# Methods of Celestial Mechanics

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Methods of Celestial Mechanics provides a comprehensive background of celestial mechanics for practical applications. Celestial mechanics is the branch of astronomy that is devoted to the motions of celestial bodies. This book is composed of 17 chapters, and begins with the concept of elliptic motion and its expansion. The subsequent chapters are devoted to other aspects of celestial mechanics, including gravity, numerical integration of orbit, stellar aberration, lunar theory, and celestial coordinates. Considerable chapters explore the principles and application of various mathematical methods. This book is of value to mathematicians, physicists, astronomers, and celestial researchers.

## Table of Contents

Preface

I. Elliptic Motion

1. Historical Introduction

2. The Laws of Motion and the Law of Gravitation

3. Equations of Motion for the Two-Body Problem

4. Motion of the Center of Mass

5. Equations of Motion about the Center of Mass

6. Equations for the Relative Motion

7. The Integrals of Area

8. The Vis Viva Integral

9. Motion in the Orbital Plane

10. Kepler's Third Law

11. The Eccentric Anomaly

12. The Mean Anomaly

13. Formulas for Obtaining the Position in the Orbital Plane

14. Motion about the Center of Mass

15. The Energy Integral

16. The Potential Energy

17. Change to a Coordinate System with the Origin at the Center of Mass

18. The Integrals of Area

19. Coordinates Referred to the Ecliptic

20. Coordinates Referred to the Equator

21. Introduction of Matrices

22. Change of Order in a Product of Matrices

23. Rotation Matrices.

24. General Rotations of Coordinate Systems

25. Use of Polar Coordinates

26. Reduction to the Ecliptic

27. Calculation of the Elements from the Coordinates and Velocity Components at a Given Time

28. Accuracy of the Elements

29. Constants for the Equator

30. Expressions in Terms of Initial Coordinates and Velocity Components

31. The Gaussian Constant

Notes and References

II. Expansions in Elliptic Motion

1. Introduction

2. Expansions in a Fourier Series

3. The True Anomaly Expressed in Terms of the Eccentric Anomaly

4. The Mean Anomaly Expressed in Terms of the True Anomaly

5. Introduction of Bessel Functions

6. Application of Bessel Functions

7. Calculation of the Bessel Functions

8. Solution of Kepler's Equation

9. Solution of the Equations of Motion in Terms of the Mean Anomaly

10. The Rotating Coordinate System

11. Complex Rectangular Coordinates

12. Expansions by Harmonic Analysis

Notes and References

III. Gravitational Attraction between Bodies of Finite Dimensions

1. Introduction

2. Attraction of a Particle by a Body of Finite Dimensions and Arbitrary Distribution of Mass

3. Legendre Polynomials

4. the Principal Parts of U

5. Introduction of Polar Coordinates

6. The Expression for U3

7. The Expression for U4

8. The Potential of a Spheroid

9. Potential for Two Bodies of Finite Dimensions

Notes and References

IV. Calculus of Finite Differences

1. Representation of Functions

2. Differences

3. Detection of Casual Errors

4. Direct Interpolation

5. Everett's and Bessel's Formulas

6. Newton's Formula

7. Lagrange's Formula for Interpolation to Halves

8. Inverse Interpolation

9. Error of an Interpolated Quantity

10. Numerical Differentiation

11. Special Formulas

12. Numerical Integration

13. Accumulation of Errors in Numerical Integration

14. Symbolic Operators

Notes and References

V. Numerical Integration of Orbits

1. Introduction

2. Equations for Cowell's Method

3. Numerical Application of Cowell's Method

4. Equations for Encke's Method

5. Numerical Application of Encke's Method

6. Equations with Origin at the Center of Mass

7. Integration with Augmented Mass of the Sun

8. Relative Advantages of Cowell'S and Encke'S Methods

Notes and References.

