ec 2 00 6 Long - term evolution of orbits about a precessing ∗ oblate planet . 2 . The case of variable precession .

We continue the study undertaken in Efroimsky (2005a) where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky & Goldreich (2004), this opinion had stemmed from an inexact interpretation of a correct result by Goldreich (1965). Though Goldreich (1965) mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition has been persistently ignored for forty years. It was explained in Efroimsky & Goldreich (2004) that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky (2005a) that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich (1965), remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in | . ~ μ | (where ~ μ is the planetary equator’s total precession rate, rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander). We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator. 1 The scope of the project 1.1 The motivation Calculation of the obliquity of a planet (Ward 1973; Laskar & Robutel 1993; Touma & Wisdom 1994) are always obtained within a simplified model based on representation of the ∗ By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet – from its long-term variations down to nutations down to the Chandler wobble and polar wander. 1 planet by a symmetrical rigid rotator, with no internal structure or dissipative phenomena taken into consideration. This model yields, via the Colombo (1966) equation, the history of the planet axis’ inclination in an inertial frame. Thence the evolution of the obliquity can be found. The Colombo (1966) equation was derived for a rigid planet in the principal rotation state. These assumptions raise questions when it comes to real physics. First, a planet is deformable and, thereby, is subject to solar tides. It also tends to yield its shape to the instantaneous axis of rotation. (This phenomenon is always acknowledged in regard to the Chandler wobble, but never in regard to the equinoctial precession.) Second, a forced rotator is never in a principal spin state, and its angular-velocity vector is always slightly off its angular-momentum vector. These three phenomena influence the equinoctial precession and, through it, the obliquity variations. On the one hand, these phenomena are feeble; on the other hand, we are interested in their accumulation over the longest time scales, and therefore we are unsure of the outcome. Last, and by no means least, the Colombo description of the equinoctial precession ignores the possibility of planetary catastrophes that might have altered the planet’s spin mode. It would be good to develop a model-independent check of whether the planet could have maintained, through its entire past, the same equinoctial precession as it has today. Such a check is offered by Mars’ two satellites. The present proximity of both moons to the Martian equatorial plane is hardly a mere coincidence. Hence, the question becomes: could Mars have maintained equinoctial precession, predicted by the Colombo model, through its entire history without pushing an initially near-equatorial satellite too far away from the equatorial plane? 1.2 The objective If it turns out that the equinoctial precession, predicted by the Colombo (1966) model, does not drive the satellites away from the equator, or drives them away at a very slow rate, then this will become an independent confirmation of this model’s applicability. If, however, it turns out that the predicted precession of the spin axis leads to considerable variations in the satellite inclination relative to the equator of date, this will mean that the Colombo model should be further improved or/and that a planetary catastrophe may have altered Mars’ spin state. According to Goldreich (1965) and Kinoshita (1993) the inclination of a near-equatorial satellite only oscillates about its initial value, provided the equinoctial precession is uniform. However, even within the simple Colombo model, the equinoctial precession is variable. Besides, in these works non-osculating elements were used, circumstance noticed by Goldreich (1965) but missed by many authors who employed and furthered his result. Whenever the disturbance depends upon velocities (like a transition from inertial axes to ones co-precessing with the planet), a mere amendment of the disturbing function makes the planetary equations render not the osculating but the so-called contact orbital elements whose physical interpretation is nontrivial (Efroimsky & Goldreich 2004). To furnish osculating elements, the equations should be enriched with extra terms, some of which will not be additions to the disturbing function. Some of them will be of the first 1 These terms will complicate both the Lagrangeand Delaunay-type equations. The Delaunay equations will no longer be Hamiltonian. This parallels a predicament with the Andoyer elements used in the theory of rigid-body rotation with angular-velocity-dependent perturbations (Efroimsky 2007; Gurfil, Elipe, Tangren, & Efroimsky 2007)