Migration Of Low-mass Planets Research Articles

TYPE I PLANET MIGRATION IN NEARLY LAMINAR DISKS

We describe two-dimensional hydrodynamic simulations of the migration of low-mass planets ( 30 M⊕) in nearly laminar disks (viscosity parameter α< 10 −3 ) over timescales of several thousand orbit periods. We consider disk masses of 1, 2, and 5 times the minimum mass solar nebula, disk thickness parameters of H/r = 0.035 and 0.05, and a variety of α values and planet masses. Disk self-gravity is fully included. Previous analytic work has suggested that Type I planet migration can be halted in disks of sufficiently low turbulent viscosity, for α ∼ 10 −4 . The halting is due to a feedback effect of breaking density waves that results in a slight mass redistribution and consequently an increased outward torque contribution. The simulations confirm the existence of a critical mass (Mcr ∼ 10M⊕) beyond which migration halts in nearly laminar disks. For α 10 −3 , density feedback effects are washed out and Type I migration persists. The critical masses are in good agreement with the analytic model of Rafikov. In addition, for α 10 −4 steep density gradients produce a vortex instability, resulting in a small time-varying eccentricity in the planet’s orbit and a slight outward migration. Migration in nearly laminar disks may be sufficiently slow to reconcile the timescales of migration theory with those of giant planet formation in the core accretion model.

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.

Numerical simulations of type I planetary migration in non-turbulent magnetized discs

Using 2D magnetohydrodynamic (MHD) numerical simulations performed with two different finite-difference Eulerian codes, we analyse the effect that a toroidal magnetic field has on low-mass planet migration in non-turbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis, we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the latter is either fixed on a circular orbit or allowed to migrate. For a 5-M ○+ planet, when the ratio β between the square of the sound speed and that of the Alfven speed at the location of the planet is equal to 2, we find inward migration when the magnetic field B Φ is uniform in the disc, reduced migration when B Φ decreases as r -1 and outward migration when B Φ decreases as r -2 . These results are in agreement with predictions from the linear analysis. Taken as a whole, our results confirm that even a subthermal stable field can stop inward migration of an earth-like planet.