Golden Elliptical Orbits in Newtonian Gravitation

In spherical symmetry with radial coordinate $r$, classical Newtonian gravitation supports circular orbits and, for $-1/r$ and $r^2$ potentials only, closed elliptical orbits [1]. Various families of elliptical orbits can be thought of as arising from the action of perturbations on corresponding circular orbits. We show that one elliptical orbit in each family is singled out because its focal length is equal to the radius of the corresponding unperturbed circular orbit. The eccentricity of this special orbit is related to the famous irrational number known as the golden ratio. So inanimate Newtonian gravitation appears to exhibit (but not prefer) the golden ratio which has been previously identified mostly in settings within the animate world.