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5.7 Impact of 50X50 Gravity Models in Orbit Determination

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5.7 Impact of 50X50 Gravity Models in Orbit Determination IMPLEMENTING A 50x50 GRAVITY FIELD MODEL IN AN ORBIT DETERMINATION SYSTEM by Daniel John Fonte, Jr. B.S. Astronautical Engineering United States Air Force Academy (1991) SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1993 @ Daniel John Fonte, Jr. uthor - I W Signature of A IT ,, Department of Aeronautics and Astronautics June 1993 Certified by F H J f I "0.41""'-Y- % w ---M. I V / Dr. Ronald J. Proulx Thesis Supervisor, CSDL Certified by / Dr. Paul J. Cefola Thesis Supervisor, CSDL , ~Lecturer, Departnent of Aeronautics and Astronautics Accepted bV 0 * F F Professor Harold Y. Wachman Chairman, Departmental Graduate Committee MASSACHUSETTS INSTITUTE OF TECHNOLOGY A , r1I 10" P%01 %# IMPLEMENTING A 50x50 GRAVITY FIELD MODEL IN AN ORBIT DETERMINATION SYSTEM by Daniel John Fonte, Jr. Submitted to the Department of Aeronautics and Astronautics on May 7, 1993 in partial fulfillment of the requirements for the Degree of Master of Science ABSTRACT The Kepler problem treats the earth as if it is a spherical body of uniform density. In actuality, the earth's shape deviates from a sphere in terms of latitude (described by zonal harmonics), longitude (sectorial harmonics), and combinations of both latitude and longitude (tesseral harmonics). Operational Orbit Determination (OD) systems in the 1960's focused on the effects of the first few zonal harmonics since (1) they represented the dominant terms of the geopotential perturbation, (2) they were well known, and (3) the use of a limited number of harmonics greatly simplified the perturbation theory used. The demand for increasingly accurate modeling of a satellite's motion, combined with an increase in knowledge of the geopotential and an advancement in computer technology, led to the inclusion of tesseral harmonics. The Draper Laboratory version of the Goddard Trajectory Determination System (R&D GTDS), one operational OD system, can currently implement up to a 21x21 gravity field model in its Cowell and Semianalytic Satellite Theory (SST) orbit generators. This thesis investigates the extension of R&D GTDS to include a 50x50 gravity field model in the Cowell and SST orbit generators. This extension would require code modifications in the following environments to support the various operational versions of R&D GTDS: IBM, VAX, Sun Workstation, and Silicon Graphics. In each of these environments, the Legendre polynomials, associated Legendre polynomials, Jacobi polynomials, Hansen coefficients, and harmonic coefficients must be investigated to determine if (1) overflow/underflow boundaries would be violated in computations or (2) a loss of accuracy would occur in computations of high degree and order. This investigation will determine whether normalized or un-normalized components of the potential must be used. Thesis Supervisor: Dr. Ronald J. Proulx Title: Technical Staff Engineer, The Charles Stark Draper Laboratory, Inc. Thesis Supervisor: Dr. Paul J. Cefola Title: Lecturer, Department of Aeronautics and Astronautics Program Manager, The Charles Stark Draper Laboratory, Inc. [This page intentionally left blank.] Acknowledgements My thanks is extended to many: Paul Cefola, for (1) ensuring I got here and (2) ensuring I got out of here. You taught me about space programs and how to always look at things from a positive perspective. Wayne McClain, for teaching me astrodynamics and about business life, in general. You were always willing to lend a hand and to lead me out of the land of the lost and confused. Ron Proulx, for being my partner in the trenches of GTDS. You gave me numerous tips concerning the handling of large software systems. David Carter, Rick Metzinger, Mark Slutsky, Moshe Cohen, and Jay Portnoy, your discussions on all topics ranging from GTDS, estimation, mathematics, physics, propulsion, to thesis preparation all added to the knowledge I captured here. Linda Leonard, for keeping the VAX raring to go. Roger Medeiros, for your generous support as group leader. John Sweeney and Joan Chiffer, for giving me the opportunity and handling countless administrative issues. My friends at Draper, you all know first hand what this is about. Keith Mertz, Diane Kim, and Hermann Rufenacht, for contributing to my knowledge about space systems. Dick Farrar, for providing TRACE results. Girish Patel (NASA GSFC) and Carl Wagner (NOAA), for sending the gravity models. Lee Myers (CSC), for updates to GTDS databases and TRAMP related discussions. My family, Stacee, and God, for unlimited support and encouragement. This thesis was prepared at The Charles Stark Draper Laboratory, Inc., with support from the Canadian Space