Radiation-driven warping. II. Nonisothermal disks

Recent work by Pringle and by Maloney, Begelman, & Pringle has shown that g eometrically thin, optically thick, accretion disks are unstable to warping driven by radiation torque from the central source. This work was cofined to isothermal (i.e., surface density Sigma proportional to R-3/2) disks. In this paper we generalize the study of radiation-driven warping to include general power-law surface density distributions, Sigma proportional to R-de lta. We consider the range from delta = 3/2 (the isothermal case) to delta = -3/2, which corresponds to a radiation-pressure-supported disk; this span s the range of surface density distributions likely to be found in real ast rophysical disks. In all cases there are an infinite number of zero-crossin g solutions (i.e., solutions that cross the equator), which are the physica lly relevant modes if the outer boundary of the disk is required to lie in a specified plane. However, unlike the isothermal disk, which is the degene rate case, the frequency eigenvalues for delta not equal 3/2 are all distin ct. In all cases the location of the zero moves outward from the steady sta te (pure precession) value with increasing growth rate; thus, there is a cr itical minimum size for unstable disks. Modes with zeros at smaller radii a re damped. The critical radius and the steady state precession rate depend only weakly on delta. An additional analytic solution has been found for de lta = 1. The case delta = 1 divides the solutions into two qualitatively di fferent regimes. For delta greater than or equal to I, the fastest growing modes have maximum warp amplitude, beta(max), close to the disk outer edge, and the ratio of beta(max) to the warp amplitude at the disk inner edge, b eta(0), is much greater than 1. For delta < 1, beta(max)/beta(0) h similar or equal to 1 and the warp maximum steadily approaches the origin as S decr eases. This implies that nonlinear effects must be important if the warp ex tends to the disk inner edge for delta greater than or equal to 1, but for delta < 1 nonlinearity will be important only if the warp amplitude is larg e at the origin. Because of this qualitative difference in the shapes of th e warps, the effects of shadowing of the central source by the warp will al so be very different in the two regimes of delta. This has important implic ations for radiation-driven warping in X-ray binaries, for which the value of delta characterizing the disk is likely to be less than unity. In real a ccretion 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 beco me optically thin before the outer disk boundary is reached, whereas in X-r ay 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.