Although it is well known that a massive planet opens a gap in a proto-planetary gaseous disk, there is no analytic description of the surface density profile in and near the gap. The simplest approach, which is based upon the balance between the torques due to the viscosity and the gravity of the planet and assumes local damping, leads to gaps with overestimated width, especially at low viscosity. Here, we take into account the fraction of the gravity torque that is evacuated by pressure supported waves. With a novel approach, which consists of following the fluid elements along their trajectories, we show that the flux of angular momentum carried by the waves corresponds to a pressure torque. The equilibrium profile of the disk is then set by the balance between gravity, viscous and pressure torques. We check that this balance is satisfied in numerical simulations, with a planet on a fixed circular orbit. We then use a reference numerical simulation to get an ansatz for the pressure torque, that yields gap profiles for any value of the disk viscosity, pressure scale height and planet to primary mass ratio. Those are in good agreement with profiles obtained in numerical simulations over a wide range of parameters. Finally, we provide a gap opening criterion that simultaneously involves the planet mass, the disk viscosity and the aspect ratio.

We evaluate the coorbital corotation torque on a migrating protoplanet. The coorbital torque is assumed to come from orbit crossing fluid elements that exchange angular momentum with the planet when they execute a U-turn at the end of horseshoe streamlines. When the planet migrates inward, the fluid elements of the inner disk undergo one such exchange as they pass to the outer disk. The angular momentum they gain is removed from the planet, and this corresponds to a negative contribution to the corotation torque, which scales with the drift rate. In addition, the material trapped in the coorbital region drifts radially with the planet, giving a positive contribution to the corotation torque, which also scales with the drift rate. These two contributions do not cancel out if the coorbital region is depleted, in which case there is a net corotation torque that scales with the drift rate and the mass deficit in the coorbital region and has the same sign as the drift rate. This leads to a positive feedback on the migrating planet. In particular, if the coorbital mass deficit is larger than the planet mass, the migration rate undergoes a runaway that can vary the protoplanet semimajor axis by 50% over a few tens of orbits. This can happen only if the planet mass is sufficient to create a dip or gap in its surrounding region and if the surrounding disk mass is larger than the planet mass. This typically corresponds to planet masses in the sub-Saturnian to Jovian mass range embedded in massive protoplanetary disks. Runaway migration is a good candidate to account for the orbital characteristics of close orbiting giant planets, most of which have sub-Jovian masses. These are known to cluster at short periods, whereas planets of greater than two Jovian masses are rare at short periods, indicating a different type of migration process operated for the two classes of object. Further, we show that in the runaway regime, migration can be directed outward, which makes this regime potentially rich in a variety of important effects in shaping a planetary system during the last stages of its formation.

Abstract. We present an efficient and simple modification of the standard transport algorithm used in explicit eulerian fixed polar grid codes, aimed at getting rid of the average azimuthal velocity when applying the Courant condition. This results in a much larger timestep than the usual procedure, and it is particularly well-suited to the description of a Keplerian disk where one is traditionally limited by the very demanding Courant condition on the fast orbital motion at the inner boundary. In this modified algorithm, the timestep is limited by the perturbed velocity and by the shear arising from the differential rotation. FARGO stands for "Fast Advection in Rotating Gaseous Objects". The speed-up resulting from the use of the FARGO algorithm is problem dependent. In the example presented here, which shows the evolution of a Jupiter sized protoplanet embedded in a minimum mass protoplanetary nebula, the FARGO algorithm is about an order of magnitude faster than a traditional transport scheme, with a much smaller numerical diffusivity.

Astrophysical Journal, 642, pp. 478-487, http://dx.doi.org./10.1086/500967International audienc