Halting migration: Numerical calculations of corotation torques in the weakly nonlinear regime

Planets in their formative years can migrate due to the influence of gravitational torques in the protoplanetary disk they inhabit. For low-mass planets in an isothermal disk, it is known that there is a strong negative torque on the planet due to its linear perturbation to the disk, causing fast inward migration. The current investigation demonstrates that in these same isothermal disks, for intermediate-mass planets, there is a strong positive nonlinear corotation torque due to the effects of gas being pulled through a gap on horseshoe orbits. For intermediate-mass planets, this positive torque can partially or completely cancel the linear (Type I) torque, leading to slower or outward migration, even in an isothermal disk. The effect is most significant for super-Earth and sub-Jovian planets, during the transition from a low-mass linear perturber to a nonlinear gap-opening planet, when the planet has opened a so-called "partial gap," though the precise values of these transition masses depend sensitively on the disk model (density profile, viscosity, and disk aspect ratio). In this study, numerical calculations of planet–disk interactions calculate these torques explicitly, and scalings are empirically constructed for migration rates in this weakly nonlinear regime. These results find outward migration is possible for planets with masses in the range 20–100 $M_{\oplus }$, though this range depends on the disk model considered. In the disk models where torque reversal occurs, the critical planet-to-star mass ratio for torque reversal was found to have the robust scaling $q_{crit}\propto \sqrt{\alpha }{\left(h/r\right)}^3$, where α is the dimensionless viscosity parameter and $h/r$ is the disk aspect ratio.