Torques on Low-mass Bodies in Retrograde Orbit in Gaseous Disks

Abstract

We evaluate the torque acting on a gravitational perturber on a retrograde circular orbit in the midplane of a gaseous disk. We assume that the mass of this satellite is so low it weakly disturbs the disk (type I migration). The perturber may represent the companion of a binary system with a small mass ratio. We compare the results of hydrodynamical simulations with analytic predictions. Our two-dimensional (2D) simulations indicate that the torque acting on a perturber with softening radius $R_{\rm soft}$ can be accounted for by a scattering approach if $R_{\rm soft}<0.3H$, where $H$ is defined as the ratio between the sound speed and the angular velocity at the orbital radius of the perturber. For $R_{\rm soft}>0.3H$, the torque may present large and persistent oscillations, but the resultant time-averaged torque decreases rapidly with increasing $R_{\rm soft}/H$, in agreement with previous analytical studies. We then focus on the torque acting on small-size perturbers embedded in full three-dimensional (3D) disks and argue that the density waves propagating at distances $\lesssim H$ from the perturber contribute significantly to the torque because they transport angular momentum. We find a good agreement between the torque found in 3D simulations and analytical estimates based on ballistic orbits. We compare the radial migration timescales of prograde versus retrograde perturbers. For a certain range of the perturber's mass and aspect ratio of the disk, the radial migration timescale in the retrograde case may be appreciably shorter than in the prograde case. We also provide the smoothing length required in 2D simulations in order to account for 3D effects.