VI. Aberration

1. Introduction

2. Stellar Aberration

3. Planetary Aberration

4. Diurnal Aberration

5. Calculation of the Annual Aberration

6. Ephemerides

7. Special Cases of Aberration

Notes and References

VII. Comparison of Observation and Theory

1. Introduction

2. Motions of the Planes of Reference

3. Precession

4. Nutation

5. Geocentric Parallax

6. Precepts

Notes and References

VIII. the Method of Least Squares

1. Introduction

2. Frequency Distribution of Errors of Observations

3. The Preferred Value of a Measured Quantity

4. Weights of Observations

5. Indirect Measurements

6. Equations of Condition

7. Weights of Equations

8. Formation of Normal Equations

9. Normal Equations

10. Formal Solution

11. Numerical Example

12. Combinations of the Unknowns

13. Correlations

14. Normal Places

Notes and References

IX. the Differential Correction of Orbits

1. Introduction

2. Use of Rectangular Equatorial Coordinates

Notes and References

X. General Integrals. Equilibrium Solutions

1. The Integrals of the Center of Mass

2. The Integrals of Area and the Energy Integral

3. The Restricted Problem of Three Bodies

4. Tisserand'S Criterion

5. Surfaces and Curves of Zero Velocity

6. The Particular Solutions of Lagrange

7. Small Oscillation about Equilibrium Solutions

8. Different Forms of the Equations of Motion

Notes and References

XI. Variation of Arbitrary Constants

1. Basic Principles of the Method

2. Lagrange's Brackets

3. Time Independence of Lagrange's Brackets

4. Whittaker'S Method for Obtaining Lagrange's Brackets

5. The Derivatives of the Keplerian Elements

6. Modification to Avoid t Outside of Trigonometric Arguments

7. Alternative Forms in Cases of Small Eccentricity Or Small Inclination

8. The Set a, eE, I, σ, W, Ω

9. A Canonical Set of Elements

10. Perturbations of the First Order. Secular and Periodic Terms

11. Perturbations of the Second Order

12. Small Divisors

13. Gauss's Form of the Equations

14. Direct Derivation of Gauss's Equations

Notes and References

XII. Lunar Theory

1. Statement of the Problem

2. The Equations of Motion

3. Development of the Disturbing Function in Terms of Elliptic Elements

4. Properties of the Disturbing Function

5. Integration of the Principal Terms by the Method of the Variation of Arbitrary Constants

6. Secular Terms

7. The Principal Periodic Terms

8. The Variation

9. The Evection

10. The Annual Equation

11. The Parallactic Inequality

12. The Principal Perturbation in the Latitude

13. Application of Kepler's Third Law to Satellite Orbits

14. Terms Without m as a Factor

15. Further Approximations

16. Comments On Delaunay's and Hansen's Theories

17. Introductory Remarks On Hill's "Researches in the Lunar Theory"

18. Hill'S Equations for the Moon's Motion

19. Introduction of u and s

20. Solution of u and s in Powers of m

21. Results for the Variation Orbit

22. The Scale Factor a

23. Transformation of the Equations

24. The Function Θ

25. The Motion of the Perigee

26. The Motion of the Node

27. Brown's Method of Differential Correction

28. Brown's Lunar Theory

Notes and References

XIII. Perturbations of the Coordinates

1. Introduction

2. Differential Equations

3. Integration

4. Hansen's Device

5. The Factors q1 and q2

6. the Superfluous Constant

7. Perturbations of the First Order

8. Secular Perturbations

9. Introduction to Brouwer's Method

10. Equations of Motion

11. Integration

12. Formal Solution

13. Explicit Solution

14. Expressions for the Perturbations

15. The Square Brackets

16. Constants of Integration

17. The Disturbing Function and Its Derivatives

Notes and References

XIV. Hansen's Method

1. Introduction

2. Principle of the Method

3. Systems of Coordinates

4. The Equations for v and r

5. The Expression for w0

6. The Equation for u

7. The Time as Independent Variable

8. Constants of Integration—Time as Independent Variable

9. Eccentric Anomaly as Independent Variable

10. Constants of Integration—Eccentric Anomaly as Independent Variable

11. The Disturbing Function and Its Derivatives

12. Perturbations of the Second Order

Notes and References

XV. the Disturbing Function

1. Introduction

2. Numerical Method

3. Numerical Method with Laplace Coefficients

4. Literal Method

5. The Indirect Portion

6. Literal Development

7. Laplace Coefficients

8. Derivatives of the Laplace Coefficients

Notes and References

XVI. Secular Perturbations

1. Introduction

2. The Secular Part of the Disturbing Function

3. Solution for Two Planets

4. Extension of the Solution to Any Number of Planets

5. Evaluation of the Constants of Integration

6. Jacobi'S Solution of the Determinant Equations

7. Secular Perturbations of Minor Planets

Notes and References

XVII. Canonical Variables

1. General Principles

2. Canonical Transformations

3. The Jacobian Determinant

4. Infinitesimal Contact Transformations

5. Examples

6. The Determining Function

7. Delaunay's Method

8. Study of a Delaunay Transformation

9. Solution of Delaunay's Problem by Finding a Determining Function

10. Example of a Delaunay Transformation

11. Solution of the Same Problem with the Aid of a Determining Function

12. the Motion of an Artificial Satellite

13. Relation to the Problem of Two Fixed Centers

14. the Atmospheric Drag Effect in the Motion of an Artificial Satellite

15. Application to the Motion of a Minor Planet Perturbed by Jupiter

16. Equations in the Delaunay Variables for the General Problem of Planetary Motion

Notes and References

Subject Index

## Product details

- No. of pages: 610
- Language: English
- Copyright: © Academic Press 1961
- Published: January 1, 1961
- Imprint: Academic Press
- eBook ISBN: 9781483225784

## About the Authors

### Dirk Brouwer

### Gerald M. Clemence

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