The Rotation of Earth Around the Sun; A Mathematical Treatment Based on Keplerian and Newtonian Celestial Mechanics

Abstract

 One of the most difficult problems of classical solar system mechanics is the orbit of the earth around the Sun. The present book addresses this question, however, by purely mathematical means, starting from Newton's law of universal gravitation and ending with a complete analytical solution which re-creates Kepler's three "empirical" laws. The two-body problem is simplified to a single body problem about the center of mass, and angle momentum as well as mechanical energy conservation is used to integrate motion equations. The shape of the orbit of the earth-sun system is an elliptical shape with a semi-major axis of approximately 1.49598 × 10¹¹ m and an eccentricity of approximately 0.0167.  Derivation of the vis-viva relation, calculation of the change in orbital speed from the perihelion to the aphelion is done analytically and verified numerically by integrating the equations of motion by the fourth-order Runge–Kutta method. The data for the eight planets of the Solar System is tested for Kepler's third law, which results in a correlation in the log–log plane of almost unity. The results restate the mathematical beauty and predictive power of the inverse-square central-force model, and give an example to show how the rotation of the Earth about the sun is a natural consequence of a single differential equation whose parameters are determined by the properties of the sun and earth.