Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

Abstract

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. 11 Physical motivation and the statement of purpose Ward (1973Ward ( , 1974 noted that the obliquity of Mars may have suffered large-angle motions at long time scales. Later, Laskar and Robutel (1993) and Touma and Wisdom (1994) demonstrated that these motions may have been chaotic. This would cause severe climate variations and have major consequences for development of life.It is a customary assumption that a near-equatorial satellite of an oblate planet would always keep up with the planet's equator (with only small oscillations of the orbit inclination) provided the obliquity changes are sufficiently slow (Goldreich 1965, Kinoshita 1993. As demonstrated in Efroimsky and Goldreich (2004), this belief stems from a calculation performed in the language of non-osculating orbital elements. A similar analysis carried out in terms of osculating elements will contain hitherto overlooked extra terms entailed by the planet's obliquity variations. These terms (emerging already in the first order over the precession-caused perturbation) will cause a secular angular drift of the satellite orbit away from the planetary equator.The existence of Phobos and Deimos, and the ability of Mars to keep them close to its equatorial plane during obliquity variations sets constraints on the obliquity variation amplitude and rate. Our eventual goal is to establish such constraints. If the satellites' secular inclination drifts are slow enough that the sate...

A resonant-term-based model including a nascent disk, precession, and oblateness: application to GJ 876

“…This result, proven by Efroimsky (2005b), is valid only in the case of uniform precession and only for a solitary satellite, in the absence of any other disturbances (like the gravitational pull of the Sun or of another satellite). However, numerical simulation has shown that even in realistic situations of variable precession, the deviations between the secular parts of the contact and the corresponding osculating elements accumulate very slowly, for a solitary satellite (Gurfil et al, 2006).…”

“…Kinoshita (1993) studied the motion of a satellite relative to the equatorial plane of its oblate parent parent, and Rubincam (2000) discusses the possibility that Pluto may be in "precession-orbit" resonance. Expressions for the precession contribution to the disturbing function are given implicitly by Goldreich (1965b) and explicitly by Kopal (1969), Brumberg et al (1970) and Efroimsky (2005b); using their results, we can write:…”

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

“…Indeed, according to (9 -10), the Cartesian components of the velocity in the precessing equatorial frame will be given by g ( C , t ) + ∂∆H precess /∂ p . However, since the second term of this sum is equal to − µ × r , then g ( C , t ) turns out to always render the velocity with respect to the inertial frame of reference (Efroimsky & Goldreich 2004, Efroimsky 2005). …”

“…Instead, we must use (9) with the substitution (12). This generic rule applied both to orbital and rotational motions.…”

“…Otherwise, they differ already in the first order of the time-dependent perturbation (Efroimsky & Goldreich 2003. Luckily, in some situations, their secular parts differ only in the second order (Efroimsky 2005), a fortunate circumstance anticipated by Goldreich (1965), who came across these elements in a totally different context unrelated to canonicity. The case of rotational motion will parallel the theory of orbits.…”