On the migration rate of giant planets protocores

We present a large set of numerical simulations aimed at investigating a behavior previously observed in earlier work, namely a significant discrepancy between the linear migration time estimate and its value as obtained from simulations, for planet masses in the range 5-15 earth masses, characteristic of the solid cores of giant planets. For such masses, numerical simulations yield a much longer migration time, and can even display, for some sets of parameters, an outward migration. Our simulations show that this offset scales with the gradient of the specific vorticity, increases with the disk viscosity, and has a maximum for a planet mass proportional to H (H being the disk thickness). These findings are compatible with non-linear effects associated to the corotation torque acting upon the planet. Numerical simulations versus analytical estimates The most recent analytical estimate of the torque acting upon a small mass planet embedded in a disk with power law surface density and temperature profiles is given by Tanaka et al. (2002), and yields the following migration timescale in a disk with uniform aspect ratio h = H/r: τ = (2.7 + 1.1α)qμhΩ p , (1) where q is the planet to star mass ratio, μ the disk to star mass ratio (namely Σa2/M∗, where Σ is the disk surface density at the planet orbit, which has radius a), and Ωp is the planet angular frequency. This expression has been obtained for a three dimensional isothermal disk, taking into account: • The differential Lindblad torque acting on the planet (i.e. the torque that stems from the planet wake). • The corotation torque (i.e. the torque that comes from the horseshoe region drag), assumed to be unsaturated. D’Angelo et al. (2003) have tried to verify the analytical expression of Tanaka et al. (2002) by means of 3D numerical simulations. They used the code NIRVANA with a hierarchy of nested grids at the planet location to achieve a high resolution in the planet vicinity. Their results are displayed on Fig. 1. They show that: • at low mass (Mp < 4 M⊕) one recovers the analytical estimate by Tanaka et al. • at large mass (Mp > 100 M⊕) the migration rate is no longer described by this linear analytical estimate. The disk response for that regime is strongly non-linear, a gap is emptied around the planet orbit, and the planet undergoes a so-called type II migration. • In between these two extreme regimes, for 4 M⊕ < Mp < 20 M⊕, the migration rate drops significantly with respect to the analytic prediction. In particular, for Mp ∼ 10M⊕, the migration rate is one order of magnitude below the analytical estimate. It is this large discrepancy that motivated our work. For brief we shall refer hereafter to this discrepancy as the “offset”. 2D versus 3D numerical simulations We investigated this behavior by means of 2D and 3D simulations. The former cannot give a reliable value for the torque, since the torque value depends on the potential smoothing length. However, 2D simulations are CPU inexpensive and allow to investigate the existence and behavior of the offset for a large set of disk parameters. 3D simulations are not affected by smoothening issues and therefore give reliable values for the torque. However, as they are much more computationally expensive, they do not allow a parameter space exploration with the same level of detail as 2D simulations.