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Techniques for analysis and design of temporary capture and resonant motion in astrodynamics

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Techniques for analysis and design of temporary capture and resonant motion in astrodynamics

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Content TECHNIQUES FOR ANALYSIS AND DESIGN OF TEMPORARY CAPTURE AND RESONANT MOTION IN ASTRODYNAMICS by Brian D. Anderson A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ASTRONAUTICAL ENGINEERING) May 2019 Copyright 2019 Brian D. Anderson To my wife and my family. Without their unwavering support and encouragement, I would have never made it this far. i Acknowledgements First and foremost, I would like to thank Dr. Martin Lo for providing guidance throughout my entire program and for being a nearly inexhaustible source of knowledge in the field of astrodyna- mics. Many interesting discussions with him led to new ideas and surprising discoveries. From the University of Southern California, I would like to thank the Chair of my committee, Prof. Daniel Erwin and all the remaining members of the committee, Prof. Mike Gruntman, Prof. Edward Rhodes, Prof. Azad Madni, and Prof. Charlie Campbell. Besides the academic guidance through both undergraduate and graduate programs, Dr. Erwin also provided my first formal introduction to astrodynamics, and I will be forever grateful that he sparked my interest in the field. Prof. Gruntman helped me appreciate the considerations required for spacecraft systems engineering, but also provided useful guidance through my graduate program. Prof. Rhodes spent long and late hours to help me make the dissertation manuscript the best it can be, and for that, I am infinitely thankful. I received many good suggestions on how to structure my oral presentations and manuscripts for both my qualification exam and dissertation from Prof. Madni, suggestions that proved invaluable for my success. Prof. Campbell provided valuable feedback on both oral exams and manuscripts, which helped make them as clear and well structured as they can be. I also cannot express how significant it was to receive the USC Doctoral Merit Fellowship from the Viterbi School of Engineering. It was an honor to receive the award and it was financially a great aid toward completing my degree. From the Jet Propulsion Laboratory, there are many who have in some way helped me through my program and it would be impossible to name them all. Most notably, I would like to thank Dr. ii Jon Sims for providing a place at JPL where I could creatively explore new ideas and collaborate with some of the greatest minds in the industry. I also want to thank Tim McElrath for inviting me to be a part of of the Europa Lander mission concept and contribute in a meaningful way to a possi- bly historical achievement in solar system exploration. Pez Zarifian worked hard to make my first internship at JPL a possibility, which opened the door to many rich collaborations and experiences, and for that I will always be grateful. I also thank the late Dr. Eric DeJong for letting me be part of a larger plan to bring science and engineering to more people. I am fortunate to have known such a driven and intelligent human being. Through several years of collaboration with JPL, I want to give thanks to the programs that made my work possible. They are the JPL Summer Internship Program, JPL Year-Round Internship Program, JPL Graduate Fellowship, JPL Strategic University Research Partnership, NASA Innovative Advanced Concepts, and NASA Center Innovation Fund. Last, but very much not least I want to thank my family. My wife, Jennifer never faltered in her support of my endeavor and gave me the encouragement I needed when I needed it most. I thank my parents, Siw and Dennis for making my entire educational journey possible in the first place, and supporting my choices without fail so I could forge my own path. I also thank my sister Emily and brother-in-law Gustaf for their unending support and encouragement, and for making many overseas visits to bring a little bit of home with them to me. I thank my wife’s parents Cliff and Ping, and sister Christine for welcoming me into their family and being a family away from home. Finally, I thank my two dogs Chewbacca and Loki. Never has someone that can’t speak provided so much comfort by simply being present. This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. This rese- arch was carried out in part at the University of Southern California. The work in Section 4.1 was funded in part by the JPL Strategic University Research Partnership Program, the Center Innova- tion Fund Program at JPL, and the JPL AMMOS/MGSS Program. For the work in Sections 4.1– 4.2, I thank Paul Chodas for discussions on the trajectory of Asteroid 2006 RH 120 and Edward iii Wright for help with the NAIF Toolkit and its MATLAB interface, MICE. The work in Section 4.2 was funded in part through the JPL Strategic University Research Partnership program, the JPL Year-Round Internship Program, and the JPL Center Innovation Fund. The work in Chapter 6 was funded in part through the JPL Summer Internship Program and JPL Year-Round Internship Program. iv Abstract Due to recent interest in asteroid rendezvous, capture and deflect missions, temporary moons of Earth have also gained interest. Temporary moons are captured by the planet in highly chaotic orbits, which means small forces can cause deflection or permanent capture. It also means that sending a spacecraft to rendezvous with such objects can have very low fuel requirements. While the outer planets are known to have many temporary moons, few examples exist for Earth. In 2006, asteroid RH 120 became the only known temporary moon of Earth as it remained in orbit for nearly a year. Previous work has explained temporary capture dynamics of Jupiter comets using invariant manifold theory, and we extend that work to explain the temporary capture of the afore- mentioned asteroid at Earth. We showed that 2006 RH 120 closely followed the invariant manifolds of halo orbits in the Sun-Earth Circular Restricted 3-Body Problem during the capture and escape phases. In addition, we showed that it performed an orbital resonance transition as a result of its Earth encounter. While studying the temporary capture phase of the asteroid, we found that the perturbation from the Moon was the dominant factor controlling the Asteroid’s motion. For that study, we applied a modified version of a signal processing method called Dynamic Time Warping in a novel approach to quantitatively compute the similarity of trajectories. This method can be useful for trajectory classification and optimization. This prompted a follow up study to examine the effect of the Moon on low energy objects approaching Earth. A comparison of transit, cap- ture and impact rates in the low energy regime of the Bicircular Problem and Circular Restricted 3-Body Problem show that the Moon reduces the average rate of Earth impact at the high end of the energy range studied. At lower energy, the Earth impact rate was similar or higher with the v Moon present, depending on the exact energy. The Moon had the effect of decreasing the average rate of transit from interior to exterior regions at most values in the low end of the energy range. The resonant behavior of Asteroid 2006 RH 120 before and after temporary capture led to a study of orbital resonance. We developed and applied a new algorithm for computing compound resonant orbits, which are finite period orbits that both orbit Lagrange points and exhibit resonant behavior. This allows for prediction of which resonant orbits can be naturally reached at a given Jacobi con- stant. We correlated those results with the estimated Jacobi constant of the asteroid, and we found that the resonances observed were in the allowable range. These compound orbits are inherently unstable due to the close encounter with the secondary body, which makes them useful for certain aspects of trajectory design. However, long term stability can also be a requirement. Therefore, we extended the study of resonance to evaluate stability in the CR3BP and more realistic models. This involved developing a predictive model for stable and unstable fixed points in a Poincar´ e map, and defining a new method for computing finite time orbit stability from perturbed Poincar´ e maps. This method was used to evaluate the stability of resonant orbits in satellite systems with inherent resonance. In the case of the Galilean satellite system, we discovered that a rule-of-thumb regar- ding the resonance ratio integers allows for better long term stability. As a computational aid for the work within, an algorithm was presented for solving ODEs in parallel on GPU hardware. First, it was used for propagating large sets of trajectories, for which it was 1 10 2 1 10 4 faster than a CPU implementation. Second, it was used to compute Poincar´ e maps, for which it was about 200 times faster. vi Table of Contents Dedication i Acknowledgements ii Abstract v List of Figures xi List of Tables xv Chapter 1 : Introduction 1 1.1 The Problem and its Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Current Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Our Approach and Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Major Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2 : Background 5 2.1 Circular Restricted 3-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Bicircular Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Elliptic Restricted 3-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Invariant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Poincar´ e Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Delaunay Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 Homoclinic and Heteroclinic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.10 Heteroclinic-Homoclinic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.11 Temporary Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 3 : Statement of the Problem 35 3.1 The Research Problem We are Addressing . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Why It Is a Research Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Impact on Astrodynamics & Dynamical Astronomy . . . . . . . . . . . . . . . . . 36 vii 3.4 Impact on Methodological Approaches . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Impact on Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6.1 Hypothesis 1: Manifolds Control Transit Dynamics . . . . . . . . . . . . . 38 3.6.2 Hypothesis 2: Dynamics Largely Persists in 4-Body . . . . . . . . . . . . . 39 3.6.3 Hypothesis 3: The Moon Significantly Affects NEO Impacts on Earth . . . 40 3.6.4 Hypothesis 4: Transit Leads to Resonance Transition . . . . . . . . . . . . 40 3.6.5 Hypothesis 5: Stable CR3BP Resonances Persist in Resonant N-Body Sys- tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Matrix of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.8 Our Plan on Addressing the Limitations . . . . . . . . . . . . . . . . . . . . . . . 42 3.9 Current Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.9.1 Approach 1: Perturbed 2-Body . . . . . . . . . . . . . . . . . . . . . . . . 43 3.9.2 Approach 2: Multibody Without Dynamical Systems Theory . . . . . . . . 44 3.9.3 Approach 3: 3-Body With Dynamical Systems Theory . . . . . . . . . . . 44 3.10 Limitations of Current Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.10.1 2-Body Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.10.2 Multibody Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.10.3 3-Body Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 4 : The Temporary Capture of Asteroid 2006 RH 120 via Invariant Manifolds 48 4.1 Part I:Pre-Capture and Escape Phases . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.3 CR3BP and Ephemeris model . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.4 Asteroid Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.5 Earth Approach ThroughL 1 Manifolds . . . . . . . . . . . . . . . . . . . 61 4.1.6 Earth Departure ThroughL 2 Manifolds . . . . . . . . . . . . . . . . . . . 63 4.1.7 Example of Complex Lunar Interactions With Asteroid-Like Trajectories . 65 4.1.8 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Part II:Temporary Capture Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.5 Section Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Chapter 5 : Effect of Moon on Transit Dynamics 96 5.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 viii Chapter 6 : Analysis of Multibody Resonance with Applications to the Galilean System 116 6.1 Resonance Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1.1 Topology in Delaunay Variables . . . . . . . . . . . . . . . . . . . . . . . 117 6.1.2 Locating Stable Island Centers . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.3 Producing Initial Guess Conditions for Stable and Unstable orbits . . . . . 135 6.2 Counting Stable Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2.1 Circular Restricted 6-Body Problem . . . . . . . . . . . . . . . . . . . . . 141 6.2.2 Finite Time Stability and Resonance Width . . . . . . . . . . . . . . . . . 146 6.2.3 Resonance Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.3 Resonance Abundance in the Galilean System . . . . . . . . . . . . . . . . . . . . 152 6.3.1 Small Exterior Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.2 Medium Exterior Resonances . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.3.3 Large Exterior Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.3.4 Summary of Galilean Resonance Abundance . . . . . . . . . . . . . . . . 191 6.4 Quarantine Orbit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.4.1 Stable Resonant Orbits in the CR3BP . . . . . . . . . . . . . . . . . . . . 193 6.4.2 Stable Resonant Orbits in a Galilean 6-Body Problem . . . . . . . . . . . . 194 6.4.3 Stable Resonant Orbits in an Ephemeris Model . . . . . . . . . . . . . . . 195 6.5 Compound Resonant Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.5.1 New Method for Computing Compound Orbits . . . . . . . . . . . . . . . 201 6.5.2 Compound Orbit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.5.3 Relationship Between Compound Orbits and Homoclinic Orbits . . . . . . 216 6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Chapter 7 : Conclusion 219 7.1 Novel Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.1.1 Manifold Control of Asteroid 2006 RH 120 . . . . . . . . . . . . . . . . . . 221 7.1.2 Temporary Capture Analysis of Asteroid 2006 RH 120 . . . . . . . . . . . . 221 7.1.3 Dynamic Time Warping for Trajectory Comparison . . . . . . . . . . . . . 221 7.1.4 The Moon’s Effect on Transit Dynamics . . . . . . . . . . . . . . . . . . . 222 7.1.5 6-Body Model for Design of Resonant Europa Quarantine Orbit . . . . . . 222 7.1.6 GPU Accelerated Mission Design . . . . . . . . . . . . . . . . . . . . . . 223 7.1.7 NIAC Interplanetary Cubesat Study . . . . . . . . . . . . . . . . . . . . . 224 7.1.8 Conley Theorem Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.2.1 DTW Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.2.2 GPU Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Publications 227 Bibliography 228 ix Chapter A: GPU Accelerated Mission Design 235 A.1 GPU Computing Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.2 Runge-Kutta-Fehlberg 7/8th Order Adaptive ODE Solver . . . . . . . . . . . . . . 238 A.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 A.2.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 A.3 Event Stopping and H´ enon Step . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.4 Poincar´ e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.4.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 A.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Chapter B: Techniques 253 B.1 Differential Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 B.1.1 Single Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 B.1.2 Multiple Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 B.2 Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 B.3 Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 B.3.1 Single Parameter Continuation . . . . . . . . . . . . . . . . . . . . . . . . 259 B.4 Ephemeris Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 B.4.1 Jet Propulsion Laboratory DE431 . . . . . . . . . . . . . . . . . . . . . . 260 B.4.2 Jet Propulsion Laboratory JUP310 . . . . . . . . . . . . . . . . . . . . . . 260 B.5 Ephemeris to CR3BP Frame Conversion . . . . . . . . . . . . . . . . . . . . . . . 260 B.5.1 Conversion Method 1: Variable Length and Time Units . . . . . . . . . . . 260 B.5.2 Conversion Method 2: Fixed Length and Time Units . . . . . . . . . . . . 262 B.6 Delaunay Variable Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 B.7 Poincar´ e Map Delaunay Variables to Cartesian Conversion . . . . . . . . . . . . . 266 x List of Figures 2.1 Earth-Moon CR3BP rotating coordinate frame. . . . . . . . . . . . . . . . . . . . 7 2.2 Position ofL 1 andL 2 vs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Velocity contours in the CR3BP with = 0:3. . . . . . . . . . . . . . . . . . . . . 10 2.4 Velocity contours in the Earth-Moon CR3BP ( 0:01215). . . . . . . . . . . . . 11 2.5 CR3BP Jacobi constant surface formed from contours of fixed Jacobi constant. . . 12 2.6 Forbidden Region surface topology. . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 Bicircular Problem Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Diagram of the Sun-Planet-Moon Bicircular Problem. . . . . . . . . . . . . . . . . 17 2.9 Diagram of the Elliptic Restricted 3-Body Problem. . . . . . . . . . . . . . . . . . 19 2.10 Examples of periodic orbits in the CR3BP. . . . . . . . . . . . . . . . . . . . . . . 22 2.11 Invariant manifold diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.12 Poincar´ e map concept diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.13 Example Poincar´ e map in the Jupiter-Europa CR3BP. . . . . . . . . . . . . . . . . 26 2.14 Delaunay variable diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.15 Comparison of Poincar´ e map coordinates. . . . . . . . . . . . . . . . . . . . . . . 29 2.16 The heteroclinic-homoclinic chain for Jupiter. . . . . . . . . . . . . . . . . . . . . 32 2.17 The Genesis homoclinic-heteroclinic chain. . . . . . . . . . . . . . . . . . . . . . 32 4.1 The trajectory of Asteroid 2006 RH 120 . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Estimated Forbidden Region surfaces for Asteroid 2006 RH 120 . . . . . . . . . . . . 52 4.3 Asteroid 2006 RH 120 rotating frame trajectory. . . . . . . . . . . . . . . . . . . . . 57 4.4 Estimated Jacobi constant of Asteroid 2006 RH 120 . . . . . . . . . . . . . . . . . . 58 4.5 Osculating semimajor axis of Asteroid 2006 RH 120 for the period 1950-2050. . . . 59 4.6 Asteroid 2006 RH 120 approach with stable manifold. . . . . . . . . . . . . . . . . . 62 4.7 Asteroid 2006 RH 120 departure with stable manifold. . . . . . . . . . . . . . . . . 64 4.8 Chaotic evolution of the Genesis trajectory. . . . . . . . . . . . . . . . . . . . . . 65 4.9 Distance between Asteroid 2006 RH 120 and the Moon during Temporary Capture. . 67 4.10 Visual definition of Segments and Phases. . . . . . . . . . . . . . . . . . . . . . . 69 4.11 State Coherence diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.12 Lunar phase angle computation diagram. . . . . . . . . . . . . . . . . . . . . . . . 79 4.13 CR3BP model results for Segment 2. . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.14 BCP model results for Segment 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.15 ER3BP model results for Segment 2. . . . . . . . . . . . . . . . . . . . . . . . . . 86 xi 4.16 CR3BP model results for Segment 1. . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.17 BCP model results for Segment 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.18 ER3BP model results for Segment 1. . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.19 CR3BP model results for Segment 3. . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.20 BCP model results for Segment 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.21 ER3BP model results for Segment 3. . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.1 Diagram of Fundamental Interval for a Sun-Earth planarL 1 Lyapunov orbit. . . . . 98 5.2 Capture, Exterior, and Interior Region definition diagram. . . . . . . . . . . . . . . 101 5.3 L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, C 0 = 3:000875. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4 L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, C 0 = 3:00075. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.5 L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, C 0 = 3:000425. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6 L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, C 0 = 3:00020. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.7 L 1 capture and impact statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, 2:99945C 0 3:0008875. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.8 L 1 interior and exterior transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, 2:99945C 0 3:0008875. . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.9 Example high energy transit rates vs. lunar phase angle in the BCP. . . . . . . . . . 114 6.1 Example Poincar´ e map in Delaunay variables (L; g). . . . . . . . . . . . . . . . . 117 6.2 One stable island from a 3:4 resonant orbit with no loops. . . . . . . . . . . . . . . 119 6.3 Three connected, indistinguishable stable islands from a 3:4 resonant orbit with loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.4 Three connected, distinguishable stable islands from a 3:4 resonant orbit with loops. 122 6.5 Three separate stable islands from a 3:4 resonant orbit with loops. . . . . . . . . . 123 6.6 Critical Jacobi constant that produces loops. . . . . . . . . . . . . . . . . . . . . . 124 6.7 Representative orbits showing alignment of extrema. . . . . . . . . . . . . . . . . 126 6.8 Resonance pattern topology, Exterior Region. . . . . . . . . . . . . . . . . . . . . 128 6.9 Resonance pattern topology, Interior Region. . . . . . . . . . . . . . . . . . . . . . 129 6.10 Approximate island centers of a Poincar´ e map, Exterior Region, Jupiter-Europa, C 0 = 3:0039. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.11 13:15 unstable orbit, predicted to be stable, Jupiter-Europa,C 0 = 3:0039. . . . . . 135 6.12 Poincar´ e map of approximated stable solutions, Jupiter-Europa,C 0 = 3:0039 . . . 138 6.13 Poincar´ e map indicating quality of approximation. Exterior Region. . . . . . . . . 139 6.14 Poincar´ e map indicating quality of approximation. Interior Region. . . . . . . . . . 140 6.15 Diagram of the CR6BP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.16 Definition of Resonance Width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.17 Resonance Width for a 12-year Poincar´ e map, Jupiter-Europa,C 0 = 3:0039. . . . . 149 6.18 CR3BP Poincar´ e map, colored by Resonance Width, Jupiter-Europa,C 0 = 3:0039. 150 xii 6.19 Stable resonant orbit Abundance, Jupiter-Europa,C 0 = 3:0039. . . . . . . . . . . . 151 6.20 Jupiter-Europa CR3BP vs. Galilean moon CR6BP 12-year Poincar´ e map, C 0 = 3:0039. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.21 Resonance Width for a Galilean moon CR6BP 12-year Poincar´ e map,C 0 = 3:0039. 156 6.22 Galilean moon CR6BP 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.23 Stable resonant orbit Abundance for a Galilean moon CR6BP 12-year Poincar´ e map,C 0 = 3:0039. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.24 Comparison of Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP,C 0 = 3:0039. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.25 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. . . . . . . . . . 161 6.26 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. . . . . . . . . . 162 6.27 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. . . . . . . . . . 163 6.28 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. . . . . . . . . . 166 6.29 Small Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.30 Approximation error in computation ofG ande fromC. . . . . . . . . . . . . . . 168 6.31 Resonant orbit variability as a function ofC. . . . . . . . . . . . . . . . . . . . . . 168 6.32 Estimate of Forbidden Region boundary forC 0 = 3:0036. . . . . . . . . . . . . . . 169 6.33 Example small (5:7) resonant orbit in Galilean moon CR6BP,C 0 = 3:0039. . . . . 171 6.34 Example small (6:7) resonant orbit in Galilean moon CR6BP,C 0 = 3:0039. . . . . 172 6.35 Example small (6:7) resonant orbit in ephemeris model,C 0 = 3:0039. . . . . . . . 173 6.36 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. . . . . . . . . . 175 6.37 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. . . . . . . . . . 176 6.38 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. . . . . . . . . . 177 6.39 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. . . . . . . . . . 178 6.40 Medium Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.41 Example medium (1:2) resonant orbit in Galilean moon CR6BP,C 0 = 3:0039. . . . 181 6.42 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. . . . . . . . . . 184 6.43 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. . . . . . . . . . 185 6.44 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. . . . . . . . . . 186 6.45 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. . . . . . . . . . 187 6.46 Large Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.47 Example large (3:7) resonant orbit in Galilean moon CR6BP,C 0 = 3:0039. . . . . 190 6.48 Example 7:8 resonant orbit in ephemeris model,C 0 = 3:0039. . . . . . . . . . . . 197 6.49 Example 4:5 resonant orbit in ephemeris model,C 0 = 3:0039. . . . . . . . . . . . 198 6.50 Example 5:6 resonant orbit in ephemeris model,C 0 = 3:0039. . . . . . . . . . . . 199 6.51 L 2 Lyapunov orbit manifold intersections with Poincar´ e section, Jupiter-Europa CR3BP,C = 3:00360. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.52 Fixedi Compound Orbits in the Jupiter-Europa CR3BP,C = 3:00330. . . . . . . . 206 xiii 6.53 Fixedk Compound Orbits in the Jupiter-Europa CR3BP,C = 3:00330. . . . . . . . 207 6.54 L 2 Compound Orbits in the Jupiter-Europa CR3BP,C = 3:00360. . . . . . . . . . 209 6.55 L 2 Compound Orbits in the Jupiter-Europa CR3BP,C = 3:00330. . . . . . . . . . 210 6.56 L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000891. . . . . . . . . . . . 212 6.57 L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000886. . . . . . . . . . . . 213 6.58 L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000500. . . . . . . . . . . . 214 6.59 L 1 Compound Orbits in the Sun-Earth CR3BP,C = 3:000500. . . . . . . . . . . . 215 A.1 Diagram of GPU thread synchronization. . . . . . . . . . . . . . . . . . . . . . . . 242 A.2 Performance metrics of GPU propagator. . . . . . . . . . . . . . . . . . . . . . . . 246 A.3 Performance metrics of GPU Poincar´ e map. . . . . . . . . . . . . . . . . . . . . . 251 xiv List of Tables 3.1 Matrix of proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1 Reference dates used for phase and segment definitions. . . . . . . . . . . . . . . . 70 4.2 Trajectory similarity between models. . . . . . . . . . . . . . . . . . . . . . . . . 83 6.1 Case selection matrix for island location algorithm. . . . . . . . . . . . . . . . . . 133 6.2 Parameters used for Galilean moon CR6BP . . . . . . . . . . . . . . . . . . . . . 146 6.3 Summary of total Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.1 Summary of contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 A.1 CPU and GPU test hardware. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 xv CHAPTER 1: INTRODUCTION There are two types of moons, stable and temporary moons. Stable moons are in stable orbits around the planet; temporary moons are captured by the planet in highly chaotic orbits. The outer planets are known to have many temporary moons. In fact, the Jupiter Family of Comets like Oterma, Gehrels 3, Helin-Roman-Crockett were temporarily captured by Jupiter and then escaped. Shoemaker-Levy 9 was a temporary moon that eventually impacted Jupiter itself. In 2006, Near Earth Object (NEO) 2006 RH 120 was captured into Earth orbit for nearly a year. It is the first temporary moon of the Earth to be observed. Previous work has explained the dynamics of the temporary capture of Jupiter comets using invariant manifold theory, see Koon et al.[54] and Howell, Lo and Marchand[47]. The same dynamics is also at work for the temporary capture of Earth’s temporary moons. However, the small mass parameter of the Sun-Earth/Moon Barycenter Circular Restricted 3-Body Problem as well as the large mass of Earth’s Moon greatly complicate the temporary capture of NEOs around the Earth as compared to the capture of comets around Jupiter or Saturn. 1.1 The Problem and its Importance Temporary capture and other low-energy planetary flyby trajectories have been shown to cause asteroids and comets to undergo resonance transitions. An asteroid may be in orbital resonance with a planet and as a result approach the planet. During this approach, the asteroid’s mechanical energy is changed such that it departs the planet on a new resonant orbit. From a heliocentric, Keplerian point of view, the asteroid has gained free energy. Due to the nature of low-energy trajectories, these approaches and departures are likely to cause resonances. The result of the resonant behavior is that the asteroid will return again to approach the planet. This return can cause a subsequent resonance transition. This process can continue indefinitely, or until the gravity of another celestial body takes over. If that happens, further resonance transitions with the new body 1 of 268 1.2. CURRENT APPROACHES are possible. This mechanism can describe natural transport of material through the solar system. Understanding the dynamics that govern temporary capture and resonance transitions will help in the fields of Planetary Defense as well as asteroid exploration and asteroid mining. Through this understanding, we may develop methods to efficiently search for these objects, deflect unwanted targets from the Earth, capture desired targets to accessible near-Earth orbits, or rendezvous with interesting science targets. Moreover, the low-energy resonance transitions are attractive from the point of view of space mission design. An example would be a tour design of a planetary system of moons. As another example, we may in some cases desire trajectories that avoid resonance transition for the purpose of stability. The knowledge of temporary capture and transition dynamics will yield an alternative approach to low-cost trajectory design. 1.2 Current Approaches Due to the well behaved nature of Keplerian orbits, analysis of resonance transitions has been mostly applied by patched conics. A resonance transition is akin to a gravity assist, where a flyby of a planet in the solar system can increase or decrease the mechanical energy of the asteroid or comet. This can therefore change the period from one resonance to another. The total heliocentric energy can change due to the rotation of the planetocentricV 1 vector. This vector represents the remaining velocity once an object has fully escaped an attracting body. When seen from a 3-body point of view, a flyby like this keeps its energy as defined in that frame, known as the Jacobi constant. The Tisserand parameter has been widely used to analyze asteroids and comets that undergo transitions and planetary flybys, and is in fact an approximation of the Jacobi constant as expressed in orbital elements. Studying temporary capture becomes very problematic with patched conics because a ballistic flyby through a planet’s sphere of influence will always be hyperbolic. Therefore it cannot complete any revolutions of the planet. A temporary capture would require at least one revolution of the planet before departing. One can begin to study the problem using a perturbed 2-body model, but this is only a simplification over studying the actual 3-body system. Due to the complex 2 of 268 1.3. OUR APPROACH AND ADV ANTAGES behavior of the 3-body dynamics, situations where asteroids or comets approach and orbit a planet for any number of revolutions before departing can exist. In fact, a theorem states that given any number of revolutions around a planet, and any other number of revolutions around the entire set of attracting bodies, there exists a solution that satisfies that characteristic. Thus an infinite number of trajectories exist that wind around the planet and exterior region in any combination. For example, given any two integersm andn, there exists a trajectory that winds around the Earthn times and then winds around the Earth’s orbit m times in the exterior region. This effect is not present in Keplerian dynamics, as chaos is not present. 1.3 Our Approach and Advantages By studying the natural flow to and from near-planet space by analyzing invariant manifolds of periodic orbits, the natural paths of the system are revealed. It has been theorized that libration point orbit manifolds control the flow of objects to and from interior/exterior regions and through the near-planet region. For the planar case (2-dimensional Restricted 3-Body Problem) and limited range of Jacobi constant, this has been shown by Conley[21]. The theory has not been shown to be valid for arbitrary Jacobi constant and does not directly extend to the spatial problem (3- dimensional Restricted 3-Body Problem). However, some numerical studies have indicated that similar controlling factors may be occurring, such as the one by Ren and Shan[74]. If we can show that the theory extends to the spatial problem in a more systematic way, we can apply it to the problem of asteroid temporary capture and resonance transition. 1.4 Major Research Goals Our goal is to provide evidence that the result of Conley retains some validity in the spatial pro- blem. Namely, that passing through a Lagrange point gateway requires following the invariant manifolds of its periodic orbits. If gateway transit is controlled by manifolds, then by extension, 3 of 268 1.4. MAJOR RESEARCH GOALS so is transit from interior to exterior regions. We also aim to show that this overall transit behavior is controlled by orbit manifolds in the 4-Body Problem. This would allow simpler manifolds from the 3-Body Problem to approximate behavior in a 4-body model. There will certainly be some differences between the models, but we expect the overall behavior to be similar. However, we do want to study those differences, and we expect that one key difference will be a significant diffe- rence in the rate of Earth impacts in the 3- and 4-body models. Finding such a large difference would imply that the Moon has a large effect upon the rate of impacts of small objects on the Earth. We also aim to give examples of how we can use transit dynamics effectively for mission design, by applying it to some choice problems. Finally, the problem of intentionally avoiding resonance transition is considered. The goal is to show that we can use the knowledge of resonance to find long term stable orbits in low- and high-fidelity models. 4 of 268 CHAPTER 2: BACKGROUND 2.1 Circular Restricted 3-Body Problem This section is aimed at describing the Circular Restricted 3-Body Problem (CR3BP) and its basic properties. For more detail, see Szebehely[84] and Koon, Lo, Marsden, & Ross[53] and references therein for the equations of motion for the CR3BP and the theory of invariant manifolds of periodic orbits. The CR3BP utilizes a coordinate system that rotates along with two large bodies that orbit their common barycenter. This coordinate system is shown in Fig. 2.1 for the Earth-Moon system, which we will use as an illustrative example henceforth. Note that in another system we will use later, the Sun-Earth system, “Earth” actually signifies the Earth-Moon Barycenter orbiting the Sun in a circular orbit because the Moon has a non-negligible mass in the planetary system and causes a non-negligible offset of the Earth’s heliocentric orbit. In this reference frame, there are five equilibrium points known as Lagrange points labeledL 1 throughL 5 . L 1 –L 3 lie on thex-axis whileL 4 andL 5 are symmetrically placed on either side of thex-axis. In fact, L 4 andL 5 lie on the vertices of equilateral triangles, with the two Primaries at the other vertices. Because of this, we will sometimes refer toL 1 -L 3 as the collinear points, and toL 4 –L 5 as the triangular points. All Lagrange points lie in the planez = 0. In the remainder of this text, we will use “Primary” and “Secondary” to refer to the smaller and larger body, respectively. Furthermore, we will also refer to the Primary and Secondary collectively as the “Primaries”. We also choose to use the coordinate system definition that places the larger body on the negative x-axis and the smaller body on the positivex-axis. Another useful feature of this model is that it is autonomous. This means that the equations of motion are not explicitly a function of time, but only a function of the instantaneous 5 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM state. This will become clear when we introduce the equations of motion next. The second order differential equations used to model this system are shown in Eqs. 2.1–2.3: x = 2 _ y +x 1 r 3 1 (x +) r 3 2 (x 1 +) y =2 _ x +y 1 r 3 1 y r 3 2 y z = 1 r 3 1 z r 3 2 z (2.1) Here, the distances to each of the Primaries (r 1 ;r 2 ) are computed as r 1 = p (x +) 2 +y 2 +z 2 r 2 = p (x 1 +) 2 +y 2 +z 2 : (2.2) This system only has one parameter,, which is the mass ratio of the secondary to that of the total system mass as defined in Eq 2.3 below. As is customary with the CR3BP, we perform a change of units in order to make them non-dimensional. First, units of mass are divided by the total mass of the system. Second, units of length are divided by the semimajor axis of the Primaries’ orbital motion. Lastly, units of time are chosen such that the mean motion of the Primaries’ orbit is 1. This has the additional effect of making the universal gravitational constant 1 in this system of units. = m 2 m 1 +m 2 (2.3) Regarding the Lagrange points, we can compute their location by solving for the positions where Eq. 2.1 is zero in all components. The triangular points are simple and will always lie at x = 1=2 andy = p 3=2. However, the collinear points are solved numerically and change position along thex-axis as a nonlinear implicit function of. Examples of the position ofL 1 and L 2 as a function of are shown in Fig. 2.2. In general, L 1 and L 2 move farther away from the Secondary as increases. 6 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM -1.5 -1 -0.5 0 0.5 1 1.5 X [NON] -1.5 -1 -0.5 0 0.5 1 1.5 Y [NON] EARTH MOON L 1 L 2 L 3 L 4 L 5 INTERIOR EXTERIOR FORBIDDEN NECK (a) 0.5 1 1.5 X [NON] -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Y [NON] MOON L 1 L 2 INTERIOR EXTERIOR FORBIDDEN NECK TO EARTH (b) Figure 2.1: Earth-Moon CR3BP rotating coordinate frame. (a) shows a global view of all the salient features while (b) is a local view near the Moon. Earth is atx = and the Moon is at x = 1. The five Lagrange points are indicated by black triangles. The Forbidden Region is shown in gray and is used to indicate the three regions of motion. The interior region is the area near the Earth, and the exterior region is outside the Moon’s orbit. The region near the Moon between L 1 andL 2 is the Neck region. The mass ratio for the Earth-Moon system is 0:01215. While we will regularly be studying the Sun-Earth system, we chose the Earth-Moon for this illustration since it is difficult to see the topology at this scale for the small of the Sun-Earth system. This dynamical system has a single constant of integration known as the Jacobi constant, C, which can at times be referred to as the “energy” in the literature as well. The Jacobi constant is computed from the position dependent quasi-potential, , and the speed,v, as shown in Eq. 2.4. An object with a given position and velocity in this frame that obeys the dynamics of the system will have a fixed Jacobi constant for all time. = 1 r 1 r 2 + x 2 +y 2 2 C = 2 v 2 = 2(1) r 1 2 r 2 +x 2 +y 2 ( _ x 2 + _ y 2 + _ z 2 ) (2.4) 7 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM (a) (b) Figure 2.2: Position of L 1 and L 2 vs. . The y-axis is logarithmic and shows the value of the mass ratio. Black indicates the position of the Secondary body (m 2 ), while red isL 1 and blue is L 2 . (a) shows the absolute position of the Lagrange points and the Secondary body on thex-axis, while (b) shows the position relative to the Secondary body. Thus, the position of the Secondary body in (b) is a vertical line at 0. Horizontal dashed lines indicate some example values of in real CR3BP systems. Note thatL 1 andL 2 are not equally distant fromm 2 , andL 2 is generally slightly farther away. However, for small, it is a close approximation to say they are nearly equidistant fromm 2 . We intentionally terminate the lines at = 0:5, since that is the case of equal masses. For> 0:5, the solutions are identical to the case wherem 1 andm 2 switch places. The energy-like behavior of the Jacobi constant is reversed however, as a low Jacobi constant is actually a high “mechanical energy”. The mechanical energy we refer to is the sum of kinetic energy and gravitational potential energy. The term (x 2 +y 2 ) is a result of the centrifugal force of the rotating frame. If we examine the three other terms in the Jacobi constant, it is clear that there are two potential energy terms and one kinetic energy term. In the 2-Body Problem, the total energy usually has a negative sign for the potential energy term and a positive sign for the kinetic energy term. In the 3-Body Problem, we choose the opposite by definition and have positive signs on both potential energy terms and a negative sign on the kinetic energy. This is the convention that has been used for the 3-Body Problem in literature and thus I choose to adopt that terminology to 8 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM be consistent with other work in the field. Thus, an action that raises the total energy in the 2-Body Problem, will lower the Jacobi constant in the 3-Body Problem. For example, moving closer to a gravitational attracting body makes the total energy lower in the 2-Body Problem, but increases the Jacobi constant. Similarly, it is clear that increasing the speed will increase the total energy in the 2-Body Problem, but lower the Jacobi constant. One way to visualize this is to examine contours of fixed speed in the planar CR3BP for a given Jacobi constant, which we show in Figs. 2.3–2.4. In those diagrams, we see that higher speeds result from approaching either of the two Primaries, as a result of trading away potential energy for kinetic energy. We also see that there are regions where the velocity contour value approaches 0, which leads us to the topic of the “Forbidden Region”. The Jacobi constant will determine the allowable regions of motion and the region where motion is not allowed, which is called the “Forbidden Region”. The Forbidden Region of an example Jacobi constant in the Earth-Moon CR3BP is shown in Fig. 2.1. Figure 2.4 shows the Forbidden Region in the same system for a few other values of the Jacobi constant. While we use the Earth-Moon system for the purpose of this discussion, the qualitative features are the same if they are replaced by another set of Primaries. Since the system is spatial, the plots here only show a slice of a 3-dimensional Jacobi constant surface. For the Jacobi constant level plotted, we divide the region of allowable motion into three subregions separated by vertical planes through the L 1 and L 2 Lagrange points: the Interior Region, the Exterior Region and the Neck Region. The Interior Region is around the Earth inside the Moon’s orbit and Forbidden Region. The Exte- rior Region is outside of the Moon’s orbit and Forbidden Region. The Neck Region is the region containing the Moon andL 1 andL 2 , connecting the Interior and Exterior Regions. For very low energy levels, all regions are completely cut off from each other. As the energy increases (Jacobi constant decreases), a hole opens up that allows for motion between the Interior and Neck Region throughL 1 . Further increase in the energy opens a hole to the Exterior Region. At an even higher energy, all in-plane motion is allowed. 9 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM (a)C = 3:95 (b)C = 3:65 (c)C = 3:45 (d)C = 3:00 Figure 2.3: Velocity contours in the CR3BP with = 0:3. The Forbidden Region is shown in gray, with contours of fixed speed as indicated by the color bar. Note that the contours represent fixed steps in the natural logarithm of the speed, since it changes rapidly near the Forbidden Region and the Primaries. (a) At this Jacobi constant, three separated regions of motion exist: One around the Primary, one around the Secondary, and the third outside the Forbidden Region. (b) Now, there are three regions, two of which are connected. TheL 1 gateway is open and motion is allowed between the interior and region near the Secondary body. (c) Here, the gateway atL 2 is also open, allowing motion between interior, exterior, and near-Secondary regions. However, transfer between interior and exterior must occur through the near-Secondary region. (d) Finally, the opening at L 3 is also open and transfer between interior and exterior is allowed there as well. As indicated by the contours, allowable speeds increase if you move away from the Forbidden Region, or if you approach either of the two Primaries. 10 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM (a)C = 3:19 (b)C = 3:18 (c)C = 3:15 (d)C = 3:01 Figure 2.4: Velocity contours in the Earth-Moon CR3BP ( 0:01215). Refer to Fig. 2.3 for description of labels, since they are identical. The same topology is found for a smaller mass ratio, but there are some quantitative changes. In (a), the isolated region near the Secondary is much smaller in contrast to Fig. 2.3(a). In (c), the thickness of the horseshoe-like forbidden region is also much smaller in contrast to Fig. 2.3(c). Note that the appropriate values of the Jacobi constant that produce similar topological behavior are different for this mass ratio. 11 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM As another way to visualize how the Forbidden Region topology changes with the Jacobi con- stant, we can plot a surface satisfyingv 2 = 0 = 2 C withC on thez-axis. We show such a diagram for the Earth-Moon system and a system with = 0:3 in Fig. 2.5. Any horizontal slice through this surface represents the Forbidden Region of that Jacobi constant. One obvious diffe- rence between Fig. 2.5(a) and (b) is that the range of C in which the topology changes is much smaller for the smaller value of . In fact, for mission design of realistic mass ratios, we often study Jacobi constants very close to 3, since that is where the topology is changing rapidly. This enables far-reaching motion of spacecraft without expending too much energy. (a) = 0:3 (b) 0:01215 Figure 2.5: CR3BP Jacobi constant surface formed from contours of fixed Jacobi constant. Thez- axis is the negative of the Jacobi constant, in order to produce higher energy upward. (a) A notional system with = 0:3. A binary star system could be an example of a system that would have a mass ratio this large. The highlighted level curves in black represent the four forbidden region boundaries shown in Fig. 2.3(a)-(b). (b) Earth-Moon CR3BP. The highlighted level curves in black represent the four forbidden region boundaries shown in Fig. 2.4(a)-(b). The orange surface has been left partially transparent in order to see the shape of the level curves match the corresponding forbidden regions. The two maxima of this surface correspond to the Lagrange pointsL 4 andL 5 , while the three saddle points are the L 1 –L 3 points. Thus we can see that increasing the Jacobi constant beyond that ofL 4 /L 5 no longer produces a Forbidden Region in the plane. 12 of 268 2.1. CIRCULAR RESTRICTED 3-BODY PROBLEM While we examined the planar component of the Forbidden Region, it is in fact 3-dimensional. For higher energies where all motion in the plane is allowed, motion between the Interior and Exterior Region is still unlikely to occur at any point along the Moon’s orbit. Trajectories with non-trivial out of plane components will still be more likely to cross between regions where the vertical opening between regions is the largest, such as the Neck Region near the Moon. An example of the topological change in the Jacobi constant surface properties is seen in Fig. 2.6. The low energy orange surface only allows motion between Interior and Exterior Regions through the Neck Region near the Moon, while the high higher surface in purple causes the top and bottom of the Forbidden Region to become two separate surfaces. (a) (b) Figure 2.6: Illustrative example of Forbidden Region surfaces showing topological change for different Jacobi constants in the Sun-Earth CR3BP. The solid regions shown are forbidden and thus motion is not allowed into them at their associated Jacobi constant. All the empty space shown is the region of allowed motion. Thus, we see that the higher Jacobi constant (lower energy) in orange only has a Neck Region near Earth, while the Forbidden Region with lower Jacobi constant (higher energy) in purple allows for motion in the entire plane z = 0. The low Jacobi constant surface is only shown for y 0 in order for the high Jacobi constant surface to be visible. (a) shows the whole surfaces while the Neck Region near Earth (black box) is shown in an expanded scale in (b). 13 of 268 2.2. BICIRCULAR PROBLEM 2.2 Bicircular Problem It is common to have a planetary system consisting of a planet and moon which in turn orbits the Sun. This instance of the 4-Body Problem is called the bicircular problem since it assumes the two smaller bodies orbit their common barycenter in circular orbits, while that barycenter and the Sun orbit the combined barycenter of all bodies in circular orbits. The motion of a particle is sometimes dominated by two of the three large bodies and can then be approximated well by a CR3BP centered on those bodies. When the effect of the 3rd major body becomes difficult to ignore in order to faithfully represent the dynamics of the system a bicircular model is useful. While it does not have the convenient property of being autonomous or having a constant integral of motion, it is still useful as a compromise between the real system and the CR3BP. We remind you that an autonomous model is not an explicit function of time, while this model has an explicit time dependence. It is less accurate than the real system, but much simpler to analyze, understand, and simulate. Since it is essentially a perturbed CR3BP, many of the tools for analysis can still be applied while gaining accuracy in the model. Note that there are two reasonable choices for rotating reference frames at this point. One frame fixes the position of the Sun and planet-moon barycenter and allows the planet and moon to rotate as shown in Fig. 2.7a. Another frame fixes the position of the planet and moon and allows the Sun to rotate around the planet moon barycenter as shown in Fig. 2.7b. The choice of system is often a result of the desired focus of the study. If we focus on the Sun-Planet-Moon model, the feature that makes the problem inherently non-autonomous is the moving moon. Therefore we define the lunar phase angle as shown in Fig. 2.8. This figure also shows inertial frame representation of the BCP. A known trajectory in the Sun-Planet CR3BP model can be easily expressed in the Sun-Planet-Moon BCP frame, since the units and coordinate system are the same as we have defined them. This is also true when expressing a Planet-Moon CR3BP frame trajectory in the Planet-Moon-Sun frame. The second order differential equations used to model this system are shown in Eqs. 2.5–2.8: 14 of 268 2.2. BICIRCULAR PROBLEM (a) (b) Figure 2.7: The bicircular problem as shown from two perspectives. Black dots indicate the three major bodies. (a) Sun-Planet-Moon rotating frame and normalized units. Blue shows the motion of the Moon around the Planet-Moon barycenter. Dashed gray shows the orbit of the Planet-Moon barycenter around the P-M-S barycenter if it were moving in this frame. (b) Planet-Moon-Sun rotating frame and normalized units. Gray shows the motion of the Sun around the Planet-Moon barycenter. Dashed blue shows the orbit of the Moon around the Planet-Moon barycenter if it were moving in this frame. x = 2 _ y +x 1 1 r 3 1 (x + 1 ) 1 (1 2 ) r 3 2 x 2 1 2 r 3 3 x 3 y =2 _ x +y 1 1 r 3 1 y 1 (1 2 ) r 3 2 y 2 1 2 r 3 3 y 3 z = 1 1 r 3 1 z 1 (1 2 ) r 3 2 z 1 2 r 3 3 z (2.5) Here, the distances to each primary (r 1 ;r 2 ;r 3 ) are computed as 15 of 268 2.2. BICIRCULAR PROBLEM ~ r 1 = 2 6 6 6 4 x + 1 y z 3 7 7 7 5 ;r 1 =j~ r 1 j ~ r 2 = 2 6 6 6 4 x (1 1 ) (1 2 )l 2 cos(n 2 t + 0 ) y (1 2 )l 2 sin(n 2 t + 0 ) z 3 7 7 7 5 ;r 2 =j~ r 2 j ~ r 3 = 2 6 6 6 4 x (1 1 ) + 2 l 2 cos(n 2 t + 0 ) y + 2 l 2 sin(n 2 t + 0 ) z 3 7 7 7 5 ;r 3 =j~ r 3 j (2.6) With the position of the moon represented by the time-dependent lunar phase angle(t), (t) =n 2 t + 0 (2.7) Here, 0 is the initial phase angle of the Moon att = 0. In this system 1 , 2 ,l 2 , andn 2 are new derived parameters that are computed from the phy- sical and orbital parameters of the three major bodies. The semimajor axis of the relative motion betweenm 2 andm 3 in non-dimensional units isl 2 . n 2 is their mean motion in non-dimensional units, corrected for the frame’s own rotation, computed as 1 = m 2 +m 3 m 1 +m 2 +m 3 2 = m 3 m 2 +m 3 n 2 = T 12 T 23 1 (2.8) 16 of 268 2.3. ELLIPTIC RESTRICTED 3-BODY PROBLEM Here, T 12 andT 23 are the periods of the relative motion form 1 -m 2 andm 2 -m 3 , respectively. Note that 1 is equivalent to in a 3-body problem where bodiesm 2 andm 3 are combined as a point mass. Also, 2 is equivalent to for a 3-body problem consisting of onlym 2 andm 3 . For more details on this system, see Cronin et al.