Abstract　  

   In an important paper, P.H. Roberts (1963, ApJ, 138, 809) 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, for the first time, the three-dimensional, finite-element, fully self-consistent, continuous solution for a rapidly rotating polytropic gaseous body without making any prior assumptions about either its outer shape or internal structure. Upon partially relaxing the Roberts' approximation by assuming that only the outer surface is in the shape of an oblate spheroid, we also derive an exact analytic solution based on the Green’s function using spheroidal wave functions 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.