The interaction of giant planets with a disc with MHD turbulence – IV. Migration rates of embedded protoplanets

We present the results of global cylindrical disc simulations and local shearing box simulations of protoplanets interacting with a disc undergoing magnetohydrodynamic (MHD) turbulence. The specific emphasis of this paper is to examine and quantify the magnitude of the torque exerted by the disc on the embedded protoplanets as a function of the protoplanet mass, and thus to make a first study of the induced orbital migration of protoplanets resulting from their interaction with magnetic, turbulent discs. This issue is of crucial importance in understanding the formation of gas giant planets through the so-called core instability model, and the subsequent orbital evolution post-formation prior to the dispersal of the protostellar disc. Current estimates of the migration time of protoplanetary cores in the 3–30 Earth mass range in standard disc models are τmig≃ 104–105 yr, which is much shorter than the estimated gas accretion time-scale of Jupiter-type planets. The global simulations were carried out for a disc with constant aspect ratio H/r= 0.07 and protoplanet masses of Mp= 3, 10, 30 Earth masses, and 3 Jupiter masses. The local shearing box simulations were carried out for values of the dimensionless parameter (Mp/M*)/(H/R)3= 0.1, 0.3, 1.0 and 2.0, with M*, R and H being the central mass, the orbital radius and the local disc semithickness, respectively. These allow both embedded and gap-forming protoplanets for which the disc response is non-linear to be investigated. In all cases the instantaneous net torque experienced by a protoplanet showed strong fluctuations on an orbital time-scale, and in the low-mass embedded cases oscillated between negative and positive values. Consequently, in contrast to the laminar disc type I migration scenario, orbital migration would occur as a random walk. Running time averages for embedded protoplanets over typical run times of 20–25 orbital periods indicated that the averaged torques from the inner and outer disc took on values characteristic of type I migration. However, large fluctuations occurring on longer than orbital time-scales remained, preventing convergence of the average torque to well-defined values or even to a well-defined sign for these lower mass cases. Fluctuations became relatively smaller for larger masses indicating better convergence properties, to the extent that in the 30-M⊕ simulation consistently inward, albeit noisy, migration was indicated. Both the global and local simulations showed this trend with increasing protoplanet mass, which is due to its perturbation on the disc increasing to become comparable to and then dominate the turbulence in its neighbourhood. The turbulence then becomes unable to produce very large long-term fluctuations in the torques acting on the protoplanet. Eventually gap formation occurs and there is a transition to the usual type II migration at a rate determined by the angular momentum transport in the distant parts of the disc. The existence of significant fluctuations occurring in the turbulent discs on long time-scales is an important unexplored issue for the lower mass embedded protoplanets, which are unable to modify the turbulence in their neighbourhood, and which have been studied here. If significant fluctuations occur on the longest disc evolutionary time-scales, convergence of torque running averages for practical purposes will not occur and the migration behaviour of low-mass protoplanets considered as an ensemble will be very different from predictions of type I migration theory for laminar discs. The fact that noise levels were relatively smaller in the local simulations may indicate the presence of long-term global fluctuations, but the issue remains an important one for future investigation.