A torque formula for non-isothermal Type I planetary migration - I. Unsaturated horseshoe drag

TL;DR

This paper derives a simple, validated formula for Type I planetary migration torque in non-isothermal protoplanetary discs, accounting for horseshoe drag and thermal effects, applicable in optically thick regions.

Contribution

It introduces a new, simple formula combining linear Lindblad torque with a non-linear adiabatic horseshoe drag model for non-isothermal discs, validated by simulations.

Findings

The formula accurately predicts migration torques in simulations.

Good agreement across three independent numerical methods.

Applicable in optically thick, viscous, and thermally diffusive regions.

Abstract

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.