Linear Corotation Torques in Non-Barotropic Disks

I derive a fully analytic expression for the linear corotation torque to first order in eccentricity for planets in non-barotropic protoplanetary disks, taking into account the effect of disk entropy gradients. This torque formula is applicable to both the co-orbital corotation torque 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 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.