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Planets migrate due to the recoil they experience from scattering solid(planetesimal) bodies. To first order, the torques exerted by the interior andexterior disks cancel, analogous to the cancellation of the torques from thegravitational interaction with the gas (type I migration). Assuming thedispersion-dominated regime and power-laws characterized by indices {\alpha}and {\beta} for the surface density and eccentricity profiles, we calculate thenet torque on the planet. We consider both distant encounters and close(orbit-crossing) encounters. We find that the close and distant encountertorques have opposite signs with respect to their {\alpha} and {\beta}dependences; and that the torque is especially sensitive to the eccentricitygradient ({\beta}). Compared to type-I migration due to excitation of densitywaves, the planetesimal-driven migration rate is generally lower due to thelower surface density of solids in gas-rich disk, although this may bepartially or fully offset when their eccentricity and inclination are small.Allowing for the feedback of the planet on the planetesimal disk throughviscous stirring, we find that under certain conditions a self-regulatedmigration scenario emerges, in which the planet migrates at a steady pace thatapproaches the rate corresponding to the one-sided torque. If the localplanetesimal disk mass to planet mass ratio is low, however, migration stalls.We quantify the boundaries separating the three migration regimes.

Migration rates of planets due to scattering of planetesimals