On the co-orbital corotation torque in a viscous disk and its impact on planetary migration

We evaluate the co-orbital corotation torque on a planet on a fixed circular orbit embedded in a viscous protoplanetary disk for the case of a steady flow in the planet frame. This torque can be evaluated just from the flow properties at the separatrix between the librating (horseshoe) and circulating streamlines. A stationary solution is searched for the flow in the librating region. When used to evaluate the torque exerted by the circulating material of the outer and inner disk on the trapped material of the librating region, this solution leads to an expression of the co-orbital corotation torque in agreement with previous estimates. An analytical expression is given for the corotation torque as a function of viscosity. Last, we show that additional terms in the torque expression can play a crucial role. In particular, they introduce a coupling with the disk density profile perturbation (the "dip" that surrounds the planet) and add to the corotation torque a small, positive fraction of the one-sided Lindblad torque. As a consequence, the migration could well be directed outward in very thin disks (aspect ratio smaller than a few percent). This two-dimensional analysis is especially relevant for mildly embedded protoplanets (sub-Saturn-sized objects).