Perihelion precession in power-law potentials: Hénon's theorem

Abstract

In 1977, Michel Hénon proved a remarkable theorem for planar N-body orbits in power-law potentials that relates the rate of change of the perihelion angle (the precession rate), the rate of change of the period evaluated at constant energy, the angular momentum of an orbit, and the power law of the potential. We provide a simple proof of this theorem for two bodies in periodic orbits that interact via a radial power-law force, which is, of course, equivalent to a one-body problem with a power-law central potential. We discuss this theorem's underlying assumptions and implications, including its relation to Bertrand's and Bohlin's theorems, and we illustrate it with several numerically calculated examples.