Linear Stability Analysis of a Magnetic Rotating Disk with Ohmic
  Dissipation and Ambipolar Diffusion
  We perform a linear analysis of the stability of isothermal, rotating,
magnetic, self-gravitating sheets that are weakly ionized. The magnetic field
and rotation axis are perpendicular to the sheet. We include a self-consistent
treatment of thermal pressure, gravitational, rotational, and magnetic
(pressure and tension) forces together with two nonideal magnetohydrodynamic
(MHD) effects (Ohmic dissipation and ambipolar diffusion) that are treated
together for their influence on the properties of gravitational instability for
a rotating sheet-like cloud or disk. Our results show that there is always a
preferred length scale and associated minimum timescale for gravitational
instability. We investigate their dependence on important dimensionless free
parameters of the problem: the initial normalized mass-to-flux ratio $\mu_0$,
the rotational Toomre parameter $Q$, the dimensionless Ohmic diffusivity
$\tilde{\eta}_{\rm OD,0}$, and the dimensionless neutral-ion collision time
$\tilde{\tau}_{\rm{ni,0}}$ that is a measure of the ambipolar diffusivity. One
consequence of $\tilde{\eta}_{\rm OD,0}$ is that there is a maximum preferred
lengthscale of instability that occurs in the transcritical ($\mu_0 \gtrsim 1$)
regime, qualitatively similar to the effect of $\tilde{\tau}_{\rm{ni,0}}$, but
with quantitative differences. The addition of rotation leads to a generalized
Toomre criterion (that includes a magnetic dependence) and modified
lengthscales and timescales for collapse. When nonideal MHD effects are also
included, the Toomre criterion reverts back to the hydrodynamic value. We apply
our results to protostellar disk properties in the early embedded phase and
find that the preferred scale of instability can significantly exceed the
thermal (Jeans) scale and the peak preferred fragmentation mass is likely to be
$\sim 10- 90 \ M_{\rm Jup}$.
