Long-Term Evolution of Orbits About a Precessing Oblate Planet: 3. A Semianalytical and a Purely Numerical Approach

Construction of an accurate theory of orbits about a precessing and nutating oblate planet, in terms of osculating elements defined in a frame associated with the equator of date, was started in Efroimsky and Goldreich 2004 and Efroimsky 2004, 2005, 2006a, b. Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors the J2 of the planet, its nonuniform equinoctial precession described by the Colombo formalism, and the gravitational pull of the Sun. This semianalytical and seminumerical theory, based on the Lagrange planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales up to 1 billion years. In parallel with the said semianalytical theory for the Keplerian elements defined in the co-precessing equatorial frame, we have also carried out a completely independent, purely numerical, integration in a quasi-inertial Cartesian frame. The results agree to within fractions of a percent, thus demonstrating the applicability of our semianalytical model over long timescales.