The Two-Body Problem

Abstract

For about two thousand years, the efforts of Western astronomers have been largely devoted to the understanding of the motion of the Sun, the Moon and the planets in the sky. More and more complex geometrical and kinematical models, both geocentric and heliocentric, were devised in order to reproduce observational data of increasing accuracy. The final outcome of this effort, and at the same time the starting point of a scientific revolution, can be traced back to J. Kepler’s theory of planetary motions. Giving up the long-standing a priori requirement of circular paths and uniform velocities (or of finite combinations of them, resulting in epicyclic trajectories), Kepler’s three laws (elliptic orbits with the Sun at one focus, constant areal velocities, and 3/2-power dependence of orbital periods on semimajor axes) fitted astronomical observations with unprecedented accuracy, and at the same time summarized in a few simple mathematical relationships most of the available observations.