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ORBIT DESIGN AND CONTROL OF
PLANETARY SATELLITE ORBITERS IN THE
HILL 3-BODY PROBLEM
by
Marci Paskowitz Possner
A dissertation submitted in partial fulfillmentof the requirements for the degree of
Doctor of Philosophy(Aerospace Science)
in The University of Michigan2007
Doctoral Committee:
Associate Professor Daniel J. Scheeres, ChairProfessor Pierre T. KabambaProfessor N. Harris McClamrochAssociate Professor Thomas Zurbuchen
Marci Paskowitz Possnerc© 2007
All rights reserved.
To my husband, Adam Possner,
and my parents, Sharon & Jerry Paskowitz
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ACKNOWLEDGEMENTS
Completing a PhD program, like exploring the farthest reaches of space, is not a
one person endeavor. Many individuals have helped to make this accomplishment
possible.
First and foremost, I’d like to thank my advisor Professor Daniel J. Scheeres.
Professor Scheeres met with me every week during my time at Michigan except while
he was on sabbatical, and that’s only because he was halfway around the world in
Japan. Without his patient guidance and technical insights, this thesis would never
have gotten off the ground. I’d also like to thank the rest of my committee members,
Professors McClamroch, Kabamba and Zurbuchen, for taking the time to evaluate
my work.
Not many graduate students have the luxury of focusing on their research without
also worrying about where their funding is going to come from. The generous
support of the FXB Foundation truly provided the environment that allowed me to
make this endeavor everything that it could be.
I’d also like to thank the members of the Project Prometheus Jupiter Icy Moons
Orbiter Mission Design team at the Jet Propulsion Laboratory. For nearly a year
and a half, I had the pleasure of collaborating with Jon Sims, Lou D’Amario,
John Aiello, Ryan Russell and Try Lam, among others. Furthermore, I’d like to
acknowledge financial support from the Prometheus Project at JPL.
Finally, I’d like to thank my family. My husband, Adam, has been a tremendous
source of strength for me. With his constant love and encouragement, he never let
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me forget that I could finish this dissertation. My parents, Sharon and Jerry, who
always tell me how proud they are of me, and who have supported me in everything
that I’ve done, even when it meant leaving Canada. And my siblings, Erin and
Robbie, who have always supported their big sister.
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PREFACE
The work described in this thesis draws upon the following papers that have been
published, presented at conferences or will be submitted for publication. However,
the organization of the dissertation does not strictly follow the organization of each
paper, as this dissertation gives a full picture of our research.
Journal papers:
• [30] Marci E. Paskowitz & Daniel J. Scheeres, “Robust Capture and Transfer
Trajectories for Planetary Satellite Orbiters,” Journal of Guidance, Control
and Dynamics, 29(2), pp. 342-353, 2006.
• [29] Marci E. Paskowitz & Daniel J. Scheeres, “The Design of Science Orbits
About Planetary Satellites: Application to Europa”, Journal of Guidance,
Control and Dynamics, 29(5), pp. 1147-1158, 2006.
• [32] Marci Paskowitz Possner and Daniel J. Scheeres “Control of Science Orbits
About Planetary Satellites”, in preparation.
Conference papers:
• [26] Marci E. Paskowitz & Daniel J. Scheeres, “Orbit Mechanics About
Planetary Satellites,” paper presented at the 2004 AAS/AIAA Space Flight
Mechanics Meeting, Maui, Hawaii, February 2004, AAS 04-244.
• [25] Marci E. Paskowitz & Daniel J. Scheeres, “Identifying Safe Zones for
Planetary Satellite Orbiters,” paper presented at the 2004 AIAA/AAS
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Astrodynamics Specialist Conference, Providence, Rhode Island, August 2004,
AIAA 2004-4862.
• [27] Marci E. Paskowitz & Daniel J. Scheeres, “Orbit Mechanics About
Planetary Satellites Including Higher Order Gravity Fields,” paper presented
at the 2005 AAS/AIAA Space Flight Mechanics Meeting, Copper Mountain,
Colorado, 2005, AAS 05-190.
• [28] Marci E. Paskowitz & Daniel J. Scheeres, “Transient Behavior of Planetary
Satellite Orbiters,” paper presented at the 2005 AAS/AIAA Astrodynamics
Specialist Conference, Lake Tahoe, California, August 2005, AAS 05-358.
• [31] Marci E. Paskowitz & Daniel J. Scheeres, “A Toolbox For Designing
Long-Lifetime Orbits About Planetary Satellites: Application to JIMO at
Europa,” invited paper presented at the 2006 AAS/AIAA Space Flight
Mechanics Meeting, Tampa, Florida, 2006, AAS 06-191.
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TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . iii
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of Results . . . . . . . . . . . . . . . . . . . . . . . 41.2 Organization of the Dissertation . . . . . . . . . . . . . . . . 7
2. OVERVIEW OF THE DYNAMICAL SYSTEM . . . . . . . . 92.1 The Three Body and Restricted Three Body Problems . . . . 102.2 The Hill 3-Body Problem . . . . . . . . . . . . . . . . . . . . 132.3 The Modified Hill 3-Body Problem . . . . . . . . . . . . . . . 18
3. FROZEN ORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Reduction of the System by Averaging . . . . . . . . . . . . . 313.2 Frozen Orbits: Equilibrium Solutions of the 1-DOF System . 373.3 Stability of Frozen Orbit Solutions . . . . . . . . . . . . . . . 453.4 Contour Plots and Frozen Orbit Integrations . . . . . . . . . 54
4. ROBUST CAPTURE AND TRANSFER TRAJECTORIES 634.1 Non-dimensional Hill 3-Body Problem and Libration Points . 644.2 Periapsis Poincare Maps . . . . . . . . . . . . . . . . . . . . . 664.3 Symmetry Between Escape and Capture Trajectories . . . . . 744.4 Safe Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Transferring to Stable Frozen Orbits . . . . . . . . . . . . . . 81
5. THE DESIGN OF LONG LIFETIME SCIENCE ORBITS . 935.1 Dynamics in the 1-DOF System . . . . . . . . . . . . . . . . 955.2 Computing Initial Conditions in the 3-DOF System . . . . . . 1005.3 Long Lifetime Orbits in the 3-DOF System . . . . . . . . . . 1095.4 A Toolbox for Computing Long Lifetime Orbits . . . . . . . . 120
6. CONTROL OF LONG LIFETIME ORBITS . . . . . . . . . . 122
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6.1 Low-Thrust Control . . . . . . . . . . . . . . . . . . . . . . . 1246.2 Resetting a Long Lifetime Orbit . . . . . . . . . . . . . . . . 1356.3 Orbit Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 1456.4 Long Lifetime Orbits with Initial Errors . . . . . . . . . . . . 149
7. CONCLUSIONS AND FUTURE DIRECTIONS . . . . . . . 1567.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . 1577.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 159
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
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LIST OF FIGURES
2.1 Schematic of the Circular Restricted 3-Body Problem . . . . . . . . 122.2 Schematic of the Hill 3-Body Problem . . . . . . . . . . . . . . . . . 142.3 Trajectory integration in the Modified Hill 3-body problem (Third
degree and order gravity field) . . . . . . . . . . . . . . . . . . . . . 222.4 Trajectory integration using the full ephemeris model, and Europa’s
gravity field up to third degree and order [3]. . . . . . . . . . . . . . 232.5 Trajectory integration in the Modified Hill 3-body problem (truncated
gravity field) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Trajectory integration in the Elliptic Modified Hill 3-body problem . 29
3.1 A frozen orbit in the 1-DOF system integrated over several orbitalperiods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Frozen orbit solutions for the tide-only case. . . . . . . . . . . . . . 403.3 Comparison of frozen orbit solutions with and without J2. . . . . . . 423.4 Frozen orbit solutions for J2 6= 0 and J3 > 0. . . . . . . . . . . . . . 433.5 Frozen orbit solutions with larger and smaller values of J3. . . . . . 453.6 Stability of frozen orbits about Europa for the tide-only case. . . . . 473.7 Stability of the frozen orbits for the tide plus J2 case. . . . . . . . . 483.8 Stability for frozen orbits about Europa that include the tidal term,
J2 and J3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.9 Characteristic times for the tide-only unstable frozen orbits with 100
km altitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.10 Characteristic times for unstable frozen orbits about Europa . . . . 513.11 Stability and characteristic times of frozen orbits about Europa with
J3 one order of magnitude smaller than Table 3.1 . . . . . . . . . . . 523.12 Stability and characteristic times of frozen orbits about Europa with
J3 one order of magnitude larger than Table 3.1 . . . . . . . . . . . 533.13 Contour plots for motion in the vicinity of a stable frozen orbit with
e∗ = 0.548, i∗ = 50o, J3 sin ω > 0. . . . . . . . . . . . . . . . . . . . . 553.14 Contour plots for motion in the vicinity of an unstable frozen orbit
with e∗ = 0.0.039, i∗ = 55o, J3 sin ω > 0. . . . . . . . . . . . . . . . . 563.15 Contour plots for motion in the vicinity of a stable frozen orbit with
e∗ = 0.047, i∗ = 30o, J3 sin ω < 0. . . . . . . . . . . . . . . . . . . . . 583.16 Contour plots for motion in the vicinity of an unstable frozen orbit
with e∗ = 0.021, i∗ = 80o, J3 sin ω < 0. . . . . . . . . . . . . . . . . . 59
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3.17 Contour plots and integrations in the 3-DOF of various frozen orbits 62
4.1 Regions of allowable motion and libration points for J = −2.15. . . . 654.2 Periapsis Poincare maps for J = −2.15 (planar case). . . . . . . . . 694.3 Poincare maps for various values of J . . . . . . . . . . . . . . . . . . 714.4 First periapsis passage region for J = −2.15 (3-d case). . . . . . . . 724.5 Inclination as a function of normalized radius for 3-D capture trajec-
tories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.6 Poincare maps showing symmetric capture and escape regions for J =
−2.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7 A capture trajectory that escapes after 4 periapsis passages. . . . . . 764.8 Poincare map showing the safe zone, impact and escape regions. . . 794.9 Poincare maps of planar safe zones for various values of J . . . . . . . 794.10 Characteristics of safe capture trajectories for J = −1.70. . . . . . . 814.11 Periapsis passages of capture trajectories for J = −1.70 with altitudes
≤ 250 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.12 Potential capture trajectories with J = −1.60 for transfers to elliptic
frozen orbits and costs. . . . . . . . . . . . . . . . . . . . . . . . . . 854.13 Contour plots and the numerical results for the orbit obtained after
the transfer maneuver. . . . . . . . . . . . . . . . . . . . . . . . . . 864.14 Contour plots and numerical results for a transfer orbit to a circular
frozen orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.15 Characteristics of the near-frozen elliptic transfer orbits. . . . . . . . 894.16 Characteristics of both possible circularized orbits over one week. . . 894.17 ω as a function of eccentricity for safe capture trajectories with J =
−1.60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 Eccentricity as a function of time for trajectories with different initialvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Frozen orbit solutions: unstable near-circular, near-polar orbits indi-cated by heavy dashed line. . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Stable and unstable manifolds for a frozen orbit, identified with an’x’, with e∗ = 0.0129, i∗ = 70o, ω∗ = −90o and a∗ = 1682.5 km. . . . 98
5.4 Expansion of the central region in 5.3(b) with Europa impact circle . 995.5 3-DOF system integration of a frozen orbit. . . . . . . . . . . . . . . 1005.6 Integration of trajectory initialized on 1-DOF manifold. . . . . . . . 1025.7 Integration of trajectories initialized at xm
0 + δxm0 in 2-DOF system
and xm0 in 1-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . 106
5.8 Integration of trajectories initialized at x0+δx0 in 3-DOF system andx0 in 2-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.9 Integration in the 3-DOF of a trajectory initialized at x0 with the1-DOF manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.10 Time history of the eccentricity for a long lifetime orbit. . . . . . . . 1115.11 1-DOF manifolds with minimum eccentricity circle (dashed curve). . 1125.12 Long lifetime orbits about Europa (70o frozen orbit inclination) . . . 117
x
5.13 Long lifetime orbit about Enceladus (70o frozen orbit inclination) . . 1195.14 Long lifetime orbit about Dione (70o frozen orbit inclination) . . . . 120
6.1 Time histories of orbital elements integrated in the controlled 1-DOFsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Time histories of orbital elements integrated in the uncontrolled 1-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3 Integration of the 3-DOF system controlled with Fr. . . . . . . . . . 1296.4 Manifold in the 1-DOF system and integration of the 3-DOF system
controlled with Fr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.5 Time histories of total thrust magnitude and components of thrust. . 1316.6 Total ∆v used to control the spacecraft using the thrust law given in
Eq.(6.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.7 Integration using constant thrust in the radial direction of −9.25×10−7.1326.8 Integration using constant thrust in the radial direction of−1.045×10−6.1336.9 Eccentricity as a function of time over one orbital period. . . . . . . 1346.10 Time histories of the orbital elements when the transverse thrust is
applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.11 Diagram of the reset scheme in the 1-DOF system in (e,ω)-space. . . 1376.12 Original (black) and target (red) long lifetime trajectories. . . . . . . 1446.13 Time histories of the orbital elements of the original (black) and target
(red) long lifetime trajectories. . . . . . . . . . . . . . . . . . . . . . 1446.14 Monte Carlo simulation results for various long lifetime trajectories. 1486.15 Schematic of Hohmann transfer between two elliptic orbits. . . . . . 151
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LIST OF TABLES
2.1 Jupiter-Europa System Parameters . . . . . . . . . . . . . . . . . . . 21
3.1 Parameters of Europa . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Comparison of costs to transfer from a capture trajectory to a circularfrozen orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Comparison of costs to transfer from capture trajectory to tightlybound circular orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1 Algorithm for computing a long lifetime orbit . . . . . . . . . . . . . 1145.2 Long Lifetime Orbits for the Europa System . . . . . . . . . . . . . 1165.3 Long Lifetime Orbits about Enceladus . . . . . . . . . . . . . . . . . 1185.4 Long Lifetime Orbits about Dione . . . . . . . . . . . . . . . . . . . 118
6.1 Examples of Long Lifetime Orbit Reset . . . . . . . . . . . . . . . . 1436.2 Monte Carlo Results for Long Lifetime Orbits . . . . . . . . . . . . . 1476.3 Average cost to correct a long lifetime trajectory after an initial po-
sition error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.4 Average cost to correct a long lifetime trajectory after a larger initial
position error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
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CHAPTER 1
INTRODUCTION
The exploration of planetary satellites by robotic spacecraft began with the Luna
1 and Pioneer 4 missions flying by the Moon in 1959[2]. Since then, spacecraft
have performed detailed studies of our moon and many other planetary satellites,
including several of the Saturnian moons by the Cassini spacecraft[33] and some
of Jupiter’s moons by the Galileo spacecraft[6]. There is a strong interest in the
scientific community to explore planetary satellites, specifically those that can teach
us more about Earth and the origins of life on Earth. To that end, planetary satellites
of primary interest are those that may have either liquid or frozen water oceans.
Specifically, moons that have ice on the surface may have liquid water underneath
if they are geologically active due to tidal heating caused by their central planet[9].
Examples of icy planetary satellites are Jupiter’s moons Europa, Ganymede and
Callisto and many of Saturn’s moons including Enceladus. Of these, there is the
most pressure from the scientific community to send a spacecraft to Europa since
evidence from the Galileo mission has shown that it has an icy surface covering a
liquid water ocean[39].
Sending a spacecraft to orbit a planetary satellite has not been performed to date,
other than our moon. Doing so is a very challenging problem due to the proximity
of the planetary satellite to the planet. The third body gravitational attraction from
the planet causes large perturbations in the system, and as a result, the two-body
model can not be used. The necessity of including third-body effects without having
1
a full analytical understanding of them (as is known for the 2-body problem) can
make mission design a difficult and open problem. However, it is possible to use
these perturbations, and hence the natural dynamics of the system, as an asset for
orbit design by working with the perturbations rather than against them.
Spacecraft missions are generally divided into several stages, and missions to
planetary satellites are no exception. Of these, the phase of the mission where
scientific data is acquired is particularly challenging as the orbit is situated in a
highly perturbed environment not usually encountered for the science phases of a
planetary mission. The orbit of the spacecraft for this part of the mission is called
the science orbit. Science orbits have, in general, low altitudes and high inclinations
so that the entire surface can be mapped, and the science requirements of the
mission can be accomplished. These requirements may include imaging the surface,
determining its composition and measuring features of the planetary satellite such
as the gravity field and the radiation environment. Another important requirement
for a science orbit is that it be stable in the sense of having a lifetime long enough
such that a maneuver failure will not cause the satellite to impact with the planetary
satellite. Aside from designing the science orbit, it is also important to consider how
to transfer to that orbit from a region far away from the planetary satellite without
consuming excessive amounts of propellant. The analysis of both of these mission
phases is the focus of this dissertation, with more emphasis placed on the science
orbit design.
Although the planet-planetary satellite-spacecraft system is a three body
problem, the complexity of the system can be significantly reduced by modelling
the system with the Hill 3-body problem. The Hill 3-body problem was originally
derived by G.W. Hill [19] in 1878 to analyze the motion of the moon in the
Sun-Earth-Moon system. The Hill problem is a simplified version of the 3-body
problem in which the masses of two of the bodies are much smaller than the third,
2
and where the goal is to investigate the motion of the two smaller masses relative
to each other. This is obviously applicable to a planetary satellite system, where
the masses of the planetary satellite and spacecraft are much smaller than the mass
of the planet, and where we wish to evaluate the motion of the spacecraft in the
vicinity of the planetary satellite. One of the main advantages of using the Hill
3-body problem is that it is a much simpler model in terms of analysis than the
3-body and restricted 3-body problems, but still retains the non-integrable dynamics
in the regions to be investigated. Aside from its application to the planet-planetary
satellite-spacecraft system, the Hill problem has many other applications such as
motion in the Earth-Moon-Sun system [35], motion in the vicinity of asteroids [36],
motion about comets [38], periodic orbits [17] and spacecraft maneuvers [41], in
addition to other problems in dynamical astronomy and astrodynamics.
Although the Hill 3-body problem is significantly less complex than other
representations of the 3-body problem, it is still a non-integrable system, which
makes analytic analysis difficult. Since the main goal of this research is to use the
perturbations inherent in the system to our advantage, it is necessary to have a very
good understanding of the underlying dynamics of the system. One method in which
to do this averaging. Specifically, the general Hill 3-body problem can be reduced to
an integrable system by averaging over the orbit of the spacecraft and the orbit of
the planetary satellite (or their equivalents if another system is being investigated).
The averaged system provides some insight into the natural dynamics of the full
system and directs the design of long lifetime science orbits in this dissertation. The
technique of averaging has been applied to planetary satellite systems [37, 34, 22],
to general systems [7], to the motion of asteroids [21, 23] and to stellar systems [16],
to name a few.
The Hill problem is also very useful for the investigation of the regions farther
away from the planetary satellite, in terms of planning the transfer of the spacecraft
3
to the science orbit. Since for this scenario, the spacecraft is still much closer to
the planetary satellite than to the planet, the Hill approximation remains valid.
The types of trajectories studied are those that come from outside the Hill region
and circle the planetary satellite at least once. These trajectories are called capture
trajectories, and their analog, escape trajectories, have been studied in [42, 41].
Capture trajectories exist not only for spacecraft orbiting planetary satellites but also
in other systems where the Hill model applies. One example is the irregular moons
of Jupiter and Saturn which might have become captured due to this effect [5].
1.1 Overview of Results
The results in this dissertation all use the Hill 3-body problem as a model. The
primary motivation is a future Europa orbiter mission, but the results are general
enough to be applied to many different planetary satellites. The two main problems
that are investigated are the design of science orbits and the design of capture
and transfer trajectories. To accomplish both of these tasks, a thorough study of
the dynamics of the system is undertaken. This begins with averaging, where the
system, which has three degrees of freedom (DOF) is averaged twice, and reduced
to an integrable 1-DOF system. The averaging takes place over the orbit of the
spacecraft about the planetary satellite and over the orbit of the planetary satellite
about the planet. The averaging and subsequent analysis is performed for the basic
(tide-only) Hill 3-body problem as well as for a modified Hill 3-body problem which
includes terms from the gravity field of the planetary satellite.
In the 1-DOF system, equilibrium solutions exist which are called frozen orbits.
These orbits have constant values of the orbital elements on average. An analysis of
the frozen orbit solutions shows that both stable and unstable frozen orbits exist,
and both types have potential applications for the goals of this research. Adding the
planetary satellite gravity field to the model slightly changes the structure of the
4
frozen orbit solutions, but the qualitative features of the near-polar frozen orbits
remain similar. Specifically, including the J3 term of the planetary satellite gravity
field causes what were circular, near-polar frozen orbits to become very slightly
elliptical. This has implications for science orbit design since the motion occurs very
close to the planetary satellite, but shows that when planning capture and transfer
trajectories, using the tide-only model is sufficient. Furthermore, it is shown that
changing the value of J3 does not affect the qualitative frozen orbit results. This
is important since an accurate value (magnitude or sign) for J3 for Europa is not
currently available. The validity of the 1-DOF system as an approximation for
motion in the 3-DOF system is verified by performing numerical simulations in the
3-DOF system and comparing the resulting motion to plots of the 1-DOF system
motion. It is found that the 1-DOF system provides a very good approximation to
the 3-DOF system, especially when the altitude of the orbit remains relatively small.
To study capture and transfer trajectories for planetary satellite orbiters, the
dynamics are reduced to a discrete map represented by periapsis passages. This
allows us to study the trajectories that enter the Hill region and orbit the planetary
satellite. Specifically, trajectories that orbit the planetary satellite for at least one
week are identified and the regions in which their periapsis passages exist are called
safe zones. These safe zones represent potential trajectories for use in a mission to
transfer the spacecraft to the vicinity of the planetary satellite. In particular, there
are trajectories that exist in the safe zones that are especially useful as gateways
to transfers to a science orbit with relatively low fuel usage. The science orbits
considered in this analysis are circular and elliptic frozen orbits.
The frozen orbits in the 1-DOF system also provide the basis for our science
orbit design. Specifically, the low-altitude, near-polar unstable frozen orbits in the
modified Hill 3-body problem have stable and unstable manifolds in the 1-DOF
system. By designing orbits such that they follow the stable to unstable manifold
5
path, it is possible to extend the lifetime of trajectories significantly compared to
simply using frozen orbits. The challenge for this method is determining how to
choose initial conditions such that the trajectory in the 3-DOF system follows the
1-DOF manifolds. This is accomplished via a linearization algorithm that relates
the 1-DOF, 2-DOF and 3-DOF systems to each other in order to predict the
corrections necessary to the 1-DOF manifold solutions. The resulting trajectories
have sufficiently long lifetimes to be very practical as science orbits for planetary
satellite orbits.
Once science orbits have been designed, some aspects of controlling them must
be considered. First, the effectiveness of using constant thrust to maintain the orbit
is studied, and found to be too costly. Then, a scheme is developed for a low cost
transfer consisting of two maneuvers that restarts a science orbit trajectory before it
impacts with the planetary satellite. The effect of orbit uncertainty on science orbit
design is also investigated to determine the sensitivity of the science orbits to initial
errors. Finally, given an initial position error, a technique is developed to return the
spacecraft with minimal fuel usage to its nominal trajectory.
In summary, the main contributions of this thesis are:
• The analysis of the 1-DOF system dynamics
• The analysis of capture trajectories in safe zones
• The design method for long lifetime science orbits
• The optimal transfer strategy for restarting a long lifetime orbit
• The control strategy to correct an initial position error
6
1.2 Organization of the Dissertation
This dissertation begins with a discussion of the dynamical model in Chapter II. The
Hill 3-body problem is derived from the 3-body problem, showing that the motion of
a planetary satellite orbiter can be modeled by this simpler problem. Additionally,
the planetary satellite’s gravity field is included in the model since its effect on the
motion is shown to be significant.
In Chapter III, frozen orbit solutions are derived and their structure is
investigated. Comparisons are made for frozen orbit solutions in models with and
without the planetary satellite gravity field, as well as for different values of J3.
Their applicability to the 3-DOF system is also examined.
Chapter IV deals with trajectories that come from outside the Hill region to the
vicinity of the planetary satellite. These are studied using Periapsis Poincare maps,
and sets of trajectories viable as capture and transfer trajectories for a planetary
satellite orbiter are identified.
The design of science orbits is undertaken in Chapter V, where the structure
of the 1-DOF system is used as a basis for the design. The desired trajectories
are identified in the 1-DOF system, and then an algorithm is developed whereby
these trajectories are translated into the 3-DOF system. Several examples of long
lifetime science orbits are presented for the Europa system, and for the Saturnian
moons Enceladus and Dione. Finally, a toolbox to compute the initial conditions
for long lifetime science orbits that was written and delivered to the Jet Propulsion
Laboratory for the Jupiter Icy Moons Orbiter Mission is described.
Chapter VI looks at the control of long lifetime science orbits by both resetting
ideal trajectories before they impact and examining the effect of orbit uncertainty
on the orbit design. Monte Carlo simulations are used to determine how science
orbits are impacted by initial position errors, and a scheme is developed to return
the spacecraft to its nominal trajectory after an initial position error.
7
Finally, a summary and possible future research directions are presented in
Chapter VII.
8
CHAPTER 2
OVERVIEW OF THE DYNAMICAL SYSTEM
The three body problem is the most general way to describe the motion of
three masses under their mutual gravitation attraction. It can be used to analyze
the motion of any system of three bodies such as the Sun, Earth and Moon, a
spacecraft orbiting a binary asteroid system, or our application to a spacecraft in
orbit about a planetary satellite. There are special cases of the three body problem
in which certain assumptions can be made that simplify the analysis of the system.
We will first briefly discuss the Restricted Three Body Problem which deals with
the case of a small mass attracted by 2 point masses revolving around each other.
This simplification can be used for the analysis of the motion of a spacecraft in the
vicinity of two bodies such as planets, a planet and the sun, two asteroids, or a
planet and a planetary satellite.
A further simplification can be made for the case where two of the three masses
are close together and small compared to the third one. This is called the Hill
3-body problem. For the general Hill 3-body problem, it is not necessary to assume
that one of the two smaller masses is extremely small, such as a spacecraft in the
Restricted 3-body problem [18]. However, since our application is to the motion of
a spacecraft in orbit about a planetary satellite, we consider what is formally called
the Restricted Hill 3-body problem. For simplicity, we refer to it as the Hill 3-body
problem.
All of our analysis in this dissertation is performed using the Hill 3-body problem
9
as the model. However, to account for the shape of the planetary satellite, we
develop a modified form of Hill’s 3-body problem that includes some of the gravity
field coefficients of the planetary satellite. The question then arises of which gravity
field terms to include. This is a difficult question since, for our main focus of Europa,
accurate values of most of the gravity field coefficients are not known. When gravity
field coefficients are included in this research, those included are J2, C22 and J3.
These are chosen since they have a large effect of the motion of the spacecraft. This
choice is justified by the demonstration that if the full third degree and order gravity
field is included, the differences in the motion are not significant. We also justify the
use of the Hill model by comparing it to results generated using the full ephemeris
model. The Hill approximation provides results that are close to those using the
full ephemeris model, but it allows for analysis that would not be possible in the
ephemeris model.
The Hill model used in this research assumes a circular orbit for the planetary
satellite. The validity of this assumption is verified for Europa by simulating the
elliptic Hill 3-body problem. It is found that the results from the two models are
not substantially different. The conclusion is therefore that using a modified form
of the Hill 3-body problem where the J2, C22 and J3 gravity field coefficients are
included gives strong results that give a good picture of the motion and can be used
for mission design.
2.1 The Three Body and Restricted Three Body
Problems
The general 3-body problem is used to describe the motion of three bodies under
their mutual gravitational attraction. The equation of motion for each body is given
by[12]:
Miri = −GMi
3∑j=1,j 6=i
Mj(ri − rj)
|ri − rj|3, (2.1)
10
where G = 6.673×10−11m3/kgs2 is the universal gravitational constant and i = 0−2.
