Streamline formalism and ring orbit determination

Abstract

The orbital parameters of the Uranian rings are commonly obtained by fitting to ring occultation data precessing inclined ellipses with constant elliptic elements and constant precession rates; these elements are referred to as “geometric” elements. On the theoretical side, many results were derived by considering the streamlines of the flow of particles and studying their dynamical evolution with the help of Gauss perturbation equations, a method initially developed by Borderies, Goldreich, and Tremaine; this approach involves some kind of “mean” elements. The purpose of this paper is to give a theoretical basis to these approaches. It is well known indeed that the mean and geometric elements involved in both approaches can be significantly different from the osculating elliptic elements. For example, although the relative difference between the mean and osculating semimajor axes is of order J 2, the mean and osculating eccentricities can be orders of magnitude different; the same remark applies to the geometric elements. However, the application of the elliptic formulæ and of Gauss variational equations to the mean and geometric elements has never been given a clear theoretical basis. Such a justification is developed here in three steps: first, it is shown that ring shapes and evolution are most appropriately described in terms of epicyclic elements; second, it is shown that the geometric elements obtained from the ring occultation data and the mean elements involved in the streamline formalism are not elliptic elements, but in fact nearly identical to a suitably defined set of epicyclic elements; finally, it is shown that to the lowest order in eccentricity, the elliptic and epicyclic elements are governed by nearly the same equations of evolution. It is finally suggested that epicyclic elements be used instead of elliptic elements in discussions of ring problems: this would improve data fits and remove all theoretical inconsistencies, at almost no cost because of the similarity between the two kinds of elements.