We study the effects of diffusion on the non‐linear corotation torque, or horseshoe drag, in the two‐dimensional limit, focusing on low‐mass planets for which the width of the horseshoe region is much smaller than the scaleheight of the disc. In the absence of diffusion, the non‐linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20 per cent of the numerical simulations.

We study the torque on low‐mass planets embedded in protoplanetary discs in the two‐dimensional approximation, incorporating non‐isothermal effects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non‐linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs type I migration in non‐isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diffusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, using three independent numerical methods, and find good agreement.

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As planets form and grow within gaseous protoplanetary disks, the mutual gravitational interaction between the disk and planet leads to the exchange of angular momentum, and migration of the planet. We review current understanding of disk-planet interactions, focussing in particular on physical processes that determine the speed and direction of migration. We describe the evolution of low mass planets embedded in protoplanetary disks, and examine the influence of Lindblad and corotation torques as a function of the disk properties. The role of the disk in causing the evolution of eccentricities and inclinations is also discussed. We describe the rapid migration of intermediate mass planets that may occur as a runaway process, and examine the transition to gap formation and slower migration driven by the viscous evolution of the disk for massive planets. The roles and influence of disk self-gravity and magnetohydrodynamic turbulence are discussed in detail, as a function of the planet mass, as is the evolution of multiple planet systems. Finally, we address the question of how well global models of planetary formation that include migration are able to match observations of extrasolar planets.

We study the structure and dynamics of the gap created by a protoplanet in an accretion disc. The hydrodynamic equations for a flat, two‐dimensional, non‐self‐gravitating protostellar accretion disc with an embedded, Jupiter‐sized protoplanet on a circular orbit are solved. To simulate possible accretion of mass on to the protoplanet we continually remove mass from the interior of the planet's Roche lobe, which is monitored. First, it is shown that consistent results independent of numerical issues (such as boundary or initial conditions, artificial viscosity or resolution) can be obtained. Then, a detailed parameter study delineates the influence of the disc viscosity and pressure on the magnitude of the accretion rate. We find that, even after the formation of a gap in the disc, the planet is still able to accrete more mass from the disc. This accretion occurs from regions of the disc that are radially exterior and interior to the planet's orbital radius. The rate depends on the magnitude of the viscosity and vertical thickness of the disc. For a disc viscosity α=10−3 and vertical thickness H/r=0.05 we estimate the time‐scale for the accumulation of one Jupiter mass to be of the order of a hundred thousand years. For a larger (smaller) viscosity and disc thickness this accretion rate is increasing (decreasing). For a very small viscosity α5×10−4 the mass accretion rate through the gap on to the planet is markedly reduced, and the corresponding accretion time‐scale becomes larger than the viscous evolution time of the disc.

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 analyzed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5% for the Eulerian simulations while 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%. The spread is larger in the tidal torques acting on the planet which agree within a factor 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.Comment: 32 pages, accepted for publication in MNRA

Aims. We investigate the response of an accretion disk to the presence of a perturbing protoplanet embedded in the disk through time dependent hydrodynamical simulations. Methods. The disk is treated as a two-dimensional viscous fluid and the planet is kept on a fixed orbit. We run a set of simulations varying the planet mass, and the viscosity and temperature of the disk. All runs are followed until they reach a quasi-equilibrium state. Results. We find that for planetary masses above a certain minimum mass, already 3 M Jup for a viscosity of ν = 10 −5 , the disk makes a transition from a nearly circular state into an eccentric state. Increasing the planetary mass leads to a saturation of disk eccentricity with a maximum value of around 0.25. The transition to the eccentric state is driven by the excitation of an m = 2 spiral wave at the outer 1:3 Lindblad resonance. The effect occurs only if the planetary mass is large enough to clear a sufficiently wide and deep gap to reduce the damping effect of the outer 1:2 Lindblad resonance. An increase in viscosity or temperature in the disk, which both tend to close the gap, have an adverse influence on the disk eccentricity. Conclusions. In the eccentric state the mass accretion rate onto the planet is greatly enhanced, an effect that may ease the formation of massive planets beyond about 5 M Jup that are otherwise difficult to reach.

Context. The migration of growing protoplanets depends on the thermodynamics of the ambient disc. Standard modelling, using locally isothermal discs, indicate an inward (type-I) migration in the low planet mass regime. Taking non-isothermal effects into account, recent studies have shown that the direction of the type-I migration can change from inward to outward. Aims. In this paper we extend previous two-dimensional studies and investigate the planet-disc interaction in viscous, radiative discs using fully three-dimensional radiation hydrodynamical simulations of protoplanetary accretion discs with embedded planets, for a range of planetary masses. Methods. We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that includes full tensor viscosity. We have added implicit radiation transport in the flux-limited diffusion approximation, and to speed up the simulations significantly we have newly adapted and implemented the FARGO-algorithm in a 3D context. Results. First, we present results of test simulations that demonstrate the accuracy of the newly implemented FARGO-method in 3D. For a planet mass of 20 M earth , we then show that including radiative effects also yields a torque reversal in full 3D. For the same opacity law, the effect is even stronger in 3D than in the corresponding 2D simulations, due to a slightly thinner disc. Finally, we demonstrate the extent of the torque reversal by calculating a sequence of planet masses. Conclusions. Through full 3D simulations of embedded planets in viscous, radiative discs, we confirm that the migration can be directed outwards up to planet masses of about 33 M earth . As a result, the effect may help to resolve the problem of inward migration of planets that is too rapid during their type-I phase.