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An Analytical Theory for the Perturbative Effect of Solar An Analytical Theory for the Perturbative Effect of Solar Radiation Pressure on Natural and Artificial Satellites by Jay W. McMahon B.S., University of Michigan, 2004 M.S.E, University of Southern California, 2006 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 2011 This thesis entitled: An Analytical Theory for the Perturbative Effect of Solar Radiation Pressure on Natural and Artificial Satellites written by Jay W. McMahon has been approved for the Department of Aerospace Engineering Sciences Daniel J. Scheeres Hanspeter Schaub George Born 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 McMahon, Jay W. (Ph.D., Aerospace Engineering Sciences) An Analytical Theory for the Perturbative Effect of Solar Radiation Pressure on Natural and Artificial Satellites Thesis directed by Professor Daniel J. Scheeres Solar radiation pressure is the largest non-gravitational perturbation for most satellites in the solar system, and can therefore have a significant influence on their orbital dynamics. This work presents a new method for representing the solar radiation pressure force acting on a satellite, and applies this theory to natural and artificial satellites. The solar radiation pressure acceleration is modeled as a Fourier series which depends on the Sun's location in a body-fixed frame; a new set of Fourier coefficients are derived for every latitude of the Sun in this frame, and the series is expanded in terms of the longitude of the Sun. The secular effects due to the solar radiation pressure perturbations are given analytically through the application of averaging theory when the satellite is in a synchronous orbit. This theory is then applied to binary asteroid systems to explain the Binary YORP effect. Long term predictions of the evolution of the near-Earth asteroid rd 1999 KW4 are discussed under the influence of solar radiation pressure, J2, and 3 body gravitational effects from the Sun. Secular effects are shown to remain when the secondary asteroid becomes non-synchronous due to a librational motion. The theory is also applied to Earth orbiting spacecraft, and is shown to be a valuable tool for improved orbit determination. The Fourier series solar radiation pressure model derived here is shown to give comparable results for orbit determination of the GPS IIR-M satellites as JPL's solar radiation pressure model. The theory is also extended to incorporate the effects of the Earth's shadow analytically. This theory is briefly applied to the evolution of orbital debris to explain the assumptions that are necessary in order to use the cannonball model for debris orbit evolution, as is common in the literature. Finally, the averaging theory methodology is applied to a class of Earth orbiting solar sail spacecraft to show the orbital effects when the sails are made to cone at orbit rates in the local horizontal frame. Dedication To my wife, Sarah, for always supporting me and keeping me on task; to my dog, Kona, for always recognizing when I have been working for too long and need to go for a walk; and to my first-born daughter, Maggie Rose, for ensuring that I finish this degree in a timely manner. v Acknowledgements In some respects, this is the hardest section to write as I would like to adequately express my thanks for everyone that helped me finish this dissertation. None of this work would have happened without Dan Scheere's guidance and support. In fact, I may not have even ended up at CU if Dan hadn't decided to come here at the same time! So many thanks to the best advisor I could have asked for. Next, I'd like to thank my colleagues in the Celestial and Spaceflight Mechanics Laboratory. There are few things more valuable than having good, smart colleagues to bounce ideas off of. In particular, Marcus, Christine, and Seth - it has been three wonderful years of working closely together. The faculty in the Aerospace Engineering Mechanics department is amazing. I