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Keplerian Ellipses

The physics of the gravitational two-body problem
Bruce Cameron Reed


The development of man's understanding of planetary motions is the crown jewel of Newtonian mechanics. This book offers a concise but self-contained handbook-length treatment of this historically important topic for students at about the third-year-level of an undergraduate physics curriculum. After opening with a review of Kepler's three laws of planetary motion, it proceeds to analyze the general dynamics of "central force" orbits in spherical coordinates, how elliptical orbits satisfy Newton's gravitational law and how the geometry of ellipses relates to physical quantities such as energy and momentum. Exercises are provided and derivations are set up in such a way that readers can gain analytic practice by filling in missing steps. A brief bibliography lists sources for readers who wish to pursue further study on their own.

About Editors

Bruce Cameron Reed is the Charles A Dana Professor of Physics Emeritus at Alma College, Michigan. He holds a PhD in physics from the University of Waterloo in Canada. In addition to a quantum mechanics text, he has published four books on the Manhattan Project, including the IOP Concise Physics volumes Atomic Bomb: The Story of the Manhattan Project and The Manhattan Project: A very brief introduction to the physics of nuclear weapons.

Table of Contents



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


Paperback ISBN: 9780750329859

Ebook ISBN: 9781643274690

DOI: 10.1088/2053-2571/ab0303

Publisher: Morgan & Claypool Publishers


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