University of Colorado, Boulder CU Scholar Aerospace Engineering Sciences Graduate Theses & Dissertations Aerospace Engineering Sciences Spring 1-1-2010 Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime Paul V. Anderson University of Colorado Boulder, anderson.paul.v@gmail.com Follow this and additional works at: http://scholar.colorado.edu/asen_gradetds Part of the Aerospace Engineering Commons This Thesis is brought to you for free and open access by Aerospace Engineering Sciences at CU Scholar. It has been accepted for inclusion in Aerospace Engineering Sciences Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact cuscholaradmin@colorado.edu. Recommended Citation Anderson, Paul V., "Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime" (2010). Aerospace Engineering Sciences Graduate Theses & Dissertations. Paper 14. Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime by Paul V. Anderson B.S., Embry-Riddle Aeronautical University, 2011 M.S., University of Colorado Boulder, 2014 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Aerospace Engineering Sciences 2015 This thesis entitled: Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime written by Paul V. Anderson has been approved for the Department of Aerospace Engineering Sciences Dr. Hanspeter Schaub Dr. Brandon Jones Dr. Jerey Parker Dr. Juan Restrepo Dr. Darren McKnight Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Anderson, Paul V. (Ph.D., Aerospace Engineering Sciences) Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime Thesis directed by Dr. Hanspeter Schaub The geosynchronous orbit (GEO) is a unique commodity of the satellite industry that is becoming increasingly contaminated with orbital debris, but is heavily populated with high-value assets from the civil, commercial, and defense sectors. The GEO arena is home to hundreds of communications, data transmission, and intelligence satellites collectively insured for an estimated 18.3 billion USD. As the lack of natural cleansing mechanisms at the GEO altitude renders the lifetimes of GEO debris essentially infinite, conjunction and risk assessment must be performed to safeguard operational assets from debris collisions. In this thesis, longitude-dependent debris congestion is characterized by predicting the number of near-miss events per day for every longitude slot at GEO, using custom de- bris propagation tools and a torus intersection metric. Near-miss events with the present- day debris population are assigned risk levels based on GEO-relative position and speed, and this risk information is used to prioritize the population for debris removal target se- lection. Long-term projections of debris growth under nominal launch trac, mitigation practices, and fragmentation events are also discussed, and latitudinal synchronization of theGEOdebris population is explained via node variations arising from luni-solar gravity. In addition to characterizing localized debris congestion in the GEO ring, this thesis further investigates the conjunction risk to operational satellites or debris removal systems applying low-thrust propulsion to raise orbit altitude at end-of-life to a super-synchronous disposal orbit. Conjunction risks as a function of thrust level, miss distance, longitude, and semi-major axis are evaluated, and a guidancemethod for evading conjunctingdebriswith continuous thrust bymeans of a thrust heading change via single-shooting is developed. Dedication To God, my Heavenly Father, To Jesus Christ, my Savior, To the Spirit, my Sanctifier, for empowering me for this work To Ashley, my wife, for showing me the Gospel To Vince and Lynn, my parents, for making all of this possible “W , ” E : () vAcknowledgements Foremost, I am honored and grateful to have had Dr. Hanspeter Schaub as my Ph.D. advisor. I have grown in an academic, technical, and professional sense under your contin- ual guidance, support, and encouragement. Thank you for your mentorship, and support when Imoved away to California after mywedding. Tomy committee, Dr. Brandon Jones, Dr. Je Parker, Dr. Juan Restrepo, and Dr. DarrenMcKnight, thank you for your academic and professional support of this work – your insights have helped to strengthen this thesis. Ashley Anderson, I thank you for your unconditional love and patience throughout this process. My life changed themoment I laid eyes on you, sitting in the back rowof orbit determination class. Now, as your husband, I look forward to our beautiful, godly future. I will always be grateful for the continual love and support of my parents, Vince and LynnAnderson. Your encouragement andmotivation have grantedme the opportunity to boldly pursue my dreams, one of which is achieving the Ph.D. I write with the knowledge that I would not be where I am so joyfully today without your selfless investments in me. I would also like to acknowledge the other students working in the AVS Lab over the past several years, especially Lee Jasper, Erik Hogan, and Steve O’Keefe. I am thankful for our enjoyable collaborations, our thought-provoking discussions about life, research, and coursework, and—of course—our too-frequentArnold Schwarzeneggermovie quotations. This dissertationwas supported through two teaching assistantships generously pro- vided by the Department of Aerospace Engineering Sciences, the Graduate Assistance in Areas of National Need (GAANN) fellowship, and the National Defense Science and En- gineeringGraduate (NDSEG) fellowship administered by theU.S. Department of Defense. Contents Chapter 1 Introduction 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Research Overview and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Research Goal 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Research Goal 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Research Goal 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Contributed Journal Publications . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Contributed Conference Publications . . . . . . . . . . . . . . . . . . 11 2 ResearchGoal 1(a): Forecasting longitude-dependent congestion atGEOarising from the current debris population 13 2.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Current RSO Population at GEO . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Methodology for Forecasting Longitude-Dependent Congestion at GEO . . 17 2.3.1 Formulation of Near-Miss Event Metric . . . . . . . . . . . . . . . . . 17 2.3.2 Propagator and Implementation . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Risk Functions for GEO-Relative Position and Velocity . . . . . . . . 23 vii 2.4 Results of Longitude-Dependent Congestion and Risk Forecast at GEO . . . 26 2.4.1 Localized Congestion, Velocity, and Risk Results . . . . . . . . . . . . 26 2.4.2 Time-Averaged Congestion, Velocity, and Risk Results . . . . . . . . 30 2.4.3 Timing of Near-Miss Events . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.4 Near-Miss Overlap Comparison . . . . . . . . . . . . . . . . . . . . . 36 2.5 Identifying Highest-Risk Debris Objects at GEO . . . . . . . . . . . . . . . . 38 2.5.1 Global Risk Accumulation Metric . . . . . . . . . . . . . . . . . . . . 39 2.5.2 Local Risk Accumulation Metric . . . . . . . . . . . . . . . . . . . . . 43 2.5.3 Absolute Worst-Case Risk Metric . . . . . . . . . . . . . . . . . . . . . 45 2.6 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 Research Goal 1(b): Investigating the synchronization of the GEO debris population 50 3.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Inclination and RAAN Variations Induced by Luni-Solar Perturbations . . . 52 3.3 Current Inclination and RAAN Distribution . . . . . . . . . . . . . . . . . . . 55 3.4 Evolution of Inclination and RAAN Distribution . . . . . . . . . . . . . . . . 60 3.5 Investigating the Unsynchronized Objects . . . . . . . . . . . . . . . . . . . . 62 3.6 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.7 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 ResearchGoal 2(a): Forecasting longitude-dependent congestion atGEOarising from nominal launch trac 67 4.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Operational GEO Orbit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Launch Instantiation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 “Business as Usual” Launch Trac without Mitigation . . . . . . . . . . . . 73 4.5 “Business as Usual” Launch Trac with Mitigation . . . . . . . . . . . . . . 75 viii 4.6 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5 ResearchGoal 2(b): Forecasting longitude-dependent congestion atGEOarising from on-orbit fragmentation events 79 5.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Fragmentation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3 Longitude-Dependent Eects of Fragmentations at GEO . . . . . . . . . . . 83 5.3.1 Harmonic Oscillator Model of Libration Motion . . . . . . . . . . . . 84 5.3.2 Rocket Body Explosions over Gravitational Wells . . . . . . . . . . . 85 5.3.3 Accuracy of Harmonic Oscillator Model . . . . . . . . . . . . . . . . . 90 5.3.4 Rocket Body Collisions over Gravitational Wells . . . . . . . . . . . . 93 5.4 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6 Research Goal 3(a): Evaluating the conjunction challenges of low-thrust GEO debris removal maneuvers 98 6.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 Analytic Results for Continuous Thrust Trajectories . . . . . . . . . . . . . . 102 6.3 Evaluating Conjunctions for GEO Re-orbit Maneuvers . . . . . . . . . . . . . 105 6.4 Results of GEO Re-orbit Conjunctions Study . . . . . . . . . . . . . . . . . . 108 6.4.1 Global Conjunction Challenge . . . . . . . . . . . . . . . . . . . . . . 108 6.4.2 Dependence on Initial Longitude and Inclination . . . . . . . . . . . 109 6.4.3 Conjunctions in SMA and Longitude Space . . . . . . . . . . . . . . . 111 6.4.4 Relative Velocity and Risk Considerations . . . . . . . . . . . . . . . . 113 6.4.5 Distribution of Conjunctions in Hill Frame . . . . . . . . . . . . . . . 116 6.4.6 Eect of Delaying Start Time of Maneuver . . . . . . . . . . . . . . . 117 6.5 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 121 ix 6.6 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7 Research Goal 3(b): Developing a guidance strategy for low-thrust debris avoidance during GEO re-orbit 124 7.1 Overview of Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.2 Development of Guidance Strategy for Debris Avoidance . . . . . . . . . . . 126 7.2.1 Nonlinear Relative Equations ofMotionwith In-Plane Chief Thrusting126 7.2.2 Eciency Factor for Re-orbit Maneuvers . . . . . . . . . . . . . . . . 128 7.2.3 Formulation of Bisection Guidance Strategy . . . . . . . . . . . . . . 129 7.3 Performance of Guidance Strategy for Debris Avoidance . . . . . . . . . . . 134 7.3.1 Development of Prototype Collision Scenarios . . . . . . . . . . . . . 134 7.3.2 Performance Bounds of Bisection Guidance Method . . . . . . . . . . 136 7.4 Conclusions from Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.5 Continuing Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8 Conclusions 142 Bibliography 146 Appendix A The Geosynchronous Orbit Environment 152 A.1 Fundamentals of GEO Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 A.2 Natural Perturbations in the GEO Environment . . . . . . . . . . . . . . . . . 154 A.2.1 Non-Spherical Earth Perturbations . . . . . . . . . . . . . . . . . . . . 154 A.2.2 Luni-Solar Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.2.3 Solar Radiation Pressure Perturbation . . . . . . . . . . . . . . . . . . 157 A.3 Why Does Libration Motion Occur? . . . . . . . . . . . . . . . . . . . . . . . 158 Tables Table 2.1 Orbit classifications for geosynchronous objects used in this thesis. . . . . . 15 2.2 Highest-risk derelicts atGEO, based onglobal risk accumulation (unweighted). 40 2.3 Highest-risk derelicts at GEO, based on global risk accumulation (weighted). 42 2.4 Highest-risk debris objects near Eastern gravitational well, based on local risk accumulation on [60,90E]. Shaded rows indicate objects that are both globally high-risk (that is, included within the unweighted top 10 ranking in Table 2.2) and locally high-risk. . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Highest-risk debris objects near Western gravitational well, based on local risk accumulation on [90,120W]. Shaded rows indicate objects that are both globally high-risk (that is, included within the unweighted top 10 ranking in Table 2.2) and locally high-risk. . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.6 Comparison between top five highest-risk objects, based on two ranking criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 GEO debris objects exhibiting unsynchronized latitude cycles on 01 March 2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1 Simulation results for three collision scenarios between two 2000 kg upper stages at 60E, where NC and C denote non-catastrophic and catastrophic collisions, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Figures Figure 1.1 Snapshot of debris population in GEO regime. . . . . . . . . . . . . . . . . . 3 2.1 Orbit classifications for 28 February 2014 TLE set. . . . . . . . . . . . . . . . 16 2.2 Libration periods for L1/L2 objects included in study. . . . . . . . . . . . . . 17 2.3 GEO torus concept for tracking near-miss events. . . . . . . . . . . . . . . . . 19 2.4 Examples of counting logic that eliminates redundancy in intersection checks. 19 2.5 Functions selected for computing combined risk factor of each near-miss event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Three portraits of longitude-dependent congestion in the GEO ring, gener- ated by the large-scale, trackable, and unclassified GEO debris population on 28 February 2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 Contributions of individual orbit classes to localized congestion at GEO. . . 29 2.8 Contributions of dominant GEO perturbations to combined risk forecast. . . 31 2.9 Average number of near-miss events per day at 50 km for each longitude slot, by orbit class and above or beneath various relative position, velocity, and risk thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.10 Standard deviation of near-miss events per day with risk factor thresholds. . 34 2.11 Equatorial crossing times for the synchronized debris population at GEO are episodic and dependent upon time of year.[1] Shaded swaths illustrate 1/2/3 corridors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 xii 2.12 Comparison of two overlapping congestion forecasting periods. . . . . . . . 36 2.13 Breakdownof number of near-misses and combined risk accumulated glob- ally around the GEO ring during the 5-year forecasting period, by respon- sible debris class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.14 Longitude weighting function based upon operational satellite densities. [2] . 42 2.15 Percent of accumulated risk total as a function of top percent of ranked debris population, which highlights accumulated risk contributions from high-ranking subsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.16 Removal of highest-risk debris objects (Tables 2.4-2.5) from 28 February 2014 TLE attenuates localized risk for operational assets positioned near the gravitational wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Longitude/latitude distribution of GEO debris population on 01 March 2014, shown in six-hour intervals to illustrate the “wave-like” behavior of the debris population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Phase space portraits illustrating the doubly-averaged Equations (3.1)-(3.2) for inclination and RAAN variations induced by luni-solar perturbations at the GEO altitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Phase space distribution of controlled GEO satellites on 28 February 2014. . 56 3.4 Phase space distribution of uncontrolled GEO derelicts on 01 March 2014. . 58 3.5 Evolution of derelict motion at GEO in inclination/RAAN space over 60 years, predicted using Equations (3.1)-(3.2) for Sun/Moon-driven inclina- tion/RAAN variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.6 RAAN histories for initially synchronized and initially unsynchronized ob- jects near GEO on 28 February 2014, predicted using the doubly-averaged Equations (3.1)-(3.2). Shaded swaths indicate ±1, ±2, and ±3 standard de- viations from the distribution mean. . . . . . . . . . . . . . . . . . . . . . . . 64 xiii 4.1 Parameter distributions for representative sampling of operational GEO or- bits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Annual launches to GEO since 1963. . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Localized congestion forecasting (50 years) using the linearly-increasing launch trac model, shown as the number of near-misses per day and increase in number of near-miss events over the idealized “no future launches” sce- nario, i.e., without augmentation. . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Eect of perfectmitigation—implemented at various levels of compliance— on long-term, localized congestion in the 30 vicinity of the gravitational wells (300 km torus). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Flow-chart for computer implementation of NASA Standard Breakup Model. 82 5.2 Characteristics of fragmentation distribution for simulated explosion at 60E. 86 5.3 Five-year longitude-dependent congestion forecasts for simulated rocket body explosions in the vicinity of the Eastern gravitational well (torus mi- nor radius of 100 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4 One-year longitude-dependent congestion forecast for simulated rocket body explosion at 60E (torus minor radius of 100 km). . . . . . . . . . . . . . . . . 88 5.5 Longitude-dependent congestion forecast (30 years) for simulated rocket body explosion at 60E (torus minor radius of 100 km), using full and Earth gravity force models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6 Five-year longitude-dependent congestion forecasts for simulated rocket body explosions in the vicinity of the Western gravitational well (torus mi- nor radius of 100 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.7 Five-year longitude-dependent congestion forecast for simulated rocket body explosion at 165E (torus minor radius of 100 km). . . . . . . . . . . . . . . . 92 xiv 5.8 Longitude histories for the 60 librating fragments captured by the gravita- tional well at 75E, computed with the harmonic oscillator model, simu- lated under J2,2 only, and simulated using full GEO forcing. . . . . . . . . . 93 5.9 Five-year longitude-dependent congestion forecasts for simulated rocket body collisions in the vicinity of the Eastern gravitational well (torus mi- nor radius of 100 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 Compliance to end-of-lifeGEOmitigation guidelines introduced by the IADC in 1997, illustrated as compliance rate by year and compliance rate by entity through 2013. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Challenges of low-thrust GEO debris re-orbit. . . . . . . . . . . . . . . . . . . 101 6.3 Duration of continuous-thrust re-orbiting maneuvers to three semi-major axes above GEO as a function of constant thrust acceleration and force, as- suming 1000 kg mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Accumulated longitudinal drift for continuous-thrust re-orbit maneu- vers to three semi-major axes above GEO, as a function of constant thrust acceleration and force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Mean number of conjunction events per trajectory as a function of thrust level and distance threshold shown with hyperbolic best-fit lines on linear and logarithmic axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.6 Number of conjunction events experienced during a continuous-thrust re- orbit to 300 km above the GEO ring, illustrated as a function of initial lon- gitude and inclination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.7 Density map illustrating regions in SMA/longitude space that experience high frequencies of conjunction events at 50 km, shown with three re-orbit trajectories at various thrust levels from an initial longitude of 105W, as predicted by Equations (6.5) and (6.9). . . . . . . . . . . . . . . . . . . . . . . 112 xv 6.8 Cumulative probability of conjunction event as function of semi-major axis and longitude during re-orbit burn, providedwith uniform cumulative dis- tribution function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.9 Number of conjunctions at 50 km shown with worst-case relative veloci- ties and combined risk factor for these trajectories, as a function of initial longitude and inclination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.10 Mean number of conjunctions per trajectory at 50 km as a function of thrust level and risk level, provided with hyperbolic best-fit lines on linear and logarithmic axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.11 Distribution of conjunction events at 50 km in the local Hill frame, colored by relative position and relative velocity magnitude (thrust acceleration of 5.0⇥ 106 m/s2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.12 Eect of delaying start time for re-orbit maneuver on the number of con- junction events experienced across all trajectories surveyed at a thrust level of 2.0⇥ 106 m/s2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.13 Eect of delaying start time for re-orbit maneuver on the worst-case con- junction speeds experienced across all trajectories surveyed at a thrust level of 2.0⇥ 106 m/s2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.14 Number of conjunctions as a function of maneuver delay (thrust accelera- tion of 2.0⇥106 m/s2) for two trajectories beginning at 75E longitude and 0/15 inclination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1 In-plane guidance concept for low-thrust debris avoidance. . . . . . . . . . . 130 7.2 Flowchart for computer implementation of bisection guidance strategy. . . . 131 7.3 Three examples of bisection guidance technique, which converges on = 0.706 to achieve a specified miss distance of 5 km at the predicted collision time of 24 hours out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 xvi 7.4 Convergence of and the eciency factor resulting from bisection method are dependent upon the lead time with which the object is detected prior to the collision time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5 Representative GEO collision geometries for testing bisection debris avoid- ance strategy, where collision speed increases from SMA case at left to in- clination case at right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6 Distribution of 50 km conjunction velocities, derived from re-orbit conjunc- tion data inChapter 6, by thrust level. Shaded region illustrates catastrophic collision regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.7 Relative Hill frame trajectories for representative GEO collision geometries. 137 7.8 Performance bounds of bisection method for derelict avoidance, illustrated for various miss distances with the prototype GEO collision geometries in Figures 7.5 and 7.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.1 Full disk views of Earth from GOES East and West satellites. . . . . . . . . . 153 A.2 Illustration of various Stokes coecients.[3] . . . . . . . . . . . . . . . . . . . 156 A.3 Top-down schematic showing ellipticity of Earth.[4] . . . . . . . . . . . . . . 159 A.4 Five-year libration profile for GEO satellite at 160W. . . . . . . . . . . . . . . 160 A.5 Daily semi-major axis and longitude drift as a function of longitude. . . . . 161 Chapter 1 Introduction 1.1 Background and Motivation The geosynchronous orbit (GEO) regime is a unique commodity of the Earth-orbiting satellite industry that is becoming increasingly contaminatedwith orbital debris, [5–7] but is heavily populatedwith high-value assets. TheGEOorbit ring is home to hundreds of com- munications, remote sensing, and intelligence satellites harnessed for applications ranging from video and data transmission to weather support and broadband applications.[8] As of 2011, the net worth of insured satellites at GEOwas valued at 18.3 billion USD.[8] As the lack of environmental cleansing mechanisms at the GEO altitude renders the lifetimes of GEOdebris objects essentially infinitely long,[9–11] conjunction andmitigation assessments must be performed to safeguard operational assets from collisions with the current debris population, which could have costly financial, legal, and geopolitical consequences. Since GEO satellites are required to maintain specific longitude slots authorized by the Interna- tional TelecommunicationUnion (ITU)[12]—and cannot simply phase shift to evadedebris— analyses of the macroscopic behavior of the GEO debris population are useful for describ- ing debris fluxes through particular longitude slots, to forecast how often operators with expensive satellites in these regions must potentially execute maneuvers to evade debris. As overcrowding of GEO is becoming a serious concern for satellite owners and operators internationally, knowledge of these debris flux patterns—termed debris weather [13]—is an imperative for space situational awareness (SSA) andmission assurance activities at GEO. 2Existing debris analysis and evolution tools[14–16] use inertially-fixed spatial cell defi- nitions to track debris cell passage events (CPE) triggered by osculating orbit intersections with cells of interest during long-term propagation of the orbital debris population. Har- nessing a variety of probability models, the spatial density and flux contributions for each CPE—for each known derelict object—over a specified time frame are computed and sub- sequently employed in collision risk assessments. For theGEO regime, these analysis tools often average over cell right ascension, providing debris fluxes as a lower-resolution func- tion of altitude and declination only.[11] Furthermore, if inertially-fixed cell definitions are used only, derelict flux contributions to particular GEO longitude slots at arbitrary times cannot be determined. Consequently, although spatially-averaged flux conditions at GEO may be estimated with existing software tools, local intersection events for arbitrary lon- gitude slots are not accessible, and the latter is of significant interest to satellite operators concernedwith the debris weather conditions in the vicinity of their operational satellites. Reference 1 emphasizes that fluxes averaged across longitude and time often grossly mis- represent the short-term collision hazard at GEO, and as a consequence, alternative GEO collision hazard depictions that employ higher temporal and spatial resolutions should be adopted by the operator community. Reference 8 illustrates that the gravitational wells at GEO render average debris flux a less relevant statistic for evaluating congestion atGEO, as the probability of collision at the center of thesewells is seven times larger than at longitudes far from these regions. Therefore, a framework for quantifying which localized regions of the GEO regime are, in general, most susceptible to rising debris fluxes over dierent time frames of interest is a necessary, albeit challenging, undertaking to rectify the too-common misconception that orbit debris congestion levels atGEOare homogeneous over longitude. Figure 1.1 depicts the unclassified and trackable (larger than approximately one me- ter in eective diameter for the GEO altitude[11]) resident space object (RSO) population at GEO for which up-to-date tracking data were available as of March 2014. Following a tax- onomy introduced in the European Space Agency’s Classification of Geosynchronous Objects 3Controlled Population Librating Population Drifting Population Indeterminate Population Figure 1.1: Snapshot of debris population in GEO regime. reports, this population is subdivided into four primary classes: (1) fully-controlled satel- lites with East-West (longitude control) and/or North-South (inclination control) station- keeping capability, (2) drifting objects that circulate along the GEO regime, as seen by an Earth-fixed observer, (3) librating objects that are “trapped” and thus oscillate in longitude about one or both of the gravitational potential wells located at 75E and 105W longitude, and (4) indeterminate objects for which the type of orbit is unclassifiable or unknown. The GEO regime is subjected to a steadily-increasing population of catalogued objects, which grows at an average rate of 30 new entries each year.[11] As of March 2014, 1145 catalogued objects satisfied the European Space Operations Centre’s (ESOC) criteria for GEO objects, and of this population, 35% is operational and under at least longitude control, but 51% is drifting above, below, or through the GEO ring, and 14% is librating about one or both of the gravitational wells. It is imperative to note that these large-scale debris objects—which consist of derelict satellites abandoned at end-of-life, Titan IIIC andProton-Kupper stages, etc.—constitute a subset of the actual debris field at GEO, which is dominated by signifi- cant populations of uncatalogued objects at sizes as small as 10-15 cm, detected in optical 4observation campaigns and indicative of undetected break-up events in this regime.[8,17,18] Localized, longitude-dependent congestion forecasting for the GEO regime is thus a fruitful and highly-applicable area of research, as it provides ametric as to how frequently satellite operators with assets in particular longitude slots will have to track nearby debris motion, perform high-precision conjunction assessments, and execute avoidance maneu- vers if absolutely necessary. The latter is of particular importance, as avoidancemaneuvers can temporarily force a satellite outside of its longitude slot, which may induce problems for the mission and be dicult to manage if neighboring assets are collocated in the same slot. Currently, the derelict population at GEO is sparse enough such that a simple time- shift of a scheduledmaintenancemaneuver is sucient for evading debris – in these situa- tions, no additional propellant is expended beyond that allocated for routine GEO station- keeping. However, as the debris population at GEO continues to increase, the amount of propellant required to maintain a specified longitude slot while simultaneously executing avoidance maneuvers—and the costs associated with analyzing conjunction events to de- termine if evasive action is even necessary—will begin increasing in tandem. Reference 8 summarizes the core implications of orbital debris, especially applicable to theGEO arena: Space debris is no longer an academic issue. Nor is it merely an “environmental” problem; rather, debris has the potential to damage or destroy high-value, opera- tional satellites with resulting revenue losses running into billions of U.S. dollars. Therefore, characterization of longitude-dependent debrisweather trends is an imperative area of research that is not only directly applicable to the interests of GEO satellite owners and operators on an international scale, but is critical to preserving the long-term “health” and sustainability of this delicate regime, which has historically been too-often overlooked in environmental debris impact studies. To this end, this thesis shall paint an illuminating portrait of localized debris congestion at GEO, to drive further research in this critical area, and—ultimately—spearhead proactive, global cooperation in support of this orbit regime. 51.2 Previous Studies As a consequence of the increasing population of derelict objects in the GEO regime, there was a growing concern as early as the 1980s regarding the overcrowding of the GEO arena and the long-term impacts on the sustainability of this regime[5,19] (for reference, the first spacecraft to be inserted into a GEO orbit was the USA’s Syncom-2 satellite in 1963[5]). Although a variety of internationalmitigation standards have been implemented to curtail debris growth in this delicate arena, includingNASA’s Technical Standard 8719.14 (Process for Limiting Orbital Debris) [20] and the re-orbiting guidelines proposed by the Inter-Agency SpaceDebris Coordination Committee (IADC),[21,22] Reference 23 indicates that of the 80% ofGEOoperators in 2001-2010 that attempted to complywithmitigation standards,merely 50%were successful in achieving compliance. Johnson highlights the implications of non- enforceablemitigation guidelines – “the viability of voluntary protectionmeasures for this regime depends upon the responsible actions of the aerospace community as a whole.”[5] Thereby, in the absence of perfect adherence to adopted mitigation measures by operators internationally, the debris population in the GEO regimewill continue steadily increasing. Naturally, a variety of studies concernedwith projecting the long-term growth of the GEO debris population began to arise in the literature. Flury assesses collision probability and long-term orbit evolution of the 1988GEOdebris population, and indicates the proba- bility of collision over the gravitational wells is a factor of two larger than in less-congested regions.[24] Based on the 1999 debris catalogue, Yasaka constructs an evolution model that incorporates launch trac, fragmentation events, and representative collision risks into a 200-year growth projection forGEO.[9] Wegener implements the European SpaceAgency’s MASTER-2001 debris environment tool to quantify the eects of mission debris, explosion fragments, solid rocket motor (SRM) slag, and end-of-life re-orbiting on forecasted spatial density in theGEO regime.[6] References 25–29 exhaustively evaluate long-term evolution, stability, and performance factors for super-synchronous disposal orbits for GEO. Again, 6these studies average over the GEO longitude slots, providing spatial density as a function of altitude or declination only, such that localdensity conditions are not available. Recently, McKnight andDi Pentino re-examine the 2010 debris population and demonstrate that the probability of collision over the gravitational wells is now a factor of seven larger than over less-congested regions, and emphasize that equatorial crossings of the derelict population are synchronized spatially and in time, such that local congestion increases are episodic.[1] Although the literature strongly focuses on the eects of debris-generating fragmen- tation events in low-Earth orbit (LEO), several previous studies have beendevoted to quan- tifying increases in collision risk driven by fragmentations in theGEO ring. Yasaka utilizes momentum and energy principles tomodel hyper-velocity impacts, and shows that the re- sulting fragment cloud remains close to theGEOaltitude, and spreads longitudinally to all regions of theGEO ring inweeks.[30] Oltrogge and Finkleman consider a probable conjunc- tion scenario at GEO to show that the fragment energy distribution resulting frombreakup is sucient for permeating all orbit regimes from re-entry through super-synchronous.[31] Hansen and Sorge indicate that although close approach velocities for the GEO regime are in general lower than in the LEO arena, a low-energy breakup over one of the gravitational wells could be aworst-case scenario in that resonance is preserved formost of the fragment population, serving to increase collision riskwith operational assets at these longitudes.[32] Though the tools that the authors of these previous studies use for characterizing breakup events at GEOdier, the independent conclusions of each study agree that fragmentations at the GEO altitude have long-term consequences that are detrimental to the sustainability and continued usefulness of this unique orbital regime and driver for space development. 1.3 Research Overview and Scope This thesis answers aspects of the following fundamental questions of interest to the GEOoperator community, each expounding a dimension of long-termGEO sustainability: 7(1) How severe is the longitude-dependent GEO congestion arising from the present- day, large-scale, unclassified, and trackable debris population? If 5-10 derelict ob- jects at GEO could be removed immediately, which derelicts would be targeted? (2) What are the eects of critical debris growthmechanisms, namely nominal launch trac and fragmentation events, on longitude-dependent congestion atGEO?How does adherence to end-of-life mitigation standards aect derelict growth at GEO? (3) What can be done now to limit derelict growth at GEO? If a low-thrust, tug removal system is envisioned, what trajectory guidance and control techniques are needed both for re-orbiting of debris and avoiding collisions, if demanded autonomously? This thesis is thus organized into three core research goals reflective of this set of questions: (1) Research Goal 1 (Chapters 2-3): Forecasting longitude-dependent congestion at GEO arising from the current debris population, and investigating the mechanisms be- hind thewave-like latitudinal synchronization exhibited by the debris population. (2) Research Goal 2 (Chapters 4-5): Forecasting longitude-dependent congestion at GEO arising fromnominal launch trac, mitigation, and on-orbit fragmentation events. (3) Research Goal 3 (Chapters 6-7): Evaluating the conjunction challenges for low-thrust GEOdebris removalmaneuvers, anddeveloping a guidancemethod for low-thrust, online debris avoidance during orbit raising to super-synchronous disposal orbits. The scope—and specific tasks—relevant to each of these research goals are detailed below. 1.3.1 Research Goal 1 In Chapter 2, this thesis forecasts longitude-dependent debris congestion trends aris- ing from the present-day, large-scale debris population in the GEO regime, using publicly- available tracking data to provide representative initial conditions for the population. This 8is illustrative of the localized congestion thatwould result in the short-term if no additional debris were generated in this regime, and can be used to identify the subset of this derelict population that is consistently themost threatening to operational assets in the GEO arena for purposes of active debris removal (ADR) target selection. Furthermore, the latitudinal synchronization exhibited by the GEO debris population is investigated in Chapter 3, and explained by inclination andnode variations driven by luni-solar perturbations at theGEO altitude. From this bottom-up, analytic theory, insights into top-down, operational obser- vations of “macroscopic”GEOdebris behavior are developed, which have implications for flight safety, anomaly resolution, sensor tasking, and space situational awareness at GEO. Specific research tasks for accomplishing this thesis objective are provided as follows: (1) Develop custom software to read TLE files, propagate GEO debris population for- ward in time, and determine near-miss events for each longitude slot in GEO ring. (2) Characterize “congestion culprits” at GEO: the derelicts that dominate debris con- gestion both globally around GEO and in a localized, longitude-dependent sense. (3) Develop risk functions to determine the most threatening subset of the large-scale GEOdebris population in terms of proximity to theGEO ring and relative velocity. (4) Merge theory of GEO orbit evolution with observational data to explain apparent anomalous behavior of derelicts not conforming to macroscopic synchronization. 