On the orbital evolution of low mass protoplanets in turbulent, magnetised disks

We present the results of MHD simulations of low mass protoplanets interacting with turbulent, magnetised protostellar disks. We calculate the orbital evolution of “planetesimals” and protoplanets with masses in the range $0 \le m_{\rm p} \le 30$ $M_{\oplus}$. The disk models are cylindrical models with toroidal net-flux magnetic fields, having aspect ratio $H/r=0.07$ and effective viscous stress parameter $\alpha \simeq 5 \times 10^{-3}$. A significant result is that the $m_{\rm p}=0$ “planetesimals”, and protoplanets of all masses considered, undergo stochastic migration due to gravitational interaction with turbulent density fluctuations in the disk. For simulation run times currently feasible (covering between 100-150 planet orbits), the stochastic migration dominates over type I migration for many models. Fourier analysis of the torques experienced by protoplanets indicates that the torque fluctuations contain components with significant power whose time scales of variation are similar to the simulation run times. These long term torque fluctuations in part explain the dominance of stochastic torques in the models, and may provide a powerful means of counteracting the effects of type I migration acting on some planets in turbulent disks. The effect of superposing type I migration torques appropriate for laminar disks on the stochastic torques was examined. This analysis predicts that a greater degree of inward migration should occur than was observed in the MHD simulations. This may be a first hint that type I torques are modified in a turbulent disk, but the results are not conclusive on this matter. The turbulence is found to be a significant source of eccentricity driving, with the “planetesimals” attaining eccentricities in the range $0.02 \le e \le 0.14$ during the simulations. The eccentricity evolution of the protoplanets shows strong dependence on the protoplanet mass. Protoplanets with mass $m_{\rm p}=1$ $M_{\oplus}$ attained eccentricities in the range $0.02 \le e \le 0.08$. Those with $m_{\rm p}=10$ $M_{\oplus}$ reached $0.02 \le e \le 0.03$. This trend is in basic agreement with a model in which eccentricity growth arises because of turbulent forcing, and eccentricity damping occurs through interaction with disk material at coorbital Lindblad resonances. These results are significant for the theory of planet formation. Stochastic migration may provide a means of preventing at least some planetary cores from migrating into the central star