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The Pennsylvania State University Schreyer Honors College THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF AEROSPACE ENGINEERING LONG TERM ORBITAL MODELING FOR OBJECTS IN GEOSTATIONARY EARTH ORBIT PHILIP CHOW SPRING 2015 A thesis submitted in partial fulfillment of the requirements for a baccalaureate degree in Aerospace Engineering with honors in Aerospace Engineering Reviewed and approved* by the following: David B. Spencer Professor of Aerospace Engineering Thesis Supervisor Robert G. Melton Professor of Aerospace Engineering Honors Adviser George A. Lesieutre Professor of Aerospace Engineering Head of Aerospace Engineering * Signatures are on file in the Schreyer Honors College. i ABSTRACT The suitability of different numerical integrators for the long term orbital modeling of a satellite in geostationary Earth orbit is examined. The integrators used are the ODE45 Runge- Kutta Method from MATLAB and the symplectic Euler method, assuming only spherical harmonics of the Earth and n-body perturbations from the Sun, Moon, and other planets. Results show that the energy drift associated with both integrators makes them unsuitable for long term modeling, and that the symplectic Euler method does not actually maintain constant energy. A possible factor in the energy drift of both integrators includes the model used for n-body perturbations; future work should focus on determining the extent that this affected the results and the steps necessary to correct the errors. ii TABLE OF CONTENTS LIST OF FIGURES ..................................................................................................... iii ACKNOWLEDGEMENTS ......................................................................................... iv NOMENCLATURE .................................................................................................... v VECTOR SUBSCRIPTS ............................................................................................. vi Chapter 1 Introduction ................................................................................................. 1 1.1 Geostationary Earth Orbit ............................................................................... 1 1.2 Orbital Debris ................................................................................................. 2 1.3 Numerical Integrators ..................................................................................... 3 1.4 Structure of the Thesis .................................................................................... 3 Chapter 2 Equations of Motion .................................................................................... 5 2.1 Third Body Perturbations ............................................................................... 5 2.2 Spherical Harmonics ....................................................................................... 7 Chapter 3 Method ........................................................................................................ 10 3.1 Standish’s Orbital Elements ........................................................................... 10 3.2 Vallado’s Lunar Orbit ..................................................................................... 11 3.3 Vector Algebra ................................................................................................ 12 3.4 Specific Orbital Energy .................................................................................. 14 Chapter 4 Analysis ....................................................................................................... 16 4.1 Two-Body Problem ........................................................................................ 16 4.2 Luni-Solar Perturbations Only ........................................................................ 18 4.3 Lunar Perturbations Only ............................................................................... 20 4.4 Solar Perturbations Only................................................................................. 21 4.5 Planetary Perturbations Only .......................................................................... 23 4.6 Spherical Harmonics Only .............................................................................. 24 4.7 All Perturbations ............................................................................................. 26 Chapter 5 Conclusion ................................................................................................... 28 REFERENCES ............................................................................................................ 29 iii LIST OF FIGURES Figure 2.1: Geometry of the Three Body Problem ...................................................... 