Linear corotation torques in non-barotropic disks

A fully analytic expression for the linear corotation torque to first order in eccentricity for planets in non-barotropic protoplanetary disks is derived, taking into account the effect of disk entropy gradients. This torque formula is applicable to both the co-orbital, corotation torques and the non-co-orbital, corotation torques—for planets in orbits with non-zero eccentricity—in disks where the thermal diffusivity and viscosity are sufficient to maintain the linearity of these interactions. While the co-orbital, corotation torque is important for migration of planets in Type I migration, the non-co-orbital, corotation torque plays an important role in the eccentricity evolution of giant planets that have opened gaps in the disk. The presence of an entropy gradient in the disk can significantly modify the corotation torque in both these cases.