have learned, and continue to learn, an enormous amount from them. These lessons, both inside and outside of the classroom, have gone a long way in shaping this dissertation. My main motivation in life is driven by my family, and as of yesterday (June 7th, 2011) that family has grown due to the birth of my daughter Maggie! Knowing she was on the way helped me complete this dissertation on time. To Sarah, my beautiful, amazing wife - I can't thank you enough for everything that you've done for me over the past three years. There is no doubt in my mind that I would be a total mess without you in my life. My parents, Jim and Kathy - without the examples you set and the sacrifices you made to support me throughout my life, I wouldn't even have the chance to pursue a PhD! Thank you so much. Finally, to my brother and sister, Cory and Gina, and to my many wonderful friends - thank you for supporting me and making my life wonderful to live. Having wonderful distractions like all of you doesn't directly help me get work done, but it does make me a much happier person. Contents Chapter 1 Introduction 1 1.1 Orbital Perturbations due to Solar Radiation Pressure . .1 1.1.1 SRP Models . .6 1.1.2 Natural Bodies . .7 1.1.3 Artificial Bodies . .9 1.2 Organization and Contributions . 11 1.2.1 Contributions . 13 1.2.2 Publications . 14 2 Solar Radiation Pressure Model 17 2.1 Coordinate Frame Definitions . 17 2.2 Solar Radiation Force Representation . 23 2.3 Comparison with other Solar Radiation Pressure Models . 25 3 Orbit Variational Equations 28 3.1 Dynamical Elements . 28 3.2 Keplerian Orbital Elements . 32 3.3 Delaunay Elements . 34 4 Attitude Dynamics Coupling in the Force Model 35 4.1 Synchronous Rotation . 35 vii 4.2 Periodic Rotation . 39 4.3 Higher Order Resonances . 40 4.4 Non-resonant Rotation . 41 5 Averaging Theory 42 5.1 Averaging Methodology . 43 5.1.1 Orbit Averaging . 43 5.1.2 Offset Due to Periodic Terms . 45 5.1.3 Year Averaging . 46 5.1.4 Long Term Averaging . 51 5.2 Application to a Circular Orbit . 52 5.2.1 Circular Orbit Average Equation Derivation . 52 5.2.2 Initial Offset Correction . 56 5.2.3 Circular Orbit Simulation . 58 5.2.4 Averaging the Circular Results Over 1 year . 62 5.3 Application to an Elliptical Orbit . 63 5.3.1 Orbit Averaged Equations . 63 5.3.2 Initial Offset Correction . 76 5.3.3 Eccentric Orbit Simulation . 76 5.3.4 Averaging Eccentric Equations Over a Year . 79 6 Application to Natural Bodies: BYORP 82 6.1 Nominal Predictions . 84 6.1.1 Non-dimensionalized BYORP Model . 85 6.1.2 BYORP on Doubly Synchronous Binary Systems . 85 6.1.3 Secular Evolution of Binary Asteroid Systems . 87 6.1.4 Application to Binary System 1999 KW4 . 97 6.1.5 Discussion of Binary Asteroid Results . 114 viii 6.2 Effects of Libration . 116 6.2.1 Librational Dynamics . 117 6.2.2 Fourier Coefficients for a Librating Secondary . 118 6.2.3 Libration Averaging . 121 7 Application to Artificial Bodies 126 7.1 Satellite Application Example: GRACE . 128 7.1.1 Shadowing . 134 7.1.2 Shadow Approach . 134 7.1.3 Orbital Effect of the Shadow . 135 7.1.4 Penumbra Models . 136 7.2 GPS SRP Model . 138 7.2.1 JPL Model . 139 7.2.2 University of Bern . 140 7.2.3 Rotating Frame Coefficients . 141 7.3 SRP Estimation for GPS Orbit Determination . 145 7.4 Orbital Debris . 152 7.4.1 Tumbling Debris at Unique Rates . 152 7.4.2 Tumbling Debris with Resonant Rates . 154 7.5 Solar Sails . 155 7.5.1 System Definition . 156 7.5.2 Attitude Motion . 157 7.5.3 Orbital Effects . 158 7.5.4 Applications . 167 7.5.5 Conclusion . 169 8 Conclusion and Future Work 173 ix Bibliography 176 Appendix A Mathematical Relationships 183 A.1 Trigonometric Functions . 183 A.2 Fourier Series . 184 A.3 Bessel Functions . 184 Tables Table 6.1 Physical properties of the KW4 system. All data is reproduced from [76], except those marked with a * which are from [73], and those marked with a # which are from [92]. 105 6.2 These values have either been derived from the previously cited.
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