It is convenient to express the equations of motion in force potential notation, where
the force potential U is[12]:
U =1
2
3∑i=1
3∑j=1,j 6=i
GMiMj
|rij|, (2.2)
and rij = rj − ri. Then, the equations of motion for k = 0− 2 are
Mkrk =∂U
∂rk
. (2.3)
The restricted three body problem describes the dynamics of a small mass
such as a spacecraft attracted by two point masses rotating about each other. We
concentrate on the circular restricted three body problem in which the two larger
masses are in a circular orbit about each other. Let P0 and P1 be the larger masses
and P2 the smaller mass (spacecraft), so that M2 = 0. Let the origin of this system
be the center of mass of the P0, P1 system (see Figure 2.1). Since we are focussing
on a planet/planetary satellite system in this dissertation, let P0 be the planetary
satellite and P1 the planet. Then, R is the vector from the planetary satellite to the
planet and r is the position of the spacecraft. Since the planetary satellite is in a
circular orbit about the planet, |R| is constant. We focus only on the motion of the
spacecraft since that is our primary interest here. Then,
r =∂U
∂r, (2.4)
where
U =GM0
|r + RM1/(M0 + M1)|+
GM1
|r−RM0/(M0 + M1). (2.5)
Let N =√
G(M0 + M1)/R3 be the mean motion of the planetary satellite,
and note that it is also a constant. Now, express the equations of motion of the
spacecraft in the frame rotating about the planetary satellite. Let rI be the position
of the spacecraft in the inertial frame and rr be the position of the spacecraft in
11
Figure 2.1: Schematic of the Circular Restricted 3-Body Problem
the rotating frame. The rotation of the frame is expressed as Ω = N z. Since both
frames have the same origin, rI = rr. The position and velocity in the rotating frame
are [14]:
rI = rr + Ω× rr , (2.6)
rI = rr + 2Ω× rr + Ω×Ω× rr . (2.7)
Since rI = rr, the subscripts on r can be removed. Then, substituting for Ω and
equating Eqs.(2.4) and (2.7), we obtain:
r + 2N z + N2z× z× r =∂U
∂r. (2.8)
This can be broken down into equations for the x, y and z positions of the spacecraft:
x− 2Ny −N2x =∂U
∂x, (2.9)
y + 2Nx−N2y =∂U
∂y, (2.10)
z =∂U
∂z, (2.11)
12
where U from Eq.(2.5) expressed in coordinate form is:
U =GM0√
(x + M1/(M0 + M1)R)2 + y2 + z2
+GM1√
(x−M0/(M0 + M1)R)2 + y2 + z2. (2.12)
To simplify the equations of motion, let
V =1
2N2(x2 + y2) + U . (2.13)
Then, Eqs.(2.9)-(2.11) can be simplified to:
x−Ny =∂V
∂x, (2.14)
y + Nx =∂V
∂y, (2.15)
z =∂V
∂z. (2.16)
Eqs.(2.14)-(2.16) are the equations of motion for a spacecraft in the Circular
Restricted 3-Body Problem.
2.2 The Hill 3-Body Problem
The conditions of the Hill 3-body problem are that the mass of the planet, M0,
dominates the system and that the planetary satellite and spacecraft are close to
each other and their center of mass is far away from the planet[35]. As previously
mentioned, for simplicity, we consider only the restricted Hill 3-body problem, in
which the mass of the spacecraft tends to zero since that is the specific problem
being addressed in this dissertation. In other words, we focus on the situation where
the motion of the spacecraft is close to the planetary satellite and the mass of the
planetary satellite is much smaller than the mass of the planet (see Figure 2.2).
First, let
µ =M1
M0 + M1
. (2.17)
13
Figure 2.2: Schematic of the Hill 3-Body Problem
The assumption is then that µ 1. Next, non-dimensionalize the system using the
length scale R and the time scale 1/N . Then, x = x/R, y = y/R, z = z/R and
t = Nt. Substituting these in Eqs.(2.14)-(2.16), the non-dimensional equations of
motion for the Circular Restricted 3-Body Problem are:
¨x− 2y =∂V
∂x, (2.18)
¨y + 2x =∂V
∂y, (2.19)
¨z =∂V
∂z, (2.20)
where
V =1
2(x2 + y2) +
1− µ√(x + µ)2 + y2 + z2
+µ√
(x− 1 + µ)2 + y2 + z2. (2.21)
Next, consider motion in the vicinity of the planetary satellite, and shift the
origin to the planetary satellite. This entails setting x = 1 − µ − x where x is the
14
new x-coordinate. Then, the equations of motion are:
¨x− 2y =∂V
∂x, (2.22)
¨y + 2x =∂V
∂y, (2.23)
¨z =∂V
∂z, (2.24)
where
V =1
2((1− µ− x)2 + y2) +
1− µ√(1− x)2 + y2 + z2
+µ√
x2 + y2 + z2. (2.25)
The condition that µ 1 is made explicit by the following change of coordinates[35]:
x = µ1/3x , (2.26)
y = µ1/3y , (2.27)
z = µ1/3z . (2.28)
Note that x, y and z are now the coordinates of the Hill 3-Body Problem, with
the following equations of motion in the rotating frame, centered on the planetary
satellite:
x− 2y =1
µ2/3
∂V
∂x, (2.29)
y + 2x =1
µ2/3
∂V
∂y, (2.30)
z =1
µ2/3
∂V
∂z, (2.31)
where we have dropped the ∼ on V. The remaining step is to expand the equations of
motion in powers of µ1/3, ignoring orders of µ2/3 and higher. This entails expanding
15
1/µ2/3V as follows:
1
µ2/3V =
1
2µ2/3
[(1− µ + µ1/3x)2 + µ2/3y2
]+
1− µ
µ2/3√
µ2/3r2 + 1 + 2µ1/3x+
1
r
=1
2µ2/3− µ1/3 +
µ4/3
2+
1
2(x2 + y2) +
1
r+
x
µ2/3− µ2/3x (2.32)
+1− µ
µ2/3√
1 + 2µ1/3x + µ2/3r2,
where r =√
x2 + y2 + z2. The term in the above expression to be expanded is:
1
µ2/3√
1 + 2µ1/3x + µ2/3r2. (2.33)
This is accomplished using the following Taylor expansion:
1√1 + ε
= 1− 1
2ε +
3
8ε2 + · · · (2.34)
Therefore,
1
µ2/3√
1 + 2µ1/3x + µ2/3r2=
1
µ2/3− x
µ1/3− 1
2r2 + 3x2
+3
2µ1/3xr2 + O(µ2/3) . (2.35)
Then, substituting into Eq.(2.32):
V = const +1
r+
3
2x2 − 1
2z2 + O(µ2/3) . (2.36)
Finally, since µ1/3 1, the orders of µ2/3 are ignored and since the equations
of motion use the derivatives of V the constant terms can also be ignored. The
potential for the Hill problem is then
VH =1
r+
3
2x2 − 1
2z2 . (2.37)
It is convenient to divide the potential for the Hill problem into two components: the
part due to the planetary satellite and the part due to the planet, which is called the
perturbing potential and denoted R. In addition, for the majority of this dissertation
16
the dimensional form of the potential is used, and so we re-dimensionalize here:
VH =µ
r+
1
2N2(x2 + y2) + R . (2.38)
with
R = −1
2N2r2 +
3
2N2x2 . (2.39)
Observe that substituting R from Eq.(2.39) into Eq.(2.38) yields the simpler form of
VH that combines the planetary satellite and the planet components:
VH =µ
r+
1
2N2(x2 + y2)− 1
2N2r2 +
3
2N2x2
=µ
r+
1
2N2(x2 + y2)− 1
2N2(x2 + y2 + z2) +
3
2N2x2
=µ
r+
3
2N2x2 − 1
2N2r2 .
Then, the general equations of motion for the Hill 3-body problem are:
x− 2Ny = − µ
r3x + N2x +
∂R
∂x, (2.40)
y + 2Nx = − µ
r3y + N2y +
∂R
∂y, (2.41)
z = − µ
r3z +
∂R
∂z. (2.42)
The Hill 3-Body Problem has the Jacobi Integral, which can be derived by
multiplying Eq.(2.40)-(2.42) by x, y and z respectively:
xx + yy + zz − 2Nxy + 2Nxy = − µ
r3(xx + yy + zz) + N2(xx− yy)
+∂R
∂xx +
∂R
∂yy +
∂R
∂zz . (2.43)
This is equivalent to
d
dt
[1
2(x2 + y2 + z2)
]=
d
dt
[1
r+ N2(x2 + y2) + R
], (2.44)
17
which can be expressed as:
d
dt
[1
2(x2 + y2 + z2)− 1
r−N2(x2 + y2)−R
]= 0 . (2.45)
Therefore, the Jacobi Integral, which is constant, is:
J =1
2(x2 + y2 + z2)− 1
r−N2(x2 + y2)−R . (2.46)
2.3 The Modified Hill 3-Body Problem
The modified Hill 3-body problem is the general Hill 3-body problem with some
additional system parameters included. The first logical additions are the higher
order terms of the gravity field of the planetary satellite. It is especially interesting
to consider the effect of including these terms when considering motion very close
to the planetary satellite. In general, for a particular planetary satellite under
consideration, only some of the gravity field terms will be known. In addition, even
if the values of many of the gravity field terms are available, including them in our
analysis may not change the results, or may make the analysis difficult. Another
possible modification to the Hill 3-body problem is removing the assumption that
the planetary satellite is in a circular orbit about the planet by constructing a model
that includes the ellipticity in the planetary satellite’s orbit.
2.3.1 Planetary Satellite Gravity Field
Including gravity field terms into the model simply entails adding them to the
perturbing potential as follows[24]:
R = −N2
2r2 +
3
2N2x2 +
µ
r
∞∑n=2
n∑m=0
Rns
rnPnm(sin φ)(Cnm cos mλ + Snm sin mλ) , (2.47)
18
where
x = r cos φ cos λ ,
y = r cos φ sin λ ,
z = r sin φ .
Rs is the radius of the planetary satellite, n is the degree and m is the order.
Pnm(sin φ) are the Legendre polynomials and the Cnm and Snm are coefficients that
describe the dependence on the internal mass distribution of the planetary satellite.
In addition, the coefficients Cn0 are generally denoted as −Jn. The equations of
motion for the model that includes the planetary satellite gravity field are:
x + 2Ny = − µ
r3x + N2x +
∂R
∂x, (2.48)
y − 2Nx = − µ
r3y + N2y +
∂R
∂y, (2.49)
z = − µ
r3z +
∂R
∂z. (2.50)
Obviously, it is not possible to include all of the gravity field terms for the
planetary satellite under consideration. First of all, the Cnm and Snm coefficients are
determined by analyzing data returned from missions to these planetary satellites.
This means that only a few gravity field coefficient values are available. Secondly,
the more gravity field terms included, the more difficult is the analysis. When it
comes to integrating the equations of motion to determine the path of a trajectory,
any number of gravity field terms can be included. However, for analysis such as
that performed in this dissertation, it is necessary to restrict the inclusion of gravity
field terms to those that have the largest effect on the motion.
An important consideration in this research is what the effect of using the Hill
model has on the motion of the spacecraft, as compared to using a full ephemeris
model which includes all of the physics in the system. It is necessary to verify
19
that the results obtained for the Hill model with fewer of the gravity field terms
are reasonable approximations. For these comparisons, we use the Jupiter-Europa
system. It is important to note that the Europa gravity field is only known for
certain up to the second degree and order [4]. The third degree values were obtained
from [3] and are very rough estimates. All of the values used in this analysis are
in Table 2.1. For this comparison, a trajectory is integrated in the modified Hill
3-Body Problem, with Europa’s third degree and order gravity field included. It is
compared to the integration in a full ephemeris model with the same Europa gravity
field of a trajectory with the same initial conditions. The time histories of the
orbital elements for both cases are shown in Figures 2.3 and 2.4, respectively. The
results in Figure 2.4 were obtained from John Aiello [3], who generated the plots
using DPTRAJ at the Jet Propulsion Laboratory. It is clear that the qualitative
features of the plots are a very good match. The periods of all of the oscillations in
the plots are identical, and the motion of all of the orbital elements agree very well.
Therefore, using a modified form of the Hill 3-body problem provides good results
and is justified for use in our analysis.
Although the modified Hill 3-body problem with the Europa third degree and
order gravity field adequately approximates motion in the vicinity of Europa, it is
not practical for analysis to include that many gravity field coefficients. For Europa,
which is the primarily application of our results in this dissertation, the J2, C22 and
J3 contributions have been shown to have a strong effect on the motion [27], and are
included in the majority of our analysis. The components of the perturbing potential
corresponding to J2, C22 and J3 are:
RJ2 = −µJ2
2
(3z2
r5− 1
r3
), (2.51)
RC22 =3µC22
r5(x2 − y2) , (2.52)
RJ3 = −µJ3
2
(5z3
r7− 3z
r5
). (2.53)
20
Table 2.1: Jupiter-Europa System Parameters
Parameter Symbol Value
Jupiter gravitational parameter µp 1.267× 108 km3/s2
Europa gravitational parameter µ 3201 km3/s2
Europa orbit rate N 2.05× 10−5 rad/sEuropa orbital period TE 3.55 daysEuropa eccentricity es 0.01Europa semi-major axis as 671100 kmEuropa radius Rs 1560.8 kmEuropa J2 J2 1041.39 km2
Europa C21 C21 −0.32 km2
Europa S21 S21 −20.47 km2
Europa C22 C22 312.97 km2
Europa S22 S22 −4.71 km2
Europa J3 J3 524117.32 km3
Europa C31 C31 120230.75 km3
Europa S31 S31 81414.49 km3
Europa C32 C32 −11938.14 km3
Europa S32 S32 −3617.07 km3
Europa C33 C33 −1765.93 km3
Europa S33 S33 3669.41 km3
21
0 20 40 60 80 100 1201550
1600
1650
1700
1750
1800ra
dius
(km
)
time (days)
0 20 40 60 80 100 1201660
1661
1662
1663
1664
1665
1666
1667
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days) 0 20 40 60 80 100 120
109.5
110
110.5
111
111.5
112
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120−120
−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)
0 20 40 60 80 100 120−200
−150
−100
−50
0
50
100
150
200
Ω (
deg)
time (days)
Figure 2.3: Trajectory integration in the Modified Hill 3-body problem (Third degree and ordergravity field)
22
Figure 2.4: Trajectory integration using the full ephemeris model, and Europa’s gravity field up tothird degree and order [3].
23
where the values of J2, C22 and J3 have been dimensionalized by their multiplications
by R2s and R3
s, respectively. Therefore, we verify that including only the J2, C22
and J3 coefficients does not drastically affect the results. In a real mission design
situation, analysis would be performed using this reduced gravity field and a
trajectory valid in the Hill problem would be computed. This trajectory could then
be used as a starting point for designing a trajectory in the full ephemeris model.
In examining results obtained using a truncated Europa gravity field, it is not
necessary to find the exact same behavior of the trajectory, especially in terms of
its lifetime. The environment around Europa in particular is highly perturbed, and
a slight change in the initial conditions can result in a drastic change in the overall
trajectory. Therefore, we are only interested in verifying that using a model with a
truncated gravity field captures the important features of the motion. Comparing
Figure 2.5 which shows the integration of a trajectory in the modified Hill 3-body
problem with the gravity field truncated to J2, C22 and J3 to Figure 2.3 which
shows an integration using the third degree and order gravity field, we see that the
overall features are very similar. Specifically, the amplitudes of the oscillations in
inclination and semi-major axis are the same in both cases, as are the trends in the
motion of all of the orbital elements. The main difference between the two cases
is the amplitude of the eccentricity and argument of periapsis oscillations. In the
third degree and order gravity field case, the size of the oscillations are larger for
both orbital elements. However, since the average motion remains the same between
the full gravity field and truncated gravity field case, the results obtained from the
truncated gravity field case are sufficient for our purposes.
2.3.2 Planetary Satellite Eccentricity
In this section the assumption that the planetary satellite is in a circular orbit about
the planet is verified. To do so, an elliptic planetary satellite orbit is added to the
24
0 20 40 60 80 100 1201550
1600
1650
1700
1750
1800ra
dius
(km
)
time (days)0 20 40 60 80 100 120
1661
1662
1663
1664
1665
1666
1667
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days)0 20 40 60 80 100 120
109.8
110
110.2
110.4
110.6
110.8
111
111.2
111.4
111.6
111.8
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)0 20 40 60 80 100 120
−200
−150
−100
−50
0
50
100
150
200
Ω (
deg)
time (days)
Figure 2.5: Trajectory integration in the Hill 3-body problem, with the Europa gravity fieldtruncated to include only the J2, C22 and J3 contributions.
25
model. In the circular model, the assumption that the rotation rate of the planetary
satellite is synchronous with its orbit rate was made. Therefore, when an elliptic
planetary satellite orbit is introduced, it is important to ensure that the rotation
rate of the planetary satellite is still constant, and does not vary as the speed of
the planetary satellite on its orbit varies. For these comparisons, the modified
Hill 3-body problem includes the J2, C22 and J3 planetary satellite gravity field
components.
The equations of motion for the elliptic Hill 3-body problem are most easily
expressed in non-dimensional coordinates. The potential of the non-dimensional
circular Hill problem is:
VC =1
r+
1
2(x2 + y2) + RC , (2.54)
where
RC = −1
2r2+
3
2x2− J2
2ε2d2
(3z2
r5− 1
r3
)+
3C22
ε2d2r5(x2−y2)− J3
2ε2d2
(5z3
r7− 3z
r5
). (2.55)
The factors used to remove the dimensions of the system are ε and the radius of the
planetary satellite in orbit about the planet, d:
ε = (µ/µp)1/3 , (2.56)
d =as(1− e2
s)
1 + es cos fs
, (2.57)
where µp is the gravitational parameter of the planet, as is the semi-major axis
of the planetary satellite, es is the eccentricity of the planetary satellite and fs is
the true anomaly of the planetary satellite. Then, the potential of the elliptic Hill
3-body problem is[13]:
VE =1
1 + es cos fs
VC . (2.58)
One final correction to the potential must be made, to ensure the constant rotation
rate of the planetary satellite. This correction only applies to the C22 component of
26
the potential since it is the only non-symmetric component. Specifically, the x and
y values in the C22 component in Eq.(2.55) are replaced by x′ and y′, where x′
y′
=
cos (Ms − fs) sin (Ms − fs)
sin (Ms − fs) cos (Ms − fs)
x
y
, (2.59)
and Ms is the mean anomaly of the planetary satellite. Then, the equations of
motion for the elliptic modified Hill 3-body problem are:
x = 2y +∂VEH
∂x, (2.60)
y = −2x +∂VEH
∂y, (2.61)
z =∂VEH
∂z, (2.62)
where
VEH =1
1 + es cos fs
[1
r+
1
2(x2 + y2) + REH
], (2.63)
and
REH = −1
2r2 +
3
2x2 − J2
2ε2d2
(3z2
r5− 1
r3
)+
3C22
ε2d2r5(x′
2 − y′2)
− J3
2ε2d2
(5z3
r7− 3z
r5
). (2.64)
Figure 2.6 shows the results of the integration of a trajectory in the elliptic
modified Hill 3-body problem. The parameters used are those for a spacecraft in
orbit about Europa, and are given in Table 2.1. Compared to Figure 2.5, it is clear
that the main features of the motion are the same. The orbital elements show the
same trends, and the sizes of the oscillations are the same. The major difference
is that in the elliptic case, the semi-major axis shows a slow decrease. This is
not a large concern, since the overall decrease is only about 2 km over 130 days.
Ignoring this effect in the circular case will not strongly affect the overall analysis.
Therefore, for ease of analysis, the Circular Modified Hill 3-body problem is used for
27
the remainder of this dissertation.
28
0 20 40 60 80 100 120 1401560
1580
1600
1620
1640
1660
1680
1700
1720
1740
1760ra
dius
(km
)
time (days)0 20 40 60 80 100 120 140
1656
1658
1660
1662
1664
1666
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days)0 20 40 60 80 100 120 140
109
109.5
110
110.5
111
111.5
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120 140−120
−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)0 20 40 60 80 100 120 140
−60
−40
−20
0
20
40
60
80
100
120
140
Ω (
deg)
time (days)
Figure 2.6: Trajectory integration in the Elliptic Modified Hill 3-body problem, with the Europagravity field truncated to include only the J2, C22 and J3 contributions.
29
CHAPTER 3
FROZEN ORBITS
The Hill 3-body problem is very complex in that it is a 3 degree of freedom (DOF)
non-integrable system. Detailed analysis of it in that form is only possible through
numerical simulations and it is difficult to obtain insight into the system except for
some special situations such as equilibrium points. Given that the ultimate goal of
the study of planetary satellite orbiters is the design of useful orbits for a mission,
it is important to learn more than what can be gained from numerical simulations.
This is accomplished with the analysis of a reduced system. By averaging over the
motion of the spacecraft about the planetary satellite and over the motion of the
planetary satellite about the planet, the 3-DOF system can be reduced to a 1-DOF
integrable system. This 1-DOF system has equilibrium solutions, denoted as frozen
orbits, which have constant values of the eccentricity, inclination and argument of
periapsis, on average. We compute frozen orbit solutions for three models: tide-only,
tide + J2 and tide + J2 + J3 and characterize the differences between them.
Previous work using averaging methods has studied the dynamics of orbits
about planetary satellites and frozen orbits. Broucke [7] studied orbit dynamics in
a model that included the tidal force. He found elliptic and circular frozen orbits
and investigated their stability. Scheeres, et al. [37] studied near-circular frozen
orbits in a model that included both the tidal force and J2. They investigated the
stability of these orbits and developed an analytic theory to postpone impact with
the surface for significant periods of time. In addition San-Juan et al. [34] and Lara
30
et al. [22] studied orbit dynamics about planetary satellites using a more rigorous
averaging method that involved Lie transforms to develop averaged equations to a
higher order. They included the tidal term and J2 and the tidal term, J2 and C22,
respectively.
Aside from computing the frozen orbit solutions, we also examine their stability
since some frozen orbits are stable and others are unstable. The concept of the
characteristic time is also introduced. Just as the distribution of the frozen orbit
solutions in orbit element space is affected by the inclusion of the gravity field
coefficients, so too is their stability. The stability of the frozen orbit solutions is
therefore computed for the three different cases.
Contour plots are a way to study the qualitative secular motion in the vicinity
of frozen orbits. The motion in the vicinity of the different types of frozen orbits
is shown using contour plots in the 1-DOF system and integrations in the 3-DOF
system. The integration of frozen orbit solutions in the 3-DOF system is used to
determine whether the stable frozen orbits are viable as trajectories in the 3-DOF
system and to show that unstable frozen orbits in the 1-DOF system are also
unstable in the 3-DOF system.
3.1 Reduction of the System by Averaging
Recall the perturbing potential of the 3-DOF system:
R =1
2N2(3x2 − r2
)− µJ2
2
(3z2
r5− 1
r3
)+
3µC22
r5(x2−y2)− µJ3
2
(5z3
r7− 3z
r5
). (3.1)
Before presenting the results of the reduction of the system, it is necessary to
introduce three regimes other than Cartesian coordinates in which the motion can
be analyzed. Orbital elements are a set of six values used frequently in celestial
mechanics and astrodynamics to describe the size, shape and orientation of an orbit.
Semi-equinoctial elements are useful for dealing with near-circular orbits. Delaunay
31
variables are a canonical set of six elements of which three are angles and three are
momenta. The equations of motion in terms of Delaunay variables are quite simple
and lead directly to the idea of the reduction of the system by averaging.
The orbital elements uniquely define a Keplerian orbit, given the Cartesian
position and velocity. For the case of the motion of a point mass in a central
gravitational field (which is integrable) they describe the specific conic section that
the path of the particle follows. In our case, that of a spacecraft in a non-central
field, the orbital elements vary as functions of time and represent a convenient way
to describe the motion. The orbits that we are primarily interested in are elliptical.
The six orbital elements most commonly used are:
• a: semi-major axis
• e: eccentricity
• i: inclination
• ω: argument of periapsis
• Ω: longitude of the ascending node
• M : mean anomaly
Some other orbital elements that we use are:
• ra = a(1 + e): radius of apoapsis
• rp = a(1− e): radius of periapsis
• E: eccentric anomaly (M = E − e sin E)
• f : true anomaly(tan f/2 =
√1+e1−e
tan E/2)
32
All of the above orbital elements are in the inertial frame. Since the motion that we
consider takes place in the rotating frame, the longitude of the ascending node in
the rotating frame is also used:
Ω = Ω−N(t− t0) , (3.2)
where t0 is an initial epoch.
The equations of motion in terms of the orbital elements are obtained from
the Lagrange Planetary Equations, where R is the perturbing potential under
consideration[11]:
da
dt=
2
na
∂R
∂M, (3.3)
de
dt=
1− e2
na2e
∂R
∂M−√
1− e2
na2e
∂R
∂e, (3.4)
di
dt=
1
na2√
1− e2
[cot i
∂R
∂ω− csc i
∂R
∂Ω
], (3.5)
dω
dt=
√1− e2
na2e
∂R
∂e− cot i
na2√
1− e2
∂R
∂i, (3.6)
dΩ
dt=
1
na2√
1− e2 sin i
∂R
∂i, (3.7)
dM
dt= n− 1− e2
na2e
∂R
∂e− 2
na
∂R
∂a. (3.8)
Semi-equinoctial elements, denoted as h and k, are defined as [24]:
h = e sin ω ,
k = e cos ω .
This set of variables is particularly useful for near-circular orbits since all of the
motion is concentrated in a small region of possible (h, k) space, making for easier
analysis and plotting.
The Delaunay variables consist of the three angles M , ω and Ω complemented by
33
three corresponding momenta, denoted as L, G and H and defined as follows:
L =√
µa ,
G = L√
1− e2 , (3.9)
H = G cos i .
The equations of motion of the 3-DOF system in terms of Delaunay variables are[8]:
dL
dt=
∂R
∂M,
dM
dt= n− ∂R
∂L,
dG
dt=
∂R
∂ω,
dω
dt= −∂R
∂G, (3.10)
dH
dt=
∂R
∂Ω,
dΩ
dt= − ∂R
∂H.
The fact that the full system has three degrees of freedom can be seen from the
Delaunay variable representation. The three degrees of freedom correspond to the
three free angular coordinates, M , ω and Ω. Note that H is the z-component of
the angular momentum. The Jacobi integral, defined in Eq.(2.46) in Cartesian
coordinates can be expressed in osculating orbital elements as:
J = − µ
2a−NH −R(a, e, i, ω, Ω, M) . (3.11)
Averaging is an approximation technique that can be used to remove degrees of
freedom from this system. In general, averaging occurs over an angle of the system
and causes its corresponding momentum to be conserved. For the first averaging, we
assume that the spacecraft’s mean motion about the planetary satellite, n =√
µ/a3,
is much greater than N , the mean motion of the planetary satellite about the
planet[37]. This means that over the time it takes for one revolution of the spacecraft
about the planetary satellite, the planetary satellite will only have moved by a small
amount around the planet. For a low-altitude orbit about Europa, the ratio of N/n
is ∼ 0.02− 0.03 which means that Europa will only have moved around Jupiter by
34
about 7-10 degrees. This condition is also satisfied by many other planetary satellites
in our solar system. Two in particular to be discussed later are Saturn’s moons
Enceladus and Dione. They have N/n ratios of ∼ 0.09 and ∼ 0.04, respecively [37].
The first averaging, to reduce the system to 2 degrees of freedom is over the
mean anomaly of the spacecraft about Europa; define
R =1
2π
∫ 2π
0
RdM . (3.12)
The computation of R requires the definition of the Cartesian coordinates x, y and
z as:
x = r[cos (ω + ν) cos Ω− sin (ω + ν) sin Ω cos i] , (3.13)
y = r[cos (ω + ν) sin Ω + sin (ω + ν) cos Ω cos i] , (3.14)
z = r sin i sin (ω + ν) , (3.15)
where t is the time. Note that for this computation, Ω is assumed to be constant.
The singly-averaged potential is found to be:
R =N2a2
4
(1− 3
2sin2 i +
3
2cos 2Ω sin2 i
)(1 +
3
2e2
)+
15
4e2 cos 2ω
[sin2 i + cos 2Ω
(1 + cos2 i
)]− 15
2e2 sin 2ω sin 2Ω cos i
+
µJ2
2a3(1− e2)3/2
(1− 3
2sin2 i
)+
3µC22
2a3(1− e2)3/2cos 2Ω sin2 i
+3µJ3
2a4(1− e2)5/2e sin ω sin i
(1− 5
4sin2 i
). (3.16)
Since we averaged over the mean anomaly, it is no longer present in the potential
defined by Eq.(3.16). Additionally, recall the equation for the time-derivative of
the semi-major axis in Eq.(3.3). Since M does not appear in R, a = 0 and so the
semi-major axis is constant and acts as another integral of motion. Since L =√
µa,
L is also constant. We have reduced the system to a 2-DOF system consisting of the
pairs (H, Ω) and (G, ω).
35
The 2-DOF system is a time-periodic function of Ω. This makes initial analysis
of the qualitative behavior of the system difficult and so we reduce it further by
performing another averaging, over the orbit of the planetary satellite about the
planet. This averaging is justified for Europa since there are order of magnitude
differences between the period of a Europa orbiter (∼ 2 hours) and the period of
Europa about Jupiter (3.5 days) and between the period of Europa about Jupiter
and the time period of the instability of the spacecraft (∼ 1 month)[37]. It is
similarly justified for Enceladus and Dione.
Since Ω is uniformly decreasing with time due to its definition with respect to
the rotating coordinate frame, it can be used as an independent variable in place of
the time. Then, the dynamics are averaged over this new independent variable to
find their simplest form; define
R =1
2π
∫ 2π
0
RdΩ . (3.17)
This defines the doubly averaged potential:
R =N2a2
4
[(1− 3
2sin2 i
)(1 +
3
2e2
)+
15
4e2 cos 2ω sin2 i
]+
µJ2
2a3(1− e2)3/2
(1− 3
2sin2 i
)(3.18)
+3µJ3
2a4(1− e2)5/2e sin ω sin i
(1− 5
4sin2 i
).
