Three-dimensional Disk-Planet Torques in a Locally Isothermal Disk

TL;DR

This paper derives an improved formula for Type I planet migration torque in locally isothermal disks, based on 3D hydrodynamical simulations, revealing the limited impact of opacity or shadowing on migration slowdown.

Contribution

It provides a new, accurate torque density formula for 3D disks that improves upon 2D approximations and accounts for radial density and temperature gradients.

Findings

Torque is sensitive to density and temperature gradients.

2D calculations with smoothed potential are inadequate for gradient effects.

Slowing or stopping migration via opacity changes is unlikely in locally isothermal disks.

Abstract

We determine an expression for the Type I planet migration torque involving a locally isothermal disk, with moderate turbulent viscosity (~0.0005 < alpha < ~0.05), based on three-dimensional nonlinear hydrodynamical simulations. The radial gradients (in a dimensionless logarithmic form) of density and temperature are assumed to be constant near the planet. We find that the torque is roughly equally sensitive to the surface density and temperature radial gradients. Both gradients contribute to inward migration when they are negative. Our results indicate that two-dimensional calculations with a smoothed planet potential, used to account for the effects of the third dimension, do not accurately determine the effects of density and temperature gradients on the three-dimensional torque. The results suggest that substantially slowing or stopping planet migration by means of changes in disk opacity or shadowing is difficult and appears unlikely for a disk that is locally isothermal. The scalings of the torque and torque density with planet mass and gas sound speed follow the expectations of linear theory. We also determine an improved formula for the torque density distribution that can be used in one-dimensional long-term evolution studies of planets embedded in locally isothermal disks. This formula can be also applied in the presence of mildly varying radial gradients and of planets that open gaps. We illustrate its use in the case of migrating super-Earths and determine some conditions sufficient for survival.