Single-Sided Shepherding

Narrow ringlets are observed to form at isolated Lindblad resonances in recent simulations (J. Hänninen and H. Salo, Icarus 108, 325-346 (1994) and 117, 435-438 (1995)). Our remarks are directed toward the interpretation of this phenomenon. Ringlet formation is a consequence of the negative angular momentum luminosity promoted by satellite perturbations of the streamlines of particle flow. Contraction halts once the surface density reaches a value such that the angular momentum luminosity vanishes. Our estimate for the formation time, t_f ∼ (M_p/M_s)^1/2Ω,^(-1), for low order resonances in optically thin rings is consistent with the results of the simulations. Ringlets drift across the width of resonance, W ∼ (M_s/M_p)^(1/2)ɑ, as a result of the unbalanced satellite torque. This occurs on the much longer timescale t_d ∼ (M_p/M_s)Ω^(-1) and cannot be observed in current simulations.