Can Hyperbolic Diffusion Help Explain Sharp Edges in the Gaps in Saturn's Rings?

Abstract

We explore whether hyperbolic diffusion may help explain sharp edges in the gaps in Saturn's rings. Sharp edges are conventionally understood to be due to angular momentum flux reversal at gap edges. We do not dispute this finding, but investigate whether non-classical diffusion may amplify this finding. We explore a simple model of hyperbolic diffusion for the radial spread of material in planetary rings. The model arises by the introduction of a relaxation time in an advection equation for the radial diffusive angular momentum flux. We show that radial secular forcing, combined with a hyperbolic diffusion equation, leads to sharp gap edges, in which the density of ring material drops precipitously down to zero at some critical distance from the moon's orbit. Additionally, we show that our simple model can produce large ``spikes'' or ``horns'' in the density profile on either side of a ring gap, mirroring results of large N-body simulations. It remains to be seen how these results may be affected by the inclusion of the well-understood angular momentum flux reversal near tidally-induced gap edges.