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

We performed linear calculations to determine the Type I planetary migration rate for 3D 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 nonremoval 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.