University of North Dakota UND Scholarly Commons Theses and Dissertations Theses, Dissertations, and Senior Projects January 2020 A Preliminary Study Of Leo To Geo Transfers For Inclination Changes Using Libration Point Orbits John Shepard Follow this and additional works at: https://commons.und.edu/theses Recommended Citation Shepard, John, "A Preliminary Study Of Leo To Geo Transfers For Inclination Changes Using Libration Point Orbits" (2020). Theses and Dissertations. 3121. https://commons.und.edu/theses/3121 This Thesis is brought to you for free and open access by the Theses, Dissertations, and Senior Projects at UND Scholarly Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UND Scholarly Commons. For more information, please contact [email protected]. A PRELIMINARY STUDY OF LEO TO GEO TRANSFERS FOR INCLINATION CHANGES USING LIBRATION POINT ORBITS by John Philip Shepard Applied Associates of Science, Community College of the Air Force, 2012 Applied Associates of Science, Community College of the Air Force, 2019 Bachelor of Science, Chadron State College, 2012 A Thesis . Submitted to the Graduate Faculty of the University of North Dakota in partial fulfillment of the requirements for the degree of Master of Science Grand Forks, North Dakota May 2020 iii This thesis , submitted by John Shepard in partial fulfillment of the requirement for the Degree of Master of Science in Space Studies from the University of North Dakota, has been read by the Faculty Advisory Committee under whom the work has been done and is hereby approved. This thesis is being submitted by the appointed advisory committee as having met all of the requirements of the school of Graduate Studies at the University of North Dakota and is hereby approved. iv PERMISSION Title A Preliminary Study of LEO to GEO Transfers for Inclination Changes Using Libration Point Orbits Department Space Studies Degree Master of Science In presenting this thesis in partial fulfillment of the requirements for a graduate degree from the University of North Dakota, I agree that the library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by the professor who supervised my thesis work or, in his absence, by the Chairperson of the department or the dean of the School of Graduate Studies. It is understood that any copying or publication or other use of this thesis or part thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of North Dakota in any scholarly use which may be made of any material in my thesis. John P. Shepard May 7th, 2020 v TABLE OF CONTENTS A PRELIMINARY STUDY OF LEO TO GEO TRANSFERS FOR INCLINATION CHANGES USING LIBRATION POINT ORBITS .............................................................................. ii PERMISSION .................................................................................................................... iv TABLE OF CONTENTS .................................................................................................... v ACKNOWLEDGEMENTS ............................................................................................... ix ABSTRACT ........................................................................................................................ x 1. INTRODUCTION ....................................................................................................... 1 1.1 Inclination Change ............................................................................................... 2 1.2 The Circular Restricted Three-Body Problem ..................................................... 3 1.3 Mission Scenario .................................................................................................. 4 1.4 Previous Contributors ........................................................................................... 7 1.4.1 Libration Point Orbits and Manifolds ........................................................... 8 1.4.2 Application in Mission Design ................................................................... 10 1.5 Current Work...................................................................................................... 11 1.6 Overview ............................................................................................................ 12 2. BACKGROUND ....................................................................................................... 14 2.1 The Circular Restricted Three-Body Problem ................................................... 14 2.2 Equilibrium Points.............................................................................................. 18 2.3 Numerical Computation Runge-Kutta Method .................................................. 19 2.4 Constructing Libration Point Orbits ................................................................... 20 2.4.1 Jacobi Constant and Forbidden Regions ..................................................... 20 2.4.2 Linear Approximation for Lyapunov Orbits ............................................... 22 2.4.3 Non-Linear Correction ................................................................................ 25 2.5 Brief introduction to Invariant Manifold Theory ............................................... 29 3. GENERATING LARGE INCLINATION ORBITAL MANEUVERS .................... 31 3.1 Exploring Capture Dynamics and Chaotic Motion ............................................ 31 3.2 Test Case 1: The Original Model & Changes for Desired Maneuvers .............. 33 vi 3.2.1 Adjusting the Equations of Motion and Numerical Method ....................... 39 3.2.2 Earth-Moon Design Transfers..................................................................... 40 3.3 Test Case 2: Orbital Transfers from LEO using Invariant Manifolds................ 44 3.4 Optimality vs Efficiency .................................................................................... 52 3.4.1 Bounding Spheres ....................................................................................... 52 3.4.2 Primer Vector Theory ................................................................................. 53 3.5 Orbital Construction ........................................................................................... 54 3.5.1 Construction of Orbit .................................................................................. 55 3.5.2 Retrograde Motion & Rendezvous Missions Explored .............................. 62 3.5.3 Results and Comparison ............................................................................. 64 4. ENVIRONMENTAL CONDITIONS & POTENTIAL APPLICATIONAL USE ... 67 4.1 Rescue and Salvage Missions ............................................................................ 68 4.2 Mission Design for Low Earth Orbit ................................................................. 71 4.2.1 Environmental Conditions .......................................................................... 72 4.3 Overall Efficiency .............................................................................................. 74 4.3.1 Economic Efficiency ................................................................................... 75 4.4 Environmental and Design Concluding Remarks .............................................. 76 5. Conclusion ................................................................................................................. 78 5.1 Summary of Results ........................................................................................... 78 5.2 Recommendation for Future Research ............................................................... 80 6. LIST OF REFERENCES........................................................................................... 82 Appendix A ....................................................................................................................... 86 Appendix B ....................................................................................................................... 91 LIST OF TABLES Table 2-2-1 Locating the Collinear Lagrange Points µ = 3.04x10-6 ................................ 18 Table 2-2-2 Initial Conditions for L2 Lyapunov Orbits ................................................... 28 vii Table 3-1 Amplitude in the z-axis vs Jacobi Constant with µ = 3.04x10-6 ..................... 45 Table 3-2 Initial Conditions for L2 Halo Orbit ................................................................. 46 Table 3-3 Δv vs Jacobi Constant for Designed Transfer Orbit ......................................... 47 Table 3-4 Time of Flight vs Jacobi Constant for Designed Transfer Orbit ...................... 49 Table 3-5 Δv vs Jacobi Constant for Designed Transfer Orbit. ........................................ 49 Table 3-6 Δi vs Jacobi Constant for Designed Transfer Orbit .......................................... 50 Table 3-7 Resulting Data from Orbital Maneuver ............................................................ 51 Table 3-8 Initial Conditions for L2 Horizontal Orbits ...................................................... 56 Table 3-9 Initial Conditions for L2 Horizontal Orbits ...................................................... 56 Table 3-10 Initial Conditions for L2 Axial Orbits ...........................................................