Keplerian Ellipses: The Physics of the Gravitational Two-Body Problem

Description

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 the missing steps. A brief bibliography lists sources for readers who wish to pursue further study on their own. PRELIMS.pdf Preface Acknowledgments Author biography Bruce Cameron Reed CH001.pdf Chapter 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 CH002.pdf Chapter 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 CH003.pdf Chapter 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 CH004.pdf Chapter 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? CH005.pdf Chapter 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 CH006.pdf Chapter 6 Kepler’s equation: anomalies true, eccentric, and mean CH007.pdf Chapter 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 BIBLIO.pdf Outline placeholder References