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Calhoun: The NPS Institutional Archive DSpace Repository Reports and Technical Reports All Technical Reports Collection 1995-02 Semianalytic satellite theory Danielson, D. A. (Donald A.).; Early, Leo W.; Sagovac, Christopher Patrick; Neta, Beny Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/24428 Downloaded from NPS Archive: Calhoun r NPS-MA-95-002 NAVAL POSTGRADUATE SCHOOL Monterey, California DTI r^ELECTE ^ JUN 0 6 19951 F SEMIANALYTIC SATELLITE THEORY by D.A. Danielson C.P. Sagovac B. Neta L.W. Early February 1995 Report for Period July 1992 - February 1995 Approved for public release; distribution is unlimited Prepared for: Naval Postgraduate School Monterey, CA 93943 DTIC QUALTT7 INSPECTED 9 19950602 016 NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943 Rear Admiral T.A. Mercer Harrison Shull Superintendent Provost This report was prepared in conjunction with research conducted for the Naval Space Command and the Naval Postgraduate School and funded by NAVSPACECOM (formerly NAVSPASUR) and the Naval Postgraduate School. Reproduction of all or part of this report is authorized. This report was prepared by: Accesion For NTIS CRA&I DTIC TAB Q D.A. Danielson Unannounced □ Professor of Mathematics Justification c By C.P. Sagoväc Distribution / Availability Codes B. Neta Dist Ava Silp eacniad/l or Professor of Mathematics M L.W. Early Reviewed by: Released by: fsjbfljj j f^rtfiX- RICHARD FRANKE PAUL/J. MARTO Chairman Dean of Research Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0188 Public reporting burden for this collection erf information is estimates to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Washington Headquarters Services. Directorate for information Operations and Reports, 1215 Jefferson Davis Highway. Suite 1204, Arlington. VA 22202-4302. and to the Office of Management and Budget. Paperwork Reduction Project (0704-0188). Washington. DC 20503. 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED February 1995 Technical Report July 1992 - February 1995 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS SEMIANALYTIC SATELLITE THEORY 6. AUTHOR(S) D.A. Danielson, C. P. Sagovac, B. Neta, L.W. Early 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Naval Postgraduate School Monterey, CA 93943-5000 NPS-MA-95-002 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY REPORT NUMBER Naval Space Command and Naval Postgraduate School 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for public release; distribution is unlimited. 13. ABSTRACT (Maximum 200 words) Modern space surveillance requires fast and accurate orbit predictions for myriads of objects in a broad range of Earth orbits. Conventional Special Perturbations orbit propagators, based on numerical integration of the osculating equations of motion, are accurate but extremely slow. Conventional General Perturbations orbit propagators, based on fully analytical theories like those of Brouwer, are faster but contain large errors due to inherent approximations in the theories. New orbit generators based on Semianalytic Satellite Theory have been able to approach the accuracy of Special Perturbations propagators and the speed of General Perturbations propagators. Semianalytic Satellite Theory has been originated by P. J. Cefola and his colleagues, but the theory is scattered throughout the published and unpublished literature. In this document the theory is simplified, assembled, unified, and extended. 14. SUBJECT TERMS 15. NUMBER OF PAGES Artifical Satellites, Orbit Propagation, Semianalytic 112 16. PRICE CODE Satellite Theory, Perturbations. 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT OF THIS PAGE OF ABSTRACT Unclassified Unclassified Unclassified UL NSN 7540-01-280-5500 Standard Form 298 (Rev 2-89) Prescribed by ANSI Sid 239-18 298-102 SEMIANALYTIC SATELLITE THEORY D. A. Danielson C. P. Sagovac, B. Neta, L. W. Early Mathematics Department Naval Postgraduate School Monterey, CA 93943 Acknowledgements Dr. Steve Knowles of the Naval Space Command gave the initial impetus for this undertaking. Dr. Paul Cefola, Dr. Ron Proulx, and Mr. Wayne McClain were always helpful and supportive of our work. Mrs. Rose Mendoza and Mrs. Elle Zimmerman, TßX typesetters at the Naval Postgraduate School, diligently worked many hours trying to decipher our handwriting. NAVSPACECOM (formerly NAVSPASUR) and the NPS Research Program provided the necessary financial support. Contents 1 Introduction 3 2 Mathematical Preliminaries 4 2.1 Equinoctial Elements 4 2.1.1 Definition of the Equinoctial Elements 4 2.1.2 Conversion from Keplerian Elements to Equinoctial Elements .... 