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Migration of a moonlet in a ring of solid particles : Theory and application to Saturn's propellers

Hundred meter sized objects have been identified by the Cassini spacecraft in Saturn's A ring through the so-called "propeller" features they create in the ring. These moonlets should migrate, due to their gravitational interaction with the ring ; in fact, some orbital variation have been detected. The standard theory of type I migration of planets in protoplanetary disks can't be applied to the ring system, as it is pressureless. Thus, we compute the differential torque felt by a moonlet embedded in a two-dimensional disk of solid particles, with flat surface density profile, both analytically and numerically. We find that the corresponding migration rate is too small to explain the observed variations of the propeller's orbit in Saturn's A-ring. However, local density fluctuations (due to gravity wakes in the marginally gravitationally stable A-ring) may exert a stochastic torque on a moonlet. Our simulations show that this torque can be large enough to account for the observations, depending on the parameters of the rings. We find that on time scales of several years the migration of propellers is likely to be dominated by stochastic effects (while the former, non-stochastic migration dominates after ~ 10^{4-5} years). In that case, the migration rates provided by observations so far suggests that the surface density of the A ring should be of the order of 700 kg/m^2. The age of the propellers shouldn't exceed 1 to 100 million years, depending on the dominant migration regime.