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

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

Edition: CD

ISBN13: 9783540407492

ISBN10: 3540407499

Format: Hardcover

Pub. Date: 2004-08-30

Publisher(s): Springer Verlag

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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.

Part I. Physical, Mathematical, and Numerical Principles

1. Overview of the Work

(16)

1.1 Part I: Theory

(6)

1.2 Part II: Applications

(5)

1.3 Part III: Program System

(5)

2. Historical Background

(26)

2.1 Milestones in the History of Celestial Mechanics of the Planetary System

(12)

2.2 The Advent of Space Geodesy

(14)

3. The Equations of Motion

(78)

3.1 Basic Concepts

(4)

3.2 The Planetary System

(11)

3.2.1 Equations of Motion of the Planetary System

(4)

3.2.2 First Integrals

(6)

3.3 The Earth-Moon-Sun-System

(35)

3.3.1 Introduction

(2)

3.3.2 Kinematics of Rigid Bodies

(8)

3.3.3 The Equations of Motion in the Inertial System

(7)

3.3.4 The Equations of Motion in the Body-Fixed Systems

(2)

3.3.5 Development of the Equations of Motion

(10)

3.3.6 Second Order Differential Equations for the Euler Angles Ψ, epsilon and Θ

(1)

3.3.7 Kinematics of the Non-Rigid Earth

(3)

3.3.8 Liouville-Euler Equations of Earth Rotation

(2)

3.4 Equations of Motion for an Artificial Earth Satellite

(20)

3.4.1 Introduction

(1)

3.4.2 Equations for the Center of Mass of a Satellite

(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

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

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