The role of dissipation in shepherding of ring particles

In the absence of any damping mechanism, a shepherd satellite would force oscillations in the motion of a ring particle (relative to circular motion) that are symmetrical with respect to the encounter geometry. No net torque would be exerted by the satellite on the rings. Only in the presence of some damping mechanism (such as density wave propagation or a dissipative medium) can a particle's response lag so as to provide the asymmetry that permits a torque. Remarkably, the standard formula for the confining torque exerted by a sheperd satellite seems to be independent of damping. Moreover, many heuristic derivations of the formula tend to obscure the role of damping. In fact, the torque on any given particle does depend on the degree of damping, but that dependence disappears when the torque is averaged over a range of orbits that span resonances if the degree of damping is within a certain range. If damping is too weak or too strong, the torque can be much less than is given by the standard formula.