[22]. Note that for the model to accurately simulate a real system, the initial lunar phase angle must be known. 1 2 3 (a) (b) Figure 2.8: Diagram of the Sun-Planet-Moon Bicircular Problem. (a) Shows the inertial frame with X and Y representing inertial coordinates and x and y representing rotating frame coordinates. The barycenter of m 2 and m 3 acts as a point mass and orbits the barycenter of all three masses in a circular orbit. m 2 and m 3 then orbit their common barycenter in circular orbits. (b) Shows the rotating frame coordinates, where m 1 and the m 2 -m 3 barycenter is stationary. m 2 and m 3 orbit each other. The lunar phase angle is labeled as, and is shown in Eq. 2.7. 2.3 Elliptic Restricted 3-Body Problem Another improvement to the CR3BP model used to more accurately model the dynamics of a real system is to allow elliptic motion of the primary bodies. We use the model with a rotating, pulsating coordinate system. This allows the Primaries to be fixed in the frame but adds fictional forces to 17 of 268 2.3. ELLIPTIC RESTRICTED 3-BODY PROBLEM the equations of motion due to the non-uniform rotation of the coordinate frame. Figure 2.9 shows how this coordinate system is determined from the inertial motion of the Primaries as well as the definition of the true anomaly used. The resulting set of second order differential equation we use to model this system is shown in Eq. 2.9. For more details on this dynamical system, see Szebehely and Giacaglia[85] and Szebehely[84]. x 00 = 2y 0 + (1 +e cosf) 1 [x 1 r 3 1 (x +) r 3 2 (x 1 +)] y 00 =2x 0 + (1 +e cosf) 1 [y 1 r 3 1 y r 3 2 y] z 00 =z + (1 +e cosf) 1 [z 1 r 3 1 z r 3 2 z] r 1 = p (x +) 2 +y 2 +z 2 r 2 = p (x 1 +) 2 +y 2 +z 2 = m 2 m 1 +m 2 (2.9) Here, the mass ratio is the same as that for the CR3BP. Note that the time dependence of these equations arises due to the Primaries’ true anomaly,f. We propagate this system usingf as the independent variable and use the resulting true anomaly to compute the associated time. The primes in Eq. 2.9 thus indicate derivatives with respect to true anomaly. This means that the correct initial true anomaly for the Primaries must be chosen for the model to be accurate. 18 of 268 2.4. PERIODIC ORBITS 1 2 Figure 2.9: Diagram of the Elliptic Restricted 3-Body Problem in the inertial frame. X and Y represent the inertial frame whilex andy are the rotating frame pulsating coordinates. The distance betweenm 1 andm 2 is normalized to 1 such that those bodies remain fixed in the rotating frame as seen in Fig. 2.1. The true anomaly,f, of the Primaries’ motion is measured from periapsis as shown above. 2.4 Periodic Orbits Ballistic trajectories in the unperturbed 2-Body Problem are well known and completely explored. All trajectories are conic sections, with the only closed, periodic orbits being ellipses with the cen- tral body at one focus. While this makes designing missions and predicting the paths of natural objects a less complex process, since the search space contains less possibilities, it also limits the options. Finding a ballistic path between two points in space and time in this system has become trivial by solving Lambert’s Problem. Various algorithms that solve Lambert’s Problem have been developed over the years, and are generally referred to as “Lambert solvers”. For more details on Lambert’s Problem see Bate, Mueller and White[11], Battin[12], Lancaster and Blanchard[56], and Gooding[37]. In more complex models such as the CR3BP, we may also find periodic solu- tions which can yield more desirable conditions for mission design. In fact, the CR3BP has been shown to have a plethora of orbit families with very different characteristics, see Doedel et al.[25]. These orbits cannot be computed in the 2-Body model. Due to the chaotic nature of the model, computing these orbits can be difficult and many of the orbits are unstable, as opposed to the 2- Body orbits, which are all completely stable if unperturbed. However, for many of these unstable orbits, minimal fuel is necessary for stationkeeping and their instability is actually beneficial when 19 of 268 2.4. PERIODIC ORBITS entering and exiting the orbit. Moreover, no general solution exists for finding ballistic trajectories between two points in space and time in the CR3BP. To provide some examples of periodic orbits in the CR3BP that will be discussed, we show some examples in Fig. 2.10. For Fig. 2.10(a)-(c), multiple orbits belonging to the same family are shown. Orbits that belong to the same family exhibit similar behavior, and the initial conditions of the entire family must form a continuous path in state space. Imagine that by changing the initial condition of one orbit infinitesimally, a new periodic orbit exists that is very similar to the first. This process can be repeated up to some limit, thus producing a continuous path of initial conditions in state space that represent all the orbits of that family. For example, Fig. 2.10(a) is displaying a discrete set of orbits from the planar Lyapunov family of orbits aroundL 1 in the Sun-Earth CR3BP. The orbits all lie in thexy-plane and loop aroundL 1 . The smallest orbits look like ellipses, but get more asymmetric as they grow in size. As the orbit size shrinks, the family degenerates into the edge case of the stationary Lagrange point itself. In the case of the planar Lyapunov orbits, the Jacobi constant shrinks monotonically with increasing orbit size. However, this is not the case for all orbit families, and the halo orbit family is one such exception. Figure. 2.10(b) shows the vertical Lyapunov family of orbits around Jupiter-Europa L 2 . These orbits typically move far out of the xy-plane and cross that plane near the Lagrange point. They too have the property that the smallest orbit degenerates to the Lagrange point itself. One orbit family that has seen much use for real missions is the halo orbit family, shown in Fig. 2.10(c). This figure only shows the northern family of halo orbits, and a symmetric southern family exists. Orbits in this southern halo family would be inclined to the ecliptic plane such that they would lie below the ecliptic plane at their outer edges. In contrast to the two Lyapunov families, this family does not degenerate to the Lagrange point, but instead has a minimum size at that end of the family. See Breakwell & Brown[17], or Howell[46] for details about halo orbits. Last, we show a single example of a Lissajous orbit in Fig. 2.10(d). Lissajous orbits are unique in that they are not perfectly periodic, but rather form a dense set of trajectories that produce a cylinder-like shape around the Lagrange point. As indicated by the red 20 of 268 2.4. PERIODIC ORBITS single revolution in the figure, each revolution misses its own initial state by some amount. This process repeats for each revolution until the trajectory has traced the path of a cylinder shape. One advantage of these orbits is that the amplitude of oscillations in they andz-directions can be chosen independently, as opposed to the halo orbits, where they are dependent parameters. This is in fact part of the definition of Lissajous orbits, since they are formed when there is an irrational relationship between two frequencies. If one linearizes the equations of motion for the CR3BP around the collinear Lagrange points, one finds that there are two uncoupled frequencies. One is associated with the in-plane (xy-plane) motion and one with the out-of-plane (z-axis) motion. When these two frequencies have an irrational ratio, the trajectory in space never perfecly repeats and is thus quasi-periodic. The motion of such orbits when viewed along thex-axis resembles a Lissajous curve, which gives the orbit type its name. The halo orbit is in fact the special case of this type of orbit when the ratio between the two frequencies is 1, resulting in perfect synchronization of the in-plane and out-of-plane motion. For more details on the topic of Lissajous orbits, refer to Farquhar and Kamel[30], or Richardson[76]. Another good resource that summarizes many orbit families in the CR3BP is Doedel et al.[26]. 21 of 268 2.4. PERIODIC ORBITS 0.95 1 1.05 X [NON] -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Y [NON] L 2 L 1 EARTH (a) 0.96 L 1 0.98 X [NON] 1 EUROPA -0.02 L 2 1.02 -0.02 0 Z [NON] Y [NON] 0.02 0 1.04 0.02 (b) L 1 -5 0.99 0 EARTH 5 Z [NON] 10 -3 -5 10 X [NON] Y [NON] 10 -3 1 0 L 2 5 1.01 (c) EARTH 1 0.995 X [NON] L 1 0.99 -2 0 2 10 -3 Z [NON] 2 0 Y [NON] 10 -3 -2 (d) Figure 2.10: Examples of periodic orbits in the CR3BP. (a) The family of planar Lyapunov orbits at L 1 of the Sun-Earth system. (a) The family of vertical Lyapunov orbits at L 2 of the Jupiter- Europa system. An observer on Europa would see these orbits as figure-eight shapes with the middle crossing point at the Lagrange point. (a) The family of northern halo orbits at L 2 of the Sun-Earth system. An observer on Earth would see these orbits as loops around the Lagrange point. (a) A Lissajous orbit atL 1 of the Sun-Earth system. This orbit is quasi-periodic, and thus does not perfectly repeat but instead draws a cylinder-like shape in space. One revolution of the orbit is highlighted in red. 22 of 268 2.5. INV ARIANT MANIFOLDS 2.5 Invariant Manifolds As mentioned earlier, an unstable orbit has the attractive characteristic that inserting into the orbit or departing from the orbit can require very little fuel. This is accomplished by use of invariant manifolds. In a chaotic dynamical system it is possible to find invariant sets of solutions called invariant manifolds which are useful in analyzing the flow of the system, see Hirsh, Pugh and Shub[45], or Roberts[77]. In the case of the CR3BP, the stable and unstable manifolds of an unsta- ble periodic orbit show how trajectories naturally approach and depart that orbit respectively. Each manifold is an infinite set of trajectories that connect to the orbit via asymptotic paths. This set of trajectories lies on an energy surface in phase space. Figure 2.11 shows the invariant manifolds of planar Lyapunov orbits around L 1 and L 2 with the same Jacobi constant. The green curves forming tubes are on the stable manifolds which approach the Lyapunov orbits. The red curves forming tubes are on the unstable manifolds which leave the Lyapunov orbit. It is clear that the manifolds reach far from the orbits and thus indicates that entering the orbit from far away beco- mes a simple matter of placing a spacecraft on the desired manifold. Similarly, leaving an orbit would allow one to travel far without any fuel expenditure. To achieve a purely ballistic transfer, approaching or departing an orbit would theoretically take an infinite amount of time due to the asymptotic nature of the manifolds. However, even extremely small perturbations can make trans- fers with a reasonable time of flight possible. These perturbations can of course be intentionally produced through the use of propellant and thrusters or other in-space propulsion methods. 23 of 268 2.6. POINCAR ´ E MAPS Figure 2.11: The invariant manifolds of the L 1 and L 2 Lyapunov orbits create the low energy routes for asteroids to approach Earth, impact the Earth or get temporarily captured by the Earth and then escape. The green trajectories (stable manifold) approach the Lyapunov orbits, while the red trajectories (unstable manifold) leave the Lyapunov orbits, like the on- and off-ramps of a highway. The tubes can intersect to form a unique set of orbits called the heteroclinic-homoclinic chain in Fig. 2.16 2.6 Poincar´ e Maps A useful tool in computing periodic orbits and analyzing global behavior of dynamical systems is the Poincar´ e map, invented by Henri Poincar´ e. It is a clever method of reducing the dimension of a dynamical system. Instead of computing trajectories and viewing them in position space, only the intersections of a trajectory with a plane are recorded. Figure 2.12(a) illustrates this idea. Fixing this section reduces the problem’s dimension by one. For a dynamical system with chaos, trajectories exist that are unstable and cross the intersection at seemingly random locations each time it returns, which is shown in Fig. 2.12(b). However, a periodic orbit will intersect in the same place every time, thus resulting in a single point on the Poincar´ e map, as shown in Fig. 2.12(c). If quasi-periodic motion exists in the dynamical system, they are constrained to travel on the surface of invariant tori. The tori therefore generate loops when intersecting the Poincar´ e section. A truly quasi-periodic orbit will produce a dense covering of the torus and would show up as a continuous loop on the section. However, for finite time, the section will show a series of points that trace 24 of 268 2.6. POINCAR ´ E MAPS the loop, as seen in Fig. 2.12(d). To use this concept, we typically propagate a large number of initial conditions in the dynamical system and record the intersections with a chosen section. For a sufficient number of initial particles and iterations of this map, a global structure of the dynamical system can be analyzed. By choosing initial states that all have the same energy (Jacobi constant in the case of the CR3BP), we have reduced the dimension of the problem again. Thus, we have reduced the dimension of the problem by a total of two. We can search for the loop structures and isolated dots in a Poincar´ e map to locate stable periodic orbits and quasi-periodic orbits, and locate regions of instability. We refer the reader to Wiggins[88] and Parker & Chua[73] for more details on Poincar´ e maps and algorithms used to compute them. As an example, we show a Poincar´ e map for the Jupiter-Europa CR3BP in Fig. 2.13. In this image, we select the negative x-axis as the Poincar´ e section (y = 0), and plot thex and _ x components. Since all the particles have the same Jacobi constant, knowing the (x; _ x)-coordinate allows for computation of the _ y-component. (a) (b) (c) (d) Figure 2.12: Poincar´ e map concept diagram. (a) The intersections of a trajectory (left) with a chosen section are recorded and turned into a map (right). (b) A trajectory will in general produce intersections that may or may not have an obvious pattern. (c) A periodic orbit will produce a single dot, as the orbit crosses the section at the same location repeatedly. (d) A quasi-periodic orbit will produce a loop on the section over time, as it is restricted to traveling on a torus, and its position on the torus at the intersection changes each time. 25 of 268 2.7. DELAUNAY V ARIABLES Figure 2.13: Example Poincar´ e map in the Jupiter-Europa CR3BP. Coordinates are x and _ x in nondimensional units. The holes in the otherwise seemingly random sea of points indicates stable, quasi-periodic motion. The sea of dots represents chaotic, unstable behavior. 2.7 Delaunay Variables It is sometimes useful to study orbits or a Poincar´ e map in alternative coordinate systems to the typical Cartesian state space. One such coordinate system we will use extensively in Chapter 6 is the Delaunay variables. This is an overview of the Delaunay variables from a set of planar, rotating frame coordinates in the CR3BP. The Delaunay variables in the 2D problem are expressed in classical orbital elements in Eq. 2.10. l =M g =! L = p a G = p a(1e 2 ) (2.10) Here, M is the mean anomaly, ! is the argument of periapsis, a is the semimajor axis, e is the eccentricity, and is the gravitational parameter of the central body. Note thatG represents the angular momentum of the orbit. The map between orbital elements and these variables is thus 26 of 268 2.7. DELAUNAY V ARIABLES relatively straightforward. A slight modification to the variableg is often used when dealing with the CR3BP rotating frame as explained by Koon et al.[55], and is shown in Eq. 2.11. If we use the non-dimensional units for the CR3BP, then = 1 and the equation simplifies. The coordinate g is thus the angular position of the periapsis relative to the rotating frame. This modification becomes useful when evaluating the stability of resonances by indicating if the periapsis has an oscillatory or secular nature in the rotating frame. A periapsis that only oscillates relative to the position of the two attracting bodies in the CR3BP can be considered stable. Figure 2.14 gives a geometric explanation for theL and g variables. l =M ggt =!t L = p a G = p a(1e 2 ) (2.11) The phase space commonly used for propagation in the CR3BP is the Cartesian position and velocity phase space. Using a Poincar´ e map at fixed energy reduces the total degrees of freedom by two. Thus, we often study a 2-dimensional representation of a Poincar´ e map in the planar CR3BP, but the choice of variables is important. We often use the (L; g)-space for Poincar´ e maps. This space is useful for analyzing resonant motion for several reasons as pointed out by Anderson[7]. First, the L-coordinate is directly related to the approximate resonance because it represents the square root of the osculating semimajor axis. If the Poincar´ e map is evaluated far from either primary, this Keplerian approximation of the period is more accurate. Second, the (L; g)-space is a symplectic set of coordinates, and thus also area preserving. This means that an area of initial conditions with a fixed boundary will retain the area inside the boundary when propagated by the map even if it changes shape. Third, the topology of resonant motion becomes much more clear in these coordinates, due in part to the resonances being partitioned along the L-axis as well as the location of the periapsis (defined by g) being key in determining stability. For a comparison 27 of 268 2.7. DELAUNAY V ARIABLES APOAPSIS PERIAPSIS ω t S/C L 2 = Earth Figure 2.14: Delaunay variable diagram. The Sun-Earth system is chosen as an illustrative exam- ple. L maps easily to the semimajor axis shown asa =L 2 . The rot axes rotate relative to the iner axes with angular rate 1 radian/time unit, and thus the angle between them becomest as shown. The inertial frame argument of periapsis is indicated by the angle!. Image modified from Koon et al.[55], with permission. between the two coordinate systems, refer to Fig. 2.15. Note the distorted shapes of the stable islands in the (x;dx=dt)-coordinates and the more obvious structure of those same islands in the (L; g)-coordinates. For clarity, we highlight a set of intersections produced by a single particle in both coordinates. The nature of the periodic orbit is more clear from the second set of coordinates. We see that theL, and therefore the semimajor axis, oscillates closely around a single value. This value is used to approximate its resonance. Moreover, the g-coordinate is also oscillating around g = , indicating that the periapsis is librating around the negativex-axis in the rotating frame. To be clear, it’s actually the osculating periapsis at the intersections of thex-axis that is librating, since the osculating ellipse will be continuously rotating in this frame. 28 of 268 2.8. RESONANCES (a) (b) Figure 2.15: Comparison of Poincar´ e map coordinates. (a) Poincar´ e map in the Jupiter-Europa CR3BP for C 0 = 3:0039, (x;dx=dt)-coordinates. (b) The same map in (L; g)-coordinates. A single particle’s intersections have been highlighted in red in both maps to show the clear difference in shape of stable solutions. The approximate resonance of this set is indicated by a green label 2.8 Resonances Any periodic orbit in the rotating frame will be in some state of resonance with the orbital motion of the Primaries. Since the orbit repeats itself perfectly in the rotating frame, the relative position of the Primaries and the 3rd body repeat exactly with a fixed interval. The period of such an orbit 29 of 268 2.9. HOMOCLINIC AND HETEROCLINIC ORBITS when compared to the period of the Primaries (which will be 2 nondimensional time units) will determine the resonance ratio. Resonance is a common phenomenon in nature. In our case, we are working with orbits in mean motion resonance, that is, resonance with respect to the periods of two orbiting bodies. By ap:q resonant orbit in a Primary-Secondary System, we mean an object movingp times around the Primary while the Secondary movesq times around the Primary where p andq are relatively prime positive integers, see Murray and Dermott[68] for details. IfP o is the period of the object andP is the period of the Primaries’ motion, the resonant condition is that pP o qP =) p q P P o : (2.12) The quantityjpqj is called the order of the resonance. More precisely, for the real number = P=P o and an error bound of, we definep=q as the best Diophantine approximation of to within error, that is, for any other rational numberp 0 =q 0 whereq 0 <q, we have p q << p 0 q 0 : (2.13) See Khinchin[51] for details on the best Diophantine approximation. The rational numberp=q is found through a continued fraction process such as implemented in the MATLAB “rat.m” function. 2.9 Homoclinic and Heteroclinic Orbits When a stable and an unstable manifold intersect in phase space, a free transfer between the orbits at the origin of those manifolds is possible. In the case where the origin of both manifolds is the same orbit, we get a homoclinic connection or homoclinic orbit. This type of connection will asymptotically depart a periodic orbit and later return asymptotically to the same orbit it began on. When the manifolds belong to two different periodic orbits, we get a heteroclinic connection. For this type, the connection will asymptotically depart the initial orbit and eventually approach the 30 of 268 2.10. HETEROCLINIC-HOMOCLINIC CHAINS second orbit asymptotically. These transfers are powerful tools in mission design as well as solar system dynamics, since they allow for free transfers between different target orbits. 2.10 Heteroclinic-Homoclinic Chains The tubes intersect one another to produce a chain of special orbits called the heteroclinic- homoclinic chain. This chain provides a template for the motion of comets and asteroids at this energy level that are temporarily captured by Jupiter. Furthermore, this chain satisfies the Conley- Moser Condition for the existence of chaos, a theorem stated in Moser[67]. Hence any object fol- lowing this chain is in a chaotic orbit by this theorem. Figures 2.16a and 2.16b show this chain for Jupiter at the Jacobi constant of the Comet Oterma, a well known member of the short period Jupi- ter Family of Comets. Figure 2.16c shows Oterma’s orbit (red curve) closely following the one of Jupiter’s heteroclinic-homoclinic chains (black curve). See Koon, Lo, Marsden, and Ross[54, 53] for details on the computation and dynamics of the chain. Chains that involve resonant orbits can also result in a phenomenon called resonance hopping. If a chain contains two different resonant orbits, then an object can “hop” back and forth between the two resonant orbits as it travels along the chain. This can be extended to include an arbitrarily large number of different resonant orbits inside a chain, which results in resonance hopping between many resonances. For more details on resonance hopping, see Belbruno and Marsden[13]. The same dynamics is present at all other planets and moons in the solar system. Figure 2.17 shows a chain for the Sun-Earth system at the Jacobi constant of the Genesis trajectory. The Conley-Moser Condition is satisfied here also which shows that the Genesis trajectory is chaotic. This explains why it required so little V , hence fuel, for its control throughout the mission. In the Jupiter problem, the homoclinic orbits (which connect the L 1 Lyapunov orbit to itself and similarly atL 2 ) are also shadowing the resonant orbits. For Jupiter, the Interior Resonance is 3:2 (the comet goes around the Sun three times for every two times Jupiter goes around the Sun), the Exterior Resonance is 2:3. For the Earth, typical of smaller planets, the resonances have many 31 of 268 2.10. HETEROCLINIC-HOMOCLINIC CHAINS (a) (b) (c) Figure 2.16: (a) The heteroclinic-homoclinic chain for Jupiter. The homoclinic orbit (blue) in the Interior Region is following closely the 3:2 resonance with Jupiter. The homoclinic orbit (gold) in the Exterior Region is shadowing the 2:3 resonance with Jupiter. (b) The two Lyapunov orbits (black) at L 1 and L 2 are the generators of this chain. The magenta trajectory connecting the Lyapunov orbits is the heteroclinic orbit. (c) The orbit of Oterma (red) from 1910 to 1980 closely follows the chain (black) showing that it is a chaotic orbit by the Conley-Moser Condition. Figure 2.17: The Genesis homoclinic-heteroclinic chain. The figure on the right shows the Genesis trajectory in black and the heteroclinic orbit connecting the two Lyapunov orbit atL 1 andL 2 in magenta. The homoclinic orbit for theL 1 Lyapunov orbit is blue, forL 2 , it’s gold. The figure on the left shows how the homoclinic orbits have many periods around the Sun before returning to the L 1 Lyapunov orbit. They are in fact shadowing resonant orbits. more revolutions. The unstable resonant orbits are the means by which asteroids and comets can quickly transport across the solar system via Mean Motion Resonances which we call Resonant Transitions. In fact, this is precisely what is going on for resonant gravity assist maneuvers that has enabled missions from V oyager to Galileo and Cassini. See Anderson[7] and Lo et al.[58] for details. Secular resonances also play a crucial role in the dynamics of transport in the solar system, 32 of 268 2.10. HETEROCLINIC-HOMOCLINIC CHAINS but these require very long time span of many millions of years. In Chapter 4, we will show the sequence of resonant and libration orbits used by Asteroid 2006 RH 120 to approach the Earth for temporary capture, and then escape the Earth through resonance transitions. For the comet Oterma and the Genesis spacecraft, the chains have been computed in the planar CR3BP model, see Koon et al.[54]. Conley[21] showed that for energies slightly above that of theL 1 andL 2 Lagrange points, the Lyapunov orbits control the dynamics around the planet. Any object at these energies can only approach and escape the planet by going through the invariant manifolds of the Lyapunov orbits. This is because for the planar CR3BP the invariant manifolds are 2D tubes in 3D energy surfaces. Since there is a well-defined inside and outside topologically for a 2D tube in 3D space, this constrains the motion of comets and asteroids in the planar CR3BP as stated above. However, actual comets and asteroids move in 3D trajectories. Howell, Marchand, and Lo[47] showed that several of the comets in the Jupiter Family (Gehrels 3, Helin-Roman-Crockett) closely follow the invariant manifolds of Jupiter halo orbits during their Temporary Capture Phase. This suggests that libration orbits around theL 1 andL 2 Lagrange points are gateways for the Tempo- rary Capture Phenomenon. However, so far, this conjecture has not been proved except for a few specific comets. In fact, even for the 2D case, the existence theorem of Conley does not specify the range of Jacobi constant for which it is true. For very high energies such as that of the asteroid Apophis, halo orbits no longer exist. It is not known what controls the dynamics of such aste- roids and comets that approach the Earth or Jupiter. Recently, based on numerical study, Ren and Shan[74] conjectured that the dynamics of objects approaching and departing a planet is controlled by the invariant manifolds of the vertical Lyapunov orbit and the planar Lyapunov orbit. This is an exciting result which requires closer study and verification. 33 of 268 2.11. TEMPORARY CAPTURE 2.11 Temporary Capture It has been long observed that comets and asteroids can be temporarily captured by planets from time to time. A tremendous amount of work has been done to study this phenomenon with a vast literature. The most famous group of temporarily captured objects are the Jupiter Family of Comets, a few of which were mentioned earlier. The moons of Mars may be asteroids captured through a similar process. The transport mechanism is very complex. We know resonances play a significant role. The resonance transitions eventually bring the small body close to theL 1 orL 2 Lagrange points of the planet. At these locations, the invariant manifolds of libration orbits such as halo orbits or Lissajous orbits are able to attract the small body and bring it into the region around the planet where it is temporarily captured. It has been shown how this occurs for Jupiter comets, see Koon et al.[54] and Howell, Lo and Marchand[47]. The same process works for other planets as well. The significance of this approach to explain the Temporary Capture Phenomenon is that it helps us to visualize the capture process and this could help to predict the location of the NEOs before and after temporary capture. Moreover, this could help us devise methods to prevent future impacts by dangerous NEOs, and it could help us design trajectories to visit the Minimoons or effect methods to change their energies and capture them around the Moon or the Earth for many applications. 34 of 268 CHAPTER 3: STATEMENT OF THE PROBLEM 3.1 The Research Problem We are Addressing We are addressing the problem of the dynamics of the temporary capture and impact/landing of objects around planets and satellites. The objects can be a spacecraft or a small body such as an asteroid or a comet. The questions we want to answer are: Do manifolds of periodic orbits in CR3BP control flow of trajectories between major regions of space?; How do manifolds of periodic orbits in CR3BP control the temporary capture, transit and impact dynamics?; Which orbit families control these dynamics? 3.2 Why It Is a Research Problem The dynamics of the temporary capture of objects around planets and satellites in the solar system is of great interests not only to the dynamical astronomy community, but also to the space programs around the world. Despite the extensive work that has been done on these problems, a great deal is still unknown about the nonlinear dynamics that control these delicate processes. This is due to the dynamics of the capture and impact/landing of objects being chaotic in nature due to the effects of the 3-Body Problem. This problem is of great interest to many communities in the world. First of all, these dynamics are what control how Near Earth Asteroids (NEA) can approach the Earth and impact the Earth. Comet Shoemaker-Levy-9 is a prime example of how a small body would impact a planet or a Moon using these chaotic dynamics. The same dynamics can also be used to design low energy captures of spacecraft around planets and satellites with a tremendous fuel savings. For example, these dynamics can be used to capture an NEA around the Earth or the Moon from a near future Asteroid Retrieval Mission. They are also 35 of 268 3.3. IMPACT ON ASTRODYNAMICS & DYNAMICAL ASTRONOMY being used currently for the design of the Europa Lander mission to land a spacecraft on Europa using low energy trajectories. We would like to answer the above questions because asteroids and comets can use these low energy transits to move through the solar system. These transits can in turn lead to close approa- ches, impacts or temporary captures. These are scenarios we are interested in for several reasons. Close approaches may require deflection for Potentially Hazardous Asteroids. Close approaches may be interesting for rendezvous if the target is scientifically interesting. Temporary capture is interesting for two reasons. First, it is interesting for rendezvous with a scientific target. Second, it may be desirable to capture for asteroid mining or scientific study. Impact is clearly interesting if the target is large enough to be potentially hazardous, since then a deflection will be necessary. Asteroids on impact trajectories can also be interesting from a mining or science perspective, and we may then be able to deflect and capture the target. It is worthy to note that the orbits around the Sun-Earth Lagrange points have unstable manifolds that are close enough to the Earth for impact to occur, so studying this problem can be very important. 3.3 Impact on Astrodynamics & Dynamical Astronomy If we solve the problem we have outlined, it would aid the field of dynamical astronomy in explai- ning the mechanism of the capture of small bodies by planets. This would help to determine the origin of some irregular moons in the solar system whose origins cannot be explained through for- mation in an accretion disk around the planet. Moreover, explaining the mechanics of resonance transition through dynamical transport is beneficial for explaining how material moves through the solar system naturally. For the field of astrodynamics, the benefits of having this problem solved will open new methods of trajectory design that exploits the known network of invariant manifolds that exists everywhere in the solar system. This is a mission-enabling technology because it can greatly reduce the fuel and cost for transporting materials in space. The propulsive capability of 36 of 268 3.4. IMPACT ON METHODOLOGICAL APPROACHES any mission is limited by current hardware, but being able to accomplish more complex trajecto- ries and farther destinations can enable new missions, or extend the scientific payload capabilities of currently feasible missions. An example of this is the work we have been doing on the landing trajectory analysis for the Europa Lander at the Jet Propulsion Laboratory. 3.4 Impact on Methodological Approaches Some of the tools we need for this project can have significant impacts by themselves. For example, the methods used to convert trajectories between various dynamical models would be very useful for both mission design and for analyses of comets and asteroids. The initial mission design can be done in the simpler models like CR3BP or BCP. The resulting trajectories can then be transformed into the JPL ephemeris model for actual operations. Known ephemerides of comets and asteroids can now be analyzed in the more structured CR3BP and BCP models to better understand the causes of their behavior. Another significant impact of my work is the development of an orbit catalog for different 3- body systems within the solar system such as the Sun-Earth, Earth-Moon or Jupiter Europa CR3BP. This combined with a tool that can patch together trajectories between these orbits allows for an advanced method for missions throughout the solar system. In addition, we are developing methods for the global study of impacts in different planetary systems. The current use case for this method is the Europa Lander mission. This method allows mission designers to evaluate the accessibility of different landing sites for a low energy approach trajectory. 3.5 Impact on Society The work of my proposed research will help explain how asteroids can reach and impact the Earth. This knowledge will help astronomers know where to look for these dangerous asteroids. Once 37 of 268 3.6. HYPOTHESES detected, using the chaotic dynamics, it requires very little energy to deflect the asteroid either for escape or capture by the Moon or the Earth. This would work for asteroids in the energy range of the Minimoons like Asteroid 2006 RH 120 . This dynamics also provides a blueprint for the future development of space around the Earth-Moon system for colonization and industrialization. The transport of cargo between the various locations likeL 1 ,L 2 of the Earth and of the Moon are nearly free using these low energy trajectories. This technology will be critical for the development of space infrastructure. 3.6 Hypotheses We list the hypotheses that we aim to prove with the current work. 3.6.1 Hypothesis 1: Manifolds Control Transit Dynamics Our first hypothesis states that manifolds of periodic orbits in the Planar CR3BP control transit, temporary capture and impact of bodies near the region around the Secondary Body. A transit completely through this region requires that a trajectory enters theL 1 gateway and then departs the region through theL 2 gateway, or moves in the opposite direction fromL 2 toL 1 . The theory pro- posed by Conley[21] suggests that certain regions in physical space, as separated by the boundaries of the invariant manifolds of planar Lyapunov orbits can be said to be guaranteed to pass through the gateway around the orbit, while other regions are guaranteed to not pass through. Within the manifold in physical space, a range of velocity directions must be satisfied for a trajectory to tran- sit the gateway. This transit theory regards the Lagrange point gateways only and supposedly is limited to be valid for Jacobi constant sufficiently close to that of the Lagrange point. We have recently studied the energy range for which this remains true, see Swenson et al.[83]. We propose that this theory in combination with heteroclinic connections betweenL 1 andL 2 orbits can show 38 of 268 3.6. HYPOTHESES the space of allowable transit between interior and exterior regions. This becomes a study of dyn- amical transport. This hypothesis is testable through numerical integration of a large selection of representative trajectories. The regions of transit and non-transit as indicated by Conley will be represented in the pool of initial conditions. The path of the integrated trajectories, along with their source region will show if the hypothesis predicted the results accurately. We are aware that a test such as this one will not rigorously confirm the hypothesis, but will strongly support it. In this case, strong support is enough to show that the hypothesis statement is useful for practical applications. We also aim to show that this extends to the spatial case. This requires some evidence that the Lagrange point gateway transit theory by Conley has an extension into the spatial problem. There has been some work toward showing this already by Ren and Shan[74]. It is more difficult to prove Conley’s theorem in the spatial CR3BP due to the increased degrees of freedom. While the phase space increases from 4 dimensional to 6 dimensional, there is still only one integral of motion. This hypothesis also requires an analogy to the heteroclinic connections betweenL 1 and L 2 orbits in the Spatial CR3BP. The existence of heteroclinic connections is still completely valid since it is a property within dynamical systems theory. As long as manifold intersections can be found, heteroclinic connections exist between spatial orbits as well as planar orbits. There are practical limitations to computing these connections however. There have been some attempts at overcoming these challenges, such as the one by Haapala and Howell[40]. We plan to investigate alternative approaches to solve this problem. We will numerically study this system as well, to show that certain orbit manifolds control flow to and from the region near the Secondary body. 3.6.2 Hypothesis 2: Dynamics Largely Persists in 4-Body We hypothesize that the structure present in the flow of the CR3BP remains mostly intact as seen in the Bicircular Problem. Because the Bicircular Problem is essentially a perturbed CR3BP, if the perturbation is small enough the natural flow that describes transit, temporary capture and impact is still mostly controlled by the 3-body effects. The dynamics of the CR3BP are already 39 of 268 3.6. HYPOTHESES chaotic, and so it is possible that small perturbations are enough to have a significant effect on the flow. However, therein lies the research. We propose that for many 4-body systems, the 3-body manifold control is sufficient to explain the overall behavior of objects. 3.6.3 Hypothesis 3: The Moon Significantly Affects NEO Impacts on Earth As a related theory to the previous hypothesis, this one concerns the rate of Earth impact. We propose that the Moon changes the dynamical environment near Earth enough to significantly change the rate of Earth impacts by NEOs. The previous hypothesis is concerned with the overall rate of transit through the Earth-Moon region and that it remains similar even when we add the perturbation of the Moon. In contrast, this hypothesis focuses specifically on Earth impact. We can make an educated guess that most NEOs approaching Earth will either transit between regions, reflect into the region the came from, or capture. Earth impacts will most likely constitute a minority of the cases. Therefore, we suggest that the presence of the Moon causes significant changes in this already small set of cases. We plan to test this by comparing the Sun-Earth CR3BP to the Sun-Earth-Moon BCP, where the first represents a fictitious case where the Moon doesn’t exist. 3.6.4 Hypothesis 4: Transit Leads to Resonance Transition This hypothesis is an extension of the first. The known existence of homoclinic connections sug- gests that libration point orbits are sources of resonant orbits and we will be exploiting this theory to find the resonant orbits and their compound orbits. If the orbit manifolds control transit between regions, and the orbit manifolds in their respective regions can link to resonant orbits, then we should be able to show that dynamical transport in the CR3BP leads to transitions between reso- nances. The transit necessitates a movement between interior and exterior regions, so the resonance 40 of 268 3.7. MATRIX OF PROOF cannot be the same before and after transit. The Jacobi constant of the libration orbit would deter- mine the accessible resonances, and we propose that for any Jacobi constant that allows transit, there exists a resonance for that transit to move onto. 3.6.5 Hypothesis 5: Stable CR3BP Resonances Persist in Resonant N-Body Systems Transit dynamics are inherently linked to resonant motion, since resonant motion can result in repeated planetary body encounters and therefore opportunities for transit. We now discuss an inversely related problem in mission design, namely stable resonances. This is an inverse problem, since it actively seeks out to avoid resonance transition and planetary encounters. While stable resonances found in the CR3BP are dynamically stable for infinite time, we propose that these resonances have practical stability in higher fidelity models. In particular, we suggest thatN-Body systems with N > 3 that are in mutual resonance result in finite time stability of orbits that are resonant with any choice of CR3BP system that is a subset of the larger system. For example, a stable resonant orbit in the Jupiter-Europa CR3BP remains stable for a long time even under the perturbation of the other Galilean moons. We define practical stability as remaining near the original orbit for a time useful for mission design considerations, such as a 10 year period. 3.7 Matrix of Proof We present a readers guide to the hypotheses and the sections of the document that supports them. This guide is presented in Table 3.1. 41 of 268 3.8. OUR PLAN ON ADDRESSING THE LIMITATIONS Hypothesis Section Summary Manifolds control transit dynamics 4.1 Asteroid 2006 RH 120 capture through L 2 halo orbit manifold (Section 4.1.5) and escape throughL 2 halo orbit mani- folds (Section 4.1.6) Dynamics largely persists in 4-Body 5.2 Transit between regions follows simi- lar trends in both CR3BP and BCP in a low energy regime, but have different absolute values. The Moon significantly affects NEO impacts on Earth 5.2 Depending on energy, Moon can decre- ase or increase impact rates signifi- cantly. Transit leads to resonance transition 4.1 Asteroid 2006 RH120 transitioned from a 29:27 resonance to a 20:21 reso- nance. 6.5 Compound resonant orbits exist with many resonances, e.g. 20:21. Com- pound orbits by definition connect peri- odic orbits together and can thus pro- duce transit. Stable CR3BP resonances persist in resonantN-Body systems 6.3 The resonances that remain stable in anN-Body system are those that have resonance with all or some of the N bodies. Table 3.1: Matrix of proof. The section(s) in which the hypothesis is proven is indicated. 3.8 Our Plan on Addressing the Limitations The hypotheses we stated are difficult to prove in a rigorous mathematical way. We are aware of this and have chosen to use numerical integration on a large scale to show correlation between hypothesis and actual solutions of the dynamical equations. The limitation of this approach comes at the cost of CPU time. To solve this, we will be integrating trajectories in parallel, for a significant decrease in necessary time. Most recently we have also implemented massively parallel propaga- tion using GPU hardware. For some cases, this can produce speedup over 1000 times faster as discussed in Appendix A. Additionally, we are choosing trajectories that are representative of the regions we wish to prove as transit or non-transit based on the hypothesis. This further limits the 42 of 268 3.9. CURRENT APPROACHES amount of necessary integration. Another limitation regards the computation of resonant orbits. One method of locating these orbits is through integration of large number of trajectories for each Jacobi constant, and analyzing well chosen Poincar´ e sections for islands of stability. However, since I know that the resonances I am searching for are ones that approach libration points, I can approximate the resonant orbit by a homoclinic connection of the libration orbit. To find these homoclinic connections, I only need to integrate the manifold of a single orbit at a given energy, instead of many randomly selected trajectories for a Poincar´ e Section. This reduces the time and effort necessary to generate resonant orbits. 3.9 Current Approaches The conventional approach for studying temporary capture and impact/landing on the bodies of the solar system typically uses 3 approaches. Approach 1 uses perturbation theory on the 2-Body Problem as a model. Approach 2 uses a Multibody Problem without using dynamical systems theory. Approach 3 uses the 3-Body Problem with dynamical systems theory. 3.9.1 Approach 1: Perturbed 2-Body Due to the well developed theory behind Keplerian motion, it is attractive to stick to analysis and design within that framework. When necessary, the system is simply perturbed and the approxi- mate effects of those perturbations are taken into account. For long term analysis, the perturbation effects are often averaged. One method of perturbation analysis in the 2-Body model that is indeed a 3-body analysis tool in disguise is the Tisserand Parameter. This quantity can be computed from orbital elements of natural objects and can help astronomers determine if newly discovered heli- ocentric objects may just be known objects that have had their orbit perturbed significantly by a planet. The Tisserand Parameter is in fact an approximation of the Jacobi constant, as expressed in orbital elements, see Murray and Dermott[68]. Since the Jacobi constant is an integral of motion 43 of 268 3.9. CURRENT APPROACHES in the CR3BP, showing that two observed objects have different Keplerian elements, but the same Tisserand Parameter can mean that the two objects are the same. 3.9.2 Approach 2: Multibody Without Dynamical Systems Theory The problem of explaining the effect that causes temporary capture has been studied in a multibody approach that disregards the structure known by dynamical systems theory. Large numbers of initial conditions are integrated in a multibody propagator. By analyzing the Keplerian orbital elements of the resulting trajectories, some intuition about what causes temporary capture was sought. Marchand[64] comments: “A widely accepted approach to study TSC [Temporary Satellite Capture] is to nume- rically integrate the equations of motion for theN-Body problem using a wide range of initial conditions obtained from the heliocentric two-body problem; then, a search ensues for instances when the Joviocentric energy becomes negative.” 3.9.3 Approach 3: 3-Body With Dynamical Systems Theory We propose to use Approach 3 because it provides the best model for understanding the chaotic dynamics that controls these phenomena. Work by Koon, Lo Marsden & Ross[54] showed that the invariant manifolds of the Lagrange pointsL 1 andL 2 seem to play a crucial role in the tem- porary capture and impacts of Jupiter comets. Lo’s work with Koon et al.[53, 54] provided the theory explaining this phenomenon using invariant manifolds of libration orbits around L 1 and L 2 . Lo’s work with Howell et al.[47] provided numerical simulations exploring this phenomenon. Marchand[64] also applied the 3-Body Problem and Dynamical Systems Theory to analyze the comets Oterma and Helin-Roman-Crockett in the Sun-Jupiter Spatial CR3BP. Marchand showed that the comets are approximately following the manifolds of 3-dimensional halo orbits in the Sun-Jupiter system. They showed that the Jupiter family of comets are controlled by the invariant manifolds for the temporary capture of the comets around Jupiter that can eventually lead to impact 44 of 268 3.10. LIMITATIONS OF CURRENT APPROACHES on Jupiter. This showed that invariant manifolds of orbits around the Lagrange pointsL 1 andL 2 play a critical role in the temporary capture and impact/landing problem. 3.10 Limitations of Current Approaches 3.10.1 2-Body Limitations The 2-Body Problem with perturbation cannot capture the chaotic dynamics in the 3-Body Pro- blem. The capture mechanism cannot be understood without using invariant manifold theory in the 3-Body Problem. Studying a perturbed 2-body model becomes challenging to analyze effecti- vely when the perturbation becomes large relative to the main forcing function. Moreover, it is hard to gain intuition about the dynamics when the orbital elements are varying quickly and signi- ficantly over the course of a single orbit. This effect is common while in the more chaotic regions of motion. In addition, designing trajectories for missions using conic sections does not allow the exploitation of chaotic effects and would even fail to find certain solutions, see Barden, Howell and Lo[10]. In fact, an example of a real mission design scenario shows how design in the CR3BP enabled missions that would have been impossible with a normal 2-body framework. The Gene- sis mission used a CR3BP design, and even had no deterministic maneuvers after launch at all. This trajectory would have been completely ballistic without orbit injection errors after launch and non-deterministic perturbations, see Koon et al.[52]. One comparison of design in the 2-body vs. CR3BP model has also been shown to prove significant improvements, see Lo and Chung[59]. Showing capture and transit in the CR3BP model by direct simulation proves existence, but does not yield intuition of the dynamical reason behind the effect. This intuition can be obtained by stu- dying manifold structures. In the real world, periodic orbits of the CR3BP become more complex quasiperiodic and almost periodic orbits. The invariant manifolds persists but becomes a much more complex structures that are difficult to compute. However, the manifolds in the CR3BP pro- vide excellent initial solutions with controls that can be rigorously moved into the high precision 45 of 268 3.10. LIMITATIONS OF CURRENT APPROACHES JPL ephemeris models. For the purpose of analysis of natural objects, the Tisserand Parameter has its limitations as well. The limitation here is that no more information is gained once two objects have the same Tisserand Parameter. Within the context of dynamical systems in the CR3BP, the object’s path can also be compared to invariant manifolds to see if a connection exists between the two observed objects for further proof that they may be the same. 3.10.2 Multibody Limitations Although the integrated trajectories in this method are highly accurate due to the model’s fidelity, restricting the analysis of the results to 2-Body parameters severely limits the understanding that one can gain. By proving that the Keplerian energy becomes slightly negative at some point along such a trajectory does not prove that capture has occurred, unless this happens once the object has exited the chaotic regime. There is no guarantee that the energy will stay negative under the effects of multibody perturbations and predicting these effects in this model is difficult. 3.10.3 3-Body Limitations The problem with this model lies in the accuracy of the model when results are moved to a higher fidelity model. The model is clearly more accurate than a 2-body model in terms of propagation, but the issues appear when modeling chaotic behavior. While the 2-body model has the limitation of not being able to accurately model chaotic effects, this one can model and exploit those effects. However, the resulting trajectories can be very sensitive, as is the very nature of chaos. This sensitivity means that the perturbations added by a higher fidelity model may break the expected behavior of a known solution in the CR3BP. These limitations are generally overcome through differential correction or optimization, which corrects the 3-body solution within the higher order model in order to satisfy the new dynamics and keep the original characteristics. One limitation of this approach is a practical one. Since the tools for analysis and design in this model can be very specialized and complex, mission designers who are unfamiliar with the 3-Body Problem are often 46 of 268 3.10. LIMITATIONS OF CURRENT APPROACHES unwilling to try their hand at design with this model. The mission design tools used for low-energy trajectories need to become more user-friendly. 