1.3.2 Research Goal 2 The eects of two dominant debris growthmechanisms—nominal launch trac and on-orbit fragmentation events—on longitude-dependent debris congestion at GEO are in- dividually assessed in Chapters 4-5, to depict (a) the consequences of improperly-executed mitigation eortswith operational satellites reaching end-of-life, and (b) the consequences of on-orbit explosions and collisions in theGEO ring from a longitude-dependent perspec- 9tive. Each of these research thrusts motivates a growing need to synthesize, demonstrate, and implement an environment remediation system tailored specifically for the GEO ring. Specific research tasks for accomplishing this thesis objective are provided as follows: (1) Use historical and projected GEO launch data to construct probabilitymodels that describe where GEO satellites are likely to reside in longitude and element space. (2) Develop representative launch tracmodels and implement population augmen- tation in software to characterize long-term debris congestion over 50 year period. (3) Evaluate longitude-dependent eects of fragmentation events (i.e., explosions and collisions) on localized congestion—and longitude slot safety—in theGEO regime. 1.3.3 Research Goal 3 InChapters 6-7, this thesis addresses twodimensions of the debris removal challenge by surveying (a) the number of conjunction events with the current debris population that could be experienced during a typical re-orbit to a super-synchronous disposal orbit, and (b) an autonomous guidance technique required by a low-thrust, tug-basedADR system to properly dispose of GEO objects while preventing collisionswith the surrounding derelict population. This is a challenging eort, because contactless ADR concepts in the literature (highlighted further in Chapter 6) are constrained by mN thrust levels, preventing a high- thrust avoidance maneuver if one were necessary to mitigate a collision in the short-term. Therefore, with such low-thrust levels, it is likely necessary to detect upcoming collisions multiple revolutions in advance, to provide the guidance and control system enough lead time to safely adjust the thrust profile of the removal tug. Furthermore, towarrant the cost- eectiveness of such an ADR system, it should be designed with simple-yet-robust auton- omy in mind, to minimize human-in-the-loop (e.g., ground intervention) where possible. Specific research tasks for accomplishing this thesis objective are provided as follows: 10 (1) Harness TLEdata and customdebris propagation tools to quantify how frequently near-miss events occur for a tug-based ADR system during a typical, spiraling re- orbit trajectory from the GEO altitude to a compliant disposal orbit 300 km above. (2) Develop a simple-yet-robust guidance strategywith the autonomyof the system in mind, to appropriately alter the magnitude and/or direction of the thrust profile to safelymitigate an upcoming conjunction eventmultiple revolutions in advance. (3) Evaluate the tradeo between tug thrust level anddetection time for the developed guidance strategy, to assess performancewith perfect tug/debris state knowledge. 1.4 Research Contributions The content of this thesis has been primarily drawn from the two lists of publications below, which were independently produced in support of the above thesis research goals. 1.4.1 Contributed Journal Publications Relevant articles submitted, peer-reviewed, and published in scholarly journals are: (1) Anderson, P. V. and Schaub, H., “Methodology for Characterizing High-Risk Or- bital Debris in the Geosynchronous Orbit Regime,” In preparation for publication in Advances in Space Research (Elsevier). (2) Anderson, P. V. and Schaub, H., “Conjunction Challenges of Low-Thrust Geosyn- chronous Debris Removal Maneuvers,” In preparation for publication in Acta Astro- nautica (Elsevier). (3) Anderson, P. V. and Schaub, H., “Longitude-Dependent Eects of Fragmentation Events in the Geosynchronous Orbit Regime,” Acta Astronautica, Vol. 105, No. 1, 2014, pp. 285-297. doi: http://dx.doi.org/10.1016/j.actaastro.2014.09.011 11 (4) Anderson, P. V. and Schaub, H., “Local Debris Congestion in the Geosynchronous Environment with Population Augmentation,” Acta Astronautica, Vol. 94, No. 2, 2014, pp. 619-628. doi: http://dx.doi.org/10.1016/j.actaastro.2013.08.023 (5) Anderson, P. V. and Schaub, H., “Local Orbital Debris Flux Study in the Geosta- tionary Ring,” Advances in Space Research, Vol. 51, No. 12, 2013, pp. 2195-2206. doi: http://dx.doi.org/10.1016/j.asr.2013.01.019 1.4.2 Contributed Conference Publications Relevant articles submitted, accepted, and presented at technical conferences are: (1) Anderson, P. V., McKnight, D., Di Pentino, F., and Schaub, H., “Operational Con- siderations of GEODebris Synchronization Dynamics,” Submitted to the 66th Inter- national Astronautical Congress, Jerusalem, Israel, October 2015. (2) Anderson, P. V. and Schaub, H., “Methodology for Characterizing High-Risk Or- bital Debris in the Geosynchronous Orbit Regime,” 25th AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, VA, January 2015. (3) Anderson, P. V. and Schaub, H., “Conjunction Challenges of Low-Thrust Geosyn- chronous Debris Removal Maneuvers,” 25th AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, VA, January 2015. (4) Schaub, H., Jasper, L., Anderson, P. V., and McKnight D., “Cost and Risk Assess- ment for Spacecraft Operation Decisions Caused by the Space Debris Environ- ment,” 65th International Astronautical Congress, Toronto, Canada, October 2014. (5) Jasper, L., Anderson, P. V., and Schaub, H., “Economic and Risk Challenges of Operating in the Current Space Debris Environment,” Third Workshop on Space Debris Modeling and Remediation (CNES-HQ), Paris, France, June 2014. 12 (6) Anderson, P. V. and Schaub, H., “Characterizing Localized Debris Congestion in the Geosynchronous Orbit Regime,” 24th AAS/AIAA Space Flight Mechanics Meeting, Santa Fe, NM, January 2014. (7) Anderson, P. V. and Schaub, H., “Longitude-Dependent Eects of Fragmentation Events in the Geosynchronous Orbit Regime,” 24th AAS/AIAA Space Flight Me- chanics Meeting, Santa Fe, NM, January 2014. (8) Anderson, P. V. and Schaub, H., “Local Debris Congestion in the Geosynchronous Environmentwith PopulationAugmentation,” Sixth EuropeanConference on Space Debris, Darmstadt, Germany, April 2013. (9) Anderson, P. V. and Schaub, H., “Local Orbital Debris Flux Study in the Geosta- tionary Ring,” AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN, August 2012. Chapter 2 Research Goal 1(a): Forecasting longitude-dependent congestion at GEO arising from the current debris population 2.1 Overview of Research Goal Using publicly-available tracking data provided by the U.S. Strategic Command, and a classification system published by the European Space Agency,[33] this chapter illustrates a five-year, localized congestion forecast at GEO for large-scale, trackable, and unclassified resident space objects (RSO)with up-to-date two-line element (TLE) sets as ofMarch 2014. Small-scale sources such as explosion and collision events, although treated in Chapter 5, are not considered in this chapter, nor are other debris growth mechanisms, such as ejec- tion of solid rocket motor (SRM) slag or shedding of multi-layered insulation (MLI). [6] The results of this congestion forecast are leveraged to characterize which classes of debris ob- jects contribute the most to longitude-dependent risk levels, globally and over a specified subset of longitude slots. In particular, methodology for identifying the derelicts that con- tribute the largest percentages to the total risk level accumulated either globally or locally during the forecasting period is presented. For ADR initiatives geared towards slot clean- up at particular longitudes, information such as that determined from debris simulations using a localized congestionmetric to evaluate longitude-dependent risk is especially use- ful in helping to identify which individual objects should be removed to maximize reduc- tion in risk—either globally or across a subset of longitude slots—namely, in the vicinity of the two gravitational wells at 75E and 105W, which are dense with operational assets. [2] 14 The results of this work should therefore be harnessed in tandemwith long-term de- bris prediction studies, such as that performed in Chapter 4, to provide recommendations for architecture and design of potential ADR demonstration missions in the GEO regime. Studies led by theNASAOrbital Debris ProgramOce use the product of debrismass and probability of collision at a desired epoch as selection criterion for identifying the topADR targets in the LEO regime.[34] ADR target selection studies for the GEO ring have received far less attention than equivalent ADR target studies for LEO.[34–36] This chapter seeks to begin filling this void in the literature by formulating methodology for identifying the top GEOADR targets, based on a torus intersectionmetric and independent of the probability of collisionmeasure often used in conjunction assessment. Note that this torus intersection metric does not claim that these high-risk derelicts might actually collide with specific op- erational satellites; rather, themethodology presented in this chapter identifies the specific derelicts that contribute the most significant macroscopic risk to the GEO orbit in general. 2.2 Current RSO Population at GEO TheRSOpopulation in theGEO ring is classifiedunder a taxonomyused by the Euro- pean Space Agency’s DISCOS database (Database and Information System Characterising Objects in Space). [33] ForGEOobjects, seven orbit categories are defined to classify the type of orbits for these objects – two controlled classes, one drifting class, three libration classes, and one indeterminate class. Table 2.1 provides a description of this classification system. Note that only uncontrolled debris objects are assumed to contribute to localized conges- tion in this thesis, as operational GEO assets have the capability to execute dedicated colli- sion avoidance maneuvers if deemed necessary. GEO objects are selected according to the three criteria given in the European Space Agency’s Classification of Geosynchronous Objects reports: [33] • Eccentricity smaller than 0.2 (e < 0.2) 15 • Inclination smaller than 70 (i < 70) • Mean motion between 0.9 and 1.1 revolutions per sidereal day (0.9 < n < 1.1)1 Table 2.1: Orbit classifications for geosynchronous objects used in this thesis. Class Type Description C1 Controlled Longitude/inclination control (E-W/N-S control) C2 Controlled Longitude control only (E-W control only) D Drifting Circulating above/below/through GEO ring L1 Librating Libration about Eastern stable point at 75E L2 Librating Libration about Western stable point at 105W L3 Librating Libration about Eastern and Western stable points IN Indeterminate Unknown status (e.g., recent TLE set not available) Orbital data is obtained from the publicly-available two-line element (TLE) sets pro- vided by theU.S. StrategicCommand (USSTRATCOM).2 For this study, a reference TLE set obtained on 28 February 2014 is employed; the class distribution for the 1145 objects meet- ing the above GEO criterion in this TLE set is shown in Figure 2.1. TLE data are provided in the form of “doubly-averaged” Keplerian elements with mean motion instead of semi- major axis, [11] transformed into Cartesian states in the true equator, mean equinox (TEME) frame using SGP-4[37,38] theory.3 Because of the limited accuracy of TLE sets,[40] these data are not intended for analyses that require highly-accurate orbit prediction capabilities. As the purpose of this research objective is to forecast longitude-dependent congestion upon a macroscopic scale—not on a specialized, satellite-by-satellite basis—the accuracy of these data is sucient for the scope of this thesis.4 For high-accuracy applications, multiple TLE setsmay be treated as pseudo-observations in an initial orbit determination routine using a filtering algorithm to recover osculating orbits suitable for higher-fidelity propagation.[11] 1 This mean motion range corresponds to the semi-major axis range of [-2596,3068] km relative to GEO, such that longitudinal drift rates for objects included in the study are at a maximum of ±36 per solar day. 2 Publicly-available TLE data sets are available for download from https://www.space-track.org/ 3 ANSI-C implementation of SGP-4 is available at http://www.sat.dundee.ac.uk/~psc/sgp4.html [39] 4 Numerical justification for using TLE sets to quantify local GEO congestion is provided in Reference 2. 16 Indeterminate/ Unknown/[IN] 1"(0.1%) Longitude/Inclination Control/[C1] 318"(27.7%) Longitude Control/[C2] 82"(7.2%) Drifting/[D] 583"(50.9%) Eastern/ Librating/[L1] 105"(9.2%) Western/ Librating/[L2] 40"(3.5%) East/West Librating/[L3] 16"(1.4%) Figure 2.1: Orbit classifications for 28 February 2014 TLE set. As only objects larger than approximately 0.8-1.0 m in eective diameter are actively tracked at theGEOaltitude,[33] only objects at least of this size are considered in this study.5 Since this research incorporates the trackable, catalogued, and unclassified GEO popula- tionwith recent TLE data, the findings of this study serve to illustrate a conservative lower bound of the actual debris congestion and risk situation at GEO. This congestion and risk information is then harnessed to identifywhich large-scale derelicts currently at GEO con- tribute themost to risk levels across this regime, both globally and in the vicinity of the two gravitationalwells. Since risk to operational geostationary satellites is the primary concern of this thesis, debris-debris collision risk among the 745 uncontrolled objects is not treated. The Eastern librating (L1) andWestern librating (L2) RSOs oscillate within the Earth- fixed frame around their respective gravitational wells with a libration period of 816 days atminimum for small amplitudes.[11] Using tabulated data in Reference 33 in tandemwith analytic libration theory from Reference 4, a histogram of the libration periods for the 145 5 USSTRATCOM collects tracking data for GEOwith the GEODSS (Ground-Based Electro-Optical Deep- Space Surveillance) and MOTIF (Maui Optical Tracking and Identification Facility) installations, part of the Space Surveillance Network (SSN). [11] 17 L1/L2 derelicts included in this study is shown in Figure 2.2. The median of this libration period distribution is approximately 2.6 years, such that a 5-year time frame is used for this study to capture two mean cycles of the libration motion. In this manner, bias introduced in both global and localized risk results towards particular classes of debris is minimized. See Appendix A for an overview of libration motion at GEO and why it physically occurs. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Libration Period [0.1 year bins] 0 5 10 15 20 25 N um be ro fO bj ec ts Class L1 Class L2 Figure 2.2: Libration periods for L1/L2 objects included in study. 2.3 Methodology for Forecasting Longitude-Dependent Congestion at GEO 2.3.1 Formulation of Near-Miss Event Metric Longitude-dependent debris congestion at GEO is investigated by tracking near-miss events for each of the GEO longitude slots over a specified forecasting interval, determined by constructing a toruswithmajor radius rGEO = 42164 km andminor radius r˜, partitioned into longitudinal cells of width = 1.0 (see Figure 2.3). Theminor radius r˜ is the radius of the circular torus cross-section, and provides a means for evaluating debris congestion levels occurring within desired distances of the GEO longitude slots, i.e., a larger minor radius r˜ captures more near-miss events. Furthermore, this torus formulation is a natural choice for counting near-misses for these non-inertial cells, as torus geometry is invariant as seen by both the Earth-centered, inertial frame and Earth-centered, Earth-fixed frame, 18 in which these longitude slots are stationary.[13] Near-miss events are detected during propagation of a derelict object by checking for the transversal of this GEO torus boundary at each time step during numerical integration, and between these primary time steps using an interpolation method if higher resolution is desired. Mathematically, a near-miss event occurs if[13]✓ rGEO q r2X + r 2 Y ◆2 + r2Z r˜2 < 0 (2.1) is satisfied,where (rX , rY , rZ)T denotes theRSOposition vector expressed in Earth-centered inertial (ECI) frame components. The longitude of intersection CPE is thus determined as CPE = arctan ✓ rY rX ◆ ↵G (2.2) where↵G is the right ascension ofGreenwich (a.k.a. Greenwich sidereal time), determined at the current simulation epoch via the well-known sidereal time approximation[38,41] J0 = 367ty floor ✓ 7 4  ty + floor ✓ tm + 9 12 ◆◆ + floor ✓ 275tm 9 ◆ + td + 1721013.5 (2.3a) T0 = J0 2451545 36525 (2.3b) ↵G,0 = 100.4606184 + 36000.77004 T0 + 0.000387933 T 2 0 2.583 108 T 30 [deg] (2.3c) ↵G = ↵G,0 + 360.98564724 ✓ th 24 ◆ [deg] (2.3d) where current propagation time is expressedwith (ty, tm, td, th), corresponding to the year, month, day, and hour (e.g., terrestrial time) of the current simulation epoch. When a torus- intersection is detected via Equation (2.1), the longitude of intersection CPE is determined with Equation (2.2), and the total near-miss count for the corresponding cell is advanced. Again, note that because of the convenient invariance of this torus geometry, inertial frame coordinates obtained during numerical integration need not be converted into the rotating Earth-centered, Earth-fixed reference frame to determine longitudinal cell intersections. To ensure that similar intersection events are not accounted for more than once dur- ing propagation—e.g., in the case of repeated intersections occurring over subsequent time 19 r˜ eˆ1 eˆ2 eˆ3 GEO Torus RSO Orbits Figure18 Orbits 17 Near-miss events are detected during propagation of a derelict object by checking for the transversal of this GEO torus boundary at each time step during numerical integration, and between these primary time steps using an interpolation method if higher resolution is desired. Mathematically, a near-miss event occurs if[11]✓ rGEO q r2X + r 2 Y ◆2 + r2Z r˜2 < 0 (2.1) is satisfied,where (rX , rY , rZ)T denotes theRSOposition vector expressed in Earth-centered inertial (ECI) frame components. The longitude of intersection CPE is thus determined as CPE = arctan ✓ rY rX ◆ ↵G (2.2) where↵G is the right ascension ofGreenwich (a.k.a. Greenwich sidereal time), determined at the current simulation epoch via the well-known sidereal time approximation[36,40] J0 = 367ty floor ✓ 7 4  ty + floor ✓ tm + 9 12 ◆◆ + floor ✓ 275tm 9 ◆ + td + 1721013.5 (2.3a) T0 = J0 2451545 36525 (2.3b) ↵G,0 = 100.4606184 + 36000.77004 T0 + 0.000387933 T 2 0 2.583 108 T 30 [deg] (2.3c) ↵G = ↵G,0 + 360.98564724 ✓ th 24 ◆ [deg] (2.3d) where current propagation time is expressedwith (ty, tm, td, th), corresponding to the year, month, day, and hour (e.g., terrestrial time) of the current simulation epoch. When a torus- intersection is detected via Equation (2.1), the longitude of intersection CPE is determined with Equation (2.2), and the total near-miss count for the corresponding cell is advanced. Again, note that because of the convenient invariance of this torus geometry, inertial frame coordinates obtained during numerical integration need not be converted into the rotating Earth-centered, Earth-fixed reference frame to determine longitudinal cell intersections. To ensure that similar intersection events are not accounted for more than once dur- ing propagation—e.g., in the case of repeated intersections occurring over subsequent time Figure 2.3: GEO torus concept for tracki g near-miss ve ts. Slot!A Slot!B (a) 1 event per cell. Slot!A Slot!B (b) 3 events per cell. Slot!A Slot!B (c) 1 event per cell. Slot!A Slot!B (d) 2 events per cell. Figure 2.4: Examples of counting logic that eliminates redundancy in intersection checks. steps—counting logic is employed before any cell intersection counter is updated to screen the event for redundancy. Figure 2.4 illustrates several examples as to how near-miss in- tersection events are counted during numerical simulations. Relative to an Earth-fixed ob- server, orbits that reside entirely within a particular torus cell, as depicted in Figure 2.4(a), are counted only once during the specified near-miss tracking interval.6 If an object exits 6 Arbitrary near-miss tracking intervals are predefined to catalog intersections in consistent, reoccurring time frames. For this thesis, near-miss events are tracked in intervals of one solar day, such that following each day of propagation, torus cell counts and information for every near-miss occurring over the previous day are output to data files. Torus cell counts are then zeroed to initialize the next near-miss tracking interval. 20 and later re-enters the same torus cell, however, additional near-miss events are counted following every re-entry, as illustrated in Figure 2.4(b). Furthermore, additional near-miss events are counted if the relative orbit drifts into a neighboring slot during the near-miss tracking interval, such as in Figures 2.4(c)-2.4(d). This counting scheme ensures that near- miss events are evaluated for each of the GEO longitude slots in a logical, consistent, and non-redundant manner over the entirety of the forecasting period. The full algorithm for quantifying near-miss events using the torus intersection metric is provided as follows.[13] 2.3.1.1 Algorithm for Implementation of Torus Intersection Metric Let ti be the current propagation epoch of the ith RSO, and let t, tint, and tf denote in- tegration time, near-miss tracking interval length, and final simulation time, respectively. DefineNRSO as the number of GEO objects extracted from the TLE set, andNCPE|bin as the number of near-miss events for the torus cell indexed by longitude bin bin. Furthermore, let C denote the subset of all class C1/C2 objects, and let D,L, and I be the subsets of all class D, L1/L2/L3, and IN objects, respectively. Implementation pseudocode is as follows: 1: while ti < tf do 2: for i = 1! NRSO do 3: Intersection flag 0 4: Longitude flag 1 5: if i 2 D [ L [ I then 6: while t < tint do 7: Propagate: t t+t) ri = (rX , rY , rZ)T 8: if (rGEO p r2X + r 2 Y ) 2 + r2Z r˜2 < 0 then 9: th ti + t/3600 10: Equations (2.3a)-(2.3d)) ↵G 11: CPE = atan2(rY , rX) ↵G 12: bin = floor(CPE) 13: if Intersection flag = 0 k Longitude flag 6= bin then 14: Increment: NCPE|bin NCPE|bin + 1 21 15: Longitude flag bin 16: end if 17: end if 18: if (rGEO p r2X + r 2 Y ) 2 + r2Z r˜2 < 0 then 19: Intersection flag 1 20: else 21: Intersection flag 0 22: end if 23: end while 24: ti ti + tint 25: Update RSO fields 26: else (i 2 C) 27: ti ti + tint 28: Update ri i maintained 29: Update RSO fields 30: end if 31: end for 32: Output NCPE|bin 8 bin 2 [0, 360) 33: NCPE|bin = 0 8 bin 2 [0, 360) 34: end while 2.3.2 Propagator and Implementation Acustom special perturbations propagator implemented inANSI-C andparallelized with OpenCL7 is called to propagate the uncontrolled RSO population forward in time to determine torus intersection events. A lower-fidelity, albeit representative force model for theGEOenvironment is used for the benefit of dramatically-reduced simulation run times. In this formulation, two-body equations of motion are numerically integrated under 4 ⇥ 4 EGM-96 gravitation, luni-solar perturbations, and solar radiation pressure (SRP),modeled with the standard cannonball assumption[38] and attenuated via the occultation algorithm 7 OpenCL 1.2 Specification is available fromKhronos Group at: http://www.khronos.org/registry/cl/. 22 given in Reference 42. Specifically, the equations of motion for each uncontrolled RSO are r¨ = µ r3 r + a + a% + a + aSRP (2.4) where the first term denotes two-body acceleration, a is the perturbation due to the non- sphericity of Earth, a% and a are the third-body perturbations from the Moon and Sun, respectively, andaSRP is the SRPperturbation. Solar pressure ismodeled using the inverse- square diusion formulation of the solar luminosityL ⇡ 3.839⇥1026 J/s, with coecient of reflectivity cr ⌘ 1.5 and a GEO-representative area-to-mass ratioA/m = 0.04m2/kg.8 This GEO forcingmodel is in agreement with the results of Reference 32, which ranks the importance of incorporating various environmental perturbations in GEO forcing models for debris analysis over time scales ranging from 1week to 10 years. Appendix A provides a detailed description of the specific computations for each perturbation in Equation (2.4). In higher-fidelity force models, coordinate transformations between Earth-fixed and Earth-inertial frames use high-accuracy Earth orientation parameters (EOP) to account for precession, nutation, and polar motion – software suites such as the SPICE toolkit can be used to perform these intensive coordinate transformations.9 In this parallel propagator, however, a lower-fidelity transformation that accounts strictly for z-axis rotation byGreen- wich sidereal time is used for increased speeds at run time. Further, instead of extracting Moon and Sun vectors from JPL’s DE-421 ephemerides, this routine uses lower-precision formulas for the geocentric positions of these bodies, given in theAstronomical Almanac. [44] Reference 2 provides verification of this lower-fidelity forcemodel by comparing localized congestion results over a 5-year period with those obtained under higher-fidelity forcing. The propagator uses an eighth-order, predictor-correctorGauss-Jackson integrator[45] initializedwith the Prince-Dormand 8(7) algorithm for integrating the equations ofmotion in Equation (2.4). During initial propagation of the uncontrolled debris field to the fore- 8 Reference 43 indicates that this ratio is representative for operational and derelict satellites at GEO; this area-to-mass ratio is thus used in the SRP computation for all large-scale derelicts considered in this chapter. 9 The SPICE toolkit is available from NASA/JPL at: http://naif.jpl.nasa.gov/naif/toolkit.html. 23 cast start date, andduring near-miss event detection over the prediction span, a ten-minute time step is specified for sucient fidelity in capturing near-miss events. To increase reso- lutionwithout significantly increasing run times, a Lagrange interpolationmethod is used to detect torus intersections in one-minute increments between primary ten-minute steps. 2.3.3 Risk Functions for GEO-Relative Position and Velocity To begin identifying the particular uncontrolled derelicts responsible for the highest levels of both global and localized risk in the GEO regime, the GEO-relative position and velocity for an object at torus intersection must be translated into a risk metric that can be applied to categorize and rank the GEO debris population based upon levels of globally- and locally-contributed risk. It is important to note that the number of near-miss events alone for each object over the duration of the forecasting period is an insucient measure for evaluating both global and localized risk, since the proximity of these near-miss events to the GEO altitude—and the GEO-relative velocity with which these near-misses occur— are not explicitly taken into account. For this study, each near-miss event is assigned a level of combined risk that is based upon the GEO-relative state of the responsible debris object at the time of torus intersection. Position and velocity risk factor functions are employed to individually weight (a) how close the object comes to the GEO altitude, and (b) how fast the object is traveling relative to local GEO orbit velocity. Specifically, the position and velocity risk functions selected for this study are: Rr(r) = ✓ r˜ r r˜ ◆2 , 0  r  r˜ (2.5) Rv(v) = 1 e3v/v¯, v 0 (2.6) where r and v are the GEO-relative position and velocity magnitudes, respectively. The position risk factor function in Equation (2.5) is of quadratic form, rising smoothly from Rr(r˜) = 0 at the torus boundary to Rr(0) = 1 at the GEO altitude. The velocity risk func- tion in Equation (2.6) is of an exponential form, selected to rapidly saturate to Rv(v) ! 1 24 0 10 20 30 40 500.0 0.2 0.4 0.6 0.8 1.0 Distance from GEO Altitude @kmD Po sit io n Ri sk Fa cto r (a) Quadratic form used for position risk factor. 0.0 0.5 1.0 1.5 2.00.0 0.2 0.4 0.6 0.8 1.0 Relative Velocity @kmêsD Ve lo cit y Ri sk Fa cto r (b) Exponential form used for velocity risk factor. Figure 2.5: Functions selected for computing combined risk factor of each near-miss event. when the GEO-relative velocity surpasses a defined threshold, beyondwhich collision ve- locities are considered catastrophic. In this manner, all relative velocities beyond this criti- cal threshold are weighted nearly equally – collisions with uncontrolled derelicts traveling with catastrophic relative speeds have serious and likelymission-impacting consequences, regardless of where these speeds lie above the critical threshold. The position and velocity risk factor functions defined in Equations (2.5)-(2.6) are illustrated in Figure 2.5. From Equation (2.1), the GEO-relative position magnitude r is given by r = s✓ rGEO q r2X + r 2 Y ◆2 + r2Z (2.7) The GEO-relative velocity is computed by dierencing the object’s inertial velocity vector at torus intersection with the local, two-body orbit velocity vector of a hypothetical GEO satellite stationed at the longitude of intersection, expressed in inertial frame components: vGEO =  r µ rGEO sin(CPE + ↵G), r µ rGEO cos(CPE + ↵G), 0 T (2.8) The GEO-relative velocity is scaled by the “time-constant-like” parameter v¯ in the velocity risk factor function, such that when v = v¯, the risk function Rv(v¯) = 1 e3 ⇡ 0.950 and begins saturating.10 For this study, v¯ is derived from the NASA Standard BreakupModel 10 The parameter n in a more general function of the form 1 env/v¯ can be selected to adjust the value of the velocity risk at v = v¯ explicitly, i.e., for a desired velocity threshold risk Rv(v¯), then n = ln(1Rv(v¯)). 25 as the relative speed threshold abovewhich collisions are “catastrophic,” assuming equiv- alent masses for the target and impactor:[46] E˜⇤p = 1 2 v¯2 ! v¯ = q 2E˜⇤p ⇡ 0.2828 km/s (2.9) where E˜⇤p ⌘ 40 kJ/kg is the defined specific energy threshold for a catastrophic colli- sion.[11] After the GEO-relative position and velocity for the near-miss event are com- puted, the corresponding risk factor functions in Equations (2.5)-(2.6) are evaluated, and the combined risk factor given by the product Rr(r)Rv(v) is evaluated. This combined risk factor for the near-miss event is on the unit interval [0,1] and provides ametric for gauging how “threatening” the near-miss is for a hypothetical GEO satellite stationed on the lon- gitude of intersection. Under this metric, uncontrolled derelicts that closely pass the GEO altitude—and have catastrophic-collision-inducing relative speeds when doing so—will, in general, have a higher combined risk for near-miss events detected during propagation. Note that both close proximity and sucient relative speed must be present in order for a near-miss event to be considered high-risk under this metric, e.g., higher-speed events near the torus boundary and lower-speed events closer to the GEO altitude are both de- weighted in the combined risk factor. It is important to note that object mass is not considered in this combined risk factor, as it is used in the NASAOrbital Debris ProgramOce’s studies to identify top active de- bris removal (ADR) targets for the LEO regime.[34] Mass information is not includedwithin the publicly-available TLE data source harnessed in this study. If approximate mass infor- mation is available, an auxiliarymass function can be used toweight the combined risk fac- tor, or each object’smass can be in-built by using a kinetic energy of the form 12mv2 in Equa- tion (2.6) directly, i.e., by replacing relative speed risk with kinetic energy risk. With these modifications, the combined risk for more massive derelicts will be emphasized, in a sim- ilar manner that probabilities of collision are scaled by LEO object mass in Reference 34. 26 2.4 Results of Longitude-Dependent Congestion and Risk Forecast at GEO As a precursor to identifyingwhich classes of GEOdebris contribute themost to both global and localized risk levels in this regime, a 5-year macroscopic congestion forecast is performedwith theminor radius r˜ = 50 km, using the baseline object population in the 28 February 2014 TLE set. Controlled assets (C1/C2) are assumed to maintain their specified longitude slots, while the 745 uncontrolled objects extracted from this set are propagated forward in time and incorporated in the near-miss study. Note that this simulation is only investigating the congestion generated by the current debris population over a 5-year time frame. Nominal population growth, fragmentation events, solid rocket motor (SRM) slag, multi-layered insulation (MLI) shedding, and other debris growth mechanisms treated in Reference 6, for example, are not treated in this simulation. The objective for this research is to investigate which classes of large-scale derelicts currently at GEO contribute the most to risk levels across this regime, and determine the highest-risk derelicts in these classes. 2.4.1 Localized Congestion, Velocity, and Risk Results The localized debris weather forecast at GEO for the 5-year analysis period is shown in Figure 2.6(a), which illustrates the number of near-miss events per day at 50 km for each of the longitude slots at GEO. Accumulation of uncontrolled derelicts around the gravita- tional wells at 75E and 105W is well-known, and discussed in References 47 and 48, for example. This is particularly troublesome, because operational GEO satellites are typically inserted into longitude slots near these debris-critical longitudes,[2] and the probability of collision in the vicinity of these gravitational wells is seven times larger than in surround- ing regions at GEO.[1,8] From Figure 2.6(a), controlled assets in the longitude slots neigh- boring the two gravitational wells are subject to 4-8 close calls per day at a miss distance of 50 km – this is a factor of four increase over less congested longitudes (e.g., Atlantic and Pacific Oceans), which experience a maximum of 1-2 near-misses per day at 50 km. Note 27 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] (a) Near-miss events per day at 50 km during 5-year forecasting period. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (b) Maximum relative velocities of near-miss events during forecasting period. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0.0 0.1 0.2 0.3 0.4 0.5 C om bi ne d R is k Fa ct or (c) Maximum combined risk factor of near-miss events during forecasting period. Figure 2.6: Three portraits of longitude-dependent congestion in the GEO ring, generated by the large-scale, trackable, and unclassifiedGEOdebris population on 28 February 2014. that the number of near-misses during the forecasting period for any longitude slot is not equivalent to the probability of collision Pc used in operational conjunction assessment.11 Regions of increased near-miss events in Figure 2.6(a) are a result of (1) multiple objects with single near-miss events per day, (2) single objects with multiple near-miss events per day, or (3) a combination of both of these possibilities, in which overlapping Earth-relative 11 For discussions as to the impact of the debris population on longitude-dependent probability of collision at GEO, see Reference 1. 28 debrismotions (cf. Figure 2.3) induce the spatially-dense regions observed in Figure 2.6(a). Individual orbit class contributions to the localized congestion forecast in Figure 2.6(a) are illustrated in Figure 2.7 for drifting (D), Eastern librating (L1), Western librating (L2), and East/West librating (L3) objects. The near-linear traces exhibited by the class D objects in Figure 2.7(a) demonstrate the longitudinal drift magnitudes of the responsible objects – traces with a positive slope indicate eastward drift (belowGEO altitude), and traces with a negative slope indicate westward drift (above GEO altitude). Curvilinear traces of objects classified as drifting in Reference 33 indicate that these objects are now exhibiting librating class behavior, or were misclassified when initially catalogued due to insucient tracking data, uncertainties in the orbit determination process, force model approximations, etc. [40] As anticipated, increased densities of near-miss events per day near the gravitational wells result fromdominant contributions by the class L1 and L2 objects, shown in Figures 2.7(b)- 2.7(c) and illustrative of the libration period distribution in Figure 2.2. Class L3 behavior in Figure 2.7(d) results from a longitude of abandonment near the unstable equilibrium at 165E – initial potential energy in this region is large enough to propel the derelict through the first gravitational well, over the “potential hill” at 15W, and through the second gravi- tational well before reversing the direction ofmotion in longitude to complete the cycle.[11] It is interesting to highlight that although the number of near-miss events for a par- ticular longitude slot may be relatively benign on a daily basis, the relative velocities—and combined risk factors—withwhich these near-misses occurmay not be. Figure 2.6(b) illus- trates worst-case GEO-relative velocities experienced in each 1 longitude slot throughout the forecasting period. This relative velocity forecast shows that even though a longitude slot could experience 1-2 near miss events per day, the maximum relative speed for these near-miss events could be upwards of⇠0.6 km/s (approximately 1340mph).12 In general, 12 Relative velocities are lower in the GEO regime when compared to LEO, since (a) orbit velocity is lower at the GEO altitude, and (b) objects at GEO are, in general, orbiting in the same direction. [32] Compared to the 2009 Iridium-33/Cosmos-2251 collision, which occurred at a relative velocity of upwards of 24,600 mph (⇠11 km/s)[49] at LEO, 1340 mph at GEO is a much smaller—but still mission-impacting—collision velocity. 29 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] (a) Contribution of drifting (D) objects to near-miss event forecast. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] (b) Contribution of Eastern librating (L1) objects to near-miss event forecast. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] (c) Contribution of Western librating (L2) objects to near-miss event forecast. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] (d) Contribution of East/West librating (L3) objects to near-miss event forecast. Figure 2.7: Contributions of individual orbit classes to localized congestion at GEO. 30 higher relative speeds can be attributed to uncontrolled derelicts with higher inclinations, such that relative speeds at the ascending anddescending nodes are increasedwith respect to equatorial GEO orbit velocity, which does not have an out-of-plane component for geo- stationary orbits. Nonzero eccentricity also serves to increase relative speeds at the nodes, such that near-miss eventswith higher-inclination andhigher-eccentricity objects aremore hazardous to operational assets, that is, these objects exhibit higher combined risk factors. Figure 2.6(c) shows worst-case combined risk factors experienced in each longitude slot throughout the forecasting period. Higher combined risk levels are visible in the vicin- ity of the gravitational wells, exhibiting a banding phenomenon driven by once-yearly os- cillations in the magnitude of the eccentricity vector, induced by the SRP perturbation.