Methods of Celestial Mechanics: Physical, Mathematical, and Numerical Principles

by Beutler, Gerhard; Mervart, Leos; Verdun, Andreas

• ISBN13: 9783540407492
• ISBN10: 3540407499
• Edition: CD
• Format: Hardcover
• Copyright: 2004-08-30
• 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 CD-ROM enables readers to employ this software themselves and also serves as to illustrate and reinforce the related theoretical concepts.

Table of Contents

Part I. Physical, Mathematical, and Numerical Principles
1. Overview of the Work	3	(16)
1.1 Part I: Theory	3	(6)
1.2 Part II: Applications	9	(5)
1.3 Part III: Program System	14	(5)
2. Historical Background	19	(26)
2.1 Milestones in the History of Celestial Mechanics of the Planetary System	19	(12)
2.2 The Advent of Space Geodesy	31	(14)
3. The Equations of Motion	45	(78)
3.1 Basic Concepts	46	(4)
3.2 The Planetary System	50	(11)
3.2.1 Equations of Motion of the Planetary System	51	(4)
3.2.2 First Integrals	55	(6)
3.3 The Earth-Moon-Sun-System	61	(35)
3.3.1 Introduction	61	(2)
3.3.2 Kinematics of Rigid Bodies	63	(8)
3.3.3 The Equations of Motion in the Inertial System	71	(7)
3.3.4 The Equations of Motion in the Body-Fixed Systems	78	(2)
3.3.5 Development of the Equations of Motion	80	(10)
3.3.6 Second Order Differential Equations for the Euler Angles Ψ, epsilon and Θ	90	(1)
3.3.7 Kinematics of the Non-Rigid Earth	91	(3)
3.3.8 Liouville-Euler Equations of Earth Rotation	94	(2)
3.4 Equations of Motion for an Artificial Earth Satellite	96	(20)
3.4.1 Introduction	96	(1)
3.4.2 Equations for the Center of Mass of a Satellite	97	(13)
3.4.3 Attitude of a Satellite	110	(6)
3.5 Relativistic Versions of the Equations of Motion	116	(4)
3.6 The Equations of Motion in Overview	120	(3)
4. The Two- and the Three-Body Problems	123	(52)
4.1 The Two-Body Problem	123	(17)
4.1.1 Orbital Plane and Law of Areas	123	(2)
4.1.2 Shape and Size of the Orbit	125	(5)
4.1.3 The Laplace Integral and the Laplace Vector q	130	(2)
4.1.4 True Anomaly upsilon as a Function of Time: Conventional Approach	132	(5)
4.1.5 True Anomaly upsilon as a Function of Time: Alternative Approaches	137	(3)
4.2 State Vector and Orbital Elements	140	(4)
4.2.1 State Vector -> Orbital Elements	142	(1)
4.2.2 Orbital elements -> State Vector	143	(1)
4.3 Osculating and Mean Elements	144	(3)
4.4 The Relativistic Two-Body Problem	147	(3)
4.5 The Three-Body Problem	150	(25)
4.5.1 The General Problem	152	(3)
4.5.2 The Problème Restraint	155	(20)
5. Variational Equations	175	(34)
5.1 Motivation and Overview	175	(1)
5.2 Primary and Variational Equations	176	(7)
5.3 Variational Equations of the Two-Body Problem	183	(12)
5.3.1 Elliptic Orbits	186	(4)
5.3.2 Parabolic Orbits	190	(2)
5.3.3 Hyperbolic Orbits	192	(1)
5.3.4 Summary and Examples	193	(2)
5.4 Variational Equations Associated with One Trajectory	195	(3)
5.5 Variational Equations Associated with the N-Body Problem	198	(4)
5.6 Efficient Solution of the Variational Equations	202	(4)
5.6.1 Trajectories of Individual Bodies	203	(2)
5.6.2 The N-Body Problem	205	(1)
5.7 Variational Equations and Error Propagation	206	(3)
6. Theory of Perturbations	209	(44)
6.1 Motivation and Classification	209	(2)
6.2 Encke-Type Equations of Motion	211	(4)
6.3 Gaussian Perturbation Equations	215	(17)
