Spin rates of small moonlets embedded in planetary rings: I. Three-body calculations

We investigate the spin rates of moonlets embedded in planetary rings, subject to collisions with surrounding small particles, using three-body integrations including friction and spins. All successive impacts of the particle with the moonlet are followed, including a possible sliding phase after the initial inelastic rebounds. Two methods for treating impacts, (1) as instantaneous velocity changes and (2) using an impact force model, are applied after Salo (1995, Icarus 117, 287). Conducting a series of integrations with various initial summed spin velocity of the moonlet and the particle, we determine the equilibrium spin rate for which the averaged torque vanishes. This equilibrium spin rate corresponds to the final spin rate of the moonlet if the moonlet is much larger than the surrounding particles; it also corresponds to the mean spin rate for a ring composed of identical particles. We find that the equilibrium spin rate is enhanced by sliding orbits as compared with the spin rate determined by considering only the first impacts of the particles with the moonlet. If the random velocities of incident particles are small enough, the resulting equilibrium spin rate of the moonlet can be larger than the synchronous rotation rate, for rp∼1, where rp denotes the sum of radii of the colliding pair normalized by their mutual Hill radius. In this special case aggregates without internal strength may become rotationally unstable. However, the equilibrium spin rate decreases with increasing random velocity, and aggregates are always rotationally stable in the more likely case where the relative velocities are comparable to the mutual escape velocity. We also compare our results with the mean spin rates found in previous N-body simulations, and find a good agreement for optically thin rings; however the spin rates for optically thick rings are significantly larger than those predicted by our three-body calculations.