Linear Analysis of the Magnetorotational Instability, Including Ambipolar Diffusion, with Application to Protoplanetary Disks

We present a linear analysis of the magnetorotational instability (MRI) in differentially rotating disks and derive the most general instability criterion to date. Our analysis improves on earlier work on this topic in that it simultaneously accounts for arbitrary geometry and the full effects of magnetic diffusion. We allow the magnetic field to have arbitrary orientation and for linear modes to propagate at an angle to the rotation axis. We also include in our analysis all three forms of magnetic diffusion: ohmic dissipation, ambipolar diffusion, and Hall currents. Previous analyses have included either arbitrary geometry or ambipolar diffusion, but never both. The simultaneous inclusion of these effects allows us to identify a new unstable mode in which ambipolar diffusion and differential rotation can couple and amplify the magnetic field. We provide a physical explanation of this mode. Our linear analysis is aimed at determining which parts of protoplanetary disks may be unstable to the MRI. Accordingly, we outline the conditions that are likely to obtain in protoplanetary disks and make estimates of the coupling between the gas and the magnetic field. We derive a linear stability criterion that can be applied to protoplanetary disks.