Keplerian Ellipses

Preface
Acknowledgements
1 Spherical Coordinates - A Review 1.1 Fundamental Definitions 1.2 Spherical Coordinate Unit Vectors 1.3 Time Derivatives of Spherical Coordinate Unit Vectors 1.4 Some Useful Integrals
2 Dynamical Quantities in Spherical Coordinates 2.1 Position, Velocity, Acceleration, Angular Momentum, Torque, and Energy 2.2 Uniform Circular Motion: A Specific Case of the Acceleration Formula
3 Central Forces 3.1 The Reduced Mass 3.2 Central Force Dynamics: The Potential 3.3 Why An Inverse-Square Law? 3.4 Central Force Dynamics: Conservation of Angular Momentum 3.5 Central Force Dynamics: Integrals of the Motion 3.6 Central Force Dynamics: Acceleration in Terms of the Azimuthal Angle
4 The Ellipse 4.1 The Ellipse in Cartesian and Polar Coordinates 4.2 Area of an Ellipse 4.3 Area as a Vector Cross Product, and Kepler's Second Law 4.4 How Did Kepler Plot the Orbits?
5 Elliptical Orbits and the Inverse-Square Law: Geometry Meets Physics 5.1 Proof By Assuming an Elliptical Orbit: Angular Momentum 5.2 Velocity, the Vis-Viva Equation, and Energy 5.3 Proof of Elliptical Orbits by Direct Integration 5.4 Kepler's Third Law 5.5 The Time-Angle Equation 5.6 Example: An Earth-Orbiting Spy Satellite
6 Kepler's Equation: Anomalies True, Eccentric, and Mean
7 Some Sundry Results 7.1 Average Distance of a Planet From the Sun 7.2 Determining Initial Launch Conditions 7.3 A Brief Lesson in Unit Conversions 7.4 Orientation of Earth's Orbit 7.5 Some Final Words
8 Bibliography
9 Glossary of Symbols