We study the effects of diffusion on the non‐linear corotation torque, or horseshoe drag, in the two‐dimensional limit, focusing on low‐mass planets for which the width of the horseshoe region is much smaller than the scaleheight of the disc. In the absence of diffusion, the non‐linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20 per cent of the numerical simulations.

We study the torque on low‐mass planets embedded in protoplanetary discs in the two‐dimensional approximation, incorporating non‐isothermal effects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non‐linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs type I migration in non‐isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diffusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and find good agreement.

We consider the angular momentum exchange at the corotation resonance between a two-dimensional gaseous disk and a uniformly rotating external potential, assuming that the disk flow is adiabatic. We first consider the linear case for an isolated resonance, for which we give an expression of the corotation torque that involves the pressure perturbation, and which reduces to the usual dependence on the vortensity gradient in the limit of a cold disk. Although this expression requires the solution of the hydrodynamic equations, it provides some insight into the dynamics of the corotation region. In the general case, we find an additional dependence on the entropy gradient at corotation. This dependence is associated to the advection of entropy perturbations. These are not associated to pressure perturbations. They remain confined to the corotation region, where they yield a singular contribution to the corotation torque. In a second part, we check our torque expression by means of customized two-dimensional hydrodynamical simulations. In a third part, we contemplate the case of a planet embedded in a Keplerian disk, assumed to be adiabatic. We find an excess of corotation torque that scales with the entropy gradient, and we check that the contribution of the entropy perturbation to the torque is in agreement with the expression obtained from the linear analysis. We finally discuss some implications of the corotation torque expression for the migration of low mass planets in the regions of protoplanetary disks where the flow is radiatively inefficient on the timescale of the horseshoe U-turns.Subject headings: accretion, accretion disks -hydrodynamics -methods: numerical -planetary systems: formation -planetary systems: protoplanetary disks 1 Send offprint request to clement.baruteau@cea.fr.