Dynamics of the Sharp Edges of Broad Planetary Rings

This paper describes a model of a broad planetary ring whose sharp edge is confined by a satellite’s mth Lindblad resonance (LR). This model uses the streamline formalism of Borderies et al. to calculate the ring’s internal forces, namely, ring gravity, pressure, and viscosity. The model also allows for the possibility of a drag force that can affect small ring particles directly, and large ring particles indirectly via collisions with the small. The model calculates the streamlines’ forced eccentricities e, their longitudes of periapse ˜ ω, and the surface density σ throughout the perturbed ring. This 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. A suite of ring models are used to illustrate how a ring’s perturbed state depends on the ring’s physical properties: its surface density, its viscosity, the ring particles’ dispersion velocity, and the strength of the hypothetical drag force. A comparison of simulations with the outer B ring’s observed properties suggests that the ring’s surface density there is 10 σ 280 gm cm −2 in the ring’s outermost ∼40 km. The ring’s sharp edge identifies the site where the ring’s viscous torque precisely counterbalances the perturbing satellite’s gravitational torque on the ring. However, an examination of several seemingly conventional viscous B ring models shows that they all fail, by wide margins, to balance these torques at the ring’s outer edge. This is partly due to the ring’s self-gravity, which tends to reduce forced eccentricities near the resonance. But this is also due to the fact that a viscous ring tends to be nearly peri-aligned with the satellite. Both effects conspire to reduce the satellite’s torque on the ring, which in turn 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 νb/νs ∼ 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 across a wider annulus in the ring, which can successfully counterbalance the ring’s viscous torque there. We also show how this streamline model can be adapted to study other interesting ring phenomena, such as narrow eccentric ringlets and nonlinear spiral density waves.