6.3.1 General Form of the Equations	215	(2)
6.3.2 The Equation for the Semi-major Axis α	217	(1)
6.3.3 The Gaussian Equations in Terms of Vectors h, q	218	(5)
6.3.4 Gaussian Perturbation Equations in Standard Form	223	(5)
6.3.5 Decompositions of the Perturbation Term	228	(4)
6.4 Lagrange's Planetary Equations	232	(8)
6.4.1 General Form of the Equations	232	(2)
6.4.2 Lagrange's Equation for the Semi-major Axis α	234	(1)
6.4.3 Lagrange's Planetary Equations	234	(6)
6.5 First- and Higher-Order Perturbations	240	(2)
6.6 Development of the Perturbation Function	242	(5)
6.6.1 General Perturbation Theory Applied to Planetary Motion	243	(4)
6.7 Perturbation Equation for the Mean Anomaly σ(t)	247	(6)
7. Numerical Solution of Ordinary Differential Equations: Principles and Concepts	253	(102)
7.1 Introduction	253	(2)
7.2 Mathematical Structure	255	(4)
7.3 Euler's Algorithm	259	(5)
7.4 Solution Methods in Overview	264	(15)
7.4.1 Collocation Methods	264	(2)
7.4.2 Multistep Methods	266	(3)
7.4.3 Taylor Series Methods	269	(2)
7.4.4 Runge-Kutta Methods	271	(4)
7.4.5 Extrapolation Methods	275	(2)
7.4.6 Comparison of Different Methods	277	(2)
7.5 Collocation	279	(33)
7.5.1 Solution of the Initial Value Problem	280	(3)
7.5.2 The Local Boundary Value Problem	283	(2)
7.5.3 Efficient Solution of the Initial Value Problem	285	(6)
7.5.4 Integrating a Two-Body Orbit with a High-Order Collocation Method: An Example	291	(4)
7.5.5 Local Error Control with Collocation Algorithms	295	(9)
7.5.6 Multistep Methods as Special Collocation Methods	304	(8)
7.6 Linear Differential Equation Systems and Numerical Quadrature	312	(18)
7.6.1 Introductory Remarks	312	(1)
7.6.2 Taylor Series Solution	313	(2)
7.6.3 Collocation for Linear Systems: Basics	315	(2)
7.6.4 Collocation: Structure of the Local Error Function	317	(3)
7.6.5 Collocation Applied to Numerical Quadrature	320	(4)
7.6.6 Collocation: Examples	324	(6)
7.7 Error Propagation	330	(25)
7.7.1 Rounding Errors in Digital Computers	332	(2)
7.7.2 Propagation of Rounding Errors	334	(7)
7.7.3 Propagation of Approximation Errors	341	(7)
7.7.4 A Rule of Thumb for Integrating Orbits of Small Eccentricities with Constant Stepsize Methods	348	(2)
7.7.5 The General Law of Error Propagation	350	(5)
8. Orbit Determination and Parameter Estimation	355	(86)
8.1 Orbit Determination as a Parameter Estimation Problem	355	(1)
8.2 The Classical Pure Orbit Determination Problem	356	(10)
8.2.1 Solution of the Classical Orbit Improvement Problem	357	(6)
8.2.2 Astrometric Positions	363	(3)
8.3 First Orbit Determination	366	(30)
8.3.1 Determination of a Circular Orbit	369	(4)
8.3.2 The Two-Body Problem as a Boundary Value Problem	373	(5)
8.3.3 Orbit Determination as a Boundary Value Problem	378	(3)
8.3.4 Examples	381	(3)
8.3.5 Determination of a Parabolic Orbit	384	(4)
8.3.6 Gaussian- vs. Laplacian-Type Orbit Determination	388	(8)
8.4 Orbit Improvement: Examples	396	(8)
8.5 Parameter Estimation in Satellite Geodesy	404	(37)
8.5.1 The General Task	405	(1)
8.5.2 Satellite Laser Ranging	406	(7)
8.5.3 Scientific Use of the GPS	413	(10)
8.5.4 Orbit Determination for Low Earth Orbiters	423	(18)
References	441	(8)
Abbreviations and Acronyms	449	(4)
Name Index	453	(2)
Subject Index	455