[3] In Section 2.5.2, this combined risk forecast is used to validate an identified subset of high- risk derelicts, that is, the combined risk in the neighborhoods of the gravitational wells is attenuated if this high-risk subset of objects is omitted from the 5-year congestion forecast. Figure 2.8 illustrates the eect of various force model contributions on the combined risk factor forecast in Figure 2.6(c). Figure 2.8(a) provides the resulting risk forecast under 4⇥4 gravitation only; luni-solar perturbations are then applied in Figure 2.8(b), and SRP is applied in Figure 2.8(c). The inclusion of luni-solar perturbations changes the phase of the L2 congestion band centered on theWesternwell, and SRP gives rise to the annual banding phenomenonmentioned above. Figure 2.8 corroborates the results of Reference 32, which ranks the importance of incorporating various environmental perturbations in GEO force models for debris analysis over time scales ranging from 1week to 10 years. For the 5-year time scale used in Figure 2.8, third-body perturbations and SRP are assigned “very high” and “high” importance, respectively.[32] 2.4.2 Time-Averaged Congestion, Velocity, and Risk Results In addition to tracking the number of near-miss events at 50 km occurring daily for each longitude slot over this 5-year analysis period (cf. Figure 2.6), it is of interest to assess 31 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0.0 0.1 0.2 0.3 0.4 0.5 C om bi ne d R is k Fa ct or (a) Maximum combined risk (4⇥4 gravitation only). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0.0 0.1 0.2 0.3 0.4 0.5 C om bi ne d R is k Fa ct or (b) Maximum combined risk (4⇥4 gravitation and luni-solar perturbations). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 03/14 03/15 03/16 03/17 03/18 03/19 M on th /Y ea r 0.0 0.1 0.2 0.3 0.4 0.5 C om bi ne d R is k Fa ct or (c) Maximum combined risk (4⇥4 gravitation, luni-solar perturbations, and SRP). Figure 2.8: Contributions of dominant GEO perturbations to combined risk forecast. which classes of uncontrolled objects contribute the most to longitude-dependent conges- tion, for purposes of recommending which types of objects to remove if a particular longi- tude slot is to be cleared of debris. Figure 2.9(a) provides the average number of near-miss events per day contributed by objects of each uncontrolled class considered in this study. Longitude slots neighboring the debris-critical gravitational wells at 75E and 105W are subject to 3-4 close calls per day on average, while less congested longitude slots experience only 0.5 close calls (at 50 km) per day on average. Congestion contributions in the regions 32 encompassing the Eastern andWestern gravitational wells are dominated by class L1 and L2 objects, which oscillate in the Earth-fixed frame around these stable equilibria with the libration periods shown in Figure 2.2. In addition, the drifting (class D) objects contribute an approximate 0.25 near-misses per day across the entirety of the GEO ring, thereby gen- erating the “background noise” observed in the debris congestion forecast in Figure 2.6(a). Interestingly, the near-miss contributions for class D objects peak at the unstable equilib- rium longitudes 165E and 15W – the longitudinal rate ˙ induced by the J2,2 harmonic is at a minimum at these locations,[4] causing drifters to linger longer at these longitudes.[1] In summary of the localized congestion, relative velocity, and combined risk results shown for the 5-year analysis period in Figure 2.6, Figures 2.9(b)-2.9(d) show the average number of near-miss events at 50 km per day, beneath or above various relative position, relative speed, and combined risk factor thresholds, respectively. For example, the Eastern well at 75E experiences an average of four near-misses per day at 50 km over the forecast (Figure 2.9(a)), two ofwhich are beneath 40 km in proximity to theGEO ring (Figure 2.9(b)) and greater than 0.75 km/s in relative velocity (Figure 2.9(c)). Although Figure 2.9(c) illus- trates that approximately all of the near-miss events occurring at 75E are above the catas- trophic velocity threshold in Equation (2.6), the combined risk summary in Figure 2.9(d) indictates that only one near-miss event per day in this longitude slot is above a combined risk factor of 0.2 on average. The position risk function in Equation (2.5) is de-weighting the risk of these higher-velocity encounters – Figure 2.9(b) shows that themajority of these near-miss events are occurring above 25 km in proximity to GEO, a distance beyondwhich the position risk function in Equation (2.5) is beneath 0.25 in magnitude (cf. Figure 2.5(a)). For purposes of slot safety assessment, it is important to consider not only the mean of the number of near-miss events occurring daily for each longitude slot, but the standard deviation for these distributions, aswell. Figure 2.10 illustrates the 1 number of near-miss events at 50 km for each longitude slot and risk threshold provided in Figure 2.9(d), high- lighting the spread of these slot-specific distributions (i.e., generated by tracking the num- 33 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay Class D Class L1 Class L2 Class L3 Class IN (a) Average number of near-miss events per day at 50 km by object class. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay < 50 km < 40 km < 25 km < 10 km (b) Average number of near-miss events per day with position thresholds. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay > 0.00 km/s > 0.50 km/s > 0.75 km/s > 1.00 km/s (c) Average number of near-miss events per day with velocity thresholds. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay > 0.0 > 0.1 > 0.2 > 0.4 (d) Average number of near-miss events per day with risk factor thresholds. Figure 2.9: Average number of near-miss events per day at 50 km for each longitude slot, by orbit class and above or beneath various relative position, velocity, and risk thresholds. 34 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 1 N ea r- M is se s pe rD ay > 0.0 > 0.1 > 0.2 > 0.4 Figure 2.10: Standard deviation of near-miss events per day with risk factor thresholds. ber of near-misses within each longitude slot, above each risk threshold, on each day over the forecast). Near-miss distributions around the two gravitational wells have a larger 1 spread for all risk thresholds, indicating that these congested regions can experiencemany more near-misses on a given day than the 5-year average of approximately 3-4 near-misses per day shown in Figure 2.9(d). In particular, the near-miss distribution for 75E extends from 0-13 near-misses per day at 50 km during the 5-year analysis period, highlighting the insuciency of the time-averagedmean in Figure 2.9(d) in forecasting slot-specific trends. 2.4.3 Timing of Near-Miss Events In addition to the number of near-miss events per day for each GEO longitude slot, and the relative velocities and combined risks with which these near-miss events occur, it is critical to study the time of day at which these near-misses occur. Reference 1 indicates that the equatorial crossingwindows of the GEOdebris population are synchronized such that collision hazard is both episodic and predictable. To illustrate this concept, Figure 2.11 provides the distribution mean and 1/2/3 corridors for the local time of all near-misses occurring across the GEO ring over each day of the first year of the forecasting period, for near-miss events during the ascending (south-to-north) equatorial pass in Figure 2.11(a), and descending (north-to-south) equatorial pass in Figure 2.11(b). The tight distributions shown in Figure 2.11 demonstrate that there exist twopredictablewindows relative to local 35 03 /14 04 /14 05 /14 06 /14 07 /14 08 /14 09 /14 10 /14 11 /14 12 /14 01 /15 02 /15 03 /15 Month/Year 20 15 10 5 0 5 10 15 20 H ou rs to Lo ca lN oo n (A sc en di ng ) (a) Equator crossing time distribution (ascending). 03 /14 04 /14 05 /14 06 /14 07 /14 08 /14 09 /14 10 /14 11 /14 12 /14 01 /15 02 /15 03 /15 Month/Year 20 15 10 5 0 5 10 15 20 H ou rs to Lo ca lN oo n (D es ce nd in g) (b) Equator crossing time distribution (descending). Figure 2.11: Equatorial crossing times for the synchronized debris population at GEO are episodic and dependent upon time of year.[1] Shaded swaths illustrate 1/2/3 corridors. noon during which observational coverage of GEO can bemaximized, regardless of longi- tudinal location around GEO. The equatorial crossing windows vary nearly linearly over the course of the year, and the ascending/descending windows occur 12 hours apart from one another at any point in the year. These time-varying equatorial crossing windows are the Earth-fixed representation of thewell-knownGEO “pinch points,” which are currently static in inertial right ascension anddeclination coordinates, and are harnessed to optimize sensor tasking for space-based space surveillance (SBSS) systems.[50,51] Relative to an Earth-fixed observer, the collective motion of the GEO debris popula- tion is qualitatively similar to a transverse wave[1] – this motion arises from a clustering in right ascension of the ascending node, driven by luni-solar perturbations dominant at the GEO altitude. The equatorial crossing windows shown in Figure 2.11 are a physical man- ifestation of this phenomenon, and can be leveraged for maximizing observational cover- age of the GEO ring, anomaly correlation, and forecasting of potential conjunction events. With this knowledge, the threat of near-miss events with higher combined risk factors can be mitigated through increased coverage of the desired slot during the episodic windows relative to local noon shown in Figure 2.11. An in-depth discussion of GEO “pinch point” 36 evolution and its impact on operational geostationary satellites is provided in Chapter 3. 2.4.4 Near-Miss Overlap Comparison To investigate howwell the near-miss eventmetric performs in forecasting longitude- dependent congestion trends at GEO in the short-term, an overlap study is conducted. In this comparison, two five-year predictions are generated – one using the debris population from the 31 August 2011 TLE set, and the other using the population in the 31 August 2013 TLE set. To ensure that controlled objects in the TLE set from 2011 are not misclassified as uncontrolled debris if using the 2014 issue ofClassification of Geosynchronous Objects, [33] the 2012 issue is instead used for classifying the 2011 TLE population. Thus, for the 31 August 2011 TLE set, 705 of 1053GEOobjects are uncontrolled, and for the 31August 2013 TLE set, 742 of 1126 are uncontrolled. Eachpopulation is propagated forward for five years, and the localized congestion trends in the September 2013-2016 overlapping period are compared. Figure 2.12 illustrates the results of this overlap comparison by providing the dier- ence in slot-dependent near-miss events per day detected between the 2011 and 2013 pre- dictions. A negative dierence indicates that the 2011 propagation under-predicted future 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/11 09/12 09/13 09/14 09/15 09/16 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [5 0 km ] M on th /Y ea r -4 -2 0 2 4 N ea r- M is s D if fe re nc e Figure 2.12: Comparison of two overlapping congestion forecasting periods. 37 localized congestion; conversely, a positive dierence indicates that the 2011 propagation over-predicted future congestion trends (i.e., more near-miss events per day in a particular longitude slot than forecasted using the 2013 debris population). As shown in Figure 2.12, near-miss dierences during the overlapping period vary widely, and are most significant in the vicinity of the gravitational wells. Large congestion dierences in certain longitude slots at particular times can be attributed to one or more of the following error sources: • Errors in initial conditions. Publicly-available TLE data sets used to derive the initial conditions of the debris population prior to propagation are not intended for high- accuracy applications, and corresponding uncertainty information for each object is not provided. Even in the absence of propagation errors, a state vector extracted from a later TLE set will dier from the prediction derived from an earlier TLE set due to measurement noise and errors inherent to the orbit determination process. • Errors in propagation. As stated in Section 2.3.2, the propagator used in this chapter includes a lower-fidelity, representative force model for the GEO environment, in which gravity terms larger than fourth degree and order are truncated. Thus, this force model does not provide a high-accuracy description of GEO dynamics, such that unmodeled/mismodeled perturbations combine with initial condition errors to augment slot-dependent congestion dierences observed in the overlap period. • Errors in population prediction. Under-prediction occurs when objects listed as con- trolled in the 2011 population are classified as uncontrolled in the 2013 population. These dierences also occur because the debris population increased by 37 objects between the 31 August 2011 and 2013 TLE populations, and the 2011 propagation does not reflect this increase in the number of debris, leading to under-prediction. Ultimately, the widely-varying congestion dierences in Figure 2.12 indicate that the com- bination of recent TLE sets, shorter prediction spans, up-to-date object classifications, and 38 higher-fidelity propagation are required to reduce the uncertainties in the number of near- miss events per day experienced in arbitrary longitude slots at arbitrary times. Such high- accuracy congestion predictions could be leveraged to informGEO operators as to specific times of year—in addition to the specific times of day shown in Figure 2.11—duringwhich conjunction events may bemore likely, due to an increase in the number of objects passing through their longitude slots (“stormmonitoring” in the debris weather forecast analogy). 2.5 Identifying Highest-Risk Debris Objects at GEO To begin identifying the classes of uncontrolled debris objects in theGEO regime that contribute themost to longitude-dependent congestion—andmore importantly, longitude- dependent risk—the results of the 5-year congestion and risk forecast shown in Figure 2.6 are partitioned by debris orbit class. Figure 2.13(a) gives the breakdown of the percentage that each uncontrolled derelict class contributes to the total number of near-miss events at 50 km occurring globally over the entire 5-year analysis period. Figure 2.13(b) provides a breakdown of the percentage that each class contributed to the total combined risk for the near-miss events accumulated globally during the 5-year analysis period. These pie charts are interesting when compared alongside the TLE orbit class breakdown in Figure 2.1. Al- though drift objects constitute 50.9% of the large-scale, trackable RSO population at GEO, they only contribute 26.3% of the near-miss events at 50 km detected during the forecast- ing period, which collectively account for 23.8% of the global risk accumulated during this period. Librating objects, on the other hand, contribute a dominating 73.1% of the number of near-misses—accounting for 76.1% of the global accumulated risk—even though these objects collectively make up a mere 14.1% of the RSO population at GEO (that is, 27.1% of the uncontrolled debris population in the 28 February 2014 TLE set). Of significant interest for ADR initiatives are the 105 Eastern librating (L1) derelicts that contribute 49.9% to the globally accumulated risk over the 5-year forecast, that is, 14.1% of the GEO debris popu- lation is collectively responsible for nearly 50% of globally accumulated risk around GEO. 39 L1 105 (47.6%) L2 40 (18.6%) L3 16 (6.9%)D 583 (26.3%) IN 1 (0.5%) (a) Breakdown of near-miss events. L1 105 (49.9%) L2 40 (18.8%) L3 16 (7.4%)D 583 (23.8%) IN 1 (0.05%) (b) Breakdown of combined risk factor. Figure 2.13: Breakdown of number of near-misses and combined risk accumulated glob- ally around the GEO ring during the 5-year forecasting period, by responsible debris class. 2.5.1 Global Risk Accumulation Metric Although Figure 2.13 gives a congestion and risk breakdown by debris class, it does not illustrate the contribution of individual objects within each orbit class to the combined risk globally accumulated during the forecasting period. One measure for identifying the highest-risk objects at GEO is the global risk accumulation metric, which utilizes the risk con- tribution of individual objects to the total risk globally accumulated around GEO during the forecasting period to rank the derelict population from highest to lowest risk level. To compute the global accumulated risk, the combined risk factors for all near-miss events occurring in all longitude slots throughout the 5-year forecasting period are summed into a grand total. Then, the risk contributions of the near-miss events for individual objects to this grand total are applied to sort these derelicts into order of descending global risk con- tribution. Table 2.2 lists the top 10 highest-risk objects from the 28 February 2014 TLE set, identified using this global risk accumulation metric. Notably, all of these objects are de- 40 funct payloads that are from the L1/L2 librating classes.13 The semi-major axes for these high-risk derelicts arewithin±15 kmof the GEO radius at the beginning of the forecasting period, and the inclinations for these derelicts range from 13-16 relative to the equatorial plane. Collectively, these top 10 derelicts are responsible for 9.39% of the global combined risk factor accumulated around the GEO ring during the 5-year forecast in Figure 2.6(c). Table 2.2: Highest-risk derelicts at GEO, based on global risk accumulation (unweighted). Rank Object Name COSPAR Class Country Slots Aected % Risk 1 Aurora 1 1982-105A L2 USA 97-110W 0.968% 2 Cosmos 1366 1982-044A L1 USSR/Russia 65-84E 0.965% 3 ASC 1 1985-076C L2 USA 79-130W 0.961% 4 Raduga 1-2 1990-116A L1 USSR/Russia 49-100E 0.958% 5 Insat 1C 1988-063A L1 India 57-92E 0.949% 6 GStar 1 1985-035A L2 USA 100-108W 0.947% 7 GStar 3 1988-081A L2 USA 101-107W 0.931% 8 Telstar 4A 1993-077A L2 USA 97-113W 0.917% 9 Raduga 26 1990-112A L1 USSR/Russia 55-94E 0.899% 10 Cosmos 1897 1987-096A L1 USSR/Russia 49-100E 0.898% TOTAL 9.39% It is critical to emphasize that this top 10 ranking is in eect for the next 5 years only, since it harnesses combined risk factor data for near-miss events detected during a 5-year forecasting period. Awell-known eect of luni-solar perturbations onGEOobjects is long- term cyclical precession of the orbital plane, in which the inclination of initially equatorial objects rises to 15 and back with a period of approximately 53 years.[3] Thus, in 26 years, the objects listed in Table 2.2will be in nearly equatorial orbits, such that the relative speeds of these objects at the nodes are decreased. As a result, the combined risk contributions for the near-miss events triggered by these currently highest-risk derelicts will diminish—and successively increase again—over the course of this 53-year inclination cycle (Chapter 3). Note that although the position and velocity relative to GEO at torus intersection are 13 As a first-order condition for stable point capture improves as proximity to the GEO radius decreases, [4] this is an intuitive result. Librating objects have semi-major axes that are closer to the synchronous altitude, such that relative position risk in Equation (2.5) is higher, in general, for the near-miss events triggered by objects from these librating classes. 41 translated into a combined risk factor using Equations (2.5)-(2.6), the longitude at which the near-miss event occurs is not taken into account in the present risk formulation. Since operational GEO satellites aremost often inserted into longitude slots in the neighborhood of the gravitational wells, [2] it is useful to consider weighting the combined risk factor by the longitude at which the near-miss event occurs. In this manner, higher-risk near-misses occurring near higher-density longitudes are emphasized, and conversely, higher-risk near- misses at longitudeswith lowdensities of operational assets are de-weighted. An example of one such longitude weighting function is drawn from the wrapped-normal probability density function fit to past, present, and projected satellite longitude data in Section 4.2.[2] This weighting function (illustrated in Figure 2.14) amplifies the risk experienced in longi- tude slots covering theUnited States and the Eurasian region, and reduces risk factors over the Atlantic and Pacific Oceans, where operational satellite densities are at a minimum. Table 2.3 lists the top 10 highest-risk objects from the 28 February 2014 TLE set, iden- tified using the global risk accumulation metric with longitudinal weighting applied. The first column provides the rank for each object under longitudinal weighting, and the num- ber of places that these objects rose or fell in the unweighted top 10 ranking in Table 2.2 as a result. The emphasis of the Eastern well over theWestern well (cf. Figure 2.14) increases the ranking of Eastern librating (class L1) objects in the ranked population – Aurora 1, the highest-risk L2 object in the unweighted top 10 ranking, falls three places in the weighted top 10 ranking. Furthermore, two L2 payloads from Table 2.2—ASC 1 and Telstar 4A—fall out of the top 10 ranking completely when longitudinal weighting is applied. Collectively, the top 10 objects under longitudinalweighting are responsible for 11.53%of the total com- bined risk factor accumulated aroundGEOduring the 5-year forecast, a higher percentage than the 9.39% collectively contributed by the top 10 objects in the unweighted ranking. To summarize the eect of the global risk accumulation metric in identifying objects with the highest cumulative risk over the analysis period—that is, the objects that consis- tently come close to theGEOaltitude at consistently high relative velocities—Figure 2.15(a) 42 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0.0 0.05 0.1 0.15 0.2 0.25 0.3 Lo ng it ud e W ei gh ti ng Fa ct or Stable 75E Unstable 165E Stable 105W Unstable 15W Figure 2.14: Longitude weighting function based upon operational satellite densities. [2] Table 2.3: Highest-risk derelicts at GEO, based on global risk accumulation (weighted). Rank Object Name COSPAR Class Country Slots Aected % Risk 1 ("1) Cosmos 1366 1982-044A L1 USSR/Russia 65-84E 1.226% 2 ("3) Insat 1C 1988-063A L1 India 57-92E 1.198% 3 ("1) Raduga 1-2 1990-116A L1 USSR/Russia 49-100E 1.183% 4 (#3) Aurora 1 1982-105A L2 USA 97-110W 1.168% 5 ("1) GStar 1 1985-035A L2 USA 100-108W 1.149% 6 ("1) GStar 3 1988-081A L2 USA 101-107W 1.131% 7 ("2) Raduga 26 1990-112A L1 USSR/Russia 55-94E 1.125% 8 ("2) Cosmos 1897 1987-096A L1 USSR/Russia 49-100E 1.117% 9 ("3) Cosmos 1961 1988-066A L1 USSR/Russia 68-82E 1.117% 10 ("3) Luch 1-1 1995-054A L1 USSR/Russia 72-77E 1.116% TOTAL 11.53% gives contributed percent of global accumulated risk factor as a function of top percent of the ranked debris population, with andwithout longitudinalweighting. Recalling that the debris population is ranked in order of descending global risk contribution, Figure 2.15(a) shows that 60% of the derelicts (453 objects) do not contribute any risk to the accumulated total, i.e., near-miss events for these derelicts are not triggered over the 5-year forecasting period. Without longitudinal weighting, the top 70 ranked objects are collectively respon- sible for 50% of global risk experienced during the forecasting period. Under longitudinal weighting, this cumulative trend line becomes steeper, such that the top 59 ranked objects 43 0 10 20 30 40 50 60 70 80 90 100 Top Percent of Ranked Debris Population 0 10 20 30 40 50 60 70 80 90 100 Pe rc en to fG lo ba lA cc um ul at ed R is k Fa ct or Unweighted (0-359E) Weighted (0-359E) Number'of'Ranked'Debris'Objects 0"""""74""""149"""223"""298""372"""447""521""596"""670""745 Unweighted:"Top"70"objects" responsible"for"50%"risk Global:"60%"objects" have"no"risk"impact Weighted:"Top"59"objects" responsible"for"50%"risk (a) Summary of global accumulated risk at GEO. 0 10 20 30 40 50 60 70 80 90 100 Top Percent of Ranked Debris Population 0 10 20 30 40 50 60 70 80 90 100 Pe rc en to fA cc um ul at ed R is k Fa ct or ar ou nd W el l Eastern (60-90E) Western (90-120W) Number'of'Ranked'Debris'Objects 0"""""74""""149"""223"""298""372"""447""521""596"""670""745 Eastern:"Top"21"objects" responsible"for"50%"risk Western:"Top"9"objects" responsible"for"50%"risk Eastern:"69%"objects"have"no"impact Western:"77%"objects"have"no"impact (b) Summary of local accumulated risk at GEO. Figure 2.15: Percent of accumulated risk total as a function of top percent of ranked debris population, which highlights accumulated risk contributions from high-ranking subsets. are responsible for 50% of global accumulated risk. Comparing Tables 2.2-2.3, this results from increased risk contributions for the top ranked objects under longitudinal weighting. 2.5.2 Local Risk Accumulation Metric If identifying the derelicts that contribute the highest accumulated risk to a particular subset of GEO longitude slots is desired, the local risk accumulation metric can be evaluated. In contrast to the global accumulationmetric, which sums the combined risk factors for all near-miss events occurring across all longitude slots during the analysis period, local risk accumulation uses the sum of combined risk for only those near-miss events occurring in a subset of longitude slots during the analysis period. The local risk accumulation metric is useful for identifying the objects that contribute the highest cumulative risk in the vicinity of the twodebris-critical gravitationalwells. Tables 2.4-2.5 delineate the top 10 highest-risk contributors to the total accumulated risk over±15 longitude intervals centered upon the Eastern andWestern gravitational wells, respectively. Note that these 10 ranked objects in 44 Table 2.4: Highest-risk debris objects near Eastern gravitational well, based on local risk accumulation on [60,90E]. Shaded rows indicate objects that are both globally high-risk (that is, included within the unweighted top 10 ranking in Table 2.2) and locally high-risk. Rank Object Name COSPAR Class Country Slots Aected % Risk 1 Cosmos 1366 1982-044A L1 USSR/Russia 65-84E 3.636% 2 Cosmos 1961 1988-066A L1 USSR/Russia 68-82E 3.290% 3 Luch 1-1 1995-054A L1 USSR/Russia 72-77E 3.279% 4 SIRIO 1 1977-080A L1 Italy 72-77E 3.115% 5 Comstar 4 1981-018A L1 USA 69-80E 3.094% 6 Cosmos 2133 1991-010A L1 USSR/Russia 67-82E 2.971% 7 Raduga 12 1983-028A L1 USSR/Russia 63-85E 2.921% 8 Raduga 14 1984-016A L1 USSR/Russia 61-87E 2.637% 9 Cosmos 2085 1990-061A L1 USSR/Russia 67-82E 2.504% 10 Raduga 26 1990-112A L1 USSR/Russia 55-94E 2.490% TOTAL 29.94% Table 2.4 contribute 29.94% of the total risk accumulated in [60E, 90E] during the 5-year period, while the 10 ranked objects in Table 2.5 contribute 57.54% of the risk accumulated in [90W, 120W] during this period. This has significant implications for ADR initiatives seeking to attenuate longitude-dependent risk in the vicinity of theWesternwell, as almost 60% of the combined risk accumulated over this region of GEO is attributed to the top 10 objects in Table 2.5 alone. Note again that this localized riskmeasure is not associatedwith the probability of collision Pc metric commonly used in conjunction assessment, and thus should not be treated as equivalentwith this statisticalmetric over any analysis time frame. Figure 2.15(b) showspercent contributed to the accumulated risk around eachwell as a function of the top percent of the ranked debris population. From this chart, 69% of dere- licts (515 objects) do not contribute to accumulated Eastern well risk, and 77% of derelicts (574 objects) do not contribute to accumulatedWestern well risk. As a result, these cumu- lative trend lines for the local accumulated risk in the vicinity of the gravitational wells in Figure 2.15(b) are steeper than those for the globally accumulated risk in Figure 2.15(a). In particular, the top 21 highest-risk derelicts near the Eastern well—and the top 9 highest- risk objects near the Western well (cf. Table 2.5)—are responsible for 50% of the localized 45 Table 2.5: Highest-risk debris objects near Western gravitational well, based on local risk accumulation on [90,120W]. Shaded rows indicate objects that are both globally high-risk (that is, included within the unweighted top 10 ranking in Table 2.2) and locally high-risk. Rank Object Name COSPAR Class Country Slots Aected % Risk 1 Aurora 1 1982-105A L2 USA 97-110W 8.215% 2 GStar 1 1985-035A L2 USA 100-108W 8.037% 3 GStar 3 1988-081A L2 USA 101-107W 7.902% 4 Telstar 4A 1993-077A L2 USA 97-113W 7.780% 5 GOES 3 1978-062A L2 USA 99-109W 5.449% 6 LES 8 1976-023A L2 USA 97-111W 5.364% 7 LES 9 1976-023B L2 USA 99-111W 4.821% 8 ASC 1 1985-076C L2 USA 79-130W 3.630% 9 NATO IIB 1971-009A L2 USA 102-108W 3.218% 10 Solidaridad 1 1993-073A L2 Mexico 100-111W 3.124% TOTAL 57.54% risk observed in the 30 longitude regions centered on each gravitationalwell, respectively. To show how removal of the top contributing objects in Tables 2.4-2.5 attenuates lo- calized risk in the vicinity of the gravitationalwells over the forecasting period, Figure 2.16 provides the mean number of near-miss events per day at 50 km above various combined risk levels, with the full GEOdebris population in the 28 February 2014 TLE (Figure 2.9(d)), with the top 5 derelicts in Tables 2.4-2.5 removed (Figure 2.16(a)), and with all 10 derelicts in Tables 2.4-2.5 removed (Figure 2.16(b)). As Figure 2.16 illustrates, removal of the highest cumulative risk objects in the vicinity of the gravitational wells serves to attenuate local- ized risk in these critical regions. In particular, Figure 2.16(b) illustrates that removal of the 10 derelicts in Table 2.5 not only generates a⇠50% reduction in the mean number of near- miss events occurring around theWesternwell, but nearly eliminates any near-miss events with combined risk factors greater than 0.4 during the five-year forecasting period. 2.5.3 Absolute Worst-Case Risk Metric The global and local risk accumulationmetrics use the sum of the combined risk fac- tors for near-miss events detected across all longitude slots—or a defined subset thereof— 46 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay Full TLE > 0.0 > 0.1 > 0.2 > 0.4 (a) Mean number of near-misses with risk factor thresholds (top 5 removed). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 0 1 2 3 4 M ea n N ea r- M is se s pe rD ay Full TLE > 0.0 > 0.1 > 0.2 > 0.4 (b) Mean number of near-misseswith risk factor thresholds (top 10 removed). Figure 2.16: Removal of highest-risk debris objects (Tables 2.4-2.5) from 28 February 2014 TLE attenuates localized risk for operational assets positioned near the gravitational wells. over the forecasting period. Though these twomeasures are useful for identifying theGEO debris objects responsible for the highest levels of cumulative risk experienced either glob- ally or locally at GEO, these metrics do not guarantee that the derelicts responsible for the highest absolute risk events will be identified in the ranking scheme. Thus, instead of rank- ing the debris population by individual risk contribution to a globally or locally accumu- lated risk total, the absolute worst-case risk metric ranks these debris objects by themaximum single-event risk factor each object contributed over the duration of the forecasting period. Table 2.6 compares the top 5 globally highest-risk objects, identified via (a) the global accumulation metric (cf. Table 2.2), and (b) the absolute worst-case metric. Although risk contributions for the accumulated risk subset are the highest of the population, the worst- case risk events for these objects are not as severe as the worst-case events for the absolute 47 risk subset. Conversely, the absolute risk subset captures the highest single-event risks ob- served during the analysis period, but the cumulative risk contributions for these objects are far beneath the contributions of the accumulated risk subset. Comparing the number of near-misses for the accumulated risk subset with the number of near-misses for the ab- solute risk subset (cf. Table 2.6), this is largely a consequence of the number of events that the derelicts in each subset are responsible for over the duration of the forecasting period. Table 2.6: Comparison between top five highest-risk objects, based on two ranking criteria. Rank Object Name COSPAR Class Events Contribution Worst Risk 1 Aurora 1 1982-105A L2 3666 0.968% 0.886 2 Cosmos 1366 1982-044A L1 3668 0.965% 0.952 3 ASC 1 1985-076C L2 3593 0.961% 0.929 4 Raduga 1-2 1990-116A L1 3400 0.958% 0.911 A cc um ul at ed 5 Insat IC 1988-063A L1 3568 0.949% 0.884 1 Syncom 2 1963-031A D 1753 0.438% 0.993 2 IUS stage 2 1991-054D L3 1293 0.264% 0.990 3 LES 9 1976-023B L2 2443 0.568% 0.981 4 Raduga 7 1980-081A L2 1960 0.311% 0.980A bs ol ut e 5 Satcom 3R 1981-114A D 2121 0.511% 0.979 As the longitude interval applied for local risk accumulation decreases in length, the percent contribution andworst risk dierences between the accumulated risk and absolute risk subsets are amplified, i.e., the worst single-event risk for the highest accumulated risk object in a single longitude slot at GEO could be significantly smaller than theworst single- event risk in that slot for the highest absolute risk object. Therefore, depending upon the objectives for a particular ADR initiative—that is, whether the eort is geared towards re- ducing cumulative or absolute risk in a particular subset of longitude slots—two dierent methodologies are presented to identify the highest-risk target objects for either objective. 2.6 Conclusions from Research Goal In this chapter, forecasting localized debris congestion in the GEO ring is performed to determine howmany near-miss events at 50 km occur upon a daily basis for each of the 48 GEO longitude slots over a 5-year analysis period. The GEO-relative position and velocity at torus intersection for each near-miss event are translated into a combined risk factor that gauges how threatening the near-miss is for a hypothetical GEO satellite positioned at the longitude of intersection. Three methods for identifying the highest-risk debris objects at GEO are presented to establish a framework for determining the top ADR targets at GEO. Twomethods use the contribution to a combined risk total, accumulated either globally or over a subset of longitude slots, to rank the debris population from highest to lowest risk; the third method considers maximum single-event risk as the metric for ranking derelicts. Simulation results show that the librating class objects—especially those of the East- ern librating class—are themost responsible for both global and localized risk levels, espe- cially in longitude slots surrounding the two gravitational wells at 75E and 105W. In the vicinity of the Eastern well, the top 21 ranked objects account for 50% of the accumulated risk in this region over the 5-year forecasting period, and in the vicinity of theWesternwell, the top 9 ranked objects are responsible for 50% of locally-accumulated risk. Therefore, the results of this chapter have important implications for ADR initiatives aimed at attenuat- ing orbit debris risk, either globally or locally, across the GEO ring. Removal of the top 10 highest cumulative risk objects by the Western well leads to a 50% reduction in the mean number of near-miss events per day occurring in this region, andnearly eliminates all near- miss events in this region occurring with critical risk factors—those that could potentially warrant evasive action by operators. This chapter serves to illustrate localized, longitude- dependent congestion arising from the current GEO debris population, investigate which classes of debris objects are most responsible for congestion contributions, and ultimately, spearhead a dialogue concerned with identifying highest-risk GEO debris for purposes of ADR target selection in this regime. 2.7 Continuing Research Questions For continuing research in this area, the following research questions are of interest: 49 (1) The torus minor radius r˜ = 50 km is used in this chapter as an approximate upper bound at which precise conjunction assessment could potentially be considered in the GEO regime. Howdoes this radius comparewith the threshold distance—and probability of collision—below and above which GEO operators plan and execute a dedicated or multi-purpose avoidance maneuver to reduce debris collision risk? Can this information be leveraged to estimate av budget from a daily near-miss event average (Figure 2.9) forecasted for the operational longitude slot of interest? (2) The near-miss eventmetric and longitude-dependent congestion results presented in this chapter illustrate inhomogeneity of asset-debris conjunction risk at theGEO altitude – debris-debris collision risk, however, is not addressed. Could particular objects not ranked highly under global risk accumulation still be considered high- risk, inasmuch as these objects consistentlymake close approaches with neighbor- ing debris? Can such “environment criticality” be incorporated into risk rankings? (3) Can the combined risk factor developed in this chapter for ranking the debris pop- ulation be further refined to take into account other factors of interest for ADR tar- get selection, such as mass, debris-debris conjunction potential, or attitude rates? Chapter 3 Research Goal 1(b): Investigating the synchronization of the GEO debris population 3.1 Overview of Research Goal Visualizations of the GEO debris population from the slot-relative perspective of the Earth-centered, Earth-fixed framedemonstrate that the collective once-dailymotion of this population is similar to a transverse wave that circulates aroundGEOwith a period of one sidereal day.[1] Thus, daily latitude cycles for uncontrolled objects neighboring one another in longitude are not, in general, out-of-phase. Figure 3.1, which provides the distribution of the 28 February 2014 TLE population in longitude/latitude coordinates at four dierent times during 01March 2014, shows this correlation in the latitude cycle – as time advances, the peak and trough of this transverse “debris wave” shift westward linearly in longitude. As will be shown in this chapter, apparent latitudinal synchronization of this debris pop- ulation is driven by the combination of inclination and clustering in right ascension of the ascending node (RAAN), the second of which results from operators leveraging naturally- occurring luni-solar perturbations to reduce costly north-south station-keeping eort. [50] Upon close inspection of Figure 3.1 (and especially if animating Figure 3.1 using one frameperminute of propagation time), eight objects in the population qualitatively appear to be out-of-sync with the transverse latitude cycle exhibited by the rest of the population. That is, when the local debris populations at the longitudes of these “outliers” are rising in latitude from south to north through the equatorial plane, these objects are descending in latitude fromnorth to south, against the flow of the synchronous patterns of life nominally 51 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Apparent Longitude [deg] 60S 30S 0 30N 60N A pp ar en tL at it ud e [d eg ] D L1 L2 L3 IN (a) Longitude/latitude distribution of debris population (00:00:00 Zulu). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Apparent Longitude [deg] 60S 30S 0 30N 60N A pp ar en tL at it ud e [d eg ] D L1 L2 L3 IN (b) Longitude/latitude distribution of debris population (06:00:00 Zulu). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Apparent Longitude [deg] 60S 30S 0 30N 60N A pp ar en tL at it ud e [d eg ] D L1 L2 L3 IN (c) Longitude/latitude distribution of debris population (12:00:00 Zulu). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Apparent Longitude [deg] 60S 30S 0 30N 60N A pp ar en tL at it ud e [d eg ] D L1 L2 L3 IN (d) Longitude/latitude distribution of debris population (18:00:00 Zulu). Figure 3.1: Longitude/latitude distribution of GEO debris population on 01 March 2014, shown in six-hour intervals to illustrate the “wave-like” behavior of the debris population. 52 observed for derelict motion at GEO. This discrepancy in the latitude cycle indicates that the descending nodes of the outliers are currently located near the ascending nodes of the derelicts clustered in nearby longitude slots. This chapter is thereby borne out of the quest to characterize the apparent anomalousmotion of these eight “outlying” objects, by study- ing whether such asynchronicity in latitude can arise under well-known RAANdynamics at GEO driven by the coupling between luni-solar perturbations and Earth’s oblateness.[3] Specifically, this chapter provides a unifying summary of bottom-up analytical the- orywith top-down observational data to highlight the common linkage between these two dimensions of GEO debris behavior. The relevance of naturally-occurring patterns of life at GEO to current and future operations in the GEO arena is discussed, and long-term im- plications of these tendencies to space situational awareness in this arena are highlighted. 3.2 Inclination and RAAN Variations Induced by Luni-Solar Perturbations The “doubly-averaged”1 dierential equations ofmotion governing inclination i and right ascension of the ascending node (RAAN) ⌦ variations induced by luni-solar pertur- bations prominent at the GEO altitude are—for near-circular orbits only—given by[3,48] di dt = 2X j=1 3 8 j h cos(i) sin(2ij) sin(⌦ ⌦j) + sin(i) sin2(ij) sin(2(⌦ ⌦j)) i (3.1) d⌦ dt = 3 2 J2 ✓ R a ◆2 n cos(i) + 2X j=1 3 16 ✓ j sin(i) ◆h sin(2i) 1 3 cos2(ij) + 2 cos(2i) sin(2ij) cos(⌦ ⌦j) + sin(2i) sin2(ij) cos(2(⌦ ⌦j)) i (3.2) where the summations are performed over the Sun and the Moon, respectively, with j ⌘ n2jRm/n for third-body mass ratios Rm = 1 for the Sun and Rm = 1/82.3 for the Moon.[48] In this formulation, i, ⌦, n, and a denote the inclination, RAAN, mean motion, and semi- major axis of the uncontrolled derelict, respectively, while ij , ⌦j , and nj denote third-body 1 Termed “doubly” averaged because the complete equations of motion for inclination and RAAN have been averaged once over the true anomaly f (a fast variable), and a second time over the argument of perigee ! (a slow variable), such that daily, monthly, and yearly periodic variations in i and⌦ have been removed. [3] 53 inclination with respect to the equatorial plane, third-body RAANwith respect to the line of Aries, and third-body mean motion (i.e., approximately one revolution per year for the Sun and one revolution per sidereal month for theMoon). Note that the coupling between luni-solar perturbations and central body oblateness is incorporated in Equation (3.2) via thewell-known term that describes secular drift in⌦ induced by the J2 zonal harmonic.[38] Equations (3.1)-(3.2) oer insight into the long-term evolution of uncontrolled orbits in the GEO regime. In particular, Equation (3.1) indicates that inclination drift will be zero inasmuch as⌦⌦j = 0 or⌦⌦j = 180, i.e., the line of nodes for the uncontrolled object is aligned with that of the third body.[3] Without routine north-south station-keeping ma- neuvers, Equations (3.1)-(3.2) combine to drive a long-term, cyclical precession of the orbit plane, in which correlated motions of RAAN and inclination over an approximate 53-year period are analogous to the precession and nutation of a gyroscope.[50] Physically, out-of- plane force components acting on the uncontrolled object torque the orbit plane along the line of nodes, resulting in the angularmomentumvector precessing about an intermediate axis displaced approximately 7.4 from Earth’s rotation axis towards the ecliptic pole.[50,52] The long-term, coupled inclination andRAANvariations described in Equations (3.1)- (3.2) are visualizedwith thewell-knownphase portrait in Figure 3.2(a), which provides the clockwise trajectories that uncontrolled objects will traverse in this inclination and RAAN phase space. Objects beginningwith low inclination and aRAANwithinQuadrants I or IV will experience a periodic, egg- or triangular-shaped (Type I) progressionwithin this phase space, while objects initializedwith a RAAN inQuadrants II and III experience a periodic, bell-shaped (Type II) progression.[50] For Type I motion, the node advances and regresses through theminimumandmaximum inclinations of the cycle, respectively, each occurring at⌦ = 0. For Type II motion, the node continually regresses fromminimum tomaximum inclinations – nodal advancement is not experienced along these bell-shapedprogressions. The thick, red line in Figure 3.2(a) highlights the “separatrix-like,” triangular-shaped boundary between Type I/II progressions,[53] informally termed the conga-line trajectory2 54 0 5 10 15 20 !150 !100 !50 0 50 100 150 Inclination !deg" Ri gh tA sc en sio n of As ce nd in gN od e!deg" )[ ] R ig ht )A sc en si on )o f)A sc en di ng )N od e) [d eg ] 180 120 60 :60 :120 :180 ;Conga)Line;) Trajectory Stable)Plane) Equilibrium Quadrant)III Nodal&Regression Inclination&Decrease Quadrant)II Nodal&Regression Inclination&Increase Quadrant)I Inclination&Increase Quadrant)IV Inclination&Decrease 90 :90 (a) Description of phase space trajectories at GEO. 1 10 00 00 00 00 1 50 00 00 00 1 10 00 00 00 02 3!55 4 !5 1 10 00 00 00 0 2 5 5 3 !5 1 50 00 00 00 2 4 !55 2!5 1 50 00 00 00 2 2 !55 5 0 5 10 15 20 !150 !100 !50 0 50 100 150 Inclination !deg" Ri gh tA sc en sio n of As ce nd in gN od e!deg" )[ ] R ig ht )A sc en si on )o f)A sc en di ng )N od e) [d eg ] 180 90 990 9180 Angular)Velocity)(Ω²)+)i²)¹/²)[deg/day] 0.0 05 0.0 09 0.0 16 0.0 30 0.0 54 0.0 99 (b) Angular rate on phase space trajectories. Figure 3.2: Phase space portraits illustrating the doubly-averaged Equations (3.1)-(3.2) for inclination and RAAN variations induced by luni-solar perturbations at the GEO altitude. because operational assets at GEO are often station-kept onto this curve to maintain near- zero inclination over their operational lifespans.[50] GEO operators minimize north-south station-keeping fuel allocation by initializing their assetswith 1-2 inclination and aRAAN withinQuadrant IV (e.g.,⌦ ⇡ 80), such that over the nominal lifespan of these satellites, luni-solar perturbations naturally decrease the inclination to nearly zero before the 53-year cycle begins increasing the inclination at a rate of approximately 0.8 per year.[9] If desired, routine station-keepingmaneuvers can be used to re-initiate this portion of the triangular- shaped progressionwhen the inclination begins to exceed the operational requirement.[50] Thus, maintaining the conga-line in Figure 3.2(a) is popular among GEO operators, as this phase progression best leverages natural perturbations to minimize expensive inclination 2 See References 1 and 54. 55 control burns, whichmake up a sizeable portion of typical station-keeping fuel budgets.[3] Figure 3.2(a) also illustrates the existence of thewell-knownGEO stable plane, which is a fixed point (i˙ = ⌦˙ = 0) of the doubly-averaged Equations (3.1)-(3.2). Objects initialized at this equilibrium configuration with i0 ⇡ 7.4 and ⌦0 ⇡ 0 exhibit dramatically-reduced inclination and RAAN variations during the 53-year cycle – inclination variations are con- fined towithin 1.2 of the initial plane in this setup, in comparison to the 12-15 excursions typical for Type I and II progressions in Figure 3.2(a). [52] Physically, the stable plane equi- librium results from luni-solar perturbations acting against the nodal regression driven by central body J2 oblateness, such that these equal-but-opposite eects cancel each other.[52] The stable plane has important operational implications for GEO satellite collocation, con- junction velocities, and mitigation alternatives, all of which are discussed in Reference 52. Since both Type I and Type II progressions in Figure 3.2(a) have a period of approxi- mately 53 years—even though inclination andRAANvariations are, in general, smaller for Type I cycles than for Type II cycles—Figure 3.2(b) shows the non-uniformity of the angular speed p ⌦˙2 + i˙2 along these progressions. As is predicted analytically with Equation (3.2), rapid nodal advancement occurs at small inclinations, when the luni-solar contributions to Equation (3.2) dominate J2-induced nodal regression.[53] Type II motion exhibits a higher angular rate than “inner” Type Imotion, such that each cyclemaintains the 53-year period. Thus, although smart orbital insertion can leverage the naturally-occurring conga line tra- jectory to reduce the orbit from several degrees of inclination to equatorial, this is themost rapid part of the progression (lasting about three years), requiring re-initiation every time the inclination begins exceeding operational limits (e.g., five times in a 15-year lifespan).[50] 3.3 Current Inclination and RAAN Distribution In light of the analytic theory presented in Section 3.2, it is instructive to consider the distribution of both controlled GEO satellites and the GEO derelict population within the inclination and RAAN phase portrait illustrated in Figure 3.2. Using the 28 February 2014 56 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] D L1 L2 L3 IN 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] C1 C2 (a) Distribution of controlled satellites. 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] 1989 1992 1995 1998 2001 2004 2007 2010 2013 La un ch Ye ar (b) Controlled distribution by launch year. Figure 3.3: Phase space distribution of controlled GEO satellites on 28 February 2014. TLE population described previously in Section 2.2, Figure 3.3 provides the phase portrait distribution for the 400 controlled GEO satellites in this TLE set, 318 of which are class C1 (longitude and inclination control), and 82 of which are class C2 (longitude control only). In particular, Figure 3.3(a) shows this controlled distribution colored by orbit class. As an- ticipated, the majority of class C1 satellites are maintaining near-zero inclination by either (a) executing routine station-keeping burns, or (b) leveraging the conga-line trajectory or a variation thereof for a naturally-occurring inclination decrease, as described in Section 3.2. Conversely, the majority of C2 satellites—those that were not equipped to perform out-of- plane station-keeping, or those that have lost this capability on-orbit—are progressing on or near the intermediate conga-line curve.3 Figure 3.3(b), which illustrates this controlled distribution colored by launch year, confirms that the class C2 satellites furthest along the 3 Recall that the conga-line trajectory and the doubly-averaged Equations (3.1)-(3.2) provide approximate, analytic, first-order descriptions of actual inclination and RAANvariations experienced at the GEO altitude. 57 clockwise, 53-year cycle (i.e., have largest inclinations in Quadrant I) are the oldest objects in the operational population. Thus, given two collocated C2 objects in this synchronized configuration, the younger objectwill lag behind the older object in the daily latitude cycle, since the node of the younger satellite has not regressed as far as that of the older satellite.4 Ultimately, the dichotomy between fully-controlled C1 satellites and partially-controlled C2 satellites in Figure 3.3 demonstrates that while out-of-plane station-keeping is used to maintain near-equatorial orbits for C1 satellites, coupled inclination and RAANvariations exhibited by C2 objects follow predictable progressions described by Equations (3.1)-(3.2). Figure 3.4 provides the phase portrait distribution for the 745 uncontrolled GEO ob- jects extracted from the 28 February 2014 TLE set (recall Figure 2.1). Prominent clustering in debris objects is observed on or near the conga-line trajectory, especially around the apex of this triangular-shaped progression, where the inclination achieves amaximum value of approximately 15 over the 53-year cycle. Reference 51 shows that in 2002 (i.e., 39 elapsed years since the first GEO utilization in 1963), the majority of uncontrolled objects at GEO had not yet crossed the apex of this systematic progression. Now, 12 years later, Figure 3.4 shows that a significant portion of this uncontrolled population has already crossed—or is currently crossing—into the second half of the conga line cycle (fromQuadrant I to Quad- rant IV), in which the RAAN continues to regress while the inclination begins decreasing. Since 51 years have now elapsed since the first GEOutilization, the first GEOdebris objects generated in the 1960s—including theworld’s firstGEOcommunications satellite, Syncom 2—are soon to complete a full Type I revolution in this inclination and RAANphase space. Figure 3.4(a), which shows the uncontrolleddebris distribution colored by orbit class, demonstrates that factors such as longitude of abandonment or orbit class—whether drift- ing or librating about one or both of the gravitational wells—do not aect the phase space progression that is followed by a debris object.5 It is important to emphasize that inclina- 4 Latitudinal lag between collocated satellites of diering on-orbit age has been confirmed in animations and numerical simulations performed by Darren McKnight and Frank Di Pentino of Integrity Applications. 5 Although orbit class does not aect the progression directly, the semi-major axis, which is critical in de- 58 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Apparent Longitude [deg] 60S 30S 0 30N 60N A pp ar en tL at it ud e [d eg ] D L1 L2 L3 IN 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] 76023A 76023B 76023F 76023J 93015A 07003A 06048A 13062B (a) Distribution of uncontrolled objects. 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Ye ar A ba nd on ed (b) Uncontrolled distribution by abandon year. Figure 3.4: Phase space distribution of uncontrolled GEO derelicts on 01 March 2014. tion and RAAN at the moment the object is abandoned dictate whether a Type I or Type II progression will result in the following cycle. Since luni-solar gravitational perturbations are conservative, the resulting cycles governed byEquations (3.1)-(3.2) are periodic, anddo not admit natural transition from Type I to Type II behavior, or vice-versa, under luni-solar perturbations alone. The apparent “outlying” objects labeled in Figure 3.4(a), discussed in Section 3.5, could result from one or more of the following possibilities or artificial means: (1) When operational, these satelliteswere not station-kept to the conga line trajectory formission-related reasons, or (for launch debris)were not inserted onto this cycle; (2) End-of-life disposal was attempted, and themaneuver sequence—purposefully or inadvertently—altered the inclination/RAAN while raising the semi-major axis; terminingwhether an object will drift or librate once uncontrolled, is a required parameter of Equation (3.2), and is assumed to be constant when propagating this dierential equation. This is a valid assumption given the magnitude of naturally-occurring semi-major axis variations at GEO, which are insignificant relative to the radius of the GEO ring (e.g., a 60 libration amplitude results from±30 km of semi-major axis variation). 59 (3) SinceGauss’s variational equations show that any out-of-plane accelerationwill si- multaneously change both the inclination and RAAN,[55] an out-of-plane collision with either environmental or artificial debris occurred, or the object has vented (or is currently venting) propellant externally, with a component out of the orbit plane. Although these possibilities are not exhaustive, they enforce the fact that initial conditions in inclination and RAAN space dictate whether a Type I or Type II progression will result. Under the doubly-averaged Equations (3.1)-(3.2), these twomodes are mutually exclusive. To illustrate howobject age on-orbit can be approximated by the phase portrait of Fig- ure 3.2, Figure 3.4(b) shows the uncontrolled debris distribution colored by estimated year of abandonment. For upper stages and other launch ormission-related debris (e.g., apogee kick motors), the year of abandonment is the launch year; for derelict payloads in the pop- ulation, the year of abandonment is approximated via a linear lifespanmodel derived from historical lifespan data, which reflects a linear increase from six years on average in 1977 to 13 years on average in 2010.[1] Similar to Figure 3.3(b), Figure 3.4(b) illustrates that derelicts further along the triangular-shaped progression are, in general, older than those near the “beginning” of the progression close to i ⇡ 0 and ⌦ ⇡ 90. Objects abandoned in the late 1980s have achieved their maximum inclination of the cycle at the apex of the progression, while objects abandoned in the 1960s (approximately 26-27 years earlier) are nowmoving through theminimum inclination of the cycle towards a full revolution in the phase space. Thus, assuming that a particular payload of interest—operational or defunct—was station- kept onto the popular conga-line trajectory, the phase portrait of Figure 3.4(b) can be lever- aged in tandemwith the angular rate information of Figure 3.2(b) to estimate either when this object was abandoned, or when this object lost north-south station-keeping capability. 60 3.4 Evolution of Inclination and RAAN Distribution In addition to studying the current distribution of derelict GEO objects in inclination andRAAN space, it is useful to predict the evolution of this distribution under the doubly- averaged Equations (3.1)-(3.2). Figure 3.5 shows this phase space distribution on 01March 2014, as compared to the approximate distributions predicted 20, 40, and 60 years after this date. As noted in Section 3.3, a significant proportion of the present-day debris population is crossing into the second half of the conga line progression, producing strong clustering in RAAN about the line of Aries (⌦ = 0). Deterministic, predictable clustering in RAAN has been widely leveraged for space-based space surveillance (SBSS), sensor tasking, and space situational awareness activities at GEO.[51] By pointing ground- or space-based sen- sors towards the well-known GEO “pinch points,” observational coverage of the resident space object population at GEO can bemaximized, and the probability of detecting “new” objects, for purposes of catalog maintenance, is improved. Reference 50 confirms that the productivity of the Space-Based Visible (SBV) sensor on the Midcourse Space Experiment (MSX) satellite improved 20-30% by taking advantage of this natural clustering in RAAN. However, as Figure 3.5 illustrates, the interplay between location in the phase space and the location-specific angular rates in Figure 3.2(b) gives rise to the periodic dispersion and eventual re-focusing in the RAANdistribution. As the Type I derelicts circulate in this phase space, the spread of this distribution expands from reflecting Quadrant I clustering on [0,90] to reflecting full, Quadrant I/IV clustering on [-90,90]. For purposes of SBSS and space situational awareness activities, this dispersion of objectswithin the phase space indicates that the GEO pinch points are spreading out, thus diminishing in usefulness for sensor tasking. Therefore, in order to establish a leak-proof search fence, a more substan- tial search region in right ascensionmust be implemented tomaintain current productivity levels achieved by SBV and other ground- or space-based sensors. Beneficially, Figure 3.5 shows that dispersion in RAAN does not continue indefinitely; rather, the distribution be- 61 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] (a) Distribution on 01 March 2014. 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] (b) Predicted distribution in 20 years. 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] (c) Predicted distribution in 40 years. 0 5 10 15 20 Inclination [deg] 180 90 0 90 180 R ig ht A sc en si on of A sc en di ng N od e [d eg ] (d) Predicted distribution in 60 years. Figure 3.5: Evolution of derelict motion at GEO in inclination/RAAN space over 60 years, predicted using Equations (3.1)-(3.2) for Sun/Moon-driven inclination/RAANvariations. 62 gins to re-focus as themajority of the population passes through theminimum-inclination portion of the 53-year cycle to complete one Type I revolution, afterwhich nodal regression rates slow. The spread in this RAAN distribution thus “ebbs and flows” as luni-solar per- turbations andEarth oblateness propel the derelict population around the conga line cycle. 3.5 Investigating the Unsynchronized Objects In light of the bottom-up, analytic theory presented in Section 3.2, and the top-down, observational portrait of the current inclination and RAAN distribution in Section 3.3, the eight derelict objects identified as being asynchronous to the transversewave-like behavior shown in Figure 3.1 are now examined. Animations of the daily latitude cycle of the GEO derelict population indicate that these “outliers” traverse latitude cycles either partially or fully out-of-phase with those exhibited by nearby derelicts in neighboring longitude slots. As noted earlier, two latitude cycles becomemore out-of-phase the farther the correspond- ing RAANangles are displaced from one another, e.g., two objects with RAANangles that are 180 apart will be completely unsynchronized in the latitude cycle. Two asynchronous latitude cycles are best visualized in right ascension and declination coordinates, in which these once-daily cycles appear as out-of-phase sinewaves instead of “figure-8” orbits. [50,51] For purposes of flight safety and on-orbit anomaly correlation, it is critical to under- standwhy the RAAN angles for these eight outliers—listed in Table 3.1—are not clustered with those of the rest of the large-scale, unclassifiedGEOdebris population extracted from the 28 February 2014 TLE set. Reference 1 highlights that the equatorial crossingwindows for the synchronized debris population are predictable and dependent upon time of day in addition to time of year. These alternating, six-hour intervals of increasing and decreasing flux relative to any given longitude slot are important for risk assessment and prediction at GEO, since conjunction events are most likely to occur in the out-of-plane direction when the longitude slot is passing through either the ascending or descending node pinch point. Derelicts that are asynchronous to this pattern of life present a special hazard in that these 63 Table 3.1: GEO debris objects exhibiting unsynchronized latitude cycles on 01March 2014. Common Name COSPAR Class Common Name COSPAR Class LES-8 76023A L2 USA 98 (UFO F1) 93015A D LES-9 76023B L2 Sinosat-2 (Xinnuo-2) 06048A D Titan 3C Transtage R/B 76023F D Beidou-1D (Beidou-4) 07003A D LES Operational Debris 76023J D Proton-M Briz-M R/B 13062B D objects cross the equatorial plane when north-south flux levels are at a daily minimum.6 As the current inclination and RAAN distribution in Figure 3.4 shows, the eight un- synchronized derelicts in Table 3.1 are currently positioned in Quadrants II and III, away from the significant RAAN clustering occurring at the boundary betweenQuadrants I and IV. These objects are not exhibiting oscillatory Type I trajectories in the phase space; rather, they are following bell-shaped Type II progressions, according to Equations (3.1)-(3.2). In contrast to the Type I cycles, which exhibit long durations of nodal regression followed by shorter periods of nodal advancement near the minimum inclination of the cycle, Type II cycles exhibit continual nodal regression over the entirety of the 53-year cycle. This critical dierence is illustrated in Figure 3.6, which provides the predicted RAAN trajectories over a 60-year interval for various synchronized and unsynchronized objects in the 28 February 2014 TLE set. Figure 3.6(a) provides the RAAN trajectories for eight initially synchronized objects, i.e., with RAAN angles within one standard deviation of the distribution mean.7 The RAANangles for these Type I derelicts never exceed two standard deviations from the mean at any point during the 60-year prediction period. Thus, from the perspective of lon- gitude/latitude coordinates, these objects are “strongly” synchronizedwith the transverse wave-like behavior in Figure 3.1, andmaintain this synchronization over the 53-year cycle. Conversely, Figure 3.6(b) provides predicted RAAN trajectories for the eight unsyn- chronized objects in Table 3.1, which have initial RAAN angles greater than two standard 6 As an analogy, consider the hazard to one’s vehicle when driving on the highway. Although risk levels during rush hour can be reasonably characterized in that they spike during predictable time frames, the risk attributed to a rogue driver speeding southbound down the northbound side is challenging to characterize. 7 The expanding and contracting 1/2/3 swaths in Figure 3.6 reflect the future distributions in Figure 3.5. 64 0 10 20 30 40 50 60 Elapsed Years from 01 March 2014 180 120 60 0 60 120 180 D ev ia ti on fr om M ea n R A A N [d eg ] 78035A (L1) 84041A (L1) 88081A (L2) 93058B (L2) 94067D (L3) 00029A (L3) 87022F (D) 08006C (D) (a) Projected 60-year RAAN profiles for initially synchronized objects. 0 10 20 30 40 50 60 Elapsed Years from 01 March 2014 180 120 60 0 60 120 180 D ev ia ti on fr om M ea n R A A N [d eg ] 76023A (L2) 76023B (L2) 76023F (D) 76023J (D) 93015A (D) 06048A (D) 07003A (D) 13062B (D) (b) Projected 60-year RAAN profiles for initially unsynchronized objects. Figure 3.6: RAANhistories for initially synchronized and initially unsynchronized objects near GEO on 28 February 2014, predicted using the doubly-averaged Equations (3.1)-(3.2). Shaded swaths indicate ±1, ±2, and ±3 standard deviations from the distribution mean. deviations away from the distributionmean. Continuous nodal regression of these Type II trajectories drives theRAANangles of these outliers—and thus their daily latitude cycles— from being unsynchronized on 01March 2014 into strong synchronization in an estimated 25-35 years. After approaching and achieving the distribution mean, however, the RAAN angles for these Type II derelicts continue to regress, until these objects once again become unsynchronized in 45-55 years, during which the node rapidly regresses as the minimum inclination of the cycle is achieved. The patterns of life for Type II objects thus dier from 65 those of Type I objects in that although Type I trajectories admit synchronized latitude cy- cles over the entirety of the 53-year progression, Type II trajectories result in latitude cycles that shift from unsynchronized to synchronized—and back—during a single progression. The apparent anomalous behavior observed in Figure 3.1 can therefore be explained by the Type II progressions that these eight derelicts are traversing in inclination/RAAN space. Again, since natural transitions from a Type I to Type II cycle (or vice versa) will not occur under conservative gravitational perturbations alone, the three possibilities highlighted in Section 3.3 apply if assessing how or why these objects are following Type II progressions. 3.6 Conclusions from Research Goal In this chapter, the wave-like synchronization exhibited by the derelict population at GEO is explained using bottom-up, analytical theory that describes inclination andRAAN variations driven by luni-solar perturbations and Earth’s oblateness over a 53-year period. Thiswell-known theory is combinedwith top-down, operational observation data to paint a portrait of the current inclination and RAAN distribution by controlled or uncontrolled class and age on-orbit, andhow this distributionwill evolve under the approximate doubly- averaged Equations (3.1)-(3.2). Lastly, derelictswith partially or fully out-of-phase latitude cycles are explained not by apparent anomalous motion, but rather in terms of the dier- ence between egg-shaped, Type I and bell-shaped, Type II progressions in the phase space. Several operational implications arising from the synchronizeddebrismotion atGEO are highlighted in this chapter for purposes of flight safety, anomaly correlation, and space situational awareness activities in this regime. Naturally-occurringType IImotion—which arises from the combination of inclination, RAAN, and semi-major axis when the satellite is abandoned—explains not onlywhy younger objects lag behind older objects in the daily latitude cycle, but also shows how the latitude cycle of a Type II object transitions into and out of synchronization with the Type I population over the course of the 53-year cycle. For sensor tasking, predictions of the combined Type I/II motion over a projected 60-year time 66 frame highlight that the GEOpinch points resulting from tight clustering in RAAN are be- ginning to spread out, achievingmaximum3 dispersion in approximately 40 years. Thus, both ground and space-based surveillance systemswill need an increasingly wider search fence in right ascension tomaintain current levels of sensor productivity in the near future. 3.7 Continuing Research Questions For continuing research in this area, the following research questions are of interest: (1) The doubly-averaged Equations (3.1)-(3.2), which are valid for near-circular orbits only, describe long-period inclination and RAAN variations driven by a combina- tion of luni-solar gravitation and central body oblateness. What does the evolution of the phase space distribution in Figure 3.5 look like under higher-fidelity forcing, particularly if the non-conservative eect of solar radiation pressure is included? (2) Figure 3.5 highlights the existence of multiple trajectories similar to the conga line that are popular amongst GEOoperators. Howmany individual “conga-line-like” Type I cycles exist? Do they arise out of a need to maintain a minimum inclination for latitudinal coverage requirements, or do they arise from orbit insertion errors? (3) Can the periodic expansion and contraction of the RAAN distribution of the GEO debris population—as illustrated in Figures 3.5-3.6—be leveraged by autonomous ground and space-based sensors, in order tomaintain desired productivity levels? Chapter 4 Research Goal 2(a): Forecasting longitude-dependent congestion at GEO arising from nominal launch trac 4.1 Overview of Research Goal In addition to assessing the longitude-dependent congestion arising from the present- day, large-scale debris population at GEO, it is also imperative to investigate the influence of critical population growthmechanisms, in particular, nominal launch trac to the GEO regime. From the perspective of orbital debris congestion atGEO, the “no future launches” scenario typically considered in various long-termdebris growth analyses ismost optimal, yet highly unrealistic for the profitable real estate at GEO. Currently, the GEO catalog is in- creasing by approximately 30 new operational satellites per year,[6,11] and aswill be shown, historical launch data indicates that this launch trac rate will only continue increasing in the years to come. Although widespread use of integrated apogee kick motors has caused the growth of upper stages in GEO orbit to begin leveling-o, this reduction in GEO object growth has been compensated for by a rapidly-intesifying number of payload insertions.[6] Population augmentation in the GEO environment has been researched briefly in the literature,[6,11] albeit, these studies provide debris fluxes averaged over longitude, altitude, and time, and thus fail to address which longitude slots are themost prone to proliferating debris populations resulting froma combination of routine launch trac, operational lifes- pan, and compliance tomitigation guidelines. To the author’s knowledge, a representative launch trac model for the GEO regime—one that characterizes not only the frequency of 68 launches to GEO, but the operational orbits that these satellites are inserted into—has not been presented in the literature. This thesis addresses this critical void in the literature by presenting a non-homogeneous method for modeling growth of the GEO catalogue, thus facilitating prediction of high-risk longitude slots for operational GEO assets over the next 50 years, and serving to further enforce that averaging debris flux is becoming increasingly misrepresentative for GEO. The objective of this research goal is to illustrateworst-case de- bris congestion at GEOunder representative launch trac and a nominal operational lifes- pan, and demonstrate that compliance to mitigation guidelines at end-of-life can serve to attenuate localized congestion – especially in the neighborhoods of the gravitational wells. It is important to emphasize that nominal launch trac is the only input to the debris growthmodel used in this chapter. On-orbit fragmentation events (investigated separately in Chapter 5), ejection of solid rocketmotor slag, shedding ofmulti-layered insulation, and other growthmechanics incorporated inReference 6, for example, are not considered here. 4.2 Operational GEO Orbit Model During long-term congestion forecasting, the RSOpopulation at GEOmust bemean- ingfully augmented to simulate typical launch trac in this regime. Realistic population augmentation requires an operational GEOorbitmodel to quantifywhere newGEOassets are typically positioned at insertion into their designated longitude slots – such a model serves to generate the initial conditions for new operational satellites created during long- term forecasting (the GEO launch instantiation procedure is highlighted in Section 4.4). To construct a first-order operational orbit model, data from the 01 January 2013 TLE set, the Space-TrackGeosynchronous Report,1 and the SatBeamsdatabase2 are compiled, giving semi- major axis, eccentricity, inclination, and geocentric longitude information for 768 past and presentGEO satellites—including all unclassified launches toGEOas of 01 January 2013— 1 Space-Track Geosynchronous Report available at: https://www.space-track.org/perl/geo_report.pl. 2 The SatBeams database provides GEO satellite longitude and launch year information, and is available at: http://www.satbeams.com/satellites. 69 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 60 120 180 240 300 3600 5 10 15 20 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 0 0.1 0.2 0.3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 421755%%%%%%%%% %%%%%%% %%%%%%%% %%%%%%%% N um be r%o f%G EO %S at el lit es Pr ob ab ili ty %D en si ty %F un ct io n Semi@Major%Axis%[0.5%km%bins] 1960%1969 1970%1979 1980%1989 1990%1999 2000%2009 2010%Now Planned 0 0.1 0.2 .3 0.4 4.2155 4.216 4.2165 4.217 4.2175 x 104 0 25 50 75 100 125 150 42155 42160 42165 42170 42175!!42155!!!!!!! !!!!!!! !!!!!!!! !!!!!!!42175 N um be r!o f!G EO !S at el lit es Pr ob ab ili ty !D en si ty !F un ct io n Semi@Major!Axis![0.5!km!bins] (a) Semi-major axis PDF for GEO orbit. 0 400 800 1200 1600 0 1 2 3 4 5 x 10−3 0 25 50 75 100 125 150 0.0010 0.002 0.003 0.004 0.005 N um be r'o f'G EO 'S at el lit es Pr ob ab ili ty 'D en si ty 'F un ct io n '''''''' . ''''' . ''''' ''''' ''''' Eccentricity'[10'''bins]B4 (b) Eccentricity PDF for GEO orbit. 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 10 10 20 30 40 50 60 70 80 Pr ob ab ili ty *D en si ty *F un ct io n N um be r*o f*G EO *S at el lit es Inclination*[0.02**bins] 0***********0.2**********0.4**********0.6**********0.8***********1* (c) Inclination PDF for GEO orbit. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 60 120 180 240 300 3600 5 10 15 20 0"""""""""60"""""""120"""""180""""""240""""""300"""""360" Longitude"[5""bins] N um be r"o f"G EO "S at el lit es Pr ob ab ili ty "D en si ty "F un ct io n " " " """ """ 0 0 0 (d) Longitude PDF for GEO orbit. Figure 4.1: Parameter distributions for representative sampling of operational GEO orbits. and 94 satellites planned through the year 2020. Employing these data, stacked histograms are constructed for the semi-major axis, eccentricity, inclination, and geocentric longitude, and appropriate probability density functions are fit to these histograms to construct rep- resentative distributions fromwhich the orbital elements of a new controlled satellite may be sampled.3 Figure 4.1 provides these parameter histograms (stacked by launch decade) and their associated probability density functions, summarized for each element below: 3 The assumption underlying this methodology is that the semi-major axis, eccentricity, inclination, and longitude of the controlled satellite are considered independent, uncorrelated parameters during sampling. 70 • Semi-major axis (a). Normal distribution with mean µ = 42164.8 km and standard deviation of = 1.0 km. The probability density function (PDF) and cumulative distribution function (CDF) for this distribution are given by (for 1 < x <1): fN (x;µ, ) = 1 p 2⇡ exp " 1 2 ✓ x µ ◆2# (4.1) FN (x;µ, ) = 1 2  1 + erf ✓ x µ p 2 ◆ (4.2) • Eccentricity (e). Half-normal distribution derived from a normal distribution with zeromean and = 5.0⇥104, for which the PDF and CDF are given by (for x 0): fN/2(x; ) = p 2 p ⇡ exp ✓ x 2 22 ◆ (4.3) FN/2(x; ) = erf ✓ x p 2 ◆ (4.4) • Inclination (i). Half-normal distribution derived from a normal distribution with zero mean and = 0.08. • Longitude (). Gaussianmixture of twowrappednormal distributionswith (µ1, 1) = (55, 65) and (µ2, 2) = (260, 25), mixed as follows:4 ⇠ 0.75fW(µ1, 1) + 0.25fW(µ2, 2) (4.5) The PDF and CDF for the wrapped normal distribution are given by (0  x  2⇡): fW(x;µ, ) = 1 p 2⇡ 1X k=1 exp  (x µ+ 2⇡k) 2 22 (4.6) FW(x;µ, ) = 1 2 1X k=1 " erf ✓ x µ+ 2⇡k p 2 ◆ erf ✓µ+ 2⇡k p 2 ◆# (4.7) For computer implementations, sampling from the distributions is performed by drawing a pseudo-random number on the interval [0, 1] that represents the cumulative probability 4 A Gaussian mixture is selected to simulate the bimodality of the longitude histogram in Figure 4.1(d), i.e., high concentrations of operational satellites above Europe/Asia andNorth America, but minimal assets above the Atlantic and Pacific Oceans. 71 sampled in the desired distribution. The appropriate CDF is then back-solved for the value of x that would yield this cumulative probability. Back-solving the wrapped normal CDF is performed by incrementally stepping through the interval [0, 2⇡] in Equation (4.7) until the sampled cumulative probability is achieved to tolerance (it is sucient to iterate k from k = 100 . . . 100). Right ascension of the ascending node⌦ is selecteduniformly on the interval [0, 360], and the true anomaly is initialized as 0, i.e., new satellites are inserted at periapsis. As a result, the argument of perigee ! is a dependent parameter that must satisfy the relation ! = ↵G + ⌦ (4.8) where ↵G denotes the right ascension of Greenwich, computed at the epoch of insertion. After the initial orbit elements have been sampled in thismanner, theseKeplerian elements are converted into aCartesian state and assigned to the created satellite. Thismethodology ensures that new controlled satellites created during long-term forecasting are initialized with orbit elements that are highly representative of the operational GEO orbits harnessed since the first GEO utilization in 1963. 