The Formation of Sharp Edges in Planetary Rings by Nearby Satellites

Sharp edges, boundaries between regions of high and low optical depth in planetary rings, are maintained by shepherd satellites which transfer angular momentum to and from the ring particles. We derive equations that govern the shapes of the perturbed streamlines near such a boundary. These equations are solved for a simple numerical model whose parameters are chosen to resemble those of the Encke division and its associated satellite. The results of our calculation faithfully reproduce the sharp edges which bound the division and imply that the ring thickness in the unperturbed regions far from the edges is of order 10 m. In particular, the angle-averaged surface density is found to vary on a much shorter radial length scale than that over which the satellite torque is applied. We demonstrate that this striking feature is related to the local reversal of the viscous transport of angular momentum in the most strongly perturbed regions.