Eq. (A.26) as before, eliminate Ax312
between the resulting equations and arrange in powers of X to produce Lagrange’s
quintic equation
(m1 + m2 )X 5 + (3m1 + 2m2 )X 4 + (3m1 + m2 )X 3
− (m2 + 3m3 )X 2 − (2m2 + 3m3 )X − (m2 + m3 ) = 0
(A.28)
Once again, we can use a quintic solver to solve for X (see Section A.5), knowing
that again there is only one real positive root. We can then use that value for X to
determine the relative location of the massless particle, that is, the location of L2 ,
with respect to the other two particles by solving for x3 in Eq. (A.26)
X=
A.4.3
x3 − x2
x2 − x1
⇒
x3 = x2 + X(x2 − x1 )
(A.29)
Case 312: Identifying the L3 point
For case 312, we are looking for a positive value of X such that
x12 ⎫
x2 − x1
⎪
=
⎬
x31
x1 − x3
x2 − x 3
x32 ⎭
⎪
X +1=
=
x31
x1 − x3
X=
(A.30)
In order to identify X, we follow a similar derivation as in case 132. We ﬁrst subtract
Ax1 from Ax2 and Ax3 from Ax1 from Eq. (A.21) to arrive at Ax12 and Ax31
m1 + m3
+ m2
x231
m1 + m2
=−
+ m3
x212
Ax31 = −
Ax12
1
1
− 2
x212
x32
1
1
− 2
x231
x32
(A.31)
We then substitute in X and (X + 1) from Eq. (A.30) as before, eliminate Ax331
between the resulting equations and arrange in powers of X to produce Lagrange’s
quintic equation
(m1 + m3 )X 5 + (2m1 + 3m3 )X 4 + (m1 + 3m3 )X 3
− (m1 + 3m2 )X 2 − (2m1 + 3m2 )X − (m1 + m2 ) = 0
(A.32)
ALGORITHMS
357
Once again, we can use a quintic solver to solve for X (see Section A.5), knowing
that again there is only one real positive root. We can then use that value for X to
determine the relative location of the massless particle, that is, the location of L3 ,
with respect to the other two particles by solving for x3 in Eq. (A.30)
X=
x2 − x1
x 1 − x3
⇒
x3 = x1 −
x2 − x1
X
(A.33)
A.5 ALGORITHMS
The quintics given in Eqs. (A.25), (A.29), and (A.33) provide analytic determinations
of the locations of the ﬁrst, second, and third Lagrange points, respectively, in
the circular restricted threebody system. Szebehely outlines a ﬁxedpoint iterative
scheme that may be implemented to identify the single positive real root of each of the
quintic equations [86]. The fourth and ﬁfth Lagrange points make equilateral triangles
with the primaries; hence, their locations are easily determined using geometry.
Sections A.5.1–A.5.3 provide pseudocode that may be used to implement a
ﬁxedpoint iterative scheme to ﬁnd the xcoordinate of L1 – L3 , respectively. The
coordinate axis and the deﬁnition of µ are deﬁned in Section 2.5.1.
A.5.1
Numerical Determination of L1
γ0 =
µ(1 − µ)
3
1/3
γ = γ0 + 1
whileγ − γ0  > tol
γ0 = γ
γ=
µ(γ0 − 1)2
3 − 2µ − γ0 (3 − µ − γ0 )
endwhile
xL1 = 1 − µ − γ
1/3
358
LOCATING THE LAGRANGE POINTS
A.5.2
Numerical Determination of L2
γ0 =
µ(1 − µ)
3
1/3
γ = γ0 + 1
whileγ − γ0  > tol
γ0 = γ
γ=
µ(γ0 + 1)2
3 − 2µ + γ0 (3 − µ + γ0 )
1/3
endwhile
xL2 = 1 − µ + γ
A.5.3 Numerical Determination of L3
γ0 =
µ(1 − µ)
3
1/3
γ = γ0 + 1
whileγ − γ0  > tol
γ0 = γ
γ=
(1 − µ)(γ0 + 1)2
1 + 2µ + γ0 (2 + µ + γ0 )
endwhile
xL3 = −µ − γ
1/3
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TERMS
ΔV
Ω
3BSOI
ACE
ACS
AAS
AAS
AIAA
AMMOS
ARTEMIS
AU
change in velocity, delta velocity
ascending node
threebody sphere of inﬂuence
Advanced Composition Explorer
attitude control system
American Astronautical Society
American Astronomical Society
American Institute of Aeronautics and Astronautics
Advanced MultiMission Operations System
Acceleration, Reconnection, Turbulence and
Electrodynamics of the Moon’s Interaction with the Sun
astronomical unit, ∼149,600,000 kilometers
BIPM
BLT
Bureau International des Poids et Mesures
ballistic lunar transfer
377
378
ACRONYMS AND NOMENCLATURE
C3
CH1
cm/s
CRTBP
launch injection energy parameter
Chandrayaan1
centimeter per second
circular restricted threebody problem
DAV
DE
deg
DLA
DPO
DRO
DSN
DUNE
declination of apogee vector
developmental ephemerides, e.g., DE421
degree
declination of launch asymptote
distant prograde orbit
distant retrograde orbit
Deep Space Network
Dust Near Earth
EDL
EL1 / EL2
EM
EME2000
EMO2000
EPO
ET
entry, descent, and landing
Sun–Earth Lagrange point 1 / 2
Earth–Moon
Earth Mean Equator and Equinox of J2000
Earth Mean Orbit of J2000
Earthphasing orbits
Ephemeris Time, also called Dynamical Time
FPA
FPAz
ﬂight path angle
ﬂight path azimuth angle
GEO
GM
GPS
GRAIL
geosynchronous Earth orbit
gravitational constant × mass
Global Positioning System
Gravity Recovery and Interior Laboratory
HGS1
name given to AsiaSat 3 after AsiaSat 3 failed to get a
correct orbit and was transferred to Hughes Global
Services, Inc.
