The extent and style of viscous relaxation of the largest crater basins on the icy satellites of Saturn can be used as a probe of the viscosity of the deep interiors of these bodies. In particular, some constraints on the size of a silicate core and the magnitude of the internal viscosity gradient can be determined if the relaxation process is sufficiently well understood. We discuss the application of a finite element model incorporating spherical symmetry to the general problem of relaxation of large craters on spherical bodies. We examine the influence of core size, geothermal gradient, and rheologic law on relaxation flow and consider the implications for the internal structures of Mimas and Tethys. Our model results show that relaxation of crater topography on a spherical satellite is qualitatively and quantitatively different from relaxation on a planar surface once the crater diameter becomes a significant fraction of the satellite radius. For small-to moderate-sized basins, relaxation flow is bidirectional, with flow both from rim to bowl and rim to surrounds.

This is the same style of flow observed for craters on planar surfaces. For large enough basins, however, the spherical geometry of the problem leads to flow that is unidirectional, toward the crater bowl throughout the satellite. The transition from bidirectional to unidirectional flow appears to be very sensitive to the basin diameter. The critical basin diameter for transition increases with core size, thermal gradient, rim height, and rheologic law stress exponent. Unidirectional and bidirectional flow produce fundamentally different stress fields in the regions surrounding the relaxing crater. Relaxation velocities for large craters on spherical bodies can be appreciably more rapid than those for equivalent craters on planar surfaces, again because of the geometric effects of relaxation on a sphere. The difference in degree of relaxation between the large basins Odysseus on Tethys (which has undergone significant modification) and Herschel on Mimas (which appears essentially unmodified) can be explained by different thermal histories for Tethys and Mimas, by the effects of stress-dependent rheology, or by a combination of the two. The influence of a core (at sizes consistent with the observed densities of the satellites) on relaxation time scales is unimportant. If the Ithaca Chasma graben system on Tethys is the result of stresses generated by relaxation of they Odysseus basin, then Tethys's core must be small enough that is allows unidirectional relaxation flow for a basin of Odysseus's size. If this assumption is correct, we estimate an upper limit of roughly 20% of Tethys's radius for the radius of the satellite's core. ¿ American Geophysical Union 1988