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, $M_{\mathrm{B}\mathrm{H}}$, as much as physical timescales in their accretion discs do. We calculate emission-weighted mean radii, $R_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$, and orbital timescales, $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$, of standard thin disc models for emission wavelengths $\lambda$ from 1000 to 10000 AA, $M_{\mathrm{B}\mathrm{H}}$ from ${10}^6$ to ${10}^{11}$ solar masses, and Eddington ratios from 0.01 to 1. At low $M_{\mathrm{B}\mathrm{H}}$, we find the textbook behaviour of $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}\propto M_{\mathrm{B}\mathrm{H}}^{-1\mathrm{/}2}$ alongside $R_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}\approx \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, but toward higher masses the growing event horizon imposes $R_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}\propto M_{\mathrm{B}\mathrm{H}}$ and thus a turnover into $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}\propto M_{\mathrm{B}\mathrm{H}}$. For quasars of $\log L_{\mathrm{b}\mathrm{o}\mathrm{l}}=47$, the turnover mass, where $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$ starts rising is $M_{\mathrm{B}\mathrm{H}}\approx 9.5$, which means that the turnover in $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}$ 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 $R_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}(\lambda ,M_{\mathrm{B}\mathrm{H}},L_{3000})$ and $t_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}(\lambda ,M_{\mathrm{B}\mathrm{H}},L_{3000})$.