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Celestial Mechanics (last updated: 2007 February 24)
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 | Newtons Law of Gravitation | 5.4 | The Gravitational Fields of Various Bodies | |
5.4.1 | Field of a Point Mass | 5.4.2 | Field on the Axis of a Ring | 5.4.3 | Plane discs | 5.4.4 | Infinite Plane Laminas | 5.4.5 | Hollow Hemisphere | 5.4.6 | Rods | 5.4.7 | Solid Cylinder | 5.4.8 | Hollow Spherical Shell | 5.4.9 | Solid Sphere | 5.4.10 | Bubble Inside a Uniform Solid Sphere |
| 5.5 | Gausss Theorem | 5.6 | Calculating Surface Integrals | 5.7 | Potential | 5.8 | The Gravitational Potentials Near Various Bodies | |
5.8.1 | Potential Near a Point Mass | 5.8.2 | Potential on the Axis of a Ring | 5.8.3 | Plane Discs | 5.8.4 | Infinite Plane Lamina | 5.8.5 | Hollow Hemisphere | 5.8.6 | Rods | 5.8.7 | Solid Cylinder | 5.4.8 | Hollow Spherical Shell | 5.8.9 | Solid Sphere |
| 5.9 | Work Required to Assemble a Uniform Sphere | 5.10 | Nabla, Gradient and Divergence | 5.11 | Legendre Polynomials | 5.12 | Gravitational Potential of any Massive Body | 5.13 | Pressure at the Centre of a Uniform Sphere |
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 | Keplers Laws | 9.3 | Keplers Second Law from Conservation of Angular Momentum | 9.4 | Some Functions of the Masses | 9.5 | Keplers First and Third Laws from Newtons 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 | |
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
(In preparation)
Chapter 13. Calculation of Orbital Elements
13.1 | Introduction | 13.2 | Triangles | 13.3 | Sectors | 13.4 | Keplers 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 | Lagranges Planetary Equations | 14.5 | Motion Around an Oblate Symmetric Top |
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 | Hookes 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 |
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