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Satellite_Orbits_Models_Methods_and_Applications.pdf
SATELLITE ORBITS: MODELS, METHODS, APPLICATIONS
Half-title
Title Page
Copyright Page
Preface
Contents
Chapter 1. Around the World in a Hundred Minutes
1.1 A Portfolio of Satellite Orbits
1.1.1 Low-Earth Orbits
1.1.2 Orbits of Remote Sensing Satellites
1.1.3 Geostationary Orbits
1.1.4 Highly Elliptical Orbits
1.1.5 Constellations
1.2 Navigating in Space
1.2.1 Tracking Systems
1.2.2 A Matter of Effort
Chapter 2. Introductory Astrodynamics
2.1 General Properties of the Two-Body Problem
2.1.1 Plane Motion and the Law of Areas
2.1.2 The Form of the Orbit
2.1.3 The Energy Integral
2.2 Prediction of Unperturbed Satellite Orbits
2.2.1 Kepler's Equation and the Time Dependence of Motion
2.2.2 Solving Kepler's Equation
2.2.3 The Orbit in Space
2.2.4 Orbital Elements from Position and Velocity
2.2.5 Non-Singular Elements
2.3 Ground-Based Satellite Observations
2.3.1 Satellite Ground Tracks
2.3.2 Satellite Motion in the Local Tangent Coordinate System
2.4 Preliminary Orbit Determination
2.4.1 Orbit Determination from Two Position Vectors
2.4.2 Orbit Determination from Three Sets of Angles
Exercises
Chapter 3. Force Model
3.1 Introduction
3.2 Geopotential
3.2.1 Expansion in Spherical Harmonics
3.2.2 Some Special Geopotential Coefficients
3.2.3 Gravity Models
3.2.4 Recursions
3.2.5 Acceleration
3.3 Sun and Moon
3.3.1 Perturbing Acceleration
3.3.2 Low-Precision Solar and Lunar Coordinates
3.3.3 Chebyshev Approximation
3.3.4 JPL Ephemerides
3.4 Solar Radiation Pressure
3.4.1 Eclipse Conditions
3.4.2 Shadow Function
3.5 Atmospheric Drag
3.5.1 The Upper Atmosphere
3.5.2 The Harris–Priester Density Model
3.5.3 The Jacchia 1971 Density Model
3.5.4 A Comparison of Upper Atmosphere Density Models
3.5.5 Prediction of Solar and Geomagnetic Indices
3.6 Thrust Forces
3.7 Precision Modeling
3.7.1 Earth Radiation Pressure
3.7.2 Earth Tides
3.7.3 Relativistic Effects
3.7.4 Empirical Forces
Exercises
Chapter 4. Numerical Integration
4.1 Runge–Kutta Methods
4.1.1 Introduction
4.1.2 General Runge–Kutta Formulas
4.1.3 Stepsize Control
4.1.4 Runge–Kutta–Nyström Methods
4.1.5 Continuous Methods
4.1.6 Comparison of Runge–Kutta Methods
4.2 Multistep Methods
4.2.1 Introduction
4.2.2 Adams–Bashforth Methods
4.2.3 Adams–Moulton and Predictor–Corrector Methods
4.2.4 Interpolation
4.2.5 Variable Order and Stepsize Methods
4.2.6 Stoermer and Cowell Methods
4.2.7 Gauss–Jackson or Second Sum Methods
4.2.8 Comparison of Multistep Methods
4.3 Extrapolation Methods
4.3.1 The Mid-Point Rule
4.3.2 Extrapolation
4.3.3 Comparison of Extrapolation Methods
4.4 Comparison
Exercises
Chapter 5. Time and Reference Systems
5.1 Time
5.1.1 Ephemeris Time
5.1.2 Atomic Time
5.1.3 Relativistic Time Scales
5.1.4 Sidereal Time and Universal Time
5.2 Celestial and Terrestrial Reference Systems
5.3 Precession and Nutation
5.3.1 Lunisolar Torques and the Motion of the Earth's Rotation Axis
5.3.2 Coordinate Changes due to Precession
5.3.3 Nutation
5.4 Earth Rotation and Polar Motion
5.4.1 Rotation About the Celestial Ephemeris Pole
5.4.2 Free Eulerian Precession
5.4.3 Observation and Extrapolation of Polar Motion
