Self-consistent internal structure of a rotating gaseous planet and its comparison with an approximation by oblate spheroidal equidensity surfaces

• We compute the first three-dimensional, finite-element, fully self-consistent, continuous solution for a rapidly rotating polytropic gaseous body with Jupiter-like parameters without making any prior assumptions about its outer shape and internal structure.
• We show that all equidensity surfaces of the fully self-consistent solution differ only slightly from the oblate spheroidal shape, suggesting that the assumption of spheroidal equidensity surfaces represents a reasonably accurate approximation for rotating polytropic gaseous bodies with Jupiter-like parameters.
• We present the three different solutions in non-spherical geometries – the fully selfconsistent numerical solution, the numerical solution with the outer spheroidal shape and the exact analytical solution – that can also serve as a useful benchmark for other solutions based on different numerical methods.

In an important paper, Roberts (1963b) studied the hydrostatic equilibrium of an isolated, self-gravitating, rapidly rotating polytropic gaseous body based on a controversial assumption/approximation that all (outer and internal) equidensity surfaces are in the shape of oblate spheroids whose eccentricities are a function of the equatorial radius and whose axes of symmetry are parallel to the rotation axis. We compute the three-dimensional, finite-element, fully self-consistent, continuous solution for a rapidly rotating polytropic gaseous body with Jupiter-like parameters without making any prior assumptions about its outer shape and internal structure. Upon partially relaxing the Roberts’ approximation by assuming that only the outer equidensity surface is in the shape of an oblate spheroid, we also compute a finite-element solution with the same parameters without making any prior assumptions about its internal structure. It is found that all equidensity surfaces of the fully self-consistent solution differ only slightly from the oblate spheroidal shape. It is also found that the characteristic difference between the fully self-consistent solution and the outer-spheroidal-shape solution is insignificantly small. Our results suggest that the Roberts’ assumption of spheroidal equidensity surfaces represents a reasonably accurate approximation for rotating polytropic gaseous bodies with Jupiter-like parameters. The numerical accuracy of our finite-element solution is checked by an exact analytic solution based on the Green’s function using the spheroidal wave function. The three different solutions in non-spherical geometries – the fully self-consistent numerical solution, the numerical solution with the outer spheroidal shape and the exact analytical solution – can also serve as a useful benchmark for other solutions based on different numerical methods.