Nonhydrostatic Effects and the Determination of Icy Satellites' Moment of Inertia

Abstract

We compare the moment of inertia (MOI) of a simple hydrostatic, two layer body as determined by the Radau-Darwin Approximation (RDA) to its exact hydrostatic MOI calculated to first order in the parameter q = w^2R^3/GM, where w, R, and M are the spin angular velocity, radius, and mass of the body, and G is the gravitational constant. RDA is in error by less than 1% for many configurations of core sizes and layer densities congruent with those of solid bodies in the Solar System. We determine the error in the MOI of icy satellites calculated with the RDA due to nonhydrostatic effects by using a simple model in which the core and outer shell have slight degree 2 distortions away from their expected hydrostatic shapes. Since the hydrostatic shape has an associated stress of order pw^2R^2 (where p is density) it follows that the importance of nonhydrostatic effects scales with the dimensionless number s/pw^2R^2, where s is the nonhydrostatic stress. This highlights the likely importance of this error for slowly rotating bodies (e.g., Titan and Callisto) and small bodies (e.g., Saturn moons other than Titan). We apply this model to Titan, Callisto, and Enceladus and find that the RDA-derived MOI can be 10% greater than the actual MOI for nonhydrostatic stresses as small as ~0.1 bars at the surface or ~1 bar at the core-mantle boundary, while for Ganymede the stresses necessary to produce the same MOI errors are an order of magnitude greater due to its faster rotation. If satellites can reorient to the lowest energy state then RDA will always give an overestimate of the true MOI. Observations have shown that small nonhydrostatic gravity anomalies exist on Ganymede and Titan. We conclude that nonhydrostatic effects could be present to an extent that allows Callisto and Titan to be fully differentiated.