47 of 268 CHAPTER 4: THE TEMPORARY CAPTURE OF ASTEROID 2006 RH 120 VIA INVARIANT MANIFOLDS This chapter is intended to add measured evidence to the theoretical hypotheses I have stated. Since the hypotheses regard the dynamics of resonance transitions and transit between interior and exterior regions, I have focused on a known asteroid with that behavior. That asteroid is the first temporary moon of the Earth to be observed, and is named 2006 RH 120 . We split the analysis into two parts. Part I 4.1 focuses on the pre- and post-capture phases of the asteroid’s trajectory, and how invariant manifolds control its motion toward and away from Earth. Part II 4.1 focuses on the temporary capture phase near Earth, and which dynamical effects were dominant in controlling the asteroid’s behavior. 4.1 Part I:Pre-Capture and Escape Phases Asteroid 2006 RH 120 was the first natural object captured by the Earth to be observed called a Minimoon. In this work, we show that the invariant manifolds of the orbits around theL 1 andL 2 Lagrange points play a significant role in the capture of the asteroid around Earth and its eventual escape from the Earth approximately one year later. This is similar to the Temporary Capture of comets around Jupiter. We determined that the asteroid was in a 27:29 mean motion resonance with the Earth and approached the Earth through the stable manifold of anL 1 northern halo orbit. After the Temporary Capture, the asteroid escaped the Earth through the unstable manifold of an L 2 southern halo orbit and into a 21:20 resonant orbit with the Earth. The asteroid traveled through a series of resonant orbits before and after the capture. These resonant transitions are similar to the orbits of Galileo and Cassini during their touring phase, using resonant orbits to reduce mission V requirements. 48 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES 4.1.1 Introduction There are two types of moons, stable and temporary moons. Stable moons are in stable orbits around the planet; temporary moons are captured by the planet in highly chaotic orbits. The outer planets are known to have many temporary moons. In fact, the Jupiter Family of Comets like Oterma, Gehrels 3, Helin-Roman-Crockett were temporarily captured by Jupiter and then escaped. Shoemaker-Levy 9 was a temporary moon that eventually impacted Jupiter itself. In 2006, Near Earth Object (NEO) 2006 RH 120 was captured into Earth orbit for nearly a year. It is the first temporary moon of the Earth to be observed. In previous work, Koon, Lo, Marsden and Ross[54], and Howell, Lo, and Marchand[47] explained the dynamics of the Temporary Capture of Jupiter comets using invariant manifold theory. The same dynamics is also at work for the Temporary Capture of Earth’s temporary moons. However, the small mass parameter of the Sun-Earth/Moon Barycenter Circular Restricted 3-Body Problem as well as the large mass of Earth’s Moon greatly complicate the Temporary Capture of NEOs around the Earth as compared to the capture of comets around Jupiter or Saturn. Although Asteroid 2006 RH 120 is Earth’s first temporary moon to be observed, recent work by Granvik, Vaubaillon, and Jedicke[38] indicates that these moons may be abundant. Their simula- tions showed that at any time there is at least one temporary moon of one meter diameter or larger orbiting the Earth. They named the Earth’s temporary moons as Minimoons. If the prediction of this population of Minimoons of the Earth is verified, this could open the door to many potentially interesting missions to NEOs right at our door step! Using CubeSats or SmallSats, these missions could be very low cost. Astronomers are currently busy at work to verify the existence of this population of NEOs. Minimoons are difficult to detect because of their small size and their lower velocity profile which distinguishes them from the much faster moving NEOs like Apophis. Given the potential for an abundance of these interesting Minimoons, a deeper understanding of their dynamics would help in locating them and in designing missions to explore them. In this section, we examine the role that invariant manifolds of libration orbits around the Earth’sL 1 andL 2 play 49 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES in the capture and escape of Asteroid 2006 RH 120 . We defer the study of the asteroid’s interaction with the Moon, which involves the Sun-Earth-Moon 4-Body Problem, to Section 4.2. (a) (b) Figure 4.1: (a) The trajectory of Asteroid 2006 RH 120 (black), the L 1 northern halo orbit (blue) and the southern halo orbit at L 2 (red). The invariant manifolds of these orbits controlled the motion of the asteroid’s Temporary Capture and escape. (b) Global view of the asteroid orbit before capture (magenta) and after escape (black). The green surface is the Forbidden Region for the estimated Jacobi constant of the pre-capture phase. The reference asteroid trajectory in these figures is computed by JPL based on observations and propagation in a high fidelity model of the solar system. We computed the halo orbits and their manifolds to show the similarity between observed and simulated behavior. Figure 4.1(a) shows the black trajectory of Asteroid 2006 RH 120 in the Sun-Earth rotating frame, the blue northern halo orbit atL 1 which controls the entry of the asteroid into Temporary Capture by the Earth-Moon System; the red halo orbit atL 2 controls the exit of the asteroid into heliocentric orbit. Note that theL 1 northern halo orbit has a smaller Jacobi constant than theL 2 southern halo orbit. The interaction of the Moon reduced the energy of the asteroid as it orbited the Earth-Moon before escaping. Alternatively, the elliptic nature of Earth’s orbit may be the reason that the Jacobi constant of the asteroid, as we have estimated it, is different before and after Tempo- rary Capture. The Jacobi constant is a quantity associated with the ideal system in the CR3BP and so introducing eccentricity to the Primaries’ motion nullifies the assumptions necessary for this quantity to be a constant, see details regarding the Elliptic Restricted 3-Body Problem in Szebe- hely and Giacaglia[85] and Szebehely[84]. Figure 4.1(b) is a global view of the asteroid trajectory 50 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES for over 50 years. The magenta spiral curve is the resonant orbit which brought the asteroid to the Earth. The black spiral curve is the current orbit of the asteroid since it escaped the Earth in 2006. The close up shows the orbit in the region around the Earth (the Earth is hidden by the surface of the Forbidden Region). The Temporary Capture orbit is in red here and with a radius around 1 million km, it is much larger than the Moon’s orbit. The many spiral loops of the trajectory prior to the capture and after the escape show that the asteroid is trapped in some mean motion resonance with the Earth. 4.1.2 Background In this section, we provide a heuristic description of invariant manifolds of periodic orbits and their significance for the Temporary Capture problem. We model the dynamics of Minimoons using the Circular Restricted 3-Body Problem (CR3BP). Please refer to Section 2.1 for details regarding the properties of this model. Although the two primary bodies of the CR3BP can be any system of co- rotating celestial bodies, this work focuses on the Sun-Earth system so we will from here on refer to the Primaries as the Sun and the Earth. Note that by “Earth” here we actually mean the Earth- Moon Barycenter orbiting the Sun in a circular orbit. In this chapter, for clarity, when we refer to “energy” it will always mean “mechanical energy”, otherwise, we will refer to the Jacobi constant. Please refer to the discussion in Section 2.1 for details on our definition of mechanical energy and its relationship to the Jacobi constant. Section 2.1 uses the Earth-Moon system to describe the properties of the CR3BP, so in this chapter, there will be some small differences. One such difference is in regard to the definition of the regions. The Interior Region is around the Sun inside the Earth’s orbit and Forbidden Region. The Exterior Region is outside of the Earth’s orbit and Forbidden Region. The Neck Region is the region containing the Earth andL 1 andL 2 , connecting the Interior and Exterior Regions. Since the Neck Region is now near Earth, that is where transit between Interior and Exterior Regions is most likely to occur, even at higher energy. The energy surfaces for the asteroid in question are shown in Fig. 4.2. Two surfaces are indicated, since the 51 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES estimated Jacobi constants for the pre- and post-capture phases are not the same. Note that the Neck Region for the pre-capture Jacobi constant is much more open and would more readily allow passage between the Interior and Exterior Regions. We will be studying this transit of the asteroid from the Interior Region, through the Neck Region and out to the Exterior Region. (a) (b) (c) Figure 4.2: Estimated Forbidden Region surfaces for Asteroid 2006 RH 120 shown for the half plane y 0 (a) pre-capture phase. (b) post-capture phase. (c) Large scale comparison of (a) and (b). Note that the surfaces are very similar for both pre- and post-capture phase, but that the opening is slightly larger for the pre-capture phase, indicating a higher energy or equivalently a smaller Jacobi constant. On a large scale, the surfaces are nearly indistinguishable. [NON] indicates nondimensional units. It has been long observed that comets and asteroids can be temporarily captured by planets from time to time. A tremendous amount of work has been done to study this phenomenon with a vast literature. The most famous group of temporarily captured objects are the Jupiter Family of Comets, a few of which were mentioned earlier. The moons of Mars may be asteroids captured through a similar process. The transport mechanism is very complex. We know resonances play a significant role. The resonance transitions eventually bring the small body close to theL 1 orL 2 Lagrange points of the planet. At these locations, the invariant manifolds of libration orbits such as halo orbits or Lissajous orbits are able to attract the small body and bring it into the region around the planet where it is temporarily captured. See Section 2.4 for the definition of these types of orbits. In the aforementioned papers, it was shown how this occurs for Jupiter comets, see Koon, Lo, Marsden and Ross[54], and Howell, Lo, and Marchand[47]. The same process works for other 52 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES planets as well. The significance of this approach to explain the Temporary Capture Phenomenon is that it helps us to visualize the capture process and this could help to predict the location of the NEOs before and after Temporary Capture. Moreover, this could help us devise methods to prevent future impacts by dangerous NEOs, and it could help us design trajectories to visit the Minimoons or effect methods to change their energies and capture them around the Moon or the Earth for many applications. We will be comparing the asteroid trajectory to invariant manifolds in the Sun-Earth CR3BP in this chapter, and we refer the reader to Section 2.5 for details on invariant manifolds. The interaction of the asteroid with Earth is similar to that of some comets with Jupiter. Those comets were shown to follow heteroclinic-homoclinic chains, which we introduce in Section 2.10. For the comet Oterma, a chain involving the 2:3 and 3:2 resonances of Jupiter has been shown to control its motion. In this chapter, we will show the sequence of resonant and libration point orbits used by Asteroid 2006 RH 120 to approach the Earth for Temporary Capture, and then escape the Earth through resonance transitions. The main results of Section 4.1 is demonstrating that the trajectory of Asteroid 2006 RH 120 is guided by a sequence of resonant and libration orbits and their invariant manifolds. This is typical of Minimoons. Hence by understanding the dynamics of these resonant transitions and temporary capture trajectories, we may be able to formulate theories and rules of thumb to locate, rendezvous, deflect, or capture NEOs at the energy levels of Minimoons. 4.1.3 CR3BP and Ephemeris model The Sun-Earth/Moon Barycenter Circular Restricted 3-Body Problem (CR3BP) is a simplification of the motion of an object around the Sun and Earth, see Szebehely[84]. Therefore, analysis within this system requires some form of transformation from the real physical system. The Jet Propulsion Laboratory DE431 ephemeris[33] will be used to represent the “real system” in this case. In the CR3BP system, the distance between the Primaries is assumed constant as is their 53 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES rotation rate about the common barycenter. This would be the case if the two primary bodies were in circular orbits around their mutual barycenter and there are no other perturbing forces. In such a perfect system, units of length are nondimensionalized by this fixed distance and the units of time are nondimensionalized by setting the mean motion of the Primaries to unity. Units of velocity and acceleration are derived from length and time. The frame of reference rotates along with the Primaries such that the x-axis lies along the line from the Sun to the Earth and thez-axis is along the angular momentum of their motion. Although there is no perfect method for converting real trajectories into the CR3BP frame, the method used for this work allows for a rough analysis and is described in detail by Anderson and Lo[5]. Two methods are used for the work here, each with its advantages and disadvantages. Conversion Method 1 uses variable length and time units, representing the current position and motion of the Primaries. The instantaneous distance between the Primaries is selected as the distance unit, while the time unit is selected such that the instantaneous angular velocity of the Primaries has unity magnitude. Conversion Method 2 uses fixed length and time units, chosen to represent the position and motion of the Primaries at a specific reference time. The distance and time units are computed as in the Conversion Method 1, but the instantaneous conditions of the Primaries at a reference time are selected and used for unit conversions. The common principles for frame conversion between both methods is the position of the right handed coordinate system. Thex-axis is based on the instantaneous vector from the Sun to the Earth. Thez-axis is aligned with the angular momentum vector of the motion of the Sun and Earth. They-axis completes the right handed system and the origin is located at the barycenter of the two Primaries. The most natural conversion using Method 2 would be to use the mean semimajor axis and mean motion of the Primaries, but this choice is not always the best one depending on the behavior one wants to analyze. For example, if Earth happens to be near its perihelion when an asteroid makes its closest approach, this conversion would place the asteroid around atx 0:983 which by theory is beyond evenL 1 in the CR3BP and would not be near Earth. Any analysis of the motion of 54 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES an asteroid in the region betweenL 1 andL 2 would be better suited by using Conversion Method 1, as it would more accurately represent the motion of the asteroid relative to Earth. This analysis is however deferred to Section 4.2 and the current work focuses on the pre- and post-capture phases of Asteroid 2006 RH 120 . For this purpose, Conversion Method 2 is used extensively with the reference time chosen appropriately to allow analysis of the asteroid’s motion as it approachesL 1 and departsL 2 , as well as the mostly heliocentric motion, far from the gravitational influence of Earth. The equations for the conversions are given in the Appendix. Conversion Method 1 has the advantage of producing an accurate visual representation of the positional trajectory in an approximate CR3BP frame of reference. Since the method displays the position of an object relative to the instantaneous position of Earth, the ellipticity effects of Earth’s orbit are minimized. Based on this work, Conversion Method 1 produces a position plot that more closely follows the dynamics of an ideal CR3BP system. However, due to the oscillatory nature of the unit conversions, the converted velocity also displays oscillations. There are several coupled frequencies, causing the velocity to not behave as expected in an ideal CR3BP model. Since the Jacobi constant is computed from the position and velocity of an object, in this case the Jacobi constant exhibits oscillations on multiple frequencies. Due to the better behaved position conversion of Conversion Method 1, it is better suited for analyzing the long term resonances of the asteroid. However, the behavior of the Jacobi constant of Method 1 is unsuitable for analyzing the libration orbits for the approach and departure phases near Earth. Method 2 is more suitable in this case. Conversion Method 2 removes the oscillations in the unit conversions and thus eliminates some of the frequencies in the Jacobi constant oscillation. The frequency that remains is an annual cycle. However, this method has the drawback that while it is a more accurate representation of the behavior close to the reference point, it may be poor far from the reference point. For this work, Conversion Method 2 is well suited for the analysis of the interaction between the invariant 55 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES manifolds of the libration orbits and the asteroid. It is less suited for the long-term analysis of the Asteroid’s resonances. 4.1.4 Asteroid Orbit The Near Earth Object (NEO) 2006 RH 120 is the first known natural celestial body to be temporarily captured by the Earth, see Williams[89] and Granvik, Vaubaillon and Jedicke[38]. The asteroid orbited Earth for approximately one year in a large, chaotic orbit outside the orbit of the Moon, but between the Sun-EarthL 1 andL 2 Lagrange points. We call this the Temporary Capture (TC) phase. Before the TC phase, Asteroid 2006 RH 120 was on a heliocentric orbit inside Earth’s orbit. After the TC encounter, the asteroid has transitioned to an orbit outside Earth’s orbit. The interaction of the asteroid with the Earth-Moon System is highly nonlinear and its orbit was significantly changed during the Temporary Capture phase. Classical orbital elements varied wildly during this phase. In the pre-capture and post-capture phases, the asteroid underwent resonance hopping which is also exhibited by the Jupiter Family of Comets. We use two methods to determine the resonances of Asteroid 2006 RH 120 . Resonance Met- hod 1 used visual inspection of the trajectory in the rotating frame as follows. Each “loop” or oscillatory cycle of the asteroid’s path in the rotating frame represents one revolution around the Sun. Starting the analysis immediately before or after Temporary Capture, a propagation of the asteroid’s trajectory backward or forward in time respectively reveals details about its resonance cycle. After propagating the orbit for a certain amount of time, the asteroid returns to the vicinity of Earth, usually nearL 1 orL 2 . At this time, the number of heliocentric orbits of the asteroid is compared to the time elapsed in Earth years. The integer ratio of these two numbers will be the mean motion resonance of the cycle. Figure 4.3(a) shows the pre-capture orbit as converted to the rotating frame using Conversion Method 1 as described earlier. Analysis of the pre-capture orbit shows that a complete resonance cycle is completed in 27 years during which time the asteroid 56 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES orbited the Sun 29 times. This suggests that the asteroid is initially in a 27:29 mean motion reso- nance with the Earth. Figure 4.3(b) shows the post-capture orbit converted to the rotating frame using Conversion Method 1. Analysis of the post-capture orbit in the rotating frame shows that a complete resonance cycle is completed in 21 years during which time the asteroid orbits the Sun 20 times. This suggests that the asteroid is in a 21:20 mean motion resonance with the Earth after TC. This initial analysis covers the time period April 1, 1979-November 1, 2028, which exhibits one full resonance cycle for both Pre- and post-capture phases. (a) (b) Figure 4.3: Asteroid 2006 RH 120 trajectory converted to CR3BP rotating frame by Conversion Method 1 (a) asteroid trajectory in pre-capture phase. (b) asteroid trajectory in post-capture phase. The fundamental change in the orbit is clearly visible in the rotating frame as seen in Fig. 4.3. If the same method of analysis is applied to an even longer time span, more interesting effects appear. As is seen in Fig. 4.4(c), there are multiple rapid changes in the behavior of the Jacobi constant of Asteroid 2006 RH 120 . Upon closer correlation of these changes to the position of the asteroid, it is found that these events are the result of near-Earth approaches that do not result in Temporary Capture. At each Earth encounter, the asteroid changes resonance with the Earth. We will later correlate these changes to heteroclinic transitions between the resonant periodic orbits around the Sun in the CR3BP. 57 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES (a) (b) (c) Figure 4.4: Estimated Jacobi constant of Asteroid 2006 RH 120 . Pre- (purple) and post-capture (black) phases used Conversion Method 2 with units chosen at their respective reference points. The Temporary Capture phase (red) used Conversion Method 1. (a) Near TC with reference points. (b) Showing one full resonance cycle on either side of TC. (c) Extended time showing several resonances 1950-2050. The dashed lines indicate Jacobi constant for the Lagrange points. Values under the upper black boundary indicate an open Neck Region near Earth, while values under the lower black boundary indicate when all motion in the plane is allowed. An alternate method for approximating the resonance of an object is through a simple Keple- rian analysis. We call this Resonance Method 2. A period resonance in a pure 2-body situation can simply be described by the osculating period of an object relative to the period of some reference object (the Earth in this case). The semimajor axis of an object can be computed from its Keple- rian energy, which in turn is determined directly from the object’s position and velocity. In an unperturbed 2-body system, this energy should remain constant. But due to the multibody effects of the real physical system, the 2-body energy changes with time. This allows us to determine the theoretical instantaneous period of an object over time. By focusing on parts of this period when it reaches a relatively constant value, a simple resonance analysis is possible. Figure 4.5 shows the semimajor axis of Asteroid 2006 RH 120 over time. It is clear that when the asteroid is far enough away from Earth, the osculating semimajor axis is nearly constant. At these reference points, the Keplerian period indicates that the pre-capture resonance is 40:43 while the post-capture resonance is 21:20. These results agree with the previous results for the post-capture trajectory, but are slig- htly different for the pre-capture trajectory. It should be noted that the ratios 40:43 and 27:29 are very close, both are nearly 1:1 resonances, small differences become difficult to tell apart. The 58 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES discrepancy between the results can be attributed to the more approximate nature of the Keplerian analysis. While the period of the asteroid stays nearly constant for a large part of its resonance cycle, the non-Keplerian effects near the beginning and end of this cycle have an effect that causes the overall resonance cycle to be slightly different. Therefore, while Resonance Method 1 may be more time consuming, it is the preferred method for more accurate results. Figure 4.5: Osculating semimajor axis of Asteroid 2006 RH 120 for the period 1950-2050. Reso- nance Method 2 was used to estimate period resonance ratios. The data shows multiple encounters and resonance transitions. As mentioned before, estimating the Jacobi constant of a real physical object will be largely affected by the choice of method for converting the trajectory into the ideal CR3BP frame. It becomes important to have a good estimate of the Jacobi constant when discussing the interaction between libration orbit manifolds and the trajectory. Every periodic orbit in the ideal CR3BP has a specific Jacobi constant and as such one should choose orbits that match the Jacobi constant of the asteroid. We chose to approximate the Jacobi constant of the asteroid at an appropriate reference point for studying its behavior. More specifically, Conversion Method 2 was used as explained 59 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES previously and the reference point was chosen to be at the crossing of the L 1 plane for the pre- capture phase and at the crossing theL 2 plane for the post-capture phase. These reference points are indicated in Fig. 4.4(a) while the Jacobi constant variation during the whole resonance cycle is shown in Fig. 4.4(b). The planes through the Lagrange points are defined by the plane passing through the Lagrange point that is normal to the vector from the Sun to the Lagrange point. The position of the Lagrange points were obtained from JPL Ephemeris DE431[33] (as objects “391” and “392”). The Jacobi constants were estimated to be C pre = 3:000228226120707 (4.1) C post = 3:000425683288712 (4.2) We realize the estimates cannot be accurate to 15 decimal places, but we have included the full value we used for finding orbits and surfaces in case anyone would like to reproduce some of our work. If the time span under consideration is expanded to the full range provided for the object within the DE431 Ephemeris, 1950-2050, the Jacobi constant and semimajor axis both exhibit signs of repeated Earth encounters as seen in Fig. 4.4(c) and Fig. 4.5. Resonance Method 2 allows us to quickly approximate the resonances between each Earth encounter and are shown in Fig. 4.5. The available data shows a trend of an increasing semimajor axis by repeated Earth encounters, with the most significant change occurring during the Temporary Capture. This behavior for an increa- sing semimajor axis is typical for solar system dynamics. What is happening is that the invariant manifolds of the resonant orbits in the CR3BP intersect, thus allowing for natural transfers of the asteroid between each resonance set. This is a basic transport mechanism of the “Interplanetary Superhighway”. This can be compared to the behavior of the Jupiter Family of short period comets such as Oterma, see Koon et al.[54]. Further work is required to find the specific resonant orbits in the CR3BP and analyze their manifolds for heteroclinic connections. However, due to the large 60 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES time periods involved, such an analysis in the ideal system may not have much application to the real physical system. Furthermore, the large time span between encounters allows for buildup of small perturbations and model inaccuracies. 4.1.5 Earth Approach ThroughL 1 Manifolds According to our theory, the asteroid approaches the Earth via the stable manifold of orbits around L 1 . We computed 4 periodic orbits aroundL 1 along with their invariant manifolds with the refe- rence Jacobi constant of the pre-capture asteroid trajectory: a Planar Lyapunov orbit, a Vertical Lyapunov orbits, a northern halo orbit, and a southern halo orbit. We used a differential correcti- ons algorithm which constrained the Jacobi constant to find these periodic orbits. This algorithm was adapted from Howell[46]. For a detailed description, refer to Appendix B.1. We found that the asteroid is following most closely to the stable manifold of a northern halo orbit aroundL 1 as shown in Fig. 4.6. 61 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES (a) (b) (c) (d) Figure 4.6: Asteroid 2006 RH 120 approach with stable manifold. (a) The stable manifold (green) of a northern halo orbit atL 1 attracting the asteroid (black) to approach Earth. The orange surface is the Forbidden Region at the pre-capture energy. This panel is looking outward diagonally away from the Sun so that the green manifold is approaching the Earth on the inside of the Earth’s orbit. (b) Near Earth close-up. The view is also rotated by about 90 degrees so that it is now looking outward but toward the direction of the Earth’s orbital motion. (c) A single trajectory on the manifold (green) selected for its similarity to the asteroid trajectory (black). The view is looking in the general direction opposite of Earth’s velocity vector, toward thexz-plane. (d) A close up view of the selected manifold trajectory (green), the halo orbit (blue) and asteroid trajectory (black). The view is looking diagonally toward the Sun, with the Sun direction toward the top right. [NON] indicates nondimensional units. 62 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES 4.1.6 Earth Departure ThroughL 2 Manifolds A similar analysis was performed on the escaping trajectory and orbits around L 2 . In this case the unstable manifolds of L 2 periodic orbits were used to compare with the departing asteroid trajectory. For the reference Jacobi constant of the asteroid’s post-capture phase, 4 periodic orbits were found: a Planar Lyapunov orbit, a Vertical Lyapunov orbits, a northern halo orbit, and a southern halo orbit. For the post-capture phase, we found the southern halo orbit and its manifolds around L 2 to provide the best fit with the departing asteroid trajectory. The halo orbit and its unstable manifold are shown in Fig. 4.7. Visual inspection of the first two plots show the asteroid closely following the surface of the unstable manifold as seen in Fig. 4.7(b) and (c). 63 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES (a) (b) (c) (d) Figure 4.7: Asteroid 2006 RH 120 departure with stable manifold. (a) The unstable manifold (orange) of a southern halo orbit at L 2 guiding the asteroid (black) away from Earth with the plot of the estimated post-capture Forbidden Region (yellow). The view is looking inward toward the inner solar system and looking eastward of the direction to the Sun. (b) Near Earth close-up of (a) with the same orientation. (c) A single trajectory on the unstable manifold (orange) selected for its similarity to the asteroid trajectory (black). The orientation is similar to that of (a), but closer to the ecliptic plane. (d) A close up view of the selected manifold trajectory (orange), the halo orbit (blue) and asteroid trajectory (black). The view here is looking approximately along the Earth’s velocity vector, with the Sun to the left. [NON] indicates nondimensional units. 64 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES 4.1.7 Example of Complex Lunar Interactions With Asteroid-Like Trajec- tories Figure 4.8: Chaotic evolution of the Genesis trajectory. Shows a small subset of the invariant manifold for the Genesis trajectory. Minor perturbations of an initial state result in very chaotic subsequent trajectory evolutions. Figure 4.8 shows a small portion of the unstable manifold (green curves) of the Genesis halo orbit (magenta orbit on the left) in the CR3BP rotating frame demonstrates the complex interactions with the Moon (in the gray orbit) and the Earth. Moving approximately from left to right, we see a lunar Flyby has torn the manifold into two pieces. One piece of the manifold resulted in orbits captured around the Moon like the Hiten trajectory. The Japanese Hiten probe accomplished the first ballistic lunar capture by traveling well beyond the orbit of the Moon to Sun-Earth L 1 , after which it slowly fell back and approached the Moon such that it captured without need of an additional maneuver. The other piece of the manifold resulted in an Earth Flyby where some of them became captured by the Earth (temporarily), others Flyby the Earth once and then escape via L 2 into a Spitzer-like Earth Trailer Orbits. The orbit used for the Spitzer mission is chosen such that it just barely escapes the gravity well of Earth, and ends up trailing the Earth around the Sun, while very slowly drifting away from Earth. In particular, one of the trajectories of the unstable manifold makes a wide excursion toL 2 and then impacts the Earth at the Utah Test and Training 65 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES Range, sometime before 9 AM in September. This is the Genesis trajectory to return to Earth. Similarly, Near Earth Asteroids can approach the Earth from far away via the stable manifold of a halo orbit atL 1 and then follow its unstable manifold into the Neck Region around the Earth-Moon where this complex range of motions are possible. What it tells us is that an asteroid Minimoon can be captured around the Moon, or Earth, flyby the Earth and escape via L 2 , or in the worst case, it can impact the Earth like the Genesis spacecraft. The analysis of the complex trajectory of Asteroid 2006 RH 120 during its year-long capture phase requires the analysis of the orbital motions near the Moon separately from the motions near the Earth. The invariant manifolds of some lunar L 2 orbit are interacting with the invariant manifolds of the periodic orbits around the Sun-Earth L 1 andL 2 described earlier. The interaction with the Moon during the Temporary Capture phase reduced the energy of the asteroid (raising its Jacobi constant) when it escaped the Earth viaL 2 . A detailed analysis of this work will be described in Section 4.2. 4.1.8 Section Summary Our goal is to show from this research that the dynamics of Minimoons can be understood using the invariant manifolds of unstable orbits in the Sun-Earth-Moon System. Although the dynamics of Minimoons is important in its own right, the real significance for understanding this dynamics is in the practical applications to science, engineering, and the industrialization of space. Knowledge and insight of the dynamics of Minimoons would enable us to detect and locate them, potentially design means to deflect them. Since they may be abundant and they are so easily accessible from the Earth or the Moon, understanding their dynamics could help us potentially design very cheap missions to rendezvous and land on them, if they are big enough, with robotic and human missions. This knowledge could help us potentially capture these Minimoons into orbit around the Earth or the Moon with minimal effort. In the near future, we could mine the Minimoons and begin a new industrial age in space. From the resources and wealth created from the commerce and industry in space, we could then truly begin to colonize the Moon and beyond. 66 of 268 4.1. PART I:PRE-CAPTURE AND ESCAPE PHASES In this section, we analyzed the dynamics of the Temporary Capture of the Minimoon 2006 RH 120 during its pre-capture and post-capture phases using the CR3BP. For the pre-capture phase, we have shown that the asteroid trajectory is shadowing closely the trajectories of the invariant manifolds of a northern halo orbit aroundL 1 . Similarly, for the post-capture phase, we have shown that the asteroid trajectory is shadowing closely the trajectories of the invariant manifolds of a southern halo orbit aroundL 2 . The next section approaches the dynamics of the Capture phase, which requires a detailed analysis of the 4-body interaction of the Sun-Earth-Moon-Minimoon System. This is much more complicated. In Fig. 4.9 we have plotted the distance between the asteroid and the Moon during the Capture phase in the Neck Region. This figure shows the asteroid moving in the Neck Region of the Earth where the invariant manifolds of the Sun-Earth and Earth-Moon 3-body systems intersect. These intersections create lobes that can trap the asteroid temporarily in highly chaotic orbits. Once we understand the dynamics of these Temporary Capture orbits, we can then begin thinking about low energy and low cost methods to convert a Temporary Capture orbit into a Long-Term Capture orbit. Figure 4.9: Distance between Asteroid 2006 RH 120 and the Moon during Temporary Capture. The R SOI is the radius of the Lunar Sphere of Influence. 67 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE 4.2 Part II:Temporary Capture Phase In this section, we use the Bicircular (BCP) and Elliptic Restricted 3-Body Problem Models (ER3BP) to study how simple models approximate the chaotic motions of the Asteroid 2006 RH 120 during the capture phase. Hypothetical scenarios of alternate lunar phasing and Sun-Earth true ano- maly and its effects are explored. Overall, the eccentricity of the Earth’s orbit seems to dominate the behavior of the Asteroid orbit. During the Capture phase, the 4-body effects are important, but not critical. By adjusting the phasing, the ER3BP can model the Capture phase nearly as accurately as the BCP. 4.2.1 Introduction In 2006, Near Earth Object (NEO) 2006 RH 120 was captured into orbit around the Earth-Moon system for nearly a year. In Fig. 4.1, we showed the trajectory of this asteroid in the rotating frame. It is the first Minimoon temporarily captured around the Earth to be observed. In the previous section (4.1), we studied the pre-capture and escape phases of Asteroid 2006 RH 120 . We showed that the manifolds of the Sun-EarthL 1 andL 2 halo orbits play a significant role in these dynamics. The goal of this work is to study the temporary capture phase around the Earth-Moon system. This is a much more complex problem because the addition of the Moon makes it a 4-body problem. Surprisingly, the 3-body models provide a very good description of the asteroid’s orbit in the capture phase. This is because the orbit is so large and it is always at least 4.4 times the radius of the lunar sphere of influence away from the Moon. On the other hand, examination of the capture orbit in the Earth-Moon rotating frame produced very erratic orbits that do not seem to follow invariant manifolds of the Earth-Moon libration orbits. These dynamics appear to be even more complex, hence we will defer a detailed study of the orbit’s interaction with the Moon to future work. Thus, in this section, when we refer to L 1 and L 2 , it will be strictly that of the Sun-Earth libration points. 68 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE This section focuses on studying the temporary capture phase through analysis of the Aste- roid’s behavior in a few dynamical systems that approximate the real phenomenon. The analysis is applied to three segments of the Asteroid’s trajectory. These segments are combinations of three phases clearly separated in time. A diagram depicting the relationship between the phases, seg- ments, events and dates is shown in Fig. 4.10, with the associated dates listed in Table 4.1. Phase 1 is the time between the last close Earth encounter in 1979 and the recent Earth encounter and L 1 approach in 2005. Phase 2 is the temporary capture phase as the Asteroid travelled fromL 1 to L 2 between 2005 and 2008. Finally, phase 3 is the time fromL 2 escape in 2008 to the next future close Earth encounter in 2028. Strictly speaking, the temporary capture period is the time interval when the asteroid crossed theL 1 andL 2 gateways in 2006 and 2007. However, in order to analyze the dynamics of the capture phase, it is necessary to include the dynamics of the precapture and postcapture phases. Thus, we have decided to pad this time interval with six months on either side. Segment 1 includes phase 1 and 2 and is used to determine the accuracy of dynamical models for the interior resonance through temporary capture. This segment revisits part of the analysis done in Anderson & Lo[6]. The focus of this section is segment 2, which only considers the temporary capture itself, which is phase 2. Segment 3 is similar to Segment 1, as it combines the temporary capture in phase 2 with the exterior resonance in phase 3. Figure 4.10: Visual definition of Segments and Phases. The Earth symbols represent close Earth encounters 1, 2 and 3. “L 1 ” and “L 2 ” represent approximate gateway transits, where we used six months beforeL 1 and six months afterL 2 as reference points. The years on the bottom correspond to the specific dates listed in Table 4.1 69 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Table 4.1: Reference dates used for phase and segment definitions. Event Date Earth Encounter 1 July 1 st , 1979 Earth Encounter 2,L 1 - six months November 23 rd , 2005 L 2 + six months January 29 th , 2008 Earth Encounter 3 November 1 st , 2028 The analysis we performed to determine which dynamical model accurately represents the behavior of Asteroid 2006 RH 120 requires us to compare known trajectories to simulated ones. Urrutxua and Scheeres[86] have studied this problem as well, but with a different approach . In their paper, they concluded that a high-fidelity model is needed in order to maintain the tempo- rary capture. Their model includes the forces of several planets as determined from an accurate ephemeris, as well as relativistic effects in certain regions of motion. However, their application is slightly different from ours. They were attempting to compute the V necessary to capture or prolong the temporary capture of the Asteroid. In order to compute accurate values of these V s, the computed Asteroid orbit must closely mirror the actual path taken. Whereas in our work, the goal is to study the role of invariant manifolds in the control of the asteroid orbit. For this, we need simpler models that approximate the temporary capture behavior without becoming cumbersome. We accomplished our study by applying a technique of path comparison from computer vision. Specifically, we use a modified version of Dynamic Time Warping which is a technique applied to problems such as shape analysis and speech recognition, see Sakoe and Chiba[80]. Alternate met- hods exist for measuring the similarity of curves, such as the discrete Fr´ echet distance, see Eiter and Mannila[27] and Alt and Godau[4]. In fact, the variant of the discrete Fr´ echet distance in Eiter and Mannila[27] that computes the sum of the distances between the curves is very similar to Dyn- amic Time Warping. The algorithm allows the time along two curves to warp, thus analyzing the shape alone for similarities. Two trajectories in a nonlinear dynamical system that follow similar paths may move forward in time at different rates. This method removes that issue and allows for comparing the shape. This novel approach may be new as we have not seen it used for comparing 70 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE space trajectories in the literature. Note that this trajectory comparison metric can also be used to classify trajectories by grouping the ones that are similar. 4.2.2 Background We model the dynamics of Minimoons using the Circular Restricted 3-Body Problem (CR3BP), Elliptic Restricted 3-Body Problem (ER3BP) as well as the Bicircular Problem (BCP). The details of these models are discussed later in this section. The Near Earth Object (NEO) 2006 RH 120 is the first known celestial object to be temporarily captured by the Earth, see Williams[89], and Granvik, Vaubaillon, and Jedicke[38]. The asteroid orbited Earth for approximately one year in a large, chaotic orbit outside the orbit of the Moon, but between the Earth’s L 1 and L 2 . Before the Temporary Capture (TC), 2006 RH 120 was on a heliocentric orbit inside the orbit of the Earth about the Sun, and it has transitioned to an orbit outside Earth’s after its TC encounter. Its interaction with the Earth system caused a significant change in the orbit as described by classical orbital elements. Previous analysis revealed that the asteroid moved from a 29:27 resonance with the Earth inside the Earth’s orbit to a 20:21 resonance outside the Earth’s orbit, see Anderson and Lo[6]. We also showed that the approach and escape trajectory of the asteroid closely follows the stable and unstable manifolds of halo orbits in the Sun- Earth CR3BP as indicated in Figs. 4.6 and 4.7. For details on how we selected the pre-capture and post-capture energies used for the forbidden regions in Figs. 4.6 and 4.7, see our previous paper[6]. In the work described within, we primarily study the temporary capture phase, approximately July 2006–September 2007. CR3BP Model The CR3BP is a convenient model to use because it represents an autonomous system and has an integral of motion. It adds fidelity relative to the Keplerian 2-body model, while not adding too much complexity. This is the simplest model we use in this chapter. In the CR3BP, two primary 71 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE bodies are assumed to orbit their common barycenter on circular orbits, while a third body of negligible mass is influenced by them. A representative diagram of the rotating frame used in this dynamical system with the fixed positions of the two Primaries is shown in Fig. 2.1. The equations of motion and basic features of the system are provided in Section 2.1. For more details, see Szebehely[84] and Koon, Lo, Marsden, and Ross[53]. BCP Model To study more complex effects during temporary capture, we also use the Bicircular Problem (BCP). The details of this model are described in Section 2.2. Note that we chose to approximate the orbital plane of the Earth-Moon as lying in the ecliptic plane to keep the models simple. We justify this approximation by noting that the inclination of the Moon’s orbit to the ecliptic oscillates in the range 4.99-5.30 degrees, which is relatively small, see Bate, Mueller and White[11]. The intent of Section 4.2 is to determine if simple models can approximate real motion well. Clearly, this model will represent the effect of lunar perturbations on the asteroid more accurately than the CR3BP or ER3BP. However, it does not consider the eccentricity of Earth’s orbit or the Moon’s orbit. ER3BP Model Another model we use to study complex effects is the Elliptic Restricted 3-Body Problem (ER3BP). This model is described in Section 2.3. The purpose of using this model is to study the effect of Earth’s orbital eccentricity on the asteroid. This model clearly cannot indicate any effects purely due to lunar perturbation as the BCP does. 72 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE 4.2.3 Method To test the accuracy of the various models, slightly different approaches were used in the different segments mentioned earlier. For Segment 1 (contains the pre-capture and capture phases), integra- ting the trajectory backwards in time tests how well a model simulates the capture and places the asteroid on the proper interior resonance. For Segment 3 (contains the capture and post-capture phases), integrating the trajectory forward in time tests how well a model simulates the capture and places the asteroid on the proper exterior resonance. Finally, for the capture phase itself (Segment 2), since the sensitivity of the capture trajectory is typically highest at perigee, starting from peri- gee and integrating backwards and forwards provided the best match of the model trajectories with the ephemeris data. Table 4.2 in Section 4.2.4 below summarizes the comparison results. In the following sections, we also explain the methods we used for comparing trajectories and converting ephemeris trajectories to a rotating frame. Converting Ephemeris to Rotating Frame The Sun-Earth/Moon Barycenter CR3BP is a simplification of the motion in the solar system in the regions dominated by the Sun and Earth. Therefore, analysis in this system requires some form of transformation from the real system. The JPL DE431 ephemeris will be used to represent the “real system” in this case (see Folkner et al.[33]). In the simplified system, the distance between the Primaries is constant as well as their rotation rate about the common barycenter. Units of length are non-dimensionalized by this distance and the units of time are non-dimensionalized by setting the mean motion of the Primaries to unity. Velocity and acceleration are derived from length and time. The frame of reference rotates along with the Primaries such that thex-axis lies along the line from the Sun to the Earth and thez-axis is along the angular momentum of their motion. Although there is no perfect method for converting real trajectories into the CR3BP frame, the method used for this work allows for a rough analysis and is described in detail by Anderson and Lo[5]. The length unit is chosen as the instantaneous distance between the Primaries, while the time unit is 73 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE fixed such that the mean motion is unity. The velocity and acceleration unit is derived from the length and time units. The choice of variable length unit allows a more accurate representation of the trajectory relative to the Earth. The x- and z-axes are based on the instantaneous vectors from the Sun to the Earth and angular momentum vectors respectively, while they-axis completes the right-handed triad. Since we are using two other dynamical models whose coordinate frames match that of the CR3BP, we use the same conversion for all three models. In the case of the ER3BP, we propagate the initial conditions with respect to true anomaly instead of time. This would imply that the trajectory as converted to the rotating frame and units must be adjusted. However, the method we use to convert takes the instantaneous frame rotation into account when computing the time unit normalization, see Anderson and Lo[6]. The time unit normalization influences the velocity unit normalization. This has the effect of producing converted velocities which represent derivatives with respect to true anomaly. The normalization factors are, andu for distance, time and velocity, respectively. We compute using the instantaneous frame rotation,!(t) in Eq. 4.3. = 1 !(t) = dt df (4.3) Using and, we then derive the velocity normalization in Eq. 4.4. u = = df dt (4.4) Finally, the normalization of the dimensional position and velocity vectors ~ R and ~ V to the non-dimensional vectors~ r and~ v is shown in Eq. 4.5. ~ r = ~ R ~ v = ~ V u = d ~ R dt 1 dt df = d~ r dt dt df = d~ r df (4.5) The result shows that the velocity as determined using this method represents the derivative with respect to true anomaly. 74 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE State Coherence When a trajectory taken from ephemeris data is converted to the CR3BP reference frame and units, the algorithm used does not guarantee the velocity coordinates to be tangent to the path of the position coordinates. This poses a problem to the analysis we are attempting, because we are choosing states on the converted trajectory to propagate in various dynamical models. If the velocity of a chosen state guides the motion in a direction very different from the expected path, it is unlikely that any chosen model will provide a close match to the real trajectory. We address this issue by defining a metric for each point on the converted trajectory called the state coherence. In simple terms, it represents the normalized dot product between the velocity and the path tangent. Thus, the coherence is 1 if they are aligned, 0 if they are perpendicular, and -1 if they are opposing. Equation 4.6 shows the basic computation for any point on a trajectory, while the diagram in Fig. 4.11 shows the idea graphically. ^ d i = ~ r i+1 ~ r i j~ r i+1 ~ r i j ^ v im = ~ v i+1 +~ v i j~ v i+1 +~ v i j q i = ^ v im ^ d i (4.6) Here, ^ d i is a unit position vector along a segment of the trajectory, ^ v im is a unit vector along the average velocity on that segment, andq i is the state coherence. Naturally, we seek to use points from the trajectory with maximum state coherence, so we compute the coherence for the trajectory in the time period of interest and locate the peaks near q=1. Each peak is a candidate for analysis, and we chose one by proximity to an event or time of interest. For example, the first peak can be chosen for a forward propagation through a chosen segment of time. 75 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Figure 4.11: State Coherence diagram. The midpoint path tangent is approximated from the seg- ment endpoint position vectors. The midpoint velocity vector is the average of the segment end- point velocity vectors. The red vectors are the normalized midpoint path and velocity vectors, from which the coherence is computed. Modified Dynamic Time Warping The algorithm for Dynamic Time Warping computes the minimum sum of the distances between two discrete input signals, see Sakoe and Chiba[80]. In our case the signals are multidimensional vectors as functions of time, and the distance is then computed as the Euclidean norm between the signals. However, the default algorithm poses a problem for our purposes. Consider a set of two trajectories with more discrete points than another set of trajectories. This could be because the trajectories cover a larger range of time, or that the curve accuracy is simply higher. In that case, the first set will have a larger DTW distance than the second set simply from summing more distances to form the total. This does not represent how “close” two trajectories are in the sense we want to know. A more useful measure for our analysis is the mean distance between the curves. This is a simple modification to the normal DTW algorithm and is seen in Eq. 4.7. x i = (x 1 ;:::;x p ) y i = (y 1 ;:::;y q ) d =dtw(x i ;y i ) d =d=n (4.7) 76 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Where dtw() is an operator representing the standard DTW algorithm, and n is the number of discrete distances computed within the DTW algorithm. For our work, we have used the DTW algorithm provided with the Signal Processing Toolbox in MATLAB [65]. As was mentioned in the previous section (“State Coherence”), a trajectory converted from ephemeris data to the rotating frame may not have a velocity that matches the path tangent in position space. Therefore, we chose to use the position coordinates only for the trajectory similarity as opposed to comparing the entire 6-dimensional state. We are interested in showing that the trajectory follows the same path in space when simulated in various models. Alternate methods exist for measuring the similarity of curves, such as the discrete Fr´ echet distance, see Eiter and Mannila[27] and Alt and Godau[4]. In fact, the variant of the discrete Fr´ echet distance in Eiter and Mannila[27] that computes the sum of the distances between the curves is very similar to Dynamic Time Warping. The Fr´ echet distance is in essence the shortest line necessary between two points (one on each curve) that are only allowed to move forward on the curves. The discrete Fr´ echet distance is simply the modification of this metric applied to polygonal curves. Although this has been used as a path similarity measure in other applications, it is not as useful for us. Consider that it would essentially determine the largest local separation between our trajectories. This skews the metric in favor of single features as opposed to the trajectory as a whole. It should be noted that while we used the DTW metric for determining similarity between individual pairs of trajectories, this extends to the problem of classifying trajectories. By using a measure of path similarity, one can determine groups of trajectories with similar behavior as sub- sets of a larger group of trajectories. This can be useful for studying complex dynamical systems through Monte Carlo analysis. In fact, DTW has been applied in a supervised machine learning algorithm to classify signals, see Ding et al.