ACRONYMS AND NOMENCLATURE
379
IAG
IAU
IBEX
ICE
ICRF
ISEE3
ISRO
ISTP
International Association of Geodesy
International Astronomical Union
Interstellar Boundary Explorer
International Cometary Explorer
International Celestial Reference Frame
International Sun–Earth Explorer3
Indian Space Research Organization
International Solar Terrestrial Physics
J2000
JPL
JSC
currently used standard equinox for January 1, 2000
Jet Propulsion Laboratory
Johnson Space Center
km
km/s
km2 /s2
km3 /s2
KSC
kilometer
kilometers per second
kilometers squared per second squared
cubic kilometer per second squared
Kennedy Space Center
L1
L2
L3
L4
Lagrange point 1, between the two primary bodies
Lagrange point 2, on the far side of the smaller primary
Lagrange point 3, on the far side of the larger primary
Lagrange point 4, leading the smaller primary in its orbit
about the barycenter
Lagrange point 5, trailing the smaller primary in its orbit
about the barycenter
Lunar Crater Observation and Sensing Satellite
low energy
lowEarth orbit
Earth–Moon lunar Lagrange point 1 / 2
lowlunar orbit
lunarorbit insertion
lunar principalaxis bodyﬁxed
L5
LCROSS
L.E.
LEO
LL1 / LL2
LLO
LOI
LPABF
380
ACRONYMS AND NOMENCLATURE
LRO
LPO
LSP
LST
LTool
LTST
Lunar Reconnaissance Orbiter
librationpoint orbit
Launch Services Program
Local Solar Time
Libration Point Mission Design Tool
Local True Solar Time
MARS
MGSS
MI
mm/s
mo
MONTE
m/s
MUSES
MidAtlantic Regional Spaceport
Multimission Ground System and Services Ofﬁce
manifold insertion
millimeters per second
month
Missionanalysis, Operations, and Navigation Toolkit
Environment
meters per second
Mu Space Engineering Spacecraft (Hiten) A
NASA
NLS
National Aeronautics and Space Administration
NASA Launch Services
OLST
Orbit Local Solar Time
PSLV
Polar Satellite Launch Vehicle
RAV
RFK78
RLA
right ascension of apogee vector
RungeKuttaFehlberg seventhorder (integrator)
right ascension of launch asymptote
SE
SELENE
SI
SKM
SMART1
Sun–Earth
Selenological and Engineering Explorer
Syste` me International
stationkeeping maneuver
Small Missions for Advanced Research in Technology 1
ACRONYMS AND NOMENCLATURE
SNOPT
SOHO
SOI
SQP
SRM
sparse nonlinear optimizer
Solar and Heliospheric Observatory
sphere of inﬂuence
sequential quadratic programming
state relationship matrix
TAI
TIP
TLC
TLI
TOF
TT
Temps Atomique International / International Atomic
Time
trajectory correction maneuver
Barycentric Dynamic Time
Time History of Events and Macroscale Interactions
during Substorms
targeting interface point
translunar cruise
translunar injection
time of ﬂight
Terrestrial Time
USA
USSR
UT
UTC
UTTR
United States of America
Union of Soviet Socialist Republics (Soviet Union)
Universal Time
Coordinated Universal Time
Utah Test and Training Range
VLBI
very long baseline interferometry
WMAP
WSB
Wilkinson Microwave Anistropy Probe
weak stability boundary
yr
year
TCM
TDB
THEMIS
381
Constants
AU
c
C
Dm
Re
Rm
G
GMe
GMm
GMem
GMs
µem
µse
382
astronomical unit
1.49597871×108
speed of light
299,792.458
Jacobi constant (see Eq. 2.6)
mean distance between the
384,400
Earth and Moon
mean equatorial radius of the
6378.1363
Earth
mean equatorial radius of
1737.4
the Moon
universal gravitational constant 6.67300×10−20
gravitational parameter of
398,600.432897
the Earth
gravitational parameter of
4902.800582
the Moon
gravitational parameter of the
403,503.233479
Earth–Moon Barycenter
gravitational parameter of the Sun 1.32712440×1011
threebody constant of the
0.0121505856
Earth–Moon system
threebody constant of the
3.04042339×10−6
Sun–Earth/Moon system
km
km/s
km
km
km
km3 /s2 /kg
km3 /s2
km3 /s2
km3 /s2
km3 /s2