G. D'Angelo

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.

Nested-grid calculations of disk-planet interaction

We study the evolution of embedded protoplanets in a protostellar disk using very high resolution nested- grid computations. This method allows us to perform global simulations of planets orbiting in disks and, at the same time, to resolve in detail the dynamics of the flow inside the Roche lobe of the planet. The primary interest of this work lies in the analysis of the gravitational torque balance acting on the planet. For this purpose we study planets of dierent masses, ranging from one Earth-mass up to one Jupiter-mass, assuming typical parameters of the protostellar disk. The high resolution supplied by the nested-grid technique permits an evaluation of the torques, resulting from short and very short range disk-planet interactions, more reliable than the one previously estimated with the aid of numerical methods. Likewise, the mass flow onto the planet is computed in a more accurate fashion. The obtained migration time scales are in the range from few times 10 4 years, for intermediate mass planets, to 10 6 years, for very low and high mass planets. These are longer than earlier assessments due to the action of circumplanetary material. Typical growth time scales depend strongly on the planetary mass. Below 64 Earth-masses, we nd this time scale to increase as the 2=3-power of the planets mass; otherwise it rises as the 4=3-power. In the case of Jupiter-size planets, the growth time scale is several times ten thousand years.

Thermohydrodynamics of Circumstellar Disks with High-Mass Planets

With a series of numerical simulations, we analyze the thermohydrodynamic evolution of circumstellar disks containing Jupiter-sized protoplanets. In the framework of a two-dimensional approximation, we consider an energy equation that includes viscous heating and radiative effects in a simplified yet consistent form. Multiple nested grids are used in order to study both global and local features around the planet. By means of different viscosity prescriptions, we investigate various temperature regimes. A planetary mass range from 0.1 to 1 MJ is examined. Computations show that gap formation is a general property that affects density, pressure, temperature, optical thickness, and radiated flux distributions. However, it remains a prominent feature only when the kinematic viscosity is on the order of 1015 cm2 s-1 or lower, although it becomes rather shallow for 0.1 MJ perturbers. Around accreting planets, a circumplanetary disk forms that has a surface density profile decaying exponentially with distance and whose mass is 5-6 orders of magnitude smaller than Jupiters mass. Circumplanetary disk temperature profiles decline roughly as the inverse of the distance from the planet, matching the values measured in the gap toward the border of the Roche lobe. Temperatures range from some 10 to ~1000 K. Moreover, circumplanetary disks are generally opaque, with optical thicknesses larger than 1 and aspect ratios around a few tenths. Nonaccreting protoplanets provide quite different scenarios, with a clockwise, i.e., reversed flow, rotation around low-mass bodies. Planetary accretion and migration rates depend on the viscosity regime, with discrepancies within an order of magnitude. Co-orbital torques increase as viscosity increases. For high viscosities, type I migration may extend to larger planetary masses. Estimates of growth and migration timescales inferred from these models are on the same orders of magnitude as those previously obtained with locally isothermal simulations, in both two and three dimensions.

Vortex generation in protoplanetary disks with an embedded giant planet

Context. Vortices in protoplanetary disks can capture solid particles and form planetary cores within shorter timescales than those involved in the standard core-accretion model. Aims. We investigate vortex generation in thin unmagnetized protoplanetary disks with an embedded giant planet with planet to star mass ratio 10 4 and 10 3 . Methods. Two-dimensional hydrodynamical simulations of a protoplanetary disk with a planet are performed using two di erent numerical methods. The results of the non-linear simulations are compared with a time-resolved modal analysis of the azimuthally averaged surface density profiles using linear perturbation theory. Results. Finite-di erence methods implemented in polar coordinates generate vortices moving along the gap created by Neptune- mass to Jupiter-mass planets. The modal analysis shows that unstable modes are generated with growth rate of order 0:3 K for azimuthal numbers m= 4; 5; 6, where K is the local Keplerian frequency. Shock-capturing Cartesian-grid codes do not generate very much vorticity around a giant planet in a standard protoplanetary disk. Modal calculations confirm that the obtained radial profiles of density are less susceptible to the growth of linear modes on timescales of several hundreds of orbital periods. Navier-Stokes viscosity of the order = 10 5 (in units of a 2 p) is found to have a stabilizing e ect and prevents the formation of vortices. This result holds at high resolution runs and using di erent types of boundary conditions. Conclusions. Giant protoplanets of Neptune-mass to Jupiter-mass can excite the Rossby wave instability and generate vortices in thin disks. The presence of vortices in protoplanetary disks has implications for planet formation, orbital migration, and angular momentum transport in disks.

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 planets 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 planets 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 planets Hill sphere, since the gas mass contained within the Hill sphere is considerably less than the planets 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 planets 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.