Timescales of Quasar Accretion Discs from Low to High Black Hole Masses and a Turnover at the High Mass End
Characteristic time scales in the stochastic UV-optical variability of quasars may depend on the mass of their black holes, $M_{\rm BH}$, as much as physical timescales in their accretion discs do. We calculate emission-weighted mean radii, $R_{\rm mean}$, and orbital timescales, $t_{\rm mean}$, of standard thin disc models for emission wavelengths $\lambda$ from 1000 to 10000 AA, $M_{\rm BH}$ from $10^6$ to $10^{11}$ solar masses, and Eddington ratios from 0.01 to 1. At low $M_{\rm BH}$, we find the textbook behaviour of $t_{\rm mean}\propto M_{\rm BH}^{-1/2}$ alongside $R_{\rm mean} \approx \mathrm{const}$, but toward higher masses the growing event horizon imposes $R_{\rm mean} \propto M_{\rm BH}$ and thus a turnover into $t_{\rm mean}\propto M_{\rm BH}$. For quasars of $\log L_{\rm bol}=47$, the turnover mass, where $t_{\rm mean}$ starts rising is $M_{\rm BH}\approx 9.5$, which means that the turnover in $t_{\rm mean}$ is well within the range of high-luminosity quasar samples, whose variability time scales might thus show little mass dependence. We fit smoothly broken power laws to the results and provide analytic convenience functions for $R_{\rm mean}(\lambda,M_{\rm BH},L_{3000})$ and $t_{\rm mean}(\lambda,M_{\rm BH},L_{3000})$.