4.3 Launch Instantiation Procedure Launches to operational GEO orbits are simulated with the probabilistic, open-loop event instantiation method implemented in the European Space Agency’s DELTA (Debris Environment Long-TermAnalysis) tool. [11] Following Reference 11, the probability Pj of j launches occurring over a given analysis interval ismodeledwith the Poisson distribution: Pj = c j j! exp(c) (4.9) where c is the average number of launches occurring over the analysis interval, computed as c = (average annual launch rate to orbit regime [years1])⇥ (length of analysis interval [years]). [11] Assuming the “business-as-usual”GEO launch rate of 30 satellites per year[6,11] 72 and using one-day analysis intervals, c ⇡ 0.082 as a first approximation for typical launch trac at GEO. For computer implementations, the probabilities Pj for j = 0, 1, . . . , k are first determined until a threshold ✏ is achieved, such thatPk+1  ✏ (for this study, ✏ ⌘ 106). The resultant probabilities are then normalized such that their sum is unity: Pˆj = PjPk i=0Pi =) kX j=0 Pˆj = 1 (4.10) A pseudo-random number ⇣ˆ on [0, 1] is drawn, and the number of launch occurrences in the analysis interval is thus determined by the largest j for which the sum of the normal- ized probabilities Pˆj is still less than ⇣ˆ , i.e., jX i=0 Pˆi  ⇣ˆ < j+1X i=0 Pˆi (4.11) Therefore, following each day of propagation, a pseudo-random number ⇣ˆ is drawn, and the number of launches during the past day is determined with Equation (4.11). If a suc- cessful launch has been initiated, a controlled GEO satellite (C1) is created, and the initial orbit elements are sampled from the representative distributions presented in Section 4.2. For the purposes of this congestion study, the satellite is injected into its orbit at midnight, and is assigned a nominal lifetime of 15 years.5 Controlled satellites that reach end-of-life (EOL) during propagation are abandonedwithout any attempt at mitigation compliance – these abandoned satellites are added to the uncontrolled RSOpopulation for propagation. Although the business-as-usual launch rate of 30 new GEO satellites per year oers a suitable first approximation for long-term congestion forecasting in this arena, it is more instructive to consider themore realistic case of an increasing launch rate to operational or- bits at GEO. Figure 4.2 illustrates the number of launches to GEO occurring each year since 1963, using data extracted from the Space-Track Geosynchronous Report. Observing that the 5 The baseline controlled population in the 01 January 2013 TLE set is also assigned this nominal lifetime (COSPAR designation gives launch year). References 1 and 6 illustrate that the average design lifetime for GEO satellites has continued to increase since 1964, but extended lifetimes are not simulated in this analysis. 73 1960 1970 1980 1990 2000 20100 5 10 15 20 25 30 35 40 NL = 0.62tL 1218 %%%%%%%% %%%%%%% %%%%%%%% %%%%%%% 0%%%%%%%% Launch%Year 40 35 30 25 20 15 10 %%5 %%0 N um be r%o f%G EO %L au nc he s Mean%Launch%Rate: Figure 4.2: Annual launches to GEO since 1963. number of launchesNL is approximately proportional to the launch year tL, linear regres- sion provides the following linearly-increasing GEO launch rate model: NL = 0.62tL 1218 [launches/year] (4.12) To implement this launch rate model, the parameter c is updated after every year of prop- agation, and the probabilities in Equations (4.9)-(4.11) are recomputed to reflect a linearly- increasing launch rate during probabilistic launch instantiation. 4.4 “Business as Usual” Launch Trac without Mitigation Figure 4.3 illustrates the forecasted localized congestion for a 50-year prediction span beginning on 01 January 2013, assuming this linearly-increasingmodel in strict absence of mitigation measures. Figure 4.3(a) illustrates the absolute number of near-miss events per day for the 50 km GEO torus, while Figure 4.3(b) quantifies increase in the mean number of near-miss events per day—averaged over each year of propagation—over the idealized “no future launches” scenario, inwhich all launch trac toGEO ceases. Similar results are illustrated in Figure 4.3 for the 100 km and 300 kmminor radius cases. As anticipated, the 74 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 1 2 3 4 5 6 7 8 9 10 N ea r- M is s E v en ts M o n th / Y ea r Cell Longitude [deg]Longitude*[deg] M on th /Y ea r N ea r> M is s* Ev en ts *[5 0* km ] * * * * *1 *1 * * * * * *3 * 0 0 0 0 0 0 (a) Near-miss events per day (50 km torus). 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 5 10 15 20 25 30 35 40 M o n th / Y ea r Cell Longitude [deg] In cr ea se in M ea n E v en ts / D ay M on th /Y ea r Longitude0[deg] 0 0 0 0 01 01 0 0 0 0 0 03 0 0 0 0 0 0 0 I nc re as e0 in 0M ea n0 Ev en ts 0p er 0D ay (b) Increase in near-miss events at 50 km. 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 1 2 3 4 5 6 7 8 9 10 N ea r- M is s E v en ts M o n th / Y ea r Cell Longitude [deg]Longitude*[deg] * * * * *1 *1 * * * * * *3 * N ea r: M is s* Ev en ts *[1 00 *k m ] M on th /Y ea r 0 0 0 0 0 0 (c) Near-miss events per day (100 km torus). 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 5 10 15 20 25 30 35 40 M o n th / Y ea r Cell Longitude [deg] In cr ea se in M ea n E v en ts / D ay M on th /Y ea r Longitude0[deg] 0 030 06 090 0120 0150 018 021 024 027 0300 0330 0360 01/ 01/ 01/ 01/ 01/ 01/ I nc re as e0 in 0M ea n0 Ev en ts 0p er 0D ay (d) Increase in near-miss events at 100 km. 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 1 2 3 4 5 6 7 8 9 10 N ea r- M is s E v en ts M o n th / Y ea r Cell Longitude [deg]Longitude*[deg] * * * * *1 *1 * * * * * *3 * N ea r: M is s* Ev en ts *[3 00 *k m ] M on th /Y ea r 0 0 0 0 0 0 (e) Near-miss events per day (300 km torus). 0 30 60 90 120 150 180 210 240 270 300 330 36001/13 01/18 01/23 01/28 01/33 01/38 01/43 01/48 01/53 01/58 01/63 0 5 10 15 20 25 30 35 40 M o n th / Y ea r Cell Longitude [deg] In cr ea se in M ea n E v en ts / D ay M on th /Y ea r Longitude0[deg] 0 030 06 090 0120 0150 018 021 024 027 0300 0330 0360 0 0 0 0 0 0 I nc re as e0 in 0M ea n0 Ev en ts 0p er 0D ay (f) Increase in near-miss events at 300 km. Figure 4.3: Localized congestion forecasting (50 years) using the linearly-increasing launch tracmodel, shown as the number of near-misses per day and increase in number of near- miss events over the idealized “no future launches” scenario, i.e., without augmentation. intensity of the resulting debris weather trends is amplified as the minor radius increases, especially the localized congestion in the vicinity of the Eastern andWestern gravitational wells. Congestion in these critical regions dramatically expands as propagation time pro- gresses, a consequence of the simulated abandonment of controlled assets achieving EOL. 75 This is a particularly troublesome notion, as operational GEO satellites are typically inserted into longitude slots near the congested gravitationalwells: studying Figure 4.1(d), it is evident that the two modes of the bimodal PDF selected for the geocentric longitude distribution are locateddirectly over these two regions ofmaximumcongestion, illustrated in Figure 4.3. At a distance of 50 km, Figure 4.3(a) predicts that assets in 30 longitudewin- dows centered on each gravitational well could experience upwards of 10 near-miss events daily by 2063 in the absence of mitigationmeasures. This is a striking outcome that begins to corroborate the hypothesis that as the debris population in the GEO regime continues to rise unchecked, the amount of propellant required for routine station-keeping will rise in tandem, to account for the increasing frequency of conjunction events and potential colli- sion avoidance maneuvers. Further, this result has significant implications for satellite in- surance in high-risk longitude slots, and rising costs associatedwith determiningwhether or not the execution of a dedicated collision avoidance maneuver is even necessary during a conjunction event. 4.5 “Business as Usual” Launch Trac with Mitigation The congestion forecasting performed thus far in this thesis has not addressed the in- fluence of properly-executed mitigation measures for operational satellites reaching end- of-life. Typical mitigation for the GEO regime incorporates re-orbiting to “graveyard” dis- posal orbits at perigee altitudes above theGEO ring, factoring the protectedGEO zone and area-to-mass-ratio of the satellite into the minimum altitude calculation,[7,10,20] per the re- orbiting guidelines of the Inter-Agency Space Debris Coordination Committee (IADC).[22] Although onemight speculate that adherence to these IADC guidelines could increase the risk of derelict collision in the IADC debris orbit that could accumulate above GEO under these guidelines, Reference 23 indicates that the variable amount of residual propellant at end-of-life has driven thewide variety of disposal orbits achieved in practice, such that the creation of a super-synchronous “debris ring” is unlikely. To emphasize the importance of 76 Longit ude [d eg] M ea n E v en ts / D ay 202 3 203 3 204 3 205 3 206 3 90 80 70 60 10 20 30 40 50 60 0 0% 30% 50% 75% 100% N ea r% M is s) Ev en ts )p er )D ay 20 2033204 20532063 9 o gitude )[deg] (a) Eect of mitigation on Eastern well congestion. Longit ude [d eg] M ea n E v en ts / D ay 202 3 203 3 204 3 205 3 206 3 270 260 250 240 10 20 30 40 50 60 0 0% 30% 50% 75% 100% N ea r% M is s) Ev en ts )p er )D ay 20 2033204 205 206 L t )[ eg] (b) Eect ofmitigation onWesternwell congestion. Figure 4.4: Eect of perfect mitigation—implemented at various levels of compliance—on long-term, localized congestion in the 30 vicinity of the gravitational wells (300 km torus). mitigation measures for GEO satellites, this study assumes perfectly-executed mitigation, in which a specified proportion of operational satellites achieving end-of-life are success- fully re-orbited to super-synchronous, circular orbits at altitudes that do not interfere with operational assets stationed in the GEO ring during the forecasting periods considered.6 Figure 4.4 illustrates the influence of mitigation on localized congestion in the vicin- ity of the Eastern andWestern stable points, for 0% (equivalent to the “business-as-usual” scenario in Figure 4.3), 30%, 50%, 75%, and 100% compliance (equivalent to the “no future launches” scenario), for a torus minor radius of 300 km. Figure 4.4 portrays how the year- averaged number of near-miss events per day at 300 km increases over this 50-year period around the gravitationalwells for each of thesemitigation scenarios. As themitigation rate increases, the congestion peaks centered around these gravitational wells successively di- minish to the level of the idealized “no future launches” scenario. It is instructive to recall that the case of 30% compliance is the most representative of the rate actually carried out in the GEO regime during the 1997-2003 time frame.[7,11] Of the 103 satellites that achieved end-of-life in this period, 34 were successfully re-orbited to IADC-compliant disposal or- 6 Reference 29 provides a statistical assessment of the perigee altitude distributions resulting from global adherence to the IADC guidelines, allowing for uncertainties in fuel measurement andmaneuver reliability. 77 bits, 35 attempted re-orbiting, but resulted in orbits with insucient perigee altitudes, and 34 were abandoned into libration orbits around the gravitational wells. [7] Low compliance may be attributed to (a) the significant propellant cost formeeting the IADC re-orbit guide- lines, (b) the diculties of estimating on-board propellant, and translating the re-orbitv into required propellant mass,[11] or (c) older satellites that were designed and built before establishment of the IADC guidelines in 1997, for which execution of the re-orbiting burns was not often achievable given, for example, residual propellant and thruster capability.[23] Recently, Reference 23 praises the strong support of theGEOoperator community for preservation of the GEO ring. Of the 160 operational GEO satellites that achieved end-of- life in the period 2001-2010, approximately 80% successfully re-orbited to disposal orbits. Of these satellites, 70% were transferred to orbits at least 200 km above GEO, and almost 50% achieved at least 300 km above GEO, in full compliance with the IADC guidelines.[23] Even with an optimistic mitigation rate of 80%, however, Figure 4.4 predicts that the mean number of near-miss events per day at 300 km in the vicinity of the two gravitational wells will double by the year 2063. Since on-orbit collisions and fragmentation events have not yet been considered, this observed factor of two increase is a lower bound of true conges- tion increase likely to occur in theGEOenvironment. Although recent re-orbiting statistics are promising in that they suggest increased adherence tomitigation guidelines—andulti- mately growing international desire to safeguard the GEO ring—Figure 4.4 indicates that mitigation alone is not enough to stabilize increasing debris congestion at GEO under cur- rent re-orbit success rates. To stabilize projected debris growth at GEO, stricter mitigation measuresmust be combinedwith global compliance andproactive remediation initiatives. 4.6 Conclusions from Research Goal Forecasting of localized debris congestion in the GEO ring is performed to quantify the frequency of near-miss events occurring daily for each longitude slot at GEO over a 50 year period, using a representative launch trac model, operational lifespan, and end-of- 78 life mitigation compliance rate. Results illustrate that congestion in the vicinity of the two gravitational wells will become severe, provided the mitigation guidelines for this regime are not globally adhered to. Fortunately, a burgeoning desire to preserve GEO is reflected in the increasing numbers of satellite operators attempting re-orbit at end-of-life, but even under this optimistic, present-daymitigation rate of 80%, near-miss events at 300 km in the vicinity of the debris-critical gravitational wells will increase by a factor of two in 50 years. Ultimately, mitigation measures must be combined with active remediation initiatives for the GEO ring, to preserve the long-term usefulness of this valuable driver for space devel- opment, and preclude a situation similar to that presently sustained in the LEO regime.[34] 4.7 Continuing Research Questions For continuing research in this area, the following research questions are of interest: (1) Given that operational GEO lifetimes have increased from⇠3 years in the 1960s, to ⇠7 years in the 1980s, to 15 years currently,[6] what is the eect of increasing satel- lite lifespans on localized congestion at GEOover the 50 year period, particularly if coupled with the linearly-increasing launch trac model illustrated in Figure 4.2? (2) What is the eect of more realistic mitigation on long-term congestion in the vicin- ity of the gravitational wells? In this chapter, perfectly-executed mitigation is sim- ulated by removing re-orbited objects from propagation – as an improvement, his- torical re-orbit data can be leveraged to construct PDFs that describe disposal orbit element distributions resulting from fuel uncertainty, maneuver reliability, etc. [29] (3) What is the eect of environmental remediation—active debris removal—on long- term congestion in the vicinity of the gravitational wells? Howmany objects need to be removed per year to stabilize congestion increases given an increasing launch trac model, increasing satellite lifespans, and stable mitigation compliance rate? Chapter 5 Research Goal 2(b): Forecasting longitude-dependent congestion at GEO arising from on-orbit fragmentation events 5.1 Overview of Research Goal Of critical importance is a thorough understanding of the consequences that on-orbit explosions anddebris-debris or debris-satellite collisions, collectively classified as fragmen- tation events, have on longitude-dependent congestion throughout the GEO regime. This knowledge is especially important, since although two historical fragmentation events at GEO have been documented—the 1978 explosion of the Ekran-2 satellite, and the 1992 ex- plosion of a Titan III-C transtage[11]—significant populations of objects with diameters as small as 10-15 cm have been detected in optical observations of the GEO ring, and are in- dicative of undocumented fragmentation events in this regime.[11,17,18] Although no known historical collision events have beendocumented atGEO, potential impact-induced anoma- lies have been observed in both the GEO ring and the super-synchronous GEO graveyard regime.[54] Fragmentations on-orbit are triggered by accidental mixing of hypergolic fuels, overheating of residual (non-vented) propellants, mechanical valve failures, hypervelocity impacts, or other unknown or unattributed causes.[56] Although the literature strongly focuses on the eects of debris-generating fragmen- tation events in low-Earth orbit (LEO), several previous studies have beendevoted to quan- tifying increases in collision risk driven by fragmentations in the GEO ring. Reference 30 harnessesmomentum and energy principles tomodel hypervelocity impacts, and predicts 80 that the resulting fragment cloud remains close to the GEO belt, spreading longitudinally to all regions of theGEO ringwithinweeks. Reference 31 considers a probable conjunction scenario at GEO to show that the fragment energy distribution resulting from breakup is sucient for permeating all orbit regimes from re-entry through super-synchronous. Re- cently, Reference 32 indicates that although close approach velocities for the GEO regime are in general lower than in the LEO regime, a low-energy breakup over one of the gravita- tionalwells could be aworst-case scenario in that resonance could be preserved formost of the fragment population, serving to increase collision risk with operational assets at these longitudes. While the tools that the authors of these previous studies harness for charac- terizing breakup events at GEO dier, the independent conclusions of each study agree that fragmentations at the GEO altitude have long-term consequences that are detrimental to the sustainability and continuing usefulness of this unique regime. In addition to these environmental consequences, a fragmentation event at GEO and the eects thereof could hinder the high-value services of operational assets, increase insurance costs for operators with assets in higher-risk longitude slots, and engender political repercussions, as well. [31] It is critical to determine whether the longitude of the parent body at the fragmenta- tion epoch renders the localized eects of the break-up more or less severe for operational assets in higher-risk longitude slots, especially those in the vicinity of the two gravitational wells at 75E and 105W.Aswill be demonstrated, on-orbit fragmentation in the GEO ring has the potential to generate significant localized congestion, dependent on both the lon- gitude and altitude at which the fragmentation occurs. This research therefore fills a void in the literature by addressing fragmentation analyses at GEO from the perspective of the Earth-fixed frame, and not inertial space, which has typically been used for fragmentation analyses in both the LEO and GEO regimes (see, for example, References 30 and 31). On- orbit fragmentations in a variety of representative GEO break-up scenarios are simulated to investigate short and long-term contributions to nominal, baseline congestion occurring on a slot-dependent basis for each of the longitude slots at GEO. As the debris population 81 continues to increase—especially as a result of fragmentation events—the amount of pro- pellant required to maintain a designated longitude slot while simultaneously executing avoidance maneuvers, and the costs associated with reviewing conjunctions to determine if evasive action is evenwarranted, will begin increasing in tandem. From the perspectives of this research, the significance and implications of on-orbit fragmentations at GEO will be cast in a new light, to impel continued research in this critical and highly relevant area. 5.2 Fragmentation Model Fragmentation particulates are generated using theNASA Standard BreakupModel, developed for NASA’s long-term debris environment modeling software EVOLVE 4.0[46] and validated against catalogueddebris clouds and ground-based experimental results for particles larger than 1mm.[11] TheNASAmodel uses an empirical power law to determine the cumulative number of fragments larger than the user-defined eective diameter lc that are generated during fragmentation. For explosions, the cumulative number of fragments Nf is independent of the mass of the parent body, and is given by Nf (d lc) = 6csl1.6c (5.1) where d is fragment diameter, lc the eective diameter defined in meters, and cs a unitless, empirical correction for catalogued fragmentation events.[11] For this study, the correction cs = 1.0 is assumed.1 For collision events,Nf is a function of the target and impactormass and collision velocity:[11] Nf (d lc) = 0.1mˆ0.75l1.71c (5.2) where the mass-dependent factor mˆ is defined as mˆ ⌘ 8><>: mt +mp [kg] for E˜p E˜⇤p [catastrophic] (mpvi)/1000 [kg m/s] for E˜p < E˜⇤p [non-catastrophic] (5.3) 1 Reference 46 indicates this relationship is valid for launch vehicle upper stages with mass 600-1000 kg, and introduces this correction factor to account for knowndebris from catalogued explosions of other parent body masses and types. 82 Define effective diameter lc Cumulative no. of fragmentsNf Power Law Cross-section areaAi Area-to-mass ratio (A/m)i Fragmentation velocityvi Particle mass mi Power Law Bi/uni-modal distribution Normal distribution vi)ECI = vi cos i cosivi cos i sini vi sin i i, i [0, 2)sampled uniformly: (A) i < Nf TO (A) Figure 5.1: Flow-chart for computer implementation of NASA Standard Breakup Model. wheremt andmp are the target and impactor masses, respectively, vi is the impact speed, E˜p = 0.5mpv2i /mt is the specific energy of the impactor, and E˜⇤p ⌘ 40 kJ/kg is the energy threshold for a catastrophic collision.[11] For all eective diameters lc > 11 cm, a bi-modal probability distribution is used for sampling the area-to-mass ratio of every fragment. For lc < 8.0 cm for spacecraft fragments, and lc < 1.7 cm for upper stage fragments, a single- modenormal distribution is employed.[11] For eective diameters in the “definition gap” of 8-11 cm for spacecraft and 1.7-11 cm for upper stages, a pseudo-random number on [0,1] is generated to determine if the bi-modal or single-mode distribution should be utilized.[11] The cross-sectional area for each fragment is then determined as an explicit function of the eective diameter lc, and the fragmentation delta-velocity vi applied to each fragment is sampled from a normal distribution on the area-to-mass ratio.[46] Figure 5.1 illustrates a flow-chart for using the NASAmodel – for more details regarding implementation, and for the parameters utilized in the bi-modal and single-mode area-to-mass ratio and delta- velocity distributions in the model, see References 11 and 46. Note that for both explosion and collision events, the delta-velocityvi is applied in a random direction sampled over the unit sphere for each fragment. For this study, right ascension↵i anddeclination i angles are sampled uniformly over the unit circle, and these angles are translated into a unit vector directionwith spherical coordinates (cf. Figure 5.1). 83 Note that the resulting unit vector distribution is not uniform – rather, it exhibits slight bias towards the poles.2 In the event of a collision, the number of fragments in each generated debris cloud is proportional to themass fractions of the target and impactor, e.g., the target cloud receives [mt/(mt + mp)]Nf fragments, and the impactor cloud [mp/(mt + mp)]Nf fragments, such that each cloud receives 0.5Nf fragments inasmuch asmt = mp. Since the objective of this study is to investigate the longitude-dependent eects of fragmentations at GEO—and not the accuracy of the fragmentation model itself—these assumptions are appropriate for the scope of this research objective. 5.3 Longitude-Dependent Eects of Fragmentations at GEO The longitude-dependent congestion generated by the fragmentation of a rocket body (R/B) over one of the gravitational wells at 75E and 105W is characterized by applying the fragmentation model outlined in Section 5.2 to a simulated rocket body positioned in various longitude slots at theGEOaltitude. Reference 32 hypothesizes that a lower-energy fragmentation over one of the two gravitationalwells could be aworst-case scenario in that resonance is preserved for the majority of the fragmentation particulates, resulting in fre- quent near-misses with operational assets in the vicinity of these critical longitudes. The results of this current studywill illustrate that this hypothesis of Reference 32 is exactly the case—not only does longitude-dependent congestion increase as a function of the number of fragments captured by the gravitational well, but these localized congestion increases are predictable to first-order both in longitude and time since fragmentation. First, theory introduced by Reference 4 that provides the mathematics necessary for predicting libra- tion parameters based on initial longitude and longitudinal drift rate ˙ is presented. These parameters are combined with a harmonic oscillator model to give a first-order, analytical formulation of the resulting librationmotion of fragments captured by a gravitationalwell. 2 Although the direction of the delta-velocity vectorvi is central to the instantaneous change in orbital elements experienced during fragmentation, it is ultimately the coupling between magnitude and direction that determines whether a fragment will circulate around the GEO arc or librate around a gravitational well. 84 5.3.1 Harmonic Oscillator Model of Libration Motion Using the change of origin ⌘ 1,2, where and 1,2 denote geocentric longitude of the object and gravitational well, respectively, the condition for stable point “capture” is expressed to first-order as[4] k ˙0k < kk cos 0k (5.4) where ˙0 = n0 nGEO is the initial longitudinal drift rate—a function of the initial semi- major axis only—and k is an angular-velocity-like parameter dependent upon the J2,2 sec- torial harmonic and defined as[4] k ⌘ 6nGEOR p J2,2 rGEO (5.5) Using the coecient J2,2 ⌘ q C22,2 + S 2 2,2 ⇡ 1.82⇥ 106 derived from the EGM-96 model,3 the parameter k ⇡ 8.92⇥ 108 rad/s.[11] Following Reference 4, the amplitude and period of libration for objects that satisfy the capture condition in Equation (5.4) are approximated by the two formulas: sin m = s sin2( 0) + ˙20 k2 (5.6) Tl = 4 k K(sin m) (5.7) where m denotes the libration amplitude, Tl is the period of libration, and K(sin m) is the complete elliptic integral of the first kind.[4] Note that Equations (5.6)-(5.7) are derived assuming the J2,2 term only – higher-order harmonics appearing in a full expansion of the Earth’s gravitational potential are neglected. Since the propagator harnessed in this thesis uses a 4⇥4 gravitationmodel, Equations (5.6)-(5.7) provide a first-order approximation for the libration characteristics observed in numerical simulations. Nonetheless, these formu- las can be used in tandemwith a harmonic oscillatormodel to rapidly predict librationmo- tion to first-order, i.e., under the J2,2 perturbation alone. If oscillatory motion is assumed, 3 Coecients for the EGM-96 model are available at: http://cddis.nasa.gov/926/egm96/egm96.html 85 the longitudinal motion of a captured object can be expressed as (t) = m cos ✓ 2⇡t Tl sgn( ˙0) ◆ + 1,2 (5.8) where the phase shift is determined with = arccos ✓ 0 m ◆ (5.9) Note that the sign of the initial longitudinal drift rate ˙0 is included to switch the sign of the phase angle accordingly, such that the resulting sinusoidal longitude profile reflects the initial east/west drifting direction of the particle. The accuracy of this harmonic oscillator model for predicting the libration motion of fragments captured by a gravitational well in the aftermath of a fragmentation is explored in Section 5.3.3. 5.3.2 Rocket Body Explosions over Gravitational Wells To assess longitude-dependent debris congestion generated by explosions at theGEO altitude, the NASA Standard Breakup Model is applied to generate fragmentation debris with lc > 5 cm resulting from the explosion of a simulated R/B. The initial conditions are selected to position this R/B at various longitudes surrounding the east/west gravitational wells at the simulated explosion epoch of 00:00:00 Zulu on 01 September 2013. FromEqua- tion (5.1), the NASA model predicts that 724 fragments larger than 5 cm are generated in the explosion. The fragmentation delta-velocity for each fragment is applied in a random direction sampled over the unit sphere, and is applied to the inertial velocity vector of the simulated R/B initial conditions a = rGEO, e = 0.001, i = ⌦ = M0 = 0, with argument of perigee ! chosen such that the R/B is positioned at the desired longitude, i.e., ! = ↵G+0, where ↵G is the right ascension of Greenwich at the explosion epoch.[2] Figure 5.2 provides the characteristics of the resulting fragment distribution for the case 0 = 60E. Figure 5.2(a) is aGabbard plot that highlights the resulting perigee/apogee radius spread as a function of semi-major axis, normalized by theGEO radius. Figure 5.2(b) 86 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Normalized Semi-Major Axis [a/rGEO] 0.6 0.8 1.0 1.2 1.4 1.6 1.8 N or m al iz ed R ad iu s [r p, a/ r G EO ] Fragment Perigee Fragment Apogee Breakup Altitude (a) Gabbard diagram for R/B explosion at 60E. 0.001 0.01 0.1 1.0 10.0 Area-to-Mass Ratio [m2/kg] 1.0 10.0 100.0 1000.0 A dd it io na lF ra gm en ta ti on V el oc it y [m /s ] Captured 0 1 2 3 4 5 M et ri c: (k y˙ 0k kk co sy 0k ) ⇥ 10 6 [r ad /s ] (b) Fragment distribution for R/B explosion at 60E. Figure 5.2: Characteristics of fragmentation distribution for simulated explosion at 60E. shows the area-to-mass ratio and fragmentation delta-velocity distribution, colored by the capture metric in Equation (5.4). In this case, 60/724 fragments are captured by the gravi- tational well at 75E, and of these, 57 are ejected at fragmentation delta-velocities less than 100m/s, and three at delta-velocities larger than 100m/s. This indicates that the direction of the fragmentation delta-velocity vector for these higher-speed fragments is opposite the direction of the parent body’s velocity vector. Furthermore, Figure 5.2(b) illustrates that area-to-mass ratios consistent with the definition of the high area-to-mass ratio (HAMR) objects observed in theGEO regime4 are sampled via theNASA Standard BreakupModel. This underlies the importance of including SRP in the force model, which is a dominating perturbation to the long-termmotion of both HAMR objects and fragmentation debris. [58] Five-year congestion forecasts for simulated R/B explosions occurring at 45E, 60E, and at the Eastern gravitational well at 75E are illustrated in Figure 5.3 for a torus minor radius of r˜ = 100 km. Note that 54, 60, and 69 of the 724 generated fragments are captured by the Eastern gravitational well in each scenario, respectively, and begin librating around 4 HAMR objects are defined as having area-to-mass ratios from 0.1-20 m2/kg and above, and are widely studied in the literature, e.g., in References 57 and 58. 87 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (a) Five-year congestion forecast for simulated explosion at 45E (r˜ = 100 km). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (b) Five-year congestion forecast for simulated explosion at 60E (r˜ = 100 km). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (c) Five-year congestion forecast for simulated explosion at 75E (r˜ = 100 km). Figure 5.3: Five-year longitude-dependent congestion forecasts for simulated rocket body explosions in the vicinity of the Eastern gravitational well (torus minor radius of 100 km). this critical longitude. As a consequence, bi-annual “debris storms”— dramatic increases in localized congestion—manifest at the longitudinal extents of the parent body’s libration motion. This outcome is predicted by Equation (5.6), which states that 0 = m if ˙0 = 0, that is, the libration amplitude is equivalent to the initial deviation from the gravitational 88 well inasmuch as a = rGEO at the explosion epoch. The congestion forecasts in Figure 5.3 illustrate that the resulting oscillations in longitude exhibited by the captured fragments synchronize in approximately one-year intervals at the libration amplitudes of the parent body over the five-year prediction span. Therefore, this is critical knowledge not only for operators with satellites in longitude slots at or near the fragmentation longitude 0, but also for those with satellites in longitude slots ±2 m away from 0 (i.e., in the direction of the gravitywell). Note that the resulting backgroundnoise from the drifting fragments not captured by the gravitational well exhibits a once-yearly banding phenomenon driven by once-yearly oscillations in the eccentricity magnitude induced by SRP, the eects of which aremore pronounced as a consequence of higher area-to-mass ratios (cf. Figure 5.2(b)). [3,59] Figure 5.4 provides a zoomed-in, one-year view of the five-year congestion forecast in Figure 5.3(b). Curvilinear traces of the fragments captured by the gravitational well are clearly visible, in addition to the linear traces of the higher-energy fragments that surpass capture and begin circulating around theGEO arc, with eastward drift for semi-major axes below the GEO ring, and westward drift for semi-major axes above GEO.[13] Figure 5.5(a) shows the 60E case in Figure 5.3(b) propagated for 30 years to illustrate localized conges- tion trends occurring over this longer time scale. Longitudinal focusing, although promi- nent for the first five years, begins to diuse during 5-10 years, leading to a lull in localized 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 11/13 01/14 03/14 05/14 07/14 09/14 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] Figure 5.4: One-year longitude-dependent congestion forecast for simulated rocket body explosion at 60E (torus minor radius of 100 km). 89 0 30E 60E 90E 120E 150E Longitude [deg] 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 09/31 09/33 09/35 09/37 M on th /Y ea r 0.0.20.4.6.81.0 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 Near-Miss Events [100 km] 0 30E 60E 90E 120E 150E Longitude [deg] 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 09/31 33 35 09/37 09/39 09/41 09/43 M on th /Y ea r 2 4 6 8 100 Near-Miss Events [100 km] Re -F oc us in g Fr ag m en t D iff us io n Pe rio d Fr ag m en t F oc us in g P er io d D iff us io n (a) Propagation under 4⇥4 gravity, luni-solar perturbations, and SRP (i.e., full force model). 0 30E 60E 90E 120E 150E Longitude [deg] 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 09/31 09/33 09/35 09/37 M on th /Y ea r 0.0.20.4.6.81.0 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 Near-Miss Events [100 km] 0 30E 60E 90E 120E 150E Longitude [deg] 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 09/31 09/33 09 35 37 09/39 09/41 09/43 M on th /Y ea r 2 4 6 8 100 Near-Miss Events [100 km] Longitude [deg] (b) Propagation under 4⇥4 gravity only. Figure 5.5: Longitude-dependent congestion forecast (30 years) for simulated rocket body explosion at 60E (torusminor radius of 100 km), using full andEarth gravity forcemodels. congestion levels during 10-20 years. From 20-25 years, the librating fragments begin refo- cusing into a characteristic oscillatory pattern, but the intensity of these residual fragment storms is beneath that observed during the first five years following the explosion. After 25 years, the librating fragments (and resulting local congestion spikes) once again diuse. Analysis of simulation results indicates the mechanism for this long-term focusing, 90 diusion, and re-focusing phenomenon is a combination of long-period luni-solar eects and SRP-induced eccentricity variations prevalent in the GEO ring. If these perturbations are removed from the forcemodel, such that the 30-year propagation uses 4⇥4 gravitation only, the localized congestion forecast in Figure 5.5(b) appears. In this situation, fragment diusion and re-focusing is not observed; rather, strong focusing at the libration amplitude persists throughout the entire 30-year simulation. This result emphasizes a conclusion of Reference 32 that luni-solar and SRP perturbations are critical for analyzing fragmentation events at GEO over durations longer than one year. Figure 5.6 illustrates that similar focusing eects can occur for fragments captured by theWestern gravitational well. For simulated R/B explosions at 75W, 90W, and at the Western gravitational well at 105W, 58, 67, and 66 of the 724 ejected fragments are cap- tured in each case, respectively. Although the number of captured fragments in each case is similar to the number captured in the Eastern well simulations illustrated in Figure 5.3, the intensities of the fragment focusing events are not as strong as the equivalent Eastern well scenarios, e.g., for 0 = 30, cf. Figures 5.3(a) and 5.6(a). For completeness, Figure 5.7 illustrates the five-year congestion resulting from an explosion at 165E, a well-known un- stable equilibrium longitude positioned halfway between the stable gravitational wells. In this situation, none of the 724 ejected fragments are captured by either gravitational well under the metric in Equation (5.4). However, curvilinear traces observed in Figure 5.7 in- dicate that several fragments begin oscillating about both the gravitational wells, behavior indicative of class L3 objects (recall Table 2.1). 5.3.3 Accuracy of Harmonic Oscillator Model It is interesting to assess if the harmonic oscillator model in Equations (5.8)-(5.9) can accurately predict the longitudinal focusing observed in the five-year congestion forecasts in Figures 5.3 and 5.6. Fragment focusing at the longitudinal extents appears predictable both in longitude and time since fragmentation, such that a first-order model for forecast- 91 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (a) Five-year congestion forecast for simulated explosion at 75W (r˜ = 100 km). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (b) Five-year congestion forecast for simulated explosion at 90W (r˜ = 100 km). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (c) Five-year congestion forecast for simulated explosion at 105W (r˜ = 100 km). Figure 5.6: Five-year longitude-dependent congestion forecasts for simulated rocket body explosions in the vicinity of theWestern gravitational well (torus minor radius of 100 km). ing these debris storms with minimal knowledge of the fragment distribution is desirable. If the semi-major axes ai for the resulting fragment orbits in the aftermath of a fragmenta- tion can be reasonably estimated, ˙0,i is a known initial condition, and Equations (5.6)-(5.7) can be harnessed to compute the amplitude and period of libration for fragments captured 92 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] Figure 5.7: Five-year longitude-dependent congestion forecast for simulated rocket body explosion at 165E (torus minor radius of 100 km). via Equation (5.4). Longitude histories for these fragments are thus predicted analytically as a function of elapsed time using Equations (5.8)-(5.9), and overlaid to forecast (to first- order) when and where fragment focusing will occur following a fragmentation at GEO. Figure 5.8 illustrates the accuracy of this harmonic oscillator model, showing com- parisons between analytical longitude histories predicted using themodel in Figure 5.8(a), simulated longitude histories under the J2,2 harmonic only in Figure 5.8(b), and simulated longitude histories under full forcing (i.e., 4⇥ 4 EGM-96 gravity, luni-solar perturbations, and SRP) in Figure 5.8(c). As expected, the harmonic oscillator approximation agrees well with simulated longitudes under J2,2 only, as this was the assumption made in the deriva- tion of Equations (5.6)-(5.7). When compared against “truth” longitude histories for the 60 librating fragments in Figure 5.8(c), however, discrepancies exist not in the longitudinal location of the fragment focusing events, but in the times atwhich these focusing events oc- cur. As errors in timing accumulate with time elapsed since fragmentation, this harmonic oscillatormodel should only be considered for short-term, 1-2 year predictions. Following this time span, dominating perturbations in the GEO environment that are not accounted for in the harmonic oscillator approximation begin influencing the libration motion of the fragments, and this first-order, J2,2-only approximation diverges. 