Collisional Planetary Rings: The Role of the Width and the Optical Depth

Abstract

We show analytically that the torque exerted by a satellite over a (collisionless) planetary ring centered at a first order resonance, is a function only of the product Wt, where W is the width of the ring and t is the time since the satellite is "created". For t<0 particles are in unperturbed keplerian orbits around the planet. The analytical work is made to lowest possible order in the perturbed variables. We also show the results of numerical integrations of the full Newtonian equations of motion for collisionless and collisional rings and show that: i) For collisionless rings, the value of the mean torque is a function of tW for t<t_r, where t_r represents the time when non-linearities are no longer negligible. ii) For collisional rings, the mean torque is a function only of tW as long as t<t_coll<t_r, where t_coll is the mean time between consecutive collisions among the particles. For t>t_coll, the mean torque stays constant or starts diminishing depending on the value of the optical depth and the depth of the gap that opens at the resonance.