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by Dr. J. B. Tatum
jtatum@uvic.ca



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Celestial Mechanics (last updated: 2014 November 20)


Part I. Mathematical Preambles

Chapter 1.    Numerical Methods

1.1Introduction
1.2Numerical Integration
1.3Quadratic Equations
1.4The Solution of f(x) = 0
1.5The Solution of Polynomial Equations
1.6Failure of the Newton-Raphson Method
1.7Simultaneous Linear Equations, N = n
1.8Simultaneous Linear Equations, N < n
1.9Nonlinear Simultaneous Equations
1.10Besselian Interpolation
1.11Fitting a Polynomial to a Set of Points. Lagrange Polynomials. Lagrange Interpolation.
1.12Fitting a Least Squares Straight Line to a Set of Observational Points
1.13Fitting a Least Squares Polynomial to a Set of Observational Points
1.14Legendre Polynomials
1.15Gaussian Quadrature - the Algorithm
1.16Gaussian Quadrature - Derivation
1.17Frequently-needed Numerical Procedures


Chapter 2.    Conic Sections

2.1Introduction
2.2The Straight Line
2.3The Ellipse
2.4The Parabola
2.5The Hyperbola
2.6Conic Sections
2.7The General Conic Section
2.8Fitting a Conic Section Through Five Points
2.9Fitting a Conic Section Through n Points


Chapter 3.    Plane and Spherical Trigonomtry

3.1Introduction
3.2Plane Triangles
3.3Cylindrical and Spherical Coordinates
3.4Velocity and Acceleration Components
3.5Spherical Triangles
3.6Rotation of Axes, Two Dimensions
3.7Rotation of Axes, Three Dimensions. Eulerian Angles
3.8Trigonometrical Formulas


Chapter 4.    Coordinate Geometry in Three Dimensions

4.1Introduction
4.2Planes and Straight Lines
4.3The Ellipsoid
4.4The Paraboloid
4.5The Hyperboloid
4.6The Cylinder
4.7The Cone
4.8The General Second Degree Equation in Three Dimensions
4.9Matrices


Chapter 5.    Gravitational Field and Potential

5.1Introduction
5.2Gravitational Field
5.3Newton's Law of Gravitation
5.4The Gravitational Fields of Various Bodies
5.4.1Field of a Point Mass
5.4.2Field on the Axis of a Ring
5.4.3Plane discs
5.4.4Infinite Plane Laminas
5.4.5Hollow Hemisphere
5.4.6Rods
5.4.7Solid Cylinder
5.4.8Hollow Spherical Shell
5.4.9Solid Sphere
5.4.10Bubble Inside a Uniform Solid Sphere
5.5Gauss's Theorem
5.6Calculating Surface Integrals
5.7Potential
5.8The Gravitational Potentials Near Various Bodies
5.8.1Potential Near a Point Mass
5.8.2Potential on the Axis of a Ring
5.8.3Plane Discs
5.8.4Infinite Plane Lamina
5.8.5Hollow Hemisphere
5.8.6Rods
5.8.7Solid Cylinder
5.4.8Hollow Spherical Shell
5.8.9Solid Sphere
5.9Work Required to Assemble a Uniform Sphere
5.10Nabla, Gradient and Divergence
5.11Legendre Polynomials
5.12Gravitational Potential of any Massive Body
5.13Pressure at the Centre of a Uniform Sphere


Part II. Celestial Mechanics

Chapter 6.    The Celestial Sphere

6.1Introduction
6.2Altazimuth Coordinates
6.3Equatorial Coordinates
6.4Conversion Between Equatorial and Altazimuth Coordinates
6.5Ecliptic Coordinates
6.6The Mean Sun
6.7Precession
6.8Nutation
6.9The Length of the Year


Chapter 7.    Time

Chapter 8.    Planetary Motions

8.1Introduction
8.2Opposition, Conjunction and Quadrature
8.3Sidereal and Synodic Periods
8.4Direct and Retrograde Motion, and Stationary Points


Chapter 9.    The Two Body Problem in Two Dimensions

9.1Introduction
9.2Kepler's Laws
9.3Kepler's Second Law from Conservation of Angular Momentum
9.4Some Functions of the Masses
9.5Kepler's First and Third Laws from Newton's Law of Gravitation
9.6Position in an Elliptic Orbit
9.7Position in a Parabolic Orbit
9.8Position in a Hyperbolic Orbit
9.9Orbital Elements and Velocity Vector
9.10Osculating Elements
9.11Mean Distance in an Elliptic Orbit


Chapter 10.    Computation of an Ephemeris

10.1Introduction
10.2Elements of an Elliptic Orbit
10.3Some Additional Angles
10.4Elements of a Circular or Near-circular Orbit
10.5Elements of a Parabolic Orbit
10.6Elements of a Hyperbolic Orbit
10.7Calculating the Position of a Comet or Asteroid
10.8Quadrant Problems
10.9Computing an Ephemeris
10.10Orbital Elements and Velocity Vector
10.11Hamiltonian Formulation of the Equations of Motion


Chapter 11.    Photographic Astrometry

11.1Introduction
11.2Standard Coordinates and Plate Constants
11.3Refinements and Corrections
11.3.1Parallaxes of the Comparison Stars
11.3.2Proper Motions of the Comparison Stars
11.3.3Refraction
11.3.4Aberration of light
11.3.5Optical Distortion
11.3.6Errors, Mistakes and Blunders


Chapter 12.    CCD Astrometry

Chapter 13.    Calculation of Orbital Elements

13.1Introduction
13.2Triangles
13.3Sectors
13.4Kepler's Second Law
13.5Coordinates
13.6Example
13.7Geocentric and Heliocentric Distances - First Attempt
13.8Improved Triangle Ratios
13.9Iterating
13.10Higher-order Approximation
13.11Light-time Correction
13.12Sector-Triangle Ratio
13.13Resuming the Numerical Example


Chapter 14.    General Perturbation Theory

14.1Introduction
14.2Contact Transformations and General Perturbation Theory
14.3The Poisson Brackets for the Orbital Elements
14.4Lagrange's Planetary Equations
14.5Motion Around an Oblate Symmetric Top


Chapter 16.    Equivalent Potential and the Restricted Three-Body Problem

16.1Introduction
16.2Motion Under a Central Force
16.3Inverse Square Attractive Force
16.4Hooke's Law
16.5Inverse Fourth Power Force
16.6The Collinear Lagrangian Points
16.7The Equilateral Lagrangian Points


Chapter 17.    Visual Binary Stars

17.1Introduction
17.2Determination of the Apparent Orbit
17.3The Elements of the True Orbit
17.4Determination of the Elements of the True Orbit
17.5Construction of an Ephemeris


Chapter 18.    Spectroscopic Binary Stars

18.1Introduction
18.2The Velocity Curve from the Elements
18.3Preliminary Elements from the Velocity Curve
18.4Masses
18.5Refinement of the Orbital Elements
18.6Finding the Period
18.7Measuring the Radial Velocity





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