6 Figure 3.1: Earth-Moon-Satellite Position Vectors ..................................................... 12 Figure 3.2: Earth-Moon Barycenter Vectors ............................................................... 13 Figure 3.3 Earth-Sun-Planet Position Vectors ............................................................. 14 Figure 4.1 Two-Body Problem: ODE45 Specific Orbital Energy ............................... 17 Figure 4.2 Two-Body Problem: Symplectic Integrator Specific Orbital Energy ........ 18 Figure 4.3 Luni-Solar Perturbation: ODE45 Specific Orbital Energy ......................... 19 Figure 4.4 Luni-Solar Perturbations: Symplectic Integrator Specific Orbital Energy . 19 Figure 4.5 Lunar Perturbations: ODE45 Specific Orbital Energy ............................... 20 Figure 4.6 Lunar Perturbations: Symplectic Integrator Specific Orbital Energy ........ 21 Figure 4.7 Solar Perturbations: ODE45 Specific Orbital Energy ................................ 22 Figure 4.8 Solar Perturbations: Symplectic Integrator Specific Orbital Energy ......... 22 Figure 4.9 Planetary Perturbations: ODE45 Specific Orbital Energy ......................... 24 Figure 4.10 Planetary Perturbations: Symplectic Integrator Specific Orbital Energy . 24 Figure 4.11 Spherical Harmonics: ODE45 Specific Orbital Energy ........................... 25 Figure 4.12 Spherical Harmonics: Symplectic Integrator Specific Orbital Energy ..... 25 Figure 4.13 All Perturbations: ODE45 Specific Orbital Energy ................................. 26 Figure 4.14 All Perturbations: Symplectic Integrator Specific Orbital Energy ........... 27 iv ACKNOWLEDGEMENTS Firstly, I would like to thank my thesis adviser Dr. David Spencer for his guidance on this thesis, both with finding a viable topic to work on and with technical assistance. Thanks also go to Dr. Paula Krisko and Dr. Philip Anz-Meador of Jacobs Technologies and Dr. Mark Matney of NASA for providing technical assistance. Secondly, I would like to thank my honors adviser Dr. Robert Melton for guiding me through the major decisions in my undergraduate education, particularly my semester abroad in the United Kingdom and my decision to graduate in three years. Thirdly, I would like to thank composers Basil Poledouris, Nick Phoenix, and Thomas J. Bergersen for composing and publishing music conducive to the coding and writing process. Lastly, I’d like to thank my family for emotionally and financially supporting my decision to study aerospace engineering despite their suggestion that I study mechanical engineering. v NOMENCLATURE 푚 2 퐺 = Universal Gravitational Constant [푁 ( ) ] 푘 푚 = Mass [푘푔] 푘푚3 휇 = Standard Gravitational Parameter [ ] 푠2 푎푒 = Reference distance [푚] or [푘푚] 푟 = Radius of the satellite’s orbit [푚] or [푘푚] 푟⃑ = Position vector [푚] or [푘푚] 푚 푘푚 푟⃑̈ = Second time derivative of the position vector (acceleration vector) [ ] or [ ] 푠2 푠2 퐹⃑ = Force vector [푁] ∇ = Gradient operator 푢̂푤 = Unit vector of the w-axis 휑 = Latitude (spherical coordinates) [°] 휆 = Longitude (spherical coordinates) [°] 푃푙(푥) = Legendre Polynomial 퐽푙 = Zonal Harmonic Term (Physical Constant of the Earth) 퐶푙푚, 푆푙푚 = Tesseral and Sectoral Harmonic Terms (Physical Constants of the Earth) 퐽 ɛ = Specific Orbital Energy [ ] 푘 vi VECTOR SUBSCRIPTS 푥/푦 = The [position, velocity, or acceleration] of X with respect to Y 푆푎푡 = Satellite 퐸푎푟 = Earth 3 = Third Body 푔 = Gravitational Forces 푆푢푛 = Sun 퐿푢푛 = Moon 퐸푀퐵푎푟 = Earth-Moon Barycenter 푃푙푎 = Generic Planet 1 Chapter 1 Introduction This thesis investigates methods of determining the orbit of an uncontrolled object in Geostationary Earth Orbit on the timespan of decades and centuries. Uncontrolled objects in orbit around the Earth – particularly in a desirable orbital band such as the geostationary orbit – pose a collision risk to many operational spacecraft, and it is in the interest of spacecraft operators to determine where such uncontrolled objects will be at points in time. A brief overview of the relationship between Geostationary Earth Orbit and orbital debris, and how suitability of the methods will be calculated, is presented. The application of Runge-Kutta and symplectic integrators in the context of this thesis is also presented. 1.1 Geostationary Earth Orbit Geostationary Earth Orbit (GEO) refers to an orbit around the Earth for which an observer on the ground will perceive the orbiting object as stationary in the sky. Because the object does not appear to move in the sky, GEO is a highly desired orbit for many purposes – particularly for communications – as it simplifies the tracking requirements on ground control stations. Consequently, many satellites are launched to this orbit, which has begun to raise concerns of orbital debris in the GEO band. GEO is a circular orbit about the Earth’s equator at an altitude of approximately 35,786
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