Nonlinear satellite wakes in planetary rings: I. Phase-space kinematics

Abstract

An explicit expression is derived for the phase-space density of a planetary ring perturbed by a nearly satellite. The derivation is facilitated by working in guiding center variables instead of local position and velocity variables and by neglecting collisions between ring particles. The usual equations for perturbed streamlines are recovered by taking first-order moments of the phase-space density. The local surface density and the local mean velocity in a nonlinear satellite wake are obtained in the form of convergent infinite series. Unlike previous estimates based on streamline crowding, this surface density is positive definite because the finite velocity dispersion of ring particles is taken into account. In other words, the phase-space density describes streamlines of finite width. The finite width of streamlines limits the maximum value of the local surface density that can result from streamline crowding. This result is consistent with numerical phase-space fluid simulations of perturbed rings reported by Brophy, Esposito, and Stewart (1991). The local mean velocity components in the satellite wake are found to deviate from the sinusoidal form of the streamline equations, and this deviation grows as the wake moves downstream from the shepherding satellite. These results suggest that collisional stresses between neighboring streamlines should depend on the second-order derivatives of the streamline parameters because the local mean velocity is itself a strong function of the first-order derivatives of the streamline parameters.