Precession of a planet with a satellite

The contribution of a satellite in the precession motion of the axis of an oblate planet has been previously studied in the approximation of a distant satellite, or in the approximation of a very close satellite. Here we study the general problem for an arbitrary semimajor axis for the satellite, without performing the usual gyroscopic approximation. We present precessional equations valid in a very general setting, and we demonstrate that this problem, after the classical expansion of the satellite potential, and averaging over the fast angles, is indeed integrable. We provide here the complete solution of this problem by quadrature, as well as some explicit approximate solutions. We also demonstrate that after averaging over the nutation motion, the pole of the spin axis, the pole of the satellite orbit, and the pole of the planet orbit remain coplanar with the total angular momentum and precess uniformly around the total angular momentum.