Test Particle Motion around an Oblate Planet

Abstract

Ring flow kinematic features, such as, e.g., the shape of the narrow rings of Uranus or the streamlines of density waves of Saturn's rings, can be analyzed by fitting ring occultation data with m-lobe shapes with constant elements and constant precession rates. The orbital parameters thus obtained (e.g., the semimajor axis, the eccentricity) are usually referred to as geometric elements. This approach raises some questions of principle: how are these geometric elements defined? Why is it justified to apply elliptic formulae (in data analyses) and Gauss perturbation equations (in theoretical analyses) to these geometric elements? In our previous papers on these issues (N. Borderies and P. Y. Longaretti, 1987, Icarus 72, 593-603, P. Y. Longaretti and N. Borderies, 1991, Icarus 94, 165-170), we have resolved these questions by showing that the geometric elements are in fact appropriately chosen epicyclic elements, and that the epicyclic formulae and perturbation equations are formally nearly identical to the more familiar elliptic theory.This paper addresses various issues that were not considered in the previous papers. First, the epicyclic solution is extended to nonequatorial motions. Second, the lowest order amplitude corrections to the frequencies of the motion are computed. These corrections allow us to derive the correct contributions to order e2J2 and I2J2 to the orbital precession rates. Also, an expression of a third nonclassical quasi-integral of the motion in a flattened axisymmetric potential is provided from the work of G. Contopoulos (1960, Zeit. für Astrophys. 49, 273-291). This third integral is a generalization of the energy in radial oscillations. Finally, the perturbation equations of the epicyclic elements are derived. These results are important for the determination of weak dynamical effects in planetary ring systems.