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Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics

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Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics ABSOLUTE AND RELATIVE MOTION SATELLITE THEORIES FOR ZONAL AND TESSERAL GRAVITATIONAL HARMONICS A Dissertation by BHARAT MAHAJAN Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, Srinivas R. Vadali Co-Chair of Committee, Kyle T. Alfriend Committee Members, John E. Hurtado Igor Zelenko Head of Department, Rodney Browersox May 2018 Major Subject: Aerospace Engineering Copyright 2018 Bharat Mahajan ABSTRACT In 1959, Dirk Brouwer pioneered the use of the Hamiltonian perturbation methods for con- structing artificial satellite theories with effects due to nonspherical gravitational perturbations in- cluded. His solution specifically accounted for the effects of the first few zonal spherical harmon- ics. However, the development of a closed-form (in the eccentricity) satellite theory that accounts for any arbitrary spherical harmonic perturbation remains a challenge to this day. In the present work, the author has obtained novel solutions for the absolute and relative motion of artificial satel- lites (absolute motion in this work refers to the motion relative to the central gravitational body) for an arbitrary zonal or tesseral spherical harmonic by using Hamiltonian perturbation methods, without resorting to expansions in either the eccentricity or the small ratio of the satellite’s mean motion and the angular velocity of the central body. First, generalized closed-form expressions for the secular, long-period, and short-period variations of the equinoctial orbital elements due to an arbitrary zonal harmonic are derived, along with the explicit expressions for the first six zonal harmonics. Next, similar closed-form expressions are obtained for the sectorial and tesseral (collectively referred to as tesserals henceforth) harmonics by using a new approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian. This approach reduces the solu- tion for the tesseral periodic perturbations to quadratures. It is shown that the existing approximate approaches for the normalization of the tesseral problem, such as the method of relegation, can be derived from the proposed exact solution. Moreover, the exact solution for the periodic variations due to the tesseral harmonics produces a unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes without any singularities for the resonant orbits. The closed- form theories developed for the absolute motion are then used to develop analytic solutions in the form of state transition matrices for the satellite relative motion near a perturbed elliptic reference orbit. The expressions for differential equinoctial orbital elements for establishing a general cir- cular orbit type satellite formation are also derived to avoid singularities for the equatorial and circular reference orbits. In order to negate the along-track drifts in satellite formations, an ana- ii lytic expression for the differential semimajor axis is derived by taking into account the secular effects due to all the zonal harmonics. The potential applications of the proposed satellite theo- ries range from fuel-efficient guidance and control algorithms, formation design, faster trade and parametric studies to catalog maintenance, conjunction analysis, and covariance propagation for space situational awareness. Two specific applications, one for solving a perturbed multiple revo- lution Lambert’s problem and the other for rapid nonlinear propagation of orbit uncertainties using point clouds, are also given. The theories presented in this work are implemented for computer simulations in a software tool. The simulation results validated the accuracy of these theories and demonstrated their effectiveness for various space situational awareness applications. iii DEDICATION To my grandfather Tara C. Gupta, whose fascinating stories about WW-II pilots and their “close encounters” piqued a young kid’s curiosity about the heavens above, although he continued to believe his grandmother went to Heaven and became one of those stars. iv ACKNOWLEDGMENTS “The mind uses its faculty for creativity only when experience forces it to do so.” -Henri Poincaré I feel lucky to have the opportunity to learn from the experience and wisdom of two great advisers, Drs. Rao Vadali and Terry Alfriend. In the past five years, I enjoyed the best of both worlds. I often feel amazed at Dr. Vadali’s clear insight into any research problem that comes my way, and Dr. Alfriend’s knack for providing instant answers using his many (almost mysterious) rules of thumb. I would like to thank both of them for their mentorship and supporting me at every step of this journey. I would also like to thank Drs. John