93 0 30E 60E 90E 120E 150E Longitude [deg] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 El ap se d Ye ar s fr om Ex pl os io n (a) Harmonic oscillator model. 0 30E 60E 90E 120E 150E Longitude [deg] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 El ap se d Ye ar s fr om Ex pl os io n (b) Under J2,2 harmonic only. 0 30E 60E 90E 120E 150E Longitude [deg] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 El ap se d Ye ar s fr om Ex pl os io n (c) Under full GEO force model. Figure 5.8: Longitude histories for the 60 librating fragments captured by the gravitational well at 75E, computedwith the harmonic oscillator model, simulated under J2,2 only, and simulated using full GEO forcing. 5.3.4 Rocket Body Collisions over Gravitational Wells Finally, the longitude-dependent congestion signature of collision events in the GEO regime is evaluated. Reference 32 indicates that since orbital velocities are lower forGEO— and objects are orbiting in the same direction—relative velocities are generally more be- nign for GEO than for LEO. Thus, according to Equation (5.2), although fewer fragments will be generated for a GEO collision event than for an equivalent event at LEO, peak risk potential increases as a result of a lower-mean fragmentationdelta-velocity distribution.[32] For this study, three collisions near the Eastern gravitational well are simulated: (a) colli- sion due to inclination dierencei, (b) collision due to eccentricity dierencee, and (c) collision between a GEO object and one in geostationary transfer orbit (GTO). For all sce- narios, two 2000 kg upper stages are assumed to collide at 60E, using the initial conditions 94 for the target R/B selected for the explosion cases shown in Section 5.3.2. Simulation re- sults are highlighted in Table 5.1, and the five-year localized congestion forecasts for each collision scenario are provided in Figure 5.9. Table 5.1: Simulation results for three collision scenarios between two 2000 kg upper stages at 60E, whereNC andC denote non-catastrophic and catastrophic collisions, respectively. Collision Scenario Description Collision Speed Nf (Captured) Inclination dierence i = 2.0 0.107 km/s (NC) 941 (61) Eccentricity dierence e = 0.05 0.157 km/s (NC) 1251 (71) At apogee of 200 km ⇥ 35744 km GTO 1.479 km/s (C) 8439 (218) The first two collision scenarios are non-catastrophic according to the specific energy defined in Equation (5.3), and a smaller proportion of fragments are captured by the gravi- tationalwell for both of these cases than for the 60E explosion (6.5% and 5.7%are captured for the inclination and eccentricity dierence collisions, respectively, compared to the 8.3% captured in the explosion). This is largely a consequence of the fragmentation velocity dis- tribution in the NASA Standard Breakup Model, which defines a higher meanv for col- lision events than for explosion events.[46] In the third simulated collision event—defined as catastrophic according to Equation (5.3)—amere 2.6% of the staggering 8439 fragments larger than 5 cm in diameter are captured by the gravitational well at 75E. Figure 5.9 shows that the localized congestion forecasts generated by collision events qualitatively appear to be similar to themacroscopic trends exhibited by explosion events, even though two separate fragment clouds exist in the former, and only a single cloud of ejecta is generated in the latter. The significant increase in the number of near-miss events at 100 km for the GEO-GTO collision (cf. Figure 5.9(c)) is a consequence of the large num- ber of fragments ejected in this catastrophic collision event. Longitudinal bunching at the libration amplitude of the target R/B is still observed with similar periodicity in all three collision scenarios, although localized congestion during a bunching event in Figure 5.9(c) is a factor of four times more intense than for the inclination and eccentricity dierence 95 scenarios in Figures 5.9(a)-5.9(b). These results indicate that longitude-dependent eects of explosions and collisions at GEO are approximately equivalent in characteristic, dier- ing strictly by the number of fragments captured by the gravitational well in each scenario. 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (a) Five-year congestion forecast for simulated collision at 60E (inclination dierence). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 2 4 6 8 10 N ea r- M is s Ev en ts [1 00 km ] (b) Five-year congestion forecast for simulated collision at 60E (eccentricity dierence). 0 30E 60E 90E 120E 150E 180 150W 120W 90W 60W 30W 0 Longitude [deg] 09/13 09/14 09/15 09/16 09/17 09/18 M on th /Y ea r 0 10 20 30 40 N ea r- M is s Ev en ts [1 00 km ] (c) Five-year congestion forecast for simulated collision at 60E (GEO-GTO collision). Figure 5.9: Five-year longitude-dependent congestion forecasts for simulated rocket body collisions in the vicinity of the Eastern gravitational well (torus minor radius of 100 km). 96 5.4 Conclusions from Research Goal The eect of on-orbit fragmentations on longitude-dependent congestion in the GEO regime is investigated by simulating explosions and collisions of uncontrolled rocket bod- ies in various orbit configurations, including libration about one or both of the stable longi- tudes at the GEO altitude. Applying the NASA Standard Breakup Model in tandem with the near-miss event metric, short- and long-term localized congestion patterns are rapidly forecasted to investigate how strongly the severity of a GEO fragmentation event depends on the longitude of the parent body at the breakup epoch. Simulation results indicate that on-orbit GEO fragmentations in the vicinity of one of the critical gravitational wells spawn bi-annual debris storms at the longitudinal extents of the parent body’s libration motion. Provided the semi-major axis distribution for a GEO fragmentation can be reasonably esti- mated using tracking data, longitudes and times atwhich dramatic increases in congestion will occur can be predicted to first-order, applying the harmonic oscillator formulation for time scales of less than five elapsed years since fragmentation. For all fragmentation cases, the level of existing TLE background noise is achieved or surpassed only during the observed longitudinal focusing. The intensity of these longitude- dependent debris storms is proportional to the number of particles captured by the nearest gravitational well, however, and thus could readily overtake background congestion levels if higher-energy collisions are considered (as in the scenario of the GTO apogee collision), or if fragment sizes smaller than 5 cm are included. Ultimately, this chapter highlights the importance of energy passivation at end-of-life – satellites and upper stages not appropri- ately prepared for disposal during decommissioning are prone to fragmenting and gener- ating a fragment cloud that threatens severe consequences for operational assets at GEO. 5.5 Continuing Research Questions For continuing research in this area, the following research questions are of interest: 97 (1) TheNASAStandardBreakupModel is an empirically-derived fragmentationmodel that does not satisfy conservation of mass or conservation of energy. What are the eects of a more realistic breakup model on the longitude-dependent congestion signature resulting fromGEO fragmentation events studied in this chapter, if any? (2) The cannonball SRPmodel is applied to the fragments propagated in this chapter, but this approximation is unrepresentative forwidely-varying fragment geometry in the aftermath of a breakup. Howwould amore accurate force model—coupled with fragment attitudemotion—aect the longitude-dependent congestion trends illustrated in this chapter, especially over longer time scales (recalling Figure 5.5)? (3) Can the harmonic oscillator model for fragment libration be modified—via a scal- ing factor, for example—to account for luni-solar and SRP forcing not modeled by this analytic formulation, i.e., can the time scale overwhich it is valid be extended? (4) Can fragments exhibiting correlated congestion signatures be back-propagated to identify a common fragmentation epoch, accounting for covariance information? Chapter 6 Research Goal 3(a): Evaluating the conjunction challenges of low-thrust GEO debris removal maneuvers 6.1 Overview of Research Goal Currently, themostwidely-implementedmitigation strategy forGEOsatellites reach- ing end-of-life is to re-orbit to a graveyard disposal orbit at a perigee altitude safely above the GEO ring, “far enough away from GEO so as not to cause interference with space sys- tems still in geostationary orbit.”[21] In 1997, the Inter-Agency Space Debris Coordination Committee (IADC) proposed the following guideline to provide a stable passivation orbit that remains safely outside the protectedGEO region, defined by a spherical shell segment within ±200 km of the GEO altitude and ±15 latitude of the equatorial plane:[22,60] hp|min 235 + 1000cr(A/m) [km] (6.1) where cr is the coecient of reflectivity andA/m is the area-to-mass ratio.[21] The 235 km minimum is the sum of the upper altitude for the protected GEO region (200 km) and the maximumdescent of a re-orbited satellite due to geopotential and luni-solar perturbations (35 km).[60] For typical GEO satellites (cr ⇡ 1.5 and A/m ⇡ 0.04 m2/kg), the second term in Equation (6.1) is approximately 60 km, such that this IADC re-orbit guideline specifies a ⇠300 km increase in perigee altitude, at minimum.1 The IADC furthermore recommends 1 The recommended mitigation re-orbit maneuver is executed in a multi-burn sequence with an approx- imate fuel cost equivalent to three months of station-keeping fuel for a typical GEO satellite, [21] correspond- ing to v ⇡ 11 m/s, that is, about 2.3% of the entire E-W/N-S station-keeping budget for a 10-year mis- sion. [11] 99 the addition of an errormargin to account for orbit determination uncertainties, maneuver execution errors, and spacecraft deterioration.[60] Although one might speculate that the IADC guidelines could increase the risk of derelict collisions in the IADC debris orbit that could accumulate above GEO under these guidelines, Reference 23 indicates that the vari- able amount of remaining propellant at end-of-life has driven the broad variety of disposal orbits achieved in practice, such that the creation of a unique “debris ring” is unlikely. Of the 160 GEO satellites that achieved end-of-life in the period 2001-2010, 80% successfully re-orbited to disposal orbits – of these satellites, 70%were transferred to orbits at least 200 kmaboveGEO, and 50%achieved at least 300 kmabove, fully compliantwith the IADC.[23] Using historical re-orbit statistics compiled from the European Space Agency’s Clas- sification of Geosynchronous Objects reports, [33] Figure 6.1 illustrates compliance to the IADC GEO re-orbit guidelines since theywere introduced to the operator community in 1997.[60] Figure 6.1(a) provides the annual number of GEO satellites that (1) reached the end of their operational lifespans, (2) attempted re-orbit to an IADC-compliant disposal orbit, and (3) successfully achieved theminimumperigee altitude increase specified in the IADC guide- lines. The margin between the number of satellites that reached end-of-life and those that re-orbited into an IADC-compliant disposal orbit has decreased since 1997, indicative of an increasing international desire to preserve the long-term utilization of GEO. Figure 6.1(b) provides a breakdown by entity of the compliance data in Figure 6.1(a). During 1997-2013, approximately 50% of GEO satellites that reached end-of-life were repositioned into com- pliant disposal orbits before deactivation; nearly 30%attemptedpost-missiondisposal, but were unsuccessful in reaching theminimumperigee altitude increase. The remaining 20% (53 GEO satellites) were abandoned without any re-orbit attempt during this time period, of which Russia contributed 33 satellites (62% of the assets abandoned during this 16-year period). As the resulting political cost to Russia for poor compliance with these guidelines has been negligible, these re-orbit data illustrate that it is less expensive in the near term to generate debris without repercussions from disregarding adoptedmitigation practices.[61] 100 19 97 19 98 19 99 20 00 20 01 20 02 20 03 20 04 20 05 20 06 20 07 20 08 20 09 20 10 20 11 20 12 20 13 Year 0 4 8 12 16 20 24 28 N um be ro fG EO Sa te ll it es Achieved End-of-Life Attempted Re-orbiting Successful Re-orbiting (a) Annual compliance to IADC re-orbit guidelines. China Intelsat Japan Russia USA Other IADC Compliant 1336(50.4%) IADC Noncompliant 786(29.5%) 6 18 Abandoned5 536(20.1%) 15 8 34 52 5 5 4 9 22 33 6 1 33 9 4 (b) Guideline compliance by entity for 1997-2013. Figure 6.1: Compliance to end-of-life GEO mitigation guidelines introduced by the IADC in 1997, illustrated as compliance rate by year and compliance rate by entity through 2013. Even under an optimistic 80% compliance rate, however, Section 4.5 predicts that the mean number of near-miss events per day at 300 km in the vicinity of the two gravitational wells will double by the year 2063. Although recent re-orbiting statistics are promising in that they suggest increasing adherence to mitigation guidelines—and ultimately growing international desire to safeguard theGEO ring—mitigation alone is not enough to stabilize the growing debris congestion at GEO under current re-orbiting success rates. To stabilize projected debris growth at GEOunder currentmitigation levels, active debris remediation (ADR)measuresmust be synthesized and implemented to remove potentially threatening derelicts that (a) failed to achieve an IADC-compliant graveyard orbit at end-of-life, or (b) were simply abandoned by their operators at end-of-life with no attempt at a re-orbit. The necessity for cost-eective ADR activities at GEO is becomingmore relevant, especially for larger derelicts (payloads, upper stages) that pose the greatest threat to operational assets. A “chief space-tug” concept is often envisioned for performing the re-orbitmaneuver once contact with the target debris object has been established. However, as rendezvous, proximity operations, and docking with an uncontrolled—and likely tumbling—derelict 101 "No-Fly" Zone "No-Fly" Zone + Covariance Re-orbiting Trajectory Position Covariance Avoidance TrajectoryTowed Derelict Ground Tracking Onboard Sensing Conjuncting Debris Orbit Re-orbit System Figure 6.2: Challenges of low-thrust GEO debris re-orbit. are challenging, several proposed methods have focused on contactless technologies such as an electrostatic tractor[62] or ion beam shepherd[63] for ADR at GEO. Each of these con- tactless ADR technologies rely on low-thrust engines for performing the required re-orbit maneuver. With lower maneuverability, however, collision avoidance for such low-thrust re-orbit systems is challenging. Potential conjunctions must be detected multiple revolu- tions in advance, to provide the tug’s guidance system enough lead time to place the tug on a sucient evasive trajectory, especially if this craft is designed to operate autonomously withminimal ground support. Figure 6.2 shows the challenges of developing a low-thrust guidance strategy for GEOdebris re-orbiting applications. Of interest is investigating how many conjunction events with the current debris population at GEO could be experienced during a typical continuous thrust re-orbitingmaneuver to an IADC-compliant graveyard orbit 300 km above the GEO ring.[7] Specifically, quantifying the conjunction challenge for a range of thrust levels—i.e., the global average of the number of conjunctions that could be expected for a re-orbit trajectory at each thrust level, regardless of the initial longitude or 102 inclination—is a useful tool for system design of an ADR demonstration mission at GEO. Evaluating the global conjunction challenge for low-thrust GEO disposal maneuvers is beneficial not only for debris removal concepts, but for operational end-of-lifemitigation activities, too. As lightweight, all-electric satellite busses are becomingmore prominent in the satellite industry,2 low-thrust electric propulsion is nowbeing harnessed both for orbit raising and station-keeping of satellites. Previously, GEO satelliteswith chemical thrusters have used a two- to three-impulse Hohmann-like transfer to re-orbit to a disposal orbit at end-of-life, [11,64] but GEO satellites equipped with low-thrust electric engines must use continuous thrust orbit raising techniques to achieve the required increase in semi-major axis. Since continuous thrust re-orbiting performed under mN levels of thrust takes from weeks tomonths to achieve a 300 km increase in semi-major axis, [62] it is critical to quantify the number of conjunctions that such satellites might experience during this phase of de- commissioning. As it is undesirable to be responsible for a debris-generating event while engaged in an act of mitigation or remediation, characterizing the conjunction challenges of low-thrust re-orbit maneuvers is critical knowledge for both all-electric satellite opera- tors and designers of contactless technologies using low-thrust engines for ADR at GEO.3 6.2 Analytic Results for Continuous Thrust Trajectories In the framework of two-body dynamics, analytic expressions that describe the semi- major axis and longitude profiles for a continuous thrust re-orbit trajectory—as a function of thrust acceleration and elapsed time since the start of the re-orbit maneuver—are now derived. Following Reference 65, the time derivative of specific two-body orbit energy is ✏ = µ 2a =) ✏˙ = µ 2a2 a˙ (6.2) 2 de Selding, P. B., “News from Satellite 2014: Boeing Electric Satellite Backlog Posed to Grow, includes Previously Undisclosed U.S. Gov’t Order,” Space News, 13 March 2014, Web. 3 In this thesis, GEO satellites using continuous thrusting during end-of-life operations, and ADR space- tug concepts that use continuous thrust to re-orbit derelict objects to disposal orbits, are collectively termed re-orbit systems for generality. 103 where a˙ denotes the time rate of change of the semi-major axis. From elementary physics, the rate of change of specific energy due to the thrust vector is ✏˙ = · v, where v is the inertial velocity vector. Assuming that the thrust acceleration is directed along the instan- taneous velocity vector direction (i.e., zero flight path angle), then ✏˙ = v, such that µ 2a2 a˙ = v =) a˙ = 2a 2 µ v ⇡ 2a 3/2p µ (6.3) where it is assumed that the osculating orbit remains near-circular during the continuous- thrust maneuver, such that the circular orbit velocity v = p µ/a is applicable. Separating variables and assuming constant , we have:Z a3/2da = 2p µ Z dt (6.4) Performing the integration and enforcing the initial condition a(0) = a0: 1p a = p µ t 1p a0 =) a(t) = a0µ ( p µ pa0t)2 (6.5) The time tf required to transfer between circular orbits of radii a0 and af > a0 is therefore given by:[65] tf = p µ ✓ 1p a0 1p af ◆ = v0 vf (6.6) As a result, vf = v0 tf , that is, the re-orbit system slows down linearly in time as local circular orbit speed is maintained.[65] Figure 6.3 illustrates Equation (6.6) as a function of thrust acceleration for three super-synchronous semi-major axes. The hyperbolic form of Equation (6.6) indicates that as the thrust acceleration increases, the length of the re-orbit maneuver decreases rapidly. Furthermore, as the target semi-major axis af increases, local orbit speed vf at af decreases, such that the length of the re-orbit maneuver increases. As will be shown, the duration of the re-orbit maneuver is strongly correlated to the number of conjunction events a trajectory at a specified thrust acceleration experiences on average. For a near-circular orbit, the rate of change of longitude ˙ is given by themeanmotion dierence ˙ = n nGEO = r µ a3 r µ a3GEO (6.7) 104 106 105 104 103 Thrust Acceleration [m/s2] 0 50 100 150 200 250 Le ng th of R e- or bi tM an eu ve r [d ay s] 100 km above GEO 300 km above GEO 500 km above GEO 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 1%week 1%month 4%months (a) Analytic re-orbit duration (linear y-axis). 106 105 104 103 Thrust Acceleration [m/s2] 102 101 100 101 102 103 Le ng th of R e- or bi tM an eu ve r [d ay s] 100 km above GEO 300 km above GEO 500 km above GEO 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 1%day 1%week 1%month (b) Analytic re-orbit duration (logarithmic y-axis). Figure 6.3: Duration of continuous-thrust re-orbiting maneuvers to three semi-major axes above GEO as a function of constant thrust acceleration and force, assuming 1000 kgmass. Substituting Equation (6.5) for the semi-major axis a(t), the longitude rate becomes ˙(t) = ( p µ pa0t)3 a3/20 µ r µ a3GEO (6.8) Integrating Equation (6.8) and enforcing the initial condition (0) = 0: (t) = µ2 (pµ pa0t)4 4a20µ r µ a3GEO t+ 0 (6.9) Given that the re-orbit system begins at longitude 0, Equation (6.9) provides an analytic prediction for the resulting longitude profile of the system as a function of elapsed time t since maneuver start, initial semi-major axis a0, and thrust acceleration . Since the semi- major axis increases above that of GEO during the re-orbit, as described by Equation (6.5), the mean motion of the system slows with respect to the rotational rate of Earth, resulting in westward longitudinal drift. To evaluate the amount of longitude drift ⌘ (tf )0 that occurs during re-orbit as a function of thrust acceleration, the maneuver duration tf in Equation (6.6) is substituted into Equation (6.9), giving = µ 4a20 " 1 ✓ a0 af ◆2# r µ a3GEO ✓ v0 vf ◆ (6.10) 105 106 105 104 103 Thrust Acceleration [m/s2] 0 100 200 300 400 500 600 700 W es tw ar d Lo ng it ud in al D ri ft [d eg ] 100 km above GEO 300 km above GEO 500 km above GEO 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 1%revolution (a) Analytic longitude drift (linear y-axis). 106 105 104 103 Thrust Acceleration [m/s2] 102 101 100 101 102 103 W es tw ar d Lo ng it ud in al D ri ft [d eg ] 100 km above GEO 300 km above GEO 500 km above GEO 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) (b) Analytic longitude drift (logarithmic y-axis). Figure 6.4: Accumulated longitudinal drift for continuous-thrust re-orbit maneuvers to three semi-major axes aboveGEO, as a function of constant thrust acceleration and force. Figure 6.4 illustrates Equation (6.10) as a function of thrust level for three super-synchronous semi-major axes. The hyperbolic form of Equation (6.10) indicates that as the thrust accel- eration increases, the total longitudinal drift accumulated over the duration of the re-orbit decreases rapidly. Further, since larger af require longer maneuver durations tf by Equa- tion (6.6), total longitudinal drift increases with af as a result. Figure 6.4 provides an in- teresting compliment to continuous-thrust trajectory representations in the inertial frame, which exhibit a “spiral-like” pattern in inertial space. For simulated maneuvers to 300 km above GEO, Figure 6.4 illustrates that the re-orbit system does not complete an entire revo- lution in the Earth-fixed frame over the range of thrust levels considered. This has implica- tions for ground station coverage of these re-orbit maneuvers, and highlights that systems beginning in less-congested longitude slots could still pass through heavily-congested lon- gitudinal regions (e.g., one or both of the gravitationalwells) over the course of the transfer. 6.3 Evaluating Conjunctions for GEO Re-orbit Maneuvers To begin developing insight into the conjunction challenges for continuous thrust re- orbit systems at GEO, this study considers the number of conjunction events that a re-orbit 106 system could encounter with the current GEO debris population during a typical re-orbit maneuver to 300 km above the GEO ring. Specifically, the research questions investigated in this GEO conjunctions study are as follows: • Howmany conjunctions within various distances can be expected, on average, for a specified thrust acceleration, ranging from 106 to 103 m/s2 (i.e., 1 mN to 1 N thrust force for a 1000 kg system), over the duration of a 300 km re-orbitmaneuver? • Is the number of conjunctions dependent on initial longitude and inclination, and if so, is this dependency sensitive to the thrust acceleration or the time atwhich the re-orbit maneuver begins? • When are conjunctions most likely to occur during the re-orbit burn, regardless of initial longitude or inclination? Does the likelihood of a conjunction increase over a range of semi-major axes, and if so, which classes of debris objects are responsi- ble for these conjunctions? Is conjunction likelihood also sensitive to longitude? • What is the distribution of conjunction events in the Hill frame for a given thrust level? What are the relative speeds and combined risk factors associated with the conjunction events for a given thrust level? The equations of motion for a re-orbit system are equivalent to those for the uncon- trolled debris objects (described in Section 2.3.2), with the addition of a continuous thrust vector at, which is oriented in the in-track (along-track) direction of the local orbit frame: r¨ = µ r3 r + a + a% + a + aSRP + at (6.11) The debris population is propagated from 01March 2014 (00:00:00 Zulu) for fivemonths in 5 minute time steps, and each continuous thrust re-orbit trajectory is propagated forward from the same epoch under Equation (6.11), until 300 km above GEO in semi-major axis is achieved. Conjunctions are then detected in post-processing by checking for debris objects 107 that comewithin a specified distance threshold of the re-orbit system at any time step over the duration of the maneuver. Logic is applied to ensure that multiple conjunctions from the same debris object detected over subsequent propagation steps are only counted once. To quantify how threatening simulated conjunction events are for a re-orbit system, each conjunction event is assigned a level of combined risk based upon the relative state of the responsible debris object at the time of conjunction. Position andvelocity risk functions are employed to individually weight (a) how close the object comes to the re-orbit system, and (b) how fast the object is traveling relative to the re-orbit system. Specifically, the posi- tion and velocity risk functions selected for this analysis are (equivalent to Section 2.3.3): Rr(r) = ✓ r˜ r r˜ ◆2 , 0  r  r˜ (6.12) Rv(v) = 1 e3v/v¯, v 0 (6.13) where r˜ denotes the conjunction distance threshold, and r and v denote the magnitude of the relative position and velocity vectors at conjunction, respectively.4 As in Section 2.3.3, the scaling velocity v¯ ⇡ 0.2828 km/s, derived from the NASA Standard BreakupModel as the relative speed threshold above which collisions are “catastrophic,” assuming equiva- lent masses for the target and impactor.[46] After the relative position and velocity for the conjunction event are computed, the risk functions defined via Equations (6.12)-(6.13) are evaluated, such that the combined risk factor given by the productRr(r)Rv(v) is evaluated. This combined risk factor for the conjunction event is on the interval [0,1] and provides a metric for gauging how “threatening” the conjunction is for the re-orbit system. Note that both close proximity and sucient relative speed must be present for a conjunction event to be consideredhigh-risk under thismetric – higher-speed events near the distance thresh- old and lower-speed events near the re-orbit systemare bothde-weighted (cf. Section 2.3.3). Note that this riskmeasure is not associatedwith the probability of collisionmetric used in 4 Given that inertial frame positions and velocities for both the re-orbit system and the conjuncting debris object are known at the time of conjunction, the position r and velocity v for the debris object, relative to the re-orbit system, can be described in the local Hill frame using the relative motion algorithm in Reference 55. 108 conjunction assessment and should not be treated as equivalentwith this statisticalmetric. 6.4 Results of GEO Re-orbit Conjunctions Study 6.4.1 Global Conjunction Challenge To assess the global conjunction challenge for a given thrust level, a two-dimensional sweep over initial longitude on [0,360) and inclination on [0,15] is performed in 5 and 1 increments, respectively. The re-orbiting system is initialized using a = aGEO, e = 0.001, ! = M0 = 0, and right ascension of ascending node ⌦ selected to enforce the initial lon- gitude at the start epoch of 01 March 2014 (00:00:00 Zulu). Figure 6.5 illustrates the mean number of conjunctions per trajectory as a function of thrust level and distance threshold, globally averaged over the 1152 trajectories propagated for each of the seven thrust accel- erations surveyed, ranging from 106 to 103 m/s2 (that is, 1 mN to 1 N of thrust for a 1000 kg system). For all distances considered, the mean number of conjunctions per trajectory rises hyperbolically as the thrust level decreases, a consequence of the result that re-orbit duration to 300 km above GEO increases hyperbolically as the thrust level decreases (see Figure 6.3). Therefore, the lower the thrust level considered for the re-orbiting system, the more “challenging” themaneuver becomes from the perspective of conjunction potential – the required burn time is lengthened, such that more conjunctions with the large-scale de- bris population are possible over the duration of the maneuver. Note that for the 500 mN and 1 N levels, no conjunctions beneath 25 km are experienced across all 1152 trajectories surveyed at these thrust levels, emphasizing that the mean number of conjunctions for a given thrust level and distance threshold becomes nearly zero as the thrust level increases. In addition to quantifying the mean number of conjunctions per trajectory, it is also of interest to evaluate the standard deviation for the conjunction distributions generated at each thrust level and distance threshold. Figure 6.5(a) illustrates 1 swaths for each thrust level and threshold, highlighting that the 1 spread of these distributions decreases as the 109 106 105 104 103 Thrust Acceleration [m/s2] 0 1 2 3 4 5 M ea n C on ju nc ti on s pe rT ra je ct or y Distance: < 50 km Distance: < 25 km Distance: < 10 km 1 1 1 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) (a) Mean number of conjunctions with 1 swaths. 106 105 104 103 Thrust Acceleration [m/s2] 106 105 104 103 102 101 100 101 M ea n C on ju nc ti on s pe r Tr aj ec to ry Distance: < 50 km Distance: < 25 km Distance: < 10 km 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) No&conjunctions (b) Mean number of conjunctions (log y-axis). Figure 6.5: Mean number of conjunction events per trajectory as a function of thrust level and distance threshold shownwith hyperbolic best-fit lines on linear and logarithmic axes. thrust increases at each distance threshold. Thus, the number of conjunctions anticipated for a given trajectory at the 1 mN level, for example, becomes more uncertain in that this trajectory could experience fewer—or conversely, many more—conjunctions at a distance threshold of 50 km than the global average of approximately 2.7 conjunctions per trajectory at this distance. Inhomogeneities in conjunction potential are explored in the next sections. 6.4.2 Dependence on Initial Longitude and Inclination Next, it is of interest to evaluate whether the number of conjunctions experienced for a particular trajectory at a particular thrust acceleration is sensitive to the initial longitude and inclination of the re-orbit system at the beginning of the maneuver. Figure 6.6 shows the number of conjunctions at a distance threshold of 50 km experienced for each of the 1152 trajectories surveyed under 1 mN, 5 mN, and 10 mN thrust accelerations. Qualitative trends in the number of conjunction events experienced at each thrust acceleration are not observed, indicating no deterministic dependence of the number of anticipated conjunc- tions on the initial longitude and inclination of the re-orbit system. As the thrust accelera- tion increases, “holes” in the initial condition space begin appearing in greater frequency, 110 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (a) Number of conjunctions at 50 km (thrust level: 1.0⇥ 106 m/s2). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (b) Number of conjunctions at 50 km (thrust level: 5.0⇥ 106 m/s2). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (c) Number of conjunctions at 50 km (thrust level: 1.0⇥ 105 m/s2). Figure 6.6: Number of conjunction events experienced during a continuous-thrust re-orbit to 300 km above the GEO ring, illustrated as a function of initial longitude and inclination. illustrating the result of Figure 6.5 that the number of conjunctions experienced globally for a given thrust acceleration decreases hyperbolically as this thrust increases. Although the global average in Figure 6.5 indicates that the number of conjunctions is sparse at all distances considered for thrust levels larger than⇠5 mN, Figure 6.6(c) illustrates that par- ticular regions of the initial condition space could still be subject to upwards of three con- junction events at 50 kmover the course of the re-orbit. Thus, it is critical to emphasize that Figure 6.5 provides a first-order—not exhaustive—design tool for forecasting the number 111 of conjunction events that could be experienced during re-orbit at a particular thrust level. 6.4.3 Conjunctions in SMA and Longitude Space In addition to studying the dependence of the number of conjunctions on the initial longitude and inclination of the re-orbit system, it is important to quantify where conjunc- tion events are most likely to occur over the duration of the maneuver. Instead of studying the frequency of conjunctions as a function of elapsed time sincemaneuver start, it is more consistent to consider the distribution of conjunction events in semi-major axis (SMA) and longitude space. Using the total number of conjunctions detected at a distance threshold of 50 km accumulated across all thrust levels considered, Figure 6.7 shows the conjunction probability for various regions in this SMA/longitude phase space.5 The contributions of the librating debris population are evident around the two gravitational wells at 75E and 105W,with GEO-relative semi-major axes from 0-50 km above the GEO ring. As the SMA increases relative toGEO, neighborhoodswith higher conjunction probabilities are experi- enced, indicative of contributions by the westward drifting population at higher altitudes. The conjunction probability map illustrated in Figure 6.7 is an important tool in that it can be combined a prioriwith the analytic SMA/longitude predictions in Equations (6.5) and (6.9) to predictwhen a re-orbit systemwill be entering a region in the phase spacewith a higher conjunction probability relative to the surrounding regions in this phase space. Figure 6.7 further shows three re-orbit trajectories beginning at a longitude of 105W,with 1mN, 5mN, and 10mN of thrust acceleration, respectively. Given that the thrust accelera- tion and initial location in this phase space are known, the resulting re-orbit trajectory can be predicted with Equations (6.5) and (6.9), such that increased ground tracking can be re- quested prior to the re-orbit system entering a regionwith a higher conjunction likelihood. In the case of an autonomous system for ADR at GEO, onboard environmental sensing can 5 “Longitude” refers here to the longitude at which each conjunction event occurs during re-orbit, not to the initial longitude of the re-orbit system, which is illustrated in Figure 6.6. 112 0 30 60 90 120 150 180 210 240 270 300 330 360 Conjunction Longitude [deg] 0 50 100 150 200 250 300 C on ju nc ti on SM A [k m ] 0.0 0.2 0.4 0.6 0.8 1.0 C on ju nc tio n) Pr ob ab ili ty% % % % % % 1#mN 5#mN 10#mN Figure 6.7: Densitymap illustrating regions in SMA/longitude space that experience high frequencies of conjunction events at 50 km, shownwith three re-orbit trajectories at various thrust levels from an initial longitude of 105W, as predicted by Equations (6.5) and (6.9). 0 50 100 150 200 250 300 Conjunction SMA Relative to GEO [km] 0 20 40 60 80 100 C um ul at iv e C on j. Pr ob ab il it y [% ] Distance: < 50 km Distance: < 25 km Distance: < 10 km (a) Cumulative probability by semi-major axis. 0 60 120 180 240 300 360 Conjunction Longitude [deg] 0 20 40 60 80 100 C um ul at iv e C on j. Pr ob ab il it y [% ] Distance: < 50 km Distance: < 25 km Distance: < 10 km (b) Cumulative probability by longitude. Figure 6.8: Cumulative probability of conjunction event as function of semi-major axis and longitude during re-orbit burn, provided with uniform cumulative distribution function. be applied or augmented when the re-orbit system enters higher conjunction risk regions. If the conjunction probabilities illustrated in Figure 6.7 are binned by semi-major axis or longitude of conjunction only, the cumulative conjunction probability distributions for these twophase space variables—shown respectively in Figure 6.8—can be constructed. In particular, Figure 6.8(a) highlights that the cumulative probability of conjunction achieves 50% at approximately 120 km in SMA above the GEO ring across all thrust levels and dis- tance thresholds considered. Therefore, the distribution of conjunctions in semi-major axis 113 is not uniform; rather, it exhibits slight bias towards the beginning of the maneuver, a con- sequence of the higher densities of librating objects in closer proximity to theGEOaltitude. If equippedwith a variable-thrust propulsion system, the re-orbiting system could reduce (or potentially eliminate) the number of anticipated conjunctions by increasing the thrust acceleration during the initial stages of the re-orbit, such that the system achieves a 120 km increase in SMA as rapidly as possible. After, the thrust acceleration could be decreased to conserve propellant until the desired re-orbit is accomplished. Interestingly, the cumula- tive probability by longitude in Figure 6.8(b) does not show strong contributions from the two gravitational wells, since these longitudes are only significant for conjunction proba- bilities at lower GEO-relative altitudes in closer proximity to the GEO ring (cf. Figure 6.7). 