Methods of Celestial Mechanics

PrefaceI. 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 ReferencesII. 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 ReferencesIII. 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 ReferencesIV. 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 ReferencesV. 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 ReferencesVII. 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 ReferencesVIII. 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 ReferencesIX. the Differential Correction of Orbits 1. Introduction 2. Use of Rectangular Equatorial Coordinates Notes and ReferencesX. 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 ReferencesXI. 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 ReferencesXII. 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 ReferencesXIII. 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 ReferencesXIV. 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 ReferencesXV. 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 ReferencesXVI. 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 ReferencesXVII. 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 ReferencesSubject Index