Agency and Spar Aerospace Limited under Spar Aerospace Purchase Order 302163 and the RADARSAT Flight Dynamics Program (Contract S.700067). Publication of this thesis does not constitute approval by The Charles Stark Draper Laboratory, Inc., or the Massachusetts Institute of Technology of the findings or conclusions contained herein. It is published for the exchange and stimulation of ideas. I hereby assign my copyright of this thesis to The Charles Stark Draper Laboratory, Inc., Cambridge, Massachusetts. Daniel John Fonte, Jr., Lt.,TSAF Permission is hereby granted by The Charles Stark Draper Laboratory, Inc., to the Massachusetts Institute of Technology to reproduce any or all of this thesis. [This page intentionally left blank.] Contents Chapter 1 Introduction ........................................................................... 19 1.1 B ackground ......................................................................... 19 1.2 Thesis Overview ................................................................... 29 Chapter 2 Mathematical Techniques ............................................................ 32 2.1 Non-Spherical Earth Gravitational Attraction ................................... 32 2.1.1 Geopotential and Spherical Harmonics..............................33 2.1.2 Zonal Harmonics (order m = 0) ....................................... 48 2.1.3 Tesseral Harmonics (order m 0) .................................... 57 2.2 Perturbation Techniques ........................................................... 62 2.2.1 Cowell Mathematical Techniques ..................................... 65 2.2.2 Variation of Parameters (VOP) .................................... 68 2.2.2.1 Lagrange's VOP Equations .............................. 75 2.2.2.2 Gauss' VOP Equations ................................... 78 2.2.3 Implementation of VOP Formulations with Numerical Methods .................................................................. 82 2.3 Semianalytic Methods ............................................................. 82 2.3.1 Semianalytic Equations of Motion and the Generalized Method of Averaging ............................... 83 2.3.2 Semianalytic Propagators and Orbit Determination ................. 96 Chapter 3 Stability Testing ............................ ...................................... 99 3.1 Background ......................................................................... 99 3.2 Cowell Truth Model Description and Test Set-Up ............................ 101 3.3 Cowell Testing for 50x50 Fields ................................................. 105 3.4 Stability Testing for Semianalytic Theory ....................................... 109 3.4.1 Jacobi Polynomial Stability Testing .................................. 110 3.4.2 Hansen Coefficient Stability Testing ............................... 117 Chapter 4 Draper R&D GTDS Description .................................................... 125 4.1 Chapter Introduction ............................................................... 125 4.1.1 GTDS Overview ...................................................... 1..25 4.1.2 GTDS Developmental History ........................................ 128 4.2 R&D GTDS Functionality Associated to Gravity Modeling ............... 1...32 4.2.1 Numerical Theories ..................................................... 133 4.2.2 Analytical Theories ..................................................... 134 4.2.3 Semianalytical Theory .................................................. 134 4.2.4 Gravity-Related Input Processing .............. 140 4.2.5 Gravity-Related Database Maintenance ................. .. 145 4.3 Code Related Changes for 50x50 Gravity Field Models ...................... 148 4.3.1 WRITHARM Replacement, DANWHARM.FOR .................. 151 4.3.2 Changes Shared by Cowell and Semianalytic Theory .............. 160 4.3.2.1 HRMCF.CMN .......... ..............................160 4.3.2.2 GEOVAR.CMN .......... ............................. 165 4.3.2.3 LEGPOL.CMN ........................................... 168 4.3.2.4 FRCBD.FOR ........... ............................... 168 4.3.2.5 SETDAF.FOR ........................................... 169 4.3.2.6 Harmonic Coefficient READ Logic ..................... 169 4.3.2.7 LUMPCS.CMN ........................................... 172 4.3.2.8 CSBLNK.CMN ........................................... 173 4.3.3 Changes Unique to the Semianalytic Theory (SST)................175 4.3.3.1 ANAV1.CMN ............................................. 176 4.3.3.2 AVEPOT.CMN .......................................... 178 4.3.3.3 GRAVITY.CMN ..........................................179 4.3.3.4 MDWRK.CMN ........................................... 180 4.3.3.5 NUKES.CMN.............................. ..... 181 4.3.3.6
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