This potential has only tidal, J2, and J3 components since the C22 component
vanishes under the averaging. Although this means that the effect of C22 is not
included during the analysis of this system, its effect is considered in later chapters
when the averaging is removed. Since Ω is now eliminated, an additional integral of
motion exists for this system, the z-component of the angular momentum defined
by Eq.(3.9). Since a was constant in the 2 degree of freedom system, it is also
constant in this system. This leads to the definition of a simpler functional form of
36
the integral, θ = H2/µa, or [7]:
θ = (1− e2) cos2 i . (3.19)
Therefore, the system governed by the doubly averaged potential in Eq.(3.18) is a
1-DOF system in (G,ω). To simplify later analysis, recall the Jacobi Integral which,
since it was an integral of motion in the full 3-DOF system, is also in integral of
motion here and has the same value as before the averaging since it is a constant.
Therefore, Eq.(3.11) implies that
J = − µ
2a−NH − R(e, i, ω) . (3.20)
Since a and H are integrals of motion in this system, the potential of the 1-DOF
system, R is also an integral of motion for the reduced 1-DOF system. Note that
this is not true for the potentials of the 2 and 3-DOF systems.
3.2 Frozen Orbits: Equilibrium Solutions of the
1-DOF System
The 1-DOF system is completely described by the variables (G,ω). The equations
of motion of these two variables can be obtained from the 1-DOF potential R as
follows:
dG
dt=
∂R
∂ω, (3.21)
dω
dt= −∂R
∂G. (3.22)
Note that since R is an integral of motion in this system, it can be formally reduced
to quadratures. However, for this analysis it is more useful to consider the time
derivatives of the osculating orbital elements e, i and ω, noting that the equations for
e and i can be combined into the equation for G. These equations are obtained by
substituting the 1-DOF potential, Eq.(3.18) into the Lagrange Planetary Equations
37
given in Eqs.(3.3)-(3.8). Before performing a full-blown analysis of the 6 equations,
some general observations can be made to simplify the analysis. First, note that
the 1-DOF potential is independent of M and Ω. This means that the other orbital
elements don’t depend on M and Ω in the 1-DOF system and so the equations for M
and Ω can be excluded from this analysis. Also, recall that the semi-major axis, a, is
constant in the 1-DOF system, and so its equation of motion can also be excluded.
The three remaining equations of motion, those for e, i and ω, are then the only
ones considered in this section:
de
dt=
15N2
8ne√
1− e2 sin2 i sin 2ω − 3J3n
2a3(1− e2)3sin i
(1− 5
4sin2 i
)cos ω , (3.23)
di
dt=− 15N2
16n
e2
√1− e2
sin 2i sin 2ω +3J3n
2a3(1− e2)3e cos i (1
−5
4sin2 i
)cos ω , (3.24)
dω
dt=
3N2
8n√
1− e2
[5 cos2 i− 1 + 5 sin2 i cos 2ω + e2(1− 5 cos 2ω)
]+
3J2n
4a2(1− e2)2
(1− 5
4sin2 i
)+
3J3n
2a3(1− e2)3
sin ω sin i
e(3.25)
·[(
1− 5
4sin2 i
)(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)].
Denote the equilibrium solutions of Eqs.(3.23)-(3.25) as frozen orbits. These
orbits are not frozen in space, but have constant values of e, i and ω on average.
Figure 3.1 shows an example of a frozen orbit integrated in the 1-DOF system over
a few orbital periods.
3.2.1 Tide-only Case
Consider first the tide-only case with J2 = J3 = 0 in Eqs.(3.23)-(3.25). First of all,
for circular orbits where e = 0, de/dt = di/dt = 0 automatically and ω is undefined.
Therefore, circular frozen orbits exist in the tide-only case for all inclination values.
Then with e 6= 0, de/dt = 0 and di/dt = 0 for ω = ±π/2, ω = 0 and ω = π. However,
for ω = 0,π the condition necessary to satisfy dω/dt = 0 is e = 1 which corresponds
38
−3000−2000
−10000
10002000
3000
−3000
−2000
−1000
0
1000
2000
3000−3000
−2500
−2000
−1500
−1000
−500
0
500
1000
1500
x (km)y (km)z
(km
)
Figure 3.1: A frozen orbit in the 1-DOF system integrated over several orbital periods.
to a rectilinear orbit. Since we are only interested in elliptic and circular orbits, we
consider only ω = ±π/2. The e and i solutions for the frozen orbits are then given
by:
5
3cos2 i− 1 + e2 = 0 (3.26)
which gives the following relationship between e and i:
e =
√1− 5
3cos2 i (3.27)
Figure 3.2 shows the frozen orbit solutions for the tide-only case. Note that the
elliptic frozen orbits bifurcate from the circular frozen orbits at i ∼ 39.23o and
i ∼ 140.77o.
An important observation of the tide-only frozen orbit solutions is that they
don’t have any dependence on the system parameters or the altitude of the desired
frozen orbit. Therefore, these frozen orbit solutions are valid for any planetary
satellite/planet system where only the tidal force is considered and for any altitude
frozen orbit.
39
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
Figure 3.2: Frozen orbit solutions for the tide-only case (ω = ±π/2 for elliptic orbits).
3.2.2 Tide and J2 Case
Next, we remove the J2 = 0 condition and evaluate the frozen orbit solutions. Once
again, circular frozen orbits exist for all values of the inclination and elliptic frozen
orbits have the ω = ±π/2 condition. The equation to be solved to obtain the
relationship between eccentricity and inclination for the frozen orbit solutions is
Eq.(3.25) set equal to zero with J3 = 0 and ω = ±π/2:
3N2
3n√
1− e2[10 cos2 i− 6(1− e2)
3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)= 0 . (3.28)
This yields the following analytic relationship between e and i for the frozen orbit
solutions:
sin2 i =2N2(4 + 6e2)a2(1− e2)3/2 + 2n2J2
20N2a2(1− e2)3/2 + 5n2J2
. (3.29)
One difference between this result and the tide-only result is that the semi-major
axis appears in this solution. In the tide-only case, the frozen orbit solutions were
independent of the semi-major axis, meaning that given either an inclination or
eccentricity, all sizes of orbits would have the same frozen orbit characteristics. In
this case, however, the frozen orbit characteristics will vary as the size of the orbit
40
Table 3.1: Parameters of Europa
Parameter Symbol Value
Europa radius RE 1560.8 km
Europa orbital period T 3.55 days
Europa orbit rate N 2.05× 10−5 rad/s
Europa gravitational parameter µ 3.201× 103 km3/s2
Non-dimensional Europa J2a J2 4.2749× 10−4
Non-dimensional Europa C22a C22 1.2847× 10−4
Non-dimensional Europa J3a J3 1.3784× 10−4
a Values obtained from John Aiello, personal communication, August2004.
varies. Since we are primarily interested in low-altitude orbits, when solving for the
frozen orbit solutions we specify a radius of periapsis and scale the semi-major axis
correspondingly, where:
a =rp
1− e. (3.30)
A periapsis altitude of 100 km is chosen and the values of the other parameters used
are given in Table 3.1.
Figure 3.3 shows the frozen orbit solutions for the tide plus J2 case. The dashed
lines represent the frozen orbit solutions for the tide plus J2 case that differ from the
tide-only case. For eccentricities larger than about 0.4 and circular orbits, the frozen
orbit solutions are not affected by the inclusion of the J2 gravity field coefficient.
However, for elliptic orbits with eccentricities smaller than about 0.4, the frozen
orbit solutions for the tide plus J2 case have larger inclinations than the solutions
for the tide-only case. This occurs because the elliptic frozen orbits bifurcate from
the circular frozen orbits at a larger inclination for the tide plus J2 case than for
the tide-only case. The amount of the inclination difference depends on the value
41
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
tide−onlytide+J
2
Figure 3.3: Comparison of frozen orbit solutions with and without J2 (ω = ±π/2 for elliptic orbits).
of J2 and the desired radius of periapsis. The results shown here are obviously for
particular values of J2 and rp. The overall importance of this result is that for
near-circular frozen orbits, including J2 leads to orbits with higher inclinations which
could be useful for the design of a science orbit about a planetary satellite since in
general, near-polar orbits are desirable for science orbits.
3.2.3 Tide, J2 and J3 Case
Consider now Eqs.(3.23)-(3.25) with J2 6= 0 and J3 6= 0. Again, Eqs.(3.23) and
(3.24) are in equilibrium for ω = ±π/2. Substituting this into Eq.(3.25) and setting
it equal to zero, we obtain:
0 =3
8
N2
n
1√1− e2
[4− 10 sin2 i + 6e2] +3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)± 3J3n
2a3(1− e2)3
sin i
e
[(1− 5
4sin2 i
)(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)], (3.31)
where the ± on the J3 terms corresponds to ω = ±π/2. This implies that the
inclusion of the J3 gravity field coefficient breaks the symmetry between the
ω = ±π/2 frozen orbit solutions. In its place, a symmetry between the signs of J3
42
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
Figure 3.4: Frozen orbit solutions for J2 6= 0, J3 > 0 and ω = ±π/2.
and ω is created, where changing the sign of J3 is equivalent to shifting ω by π. This
is useful in terms of mission design since at this time only the degree two terms of
the Europa gravity field are known [4]. An accurate value of J3 for Europa is not
currently available, including its sign. Due to the existence of the symmetry between
the signs of J3 and ω any desired frozen orbit can be attained, regardless of the sign
of J3 by adjusting ω appropriately.
Eq.(3.31) cannot be solved analytically for the relationship between e and i for
frozen orbit solutions in this case and must be solved numerically. One possible
solution method is to use the θ integral and eliminate either e or i from Eq.(3.31).
However, since the range of θ is not immediately apparent, it is easier to choose
values of either e or i and solve Eq.(3.31) for the other orbital element.
Figure 3.4 shows the frozen orbit solutions for J2 6= 0 and J3 > 0. It is
immediately apparent that including the effect of J3 significantly increases the
complexity of the structure of the frozen orbit solutions. First of all, it is not
surprising that the structure of the frozen orbit solutions changes depending on the
sign of J3 sin ω. However, the symmetry returns for large values of the eccentricity
where the solutions are the same as in the tide-only case. This occurs since as the
43
eccentricity increases, the apoapsis of the orbit also increases and the influences of J2
and J3 on the dynamics decrease. Therefore, we conclude that the high-eccentricity
frozen orbits are not significantly changed by adding J2 and J3.
One of the new features apparent in Figure 3.4 is near-equatorial frozen orbits
that exist for all values of eccentricity for J3 sin ω > 0. We also observe that
circular frozen orbits exist only at four isolated points, which in this case have
inclinations of 0o, 180o, 63.43o and 116.56o. In contrast, in the tide-only and tide
plus J2 cases, circular frozen orbits existed over all inclinations. The exact values
of the non-equatorial inclinations for the four circular frozen orbits depend on the
parameters of the system. Those values will always be close to but not exactly equal
to i = sin−1(√
4/5). Their deviation from that value is due to the existence of the
tidal term. In the case where only J2 and J3 are included, these circular frozen
orbits exist independently of the parameter values [37]. Including the tidal term as
we do here breaks this relationship and the frozen orbits with i = sin−1(√
4/5) have
non-zero eccentricities.
The frozen orbits with eccentricities smaller than about 0.2 have very different
characteristics depending on whether J3 sin ω > 0 or J3 sin ω < 0. In terms of
Figure 3.4 where J3 > 0, frozen orbits with ω = π/2 have inclinations in the middle
range, between about 45o − 65o and 105o − 135o. Conversely, for frozen orbits with
ω = −π/2, near-circular frozen orbits exist for inclinations between about 65o− 105o
(i.e. near-polar). In addition, for eccentricities up to about 0.1 there exist frozen
orbits with inclinations ranging from circular to about 40o for direct orbits and in
the 140o − 180o range for retrograde orbits.
3.2.4 Variations in J3
The results discussed above were generated for specific values of J2 and J3. Since an
accurate value for J3 for Europa is not currently known, it is important to verify that
44
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
(a) Order of magnitude larger J3
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
(b) Order of magnitude smaller J3
Figure 3.5: Frozen orbit solutions with larger and smaller values of J3 (compared to Table 3.1).
the qualitative features of the results stay the same for different values of J3. This
would demonstrate that these results are applicable not only to Europa, whatever
its gravity coefficients are, but also to other planetary satellites. Figure 3.5 shows
the frozen orbits solutions using values of J3 an order of magnitude larger and an
order of magnitude smaller than the value given in Table 3.1. Although the shape of
the curves is slightly different from the curves in Figure 3.4, in both cases the same
classes of frozen orbits still exist. Therefore, even though the exact value of J3 for
Europa is not known at this time, the analysis of the system shown thus far should
be valid no matter what the value is and these results can be used in subsequent
analysis.
3.3 Stability of Frozen Orbit Solutions
To determine the stability of the frozen orbit solutions, linearize Eqs.(3.23)-(3.25)
about the frozen orbit solutions. First, express Eqs.(3.23)-(3.25) as:
de
dt= fe(e, i, ω) , (3.32)
di
dt= fi(e, i, ω) , (3.33)
45
dω
dt= fω(e, i, ω) . (3.34)
Define i = i∗ + δi, e = e∗ + δe and ω = ω∗ + δω where e∗, i∗ and ω∗ are the frozen
orbit values of these orbital elements. Then,
d(e∗ + δe)
dt= fe(e
∗ + δe, i∗ + δi, ω∗ + δω) , (3.35)
d(i∗ + δi)
dt= fi(e
∗ + δe, i∗ + δi, ω∗ + δω) , (3.36)
d(ω∗ + δω)
dt= fω(e∗ + δe, i∗ + δi, ω∗ + δω) . (3.37)
Expanding up to first order, we obtain:
e∗ + δe = fe(e∗, i∗, ω∗) +
∂fe
∂e
∣∣∣∣∗δe +
∂fe
∂i
∣∣∣∣∗δi +
∂fe
∂ω
∣∣∣∣∗δω , (3.38)
i∗ + δi = fi(e∗, i∗, ω∗) +
∂fi
∂e
∣∣∣∣∗δe +
∂fi
∂i
∣∣∣∣∗δi +
∂fi
∂ω
∣∣∣∣∗δω , (3.39)
ω∗ + δω = fω(e∗, i∗, ω∗) +∂fω
∂e
∣∣∣∣∗δe +
∂fω
∂i
∣∣∣∣∗δi +
∂fω
∂ω
∣∣∣∣∗δω . (3.40)
First, since the frozen orbit solutions are equilibrium solutions to Eqs.(3.32)-(3.34)
e∗ = fe(e∗, i∗, ω∗) = 0, i∗ = fi(e
∗, i∗, ω∗) = 0 and ω∗ = fω(e∗, i∗, ω∗) = 0. In addition,
since ω∗ = ±π/2:
∂fe
∂e
∣∣∣∣∗
=∂fe
∂i
∣∣∣∣∗
=∂fi
∂e
∣∣∣∣∗
=∂fi
∂i
∣∣∣∣∗
=∂fω
∂ω
∣∣∣∣∗
= 0 .
Therefore, Eqs.(3.38)-(3.40) simplify to the following:
δe =∂fe
∂ω
∣∣∣∣∗δω , (3.41)
δi =∂fi
∂ω
∣∣∣∣∗δω , (3.42)
δω =∂fω
∂e
∣∣∣∣∗δe +
∂fω
∂i
∣∣∣∣∗δi . (3.43)
46
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.6: Stability of frozen orbits about Europa for the tide-only case.
Next, taking the second derivative of Eq.(3.43) yields:
δω =
(∂fω
∂e
∣∣∣∣∗
∂fe
∂ω
∣∣∣∣∗+
∂fω
∂i
∣∣∣∣∗
∂fi
∂ω
∣∣∣∣∗
)δω . (3.44)
For each frozen orbit the partial derivatives evaluated at that frozen orbit are
constants. Therefore, δω is just a constant multiplied by δω. This second order
linear differential equation has the simple form:
δω = Λ2(e∗, i∗, ω∗)δω , (3.45)
where Λ2 is a constant for each frozen orbit.
The evaluation of Λ2(e∗, i∗, ω∗) for a particular frozen orbit determines its
stability. If Λ2 > 0 the frozen orbit is unstable and if Λ2 < 0 the frozen orbit is
stable. Figure 3.6 shows the stability of the frozen orbits for the tide-only case. Note
that all of the elliptic frozen orbits are stable and the circular frozen orbits are stable
for inclinations less than about 39.23o and greater than about 140.77o. The circular
frozen orbits with inclinations between ∼ 39.27o − 140.77o are unstable.
Figure 3.7 shows the stability of the frozen orbits for the tide plus J2 case. The
similarities between this and the tide-only case are that the elliptic frozen orbits
47
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.7: Stability of the frozen orbits for the tide plus J2 case.
with eccentricities larger than about 0.2 are stable. As well, the near-equatorial
circular frozen orbits are also stable. The major difference between this case and
the tide-only case is that the elliptic frozen orbits with eccentricities smaller than
about 0.2 are unstable as are the circular frozen orbits with inclinations ranging
from about 47o − 133o. The circular frozen orbits change from stable to unstable at
the bifurcation points (i.e., the locations where the elliptic frozen orbits bifurcate
from the circular frozen orbits).
The stability of the frozen orbits for the case including the tidal term, J2 and
J3 is similar to the tide plus J2 case in that the higher eccentricity frozen orbits are
stable. However, for the lower-eccentricity frozen orbits, the stability characteristics
of this case depend on the sign of J3 sin ω. For J3 sin ω > 0, the frozen orbits with
eccentricities less than about 0.2 are unstable. These are the mid-inclination frozen
orbits. In addition, the near-equatorial frozen orbits that exist for all inclinations are
also stable. On the other hand, for J3 sin ω < 0, the low-eccentricity, near-equatorial
frozen orbits are stable while the near-circular, near-polar frozen orbits are unstable.
The strength of the instability of an unstable frozen orbit can be measured by the
characteristic time. The characteristic time is defined as τ = 1/Λ, and is the time
48
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.8: Stability for frozen orbits about Europa that include the tidal term, J2 and J3.
interval it takes the eccentricity to increase by a factor of exp (1) ∼ 2.718[37]. In
general, the eccentricity will increase by an order of magnitude after exp (1) ∼ 2.718
characteristic times. Even though the tide-only frozen orbit solutions themselves do
not depend on the system parameters, the characteristic times of the unstable frozen
orbits are parameter-dependent. Consider Figure 3.9 which shows the characteristic
times for the 100 km altitude unstable circular frozen orbits for the tide-only cases of
Europa, Dione and Enceladus. Observe that the characteristic times for Europa are
the largest, followed by Dione and then Enceladus. This means that the frozen orbits
about Enceladus are the most unstable. In particular, the Enceladus characteristic
times are about an order of magnitude smaller than the Europa case. This has
strong implications for orbit design since a smaller characteristic time implies a more
highly unstable system in the sense that it will grow along its unstable manifold
more rapidly. All three of these planetary satellites are highly perturbed systems
in which science orbit design is very difficult, and the goals for the orbit design
will have to take the characteristic times into consideration. Observe in Figure 3.9
that the more polar the frozen orbit, the more unstable it is since it has a smaller
characteristic time.
49
40 50 60 70 80 90 100 110 120 130 1400
10
20
30
40
50
60
70
80
90
inclination (degrees)
time
(day
s)
(a) Europa
40 50 60 70 80 90 100 110 120 130 1400
5
10
15
20
25
30
inclination (degrees)
time
(day
s)
(b) Dione
40 50 60 70 80 90 100 110 120 130 1400
1
2
3
4
5
6
7
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Enceladus
Figure 3.9: Characteristic times for the tide-only unstable frozen orbits with 100 km altitudes.
50
60 70 80 90 100 110 12013
13.5
14
14.5
15
15.5
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(a) Circular, near-polar
40 50 60 70 80 90 100 110 120 130 14010
20
30
40
50
60
70
80
90
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Elliptic mid-inclination
Figure 3.10: Characteristic times for unstable frozen orbits about Europa
For the case including the tidal term, J2 and J3, there are two classes of unstable
frozen orbits, namely the near circular near polar frozen orbits with J3 sin ω < 0
and the mid-inclination frozen orbits with J3 sin ω > 0. Figure 3.10 shows the
characteristic times as functions of the inclination for both of these classes of
unstable frozen orbits about Europa. The characteristic times for the near-circular,
near-polar frozen orbits have the same structure as the circular frozen orbits for
the tide-only case, with the minimum characteristic time corresponding to a polar
orbit. The same trend is also apparent for the mid-inclination frozen orbits where
the closest to polar inclination also corresponds to the minimum characteristic time.
Just as we verified the frozen orbit solutions for different possible values of J3,
we also look at the stability and characteristic times for values of J3 one order of
magnitude smaller and one order of magnitude larger than the value in Table 3.1.
Figure 3.11 shows the stability and characteristic times for frozen orbits computed
with a J3 value one order of magnitude smaller and Figure 3.12 shows the same
plots computed with a J3 value one order of magnitude larger. We see that when
compared the the results for our nominal J3 case, the stability properties follow the
same trends. For the larger J3 value, the characteristic times are smaller, meaning
51
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
(a) Stability
65 70 75 80 85 90 95 100 105 110 11513.5
14
14.5
15
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Characteristic times for circular, near-polar orbits
40 50 60 70 80 90 100 110 120 130 14010
20
30
40
50
60
70
80
90
100
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Characteristic times for elliptic, mid incli-nation orbits
Figure 3.11: Stability and characteristic times of frozen orbits about Europa with J3 one order ofmagnitude smaller than Table 3.1
that the orbits are more unstable. This is expected since the J3 perturbation is
acting more strongly. For the smaller J3 value, the characteristic times are larger
than for the nominal J3 value case meaning that the orbits are more stable.
52
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
(a) Stability
65 70 75 80 85 90 95 100 105 110 1157
8
9
10
11
12
13
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Characteristic times for circular, near-polar orbits
40 50 60 70 80 90 100 110 120 130 14010
12
14
16
18
20
22
24
26
28
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Characteristic times for elliptic, mid-inclination orbits
Figure 3.12: Stability and characteristic times of frozen orbits about Europa with J3 one order ofmagnitude larger than Table 3.1
.
53
3.4 Contour Plots and Frozen Orbit Integrations
In this section, the secular motion in the vicinity of frozen orbits will be examined
using contour plots. Then, results of integrations in the 3-DOF system of frozen
orbit solutions will be compared with the contour plots to determine if the stable
frozen orbits are viable orbits in the full 3-DOF system and how unstable frozen
orbits behave in the 3-DOF system. All of the results in this section are computed
for the Europa system, with the tidal component, J2, C22 and J3 gravity field terms,
but are applicable to other planetary satellite systems.
3.4.1 Contour Plots
Contour plots are a tool that can be used to visualize the secular motion of an
orbiter. They are plotted in the 1-DOF system which is integrable and a function
of only two variables, (G,ω) and so a plot in terms of those two variables can
completely describe the motion. It is useful, however, to produce contour plots for
other sets of variables in order to get a better picture of the motion. Recall that
G =√
µa(1− e2). Therefore, a plot in terms of (e,ω) can also be used to describe
the motion. The final contour plot that we use is in terms of the semi-equinoctial
variables (h,k). Although we initially compute these plots for a range of frozen
orbits, they are most useful to describe near-circular motion, as will be shown later.
Note that contour plots of the set (i,ω) are also possible. However, since the motion
is integrable, the inclination can be determined from the integral of motion θ and
the eccentricity. We are, in general, more interested in the motion of the eccentricity
than the inclination since both remain relatively constant for stable orbits, and
for unstable orbits it is the growing eccentricity that causes an impact with the
planetary satellite.
The contour plots are made up of curves with constant values of the 1-DOF
potential, R. The curves define the path that the motion will follow on average.
54
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−3000 −2000 −1000 0 1000 2000 3000
−3000
−2000
−1000
0
1000
2000
3000
Gcosω
Gsi
nω
Figure 3.13: Contour plots for motion in the vicinity of a stable frozen orbit with e∗ = 0.548,i∗ = 50o, J3 sinω > 0.
55
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.14: Contour plots for motion in the vicinity of an unstable frozen orbit with e∗ = 0.0.039,i∗ = 55o, J3 sinω > 0.
56
Figures 3.13-3.16 show contour plots in all three regimes for motion in the vicinity
of four different types of frozen orbits. Figure 3.13 is an example of motion in the
vicinity of a stable frozen orbit with an inclination of 50o, an eccentricity of about
0.548 and J3 sin ω > 0. It is, therefore, a mid-inclination, mid-eccentricity frozen
orbit. We see that for this case the motion is symmetric for ω = ±π/2, showing
that the tidal effect dominates over the J2 and J3 effects. Figure 3.14 shows motion
in the vicinity of an unstable frozen orbit with an inclination of 55o, an eccentricity
of about 0.039 and J3 sin ω > 0. In this case, the motion is not symmetric about
ω = ±π/2 since the eccentricity is small and the J2 and J3 effects are important.
Note that there appears to be two stable regions in the plot, other than the unstable
frozen orbits. These regions correspond to equilibrium points in the 1-DOF system
that do not have the altitude of periapsis of 100 km that we consider. Their locations
correspond to motion below the surface of the planetary satellite since the contour
plots have a constant value of the semi-major axis. For this case, the semi-major
axis is a = rp/(1 − e) = 1660.8/(1 − 0.039) = 1728 km and so motion below the
surface of the planetary satellite occurs for eccentricities greater than 0.097. The
consequence of this result is that the only motion that we are interested in from a
contour plot is that in the vicinity of the frozen orbit.
Now, consider contour plots for frozen orbits with J3 sin ω < 0. First, Figure 3.15
shows motion in the vicinity of a stable frozen orbit with an inclination of 30o and
an eccentricity of about 0.047. Again, the motion is not symmetric about ω = ±π/2
since this is a low-altitude orbit where the effects of J2 and J3 are significant. The
last example, in Figure 3.16, is an unstable frozen orbit with an inclination of 80o
and an eccentricity of 0.021. Note that the plot for (e,ω) is not over the entire range
of eccentricity but just in the region of the frozen orbit. This gives a better picture
of the motion in the vicinity of the frozen orbit and as previously noted, motion
for larger values of the eccentricity takes place under the surface of the planetary
57
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.15: Contour plots for motion in the vicinity of a stable frozen orbit with e∗ = 0.047,i∗ = 30o, J3 sinω < 0.
58
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
ω (degrees)
ecce
ntric
ity
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.16: Contour plots for motion in the vicinity of an unstable frozen orbit with e∗ = 0.021,i∗ = 80o, J3 sinω < 0.
59
satellite. The frozen orbit in this example is near-circular and near-polar, and it is
the type that will be considered later as a possible science orbit for a mission to a
planetary satellite.
3.4.2 Frozen Orbit Integrations
Results of integrations of frozen orbit solutions are presented overlaying contour
plots in (e,ω) space for both stable and unstable frozen orbits. For stable frozen
orbits, the goal is to determine whether they are viable long-term stable orbits in
the 3-DOF system. For unstable frozen orbits, we are interested in comparing the
motion in the 3-DOF system with the contour plots to see if the motion follows the
contours well.
Figure 3.17(a) shows a contour plot in (e,ω) space along with the results of the
integration in the 3-DOF system of a stable frozen orbit with an inclination of 134o,
an eccentricity of 0.387 and J3 sin ω > 0. The integration takes place over a period
of 150 days and it is clear that the orbit is stable since it stays in the libration
region. Note that the center of the circle representing the integrated trajectory is
not exactly at the frozen orbit location. This is due to the perturbations that are
averaged out in the 1-DOF system. Even though the integrated trajectory does not
follow the curves in the contour plot exactly, the frozen orbit conditions still yield a
stable orbit in the 3-DOF system.
The next example, shown in Figure 3.17(b), shows the integration of a stable
frozen orbit with an inclination of 62o, an eccentricity of 0.795 and J3 sin ω > 0.
The integrated trajectory, shown by the red curve, does not follow the curves in the
contour plot at all and reaches a very high eccentricity which leads to an impact with
the planetary satellite. This occurs since although the orbit is stable in the 1-DOF
system, it has a large eccentricity which means that it has a large apoapsis radius.
Since the motion takes place farther away from the planetary satellite, perturbations
60
not taken into account in the derivation of the 1-DOF arise and cause the 3-DOF
trajectory to be unstable.
Finally, Figure 3.17(c) is the integration of an unstable frozen orbit with an
inclination of 80o, an eccentricity of 0.021 and J3 sin ω < 0. Since this orbit is
unstable, all that we expect is the trajectory to follow the contour curve starting at
the frozen orbit location. This is observed, allowing us to conclude that for small
eccentricities the contour plots give a very good estimate of what motion to expect in
the 3-DOF system. Therefore, the motion in the 1-DOF system can be used to plan
orbits for a mission to a planetary satellite, as will be seen in subsequent chapters.
61
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
(a) e∗ = 0.387, i∗ = 134o, J3 sinω > 0
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
(b) e∗ = 0.795, i∗ = 62o, J3 sinω > 0
0 50 100 150 200 250 300 3500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
ω (degrees)
ecce
ntric
ity
(c) e∗ = 0.021, i∗ = 80o, J3 sinω < 0
Figure 3.17: Contour plots and integrations in the 3-DOF of various frozen orbits
62
CHAPTER 4
ROBUST CAPTURE AND TRANSFER
TRAJECTORIES
In this chapter, we study capture trajectories in the Hill 3-body problem.