Hurtado and Igor Zelenko for serving on my committee, and reviewing my dissertation. I am grateful to my colleagues and friends, Drs. Kirk Johnson, Hui Yan, Kohei Fujimoto, and Inkwan Park, for all the discussions and idea exchanges we had. A few more names deserve mention for being my mentors over the years: Drs. Hank Pernicka and S. N. Balakrishnan, Ankur Gupta from my undergraduate days, and Vaibhav Khanduja from my time in Hewlett-Packard. And my friends for all the good times, you know who you are. A special thanks to my mother, Shashi, for learning LATEX and offering me her help in typeset- ting the equations for this dissertation. I might start using this catchphrase from now on: My mom knows LATEX, how about yours! Looking back, it was surely an odyssey of ten years since my baccalaureate, including my time in the industry and all the scheming to go back to school. I wouldn’t have come this far if not for you: Renu, Ravi, and Shashi (the three sisters, the three mothers, and literally, my three very first teachers). v CONTRIBUTORS AND FUNDING SOURCES Contributors This work was supported by a dissertation committee consisting of Professors Srinivas R. Vadali (advisor), Kyle T. Alfriend (co-advisor) and John E. Hurtado of the Department of Aerospace Engineering, and Professor Igor Zelenko of the Department of Mathematics. All the work con- ducted for the dissertation was completed by the author independently. Funding Sources The author’s graduate study was supported by Air Force Research Lab (AFRL) contract No. FA9453-13-C-0202 along with teaching assistantships from the Department of Aerospace Engi- neering. vi NOMENCLATURE AST Artificial Satellite Theories GCO General Circular Orbit PCO Projected Circular Orbit GA-STM Gim-Alfriend State Transition Matrix RM-STM Relative Motion State Transition Matrix (Φ) MC Monte Carlo PRINCE Perturbation Relegation In New Canonical Elements PBJ Poisson Bracket Jacobian (A; B) Poisson Bracket of A and B LVLH Local Vertical Local Horizontal RSS Root Sum Square U Gravitational potential energy V Gravitational potential H Hamiltonian K Transformed (or averaged) Hamiltonian S Generating function for canonical transformations W Generating function for Lie-series based near-identity canon- ical transformations t Time C Direction cosine matrix Σ Geometric transformation matrix φm Differential mean state transition matrix vii DSP Differential long-period to short-period transformation matrix DLP Differential mean to long-period transformation matrix D Differential mean to osculating transformation matrix (= DSP DLP ) µ Gravitational parameter n Spherical harmonic degree m Spherical harmonic order φ Latitude (planetocentric) λ Longitude (planetocentric) θ Greenwich sidereal time Re Radius of the central body (Earth) wE Rotational speed of the central body (Earth) Pn Legendre polynomial of degree n Jn Zonal harmonic coefficient of degree n Cnm Spherical harmonic coefficient of degree n and order m Snm Spherical harmonic coefficient of degree n and order m r radial distance n~ Mean motion p Semilatus rectum a Semimajor axis e Eccentricity i Inclination h (also Ω) Right ascension of the ascending node g (also !) Argument of periapsis l Mean anomaly viii f True anomaly Λ Mean longitude (l+g+h) Ψ True longitude (f+g+h) # True argument of latitude (f+g) p η 1 − e2 si sin i ci cos i i p1 tan 2 cos(h) i p2 tan 2 sin(h) q1 e cos(g + h) q2 e sin(g + h) [x; y; z] Curviliear coordinate vector in the radial, along-track, and cross-track directions Z Set of all integers ix TABLE OF CONTENTS Page ABSTRACT ......................................................................................... ii DEDICATION....................................................................................... iv ACKNOWLEDGMENTS .......................................................................... v CONTRIBUTORS AND FUNDING SOURCES ................................................. vi NOMENCLATURE ................................................................................. vii TABLE OF CONTENTS ........................................................................... x LIST OF FIGURES ................................................................................. xiii LIST OF TABLES...................................................................................xviii 1. INTRODUCTION AND LITERATURE REVIEW ........................................... 1 1.1 Introduction................................................................................ 1 1.2 Literature Review ......................................................................... 2 1.2.1 Satellite Theories for Nonspherical Gravitational Perturbations ............... 3 1.2.1.1 Before 1959...........................................................
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