6 2.1.3 Conversion from Equinoctial Elements to Keplerian Elements .... 7 2.1.4 Conversion from Equinoctial Elements to Position and Velocity ... 7 2.1.5 Conversion from Position and Velocity to Equinoctial Elements ... 9 2.1.6 Partial Derivatives of Position and Velocity with Respect to the Equinoc- tial Elements 10 2.1.7 Partial Derivatives of Equinoctial Elements with Respect to Position and Velocity 11 2.1.8 Poisson Brackets 12 2.1.9 Direction Cosines (of, ß,-y) 13 2.2 Variation-of-Parameters (VOP) Equations of Motion 14 2.3 Equations of Averaging 16 2.4 Averaged Equations of Motion 20 2.5 Short-Periodic Variations 26 2.5.1 General 7; Expansions in A 27 2.5.2 General rn Expansions in F 29 2.5.3 General r/i Expansions in L 31 2.5.4 General rn Expansions in A,# 36 2.5.5 First-Order r}ia for Conservative Perturbations 37 2.5.6 Second-Order rjiaß for Two Perturbations Expanded in A 38 2.5.7 Second-Order r)ia0 for Two Perturbations Expanded in L 39 2.6 Partial Derivatives for State Estimation 43 2.7 Central-Body Gravitational Potential 46 2.7.1 Expansion of the Geopotential in Equinoctial Variables 46 2.7.2 Calculation of V£ Coefficients 49 2.7.3 Calculation of Kernels Kf of Hansen Coefficients 49 v w 2.7.4 Calculation of Jacobi Polynomials P t 51 3 2.7.5 Calculation of G\ns and H ms Polynomials 52 2.8 Third-Body Gravitational Potential 52 2.8.1 Expansion of Third-Body Potential in Equinoctial Variables 53 2.8.2 Calculation of Vns Coefficients 53 2.8.3 Calculation of Qns Polynomials 54 3 First-Order Mean Element Rates 54 3.1 Central-Body Gravitational Zonal Harmonics 54 3.2 Third-Body Gravitational Potential 60 3.3 Central-Body Gravitational Resonant Tesserals 61 3 3.4 Atmospheric Drag " 3.5 Solar Radiation Pressure 66 68 First-Order Short-Periodic Variations 4.1 Central-Body Gravitational Zonal Harmonics 68 8 4.2 Third-Body Gravitational Potential ? 4 3 Central-Body Gravitational Tesserals °3 4.4 Atmospheric Drag °^ 5 4.5 Solar Radiation Presure ° 86 Higher-Order Terms 5.1 Second-Order Aiaß and rjia0 Due to Gravitational Zonals and Atmospheric Drag 86 5 2 Second-Order maß Cross-Coupling Between Secular Gravitational Zonals and Tesseral Harmonics °' 88 Truncation Algorithms 6.1 Third-Body Mean Gravitational Potential 90 6.2 Central-Body Mean Zonal Harmonics 91 6.3 Central-Body Tesseral Harmonics 92 6.4 Central-Body Zonal Harmonics Short-Periodics 93 6.5 Third-Body Short-Periodics 97 6.6 Nonconservative Short-Periodics and Second-Order Expansions 98 10 Numerical Methods ° 7.1 Numerical Solution of Kepler's Equation 100 7.2 Numerical Differentiation iOO 7.3 Numerical Quadrature !00 7.4 Numerical Integration of Mean Equations 101 103 7.5 Interpolation 1 Introduction Modern space surveillance requires fast and accurate orbit predictions for myriads of objects in a broad range of Earth orbits. Conventional Special Perturbations orbit propagators, based on numerical integration of the osculating equations of motion, are accurate but ex- tremely slow (typically requiring 100 or more steps per satellite revolution to give good predictions). Conventional General Perturbations orbit propagators, based on fully analyt- ical orbit theories like those of Brouwer, are faster but contain large errors due to inherent approximations in the theories. New orbit generators based on Semianalytic Satellite Theory (SST) have been able to approach the accuracy of Special Perturbations propagators and the speed of General Perturbations propagators. SST has been originated by P. J. Cefola and his colleagues, whose names are in the refer- ences at the end of this document. The theory is scattered throughout the listed conference preprints, published papers, technical reports, and private communications. Our purpose in this document is to