Capture trajectories begin in the exterior region of Hill’s problem, enter the interior
region and orbit the planetary satellite at least once. One particular feature of
capture trajectories that we investigate is their lifetime. Uncontrolled, these orbits
can impact with the planetary satellite or exit the interior Hill region after very
short time spans. Koon et al. [20] investigated orbits that travel between the interior
and exterior Hill regions and showed that the amount of time a trajectory spends
orbiting the planetary satellite is determined by chaotic dynamics.
A method is developed which identifies sets of capture trajectories that do
not impact or escape the planetary satellite for extended time periods. These are
called ‘safe trajectories’ and the regions in which they lie ‘safe zones’. These safe
low-energy trajectories may be useful for a future mission to a planetary satellite
such as Europa. With the science goals of that type of mission being such that a
low-altitude, high inclination stable orbit about Europa is desirable, we examine
low-cost methods to transfer from a safe capture trajectory to a long-term stable
orbit such as the elliptic frozen orbits developed in the last chapter.
Criteria on the safe capture trajectories that result in the lowest cost transfers
are determined and we develop schemes to transfer to elliptic frozen orbits and
circular frozen orbits. These schemes are analyzed based on their costs and specific
63
examples are shown. The end result is a method that identifies trajectories that
enter into orbit about Europa on low-energy capture trajectories and which do not
impact or escape for at least one week, and from which it is possible to transfer to a
more stable orbit with relatively low cost.
4.1 Non-dimensional Hill 3-Body Problem and
Libration Points
In this chapter, only the tidal term is included in the perturbing potential R, given
in Eq.(3.1), yielding the following potential:
R =1
2N2(3x2 − r2
). (4.1)
This is an acceptable simplification since we are primarily dealing with motion
farther away from Europa than the low-altitude frozen orbits in which the gravity
coefficients are important. Although the capture trajectories to be presented in this
chapter are in the vicinity of Europa, if they do pass very closely to Europa, it is
only for a short period of time. In addition, when we consider transfers from capture
trajectories to frozen orbits, we consider either the elliptic frozen orbits or higher
altitude circular orbits, both of which are not affected very strongly by J2, C22 or J3.
The equations of motion for this system are:
x = 2Ny − µ
r3x + 3N2x , (4.2)
y = −2Nx− µ
r3y , (4.3)
z = − µ
r3z −N2z , (4.4)
and they are in dimensional form. However, to simplify the computations in this
chapter we make the model non-dimensional. To do this, take l = (µ/N2)1/3
as the
unit length and τ = 1/N as the unit time. The equations then take on the following
64
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
y
Forbidden Region
Forbidden Region
L2 L
1
Figure 4.1: Regions of allowable motion and libration points for J = −2.15. The dotted linerepresents a planar capture trajectory that originates near L2.
form
x = 2y − x
r3+ 3x, , (4.5)
y = −2x− y
r3, (4.6)
z = − z
r3− z , (4.7)
and have no free parameters.
Equilibrium solutions exist in the Hill 3-body problem, analogous to the L1 and
L2 libration points in the CR3BP. There are two libration points that are symmetric
about the origin with coordinates x = ± (µ/3N2)1/3
, y = z = x = y = z = 0
where the x-coordinate in nondimensional form is x = ± (1/3)1/3 = ±0.693 · · · (see
Figure 4.1).
Recall that Eqs. (4.5)-(4.7) have the Jacobi integral, which in non-dimensional
form for the tide-only case is:
J =1
2v2 − 1
r− 1
2(3x2 − z2) , (4.8)
where v =√
x2 + y2 + z2 is the speed of the spacecraft in the rotating frame. The
condition v ≥ 0 in Eq.(4.8) places a restriction on the position of the particle for
a given value of J . Setting v = 0 defines the zero-velocity surface, which sets a
65
physical boundary for the motion (see Figure 4.1).
For values of the Jacobi constant above a critical value, it is possible for both
escaping and capture trajectories to exist in this problem. This critical value of J
defines the energy at which the zero-velocity surfaces open at L1 and L2 and is equal
to
Jcrit = −1
292/3 = −2.16337 . (4.9)
Since our goal is to characterize capture trajectories, we restrict ourselves to
trajectories with a Jacobi constant greater than Jcrit.
4.2 Periapsis Poincare Maps
A Poincare map associates a continuous time dynamical system to a discrete time
system. The use of a Poincare map reduces the dimensionality of a system by at
least one, and by two if there exists a first integral in the system, as is the case in the
Hill 3-body problem. In general, Poincare maps are defined such that the surface of
section is a plane in position space, such as x = 0. However, since all that is needed
to define a Poincare map are two surfaces of section that are transversal to the flow,
they are defined differently here and periapsis passages are used as the surface of
section, following [42].
4.2.1 Definition of the Map
Our Poincare map relates capture trajectories that start near the libration point
L2 to the periapsis passages of these trajectories. Following [41], the initial surface
of section is defined as the surface of a sphere with radius (13)1/3 bounded by the
zero-velocity surface. This surface passes through the libration point region and
extends as far as the zero-velocity surface of the Hill problem. The image surface is
defined by the periapsis condition r = 0 and r > 0. For the 3-D Hill problem, the
phase space is six-dimensional. The use of the Poincare map restricts the problem
66
to the surface of section, and hence reduces the problem by one dimension. The
Poincare map is then computed at a given value of the Jacobi integral J , (and so
the map is a function of J), reducing the problem by one further dimension to four
dimensions.
Due to the symmetry between the L1 and L2 libration points, analysis is only
performed on capture trajectories that originate near L2; however, the results
can be directly related to trajectories that originate near L1 by the following
transformation[42]
(x, y, z, x, y, z, t)Gµ−→ (−x,−y,−z,−x,−y,−z, t) (4.10)
since the equations of motion remain unchanged. By a capture trajectory, we mean a
trajectory that remains in the Hill region of the planetary satellite for a finite period
of time. These trajectories enter the region of the planetary satellite in the vicinity
of the libration point and orbit the planetary satellite at least once. Figure 4.1 shows
an example of a capture trajectory.
In the 3-D case, the periapsis Poincare map is four dimensional and four
parameters are needed to characterize it. For the initial surface of section, we use
the (x, z) coordinates and two angles (δ, φ) for the direction of the velocity vector.
The (x, z) coordinates are chosen randomly on the section of the surface of the
sphere that falls within the allowable region. Since the surface of section is a sphere,
the y coordinate can be computed from the relationship r =√
x2 + y2 + z2, where
r = (1/3)1/3 and the sign for y is chosen randomly. The initial conditions are then:
x0 = x x0 = v cos φ cos δ
y0 = ±√
r2 − x2 − z2 y0 = v cos φ sin δ (4.11)
z0 = z z0 = v sin φ
67
where (φ, δ) ∈ [π/2, 3π/2], and v is computed from the Jacobi integral:
v =
√2
(J +
1
r
)+ (3x2 − z2) . (4.12)
The parameters used to represent the image map are the periapsis position vector
(x, y, z) and the inclination.
With the initial conditions given by Eq.(4.11), an 8(7) order Runge-Kutta-
Fehlberg integration routine is used to integrate the trajectories. They are integrated
for multiple passages through the image map, meaning that they have multiple
periapsis passages. The phase space is studied in terms of these multiple periapsis
passages for various values of J .
4.2.2 Poincare Map Results
As an initial study, consider the planar case of the Hill 3-body problem. In this
model, motion is restricted to lie in the x-y plane, and the periapsis Poincare
map reduces to two dimensions. The initial surface, which consists of the surface
of a sphere in the three-dimensional case reduces here to an arc of circle. It is
parameterized with the x coordinate and an angle δ which defines the direction of
the velocity vector. This is equivalent to the initial conditions defined by Eq.(4.11)
with z = φ = 0. On the image map, the coordinates (x, y) of the periapsis position
vector are used.
Although the periapsis Poincare map is computed in non-dimensional coordinates
to allow for its application to many different physical systems, we primarily consider
Europa. On many of the periapsis Poincare maps, the surface of Europa is indicated
to give perspective to the plot. Although some of the periapsis passages of these
trajectories lie beneath the surface of Europa, indicating that they impact with it, for
the time being they are included in the analysis since they are not necessarily impact
trajectories when a different planetary satellite is considered. In a later section of
68
(a) First periapsis passage region
(b) First 4 periapsis passage regions: Black - 1st, blue- 2nd, red - 3rd, green - 4th
Figure 4.2: Periapsis Poincare maps for J=-2.15 (planar case) where the circle represents thesurface of Europa.
this chapter, we distinguish between impacting and non-impacting trajectories.
We first consider the periapsis Poincare map for the first periapsis passage of
trajectories with J = −2.15, as was done in [42]. As shown in Figure 4.2(a) the first
periapsis passages are divided into two disjoint regions. The first, located near the
planetary satellite, corresponds to trajectories that immediately enter the region in
the vicinity of the planetary satellite and have their first periapsis passage there.
The second region, located near the L2 libration point, corresponds to trajectories
that do not immediately leave the vicinity of L2 and have their first periapsis passage
in that region. These can be associated with the periodic orbit that exists about L2,
as shown in [41]. Some of these trajectories will subsequently enter the region of the
planetary satellite and some will escape from the system entirely. Note that by our
current definition of capture trajectory, there is no guarantee that the trajectory
actually comes from outside the Hill region and not from inside the region itself.
To remedy this, an additional integration is performed whereby the trajectory is
integrated backwards in time for four time units. This is a sufficient amount of
69
time for a true capture trajectory to return to its starting location outside of the
Hill region. Therefore, the set of capture trajectories is restricted to those that lie
beyond the boundary of the Hill region after a backwards-time integration of four
time units. If after the backwards-time integration a trajectory was located inside
the Hill region, we could conclude that it is not a true capture trajectory, but just
a trajectory that happened to have an inward velocity at the opening of the Hill
region at the initial time.
We now extend the periapsis Poincare map to four periapsis passages by plotting
the first four periapsis passages of each trajectory for J = −2.15. Each periapsis
passage is assigned a different color, as shown in Figure 4.2(b). Observe that
each subsequent periapsis passage region becomes more spread out. This occurs
since symplectic coordinates are not used on our map. In addition, some of the
colors overlap (i.e, there is a small blue region inside the black region). These
overlaps occur because, as previously mentioned, some of the trajectories do not
immediately have a periapsis passage in the vicinity of the planetary satellite but
instead have their first periapsis passage near the L2 region. Therefore it is not until
the second periapsis passage that these trajectories are near the planetary satellite.
The dynamics of these trajectories, once they enter the region near the planetary
satellite, are the same as the dynamics of the trajectories that enter that region
immediately. It is also important to note that the gaps in some of the regions are not
because trajectories cannot have periapsis passages there, but because these regions
were generated numerically and, as such, only a finite number of trajectories can be
considered. From this point forward, the region of first periapsis passages that occur
in the vicinity of the L2 libration point will be ignored. The first periapsis passage
of a particular trajectory is defined to be its first periapsis passage that occurs in
the vicinity of the planetary satellite.
To illustrate how the periapsis Poincare maps depend on J , Figure 4.3 shows
70
Figure 4.3: Poincare maps for various values of J . The circle in each plot denotes the surface ofEuropa.
71
−0.12−0.11
−0.1−0.09
−0.08−0.07
−0.06−0.05
0
0.01
0.02
0.03
0.04
0.05−0.015
−0.01
−0.005
0
0.005
0.01
0.015
xy
Figure 4.4: First periapsis passage region for J=-2.15 (3-d case). The position is indicated by thebase of the arrow, the velocity direction by the direction of the arrow, and the magnitude of thevelocity by the length of the arrow.
a series of Poincare maps for increasing values of J . The regions become larger
as J increases, which is expected since the measure of the initial condition region
increases with J . Only the first periapsis passage regions are shown, since the trends
shown by these regions also apply to subsequent periapsis passages. Note that the
plots in Figure 4.3 are mirror images of the corresponding plots shown in [42].
It is much more difficult to visualize the periapsis Poincare maps for the 3-D
problem since the Poincare map itself is four dimensional. In general, the trends we
see in the planar case apply to the non-planar case, such as the periapsis passage
regions becoming larger in position space as J is increased as well as for subsequent
periapsis passages. One way to visualize the Poincare map for the non-planar case
is to plot each periapsis passage’s three-dimensional position vector, with an arrow
indicating the direction of its velocity (representing the inclination at the periapsis
passage). An example of this is shown in Figure 4.4. Another way to visualize
the non-planar case is by considering the inclination of the periapsis passage as
a function of its radius. Figure 4.5(a) shows the inclination as a function of the
radius for the first periapsis passage of trajectories with various values of J . Observe
that as J increases, the periapsis passages reach higher inclinations. Figure 4.5(b)
72
(a) First periapsis passage regions, various values ofJ
(b) First 4 periapsis passages, J = −2.15
Figure 4.5: Inclination as a function of normalized radius for 3-d capture trajectories.
73
shows the inclination as a function of the radius for multiple periapsis passages of
trajectories with J=-2.15. In this case, the maximum obtainable inclination does
not increase for subsequent periapsis passages, but the periapsis passages do occur
over larger ranges of radii.
We now introduce a more systematic description of these maps and regions.
Define the set of trajectories that lie in the first periapsis passage region close to
Europa as the set S1,
S1 = x|x ∈ Initial Region . (4.13)
Then, under the flow dynamics of the Hill problem the Poincare map can be
represented as Φ, where
Sn+1 = Φ(Sn) = Φ2(Sn−1) = · · · = Φn(S1) . (4.14)
Thus, it is clear that any point in Sn+1 has a unique image in S1. If we use the
symmetry operator defined by Eq.(4.10), Gµ, we can also associate the ‘mirror image’
trajectories as S ′n = Gµ(Sn) defined by reflecting all coordinates and velocities about
the origin.
4.3 Symmetry Between Escape and Capture
Trajectories
Aside from the symmetry Gµ between L1 and L2 which was previously discussed,
additional symmetries exist in this problem (and in the CR3BP). Two that are
considered relate to the symmetry between escape and capture trajectories. If
(x, y, z, x, y, z, t) is a solution of the equations of motion, then the trajectories
obtained by applying the following transformations are also solutions [42]:
(x, y, z, x, y, z, t)G1−→ (−x, y, z, x,−y,−z,−t) , (4.15)
(x, y, z, x, y, z, t)G2−→ (x,−y, z,−x, y,−z,−t) . (4.16)
74
(a) First periapsis passage regions for capture trajec-tories and symmetric escape regions
(b) First 4 periapsis passage regions with symmetricescape regions, black - 1st, blue - 2nd, red - 3rd, green- 4th
Figure 4.6: Poincare maps showing symmetric capture and escape regions for J = −2.15.
Thus, applying either of the two above symmetries to a capture trajectory will
yield an escaping trajectory. An escaping trajectory is a trajectory that crosses the
initial surface of sphere with an outward velocity and exits the Hill region. Figure
4.6 shows the symmetry between the capture and escape regions on the Poincare
periapsis map for the first periapsis passage in the planar case. This figure shows the
first periapsis passage region for the capture trajectories, and hence for the escaping
trajectories this corresponds to the periapsis passage immediately preceding escape
(or transfer to the region in the vicinity of L1,2).
This symmetry between capture and escape trajectories can be extended past
75
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
y L2 L
1
Forbidden Region
Forbidden Region
Figure 4.7: A capture trajectory that escapes after 4 periapsis passages. The periapsis passages areindicated by ‘x’.
the first periapsis passage. This allows us to determine when capture trajectories
can escape from the Hill region. A capture trajectory escapes the Hill region when
it passes through the initial surface of sphere with an outward velocity. However,
a captured trajectory can not escape arbitrarily. We found that it will only escape
when it passes through the ‘first periapsis passage’ escape region, which is the
escaping region symmetric to the first periapsis passage capture region as shown on
Figure 4.6(a). For this example (J=-2.15), we found that the earliest that capture
trajectories pass through this region is during their third periapsis passage, and so
we conclude that once a capture trajectory with J=-2.15 is found, it is guaranteed
to have at least 3 periapsis passages prior to escape. Such a trajectory must also
lie in the pre-image of the escape region. Due to this, these transfer regions can
be easily found by identifying regions where these symmetric regions intersect with
each other. Figure 4.6(b) is a periapsis Poincare map which shows the first four
periapsis passages of capture trajectories as well as one set of their symmetric
regions corresponding to escaping trajectories. We see that there is an overlap
of these regions, demonstrating the mechanism by which capture trajectories can
subsequently escape. An example of this is shown in Figure 4.7. This trajectory has
its third periapsis passage in the portion of the red region that overlaps with the blue
symmetric region. Therefore, its fourth periapsis passage will be in the portion of the
76
green region that overlaps with the black symmetric region, as seen in Figure 4.6(b).
Any capture trajectory which has a periapsis passage in the black symmetric region
will not have any further periapsis passages and will escape. Thus, from the fact
that this trajectory had its third periapsis passage in a region that overlaps with the
blue symmetric escaping region, it will have a total of four periapsis passages and
then escape. As seen in Figure 4.7, this trajectory does indeed escape from the Hill
region after the fourth periapsis passage.
The symbolic notation defined earlier is used to give a clear description of this
result. First, let the time reversal symmetry be the operator Gt. From the property
of this symmetry:
S−(n+1) = Gt(Sn+1) = Gt(Φn(S1)) = Φ−nGt(S1) = Φ−n(S−1) . (4.17)
The condition for a trajectory x1 ∈ S1 to escape is then that it eventually lies in S−1
or Φn(x1) ∈ S−1 for some n. If this occurs Φn(S1) ∩ S−1 6= 0, or is not empty for
some n, and these trajectories escape in n − (−1) = n + 1 periapsis passages. In
Figure 4.6(b) this region is represented by the overlap of the green points and the
black points on the right side of the plot. Φ3(S1) ∩ S−1 is not empty and thus they
escape in 4 periapsis passages from their initial location in S1. Due to the uniqueness
of solutions, we also infer that the image of this set is invariant:
Φ(Φn(S1) ∩ Φ−m(S−1)) = Φn+1(S1) ∩ Φ−(m+1)(S−1) . (4.18)
In Figure 4.6(a), we immediately note that for low values of J , S1 ∩ S−1 =
S1 ∩Gt(S1) = 0 and no immediate escapes occur. We also note, however, that when
J is large enough, the set S1 crosses the y = 0 axis, meaning that S1∩Gµ(Gt(S1)) 6= 0,
where Gµ is the symmetry operator defined by Eq.(4.10), and thus direct escapes
can occur after one periapsis passage for higher J values.
Although all of the figures in this section were computed for the planar case, the
77
conclusions and the symbolic notation also apply to the non-planar case where the
sets are four dimensional. In the non-planar case, analogous symmetric escaping
regions exist, and we find that capture trajectories must have a minimum of three
periapsis passages before escaping for low values of J . Figures computed for the
planar case are used to visualize the results graphically.
4.4 Safe Zones
We have previously discussed the possibility of capture trajectories eventually
escaping from the Hill region; however, we have not yet considered the trajectories
that impact with the planetary satellite. In the Poincare maps presented thus far all
trajectories have been considered, even those whose periapsis passages clearly fall
beneath the surface of the planetary satellite. To differentiate between trajectories
that impact and trajectories that don’t we introduce the term ‘safe trajectory’
to denote a trajectory that doesn’t impact with the planetary satellite or escape
from the Hill region for some specified period of time. The regions in which these
trajectories lie are denoted as ‘safe zones’. In particular, consider the first periapsis
passage region S1, and denote the set of all safe trajectories in the set the safe zone
in S1. Then a trajectory can be identified as safe based on where its first periapsis
passage occurs. Although safe zones are defined here by the first periapsis passage of
the trajectory, they could be defined by any of the periapsis passages. However, since
the periapsis passage regions become distended for subsequent periapsis passages,
the first periapsis passage region is the smallest, and hence is a logical choice for
defining the safe zones.
We first consider safe zones in the planar case, and will continue to use Europa
as our example. For this discussion, a trajectory is considered to be safe if it persists
for a non-dimensional time period of 4π, which for Europa corresponds to 7.1 days.
Figure 4.8 shows the safe zone projected into the first periapsis passage of capture
78
Figure 4.8: Poincare map showing the safe zone, impact and escape regions. Red - impact, Blue -escape, Black - safe.
−0.17 −0.16 −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.09 −0.08
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
x
y
surface of Europa
J=−2.15
J=−2.14
J=−2.13
J=−2.12
J=−2.11
J=−2.10
Figure 4.9: Poincare maps of planar safe zones for various values of J .
trajectories (S1) with J=-2.15. Capture trajectories with J = −2.15 and their first
periapsis passage in the black region will not impact or escape for at least one week.
The trajectories that lie in the red region will impact and the trajectories in the blue
region will escape.
Next, consider the safe zones for various values of J . The results are shown in
Figure 4.9 and are presented in a slightly different fashion. Only the safe zone is
plotted, not the entire region of first periapsis passages. Once again, these maps are
computed for 4π time units and so do not impact or escape over that period of time.
79
Observe that as the Jacobi integral value increases, the safe zone regions become
more distended and have a larger y value.
By characterizing the safe zones in this way, we can conclude that if a trajectory
has its first periapsis passage in the safe zone, if will be safe for at least one week. It
is then possible, for a trajectory with a particular Jacobi integral value, to determine
where its first periapsis passage should be. A trajectory that falls in this region is
potentially useful. If the goal is to perform a maneuver to place it in a more stable
trajectory but the maneuver fails, the spacecraft will remain safe for a period of
time, hopefully long enough to perform another maneuver. Another motivation for
characterizing these safe zones arises when considering low-thrust maneuvers. Since
we know where in phase space the safe zones are located, low-thrust maneuvers can
be designed such that the spacecraft travels through safe zones over the course of its
maneuver, thus guaranteeing that the spacecraft will not impact or escape.
The safe zones discussed above were computed for the planar case of the Hill
3-body problem. Analogous regions can be computed for the non-planar case;
however, it is necessary to study them in a different way. In this case, it is not
only where the periapsis passages are located in position space that is important,
but their inclinations. It is also possible to characterize the three dimensional safe
zones in terms of the radius of the trajectory at its periapsis passage along with
its inclination. However, it is not the visualization of the safe zones in the three
dimensional problem that are important, but their existence. The method used to
find safe zones for non-planar trajectories is essentially the same as the method
used in the planar case. First, capture trajectories for a particular value of J are
identified. Then, they are integrated for a period of 4π to determine whether or not
they escape from the region or impact with the planetary satellite over a one week
period for the Europa case. The capture trajectories that do not impact or escape
during that time period are defined to be the safe trajectories.
80
(a) Inclination as a function of normalized ra-dius
(b) Eccentricity as a function of inclination
Figure 4.10: Characteristics of safe capture trajectories for J = −1.70.
4.5 Transferring to Stable Frozen Orbits
The preceding section defined a method to compute safe trajectories over a given
period of time. For science operations, it is necessary to transfer the spacecraft into
an orbit that is long-term stable. Recall from the previous chapter that for the
tide-only case, stable circular frozen orbits exist for inclination between 0 − 39.23o
and 140.77− 180o and stable elliptic frozen orbits exist over all inclinations. These
are therefore the stable orbits that we target for transfers.
4.5.1 Using a Safe Capture Trajectory to Initiate a Transfer
We first consider the elliptic frozen orbits. In order for this type of transfer to
be affordable in terms of ∆v, the safe trajectories and the frozen orbits must lie
close to each other in phase space. The analysis of capture trajectories so far has
been in terms of their periapsis passages. This is useful here since an ideal location
from which to initiate a transfer to a more stable orbit is at periapsis. Good
science trajectories should have low periapsis altitudes and this can be ensured by
performing the transfer at periapsis. Additionally, periapsis is the optimal place to
make changes to the orbital energy.
81
When considering periapsis passages of capture trajectories from which to initiate
a maneuver, we look at all of the periapsis passages of these trajectories. Ideally,
it would be better to use a periapsis passage that occurs near the beginning of
the defined safe period, for obvious reasons. However, for simplicity, any periapsis
passage that occurs during the safe period is allowed. First consider the safe zone
for capture trajectories with J=-1.70 to see over what range of radii they occur.
Figure 4.10(a) shows that the periapsis passages occur over a wide range of radii,
all the way up to a normalized radius of almost 0.7, which corresponds to 13,773
km (the radius of Europa is 1561 km). It is therefore necessary to restrict periapsis
passages to those which have altitudes much closer to the surface. In Figure 4.10(b)
observe that the periapses of the safe capture trajectories occur over all eccentricities,
up to about 0.72.
4.5.2 Transfer to Elliptic Frozen Orbits
For transfers to elliptic frozen orbits, we restrict the periapsis passages of capture
trajectories to those that are within 250 km of the surface of Europa. Figure 4.11
shows the eccentricity as a function of inclination for periapsis passages of safe
capture trajectories that satisfy this condition for different values of J . As J
increases, the periapsis passages reach higher inclinations. In addition, restricting
the periapsis passages to fall within 250 km of the surface of Europa causes a
large reduction in the range of eccentricities over which they occur. For example,
comparing the part of Figure 4.11 that corresponds to the capture trajectories for
J=-1.70 to Figure 4.10(b), note that when the radius of periapsis is not restricted,
the periapsis passages had eccentricities in the range of e = 0 → 0.72. On the other
hand, when the radius of periapsis is restricted, the eccentricity only ranges over
about e = 0.68 → 0.72. Thus we conclude that the lower eccentricity periapsis
passages of the safe capture trajectories occur at larger radii and hence are farther
82
40 50 60 70 80 90 100 110 120 130 140 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
inclination (degrees)
ecce
ntric
ity
J=−1.00J=−1.20J=−1.35J=−1.50J=−1.70J=−1.85frozen orbits
Figure 4.11: Periapsis passages of capture trajectories for J = −1.70 with altitudes ≤ 250 km.
from the surface of Europa.
Recall that the elliptic frozen orbits for the tide-only case have orbit elements
ω = ±π/2, and e =√
1− 5/3 cos2 i. Also note that for the larger eccentricity frozen
orbits being considered here, the gravity field of the planetary satellite does not
strongly affect the frozen orbit solutions, justifying our use of the tide-only case.
Since the goal is to transfer from a safe capture trajectory to a frozen orbit, it is
necessary to determine what characteristics the periapsis passages of the capture
trajectories must have in order to perform this maneuver. The first step is to only
consider periapsis passages of capture trajectories that have an argument of periapsis
close to the frozen orbit value. Since the frozen orbits oscillate in the unaveraged
system, the condition ω = ±90o can be relaxed somewhat to consider periapsis
passages of safe capture trajectories that fall within 5 degrees of ω = ±90o. Then, if
the periapsis passage of the safe capture trajectory has an inclination for which an
elliptic frozen orbit exists, a maneuver can be performed to change the eccentricity
so it corresponds with the inclination based on the frozen orbit relation, placing the
spacecraft in an elliptic frozen orbit.
Figure 4.12 shows the set of periapsis passages of safe capture trajectories with
83
J=-1.60 that lie within 250 km of the surface of Europa and have an argument of
periapsis within 5 degrees of ±90o. A line denoting where the elliptic frozen orbits
lie is shown, which makes apparent the change in eccentricity required to transfer
from a capture trajectory to a frozen orbit. Note that the frozen orbit line actually
passes through the safe zone. This means that capture trajectories that lie on the
frozen orbit line have orbital elements corresponding to a frozen orbit at that instant.
However, a true elliptic frozen orbit is stable and bounded and would not arrive from
outside of the Hill sphere. This implies that the frozen orbit assumptions are not
valid at this point. Frozen orbits at higher eccentricities would be subject to larger
perturbations and so would not be valid either. We do not know precisely where the
averaging assumptions fail, however, we have evidence from numerical integrations
that points to the left of the frozen orbit line are still governed by the averaged
equations of motion. Therefore, when considering safe capture trajectories from
which to initiate a transfer, we do not consider periapsis passages that fall to the
right of the frozen orbit line, since this is beyond the point at which the averaging
assumptions break down.
Figure 4.12 also shows the cost, in meters per second, to transfer from the safe
capture trajectory to an elliptic frozen orbit. The periapsis passages of safe capture
trajectories used to initiate the transfer are those shown in Figure 4.12. The region
in the figure denoted ‘not valid’ refers to the region where the frozen orbits have
broken down. The region denoted ‘valid’ refers to the fact that these transfers are
practical since the frozen orbits exist. The gray region is present since we do not
have an absolute boundary on where the frozen orbits cease to exist, and so some
transfers in the gray region may be valid while others may not.
To prove that our scheme to transfer to an elliptic frozen orbit works, we
computed the transfer for one of the periapsis passages of the safe capture
trajectories and then continued the integration to show that the resulting trajectory
84
48 50 52 54 56 58 60 62 64
0.6
0.65
0.7
0.75
0.8
0.85
inclination (degrees)
ecce
ntric
ity
frozen orbits
∆v
(a) (b)
Figure 4.12: Potential periapsis passages with J = −1.60 for transfers to elliptic frozen orbits andcosts. (a) Periapsis passages with altitudes ≤ 250 km ω = ±90o ± 5o. (b) Cost to transfer to acorresponding elliptic frozen orbit.
is an elliptic frozen orbit. Starting from the safe capture trajectories with J = −1.60,
as shown in Figure 4.12, we choose the periapsis passage that has the following
parameters: e = 0.718, ω = −93.66o, i = 51.14o, and a periapsis altitude of 166 km.
