Magnetorotational Instability in Protoplanetary Disks. II. Ionization   State and Unstable Regions
  We investigate where in protoplanetary disks magnetorotational instability operates, which can cause angular momentum transport in the disks. We investigate the spatial distribution of various charged particles and the unstable regions for a variety of models for protoplanetary disks taking into account the recombination of ions and electrons at grain surfaces, which is an important process in most parts of the disks. We find that for all the models there is an inner region which is magnetorotationally stable due to ohmic dissipation. This must make the accretion onto the central star non-steady. For the model of the minimum-mass solar nebula, the critical radius, inside of which the disk is stable, is about 20 AU, and the mass accretion rate just outside the critical radius is 10^{-7} - 10^{-6} M_{\odot} yr^{-1}. The stable region is smaller in a disk of lower column density. Dust grains in protoplanetary disks may grow by mutual sticking and may sediment toward the midplane of the disks. We find that the stable region shrinks as the grain size increases or the sedimentation proceeds. Therefore in the late evolutionary stages, protoplanetary disks can be magnetorotationally unstable even in the inner regions. 
Takayoshi Sano (U. of Maryland), Shoken M. Miyama (NAO, Japan),   Toyoharu Umebayashi (Yamagata U.), Takenori Nakano (Kyoto U.)