simplify, assemble, unify, and extend the theory. This document includes truncation algorithms and corrects misprints in our earlier work [Danielson, Neta, and Early, 1994]. SST represents the orbital state of a satellite with an equinoctial element set (ai,..., a6). The first five elements a\,..., as are slowly varying in time. The sixth element a& is the mean longitude A and so is rapidly varying. SST decomposes the osculating elements a, into mean elements a,- plus a small remainder which is 2n periodic in the fast variable: hi = ai + rn(a1,...,a6,t) (1) (Here we use hats to distinguish the elements of the osculating ellipse from the elements of the averaging procedure. The values of a free index are assumed to be obvious from the context; e. g. , here i can have the values 1, 2, 3, 4, 5, or 6, so (1) represents 6 equations.) The mean elements a; are governed by ordinary differential equations of the form —'- = n8i6 + Ai(a1,...,a5, t) (2) Here t is the time, n is the (mean) mean motion, and 5i6 is the Kronecker delta (i. e. , ^16 = <^26 = <^36 = <^46 = ^56 = 0, ößs = 1)- The short-periodic variations r?; are expressable in Fourier series of the form 00 a a Vi = £[C/( i>---> 5, t)cosj\ + Sj(au...,a5, t)s'mjX] (3) i=i Having formulas for the mean element rates Ai, we can integrate the mean equations (2) numerically using large step sizes (typically 1 day in length). The formulas for the Fourier coefficients C{ and S\ in (3) also only need to be evaluated at the integrator step times. Values of the osculating elements a, at request times not coinciding with the integrator step times can be computed from (1) using interpolation formulas. In subsequent chapters we will outline the methods of derivation and give explicit formulas for the terms A{, C/, SI corresponding to various perturbing forces. 2 Mathematical Preliminaries 2.1 Equinoctial Elements The generalized method of averaging can be applied to a wide variety of orbit element sets. The equinoctial elements were chosen for SST because the variational equations for the equinoctial elements are nonsingular for all orbits for which the generalized method of averaging is appropriate-namely, all elliptical orbits. In this chapter we give an overview of the equinoctial elements, which are osculating (even though they do not have hats). They are discussed in more detail in [Broucke and Cefola, 1972], [Cefola, Long, and Holloway, 1974], [Long, McClain, and Cefola, 1975], [Cefola and Broucke, 1975], [McClain, 1977 and 1978], and [Shaver, 1980]. 2.1.1 Definition of the Equinoctial Elements There are six elements in the equinoctial element set: a\ = a semimajor axis a2 = h \ components of the eccentricity vector «3 = k a4 = p } components of the ascending node vector «5 = <7 a6 = A mean longitude The semimajor axis a is the same as the Keplerian semimajor axis. The eccentricity vector has a magnitude equal to the eccentricity and it points from the central body to perigee. Elements h and k are the g and f components, respectively, of the eccentricity vector in the equinoctial reference frame defined below. The ascending node vector has a magnitude which depends on the inclination and it points from the central body to the ascending node. Elements p and q are the g and f components, respectively, of the ascending node vector in the equinoctial reference frame. There are actually two equinoctial element sets: the direct set and the retrograde set. As the names imply, the direct set is more appropriate for direct satellites and the retrograde set is more appropriate for retrograde satellites. It is possible, however, to use direct elements for retrograde satellites and vice versa, and for non-equatorial satellites this presents no problem. For equatorial satellites there are singularities which must be avoided by choosing the appropriate equinoctial element set. For direct elements 2 2 lim \Jp + q = oo (1) while for retrograde elements 2 2 lim\/p + q — oo (2) t'-+0 * For each equinoctial element set there are three associated vectors (f, g, w) which define the equinoctial reference frame. These vectors form a right-handed orthonormal triad with the following properties:

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