The eccentricity that corresponds to a frozen orbit at this inclination is 0.586, and
so the cost of performing this eccentricity change is 69 m/s. To show that the new
orbit is an elliptic frozen orbit, we plot the trajectory on contour plots of eccentricity
as a function of argument of periapsis and inclination as a function of argument of
periapsis. Figure 4.13 shows that the new orbit does stay inside the libration region
of the contour plot,verifying the successful transfer from a safe capture trajectory to
a bounded orbit close to frozen orbit conditions.
4.5.3 Transfers to Circular Frozen Orbits
Two approaches to transferring to a circular frozen orbit are considered: two-impulse
and one-impulse maneuvers. The two-impulse scheme involves performing a
maneuver at a periapsis passage of the capture trajectory that allows us to use
the dynamics of elliptic frozen orbits to achieve a lower eccentricity (which in turn
increases the inclination). Following that, a second maneuver is performed at the
85
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
e
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
10
20
30
40
50
60
70
80
90
ω (degrees)
i (de
gree
s)
Figure 4.13: Contour plots and the numerical results for the orbit obtained after the transfermaneuver.
much lower eccentricity to circularize the orbit. The one-impulse maneuver involves
circularizing directly from the periapsis passage of the capture trajectory. Note that
this analysis assumes that the circular frozen orbits in the tide-only case exist. For
low-altitude orbits, we showed in Chapter 3 that the planetary satellite’s gravity field
has a large effect on the motion. When the effect of the planetary satellite’s gravity
field is included, the previously circular frozen orbits become near-circular, and have
small eccentricities. However, this does not impact our analysis very strongly since
we compute costs to circularize, which can be considered as a worst-case assumption
that is more costly than transferring to a near-circular orbit.
Consider first the two-impulse approach. Previously, in order to transfer to an
elliptic frozen orbit, a maneuver was chosen to achieve the frozen orbit eccentricity.
By examining the left plot in Figure 4.13 we see that if a slightly higher eccentricity
is chosen (i.e. an eccentricity near the top of the libration region), much larger
variations in eccentricity are achieved, to the point where the eccentricity occasionally
gets very close to 0. Figure 4.14 shows this phenomenon, where instead of performing
a maneuver to change the eccentricity to 0.586 as above, the eccentricity is changed
to 0.687. The same periapsis passage as in the elliptic frozen orbit transfer example
is used (i.e, e = 0.718, i = 51.14o, ω = −93.66o, altitude=166km). The cost of this
86
maneuver is 38.6 m/s. The next step in this approach is to circularize the orbit.
Depending on the altitude of the orbit after it is circularized, some control maneuvers
may be necessary to keep the orbit from drifting too much. The frequency and size
of these maneuvers will depend on how much of a drift we can tolerate and how
quickly the orbit is drifting (depending on its altitude).
The circularization maneuver is performed at a periapsis passage of the orbit
shown in Figure 4.14. In order to understand how the radius, eccentricity and
inclination of the orbit vary and where the periapsis passages occur, refer to
Figure 4.15 which shows the radius, eccentricity and inclination as functions of
time. The periapsis passages are indicated with a red ‘x’ and the solid blue line in
the top plot represents the surface of Europa. Observe that a lower eccentricity is
correlated to a higher radius and higher inclination. Higher inclinations are more
desirable; however, the higher radius associated with the higher inclination will lead
to an initially large circular orbit. To compare these issues, two examples will be
provided: one where the circularization maneuver is performed from the minimum
eccentricity of the orbit shown in Figure 4.14 and one where the circularization is
performed from a higher eccentricity. In both cases, the radius of the orbit will be
relatively large and some control maneuvers will be necessary to maintain this orbit.
Table 4.1 shows the cost of achieving both of these circular orbits as well as their
average radius and initial inclination. Figure 4.16 shows their characteristics.
We now consider the one-impulse approach for transferring to a circular orbit.
This approach involves directly circularizing the orbit at the periapsis passage of
the capture trajectory. The cost of this maneuver is 444.5 m/s and the resulting
orbit has a radius of 1731 km which corresponds to an altitude of 166 km and an
inclination of 51.14o.
All of the transfers to circular frozen orbits computed thus far result in orbits
that have different radii and inclinations. To compare them, it is necessary to
87
Table 4.1: Comparison of costs to transfer from a capture trajectory to a circular frozen orbit.
Example 1a Example 2b
From capture trajectory to near-frozen elliptic orbit 38.6 m/s 38.6 m/s
Circularization 10.3 m/s 230.3 m/s
Total Cost 48.9 m/s 268.9 m/s
Average Radius 5300 km 2875 km
Initial Inclination 61.6o 57.9o
a Circularizing at the minimum eccentricity of the near-frozen elliptic orbitb Circularizing at higher eccentricity
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
e
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
10
20
30
40
50
60
70
80
90
ω (degrees)
i (de
gree
s)
Figure 4.14: Contour plots and numerical results for a transfer orbit to a circular frozen orbit.
88
0 5 10 15 20 250
0.2
0.4
0.6
0.8
e
0 5 10 15 20 2545
50
55
60
65
i (de
gree
s)
0 5 10 15 20 250
5000
10000
r (k
m)
time (days)
Figure 4.15: Characteristics of the near-frozen elliptic transfer orbit as functions of time. Eachperiapsis passage is represented by an ‘x’ and the solid line in the top plot represents the surface ofEuropa.
0 1 2 3 4 5 6 70
0.05
0.1
0.15
0.2
e
0 1 2 3 4 5 6 758
60
62
64
i (de
gree
s)
time (days)
0 1 2 3 4 5 6 74500
5000
5500
6000
r (k
m)
(a)
0 1 2 3 4 5 6 70
0.01
0.02
0.03
e
0 1 2 3 4 5 6 755
56
57
58
i (de
gree
s)
time (days)
0 1 2 3 4 5 6 72800
2850
2900
2950
r (k
m)
(b)
Figure 4.16: Characteristics of both possible circularized orbits over one week. (a) Transfer fromthe minimum eccentricity of the near-frozen elliptic orbit. (b) Transfer from a higher eccentricityof that same orbit.
89
Table 4.2: Comparison of costs to transfer from capture trajectory to tightly bound circularorbit
Method
Using Dynamics to Circularize Direct Circularization
Example 1 (i = 61.6o) 591.5 m/s 692.2 m/s
Example 2 (i = 57.9o) 586.6 m/s 604.7 m/s
a Circularization takes place at the minimum eccentricity of near-frozen elliptic orbitb Circularization takes place at a higher eccentricity
consider additional transfers so that a common orbit is achieved for both approaches.
Two common orbits are considered, with inclinations of 61.6o and 57.9o and a radius
of 1731 km. The two inclinations correspond to the inclinations achieved by the
two examples of the two-impulse approach and the radius corresponds to the radius
achieved by the one-impulse approach. The orbit obtained by the one-impulse
approach has an inclination of 51.14o and so to transfer to inclinations of 61.6o and
57.9o costs 247.7 km/s and 160.2 km/s, respectively. The radii of the orbits achieved
by the examples for the two-impulse approach are about 5300 km and 2875 km and
the costs of the Hohmann transfers to reduce those radii to 166 km are 542.6 m/s
and 299.7 m/s respectively. Table 4.2 shows a comparison of the costs of the two
approaches. We see that the method which uses the dynamics of the system in the
vicinity of the elliptic frozen orbits is more efficient, and the difference between the
two methods is larger when the final inclination is larger.
If the total costs of only the two-impulse approach are compared, we see that
although the second example is slightly cheaper, an inclination of almost 4o lower
is achieved. If the Hohmann transfer component of this approach is neglected, we
see that while the second example achieves a slightly smaller circular frozen orbit, it
has a cost of 268.9 km/s which is 5.5 times greater than the transfer via the first
example (see Table 4.1). Both circular orbits are stable there so is a clear advantage
90
to following the first example. Since the cost to achieve the stable circular orbit is
quite small (less than 50 m/s) it is possible that a low-thrust vehicle could achieve
this orbit. Following that, the low-thrust vehicle could transition down to a tighter
circular orbit with a spiral maneuver.
As another approach to transferring from a safe capture trajectory to a circular
frozen orbit, we examine the possibility of performing a simultaneous circularization
and plane change maneuver. It is not possible to do this maneuver with the current
example, since for this maneuver to be feasible the argument of periapsis at the
periapsis passage must be either 0o or 180o and for our current example the argument
of periapsis is approximately −90o. Therefore, we look back to our data of all
the periapsis passages of capture trajectories at this energy level, J = −1.60, and
examine their arguments of periapsis. Figure 4.17(a) shows argument of periapsis
plotted as a function of eccentricity for all of the periapsis passages of the capture
trajectories with J = −1.60. This plot includes periapsis passages with any radius
of periapsis. However, as before, our goal is a final orbit with a low altitude.
Therefore, restrict the periapsis passages to those with an altitude less than 250 km.
Figure 4.17(b) also shows the argument of periapsis as a function of eccentricity
for the periapsis passages that satisfy this condition. We see that the argument of
periapsis values lie between about 50 to 150o and −150 to −50o. Therefore, there
are no periapsis passages with altitudes close to the surface for which a simultaneous
circularization and plane change maneuver can be performed.
We conclude, from comparing the two plots in Figure 4.17, that the periapsis
passages with lower arguments of periapsis must have a large radius. We also see
that these periapsis passages have lower eccentricities. This is not surprising with the
doubly averaged dynamics in mind. In the doubly averaged system, the semi-major
axis is constant on average. Therefore, since the capture trajectories originate far
from the planetary satellite, for them to have a periapsis passage with a low altitude,
91
(a) (b)
Figure 4.17: ω as a function of eccentricity for safe capture trajectories with J = −1.60. (a) Allperiapsis passages. (b) Periapsis passages with altitude ≤ 250 km.
the eccentricity must be large. The numerical results support this since when all
periapsis passages are considered, they range in eccentricity from ∼ 0.32 → 0.75, but
when only those with altitudes less than 250 km are considered, the eccentricity only
ranges between about 0.705 → 0.75. Therefore, by limiting the radius of periapsis,
only periapsis passages with large eccentricities remain. Considering the averaged
dynamics once again, note that for a trajectory to reach a large eccentricity it must
be circulating about a libration region and libration regions are centered about
±90o. Then, examining Figure 4.17(b) it is clear this is the case. The arguments
of periapsis in this plot extend about ±50o on either sides of ±90o. We therefore
conclude that it is not dynamically possible to have a periapsis passage of a capture
trajectory that will allow a simultaneous circularization and plane change maneuver
to place the trajectory in a tightly bound circular orbit.
92
CHAPTER 5
THE DESIGN OF LONG LIFETIME SCIENCE
ORBITS
The requirements for a science orbit about a planetary satellite are generally a
near-polar, low-altitude orbit so that the science requirements of the mission, such
as imaging the surface and measuring various features of the planetary satellite,
are satisfied. Since the environment in the vicinity of many planetary satellites,
such as Europa, is highly perturbed, it is important for a science orbit to have a
long lifetime. For example, the nominal length of the science phase of a mission to
Europa could be one month, but it is essential to ensure that the spacecraft will not
impact with the planetary satellite if a particular control maneuver fails. Therefore,
the goal of long lifetime orbit design for Europa is lifetimes of around 100 days. This
will ensure that the spacecraft will be safe while awaiting the next possible maneuver
opportunity.
From Chapter 3, we have seen that stable frozen orbits exist at small inclinations
for low-altitude circular orbits (in the tide + J2 + C22 + J3 model) and at larger
altitudes (since the orbits are elliptic) for near-polar orbits (in tide-only and non
tide-only models). However, neither of these types of stable frozen orbits satisfy the
science requirements. Thus, we look towards unstable frozen orbits as a means for
designing long lifetime orbits that satisfy the science requirements. If a spacecraft is
arbitrarily placed in a near-polar, low-altitude orbit its lifetime is somewhat random
and can range from very short to very long. See, for example, Figure 5.1 which
93
0 20 40 60 80 100 120 140 160 1800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
time (days)ec
cent
ricity
impact
Figure 5.1: Eccentricity as a function of time for trajectories with an initial altitude of 100 km,initial inclination of 70o and different initial argument of periapsis values.
shows the eccentricity as a function of time for a few trajectories in the Europa
system. These trajectories have an initial altitude of 100 km, an inclination of 70o
and different initial argument of periapsis values. Their lifetimes vary from 40-165
days, demonstrating that arbitrarily placing a spacecraft in a low-altitude near-polar
orbit can result in impact after a relatively short period of time.
It will be shown later in this chapter that integrating a near-circular, near-polar
unstable frozen orbit in the 3-DOF system (including J2, C22 and J3) does not yield
a very long lifetime orbit. However, the structure of the dynamics in the vicinity
of these unstable frozen orbits helps in the design of long lifetime trajectories.
Specifically, since they have non-zero eccentricities, their stable and unstable
manifolds in the 1-DOF system are shifted and this causes variations in the rate of
change of the eccentricity along their paths. These variations can be exploited by
choosing the path that remains in the vicinity of a frozen orbit for the longest period
of time. If a trajectory in the full 3-DOF system follows this path, it can have a
relatively long lifetime on the order of at least 100 days, as compared to lifetimes as
short as 40 days for randomly chosen initial conditions.
After identifying the desired trajectory in the 1-DOF system, we develop a
94
method to systematically compute initial conditions in the 3-DOF system such that
the trajectory in that system will follow the 1-DOF system on average. To do this,
we use all three systems identified thus far, namely, the 1, 2 and 3-DOF systems.
We start with a point on the stable manifold of a frozen orbit in the 1-DOF system.
We then linearize the 1-DOF system about that point on the manifold and obtain
corrections such that the initial conditions lead to a trajectory in the 2-DOF system
that has motion in the 1-DOF system as its average. This procedure is repeated on
the 3-DOF system, by linearizing about the corrected point in the 2-DOF system.
The result is a set of initial conditions in the 3-DOF system that, when integrated,
follow the 1-DOF system on average.
A toolbox to compute long lifetime science orbits using the methods described in
this chapter was written in Matlab and delivered to the Jet Propulsion Laboratory
so they could use it to design these orbits. It is described at the end of this chapter.
5.1 Dynamics in the 1-DOF System
The frozen orbits that will be the basis of our analysis in this chapter are the
near-polar, near-circular unstable frozen orbits that exist in the tide + J2 + C22 +
J3 model. They are indicated by the heavy dashed line in Figure 5.2. Although
these frozen orbits have constant orbital elements on average in the 1-DOF system,
they are unstable, and so initializing at one in the 3-DOF system results in the
trajectory diverging from the frozen orbit location. Recall Figure 3.16, where we see
the trajectory in the (e,ω) plot diverge from its initial frozen orbit location. The
trajectory will impact with the planetary satellite after a short period of time since,
as seen in the plot, the eccentricity increases very rapidly leading to a decrease in
the radius of periapsis. Thus, the unstable frozen orbits can’t be used directly as
long lifetime orbits.
Since the 1-DOF system is Hamiltonian and the near-circular, near-polar frozen
95
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
Figure 5.2: Frozen orbit solutions: unstable near-circular, near-polar orbits indicated by heavydashed line.
orbits are unstable, they have stable and unstable manifolds. The stable manifolds
approach the frozen orbit and the unstable manifolds depart from the frozen orbit.
If a spacecraft is initialized on the stable manifold of a frozen orbit, we expect
that it will drift towards the frozen orbit location on the stable manifold, and then
follow the unstable manifold towards impact with the planetary satellite. A good
understanding of the manifolds can help us determine where exactly to initialize
a spacecraft in order to obtain the maximum possible lifetime. Note that these
manifolds exist only in the 1-DOF system, but their existence here impacts dynamics
at higher degrees of freedom.
The manifolds of the frozen orbits are determined by computing the contour
of constant potential, R, that passes through the frozen orbit. This represents the
motion in the 1-DOF system corresponding to a trajectory that passes through the
frozen orbit. The manifolds are plotted in the three different regimes, (h, k), (G, ω)
and (e, ω). Figure 5.3 shows the stable and unstable manifolds for a frozen orbit
with a 70 degree inclination. In all three plots, the unstable manifolds are numbered
I and III and the stable manifolds are numbered II and IV. We can see most clearly
from Figure 5.3(b) that there are four possible paths along the manifolds that pass
96
through the frozen orbit, identified with an ‘x’. By examining the possible paths
along the manifolds in Figure 5.3(c) the number of degrees of ω covered along each
path can be determined. For this computation, only the portion of the trajectory
that is near-circular is considered. Once the trajectory departs from its near-circular
value, the radius of periapsis will decrease to below the surface of the planetary
satellite since the semi-major axis remains constant. The four possible paths and
approximate number of degrees of ω covered along each path are as follows:
1. II→ I: 270 degrees
2. II→ III: 180 degrees
3. IV→ I: 180 degrees
4. IV→ III: 90 degrees
Therefore, by beginning on the stable manifold labelled II, passing through the
frozen orbit and then continuing on the unstable manifold labelled I, 270 degrees of
ω are covered while the orbit is near-circular. This is the path that will produce
the longest lifetime orbit in the 3-DOF system. Different ranges of ω are covered
along different paths because the frozen orbits are not circular. Figure 5.4 shows
the manifolds in (h,k) space with a circle denoting where the radius of periapsis is
at the surface of Europa. Therefore, values of (h,k) outside of the circle represent
radii of periapsis below the surface of Europa and hence impact. Different distances
along the manifold are covered inside of the circle along the different paths since the
frozen orbit located at the center of the ‘x’ is not at (0,0). A longer distance inside
the circle means a slower rate of change of the eccentricity which leads to a longer
lifetime. In the 3-DOF system, the trajectory will not follow the manifolds exactly.
It is possible, however, to determine initial conditions such that this path along the
manifold is followed on average and very long lifetime orbits are attained.
97
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ω (degrees)
ecce
ntric
ityII I
IV III
(a) (e,ω)-space
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
I II
III IV
(b) (h,k)-space
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
II I
IV III
(c) (G,ω)-space
Figure 5.3: Stable and unstable manifolds for a frozen orbit, identified with an ’x’, withe∗ = 0.0129, i∗ = 70o, ω∗ = −90o and a∗ = 1682.5 km.
98
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
k=ecosω
h=es
inω
I II
III IV
impact
Figure 5.4: Expansion of the central region in 5.3(b) with Europa impact circle
In the tide-only model, the manifolds of the circular unstable frozen orbits have
the same structure, with the only difference being that the frozen orbit is located at
the origin. Therefore, there is no advantage to choosing the II→I path as opposed to
the II→III path. When this design method is applied to the Saturn moons Enceladus
and Dione later in this chapter, the tide-only model is used since the gravity fields of
those moons are not known. This means that the advantage in terms of lifetime of
a non-circular frozen orbit doesn’t apply. Therefore, the lifetimes obtained for these
moons are conservative estimates of what would be possible for a real mission where
the gravity field coefficients would probably be known by the time the science phase
of the mission takes place.
These manifolds explain why when a trajectory with initial conditions
corresponding to a frozen orbit is integrated in the unaveraged system, it does not
have a particularly long lifetime. As Figure 5.5 shows, when the trajectory that
is initialized at the frozen orbit location is integrated, it immediately follows the
unstable manifold path III since the equilibrium does not exist in the 3-DOF system.
Additional lifetime is not gained since the trajectory is not initialized on the stable
manifold. This trajectory has a lifetime of 44 days.
99
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
k=ecosω
h=es
inω
Figure 5.5: 3-DOF system integration of a frozen orbit with e∗ = 0.0129, i∗ = 70o, ω∗ = −90o anda∗ = 1682.5 km with its stable and unstable manifolds.
5.2 Computing Initial Conditions in the 3-DOF
System
To this point, we have discussed reducing the 3-DOF system to a 1-DOF system
in order to have a better qualitative understanding of the overall dynamics. From
this system, equilibrium solutions are identified and motion in their vicinity is
studied. However, as seen in the previous section, using the frozen orbits as initial
conditions in the full 3-DOF unaveraged system does not produce very long lifetime
trajectories. We have already shown that the longest lifetime orbits begin on the
stable manifold of the frozen orbits, move towards the frozen orbits and then move
off on the unstable manifold. The question then becomes how to determine initial
conditions in the 3-DOF system such that the trajectory will follow the manifolds in
the 1-DOF system. The easiest first guess would be to initialize the trajectory with
initial conditions that lie exactly on the 1-DOF manifold.
Computing the point on the manifold is fairly straightforward. First a frozen
orbit is chosen and the integrals of motions in the 1-DOF system, θ = cos2 i∗(1−e∗2),
R∗ = R(e∗, i∗, ω∗) are computed. To compute a point on the manifold, start with
a desired initial eccentricity em. Then, having already computed θ from the frozen
100
orbit conditions, the inclination on the manifold, im is obtained from:
im = cos−1
√θ
1− e2m
. (5.1)
Then, substituting em and im into
R∗ = R(em, im, ωm) , (5.2)
we obtain the following quadratic equation in sin ωm:
0 =− 15N2a2
8e2
m sin2 im sin2 ωm +3µJ3
2a4(1− e2m)5/2
em sin im (1
−5
4sin2 im
)sin ωm +
N2a2
4
[(1− 3
2sin2 im
)(1 +
3
2e2
m
)(5.3)
+15
4e2
m sin2 im
]− R∗ ,
which can be easily solved for ωm. Continuing with the frozen orbit considered
thus far with e∗ = 0.0129, i∗ = 70o and ω∗ = 270o, we compute a point on the
1-DOF manifold with the following characteristics: em = 0.02, im = 69.998o and
ωm = 170.97o.
As Figure 5.6 shows, although the trajectory initialized with the above conditions
and integrated in the 3-DOF system does follow the manifolds, it is not centered on
them and its lifetime is 87 days. This occurs because there are oscillations due to
both the motion of Europa about Jupiter and the spacecraft about Europa that are
not accounted for in the 1-DOF system. As well, the effect of C22 is not accounted
for in the 1-DOF system since averaging cancels it. The ideal trajectory would be
centered on the manifold since such a trajectory would spend the longest time in
the vicinity of the frozen orbit, giving it the longest possible lifetime. Therefore, we
now discuss a method that starts with a point in the 1-DOF system, pulls the point
back to a point in the 2-DOF system and concludes by pulling back that point to
initial conditions to be used in the 3-DOF system. These initial conditions take into
101
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
(a)
−0.021 −0.02 −0.019 −0.018 −0.017 −0.016 −0.015 −0.014 −0.013
−4
−2
0
2
4
6
8
x 10−3
k=ecosω
h=es
inω
(b)
Figure 5.6: (a) Integration of trajectory in 3-DOF system, initialized on 1-DOF stable manifold.(b) Expansion of the initial part of the trajectory.
account all of the oscillations and produce a trajectory that follows the manifolds in
the 1-DOF system very closely.
The method used to relate both the 2-DOF system to the 1-DOF system and the
3-DOF system to the 2-DOF system is a linearization method. It is performed two
times, first on the 2-DOF system and then on the 3-DOF system. The method will
first be described in general terms, and then some specifics for each system will be
discussed.
Let x be a vector containing the orbital elements under consideration. Then,
the time derivatives of the orbital elements, obtained from the Lagrange Planetary
Equations, can be expressed as follows:
x′ = g0(x) + g1(x, t) , (5.4)
where ′ denotes the time derivative, g0(x) is a vector containing secular (averaged)
components of the equations, which are time-invariant and g1(x, t) is a vector
containing the time-periodic components of the equations, with period T. We assume
that |g1| |g0|. Eq.(5.4) is linearized about a point x0 in the time-invariant system
102
such that x′0 = g0(x0). Then, let x = x0 + δx where δx is a small perturbation from
the point. Substituting this into Eq.(5.4), we obtain:
(x0 + δx)′ = g0(x0 + δx) + g1(x0 + δx, t) . (5.5)
We perform a Taylor series expansion on the right hand side of Eq.(5.5), to first
order in the time-invariant terms and to zeroth order in the time-periodic terms and
obtain:
x′0 + δx′ = g0(x0) +∂g0(x0)
∂x
∣∣∣∣x0
δx + g1(x0, t) + · · · . (5.6)
The expansion is carried to only the zeroth order in the time-periodic terms since
the first order terms are much smaller than the zeroth order terms. This allows for a
straightforward analysis of the system. The linearized system can be reduced to the
following since x′0 = g0(x0):
δx′ = Aδx + B(t) , (5.7)
where higher order terms are ignored and
A =∂g0(x0)
∂x
∣∣∣∣x0
, (5.8)
B(t) = g1(x0, t) . (5.9)
Note that the motion of x0 is ignored since we are only interested in evaluating the
system over a short time span. Therefore, A is a constant matrix and B(t) is a
time-periodic vector. We can solve for δx as follows:
δx = eA(t−t0)δx0 +
∫ t
t0
eA(t−τ)B(τ)dτ . (5.10)
Our goal for the linearization is to obtain initial conditions δx0 such that the
motion in the original system has the motion in the time-invariant system as its
average. To do this, we take the average of Eq.(5.10) over one period T and set it
103
equal to zero.
0 =1
T
∫ T
0
eA(t−t0)dtδx0 +1
T
∫ T
0
∫ t
t0
eA(t−τ)B(τ)dτdt . (5.11)
Let
D =1
T
∫ T
0
eA(t−t0)dt , (5.12)
E =1
T
∫ T
0
∫ t
t0
eA(t−τ)B(τ)dτdt . (5.13)
Then, we solve for δx0 from Eq.(5.11) to obtain
δx0 = −D−1E . (5.14)
Therefore, if the original system is integrated with initial conditions x0 + δx0, the
motion should be, on average, the time-invariant system initialized at x0.
5.2.1 From 1-DOF System to 2-DOF System
In this section the algorithm is used to compute initial conditions δx0 in the
2-DOF system that will produce motion that follows the 1-DOF system on average.
Since the semi-major axis is constant, consider only x =
[e i ω
]T
. The time
derivatives of the orbital elements in the 2-DOF system are obtained by using R in
Eqs.(3.4)-(3.6):
de
dt=
1− e2
na2e
∂R
∂M−√
1− e2
na2e
∂R
∂ω, (5.15)
di
dt=
1√1− e2na2
(cot i
∂R
∂ω− csc i
∂R
∂Ω
), (5.16)
dω
dt=
√1− e2
na2e
∂R
∂e− cot i√
1− e2na2e
∂R
∂i. (5.17)
These equations can be broken up into two components where the g0 components
correspond to the LPE applied to R and the g1 components correspond to the
LPE applied to R − R. Then, the g0 components are time-invariant and the g1
104
components are time-periodic. The time periodicity appears as periodicity in Ω
with a period of π. Therefore, to simplify the system, rewrite Eqs. (5.15)-(5.17)
as derivatives with respect to Ω rather than derivatives with respect to time by
multiplying the right-hand sides of the equations by −(1/N).
Since our goal is to obtain long lifetime orbits, the 2-DOF system is linearized
about a point on the stable manifold of a frozen orbit in the 1-DOF system. Let this
point be xm0 =
[e0 i0 ω0
]T
, with Ω0. Then, A and B(Ω) as defined in Eqs.(5.8)
and (5.9) are computed fairly easily analytically. Following that, D is computed
using the Romberg quadrature algorithm [10] and E is computed using the Gaussian
quadrature for double integrals algorithm [10] to determine the corrections to xm0 ,
δxm0 . The Romberg quadrature algorithm is described in Burden and Faires[10] for a
scalar. In this case however, the integral that needs to be computed is the integral
of each term in the matrix D. Therefore, the algorithm is repeated for each entry in
D. The specific algorithms used to to compute each term in D and each term in E
are given in Appendix A.
We illustrate this method with an example. First, choose the frozen orbit with
i∗ = 70o, e∗ = 0.0129, ω∗ = 270o and Ω0 = 0. Then, compute the point on the stable
manifold of this frozen orbit with the following characteristics:
xm0 =
0.02
69.99756766o
170.97339256o
Then, by following the linearization algorithm, we obtain the following corrections
to the initial conditions:
δxm0 =
0.00027136
0.86310567o
0.21413018o
To verify that the 2-DOF trajectory, initialized at xm
0 + δxm0 , has as its average the
105
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.017
0.0175
0.018
0.0185
0.019
0.0195
0.02
0.0205
time (days)
ecce
ntric
ity
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.869
69.5
70
70.5
71
time (days)in
clin
atio
n (d
eg)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8170
171
172
173
174
175
176
177
time (days)
ω (
deg)
Figure 5.7: Eccentricity, inclination and argument of periapsis as functions of time. Solid lines -2-DOF trajectory, initialized at xm
0 + δxm0 , dashed lines - 1-DOF trajectory, initialized at xm
0 .
1-DOF trajectory initialized at xm0 , both the 2-DOF and 1-DOF trajectories are
plotted. As Figure 5.7 shows, the dashed lines representing the 1-DOF trajectory
pass through the center of the solid curves which represent the 2-DOF trajectory.
