The dependence of protoplanet migration rates on co-orbital torques

We investigate the migration rates of high-mass protoplanets embedded in accretion discs via two- and three-dimensional hydrodynamical simulations. The simulations follow the planet's radial motion and employ a nested-grid code that allows for high resolution close to the planet. We concentrate on the possible role of the co-orbital torques in affecting migration rates. We analyse two cases: (a) a Jupiter-mass planet in a low-mass disc; and (b) a Saturn-mass planet in a high-mass disc. The gap in case (a) is much cleaner than in case (b). Planet migration in case (b) is much more susceptible to co-orbital torques than in case (a). We find that, for both cases, the co-orbital torques do not depend sensitively on whether the planet is allowed to migrate through the disc or is held on a fixed orbit. We also examine the dependence of the planet's migration rate on the numerical resolution near the planet. For case (a), numerical convergence is relatively easy to obtain, even when including torques arising from deep within the planet's Hill sphere, since the gas mass contained within the Hill sphere is considerably less than the planet's mass. The migration rate in this case is numerically of the order of the Type II migration rate and much smaller than the Type I rate, if the disc has 0.01 solar masses inside 26 au. Torques from within the Hill sphere provide a substantial opposing contribution to the migration rate. In case (b), the gas mass within the Hill sphere is greater than the planet's mass and convergence is more difficult to obtain. Torques arising from within the Hill sphere are strong, but nearly cancel. Any inaccuracies in the calculation of the torques introduced by grid discretization can introduce spurious torques. If the torques within the Hill sphere are ignored, convergence is more easily achieved but the migration rate is artificially large. At our highest resolution, the migration rate for case (b) is much lower than the Type I rate, but somewhat larger than the Type II rate.