The effect of collisions on the resonant satellite torque distributionfn2

We present numerical results of the behavior of the torque exerted by a satellite over a particulate annulus centered at a first order resonance. Results for both collisionless and collisonal annuli are presented and compared. Our results show that in the absence of particle interactions, the time required for this problem to become nonlinear scales with the satellite to planet mass ratio as (Ms/Mp)−2/3. In the absence of collisions, the total initial (transient) torque over the annulus is a strong function of the width of the annulus, W′, around the resonance (located at rL), so that if W′ ⩽ rL(Ms/Mp)23, then the torque value always remains significantly below the linear standard torque. On the other hand, if W′ > rL(Ms/Mp)23 the annulus reaches ⩾95% of the linear standard torque before the problem becomes nonlinear. These simulations show that when collisions are present, the critical width to obtain 95% of the linear standard torque is substantially increased, even for very low optical depth annuli. The torque value for a fixed width also depends on the time; it asymptotically approaches the linear standard torque as long as nonlinearities remain small. When collisions are introduced, initially the torque value grows up to the value it would have in the absence of collisions at time t = tcoll (for the given width), where tcoll is the mean time between collisions; for t > tcoll the torque becomes independent of the time (although not independent of the width) for the duration of the simulations.