5.2.2 From 2-DOF System to 3-DOF System
In this section, initial conditions in the 3-DOF system that will produce motion
that follows the 2-DOF system on average are computed. Since the results from the
previous section are incorporated, the resulting motion also follows the 1-DOF system
on average. In this case, there are six equations and x =
[a e i ω Ω σ
]T
where σ = M − nt and n is the mean motion of the spacecraft about the planetary
satellite. The time derivatives of these orbital elements are those stated in
106
Eqs.(3.4)-(3.7) plus the equation for σ taken from [8]:
da
dt=
2
na
∂R
∂M, (5.18)
de
dt=
1− e2
na2e
∂R
∂M−√
1− e2
na2e
∂R
∂ω, (5.19)
di
dt=
1√1− e2na2
(cot i
∂R
∂ω− csc i
∂R
∂Ω
), (5.20)
dω
dt=
√1− e2
na2e
∂R
∂e− cot i√
1− e2na2e
∂R
∂i, (5.21)
dΩ
dt=
1√1− e2na2 sin i
∂R
∂i, (5.22)
dσ
dt= − 2
na
∂R
∂a− 1− e2
na2e
∂R
∂e. (5.23)
For this system, the g0 components correspond to the LPE applied to R and the g1
components correspond to the LPE applied to R− R. Therefore, the g0 components
are time-invariant and the g1 components are time-periodic. In this case, the time
periodicity appears in terms of the true anomaly, ν, with a period of 2π which
corresponds to one period of the spacecraft about Europa. Since the true anomaly
is a function of the mean anomaly, the equations are expressed as derivatives with
respect to mean anomaly rather than with respect to time to simplify the analysis.
To transform from time derivatives to derivatives with respect to mean anomaly, the
equations must be multiplied by 1/n.
Since our goal is to determine initial conditions such that the 3-DOF trajectory
has the 2-DOF trajectory as its average, we linearize about a point in the 2-DOF
system. In particular, we linearize about the point in the 2-DOF system that
we obtained in the previous section and let this point be x0 = xm0 + δxm
0 . The
constant matrix A is still straightforward to compute analytically even though it is
a 6x6 matrix, and D is once again computed using the Romberg algorithm given
in Appendix A. However, the vector B, which in this case is B(ν(M)), cannot be
computed analytically since Kepler’s equation must be solved to determine the true
107
anomaly, ν, as a function of the mean anomaly, M . Therefore, we turn to the
computation of E. The computation of E where
E =1
2π
∫ 2π
0
∫ M
0
eA(M−τ)B(ν(τ))dτdM (5.24)
can still be accomplished using the double integral version of Gaussian quadrature
again, but the algorithm must be modified to solve Kepler’s equation, and hence
compute B at each step. This modified algorithm to compute E term-by-term is
also given in Appendix A. The corrections to x0, δx0, are then computed.
We now continue with the example from the previous section to show that when
integrated with the initial conditions obtained from the linearized system, the 3-DOF
system follows the 2-DOF system on average. The point that is linearized about,
which is also the initial condition for the 2-DOF system, is
x0 = xm0 + δxm
0 =
1682.53057218 km
0.02027136
70.860673330
171.18752273o
0o
0o
,
The corrections to these initial conditions, δx0 are found to be:
δx0 =
2.98938278 km
0.00219834
0.00239633o
0.17321674o
−0.00679549o
−0.07757915o
.
Figure 5.8 shows plots of the integrated 3-DOF trajectory, initialized at x0 +δx0, and
108
0 50 100 150 200 250 300 350 4001679
1680
1681
1682
1683
1684
1685
1686
sem
i−m
ajor
axi
s (k
m)
mean anomaly (deg)
0 50 100 150 200 250 300 350 4000.017
0.018
0.019
0.02
0.021
0.022
0.023
ecce
ntric
itymean anomaly (deg)
0 50 100 150 200 250 300 350 40070.81
70.82
70.83
70.84
70.85
70.86
70.87
70.88in
clin
atio
n (d
eg)
mean anomaly (deg)
0 50 100 150 200 250 300 350 400164
166
168
170
172
174
176
178
ω (
deg)
mean anomaly (deg)
Figure 5.8: Semi-major axis, eccentricity, inclination, and argument of periapsis as functions ofmean anomaly. Solid lines - 3-DOF trajectory, initialized at x0 + δx0. Dashed lines - 2-DOFtrajectory, initialized at x0.
the integrated 2-DOF trajectory, initialized at x0. In all four orbital element plots,
the 2-DOF trajectory cuts through the center of the 3-DOF trajectory. Therefore,
the goal of the 3-DOF motion following the 2-DOF motion on average has been
accomplished.
5.3 Long Lifetime Orbits in the 3-DOF System
Now that we have generated initial conditions in the 3-DOF system that follow
the manifold in the 1-DOF system on average, it is necessary to verify that these
initial conditions actually generate long lifetime orbits. Consider, for example,
Figure 5.9 which shows the integration, in the 3-DOF system, of the initial conditions
109
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
impact
Figure 5.9: Integration in the 3-DOF of a trajectory initialized at x0 with the 1-DOF manifolds.
generated in the previous section. Although it was shown in the previous section
that the 3-DOF motion did follow the 1-DOF on average, the trajectory will not, in
general, follow the 1-DOF manifold exactly since the linearization used is only an
approximation to the true motion. When several examples of potential long lifetime
orbits are computed, we find that sometimes the correct manifold path is followed
and sometimes it is not. In all of the cases, the trajectory does not have the desired
manifold path as its exact average. For the case in Figure 5.9, the trajectory follows
the wrong manifold path and has a lifetime of 84 days. This is not unexpected since
there should about a 50/50 chance of the trajectory lying on one side of the manifold
or the other.
To ensure that the trajectory always follows the desired manifold path a small
correction in the argument of periapsis, called the bias, is introduced. We show that
it is possible to find a particular argument of periapsis correction that gives the
maximum long lifetime orbit for a specific set of initial conditions. For the example
in Figure 5.9, the bias in the argument of periapsis that produces the longest lifetime
orbit is -0.0067 radians (−0.38o). This maximum lifetime orbit follows the 1-DOF
manifold exactly. However, from a practical mission standpoint, it is not that useful
110
0 20 40 60 80 100 120 140 160 1800
0.02
0.04
0.06
0.08
time (days)
ecce
ntric
ity
Figure 5.10: Time history of the eccentricity for a long lifetime orbit.
to find the maximum lifetime orbit. This is because the uncertainty in the orbit’s
position is, in general, greater than the accuracy with which the maximum lifetime
orbit is determined. In addition, since the maximum lifetime orbit is one that follows
the 1-DOF manifold very closely, a very small error in the initial position could
push the trajectory to the wrong manifold path. The details regarding the orbit
uncertainty will be discussed in the next chapter. For now, we focus on finding long
lifetime orbits that follow the desired manifold path, but do not necessarily have
maximum lifetimes.
5.3.1 Minimum Eccentricity on the Manifold
In the example that has been discussed thus far, the initial eccentricity on the
manifold was chosen arbitrarily to be 0.02 since the only requirement for an initial
eccentricity is that it be on the stable manifold of the long lifetime orbit. There
is however, a more systematic way to choose the initial eccentricity. Consider
Figure 5.10 which shows the time history of the eccentricity for a long lifetime orbit
corresponding to a 70o inclination frozen orbit. Note in this plot that the eccentricity
decreases rapidly before it settles around the frozen orbit value where it stays almost
constant for the majority of the orbit. Observe further that just before the constant
value is reached, the eccentricity goes through a minimum. It is this minimum
eccentricity value that will be used from now on to initialize long lifetime orbits.
One advantage of using this minimum eccentricity value is that the period of time
111
−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.03
−0.02
−0.01
0
0.01
0.02
0.03
k=ecosω
h=es
inω
emin
frozenorbit
manifoldpoint
Figure 5.11: 1-DOF manifolds with minimum eccentricity circle (dashed curve).
before the trajectory reaches this value is quite short compared the amount of time
it spends oscillating about the frozen orbit. Thus, not much orbital lifetime is gained
by starting at a higher eccentricity value. In addition, always initializing trajectories
at the minimum eccentricity for the frozen orbit under consideration provides a good
way to compare the lifetimes of the orbits. Finally, in terms of controlling the orbit,
initializing at the minimum eccentricity provides advantages that will be shown in
the following chapter.
The minimum eccentricity on the stable manifold can be computed geometrically.
First, consider Figure 5.11. The line connecting the frozen orbit point to the
manifold point is the linear approximation of the manifold in that region. Although
the manifold in the 1-DOF system is governed by a very complicated function, it can
be approximated by a line in the vicinity of the frozen orbit, as seen in Figure 5.3(b).
The manifold point is a point that has the same eccentricity as the frozen orbit and
an argument of periapsis consistent with the 1-DOF manifold (computed using the
method in section 5.2). The minimum eccentricity then lies on this line and is the
point with the minimum distance from the origin, as shown in Figure 5.11. Let the
frozen orbit point be (k∗,h∗) and the manifold point be (km,hm). Then, the equation
112
for the line connecting the two points is:
h =
(hm − h∗
km − k∗
)(k + km)− hm , (5.25)
and the minimum distance from the origin to this line is[43]:
d =
∣∣∣∣∣∣∣det
km − k∗ k∗
hm − h∗ h∗
∣∣∣∣∣∣∣√
(hm − h∗)2 + (km − k∗)2. (5.26)
Since, in the (k,h) plot, the eccentricity is equivalent to the radial distance from the
origin to a point, the minimum eccentricity is given by the distance d in Eq.(5.26).
Then, the corresponding argument of periapsis on the manifold can be computed
by the method in section 3.2. Note that since the line used to the approximate the
manifold is not an exact solution of the manifold, it is only used to compute the
minimum eccentricity (distance to the line) and not the argument of periapsis (angle
between the origin and the line). Thus, the (e,ω) pair actually sits on the manifold.
If the approximation was used to compute both values, the resulting pair would not
lie on the manifold, but on the approximation to the manifold.
5.3.2 Examples of Long Lifetime Orbits
The general algorithm used to compute long lifetime orbits is given in Table 5.1.
As previously noted, it is not always necessary to find the maximum lifetime orbit
corresponding to a particular frozen orbit. This maximum lifetime orbit will be the
trajectory that follows the desired manifold path exactly (i.e. it has the manifold
as its average). The goal here is to show that it is possible to design long lifetime
trajectories that follow in general the II→I path as in Figure 5.4. They do not need
to have the manifold has their exact average, but must lie relatively close to the
manifold. The first examples presented are for the Europa system, with the tide, J2,
C22 and J3 components included in the model. Following that, we also show that it
113
Table 5.1: Algorithm for computing a long lifetime orbit
1. Choose frozen orbit in the 1-DOF system based on its inclination.2. Compute the minimum eccentricity point on the stable manifold of the
frozen orbit.3. Compute the corrected initial conditions to the 2-DOF system by using the
linearization of the 2-DOF system about the point on the manifold.4. Compute the corrected initial conditions to the 3-DOF system by using the
linearization of the 3-DOF system about the corrected 2-DOF initial conditions.5. Find an argument of periapsis value that produces a long lifetime orbit by
numerical search, if necessary.
is possible to design orbits for other planetary satellites such as the Saturnian moons
Enceladus and Dione. Since, as demonstrated in Chapter 3, the environments in the
vicinity of those moons are much more unstable than the Europa environment, the
long lifetime orbits designed for those systems are not nearly as long-lived as those
in the Europa system.
Starting with the Europa system, trajectories are computed using the minimum
eccentricity point on the manifold in step 2 of the algorithm in Table 5.1. We find
that the trajectory follows the desired manifold path without a bias correction in all
the cases, which cover the entire range of unstable, near-polar, near-circular frozen
orbits. The lifetimes of these trajectories are:
Frozen Orbit Lifetime
Inclination (days)
70o 105
75o 119
85o 110
95o 96
105o 95
110o 99
Note that the lifetimes of the direct orbits are all larger than the lifetimes of
114
the retrograde orbits. It has been shown that direct and retrograde orbits have
different properties in the 3-DOF system and that this difference disappears during
the averaging technique used here [22]. Therefore, it is possible that, due to the
averaging used for this analysis, the initial condition computation algorithm provides
better results for direct than for retrograde orbits. These trajectories fulfill the goal
of designing a long lifetime orbit that follows the desired manifold path, without
having it as an exact average. It is an interesting exercise to demonstrate that the
algorithm brings us very close to a trajectory that follows the manifold exactly (i.e.
the maximum lifetime orbit). To find the maximum lifetime orbit for each case, a
numerical search over argument of periapsis values is performed to find the necessary
bias (step 5 of the algorithm in Table 5.1). Also, note that although all of the
trajectories initialized at the minimum eccentricity point follow the correct manifold
path in the Europa case, as shown above, that will not necessarily be true for all
initial points on the manifold. For those cases, a numerical search over ω values can
be performed to either find the maximum lifetime orbit or any long lifetime orbit
that follows the correct manifold path.
Table 5.2 shows the bias in the argument of periapsis that produces the longest
lifetime orbit to an accuracy of 0.0002 radians (0.011o) for each frozen orbit
inclination case discussed above. Note that all of the bias values are less than about
one and a half degrees and the lifetimes increase by between 45 and 90 percent. This
large variation in improvement in lifetime occurs since the maximum lifetime orbit is
only being computed to an accuracy of 0.0002 radians in the argument of periapsis
bias. If a smaller step size in argument of periapsis was used, the lifetimes would all
increase by percentages in the higher range. Also note that the bias values are larger
for the retrograde orbits than for the direct orbits. This is probably because the
original lifetimes of the retrograde orbits, using the unbiased value of ω, are smaller
than the original lifetimes for the direct orbits. Although the existence of these
115
Table 5.2: Long Lifetime Orbits for the Europa System
Frozen Orbit Unbiased Lifetime Maximum Lifetime Bias (rad)Inclination (days) (days)
70o 105 182 0.009575o 119 174 0.002685o 110 171 0.004695o 96 182 0.0169105o 95 172 0.0254110o 99 188 0.0242
maximum lifetime orbits is interesting in a theoretical sense, from a mission planning
point of view, actually obtaining the maximum lifetime orbit is not very practical.
This is because, as we will show in the next chapter, the accuracy with which the
position and velocity of an orbit can be obtained is larger than the accuracy needed
for the maximum lifetime orbit. Figure 5.12 shows the unbiased long lifetime orbit
and the maximum long lifetime orbit for the 75o inclination case. Observe that the
maximum lifetime orbit has the manifold as its average, while the unbiased long
lifetime orbit sits slightly above the manifold.
Now, proceeding with the Saturnian moons, first note that the models for these
systems only include the tidal force since gravity field information is not currently
available. Therefore, the frozen orbits under consideration are the near-polar,
circular unstable frozen orbits. The goal here is simply to demonstrate that although
Europa has been the primary focus of this dissertation, the method for designing
long lifetime orbits described here is applicable to other planetary satellite systems.
For a real mission, the gravity fields of these planetary satellites would likely be
measured during the mission, prior to the science phase, and our design technique
including gravity field coefficients could be used. This would improve the lifetimes
of the orbits computed due to the advantage of having a non-circular frozen orbit.
The minimum eccentricity point on the manifold is actually the frozen orbit
in this case, since the frozen orbit is circular. Therefore, an arbitrary starting
116
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
impact
(a) General long lifetime orbit
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
impact
(b) Maximum lifetime orbit
Figure 5.12: Long lifetime orbits about Europa (70o frozen orbit inclination)
117
Table 5.3: Long Lifetime Orbits about Enceladus
Frozen Orbit Lifetime BiasInclination (days) (rad)
70o 10.09 085o 9.79 0.0315105o 11.18 0.0715110o 11.18 0.0890
Table 5.4: Long Lifetime Orbits about Dione
Frozen Orbit Lifetime BiasInclination (days) (rad)
70o 53.79 0.012685o 62.73 0.00725105o 53.46 0.02688110o 62.13 0.03418
eccentricity on the manifold is chosen in each case. The long lifetime orbits computed
are trajectories that follow the desired manifold path, without necessarily being the
maximum lifetime orbit. Bias corrections are included where necessary to produce
a trajectory that follows the desired path. Since Enceladus has a radius of only 252
km, it was necessary to set the frozen orbit altitude to 50 km instead of 100 km as
in the Europa case. The perturbations on the orbits about Enceladus with 100 km
altitudes were too large to make meaningful design possible. Dione has a radius of
560 km, and so frozen orbit altitudes of 100 km were feasible for the design.
Tables 5.3 and 5.4 show the results of the the long lifetime trajectory design for
Enceladus and Dione. Figures 5.13 and 5.14 show the trajectories in (h,k) space for
both planetary satellites for the 70o frozen orbit inclination case. First, observe the
the lifetimes of the orbits for both planetary satellites are significantly less than the
lifetimes obtained for the Europa system. The orbits about Enceladus have lifetimes
in the vicinity of 10 days, and the orbits about Dione have lifetimes in the vicinity
of 50-60 days. This is not surprising since the environments in the vicinity of both
118
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
k=ecosω
h=es
inω
impact
Figure 5.13: Long lifetime orbit about Enceladus (70o frozen orbit inclination)
planetary satellites are more highly perturbed than the environment about Europa.
These shorter lifetimes follow from the characteristic time results in Chapter 3,
Figure 3.9.
The main result of this section is that the long lifetime orbit design technique is
a feasible method for designing long duration orbits about planetary satellites. The
maximum possible length of the designed orbit depends on the planetary satellite
system under consideration, where the more highly perturbed a system is, the
shorter the lifetimes will be. In addition, it is possible to do an initial design using
a tide-only model to get an estimate of the lifetimes of trajectories possible for a
particular system. These orbits can then be refined in a real mission situation once
the gravity field has been measured. For Europa in particular, which is the main
motivation of this dissertation, it is possible to design orbits with lifetimes of at least
100 days using this method.
119
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
k=ecosω
h=es
inω
impact
Figure 5.14: Long lifetime orbit about Dione (70o frozen orbit inclination)
5.4 A Toolbox for Computing Long Lifetime
Orbits
The main goal of the toolbox is to provide a set of functions, with inputs modifiable
by the user, to perform analysis of the frozen orbit solutions and compute initial
conditions of the long lifetime science orbits. It can be used for science orbit design
for any planetary satellite that can be modeled by the Hill 3-body problem. This
toolbox was written for the Jupiter Icy Moons Orbiter Mission and delivered to the
Jet Propulsion Laboratory.
The toolbox, written in Matlab, integrates all of the results in this chapter and
provides three main functions:
1. Plot the frozen orbit solutions in the 1-DOF system. This plot gives an overall
picture of the underlying dynamics of the full 3-DOF system.
2. Compute initial conditions in the 3-DOF system that yield a trajectory that
follows the manifolds in the 1-DOF system on average.
3. Integrate a set of initial conditions in the 3-DOF system and produce a series
of plots.
120
This toolbox not only provides initial conditions for long lifetime science orbits, but
also allows for analysis. Since one of the input groups is the physical parameters of
the system, it can be used to compare frozen orbit solutions for different planetary
satellites, or to understand how the frozen orbit solutions depend on the physical
parameters (i.e., the gravity field terms) for a particular planetary satellite. The
user can also choose the approximate initial eccentricity of the science orbit through
the choice of the eccentricity on the 1-DOF manifold, and the desired frozen orbit
inclination. The toolbox produces initial conditions in the 3-DOF system that have
not been adjusted with a bias in the argument of periapsis. Therefore, the user
can also study how changing the initial argument of periapsis affects the science
orbit duration. This is especially important when considering the effect of orbit
uncertainty. This analysis is done with the third part of the toolbox, which also has
the capability to integrate, in the 3-DOF system, of any set of initial conditions.
121
CHAPTER 6
CONTROL OF LONG LIFETIME ORBITS
Thus far, we have simplified the original 3-DOF system to analyze the underlying
motion of the system and used this information to design long lifetime orbits in the
full 3-DOF system. However, as shown in the previous chapter, long lifetime orbits
have finite lifetimes. It is not only important to discuss the control of the orbits
in the 3-DOF system used here, but also the control of orbits in a real-life system
where it is impossible to attain the desired orbit exactly due to uncertainty errors in
the position and velocity. The specific control issues that will be discussed in this
chapter are:
• Low thrust control to keep the spacecraft at a specific location
• Impulsive maneuvers to reset the long lifetime orbit once it reaches the unstable
manifold
• Impulsive maneuvers to relocate the spacecraft to the desired position on the
manifold after an initial error
The goal of our low-thrust control technique is to continually thrust to keep
the spacecraft at a location that satisfies the requirements of a science orbit. In
theory, this seems reasonable and has the advantage of keeping the orbital elements
constant. This allows for an indefinite lifetime as long as the thrust does not stop.
Thrust laws are developed for purely radial and and purely transverse thrust. The
results show that attempting to control the spacecraft in this way requires very
122
large thrust values that are not practical. Although the transverse thrust technique
requires an order of magnitude thrust levels less than the radial thrust technique,
neither of the methods are feasible in the practical sense. Recalling the manifolds of
the unstable frozen orbits, it is not surprising that attempting to keep a spacecraft
at one of those locations is very costly. As we saw, an uncontrolled spacecraft placed
at a frozen orbit location will diverge from the frozen orbit quite rapidly along
the unstable manifold. By attempting to keep the spacecraft at the frozen orbit
location, we do not take advantage of the dynamics of the system. It is therefore
more practical to allow the spacecraft to follow the stable and unstable manifolds
and make corrections when the spacecraft gets to a certain point on the unstable
manifold past the frozen orbit.
Therefore, the next technique considered is to allow the orbit to evolve from
its initial point at the minimum eccentricity location on the stable manifold to the
minimum eccentricity point on the unstable manifold. Then, two maneuvers are
performed to reset the orbit back to the stable manifold. Instead of requiring that the
spacecraft return to the exact same initial point, some of its current characteristics
such as the radius of periapsis and inclination are used to design a new long lifetime
orbit. Finally, the two maneuvers are optimized for the lowest cost transfer to the
new target. The ideal transfer is a circularization maneuver followed by a maneuver
to increase the eccentricity at a location on the transfer orbit that would set the
argument of periapsis appropriately. That sequence is used as a starting guess for the
optimization routine that optimizes over the total cost of both transfer maneuvers.
We find that the cost to do both maneuvers is fairly low, and is practical for a real
mission.
The final control issue discussed in this chapter is correcting the spacecraft’s
position on the designed long lifetime trajectory if it drifts away due to initial errors.
Monte Carlo analysis of trajectories with initial errors show that errors within the
123
range of uncertainty for a mission to Europa can cause the trajectory to follow the
wrong manifold path. The uncertainty limits for a mission to a planetary satellite
such as Europa were taken from [40]. Since initial errors in the spacecraft’s position
can destroy the long-lifetime properties of the trajectory, it is important to determine
ways to get the spacecraft back on the correct orbit with a relatively low cost. We
show that if the erroneous trajectory is integrated along with the correct trajectory
for several days and then a sequence of maneuvers is designed to place the spacecraft
at the correct location corresponding to the time of the correction, the cost of the
correction is very low. The transfer is a two-maneuver sequence, optimized for the
total cost. The ideal transfer is a Hohmann-type transfer between two elliptic orbits
and this is used as the initial guess of the optimization. This technique results in
very low costs and is a realistic technique for a real mission.
6.1 Low-Thrust Control
Since the unstable near-polar, near-circular unstable frozen orbits have characteristics
that make them well-suited for a science orbit about a planetary satellite, it is logical
to attempt to use low-thrust control to keep the spacecraft at a frozen orbit location.
In terms of a real mission, it is easier to consider thrust along only one axis. The
radial thrust law is developed by analyzing the averaged Gauss equations and the
doubly averaged Lagrange Planetary Equations in tandem. Although the thrust law
is developed to control the orbit in the 1-DOF system, it also controls the orbit in
the 3-DOF system by preventing impact with the planetary satellite for very long
periods of time. Thrusting in the radial direction was the preferred orientation for
the proposed Jupiter Icy Moons Orbiter spacecraft as it combined gravity gradient
stability of its attitude with thrusting along its long axis. Unfortunately, the thrust
required to control the spacecraft using this thrust law is much too large to be
practical for a real mission.
124
The transverse thrust law is developed by determining the thrust required to
control the eccentricity of the orbit in the transverse direction. For this case, a
circular orbit rather than a near-circular frozen orbit is used since circular orbits are
also very useful as science orbits for mission to planetary satellites. The thrust law
developed controls the eccentricity of the orbit while maintaining the semi-major
axis of the orbit. The thrust required, while being an order of magnitude smaller
than that required for the radial thrust law, is still too large for a mission. We
demonstrate these thrust laws here to motivate the next section in which control
schemes with more reasonable thrust requirements are developed for the long lifetime
orbits developed in Chapter 5 that take advantage of the natural dynamics of the
system.
6.1.1 Thrust in the Radial Direction
The 1-DOF system is the simplest system under consideration and gives a good
description of the underlying dynamics. The thrust law in the radial direction is
therefore developed using this system and then tested in the full 3-DOF system.
The Gauss Equations describe the motion of a spacecraft subject to perturbing
accelerations in terms of the orbital elements. The important orbital elements in the
1-DOF are the semi-major axis, eccentricity, inclination and argument of periapsis.
Although the semi-major axis is constant in the 1-DOF system, it does affect the
other orbital elements and so its motion must be included in the analysis when
125
thrusting is added. The Gauss equations for these orbital elements are [11]:
daG
dt=
2e sin ν
n√
1− e2Fr +
2a√
1− e2
nrFs , (6.1)
deG
dt=
√1− e2 sin ν
naFr +
√1− e2
na2eFs , (6.2)
diGdt
=r cos (ω + ν)
na2√
1− e2Fω , (6.3)
dωG
dt= −
√1− e2 cos ν
naeFr +
√1− e2
nae
[sin ν
(1 +
1
1 + e cos ν
)]Fs
− r cot i sin (ω + ν)
na2√
1− e2Fω . (6.4)
The three thrust components are Fr, in the radial direction; Fs, 90o from r in the
velocity-increasing direction; and Fω, normal to the orbit plane.
In the 1-DOF system, an increasing argument of periapsis value causes the
eccentricity to increase. Therefore, for the spacecraft to remain at the frozen orbit
location on average, the argument of periapsis must be constant on average. This
can be accomplished with the proper choice of a thrust law. To ensure that the
thrust does not cause changes in the other orbital elements, it must be chosen such
that it has no contribution to the motion of the semi-major axis, eccentricity and
inclination. Since it has already been shown that averaging over the mean anomaly
of the orbit of the spacecraft is a reasonable approximation, the Gauss equations are
also averaged in that way, where constant thrust is assumed. Let
dxG
dt= f(a, e, i, ω, ν) (6.5)
be one of Eqs.(6.1)-(6.4). Then, the average of that orbital element over the mean
anomaly is
dxG
dt=
1
2π
∫ 2π
0
f(a, e, i, ω, ν)dM . (6.6)
Since Eq.(6.1)-(6.4) are functions of the true anomaly, the following transformation
126
is used to integrate over ν:
dM =(1− e2)3/2
(1 + e cos ν)2dν . (6.7)
Then, the averaged Gauss equations are:
daG
dt=
2√
1− e2
nFs , (6.8)
deG
dt=
3e(1− e2)
2naFs , (6.9)
diGdt
= −e√
1− e2 cos ω
naFω , (6.10)
dωG
dt=
1
naFr +
4− e2
naeFs +
3ae cot i sin ω
2√
1− e2Fω . (6.11)
The requirement that ˙ag = ˙eG = ˙iG = ˙ωG = 0 entails letting Fs = Fω = 0. To
ensure that the argument of periapsis is constant on average, use Eq.(3.25) which
describes the motion of the argument of periapsis in the 1-DOF system to determine
the thrust component Fr. The motion of the argument of periapsis in the controlled
1-DOF system is given by the addition of Eq.(3.25) and Eq.(6.11):
dω
dt=
3N2
8n
1√1− e2
[5 cos2 i− 1 + 5 sin2 i cos 2ω + e2(1− 5 cos 2ω)]
+3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)+
3J3n
2a3(1− e2)3
sin ω sin i
e[(1
−5
4sin2 i
)(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)]+
1
naFr . (6.12)
Then, the radial thrust law necessary for ω = 0 is:
Fr = −nadω
dt,
where dω/dt is given in Eq.(6.12).
The first step in testing this thrust law is make sure that it stabilizes the
spacecraft in the 1-DOF system. The doubly averaged Lagrange Equations plus the
unaveraged Gauss equations, with Fr as defined in Eq.(6.13) and Fs = Fω = 0 are
127
integrated:
da
dt=
2e sin ν
n√
1− e2Fr , (6.13)
de
dt=
15N2
8ne√
1− e2 sin2 i sin 2ω − 3J3n
2a3(1− e2)3sin i
(1− 5
4sin2 i
)cos ω
+
√1− e2 sin ν
naFr , (6.14)
di
dt= −15N2
16n
e2
√1− e2
sin 2i sin 2ω +3J3n
2a3(1− e2)3e cos i
(1− 5
4sin2 i
)cos ω , (6.15)
dω
dt=
3N2
8n
1√1− e2
[5 cos2 i− 1 + 5 sin2 i cos 2ω + e2(1− 5 cos 2ω)]
+3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)+
3J3n
2a3(1− e2)3
sin ω sin i
e
[(1− 5
4sin2 i
)·(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)]−√
1− e2 cos ν
naeFr , (6.16)
dΩ
dt= −3N2
8n
cos i√1− e2
(2 + 3e2 − 5e2 cos 2ω)− 3nJ2
2a2(1− e2)2cos i
+3nJ3
2a3(1− e2)3
e sin ω cos i
sin i
(1− 15
4sin2 i
). (6.17)
Figure 6.1 shows the time histories of the controlled orbital elements over a 6 hour
period. Observe that the argument of periapsis oscillates and has a slow drift.
