TEST PARTICLE MOTION AROUND AN OBLATE PLANET

Ring flow kinematic features, such as, e.g., the shape of the narrow r ings of Uranus or the streamlines of density waves of Saturn's rings, can be analyzed by fitting ring occultation data with m-lobe shapes wi th constant elements and constant precession rates. The orbital parame ters thus obtained (e.g., the semimajor axis, the eccentricity) are us ually referred to as geometric elements. This approach raises some que stions of principle: how are these geometric elements defined? Why is it justified to apply elliptic formulae (in data analyses) and Gauss p erturbation equations (in theoretical analyses) to these geometric ele ments? In our previous papers on these issues (N. Borderies and P. Y. Longaretti, 1987, Icarus 72, 593-603, P. Y. Longaretti and N. Borderie s, 1991, Icarus 94, 165-170), we have resolved these questions by show ing that the geometric elements are in fact appropriately chosen epicy clic elements, and that the epicyclic formulae and perturbation equati ons 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 nonequat orial motions. Second, the lowest order amplitude corrections to the f requencies of the motion are computed. These corrections allow us to d erive the correct contributions to order e2J2 and I2J2 to the orbital precession rates. Also, an expression of a third nonclassical quasi-in tegral of the motion in a flattened axisymmetric potential is provided from the work of G. Contopoulos (1960, Zeit. fur Astrophys. 49, 273-2 91). This third integral is a generalization of the energy in radial o scillations. Finally, the perturbation equations of the epicyclic elem ents are derived. These results are important for the determination of weak dynamical effects in planetary ring systems. (C) 1994 Academic P ress, Inc.