[24]. 77 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Segment 2: Temporary Capture Phase The purpose of this analysis is to test the capability of the different models to model the dyna- mics of the asteroid during temporary capture alone. For this segment, the time range is chosen as starting approximately six months before crossing theL 1 plane and ending approximately six months after crossing the L 2 plane. This results in the date range being Nov 23, 2005-Jan 29, 2008. The time step for the ephemeris trajectory is exactly one solar day starting at midnight UTC on the selected day. Therefore, the exact timestamp for each point on the trajectory varies by a few seconds from midnight across the analysis period, due only to leap seconds. However, for the current analysis these differences are negligible. We chose a reference point on the trajectory of the asteroid as converted into the rotating frame. The reference point is chosen as a location where the state coherence is nearly 1. The coherence exhibits an oscillatory nature and has several peaks, some of which approach 1. We chose the peak that is near the center of the analysis time period, which in this case was Jan 3, 2007. An example of the coherence is seen in Fig. 4.13(a), from which the aforementioned date was selected. The converted state at the reference point is now integrated numerically in the several dynamical models. We use the Sun-Earth CR3BP, Sun-Earth ER3BP and Sun-Earth-Moon BCP models. Since the reference point is nearly guaranteed to not be at either endpoint of the converted trajectory, we must integrate forward and backward in time to cover the entire analysis period. Unless noted otherwise, all integration is accomplished with a 7-8th order Runge-Kutta-Fehlberg method using an absolute tolerance of 1 10 12 . Once we have the integrated trajectories, the similarity between them and the asteroid trajectory is compu- ted using the modified DTW as described earlier. We compute the similarity for the backward and forward integrations as well as the entire segment overall. Figure 4.13(b) shows an example result of such an integration for the CR3BP model. CR3BP Model This is the simplest model we use, because it is autonomous. This means that no additional considerations are required once a state is already converted to the rotating frame. 78 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE BCP Model The BCP model uses time as the independent variable but is also non-autonomous like the ER3BP model. The key parameter that defines the initial state of the system is the lunar phasing angle which is related to the lunar elongation. It is thus necessary to determine the proper lunar phasing at the reference point. We use the relative position of the Sun, Earth and Moon as determined from the ephemeris to compute the lunar phasing. The process is indicated in Eq. 4.8. cos = ^ r EMB S ^ r M EMB ~ n = ^ r EMB S ^ r M EMB sin =sign(~ n ^ z)j~ nj (4.8) Here,r EMB S is a unit vector from the Sun to the Earth-Moon barycenter andr M EMB is a unit vec- tor from the Earth-Moon barycenter toward the Moon. These vectors are achieved from ephemeris data in ecliptic J2000 coordinates for the appropriate time. The lunar phase angle is then computed from the sine and cosine components. A diagram of this is shown in Fig. 4.12. Figure 4.12: Diagram showing the computation of the lunar phase angle from known positions of the Sun, Earth and Moon. S is the Sun, M is the Moon and EMB is the Earth-Moon barycenter. In addition to using the phasing as determined from the ephemeris, we also use manually adjusted values to study if that can improve the trajectory similarity. 79 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE ER3BP Model Since the ER3BP model uses the true anomaly of the Primaries’ motion as the independent variable instead of time, a conversion is necessary before we integrate. A conver- sion from Cartesian state to classical orbital elements is used to determine the true anomaly for each point in time during the analysis period. The reference time is thus converted to a reference true anomaly for the integration. Similarly, the endpoints of the time range are converted to true anomaly and used as the stopping conditions for the forward and backward integration. Since the model is non-autonomous, the initial true anomaly must be known to properly integrate the trajec- tory. We determine the true anomaly at the reference time from ephemeris positions of the Sun and Earth. In particular, we compute the true anomaly from the Cartesian state of the Earth-Moon barycenter relative to the Sun. Conversion from Cartesian coordinates to Keplerian elements is a well-known process and we refer to any astrodynamics textbook for details. In addition to using the true anomaly as determined from the ephemeris, we also use manually adjusted values to study if that can improve the trajectory similarity. Segment 1: Pre-Capture Phase & Temporary Capture Phase The purpose of this analysis is to determine how well the different models can model the tem- porary capture along with the initial resonance in the interior region. For this segment, the time range is chosen as starting approximately on the date when the asteroid has its previous near Earth encounter and ending approximately six months after crossing the L 2 plane. This results in the date range being Jul 1, 1979-Jan 29, 2008. The reference point is chosen as a location where the state coherence is nearly 1, as close to the end as possible. In this case the reference point was Jan 4, 2008. The state coherence for this period, which also indicates the selected reference point is shown in Fig. 4.16(a). The reason we chose the reference as close to the end as possible is because we chose to only propagate backward in this segment. Furthermore, the reason behind that is to study the trajectory as it passes “through” capture and into the interior region resonance. A test we determined as a good indicator of the model’s accuracy would be its ability to propagate through 80 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE the temporary capture accurately enough that the exiting state places the asteroid on the proper interior resonance. This requires backward integration from the end of temporary capture and to the previous Earth encounter in 1979. We compute the similarity for the entire segment overall using the Modified DTW. If the integrated trajectory follows the real trajectory closely enough, it will transit from the exterior to the interior region and return to Earth at the beginning of the analy- sis period. As with the previous segment, the initial lunar phasing and true anomaly is determined from the ephemeris. We also manually modify the initial model parameters to study the effect on the trajectory similarity. Segment 3: Temporary Capture Phase & Escape Phase The purpose of this analysis is to determine how well the different models can model the temporary capture along with the resulting resonance in the exterior region. For this segment, the time range is chosen as starting approximately six months before crossing theL 1 plane and ending approx- imately on the date when the asteroid has its next near-Earth encounter. This results in the date range being Nov 23, 2005-Nov 1, 2028. The reference point is chosen as a location where the state coherence is nearly 1, as close to the beginning as possible. In this case the reference point was Jan 3, 2006. A similar test was used for segment 3 as for segment 1. We test the model’s ability to propagate through the temporary capture accurately enough that the exiting state places the asteroid on the proper exterior resonance. This requires forward integration from the begin- ning of temporary capture and to the next Earth encounter in 2028. We compute the similarity for the entire segment overall using the Modified DTW. If the integrated trajectory follows the real trajectory closely enough, it will transit from the interior to the exterior region and return to Earth at the end of the analysis period. As with the previous segment, the initial lunar phasing and true anomaly is determined from the ephemeris. We also manually modify the initial model parameters to study the effect on the trajectory similarity. 81 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Required Hand Tuning The initial study intended for the parameters of the BCP and ER3BP models to be set based on the ephemeris data. For example, the true anomaly of the mutual Earth-Sun elliptical orbit was computed from the Cartesian states from ephemeris data. The true anomaly corresponding to the converted trajectory point chosen for analysis would be set as the initial true anomaly for the propa- gation in the ER3BP. Similarly, the relative angle between the Sun, Earth and Moon obtained from ephemeris data was used to compute the initial lunar phase angle when propagating in the BCP. However, during the study we found that by manually adjusting the computed initial parameters, the models achieved much better representation of the real trajectory. This is discussed further in results. Since the CR3BP is an autonomous system, it is not possible to tune any parameters unless one decides to modify the mass ratio. 4.2.4 Results After examining each of the three segments described in the “Method” section (4.2.3) using each of the three dynamical models, we found that the BCP has the best overall approximation of the Astroid’s behavior when using computed parameters. A notable exception is Segment 3, where agreement is very poor. If we allow for manual adjustment of the parameters, we find that the ER3BP is now the overall best approximation across all three segments. For Segment 2 the BCP is slightly better than the ER3BP. However, for all cases, it was necessary to manually tune the initial model parameters to achieve reasonable agreement between the models and the real system. Using the initial parameters as computed from ephemeris data did not result in good agreement between the models and the real system. These results seem to indicate that lunar perturbations were domi- nant for the temporary capture of asteroid 2006 RH 120 . However, the BCP model did not perform well in terms of modeling the behavior of the Asteroid before and after temporary capture. All pertinent numerical results are displayed in Table 4.2. Detailed results for each segment follows. 82 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Table 4.2: Resulting trajectory similarity between models. The values indicate the average dis- tance between the trajectory propagated in a dynamical model and the ephemeris solution. Green cells indicate the best model agreement for a given column. Segment 1 was only propagated for- ward and Segment 3 backward, so that leaves some cells unpopulated. Segment 2 was propagated forward and backwards from the perigee. See Sections 4.2.3–4.2.3 for justification of these choi- ces. Additionally, the CR3BP model has no tuning parameters, so those cells are also unpopulated. The last column indicates the value of the respective angle offset used for the tuned parameters. Segment 2: Temporary Capture Phase This segment is the focus of Section 4.2. As mentioned in the “Method” section, we first must select an appropriate reference state on the converted trajectory to propagate in the various models. We use the state coherence measure to decide which state to choose as a reference. The state cohe- rence of the converted trajectory for Segment 2 is shown in Fig. 4.13(a). This figure indicates the chosen reference point near the center of the time interval, from which we propagate forward and backward in time. First, we use the CR3BP model. The resulting trajectory is shown in Fig. 4.13(b) along with the converted trajectory. The resulting trajectory has visually similar behavior, but the distance measure indicates and average separation of nearly 1 million km. If we apply the BCP model to the same initial condition, the separation improves slightly. First, we use the computed initial model parameters, which yields a small improvement as seen in Fig. 4.14(a). Next, we note that tuning the lunar phasing angle reduces the overall separation nearly by an order of magnitude, as seen in Fig. 4.14(b). The trajectory is now visually very similar and the DTW distance metric confirms this. Note that the scale of the plot covers around 6:0 10 6 km in dimensional units, compared to the 1:810 5 km average separation. The choice of the lunar 83 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) Figure 4.13: CR3BP model results for Segment 2 (a) The coherence, q, of the converted trajectory with the red dot indicating the chosen reference point. The value in red indicates the coherence of that point. (b) The converted asteroid trajectory (black), integrated trajectories forward (red) and backward (blue). The light blue and red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as numbers in red, blue and black for the forward, backward and entire trajectory respectively. phasing angle adjustment comes from a grid search across a large number of angular offsets in the range 0 2 [0; 2). From the grid search, we selected a lunar phase angle offset that minimizes the separation. Next, we apply the ER3BP model to the same initial conditions. When we use the ephemeris to determine the initial true anomaly of Earth, the resulting asteroid trajectory is not similar to the converted ephemeris trajectory as seen in Fig. 4.15(a). However, tuning the initial true anomaly greatly improves the result, as seen in Fig. 4.15(b). Here, the average separation is on the same order of magnitude as the tuned BCP model results. As with the tuning in the BCP model, we chose the true anomaly offset from a grid search across the range f 0 2 [0; 2). From the grid search, we selected a true anomaly offset that minimizes the separation. We want to note that the phasing offsets required for both BCP and ER3BP models to achieve best performance were 84 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) Figure 4.14: BCP model results for Segment 2 (a) The converted asteroid trajectory (black), inte- grated trajectories forward (red) and backward (blue). The light blue and red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as numbers in red, blue and black for the forward, backward and entire trajectory respectively. (b) Here the initial lunar phasing angle has been manually adjusted by +89.388 deg. significant at nearly 90 degrees. Also note that the ER3BP had a very poor fit in its unadjusted mode, while the BCP was good enough to model most of the temporary capture. 85 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) Figure 4.15: ER3BP model results for Segment 2 (a) The converted asteroid trajectory (black), integrated trajectories forward (red) and backward (blue). The light blue and red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as numbers in red, blue and black for the forward, backward and entire trajectory respectively. (b) Here the initial true anomaly has been manually adjusted by +86.1 deg. Segment 1: Pre-Capture Phase & Temporary Capture Phase Next, we study segment 1 in the CR3BP model. We once again use the state coherence to choose an appropriate reference state. As discussed in Section 4.2.3, the reference point for this segment is chosen near the end of the segment, as shown in Fig. 4.16(a). The integrated trajectory deviates significantly from the converted ephemeris trajectory during temporary capture, and maintains a large average separation after entering the interior region. This is shown in Fig. 4.16(b). One positive aspect of the model is that it at least allowed the asteroid to pass throughL 2 andL 1 and enter the interior region, instead of being reflected atL 2 back into the exterior region. If we apply the BCP to segment 1, we improve the similarity as seen in Fig. 4.17(a) and (b). However, tuning the initial lunar phase angle only improves the result slightly. This result is shown in Fig. 4.17(c) and (d), with the interior resonance clearly deviating from the converted ephemeris 86 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) Figure 4.16: CR3BP model results for Segment 1. (a) The coherence, q, of the converted trajectory with the red dot indicating the chosen reference point. The value in red indicates the coherence of that point. (b) The converted asteroid trajectory (black) and integrated trajectory backward (blue). The light blue lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number in black for the entire trajectory. trajectory. As with segment 2, a grid search was used to find a suitable offset for the lunar phase angle. We finally apply the ER3BP to segment 1 and note that the similarity can be significantly improved. The computed initial true anomaly result in Fig. 4.18(a) and (b) show that the model even fails to transit to the interior region. However, tuning the initial model parameter, produces close agreement as seen in Fig. 4.18(c) and (d). Again, the average separation is three orders of magnitude less than the range of the trajectory. Once again, a grid search was used to determine an appropriate true anomaly offset. Note here that while the BCP model had a poor fit even when adjusted, the ER3BP yielded a good fit with a relatively small adjustment of phasing. Both, however managed to simulate the general behavior of the temporary capture, with a topologically similar trajectory shape. 87 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) (c) (d) Figure 4.17: BCP model results for Segment 1. (a) The converted asteroid trajectory (black) and integrated trajectory backward (blue). The thin blue lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number black for the entire trajectory. (b) shows the similarity for the entire resonance period back to the previous Earth encounter in 1979. (c) shows an improved solution by adjusting the initial lunar phasing angle by 1.400 deg. (d) shows the entire resonance period for trajectory in (c). 88 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) (c) (d) Figure 4.18: ER3BP model results for Segment 1. (a) The converted asteroid trajectory (black) and integrated trajectory backward (blue). The light blue lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number in black for the entire trajectory. (b) shows the similarity for the entire resonance period back to the previous Earth encounter in 1979. In (c), the initial true anomaly has been manually adjusted by -10.42 deg. (d) shows the entire resonance period for trajectory in (c). 89 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE Segment 3: Temporary Capture Phase & Post-Capture Phase Finally, we study segment 3. We first apply the CR3BP model to this segment. As before, we must first select an appropriate reference state. Justification for our chosen state is seen in Fig. 4.19(a) and the resulting trajectory is seen in Fig. 4.19(b). It is clear that the trajectory is not similar to the converted ephemeris trajectory with the average separation confirming this result. (a) (b) Figure 4.19: CR3BP model results for Segment 3. (a) The coherence, q, of the converted trajectory with the red dot indicating the chosen reference point. The value in red indicates the coherence of that point. (b) The converted asteroid trajectory (black) and integrated trajectory forward (red). The light red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number in black for the entire trajectory. Integrating the same initial state in the BCP model does not improve the trajectory similarity, even if the initial lunar phase angle is hand tuned. This is shown in Fig. 4.20. A grid search was performed across the lunar phase angle, but the variation in similarity across the grid was negligible. When we apply the ER3BP to segment 3, the results improve. As shown in Fig. 4.21(a) and (b), using the computed true anomaly, the similarity is better than both the CR3BP and BCP models. 90 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) Figure 4.20: BCP model results for Segment 3. (a) The converted asteroid trajectory (black) and integrated trajectory forward (red). The light red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number black for the entire trajectory. (b) shows the similarity for the entire resonance period up to the next Earth encounter in 2028. Adjusting the initial lunar phasing had no significant effect on the trajectory similarity. Moreover, if we hand tune the true anomaly, we achieve similarity across the entire segment. This is shown in Fig. 4.21(c) and (d). Note that the average separation is 7:2 10 5 km while the trajectory covers 3 10 8 km during the exterior resonance. The tuned value of the true anomaly is once again determined from a grid search. We find that similar to our results for Segment 1, the BCP is a poor fit in spite of adjusted phase angle, while the ER3BP yields a good fit when unadjusted and better fit when adjusted. The adjustment required was once again relatively small. 91 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE (a) (b) (c) (d) Figure 4.21: ER3BP model results for Segment 3. (a) The converted asteroid trajectory (black) and integrated trajectory forward (red). The light red lines represent the distance measuring points used by the DTW algorithm. The resulting average trajectory distances are indicated as the number black for the entire trajectory. (b) shows the similarity for the entire resonance period up to the next Earth encounter in 2028. In (c), the initial true anomaly has been manually adjusted by -12.952 deg. (d) shows the entire resonance period for trajectory in (c). 92 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE 4.2.5 Section Summary We have applied a new method for comparing the shape of space trajectories with a quantifiable metric called Modified Dynamic Time Warping. We used the MDTW metric to test the fidelity of simulated trajectories from several simple dynamical models to trajectories in the JPL ephemeris model. Our results indicate that the ER3BP can closely model the trajectory of Asteroid 2006 RH 120 for all 3 segments studied. However, this required the manual modification of the initial true anomaly of the Earth. During the pre-capture and post-capture phases, the Asteroid is far away from the Earth-Moon system. Hence the eccentricity of the Earth’s orbit dominates the control of the dynamics of these phases and associated segments. This is confirmed by the following observation. Phasing adjustments to the BCP provided only minor to nonexistent improvements to its model of the pre-capture and post-capture phases. This indicates that these phases are not significantly affected by the 4-Body Problem, which was expected. Whereas for the ER3BP, minor adjustments to the phasing provided significant improvements to its models of the pre-capture and post-capture phases. During the Capture phase, the Asteroid is now close to the Earth-Moon system. Here the influence of the Moon becomes important. The unadjusted BCP model provided a much better model of the Capture phase than the unadjusted ER3BP model by an order of magnitude. Once the phasing has been adjusted for both models, their performance is nearly the same, with the BCP only slightly better than the ER3BP model. We concluded that in general, the BCP provides a better model than the ER3BP during the Capture phase. However, the improvement of the adjusted BCP model is relatively small. This is understandable as the Asteroid’s closest approach is 4.4 times the radius of the lunar sphere of influence. Hence, one would expect only a small effect on the Asteroid orbit from the Moon. This was surprising since the motion of the Asteroid was in the chaotic region of the Sun-Earth-Moon system. One would expect even a small perturbation from the Moon can result in large deviations in the Asteroid orbit from the 3-body models. But this proved to be false. However, the BCP is still 93 of 268 4.2. PART II:TEMPORARY CAPTURE PHASE the best choice for modeling the temporary capture of Asteroid 2006 RH 120 when using computed parameters. Although the distances computed between trajectory solutions were at times small compared to the scale of the problem, they are not suitable for high accuracy predictions. The intent is to study overall dynamics of simpler dynamical systems as an approximate representation of the real system. We believe this study will allow others to determine if these simple models are adequate for their purposes. For future studies, the sensitivity and delicate effects of the lunar phasing require careful exa- mination. For instance, the manual adjustments of the phasing parameters required to fit the models to the ephemeris data of the Asteroid can be quite large, e.g. 90 degrees during the Capture phase. Whereas the phasing adjustments for the pre-capture and post-capture phases were rela- tively minor, e.g. on the order of 10 degrees. This indicates a significant mismatch between these simple models and the real trajectory due to the phasing. We recall a feature of the Genesis trajec- tory involving the Moon which is related to these phasing effects. It was noted that in order for a Genesis solution to reach the required landing site on Earth, the phasing of the system is severely restricted within a narrow range. From the dynamical systems point of view, this relates to the presence of invariant structures acting as separatrices of the different regions of motion. Since the effects are clearly due to the 4-Body Problem, this dynamics is much more subtle. Understanding this dynamics may help provide new insight for controlling the motions of asteroids using low energy techniques for retrieval, capture, and deflection. The novel use of the Modified Dynamic Time Warping algorithm to classify trajectories accor- ding to shape have many applications to mission design. It could form the basis for various machine learning algorithm for automation and optimization of mission design and analysis tasks. Combi- ned with parallel computing such as on the GPU, this could enable real time analysis of complex mission design problems. This in turn, can enable mission design support of concurrent engineer- ing, on-board autonomous mission design, and new classes of mission in the near future. 94 of 268 4.3. CHAPTER SUMMARY 4.3 Chapter Summary This chapter aimed to study the dynamics of transit and temporary capture at the Earth. We studied the particular case of asteroid 2006 RH 120 , which is the only observed temporarily captured natural object at the time of writing. The analysis in Section 4.1 showed that the Asteroid approached along the stable manifold of anL 1 halo orbit on the Sun-Earth CR3BP and departed on the unstable manifold ofL 2 halo orbit. Moreover, a resonance transition occurred which moved the Asteroid from a 27:29 interior resonance to a 21:20 exterior resonance with Earth. This transition means that the Asteroid has experienced close encounters with Earth in the past and is expected to do so in the future. Section 4.2 focused on the temporary capture phase and indicated that the Moon is the dominant factor controlling the Asteroid’s motion during temporary capture. This followed from the BCP being the model with the best overall agreement between reality and model when using computed parameters. When allowing for manual adjustment of the system parameters, we were able to achieve better overall performance using the ER3BP. However, it is be unlikely that such a manual adjustment can be known in another case. We also applied a new method for comparing trajectories in space, which we label Modified Dynamic Time Warping. 95 of 268 CHAPTER 5: EFFECT OF MOON ON TRANSIT DYNAMICS In this chapter, we study transit in the Sun-Earth system by comparing the effects of the CR3BP and BCP models. While we have in Chapter 4 studied an example of a known object that attained temporary capture at Earth as well as transit from interior to exterior regions, we now study how the Moon affects these dynamics in general. Under the assumption that objects are approaching on stable invariant manifolds, we compare the rates of escape, capture and impact of either the Earth or Moon in both models. Previous work in this field has used slightly different approaches. Cline[20] studied the mechanism of gravitational capture at planets that have their own natural satellites. For the specific case of the Earth-Moon system, his results indicate that objects already in orbit around the Sun can be ballistically captured at Earth due to the Moon, with some requirements on the initial heliocentric orbit. Moreover, under the assumptions of his study, objects with an Earth-centeredV 1 as high as 1.85 km/s can be captured. Later work by Granvik, Vaubaillon and Jedicke[38] computed the rate of asteroid capture in two models, one where the Earth and Moon are modeled as a combined point mass, and another where they are modeled separately. In both models, the positions of the Earth, Moon and Sun are computed from high fidelity ephemerides. They found that in general, the model with separate Earth and Moon allowed for more temporarily captured objects, and that objects with higher Earth-centric speed can be captured in that model. We note that they studied a larger range of energies. When expressed in Jacobi constant, they used test particles with 2:984 C 3:008, where we used 2:999450 C 3:0008875. However, nearly all of their resulting temporary capture objects fall in the range of 2:998 C 3:002, which is closer to our range limits. 96 of 268 5.1. APPROACH 5.1 Approach The goal is to study the difference in transit behavior between the Sun-Earth CR3BP and Sun- Earth-Moon BCP. Therefore, we need to establish some common features between the systems. The first and most obvious is that the two systems have aligned coordinate frames and units. The only difference is replacing a combined stationary Earth-Moon point mass with the individual point masses moving in circular orbits around their common barycenter. While the BCP is non- autonomous, it is periodic, so sampling the range of possible initial states of the system itself is reduced to sampling the initial lunar phase angle, 0 . We choose a discrete sampling of 0 with 0.5 degree spacing, corresponding to a separation of approximately one hour of the Moon’s motion. Refer to Section 2.2 and specifically Fig. 2.8(b) for the definition of the initial lunar phase angle, 0 . We focus on the behavior of objects approaching Earth at low energy from the interior region. The method we choose to represent this behavior is to place a set of fictitious asteroids on invariant manifolds of planar LyapunovL 1 orbits. We know that objects on the stable manifold of an orbit will approach and wind onto the orbit, after which it will depart along an unstable manifold of the same orbit. Therefore, we make the simplification that the asteroids are already on the unstable manifold of theL 1 orbit that is heading toward the Earth. This prevents wasting computation on trajectories that are already heading away from the Earth back into the interior region, which is what one side of the unstable manifold will do. One issue is that the location of the manifold initial conditions for a given orbit will be spread evenly over the orbit. In terms of relative orientation of the initial asteroid states and the Moon, each asteroid will be very different from the next. We address this issue by repositioning the initial asteroid states such that their relative orientation is similar. The technique we apply to perform this repositioning is called a Fundamental Interval. The Fundamental Interval of an invariant manifold is a small continuous strip of states near a single point on the orbit that represents the entire manifold. To generate this interval, we propagate all the states that are spread over the orbit until they return to thex-axis aty = 0, which is near the initial state of the orbit. The orbit’s initial state is on the right side of the loop and heading 97 of 268 5.1. APPROACH downward, resulting in an orbit with a clockwise motion. This is illustrated in Fig. 5.1(a), where 100 discrete original manifold states are highlighted with black circles and the manifold is the set of red trajectories. The endpoints of all the manifold trajectories now represents the entire manifold of the orbit, but localized to near the initial orbit state. For clarity, we highlight a single manifold trajectory in Fig. 5.1(b), showing its initial state near the orbit and the location where it intersects the Fundamental Interval. We denote the number of discrete manifold trajectories used byn, and for this study we chose to usen = 500. By choosing this Fundamental Interval to represent the manifold, we now have a common angle between the asteroid’s initial states and the Moon if we use a triangle with the vertices asteroid-(Earth-Moon barycenter)-Moon to define the angle. (a) (b) Figure 5.1: Diagram of Fundamental Interval for a Sun-Earth planarL 1 Lyapunov orbit. (a) shows all the trajectories on the unstable invariant manifold, their initial states, and their endpoints on thex-axis, which is the Fundamental Interval. The light blue is the orbit, the black circles are the initial states of the invariant manifold, and the red trajectories are the set of invariant manifolds propagated until they return toy = 0 near the initial state of the orbit. The Fundamental Interval is the black line at the end of the manifold shown. (b) shows just one of the unstable manifold trajectories as a red dashed line, its associated initial state on the orbit as a circle, and where it intersects the Fundamental Interval. 98 of 268 5.1. APPROACH Now that we have defined the initial states of the asteroids, we outline the method of analyzing the transit dynamics. The asteroids are propagated forward in time for a fixed time t, and we analyze the behavior of the resulting trajectory. The propagation is first performed in the Sun- Earth CR3BP, and then we use the same initial states in the Sun-Earth-Moon BCP for each value of 0 listed above. Each individual asteroid is then categorized. We define five different categories of behavior: interior escape, exterior escape, capture, Earth impact and Moon impact. Defining both impact categories is relatively straightforward. The condition for impact is simply that the minimum radius relative to the given body is less than the radius of that body as shown in Eq. 5.1. S Earth =fi : min t (j~ r i (t)~ r Earth (t)j)R Earth g S Moon =fi : min t (j~ r i (t)~ r Moon (t)j)R Moon g S impact =S Earth [S Moon (5.1) Where~ r i (t) is the trajectory of the i-th manifold, while~ r Earth (t) and~ r Moon (t) are the positions of the Earth and Moon, respectively. S Earth andS Moon are the sets representing Earth and Moon impacts, whileS impact represents all impacts. Lastly,R Earth andR Moon are the equatorial radii of the Earth and Moon. For the CR3BP, the position of the Earth is not time dependent. Additionally, the position of the Moon in the CR3BP has no definition and thus we must use the assumption that there are no lunar impacts in the CR3BP. The other 3 categories are defined geometrically. Refer to the diagram in Fig. 5.2 for the definition of the interior, exterior, and capture regions. The position of the final state relative to those 3 regions decides which category an asteroid belongs to. To qualify for any of those 3 categories, the asteroid must not satisfy the condition for impact. 99 of 268 5.1. APPROACH These rules are summarized in Eq. 5.2, where we take the complement of the three sets with the set of all impacts in order to remove them. S capture =fi :j~ r i (t f )~ r EMB jR capture gnS impact S interior =fi :j~ r i (t f )j< (1)gnS impact S exterior =fi :j~ r i (t f )j (1)gnS impact (5.2) Where~ r EMB is the position of the Earth-Moon barycenter (recall that it is fixed) andR capture is a radius parameter used to define capture. We have chosen to use the following value for the parameter: R capture =x L 2 x L 1 = 0:020089217647244 (5.3) Wherex L 1 andx L 2 are thex-coordinates of the two Lagrange points. This definition draws a cir- cular region around the Earth-Moon barycenter that reaches about twice as far as the two Lagrange points. We chose this since it will encompass even Lagrange point orbits of relatively low Jacobi constant (high energy) while still remaining near Earth. If part of the orbit itself falls outside the capture region, then even objects that stay on the orbit would be considered escaped, which is not our desired definition. Refer to Fig. 5.2 for an example of such an orbit enclosed inside the region. We must compute each of these sets once for the CR3BP andm times for the BCP, one for each value of 0 . In our case, the choice of a 0.5 degree spacing of 0 implies thatm = 720. The sets we just defined allow us to estimate the probability of transit, capture and impact by using the fraction of the members belonging to each category as an estimate. Taking an average of these probabilities over the whole range of 0 allows us to determine the overall effect the Moon has on transit dynamics. Moreover, analyzing the probabilities as a function of 0 allows us to determine how sensitive the rates are to the position of the Moon, and if some relative orientations 100 of 268 5.1. APPROACH have extreme behavior. We compute the estimated probabilities as indicated in Eq. 5.4, where “category” is replaced with each category listed in Eqs. 5.1 and 5.2 above in turn. f category = 1 n X i2Scategory 1 (5.4) Once we have computed the category probabilities for every lunar phase angle, we repeat the procedure for a range of energy levels. We present the results of this analysis in the following section. (a) (b) Figure 5.2: Capture, Exterior, and Interior Region definition diagram. (a) Close-up near Earth. The forbidden region is shown as a gray area, the triangles areL 1 andL 2 , and the red lines separate the 3 regions. The capture region is inside the ring around the Earth, the interior region is to the left, toward the Sun, and the exterior is to the right. A largeL 1 orbit is shown for scale and to indicate why the size of the capture region is chosen to be so large. (b) Large scale view. 101 of 268 5.2. RESULTS 5.2 Results For a range of energies satisfying 2:999450 C 0 3:0008875 in the Sun-Earth CR3BP, we present the transit, capture and impact rates of the same initial states in the Sun-Earth-Moon BCP. The upper limit was chosen since it is very close to the lowest energy at which orbits can exist atL 1 . The Lagrange point itself has a Jacobi constant ofC L 1 = 3:000897941481294. We used a fine step inC 0 for lower energy to capture the chaotic behavior, and a coarse step at high energy. Specifically, C = 0:0000125 for 3:000675<C 0 3:0008875, C = 0:000025 for 3:000200< C 0 3:000675, and C = 0:000050 for 2:999450 C 0 3:000200. This choice results in 52 discrete values of Jacobi constant. If we take the number of manifold trajectories and lunar phase angles into account, we require a total of approximately 1:8710 7 numerical propagations for this study. We choose to define capture based on the fixed propagation time t, which does not indicate capture for an infinite amount of time. The value we chose to use for this analysis is t = 6, which corresponds to 3 years. An object that remains in the vicinity of Earth for that long after passing byL 1 , should be considered as a temporarily captured object. Keep in mind that Asteroid 2006 RH 120 was only captured for one year. However, we should note that extending the upper time limit can allow a object that is captured by our definition to impact or escape. Thus the rates we present are inherently tied to the choice of propagation time. We indicate the rates as a function of initial lunar phase angle 0 for some choice values of the Jacobi constant in Figs. 5.3–5.6. We also present the the overall rates across all 0 and the rates in the CR3BP, both as functions of the Jacobi constant in Figs. 5.7 and 5.8. At the lowest energy, the majority of asteroids are reflected back into the interior or transit into the exterior as seen in Fig. 5.3. However, a significant fraction become captured in the CR3BP. This can be explained by the small gateways at this energy, with asteroids repeatedly reflecting off the forbidden region and some of them never approaching the gateways perfectly in order to escape. The asteroids are already on manifolds leaving L 1 and approaching Earth, so the only 102 of 268 5.2. RESULTS path to leave is through the gateways, which becomes less likely when they are small. What is interesting is that the amount of captures are significantly reduced in the BCP. On the other hand, there are no Earth impact at all in the CR3BP, while that rate increases for the BCP. We can explain the absence of impacts in the CR3BP with the natural path of the invariant manifolds. It must be that the manifolds at this energy level approach Earth with the entire manifold away from the surface and naturally maintains this minimum separation. This avoidance of impacts, combined with the small gateway, results in a high rate of capture. In contrast, the manifolds do not behave that well in the BCP, explaining the increase in Earth impacts. It is likely that even a single near- encounter with the Moon is enough to lower the perigee of an asteroid to cause an impact. The rate of lunar impacts in the BCP are high relative to the higher energy levels, which can be explained by the low energy level and small gateways. The same reason for captures being more likely results in a longer time spent near the Earth and Moon, thus increasing the likelihood of lunar impact. While the average rate of exterior transit is increased in the BCP relative to the CR3BP, the rate of reflection into the interior is significantly reduced in the BCP. Thus, the effect of the Moon at this energy is to trade interior reflections for higher exterior transit, and impacts of both Earth and Moon. The dependence of the rates on the initial position of the Moon is not significant at this energy, as indicated by the polar plot in Fig. 5.3. If we increase the energy slightly, the rates of capture are severely reduced in both models, as seen in Fig. 5.4. The obvious change is the large increase in Earth impacts in both models and across all lunar orientations, with some being particularly extreme. The approximate mirror sym- metry between the exterior transit and Earth impact indicates that the Earth impacting asteroids were taken from a group that would have escaped through L 2 . The high rate of impacts in the CR3BP is an indication that the manifolds naturally intersect the Earth at this energy. Additio- nally, the addition of the Moon does not significantly change the average rate of Earth impact, but between 1st quarter and new Moon there is a peak. It is likely that the manifolds naturally cross the orbit of the Moon, and at this phasing, the Moon is at the crossing point at the same time the 103 of 268 5.2. RESULTS 1st Quarter 3rd Quarter Full Moon New Moon BCP BCP MEAN CR3BP MEAN 1st 3rd Figure 5.3:L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP,C 0 = 3:000875. The three linear graphs shows the percentage rates vs. initial lunar phase angle as solid lines of data. Solid, thick horizontal bars indicate CR3BP rates, which are not dependent on 0 , while dashed horizontal bars indicate average BCP rates. The bottom left panel has the Moon phases indicated on thex-axis for reference. The bottom right panel shows BCP rates on a polar plot of 0 . The center represents 0% and the circle’s edge is 100%. For a given 0 , the percentage rates are stacked on top of each other. At this energy, the rate of capture is significant with a nonexistent Earth impact rate without the Moon. Adding the Moon has the effect of significantly increasing the rate of Earth impact and introduces a high rate of lunar impact, while reducing the rate of capture. Additionally, the rate of interior escape is highly reduced, while exterior escape is increased. Make note of the relatively small dependence on initial lunar phase angle on the rates. It would seem that we have traded away some interior escape for exterior escape and impacts of both types. It is likely that the lunar and Earth impact cases in the BCP come from some of the capture cases in the CR3BP. asteroids pass through the manifold there. As before, the exterior transit rates are higher, while the interior reflection rates are lower when comparing the BCP to the CR3BP. The overall decrease 104 of 268 5.2. RESULTS in interior reflection also fuels the lunar impacts, with a small range of lunar orientations causing significant impacts. 1st Quarter 3rd Quarter Full Moon New Moon BCP BCP MEAN CR3BP MEAN 1st 3rd Figure 5.4:L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP,C 0 = 3:00075. The three linear graphs shows the percentage rates vs. initial lunar phase angle as solid lines of data. Solid, thick horizontal bars indicate CR3BP rates, which are not dependent on 0 , while dashed horizontal bars indicate average BCP rates. The bottom left panel has the Moon phases indicated on thex-axis for reference. The bottom right panel shows BCP rates on a polar plot of 0 . The center represents 0% and the circle’s edge is 100%. For a given 0 , the percentage rates are stacked on top of each other. At this energy, the average rate of Earth impact in both models is similar, and very high. Similarly to Fig. 5.3, exterior escape is increased, while interior escape is reduced when adding the Moon. The mirror symmetry between exterior escape and Earth impact suggests that some asteroids that would have transited are instead impacting Earth. The rate of lunar impact is present but low. However, the BCP adds a measurable rate of capture in contrast to the nonexistent rate for the CR3BP. Moreover, we can now start seeing a more pronounced effect of initial lunar phase angle on the rates. The significant reduction in interior escape must be providing the material for capture, lunar impact, as well as the increase in exterior escape. 105 of 268 5.2. RESULTS We next study a Jacobi constant similar to that of Asteroid 2006 RH 120 during its escape from Earth. Figure 5.5 makes it clear that the lunar orientation is much more important at this energy. While the overall rate of interior reflection is similar between the two models, some orientations result in very high rates of reflection. Those same orientations reduce the exterior transit rates. It is interesting to note that the range of orientations that have higher lunar impact rates are also those that result in a more even distribution of the rates between the five categories. This can be explained by close encounters with the Moon causing a variety of orbital changes that can place the asteroid on a path to several places. We also note that the overall rate of Earth impacts is slightly higher in the BCP. This is accounted for by a large peak in Earth impacts when the Moon is between 1st quarter and full Moon. Other orientations generally have lower rates of Earth impact, with a notable peak soon after 3rd quarter Moon. The final Jacobi constant is representative of the approach of Asteroid 2006 RH 120 and is seen in Fig. 5.6. It is now clear that certain orientations of the Moon result in 100% of asteroids belonging to a single category. A very large range of lunar orientations result in complete reflection into the interior, and the overall rate is nearly 3 times higher in the BCP. We achieve 100% lunar impact near 0 = with some others occurring around 0 = 0. The first can be explained by the proximity to the Moon of the Fundamental Interval we are using. At this energy, theL 1 orbit has a perigee that is near the path of the Moon’s orbit. Therefore, the asteroids will all have positions near the Moon and low velocities relative to the Moon, resulting in a high rate of lunar impacts. Once again, the orientations that cause many lunar impacts also cause many Earth impacts. The reasoning is the same as before, with lunar encounters producing a perturbation that reduces the perigee, which in turn produces Earth impact. However, the overall rate of Earth impacts are significantly reduced at this energy, as is the rate of exterior transit. We can also draw global conclusions from the complete results presented in Figs. 5.7 and 5.8, starting with the first figure that focuses on capture and impact. Most obvious is that the rate of capture is higher in the BCP for C 0 > 3:0001, corresponding to energies where the forbidden 106 of 268 5.2. RESULTS 1st Quarter 3rd Quarter Full Moon New Moon BCP BCP MEAN CR3BP MEAN 1st 3rd Figure 5.5:L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP,C 0 = 3:000425. The three linear graphs shows the percentage rates vs. initial lunar phase angle as solid lines of data. Solid, thick horizontal bars indicate CR3BP rates, which are not dependent on 0 , while dashed horizontal bars indicate average BCP rates. The bottom left panel has the Moon phases indicated on thex-axis for reference. The bottom right panel shows BCP rates on a polar plot of 0 . The center represents 0% and the circle’s edge is 100%. For a given 0 , the percentage rates are stacked on top of each other. At this energy, the Moon reduces the rate of exterior transit in order to increase the rate of interior transit and Earth impact equally. While capture rates are low in both models, the Moon seems to increase the rate nonetheless. We also note that the initial position of the Moon now has a very strong effect on the rates, with a very high rate of Earth impact between 1st quarter and full Moon as an example. The rates of capture and lunar impact similarly have peaks between 3rd quarter and new Moon. region still exists. Lunar impacts remain similar across most energies, with the exception being higher impact rate at very low and very high energy. This was driven at low energy by the smaller gateways increasing the length of time spent near the Earth and Moon. Another important finding 107 of 268 5.2. RESULTS 1st Quarter 3rd Quarter Full Moon New Moon BCP BCP MEAN CR3BP MEAN 1st 3rd Figure 5.6:L 1 transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP,C 0 = 3:00020. The three linear graphs shows the percentage rates vs. initial lunar phase angle as solid lines of data. Solid, thick horizontal bars indicate CR3BP rates, which are not dependent on 0 , while dashed horizontal bars indicate average BCP rates. The bottom left panel has the Moon phases indicated on thex-axis for reference. The bottom right panel shows BCP rates on a polar plot of 0 . The center represents 0% and the circle’s edge is 100%. For a given 0 , the percentage rates are stacked on top of each other. At this energy, a combination of a decrease in Earth impact and much larger decrease in exterior escape rates fuels a large increase in interior escape in the BCP. Also, capture rates are increased slightly by the Moon. The lunar phase dependency is extreme in this case and the averages aren’t as useful, as indicated by one orientation resulting in 100% Earth impact rate, in spite of having a low average rate. is that there is a small range of energies for which both capture and lunar impact are slightly more likely in the BCP nearC 0 = 3:0005, with an associated decrease in Earth impacts. This knowledge can be useful for choosing asteroids for the purpose of asteroid capture missions. Asteroids near this energy level may have more feasible solutions for capture in the Earth-Moon system. For most 108 of 268 5.2. RESULTS energies, the rate of Earth impacts is slightly higher in the BCP, with the exception being the very lowest possible energies and very high energies withC 0 2:9998. The higher Earth impact at low energies can be explained by the natural tendency of manifolds in the CR3BP to avoid impact, and the small perturbations in the BCP interfering with that natural impact avoidance. It is interesting that the rate of Earth impacts is significantly lower in the BCP at very high energy. One of our hypotheses was that the Moon significantly affects the rate of asteroid impacts on Earth. It would seem that the evidence presented here shows an increase at the lower end of the energies studied, and a decrease at higher energies. This is a significant finding. These higher energies are the home of most potentially hazardous asteroids (PHAs). Thus it is the impact rate at that energy that may be more significant for protecting the Earth from cataclysmic events. Keep in mind that PHAs are asteroids that have a non-negligible probability of Earth impact, and would generate a significant amount of energy on impact. The impact energy is mostly a function of the asteroid’s mass and speed relative to Earth at impact. While we make no assumptions about the mass of the asteroid’s in our analysis, we can make some comments on the speed relative to Earth. The energies studied here are still considered “low energy” for an astrodynamics perspective. Thus, the velocity once such an asteroid reaches the Earth’s surface will be relatively low. As an example, the surface velocity of the highest energy we studied would be approximately 11.27 km/s, which corresponds to aV 1 1:5 km/s relative to Earth. Very few currently considered PHAs have an energy approaching that of our analysis, with most having much higher Earth-relative energy. Two PHAs were indicated by Chesley et al.