Compare this to Figure 6.2 which is the integration of uncontrolled doubly averaged
Lagrange Equations. The radial thrust slows the drift of the argument of periapsis
which in turn slows the drift of the eccentricity which will delay the impact of the
spacecraft with the planetary satellite.
Now that we have verified that the radial thrust law controls the spacecraft in the
1-DOF system, it is tested in the 3-DOF system. Figure 6.3 shows the time histories
of the orbital elements integrated in the 3-DOF system with the radial thrust law
Fr as defined in Eq.(6.13), over a 250 day period. It is clear that this thrust law
adequately controls the spacecraft since it has not impacted with the planetary
satellite after 250 days. Figure 6.4 shows the manifold in the 1-DOF system and the
integrated trajectory. Observe that the trajectory remains close to the frozen orbit.
128
0 1 2 3 4 5 61694.69
1694.7
1694.71
1694.72
a (k
m)
0 1 2 3 4 5 60.02
0.0202
0.0204
ecce
ntric
ity
0 1 2 3 4 5 669.999
69.9995
70
70.0005
70.001
i (de
g)
time (hours)
0 1 2 3 4 5 6−91
−90.5
−90
−89.5
ω (
deg)
0 1 2 3 4 5 6−0.8
−0.6
−0.4
−0.2
0
Ω (
deg)
time (hours)
Figure 6.1: Time histories of orbital elements integrated in the controlled 1-DOF system.
0 1 2 3 4 5 61693
1694
1695
1696
a (k
m)
0 1 2 3 4 5 60.0199
0.02
0.0201
ecce
ntric
ity
0 1 2 3 4 5 669.999
69.9995
70
70.0005
70.001
i (de
g)
time (hours)
0 1 2 3 4 5 6−91
−90.5
−90
ω (
deg)
0 1 2 3 4 5 6−0.8
−0.6
−0.4
−0.2
0
Ω (
deg)
time (hours)
Figure 6.2: Time histories of orbital elements integrated in the uncontrolled 1-DOF system.
0 50 100 150 200 2501680
1685
1690
1695
1700
a (k
m)
0 50 100 150 200 2500
0.005
0.01
0.015
0.02
ecce
ntric
ity
0 50 100 150 200 25068
69
70
71
i (de
g)
time (days)
0 50 100 150 200 2500
100
200
300
400
ω (
deg)
0 50 100 150 200 250−200
−100
0
100
200
Ω (
deg)
time (days)
Figure 6.3: Integration of the 3-DOF system controlled with Fr.
129
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
Figure 6.4: Manifold in the 1-DOF system and integration of the 3-DOF system controlled with Fr.
Although the radial thrust law, Fr adequately controls the spacecraft it requires
instantaneous values of the orbital elements which results in constant computation.
The total thrust magnitude and the x, y and z components of the thrust are shown
in Figure 6.5. Observe that the thrust is of the order 10−6km/s2. The total ∆v
can also be computed by integrating the thrust magnitude (see Figure 6.6). The
total ∆v required over a 150 day period is 12 km/s which is an average of 80 meters
per second per day. Then, instead of using the thrust law Fr in Eq.(6.13), we try
controlling the spacecraft using a constant thrust in the radial direction equal to the
average of 80 m/s/day which is −9.26× 10−7km/s2. The results of this integration
are shown in Figure 6.7. We see that the thrust used is not enough to prevent the
spacecraft from impacting. It impacts with Europa after about 50 days. It is possible
however, to control the spacecraft using a slightly larger constant radial thrust. We
find that the minimum constant thrust required is −1.045× 10−6km/s2 which is 90.2
meters per second per day. Figure 6.8 shows the results of this integration where
the spacecraft has not impacted with Europa after 150 days. The problem with the
radial thrust technique is that although in theory it is possible to keep the spacecraft
at the frozen orbit location by either constant or time-varying radial thrust, the
thrust required is much too large to be practical for a mission.
130
0 50 100 150 200 2500
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
−6
time (days)
mag
nitu
de o
f thr
ust (
km/s
2 )
50 100 150 200 250
−2
−1
0
1
2
x 10−6
Fx (
km/s
2 )
50 100 150 200 250
−2
0
2x 10
−6
Fy (
km/s
2 )
50 100 150 200−4
−2
0
2
4x 10
−6
Fz (
km/s
2 )
time (days)
Figure 6.5: Time histories of total thrust magnitude and components of thrust.
0 50 100 1500
2
4
6
8
10
12
14
∆ v
(km
/s)
time (days)
Figure 6.6: Total ∆v used to control the spacecraft using the thrust law given in Eq.(6.13).
131
0 5 10 15 20 25 30 35 40 45 501680
1685
1690
a (k
m)
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
e
0 5 10 15 20 25 30 35 40 45 5068
69
70
71
i (de
g)
time (days)
0 5 10 15 20 25 30 35 40 45 50150
200
250
300
ω (
deg)
0 5 10 15 20 25 30 35 40 45 50−100
−50
0
50
Ω (
deg)
time (days)
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
Figure 6.7: Integration using constant thrust in the radial direction of −9.26× 10−7km/s2. Thespacecraft impacts with Europa after about 50 days.
132
0 50 100 1501680
1685
1690
a (k
m)
0 50 100 1500
0.02
0.04
0.06
0.08
e
0 50 100 15068
69
70
71
i (de
g)
time (days)
0 50 100 150150
200
250
300
ω (
deg)
0 50 100 150−200
−100
0
100
200
Ω (
deg)
time (days)
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
Figure 6.8: Integration using constant thrust in the radial direction of −1.045× 10−6km/s2. Thespacecraft has not impacted with Europa after 150 days.
133
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
−3
time (days)
ecce
ntric
ity
Figure 6.9: Eccentricity as a function of time over one orbital period.
0 50 100 1501680
1685
1690
1695
1700
a (k
m)
0 50 100 1500
2
4
6x 10
−3
e
0 50 100 15068
69
70
71
i (de
g)
time (days)
0 50 100 1500
100
200
300
400
ω (
deg)
0 50 100 150−200
−100
0
100
200
Ω (
deg)
time (days)
Figure 6.10: Time histories of the orbital elements when the transverse thrust is applied.
6.1.2 Thrust in the Transverse Direction
Since the radial thrust technique resulted in very large thrust levels, we try
developing a transverse thrust law to control the spacecraft. For this case, the goal
is to control the spacecraft such that it stays in a circular, highly inclined orbit. The
control is therefore in place to prevent the eccentricity of the orbit from increasing.
The first step is to initialize a trajectory in a circular orbit with an inclination of 70o
and integrate it for one orbital period. Figure 6.9 shows the result of this integration.
The net change in the eccentricity is very small at 2.32× 10−4.
134
The next step is the computation of the ∆v required to circularize the orbit after
one orbital period. In this example, the ∆v required is 0.8395 m/s. Therefore, to
prevent the eccentricity of the orbit from increasing a first guess at the required
transverse thrust is the ∆v divided by the orbital period. The orbital period
is 7.454 × 103 (2.07 hours) and the transverse thrust is 1.126 × 10−7 km/s2
(9.73 m/s/day). In order to prevent the semi-major axis from increasing, the thrust
is applied in the velocity direction for half of the orbit and against the velocity
direction for the other half, symmetrically about periapsis. Figure 6.10 show the
time histories of the orbital elements for the integration with the transverse thrust
law used to control the orbit. During the 150 day period of the integration, the
eccentricity and semi-major axis stay relatively constant and the spacecraft does not
impact with Europa.
Although the transverse thrust required to control the spacecraft is about an
order of magnitude less than the radial thrust required, it is still much too large in
the practical sense. These exercises in developing purely radial and purely transverse
low-thrust laws to keep some of the orbital elements of the spacecraft constant show
that due to the strong perturbations present in the system, this is not an efficient
way to control a spacecraft. The following sections detail control methods that take
advantage of the natural dynamics of the system to reduce the thrust required to
prevent the spacecraft from impacting with the planetary satellite.
6.2 Resetting a Long Lifetime Orbit
Recall the long lifetime orbits developed in Chapter 5. They were designed by
computing initial conditions such that the spacecraft follows the stable and unstable
manifolds of the 1-DOF system. Although these orbits have long lifetimes, they
are finite and the spacecraft will eventually follow the unstable manifold to impact
with the planetary satellite. It is important to develop a control strategy to delay
135
impact with the planetary satellite. Since an algorithm to design long lifetime orbits
already exists, it is natural to use this algorithm during the development of the
control scheme. The technique used to control the long lifetime orbit is to allow the
trajectory to reach the unstable manifold and then, at a certain point, reset it back
to the stable manifold so another long lifetime orbit can evolve.
6.2.1 Designing the Target
The initial conditions of a long lifetime orbit are computed by starting at the
minimum eccentricity point on the manifold and are a specific 6-state. The minimum
eccentricity point on the manifold corresponds to a particular frozen orbit which in
turn corresponds to a specific radius of periapsis. As the orbit evolves, the orbital
elements oscillate, and in particular, the radius of periapsis oscillates. Therefore,
when the orbit is stopped on the unstable manifold, resetting it back to the original
long lifetime orbit initial conditions is very difficult since the modification of all of
the parameters of the orbit would be necessary. Since the goal is simply to prevent
the impact of the spacecraft with the planetary satellite by resetting the trajectory
to the stable manifold, it is not necessary to reset it to the exact same long lifetime
orbit. Therefore, at the point that the orbit is stopped on the unstable manifold a
new long lifetime orbit is computed, with parameters consistent with the current
orbital parameters.
Denote the initial conditions of the new long lifetime orbit to be designed the
target. The first step in designing the target is to specify its corresponding frozen
orbit. Since the algorithm used to design long lifetime orbits places the spacecraft
at the minimum eccentricity point on the stable manifold, it is logical to stop the
orbit at the minimum eccentricity point on the unstable manifold. This minimizes
the eccentricity changes that will be required during the transfer. For simplicity in
the design, the stopping point used is actually the first periapsis passage after the
136
180 200 220 240 260 280 300 320 340
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
ω (degrees)
ecce
ntric
ity
target
∆ v1 ∆ v
2
frozen orbit
stopping point
transfer orbit
Figure 6.11: Diagram of the reset scheme in the 1-DOF system in (e,ω)-space.
minimum eccentricity point on the unstable manifold. In the ideal 1-DOF system,
there are no oscillations while the trajectory moves from its initial point on the
stable manifold to the stopping point on the unstable manifold. Thus, to reset the
orbit to the stable manifold, only the argument of periapsis needs to be changed.
The easiest way to do this is a two maneuver sequence consisting of a circularization
at the stopping point and an a maneuver to increase the eccentricity at ω + ν = ωll,
where ωll is the initial argument of periapsis of the original long lifetime orbit.
Since the ideal transfer orbit is circular, this resets the orbit to its original point
on the stable manifold. See Figure 6.11 for a diagram of this reset scheme in the
1-DOF system. Obviously this system is not the ideal 1-DOF system and there are
perturbations present. However, the ideal transfer can be used to generate a target
in the 3-DOF system.
The first step is to circularize the orbit at the stopping point and let the orbit
evolve in the 3-DOF system until ω + ν = ωll. Denote this point with the subscript
‘IT’ (ideal target). Then, the frozen orbit to be used in the actual target design has
a radius of periapsis equal to rIT and an inclination equal to iIT . The frozen orbit
eccentricity can then be computed along with the minimum eccentricity on the new
137
manifold. Using all of these values, the initial conditions of a new long lifetime orbit
can be computed using the algorithm in Chapter 5. Note that ΩIT = ΩIT − NtIT .
Since the target is always designed for time equal to zero, Ω0 = ΩIT . The orbital
elements of the target are denoted with the subscript ‘T’. A summary of the
procedure for computing the target is as follows:
1. Stop the original long lifetime trajectory at the first periapsis after the
minimum eccentricity point on the unstable manifold.
2. Perform the ‘ideal transfer’ which consists of a circularization at the stopping
point and integrate until ω + ν = ωoriginal.
3. Let the frozen orbit corresponding to the target have a radius of periapsis and
inclination of rIT and iIT respectively and Ω0 = ΩIT .
4. Compute the target position using the algorithm for computing long lifetime
orbits in Chapter 5.
6.2.2 Transferring to the Target
The next step is determining how to transfer to the target, knowing that there are
perturbations present which prevent the use of the ideal transfer described above.
Note however, that the ideal transfer is a good first guess. Therefore, we start with
the circularization burn, and integrate until ω + ν = ωT . Denote the transfer time
tf . Aside from reaching the target, our goal is also to minimize the total cost of the
transfer, which consists of two burns – one at periapsis on the unstable manifold and
one at the target. First, we focus on reaching the target. Let rT denote the target
position and vT denote the target velocity. Then,
rT = r(tf , r0,v0 + δv) , (6.18)
vT = v(tf , r0,v0 + δv) , (6.19)
138
where r0 and v0 correspond to the stopping point (at periapsis) on the unstable
manifold and δv is the first burn. A Taylor expansion of Eq.(6.18) yields:
rT = r(tf ) +∂r
∂v0
∣∣∣∣tf
δv + · · · (6.20)
Then, keeping only first order terms and rearranging for δv:
δv =
(∂r
∂v0
)−1∣∣∣∣∣tf
(rT − r0) . (6.21)
Denote the state transition matrix as
Φ(0, tf ) =
φrr φrv
φvr φvv
. (6.22)
Then, ∂r∂v0
∣∣∣tf
= φrv. Also, let δr = (rT − r(tf )). We can converge on the δv necessary
for the transfer orbit to end at the target rT using the following iteration until
δr = 0:
δv = δv0 + δv1 + δv2 + · · · , (6.23)
where
δvi+1 = φ−1rv δri . (6.24)
The above method ensures that the spacecraft reaches the target position by the
end of the transfer orbit. At the target position a second burn is executed such that
the spacecraft velocity is equal to the target velocity. The first burn is δv and the
second burn is vT − v(tf ). However, this method does not optimize the total cost.
Therefore, the next step is refining the transfer orbit and hence the maneuvers to
reduce the cost of both burns. Let ∆v∗1 = δv and ∆v∗2 = vT − v(tf ). Then, consider
a correction to the transfer time tf , denoted as δt, needed to optimize the total cost.
This necessitates the introduction of corrections to both burns δv1 and δv2. The
139
target position and velocity are now:
rT = r(tf + δt, r0,v0 + ∆v∗1 + δv1) , (6.25)
vT = v(tf + δt, r0,v0 + ∆v∗1 + δv1) + ∆v∗2 + δv2 . (6.26)
A Taylor series expansion of Eqs.(6.25) and (6.26) yields
rT = r(tf ) +∂r
∂t
∣∣∣∣tf
δt +∂r
∂v0
∣∣∣∣tf
δv1 + · · · (6.27)
vT = v(tf ) +∂v
∂t
∣∣∣∣tf
δt +∂v
∂v0
∣∣∣∣tf
δv1 + · · ·+ ∆v∗2 + δv2 . (6.28)
Then, truncating at first order in the Taylor expansions and noting that rT = r(tf )
and vT = v(tf ) + ∆v∗2, we obtain the following expressions for δv1 and δv2:
δv1 = −(
∂r
∂v0
)−1∣∣∣∣∣tf
v(tf )δt , (6.29)
δv2 = −
(∂v
∂t
∣∣∣∣tf
+∂v
∂v0
∣∣∣∣tf
δv1
)δt . (6.30)
Then, using the definition of the state transition in Eq.(6.22), φrv = ∂r∂v0
∣∣∣tf
and
φvv = ∂v∂v0
∣∣∣tf
. The expressions for δv1 and δv2 are then:
δv1 = −φ−1rv v(tf )δt , (6.31)
δv2 = −[a(tf )− φvvφ
−1rv v(tf )
]δt , (6.32)
where a(tf ) is the acceleration at tf .
The goal of this scheme is to minimize the total cost. Let the cost function be
J = (∆v∗1 + δv1)2 + (∆v∗2 + δv2)
2 . (6.33)
Since δv1 and δv2 are functions of δt, denote them as δv1 = α1δt and δv2 = α2δt.
140
Then, J can be expressed as
J = |∆v∗1|2 + |∆v∗2|2 + 2[(∆v∗1)T α1 + (∆v∗2)
T α2]δt + (|α1|2 + |α2|2)δt2 (6.34)
= Aδt2 + 2Bδt + C . (6.35)
Since the cost J is a function of δt, take the derivative of J with respect to δt
and set it equal to zero to minimize the cost:
∂J
∂t= 2Aδt + 2B = 0 . (6.36)
Therefore, the δt required to minimize the total cost is:
δt = −B
A, (6.37)
where
A = v(tf )T φ−T
rv φ−1rv v(tf ) + [a(tf )− φvvφ
−1rv v(tf )]
T [a(tf )− φvvφ−1rv v(tf )] , (6.38)
B = −(∆v∗1)−1φ−1
rv v(tf )− (∆v∗2)T [a(tf )− φvvφ
−1rv v(tf )] . (6.39)
The overall procedure to compute the maneuvers necessary to transfer the
spacecraft from a periapsis location on the unstable manifold to a new long lifetime
orbit initialized on its stable manifold is as follows:
1. Compute the ideal transfer consisting of a circularization at periapsis on the
unstable manifold and an integration until ω + ν = ωT . Let the time required
for this transfer be tf .
2. Use the iteration scheme
δv = δv0 + δv1 + δv2 + · · · (6.40)
where
δvi+1 = φ−1rv δri (6.41)
141
to compute the necessary correction to the first burn such that δr = 0.
3. Let ∆v∗1 = δv and ∆v∗2 = vT − v(tf ) and compute new corrections to the
burns, δv1 and δv2, to minimize the cost using δt = −BA.
4. Repeat steps 2 and 3 until δti+1 − δti < 10−9.
The optimization scheme described above doesn’t actually minimize the total
cost, but minimizes the the sum of the squares of the two burns. To minimize the
total cost would require J = |∆v∗1 + δv1| + |∆v∗2 + δv2|. However, testing of both
methods results in approximately the same total cost for both cases and so the sum
of the squares method is used since it is computationally simpler.
6.2.3 Examples
The examples detailed in this section show that the technique described above
provides a practical way to reset a long lifetime orbit with a relatively low cost. As
previously mentioned, it is not necessary to find the maximum lifetime orbit on a
particular manifold since it would not be achievable in a real mission. However it is
important to ensure that the long lifetime orbit follows the correct manifold. The
long lifetime orbits used in these examples are those in Table 5.2 (the unbiased long
lifetime orbits, not the maxima). These unbiased long lifetime orbits all follow the
correct manifold path. However, when the target long lifetime orbit is computed, we
must verify that it too follows the correct long lifetime path. We found that of the
six examples computed, the 85o and 95o orbits follow the correct manifold path and
the others do not. For the four that do not follow the correct manifold path, a bias in
the argument of periapsis is included. This bias does not correspond to the longest
lifetime orbit, but to a long lifetime orbit that follows the correct manifold path and
has a lifetime in the 100-110 day vicinity. This range was chosen so that all of the
target orbits have approximately the same lifetimes. This eases the comparison of
the costs associated with the transfers.
142
Table 6.1: Examples of Long Lifetime Orbit Reset
Frozen Orbit ∆v1 ∆v2 Target Trajectory Target Trajectory Cost/dayInclination (m/s) (m/s) Lifetime (days) Bias (rad) (cm/s)
70o 16.30 17.94 110 0.06 31.175o 19.07 21.33 106 0.06 38.185o 11.20 15.67 99 – 27.195o 15.45 19.76 103 – 34.2105o 12.32 16.19 106 -0.04 26.9110o 14.36 18.62 107 -0.095 30.8
Table 6.1 shows the details of the examples, identified by the inclination of the
associated frozen orbit of the original long lifetime trajectory. For each orbit, the
magnitude of each of the two burns, the lifetime of the target trajectory and the
bias used for the target trajectory (if necessary) are given. The point of comparison
for each example is the average cost per day. This is computed by dividing the total
cost of the two maneuvers by the lifetime of the target trajectory. It is clear that
this method has a much lower cost than both the radial and transverse low-thrust
methods. Specifically, the average cost per day here is an order of magnitude less
than the transverse thrust law and two orders of magnitude less than the radial
thrust law. Figure 6.12 shows the 1-DOF manifold of an original long lifetime
trajectory, the integration of an original long lifetime trajectory and the integration
of a target trajectory. Observe that the target trajectory and original trajectory are
very close to each other. Figure 6.13 shows the time histories of the orbital elements
for the original long lifetime trajectory and the target trajectory. Observe that the
orbital elements of both trajectories are very close to each other. This is important
since resetting the orbit is only useful if the extension of the orbit still satisfies the
science requirement of the mission, which it should if this scheme is used.
143
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
Figure 6.12: Original (black) and target (red) long lifetime trajectories.
0 20 40 60 80 100 1201695
1700
1705
1710
a (k
m)
0 20 40 60 80 100 1200
0.05
0.1
e
0 20 40 60 80 100 12084
85
86
87
i (de
g)
time (days)
0 20 40 60 80 100 1201500
1600
1700
1800
1900
radi
us (
km)
0 20 40 60 80 100 1200
100
200
300
400
ω (
deg)
0 20 40 60 80 100 1200
100
200
300
400
Ω (
deg)
time (days)
Figure 6.13: Time histories of the orbital elements of the original (black) and target (red) longlifetime trajectories.
144
6.3 Orbit Uncertainty
Up to this point, we have assumed that it is possible to attain the desired long
lifetime orbits. However, in a real mission, it is important to take the uncertainty
of the orbiter’s position and velocity into account. In [40], Thompson, et al studied
orbit determination for a 30-day mission to Europa. They performed simulations
using a science orbit with similar characteristics to our long lifetime orbits. In
particular, their orbit is near-polar, near circular and has an average 100 km altitude.
One of the biggest drivers of the uncertainty of orbits about Europa is the lack of
knowledge of the Europa gravity field. Only the degree two terms of the Europa
gravity field are currently known, and they were determined from Galileo flybys of
Europa [4].
The first study performed in [40] uses a worst-case gravity error, which assumes
that no additional knowledge of Europa’s gravity field is known during the science
phase of the mission. This results in errors in the spacecraft’s radial, crosstrack
and alongtrack position components on the order of kilometers. These results agree
with those in [15]. These errors are judged to be unacceptable for a Europa orbiter
mission, and so a further assumption is made. It is assumed that prior to the science
orbit phase of the mission, two weeks of tracking data during orbit conditions similar
to the science orbit are available. This data reduces the error in the gravity field.
The best-case scenario, which is a lower bound on the gravity error, is computed to
provide a gravity field estimate up to degree 20. This results in predicted position
errors of 5.1m, 7.3m and 5.7m (3-sigma) for radial, crosstrack and alongtrack position
components respectively. The radial component points from the planetary satellite
to the spacecraft, the alongtrack component points in the direction of the velocity
and the crosstrack component completes the triad. However, they assume that a
more realistic assumption for the gravity error is in between the best case and worst
case (lower and upper) bounds. Using this gravity error results in predicted position
145
errors 24 hours after a data cutoff of 48m, 60m and 35m for radial, crosstrack and
alongtrack positions respectively (also 3-sigma values).
Using the results in [40] as a guide, we study how errors in the spacecraft’s
position affects the designed long lifetime orbits. To study the orbit uncertainty
of long lifetime orbits, Monte Carlo simulations are performed. A large number of
trajectories are integrated, each with a random initial position error. The initial
position error is randomly determining for radial, crosstrack and alongtrack errors
using a Gaussian distribution with a particular standard deviation. The first set
of results are for 3-sigma values corresponding to the realistic gravity error noted
above. Each of the six long lifetime orbits with different inclinations from section
5.3.2 are considered, with 200 trajectories integrated for each. Recall that the six
long lifetime orbits are initialized at the minimum eccentricity point and follow the
correct unstable manifold but are not the maximum lifetime orbits for each case.
Figure 6.14 shows the Monte Carlo simulation results for the minimum
eccentricity long lifetime orbits. Each cluster contains several points, each of which
represents the stopping point of a trajectory integrated to determine which manifold
path it follows. Observe that in four of the six cases, all of the 200 trajectories follow
the correct manifold path. For the 75o inclination case, 44 of 200 trajectories (22%)
follow the wrong manifold path and for the 85o inclination case, 15 of 200 trajectories
(7.5%) follow the wrong manifold path. These results are explained by taking into
account the lifetimes of the six original long lifetime trajectories (Table 5.2). The
two trajectories with the longest lifetimes are the 75o and 85o inclination cases. A
longer lifetime means that the trajectory follows the 1-DOF manifold more closely.
The shorter lifetimes that follow the correct manifold path therefore correspond
the trajectories that lie slightly above the manifold. Then, the more closely the
correct manifold path is followed, the more likely it is that an initial position error
will cause the trajectory to follow the wrong manifold path. Since the 75o and 85o
146
Table 6.2: Monte Carlo Results for Long Lifetime Orbits
Frozen Orbit Original Orbit Maximum OrbitInclination Lifetime Correct % Lifetime Correct %
70o 105 days 100 182 days 47.275o 119 days 78.0 174 days 50.585o 110 days 92.5 171 days 52.595o 96 days 100 182 days 56.5105o 95 days 100 172 days 51.0110o 99 days 100 188 days 48.0
inclination orbits have the longest lifetimes, they follow the 1-DOF manifold more
closely and therefore it is not surprising that in some cases an initial position error
results in a trajectory that follows the wrong manifold path. Further, note that the
75o inclination orbit has a longer lifetime than the 85o inclination orbit. It follows
then that a larger percentage of the trajectories in the 75o case follow the wrong
path than the 85o case.
The theory that the closer a long lifetime trajectory is to the 1-DOF manifold,
the more susceptible it is to errors in the initial position can be further verified by
performing Monte Carlo analysis of some maximum long lifetime trajectories. The
same maximum lifetime trajectories as those in Table 5.2 are used, and 200 Monte
Carlo simulations are run for each case. The initial position errors are once again
determined randomly via a Gaussian distribution with the same 3-sigma values as
above. Table 6.2 shows the results, along with the lifetimes of the maximum lifetime
orbits. Observe that the number of trajectories that follow the correct manifold is
approximately 50% in each case. This is exactly what is expected since the maximum
lifetime trajectories follow the 1-DOF manifolds very closely. Therefore, a small
initial position error can bump the trajectory either just above the manifold or just
below the manifold. If it is just above the manifold, it will follow the correct manifold
path and if it is just below the manifold, it will follow the incorrect manifold path.
It is important when planning a mission to a planetary satellite such as Europa
147
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02
−0.025
−0.02
−0.015
−0.01
−0.005
0
k=ecosω
h=es
inω
(a) 70o
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
k=ecosω
h=es
inω
(b) 75o
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
k=ecosω
h=es
inω
(c) 85o
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
k=ecosω
h=es
inω
(d) 95o
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
k=ecosω
h=es
inω
(e) 105o
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02
−0.025
−0.02
−0.015
−0.01
−0.005
0
k=ecosω
h=es
inω
(f) 110o
Figure 6.14: Monte Carlo simulation results for various long lifetime trajectories (200 trajectoriesfor each case). Each point in the plot represents the stopping point of a trajectory once it reachesthe desired (top right) or not desired (bottom left) manifold.
148
to ensure that the uncertainty error in the position of the spacecraft will not lead
to impact with the planetary satellite. Therefore, using the maximum lifetime orbit
as the nominal science orbit is not the best course of action since there is about
a 50% chance that the spacecraft will deviate too far from the nominal trajectory.
The further above the manifold the long lifetime trajectory is, the more likely it
is to follow the correct manifold path. However, the farther above the manifold
the trajectory is, the shorter lifetime it has. Choosing the nominal trajectory is
therefore a trade off between guaranteeing that it will follow the correct manifold
path and having a long lifetime. In addition, before the mission starts, the orbit
accuracy possible will not be known exactly. The values used here are based on the
assumption that prior to the science portion of the mission the gravity field will be
determined with some level of certainty. Even if a trajectory is chosen such that it
will always follow the correct manifold path for the error levels used here, it is not
certain that this level of orbit accuracy is attainable. Therefore, we discuss in the
next section how to return the spacecraft to its nominal orbit once it has drifted
away due to an initial error.
6.4 Long Lifetime Orbits with Initial Errors
It is inevitable that during the science phase of a mission to a planetary satellite that
the spacecraft will drift away from the nominal trajectory. This will most likely be
due to initial errors in the spacecraft’s position and velocity. Once it is determined
that the spacecraft is no longer on the nominal trajectory, it is important to devise
a method for returning it to the nominal trajectory so that the mission can proceed
as planned. For example, recall in section 6.1 the method to reset the long lifetime
orbit once the spacecraft reaches the unstable manifold. As shown in section 6.3, due
to initial position errors, it is possible that a long lifetime trajectory will follow the
wrong manifold path. If it does so, it is not possible to use the algorithm in section
149
6.1 to reset the trajectory. Therefore, it is important to return to spacecraft to its
nominal trajectory before it has the opportunity to follow the wrong manifold path.
