Phoebe’s differentiated interior from refined shape analysis

Context. Phoebe is an irregular satellite of Saturn, and its origin, from either between the orbits of the giant planets or the Kuiper Belt, is still uncertain. The extent of differentiation of its interior can potentially help inform its formation location because it is mainly determined by heat from 26-aluminum. The internal structure is reflected in the shape, assuming the body is relaxed to hydrostatic equilibrium. Although previous data analysis indicates Phoebe is close to hydrostatic equilibrium, its heavily cratered surface makes it difficult to tease out its low-order shape characteristics. Aims. This paper aims to extract Phoebe’s global shape from the observations returned by the Cassini mission for comparison with uniform and stratified interior models under the assumption of hydrostatic equilibrium. Methods. The global shape is derived from fitting spherical harmonics and keeping only the low-degree harmonics that represent the shape underneath the heavily cratered surface. The hydrostatic theoretical model for shape interpretation is based on the Clairaut equation developed to the third order (although the second order is sufficient in this case). Results. We show that Phoebe is differentiated with a mantle density between 1900 and 2400 kg m−3. The presence of a porous surface layer further restricts the fit with the observed shape. This result confirms the earlier suggestion that Phoebe accreted with sufficient 26-aluminium to drive at least partial differentiation, favoring an origin with C-type asteroids.

A New Analytical Procedure for Solving the Non-Linear Differential Equation Arising in the Stretching Sheet Problem

The paper discusses a new analytical procedure for solving the non-linear boundary layer equation arising in a linear stretching sheet problem involving a Newtonian/non-Newtonian liquid. On using a technique akin to perturbation the problem gives rise to a system of non-linear governing differential equations that are solved exactly. An analytical expression is obtained for the stream function and velocity as a function of the stretching parameters. The Clairaut equation is obtained on consideration of consistency and its solution is shown to be that of the stretching sheet boundary layer equation. The present study throws light on the analytical solution of a class of boundary layer equations arising in the stretching sheet problem