Radiation‐driven Warping. II. Nonisothermal Disks

Abstract

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. This work was confined to isothermal (i.e., surface density Σ ∝ R-3/2) disks. In this paper we generalize the study of radiation-driven warping to include general power-law surface density distributions, Σ ∝ R-δ. We consider the range from δ = 3/2 (the isothermal case) to δ = -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. In all cases there are an infinite number of zero-crossing solutions (i.e., solutions that cross the equator), which are the physically 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 degenerate case, the frequency eigenvalues for δ ≠ 3/2 are all distinct. In all cases the location of the zero moves outward from the steady state (pure precession) value with increasing growth rate; thus, there is a critical minimum size for unstable disks. Modes with zeros at smaller radii are damped. The critical radius and the steady state precession rate depend only weakly on δ. An additional analytic solution has been found for δ = 1. The case δ = 1 divides the solutions into two qualitatively different regimes. For δ ≥ 1, the fastest growing modes have maximum warp amplitude, βmax, close to the disk outer edge, and the ratio of βmax to the warp amplitude at the disk inner edge, β0, is 1. For δ < 1, βmax/β0 1, and the warp maximum steadily approaches the origin as δ decreases. This implies that nonlinear effects must be important if the warp extends to the disk inner edge for δ ≥ 1, but for δ < 1 nonlinearity will be important only if the warp amplitude is large at the origin. Because of this qualitative difference in the shapes of the warps, the effects of shadowing of the central source by the warp will also be very different in the two regimes of δ. This has important implications for radiation-driven warping in X-ray binaries, for which the value of δ characterizing the disk is likely to be less than unity. 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, whereas 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.