Timescales of Quasar Accretion Discs from Low to High Black Hole Masses and a Turnover at the High Mass End

Abstract

Characteristic time scales in the stochastic UV-optical variability of quasars may depend on the mass of their black holes, $M_{\mathrm{BH}}$, as much as physical timescales in their accretion discs do. We calculate emission-weighted mean radii, $R_{\mathrm{mean}}$, and orbital timescales, $t_{\mathrm{mean}}$, of standard thin disc models for emission wavelengths $\hat{I}»$ from 1000 to 10000 AA, $M_{\mathrm{BH}}$ from $10^6$ to $10^{11}$ solar masses, and Eddington ratios from 0.01 to 1. At low $M_{\mathrm{BH}}$, we find the textbook behaviour of $t_{\mathrm{mean}}\propto M_{\mathrm{BH}}^{-1/2}$ alongside $R_{\mathrm{mean}}\approx \mathrm{const}$, but toward higher masses the growing event horizon imposes $R_{\mathrm{mean}}\propto M_{\mathrm{BH}}$ and thus a turnover into $t_{\mathrm{mean}}\propto M_{\mathrm{BH}}$. For quasars of $\log L_{\mathrm{bol}}=47$, the turnover mass, where $t_{\mathrm{mean}}$ starts rising is $M_{\mathrm{BH}}\approx 9.5$, which means that the turnover in $t_{\mathrm{mean}}$ 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 $R_{\mathrm{mean}}(\hat{I}»,M_{\mathrm{BH}},L_{3000})$ and $t_{\mathrm{mean}}(\hat{I}»,M_{\mathrm{BH}},L_{3000})$.

11 pages, 6 figures, submitted to MNRAS