6.4.1 Design of the Correction Maneuvers
The original long lifetime orbit under consideration is determined as usual from the
minimum eccentricity point. Initial errors are introduced by randomly determining
position errors as in the previous section. A time (in days) is chosen to represent the
amount of time until the correction to the trajectory will be made. The amount of
time (TE) is long enough for the spacecraft to drift away from the nominal trajectory.
The maneuvers to be used for the correction are two burns corresponding to a
Hohmann-type transfer. Therefore, the initial transfer point should be at periapsis
and the target at apoapsis. The initial transfer point is determined by integrating
the erroneous trajectory until the closest periapsis passage to TE (either before or
after). Its initial position and velocity are denoted as r0 and v0, respectively. In
addition, denote its initial longitude of the ascending node as Ω0.
The nominal trajectory is the long lifetime orbit with no initial errors. The goal
of this correction scheme is to transfer the spacecraft to the nominal trajectory. The
difficulty lies in determining how to designate the target on the nominal trajectory.
Unfortunately, it is not as simple as integrating the nominal trajectory for the
same amount of time as the erroneous trajectory and setting the target accordingly.
Using this method creates inconsistencies between the initial and target longitude of
ascending node values which creates difficulties in converging to the transfer orbit.
The method that produces the best results is to integrate the nominal trajectory
until its longitude of ascending node value is equal to Ω0 and then finding the
closest periapsis passage to that point (before or after). This periapsis passage on
the nominal trajectory therefore corresponds to point where the spacecraft on the
erroneous trajectory should be. Then, the target is set to be the first apoapsis
150
Figure 6.15: Schematic of Hohmann transfer between two elliptic orbits.
passage following that periapsis passage on the nominal trajectory. Denote the
target position and velocity as rT and vT , respectively.
As in section 6.2, the goal of the transfer is to reach the target and minimize
the sum of the squares of the costs of the two burns (Eq.(6.33)). The exact same
algorithm is used, with the only change being the ideal, first guess transfer orbit.
The first guess for the first maneuver is an eccentricity change at periapsis such
that the transfer orbit arrives at the target at periapsis. Since, as before, there
are perturbations in the system that prevent the target from being reached by this
simple method, it is used as a first guess and corrections are made. Figure 6.15 is a
diagram of a Hohmann transfer between two elliptic orbits. Let the semi-major axis
of the transfer orbit, atr be given by
atr =r0 + rt
2. (6.42)
Then, the velocity of the first guess transfer orbit is:
vg =
õ
(2
r0
− 1
atr
)v0 , (6.43)
where v0 is the direction of the initial velocity v0.
151
The overall procedure to compute the maneuvers necessary to return the
spacecraft to its nominal trajectory mirrors the procedure in Section 6.2 and is as
follows:
1. Compute the ideal transfer consisting of an eccentricity change at periapsis
where the velocity is vg and integrate to apoapsis. Let the time required for
this transfer be tf .
2. Use the iteration scheme
δv = δv0 + δv1 + δv2 + · · · , (6.44)
where
δvi+1 = φ−1rv δri (6.45)
to compute the necessary correction to the first burn such that δr = 0.
3. Let ∆v∗1 = δv and ∆v∗2 = vt − v(tf ) and compute new corrections to the
burns, δv1 (Eq.(6.31)) and δv2 (Eq.(6.32)), to minimize the cost using δt = −BA
(Eqs.(6.38),(6.39)).
4. Repeat steps 2 and 3 until δti+1 − δti < ε where ε = 10−9 for our simulations.
6.4.2 Results
The algorithm described above is designed to correct the position and velocity of
a spacecraft after it has drifted away from its nominal trajectory due to an initial
error. The question is, for how long after the initial error is it possible to correct
the position? After a certain amount of time, the correction algorithm will not
converge to a transfer orbit with maneuvers that have acceptable costs. The longer
a spacecraft follows an erroneous trajectory, the more difficult is it to correct.
Therefore, after a certain point, as long as the spacecraft follows the correct manifold
152
path, it makes more sense to reset the orbit once it reaches the unstable manifold,
as in section 6.1, instead of using the technique described in this section.
The first set of examples presented have initial errors that follow the realistic
model of gravity field uncertainty described in [40]. For each long lifetime orbit case,
random initial position errors in the radial, crosstrack and alongtrack directions
are introduced, and the trajectory is integrated for 1, 5, 10 and 20 days before the
correction is made. For each case, 40 simulations are performed, and the average
cost for each maneuver is computed. Corrections are not made after periods of time
longer than 20 days since after that length of time, the trajectory has drifted far
enough away from the nominal trajectory such that this method is not feasible. This
is likely because the method is based on an initial guess which is the ideal transfer
occurring in the absence of all perturbations. It also assumes that the initial and final
orbits are co-planar which is, in general, not the case. However for corrections done
within 20 days of the initial error, these assumptions are valid and the algorithm
converges to a transfer orbit with low-cost maneuvers. After the trajectory has
evolved for longer, the algorithm does not always converge to a transfer orbit, and
when it does, the costs are sometimes very large. Therefore, this method is only
valid when the spacecraft has not drifted too far from the nominal trajectory.
The resulting costs for the two maneuvers with the error model described above
are shown in Table 6.3. The factors to be considered when determining when the
correction should be made are: how quickly after discovering that the trajectory is
not on the nominal trajectory the correction can be performed and when it is most
cost-effective to perform the correction. In examining Table 6.3, we see that the
lowest cost correction occurs at the 1 day mark. However, depending on when it
becomes known that the spacecraft is not on the nominal trajectory, it might not be
possible to perform the maneuver after 1 day. Therefore, the fact that the cost is
still relatively low for corrections after 5 and 10 days is a useful result showing that
153
Table 6.3: Average cost to correct a long lifetime trajectory after an initial position error
1 day 5 days 10 days 20 daysFrozen Orbit ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2
Inclination m/s m/s m/s m/s70o 0.0989 0.1441 0.3733 0.544985o 0.2190 0.3589 0.8830 2.388595o 0.1642 0.2986 0.6010 3.7737105o 0.1547 0.2949 0.3299 1.8024
it is not necessary to perform the correction very close to the time when the error is
discovered. These results show that performing a correction maneuver within about
a week or two from the time of the initial error is feasible.
The distribution used to compute initial errors for the results in Table 6.3 are
those designated in [40] as the most likely orbit uncertainty values for the science
portion of a mission to Europa. As previously noted, these orbit uncertainty values
were obtained by assuming that prior to the science phase of the mission, the
gravity field of Europa could be determined more accurately than the currently
available values. Therefore, it is important to study the effect of larger levels of
orbit uncertainty on the ability to make corrections to the spacecraft’s position.
Table 6.4 shows results corresponding to initial error values double the size of those
in Table 6.3. In other words, the standard deviations of the error distributions
are two times the values in [40]. It is not surprising that an increase in the initial
position error leads to a decrease in the length of time that can elapse for a correction
to be possible. For the results in Table 6.4, attempts to do a correction after 20
days caused errors in the convergence of the transfer orbit, and large costs when a
transfer orbit was found. Therefore, the results shown are for corrections after 1 day,
5 days, 10 days and 15 days. First, note that the total cost to return the spacecraft
to its nominal trajectory is larger in this case than for the results in Table 6.3.
This is not surprising since the initial errors are larger in this case, and so larger
154
Table 6.4: Average cost to correct a long lifetime trajectory after a larger initial position error
1 day 5 days 10 days 15 daysFrozen Orbit ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2
Inclination m/s m/s m/s m/s70o 0.2256 0.2466 0.8433 0.796685o 0.3746 0.7773 1.4339 1.329495o 0.4414 0.6415 1.3459 2.4245105o 0.2733 0.3783 0.7837 1.8184
maneuvers are needed. However, the costs for the corrections after 1 day, 5 days and
sometimes 10 days are still relatively low, and corrections are feasible once again
even if it is necessary to wait for a few days after the initial error. The results for
both error models are consistent since as time goes on, the deviation between the
erroneous trajectory and the nominal trajectory gets larger, increasing the cost of
the correction in the same way that a larger initial error also increases the cost of
the correction.
155
CHAPTER 7
CONCLUSIONS AND FUTURE DIRECTIONS
The main focus of this dissertation is to develop an understanding of spacecraft
dynamics in the vicinity of planetary satellites and to use these dynamics for mission
design. The main contributions are:
• The analysis of the 1-DOF system dynamics including the frozen orbits, their
stability and their dependence on the higher order gravity field terms of the
planetary satellite
• The analysis of capture trajectories in safe zones that allow an uncontrolled
spacecraft to remain in orbit and from which transfers to lower altitude orbits
are possible
• The design method for long lifetime science orbits using the stable and unstable
manifolds of the frozen orbits
• The optimal transfer strategy for restarting a long lifetime orbit using the fact
that the manifolds are offset from the origin
• The demonstration that it is possible, with relatively low costs, to return the
spacecraft to its nominal trajectory after initial position errors
156
7.1 Summary of the Results
The first main result obtained is the derivation of frozen orbit solutions in the
1-DOF system that are relevant for planetary satellites. When the J3 planetary
satellite gravity field term is included in the model, the structure of the frozen
orbit solutions change, qualitatively independent of the size of J3. Therefore, this
analysis is applicable to any planetary satellite system that can be approximated by
the Hill problem, no matter what its J3 value is. Both stable and unstable frozen
orbit solutions exist, and the stable frozen orbit solutions are long-term stable in
the 3-DOF system as long as their altitude is not too large. These stable orbits are
not of interest from a scientific perspective, however, due to their low inclinations.
The unstable frozen orbit solutions are of interest to scientific observations due to
their high (near-polar) inclinations. Thus these orbits are used to motivate science
orbit design. These are especially challenging as an uncontrolled, or poorly designed,
science orbit will impact on the planetary satellite surface in a relatively short time.
The investigation of capture trajectories in the Hill 3-body problem is
accomplished using a Periapsis Poincare map and a specific application is made to
trajectories about Jupiter’s moon Europa. However, the same analysis could be
repeated for other planetary satellites of interest. Trajectories that don’t impact
with Europa or escape for one week time periods are identified and the regions in
which they occur denoted as safe zones. These regions are characterized by the first
periapsis of each trajectory. It is an important result that the determination that
a trajectory will be safe for one week can be made from its first periapsis passage
location. These safe zones are evaluated to find trajectories from which it is possible
to transfer to long-term stable orbits with a relatively low cost. Orbits considered
as possible targets are both elliptic and circular frozen orbits which are either
long-term stable or can be stabilized with small control maneuvers. We found that
the lowest-cost method to transfer to a circular orbit is by using the dynamics of the
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system to decrease the eccentricity rather than circularizing the orbit directly. In
particular we describe a low cost sequence that results in a circular orbit reachable
by a low-thrust spacecraft. From this orbit it could then spiral down into a lower
altitude circular orbit.
The design of science orbits with long lifetimes begins with the identification of
the stable and unstable manifolds of the low-altitude, near-polar unstable frozen
orbits in the 1-DOF system. The stable to unstable manifold path that has the
largest change in argument of periapsis while the orbit stays near-circular is identified
as the desired path for a long lifetime orbit. Following that, an algorithm is derived
that allows us to relate initial conditions on the 1-DOF manifold to initial conditions
in the 3-DOF system with an intermediate step in the 2-DOF system. This
algorithm allows for the semi-analytic computation of the desired initial conditions,
and easily incorporates changes in the system parameters. In addition, it utilizes the
main dynamical features of the system to extend the lifetimes as compared to the
unstable frozen orbits. The use of this method for Europa results in science orbits
with lifetimes of at least 100 days. Therefore, the use of this orbit protects the
spacecraft in the case of control maneuver failures. This method is also applied to
Saturn’s moons Enceladus and Dione to show that it is applicable to other planetary
satellites. Viable science orbits are designed in those cases, but with shorter lifetimes
since their environments are more highly perturbed. A toolbox for computing initial
conditions for long lifetime science orbits that was written and delivered to the Jet
Propulsion Laboratory is also described.
The control of long lifetime science orbits has been investigated for a few different
scenarios. The first attempt maintains constant orbital elements using continuous
thrust. However, the thrust levels required for this type of control are found to be
too large to be practical for a mission. Next, scenarios where a long lifetime science
orbit reaches the unstable manifold and needs to be reset to the stable manifold
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is studied. For this scenario, an algorithm to redesign a long lifetime orbit based
on the parameters of the trajectory at the unstable manifold is derived. Then, a
two-maneuver sequence is computed to execute a transfer to the new transfer orbit.
The total cost of these maneuvers is optimized, and fuel costs of around 30 cm/s/day
are obtained. Next, the effect of orbit uncertainty on the long lifetime science
orbits is investigated using Monte Carlo simulations. We found that the closer the
trajectory is to the 1-DOF manifold, the more susceptible it is to veering off the
desired manifold path due to an initial position error. Finally, a scheme is developed
to return the spacecraft to its nominal trajectory following an initial position error.
A transfer orbit is optimized by the total cost of the two maneuvers necessary, and
comparisons are made based on how many days after the initial error the correction
is executed. We found that even if it is necessary to wait 5 days after the initial
error, the cost to return the spacecraft to the nominal trajectory is less than 0.4 m/s.
7.2 Future Directions
In this section, a few possible extensions of the research described in this dissertation
are discussed.
7.2.1 Transfers from Capture Trajectories
In our discussion of safe zones for captured trajectories, the safe zones are identified
by their energy level (i.e. their Jacobi integral value). Transfers from safe capture
trajectories to stable frozen orbits are then computed using impulsive maneuvers
whereby the entire set of trajectories remain at the same energy level. An extension
of this work is to derive low-cost transfers that traverse safe zones at different energy
levels rather than staying within one energy level. This is of particular interest since
low-thrust trajectories are becoming more desirable with the increased use of electric
propulsion in spacecraft. In addition, rather than stopping the scheme at a stable
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frozen orbit at a higher altitude, the low-thrust trajectory could be continued using
a spiraling-in approach to transfer the spacecraft to a low-altitude science orbit.
7.2.2 Science Orbit Design
The most logical extension for the science orbit design is to incorporate the science
orbits designed in this dissertation into precision navigation and mission design
programs that include detailed models of all of the physics acting on the system.
This is probably the most effective way to design science orbits for a real mission
since the advantage of our orbit design scheme is its semi-analytic results that are
only possible since the system is not too complex. However, since the system we use
is an approximation to the true system, at some point the more complicated true
system must be incorporated into the orbit design process.
7.2.3 Control of Long Lifetime Science Orbits
There are many issues that could be explored regarding the control of long lifetime
science orbits. Perhaps the most important is ensuring that the spacecraft remains
on the nominal trajectory. We have shown that it is possible to return the spacecraft
to the nominal trajectory after a few days following an initial position error. A more
robust technique would be to devise active controls such that the spacecraft always
stays on the nominal trajectory. This is particularly important in cases where the
system is so unstable that even leaving the trajectory uncontrolled for one day would
be disastrous. One such example is Enceladus, where we showed that long lifetime
science orbits have lifetimes of only 10 days.
Another possible extension concerns resetting the long lifetime science orbit
before impact with the planetary satellite. The transfer method described in this
dissertation involves computing the target (initial conditions of a new long lifetime
orbit) and then optimizing the transfer over the total cost. An improvement to
this method would have the optimization take place over both the total cost and
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the target itself. The goal of this resetting scheme is to transfer the spacecraft to
another long lifetime trajectory, but the exact location where the spacecraft ends
up on the new trajectory is not important. Therefore an optimization procedure
that allowed for variations of the target on the desired trajectory would most likely
produce lower-cost maneuvers than the current one which sets the target before the
optimization takes place.
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APPENDIX
162
APPENDIX A
NUMERICAL ALGORITHMS
This appendix details the algorithms used for the numerical computations of
initial conditions for science orbits in Chapter 5.
A.1 Romberg Algorithm
The number of iterations performed in this algorithm is defined by a number
n. Larger values of n correspond to a higher accuracy in the result, but also
longer computation times since more computations are being performed. We chose
n = 10 since we found that that value gives us a good estimate while keeping the
computation time not too large.
Purpose: Approximate the integral Ij,k which is the entry in the jth row and kth
column of:
I =
∫ b
a
exp (At)dt
1. h = b− a
2. R1,1 = [exp (Aa)]j,k + [exp (Ab)]j,k/2h
3. for i=2 to n
(a) sum = 0
(b) m = 2(i−2)
(c) for k=1 to m
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i. sum = sum + [exp (A(a + (k − 0.5)h))]j,k
(d) R2,1 = (R1,1 + h ∗ sum)/2
(e) for j=2 to i
i. L = 22(j−1))
ii. R2,j = R2,j−1 + (R2,j−1 −R1,j−1)/(L− 1)
(f) h = h/2
(g) for j=1 to i
i. R1,j = R2,j
4. Ij,k = R2,n
A.2 Gaussian Quadrature
For the Gaussian algorithm for double integrals, a set of weights and abscissas
must be specified. The values that we use[1] are denoted rj,k and coj,k, respectively.
The number of iterations performed is given by two numbers n and m. We chose
n = m = 12, which gives us a good approximation of the integral while ensuring not
too long of a computation time.
Purpose: Approximate the integral Ik which is the kth entry of:
I =
∫ b
a
∫ τ
0
exp (A(t− τ))B(τ)dτdt
1. h1 = (b− a)/2
2. h2 = (b + a)/2
3. aj = 0
4. for i=1 to m
(a) t = h1 ∗ rm−1,i + h2
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(b) jx = 0
(c) c1 = 0
(d) d1 = t
(e) k1 = (d1− c1)/2
(f) k2 = (d1 + c1)/2
(g) for j=1 to n
i. τ = k1 ∗ rn−1,j + k2
ii. q = [exp (A(t− τ))B]k
iii. jx = jx + con−1,jq
(h) aj = aj + com−1,ik1 ∗ jx
5. aj = aj ∗ h1
6. Ik = aj
A.3 Gaussian Quadrature with Kepler Equation
Modification
For the Gaussian algorithm for double integrals, a set of weights and abscissas
must be specified. The values that we use[1] are denoted rj,k and coj,k, respectively.
The number of iterations performed is given by two numbers n and m. We chose
n = m = 10 in this case, which gives us a good approximation of the integral while
ensuring not too long of a computation time. The number of iterations is smaller
than in Appendix B since this algorithm is used on a much more complex system
and increasing the number of iterations any larger than 10 resulted in prohibitively
long run times.
The function kepler(τ, e) computes the true anomaly ν given a mean anomaly τ
and an eccentricity e. This is done by using the built-in Matlab function ‘solve’ to
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find the roots of
τ − E + e sin E = 0 (A.1)
where E is the eccentric anomaly. Then,
ν = cos−1 ((cos(E)− e)/(1− e cos(E))) (A.2)
Purpose: Approximate the integral Ik which is the kth entry of:
I =1
2π
∫ 2π
0
∫ M
0
eA(M−τ)B(ν(τ))dτdM
1. h1 = (b− a)/2
2. h2 = (b + a)/2
3. aj = 0
4. for i=1 to m
(a) M = h1 ∗ rm−1,i + h2
(b) jx = 0
(c) c1 = 0
(d) d1 = M
(e) k1 = (d1− c1)/2
(f) k2 = (d1 + c1)/2
(g) for j=1 to n
i. τ = k1 ∗ rn−1,j + k2
ii. ν = kepler(τ, e)
iii. q = [exp (A(M − τ))B(ν)]k
iv. jx = jx + con−1,jq
(h) aj = aj + com−1,ik1 ∗ jx
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5. aj = aj ∗ h1
6. Ik = aj
167
BIBLIOGRAPHY
[1] Abscissas and weights of gauss-legendre integra-tion. From efunda: engineering fundamentals.http://www.efunda.com/math/num integration/findgausslegendre.cfm, 2005.
[2] Pioneer 3, 4. From JPL Mission and Spacecraft Library.http://leonardo.jpl.nasa.gov/msl/QuickLooks/pioneer3QL.html, 2006.
[3] J. Aiello. Personal Communication, August 2004.
[4] J.D. Anderson, G. Schubert, R.A. Jacobson, E.L. Lau, W.B. Moore, and W.L.Sjogren. Europa’s differentiated internal structure: Inferences from four Galileoencounters. Science, 281:2019–2022, 1998.
[5] S.A. Astakhov, A.D. Burbanks, S. Wiggins, and D. Farrelly. Chaos-assistedcapture of irregular moons. Nature, 423:264–267, 2003.
[6] et al. Belton, M.J.S. Galileo’s first images of Jupiter and the Galilean satellites.Science, 274(5286):377–385, 1996.
[7] R.A. Broucke. Long-term third-body effects via double averaging. Journal ofGuidance, Control and Dynamics, 26(1):27–32, 2003.
[8] D. Brouwer and G.M. Clemence. Methods of Celestial Mechanics. AcademicPress Inc., New York, NY, 1961.
[9] L. Buck, C.F. Chyba, M. Goulet, A. Smith, and P. Thomas. Persistenceof thin ice regions in Europa’s ice crust. Geophysical Research Letters,29(22):2055–2058, 2002.
[10] R.L. Burden and J.D. Faires. Numerical Analysis, 6th ed. Brooks/ColePublishing Company, Pacific Grove, CA, 1997.
[11] V.A. Chobotov, editor. Orbital Mechanics, Second Edition. American Instituteof Aeronautics and Astronautics, Inc., Reston, VA, 1996.
[12] J.M.A. Danby. Fundamentals of Celestial Mechanics, 2nd Ed. Willmann-Bell,Richmond, VA, 1992.
168
[13] R. Dvorak. Progress in the elliptic restricted 3-body problem: Asteroids in the3/1, 2/1 and 1/1 resonance. Celestial Mechanics and Dynamical Astronomy,54(1-3):195–205, 1992.
[14] D.T. Greenwood. Principles of Dynamics, Second Edition. Prentice-Hall, Inc.,Upper Saddle River, New Jersey, 1988.
[15] M.D. Guman, D.C. Roth, and B.G Williams. Navigation feasibility studies forthe Europa orbiter mission. Part I, Advances in the Astronautical Sciences,99:161–174, 1998.
[16] R.S. Harrington. The stellar three-body problem. Celestial Mechanics,1(1):200–209, 1969.
[17] M. Henon. New families of periodic orbits in Hill’s problem of three bodies.Celestial Mechanics and Dynamical Astronomy, 85(3):223–246, 2003.
[18] M. Henon and J.-M. Petit. Series expansions for Hill’s problem. CelestialMechanics, 38:67–100, 1986.
[19] G.W. Hill. Researches in the lunar theory. American Journal of Mathematics,1:5–26, 1878.
[20] W.S. Koon, M.W. Lo, J.E. Marsden, and S.D. Ross. Heteroclinic connectionsbetween periodic orbits and resonance transitions and celestial mechanics.Chaos, 10(2):427–469, 2000.
[21] Y. Kozai. Secular perturbations of asteroids with high inclination andeccentricity. Astronomical Journal, 67(9):591–598, 1962.
[22] M. Lara, J.F. San-Juan, and S. Ferrer. Secular motion around tri-axial,sychronously rotating planetary satellites: Application to Europa. Part II,Advances in the Astronautical Sciences, 120:1147–1166, 2005.
[23] M.L. Lidov and S.L. Ziglin. Non restricted double-averaged three body problemin Hill’s case. Celestial Mechanics, 13(4):471–489, 1976.
[24] O. Montenbruck and E. Gill. Satellite Orbits: Models, Methods, andApplications. Springer-Verlag, Germany, 2001.
[25] M.E. Paskowitz and D.J. Scheeres. Identifying safe zones for planetary satelliteorbiters. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conferenceat Providence, RI, paper AIAA 2004-4862, 2004.
[26] M.E. Paskowitz and D.J. Scheeres. Orbit mechanics about planetary satellites.In Proceedings of the AAS/AIAA Space Flight Mechanics Meeting at Maui, HI,paper AAS 04-244, 2004.
169
[27] M.E. Paskowitz and D.J. Scheeres. Orbit mechanics about planetary satellitesincluding higher order gravity fields. In Proceedings of the AAS/AIAA SpaceFlight Mechanics Meeting at Copper Mountain, CO, paper AAS 05-190, 2005.
[28] M.E. Paskowitz and D.J. Scheeres. Transient behavior of planetary satelliteorbits. In Proceedings of the AAS/AIAA Astrodynamics Specialist Conferenceat South Lake Tahoe, CA, paper AAS 05-358, 2005.
[29] M.E. Paskowitz and D.J. Scheeres. Design of science orbits about planetarysatellites: Application to Europa. Journal of Guidance, Control, and Dynamics,29(5):1147–1158, 2006.
[30] M.E. Paskowitz and D.J. Scheeres. Robust capture and transfer trajectoriesfor planetary satellite orbiters. Journal of Guidance, Control, and Dynamics,29(2):342–353, 2006.
[31] M.E. Paskowitz and D.J. Scheeres. A toolbox for designing long lifetime orbitsabout planetary satellites: Application to JIMO at Europa. In Proceedings ofthe AAS/AIAA Space Flight Mechanics Meeting at Tampa, FL, paper AAS06-191, 2006.
[32] M. Paskowitz Possner and D.J. Scheeres. Control of science orbits aboutplanetary satellites. in preparation, 2006.
[33] L. Rowan. Cassini drops in. Science, 307(5713):1222, 2005.
[34] J.F. San-Juan, M. Lara, and S. Ferrer. Phase space structure around oblateplanetary satellites. Journal of Guidance, Control and Dynamics, 29(1):113–120,2006.
[35] D.J. Scheeres. The restricted hill four-body problem with applications tothe earth-moon-sun system. Celestial Mechanics and Dynamical Astronomy,70:75–98, 1998.
[36] D.J. Scheeres and J. Bellerose. The restricted hill full 4-body problem:application to spacecraft motion about binary asteroids. Dynamical Systems:An International Journal, 20(1):23–44, 2005.
[37] D.J. Scheeres, M.D. Guman, and B.F. Villac. Stability analysis of planetarysatellite orbiters: Application to the Europa orbiter. Journal of Guidance,Control and Dynamics, 24(4):778–787, 2001.
[38] D.J. Scheeres and F. Marzari. Spacecraft dynamics in the vicinity of a comet.Journal of Astronautical Sciences, 50(1):35–52, 2002.
[39] D. Stevenson. Europa’s ocean – the case strengthens. Science, 289(5483):1305–1307, 2000.
170
[40] P.F. Thompson, S. Nandi, and M.C. Wong. Orbit determination studiesfor a low-altitude Europa orbiter. paper presented at the 2006 Space FlightMechanics Meeting, Tampa, Florida, AAS 06-192, 2006.
[41] B.F. Villac. Dynamics in the Hill Problem with Application to SpacecraftManeuvers. PhD thesis, The University of Michigan, 2003.
[42] B.F. Villac and D.J. Scheeres. Escaping trajectories in the Hill 3-body problemand applications. Journal of Guidance, Control and Dynamics, 26(2):224–232,2003.
[43] Eric Weisstein. Point-line distance–two-dimensional. From MathWorld–AWolfram Web Resource. http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html, 2003.
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ABSTRACT
ORBIT DESIGN AND CONTROL OF PLANETARY SATELLITE
ORBITERS IN THE HILL 3-BODY PROBLEM
by
Marci Paskowitz Possner
Chair: Daniel J. Scheeres
The exploration of planetary satellites by robotic spacecraft is currently of strong
scientific interest. However, sending a spacecraft to a planetary satellite can be
challenging due to strong perturbations from the central planet. The primary goal of
this dissertation is to identify and utilize the main dynamical features of the system
in the orbit design process.
The system is modeled using a modified form of Hill’s 3-body problem, where
the effect of the planetary satellite’s gravity field is included in the low-altitude
analysis. A thorough study of the dynamics of the system is performed by applying
averaging theory to reduce the complexity and degrees of freedom of the system.
The reduced system has one degree of freedom (DOF) and has equilibrium solutions
called frozen orbits. These frozen orbits are first used as targets for transfers from
capture trajectories in ‘safe zones’. The ‘safe zones’ in phase space are numerically
determined; they contain trajectories that enter the Hill region and allow an
uncontrolled spacecraft to remain in orbit without impact or escape for specified
time periods. Transfers from safe trajectories to frozen orbits are identified and
criteria on their costs evaluated.
Unstable low-altitude, near-polar frozen orbits are the basis for the design of long
lifetime science orbits. The stable and unstable manifolds of these frozen orbits in
the 1-DOF system are investigated and the desired path for long lifetime orbits is
identified. An algorithm is developed to systematically compute initial conditions in
the full system such that the orbits follow the desired path and have sufficiently long
lifetimes to be practical as science orbits about planetary satellites. The analysis of
the control of a planetary satellite orbiter begins with the evaluation of the effect
of orbit uncertainty on the science orbits and the identification of criteria to ensure
that the orbits have the desired behavior. Then, two control schemes are developed:
a) given the terminal conditions of a science orbit, redesign a new science orbit and
execute a low-cost transfer to it, b) return the spacecraft to its nominal trajectory
via a two-sequence set of maneuvers.
2