5.4.4 Transformation to the International Reference Pole
5.5 Geodetic Datums
Exercises
Chapter 6. Satellite Tracking and Observation Models
6.1 Tracking Systems
6.1.1 Radar Tracking
6.1.2 Laser Tracking
6.1.3 The Global Positioning System
6.2 Tracking Data Models
6.2.1 Transmitter and Receiver Motion
6.2.2 Angle Measurements
6.2.3 Range Measurements
6.2.4 Doppler Measurements
6.2.5 GPS Measurements
6.3 Media Corrections
6.3.1 Interaction of Radiation and Atmosphere
6.3.2 Tropospheric Refraction
6.3.3 Ionospheric Refraction
Exercises
Chapter 7. Linearization
7.1 Two-Body State Transition Matrix
7.1.1 Orbital-Elements Transition Matrix
7.1.2 Keplerian-to-Cartesian Partial Derivatives
7.1.3 Cartesian-to-Keplerian Partial Derivatives
7.1.4 The State Transition Matrix and Its Inverse
7.2 Variational Equations
7.2.1 The Differential Equation of the State Transition Matrix
7.2.2 The Differential Equation of the Sensitivity Matrix
7.2.3 Form and Solution of the Variational Equations
7.2.4 The Inverse of the State Transition Matrix
7.3 Partial Derivatives of the Acceleration
7.3.1 Geopotential
7.3.2 Point-Mass Perturbations
7.3.3 Solar Radiation Pressure
7.3.4 Drag
7.3.5 Thrust
7.4 Partials of the Measurements with Respect to the State Vector
7.5 Partials with Respect to Measurement Model Parameters
7.6 Difference Quotient Approximations
Exercises
Chapter 8. Orbit Determination and Parameter Estimation
8.1 Weighted Least-Squares Estimation
8.1.1 Linearization and Normal Equations
8.1.2 Weighting
8.1.3 Statistical Interpretation
8.1.4 Consider Parameters
8.1.5 Estimation with A Priori Information
8.2 Numerical Solution of Least-Squares Problems
8.2.1 QR Factorization
8.2.2 Householder Transformations
8.2.3 Givens Rotations
8.2.4 Singular Value Decomposition
8.3 Kalman Filtering
8.3.1 Recursive Formulation of Least-Squares Estimation
8.3.2 Sequential Estimation
8.3.3 Extended Kalman Filter
8.3.4 Factorization Methods
8.3.5 Process Noise
8.4 Comparison of Batch and Sequential Estimation
Exercises
Chapter 9. Applications
9.1 Orbit Determination Error Analysis
9.1.1 A Linearized Orbit Model
9.1.2 Consider Covariance Analysis
9.1.3 The GEODA Program
9.1.4 Case Studies
9.2 Real-Time Orbit Determination
9.2.1 Model and Filter Design
9.2.2 The RTOD Program
9.2.3 Case Studies
9.3 Relay Satellite Orbit Determination
9.3.1 Mathematical Models
9.3.2 The TDRSOD Program
9.3.3 Case Study
Appendix A
A.1 Calendrical Calculations
A.1.1 Modified Julian Date from the Calendar Date
A.1.2 Calendar Date from the Modified Julian Date
A.2 GPS Orbit Models
A.2.1 Almanac Model
A.2.2 Broadcast Ephemeris Model
Appendix B
B.1 Internet Resources
B.2 The Enclosed CD-ROM
B.2.1 Contents
B.2.2 System Requirements
B.2.3 Executing the Programs
B.2.4 Compilation and Linking
B.2.5 Index of Library Functions
List of Symbols
References
Index
Back Cover
Montenbruck - Gill
Satellite Orbits
Springer
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Physics and Astronomy
http://www.springende/phys/
Oliver Montenbruck Eberhard Gill
Satellite Orbits
Models, Methods, and Applications
With 97 Figures
Including io Color Figures,
47 Tables, and CD-ROM
Springer
Dr. Oliver Montenbruck
Dr. Eberhard Gill
Deutsches Zentrum fiir Luft- und Raumfahrt (DLR) e.V.
Oberpfaffenhofen
Postfach mó
82230 WeBling, Germany
e-mail:
oliver.montenbruck@d1r.de
eberhard.gill@d1r.de
Cover picture: Designed for a mission time of two years; on duty for eight years. Built by Dornier
SatelLitensysteme GmbH, the German X-ray satellite Rosat is an ongoing success story. © DSS
Library of Congress Cataloging-in-Publication Data. Montenbruck, Oliver, 1961-. Satellite orbits : mod-
els, methods, and applications/Oliver Montenbruck, Eberhard GiLl. p.cm. Includes bibliographical ref-
erences and index. ISBN354067280X (alk. paper) 1. artificial satellites-Orbits. L Eberhard, Gill, 1961-
II. Title. TL io80.M66 2000 629.4'113-dc21 00-038815
Corrected 2nd Printing 2001
1st Edition 2000
ISBN 3-540-67280-X Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-
casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of
this publication or parts thereof is permitted only under the provisions of the German Copyright Law
of September 9, 1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
Springer-Verlag Berlin Heidelberg New York
a member of BertelsmannSpringer Science+Business Media GmbH
http://www.springer.de
© Springer-Verlag Berlin Heidelberg 2000
Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant pro-
tective laws and regulations and therefore free for general use.
Please note: Before using the programs in this book, please consult the technical manuals provided by
the manufacturer of the computer - and of any additional plug-in boards - to be used. The authors and
the publisher accept no legal responsibility for any damage caused by improper use of the instructionA,
and programs contained herein. Although these programs have been tested with extreme care, we can
offer no formal guarantee that they will function correctly. The programs on the enclosed CD-ROM are
under copyright protection and may not be reproduced without written permission by Springer-Verlag.
One copy of the programs may be made as a back-up, but all further copies violate copyright law.
Typesetting: camera-ready copy from the authors
Cover design: Erich Kirchner, Heidelberg
Printed on acid-free paper
SPIN: 10855805
55/3141/ba - 5 4 3 2 1 0
Preface
Satellite Orbits — Models, Methods, and Applications has been written as a compre-
hensive textbook that guides the reader through the theory and practice of satellite
orbit prediction and determination. Starting from the basic principles of orbital
mechanics, it covers elaborate force models as well as precise methods of satellite
tracking and their mathematical treatment. A multitude of numerical algorithms
used in present-day satellite trajectory computation is described in detail, with
proper focus on numerical integration and parameter estimation. The wide range of
levels provided renders the book suitable for an advanced undergraduate or gradu-
ate course on spaceflight mechanics, up to a professional reference in navigation,
geodesy and space science. Furtheimore, we hope that it is considered useful by
the increasing number of satellite engineers and operators trying to obtain a deeper
understanding of flight dynamics.
The idea for this book emerged when we realized that documentation on the
methods, models and tools of orbit determination was either spread over numerous
technical and scientific publications, or hidden in software descriptions that are not,
in general, accessible to a wider community. Having worked for many years in the
field of spaceflight dynamics and satellite operations, we tried to keep in close touch
with questions and problems that arise during daily work, and to stress the practical
aspects of orbit determination. Nevertheless, our interest in the underlying physics
motivated us to present topics from first principles, and make the book much more
than just a cookbook on spacecraft trajectory computation.
With the availability of powerful onground and onboard computefs; as well as
increasing demands for precision, the need for analytical perturbation theories has
almost been replaced by a purely numerical treatment of the equations of motion.
We therefore focus on models and methods that can be applied within a numerical
reconstruction of the satellite orbit and its fbrecast. As a consequence, topics like
orbit design, long-term orbit evolution and orbital decay are not addressed specifi-
cally, although the required fundamentals are provided. Geodesic satellite missions,
on the other hand, have reached an unprecedented level of position accuracy with a
need for very complex force and measurement models, which could not always be
covered in full detail. In any case, references to background information are given,
so as to allow the reader easy access to these specific areas.
Each chapter includes exercises at varying levels of complexity, which aim at
an additional practice of the presented material, or address supplementary topics
of practical interest. Where possible, we have tried to focus on problems that high-
VI
Preface
light the underlying physicals models or algorithmic methods, rather than relying
on purely numerical reference examples. In most cases, the exercises include a
comprehensive description of the suggested solution, as well as the numerical re-
sults. These are either derived directly from equations given in the text, or based
on sample computer programs.
This book comes with a CD-ROM that contains the C++ source code of all
sample programs and applications, as well as relevant data files. The software is
built around a powerful spaceflight dynamics library, which is likewise provided as
source code. For the sake of simplicity we have restricted the library to basic mod-
els, but emphasized transparent programming and in-code documentation. This, in
turn, allows for an immediate understanding of the code, and paves the way for easy
software extensions by the user. Free use of the entire software package including
the right for modifications is granted for non-commercial purposes. Readers, stu-
dents and lecturers are, therefore, encouraged to apply it in further studies, and
to develop new applications. We assume that the reader is familiar with computer
programming, but even inexperienced readers should be able to use the library func-
tions as black boxes. All source code is written in C++, nowadays a widely used
programming language and one which is readily available on a variety of different
platforms and operating systems.
We would like to thank Springer-Verlag for their cordial cooperation and in-
terest during the process of publishing this book. Our thanks are also due to all our
friends and colleagues, who, with their ideas and advice, and their help in correct-
ing the manuscript and in testing the programs, have played an important role in
the successful completion of this book. Real mission data sets for the application
programs have kindly been provided by the GPS/MET project and the Flight Dy-
namics Analysis Branch of the Goddard Space Flight Center. Numerous agencies
and individuals have contributed images for the introduction of this book, which is
gratefully acknowledged.
May 2000
Oliver Montenbruck and Eberhard Gill
Contents
1 Around the World in a Hundred Minutes
1.1 A Portfolio of Satellite Orbits
1.1.1 Low-Earth Orbits
1.1.2 Orbits of Remote Sensing Satellites
1.1.3 Geostationary Orbits
1.1.4 Highly Elliptical Orbits
1.1.5 Constellations
1.2 Navigating in Space
1.2.1 Tracking Systems
1.2.2 A Matter of Effort
2
Introductory Astrodynatnics
2.1 General Properties of the Two-Body Problem
2.1.1 Plane Motion and the Law of Areas
2.1.2 The Form of the Orbit
2.1.3 The Energy Integral
2.2 Prediction of Unperturbed Satellite Orbits
2.2.1 Kepler's Equation and the Time Dependence of Motion. .
2.2.2 Solving Kepler's Equation
2.2.3 The Orbit in Space
2.2.4 Orbital Elements from Position and Velocity .
2.2.5 Non-Singular Elements
....
2.3 Ground-Based Satellite Observations
2.3.1 Satellite Ground Tracks
2.3.2 Satellite Motion in the Local Tangent Coordinate System .
2.4 Preliminary Orbit Determination
2.4.1 Orbit Determination from Two Position Vectors
2.4.2 Orbit Determination from Three Sets of Angles
Exercises
3 Force Model
3.1 Introduction
3.2 Geopotential
3.2.1 Expansion in Spherical Harmonics
3.2.2 Some Special Geopotential Coefficients
1
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3
4
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