Viscous torque and dissipation in the inner regions of a thin accretion disk: Implications for measuring black hole spin

We consider a simple Newtonian model of a steady accretion disk around a black hole. The model is based on height-integrated hydrodynamic equations, alpha-viscosity, and a pseudo-Newtonian potential which results in an innermost stable circular orbit (ISCO) that closely approximates the one predicted by general relativity. We find that, as the disk thickness H/R or the value of alpha increases, the hydrodynamic model exhibits increasing deviations from the standard thin disk model of Shakura and Sunyaev. The latter is an analytical model in which the viscous torque is assumed to vanish at the ISCO. We consider the implications of the results for attempts to estimate black hole spin by using the standard disk model to fit continuum spectra of black hole accretion disks. We find that the error in the spin estimate is quite modest so long as H/R <= 0.1 and alpha <= 0.2. At worst, the error in the estimated value of the spin parameter is 0.1 for a nonspinning black hole; the error is much less for a rapidly spinning hole. We also consider the density and disk thickness contrast between the gas in the disk and that inside the ISCO. The contrast needs to be large if black hole spin is to be successfully estimated by fitting the relativistically broadened X-ray line profile of fluorescent iron emission from reflection off an accretion disk. In our hydrodynamic models, the contrast in density and thickness is low when H/R greater than or similar to 0.1, suggesting that the iron line technique may be most reliable in extremely thin disks. We caution that these results have been obtained with a viscous hydrodynamic model. While our results are likely to be qualitatively correct, quantitative estimates of, e.g., the magnitude of the error in the spin estimate, need to be confirmed with MHD simulations of radiatively cooled thin disks.