Gravitational instabilities in a proto-planetary disk including the effects of magnetic fields

We investigate the gravitational instability of a thin, Keplerian protoplanetary disk including the effects of a largely azimuthal magnetic field. The model follows that of our previous work (Noh, Vishniac, & Cochran 1991) except for the inclusion of a magnetic field. The disk is assumed to consist of neutral and ionized gas and neutral dust which are coupled by gravity and friction. The growth rates and eigenfunctions are calculated numerically using nonaxisymmetric linear perturbation methods. The results show that the growth rate has a maximum at some intermediate azimuthal number m, but for each value of m it is reduced relative to the unmagnetized case. The effects of the magnetic field appear more strongly on small scales. As the strength of the equilibrium magnetic field increases the growth rates decrease, and the maximum instability occurs at a lower value on m due to the increasing magnetic pressure. The response of each component to the magnetic field is discussed using the behavior of the eigenfunctions in the radial direction. With the inclusion of the magnetic field, the effects of the ionization fraction and friction on the growth rates also appear to be important for high m modes. Increasing the ionization fraction or the friction suppresses instability, but only slightly changes the maximally unstable azimuthal scales. The enhanced growth rates due to a dust component for which thermal pressure is negligible are somewhat reduced by the inclusion of a magnetic field. The effects of different boundary conditions (reflecting and transmitting) on the growth rates are also shown.