The Two-Body Problem

Abstract

For about two thousand years the efforts of western astronomers have been largely devoted to the purpose of understanding the motion in the sky of the sun, the moon and the planets. 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 for an outstanding scientific revolution, can be traced back to Kepler’s theory of planetary motions. Giving up the long-standing a priori requirement of circular paths covered with unifom velocities (or of finite combinations of them, resulting into epicyclic trajectories), Kepler’s three laws elliptic orbits with the sun at one focus; constant areal velocities; 3/2-power dependence of orbital periods on semimajor axes fitted the astronomical observations with unprecedented accuracy, and at the same time summarized into a few simple mathematical relationships most of the available kinematical observations. Within less than a hundred years, the Newtonian synthesis exploited Kepler’s results starting from a completely new perspective, that of producing a conceptually simple, but very general dynamical theory of which Kepler’s laws (as well as many other physical phenomena) could be seen as specific — and approximate consequences.