Thermal hysteresis and front propagation in dense planetary rings

Saturn's rings are composed of icy grains, most in the mm to m size ranges, undergoing several collisions per orbit. Their collective behaviour generates a remarkable array of structure over many orders of magnitude, much of it not well understood. On the other hand, the collisional properties and parameters of individual ring particles are poorly constrained; usually N-body simulations and kinetic theory employ hard-sphere models with a coefficient of restitution $\epsilon$ that is constant or a decreasing function of impact speed. Due to plastic deformation of surface regolith, however, it is likely that $\epsilon$ will be more complicated, at the very least a non-monotonic function. We undertake N-body simulations with the REBOUND code with non-monotonic $\epsilon$ laws to approximate surfaces that are friable but not sticking. Our simulations reveal that such ring models can support two thermally stable steady states for the same (dynamical) optical depth: a cold and a warm state. If the ring breaks up into radial bands of one or the other state, we find that warmer states tend to migrate into the colder states via a coherent travelling front. We also find stationary `viscous' fronts, which connect states of different optical depth, but the same angular momentum flux. We discuss these preliminary results and speculate on their implications for structure formation in Saturn's B and C-rings, especially with respect to structures that appear in Cassini images but not in occultations.