The orbital dynamics of most planetary satellites is governed by the quadrupole moment from the equatorial bulge of the host planet and the tidal field from the Sun. On the Laplace surface, the long-term orbital evolution driven by the combined effects of these forces is zero, so that orbits have a fixed orientation and shape. The "classical" Laplace surface is defined for circular orbits, and coincides with the planet's equator at small planetocentric distances and with its orbital plane at large distances. A dissipative circumplanetary disk should settle to this surface, and hence satellites formed from such a disk are likely to orbit in or near the classical Laplace surface. This paper studies the properties of Laplace surfaces. Our principal results are: (1) if the planetary obliquity exceeds 68.• 875, there is a range of semimajor axes in which the classical Laplace surface is unstable; (2) at some obliquities and planetocentric distances, there is a distinct Laplace surface consisting of nested eccentric orbits, which bifurcates from the classical Laplace surface at the point where instability sets in; (3) there is also a "polar" Laplace surface perpendicular to the line of nodes of the planetary equator on the planetary orbit; (4) for circular orbits, the polar Laplace surface is stable at small planetocentric distances and unstable at large distances; (5) at the onset of instability, this polar Laplace surface bifurcates into two polar Laplace surfaces composed of nested eccentric orbits.
We continue the investigation of the dynamics of retrograde resonances
initiated in Morais & Giuppone (2012). After deriving a procedure to deduce the
retrograde resonance terms from the standard expansion of the three-dimensional
disturbing function, we concentrate on the planar problem and construct
surfaces of section that explore phase-space in the vicinity of the main
retrograde resonances (2/-1, 1/-1 and 1/-2). In the case of the 1/-1 resonance
for which the standard expansion is not adequate to describe the dynamics, we
develop a semi-analytic model based on numerical averaging of the unexpanded
disturbing function, and show that the predicted libration modes are in
agreement with the behavior seen in the surfaces of section.Comment: Celestial Mechanics and Dynamical Astronomy, in pres
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.