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Study of Spacecraft Orbits in the Gravity Field of the Moon

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Study of Spacecraft Orbits in the Gravity Field of the Moon STUDY OF SPACECRAFT ORBITS IN THE GRAVITY FIELD OF THE MOON -PROJECT REPORT - by EDGAR CARDOSO VILANA A dissertation submitted to the Department of Aerospace Engineering, ETSEIAT – Universitat Politècnica de Catalunya , in partial fulfillment of the requirements for the degree of Aeornautical Engineer Tutor: Dr. Elena Fantino January 2012 ESCOLA TÈCNICA SUPERIOR D ’E NGINYERIES INDUSTRIAL I AERONÀUTICA DE TERRASSA ENGINYERIA SUPERIOR AERONÀUTICA STUDY OF SPACECRAFT ORBITS IN THE GRAVITY FIELD OF THE MOON This page is intentionally left blank. 1 E.C ARDOSO ABSTRACT The objective of the present study is to analyze the evolution of low lunar orbits under the gravitational potential of the Moon and additional perturbations of gravitational (Earth and Sun gravity) and non-gravitational (radiation pressure) nature. Such evolution can be used to estimate the amount of propellant required to maintain a certain orbit during a given time or, alternatively as in the present case, to identify orbits which may offer a wide surface coverage if no orbit correction maneuvers are applied. The lunar gravitational potential and its first-order gradient, the acceleration, are given through spherical harmonics expansions. The traditional representation based on Associated Legendre Functions (ALFs) of the first kind has been abandoned due to the inherent singularity at the poles, where interesting orbits pass. The chosen approach is based on the functions introduced by Pines [24] and consists in representing the gravitational potential and its gradients in Cartesian coordinates. Thanks to this, the singularity is overcome and the various functionals of the gravity field can be computed all over the sphere without loss of precision. The implementation of the so-called lumped coefficients in the treatment of the series expansions allows considerable memory and computing time savings over the traditional representation. Perturbations other than the lunar mass distribution are considered when their average magnitude is an appreciable fraction of the gravitational acceleration: this includes third body and radiation pressure effects. They have been modeled by assuming that the Sun, the Earth and the Moon stay in a common plane and are aligned at the beginning of the simulations. Several orbits of interest in a general context of remote sensing have been identified and simulated over significant time intervals: some of them, such as the so-called frozen orbits or the repeat ground track orbits, are well known for their stability; other types, such as the general class of polar orbits, are characterized by rapid evolution. The former offer the possibility of continuously observing the same surface area, thanks to the fact that the perturbations are 1:1 resonant with the orbit (same period). Polar orbits, instead, could be the choice for global mapping of the lunar surface or in the context of lunar observation or reconnaissance. The evolution of the orbits has been computed by numerical integration of the equations of motion of the spacecraft, accounting for the lunar gravitational acceleration (with the LP100K spherical harmonic model complete to degree and order 100) and the additional, relevant perturbing accelerations, by means of a Runge-Kutta-Fehlberg 7(8) scheme with variable step size. 2 STUDY OF SPACECRAFT ORBITS IN THE GRAVITY FIELD OF THE MOON We have simulated the evolution of many orbits, we have observed how the perturbations modify them and we have related such modifications to a suitable type of lunar mission, one that could take advantage of the natural evolution of the orbit for its scientific or technical objectives. Keywords: Moon, Gravity field, Spherical Harmonics, Orbital perturbations, Frozen Orbits 3 E.C ARDOSO ACKNOWLEDGEMENTS Con este apartado me gustaría expresar mi más sincero agradecimiento a todas esas personas que han contribuido directa o indirectamente a la realización de este proyecto final de carrera. En primer lugar debo mencionar a mi tutora, Dra. Elena Fantino. Sin su ayuda el presente trabajo no hubiese sido posible. Su apoyo y sus enseñanzas han sido vitales. También me ha demostrado que es una docente muy cualificada y de gran profesionalidad, y no sólo eso, sino que además he podido comprobar que es una persona admirable cuya dedicación por los alumnos es totalmente encomiable. Gracias por transmitirme dichos valores y guiarme durante estos últimos meses en la elaboración de este estudio. Y como el presente trabajo representa la culminación a cinco años y medio de carrera, tampoco quiero dejar en el olvido a todos los compañeros de clase que me han apoyado, aguantado y animado durante esta etapa de mi vida en la universidad. Asimismo también mencionar a todos los docentes que durante estos años me han brindado sus conocimientos, los cuales espero que se vean reflejados de forma pertinente a lo largo de este documento. Por último tampoco puedo olvidarme de la familia ni de esas personas que durante más o menos tiempo han estado a mi lado y que me han apoyado en los momentos difíciles y que han celebrado conmigo los éxitos y los momentos de alegría. Gracias a todos. 4 STUDY OF SPACECRAFT ORBITS IN THE GRAVITY FIELD OF THE MOON TABLE OF CONTENTS ABSTRACT ................................................................................................................................... 2 ACKNOWLEDGEMENTS ............................................................................................................... 4 TABLE OF CONTENTS ................................................................................................................... 5 LIST OF FIGURES .......................................................................................................................... 7 LIST OF TABLES ............................................................................................................................ 9 LIST OF SYMBOLS ...................................................................................................................... 10 OPERATORS................................................................................................................................. 11 SUBSCRIPTS ................................................................................................................................. 12 ACRONYMS ................................................................................................................................. 12 CHAPTER ONE. INTRODUCTION............................................................................................ 14 CHAPTER TWO. PROJECT SCOPE ............................................................................................ 20 CHAPTER THREE. LUNAR GRAVITATION MODEL .................................................................. 21 3.1 GRAVITATIONAL FIELD AND POTENTIAL THEORY .......................................................... 21 3.2 SPHERICAL HARMONICS ................................................................................................ 23 3.3 THE GRAVITATIONAL POTENTIAL IN SH S ........................................................................ 24 3.4 THE GRAVITATIONAL ACCELERATION ............................................................................ 25 3.5 PINES’ REPRESENTATION ............................................................................................... 26 CHAPTER FOUR. PERTURBATIONS ...................................................................................... 30 4.1 THIRD-BODY ................................................................................................................... 31 4.1.1 Earth-induced acceleration........................................................................................ 32 4.1.2 Sun-induced acceleration .......................................................................................... 33 4.2 SOLAR RADIATION PRESSURE ........................................................................................ 33 4.2.1 Eclipse conditions ...................................................................................................... 35 4.3 LUNAR ALBEDO .............................................................................................................. 36 4.4 LUNAR THERMAL RADIATION PRESSURE ....................................................................... 38 4.5 PERTURBATIONS AS A FUNCTION OF. ALTITUDE ........................................................... 39 CHAPTER FIVE. DEVELOPMENT OF THE CODE ....................................................................... 41 5.1 INPUT PARAMETERS ...................................................................................................... 41 5.1.1 Perturbation parameters ........................................................................................... 41 5.1.2 Initial position of the satellite .................................................................................... 42 5.1.3 Parameters related to integration ............................................................................. 44 5.1.4 Parameters related to celestial bodies ...................................................................... 45 5 E.C ARDOSO 5.1.5 Other parameters ...................................................................................................... 45 5.2 OUTPUT PARAMETERS ................................................................................................... 46 5.3 CODE PERFORMANCE ...................................................................................................
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