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We study the effect of poloidal magnetic field on type I planetary migrationby linear perturbation analysis in the shearing-sheet approximation and theanalytic results are compared with numerical calculations. In contrast to theunmagnetized case, the basic equations that describe the wake due to the planetin the disk allow magnetic resonances at which density perturbation diverges.In order to simplify the problem, we consider the case withoutmagneto-rotational instability. We perform two sets of analyses:two-dimensional and three-dimensional. In two-dimensional analysis, we find thegeneralization of the torque formula previously known in unmagnetized case. Inthree-dimensional calculations, we focus on the disk with very strong magneticfield 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, incontrast to the unmagnetized case.

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