6.4.4 Relative Velocity and Risk Considerations In addition to the average number of conjunctions experienced at a given thrust level, the dependence of this number of conjunctions on the initial conditions of the re-orbiting system, and where these conjunctions occur in SMA and longitude space, it is important to consider the relative velocities and combined risk factors for these conjunction events. For the 1mN thrust level, Figure 6.9 illustrates three portraits of the conjunctions observed at a distance threshold of 50 km across the 1152 trajectories surveyed at this thrust level – the number of conjunction events in Figure 6.9(a), the maximum relative velocities for the conjunctions detected for each trajectory in Figure 6.9(b), and themaximum combined risk of the conjunctions detected for each trajectory in Figure 6.9(c). Figure 6.9 highlights that although a particular re-orbit trajectory could experience multiple conjunction events be- neath 50 km over the duration of the re-orbit burn, the worst-case relative velocity of these conjunctions—or the worst-case combined risk thereof—could be benign, indicating that evasive action by the re-orbit system may not be necessary. Conversely, Figure 6.9 shows cases inwhich although only 1-2 conjunctions are experienced during themaneuver, these conjunctions carry a worst-case risk that may be significant enough to warrant evasive ac- 114 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (a) Number of conjunctions at 50 km (thrust level: 1.0⇥ 106 m/s2). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (b) Worst-case conjunction velocities (thrust level: 1.0⇥ 106 m/s2). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.1 0.2 0.3 0.4 0.5 C om bi ne d R is k Fa ct or (c) Worst-case conjunction risk factor (thrust level: 1.0⇥ 106 m/s2). Figure 6.9: Number of conjunctions at 50 km shownwithworst-case relative velocities and combined risk factor for these trajectories, as a function of initial longitude and inclination. tion (recall that conjunctions are threatening via the combination of proximity and speed). To complement the global conjunction challenges illustrated in Figure 6.5, Figure 6.10 provides the average number of conjunctions per trajectory at various thrust levels and risk factor thresholds, globally averaged over the 1152 trajectories propagated for each thrust acceleration. For each risk threshold, themean number of conjunctions per trajectory rises hyperbolically as the thrust level decreases. As the risk threshold increases, however, the mean number of conjunctions per trajectory for a given thrust level decreases in a manner 115 106 105 104 103 Thrust Acceleration [m/s2] 0 1 2 3 4 5 M ea n C on ju nc ti on s pe rT ra je ct or y Risk: > 0.0 Risk: > 0.1 Risk: > 0.2 Risk: > 0.4 1 1 1 1 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) (a) Mean number of conjunctions with 1 swaths. 106 105 104 103 Thrust Acceleration [m/s2] 106 105 104 103 102 101 100 101 M ea n C on ju nc ti on s pe r Tr aj ec to ry Risk: > 0.0 Risk: > 0.1 Risk: > 0.2 Risk: > 0.4 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) No&conjunctions (b) Mean number of conjunctions (log y-axis). Figure 6.10: Mean number of conjunctions per trajectory at 50 km as a function of thrust level and risk level, provided with hyperbolic best-fit lines on linear and logarithmic axes. that is analogous to decreasing the conjunction distance threshold (cf. Figure 6.5). Thus, at the 1 mN level, though the global conjunction challenge is approximately 2.7 conjunctions per trajectory on average at a distance threshold of 50 km, less than 0.5 conjunctions per trajectory on average are contributed by conjunctions with combined risk greater than 0.2. This global average is similar to that for a conjunction distance threshold of 25 km for the 1 mN thrust level in Figure 6.5. To highlight the 1 spread of the conjunction distributions generated for each thrust acceleration and risk threshold, Figure 6.10(a) illustrates 1 swaths in addition to themean number of conjunctions per trajectory determined for each thrust level and risk threshold. Analogous to Figure 6.5(a), the standard deviation decreases as the thrust level increases at a given risk threshold, and furthermore decreases as the risk threshold increases at a given thrust level. Thus, the average number of conjunctionswith a combined risk factor above a given risk threshold—and the spread of the distribution that this mean conjunctionmetric describes—decreases significantly as the thrust level increases beyond ⇠10 mN. Re-orbit maneuvers beneath this thrust level become more challenging in that larger uncertainties make the number of conjunctions for any set of initial conditions more dicult to predict. 116 50 40 30 20 10 0 10 20 30 40 50 Radial [km] 50 40 30 20 10 0 10 20 30 40 50 In -T ra ck [k m ] 0 10 20 30 40 50 R el at iv e Po si ti on [k m ] (a) Relative position distribution for conjunctions. 50 40 30 20 10 0 10 20 30 40 50 Radial [km] 50 40 30 20 10 0 10 20 30 40 50 In -T ra ck [k m ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (b) Relative velocity distribution for conjunctions. Figure 6.11: Distribution of conjunction events at 50 km in the local Hill frame, colored by relative position and relative velocity magnitude (thrust acceleration of 5.0⇥ 106 m/s2). 6.4.5 Distribution of Conjunctions in Hill Frame For each of the conjunction events detected across all thrust accelerations considered in this study, it is beneficial to consider the distribution of these conjunctions in the Hill frame, which is centered on the re-orbit system and has axes directed in the orbit radial, in-track, and cross-track directions.[55] For all conjunctions at a distance threshold of 50 km detected across all 1152 trajectories at a thrust acceleration of 5 mN, Figure 6.11 illustrates the conjunction distribution in the local radial/in-track plane by relative proximity in Fig- ure 6.11(a) and relative velocity in Figure 6.11(b). As Figure 6.11 highlights, trends in these relative position and velocity distributions are not observed, that is, the events detected are uniformly distributed (qualitatively) in the radial/in-track and radial/cross-track planes. Furthermore, these conjunctions occur at a broad range of relative speeds that are uncor- relatedwith the location of these conjunctions in theHill frame. No correlations in theHill framedistributionswith thrust acceleration, initial longitude or inclination, or conjuncting debris object class are observed. Figure 6.11 has important implications for guidance strategies for continuous thrust 117 re-orbit systems that are equipped tomitigate conjunction risk by adjusting the thrust pro- file, such that a suciently evasive trajectory relative to a conjuncting derelict is achieved. Since no correlations are qualitatively observed in the Hill frame position and velocity dis- tributions, guidance strategies for debris avoidance must be flexible and robust, designed to handle any given relative position and velocity in the Hill frame. Note that the conjunc- tions shown in Figure 6.11 are the positions of conjuncting debris when these objects first cross the 50 km distance threshold, which is not necessarily the time of closest approach to the re-orbit system for each of these derelicts. Positions at the times of “first approach” are used to simulate the environmental sensing and detection performed by an autonomous re-orbit system, activities that would determine whether evasive action is even necessary. 6.4.6 Eect of Delaying Start Time of Maneuver All of the re-orbiting trajectories considered thus far in this study are initialized at the start epoch 01March 2014 (00:00:00 Zulu), but “holes” present in the initial condition space in Figure 6.6 for each thrust level suggest that the number of conjunction events for a given trajectory can be reduced, or even eliminated, if the start time of the maneuver is delayed. To evaluate the eect of delaying the start time for the 1152 trajectories propagated at each thrust level, Figure 6.12 provides the number of conjunction events at a distance threshold of 50 km, for the 2mN thrust level, for start times in 6-hour increments over 01March 2014. Studying Figure 6.12, it is evident that the number of conjunction events for the majority of trajectories can indeed be minimized or eliminated at particular times of day, which are dependent on the initial longitude—and therefore local time—of the re-orbit system. Relative to an Earth-fixed observer, the collective motion of the debris field at GEO is qualitatively similar to a transverse wave with a period of one sidereal day, resulting from a clustering in right ascension of the ascending node driven by luni-solar perturbations.6 Section 2.4.3 confirms that equatorial crossing windows of the GEO debris population are 6 For an in-depth reviewof “pinchpoint” evolution resulting from luni-solar perturbations, seeChapter 3. 118 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (a) Number of conjunctions at 50 km (re-orbit starts at 00:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (b) Number of conjunctions at 50 km (re-orbit starts at 06:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (c) Number of conjunctions at 50 km (re-orbit starts at 12:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0 1 2 3 N um be ro fC on ju nc ti on s (d) Number of conjunctions at 50 km (re-orbit starts at 18:00:00 Zulu). Figure 6.12: Eect of delaying start time for re-orbit maneuver on the number of conjunc- tion events experienced across all trajectories surveyed at a thrust level of 2.0⇥106 m/s2. 119 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (a) Worst-case 50 km conjunction speeds (re-orbit starts at 00:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (b) Worst-case 50 km conjunction speeds (re-orbit starts at 06:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (c) Worst-case 50 km conjunction speeds (re-orbit starts at 12:00:00 Zulu). 0 30 60 90 120 150 180 210 240 270 300 330 360 Initial Longitude [deg] 0 4 8 12 16 In it ia lI nc li na ti on [d eg ] 0.0 0.2 0.4 0.6 0.8 1.0 R el at iv e V el oc it y [k m /s ] (d) Worst-case 50 km conjunction speeds (re-orbit starts at 18:00:00 Zulu). Figure 6.13: Eect of delaying start time for re-orbit maneuver on the worst-case conjunc- tion speeds experienced across all trajectories surveyed at a thrust level of 2.0⇥106 m/s2. 120 synchronized such that collision hazard to any longitude slot around GEO is episodic and predictable, dependent on time of year in addition to time of day. Therefore, delaying the start time for the re-orbit maneuver is analogous to shifting the latitudinal positions of the synchronized debris population located at the initial longitude of the re-orbiting system. Recalling that the re-orbit trajectories surveyed in this study are initialized at the ascending node in the equatorial plane, this suggests that trajectories beginning “in-sync” with local debris motion will, on average, experience more conjunctions than those beginning asyn- chronous to this transversewave of debris, albeitwith lower relative conjunction velocities. Figure 6.13 shows themaximum relative velocities for the conjunction events at a dis- tance threshold of 50 km illustrated in Figure 6.12. A prominent “wave-like” phenomenon that shifts westward nearly linearly across the longitude space as the start time is delayed over the course of one day is exhibited in Figure 6.13—a physical manifestation of the GEO debris synchronization eect on the conjunction challenges investigated in this thesis. As anticipated, the wave-like regions of minimum relative speed in Figure 6.13 correspond to those longitudes at which the local debris population is ascending through the equatorial plane from south to north during that time of day.[1] Since all propagated trajectories begin at the ascending node in the equatorial plane, these longitudes of reduced relative speed are those atwhich the re-orbiting systembegins either partially or fully in-phasewith local debris motion at the beginning of the maneuver. Conversely, the darker regions of higher relative velocities in Figure 6.13 are those at which the re-orbit system begins out-of-phase with the local debris population – the re-orbit system rises in latitude from the ascending node as the local debris population descends in latitude from north to south through the equatorial plane, leading to larger out-of-plane relative velocities at conjunction. To emphasize the importance of the start time of the re-orbitmaneuver in reducing or eliminating the number of conjunction events for a given initial longitude and inclination, Figure 6.14 provides the number of conjunctions at a distance threshold of 50 km for two trajectories—(0, i0)= (75,0) in Figure 6.14(a) and (0, i0)= (75,15) in Figure 6.14(b)—as 121 0 2 4 6 8 10 12 14 16 18 20 22 24 Delay in Start Time for Re-orbit Manuever [h] 0 2 4 6 8 10 12 C on ju nc ti on Ev en ts [< 50 km ] a 0.0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 1.0 Risk%Threshold (a) Conjunctions at 50 km (0 = 75, i0 = 0). 0 2 4 6 8 10 12 14 16 18 20 22 24 Delay in Start Time for Re-orbit Manuever [h] 0 2 4 6 8 10 12 C on ju nc ti on Ev en ts [< 50 km ] a 0.0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 1.0 Risk%Threshold (b) Conjunctions at 50 km (0 = 75, i0 = 15). Figure 6.14: Number of conjunctions as a function of maneuver delay (thrust acceleration of 2.0⇥106 m/s2) for two trajectories beginning at 75E longitude and 0/15 inclination. a function of maneuver delay in half-hour increments. The number of conjunction events at each delay time is further subdivided by combined risk level, to highlight that not only is the number of conjunction events subject to change under a delayed start time, but the risk factors for these anticipated conjunctions, as well. For both cases in Figure 6.14, there exist optimal delay windows at which the number of conjunction events at a distance threshold of 50 km experienced over the entirety of the re-orbit maneuver are eliminated. Thus, the timing for the re-orbit maneuver is critical, and should be selected such that asynchroniza- tion with local debris motion at the beginning of the maneuver is leveraged to reduce—or even eliminate—the number of conjunction events experienced at a particular thrust level. 6.5 Conclusions from Research Goal The conjunction challenges of using low-thrust propulsion for continuous thrust re- orbit of GEO objects to super-synchronous disposal orbits are investigated in this chapter. Lower maneuverability of re-orbit systems equipped with lower-thrust engines increases the challenges of collision avoidance, such that investigating the number of potential con- junction events experiencedwith the current debris population at GEOduring a typical re- 122 orbit to 300 km above GEO is an imperative task. In particular, a two-dimensional sweep over initial longitude and initial inclination is performed, inwhich surveyed re-orbit trajec- tories are propagated from the GEO ring under various thrust levels until 300 km increase in semi-major axis is achieved. Then, conjunctionswith large-scale GEOderelicts from the publicly-available TLE population are detected at various conjunction distance thresholds in post-processing, as described in the methodology outlined in Section 6.3. Results indicate that on average, the number of conjunction events experienced for a given thrust level and distance threshold rises hyperbolically as the thrust level decreases, in correlation with the maneuver duration for a given thrust level. Sensitivity to the initial longitude and inclination of the re-orbit system at a particular start time, thrust level, and distance threshold are not observed. Examination of the conjunction events in semi-major axis and longitude space, however, highlights locations of this phase space in which con- junctions aremore likely, e.g., around the gravitational wells within 50 km of the GEO ring in semi-major axis. Advantageously, timing the re-orbit maneuver appropriately—that is, such that the re-orbit system is asynchronous at the start of the transfer with the wave-like behavior of the local derelict motion—serves to reduce, or possibly eliminate, the number of conjunction events anticipated for a given trajectory at a given thrust level. No conjunc- tions beneath the distance threshold of 25 km are detected across all surveyed trajectories at the 5⇥ 104 m/s2 and 103 m/s2 levels, indicating that timing may not be necessary in these cases. However, inasmuch as the debris population at GEO continues to increase in the absence of perfect, global adherence tomitigation guidelines—and since debris objects smaller than approximately 0.8-1mare not included in this study—it is critical that re-orbit systems be equippedwith a robust guidance system to adjust the thrust profile in the event that timely evasion of conjuncting debris is warranted. 6.6 Continuing Research Questions For continuing research in this area, the following research questions are of interest: 123 (1) Howmany conjunction events for a given thrust level could the re-orbiting system experience in a realistic scenario, using chief and deputy covariance information? How many of these conjunctions might warrant evasive action to mitigate risk? (2) In the case of an autonomous re-orbit system using environmental sensing in tan- demwith a guidance strategy for collision avoidance, what is a realistic distance at which conjuncting debris objects could be detected using state-of-the-art sensors? (3) Could a thrust profile be implemented such that the re-orbit system remains asyn- chronizedwith the transverse debris wave throughout the entirety of the transfer? Chapter 7 Research Goal 3(b): Developing a guidance strategy for low-thrust debris avoidance during GEO re-orbit 7.1 Overview of Research Goal In Chapter 6, the conjunction challenges for a GEO re-orbit system using low-thrust propulsion to perform orbit raising at end-of-life, in compliance with the IADCmitigation guidelines, are investigated by quantifying the number of conjunction events possiblewith the current TLE population during a typical re-orbit to 300 km above GEO. In this study, it is shown that although the average number of conjunctions per trajectory rises hyperboli- cally as the thrust acceleration decreases, the global conjunction challenge can be reduced by (a) increasing the thrust acceleration, and/or (b) timing the start of the maneuver such that the re-orbit system begins out-of-sync with the north-south, latitudinal motion of the local debris population. Depending uponmission constraints, satellite bus design, and the particular low-thrust propulsion used, option (a) may not be possible. Option (b) guaran- tees that the potential conjunction events for a given re-orbit trajectorywill be reduced, but not necessarily eliminated. Therefore, it is of critical importance that such re-orbit systems be equippedwith a GNC subsystem specifically for steering the vehicle onto an avoidance trajectory in the event a predicted conjunction warrants timely, evasive action. In this way, any remaining collision risk not mitigated under options (a) or (b) can be further reduced. Although such aGNC subsystem is beneficial for operational GEO satellites engaged inmitigation activities at end-of-life, the application to autonomous, “space-tug” concepts 125 for environment remediation is emphasized in this chapter. Conceptualized in Figure 6.2, an autonomous re-orbit system forADRatGEOwould approach a high-risk target derelict, dock with it via either physical or contactless means, perform a continuous-thrust burn to increase the perigee altitude per the IADCguidelines, and lastly return to theGEOaltitude to repeat with other debris objects until fuel has been depleted. With lower maneuverabil- ity, however, collision avoidance for such a low-thrust re-orbit system during orbit raising is a challenging task. Conjunctions must be detected with enough lead time to suciently alter the trajectory of the re-orbit system tomitigate collision risk bymaintaining a desired miss distance, given covariance information and a specified “no-fly” (or “keep-out”) zone surrounding the derelict. Intuitively, re-orbit systems with higher thrust levels (more ma- neuverability) can detect a conjuncting object later than systems with lower thrust levels (lessmaneuverability), and still miss this object by the same distance at the collision epoch. Thus, in addition to a guidance strategy for online avoidance of conjuncting debris, the re- orbit systemmust have the capability to sense the surrounding environment, and forecast potential conjunction events using relative state estimation and two-body orbit prediction. Concepts for autonomous “sense and avoid” are widespreadwithin the robotics and unmanned aerial vehicle (UAV) communities, and have recently been applied to the chal- lenges of spaceflight, in particular, satellite cluster flight and rendezvous anddocking. The artificial potential function concept for autonomous guidancewas originally developed for terrestrial robotics applications,[66,67] andwas later applied to the spacecraft formation fly- ing problem for optimal reconfiguration,[68] distributed motion planning,[69] pattern tran- sition,[70] and autonomous operations.[71] In this method, an artificial potential field (that is, non-physical and for purposes of control only) is superimposed around a neighboring object, and this potential field is activated to either attract or repel the controlled vehicle to or from this object. [67] Alternatively, the problemof continuous-thrust, on-orbit rendezvous can be treated by minimizing a path-dependent cost function—subject to path constraints enforcing collision avoidance amongmultiple vehicles—using nonlinear optimization[72,73] 126 or mixed-integer linear programming.[74] All of these intensive path-planning strategies, however, are notwell-suited for autonomousGEO re-orbit systems,whichdemand simple- yet-robust control laws that are eective for on-orbit collision avoidance, whileminimizing (1) human-in-the-loop interactions and (2) computational load required for the avoidance. The guidance strategy for debris avoidance developed and simulated in this chapter is analogous to the conflict resolutionmaneuver proposed in Reference 75 for approaching aircraft in a congested control volume. It is shown that an aircraft entering the volume can always execute a lateral displacement maneuver—via a single heading change decision— that results in a conflict-free (collision-free) trajectory, as long as thewidth of themaneuver corridor is suciently large.[75] The guidance strategy for collision avoidance presented in this chapter uses a comparable heading change, i.e., the angle of the thrust vector, to result in a conflict-free trajectory inasmuch as the lead timewithwhich the conjuncting object has been detected is suciently large for the particular maneuverability of the re-orbit system. 7.2 Development of Guidance Strategy for Debris Avoidance 7.2.1 Nonlinear Relative Equations of Motion with In-Plane Chief Thrusting Prior to formulating the guidance strategy for debris avoidance, equations of motion that describe the trajectory of a deputy debris object relative to the chief re-orbit system— which is thrusting in the plane of the local orbit frame—are now derived. Using Cartesian coordinates in the rotating Hill frame centered on the re-orbit system, the inertial position vector of the deputy object is written as rd = rc + ⇢ = (rc + x)oˆr + yoˆ✓ + zoˆh (7.1) where rc is the orbital radius of the chief, and (x, y, z) are the Cartesian coordinates of the deputy object along the radial (oˆr), in-track (oˆ✓), and cross-track (oˆh) axes of theHill frame. The angular velocity vector of the Hill frameO relative to the inertial frameN is given by !O/N = f˙ oˆh (7.2) 127 where f˙ = h/r2 is the rate of true anomaly. Note that in-plane thrusting does not aect the angular velocity of the orbit frameO directly – rather, as will be shown, in-plane thrusting in the in-track direction aects the true anomaly acceleration, which in turn aects the true anomaly rate (accelerationdoes not instantaneously alter velocity). Applying the transport theorem to take two derivatives of rd with respect to inertial frame:[55] r¨d = h r¨c + x¨ f¨y 2f˙ y˙ f˙ 2(rc + x) i oˆr + h y¨ + 2f˙(r˙c + x˙) + f¨(rc + x) f˙ 2y i oˆ✓ + z¨oˆh (7.3) This kinematic expression can be simplified bydeveloping expressions for the true anomaly acceleration f¨ and chief radial acceleration r¨c. Recalling that the chief angular momentum magnitude is given by h = r2c f˙ , dierentiate and apply Euler’s equation h˙ = rca✓ to show that the true anomaly acceleration is f¨ = 2r˙cf˙ rc + a✓ rc (7.4) where a✓ is the in-track component of the chief thrust vector at = aroˆr + a✓oˆ✓. Writing the chief position in the orbit frame as rc = rcoˆr, and applying the transport theorem twice: r¨c = (r¨c rcf˙ 2)oˆr + a✓oˆ✓ (7.5) Assuming two-body dynamics, the inertial acceleration vector of the chief is written as: r¨c = µ r3c rc + at = ✓ ar µ r2c ◆ oˆr + a✓oˆ✓ (7.6) Equating the radial components of Equations (7.5)-(7.6), the chief radial acceleration is: r¨c = rcf˙ 2 µ r2c + ar (7.7) Substituting Equations (7.4) and (7.7) into Equation (7.3), the inertial acceleration vector of the deputy becomes: r¨d =  x¨ 2f˙ ✓ y˙ y r˙c rc ◆ f˙ 2x µ r2c + ar y rc a✓ oˆr +  y¨ + 2f˙ ✓ x˙ xr˙c rc ◆ f˙ 2y + (rc + x)a✓ rc oˆ✓ + z¨oˆh (7.8) 128 The unforced, two-body deputy equations of motion are given by r¨d = µ r3d rd = µ r3d (rc + x)oˆr µ r3d yoˆ✓ µ r3d zoˆh (7.9) where the deputy radius is rd = p (rc + x)2 + y2 + z2. Finally, equating the kinematics in Equation (7.8) with the dynamics in Equation (7.9) gives the exact nonlinear relative equa- tions of motion of the deputy with respect to the chief, which admits in-plane thrusting: x¨ 2f˙ ✓ y˙ y r˙c rc ◆ f˙ 2x µ r2c + ar y rc a✓ = µ r3d (rc + x) (7.10a) y¨ + 2f˙ ✓ x˙ xr˙c rc ◆ f˙ 2y + (rc + x)a✓ rc = µ r3d y (7.10b) z¨ = µ r3d z (7.10c) Note that if thrusting is turned-o, i.e., at = 0, the relativemotion equations reduce to the unforced form derived in Reference 55. Since linearization is not applied, Equations (7.10) are valid for arbitrarily large relativemotion. Furthermore, Equations (7.10) allow the chief orbit to be eccentric – the only assumptionmade in the derivation of these equations is that two-body dynamics apply for both the deputy debris object and chief re-orbit system, and the only perturbation to the chief’s orbital motion is that of the in-plane thrust vector at. 7.2.2 Eciency Factor for Re-orbit Maneuvers It is well-known that thrusting along the instantaneous velocity vector direction gen- erates maximum increase in the semi-major axis. [55,65] For the near-circular, low-thrust re- orbit trajectories considered in this thesis, the in-track direction is approximately oriented along the instantaneous velocity direction (i.e., small flight path angle). Assuming that the maximum available thrust acceleration amax is continuously applied during orbit raising, an eciency factor for the maneuver may be defined such that the thrust vector becomes: at = amax p 1 2oˆr + amaxoˆ✓ (7.11) 129 In this formulation, = 1 yields at = amaxoˆ✓, that is, maximum available thrust is directed along the in-track direction formaximum increase in SMA.1 Applyingmaximumavailable thrust in the in-track direction is appropriate for general re-orbit maneuvers at GEO, since this minimizes the duration of the transfer, which is lengthy for mN levels of thrust (recall Figure 6.3). Conversely, = 0 yieldsat = amaxoˆr, i.e., maximumavailable thrust is directed along the radial direction. Although this thrust direction still provides an increase in SMA, this rate of increase is minimal, since the Gauss variational equation for the SMA is[55] da dt = 2a2 h h e sin(f)ar + p r a✓ i (7.12) which shows that the eectiveness of the radial thrust component is reduced first by sin(f), then by the near-zero eccentricity (or, any thrust component not directed along the velocity vector direction does nowork on the re-orbit system to increase its two-body orbit energy). Letting↵ be the angle in the orbit plane directed from the local radial direction oˆr, the thrust vector of the re-orbit system can also be expressed in the form: at = amax cos(↵)oˆr + amax sin(↵)oˆ✓ (7.13) Equating the components of Equations (7.11) and (7.13), the eciency factor is = sin(↵), and is thus dependent on the angle ↵ of the thrust vector in the orbit plane. In this chapter, ↵ is restricted to 0  ↵  180, such that the thrust vector at never has a component in the oˆ✓ direction, whichwould serve to decrease the SMAduring re-orbit via Equation (7.12). 7.2.3 Formulation of Bisection Guidance Strategy In the event of a conjunction during re-orbit that warrants evasive action, a guidance strategymust be applied to tune the eciency factor —by adjusting the in-plane angle of the thrust vector—such that the re-orbit systemmisses the conjuncting object by a specified distance at the predicted collision epoch, as conceptualized in Figure 7.1. During nominal 1 Recall that this is the manner in which all surveyed re-orbit trajectories are propagated in Section 6.3. 130 oˆr oˆ oˆh at rc Re-orbit System Debris Object Predicted Collision Current: = 1 Adjusted: 1 < 1 Adjusted: 2 < 1 Figure 7.1: In-plane guidance concept for low-thrust debris avoidance. orbit raising, the re-orbit system maintains = 1, but if a potential collision is detected at a future time, the guidance systemmust decrease the eciency factor by applying a larger thrust component ar in the radial direction. In this manner, the rate of increase in SMA is temporarily slowed, such that the conjuncting objectmisses by a specified in-track distance y 6= 0 when x = z = 0 (recall that x = y = z = 0 indicates a collision in that the deputy is at the location of the chief, i.e., the origin of the Hill frame). This desiredmiss distance can be drawn from the largest position coordinate uncertainty for the debris object (the major axis of its covariance ellipsoid), a spherical “keep-out” zone, or the superposition thereof. Inasmuch as sucient lead time is available prior to the collision epoch given thema- neuverability of the re-orbit system, a predictor-correctormethod that uses Equations (7.10) in tandemwith a bisection root-findingmethod can be applied to achieve the desiredmiss distance. In this guidance strategy, the relative equations of motion in Equations (7.10) are propagated forward to the collision epoch using the current thrust direction. The in-plane thrust angle↵ is then iteratively adjusted via root bisection to converge on the angle↵⇤ that 131 Propagate)to) collision)epoch )using)current Evaluate)miss distance)rᵐ"# Is)rᵐ"#)=)desired) miss)distance? Yes No Is)rᵐ"#)<)desired?) (i.e.,)undershoot) )Decrease)) by)!!!!!!!step Is)rᵐ"#)>)desired?) (i.e.,)overshoot) Yes No Yes )Increase by)))))))step FINISH Root%is%bracketed%in permissible%interval: Reduce)!!!!!!!step (e.g.,)))))!!=!!!!!!!/10) START = 1 Figure 7.2: Flowchart for computer implementation of bisection guidance strategy. achieves the desiredmiss distance at the collision epoch. This “single-shooting”method is fast, robust, andwell-suited to an autonomous re-orbiting system that uses environmental sensing to detect the debris environment prior to a potential conjunction event. Aflowchart outlining the bisection guidance strategy for active debris avoidance is shown in Figure 7.2. Since numerical integration of Equations (7.10) requires initial conditions for the con- juncting debris object, the guidance systemmust have knowledge of the relative trajectory of the conjuncting object in the Hill frame, which can be obtained with (a) traditional, on- ground sensing and inertial orbit determination, or (b) onboard sensing and relative orbit determination using a batch least-squares or sequential state filter, in tandemwith relative range and anglesmeasurements. Furthermore, Equations (7.10) contain the chief radius rc and radial rate r˙c = r˙c ·oˆr, such that the re-orbiting systemmust have knowledge of its own inertial position and velocity vectors, in addition to the thrust vectorat and in-plane thrust angle ↵. This state knowledge can be obtainedwith (a) traditional, on-ground sensing and inertial orbit determination, (b) by applying a parametric form of position and velocity as a function of elapsed time, using the analytic results in Equations (6.5)-(6.6), for example, or (c) a combination thereof. Although not simulated in this thesis, the in-plane thrust angle must be updated at a predetermined frequency during the entirety of the debris approach, to account for navigation and force model errors, thruster performance uncertainties, and 132 other error sources not modeled in the two-body framework of Equations (7.10).2 Figure 7.3 shows several example approach geometries in the Hill frame to illustrate the eectiveness of the bisection guidance strategy for evading a conjuncting debris object with low thrust propulsion. Using a 1mN thrust level with a desiredmiss distance of 5 km at the simulated collision time of 24 hours from the start epoch, Figure 7.3 provides propa- gated trajectories without guidance (i.e., in-track thrusting only), and with bisection guid- ance applied. In all three cases, the bisectionmethod converges on = 0.706 to achieve the specifiedmiss distance of 5 km at the predicted collision epoch. Without the guidance ap- plied, a collision occurs at the origin of the Hill frame – with bisection guidance activated, however, the relative trajectory of the conjuncting object is altered such that it achieves zero radial positionwith nonzero in-track position at the collision epoch (i.e., x = 0with y 6= 0). To illustrate the eect of delaying the start time of the avoidancemaneuver, Figure 7.4 provides the trajectory example of Figure 7.3(b), if the conjuncting object is detected not 24 hours in advance of the predicted collision, but 18, 12, and 6 hours prior to the collision. In Figure 7.3(b), the bisectionmethod converges on = 0.706 to achieve themiss distance of 5 km. If, as in Figure 7.4(a), the object is detected 18 hours prior to the collision instead of 24 hours, an eciency of = 0.471 is required tomaintain this 5 km separation at the collision epoch. In Figures 7.4(b)-7.4(c), this guidance strategy does not have enough lead time to al- ter the relative trajectory such that 5 km separation is achieved, i.e., bisection fails because the root ↵⇤ cannot be bracketed in the allowable range 0  ↵  180. Applying maximum available thrust in the negative radial direction ( = 0) yields amiss distance of 2.92 km for the 12-hour detection in Figure 7.4(b), and⇠430m for the 6-hour detection in Figure 7.4(c). It is thus of critical importance to evaluate the tradeos among thrust acceleration, desired miss distance, and required detection time, assuming perfect knowledge in the absence of representative error sources, to establish “best-case” performance limits for this method. 2 In thismanner, the open-loop, “single-shooting” guidance strategy becomes closed-loop and robust in that the desiredmiss distance can be achieved even in the presence of error sources and unmodeled disturbances. 133 10 5 0 5 10 15 Radial Direction [km] 30 20 10 0 10 20 30 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (a) First example trajectory. 15 10 5 0 5 10 Radial Direction [km] 60 50 40 30 20 10 0 10 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (b) Second example trajectory. 20 10 0 10 20 30 40 Radial Direction [km] 20 0 20 40 60 80 100 120 140 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (c) Third example trajectory. Figure 7.3: Three examples of bisection guidance technique, which converges on = 0.706 to achieve a specified miss distance of 5 km at the predicted collision time of 24 hours out. 15 10 5 0 5 10 Radial Direction [km] 60 50 40 30 20 10 0 10 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (a) Detection 18 hours prior. 15 10 5 0 5 10 Radial Direction [km] 60 50 40 30 20 10 0 10 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (b) Detection 12 hours prior. 15 10 5 0 5 10 Radial Direction [km] 60 50 40 30 20 10 0 10 In -T ra ck D ir ec ti on [k m ] Without Guidance With Guidance Object'at Detection Re.orbit System (c) Detection 6 hours prior. Figure 7.4: Convergence of and the eciency factor resulting from bisection method are dependent upon the lead time with which the object is detected prior to the collision time. 134 7.3 Performance of Guidance Strategy for Debris Avoidance 7.3.1 Development of Prototype Collision Scenarios To quantify “best-case” performance bounds for the bisection guidancemethod, rep- resentative approach geometries in the local Hill frame, i.e., prototype conjunction geome- tries that a re-orbit system could encounter during orbit raising, must first be constructed. Then, these baseline scenarios are applied with a two-dimensional sweep over thrust level andmiss distance, to assess theminimumdetection times required tomaintain the desired miss distances at the collision epoch for each surveyed thrust acceleration. Assuming two- body dynamics, Figure 7.5 provides three collision scenarios that could occur for a re-orbit system during orbit raising from the GEO altitude to a super-synchronous disposal orbit: (1) Case 1. Collision due to a semi-major axis dierence, in which the re-orbit system asymptotically approaches the object at the collision pointwith nearly zero relative velocity at impact (i.e., re-orbit trajectory “spirals-up” to the conjunction altitude). (2) Case 2. Collision due to an eccentricity dierence, in which the system approaches the object at a point of intersection between two, coplanar orbits with e1 6= e2, with 10’s of m/s of relative velocity at impact (assuming typical eccentricities at GEO). (3) Case 3. Collision due to an inclination dierence, in which the system approaches the object at one of the orbital nodes, i.e., points of intersection between two, non- coplanar orbitswith i1 6= i2, with 100’s - 1000’s ofm/s of relative velocity at impact. Cases 1 and 2 involve in-plane relative motion only, while Case 3 has out-of-plane relative motion. Noting that relative velocity at impact increases by an order ofmagnitude between each of these cases, the prototype scenarios in Figure 7.5 serve as themeans for comparing bisection performance bounds under successively-increasing collision velocity. Usingdata from the GEO re-orbit conjunctions study in Chapter 6, Figure 7.6 provides a histogram of the relative velocity for all conjunction eventswithin 50 kmdetected across all seven thrust 135 Collision Point (a) Case 1: SMA dierence. Collision Points (b) Case 2: Eccentricity dierence. Collision Point Collision Point (c) Case 3: Inclination dierence. Figure 7.5: Representative GEO collision geometries for testing bisection debris avoidance strategy, where collision speed increases from SMA case at left to inclination case at right. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Relative Velocity at Time of Conjunction [km/s] 0 50 100 150 200 250 300 C on ju nc ti on Ev en ts [< 50 km ] 1 mN 5 mN 10 mN 50 mN 100 mN 500 mN 1 N Figure 7.6: Distribution of 50 km conjunction velocities, derived from re-orbit conjunction data in Chapter 6, by thrust level. Shaded region illustrates catastrophic collision regime. accelerations surveyed. This histogram shows an extensive range of conjunction velocities are possible for simulated continuous-thrust re-orbit trajectories. In particular, of the 4099 conjunction events detected within 50 km across all surveyed thrust levels, 80.2% are at or above the catastrophic relative velocity threshold of 0.2828 km/s, detailed in Section 6.3.