F. Masset

On the width and shape of gaps in protoplanetary disks

Although it is well known that a massive planet opens a gap in a protoplanetary 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.

The migration and growth of protoplanets in protostellar discs

ABSTRA C T We investigate the gravitational interaction of a Jovian-mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions are investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Non-linear two-dimensional hydrodynamic simulations are employed using three independent numerical codes. A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches the central star in a near-circular orbit after a characteristic viscous time-scale of ,10 4 initial orbital periods. This is found to be independent of whether the protoplanet is allowed to accrete mass or not. Inward migration is helped by the disappearance of the inner disc, and therefore the positive torque it would exert, because of accretion on to the central star. Maximally accreting protoplanets reach about 4 Jovian masses on reaching the neighbourhood of the central star. Our results indicate that a realistic upper limit for the masses of closely orbiting giant planets is ,5 Jupiter masses, if they originate in protoplanetary discs similar to the minimum-mass solar nebula. This is because of the reduced accretion rates obtained for planets of increasing mass. Assuming that some process such as termination of the inner disc through a magnetospheric cavity stops the migration, the range of masses estimated for a number of close orbiting giant planets as well as their inward orbital migration can be accounted for by consideration of disc‐protoplanet interactions during the late stages of giant planet formation.

Runaway migration and the formation of hot jupiters

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.

A comparative study of disc–planet interaction

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

DISK SURFACE DENSITY TRANSITIONS AS PROTOPLANET TRAPS

The tidal torque exerted by a protoplanetary disk with power-law surface density and temperature profiles onto anembedded protoplanetary embryois generally anegative quantity that leads to the embryoinward migration. Here we investigate how the tidal torque balance is affected at a disk surface density radial jump. The jump has two consequences:(1)Itaffects thedifferential Lindbladtorque.Inparticular,ifthediskismerelyemptyontheinnerside,the differential Lindblad torque almost amounts to the large negative outer Lindblad torque. (2) It affects the corotation torque, which is a quantity very sensitive to the local gradient of the disk surface density. In particular, if the disk is depleted on the inside and the jump occurs radially over a few pressure scale heights, the corotation torque is a positive quantity that ismuchlargerthan inapower-lawdisk.Weshow bymeans ofcustomized numerical simulations of low-massplanetsembedded inprotoplanetarynebulaewithasurfacedensity jump that thesecond effect isdominant; that is, that the corotation torque largely dominates the differential Lindblad torque on the edge of a central depletion, even a shallow one. Namely, a disk surface density jump of about 50% over 3–5 disk thicknesses suffices to cancel out the total torque. As a consequence, the type I migration of low-mass objects reaching the jump should be halted, and all these objects should be trapped there provided some amount of dissipation is present in the disk to prevent the corotation torque saturation. As dissipation is provided by turbulence, which induces a jitter of the planet semimajor axis, we investigate under which conditions the trapping process overcomes the trend of turbulence to induce stochastic migration across the disk. We show that a cavity with a large outer to inner surface density ratio efficiently traps embryos from 1 to 15 M� , at any radius up to 5 AU from the central object, in a disk that has same surface density profile as the minimum mass solar nebula (MMSN). Shallow surface density transitions require light disks to efficiently trap embryos. In the case of the MMSN, this could happen in the very central parts (r < 0:03 AU). We discusswhereinaprotoplanetarydiskonecanexpectasurfacedensityjump.Thiseffectcouldconstituteasolutionto the well-known problem that the buildup of the first protogiant solid core in a disk takes much longer than its type I migration toward the central object. Subject headings: accretion, accretion disks — hydrodynamics — methods: numerical — planetary systems: formation — planetary systems: protoplanetary disks

Reversing type II migration: resonance trapping of a lighter giant protoplanet

We present a mechanism related to the migration of giant protoplanets embedded in a protoplanetary disc whereby a giant protoplanet is caught up, before having migrated all the way to the central star, by a lighter outer giant protoplanet. This outer protoplanet may get captured into the 2:3 resonance with the more massive one, in which case the gaps that the two planets open in the disc overlap. Two effects arise, namely a squared mass-weighted torque imbalance and an increased mass flow through the overlapping gaps from the outer disc to the inner disc, which both play in favour of an outwards migration. Indeed, under the conditions presented here, which describe the evolution of a pair of protoplanets respectively Jupiter- and Saturn-sized, the migration is reversed, while the semimajor axis ratio of the planets is constant and the eccentricities are confined to small values by the disc material. The long-term behaviour of the system is briefly discussed, and could account for the high eccentricities observed for the extrasolar planets with semimajor axis

On the Corotation Torque in a Radiatively Inefficient Disk

We consider the angular momentum exchange at the corotation resonance between a two-dimensional gaseous disk and a uniformly rotating external potential, assuming that the disk flow is adiabatic. We first consider the linear case for an isolated resonance, for which we give an expression of the corotation torque that involves the pressure perturbation and which reduces to the usual dependence on the vortensity gradient in the limit of a cold disk. Although this expression requires the solution of the hydrodynamic equations, it provides some insight into the dynamics of the corotation region. In the general case, we find an additional dependence on the entropy gradient at corotation. This dependence is associated with the advection of entropy perturbations. These are not associated with pressure perturbations. They remain confined to the corotation region, where they yield a singular contribution to the corotation torque. In a second part, we check our torque expression by means of customized two-dimensional hydrodynamical simulations. In a third part, we contemplate the case of a planet embedded in a Keplerian disk, assumed to be adiabatic. We find an excess of corotation torque that scales with the entropy gradient, and we check that the contribution of the entropy perturbation to the torque is in agreement with the expression obtained from the linear analysis. We finally discuss some implications of the corotation torque expression for the migration of low-mass planets in the regions of protoplanetary disks where the flow is radiatively inefficient on the timescale of the horseshoe U-turns.

Orbital eccentricity growth through disc-companion tidal interaction

We investigate the driving of orbital eccentricity of giant protoplanets and brown dwarfs through disc-companion tidal interactions by means of two dimensional numerical simulations. We consider disc models that are thought to be typical of protostellar discs during the planet forming epoch, with characteristic surface densities similar to standard minimum mass solar nebula models. We consider companions, ranging in mass between 1 and 30 Jupiter masses

Saturated Torque Formula for Planetary Migration in Viscous Disks with Thermal Diffusion: Recipe for Protoplanet Population Synthesis

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Long Range Outward Migration of Giant Planets, with Application to Fomalhaut b

that are initially embedded within the discs on circular orbits about a central solar mass. We find that a transition in orbital behaviour occurs at a mass in the range 10-20