The geometry of isochrone orbits: from Archimedes’ parabolae to Kepler’s third law

The Kepler potential ∝ - 1 / r and the harmonic potential ∝ r² share the following remarkable property: In either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and harmonic potentials are called isochrone. In this paper, we solve the following general problem: Are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Hénon (Annales d’Astrophysique 22:126–139, 1959a, 22:491–498, 1959b), in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometrical generalization of Kepler’s third law, and give a similar law for the apsidal angle, of any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations and use this to derive an analytic parameterization of any isochrone orbit. Along the way, we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem and argue that our geometrical methods can be exported to more generic orbits in potential theory.