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Celestial Mechanics (last updated: 2023 January 16)
Part I. Mathematical Preambles
Chapter 1. Numerical Methods
| 1.1 | Introduction |
| 1.2 | Numerical Integration |
| 1.3 | Quadratic Equations |
| 1.4 | The Solution of f(x) = 0 |
| 1.5 | The Solution of Polynomial Equations |
| 1.6 | Failure of the Newton-Raphson Method |
| 1.7 | Simultaneous Linear Equations, N = n |
| 1.8 | Simultaneous Linear Equations, N < n |
| 1.9 | Nonlinear Simultaneous Equations |
| 1.10 | Besselian Interpolation |
| 1.11 | Fitting a Polynomial to a Set of Points. Lagrange Polynomials. Lagrange Interpolation. |
| 1.12 | Fitting a Least Squares Straight Line to a Set of Observational Points |
| 1.13 | Fitting a Least Squares Polynomial to a Set of Observational Points |
| 1.14 | Legendre Polynomials |
| 1.15 | Gaussian Quadrature - the Algorithm |
| 1.16 | Gaussian Quadrature - Derivation |
| 1.17 | Frequently-needed Numerical Procedures |
Chapter 2. Conic Sections
| 2.1 | Introduction |
| 2.2 | The Straight Line |
| 2.3 | The Ellipse |
| 2.4 | The Parabola |
| 2.5 | The Hyperbola |
| 2.6 | Conic Sections |
| 2.7 | The General Conic Section |
| 2.8 | Fitting a Conic Section Through Five Points |
| 2.9 | Fitting a Conic Section Through n Points |
Chapter 3. Plane and Spherical Trigonomtry
| 3.1 | Introduction |
| 3.2 | Plane Triangles |
| 3.3 | Cylindrical and Spherical Coordinates |
| 3.4 | Velocity and Acceleration Components |
| 3.5 | Spherical Triangles |
| 3.6 | Rotation of Axes, Two Dimensions |
| 3.7 | Rotation of Axes, Three Dimensions. Eulerian Angles |
| 3.8 | Trigonometrical Formulas |
Chapter 4. Coordinate Geometry in Three Dimensions
| 4.1 | Introduction |
| 4.2 | Planes and Straight Lines |
| 4.3 | The Ellipsoid |
| 4.4 | The Paraboloid |
| 4.5 | The Hyperboloid |
| 4.6 | The Cylinder |
| 4.7 | The Cone |
| 4.8 | The General Second Degree Equation in Three Dimensions |
| 4.9 | Matrices |
Chapter 5. Gravitational Field and Potential
| 5.1 | Introduction |
| 5.2 | Gravitational Field |
| 5.3 | Newton's Law of Gravitation |
| 5.4 | The Gravitational Fields Around Various Bodies |
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| 5.4.1 | Gravitational Field Around a Point Mass |
| 5.4.2 | Gravitational Field on the Axis of a Ring |
| 5.4.3 | Plane discs |
| 5.4.4 | Infinite Plane Laminas |
| 5.4.5 | Rods |
| 5.4.6 | Solid Cylinder |
| 5.4.7 | Hollow Spherical Shell |
| 5.4.8 | Uniform Solid Sphere |
| 5.4.9 | Bubble Inside a Uniform Solid Sphere |
| 5.4.10 | Field Inside a Nonuniform Sphere |
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| 5.4.10.A | Differentiated Planet |
| 5.4.10.B | Undifferentiated Planet |
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| 5.4.10.B.i | Density Increases Uniformly With Depth |
| 5.4.10.B.i | Density Increases Nonuniformly With Depth |
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| 5.5 | Gauss's Theorem |
| 5.6 | Calculating Surface Integrals |
| 5.7 | Pressure at the Centre of a Uniform Solid Sphere |
| 5.8 | Gravitational Potential |
| 5.9 | Nabla, Gradient and Divergence. Poisson's and Laplace's Equations |
| 5.10 | The Gravitational Potential Near Various Bodies |
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| 5.10.1 | Potential Near a Point Mass |
| 5.10.2 | Potential on the Axis of a Ring |
| 5.10.3 | Plane Discs |
| 5.10.4 | Infinite Plane Lamina |
| 5.10.5 | Hollow Hemisphere |
| 5.10.6 | Rods |
| 5.10.7 | Solid Cylinder |
| 5.10.8 | Hollow Spherical Shell |
| 5.10.9 | Uniform Solid Sphere |
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| 5.10.9.A | Potential Inside and Outside |
| 5.10.9.B | Work Required to Assemble a Uniform Sphere |
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| 5.11 | Legendre Polynomials |
| 5.12 | Gravitational Potential in the Vicinity of any Massive Body |
Part II. Celestial Mechanics
Chapter 6. The Celestial Sphere
