Astrophysics, abstract
astro-ph/0607522

From: Michael Efroimsky [view email]

Date: Sun, 23 Jul 2006 23:16:05 GMT (40kb)

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

Authors: Michael Efroimsky

Comments: Extended version of a paper to be published in ``Celestial Mechanics and Dynamical Astronomy."

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 and 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 and 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 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.

http://arxiv.org/abs/astro-ph/0607522

Astrophysics, abstract
astro-ph/0607530

From: Michael Efroimsky [view email]

Date: Sun, 23 Jul 2006 22:57:29 GMT (425kb)

Long-term evolution of orbits about a precessing oblate planet. 3. A semianalytical and a purely numerical approach

Authors: Valery Lainey, Pini Gurfil, Michael Efroimsky

Comments: Submitted to "Celestial Mechanics and Dynamical Astronomy."

Construction of a theory of orbits about a precessing oblate planet, in terms of osculating elements defined in a frame of the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2005, 2006). We now combine that analytical machinery with numerics. The resulting semianalytical theory is then applied to Deimos over long time scales. In parallel, we carry out a purely numerical integration in an inertial Cartesian frame. The results agree to within a small margin, for over 20 Myr, demonstrating the applicability of our semianalytical model over long timescales. This will enable us to employ it at the further steps of the project, enriching the model with the tides, the pull of the Sun, and the planet's triaxiality. Another goal of our work was to check if the equinoctial precession predicted for a rigid Mars could have been sufficient to repel the orbits away from the equator. We show that, both for high and low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform 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" confirms that an oblate planet can, indeed, afford a large equinoctial precession for dozens of millions of years, without repelling its near-equatorial satellites away from the equator of date: the inclination oscillates but does not show a secular increase. Nor does it show a secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos.

http://arxiv.org/abs/astro-ph/0607530

Astrophysics, abstract
astro-ph/0408168

From: Michael Efroimsky [view email]

Date (v1): Mon, 9 Aug 2004 22:28:03 GMT (48kb)

Date (revised v2): Wed, 25 Aug 2004 01:24:31 GMT (47kb)

Long-term evolution of orbits about a precessing oblate planet: 1. The case of uniform precession

Authors: Michael Efroimsky

Subj-class: Astrophysics; Classical Physics; Dynamical Systems; Exactly Solvable and Integrable Systems

Journal-ref: Celest.Mech.Dyn.Astron. 91 (2005) 75-108

It was believed until very recently that a near-equatorial satellite would always keep up with the planet's equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich (2004), this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet's obliquity change. Some of these terms will be nontrivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet's obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite's orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in the subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics.

http://arxiv.org/abs/astro-ph/0408168