[18] as having very lowV 1 relative to Earth. The objects in question were 2000 SG 344 and 2001 GP 2 , with an Earth surface speed of 11.34 and 11.39 km/s, respectively. Their low energy raised the question if they may be man-made objects, for which evidence was provided in Chodas and Chesley[19], but not fully confirmed. At these speeds the asteroid needs to be very large to produce enough energy to be considered hazardous. Thus, our analysis shows that the asteroids that are capable of causing more damage to the Earth and our civilization are less likely to impact when the Moon is taken into account. 109 of 268 5.2. RESULTS Another overall trend regards the rate of interior and exterior transit. Figure 5.8 indicates a general agreement between the CR3BP and BCP model exterior and interior transit rates for lower energies. ForC 0 3:0004, the peaks and valleys of both BCP and CR3BP average rates line up. However, as we increase the energy slightly, the two models no longer behave similarly. In the range 3C 0 3:0004, the Moon seems to have the effect of reducing exterior escape rates while increasing interior escape rates. This seemingly indicates that the Moon is preventing asteroids from transiting to the exterior region when compared to a fictitious case where the Earth has no moon. The effect of the Moon on these rates becomes more erratic for C 0 3. Keep in mind that the interior and exterior rates have less meaning for C 0 3, since the forbidden region no longer exists. Our geometric definition of the regions is appropriate when there is a barrier that prevents motion between interior and exterior, but at high energy, and orbit can cross this barrier repeatedly. A definition better suited for this energy would be to use an osculating semimajor axis. We started this study not intending to examine these energies, and thus chose our definition based on that. After deciding to extend the energy, we chose to keep the same definition for continuity. However, the sum of exterior and interior still indicates the total rate of escape from the Earth- Moon system at high energy. It is only their individual values that are less meaningful. Thus, the above discussion regarding the trend of interior and exterior transit applies to the energy range where they have more significant meaning. If we study the distribution of impact and capture at some of the more interesting cases at high energy in the BCP, there is a pattern that emerges. Refer to Fig. 5.9 for the following discussion. As mentioned earlier, the green and light brown regions can be considered a combined region signifying “escape”, so we are less concerned with those. However, forC 0 = 2:9996, we find a significant rate of lunar impact near 0 = 0. Thus, while the overall rate of lunar impacts are low for this energy, as indicated by Figs. 5.7 and 5.8, some orientations cause very high rates. As we increase the energy, the same orientations causing lunar impact are now combined with high rates of Earth impact and nearly zero rate of escape. At the highest energy studied, the lunar and Earth 110 of 268 5.2. RESULTS impacts are joined by a high rate of capture at certain orientations. We can therefore draw the conclusion that asteroids approachingL 1 at these energies while the Moon is near its full Moon phase should be carefully studied. These asteroids are potentially good targets for capture, but also potentially dangerous if they are large enough. 111 of 268 5.2. RESULTS Higher Energy Figure 5.7: L 1 capture and impact statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, 2:99945 C 0 3:0008875. Solid lines indicate rates in the CR3BP, while dashed lines indicate rates in the BCP averaged over 0 . Lower energy to the right, and higher toward the left. The lowest energy possible for this study is that ofL 2 , indicated by the vertical dashed black line. A few sample polar plots are shown to indicate how the lunar phase angle dependency changes with energy. These polar plots do not represent the same Jacobi constants shown in Figs. 5.3–5.6. There are no lunar impact rates for the CR3BP. The rate of Earth impacts in both models follow a similar overall trend forC 0 3, with a sharp increase on the right, followed by a slow decline as energy increases to the left. Interestingly, average rates of Earth impact are actually higher with the Moon present for the lower energies. However, as we approach higher energy, the rate of Earth impact is lower in the BCP starting aroundC 0 2:9998. There is also a large peak in Earth impacts on the left for the CR3BP that is not present in the BCP. Lunar impacts are relatively low for all energies. For most energies, the capture rate is low in both models. Two exceptions occur in the CR3BP, at the lowest energy and aroundC 0 2:9998. At the extremely low energy, the gateway is almost closed, so it is more likely that asteroids already approaching Earth from L 1 will be trapped for a long time. Moreover, forC 0 3:0001, the rate of capture in the BCP is higher, but lower for C 0 3:0001. 112 of 268 5.2. RESULTS Higher Energy Figure 5.8: L 1 interior and exterior transit statistics of the Sun-Earth-Moon BCP vs. Sun-Earth CR3BP, 2:99945 C 0 3:0008875. Data layout is the same as in Fig. 5.7. Note that the interior escape follows a similar trend forC 0 3:0004, with peaks and valleys in the same places. However, the sharp peak at very low energy in the CR3BP is erased by the Moon. Exterior transit rates also behave similarly for C 0 3:0004. For 3 C 3:0004, it is clear that the Moon increases interior escapes, while reducing exterior escape, thus reflecting asteroids back to the direction they approached. C 0 = 3 is a critical value, since that is where the Forbidden Region disappears and all motion is allowed in the plane. 113 of 268 5.2. RESULTS 3rd Quarter 1st Quarter Full Moon New Moon (a)C 0 = 2:999600 3rd Quarter 1st Quarter Full Moon New Moon (b)C 0 = 2:999550 3rd Quarter 1st Quarter Full Moon New Moon (c)C 0 = 2:999500 3rd Quarter 1st Quarter Full Moon New Moon (d)C 0 = 2:999450 Figure 5.9: Example high energy transit rates vs. lunar phase angle in the BCP. The angle around the polar plot represents the initial lunar phase angle 0 . (a)C 0 = 2:999600. (b)C 0 = 2:999550. (c)C 0 = 2:999500. (d)C 0 = 2:999450. Note the significant lunar impact rates near 0 = 0 for all these energies. For the three higher energies, we also see a high rate of Earth impacts along with the lunar impacts. Finally, the highest energy produces a high rate of capture for two specific orientations of the Moon. 114 of 268 5.3. CHAPTER SUMMARY 5.3 Chapter Summary This chapter aimed to study the more general effects of the Moon on transit dynamics. Asteroids are assumed approaching from the interior region on the manifold of anL 1 Lyapunov orbit. We determined that for lower energies, the overall effect of the the Moon reduced reflection back into the interior region with a slight increase in exterior region transit, lunar impact, and capture. At medium energy, interior reflection is greatly increased and exterior transit greatly decreased, indicating that the Moon is preventing transit. Capture rates in the BCP were higher in the range of energies considered “low”, while becoming much lower than the CR3BP for high energy. Thus the Moon is effectively preventing asteroid capture at higher energy. Lunar impacts are not radically different across the energies considered. We do however see an upward trend as we approach even higher energies, possibly indicating that more “typical” asteroid approaches are stopped by impact with the Moon. For medium energies, the overall rate of Earth impact increased in the BCP along with a slightly increase in capture and lunar impact rates. At high energies, the rate of escape into the interior region significantly increases, with some orientations of the Moon causing 100% interior escape. The capture rate also increases at these energies with a range of lunar orientations resulting in 100% capture rate. Earth overall impact rates are similar or slightly higher in the BCP at the lower energies studied, but there are some lunar orientations that cause a high rate of impacts. Most significantly, we found that Earth impacts are reduced by the Moon at high energy. Asteroids in this range are ones more likely to pose a threat to Earth, due to larger kinetic impact energy. We studied some high energy cases in detail with respect to orientation and found that asteroids approachingL 1 at these energies while the Moon is near its full Moon phase should be carefully studied. These asteroids are potentially good targets for capture, but also potentially dangerous if they are large enough. 115 of 268 CHAPTER 6: ANALYSIS OF MULTIBODY RESONANCE WITH APPLICATIONS TO THE GALILEAN SYSTEM This chapter is dedicated to the study of orbital resonance, with some general concepts and some specifically applied to the Jupiter-Europa CR3BP and the system of Galilean moons. The main goal of this chapter is to discuss the stability of resonances in the system of Galilean moons, which is the focus of Section 6.3 and 6.4. The work in Section 6.4 approaches the problem of spacecraft disposal at end of mission for a notional mission to land on Jupiter’s moon Europa and is motivated by planetary protection requirements. We require some analysis techniques to accomplish this, and they are presented in Sections 6.1 and 6.2. Section 6.1 explores a method of predicting where stable and unstable resonances will occur in phase space, and thus being able to predict the topology of a Poincar´ e map. Section 6.2 derives a method for estimating finite time stability of resonant trajectories in both the CR3BP and perturbed models. Finally, we discuss a new method for computing “Compound Orbits” in Section 6.5. We define Compound Orbits as trajectories that exhibit behavior mirroring that of multiple other periodic orbits. They are in a way constructed using other periodic orbits as building blocks. These orbits are useful since they can make loops around multiple different orbital centers in a given itinerary. 6.1 Resonance Prediction In this section, we discuss the topology we can expect to see in a Delaunay projection of the Poincar´ e map, see Section 6.1.1. Additionally, we show in Section 6.1.3 that we can use this predicted topology to efficiently seed initial conditions for a Poincar´ e map, based on the desired study. For example, if one is only interested in a specific stable resonance, then only certain initial conditions are required to produce the appropriate Poincar´ e map. This minimizes the amount of computation by limiting the number of particles that may be in unstable resonance or in other 116 of 268 6.1. RESONANCE PREDICTION resonances. We also show an iterative method for locating the periodic resonant orbit at the center of a given stable island in Section 6.1.2. For this chapter, we will assume that a Poincar´ e map is generated withy = 0;x< 0, or alternatively =. 6.1.1 Topology in Delaunay Variables In general the topology one finds in a Poincar´ e map in the CR3BP will have two features. First, there will be islands of stable resonant motion, which will appear very structured, even if they may seem complex. Second, there will be a chaotic “sea” of unstable motion which cannot be further quantified visually. For more details on this structure, see Malhotra[61], Murray and Dermott[68], and Anderson[7]. The phase space commonly used for propagation in the CR3BP is the Cartesian position and velocity phase space. For a large part of this chapter, we will use the (L; g)-space for Poincar´ e maps, which we defined in Section 2.7. We remind you that this space is useful for analyzing resonant motion for several reasons, which are outlined in that section. If we examine an example Poincar´ e map in these variables in Fig. 6.1, we note the structured shapes of the stable islands in the (L; g)-coordinates. Figure 6.1: Example Poincar´ e map in Delaunay variables (L; g) in the Jupiter-Europa CR3BP for C 0 = 3:0039. A single particle’s intersections have been highlighted in red. The approximate resonance of this set is indicated by a green label. 117 of 268 6.1. RESONANCE PREDICTION You will also notice from the plot that there are varying numbers of islands at different values of L, and that the position along g is shifted. We will now discuss the cause behind this phenomenon, and will initially limit the discussion to the exact periodic resonant orbits located at the center of stable islands. A Poincar´ e map represents discrete points at which a trajectory pierces a plane. Therefore, a periodic orbit that closes on itself only after multiple crossings of the plane will necessarily produce multiple points in the map. In the case of resonant orbits, the number of intersections produced will be equal or greater than the number of full revolutions in the rotating frame. The number of full revolutions in the rotating frame is easily computed as =jpqj (6.1) where is known as the order of the resonance, also defined by Murray and Dermott[68]. The rea- son you may see more intersections of thex-axis than there are complete revolutions in the rotating frame is because the orbit can at times reverse angular direction. When the angular direction is switched, loops are formed in the position space representation in the rotating frame. These loops can be counted to determine the resonance, as discussed by Vaquero[87]. Loops can happen at the periapsis in the exterior region, where the local inertial angular rate exceeds the fixed inertial angular rate of the Primaries’ rotation. It can also occur at the apoapsis in the interior region, where the local inertial angular rate temporarily becomes slower than the Primaries’ rotation. . To illustrate this, refer to Figs. 6.2–6.5. In Fig. 6.2, we show a periodic orbit that crosses the negative x-axis only once, with no loops. The corresponding intersection on the Poincar´ e section is a single point at (L; g) = (1:100665971042604;). Since the g coordinate is cyclical, the top and bottom in this plot are actually connected. Note that the exact value of L for a 3:4 resonance would be L 3:4 = (4=3) 1=3 = 1:100642416298209, which is extremely close to the periodic orbit we show. If we add a small perturbation to the periodic orbit while maintaining the same Jacobi constant, we achieve a quasi-periodic orbit. Since the original orbit is stable, a small enough perturbation will result in a solution that remains nearby for all time. This is clear from the Poincar´ e map of the 118 of 268 6.1. RESONANCE PREDICTION quasi-periodic orbit, which seems to follow a loop around the periodic orbit. These quasi-periodic orbits can be also be found by reducing the periodic orbit to the center manifold as described by Jorba and Masdemont[49]. A real-world example of stable resonant motion in the solar system is Pluto. By considering Pluto an infinitesimal mass in the Sun-Neptune CR3BP, Malhotra and Williams[63] showed that it is permanently locked in a stable 2:3 resonance with Neptune. Pluto has a periapsis inside Neptune’s orbit, but due to this stable resonance, it never experiences a close encounter with Neptune. The example discussed above is a case where the order of the resonance is equivalent to the number of stable islands for theL that corresponds to the considered resonance ratio. Next, we consider the cases where loops form in the periodic orbit, causing more stable islands than than the value of. (a) (b) Figure 6.2: One stable island from a 3:4 resonant orbit with no loops in the Jupiter-Europa CR3BP forC = 3:01 (a) Trajectory plot, the black line shows the stable periodic orbit, the red lines are the quasi-periodic orbit and the gray is the forbidden region. (b) Poincar´ e map of the orbit intersections on the negativex-axis in (L; g)-coordinates. The black dots are the periodic orbit and the red dots are the quasi-periodic orbit. Since the loops form when periapsis velocities are very different from the frame rotation, the eccentricity determines the existence of loops. In Section 6.3 we show from Eqs. 6.27 and 6.28, that 119 of 268 6.1. RESONANCE PREDICTION the Jacobi constant controls the value of the eccentricity for a givenL. This means that generating an orbit with the same 3:4 resonance but with loops will require changing the energy level. An example is shown in Fig. 6.3. Here we see that the periodic orbit does have three intersections, but we still only achieve one large stable island when producing a quasi-periodic orbit through a small perturbation. If we attempt to separate the stable islands by further increasing the energy level, we achieve an approximate critical point for this specific perturbation as seen in Figs. 6.4 and 6.5. We note that this does not guarantee that the entire stable island surrounding the fixed point has separated, it only holds for this specific quasi-periodic orbit. This means that a chaotic region, along with unstable fixed points may or may not exist between these two stable fixed points. The effect of the stable islands splitting as eccentricity is increased has been described by Malhotra[61] and attributed to the loops forming in the rotating frame trajectory. Note that the energy required is high enough that the forbidden region in the plane has completely disappeared. Considering the relatively high energy necessary to cause this separation, we will henceforth make the following assumption: For most low energy purposes, the intersections will produce a number of stable islands equal to the number of full revolutions. Interestingly enough, we can approximate the critical value ofC for which loops will occur in the rotating frame resonant orbit. As mentioned before, loops occur when the angular rate switches sign in the rotating frame. This will typically happen near periapsis for exterior resonances and near apoapsis for interior resonances. Therefore, if we can approximate the angular rate at periapsis and apoapsis in the rotating frame from a 2-body approximation, we can determine the Jacobi constant for which this value becomes 0. By setting _ p =v p =r p 1 = 0 forp<q and _ a =v a =r a 1 = 0 120 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.3: Three connected, indistinguishable stable islands from a 3:4 resonant orbit with loops in the Jupiter-Europa CR3BP forC = 3:0005 (a) Trajectory plot, the black line shows the stable periodic orbit, the red lines are the quasi-periodic orbit and the gray is the forbidden region. (b) Poincar´ e map of the orbit intersections on the negativex-axis in (L; g)-coordinates. The black dots are the periodic orbit and the red dots are the quasi-periodic orbit. forp>q, we can solve for the critical eccentricity using Eq. 6.2. This relationship has been shown in a different form in Murray and Dermott[68]. 0 = 8 > > > > < > > > > : e 3 3e 2 + 3 + p q 2 ! e + p q 2 1 ! ; ifp<q e 3 + 3e 2 + 3 + p q 2 ! e + 1 p q 2 ! ; ifp>q (6.2) or equivalently, 0 = 8 > < > : e 3 3e 2 + (3 +L 6 )e + (L 6 1); ifL> 1 e 3 + 3e 2 + (3 +L 6 )e + (1L 6 ); ifL< 1 (6.3) The equation generally has one real root and satisfies 0e 1. Once we solve for the critical loop eccentricity, approximating the Jacobi constant fromL,G ande is straightforward using Eqs. 6.27 121 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.4: Three connected, distinguishable stable islands from a 3:4 resonant orbit with loops in the Jupiter-Europa CR3BP forC = 2:9849 (a) Trajectory plot, the black line shows the stable periodic orbit, the red lines are the quasi-periodic orbit and the gray is the forbidden region. (b) Poincar´ e map of the orbit intersections on the negativex-axis in (L; g)-coordinates. The black dots are the periodic orbit and the red dots are the quasi-periodic orbit. and 6.28. We show example results in Fig. 6.6 and note that these results are not dependent on the system parameter, but are global. Using a plot like this, we can use a given Jacobi constant to determine the coordinate forL at which stable islands may distort into multiple sub-islands in a Poincar´ e map. For example, ifC = 3:025, we would expect the stable islands for all exterior resonances smaller than 1:2 to be simple, with single fixed points in the center. One consequence of this result is that systems with larger, such as Sun-Jupiter and Earth-Moon, will have a larger range of stable resonances with single fixed point islands for an energy level near the Lagrange points. If we move forward with the assumption regarding the number of stable islands at a given resonance, we can construct a theory for the topology we expect to find in a Poincar´ e map in (L; g)- coordinates. Determining stability of an orbit in this case reduces to the problem of whether a periapsis falls near the Secondary. Periapses near the Secondary cause gravity assist-like scenarios 122 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.5: Three separate stable islands from a 3:4 resonant orbit with loops in the Jupiter-Europa CR3BP forC = 2:9845 (a) Trajectory plot, the black line shows the stable periodic orbit, the red lines are the quasi-periodic orbit and the gray is the forbidden region. (b) Poincar´ e map of the orbit intersections on the negativex-axis in (L; g)-coordinates. The black dots are the periodic orbit and the red dots are the quasi-periodic orbit. and can drastically change the orbit, resulting in unstable behavior. This stability requirement is discussed by Malhotra[61], where they explain that two orbits of the same size, but different orientation can have drastically different stability characteristics. With this as a basis, we analyze the various cases ofp:q for periapsis locations in the rotating frame. Looking first atp, it represents the number of radial oscillations in the rotating frame because it’s the number of inertial orbits completed before a resonant cycle repeats. We will refer to a cycle of this radial oscillation as “arcs”. Therefore, if p is an even number, a periapsis will be paired with another periapsis radians ahead. This holds for apoapsis as well. The reasoning is that for evenp, must be odd. An even with evenp implies an evenq. In that scenario, the ratiop=q can be reduced to a case where p andq are not both even and thus will be odd. With an even number of arcs (p) in an odd number of full revolutions (), the halfway point of the orbit must fall on the side opposite to the initial point and will be at the same extremum as it started at. In simpler terms: if it starts at periapsis, 123 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.6: Critical Jacobi constant that produces loops. Approximate positions of common 1st order resonances are indicated as well. The region ofC below the line produces loops, and above does not. (a) Interior region withL< 1. (a) Exterior region withL> 1. there will be a periapsis on the opposite side. As a corollary, an odd p will pair a non-periapsis condition radians ahead of a periapsis. Note that this doesn’t guarantee an apoapsis, but does prevent a periapsis. The reasoning can be presented in a mathematically more rigorous fashion as follows. Assume a periapsis occurs at P , where is an angle measured counter-clockwise from the positivex-axis in the rotating frame. If we neglect 3-body perturbations of a Keplerian orbit with resonance p:q, then the complete orbit will rotate through 2 radians in the rotating frame. Thus each arc will cover 2=p radians and apoapses will be offset from periapses by half 124 of 268 6.1. RESONANCE PREDICTION that angle, or=p radians. We can then define the relative angular distances between an initial periapsis and subsequent extrema as follows. Pk P = p (2k) = Ak A ;k = 1;:::;p 1 Ak P = p (1 + 2k) (6.4) Note that we have neglected thek = 0 andk = p members, since they correspond to the initial periapsis itself and the periapsis which returns to the initial state. Also note that the separation between subsequent apoapses is the same as that for periapses. Next, we consider the apoapsis that occurs halfway through the resonant cycle with an odd p, for which k = (p 1)=2 m. The angular separation becomes Ak P = o p (1 + 2m) =o (6.5) Since is always an integer, this proves that the midpoint apoapsis will occur either on the same side as the initial periapsis or exactly opposite to it for oddp. This rule can be applied recursively to any periapsis, so every periapsis will be paired with an apoapsis either on the same side or exactly opposite. In fact, the parity of determines the side of the paired apoapsis. An odd means a periapsis is paired with an opposite apoapsis, while an even means a periapsis is paired with a same side apoapsis. Representative examples are shown in Fig. 6.7 with 3:5 satisfying (p;) = (odd; even) and 3:8 satisfying (p;) = (odd; odd). Therefore, the 3:5 resonance has an apoapsis on the same side as every periapsis, and the 3:8 has an apoapsis opposite to each periapsis. Next, we consider the periapsis that occurs halfway with evenp for whichk =p=2 n. The angular separation becomes Pk P = p (2n) = (6.6) 125 of 268 6.1. RESONANCE PREDICTION (a) (b) (c) Figure 6.7: Representative orbits showing alignment of extrema. (a) Oddp, even, apoapsis same side as periapsis. (b) Oddp, odd, apoapsis opposite to periapsis. (c) Evenp, odd, periapsis opposite to periapsis, and apoapsis opposite to apoapsis. This case has another caveat in that must be odd. It was argued previously why this must be true for evenp. This then reduces to be an angles opposite to the original periapsis. This proves that the midpoint periapsis will occur exactly opposite to the initial periapsis for evenp. We showed before that this holds for apoapses as well. Applying this recursively means that every periapsis is paired with a periapsis on the exactly opposite side, and every apoapsis is paired with a apoapsis on the exactly opposite side. An example is shown in Fig. 6.7 with 2:5 satisfying (p;) = (even; odd). We have now shown certain pairs of periapsis and apoapsis that will occur, but have not studied what is guaranteed to not occur. This is important since it will allow us to determine for example when a periapsis is guaranteed not to occur near the Secondary, thus yielding a stable orbit. We return to Eq. 6.4, and examine the periapsis to periapsis distance for oddp. The term (2k=p);k = 1;:::; (p 1) will not have any members in the set of integers. Moreover, cannot be further reduced from division by p. Therefore, the distance between periapses will never be an integer multiple of radians. Since this holds for apoapsis to apoapsis distances as well, we can conclude the following: Ifp is odd, two periapses cannot occur on the same side, or exactly opposite, neither can two apoapses. Now we examine the apoapsis to periapsis distance for even p from Eq. 6.4. Keep in mind that must be odd for this case. The terms excluding have the structure (odd) 126 of 268 6.1. RESONANCE PREDICTION (odd)=(even) which reduces to (odd)=(even), which will never be an even integer. Therefore, for an evenp, periapsis and apoapsis cannot occur on the same side or exactly opposite. Using the conclusions we formed regarding the relative positions of periapsis and apoapsis for resonant orbits in the rotating frame, we can now form a predictive model for the topology of a Poincar´ e map in the exterior region. We use the rules to construct cases where a periapsis occurs near the Secondary and label them as unstable, and cases where periapsis is prevented near the Secondary and label them as stable. If we examine the structure along the g-coordinate for a given L, the centers of the stable islands will fall at either g = 0, g = , or both. The configuration is driven by p and q. A graphical diagram illustrating the rules for the exterior region is shown in Fig. 6.8. The number of islands shown in the diagram is only representative, since the actual number is equal to the order of the resonance, but they do have the correct parity. Note that these rules should mean that you cannot have a stable resonant orbit with a chaotic region at both g = 0 and g =, and one of those is guaranteed to contain a stable island. In essence, this would imply that no asymmetric resonant orbits are stable. Symmetry here refers to symmetry across thex-axis. The statement regarding asymmetric stable orbits follows from the implied rule that an apoapsis or periapsis must occur at = 0 in the rotating frame, which guarantees a perpendicular crossing of the x-axis. However, we must be careful with assuming these rules are absolute. The rules were formed based on the relative positions of periapses and apoapses in the rotating frame. If we assume that asymmetric orbits exist, while not considering their stability, then such orbits can be oriented in such a way that no perpendicular crossing exists and no periapsis occurs near the Secondary. These scenarios are more likely for low-integer, low-order resonances, such as a 1:2 orbit. We make this statement because the angular distance between periapses in the rotating frame becomes large. Hence, a small rotation away from a symmetric orientation can still result in an orbit that avoids periapsis near the Secondary. In fact, it has been shown that these stable, asymmetric orbits exist in the CR3BP, see Malhotra[61]. We note that while stable islands will most likely exist at either g = 0 or g = at a value ofL for which there exists a stable resonant 127 of 268 6.1. RESONANCE PREDICTION orbit, this does not mean stable islands must exist at all values ofL. This follows naturally from the chaotic regions between stable resonant orbits along theL- coordinate. Figure 6.8: Resonance pattern topology, Exterior Region. The gray area represents the chaotic sea, while the white patches are stable islands. The number of islands will be equal to , but the diagram correctly represents the parity. Note that the case with bothp andq even should be impossible under our derived rules since it can be reduced to one of the other three cases. See the discussion in the text for why this is not an absolute rule. The topology of a Poincar´ e map in the interior region is also easily derived from the same rules. However, now we use the rules to construct cases where an apoapsis occurs near the Secondary and label them as unstable, and cases where apoapsis is prevented near the Secondary and label them as stable. The reason is simply that an interior resonance will have a close approach when an apoapsis occurs near the Secondary, resulting in unstable behavior. A graphical diagram illustrating the rules for the interior region is shown in Fig. 6.9. Note that the (p;q) = (odd; odd) is identical, since the apoapsis and periapsis share the same side. Therefore by placing either periapsis or apoapsis at = 0, you will prevent either to occur at = 0. Also note that the topology for the (p;q) = (even; odd) and (p;q) = (odd; even) cases have switched places. This is of course a result of the change in the rules of preventing apoapsis instead of periapsis near the secondary. For these mixed parity cases, the orientation as defined by g that was stable in the exterior region becomes unstable in the interior region, and vice versa. An interesting observation is that the original topology can hold universally, if we simply definep as the smaller integer in the set (p;q), regardless of the region. 128 of 268 6.1. RESONANCE PREDICTION The (p;q) = (odd; odd) case remains unchanged under this transformation. This may however be an inconvenient definition to use in practice, since it requires an additional parameter to separate interior and exterior resonances. Figure 6.9: Resonance pattern topology, Interior Region. The gray area represents the chaotic sea, while the white patches are stable islands. The number of islands will be equal to , but the diagram correctly represents the parity. Note that the case with bothp andq even should be impossible under our derived rules since it can be reduced to one of the other three cases. See the discussion in the text for why this is not an absolute rule. Compared to the exterior region cases, the topology for the (p;q) = (even; odd) and (p;q) = (odd; even) cases have simply switched places. 6.1.2 Locating Stable Island Centers We have shown the topology of Poincar´ e maps in Delaunay variables qualitatively, with some quantitative specifications. We quantified the locations of stable islands at g = 0 or g = for certain combinations of p and q. However, the non-perpendicular crossings of the x-axis have centers that can be predicted as well. We already know how many of them will occur and that their centers lie in the ranges< g< 0 and 0< g<. In order to compute the approximate centers, we can simply solve for the set of g that satisfies = 0 during one complete period of the orbit in the rotating frame. We will use 2-body approximations in order to achieve an algebraic solution. We will refer to these islands as “stable off-axis island centers”. 129 of 268 6.1. RESONANCE PREDICTION First, we focus on the exterior resonances for whichp<q. As we will show later in this section, it is a simple task to adapt this strategy for the interior resonances. To locate the island centers, we found that two separate cases must be handled, which follows from the two possibilities of the inertial frame argument of perigee,!. We will briefly move to an inertial frame to discuss some initial orientations of the orbit. If we focus on timet = 0 and assume this occurs when = 0, the rotating and inertial frames will be aligned, and the Secondary will be on the positivex-axis with the particle on the negativex-axis. The two orbit orientations that can yield a stable resonant orbit satisfy case 1:! =) g(t = 0) =!t =)p = odd case 2:! = 0) g(t = 0) =!t = 0)q = odd (6.7) Note that only defining the parity of eitherp orq allows two cases for the parity of the complemen- tary integer. This means that the case (p;q) = (odd; odd) actually belongs to both cases 1 and 2. This is confirmed by the topological rules we derived and the diagram in Fig. 6.8, where we see that case having stable islands at both g = 0 and g =. This has no consequence, as it simply means you can use the solution method for case 1 or case 2 to find the centers when (p;q) = (odd; odd). Since we are only interested in the states when = 0, we can find a relationship between the true anomaly and the time for all states that satisfy that. This follows from the angle in the rotating frame being a function of both the inertial angle and the time. = (! +)t (6.8) Where is the inertial frame true anomaly. This leads to case 1: =! +t =t case 2: =! +t =t + (6.9) 130 of 268 6.1. RESONANCE PREDICTION which substitutes into the explicit relationship between the true an eccentric anomaly (E), case 1: tan E 2 = r 1e 1 +e tan t 2 case 2: tan E 2 = r 1e 1 +e cot t 2 (6.10) This is now an explicit relationship between time,t and eccentric anomalyE which we will refer to asE(t). Next, we can find solutions that satisfy Kepler’s Equation at these locations. M =E(t)e sin(E(t)) (6.11) However, in the case of a resonant orbit, M = (p=q)t, which is monotonically increasing if we allowM to exceed the bounds of the unit circle. The right hand side of Eq. 6.11 is overall increa- sing at a faster rate. In one full period of the resonant orbit,M will traverse 2p radians, while the right hand side will traverse 2q radians. Therefore, we will need a term adjusting for periodicity if we intend to locate all solutions to the relationship. For exterior resonances wherep < q, the correction term subtracts integer multiples of 2 from the right hand side. case 1: p q t =E(t)e sin(E(t))(2k);k = 0;:::; (p 1) case 2: p q t + =E(t)e sin(E(t))(2k);k = 0;:::; (p 1) (6.12) where we have adjusted the mean anomaly in case 2, since the orbit is at apoapsis att = 0. When applied in a root finding method, the final form becomes f k = 0 = 8 > > < > > : E(t)e sin(E(t)) p q t(2k); k = 0;:::; (p 1); case 1 E(t)e sin(E(t)) p q t(2k + 1); k = 0;:::; (p 1); case 2 (6.13) 131 of 268 6.1. RESONANCE PREDICTION Solving for the roots of thesep equations will provide the set of timest j at which the orbit crosses thex-axis. The final step is simply to determine the value of g j that corresponds to those times. We use the simple relationship with the known argument of periapsis for the two cases as follows, g j =!t j = 8 > < > : t j ; j = 0;:::;; case 1 t j ; j = 0;:::;; case 2 (6.14) Here, is the number of intersections found, and will have one of two values depending on the eccentricity: = 8 > < > : ; ee loop + 2; e>e loop (6.15) When the eccentricity is greater than the critical eccentricity, loops form and two additional inter- sections occur. In extreme cases, loops neighboring the loop centered at = 0 may intersect the x-axis as well. This would cause to not satisfy Eq. 6.15, but we dismiss these cases as extreme and maintain that this relationship holds for most cases that we are currently studying. Cases such as these would occur for high energies withC 3. Once the g j -coordinates has been solved for, we approximate theL-coordinate easily throughL j = (q=p) ( 1=3). Recall that these two solution methods were developed under the assumption of exterior reso- nances, and we now briefly show how to adapt these to find the stable islands for interior resonan- ces. The solution is in fact quite simple, only requiring that we switch the conditions for case 1 and 2. This follows from the fact that the stable configuration switches rules in the interior region the mixed parity case as seen in Fig. 6.9, and therefore the rules for a stable orbit switch here as well. We can write the case selection matrix as shown in Table 6.1, or more consisely as seen in Eq. 6.16. Note once again that if both p and q are odd, either case 1 or 2 will yield the correct locations of the islands. case = 8 > < > : 1; min(p;q) = odd 2; max(p;q) = odd (6.16) 132 of 268 6.1. RESONANCE PREDICTION p = odd q = odd p<q case 1 case 2 p>q case 2 case 1 Table 6.1: Case selection matrix for island location algorithm. We now have an algorithm for locating all the island centers for a given pair of integers p andq. Therefore, we can examine a whole set ofp i andq i satisfying 1 (q i =p i ) ( 1=3) L max , withL max being a user chosen upper limit. If we compute the island centers and compare them to a Poincar´ e map, we can verify that the approximation closely matches the islands found in the nonlinear map. Figure 6.10 shows one example of exterior region resonances in a Poincar´ e map, and the approximated center locations. We find a very close agreement between the nonlinear map and the approximation. There are cases where we find islands in the Poincar´ e map, but not from the approximation shown. However, this is only due to the chosen upper limit q 15. There is also one case where the approximation indicates islands, but the Poincar´ e map does not, such as the blue diamonds at L = 1:049, representing a 13:15 resonance. This stable resonance is predicted to exist, but a quick analysis shows why such an orbit is unstable. Refer to Fig. 6.11 for a differentially corrected orbit at this location. Although the two periapses are off-axis near = 0, as predicted, they fall too close to the Secondary to maintain stability. We can infer from this that for (p;q) = (odd; odd), instability can occur if there are many loops, as defined by p. Having many loops makes the spacing between periapses smaller, so the 3-body perturbations can cause instability at the two periapses that bracket the Secondary. Also indicated in Fig. 6.10 is the approximate value ofL for which loops occur. This value is determined by numerically solving for L that satisfies Eq. 6.3 with a given C. This is the inverse of the original problem, where L was given and C was sought. Although the original problem is also solved numerically, we have verified that wrapping a numerical solver with the original method represented as a standard function produces accurate results. 133 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.10: Approximate island centers of a Poincar´ e map, Exterior Region, Jupiter-Europa, C 0 = 3:0039. The diamonds represent the approximated island centers, and their color indicates the resonance order. The dashed black line indicates the minimum value ofL for which loops will occur. (a) The range ofL up to and including the 1:2 resonance. (b) The range ofL from the 1:2 resonance to the 1:4 resonance. 134 of 268 6.1. RESONANCE PREDICTION Figure 6.11: 13:15 unstable orbit, predicted to be stable, Jupiter-Europa,C 0 = 3:0039. Note the two periapses very close to the Secondary, causing the instability. 6.1.3 Producing Initial Guess Conditions for Stable and Unstable orbits Since we have a method for approximating the centers of stable islands in a Poincar´ e map, it follows that we should be able to compute initial conditions that produce periodic or quasi-periodic motion. Based on the method described in Appendix B.7, we can take the solutions to Eqs. 6.13 and 6.14 and convert them into rotating Cartesian coordinates. However, we found that this numerical method does occasionally fail, and a more clever conversion is possible which will be discussed later. These states should then approximate stable resonant orbits in the CR3BP. However, since they are approximations, they are more likely to produce stable quasi-periodic orbits. As long as the approximation is close enough to a periodic stable resonant orbit, it will be within the associated stable island and thus generate quasi-periodic motion. It is also possible that the approximation generated an initial condition for an orbit that is too perturbed by the 3-body forces to remain stable, and thus the trajectory can become chaotic. We have shown that this can happen, and explained the reason behind it when discussing Fig. 6.11. The theory of the initial guesses is applied in the Jupiter-Europa system in the exterior region for 1=4 p=q < 1 andq 15, and 135 of 268 6.1. RESONANCE PREDICTION we produce a Poincar´ e map using t = 1000. The resulting Poincar´ e map is shown in Fig. 6.12, where the approximation shows the island centers as before, and the propagated initial conditions in most cases produce small groups near those centers. The few exceptions that produce chaotic motion seem to haveL approaching 1. This is an indication that the approximation we developed becomes perturbed enough that it falls outside the stable island for orbits that spend more time near the radius of the Secondary. It is also worth noting that some of the predicted stable resonances may actually become unstable and therefore there will be no associated stable islands, as shown in Fig. 6.10. Next, we examine some representative candidates of this behavior for the exterior region in Fig. 6.13. In Fig. 6.13(a), all the approximations fall neatly within the boundaries of the stable islands and thus generate several quasi-periodic orbits as indicated by the concentric loops formed at each approximate center. Since the true center of each island would produce intersections at each of the other stable island centers, one could produce the Poincar´ e map of a periodic orbit with only a single known exact island center. In our case, we only have an approximation and thus place an initial condition at the center of each island, which is why there are multiple quasi-periodic loops produced for a given resonance. Each approximate center produces its own loop which intersects the Poincar´ e section inside each associated island. In contrast, Fig. 6.13(b) shows two approximations that failed to produce stable motion. In these cases, we see that only one island for the given resonance was poorly approximated, since we still produced quasi-periodic motion from the other associated islands. These islands in turn generate loops inside the true stable island at g = 0, which indicates how the approximation failed for that island. The same behavior is apparent from the propagated approximate centers in the interior region with 1=4 p=q > 1,p 15 and t = 1000 as shown in Fig. 6.14. Once again, we see good approximations with quasi-periodic motion in Fig. 6.14(a) and some poor approximations in Fig. 6.14(b). As in the exterior region case, the poor approximation failed to produce an initial condition inside one of the stable islands associated with a given resonance. 136 of 268 6.1. RESONANCE PREDICTION We have shown that we can easily compute large sets of stable, quasi-periodic orbits in the CR3BP by applying knowledge of expected topology in a Poincar´ e map. This can be useful for studying stability in general by producing targeted Poincar´ e maps that contain almost exclusively stable solutions. Additionally, it’s useful for mission design where stable orbits are a mission requi- rement, as it allows to explore a large range of stable resonances. While we did not explore that option in the scope of the current work, it would be straightforward to also generate initial conditi- ons for unstable orbits. We already have the topological rules for unstable orbits from Section 6.1.1, so it is a simple matter of deriving the initial conditions as was done in Section 6.1.2 but for unstable orbits. This would allow us to generate Poincar´ e maps with only unstable solutions. The advan- tage of this is clearly to reduce the computational effort required to produce a Poincar´ e map of the desired density while focusing on the desired orbit type. Moreover, the approximate solutions from this method can be used as initial guesses for numerical methods to compute exact periodic orbits. We have in fact applied this, and produced an algorithm that computes resonant orbits of arbitrary p:q and desired boolean stability. The tool does not control the stability quantitatively, only qualitatively. 137 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.12: Poincar´ e map of approximated stable solutions, Jupiter-Europa,C 0 = 3:0039, Exte- rior Region. The diamonds represent the approximated island centers, and their color indicates the resonance order. The dashed black line indicates the minimum value ofL for which loops will occur. The black dots represent the Poincar´ e map of the initial conditions. Most initial conditions produce stable islands near their associated island center, but some withL approaching 1 produce chaotic orbits. (a) Exterior region. (b) Interior region. 138 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.13: Poincar´ e map indicating quality of approximation. Exterior Region. The diamonds represent the approximated island centers, and their color indicates the resonance order . (a) Well-behaved approximation, each approximate island produces quasi-periodic motion. (b) Region with two selected ill-behaved approximations where at least one approximate solution for a given resonance falls outside the stable island. Note the islands with g = 0;L 1:09; = 3 and g = 0;L 1:11; = 4 for which the actual island center is displaced from the approximated location. 139 of 268 6.1. RESONANCE PREDICTION (a) (b) Figure 6.14: Poincar´ e map indicating quality of approximation. Interior Region. The diamonds represent the approximated island centers, and their color indicates the resonance order . (a) Well-behaved approximation, each approximate island produces quasi-periodic motion. (b) Region with two selected ill-behaved approximations where at least one approximate solution for a given resonance falls outside the stable island. Note the island with g = 0;L 0:902 for which the actual island center is displaced from the approximated location. Also note the islands with g = 0;L 0:945; = 2 and g = 0;L 0:953; = 2 for which it seems there are no associated stable islands in the 3-body system. 140 of 268 6.2. COUNTING STABLE RESONANCES 6.2 Counting Stable Resonances While the stability of resonances can be absolute in the autonomous CR3BP, as discussed in Section 6.1.1, this is no longer true for non-autonomous systems such as a trueN-Body problem (e.g. ephemeris model). We define a new measure specifically to quantitatively measure the sta- bility of resonant orbits in non-autonomous systems. This measure guarantees not a mathematical infinite time stability, but rather a finite time practical stability under certain boundary restrictions. We apply this measure to study the stability of resonant orbits in the Galilean satellite system. To accomplish this, we have defined a new, simplified dynamical model of the Galilean satellite system which places the moons in circular, coplanar orbits with their periods in exact resonance. We call this System the Circular Restricted 6-Body Problem (CR6BP). The section immediately following (6.2.1) will define this model in detail. While this is an approximation of reality, it is a model with interesting characteristics and is easier to analyze and faster to propagate numerically. We also find that in spite of its approximate nature, resonance stability estimates in this CR6BP model carry over well to the ephemeris model. We note that there have been other approaches to computing stability in higher fidelity models. One example by Lara, Russell and Villac[57] uses the Fast Lyapunov Indicator (FLI) in Hill’s Problem and shows that the stability estimate carries over to ephemeris solutions. Another example is the work by Froeschl´ e, Gonczi and Lega[34], who apply the FLI in a partial ephemeris model of the solar system to compute long term stability of asteroids in the main belt. Computing the FLI requires propagating the variational equations along with the state, while our method only requires propagating the state and is thus faster and simpler. However, the FLI method can be applied to any dynamical system and is thus more general. 6.2.1 Circular Restricted 6-Body Problem We define the model which we call the Circular Restricted 6-Body Problem (CR6BP) and its parameters. We have applied it to the specific case of the system of Galilean moons, with the Jupiter-Europa rotating frame as the basis, and the other moons in exact mean motion resonance. 141 of 268 6.2. COUNTING STABLE RESONANCES Equations of Motion The model has six total bodies, with the first three being the same as those in the Circular Restricted 3-Body Problem (CR3BP). Therefore, there are five attracting bodies, three of which are added to the original CR3BP model. It is assumed these three bodies do not affect the motion of the two primaries of the CR3BP and orbit the barycenter of the CR3BP in circular orbits. Units are still nondimensionalized according to the CR3BP model of the first three bodies. Our notation assumes the 3rd body has infinitesimal mass, and is the body whose motion is studied. A diagram illustrating the geometrical arrangement of this model is shown in Fig. 6.15. The parameters that define this system specify the relative masses, initial positions and angular speeds of the attracting bodies. First we define the nondimensional mass i for bodyi as 1 = m 2 m 1 +m 2 = 2 = m 2 m 1 +m 2 = 1 4 = m 4 m 1 +m 2 5 = m 5 m 1 +m 2 6 = m 6 m 1 +m 2 ; (6.17) wherem i is the dimensional mass of bodyi. Recall that the 3rd body is assumed to have negligible mass, and is the standard mass ratio from the CR3BP. Next, we define the nondimensional semimajor axis i for bodyi as 4 = a 4 a 2 5 = a 5 a 2 6 = a 6 a 2 : (6.18) 142 of 268 6.2. COUNTING STABLE RESONANCES Once again, a i is the dimensional semimajor axis of bodyi. The initial angular positions of the added bodies will now be relative to them 1 -m 2 rotating frame and thus we must subtract the initial angular position of m 2 . The angular positions in the inertial frame we use are the initial mean longitude of each body att = 0, indicated byL 0i . We can then easily compute the rotating frame initial angles, 0i as 04 =L 04 L 02 05 =L 05 L 02 06 =L 06 L 02 : (6.19) The angular velocity of the added bodies requires both unit conversion and frame transformation. If we labeln i the mean motion of bodyi in the inertial frame and i the angular velocity in the rotating frame, then we have 4 = n 4 n 2 n 2 = n 4 n 2 1 5 = n 5 n 2 n 2 = n 5 n 2 1 6 = n 6 n 2 n 2 = n 6 n 2 1: (6.20) The numerators of the middle equation in Eq. 6.20 represent the frame conversion by subtracting the rotation of the frame, while the denominator represents the unit conversion. We seek the equations that describe the motion of m 3 , whose position we will label ~ r = [x;y;z]. The set of second order differential equations is shown in Eqs. 6.21–6.22: x = 2 _ y +x 1 r 3 1 (x +) r 3 2 (x 1 +) 4 r 3 4 x 4 5 r 3 5 x 5 6 r 3 6 x 6 y =2 _ x +y 1 r 3 1 y r 3 2 y 4 r 3 4 y 4 5 r 3 5 y 5 6 r 3 6 y 6 z = 1 r 3 1 z r 3 2 z 4 r 3 4 z 4 5 r 3 5 z 5 6 r 3 6 z 6 : (6.21) 143 of 268 6.2. COUNTING STABLE RESONANCES m 4 m 1 m 2 l 04 a 4 L 3 L 1 L 3 L 2 L 4 L 5 (a) Body 4 m 5 m 1 m 2 l 05 a 5 L 3 L 1 L 3 L 2 L 4 L 5 (b) Body 5 m 6 m 1 m 2 l 06 a 6 L 3 L 1 L 3 L 2 L 4 L 5 (c) Body 6 Figure 6.15: Diagram of the CR6BP. In this model, 1 and 2 are stationary and are equivalent to the same bodies in the CR3BP. The three added bodies ( 4 , 5 , 6 ) are indicated by the blue, red and green circles. Their respective semimajor axes and initial phase angles are also indicated. (a) shows 4 , (b) shows 5 , and (c) shows 6 . Added bodies inside the orbit of 2 (yellow circle) will move counter-clockwise, while those outside will move clockwise. We denotex i ;y i andz i as the components of the position vectors ofm 3 relative to bodyi. These distances are computed as 144 of 268 6.2. COUNTING STABLE RESONANCES ~ r 1 = 2 6 6 6 4 x + y z 3 7 7 7 5 ;r 1 =j~ r 1 j ~ r 2 = 2 6 6 6 4 x 1 + y z 3 7 7 7 5 ;r 2 =j~ r 2 j ~ r 4 = 2 6 6 6 4 x 4 cos( 4 t + 04 ) y 4 sin( 4 t + 04 ) z 3 7 7 7 5 ;r 4 =j~ r 4 j ~ r 5 = 2 6 6 6 4 x 4 cos( 5 t + 05 ) y 4 sin( 5 t + 05 ) z 3 7 7 7 5 ;r 5 =j~ r 5 j ~ r 6 = 2 6 6 6 4 x 4 cos( 6 t + 06 ) y 4 sin( 6 t + 06 ) z 3 7 7 7 5 ;r 6 =j~ r 6 j: (6.22) To give an example of a physical system, we provide the parameters for the Galilean moon CR6BP in Table 6.2. This is the system we will be using this chapter. 