| 6.1 | Introduction |
| 6.2 | Altazimuth Coordinates |
| 6.3 | Equatorial Coordinates |
| 6.4 | Conversion Between Equatorial and Altazimuth Coordinates |
| 6.5 | Ecliptic Coordinates |
| 6.6 | The Mean Sun |
| 6.7 | Precession |
| 6.8 | Nutation |
| 6.9 | The Length of the Year |
Chapter 7. Time
Chapter 8. Planetary Motions
| 8.1 | Introduction |
| 8.2 | Opposition, Conjunction and Quadrature |
| 8.3 | Sidereal and Synodic Periods |
| 8.4 | Direct and Retrograde Motion, and Stationary Points |
Chapter 9. The Two Body Problem in Two Dimensions
| 9.1 | Introduction |
| 9.2 | Kepler's Laws |
| 9.3 | Kepler's Second Law from Conservation of Angular Momentum |
| 9.4 | Some Functions of the Masses |
| 9.5 | Kepler's First and Third Laws from Newton's Law of Gravitation |
| 9.6 | Position in an Elliptic Orbit |
| 9.7 | Position in a Parabolic Orbit |
| 9.8 | Position in a Hyperbolic Orbit |
| 9.9 | Orbital Elements and Velocity Vector |
| 9.10 | Osculating Elements |
| 9.11 | Mean Distance in an Elliptic Orbit |
Chapter 10. Computation of an Ephemeris
| 10.1 | Introduction |
| 10.2 | Elements of an Elliptic Orbit |
| 10.3 | Some Additional Angles |
| 10.4 | Elements of a Circular or Near-circular Orbit |
| 10.5 | Elements of a Parabolic Orbit |
| 10.6 | Elements of a Hyperbolic Orbit |
| 10.7 | Calculating the Position of a Comet or Asteroid |
| 10.8 | Quadrant Problems |
| 10.9 | Computing an Ephemeris |
| 10.10 | Orbital Elements and Velocity Vector |
| 10.11 | Hamiltonian Formulation of the Equations of Motion |
Chapter 11. Photographic Astrometry
| 11.1 | Introduction |
| 11.2 | Standard Coordinates and Plate Constants |
| 11.3 | Refinements and Corrections |
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| 11.3.1 | Parallaxes of the Comparison Stars |
| 11.3.2 | Proper Motions of the Comparison Stars |
| 11.3.3 | Refraction |
| 11.3.4 | Aberration of light |
| 11.3.5 | Optical Distortion |
| 11.3.6 | Errors, Mistakes and Blunders |
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Chapter 12. CCD Astrometry
Chapter 13. Calculation of Orbital Elements
| 13.1 | Introduction |
| 13.2 | Triangles |
| 13.3 | Sectors |
| 13.4 | Kepler's Second Law |
| 13.5 | Coordinates |
| 13.6 | Example |
| 13.7 | Geocentric and Heliocentric Distances - First Attempt |
| 13.8 | Improved Triangle Ratios |
| 13.9 | Iterating |
| 13.10 | Higher-order Approximation |
| 13.11 | Light-time Correction |
| 13.12 | Sector-Triangle Ratio |
| 13.13 | Resuming the Numerical Example |
Chapter 14. General Perturbation Theory
| 14.1 | Introduction |
| 14.2 | Contact Transformations and General Perturbation Theory |
| 14.3 | The Poisson Brackets for the Orbital Elements |
| 14.4 | Lagrange's Planetary Equations |
| 14.5 | Motion Around an Oblate Symmetric Top |
Chapter 15. Special Perturbations
| 15.1 | Introduction |
| 15.2 | Orbital Elements and the Position and Velocity Vector |
| 15.3 | The Equations of Motion |
Chapter 16. Equivalent Potential and the Restricted Three-Body Problem
| 16.1 | Introduction |
| 16.2 | Motion Under a Central Force |
| 16.3 | Inverse Square Attractive Force |
| 16.4 | Hooke's Law |
| 16.5 | Inverse Fourth Power Force |
| 16.6 | The Collinear Lagrangian Points |
| 16.7 | The Equilateral Lagrangian Points |
Chapter 17. Visual Binary Stars
| 17.1 | Introduction |
| 17.2 | Determination of the Apparent Orbit |
| 17.3 | The Elements of the True Orbit |
| 17.4 | Determination of the Elements of the True Orbit |
| 17.5 | Construction of an Ephemeris |
Chapter 18. Spectroscopic Binary Stars
| 18.1 | Introduction |
| 18.2 | The Velocity Curve from the Elements |
| 18.3 | Preliminary Elements from the Velocity Curve |
| 18.4 | Masses |
| 18.5 | Refinement of the Orbital Elements |
| 18.6 | Finding the Period |
| 18.7 | Measuring the Radial Velocity |
Appendix A. Miscellaneous Problems
Appendix B. Solutions to Miscellaneous Problems
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