Linear stability of ring systems around oblate central masses

In this paper, we consider the stability of a ring of bodies of equal mass uniformly distributed around a large oblate central mass. The purpose of this and previous papers is to shed light on the stability of Saturn’s rings. Previous papers have been limited by the assumptions that (1) all ring bodies are at the same distance from the central body, (2) the central body acts like a point mass (i.e., is a perfect sphere), and (3) the ring bodies all have the same mass and are evenly spaced around the ring. The third limitation is probably the least important; as long as there are a large number of masses and the mass distribution is approximately uniform then the system should behave as a system of equally-spaced, equal-mass bodies. The first main purpose of this paper is to remove the second limitation. But, the paper also aims to address limitation (1). Recent computer simulations of single-ring systems have shown that the threshold for stability, as determined by a linear stability analysis, matches precisely the stability threshold predicted by simulation. In other words, a linear stability analysis while presumably just a mathematical abstraction actually tells us something quite real. Furthermore, simulations of multi-ring systems suggest that instability comes from azimuthal perturbations; small azimuthal changes are more destabilizing that small radial perturbations. Hence, in this paper, we also consider the situation where the central body consists not just of an oblate central mass but also incorporates a flat ring representing in aggregate all ring bodies at radii other than the one under consideration. The central oblate body together with a flat ring is modeled simply by introducing two oblateness terms to the gravitational potential associated with the central mass. The subsequent analysis is almost identical to the case of a single oblateness term. For Saturn, the oblateness of the central mass is six orders of magnitude more significant than the rings at other radii as a destabilizing influence.