145 of 268 6.2. COUNTING STABLE RESONANCES Parameter Value 2:528017682687079 10 5 4 4:704339952562368 10 5 5 7:804763109162038 10 5 6 5:666827531410095 10 5 4 0:628520339740724 5 1:594993294590970 6 2:805394129041872 4 2 1 = 1 5 1=2 1 =0:5 6 14=3 1 =0:785714285714286 04 05 0 06 0 Table 6.2: Parameters used for Galilean moon CR6BP. In our case, i = 4 represents Io, i = 5 represents Ganymede, and i = 6 represents Callisto. Note the exact mean motion resonance we selected for those moons, as well as their alignment at the initial time. This alignment is approximately achieved on April 3rd, 2001, 22:34 TDB. 6.2.2 Finite Time Stability and Resonance Width In order to define the concept of finite time stability, we will first demonstrate it as applied to the more structured CR3BP, for which we already have an understanding of the expected topology. Assume we have a Poincar´ e map computed, with the results in the Delaunay variablesL and g. It is impossible to compute a Poincar´ e map to infinite time, so when one is produced, the numerical upper limit of time must be set. This time limit becomes part of the definition for finite time stability. The method we have developed can only prove stability up to the propagation limit used to create the Poincar´ e map, so any mention of finite time stability must also come with a statement about the length of time it is valid for. As an example, we could have a resonance that we determine is stable from a Poincar´ e map produced with a time span of five years and we would then call that resonance “Finite Time Stable for five years”. Shorthand can be used as well, such as calling the resonance “FT 5 -stable”. As we have discussed in Section 6.1.1, stable CR3BP orbits are confined to “islands” in the Poincar´ e map. When analyzed in the (L; g)-space, these islands have a relatively narrow range in 146 of 268 6.2. COUNTING STABLE RESONANCES theL-dimension. Moreover, they will typically not share that range ofL with any other resonances. Based on this knowledge, we define the “Finite Time Resonance Width” of a given particle’s intersections as the maximum deviation inL of any one set member from the meanL of that set. There exist other definitions of Resonance Width in the literature for other contexts. Malhotra[60, 62] provides an analytic estimate of the Resonance Width that is a function of the orbit semimajor axis, eccentricity and hte mass ratio of the Primaries. That is why we specify our definition as Finite Time Resonance Width, and for the remainder of this document “Width” will refer to this concept. W k = max 8i jL ki hL ki ij;k = 1;:::;K;i = 1;:::;I k (6.23) Here,K is the total number of sets in the Poincar´ e map, which is the equivalent to the number of initial conditions used to produce the map. I k is the number of members of set k, which is equivalent to the number of intersections produced by setk in the time simulated. Visually, this is depicted in Fig. 6.16, where we show the Width for a stable and unstable orbit. It is clear that in the CR3BP, the Width of the stable set is limited for all time to remain within some deterministic limit. This is not true for the unstable orbit, as it is not bound to remain in any given range ofL given infinite time. The unstable orbit is theoretically free to roam to any region of the chaotic sea that lies between the stable islands. We acknowledge that the Resonance Width has had other definitions in the literature with no single clear choice, and we do not seek to replace those definitions. Given this measure of Resonance Width, one can imagine that a numerical limit could be selected such that a set with a Width less than that limit would be considered stable. As an initial approximation we will call this valueW , or the “critical Width”. Assuming this value is given, we can now take theK sets and group them into stable and unstable subsets,S andU: S =fk :W k W g U =fk :W k >W g (6.24) 147 of 268 6.2. COUNTING STABLE RESONANCES (a) (b) Figure 6.16: Definition of Resonance Width in an (L; g)-Poincar´ e map in the CR3BP. (a) A stable resonance, showing the limited Width of the set. (b) An unstable orbit, showing the Width of the set as it stretches far beyond its mean value and enters regions where other stable islands exist. While it may seem arbitrary that a single value forW would be adequate for separating stable from unstable orbits, a visual examination shows it working quite well. In Fig. 6.17, the Resonance Width for all the orbits in the Poincar´ e map is compared to its mean L. This figure shows why the Resonance Width we defined can be used to indicate which orbits are stable and which are unstable. There is a large group of orbits that have a nearly uniform distribution alonghLi and there are individual groups of orbits whose distribution is very focused alonghLi. The first group has a significantly higher Resonance Width than the latter groups do, indicating that stable orbits cluster around specific values of L, corresponding to integer ratios of resonance. The value of W was selected manually after visual inspection of the data in Fig. 6.17. However, as a future improvement to this technique it may be possible to automatically select the appropriate stability criterion. While we have assumed thatW is a constant here, it may be more appropriate to have it be a function of the meanL. An argument to support this can be seen visually in Fig. 6.17, where the upper limits onW for some stable groups are clearly smaller than others. In fact, we see an oscillatory behavior, with the upper limit alternating between high and low ashLi increases. There is also a more subtle trend of increasingW for increasinghLi, but the relationship is most likely not linear and would have to be estimated. If one compares the resonance ratios for the groups with low upper bounds on the Resonance Width, a pattern emerges. Namely, the resonances of 2nd or 148 of 268 6.2. COUNTING STABLE RESONANCES higher order have smaller upper bounds onW . It is thus likely that the order of a resonance plays an important part in its Resonance Width. Figure 6.17: Resonance Width for a 12-year Poincar´ e map, Jupiter-Europa, C 0 = 3:0039. The Width is plotted vs. the average L for the given orbit. Note the rather clear vertical separation between stable orbits and chaotic, unstable orbits. Also note the very focused grouping of the stable orbits along thehLi dimension. A selected value forW that appropriately separates stable from unstable orbits is indicated by a black horizontal line. Once we have computed the Resonance Width for every set, we can use that information to graphically indicate stability in the Poincar´ e map. An example of such an augmented Poincar´ e map is shown in Fig. 6.18. Due to the large range of the Resonance Width values, we opted to use a base-10 logarithmic scale to map Resonance Width to color. This only makes the difference between the stable islands and the chaotic sea more striking, as we can see clear changes in color at the edges of each stable island. 149 of 268 6.2. COUNTING STABLE RESONANCES (a) Figure 6.18: CR3BP Poincar´ e map, colored by Resonance Width, Jupiter-Europa,C 0 = 3:0039. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the stable islands in purple are clearly separated from the chaotic sea in orange, as expected. 6.2.3 Resonance Abundance Now that we have shown that Resonance Width can be used to separate stable from unstable resonant orbits, we can analyze the stable groups that are found. In particular, we can count the number of orbits that belong to the same mean motion resonance, as determined from their mean L. Since the number of orbits in any given resonance is necessarily proportional to the number of initial conditions used to produce the Poincar´ e map, we divide that number by the total number of orbits to produce a dimensionless value. We call this value the Abundance of a given mean motion resonance and label itR p:q , wherep:q is the integer ratio representing a given mean motion resonance. Mathematically, the definition for the Abundance becomes R p:q = 1 K X k2S [p k $p^q k $q] (6.25) 150 of 268 6.2. COUNTING STABLE RESONANCES , whereS is the stable subset from Eq. 6.24 and [] are Iverson brackets that obey [s] = 8 > < > : 1; ifs = true 0; ifs = false (6.26) A graphical example of computed Abundance for a Poincar´ e map is shown in Fig. 6.19. From this plot it is clear that some resonances have more members in the Poincar´ e map than others, and that each stable resonance holds a relatively small percentage of the total set of orbits used to produce the map. For this example, the stable orbits account for approximately 32% of all the orbits. Figure 6.19: Stable resonant orbit Abundance, Jupiter-Europa,C 0 = 3:0039. A 12-year Poincar´ e map is the basis. The percent Abundance is plotted vs. the average L for a given mean motion resonance. The Resonance Abundance was named such because it indicates how many stable orbits exist that belong to a given mean motion resonance. Since the Poincar´ e map method is numerical and is based on a finite set of initial conditions, the Abundance computed from it must ultimately be an approximation. As the number of initial conditions approaches infinity, and the entire phase space is explored, we would expect to approach a “true value” of the Abundance. We also note that the initial conditions were uniformly distributed in the Cartesian phase space and so may not 151 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM be uniform in the space of Delaunay variables. This means that the initial conditions may be unevenly distributed among the stable mean motion resonances. An uneven initial distribution will in turn lead to an uneven distribution in the computed Poincar´ e map. In order to make a stronger statement regarding the Abundance of stable resonant orbits it would be better to produce the initial conditions as to meet a uniform distribution in the Delaunay variableL. By “stronger statement” we mean that the Abundance of one resonance relative to that of another would have more meaning. However, the goal of our work is to compare Abundance of the same resonances between different dynamical systems, and for that purpose, this method is valid. We would like to note that after the current analysis was completed, we successfully generated Poincar´ e maps using initial conditions selected with a uniform random distribution in Delaunay variables (L; g). We required the reverse transformation method to accomplish this, which we derive in in Appendix B.7. We leave the results and analysis of these Poincar´ e maps for future work. 6.3 Resonance Abundance in the Galilean System The Galilean system is very interesting dynamically, due to the natural resonances that already exist there. It is well known that the first three Galilean moons (Io, Europa, and Ganymede) are in a 1:2:4 mean motion resonance, see Murray and Dermott[68]. This means that in one revolution of Ganymede’s orbit, Europa has completed two orbits and Io four, resulting in their relative angular in-plane orientation repeating every Ganymede orbit (about 7.155 days). However, the 4th moon Callisto is also in a near 3:7 resonance with Ganymede. This near resonance has been studied by Goldreich and Sciama[36], and Noyelles and Vienne[70, 71, 72]. They show that it is not a true resonance, since the relative orientation between Ganymede and Callisto has a slow secular drift. Over long periods of time, this drift will cause the two moons to fall out of phase. However, for the present, the times considered for mission design are shorter than those considered for celestial mechanics. Therefore, this near-resonance is interesting dynamically for our analysis. In this section we will approach the problem of determining which resonances are 152 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM stable in the Jupiter-Europa system, when perturbed by the other Galilean moons. This application is important, since missions to Europa often have strict planetary protection requirements, see Rummel et al.[78]. Therefore, having knowledge of stable disposal orbits for spacecraft is valuable. Even if these orbits are only stable for a finite time, a sufficient time would allow the spacecraft to become sterilized by the intense radiation environment around Jupiter. Resonant orbits are only one candidate for such disposal orbits, but they constitute the one we focus on here. This is where the relative resonance of the four Galilean moons comes into context, since we predict that some resonances in the Jupiter-Europa system may be more stable than others under the perturbation of the other Galilean moons. The reasoning is that stable orbits would need to avoid close encounters with all of the moons, requiring that they are in some resonance with each moon and phased properly to avoid it. From this, we predict thatp:q resonant orbits with low integersp andq relative to each moon are more likely to remain stable under perturbation. We verify this by studying the Poincar´ e maps of the perturbed system and finally verify that this stability can carry over to a full ephemeris model of the solar system. Now that we have introduced the concepts of Resonance Width and Abundance, their use- fulness will be shown by applying it to a non-autonomous system. We have not encountered many studies that have used Poincar´ e maps for perturbed systems, but some examples exist, see Borderes-Motta & Winter[16], and Andreu[8]. The system we will use is the Circular Restricted 6-Body Problem discussed in Section 6.2. Since this system has coordinates and units that are equivalent to the Jupiter-Europa CR3BP, we can use the same initial conditions used for Poincar´ e maps in that dynamical system. The system is essentially a Jupiter-Europa CR3BP perturbed by the additional Galilean moons in exact resonance with Europa. Since the topology and structure of the CR3BP Poincar´ e map must obey certain rules due to its autonomous nature, we know that a non-autonomous system has no guarantee of obeying the same rules. This does not mean that all the structure seen in a CR3BP Poincar´ e map will necessarily disappear, and it is upon this basis we perform the analysis that follows. 153 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Examples of Poincar´ e maps subject to the same initial conditions and time limit, but propagated in the CR3BP and CR6BP systems are shown in Fig. 6.20. While it is clear that the CR6BP is more chaotic, with no sharp borders visible between stable islands and chaotic seas, there are still remnants of structure. Moreover, the structures we see correspond directly to structures seen in the same location in the CR3BP Poincar´ e map. Simply seeing the density of intersections in a given region does unfortunately not give a quantifiable measure of stability, which is why we now apply the Resonance Width and Abundance. When we apply the same method as described in Section 6.2.2, we arrive at a similar map in Fig. 6.21. Just as in the CR3BP case, we find tightly focused groups with small Resonance Width, and large chaotic group with a distribution across all values ofL. It is surprising that the stable groups are still very tightly focused alongL and with orders of magnitude smaller Widths than those in the chaotic group, in spite of the long propagation time and additional perturbations. Naturally, the separation between stable and unstable groups is slightly less obvious. However, it is surprising that we can still easily see which orbits are stable and which are not. The additional perturbations in the CR6BP could easily have broken the dynamical stability that exists in the CR3BP, but that does not seem to be the case. Now using the Resonance Width, we produce an augmented Poincar´ e map like the one shown in Fig. 6.18, but in the Galilean moon CR6BP. The result is shown in Fig. 6.22. The more chaotic nature of the map is still obvious, as in Fig. 6.20, as indicated by the overall more orange colors in the augmented Poincar´ e map. However, it is also more clear that some sets are still very stable, as indicated by the multiple purple groupings. These islands of finite stability seem to shadow some of the stable islands in the corresponding CR3BP map. For this reason, we will refer to these orbits and corresponding islands as “Shadow Resonances”. While some resonances remain stable, some are visually less dense and others are nearly erased. We will study these results in more detail later in this section when we study several ranges of energy and resonance, see Sections 6.3.1, 6.3.2, and 6.3.3. 154 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.20: Jupiter-Europa CR3BP vs. Galilean moon CR6BP 12-year Poincar´ e map, C 0 = 3:0039. (a) CR3BP. (b) CR6BP. Note that some islands seem to have nearly disappeared, such as the set of three islands atL 1:11 (8:11) and two islands atL 1:12 (5:7) Once more, we compute the Abundance as described in Section 6.2.3 and arrive at the plot in Fig. 6.23. While the overall Abundance has decreased for all resonances, some retain their Abun- dance better than others. To further study this it is more informative if we graphically compare 155 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Figure 6.21: Resonance Width for a Galilean moon CR6BP 12-year Poincar´ e map,C 0 = 3:0039. The Width is plotted vs. the averageL for the given orbit. Note the rather clear vertical separation between chaotic, unstable orbits and stable orbits. Also note the very focused grouping of the stable orbits along thehLi dimension. A selected value forW that appropriately separates stable from unstable orbits is indicated by a black horizontal line. the Abundance of the CR3BP to the CR6BP as shown in Fig. 6.24. Now we can clearly see the effect of the added perturbations due to the Galilean moons. Some resonances (such as 9:10, 8:9, 7:8, 6:7, and 4:5) seem to be nearly unaffected by the perturbations while others (such as 5:6, 3:4, and 5:7) have a heavily reduced Abundance. Next, we will study the reason for this uneven effect among the resonances. We split the analysis into different ranges of resonance ratios, which is equivalent to choo- sing a range of L or orbit size. Three sets of exterior resonant orbit sizes were considered, and each is addressed in the following sections. By also expanding the analysis to cover hig- her energies (while remaining in the “low energy” regime of chaotic motion), we can study if the energy level has any influence on the effects we’ve seen. Four energy levels are considered, C =f3:0039; 3:0036:0033; 3:0030g. The energy levels were chosen specifically to cover qualita- tively different scenarios of the CR3BP. In the first case, the Jacobi constant is higher than that of L 2 and so the interior and exterior regions are separated. The second case has a Jacobi constant 156 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) Figure 6.22: Galilean moon CR6BP 12-year Poincar´ e map, colored by Resonance Width, C 0 = 3:0039. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that some stable island “Shadow Resonances” in purple still exist in this perturbed system and are separated from the chaotic sea in orange. that is just barely lower thanL 2 , thus allowing motion between the two regions through a small gateway. Finally, the last two energies open the gateway even more and makes motion between the two regions more likely. We reason that different energy levels will more readily allow trajectories to approach the Secondary and therefore will allow us to study the effects of its perturbation. In the sections that follow, each section focuses on a set range of resonances and covers all four cases of the energy. 157 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Figure 6.23: Stable resonant orbit Abundance for a Galilean moon CR6BP 12-year Poincar´ e map, C 0 = 3:0039. The percent Abundance is plotted vs. the average L for a given mean motion resonance. Figure 6.24: Comparison of Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP, C 0 = 3:0039. A 12-year Poincar´ e map is the basis. Thex-axis is labeled with a given resonance ratio but does not indicate relative scale along L, other than the fact that resonances toward the right have largerL. 158 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM 6.3.1 Small Exterior Resonances This section focuses on small exterior resonances up to and including 5:7, for which 1 L 1:1187. The augmented Poincar´ e maps that were the basis for the analysis in this section are shown in Figs. 6.25–6.28. The relative Abundance shown in Fig. 6.24, was taken from this data set for a single value of the Jacobi constant. From that data we concluded that certain resonances were highly affected by the additional perturbations present in the CR6BP, while others were nearly unaffected. Figure 6.29 shows the relative Abundance for all energy levels, and overall indicates a general agreement between them. While there is some differences between the energy levels, the sets of highly affected and less affected resonances are the same. Therefore, the energy level is not a major factor in resonance stability for resonances of this size, at least in the range of energies considered. There is an interesting relationship to consider that may give insight to this, namely one that relates the eccentricity to the Jacobi constant from Eq. 6.27. G = 1 2 T 1 L 2 1 2 C 1 L 2 e G (6.27) The approximate relationship between G and C is due to Tisserand’s Parameter, T , being an approximation of the Jacobi constant. This in turn yields an approximate relationship between C,L, and the eccentricity,e by Eq. 6.28. e = r L 2 G 2 L 2 = s 1 G L 2 (6.28) It’s worth noting that the error inG due to the approximationC T is very small given certain circumstances. The approximation is best when the measurement ofT based on orbital elements is taken far from both Primaries, which is the case in our chosen Poincar´ e map since the section is on the negative x-axis (y = 0 and x <1). Subsequently, the error in e may be small as well. A numerical verification is indicated in Fig. 6.30, where the initial conditions were used to compute the Delaunay variables exactly with the method in Appendix B.6 and Eq. 6.27 was 159 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM used to approximateG usingC. The relative error inG is on the orderO(10 6 ), while the error in e becomes significant only for very small e. For example, the error in e reaches O(10 3 ) for e < 0:01, which is a very small eccentricity. Therefore, on this Poincar´ e map we can adequately approximate the relationship betweenC,G ande with the relationships in Eqs. 6.27 and 6.28. Moreover, the variation in the orbit across a range of energies can be quite small in the chaotic region we are considering. Consider Fig. 6.31, where we compute the osculating eccentricity and apoapsis radius as functions ofL for a larger range of Jacobi constant values. We expanded the range to cover 3C 3:0039, where the lower limit approximately represents the energy where the forbidden region disappears and all motion in the plane is allowed. This range of energies is still what we would consider part of the low energy regime. What the results in Fig. 6.31 indicate is that the variation in eccentricity and apoapsis by changing C is minor even over this wider range of energies. Any vertical cut through the curve contours for a fixed 1:04 < L has a range of apoapses satisfying r a < 0:05, and this boundary becomes even more strict asL increases. ForL near the 5:7 resonance, we have r a < 0:012, which indicates that any energy change in this range is unlikely to have a large effect through causing close encounters with Ganymede. It is true that the variation is much larger for small L, but at those orbit sizes, the apoapsis is so small that even a larger change will not cause Ganymede close approaches. Keep in mind that we are discussing the osculating apoapsis radius and that this is not the same as a guaranteed upper limit on orbital radius for a general trajectory in the CR3BP. However, the estimate is locally accurate due to the measurement happening far from both bodies and we are specifically studying stable resonances which do not approach the Secondary. This last point is discussed in detail in Section 6.1.1. Since we are studying stable resonances, the osculating apoapsis radius is a good estimate of the maximum radius achieved over a full cycle in the CR3BP. We therefore conclude that we can expect to see very similar results in regards to resonance stability over the range of Jacobi constants we have considered, and that an even larger range of energies would not significantly add to the analysis. If we decide to study the effects of much higher energies, such as 160 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.25: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has a larger Resonance Width than the chaotic region in the CR3BP, as indicated by its slightly more yellow color. C < 2:99, then we should expect some differences, but that is beyond the scope of this analysis of low energy trajectories. 161 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.26: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has a larger Resonance Width than the chaotic region in the CR3BP, as indicated by its slightly more yellow color. From the analysis of the variation ofe, we also notice an interesting quality in the allowed range ofL. For a certain Jacobi constant, the minimum eccentricity of 0 occurs at some valueL min > 1. 162 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.27: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has a larger Resonance Width than the chaotic region in the CR3BP, as indicated by its slightly more yellow color. 163 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM This essentially indicates that a given energy has a minimum circular orbit it can support. If we consider the boundary of the forbidden region for a given Jacobi constant, we would expect that this lower limit is a circular orbit that lies just outside this border. We can see from Eq. 6.28 that there is some value ofL for whichG>L for a givenC and therefore causes non-real results. We can solve for thisL min value as a function of C. We must satisfy e G = 1 2 C 1 L 2 L (6.29) which results in the cubic relationship f(L) =L 3 C 2 L 2 + 1 2 0 (6.30) For appropriate energy levels, this equation has three real roots in L for f(L) = 0. The roots satisfyL min;exterior > 1, 0<L min;interior < 1, andL min;3 < 0. We are in this case only interested in the two first solutions, as the 3rd one indicates a negativeL, which is non-physical. The two first solutions should approximate the inner and outer boundary of the forbidden region. In Fig. 6.32, we can confirm that this is true. While that figure is only shown for a single energy level, we tested a large range of energies and the solution held in each case. Since the forbidden region is not a perfect circle, the estimate puts a bound on the region’s annulus. This is clear from Fig. 6.32(b), where the estimated bounds detach from the forbidden region boundary near the Secondary. With the understanding that the energy levels are not greatly affecting results regarding resonant stability, we move to study the effect on individual resonances when switching from CR3BP to CR6BP. We can reasonably assume that the 3:4 and 5:7 resonances have reduced Abundance due to their size approaching that of a possible Ganymede crossing orbit. If we assume that an orbit has a periapsis at theL 2 point of Jupiter-Europa (x 1:02) and an apoapsis at Ganymede’s orbit with a G = ( P G P E 2=3 = 2 2=3 1:5874 (6.31) 164 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM , then the orbit would have a semimajor axis of a = a G + 1:02 2 1:3037 (6.32) This translates to a Delaunay variableL value of L = p a 1:1418 (6.33) , which we will call the “criticalL”. Now we turn to the 5:7 resonance and see that thatL 5:7 is close toL , see Eq. 6.34 . This accounts for the instability seen in the 5:7 family of resonances, since even approaching Ganymede’s orbit at distance repeatedly can cause perturbations that break stability. This is of course true unless close approaches to Ganymede are avoided by proper phasing and resonance with Ganymede, which we will discuss in detail later. L 3:4 is not as close toL as L 5:7 , but it must have apoapses close enough to Ganymede that many stable members of the group from the CR3BP turn unstable in the CR6BP. Finally,L 2:3 falls almost exactly onL , and we will see in Section 6.3.2 that this resonance in particular is nearly completely erased in the CR6BP. We can now attribute that to a nearly guaranteed close encounter with Ganymede over long periods of time. L 3:4 = 4 3 1=3 1:1006; L 5:7 = 7 5 1=3 1:1187; L 2:3 = 3 2 1=3 1:1447 (6.34) 165 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.28: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has a larger Resonance Width than the chaotic region in the CR3BP, as indicated by its slightly more yellow color. Also note, that the energy has increased enough to allow the existence of a stable resonance on the far left edge, corresponding to a Jupiter-Europa 10 : 11 resonance. In addition the other resonances move to slightly larger values ofL. 166 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a)C 0 = 3:0039 (b)C 0 = 3:0036 (c)C 0 = 3:0033 (d)C 0 = 3:0030 Figure 6.29: Small Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. The green bars represent the CR3BP and the orange bars represent the CR6BP. Thex-axis is labeled with the resonance ratio. Note the drop in stable CR6BP orbits for resonances greater than 3:4. The orbits starting with 5:7 and larger are those that will cross Ganymede’s orbit, and are thus more likely to be chaotically perturbed. 167 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.30: Approximation error in computation ofG ande fromC. (a) Error inG. (b) Error in e. (a) (b) Figure 6.31: Resonant orbit variability as a function of C. The color of the lines indicate the Jacobi constant, while the upper and lower limits of the range are highlighted in red dashed lines. (a) Range of possible apoapsis values for a range ofC. (b) Range of possible eccentricity values for a range ofC. Note that the vertical range of the contours becomes small for higher values of L. A Ganymede-crossing orbit would needr a > 1:6. 168 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.32: Estimate of Forbidden Region boundary for C 0 = 3:0036. The gray area is the forbidden region, the Primaries indicated by dots and the Langrange points indicated by triangles. The green and red circles are the estimated limits 0 < L min;interior < 1 and 1 < L min;exterior , respectively. (a) Shows entire region of interest. (b) Shows region near Secondary body. 169 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Examples of Small Exterior Resonances While the Poincar´ e map can indicate how an orbit is affected by the CR6BP, it is often informative to examine a few select trajectories in position space. Figure 6.33 shows a 5:7 resonant orbit that is stable in both the CR3BP and CR6BP. We of course refer to Finite Time Stability when referring to stability in the CR6BP. From Fig. 6.33, we can now see that the perturbations from the additional three bodies has caused an offset and increased oscillation in the periapsis of the stable orbit. From this figure with one of the few stable members of the 5:7 family in the CR6BP, we can confirm our analysis of the apoapsis approaching the orbit of Ganymede. Due to the nature of the apoapsis, this is where the object lingers for the majority of the orbit period, so it is not difficult to see how an orbit such as this one could be very perturbed by Ganymede. In Fig. 6.34 we also look at a 6:7 resonance in the Galilean moon system. Here, the orbits are nearly identical visually. This indicates that the perturbations from bodies other than Jupiter and Europa in this region are either small, or applied at a frequency and phase such that major orbital changes are avoided. To test this further, we propagated the initial condition of the same orbit in an ephemeris model, with the Sun, Jupiter, Io, Europa, Ganymede and Callisto as perturbing bodies and based on the JPL DE431 and JUP310 ephemerides. We selected the epoch for this particular propagation as April 3rd, 2001, 22:34 TDB, which represents an alignment of the Galilean satellites. The alignment has Io on one side of Jupiter, and the other three moons aligned with each other approximately 180 degrees away. The resulting orbit is shown in Fig. 6.35, and we can see that the orbit remains stable for 12 years in the ephemeris model as well. In fact, the orbit does not seem very perturbed by the change of model, and has very similar periapsis oscillation amplitude and radial distance bounds. 170 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) (c) Figure 6.33: Example small (5:7) resonant orbit in Galilean moon CR6BP, C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in CR6BP. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue and red circles are the orbits of Io and Ganymede. Both orbits were propagated for 12 years. For this 5:7 resonant orbit, the orbit is much less regular in the CR6BP but keeps a five-point symmetry as in the CR3BP. The apoapsis and periapsis distances are not significantly different between the two models. 171 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) (c) Figure 6.34: Example small (6:7) resonant orbit in Galilean moon CR6BP, C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in CR6BP. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue and red circles are the orbits of Io and Ganymede. Both orbits were propagated for 12 years. In this case, visual inspection shows that the CR6BP orbit is indistinguishable from the CR3BP orbit. 172 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) (c) Figure 6.35: Example small (6:7) resonant orbit in ephemeris model,C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. This Poincar´ e map has identical data to the one in Fig. 6.34(a), we are only selecting a different orbit from it. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in an Ephemeris model. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue and red circles are the orbits of Io and Ganymede. Both orbits were propagated for 12 years. In spite of moving to a full ephemeris model, the orbit remains very similar visually to the CR3BP orbit. 173 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM 6.3.2 Medium Exterior Resonances This section focuses on medium exterior resonances up to and including 1:2, for which 1 L 1:2599. The augmented Poincar´ e maps that were the basis for the analysis in this section are shown in Figs. 6.36–6.39. The relative Abundance for all energies and this range of resonances is shown in Fig. 6.40. We can see that the trend seen in the small resonances continues, with resonances larger than 5:7 also having very low Abundance in the CR6BP. The 3:5 resonance is one exception that retains at least some stable members. The reason for this is likely the 3k:q format of the resonance, for which we will discuss the reasoning later. One clear difference is that the 1:2 resonance is quite unaffected across all energies, in spite of being a guaranteed Ganymede crossing orbit. The expla- nation is quite straightforward, since the Jupiter-Europa 1:2 resonance is the Jupiter-Ganymede 1:1 resonance. Given an appropriate phasing, and orbit with this period can easily avoid close appro- aches with Ganymede and remain stable. The members that did become unstable are most likely those that had a phasing where they crossed Ganymede’s orbit near Ganymede itself. This would of course not have any effect in the CR3BP. 174 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.36: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has an order of magnitude larger Resonance Width than the chaotic region in the CR3BP. Also note the nearly erased 2:3 resonance at L 1:145, the 3:5 shadow resonance at L 1:185, and the 1:2 shadow resonance atL 1:26. 175 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.37: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has an order of magnitude larger Resonance Width than the chaotic region in the CR3BP. Also note the nearly erased 2:3 resonance at L 1:145, the 3:5 shadow resonance at L 1:185, and the 1:2 shadow resonance atL 1:26. 176 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.38: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has an order of magnitude larger Resonance Width than the chaotic region in the CR3BP. Also note the nearly erased 2:3 resonance at L 1:145, the 3:5 shadow resonance at L 1:185, and the 1:2 shadow resonance atL 1:26. 177 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.39: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note that the chaotic region in the CR6BP has an order of magnitude larger Resonance Width than the chaotic region in the CR3BP. Also note the nearly erased 2:3 resonance at L 1:145, the 3:5 shadow resonance at L 1:185, and the 1:2 shadow resonance atL 1:26. 178 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a)C 0 = 3:0039 (b)C 0 = 3:0036 (c)C 0 = 3:0033 (d)C 0 = 3:0030 Figure 6.40: Medium Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. The green bars represent the CR3BP and the orange bars represent the CR6BP. The x- axis is labeled with the resonance ratio. We draw your attention to the region between the 3:4 and 1:2 resonances, with nearly no stable orbits in the CR6BP. The exceptions seem to be the 3:5 resonance and a very faint Abundance of the 5:7 resonance. The orbits starting with 5:7 and larger are those that will cross Ganymede’s orbit, and are thus more likely to be chaotically perturbed. The 3:5 crosses Ganymede’s orbit but still has stable orbits, indicating that with proper phasing, it can avoid significant perturbations from the other moons. The 3:5 resonance satisfies a pattern which we will discuss later, namelyp:q withp = 3k for any positive integerk. 179 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Examples of Medium Exterior Resonances While the Poincar´ e map can indicate how an orbit is affected by the CR6BP, it is often informative to examine a few select trajectories in position space. Figure 6.41 shows a 1:2 resonant orbit that is stable in both the CR3BP and CR6BP. We of course refer to Finite Time Stability when referring to stability in the CR6BP. Surprisingly, the periapsis oscillation is in fact smaller in the CR6BP for this orbit. Moreover, it seems the center of said oscillation is no longer aligned on thex-axis, but rather oscillates asymmetrically around g 240 degrees. 180 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) (c) Figure 6.41: Example medium (1:2) resonant orbit in Galilean moon CR6BP,C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in CR6BP. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue, red and green circles are the orbits of Io, Ganymede and Callisto. Both orbits were propagated for 12 years. In this case, we note that the CR3BP orbit has an oscillating periapsis relative to g = 180 degrees. In contrast, the CR6BP orbit periapsis seems to oscillate asymmetrically relative to g= 240 degrees. In addition, we also note that the oscillation amplitude is actually smaller in the CR6BP system. 181 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM 6.3.3 Large Exterior Resonances This section focuses on large exterior resonances between 1:2 and 1:22, for which 1:2599L 2:8021. The augmented Poincar´ e maps that were the basis for the analysis in this section are shown in Figs. 6.42–6.45. The Abundance of stable orbits in this range is shown for all studied energy levels in Fig. 6.46. The difference between the CR3BP and CR6BP is much more striking with orbits of this size. This is clear from both the augmented Poincar´ e maps and the Abundance. We can see that the chaotic region in general has a larger Resonance Width, as indicated by a more yellow color in the Poincar´ e maps. Additionally, only a few select resonances remain stable as indicated by the few purple shadow resonances in the augmented Poincar´ e maps and the only abundant orbits of the CR6BP in the Abundance plots. Keep in mind that most resonances in this size range will generally cross both Ganymede’s and Callisto’s orbit. The criticalL for crossing Callisto’s orbit is approximatelyL Callisto 1:3807, which is near the Europa 3:8 resonance. The remaining stable resonances seem to follow a pattern, if we ignore the resonances that don’t exist across all the energy levels studied. The pattern we find is that stable resonances seem to follow p:q withp = 3k orq = 7k, for positive integersk. Some examples from the Abundance plot are: 3:7, 3:10, 3:14, 1:7, 3:28, 1:14, and 1:21. We can explain this phenomenon by considering the fact that these are mostly Ganymede- and Callisto-crossing orbits. In order to prevent perturbations that can lead to chaotic behavior, close encounters with all moons must be avoided. To accomplish this some form of resonance must exist with each body. Otherwise, the point where the orbit crosses the orbit of the moon in the rotating frame fixed to that moon will become a nearly dense set and eventually result in an encounter. The convenient feature of the Galilean moons is that the 1:2:4 resonance of Io:Europa:Ganymede causes a resonance with any of those moons to also be a resonance with the other 2. This opens up the possibility for stability (avoiding encounters) as well as low energy transfers (targeting encounters). However, with the addition of a 4th moon in a quite 182 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM different resonant ratio, one must now target a resonance with that moon as well. That is how we arrive at the following rule-of-thumb for stable resonances in the Galilean system. p :q s.t. p = 3k; k2N OR p :q s.t. q = 7k; k2N (6.35) In Eq. 6.35,N is the set of natural numbers, or positive integers. This rule is not a guarantee and thus is neither a sufficient or necessary condition of stability. Exclusions to the rule exist, both resonances that satisfy the rule while being unstable, and those that don’t satisfy the rule while being stable. However, for large orbits that cross the orbit of Callisto, this rule can be seen as a necessary condition for stability. This follows from the previous discussion about preventing close approaches. For smaller orbits that don’t cross Callisto’s orbit, the rule still has an effect, due to the nature of resonance. The cumulative effect of perturbation due to Ganymede, Io and Callisto can be oscillatory in nature if there is an orbital resonance with each moon and the proper phasing is used. On the other hand, resonance can just as easily cause exponential instability if the phasing is such that the perturbations cause secular changes in the orbit. 183 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.42: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0039. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note the shadow resonances at L 1:326 (3:7),L 1:671 (3:14),L 1:913 (1:7), andL 2:759 (1:21). 184 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.43: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0036. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note the shadow resonances at L 1:326 (3:7),L 1:671 (3:14),L 1:913 (1:7), andL 2:759 (1:21). 185 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.44: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0033. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note the shadow resonances at L 1:326 (3:7),L 1:671 (3:14),L 1:913 (1:7), andL 2:759 (1:21). 186 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) Figure 6.45: 12-year Poincar´ e map, colored by Resonance Width,C 0 = 3:0030. (a) Jupiter-Europa CR3BP. (b) Galilean moon CR6BP. Every set of intersections from a given particle will have the same color. The color indicates the base-10 logarithm of the Resonance Width of that set, with purple indicating more stable and orange indicating more unstable. Note the shadow resonances at L 1:326 (3:7),L 1:671 (3:14), andL 1:913 (1:7). 187 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a)C 0 = 3:0039 (b)C 0 = 3:0036 (c)C 0 = 3:0033 (d)C 0 = 3:0030 Figure 6.46: Large Exterior Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. The green bars represent the CR3BP and the orange bars represent the CR6BP. The x- axis is labeled with the resonance ratio. In this plot, we have chosen to only label the resonances that have stable orbits in the CR6BP. The pattern to note here is that many of the resonances that remain in the CR6BP, across several energies follow a pattern. While there are exceptions, the pattern hasp:q withp = 3k orq = 7k for any positive integerk. Pay particular attention to the large Abundance of the 3:7 resonance, which is also the Callisto 2:1 resonance. 188 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM Examples of Large Exterior Resonances While the Poincar´ e map can indicate how an orbit is affected by the CR6BP, it is often informative to examine a few select trajectories in position space. Figure 6.47 shows a 3:7 resonant orbit that is stable in both the CR3BP and CR6BP. We of course refer to Finite Time Stability when referring to stability in the CR6BP. Just as the case of the 1:2 resonant orbit, we find that the periapsis oscillation is in fact smaller in the CR6BP for this 3:7 orbit. The center of the oscillation is also shifted, but to a lesser degree. 189 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM (a) (b) (c) Figure 6.47: Example large (3:7) resonant orbit in Galilean moon CR6BP, C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in CR6BP. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue, red and green circles are the orbits of Io, Ganymede and Callisto. Both orbits were propagated for 12 years. In this case, we note that the CR3BP orbit has an oscillating periapsis relative to g = 180 degrees. In contrast, the CR6BP orbit periapsis seems to oscillate asymmetrically relative to g= 170 degrees. In addition, we also note that the oscillation amplitude is actually smaller in the CR6BP system. 190 of 268 6.3. RESONANCE ABUNDANCE IN THE GALILEAN SYSTEM 6.3.4 Summary of Galilean Resonance Abundance We have studied the effect on stability of resonances in a CR6BP model, representing the Galilean moon system. A range of energies was chosen to represent orbits classified as “low energy” orbits, in order to make the results more applicable to quarantine orbits for low energy missions to Europa. The Abundance of stable resonant orbits was examined for a large range of resonance ratios in the exterior region of the Jupiter-Europa system, and we now turn to looking at the overall Abundance of stable resonant orbits. Table 6.3 shows the total Abundance for each system and Jacobi constant across all orbit sizes. The difference between energy levels seems negligible at this scale, and is most likely a result of the relatively small range of energies we chose to study. The behavior of a certain resonant orbit doesn’t drastically change between these energy levels. What we can tell is that the CR6BP has approximately one quarter the total Resonance Abundance as the CR3BP. Some resonances disappear alltogether in the CR6BP, and for the ones that remain stable, some choices of phasing become unstable. These two effects are what collectively reduce the total Abundance. Model C= 3.0039 3.0036 3.0033 3.0030 CR3BP 31.90% 31.36% 31.86% 30.50% CR6BP 8.58% 8.04% 8.08% 8.40% Table 6.3: Summary of total Resonance Abundance, Jupiter-Europa CR3BP vs. Galilean moon CR6BP. Each case has 5000 initial conditions and produces a 12-year Poincar´ e map. Listed is the fractional count of stable resonant orbits to total orbits simulated for the given energy levels and models. For cases with the same Jacobi constant, identical initial conditions were used. The values indicate a significant decrease in stable resonances when adding the additional Galilean moons. We also see that the dependence on Jacobi constant is negligible, at least in the energy range considered. 191 of 268 6.4. QUARANTINE ORBIT ANALYSIS 6.4 Quarantine Orbit Analysis As a sample application of this new concept of finite time stability and resonance abundance, we present a notional mission to land on Jupiter’s moon Europa. Missions like this have been studied in the past and are currently of interest for both NASA and ESA. The theoretical work in this section was performed to support studies for a notional Europa lander mission concept at JPL. The concept involves the use of low energy trajectories for the end game phase of the mission concept after the last flyby of Ganymede to approach and land on Europa. We assume two spacecraft, a carrier and a lander, joined as one from launch to arrival at Europa. The lander then separates from the carrier to land on Europa, while the carrier is placed in a disposal orbit. This architecture is only one option for such a mission concept. The disposal orbit, or alternatively “quarantine” orbit, is necessary for the purpose of planetary protection. Planetary protection policy places requirements on that carrier spacecraft to minimize the risk of Europa surface contamination by biological material from Earth. As we explore the solar system in search for signs of biological life, it is imperative that we take great care to avoid contaminating extraterrestrial bodies with terrestrial life. For details on these policies, we refer the reader to Rummel et al.[78]. One option is to incinerate the carrier as a means of sterilization. However, to incinerate the carrier is impractical because it would require carrying substantial additional mass for the fuel and mechanism for incineration. Leaving the carrier spacecraft in orbit about Jupiter for 12 years should guarantee that Jupiter radiation will have reached the innermost parts of the spacecraft, greatly reducing the number of microbes that might have been brought along from Earth. Thus, we study the option of leaving the carrier in a quarantine orbit that would not impact Europa for at least 12 years after the end of the mission. Two variations of this solution are examined in this chapter. In Section 6.4.1, 6.4.2, and 6.4.3 we consider placing the carrier in an orbit around Jupiter that is bounded away from Europa for 12 years. Much of the analysis in this section will use the methods from Section 6.2 and refer to results from Section 6.3. 192 of 268 6.4. QUARANTINE ORBIT ANALYSIS 6.4.1 Stable Resonant Orbits in the CR3BP Before we examine the more complex dynamical environments, we would like to gain an under- standing of the availability of stable resonant orbits in the simpler CR3BP model. As we discussed in Chapter 6, periodic and quasi-periodic resonant orbits in the CR3BP that exist inside the stable islands in a Poincar´ e map have guaranteed stability for infinite time. We did however develop a method in Section 6.2 for locating finite time stable resonant orbits from a computed Poincar´ e map. The method was able to locate the permanently stable resonant orbits automatically, by identifying them as finite time stable. Naturally, a permanently stable orbit should meet the requirements of a finite time stable orbit. From this analysis, it was determined that many options for stable resonant orbits exist in the Jupiter-Europa CR3BP. In Section 6.3, we presented the abundance of these sta- ble resonant orbits. These orbits will serve as the basis for analysis in higher fidelity models. The existence of the orbits is not the only requirement, of course. We also must determine if they are easy to enter. Section 6.5 focuses on that aspect of the problem, by locating compound orbits that connect orbits near Europa to resonant orbits around Jupiter. Our focus for the mission is to design low energy options for landing. This naturally leads to the carrier following low energy pathways near Europa as well. Therefore, the solution to place the carrier on a stable resonant orbit requires that the orbit be reachable at an energy not too different from its own at end of mission. The results shown in Figs. 6.54–6.55 indicate that even at low energy levels, a set of reachable resonant orbits exist when exiting through theL 2 gateway. We find that the only resonant orbit reachable at the lowest energy is a 6:7 resonance. However, at a slightly higher energy, it is also possible to enter a 7:8 and 5:6 orbit. Therefore, the stability results for these specific orbits will be most interesting in the analysis that follows. 