The dynamics of orbits in a potential field of a solid circular ring

We are interested in studying the properties and perturbations of orbits around a central planet surrounded by a ring. The problem has been studied a long time ago by Laplace, Maxwell and others. Maxwell considered a ring composed of a number of discrete masses orbiting in a circular orbit. Gauss also derived the potential due to a solid circular ring and its derivation is reproduced in Kellog's textbook on Potential Theory. The potential can be evaluated in terms of a complete elliptic integral of the first kind. Computing the accelerations also requires a second kind elliptic integral. We have experimented with at least three different methods for computing the potential and its first partial derivatives: Gauss quadrature, the Carlson functions and the Arithmetico-Geometric mean. The standard formulation breaks down near the center of the ring which is an unstable equilibrium point but a linearization can be made near this point. We have also studied the efficiency of the Spherical Harmonic expansion of the ring potential. This expansion has only the even zonal terms and thus no tesserals. In a preliminary study, we have looked at planar periodic orbits (and their stability), around a ring without a central body, both in the plane of the ring and the plane orthogonal to it. We find nearly a dozen types and families of periodic orbits. Several of these families seem to end with an orbit that collides with the ring. One of the goals of this preliminary study is to understand the effect of the singularity at the ring itself, (where the potential and the accelerations become infinite).