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 & Goldreich (2004) and Efroimsky (2004bEfroimsky ( , 2005Efroimsky ( , 2006. Here we continue this line of research by combining that analytical machinery with numerical tools. Our model includes three factors: the J 2 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-type planetary equations for the Keplerian elements, is then applied to Deimos on very long time scales (up to 1 billion of 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 fixed inertial Cartesian frame. The results agree to within 10 −3 − 10 −2 (from 0.3%, for near-polar orbits, through 1.3% for near-equatorial ones), for over 20 Myr, thus demonstrating the applicability of our semianalytical model over long * We use the term "precession" in its general meaning, which includes any change of the instantaneous spin axis. So generally defined precession embraces the entire spectrum of spin-axis variations -from the polar wander and nutations through the Chandler wobble through the equinoctial precession.
1timescales. This will enable us to employ the semianalytical model at the further stages of the project, enriching this model with the tides and the planet's triaxiality. (Lainey, Efroimsky & Gurfil 2007) Another goal of this work was to make an independent check of whether the equinoctial precession of Mars, as predicted for a rigid Mars, could have been sufficient to repel the orbits away from the equator. The answer to this question, in combination with our knowledge of the current position of Phobos and Deimos, will help us to understand whether this precession could indeed have always been as large as predicted or whether it ought to have been less in the past. It has turned out that, both for high and low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations (from about 10 −3 deg, for near-equatorial satellites, to about 2 deg, for polar ones). This is an extension, to non-uniform equinoctial precession given by the Colombo model and to an arbitrary initial inclination, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. Such "inclination locking", or the "Goldreich lock", confirms that an oblate planet can, indeed, afford a large equinoctial precession for billions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discu...