193 of 268 6.4. QUARANTINE ORBIT ANALYSIS 6.4.2 Stable Resonant Orbits in a Galilean 6-Body Problem We apply the same method that we used to find stable orbits in the CR3BP to search for finitely stable orbits in a model of the Galilean moon system, which we label the Circular Restricted 6- Body Problem (CR6BP). This model was defined explicitly in Chapter 6. Orbits appropriate for quarantine will avoid impact with Europa for 12 years. When we performed the study, we based the resonance abundance on Poincar´ e maps propagated for 12 years. Therefore, we produce orbits that exceed the requirements for quarantine by being stable for two years longer. For the remainder of this chapter, when we use the word “stable”, we refer to the 12-year finite time stability and not true dynamical stability. When comparing the abundance of stable orbits in the CR6BP against those in the CR3BP, we found that some resonances were clearly less affected by the additional perturbations. The complete results were presented in Section 6.3, and we summarize the results most important to the design of a Europa lander mission here. We determined that resonant orbits that follow a specific format remained relatively stable in the CR6BP model. p:q resonant orbits with p = 3k or q = 7k tend to retain stability in the 6-body model. We attributed this to the 3:7 near-resonance between Callisto and Ganymede. Orbits following those rules will have a low integer resonance with Callisto as well as the other Galilean moons, allowing the orbit to avoid secular, repeated perturbations from the moons if an appropriate phasing is chosen. This rule seems to be much more significant for the large resonant orbits examined in Section 6.3.3, but even smaller orbits were affected. From the three resonances that the previous section indicated as reachable at low energy, some are more stable in the CR6BP. Both the 7:8 and 6:7 resonances remain relatively stable in the CR6BP, while the 5:6 resonance seems significantly less stable. The most interesting result here is that the 6:7 orbit satisfies both of the rules listed above. For an increase in the energy level, it would be possible to also reach a 4:5 or 3:4 resonant orbit. However, of those, only the 4:5 seems like a viable option for long term stability. The abundance of stable 3:4 orbits in the CR6BP is significantly reduced. We determined that this is a result of the 3:4 orbit having apoapses near the 194 of 268 6.4. QUARANTINE ORBIT ANALYSIS orbit of Ganymede, which results in many different orbit phasing angles causing repeated Gany- mede encounters. Only the 3:4 orbits with the proper phasing successfully avoid close encounters and thus remain stable. This effectively reduces the options for designing a useful quarantine orbit for the carrier spacecraft. Thus, from the perspective of a simplified 6-body model, we determine that a few resonances are appropriate design choices for a low energy quarantine orbit, including the 7:8, 6:7 and 4:5 resonant orbits. 6.4.3 Stable Resonant Orbits in an Ephemeris Model While simple models are useful for finding a general understanding of the dynamics and easier to understand, we are ultimately searching for realistic solutions for a quarantine orbit. Thus we must extend the search into a dynamical model using a planetary ephemeris. For this purpose, we have used DE430 and JUP310 ephemerides in the NAIF SPICE toolkit. We model the point mass gravity of the Sun, Jupiter, Io, Europa, Ganymede and Callisto. Since we are studying orbits near Jupiter, we do not model the forces due to other planets in the solar system. Producing a Poincar´ e map in an ephemeris model would be very compute intensive and require another non- cyclic search dimension due to the epoch having a significant effect on results. Therefore we instead take individual solutions known to be stable in the simpler CR3BP and CR6BP models and propagate them in an ephemeris model. By sampling this smaller set, the compute time is reduced and we are more likely to find stable solutions. Of course, we still need to choose an epoch. For the current work, we use the epoch April 3rd, 2003, 22:34 TDB, because it represents a temporary alignment of the Galilean satellites. We have applied this method to one example already in Section 6.3.1, and found a 6:7 resonant orbit that was stable for 12 years in an ephemeris model. See Fig. 6.35 for that result. We now attempt to compute more examples of stable orbits. Using the resonance abundance of orbits in the CR6BP as a guide, we chose some resonances that we expect may have solutions in an ephemeris model. The most obvious ones are the 7:8, 6:7, and 4:5 orbits mentioned in the previous section. A 195 of 268 6.4. QUARANTINE ORBIT ANALYSIS 6:7 orbit has already been found, and without much difficulty we are able to compute stable orbits with a 7:8 and 4:5 resonance. These orbits are presented in Figs. 6.48–6.49. In Section 6.3.2 and specifically Fig. 6.40, we showed that the 5:6 resonance has a significantly reduced abundance in the 6-body model but still has some solutions. The search in here confirms that analysis. We were able to find a few examples of stable 5:6 resonant orbits in the ephemeris model, but many were unstable. One example is shown in Fig. 6.50. Finally, we determined that 3:4 resonant orbits would be highly perturbed by the proximity of their apoapses to Ganymede. Some 3:4 resonances may still be present in the ephemeris, so an attempt was made to compute such an orbit. However, we were note able to compute a stable 3:4 resonant orbit, confirming our suspicions of their instability. Thus we have shown that several orbits that were found to be stable in a 6-body model of the Galilean system are also stable in an ephemeris model. Moreover, the resonances with low reso- nance abundance in a 6-body model are either difficult or impossible to locate as stable resonant orbits in the ephemeris model. 196 of 268 6.4. QUARANTINE ORBIT ANALYSIS (a) (b) (c) Figure 6.48: Example 7:8 resonant orbit in ephemeris model, C 0 = 3:0039. (a) Poincar´ e map in CR3BP, L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in an Ephemeris model. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue, red and green circles are the orbits of Io, Ganymede and Callisto. Both orbits were propagated for 12 years. Transfer to an ephemeris model results in a slightly larger oscillation of the periapsis. 197 of 268 6.4. QUARANTINE ORBIT ANALYSIS (a) (b) (c) Figure 6.49: Example resonant orbit in ephemeris model, C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in an Ephemeris model. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue, red and green circles are the orbits of Io, Ganymede and Callisto. Both orbits were propagated for 12 years. In spite of moving to a full ephemeris model, the orbit remains very similar visually to the CR3BP orbit. 198 of 268 6.4. QUARANTINE ORBIT ANALYSIS (a) (b) (c) Figure 6.50: Example 5:6 resonant orbit in ephemeris model,C 0 = 3:0039. (a) Poincar´ e map in CR3BP,L vs. g, with selected orbit highlighted in red. (b) Trajectory of selected orbit in CR3BP. (c) Trajectory of selected orbit in an Ephemeris model. The orange disc is Jupiter, and the forbidden region is shown in gray. The blue, red and green circles are the orbits of Io, Ganymede and Callisto. Both orbits were propagated for 12 years. This resonance has an overall lower abundance of stability in a 6-body model, and this is exhibited by a much larger periapsis oscillation. It was also difficult to compute this orbit when compared to the previous resonances. 199 of 268 6.5. COMPOUND RESONANT ORBITS 6.5 Compound Resonant Orbits Periodic resonant orbits typically have some form of angular symmetry (we are not considering quasi-periodic resonances in this section). For example, a 3:4 exterior resonance would have three loops over its 2 radian cycle, as discussed in Section 6.1.1. In a 2-body problem this symmetry is exact due to the central force, where the orbit in the rotating frame can be split into three iden- tical arcs. However, in the CR3BP the symmetry can be slightly skewed due to the two centers of attraction, with some loops having a slightly different shape than the others. Often, the perturba- tion to the orbit’s shape occurs near the Secondary. In some cases this perturbation is significant enough to cause the orbit to make one or more loops around eitherL 2 for exterior resonances, and L 1 for interior resonances. Moreover, an exterior resonance can even pass throughL 2 without loo- ping around it, and loop around either the Secondary orL 1 before returning to the exterior region resonance. Since we know that periodic orbits exist that loop around those Lagrange points and the Secondary naturally, these loops in the resonant orbits can be likened to partial revolutions of a local Lagrange point orbit. Using this idea, we can construct orbits that exhibit resonant behavior but also connect to local Lagrange point orbits. While these orbits will be similar to homoclinic connections of Lagrange point orbits, there is a fundamental difference. These orbits have finite period and are perfectly periodic while homoclinic connections have a theoretically infinite period with no well-defined start or end point other than it being on the periodic Lagrange point orbit. Orbits such as this can maken loops on the resonant branch andm loops around a Lagrange point orbit, wheren andm can theoretically be any positive integers or zero. In fact, the Conley-Moser Theorem shows that given a set of orbital centersm i (e.g.L 1 ,L 2 , the Secondary, and the Primary), you can assign an integern i to each center and an orbit will exist that loopsn i times around center m i , see Koon et al.[53]. Any itinerary between the centers can be constructed, with loops alterna- ting between different centers. We call orbits that exhibit this behavior “Compound Orbits” (COs), since they are essentially constructed from other periodic orbits as building blocks. This concept has been introduced in the literature recently by Restrepo and Russell[75] as “Patched Periodic 200 of 268 6.5. COMPOUND RESONANT ORBITS Orbits” (PPO). The method used to compute PPOs differs from our method since their method is more general and connects any two periodic orbits, while COs assume the basis of the orbit is a homoclinic connection with a symmetry when crossing the x-axis. In essence, the input of the PPO method is two orbits and the input to our method is only one orbit. The method by which we propose to compute these orbits is explained in Section 6.5.1, which follows. 6.5.1 New Method for Computing Compound Orbits In this section, we introduce a method for using invariant manifolds to search for homoclinic connections and then differentially correct the result to obtain finite period compound orbits. We define compound orbits as orbits with parts that resemble other periodic orbits. We remind the reader that homoclinic connections occur when the stable and unstable manifolds of a periodic orbit intersect. For details refer to Koon, Lo, Marsden & Ross[53]. Locating intersections in general is nontrivial, but the problem is easily reduced if we look for intersections on a Poincar´ e section. Keep in mind that stable and unstable manifolds of a given periodic orbit by definition have the same Jacobi constant, so that reduces the problem by one dimension. The Poincar´ e section reduced the dimension once more, and thus we only search for an intersection of two curves in a 2D space. We can simplify the problem yet again if we restrict ourselves to only locating symmetric homoclinic connections. In that case, we would choose a section withy = 0 and all homoclinic connections will now have _ x = 0 on that section. The problem is thus reduced to one dimension. An example of a Poincar´ e section of a stable and unstable manifold of a planarL 2 Lyapunov orbit in the Jupiter-Europa CR3BP is shown in Fig. 6.51(a). When we generate manifolds numerically, we must choose a discrete set of points on the ori- ginal orbit and offset them by a very small amount along the stable or unstable eigenvector. This means each individual trajectory on the manifold, which we will call a filament, is associated with an initial time on the orbit. We can therefore parametrize the curve produced by the intersection of the manifold with by a parameter 0 <T , which represents the initial time on an orbit with 201 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) Figure 6.51: L 2 Lyapunov orbit manifold intersections with Poincar´ e section, Jupiter-Europa CR3BP, C = 3:00360. (a) _ x vs. x plot of the manifold at the section . The stable manifold is green and the unstable is orange. Bisected intersections are highlighted with black squares. (b) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. periodT . Since we have reduced the problem to locating _ x = 0, we only require a one-dimensional parametrized curve,f _ x (). The function represents the value of _ x when the manifold with initial time on orbit reaches the section , and for the sake of brevity we will define f() f _ x (). An example of such curve is shown in Fig. 6.51(b), which in this case has two intersections with _ x = 0, indicated by filled black squares. Since the curve has two roots, we know this periodic orbit has two symmetric homoclinic connections passing through this section. Due to the discrete mesh of manifold filaments, we need to locate the exact manifold filaments that represents the homocli- nic connections. To accomplish this we opted to use a bisection method to compute the values of that yield _ x = 0. We are given a periodic orbit in the planar CR3BP with initial state ~ X 0 , period 202 of 268 6.5. COMPOUND RESONANT ORBITS T , and Jacobi constantC. If we haven j unstable manifold filaments from that orbit, we produce _ x j = _ x( j );j = 1;:::;n j . Next, we locate the valuesm ofj for which the following is true: mj; s.t. sign(f( m ))6= sign(f( j+1 )) (6.36) That set now defines ~ i = m ;i = 1;:::;n i (6.37) , wheren i is the number of members inm. ~ i now represents the location in the existing discrete manifold mesh that brackets the exact solution. Please note that in some cases, the entire orbit manifold does not reach the section , and in that case we must only locate the solutions to Eq. 6.36 in continuous segments of the parameter. A continuous segment is represented by consecutive integers in j that have states that reach . Next, we use a bisection method with each value of ~ i as an initial guess to find the values i such thatjf( i )j < , where is a small tolerance to define a perpendicular crossing. Note thatf() requires numerical propagation of the whole state to achieve the _ x-component at , but this is possible because completely defines the state for a given orbit. The unstable manifold trajectories with i ;i = 1;:::;n i are now connected to their respective stable manifold within some numerical tolerance at the section , but the beginnings will be atta- ched to some arbitrary locations near the original periodic orbit. We address this by reversing the problem and making the section the initial point of the orbit. We will apply a differential half- orbit corrector starting at and ending near the original orbit. Please refer to Appendix B.1 for details on the algorithm. The corrector we use requires the number of intersections with thex-axis, and thus we must find that number for the initial guesses produced by the bisection. We get this information by propagating the initial conditions of the manifolds for each i to and counting the intersections for each filament,p i . We now label the states on that result from the propagation of the bisected values for i as ~ ~ X 0 i . The superscript “0” indicates this is an initial condition at time 203 of 268 6.5. COMPOUND RESONANT ORBITS 0, and the tilde indicates they have not been corrected yet. These states now represent candidates for fundamental solutions to the compound orbit. We call them fundamental because they are very close to the true homoclinic connection. Using the bisection method in this way may not yield a perfectly symmetric homoclinic connection, since we have only propagated half of the trajectory with numerical methods in a chaotic system. This is why we must use a differential corrector to adjust the orbit to be periodic to a desired tolerance. Next, we correct fundamental compound orbits by using a symmetric differential corrector with Jacobi constant constraint and search for an orbit withp i intersections. The result is ~ X 0 ik , withk indicating the number of intersections. Thus the fundamental solution will havek = p i . The fundamental solutions will typically loop around the original periodic orbit, and we can exploit this to produce a family of orbits from each funda- mental compound orbit. For a given fundamental solution, we use the solution ~ X 0 ik as an initial guess and apply the same differential corrector. However, now we search for eitherk + 1 ork 1 intersections, which means adding or subtracting loops near the original orbit respectively. We can repeat this step and each time using the solution from the previous corrector as an initial guess to find another compound orbit. The process of removing loops can be repeated untilk = 1, which means the compound orbit simply passes by the original orbit without looping around. However, the process of adding loops can technically continue indefinitely ask!1. The limiting factor in this case becomes the capability of the differential corrector to converge on orbits with an arbitrary number of revolutions around an unstable orbit. Once the process of searching for new orbits has completed, we are left with a set of compound orbits with initial states ~ X 0 ik and periodsT ik , wherei indicates the fundamental solution from which it was generated, andk indicates the number of intersection near the original orbit. We demonstrate the result varying the value k for a fixed value of i in Fig. 6.52. These are all compound orbits from the same fundamental solution, but with different number of intersections. Note that each additional intersection adds another loop either near L 2 or around it. We also find that all the solutions for a given fundamental solution have the same resonance. This is to be expected, since 204 of 268 6.5. COMPOUND RESONANT ORBITS each one had the same state as an initial guess for the corretor, only adding intersections nearL 2 . In Fig. 6.52(b), the red orbit fork = 3 is very close to the greenk = 2 orbit, up until it adds another loop aroundL 2 . A similar effect is seen in Fig. 6.52(d), but the compound orbits are very close together in this case. Thek = 2 green orbit follows directly behind thek = 1 blue orbit to the 1st intersection, at which point it departs and loops aroundL 2 . Additionally, thek = 3 red orbit is nearly invisible since it follows the greenk = 2 orbit so closely, apart from the additionaly loop aroundL 2 . We also show the result for varying values ofi for fixedk in Fig. 6.53. The orbits shown all have the same number of intersections, but are generated from different fundamental solutions. In this case, we draw the reader’s attention to the varying shapes and topology of the solutions and that they enter different resonances when departingL 2 . Some of the compound orbits behave very differently when near theL 2 orbit, but enter the same resonance when they depart, such as the blue and magenta 7:8 compound orbits in Fig. 6.53(d). One important distinction of this method is that the choice of original intersections with before searching for connections decides the resonance order. For the work within, we have only analyzed manifolds on their first intersection with . The result is that any compound orbits found will have exactly one revolution around the barycenter. This in turn means that only first order resonant orbits will be found with the form p:(p 1). However, there is no theoretical limit to producing higher order resonances with this method. One simply allows the manifolds to reach their second intersection with to search for second order resonances, or allow more intersection for yet higher order resonances. 205 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.52: Fixed i Compound Orbits in the Jupiter-Europa CR3BP, C = 3:00330. (a) Large view of compound orbits fori = 1. The forbidden region is the gray area and the Jupiter is the orange dot. (b) Close up of (a) near theL 2 orbit, with Europa shown as a blue dot. Each additional intersection produces another loop around or near theL 2 orbit, with the red trajectory following the green very closely until the time when it loops aroundL 2 one more time. (c) Large view of compound orbits fori = 3. (d) Close up of (c) near theL 2 orbit. Here too, the red orbit follows very closely to the green orbit until it loops once more around theL 2 orbit. Note that all compound orbits generated from a given fundamental solution seem to have the same resonant ratio after departing the region near the Lagrange point. The resonance ratio is indicated in the figure title, and the loops shown in the large scale plots indicate that all orbits follow similar behavior away from the Secondary. 206 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.53: Fixed k Compound Orbits in the Jupiter-Europa CR3BP, C = 3:00330. (a) Large view of compound orbits fork = 1, the forbidden region is the gray area. The resulting resonance of the compound orbit is indicated by the legend. (b) Close up of (a) near theL 2 orbit. (c) Large view of compound orbits fork = 2, the forbidden region is the gray area. The resulting resonance of the compound orbit is indicated by the legend. (d) Close up of (c) near theL 2 orbit. Note that the different fundamental solutions produce compound orbits with different resonance ratios for a fixedk. Some of the resonances are the same, such as the magenta and blue orbits. This indicates that their behavior is only different near the Lagrange point, and both fall into the same resonance when they depart that region. 207 of 268 6.5. COMPOUND RESONANT ORBITS 6.5.2 Compound Orbit Results In this section, we present some example results of the compound orbit tool we developed. We have applied the tool to planarL 2 Lyapunov orbits of two different energies in the Jupiter-Europa CR3BP, representing behavior when theL 2 gateway is just barely open and for a relatively high energy. These results are shown in Figs. 6.54 and 6.55. We also apply the tool to the Sun-Earth system for bothL 1 andL 2 orbits at low and high energies, and present a summary of the results in Figs. 6.56–6.58. For the case of C = 3:0036 in the Jupiter-Europa CR3BP, the L 2 orbit is very small, and seemingly all the compound orbits reach a single resonance of 6:7, as indicated by Fig. 6.54(a)– (b). The Poincar´ e section of the manifold at is seen in Fig. 6.51, and indicates that only two intersections exist at this energy. This means that only two true symmetric homoclinic connections exist with one revolution around the barycenter. However, it is encouraging that so many compound resonant orbits with varying behavior were found in spite of this limitation. If we examine a slightly higher energy in Fig. 6.55, we find many more compound orbits, as a result of now having six symmetric manifold intersections. At this energy, it is possible to reach three separate resonances including 5:6, 6:7 and 7:8, with several compound orbits belonging to each resonance. Moreover, this energy is still considered “low energy” for the purposes of classical mission design. Therefore, having access to these compound orbits is beneficial for applications such as tour design, with the end goal of a flyby, capture or landing. We now consider the Sun-Earth CR3BP and apply the same technique to compute compound resonant orbits for a planarL 2 Lyapunov orbit. Initially, we choose an energy very close to that of theL 2 Lagrange point, and present the resulting orbits in Fig. 6.56. We find that only two sym- metric intersections exist at this energy, and they result in seven different compound orbits, all of which connect to a 13:14 resonance. By increasing the energy slightly, we gain two intersections, and with it access to the 12:13 resonance as seen in Fig. 6.57. However, if we significantly increase 208 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) Figure 6.54: L 2 Compound Orbits in the Jupiter-Europa CR3BP, C = 3:00360. (a) Large scale plot of the set of compound orbits computed. Jupiter is the orange disc and the forbidden region is the gray area. The legend indicates the resonances and number of intersections,k. (b) Close up of the orbits nearL 2 , with Europa as a blue dot. Note that some orbits follow very closely together apart from the very end where it may loop aroundL 2 an additional time, e.g. the green trajectory. The manifold intersections for these orbits are shown in Fig. 6.51 the energy, a large variety of compound resonant orbits are found. In Fig. 6.58, the algorithm com- puted 77 orbits ranging from 9:10 to 25:26 resonances. We also see that the orbits closer to a 1:1 resonance remain closer to the orbit of the Earth, and its forbidden region as indicated by the more reddish orbits in Fig. 6.58(c). In contrast, the orbits with larger semimajor axes are seen as more blue and magenta orbits with larger apoapses in the same figure. The interesting feature to note is that at very low energy, there seems to be a fundamental first order resonance that can be achie- ved. As the energy increases, orbits with larger periods can be reached, which is to be expected. However, what is surprising is that orbits with periods smaller than the fundamental resonance aren’t reachable at very low energy. These orbits are closer to a 1:1 with the Secondary, so one would expect that they are easily achievable from an Keplerian energy standpoint. From the view of the CR3BP, we know that the forbidden region covers most of the orbit of the Secondary for low 209 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.55: L 2 Compound Orbits in the Jupiter-Europa CR3BP,C = 3:00330. (a) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. (b) Close up ofL 2 for all single intersection orbits. The colors represent the resonance, as indicated by the legend. (c) Large scale plot of all orbits, grouped by resonance. The count in the legend indicates how many solutions belong to each resonance. Jupiter is the orange disc and the forbidden region is the gray area. (d) Close up (c) nearL 2 , with Europa as a blue dot. 210 of 268 6.5. COMPOUND RESONANT ORBITS energy. This means that orbits in the exterior or interior regions cannot approach the orbit of the Secondary except to within some minimum distance. A nearly 1:1 orbit would have to approach this orbit closely, and thus at low energies, these resonances cannot exist. In the limiting case of an exact 1:1 orbit, it must either lie directly on the Secondary orbit, or cross it multiple times. That would only be possible at an energy high enough so the forbidden region allows such passage. The energy studied here is similar to that of asteroid 2006 RH120, which we studied in Chap- ter 4. We found that the Asteroid entered a 20:21 resonance after departing Sun-EarthL 2 , and we see from our current analysis that such a resonance is easily obtainable at the appropriate energy level. We also compute the compound resonant orbits for a planarL 1 Lyapunov orbit at the same energy and show the results in Fig. 6.59. Once again, we compare the resulting resonances to that of the Asteroid. The Asteroid was found to have been on a 29:27 resonance before approaching L 1 , and we see that both 14:13 and 15:14 compound resonant orbits exist. A 29:27 resonant orbit has a period that lies in between that of 14:13 and 15:14 resonances, thus we expect that the 29:27 resonant compound orbits exist at this energy. The reason our current algorithm did not locate a 29:27 solution is simply because it is a second order resonance, and we only searched for those of first order. As mentioned before, we would need to allow an extra revolution of the manifolds around the barycenter to search for those solutions. 211 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.56: L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000891. (a) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. (b) Close up ofL 2 for all single intersection orbits. The colors represent the resonance, as indicated by the legend. (c) Large scale plot of all orbits, grouped by resonance. The count in the legend indicates how many solutions belong to each resonance. The Sun is the orange disc and the forbidden region is the gray area. (d) Close up (c) nearL 2 , with the Earth as a blue dot. 212 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.57: L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000886. (a) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. (b) Close up ofL 2 for all single intersection orbits. The colors represent the resonance, as indicated by the legend. (c) Large scale plot of all orbits, grouped by resonance. The count in the legend indicates how many solutions belong to each resonance. The Sun is the orange disc and the forbidden region is the gray area. (d) Close up (c) nearL 2 , with the Earth as a blue dot. 213 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.58: L 2 Compound Orbits in the Sun-Earth CR3BP,C = 3:000500. (a) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. (b) Close up ofL 2 for all single intersection orbits. The colors represent the resonance, as indicated by the legend. (c) Large scale plot of all orbits, grouped by resonance. The count in the legend indicates how many solutions belong to each resonance. The Sun is the orange disc and the forbidden region is the gray area. (d) Close up (c) nearL 2 , with the Earth as a blue dot. 214 of 268 6.5. COMPOUND RESONANT ORBITS (a) (b) (c) (d) Figure 6.59: L 1 Compound Orbits in the Sun-Earth CR3BP,C = 3:000500. (a) _ x vs. plot of the unstable manifold intersection at . Bisected intersections are highlighted with black squares, with the initial guess for the bisection as a red square. (b) Close up ofL 1 for all single intersection orbits. The colors represent the resonance, as indicated by the legend. (c) Large scale plot of all orbits, grouped by resonance. The count in the legend indicates how many solutions belong to each resonance. The Sun is the orange disc and the forbidden region is the gray area. (d) Close up (c) nearL 2 , with the Earth as a blue dot. 215 of 268 6.6. CHAPTER SUMMARY 6.5.3 Relationship Between Compound Orbits and Homoclinic Orbits The method we propose can in the limit approach the true homoclinic connection of the original orbit. We compute resonances that return to the vicinity of the original orbit and iteratively add revolutions around the Lagrange point about which the orbit is centered. Every added loop in the vicinity of the original orbit can be seen as another winding around the manifold that connects to the orbit. A true homoclinic connection will take infinite time to wind off of the original orbit, then take infinite time to wind back onto it once it returns. Our compound orbits are finite period orbits that make loops around the original orbit, but if the process of adding revolutions was continued ad infinitum, one would expect to approach an orbit with infinite time and infinite revolutions as it winds onto and off the original orbit. For the reason of having infinite periods, homoclinic orbits are not useful for spacecraft missions. Compound orbits, on the other hand provide flexibility in choosing an orbit period for leaving and returning to the original orbit, within reason of course. There is naturally a lower limit on the time it will take to complete an exterior resonant orbit cycle, but the flexibility lies in the addition or subtraction of loops at the origin. 6.6 Chapter Summary We have successfully developed an approach to predict where stable and unstable resonant orbits will occur in a Poincar´ e map in the CR3BP. This also resulted in a prediction of the topology of the Poincar´ e map around the fixed points that represent those orbits. This discussion was presented in Section 6.1. A new method for evaluating the finite time stability of resonant orbits was presen- ted in Section 6.2, which we called Resonance Abundance. This method did not require that the dynamical system follows the strict rules in the CR3BP, and allowed for analysis of more com- plex systems. We found that this becomes useful for designing missions with long term stability requirements, and the model is more complex than the CR3BP. 216 of 268 6.6. CHAPTER SUMMARY We then applied the concept of Resonance Abundance to a 6-body model of the Galilean system in Section 6.3 to find resonant orbits appropriate for missions with long term stability requirements. The most significant finding was that for the system of Galilean moons, the resonances that meet specific requirements were more abundant in the 6-body model. The requirement follows from having resonant motion with respect to all four Galilean satellites, and not just the first three, which have a clean 1:2:4 resonant relationship. The final satellite, Callisto, has a near 3:7 resonance with Ganymede, and designing orbits to form resonances with this moon results in higher stability. We also studied some solutions to the planetary protection problem at end-of-mission for a notional mission to land on Europa in Section 6.4. We examined the option of using stable reso- nant orbits around Jupiter as a quarantine orbit. The spacecraft would remain there for 12 years without impacting Europa and the assumption is that it will be sterilized by the intense Jupiter radi- ation environment. We approached the problem in three stages: Section 6.4.1 focused on stable resonant orbits in the CR3BP, Section 6.4.2 focused on stable resonant orbits in a 6-body approxi- mation of the Galilean system, and Section 6.4.3 extended some of the previous solutions to a full ephemeris. It was determined that a few resonances are appropriate for long term stability, as well as being feasible for mission fuel requirements from an energy standpoint. The 7:8, 6:7 and 4:5 resonances were deemed most feasible. This analysis was made possible by the work completed in Sections 6.2 and 6.3. Lastly, we studied a new method for computing Compound Orbits in Section 6.5. These orbits are useful as they approximate homoclinic connections and provide insight into the natural motion for approaching and departing planets and moons. Specifically, we can predict which resonant orbits can be naturally reached at a given energy. Another useful feature of this class of orbits is the ability to add or subtract loops near the Lagrange point gateways, while remaining on the same resonance. The reason this is useful is to have flexibility with arrival or departure times on approach to a planet or moon. This opens up the design space. We can also expand this concept to 217 of 268 6.6. CHAPTER SUMMARY build itineraries of orbits that loop any number of times around a set of orbital centers, such as a planet or Lagrange point. 218 of 268 CHAPTER 7: CONCLUSION 7.1 Novel Contributions As a guide to the reader, we summarize the contributions presented by our work in Table 7.1. 219 of 268 7.1. NOVEL CONTRIBUTIONS Contribution Summary Section Manifold Control of Asteroid 2006 RH 120 Showed that the only known temporarily captu- red asteroid closely followed the invariant mani- folds of halo orbits in the Sun-Earth CR3BP during the capture and escape phases. 4.1 Temporary Capture Analysis of Asteroid 2006 RH 120 The perturbations from the Moon seem to domi- nate the behavior of the Asteroid during tempo- rary capture. 4.2 Dynamic Time Warping for trajec- tory comparison Presented a new approach for quantitatively com- paring trajectories in shape space. 4.2.3 The Moon’s effect on transit dyn- amics The Moon reduces the overall rate of Earth impact at the high end of the low energy range studied. At low energies it decreases the rate of transit from the interior to the exterior region. 5 6-Body model for design of reso- nant Europa quarantine orbit Developed predictive model for stable and unsta- ble points in Poincar´ e map; developed method for computing finite time stability from perturbed Poincar´ e map; developed model for computing stable resonances in satellite systems with inhe- rent resonance; developed compound orbit algo- rithm. 6 GPU accelerated mission design GPU propagator 1 10 2 –1 10 4 faster than a CPU implementation; H´ enon step GPU Poincar´ e map algorithm about 200 times faster. A NIAC interplanetary Cubesat study Showed that exploring the solar system with Cubesats is feasible, through use of advanced propulsion and low-energy astrodynamics. [81, 82] Conley Theorem extension Extended the range of energies for which the Conley Theorem is valid in the planar CR3BP (collaboration). [83] Table 7.1: Summary of contributions. The section(s) in which the contribution is explained is indicated. 220 of 268 7.1. NOVEL CONTRIBUTIONS 7.1.1 Manifold Control of Asteroid 2006 RH 120 We studied the particular case of asteroid 2006 RH 120 , which is the only observed temporarily captured natural object at the time of writing. The analysis in Section 4.1 showed that the Asteroid approached along the stable manifold of anL 1 halo orbit on the Sun-Earth CR3BP and departed on the unstable manifold ofL 2 halo orbit. Moreover, a resonance transition occurred which moved the Asteroid from a 27:29 interior resonance to a 21:20 exterior resonance with Earth. This transition means that the Asteroid has experienced close encounters with Earth in the past and is expected to do so in the future. 7.1.2 Temporary Capture Analysis of Asteroid 2006 RH 120 In Section 4.2 we focused on the temporary capture phase of the Asteroid and indicated that the perturbation from the Moon is the dominant factor controlling the Asteroid’s motion. This followed from the BCP being the model with the best overall agreement between reality and model when using computed parameters. We concluded that in general, the BCP provides a better model than the ER3BP during the Capture phase alone. The situation becomes a bit more obvious when considering the phases before and after capture. The ER3BP is generally better at modeling the behavior in those regions, as can be expected due to close approaches to the Earth system being more accurately modeled. At those distances, the Earth and Moon are nearly in the same place from the point of view of an asteroid, so modeling the exact position of the Moon and Earth separately will have little effect. 7.1.3 Dynamic Time Warping for Trajectory Comparison The same section that examined the temporary capture phase of Asteroid 2006 RH 120 also intro- duced a new method for comparing trajectories in space, which we label Modified Dynamic Time Warping. This method will be useful for the purpose of orbit classification or possibly optimi- zation. By being able to compare the shape of trajectories, it becomes possible to classify them 221 of 268 7.1. NOVEL CONTRIBUTIONS into groups based on shape. It also becomes possible to determine how close to an “ideal path” a computed solution is with a quantitative measure. Expressing topology of curves in a 2D plane has been studied in the past, but computing similarity of curves in 3D or higher dimensional space is not well studied. 7.1.4 The Moon’s Effect on Transit Dynamics The more general effects of the Moon on transit dynamics were examined in Chapter 5. Asteroids were assumed to be approaching from the interior region on the manifold of anL 1 Lyapunov orbit. We determine that the effect of the Moon varies with the energy of the approaching asteroid. At most energies, we found lower rate of transit from interior to exterior regions with the Moon present in the model. This indicates that the overall effect of the Moon is to prevent transit in the range of energy studied. We also noted that the position of the Moon had a smaller effect on the rates at lower energy. At higher energies, the position of the Moon has a large effect on the rates. The average rate of Earth impact is similar or slightly higher in the BCP at the lower energies studied, and there are some lunar orientations that cause a high rate of impacts. Furthermore, there was a significant reduction in Earth impact at high energy, indicating that the Moon acts to protect the Earth from asteroids with a larger potential for catastrophic damage. We also noted that the high range of orbital energy studied is still a relatively low energy from the perspective of all PHAs. This means that these objects would need to be more massive to pose a significant threat. 7.1.5 6-Body Model for Design of Resonant Europa Quarantine Orbit In order to search for solutions to the planetary protection problem at end-of-mission of a lander mission to Europa, we studied some aspects of resonance. First, we examined the option of using stable resonant orbits around Jupiter as a quarantine orbit. The spacecraft would remain there for 12 years without impacting Europa and the assumption is that it will be sterilized by the intense 222 of 268 7.1. NOVEL CONTRIBUTIONS Jupiter radiation environment. First, we developed an approach to predict where stable and unsta- ble resonant orbits will occur in a Poincar´ e map in the CR3BP. This also resulted in a prediction of the topology of the Poincar´ e map around the fixed points that represent those orbits. This discus- sion was presented in Section 6.1. A new method for evaluating the finite time stability of resonant orbits was presented in Section 6.2, which we call resonance abundance. We then applied the concept of resonance abundance to a 6-body model of the Galilean system in Section 6.3 to find resonant orbits appropriate for missions with long term stability requirements. The most significant finding was that for the system of Galilean moons, the resonances that meet specific requirements are more abundant in the 6-body model. The requirement follows from having resonant motion with respect to all four Galilean satellites, and not just the first three, which have in a clean 1:2:4 resonant relationship. The final satellite, Callisto, has a near 3:7 resonance with Ganymede, and designing orbits to form resonances with this moon resulted in higher stability. It was determined that a few resonances are appropriate for long term stability, as well as being feasible for mission fuel requirements from an energy standpoint. The 7:8, 6:7 and 4:5 resonances were deemed most feasible. This analysis was made possible by the work completed in Chapter 6. 7.1.6 GPU Accelerated Mission Design An algorithm was presented for solving ODEs in parallel on GPU hardware. We also presented a novel method for propagating to user defined events as opposed to desired time. This event stop- ping algorithm makes use of the “H´ enon step”, which is very well suited for implementation on GPU hardware. We tested the solver using a CR3BP model for normal propagation as well as computing Poincar´ e maps, and presented the relative speedup of the GPU implementation. This analysis revealed that regular propagation is in the range of 1 10 2 1 10 4 faster than a CPU implementation for normal propagation. On the other hand, the Poincar´ e map algorithm proved to be about 200 times faster when applied to problem sets reasonable for Poincar´ e maps. With the more widespread accessibility of GPUs, even on laptops, GPU algorithms can be more widely 223 of 268 7.1. NOVEL CONTRIBUTIONS used in industry. Additionally, speedup on this order can allow more rapid design cycles and can enable interactive, concurrent trajectory design. Trajectory design has always been a subsystem of a complete mission that requires some lead time. The lead time is partially due to numerical propagation as an inherent road block, since it is often a fundamental operation required by opti- mizers and Monte Carlo analysis. We hope that GPU methods become more prevalent and further enhanced to reach this goal. 7.1.7 NIAC Interplanetary Cubesat Study While the work is not explicitly included in this document, we aided in a NASA Innovative Advan- ced Concepts phase I study that aimed to evaluate the possibility of using cheap Cubesat spacecraft for interplanetary exploration. Our part involved examining the orbital mechanics techniques capa- ble of saving enough V to make these missions feasible. We also studied trajectories exploiting advanced propulsion concepts for Cubesats, such as solar sails. This study is one element that lead to the recent launch (May 5th, 2018) of the first interplanetary Cubesat mission, MarCO. The NIAC final report was presented to NASA in 2012[81] and was also submitted to a journal[82]. 7.1.8 Conley Theorem Extension Another contribution which we do not include in this document concerns the energy levels for which Conley’s Theorem holds. The work was presented jointly in Swenson et al.[83]. The original theorem presented by Conley[21] roughly states that trajectories passing through the Lagrange point gateways must in a sense pass “through” the Lyapunov orbit, but for some range of Jacobi constant near that of the Lagrange point. The extension consists of showing that this theorem holds to much higher energy than initially implied. Namely, the transit orbits seem to obey the theorem for Jacobi constant equivalent to that of theL 5 Lagrange point, which corresponds to the forbidden region disappearing in the planar problem. 224 of 268 7.2. FUTURE WORK 7.2 Future Work We present some goals we have for extension of the work presented within. 7.2.1 DTW Classification As we mentioned before, Dynamic Time Warping would be very useful for classifying trajectories by shape. This is promising for mission design applications involving studies of large sets of trajec- tories. Often, many trajectory options are presented for a given mission architecture, and they are sometimes manually classified based on their behavior. In some problems, the database of trajec- tories can get extremely large, and manual classification of their behavior can become impractical. Having a method for automatic clustering of these databases will give mission designers an over- view of the design space that is more understandable and easy to work with. In addition, artificial intelligence methods can make use of curve similarity for computing trajectories with desired cha- racteristics. By providing a quantitative measure of similarity, one can design trajectories that behave “similarly” to some desired path, while optimizing other parameters such as fuel and time of flight. 7.2.2 GPU Methods The speedup experienced by GPU propagation is on such a high order that new algorithms can be considered that were not feasible in the past. For example, computing partial derivatives along trajectories can be more convenient and fast. Automatic partial derivatives no longer requires derivation and propagation of complex variational equations that may be different for every model and application. Now, a brute force approach of propagation of many perturbed states allows for estimates of the partials for nearly no cost over propagation of the reference trajectory. These partials are required in many applications such as optimization methods, sensitivity analysis for navigation, and differential correctors. Additionally, computing Poincar´ e maps on-the-fly will 225 of 268 7.2. FUTURE WORK make that dynamical tool more useful for mission design. There are surely many more algorithms used in trajectory design that may have alternate GPU implementations that can drive design closer to a true concurrent design cycle. 226 of 268 PUBLICATIONS Publications [1] B. D. ANDERSON AND M. LO, Dynamics of asteroid 2006 RH120: Temporary capture phase, in 28th AIAA/AAS Space Flight Mechanics Meeting, Kissimmee, FL, 2018, p. 1689. [2] B. D. ANDERSON AND M. W. LO, Dynamics of asteroid 2006 RH120: Pre-capture and escape phases, in 26th AAS/AIAA Space Flight Mechanics Meeting, vol. 158 of Advances in the Astronautical Sciences, Napa, CA, 2016, Univelt, Inc., pp. 777–795. [3] B. D. ANDERSON, M. W. LO, AND M. VAQUERO, The stability of orbital resonances for europa quarantine design: Escape orbit case, in 29th AAS/AIAA Space Flight Mechanics Meeting, Ka’anapali, HI, 2019. unpublished. [4] P. C. LIEWER, A. T. KLESH, M. W. LO, N. MURPHY, R. L. STAEHLE, V. ANGELOPOULOS, B. D. ANDERSON, M. ARYA, S. PELLEGRINO, J. W. CUTLER, ET AL., A fractionated space weather base at L 5 using cubesats and solar sails, in Advances in Solar Sailing, Springer, 2014, pp. 269–288. [5] R. STAEHLE, B. ANDERSON, B. BETTS, D. BLANEY, C. CHOW, L. FRIEDMAN, H. HEM- MATI, D. JONES, A. KLESH, P. LIEWER, J. LAZIO, M. LO, P. MOUROULIS, N. MURPHY, P. PINGREE, J. PUIG-SUARI, T. SVITEK, A. WILLIAMS, AND T. WILSON, Interplanetary cubesats: opening the solar system to a broad community at lower cost, Tech. Rep. Final Report on Phase 1 to NASA Office of the Chief Technologist, NASA Innovative Advanced Concepts, December 2012. [6] R. STAEHLE, D. BLANEY, H. HEMMATI, M. LO, P. MOUROULIS, P. PINGREE, T. WILSON, J. PUIG-SUARI, A. WILLIAMS, B. BETTS, ET AL., Interplanetary cubesats: opening the solar system to a broad community at lower cost, Journal of small satellites, 2 (2013), pp. 161– 186. [7] T. SWENSON, M. LO, B. D. ANDERSON, AND T. GORORDO, The topology of transport through planar lyapunov orbits, in 28th AIAA/AAS Space Flight Mechanics Meeting, Kis- simmee, FL, 2018, p. 1692. 227 of 268 BIBLIOGRAPHY Bibliography [1] C. ACTON, N. BACHMAN, B. SEMENOV, AND E. WRIGHT, A look towards the future in the handling of space science mission geometry, Planetary and Space Science, 150 (2018), pp. 9–12. [2] C. H. 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We want to note that in this document, the algorithm development, testing and applications are the result of a group effort. A.1 GPU Computing Fundamentals GPUs have long been used for rendering real time graphics for interactive visualization. However, in recent years the advantage of the GPU computing architecture is beginning to be exploited for scientific compute applications. A standard CPU works under the MIMD or “Multiple Instruction, Multiple Data” architecture. This means that each core on a CPU can take some input data and perform a set of instructions on that data, while another core can be performing completely diffe- rent operations on different input data. These operations can also be asynchronous. On the other hand, a GPU operates with the SIMD or “Single Instruction, Multiple Data” architecture. As you can imagine this means that the cores on a GPU take different input data but perform the same operations on each. GPUs were designed to do this since rendering a large array of pixels based on scene geometry is a very parallel task where each pixel goes through the same operations as the next. The SIMD architecture can perform parallel tasks much faster than a CPU, but the trade-off is that they must often be synchronous. This results in some particular considerations when desig- ning algorithms to make use of GPU architecture. It also means that some problems are highly unsuited for GPU implementation, while others simply need to be designed properly. There is a concept within GPU computing called “thread divergence” which is something one needs to avoid 235 of 268 A.1. GPU COMPUTING FUNDAMENTALS when designing GPU algorithms. Thread divergence simply means that the same code running on two different threads reaches a branching point and don’t end up on the same branch. This is inherent to “if”-statements. The way divergence manifests on GPUs is that thread 2 will need to wait while thread 1 computes its branch and then thread 1 waits for thread 2, since they share instruction cores. Splitting into 2 branches can sometimes not cause a large performance decrease, but repeated branching is highly undesirable and can cripple GPU performance. The application we first chose to tackle to accelerate mission design was massively parallel propagation of trajectories. There are many applications where we need to perform this task, such as when computing Poincar´ e maps, or when performing Monte Carlo statistical analysis. Since propagating individual particles is a highly parallel task, with each particle is completely uncoupled from the others, we found this to be a good first problem to solve. It was also a problem that would benefit the field of trajectory design as a whole. We note that Ordinary Differential Equation (ODE) solvers on GPUs have been made by others in recent years, but often for specific applications or without some of the features we require. After a review of the literature and existing software, two existing packages were found that use GPUs for numerical integration: “ODEINT” and “Swarm-NG”. ODEINT is more general and can take any equation of motion but the package lacks event function stopping conditions, which is a requirement for our task, see Ahnert and Mulansky[3]. There is a method of finding events if one chooses an integration method with dense output, of the several methods available within ODEINT. However, only two methods currently allow dense output (Euler’s method and Dormand- Prince 5th order) and neither has an order high enough to meet our accuracy requirements. The Swarm-NG package is highly optimized for the task of simulating a large numberM ofN-body systems with small N, see Dindar et al.