We compare the moment of inertia (MOI) of a simple hydrostatic, two layerbody as determined by the Radau-Darwin Approximation (RDA) to its exacthydrostatic MOI calculated to first order in the parameter q = w^2R^3/GM, wherew, R, and M are the spin angular velocity, radius, and mass of the body, and Gis the gravitational constant. RDA is in error by less than 1% for manyconfigurations of core sizes and layer densities congruent with those of solidbodies in the Solar System. We determine the error in the MOI of icy satellitescalculated with the RDA due to nonhydrostatic effects by using a simple modelin which the core and outer shell have slight degree 2 distortions away fromtheir expected hydrostatic shapes. Since the hydrostatic shape has anassociated stress of order pw^2R^2 (where p is density) it follows that theimportance of nonhydrostatic effects scales with the dimensionless numbers/pw^2R^2, where s is the nonhydrostatic stress. This highlights the likelyimportance 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 toTitan, 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 barsat the surface or ~1 bar at the core-mantle boundary, while for Ganymede thestresses necessary to produce the same MOI errors are an order of magnitudegreater due to its faster rotation. If satellites can reorient to the lowestenergy state then RDA will always give an overestimate of the true MOI.Observations have shown that small nonhydrostatic gravity anomalies exist onGanymede and Titan. We conclude that nonhydrostatic effects could be present toan extent that allows Callisto and Titan to be fully differentiated.
展开▼
