  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.