[23]. It was essentially made for simulating planetary systems with many different initial conditions. Our task is similar to Swarm-NG, but may benefit greatly from adapting the code to optimize for our specific purpose. The reason being that we have many particles, and each is completely independent from the others. Additionally, each particle 236 of 268 A.1. GPU COMPUTING FUNDAMENTALS is perturbed by several other bodies, either determined from an ephemeris or a simpler dynamical model. This is different from the multipleN-body systems problem, where every force comes from bodies that are integrated internally by the equations of motion. This also means that our system can be non-autonomous (for the ephemeris and some other models such as the Elliptic Restricted 3-Body Problem), while the multipleN-body problem is autonomous. Moreover, the idea behind Swarm-NG is to simulate many planetary systems each with different parameters. In our case, the dynamical system and its parameters is a constant that all particles share. In addition to the two aforementioned packages, an adaptive step solver was implemented on the GPU by Garcia et al.[35] for the specific purpose of simulating electric cardiac activity. However, this algorithm is too specialized to be useful to our application. The conclusion from the above review was that we must create our own high order, adaptive step ODE solver for the GPU with event stopping functionality. The closest relative to our algorithm is the one developed by Arora[9]. Arora’s implementation uses a Runge-Kutta 5/4th order adaptive step method. The dynamical model is an Earth-centered inertial frame, with geopotential perturbations and perturbations due to the Sun and Moon. The orientation of the Earth and the positions of the Sun and Moon are computed from a custom ephe- meris. When compared to serial CPU equivalent algorithms, they recorded speedup anywhere between 50 to 70000x faster on the GPU. The large variation in speedup was found to be a result of a combination of factors, such as the number of particles simulated, the distribution of the initial states, and the order of the geopotential field. A major reason for the lower bound on speedup was a result on the smaller particle sets, where the GPUs have a lower duty cycle. GPUs can work most efficiently when saturated with workloads, so that there are always operations waiting to be completed. After review of existing software we concluded that we must create our own GPU ODE solver. The reasoning was that the chaotic systems we will be propagating requires hig- her order methods than those available, and to compute Poincar´ e maps we require event stopping functionality. Our algorithm will however share many features with that of Arora. 237 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER The speedup of our algorithm is similar to the results of Arora. Another similarity between our propagator and that of Arora is the use of a custom ephemeris solution. In our case, we use FastSPICE, developed at JPL by Paul Finlayson[32]. FastSPICE is based on the algorithms in the freely available and widely used SPICE toolkit[2, 1], but is much faster due to keeping only the capabilities required for numerical propagation. We use FastSPICE for the propagator when the ODE system is an ephemeris model. The most significant additions our algorithm makes is the addition of a higher order method, and the novel approach to event stopping. This feature enables propagation not to a desired final time (or other independent variable), but rather to a user-defined condition that is a function of the state. An example would be to stop the integration when a particle impacts the surface of a planet or reaches defined orbit insertion plane. This was one of the key obstacles that we had to overcome for making the algorithm viable for GPU implementation. The process of event stopping with numerical ODE solvers typically involves iterative refinement with a tolerance as a stopping condition. That would require a while loop, which can severely hamper the synchronicity required for GPUs to perform better than a CPU counterpart algorithm. We use a “H´ enon step” to turn this iterative process into a single operation, and we discuss the details in Section A.3. This event stopping is also what enables use to compute Poincar´ e maps using the GPU propagator, since stopping on a defined plane is a necessary feature of that method. The particulars for that are discussed in Section A.4. A.2 Runge-Kutta-Fehlberg 7/8th Order Adaptive ODE Solver In this section, we describe the adaptive step ODE solver, and how we overcame the asynchronicity inherent to adaptive stepping. The ODE solver algorithm we chose to implement is a Runge-Kutta- Fehlberg 7/8th order adaptive step method, as described by Fehlberg[31]. We chose this method because it has been used with great success for the purpose of propagating dynamical systems within the field of astrodynamics. Additionally, it is one of the methods recommended for this purpose in a survey of ODE solver methods by Montenbruck[66]. We note that an adaptive step 238 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER solver was a necessity due to the chaotic nature of the systems we typically study. There are essentially 2 major choices when it comes to programming languages for GPUs: OpenCL and CUDA. OpenCL is hardware agnostic, and thus more versatile. On the other hand, CUDA is developed by Nvidia and only compatible with their own GPU hardware, see Nickolls et al.[69]. It has been shown that in many applications CUDA is significantly faster than OpenCL. Karimi et al.[50] found that OpenCL was between 16-67% slower than CUDA, while Fang et al.[29] found that CUDA was at most 30% faster than OpenCL in most applications. We also note that the downside of being limited to Nvidia hardware if choosing CUDA is not that significant, as Nvidia holds the majority of the market share for GPU hardware. We chose to use CUDA for our GPU propagator implementation for its higher performance and existing access to the appropriate hardware. A.2.1 Algorithm In order to implement an adaptive step ode solver on a GPU, we had to overcome some limitations inherent to GPU architecture. GPUs are good at computing many identical tasks on different data inputs. This means that an efficient GPU algorithm must make sure that all threads perform the same operations in lock step. This also means that a na¨ ıve GPU implementation of an adaptive step ODE solver will provide little to no speed up. To overcome this, we decided on the following. The step size of each particle is determined individually, disregarding the error estimates of other particles. This results in particles becoming asynchronous in dynamical time, but taking steps optimal to each particle. The diagrams shown in Fig. A.1 illustrate our current solution for keeping threads synchronized. We consider three different particles. If those three particles were integrated independently on a CPU, it may take one particle 2 ms to reach a specific ODE time, which we label , while another takes 5 ms seen in the left diagram in Fig. A.1. This particle may be approaching a singularity and thus requires smaller time steps. The middle diagram shows what may happen in a na¨ ıve GPU implementation. The particle with the smallest necessary time step determines the time 239 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER step for all particles. This results in all particles advancing very slowly and provides little or no advantage to a CPU implementation. Our solution is shown in the right diagram, where we allow the individual particles to take different time steps, appropriate to their individual requirements for accuracy. The gray line indicates the first step computed, and the red indicates the second. We call this method step synchronization to differentiate it from the time synchronization implied by the na¨ ıve method. To further increase our options for performance, we chose to implement variable levels of output from the GPU memory to CPU memory. Internally, the states of the particles are stored on GPU memory and overwritten each time a propagator step meets the integration tolerance. We will call these steps “valid steps”. However, we can choose to output all valid steps for all particles, only the states at final timet f , or the states at event function triggers. Thus at each valid step, a function is executed that “checks for an event” to determine if the current state should be copied to the CPU memory. This function is generic and the “event” can be any valid state, final state, or some user defined event function. As you can imagine, outputting all states will be the slowest and we have found that this impacts GPU performance much more than it does CPU performance. It is not always necessary to know the full state history of a particle, and so this variable output approach is a way for the user to trade output completeness for performance, depending on their needs. For the purpose of a Poincar´ e map, only the states when crossing a plane is required, and so performance can be significantly higher than with CPUs, as we will discuss more in Section A.4. In addition to de-synchronizing particles, another GPU specific workaround was implemen- ted to improve performance. A sifting algorithm performs a partial sort of the particles at regular intervals. The reasoning behind this is to group particles that behave similarly, and thus minimize divergence of threads. The credit for this partial sort idea goes to Ryan Burns, who was instrumen- tal in developing the fundamental GPU ODE algorithm. The implementation chosen performs 1 partial sort after everyk ODE solver steps, withk being a tunable parameter for which we have chosenk = 16. The partial sort compares the progress in ODE time of all particles relative to their 240 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER neighbors and swaps places in order to move slower particles toward smaller indices and faster particles toward higher indices. A neighbor refers to a particle with an index one greater or one less than itself. The event stopping that we refer to in this section is only an approximate event. One common method by which event stopping is computed is to check each valid step for a sign change in a chosen event function s(t; ~ X). The exact event occurs when s(t; ~ X) = 0, but finding sign changes and reducing the step-size iteratively will approach the event to within a user selected numerical tolerance. This approach inherently requires a “while-loop” structure, which promotes thread divergence on GPUs. Therefore, we use a different approach to this problem, which we discuss in detail in Section A.3. For the purpose of this section, the event output is simply every time a sign change is detected in the event function, without further numerical refinement. We can thus think of this as an approximate event stopping. 241 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER Figure A.1: Diagram of GPU thread synchronization. The left figure shows the CPU stepping method and expected performance. “ODE Time” on the x-axis represents the dynamical time of the ODE, while the labeled times indicate CPU time in milliseconds. The three particles in this case the particles behave very differently, requiring much shorter steps for one of them. For simplicity, we assume a propagator step takes 1 ms of CPU time. Since each thread is independent, they complete as fast as they can and do not need to wait for the others to complete. The middle figure shows a na¨ ıve GPU implementation and its performance. In this case, the ODE time is synchronized across particles, so that the threads all perform the same task at the same time. This means that all particles take steps to match the “slowest” particle. Finally, the right figure indicates the stepping method chosen for our task. We chose to propagate the particles asynchronously in ODE time, which means the threads are now still performing the same actions at the same time, but “fast” particles are not hampered by “slow’ ones. A.2.2 Performance In this section we present the performance of numerical propagation in the CR3BP as a test model for varying numbers of particles and dynamical propagation time. The test case is a set of random initial conditions in the Jupiter-Europa CR3BP exterior region satisfyingC = 3:0036, g and 1:0355 L 1:442. We did not have access to an identical algorithm in C, FORTRAN or C++, and thus used a MATLAB implementation in its place. Take this into consideration when ana- lyzing the speedup factors. Additionally, this comparison assumes an output of the final state only. We found that this distinction has very little effect on the performance of the CPU algorithm when propagating for shorter time spans. For such cases, generating states at all valid steps only takes a small amount of extra computation time. If the propagation time becomes large, the compute time 242 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER per propagation time increases significantly. We believe that the large amount of memory required to keep the complete state history for all particles is causing this reduction in performance for these cases. In contrast, the effect on GPU performance when changing the desired output is significant for most data sets. The reason is that the threads can compute updated states very quickly while the state remains in GPU memory, but the memory transfer rate from GPU to CPU is not always able to keep up with the rate of propagation. This is called a “memory limited” application, where the compute cores cannot proceed because they are waiting for a memory transfer operation. The hardware used for this test is summarized in Table A.1. Note that the RAM refers to system memory for the CPU and on-board video memory for the GPU. Additionally, the core count for the GPU represents “CUDA” cores, and thus are not completely independent compute units. We note that the CPU is a 6-core chip, but for comparison purposes we executed the code in serial on one core. Further testing shows that the performance increase by executing the code in parallel on the CPU scales very close to linearly by core count. This means that the speedup factor in the Fig A.2(e)-(f), would be reduced by a factor of 6 if we compare to this specific CPU while executing in parallel. The decision to use a serial execution for the CPU is to allow for easy comparison to other CPUs with varying numbers of cores. We found that the CPU compute time scales very linearly with the number of particles for the sets studied. Therefore, we use a single combination of particle count and propagation time and scale the compute time appropriately. For many of the data sets considered, the CPU compute time becomes prohibitively expensive, which is another reason for using scaled values. The hardware used for this test is summarized in Table A.1. Note that the RAM refers to system memory for the CPU and on-board video memory for the GPU. Additionally, the core count for the GPU represents “CUDA” cores, and thus are not completely independent compute units. We note that the CPU is a 6-core chip, but for comparison purposes we executed the code in serial on one core. Further testing shows that the performance increase by executing the code in parallel on the CPU scales very close to linearly by core count. This means that the speedup factor in the Fig A.2(e)-(f), would be reduced by a factor of 6 if we compare to this 243 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER System CPU Test Hardware GPU Test Hardware Make & Model Intel Core i7-4930K Nvidia Titan V Core Count 6 5120 Clock Speed [GHz] 4.400 1.455 RAM [GB] 32 12 Table A.1: CPU and GPU test hardware. specific CPU while executing in parallel. The decision to use a serial execution for the CPU is to allow for easy comparison to other CPUs with varying numbers of cores. We found that the CPU compute time scales very linearly with the number of particles for the sets studied. Therefore, we use a single combination of particle count and propagation time and scale the compute time appropriately. For many of the data sets considered, the CPU compute time becomes prohibitively expensive, which is another reason for using scaled values. We display the performance metrics comparison for the test case in Fig. A.2. First, we note that the compute time for the CPU is at least 2 orders of magnitude greater than the GPU compute time. Another interesting result is that the GPU has a clear overhead, and a nonlinear relationship between problem size and compute time. The overhead is indicated in Figs. A.2(a)-(b) by the lower bound on the compute time at just under 1 second, regardless of the problem size. Those same figures also indicate the nonlinear relationship, with an increasing slope on a log-log scale. We can see from Fig. A.2(c) that the compute time per particle seems to reach an asymptote approximately when N 10 4 . Figure A.2(d) similarly indicates that we require a propagation time of approximately 5 years to reach near the asymptote. This is of course subjective to each dynamical model, and 5 years in this case represents approximately 3231 non-dimensional time units in the CR3BP. Both of these results indicate that keeping the GPU occupied with workload is beneficial to performance. These results are similar to those found by Arora[9]. Increasing the propagation time also means that each thread spends more time on computation than it does on initialization and output, which helps increase performance. Finally, the relative speedup of GPU 244 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER vs. CPU is shown in Fig. A.2(e)-(f), where it’s clear that the relative performance approaches an asymptote just over 1 10 4 faster for the fixed time and 6 10 3 faster for the fixed particle count. 245 of 268 A.2. RUNGE-KUTTA-FEHLBERG 7/8TH ORDER ADAPTIVE ODE SOLVER (a) (b) (c) (d) (e) (f) Figure A.2: Performance metrics of GPU propagator. (a) Total compute time vs. number of particles for CPU (red) and GPU (black). (b) Total compute time vs. ODE propagation time for CPU (red) and GPU (black). (c) Compute time per particle vs. number of particles for GPU. (d) Compute time per ODE time vs. ODE propagation time for GPU. (e) GPU speedup factor vs. number of particles. (f) GPU speedup factor vs. ODE propagation time. The propagation time was fixed to 12 years for all the plots in the left column, while the number of particles were fixed to 5000 for the plots in the right column. 246 of 268 A.3. EVENT STOPPING AND H ´ ENON STEP A.3 Event Stopping and H´ enon Step In this section, we explain how integration event stopping is typically accomplished, and our appro- ach to make it viable for GPU implementation. A.3.1 Algorithm To compute the time when an event happens during numerical propagation in a dynamical system, one must first define the event in question. In a generaln-dimensional ODE satisfying d~ x dt = ~ f(t;~ x) (A.1) an event would be defined in the form s(~ x) = 0 (A.2) Therefore, events occur when some arbitrary function of the current state vector reaches 0. To locate these events during numerical propagation, one computes the value of s(~ x) each time a valid step is taken and searches for sign changes. After a sign change ins(~ x) is detected between two consecutive steps, some method must be used to converge on the true event. In the case of an iterative approach, a bisection method is one example for accomplishing this. An approach of this type is very non-ideal for GPU implementation. To perform this computation in a way that is efficient on GPU hardware, we chose to make use of an integration trick developed by H´ enon[42]. As outlined in the original paper, the trick involves changing the independent variable of integration from time, to the value of the event functions. The result is an augmented dynamical system. This new system has the time as an added state variable, ~ = 2 4 ~ x t 3 5 (A.3) 247 of 268 A.3. EVENT STOPPING AND H ´ ENON STEP and has the ODE form d ~ ds =~ g(s; ~ ) (A.4) It is now trivial to propagate to s = 0 by simply choosing an integration step size of h =s 0 , where s 0 is the value of the event function at the current state. One important change from the original ODE in Eq. A.1 is that the system is now (n + 1)-dimensional. We note that this method requires that the flow is locally transverse to the event, which is true if the event is already “nearby”. We can guarantee this by only performing this “H´ enon step” on states that have already been determined to trigger events using the approximate event outlined previously. For details of how to arrive at the form in Eqs. A.3–A.4, we refer the reader to the original paper[42]. However, we can mention that it requires the derivative of the event function with respect to the state. Performing this automatically for arbitrary event functions may not be easy to implement, but there is an alternative for one common form fors(~ x). Fors(~ x) of the form s(~ x) =x i c (A.5) wherec is a constant andx i is thei-th component of~ x, we no longer need to add a component to the state or compute derivatives ofs(~ x). The state is stilln-dimensional, with thei-th component of the state replaced witht. ~ = 2 6 6 6 6 6 6 6 6 6 6 4 x 1 . . . t . . . x n 3 7 7 7 7 7 7 7 7 7 7 5 (A.6) The ODE is now a derivative with respect to thei-th component of the state. d ~ dx i =~ g(x i ; ~ ) (A.7) 248 of 268 A.4. POINCAR ´ E MAP The form shown in Eqs. A.6–A.7 are the ones we have chosen to implement for the current iteration of the algorithm, since this simple event function is the type used to compute Poincar´ e maps. Our current approach to computing events follows a 2 step process. We first compute all the approximate events using the GPU propagator with the normal ODE. Once all the event candidates have beeen computed, the states near the events are transformed according to Eq.A.6. Then, the modified ODE in Eq. A.7 is used in the GPU propagator to move all approximate events directly to the events(~ x) = 0. You can imagine why this is an efficient GPU operation. The H´ enon step is an identical operation across all particles, with no possibility of thread divergence. We believe this is our most novel contribution to ODE solvers for GPU hardware. A.4 Poincar´ e Map Here, we show how the propagator is used to generate Poincar´ e maps, and indicate the performance speedup when compared to computing them on a CPU. A.4.1 Algorithm As described in Section 2.6, we compute Poincar´ e maps by propagating large numbers of particles until they’ve intersected the desired section several times. Thus, we can use the algorithm we outlined previously to compute these maps. As you have seen in Chapter 6, a common choice of Poincar´ e map is one with a section aty = 0 in the CR3BP. We have applied the GPU propagator to compute these maps, with a significant speedup. Propagating an ODE with an event function is fundamentally the same as producing a Poincar´ e map of that ODE. A Poincar´ e map is just a more specific application, as the choice of event functions is generally more limited. Another difference is the analysis of the output. One may not perform analysis of the Poincar´ e map in the normal state components, but possibly Delaunay variables. This is simply a post-processing step however, 249 of 268 A.4. POINCAR ´ E MAP and is not necessarily part of the Poincar´ e map algorithm itself. For the graphical output of the Poincar´ e map algorithm, we refer the reader to the numerous examples in Chapter 6. We must note that the algorithm in its current form, relies on the CPU for state and time transformation to satisfy the modified state required by the H´ enon step. This part of the algorithm could be performed on the GPU, but we have yet to implement it. The significance is that this conversion step takes a significant amount of time when performed on the CPU, and can on average double the total run-time required to compute a Poincar´ e map. Therefore, we hope to improve the algorithm with this addition in the future, and hope to see even larger performance gains when we do so. A.4.2 Performance We present a performance comparison of the GPU Poincar´ e map algorithm when compared to a CPU implementation. We did not have access to an identical algorithm in C, FORTRAN or C++, and thus used a MATLAB implementation in its place. Since the Poincar´ e map algorithm inhe- rently requires the use of the H´ enon step described previously, this section serves as a performance evaluation of that method as well. The initial conditions and hardware used for this comparison are the same as those used for Section A.2. As before, the GPU algorithm seems to have an overhead, as indicated by the lower bound for small particle counts in Fig. A.3(a). It also exhibits the same nonlinear behavior with an increasing slope of compute time until it approaches nearly linear behavior for large number of particles. Once again, it seems as if N 10 4 is required to reach linear behavior, as seen in Fig. A.3(c). Moreover, we also find a minimum propagation time to approach approximately linear behavior at around 10 years of propagation, as seen in Fig. A.3(d). The relative performance of the GPU algorithm with respect to the CPU algorithm are indicated in Figs. A.3(e)-(f). We can see that both curves are seemingly approaching an asymptote at just over 200 times faster, with larger particle count and longer propagation time being favored. 250 of 268 A.4. POINCAR ´ E MAP (a) (b) (c) (d) (e) (f) Figure A.3: Performance metrics of GPU Poincar´ e map. (a) Total compute time vs. number of particles for CPU (red) and GPU (black). (b) Total compute time vs. ODE propagation time for CPU (red) and GPU (black). (c) Compute time per particle vs. number of particles for GPU. (d) Compute time per ODE time vs. ODE propagation time for GPU. (e) GPU speedup factor vs. number of particles. (f) GPU speedup factor vs. ODE propagation time. The propagation time was fixed to 1/2 year for all the plots in the left column, while the number of particles were fixed to 5000 for the plots in the right column. 251 of 268 A.5. SUMMARY A.5 Summary In this appendix, we present an algorithm for solving ODEs in parallel on GPU hardware. We also present a novel method for propagating to user defined events as opposed to desired time. This event stopping algorithm makes use of the “H´ enon step”, which is very well suited for implemen- tation on GPU hardware. We tested the solver using a CR3BP model for normal propagation as well as computing Poincar´ e maps, and present the relative speedup of the GPU implementation. This analysis revealed that regular propagation is in the range of 110 2 –110 4 faster than a CPU implementation for normal propagation. On the other hand, the Poincar´ e map algorithm proved to be about 200 times faster when applied to problem sets reasonable for Poincar´ e maps. With the more widespread accessibility of GPUs, even on laptops, GPU algorithms can be more widely used in industry. Additionally, speedup on this order can allow more rapid design cycles and can enable interactive, concurrent trajectory design. Trajectory design has always been a subsystem of a complete mission that requires some lead time. The lead time is partially due to numerical propagation as an inherent road block, since it is often a fundamental operation required by opti- mizers and Monte Carlo analysis. We hope that GPU methods become more prevalent and further enhanced to reach this goal. 252 of 268 APPENDIX B: TECHNIQUES B.1 Differential Correction The process of differential correction uses the variational equations of the equations of motion in a dynamical system. Through knowledge of how variations propagate along with a given solution, the effect of perturbations to an initial state on a subsequent state can be approximated. Since this approximate relationship becomes linear, it can systematically be used to solve for necessary initial state perturbations to yield desired changes in the final state. By applying this technique, trajectories with specific desired qualities can be generated numerically, such as period orbits in systems which do not have closed form solutions. B.1.1 Single Shooting Due to this algorithm searching for symmetric solutions, 3 of the initial 6 state components are fixed asy 0 = _ x 0 = _ z 0 = 0. Although many types of symmetric orbits exist with respect to the xz-plane, the simplest ones are those that intersect thexz-plane for the first time after half an orbit period,T=2. Thus the Howell algorithm attempts to eliminate the velocities in thexz-plane at the 1st point of crossing (y=0). Howell derives the original version of the algorithm, which fixes one of the three available initial state components (for symmetric solutions). This results in a square system, where 2 constraints ( _ x(T=2) = 0; _ z(T=2) = 0), are used to solve for 2 variables ( _ y 0 , and eitherx 0 orz 0 ). If (t 1 ;t 0 ) is the state transition matrix that estimates the propagation of variations fromt 0 to t 1 then the estimated change in the final state can be computed by X = (T=2; 0)X 0 + dX dt T=2 (T=2) (B.1) 253 of 268 B.1. DIFFERENTIAL CORRECTION let ij represent the i;j component of (T=2; 0). Assuming the previous solution yielded y(T=2) = 0, we require thaty = 0. This results in y = 0 = 21 x 0 + 23 z 0 + 25 _ y 0 + _ y(T=2) =)(T=2) = 1 _ y ( 21 x 0 + 23 z 0 + 25 _ y 0 ) (B.2) From Eq. B.1 and replacing(T=2) using Eq. B.2, _ x = 41 x 0 + 43 z 0 + 45 _ y 0 x _ y ( 21 x 0 + 23 z 0 + 25 _ y 0 ) (B.3) _ z = 61 x 0 + 63 z 0 + 65 _ y 0 z _ y ( 21 x 0 + 23 z 0 + 25 _ y 0 ) (B.4) This results in 2 4 _ x _ z 3 5 = 0 @ 2 4 41 43 45 61 63 65 3 5 1 _ y 2 4 x z 3 5 h 21 23 25 i 1 A 2 6 6 6 4 x 0 z 0 _ y 0 3 7 7 7 5 (B.5) Here, the desired changes are _ x = _ x and _ z = _ z in order to yield a perpendicular xz- plane crossing. Eq. B.5 is under-constrained as it has 3 unknowns and 2 known constraints. This equation, although having infinite solution sets inx 0 ;z 0 and _ y 0 , could be used with a minimum norm solution to yield a unique correction step. Howell simplifies this result into two different methods by either fixing the initial x- or z- component, thus setting eitherx 0 = 0 orz 0 = 0 and eliminating one of the columns in the right hand side matrix of Eq. B.5. Either one of those simplifications make the system square and thus yields a unique solution. However, this specific constraint on the initial state can sometimes yield unpredictable results. In particular, it can prevent the corrector from being used in a continuation method since two different orbits of the same family may have the same initialx- orz-component. 254 of 268 B.1. DIFFERENTIAL CORRECTION The modified algorithm seeks to replace the necessary additional constraint on the initial state by a functional constraint. This would allow all three variables that control the initial state to be computed instead of just two. The example constraint discussed here will fix the Jacobi constant of the trajectory. Finding a periodic solution with a specific Jacobi constant (energy) is a common requirement, and as such this particular choice of constraint should be a useful one. Since the dynamics of the CR3BP conserves the Jacobi constant, the choice of the trajectory point at which we constrain the Jacobi constant is arbitrary. Since we are correcting the initial state variables, the obvious simple choice is to constrain the Jacobi constant as computed from the initial state. The variation in the Jacobi constant is C = @C @X X 0 X 0 (B.6) However, 3 of the initial state components have zero variation and thus C = @C @x X 0 @C @z X 0 @C @ _ y X 0 2 6 6 6 4 x 0 z 0 _ y 0 3 7 7 7 5 (B.7) We define the Jacobi constant as C = 2(1) r 1 + 2 r 2 +x 2 +y 2 ( _ x 2 + _ y 2 + _ z 2 ) (B.8) and the primary and secondary distances are r 1 = p (x +) 2 +y 2 +z 2 r 2 = p (x 1 +) 2 +y 2 +z 2 (B.9) 255 of 268 B.1. DIFFERENTIAL CORRECTION From Eq. B.8, the partial derivatives required are @C @x = 2(1)(x +) r 3 1 2(x 1 +) r 3 2 + 2x @C @z = 2(1)z r 3 1 2z r 3 2 @C @ _ y =2 _ y (B.10) Of course, these derivatives can be computed by numerical algorithms as well, such as the complex step derivative or finite difference. The complex step is a particularly effective choice due to its accuracy and simplicity. The derivative of component x would be computed in a two step algorithm. First evaluate Eq. B.8 atx +i with 1 (for example = 1 10 20 ) C c =C(x +i;y;z; _ x; _ y; _ z) (B.11) Then the approximate derivative will simply be the imaginary part of the result. @C @x ImfC c g (B.12) Repeat this process for the other desired partial derivatives. If the desired Jacobi constant is C and the current computed Jacobi constant is C, then the desired change in Jacobi constant is C =(CC ) (B.13) We label the constraint vectorb, the variable vectorq, then the square system has the form Mq =b (B.14) 256 of 268 B.1. DIFFERENTIAL CORRECTION with b = 2 6 6 6 4 _ x _ z C C 3 7 7 7 5 q = 2 6 6 6 4 x 0 z 0 _ y 0 3 7 7 7 5 M = 2 6 6 6 6 4 41 43 45 61 63 65 @C @x X 0 @C @z X 0 @C @ _ y X 0 3 7 7 7 7 5 1 _ y 2 6 6 6 4 x z 0 3 7 7 7 5 h 21 23 25 i (B.15) Eq. B.14 can be solved with any method for a system of this type, but it has a trivial solution by direct matrix inversion: q =M 1 b (B.16) Using Eq. B.16 iteratively to update the appropriate components of X 0 , a sufficiently close initial guess should converge to a periodic solution with the desired Jacobi constant, and will adjust both initialx- andz-components of the state in order to achieve that. B.1.2 Multiple Shooting Due to the sensitive nature of chaotic dynamics, the variational equations do not accurately reflect the propagation of state perturbations for long periods of time. For design of larger orbits and trajectories with longer timespans, it becomes necessary to patch together several separate single shooting problems. This reduces the inaccuracy of the linearized approximation to short segments of time, but increases the complexity of the algorithm. However, this algorithm can be parallelized for an advantageous speedup, as opposed to the single shooting method. The inherent constraint 257 of 268 B.2. COLLOCATION in this method is the continuity of the trajectory in position and velocity, but additional constraints can be added. It is often desirable to enforce periodic solutions, solutions with specific periods, or conditions on the trajectory boundaries when solving Two Point Boundary Value Problems. B.2 Collocation Collocation is similar to multiple shooting in that it approximates a trajectory by a set of intervals. However, it is different in that each interval is then assumed to follow a polynomial form. The piecewise polynomial represents the full trajectory, and the coefficients are solved for such that the equations of motion are satisfied along specific test points along the polynomial. This technique has the advantage of being more robust than the shooting techniques and making addition of constraints or modification of the underlying dynamics much simpler. Additionally, the resulting solution is an analytic trajectory instead of a discrete mesh of points. However, the technique has the limitation of often yielding solutions that are less accurate than those from shooting methods. This may require shooting methods to refine the solution from a collocation method. For more details regarding collocation and its various implementations, we refer the reader to a collection of resources[79, 41, 28, 15, 43, 44, 14, 39]. B.3 Continuation The theory of continuation is used to explore families of solutions with the same properties in dynamical systems. Once one solution is known, either a fixed point or periodic orbit, continuation methods can be applied to find nearby, similar solutions once one or more parameters are varied. One example would be to find a halo orbit in the CR3BP and then vary the Jacobi constant and attempt to find another halo orbit with that new Jacobi constant. Continuation is also useful to locate new, unknown families of solutions once one family “connects” with another. This behavior is called bifurcation. One example is the Planar Lyapunov Orbit family and Halo orbit family, at 258 of 268 B.4. EPHEMERIS MODEL one specific energy, there exists an orbit that is part of both families. Continuation has methods of detecting these bifurcations and switching branches to the new family. Through this method, several families with different characteristics can be found in a system. B.3.1 Single Parameter Continuation The simplest form of continuation uses a single defining parameter of a solution as the “continua- tion parameter”. The user selects a physical parameter to use and allows the algorithm to search for families of solutions. This method is simple, and works for well behaved systems or for short secti- ons in more complex systems. However, this method can fail if the value of the parameter across the family is not monotonic. A more sophisticated method called “pseudo-arclength” continuation computes the necessary search direction at all points along the solution curve by approximating the arclength. B.4 Ephemeris Model When the highest level of fidelity is required, such as for real mission trajectories, solutions are integrated in an ephemeris model. Here, the position of the celestial bodies in the solar system are generated at an extremely high accuracy based on verified models. To integrate a trajectory using this data, the position of the planets and other perturbing bodies are used to determine the exact gravitational force on any small body. There are several versions of the ephemeris from different sources, but the ones we choose to use are all from the Jet Propulsion Laboratory. Clearly, when high accuracy planetary positions are required for reasons other than integration, the ephemeris is also used. We use the SPICE software[2, 1] to interface with the ephemerides, which is publicly available. 259 of 268 B.5. EPHEMERIS TO CR3BP FRAME CONVERSION B.4.1 Jet Propulsion Laboratory DE431 This ephemeris model is the latest official release as of this writing, created in April 2013, see Folkner et al.[33]. DE is short for Developmental Ephemeris. This ephemeris is appropriate for propagation near the Earth-Moon system and through the Solar System, except when very near other planets. It has the positions of the barycenters of the other planetary systems in the Solar System, but not the positions of the moons of those planetary systems. Unless otherwise noted, this is the ephemeris used when we require high accuracy planetary positions. B.4.2 Jet Propulsion Laboratory JUP310 This ephemeris model is the latest official release for the position of Jupiter and its larger moons as of this writing, created in 2009, see Jacobson[48]. This ephemeris is appropriate for propagation near the Jupiter system. It has the positions of Jupiter and its larger moons. This ephemeris is often combined with a base model of the solar system such as DE431. The more accurate position of Jupiter from JUP310 is then prioritized over the data from the base ephemeris, and the positions of Jupiter’s moons become available. B.5 Ephemeris to CR3BP Frame Conversion B.5.1 Conversion Method 1: Variable Length and Time Units The conversion method assumes the trajectory to be converted is an ephemeris state in the Ecliptic J2000 coordinate system relative to the larger primary body. In this case, the length unit is the instantaneous distance between the Primaries and the time unit is such that the instantaneous angu- lar velocity of the Primaries motion is 1. The conversion method uses the following steps for each dimensional time,t D : 260 of 268 B.5. EPHEMERIS TO CR3BP FRAME CONVERSION 1.) Use the ephemeris to determine the position, ~ R, and velocity, ~ V of the smaller primary relative to the larger primary in the ecliptic J2000 coordinate frame. 2.) Set length unit. LU =j ~ Rj 3.) Compute the current angular velocity vector of the Primaries’ motion. ~ ! = ~ R ~ V LU 2 4.) Set time and velocity units. a)TU = 1 j~ !j b)VU = LU TU 5.) Select the 1st rotating frame axis as a unit vector along ~ R. ^ e 1 = ~ R j ~ Rj 6.) Select the 3rd rotating frame axis as a unit vector along~ !. ^ e 3 = ~ ! j~ !j 7.) Compute the final rotating axis to complete the right handed triad. ^ e 2 = ^ e 3 ^ e 1 8.) Assemble the rotation matrix with each unit vector on a row. Q = 2 6 6 6 4 ^ e 1 ^ e 2 ^ e 3 3 7 7 7 5 9.) Rotate the position vector into the rotating frame. ~ r D =Q~ r I D , where~ r I D is the position of the object in the inertial frame and dimensional units and~ r D is the position in the rotating frame and dimensional units. 261 of 268 B.5. EPHEMERIS TO CR3BP FRAME CONVERSION 10.) Rotate the velocity vector into the rotating frame and subtract the velocity of the frame ~ v D =Q~ v I D ~ ! ~ r I D , where~ v I D is the velocity of the object in the inertial frame and dimen- sional units and~ v D is the velocity in the rotating frame and dimensional units. 11.) Convert to nondimensional units. ~ r = ~ r D LU ~ v = ~ v D VU , where~ r and~ v are the position and velocity in the rotating frame and nondimensio- nal units. t = t D t D;0 TU , wheret D;0 is the initial time in dimensional units. This simply ensures that the nondimensional time starts at zero. 12.) Adjust position vector origin to the Primaries’ barycenter. ~ r =~ r + 2 6 6 6 4 0 0 3 7 7 7 5 B.5.2 Conversion Method 2: Fixed Length and Time Units The conversion method assumes the trajectory to be converted is an ephemeris state in the ecliptic J2000 coordinate system relative to the larger primary. In this case, the length unit is selected to be the distance between the Primaries at a selected reference time while the time unit is selected such that the angular velocity of the Primaries’ motion is unity at the selected reference time. One first selects the reference time t D;ref and determines the appropriate length and time units. The ephemeris is used to determine the position, ~ R = ~ R(t D;ref ), and velocity, ~ V = ~ V (t D;ref ), of the smaller primary relative to the larger primary in the ecliptic J2000 coordinate frame at this reference time. The units are then chosen as LU =j ~ R j ~ ! = ~ R ~ V LU 2 TU = 1 j~ ! j 262 of 268 B.5. EPHEMERIS TO CR3BP FRAME CONVERSION VU = LU TU The conversion method then uses the following steps for each dimensional time,t D : 1.) Use the ephemeris to determine the position, ~ R, and velocity, ~ V of the smaller primary relative to the larger primary in the ecliptic J2000 coordinate frame. 2.) Select the 1st rotating frame axis as a unit vector along ~ R. ^ e 1 = ~ R j ~ Rj 3.) Select the 3rd rotating frame axis as a unit vector along~ !. ^ e 3 = ~ ! j~ !j 4.) Compute the final rotating axis to complete the right handed triad. ^ e 2 = ^ e 3 ^ e 1 5.) Assemble the rotation matrix with each unit vector on a row. Q = 2 6 6 6 4 ^ e 1 ^ e 2 ^ e 3 3 7 7 7 5 6.) Rotate the position vector into the rotating frame. ~ r D =Q~ r I D , where~ r I D is the position of the object in the inertial frame and dimensional units and~ r D is the position in the rotating frame and dimensional units. 7.) Rotate the velocity vector into the rotating frame and subtract the velocity of the frame ~ v D =Q~ v I D ~ ! ~ r I D , where~ v I D is the velocity of the object in the inertial frame and dimen- sional units and~ v D is the velocity in the rotating frame and dimensional units. 8.) Convert to nondimensional units. ~ r = ~ r D LU ~ v = ~ v D VU , where~ r and~ v are the position and velocity in the rotating frame and nondimensio- nal units. 263 of 268 B.6. DELAUNAY V ARIABLE COMPUTATION t = t D t D;0 TU , wheret D;0 is the initial time in dimensional units. This simply ensures that the nondimensional time starts at zero. 9.) Adjust position vector origin to the Primaries’ barycenter. ~ r =~ r + 2 6 6 6 4 0 0 3 7 7 7 5 B.6 Delaunay Variable Computation This is an outline of the algorithm used to compute Delaunay variables from a set of planar, rotating frame coordinates in the Circular Restricted 3 Body Problem (CR3BP). Modified algorithm based on Koon et al.[55]. Given a state and time in planar, rotating CR3BP frame and coordinates, t; ~ X = [~ r;~ v] = [x;y; _ x; _ y], compute the Delaunay variables, [l; g;L;G]. 1. Set = 1 for normal rotating frame coordinates. 2. Compute polar coordinates (a) r =j~ rj (b) tan() = y x Keep in mind you need to get the quadrant appropriate output, so use the correct inverse tangent (e.g. “atan2” in MATLAB) (c) _ r = _ x x r + _ y y r (d) _ = _ x y r 2 + _ y x r 2 3. Compute Delaunay variable G (angular momentum) G =r 2 (1 + _ ) 264 of 268 B.6. DELAUNAY V ARIABLE COMPUTATION 4. Compute semimajor axis, a (relative to barycenter) (a) V 2 = ( _ r 2 + G 2 r 2 ) Leave the velocity squared in any algorithm, no need to take square root. (b) U =V 2 =2 mu r , Keplerian energy (c) a = mu 2U 5. Compute eccentricity, e (relative to barycenter) e = q 1 + 2UG 2 2 6. Compute Delaunay variable L L = p a 7. Compute true anomaly, (a) cos() =c = G 2 r 1 e Leave the cosine for now, don’t invert (b) Handle numerical errors to avoid imaginary output i. IFc > 1&jc 1j<, then setc = 1. Use = 10 10 ii. IFc <1&jc + 1j<, then setc =1. Use = 10 10 (c) =sign( _ r) cos 1 (c ). Use the principal output of inverse, cosine such that 0. 8. Compute Delaunay variable g (argument of perigee relative to rotating frame). (a) g = (b) take modulus of g such that g< 9. Compute eccentric anomaly, E E = 2 arctan( q 1e 1+e tan( 2 )). Principal output of inverse tangent is correct, as the equation is set up to avoid quadrant ambiguities 10. Compute Delaunay variablel (mean anomaly). l =Ee sin(E) 265 of 268 B.7. POINCAR ´ E MAP DELAUNAY V ARIABLES TO CARTESIAN CONVERSION B.7 Poincar´ e Map Delaunay Variables to Cartesian Conver- sion To interactively produce solutions from within a fixed jacobi constant Poincar´ e map in the variables (L; g), we need an algorithm to convert a given state in (L; g)-coordinates to a Cartesian state in the inertial frame. If we were given the complete set of Delaunay variables [l; g;L;G], the conversion would be trivial. Initially, it may seem that only having 2 of the 4 variables is an underconstrained problem since we are solving for 4 parameters. However, we have 2 additional parameters based on the chosen Poincar´ e section () and the Jacobi constant (C). We show that the inverse transformation can be written as an implicit relationship in only 1 variable, after which solving for the full state is straightforward. The known parameters are [L ; g ;C ; ] and the solution we seek is the state in the rotating frame, [x;y; _ x; _ y]. We will choose to seek the solution initially in polar coordinates [r;; _ r; _ ] as opposed to Cartesian coordinates, and will use the radius r as the unknown variable. A trivial observation is that is a known parameter, and thus one component of the polar state is determined immediately. = (B.17) Next, we show that we can explicitly express _ in terms of onlyr and the known parameters. _ = 1 r 2 1 2 C 1 (L ) 2 + 1 r 1 r 1 r 2 1 r 2 1 = (x) 2 +y 2 = (r cos ) 2 + (r sin ) 2 r 2 2 = (x 1 +) 2 +y 2 = (r cos 1 +) 2 + (r sin ) 2 (B.18) 266 of 268 B.7. POINCAR ´ E MAP DELAUNAY V ARIABLES TO CARTESIAN CONVERSION Eq. B.18 was derived from the constraints on L and C , with the latter constraint yielding the result that _ r is an explicit function ofr, _ , andL . Moreover, since _ is in turn only a function ofr and known parameters, _ r is also an explicit function ofr and known parameters. _ r 2 = 2(1) r 1 + 2 r 2 +r 2 r 2 _ 2 C (B.19) The constraint that remains to be satisfied is in the g -term. Since the solution to g requires an inverse cosine, we choose to apply the constraint instead in the following form. cos() = cos( g ) = r 3 (1 + _ ) 2 1 q 1 r 4 (1+ _ ) 2 (L ) 2 (B.20) By combining Eq. B.18 with Eq. B.20, we arrive at a implicit relationship defining the radiusr only as a function of the known parameters [L ; g ;C ; ]. By numerically solving this equation forr, the remaining 3 components of the polar state are easily computed from the constraint relationships in Eqs. B.17, B.19, and B.18. After that, conversion to Cartesian state is straightforward. To solve Eq. B.20, we choose to apply a Newton-Raphson method, and therefore need an initial guess. 267 of 268 B.7. POINCAR ´ E MAP DELAUNAY V ARIABLES TO CARTESIAN CONVERSION For the initial guess, we can use approximations that relate the Jacobi constant to the Delaunay variables. To do so, we use the following steps: G e G = 1 2 C 1 L 2 e = r 1 G 2 L 2 = g r = G 2 1 +e cos() V 2 = 2 r 1 L 2 V 2 = G 2 r 2 _ r = 8 > < > : q V 2 V 2 ; if 0 < q V 2 V 2 ; if < 2 _ = G r 2 1 x =r cos() y =r sin() _ x = _ rx r _ y _ y = _ ry r + _ x (B.21) Since this uses the approximation to computeG and all subsequent derived quantities fromC, the resulting state is not guaranteed to satisfy the Jacobi constant constraint. 268 of 268

Abstract (if available)

Abstract Due to recent interest in asteroid rendezvous, capture and deflect missions, temporary moons of Earth have also gained interest. Temporary moons are captured by the planet in highly chaotic orbits, which means small forces can cause deflection or permanent capture. It also means that sending a spacecraft to rendezvous with such objects can have very low fuel requirements. While the outer planets are known to have many temporary moons, few examples exist for Earth. In 2006, asteroid RH₁₂₀ became the only known temporary moon of Earth as it remained in orbit for nearly a year. Previous work has explained temporary capture dynamics of Jupiter comets using invariant manifold theory, and we extend that work to explain the temporary capture of the aforementioned asteroid at Earth. We showed that 2006 RH₁₂₀ closely followed the invariant manifolds of halo orbits in the Sun-Earth Circular Restricted 3-Body Problem during the capture and escape phases. In addition, we showed that it performed an orbital resonance transition as a result of its Earth encounter. While studying the temporary capture phase of the asteroid, we found that the perturbation from the Moon was the dominant factor controlling the Asteroid’s motion. For that study, we applied a modified version of a signal processing method called Dynamic Time Warping in a novel approach to quantitatively compute the similarity of trajectories. This method can be useful for trajectory classification and optimization. This prompted a follow up study to examine the effect of the Moon on low energy objects approaching Earth. A comparison of transit, capture and impact rates in the low energy regime of the Bicircular Problem and Circular Restricted 3-Body Problem show that the Moon reduces the average rate of Earth impact at the high end of the energy range studied. At lower energy, the Earth impact rate was similar or higher with the Moon present, depending on the exact energy. The Moon had the effect of decreasing the average rate of transit from interior to exterior regions at most values in the low end of the energy range. The resonant behavior of Asteroid 2006 RH₁₂₀ before and after temporary capture led to a study of orbital resonance. We developed and applied a new algorithm for computing compound resonant orbits, which are finite period orbits that both orbit Lagrange points and exhibit resonant behavior. This allows for prediction of which resonant orbits can be naturally reached at a given Jacobi constant. We correlated those results with the estimated Jacobi constant of the asteroid, and we found that the resonances observed were in the allowable range. These compound orbits are inherently unstable due to the close encounter with the secondary body, which makes them useful for certain aspects of trajectory design. However, long term stability can also be a requirement. Therefore, we extended the study of resonance to evaluate stability in the CR3BP and more realistic models. This involved developing a predictive model for stable and unstable fixed points in a Poincaré map, and defining a new method for computing finite time orbit stability from perturbed Poincaré maps. This method was used to evaluate the stability of resonant orbits in satellite systems with inherent resonance. In the case of the Galilean satellite system, we discovered that a rule-of-thumb regarding the resonance ratio integers allows for better long term stability. As a computational aid for the work within, an algorithm was presented for solving ODEs in parallel on GPU hardware. First, it was used for propagating large sets of trajectories, for which it was 1×10² − 1×10⁴ faster than a CPU implementation. Second, it was used to compute Poincaré maps, for which it was about 200 times faster.

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Creator Anderson, Brian Danny (author)

Core Title Techniques for analysis and design of temporary capture and resonant motion in astrodynamics

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Astronautical Engineering

Publication Date 05/06/2019

Defense Date 12/06/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 2006 RH₁₂₀,asteroid,astrodynamics,bicircular problem,dynamic time warping,dynamical systems,four-body problem,GPU,long-term orbital stability,minimoon,moon systems,OAI-PMH Harvest,parallel ODE,Poincare map,resonant orbits,six-body problem,temporary capture,three-body problem

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Language English

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Advisor Erwin, Daniel (committee chair), Campbell, Charles (committee member), Gruntman, Mike (committee member), Madni, Azad (committee member), Rhodes, Edward (committee member)

Creator Email brian.d.anderson@jpl.nasa.gov,brian.danny.anderson@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-167024

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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