[46] Although the simulated conjunctions inChapter 6 cannot be categorized exclusively under one of the prototype cases in Figure 7.5—since conjunctions due to a combination of orbit element dierences occur in practice—the histogram in Figure 7.6 shows that the majority of simulated conjunctions fall under Cases 2-3 in terms of relative velocity at conjunction. 136 To derive specific eccentricity and inclination dierences to apply to these prototype collision scenarios, Chapter 6 conjunction events within 10 km at the 1 mN thrust level are examined. Of the conjunction events detected at this 10 km distance threshold and thrust level, one conjunction results from an eccentricity dierencee = 0.028 (relative speed of 86 m/s), and another results from an inclination dierence ofi = 12.1 (relative speed of 649m/s). The semi-major axis dierence scenario is simulated by propagating the re-orbit system forward from the GEO radius under continuous, in-track thrust until the specified collision epoch; the Cartesian state of the re-orbit system is then assigned to the conjunct- ing object, which is then reverse-propagated to the simulation start epoch under two-body dynamics to provide the required initial conditions. Figure 7.7 illustrates the resulting ap- proach trajectories in the Hill frame for each of the three prototype scenarios in Figure 7.5, using a 1mN thrust level and specifying that the collisionwill occur 24 hours in the future, inasmuch as the bisection guidance method is not applied to alter the approach geometry. 7.3.2 Performance Bounds of Bisection Guidance Method Performance bounds for the bisection guidance method are investigated with a two- dimensional sweep over thruster acceleration andmiss distance at the collision epoch. For each thrust acceleration and desiredmiss distance, an auxiliary bisection technique is used to evaluate the minimum detection time required to maintain the surveyed miss distance at the collision time, i.e., the latest possible time that the debris object can be detected such that the re-orbit system can still evade the objectwhile achieving the desiredmiss distance. As it is more advantageous for the re-orbit system the earlier it detects a conjuncting object —since resulting eciency factorswill be closer to the = 1maximum—this performance study illustratesworst-case, full-radial performance ( = 0) under “best-case” knowledge, that is, perfect chief and deputy initial conditions, no navigation or forcemodel errors, etc. Figure 7.8 shows bisection performance bounds for each of the three representative collision scenarios in Figure 7.7. Thrust accelerations ranging from 106m/s2 to 103m/s2 137 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 0.04 0.02 0.00 0.02 0.04 Collision Point C ro ss *T ra ck /D ir ec tio n/ [k m ] ! . ! . 0 2 4 6 8 10 In*Tr ack/D irecti on/[k m] 0 0.5 1.0 1.5 2.0 Radial/D irection/[km ] (a) Case 1: Asymptotic SMA approach. 1000 500 0 500 1000 4000 3000 2000 1000 0 0.04 0.02 0.00 0.02 0.04 Collision Point C ro ss *T ra ck /D ir ec tio n/ [k m ] 0 "1000 "2000 "3000 "4000 In*Tr ack/D irecti on/[k m] 1000 500 0 "500 "1000 Radial/D irection/[km ] " . " (b) Case 2: Eccentricity dierencee = 0.028. 1000 800 600 400 200 0 400 200 0 200 400 10000 5000 0 5000 10000 Radial&D irection&[km ] C ro ss 4T ra ck &D ir ec tio n& [k m ] Collision Point !200 !400 !600 !800 !1000 0 !200!400 0 200 400 In4Tr ack&D irecti on&[k m] ! ! (c) Case 3: Inclination dierencei = 12.1. Figure 7.7: Relative Hill frame trajectories for representative GEO collision geometries. (i.e., 1 mN to 1N of thrust force for a 1000 kg re-orbit system) are surveyed in tandemwith miss distances ranging from 10 m to 10 km at the collision epoch. For example, if a Case 1 approach geometry is assumed, Figure 7.8(a) shows that a re-orbit systemwith 106 m/s2 of thrust acceleration requires 6 hours prior to the predicted collision epoch, at minimum, tomiss this object by⇠100musing the bisection guidance strategy. If the object is detected 138 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 106 105 104 103 Thrust Acceleration [m/s2] 10 m 100 m 1 km 10 km D es ir ed M is s D is ta nc e at C ol li si on Ep oc h [k m ] 0 0.5 hr 1 hr 2 hr 6 hr 12 hr 1 day 2 days M in im um D et ec ti on pr io rt o C ol li si on [h ] 1!h ou r 6!h ou rs 12! ho urs 2!h ou rs 1!d ay 30! mi n (a) Case 1 performance: Asymptotic SMA approach. 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 106 105 104 103 Thrust Acceleration [m/s2] 10 m 100 m 1 km 10 km D es ir ed M is s D is ta nc e at C ol li si on Ep oc h [k m ] 0 0.5 hr 1 hr 2 hr 6 hr 12 hr 1 day 2 days M in im um D et ec ti on pr io rt o C ol li si on [h ] 1!h ou r 6!h ou rs 12! hou rs 2!h ou rs 30! mi n (b) Case 2 performance: Eccentricity dierencee. 1!mN 10!mN 100!mN 1!N Thrust!Force!(Assuming!1000!kg!System!Mass) 106 105 104 103 Thrust Acceleration [m/s2] 10 m 100 m 1 km 10 km D es ir ed M is s D is ta nc e at C ol li si on Ep oc h [k m ] 0 0.5 hr 1 hr 2 hr 6 hr 12 hr 1 day 2 days M in im um D et ec ti on pr io rt o C ol li si on [h ] 1!h ou r 6!h ou rs 12! ho urs 2!h ou rs 30! mi n (c) Case 3 performance: Inclination dierencei. Figure 7.8: Performance bounds of bisection method for derelict avoidance, illustrated for various miss distances with the prototype GEO collision geometries in Figures 7.5 and 7.7. 139 earlier than this bound, the resulting eciency factorwill improve; conversely, if the object is detected any later than this bound, bisection root-bracketing fails, and the specifiedmiss distance is no longer achievable given themaneuverability of the re-orbit system. Note the latter does not state a collisionwill occur; rather, the desiredmiss distance is not achievable. The contours in Figure 7.8 illustrate that as the thrust acceleration decreases, and the desiredmiss distance increases, the minimum required detection time rises exponentially. This is an intuitive result, for the less maneuverability a re-orbit system has, the sooner its guidance subsystemmust know in advance of a predicted conjunction to begin altering the in-plane thrust angle such that the desiredmiss distance is achieved at the collision epoch. Thewhite triangular regions in the southeast corners of Figure 7.8 are those regions in this parameter space inwhich less than 30minutes of lead time are required tomiss the derelict using a full radial burn ( = 0). Interestingly, performance bounds improve as the relative velocity of the encounter increases, e.g., using 106m/s2 of thrust acceleration and a 100m miss distance, Case 1 in Figure 7.8(a) requires aminimumdetection time of approximately 6 hours, but this is reduced to approximately 3 hours in Figure 7.8(c) for Case 3 geometry. This is a consequence of faster relativemotion between the re-orbit system and conjuncting object at the collision epoch. Using a least-squares curve-fitting tool, the performance plots for Cases 1 and 3 in Figures 7.8(a) and 7.8(c) can be approximated by the analytic functions tmin(x˜) ⇡ exp(0.2396x˜2 + 0.4124x˜+ 0.2184) 0.5572 [Case 1] (7.14) tmin(x˜) ⇡ exp(0.2075x˜2 + 0.4201x˜+ 0.2583) 0.9666 [Case 3] (7.15) where x˜ ⌘ 1p 2 [log10(amax) log10(dmiss)]. In this formulation, amax is the thrust acceleration, dmiss the desired miss distance, and tmin the minimum detection time required to maintain dmiss at the collision time. The coecient of determination that measures goodness of fit is R2 = 0.976 for Case 1 and R2 = 0.980 for Case 3, indicative of a good fit to the minimum detection times evaluated in the thrust/distance parameter space illustrated in Figure 7.8. Note that although bisection performance bounds for Cases 2 and 3 in Figures 7.8(b)- 140 7.8(c) are qualitatively similar, a bump observed on the 6-hour contour line in Figure 7.8(b) is not exhibited in Figure 7.8(c). When computing minimum detection times in Figure 7.8, trajectories are propagated usingmaximum available thrust in both the positive radial (oˆr) and negative radial (oˆr) directions, since these directions both correspond to = 0 (i.e., ↵ = 0 and ↵ = 180, respectively). As a consequence of the approach geometry for Case 2 in Figure 7.7(b), thrusting in theoˆr direction gives larger miss distances southwest of the bump, and thrusting in the oˆr direction gives larger miss distances northeast of the bump. This observed “bump discontinuity” does not appear in Cases 1 and 3, since thrusting in theoˆr direction provides largermiss distances across the entirety of the parameter space. 7.4 Conclusions from Research Goal Acomputationally ecient guidance strategy for debris avoidance during low-thrust re-orbit to a super-synchronous disposal orbit is investigated in this chapter. The guidance strategy employs a bisection root-finding method to converge on the in-plane thrust angle that gives the desiredmiss distance at the predicted collision epoch, inasmuch as sucient lead time is available prior to the conjunction. Designedwith simple-yet-robust autonomy in mind, this bisection guidance strategy is ultimately a component of a larger, “sense and avoid” system tomitigate collision risk during orbit raising at end-of-lifewhileminimizing human-in-the-loop requirements. When coupled with environmental sensing and an on- board, relative state estimator, bisection guidance can be applied to safely avoid specified “no-fly” regions around conjuncting derelicts, while continuing to increase the semi-major axis during a re-orbit. The performance bounds quantified in this chapter illustrate that for typical conjunction velocities probable during GEO re-orbit, a desiredmiss distance of 1-2 km requires less than 12 hours of lead time for all surveyed thrust levels ranging from 106 m/s2 to 103 m/s2. To compliment Chapter 6—which shows that conjunction challenges during GEO re-orbit can be reduced by increasing the thrust level and/or timing the start of the re-orbit burn appropriately—this chapter demonstrates that any remaining collision 141 risk can be safely mitigated with bisection guidance for online avoidance of debris objects. 7.5 Continuing Research Questions For continuing research in this area, the following research questions are of interest: (1) The bisectionmethod developed in this chapter appliesmaximumavailable thrust in the radial/in-track plane of the local Hill frame to evade a conjuncting derelict. Can related guidancemethods be developed that allow for (a) variable thrustmag- nitude, (b) an out-of-plane thrust vector component, (c) coasting periods, in which the thrust vector is turned-o (i.e., at = 0), or (d) an optimal combination thereof? Investigating how to incorporate coasting periodswould be especially useful, since satellites with thrusters fixed in the body frame would have to execute an attitude maneuver to achieve the thrust vector heading commanded by bisection guidance. (2) Can optimal control theory be leveraged to develop an optimal, continuous-thrust guidance law for derelict avoidance, in which the thrust profile is adjusted to min- imize a cost function J while simultaneously enforcing various path constraints? (3) What are the eects of various error sources—such as chief anddeputy state knowl- edge, navigation and forcemodel errors, thruster pointing/execution errors, etc.— on convergence, robustness, and applicability of the bisection guidance technique? (4) Using themean conjunctions per trajectorymetric in Chapter 6 in tandemwith the bisectionmethoddeveloped in this chapter, can a lifetime fuel budget be estimated for an ADR system at GEO that uses low-thrust propulsion to perform the re-orbit for a variety of derelict GEO objects? Note that once in super-synchronous orbit, a re-orbit systemmust burn against the velocity vector to return to theGEO altitude. Chapter 8 Conclusions In this thesis, entitled “Characterizing Longitude-DependentOrbital Debris Conges- tion in the Geosynchronous Orbit Regime,” multiple dimensions of the problem of orbital debris at GEO are investigated. The GEO regime is home to hundreds of large, high-value satellites from the civil, commercial, and defense sectors, leveraged for applications rang- ing from communications anddata relay toweathermonitoring and classified surveillance activities. As the lack of environmental cleansingmechanisms at the GEO altitude renders the lifetimes of objects in this regime indefinitely long, debris generated near GEO remains there permanently – until environment remediation via a cost-eectiveADR system is syn- thesized, demonstrated, and implemented to remove large-scale derelicts and small-scale, undetectable fragmentation particulates that threaten the integrity of on-orbit GEO assets. Although recent re-orbit statistics are promising in that they suggest increased adherence to end-of-lifemitigation guidelines—and ultimately, a growing international desire to pre- serve the long-termusability of this regime—complete global compliance is challenging to reach without an internationally-recognized legal framework for enforcing space sustain- ability laws (especially at GEO, where profit is proportional to operational lifetime). Thus, it will become increasingly apparent that reactive mitigation initiatives must be augmented by proactive remediation initiatives to ultimately “solve” the problem of orbit debris at GEO. Since operational GEO satellites are required tomaintain designated longitude slots, operators cannot execute a phase shift in true anomaly (as can be done at LEO) to mitigate 143 risk incurred by conjuncting debris. Currently, the large-scale debris population at GEO is sparse enough that collision avoidance maneuvers can be built-in to routine maintenance burns. As the GEO debris population continues to increase, however, due to (a) mitigation noncompliance, (b) lack of an operational ADR system, or (c) older, non-passivated rocket bodies and payloads that threaten to explode, the costs to operate in the GEO environment will continue to rise in proportion. This thesis research thus serves to build context for the problem of orbital debris at GEO, by identifying and characterizing longitude-dependent congestion trends to “provide a foundation for future active risk reduction options.”1 In Chapter 2, longitude-dependent congestion generated by the large-scale, unclassi- fied, and trackable debris population currently atGEO is forecasted and characterized over a five-year prediction period. The near-miss event metric leveraged for quantifying local- ized congestion is introduced, and position and velocity risk functions useful for assigning a threat level to each near-miss event are presented. Congestion predictions indicate that controlled satellites in the longitude slots neighboring the two gravitational wells are sub- ject to 4-8 near-misses per day at 50 km, four times the 1-2 near-misses per day experienced in the least congested longitude slots (i.e., covering the Atlantic and Pacific Oceans). Risk data—accumulated either globally or locally across a subset of longitude slots—is used to prioritize the debris population for ADR target selection. In particular, the librating class derelicts collectively account for 76% of globally accumulated risk over a five-year period, even though this subset comprises only 14% of the RSO population at GEO. In the vicinity of theWestern gravity well, the top 9 class L2 derelicts alone are responsible for 50% of the risk locally accumulated in that region during the five-year congestion forecasting period. InChapter 3, analytical theory describing inclination andRAANvariations driven by luni-solar perturbations ismergedwith observational data to explain apparent anomalous behavior exhibited by eight derelicts with unsynchronized latitude cycles. The phase por- 1 See D. McKnight, “Pay Me Now or Pay Me More Later: Start the Development of Active Orbital Debris Removal Now,”White Paper, Integrity Applications, Inc., 2010. 144 trait for the doubly-averaged equations describing inclination and RAANvariations at the GEO altitude admit two types of behavior, one in which the RAAN circulates in the phase space (Type I), and the other in which the RAAN continually regresses (Type II). “Outlier” objects exhibiting either partially or fully out-of-phase latitude cycles followType II curves, such that these objects shift from asynchronous (with the rest of the debris population) to synchronous—and back—over the course of this 53-year inclination/RAAN progression. In Chapter 4, longitude-dependent congestion trends under the eect of a represen- tative, linearly-increasingGEO launch tracmodel are evaluated. Historical, present, and projected operational orbit data are applied to construct probability density functions that sample the semi-major axis, eccentricity, inclination, and longitude slot for new controlled satellites artificially created during long-term forecasting. In tandemwith a launch instan- tiation procedure, this population augmentation method is leveraged to simulate conges- tion trends over a 50-year forecast –with andwithoutmitigation compliance at end-of-life. Results demonstrate that even under optimistic, 80% re-orbit compliance, daily near-miss events at 300 km in the vicinity of the gravitational wells will double in 50 years, illustrat- ing the need for a coupling between aggressivemitigation compliance and active removal. In Chapter 5, longitude-dependent congestion generated by fragmentation events— explosions and collisions—in the GEO regime is evaluated. The NASA Standard Breakup Model is applied to sample initial conditions for the fragment distribution resulting from a simulated breakup event, and the longitude of the parent body at the epoch of fragmen- tation is varied to demonstrate how the severity of a breakup event at GEO depends upon initial longitude. A breakup occurring in the vicinity of one gravitational well is a “worst- case” scenario in that fragments are more likely to begin librating about the well, resulting in bi-annual bunching at the amplitude of the parent body’s librationmotion. Inasmuch as the semi-major axis distribution resulting froma fragmentation event can be estimated, the harmonic oscillator model presented in this chapter can be applied to predict to first order the specific times and longitudes at which congestion spikes will occur (in the short term). 145 In Chapter 6, the conjunction challenges for operational satellites or space-tug ADR concepts using low-thrust propulsion to achieve a super-synchronous graveyard orbit are quantified. A two-dimensional sweep over initial longitude and inclination is performed, and conjunctions with the TLE debris population are detected at various distance thresh- olds for each continuous-thrust re-orbit trajectory surveyed. Results show that on average, the number of conjunction events expected for a given thrust level and distance threshold rises hyperbolically as the thrust level decreases, in correlationwith the burn duration for a given thrust level. Furthermore, examination of the conjunction events in semi-major axis and longitude space indicates locations in this space in which conjunction events aremore likely. Beneficially, timing the maneuver appropriately—that is, such that the re-orbit sys- tem is out-of-phase at the start of the transfer with the synchronized latitudemotion of the local debris population—reduces or even eliminates the expected number of conjunctions. In Chapter 7, a computationally-ecient guidance strategy for debris avoidance dur- ing low-thrust orbit raising to a super-synchronous configuration is presented. The guid- ance strategy uses a bisection root-findingmethod to converge on the in-plane thrust angle that gives the desiredmiss distance at the predicted collision epoch, inasmuch as sucient lead time is available prior to the conjunction. When coupled with environmental sensing and an onboard, relative state estimator, bisection guidance can be applied to safely avoid specified “no-fly” zones around conjuncting debris, while continuing to increase the semi- major axis during re-orbit. Performance bounds assessed in this chapter show that for con- junction speeds probable during a re-orbit, a desired miss distance of 1-2 km requires less than 12 hours of advance notice for all surveyed thrust levels from 106m/s2 to 103m/s2. The research presented in this thesis is by no means exhaustive in scope; rather, the intent of this work is to tackle core aspects of the fundamental research questions posed in Chapter 1, and ultimately, demonstrate the increasingly relevant need for ADR in the GEO ring. Withmore aggressivemitigation standards, timelyADR implementation, and global cooperation, this unique and delicate regime can be preserved for the generations to come. Bibliography [1] D. S. McKnight and F. R. 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Epenoy, “Fuel Optimization for Continuous-Thrust Orbital Rendezvous with Col- lision Avoidance Constraint,” Journal of Guidance, Control and Dynamics, Vol. 34, March-April 2011. [74] A. Richards, T. Schouwenaars, J. P. How, and E. Feron, “Spacecraft Trajectory Plan- ningwithAvoidanceConstraints usingMixed-Integer Linear Programming,” Journal of Guidance, Control and Dynamics, Vol. 25, July-August 2002. [75] Z.-H. Mao, E. Feron, and K. Bilimoria, “Stability of Intersecting Aircraft Flows un- der Decentralized Conflict Avoidance Rules,” Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA, 2000, pp. 1042–1052. Appendix A The Geosynchronous Orbit Environment A.1 Fundamentals of GEO Orbit Geosynchronous orbit (GEO) is a unique orbit about the Earth inwhich the period of motion is approximately one sidereal day (86164 seconds), such that a GEO satellitemoves within its orbit at the same rate that the Earth rotates about its spin axis. [38] In this manner, GEO satellites appear from the Earth-centered, Earth-fixed (ECEF) frame to be stationary in longitude; those synchronous satellites that are furthermore in circular (e = 0), equato- rial (i = 0) orbits are geostationary in that these satellites are completely motionless in the sky from the perspective of a ground observer fixed to Earth. The rotation rate of Earth is approximately ! ⇡ 72.9217⇥ 106 rad/s,[41] which is equal to the mean motion for GEO: ! = r µ r3GEO =) rGEO = 42,164.17 km (A.1) Thus, the GEO ring is approximately 35,786 km (or 22,236 miles) above the surface of the Earth,1 an altitude atwhich a single GEO satellite can observe⇠42% of the Earth’s surface, with latitudinal coverage extending from 81S to 81N.[41] The GEO ring is therefore very attractive for communications, data relay, and Earth-observation applications because of such extensive on-ground coverage, and furthermore because it is not necessary to accu- rately follow GEO satellites to determine where to point antennas (e.g., DIRECTV dishes). Figure A.1 shows two views of Earth from the GEO altitude, captured in late January 2015 1 Put in perspective, this altitude is roughly nine times the distance from New York City to Los Angeles, which is approximately one-tenth of the average distance from the Earth to its nearest neighbor, the Moon. 153 Figure A.1: Full disk views of Earth from GOES East and West satellites. by the National Oceanic and Atmospheric Administration’s (NOAA) Geostationary Oper- ational Environmental Satellite system (GOES), currently stationed at 105W and 135W.2 Relative to the rotating ECEF frame, true geostationary satellites appear motionless, i.e., the ground track over one revolution appears as a single point in the equatorial plane. Inclined, geosynchronous objects in circular orbits have ground tracks that appear as ver- tical lines, reflecting the short-period change in latitude experienced over each revolution. Inclined, geosynchronous objects in elliptic orbits exhibit “figure-8” analemmas as a result of non-zero eccentricity. Although the meanmotion for these objects matches the rotation rate of Earth, non-zero true anomaly acceleration induces eastward drift during periapsis passage that is counteracted bywestward drift accumulated during apoapsis passage. The combination of this daily longitude pattern with north-south latitude motion gives the re- sulting analemmaa “double teardrop” or “figure-8” shape. Inclined, near-geosynchronous objects have sinusoidal ground tracks, which result from the combination of north-south latitude motion with continuous drift in longitude (eastward for objects beneath the GEO altitude, and westward for objects above the GEO altitude). Figure 2.3 in Chapter 2 shows 2 Full disk, hemispheric, and regional images and visualizations are available on NOAA’s Geostationary Satellite Server at: http://www.goes.noaa.gov. 154 several examples of these once-daily ECEF trajectories relative to the GEO longitude slots. A.2 Natural Perturbations in the GEO Environment Asdescribed inChapter 2, the two-body equations ofmotion for uncontrolled objects at GEO are augmented by various environmental perturbations that naturally occur at the GEO altitude. Specifically, the equations of motion for each uncontrolled GEO object are r¨ = µ r3 r + a + a% + a + aSRP (A.2) where the first term denotes two-body acceleration, a is the perturbation due to the non- sphericity of Earth, a% and a are the third-body perturbations from the Moon and Sun, respectively, and aSRP is the SRP perturbation. In the following sections, the specific com- putations applied for evaluating each of the perturbations in Equation (A.2) are presented, as implemented in customANSI-C software tools developed for this dissertation research. A.2.1 Non-Spherical Earth Perturbations Non-spherical Earth perturbations are commonlymodeled via a spherical harmonics expansion of the Earth’s gravitational field, inwhich the retention of higher-order terms in the series expansion provides a higher-fidelitymodel for the “lumpy” topography of Earth (recall that the two-body acceleration in Equation (A.2) is derived under the uniformity of a point-mass assumption). Using a potential functionU tomodel the irregular distribution of Earth’s mass, a is expressed in Earth-centered, Earth-fixed frame components as:[38] 3 ax = 1 r @U @r rz r2 p r2x + r 2 y @U @ ! rx ✓ 1 r2x + r 2 y @U @ ◆ ry (A.3) ay = 1 r @U @r rz r2 p r2x + r 2 y @U @ ! ry + ✓ 1 r2x + r 2 y @U @ ◆ rx (A.4) az = rz r @U @r + p r2x + r 2 y r2 @U @ (A.5) 3 Note that because Equations (A.3)-(A.5) provide the perturbing acceleration in non-inertial ECEF com- ponents, the acceleration vectormust be converted into inertial frame components for numerical integration. 155 where (rx, ry, rz)T denotes the satellite position vector in ECEF components, anddenote geocentric latitude and longitude, respectively, and the partial derivatives of the potential with respect to spherical coordinates are evaluated to the desired degree and orderwith[38] @U @r = µ r2 1X l=2 lX m=0 ✓ R r ◆l (l + 1)Pl,m(sin) h Cl,m cos(m) + Sl,m sin(m) i (A.6) @U @ = µ r 1X l=2 lX m=0 ✓ R r ◆l h Pl,m+1(sin)m tan()Pl,m(sin) i ⇥ h Cl,m cos(m) + Sl,m sin(m) i (A.7) @U @ = µ r 1X l=2 lX m=0 ✓ R r ◆l mPl,m(sin) h Sl,m cos(m) Cl,m sin(m) i (A.8) whereR is the Earth’s equatorial radius, Pl,m denote the associated Legendre functions of degree l and orderm, Cl,m and Sl,m are the un-normalized Stokes coecients for the EGM- 96 gravitymodel,4 and the geocentric latitude and longitude coordinates are computed by = sin1 rzp r2x + r 2 y + r 2 z ! (A.9) = tan1 ✓ ry rx ◆ (A.10) As shown in Figure A.2, three classes of spherical harmonics are described by the Stokes coecients: (a) zonal harmonics, defined by l 6= 0 and m = 0, which represent symmetric latitude bands, (b) sectorial harmonics, defined by l = m, which represent longitude bands, and (c) tesseralharmonics, defined by l 6= m 6= 0, which describe specific, “tile-like” regions on the surface of the Earth.[3,38] The dominant zonal harmonic J2 ⌘ C2,0 describes the gravitational perturbations resulting from the oblateness of the Earth, while the coecient J2,2 ⌘ q C22,2 + S 2 2,2 describes the longitudinal “bulges” responsible for libration motion— and consequently, the need for weekly, east-west station-keeping—of satellites near GEO. Section A.3 gives a high-level overview of why libration motion physically occurs at GEO. 4 Normalized Stokes coecients for the EGM-96 gravity model are available from: http://cddis.nasa. gov/926/egm96/egm96.html. Reference 38 gives the normalization process and showshow toun-normalize. 156 You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com) Figure A.2: Illustration of various Stokes coecients.[3] A.2.2 Luni-Solar Perturbations Direct and indirect perturbations from the gravitation of the Sun andMoon aremod- eled using the framework of the three-body Newtonian gravitation problem as follows:[38] a% = µ% ✓ rj,% krj,%k3 r,% kr,%k3 ◆ (A.11) a = µ ✓ rj, krj,k3 r, kr,k3 ◆ (A.12) where rj,%/ = r%/rj is the inertial position vector from the satellite to theMoon/Sun, and r,%/ = r%/r is the inertial position vector from the Earth () to theMoon/Sun, respectively. The constants µ% ⌘ GM% and µ ⌘ GM are the gravitational parameters of 157 theMoon and Sun. Luni-solar gravitation therefore exerts a direct, two-body perturbation on the satellite’s motion, and furthermore exerts an indirect perturbation via gravitational interactions with Earth. As discussed in detail in Chapter 3, these luni-solar perturbations couple with the J2 harmonic to drive a 53-year, cyclical progression of the orbit plane that is reflected in the phase portrait for doubly-averaged inclination and RAAN variations.[3] A.2.3 Solar Radiation Pressure Perturbation The solar radiation pressure acceleration is modeled with the classical “cannonball” formulation, in which the geometry of the propagated satellite is simplified as a sphere.[38] Thismodel serves as a convenient approximation for high-fidelity SRP acceleration, which is dependent not only on accurate knowledge of the Sun’s location relative to the satellite, but the satellite’s attitude, time-varying cross-sectional area exposed to the Sun, and time- varying reflectivity coecients, which govern the degree to which individual surfaces are translucent, absorptive, or reflective.[38] In this model, the perturbative SRP acceleration is aSRP = pSRPcr ✓ rj, krj,k ◆ (A.13) where pSRP is the solar radiation pressure at the satellite’s location, cr is themean coecient of reflectivity, and ⌘ A/m is the exposed-area-to-mass ratio, for which ⇡ 0.04 m2/kg is representative for typical GEO satellites. [43] The term rj, = rrj denotes the position vector of the Sun relative to the satellite, expressed in inertial frame (i.e., ECI) coordinates. Solar radiation pressure pSRP is modeled using the inverse-square “diusion” formulation pSRP = L 4⇡krj,k2c (A.14) where L ⇡ 3.839⇥1026 W is themean solar luminosity, and c ⇡ 3⇥108 m/s2 is the speed of light, such that pSRP ⇡ 4.55⇥ 106 N/m2 at 1 AU. Per Equation (A.13), this value for the solar pressure yields kaSRPk ⇡ 2.73⇥107 m/s2 at 1 AU assuming the reflectivity cr = 1.5. To account for eclipsing at GEO, the magnitude of this acceleration is attenuated by using the scaling factor derived from the geometric occultation algorithm given in Reference 42. 158 Physically, the SRP perturbation accelerates a satellite as it moves opposite the local Sundirectionduring its orbitalmotion, anddecelerates the satellite as itmoves towards the Sun. The cumulative end-result over the course of one year is a periodic oscillation in both themagnitude and direction of the eccentricity vector, which isminimized by operators by choosing the perigee of the operational orbit such that it points in the Sun direction.[3,47,59] A.3 Why Does Libration Motion Occur? As stated in SectionA.2.1, the second degree and second order sectorial harmonic, as represented by the Stokes coecientsC2,2 and S2,2, quantifies the gravity perturbations in- duced by Earth’s longitudinal bulges, which are responsible for driving periodic, libration motion at GEO. To gain intuition as towhy these oscillations in longitude physically occur, Figure A.3 shows a top-down schematic of Earth as represented by the dominantC2,2 term only, which corresponds to equatorial ellipticity.[38] By symmetry, the gravitational forceF will be purely radial when the satellite is on an extension of either the major or minor axes of this ellipse. When the satellite is o-axis, however, a tangential component ofF directed towards the nearest major axis arises.[38] By Newton’s second law, does this induce longi- tudinal acceleration in the direction of the tangential component, or against this direction? Consider a circular orbit of semi-major axis a, and let F = F✓oˆ✓ be expressed in the local tangential direction. The radial component ofF is excluded here becausewe are only interested in the change in orbital energy arising from the tangential componentF✓, i.e., the deviation from a purely-radial gravitational attraction assumed in the two-body problem. The position vector is r = aoˆr and, by application of the transport theorem, the velocity is r˙ = a˙oˆr +anoˆ✓. Dierentiating the well-known, two-body orbit energy equation, we have ✏ = µ 2a =) ✏˙ = µ 2a2 a˙ (A.15) From mechanics, the energy rate ✏˙ is given by the product of force and velocity, such that ✏˙ = F · r˙ = anF✓ (A.16) 159 1330 R. R. ALLAN 4 THE EILIPTIClTY OF THEZ FiQUATOR So far the general form of a sum over all the tesseral harmonics has been retained, mainly for completeness, although the integration can only be carried further numerically. How- ever we can give a full solution if we restrict to the dominant Js,o term. The equation (14) can then be written q = ikB sin 2(q1 - cps,$, (22) where ks = 36 naJ,,, (R/b)g. (23) The actual tangential force on the satellite is Fq = -_8iz+ = -&S sin 2(q - q.z&. (24) / (b INCREASINGj EASTWARDS Fxo. 1. LaBnATIoN IN w- The force vanishes on both the major and minor axes of the equatorial ellipse. In general the force is dire&d towards the nearer end of the major axis as one would expect (Fig. l), and the maximum value is QuP at the 45” points between the axes. With the assumed value of Js,* we have k = 1~32 x 10-’ set-l, and k* = 1.75 x lo-l4 see-a, so that the maximum disturbing acceleration is la23 x 10-s cm EC-*, that is to say a force of 1.23 dynes on a satellite of 100 kg. It is clear from (22) that the two points of equilibrium on the major axis are unstable, while the two points on the minor axis are stable. The points of equilibrium and of maximum force are set out in Table 2, assuming ~18~ = 25W. To proceed further it is more convenient to change the origin to the minor axis by writing P = 9J - 9s.s rfr: *I29 Figure A.3: Top-down schematic showing ellipticity of Earth.[4] Equating Equations (A.15)-(A.16) and using the identity µ = n2a3, the semi-major axis rate is directly proportional to the tangential force component: a˙ = 2 n F✓ (A.17) Thus, as the satellite is approaching an extension of themajor axis of the equatorial ellipse, two-body energy and the semi-major axis increase, since a˙ > 0 becauseF✓ > 0. Conversely, as the satellite is approaching an extension of the minor axis, two-body energy and semi- major decrease, since a˙ < 0 because F✓ < 0 (i.e., opposite to the local tangential direction). Noting that the longitudinal drift rate is given by ˙ = n !, then the longitudinal acceleration is given by ¨ = n˙, or ¨ = d dt ✓r µ a3 ◆ = 3n 2a a˙ = 3 a F✓ (A.18) Thus, acceleration in longitude opposes the tangential force component.[4] As the satellite is approaching an extension of the major axis of this ellipse, a˙ > 0 but ¨ < 0, such that the satellite slows in longitude until reversing direction once ˙ = 0 is reached. Conversely, the satellite accelerates in longitude as it approaches a minor axis of the ellipse, because a˙ < 0 160 but ¨ > 0. Since momentum is gained as the longitude accelerates towards the minor axis, the satellite drifts past this point, the tangential force component reverses direction in the local orbit frame, and the process repeats, producing the libration eect. [38] As a result, the two extensions of the minor axis are called the stable equilibrium points (or potential wells), which lie at 75E and 105W,while the extensions of the major axis are termed the unstable equilibrium points (or potential hills), which lie at 165E and 15W by ellipse symmetry.5 To illustrate the coupling between semi-major axis and longitude that occurs during libration about one of the stable equilibrium points, Figure A.4 shows a five-year libration profile for an uncontrolled GEO object starting at 160W (55W of the well at 105W). The semi-major axis decreases until the satellite reaches the stable point at approximately nine months, after which it begins increasing until the satellite slows down, reverses direction at 50W (55E of the well at 105W), and once again reaches the stable point on the reverse pass. According to FigureA.4, the simulated object has a libration period of approximately three years, which is proportional to the amplitude of the libration motion (Chapter 5). [4] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0200 220 240 260 280 300 320 Ea st Lo ng it ud e [d eg ] Elapsed Years 30 20 10 0 10 20 30 SM A D ev ia ti on fr om G EO [k m ] SMA$<$GEO (Drift'East) SMA$>$GEO (Drift'West) Longitude Proﬁle SMA Proﬁle Figure A.4: Five-year libration profile for GEO satellite at 160W. In the same manner than inclination drift driven by luni-solar perturbations creates the need for north-south station-keeping, longitude drift driven by the second degree and 5 The longitudes for the stable and unstable points in Figure A.3 dier by 10 because of the uncertainties in the exact values for the C2,2 and S2,2 coecients in 1963, the year of Figure A.3’s original publication. [4] 161 order sectorial harmonic creates the need for routine, east-west station-keeping. If higher- order terms in the spherical harmonics expansion are truncated, an exact analytic solution for the semi-major axis drift rate—and consequently, the longitudinal acceleration—can be derived as a function of longitudinal deviation ⌘ 2,2 from the nearestmajor axis: [3] a˙() = 12rGEOnGEOJ2,2 ✓ R rGEO ◆2 sin(2) ⇡ 132 sin(2) [m/day] (A.19) ¨() = 3n 2rGEO a˙() ⇡ 0.0017 sin(2) [deg/day2] (A.20) These are beneficial for estimating annual east-west station-keeping budgets based on the operational longitude slot. Figure A.5 illustrates Equations (A.19)-(A.20) by providing the semi-major axis drift and longitude acceleration accumulated over the course of one day. Since the gravitational force is completely radial on either the major or minor axes of the equatorial ellipse, thenF✓ = 0 and a˙ and ¨ vanish at these locations. Maximum semi-major axis and longitude drift occur midway between each of the four stable/unstable pairs, [3,4] such that the annual east-west station-keeping cost ismaximized at these longitudes. GEO satellites located at or near the stable points incurminimal east-west station-keeping costs, and have the benefit of passive stability (i.e., minimal libration amplitude) if thrusters fail. 0 30 60 90 120 150 180 210 240 270 300 330 360 Longitude [deg] 200 150 100 50 0 50 100 150 200 Se m i-M aj or A xi s D ri ft R at e [m /d ay ] 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Lo ng itu de A cc el er at io n [d eg /d ay 2 ] 0"""""""30E"""""60E""""""90E""""120E"""150E""""180""""150W""120W"""90W"""60W""""30W""""""0 E E E E 0.5W 1.0W 1.5W 2.0W Longitude*[deg] x"10 .3Stable 75E Unstable 165E Stable 105W Unstable 15W Longitude Acceleration SMA Rate Figure A.5: Daily semi-major axis and longitude drift as a function of longitude.

SGP4 Propagation of GEO Satellites and Debris

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SGP4 Propagation of GEO Satellites and Debris

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