Three-dimensional Interaction between a Planet and an Isothermal Gaseous Disk. III. Locally Isothermal Cases

TL;DR

This paper presents linear calculations of Type I planetary migration in 3D locally isothermal disks with radial temperature gradients, deriving torque formulas that align with previous simulations despite singularities at corotation.

Contribution

It introduces a method to handle singularities in the wave equation and derives torque formulas for 3D disks with temperature gradients, matching simulation results.

Findings

Derived torque formulas for locally isothermal disks with temperature gradients.

Confirmed the agreement of linear torque calculations with previous 3D hydrodynamical simulations.

Provided insights into the behavior of corotation and Lindblad torques in 3D disks.

Abstract

We performed linear calculations to determine the Type I planetary migration rate for three-dimensional locally isothermal disks with radial temperature gradients. For 3D disks with radial temperature gradients, the linear wave equation has a divergent term of the third pole, which makes corotation a non-removal singularity. We suppressed the divergence with the Landau prescription to obtain the wave solutions. Despite the singularity at corotation, we derived a definite torque on the planet because the divergent term amplifies the waves only in the neighborhood of corotation and has little effect on the planetary torque. Consequently, we derived the formulas for the total, Lindblad, and corotation torques for locally isothermal disks. The resulting torque term due to the disk temperature gradient agrees well with the results of previous 3D hydrodynamical simulations for locally isothermal disks. Our linear calculation also provides the 3D horseshoe torque, which is close to the results of previous 3D hydrodynamical simulations.