Radiation-Driven Warping. II. Non-Isothermal Disks
  Recent work by Pringle and by Maloney, Begelman & Pringle has shown that geometrically thin, optically thick, accretion disks are unstable to warping driven by radiation torque from the central source. In this paper we generalize the study of radiation-driven warping to include general power-law surface density distributions, $\Sigma\propto R^{-\delta}$. We consider the range $\delta=3/2$ (isothermal disks) to $\delta=-3/2$, which corresponds to a radiation-pressure-supported disk; this spans the range of surface density distributions likely to be found in real astrophysical disks. There is a critical minimum size for unstable disks. The critical radius and the steady-state precession rate depend only weakly on $\delta$. The case $\delta=1$ divides the solutions into two qualitatively different regimes. Nonlinear effects must be important if the warp extends to the disk inner edge for $\delta \ge 1$, but for $\delta < 1$ nonlinearity will be important only if the warp amplitude is large at the origin. The effects of shadowing of the central source by the warp will thus be very different in the two regimes of $\delta.$ In real accretion disks the outer boundary condition is likely to be different from the zero-crossing condition that we have assumed. In accretion disks around massive black holes in active galactic nuclei, the disk will probably become optically thin before the outer disk boundary is reached, while in X-ray binaries, there will be an outer disk region (outside the circularization radius) in which the inflow velocity is zero but angular momentum is still transported. We show that in both these cases the solutions are similar to the zero-crossing eigenfunctions. 
Philip R. Maloney, Mitchell C. Begelman, and Michael A. Nowak