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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