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MASTER's THESIS Behavior and Relative Velocity of Debris Near 2010:048 MASTER'S THESIS Behavior and Relative Velocity of Debris near Geostationary Orbit Lin Gao Luleå University of Technology Master Thesis, Continuation Courses Space Science and Technology Department of Space Science, Kiruna 2010:048 - ISSN: 1653-0187 - ISRN: LTU-PB-EX--10/048--SE CRANFIELD UNIVERSITY SCHOOL OF ENGINEERING MSc THESIS Academic Year 2009-10 Lin Gao Behavior and Relative Velocity of Debris near Geostationary Orbit Supervisor: Dr. S.E.Hobbs May 2010 This thesis is submitted in partial (45%) fulfillment of the requirements for the degree of Master of Science ©Cranfield University 2010. All rights reserved. No part of this publication may be reproduced without the written permission of the copyright owner. i Abstract A general model is developed describing third-body gravity perturbation to debris’ orbit. Applying this model to debris released from geostationary orbit tells their motion in both short and long term. Without considering the Moon’s precession around the solar pole, the relative velocity between GEO debris can be calculated. This is an important coefficient for simulating the GEO debris environment and can serve as an input to break up models. ii iii Acknowledgements To my parents and dear friend M.Z. for her support. Sincerely thank you to Dr. S.E.Hobbs for his supervision. iv CONTENTS v Contents Contents v List of figures viii Abbreviations x 1 Introduction 1 1.1Background................................ 1 1.2AimoftheThesis............................. 1 1.3DocumentStructure........................... 2 2 Literature Review 3 2.1GeosynchronousRegion......................... 3 2.2SpaceDebrisModels........................... 4 2.3PerturbationSource........................... 5 2.3.1 NonhomogeneityandOblatenessoftheEarth......... 6 2.3.2 AtmosphereDrag......................... 6 2.3.3 GravitationalPerturbation.................... 7 2.3.4 SolarRadiationandSolarWind................. 7 2.4MotionoftheMoon........................... 8 2.4.1 Cassini’sLaws........................... 8 2.4.2 Relative Motion between the Moon, Sun, and Earth ...... 8 2.4.3 LunarStandstill.......................... 9 vi CONTENTS 3 Gravity Perturbation Model Development 11 3.1GeneralAnalysis............................. 11 3.2GravityPerturbationModel....................... 12 3.2.1 AnalyticalApproach....................... 12 3.2.2 QuantitativeApproach...................... 13 3.2.3 Validation............................. 17 3.2.4 AmendmentofTheModel.................... 17 4 Relative Velocity Between GEO Debris 23 4.1VerificationoftheModel......................... 23 4.1.1 ComparisonBetweenOtherModels............... 23 4.1.2 Application of the Model to the Precession of Lunar Orbit . 26 4.2RelativeVelocityoftheGEODebris.................. 28 4.2.1 Motion of GEO Debris with Sun’s Perturbation ........ 28 4.2.2 CombinationofLunarandSolarGravityPerturbation.... 33 4.3TheRelativeVelocity........................... 35 5 Discussion of the Results 39 5.1AboutTheModel............................. 39 5.1.1 TheValidityoftheModel.................... 39 5.1.2 TheMeaningofDevelopingsuchaModel........... 40 5.2AboutTheResult............................. 40 6 Conclusions 42 References 43 A Calculation of iR 46 B Matlab code 48 B.1Numericallysolvethegroupofdifferentialfunctions.......... 48 Contents vii B.2PlotthePrecessionofLunarOrbitalPole................ 50 B.3 Plot Inclination and the Precession of GEO Debris’ Orbital Pole . 51 B.4PlotoftheRevolvingAscendingNode................. 53 B.5SimulationoftheDistributionofDebris................ 54 B.6SimulationoftheDistributionofRelativeVelocities.......... 54 C Application of Matlab cftool toolbox 57 viii LIST OF FIGURES List of Figures 2.1DistributionofGEODebris[1]...................... 3 2.2PayloadsandupperstageslaunchedintoGEO[2]........... 4 2.3GEOregion................................ 5 2.4ApparentpathsoftheSunandMoononthecelestialsphere...... 9 2.5ApparentmotionoftheSunandtheMoon.[3]............. 9 2.6Parametersofthelunarorbit....................... 10 3.1MaximumrelativevelocitybetweenGEOdebris............. 12 3.2TypicalpositionbetweenSunandEarth................ 13 3.3MomentappliedbySuntoGEOdebrisatSolstice........... 13 3.4GeneralsituationforSolstice....................... 14 3.5 Position of M and m in (x,y,z)and(x,y,z) .............. 14 3.6 iR is in the direction of OA when the inclination of orbital plane i =0 18 4.1 The New Defined i ............................ 24 4.2ComparisonofCoordinateSystems................... 25 4.3ThePrecessionofLunarOrbitalPole.................. 27 4.4Moon’sOrbitalInclinationtoEclipticPlane.............. 27 4.5TrackoftheOrbitalPoleofGEODebris................ 29 4.6TheInclinationoftheOrbit....................... 