  "Propellers" are features in Saturn's A ring associated with moonlets that
open partial gaps. They exhibit non-Keplerian motion (Tiscareno 2010); the
longitude residuals of the best-observed propeller, "Bl\'eriot," appear
consistent with a sinusoid of period ~4 years. Pan and Chiang (2010) proposed
that propeller moonlets librate in "frog resonances" with co-orbiting ring
material. By analogy with the restricted three-body problem, they treated the
co-orbital material as stationary in the rotating frame and neglected
non-co-orbital material. Here we use simple numerical experiments to extend the
frog model, including feedback due to the gap's motion, and drag associated
with the Lindblad disk torques that cause Type I migration. Because the moonlet
creates the gap, we expect the gap centroid to track the moonlet, but only
after a time delay t_diff, the time for a ring particle to travel from
conjunction with the moonlet to the end of the gap. We find that frog
librations can persist only if t_diff exceeds the frog libration period P_lib,
and if damping from Lindblad torques balances driving from co-orbital torques.
If t_diff << P_lib, then the libration amplitude damps to zero. In the case of
Bl\'eriot, the frog resonance model can reproduce the observed libration period
P_lib ~ 4 yr. However, our simple feedback prescription suggests that
Bl\'eriot's t_diff ~ 0.01P_lib, which is inconsistent with the observed
libration amplitude of 260 km. We urge more accurate treatments of feedback to
test the assumptions of our toy models.
