Dynamics of the Sharp Edges of Broad Planetary Rings

Abstract

(Abridged) The following describes a model of a broad planetary ring whose sharp edge is confined by a satellite's m^th Lindblad resonance (LR). This model uses a streamline formalism to calculate the ring's internal forces, namely, ring gravity, pressure, viscosity, as well as a hypothetical drag force. The model calculates the streamlines' forced orbit elements and surface density throughout the perturbed ring. The model is then applied to the outer edge of Saturn's B ring, which is maintained by an m=2 inner LR with the satellite Mimas. Ring models are used to illustrate how a ring's perturbed state depends on the ring's physical properties: surface density, viscosity, dispersion velocity, and the hypothetical drag force. A comparison of models to the observed outer B ring suggests that the ring's surface density there is between 10 and 280 gm/cm^2. The ring's edge also indicates where the viscous torque counterbalances the perturbing satellite's gravitational torque on the ring. But an examination of seemingly conventional viscous B ring models shows that they all fail to balance these torques at the ring's edge. This is due ring self-gravity and the fact that a viscous ring tends to be nearly peri-aligned with the satellite, which reduces the satellite's torque on the ring and makes the ring's edge more difficult to maintain. Nonetheless, the following shows that a torque balance can still be achieved in a viscous B ring, but only in an extreme case where the ratio of the ring's bulk/shear viscosities satisfy ~10^4. However, if the dissipation of the ring's forced motions is instead dominated by a weak drag force, then the satellite can exert a much stronger torque that can counterbalance the ring's viscous torque.