Approximate solution of third-order Clairaut theory of rotational sectorial figures of Jupiter and Saturn

Abstract

The aim of this paper is to investigate numerical solutions of third-order Clairaut theory, under the boundary conditions given in our previous work (El-Shaarawy, 1974). This solution gives an explicit form of the shape and rotational distortion, due to third-order sectorial harmonic terms, of the equipotential surfaces of the two rapidly rotating planets, Jupiter and Saturn at the different levels inside these planets owing to a certain internal density distribution model (Zharkov, 1975). We considered each of them as a heterogeneous self-gravitating fluid mass in hydrostatic equilibrium.