29 4.7Debris’RelativeMotiontotheEarthSurface.............. 30 LIST OF FIGURES ix 4.8 The Ascending Node is perpendicular to the Projection of the Orbital Pole..................................... 31 4.9 The Ascending Node Evolves with the Same Period as ω ....... 31 4.10 Curves have different amplitude and phase. They are all moving westwardsbecauseoftheEarthrotation................. 33 4.11DistributionofGEOdebris....................... 33 4.12 The simulation of debris distribution is quite similar to the observation. 34 4.13TheDistributionofRelativeVelocities................. 37 A.1 Direction of iR changswithL ...................... 46 C.1 Curve Fitting Toolbox Interface. Thetaz vs Time is plotted as blue andcurvefittingresultistheredline................... 57 C.2CurveFittingResult........................... 58 x Abbreviations Acronyms and definitions GEO Geostationary Earth Orbit IADC Inter-Agency Space Debris Coordination Committee SHM Simple-Harmonic-Motion Lunar Orbital Pole the pole perpendicular to the lunar orbit and it points to the north Ecliptic Pole the pole perpendicular to the ecliptic plane and it points to the north Introduction 1 Chapter 1 Introduction 1.1 Background Geostationary Earth Orbit (GEO) is widely used because of its unique characteris- tics. Its orbital period is exactly one sidereal day with the altitude of 35,786 km. Satellites using this orbit can thus stay above a fixed point relative to the Earth, which is very important for some satellites like communication satellites. But after their lifetimes these satellites are no longer actively controlled and become orbital debris. Break-ups and explosions would further contribute to the debris population. Since there is no nature removal mechanism in GEO region, these debris would have lifetimes exceeding a million years. Their orbit would drift from 15 degrees north to 15 degrees south of the equatorial plane and 52 km above and below the geosynchronous arc because of the gravity perturbation from the Sun and the Moon [4]. The nonhomogeneity and oblateness of the Earth cause migration west and east around the Earth. Net effect of these motions leads to a torus around the Earth, where only 32% of the 1,124 known objects (2004) are under active control [1]. Because of the threat placed by these debris, risk and damage assessment are in- dispensable in spacecrafts design. In breakup models the initial condition, e.g., the relative velocity, is of extreme importance in the case of collision (rather than explo- sion or rupture) [5]. And the distribution of relative velocity between GEO debris is exactly what this thesis is pursuing. 1.2 Aim of the Thesis The speed of any GEO objects is around 3.07 km/s with no more than 0.48% vari- ation. So the relative velocity is decided mainly by the direction, i.e., the orbit inclination. This places the necessity of a thorough understanding of third-body 2 Introduction gravity perturbation on debris’ orbit. After developing a model of gravity pertur- bation, the motion of GEO debris can be studied and relative velocity would be a direct result. 1.3 Document Structure The key to know the motion of GEO debris is to understand the mechanism how the Sun and the Moon apply gravity perturbation to them. Most of the theories describing third-body gravity perturbation were done under the assumption that the orbital inclination i ≈ 0. Chapter 3 is the development of a perturbation model without this assumption. In Section 4.1, solution of the model was compared to the work by Alby.F. It was further applied to the lunar orbit and results were quite optimistic. These prove that my model is reasonable and correct. The motion of GEO debris in short and long term considering only the Sun is discussed in Section 4.2.1. SHM for GEO debris in short term is assumed to be a good approximation because of the similarity between the actual and computed result from the model considering only the Sun. In Section 4.2.2 detailed discussion about the approximation is provided. Calculation of the relative velocity between debris is then listed in Section 4.3. Literature Review 3 Chapter 2 Literature Review Geosynchronous region is fairly crowded with retired satellites and fragments that were generated by break-ups, explosions and collisions. And the number of launches each year is generally increasing (Figure 2.2). In all the cataloged GEO objects only 31% are under control (Figure 2.1 (a)). Break-ups of spacecrafts contribute about 43% of catalogued objects, and 85% of all space debris larger than 5 cm in diameter [6]. Explosions and collisions happen less frequently but contribute about 50 percent of all tracked objects [1]. Figure 2.1 (b) is a plot of the GEO debris torus. (a) cataloged GEO objects (b) debris torus around GEO Figure 2.1: Distribution of GEO Debris [1]. 2.1
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