The Effect of Poloidal Magnetic Field on Type I Planetary Migration: Significance of Magnetic Resonance

We study the effect of poloidal magnetic field on type I planetary migration by linear perturbation analysis in the shearing-sheet approximation and the analytic results are compared with numerical calculations. In contrast to the unmagnetized case, the basic equations that describe the wake due to the planet in the disk allow magnetic resonances at which density perturbation diverges. In order to simplify the problem, we consider the case without magneto-rotational instability. We perform two sets of analyses: two-dimensional and three-dimensional. In two-dimensional analysis, we find the generalization of the torque formula previously known in unmagnetized case. In three-dimensional calculations, we focus on the disk with very strong magnetic field and derive a new analytic formula for the torque exerted on the planet. We find that when Alfven velocity is much larger than sound speed, two-dimensional torque is suppressed and three-dimensional modes dominate, in contrast to the unmagnetized case.