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118 Pages·1994·4.47 MB·English

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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
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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
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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
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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published and unpublished literature.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: