Methods of Celestial Mechanics

by Beutler, Gerhard; Mervart, Leos (CON); Verdun, Andreas (CON)

• ISBN13: 9783642148576
• ISBN10: 3642148573
• Format: Paperback
• Copyright: 2010-09-30
• Publisher: Springer Verlag

Summary

G. Beutler's Methods of Celestial Mechanics is a coherent textbook for students as well as an excellent reference for practitioners. The first volume gives a thorough treatment of celestial mechanics and presents all the necessary mathematical details that a professional would need. The reader will appreciate the well-written chapters on numerical solution techniques for ordinary differential equations, as well as that on orbit determination. In the second volume applications to the rotation of earth and moon, to artificial earth satellites and to the planetary system are presented. The author addresses all aspects that are of importance in high-tech applications, such as the detailed gravitational fields of all planets and the earth, the oblateness of the earth, the radiation pressure and the atmospheric drag. The concluding part of this monumental treatise explains and details state-of-the-art professional and thoroughly-tested software for celestial mechanics. The accompanying online files enable readers to employ this software themselves and also serves as to illustrate and reinforce the related theoretical concepts.

Table of Contents

Physical, Mathematical, and Numerical Principles
Overview of the Work	p. 3
Part I: Theory	p. 3
Part II: Applications	p. 9
Part III: Program System	p. 14
Historical Background	p. 19
Milestones in the History of Celestial Mechanics of the Planetary System	p. 19
The Advent of Space Geodesy	p. 31
The Equations of Motion	p. 45
Basic Concepts	p. 46
The Planetary System	p. 50
Equations of Motion of the Planetary System	p. 51
First Integrals	p. 55
The Earth-Moon-Sun-System	p. 61
Introduction	p. 61
Kinematics of Rigid Bodies	p. 63
The Equations of Motion in the Inertial System	p. 71
The Equations of Motion in the Body-Fixed Systems	p. 78
Development of the Equations of Motion	p. 80
Second Order Differential Equations for the Euler Angles ¿, ¿ and ¿	p. 90
Kinematics of the Non-Rigid Earth	p. 91
Liouville-Euler Equations of Earth Rotation	p. 94
Equations of Motion for an Artificial Earth Satellite	p. 96
Introduction	p. 96
Equations for the Center of Mass of a Satellite	p. 97
Attitude of a Satellite	p. 110
Relativistic Versions of the Equations of Motion	p. 116
The Equations of Motion in Overview	p. 120
The Two- and the Three-Body Problems	p. 123
The Two-Body Problem	p. 123
Orbital Plane and Law of Areas	p. 123
Shape and Size of the Orbit	p. 125
The Laplace Integral and the Laplace Vector q	p. 130
True Anomaly v as a Function of Time: Conventional Approaches	p. 132
True Anomaly v as a Function of Time: Alternative Approaches	p. 137
State Vector and Orbital Elements	p. 140
State Vector → Orbital Elements	p. 142
Orbital elements → State Vector	p. 143
Osculating and Mean Elements	p. 144
The Relativistic Two-Body Problem	p. 147
The Three-Body Problem	p. 150
The General Problem	p. 152
The Problème Restreint	p. 155
Variational Equations	p. 175
Motivation and Overview	p. 175
Primary and Variational Equations	p. 176
Variational Equations of the Two-Body Problem	p. 183
Elliptic Orbits	p. 186
Parabolic Orbits	p. 190
Hyperbolic Orbits	p. 192
Summary and Examples	p. 193
Variational Equations Associated with One Trajectory	p. 195
Variational Equations Associated with the N-Body Problem	p. 198
Efficient Solution of the Variational Equations	p. 202
Trajectories of Individual Bodies	p. 203
The N-Body Problem	p. 205
Variational Equations and Error Propagation	p. 206
Theory of Perturbations	p. 209
Motivation and Classification	p. 209
Encke-Type Equations of Motion	p. 211
Gaussian Perturbation Equations	p. 215
General Form of the Equations	p. 215
The Equation for the Semi-major Axis a	p. 217
The Gaussian Equations in Terms of Vectors h, q	p. 218
Gaussian Perturbation Equations in Standard Form	p. 223
Decompositions of the Perturbation Term	p. 228
Lagrange's Planetary Equations	p. 232
General Form of the Equations	p. 232
Lagrange's Equation for the Semi-major Axis a	p. 234
Lagrange's Planetary Equations	p. 234
First- and Higher-Order Perturbations	p. 240
Development of the Perturbation Function	p. 242
General Perturbation Theory Applied to Planetary Motion	p. 243
Perturbation Equation for the Mean Anomaly ¿(t)	p. 247
Numerical Solution of Ordinary Differential Equations: Principles and Concepts	p. 253
Introduction	p. 253
Mathematical Structure	p. 255
Euler's Algorithm	p. 259
Solution Methods in Overview	p. 264
Collocation Methods	p. 264
Multistep Methods	p. 266
Taylor Series Methods	p. 269
Runge-Kutta Methods	p. 271
Extrapolation Methods	p. 275
Comparison of Different Methods	p. 277
Collocation	p. 279
Solution of the Initial Value Problem	p. 280
The Local Boundary Value Problem	p. 283
Efficient Solution of the Initial Value Problem	p. 285
Integrating a Two-Body Orbit with a High-Order Collocation Method: An Example	p. 291
Local Error Control with Collocation Algorithms	p. 295
Multistep Methods as Special Collocation Methods	p. 304
Linear Differential Equation Systems and Numerical Quadrature	p. 312
Introductory Remarks	p. 312
Taylor Series Solution	p. 313
Collocation for Linear Systems: Basics	p. 315
Collocation: Structure of the Local Error Function	p. 317
Collocation Applied to Numerical Quadrature	p. 320
Collocation: Examples	p. 324
Error Propagation	p. 330
Rounding Errors in Digital Computers	p. 332
Propagation of Rounding Errors	p. 334
Propagation of Approximation Errors	p. 341
A Rule of Thumb for Integrating Orbits of Small Eccentricities with Constant Stepsize Methods	p. 348
The General Law of Error Propagation	p. 350
Orbit Determination and Parameter Estimation	p. 355
Orbit Determination as a Parameter Estimation Problem	p. 355
The Classical Pure Orbit Determination Problem	p. 356
Solution of the Classical Orbit Improvement Problem	p. 357
Astrometric Positions	p. 363
First Orbit Determination	p. 366
Determination of a Circular Orbit	p. 369
The Two-Body Problem as a Boundary Value Problem	p. 373
Orbit Determination as a Boundary Value Problem	p. 378
Examples	p. 381
Determination of a Parabolic Orbit	p. 384
Gaussian- vs. Laplacian-Type Orbit Determination	p. 388
Orbit Improvement: Examples	p. 396
Parameter Estimation in Satellite Geodesy	p. 404
The General Task	p. 405
Satellite Laser Ranging	p. 406
Scientific Use of the GPS	p. 413
Orbit Determination for Low Earth Orbiters	p. 423
References	p. 441
Abbreviations and Acronyms	p. 449
Name Index	p. 453
Subject Index	p. 455
Table of Contents provided by Ingram. All Rights Reserved.