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Stability of rings around a triaxial primary

Generally, the oblateness of a planet or moon is what causes rings to settle into its equatorial plane. However, the recent suggestion that a ring system might exist (or have existed) about Rhea, a moon whose shape includes a strong prolate component pointed toward Saturn, raises the question of whether rings around a triaxial primary can be stable. We study the role of prolateness in the behavior of rings around Rhea and extend our results to similar problems such as possible rings around exoplanets. Using a Hamiltonian approach, we point out that the dynamical behavior of ring particles is governed by three different time scales: the orbital period of the particles, the rotation period of the primary, and the precession period of the particles' orbital plane. In the case of Rhea, two of these are well separated from the third, allowing us to average the Hamiltonian twice. To study the case of slow rotation of the primary, we also carry out numerical simulations of a thin disk of particles undergoing secular effects and damping. For Rhea, the averaging reduces the Hamiltonian to an oblate potential, under which rings would be stable only in the equatorial plane. This is not the case for Iapetus; rather, it is the lack of a prolate component to its shape that allows Iapetus to host rings. Plausible exoplanets should mostly be in the same regime as Rhea, though other outcomes are possible. The numerical simulations indicate that, even when the double averaging is irrelevant, rings settle in the equatorial plane on an approximately constant time scale.