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

TL;DR

This work develops a physically grounded, emission-weighted thin-disc framework to quantify how quasar disc sizes and orbital timescales $R_{ m mean}$ and $t_{ m mean}$ depend on black-hole mass $M_{ m BH}$ and monochromatic luminosity proxy $L_{3000}$ across rest-frame wavelengths $\lambda$ in $[1000,10000]$ Å. By including GR corrections and exploring spins $a=+0.78,0,-1$, the authors reveal a smoothly broken power-law dependence on $M_{ m BH}$ with two regimes: at low masses, $t_{ m mean} \propto M_{ m BH}^{-1/2}$ with $R_{ m mean}$ nearly constant, while at high masses, horizon-ISCO effects push emission outward, yielding $R_{ m mean} \propto M_{ m BH}$ and $t_{ m mean} \propto M_{ m BH}$. The turnover mass is around $\log M_{ m BH} \approx 9.5$ for $\log L_{bol} \approx 47$, implying high-luminosity quasars may show little disc-timescale mass dependence, consistent with some variability studies. The paper provides smoothly broken power-law analytic approximations and explicit fits for $R_{ m mean}(\lambda,M_{ m BH},L_{3000})$ and $t_{ m mean}(\lambda,M_{ m BH},L_{3000})$, facilitating direct application to observational data and future variability analyses. These results refine how we interpret UV–optical variability in AGN and aid in the development of disc-based cosmological probes.

Abstract

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 $λ$ 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 $R_{\rm mean}(λ,M_{\rm BH},L_{3000})$ and $t_{\rm mean}(λ,M_{\rm BH},L_{3000})$.

Figures (6)

• Figure 1: Disc temperature profiles derived from Eq. \ref{['2Dfamily']} using $T_{\rm ref}=10^4$ K. For a spin-free black hole of $\log (M_{\rm BH}/M_\odot)=9$, which has $\log (R_{\rm ISCO}/\mathrm{m})=12.947$, $\log (L_{\rm Edd}/(\mathrm{erg~s^{-1}}))=47.10$ and $\dot{M}_{\rm Edd}=22.2M_\odot~\mathrm{yr}^{-1}$, the three curves represent the cases $R_{\rm Edd}=(1,0.1,0.01)$ in order of increasing $r_0$. The given $r_0$ values correspond to $R_{\rm 10^4\mathrm{K}}/R_{\rm S}\simeq (236,107,47)$. While the curves are universal for chosen values of $T_{\rm ref}$ and $r_0$, $R_{\rm Edd} \propto M_{\rm BH}$ at any fixed $r_0$, and thus e.g., for $R_{\rm Edd}=0.1$ the curves represent $\log (M_{\rm BH}/M_\odot)=(8,9,10)$. The arrows mark the mean emission radii $R_{\rm mean}$ for $\log (\lambda/$Å $)=3.5$, which are always $>R_{\rm ISCO}$ but converge to a constant for vanishing $R_{\rm ISCO}$ (or $r_0$).
• Figure 2: Flux-weighted size scale (mean emission radius) $R_{\rm mean}$ ( left) and orbital time scale $t_{\rm mean}$ ( right) at $\log (\lambda /\textup{\AA})=3.5$ for a range of accretion discs (GR approximation with spin $a=0$) using three colour codes: Eddington ratio $R_{\rm Edd}$ ( top), true bolometric luminosity $L_{\rm bol}$ (centre), and monochromatic luminosity $L_{3000}$ ( bottom), which often acts as a proxy for $L_{\rm bol}$. Dashed lines show the power law scaling suggested by the approximation in Eq. \ref{['eq:r_ana_dep']}.
• Figure 3: Left: Flux-weighted size scale $R_{\rm mean}$ vs. black hole mass of four groups of discs with $\log L_{3000}/(\mathrm{erg\,s^{-1}}\,$Å$^{-1})$ values of 40, 41, 42 and 43 ( from low $L_{3000}$ at low $M_{\rm BH}$ on the left to high $L_{3000}$ at high $M_{\rm BH}$ on the right); the black hole masses range for each value of $L_{3000}$ is driven by the Eddington ratio limits of the calculated grid. Right: Flux-weighted orbital time scale $t_{\rm mean}$ vs. black hole mass of the same discs.
• Figure 4: Luminosity-scaled size scales (left) and time scales (right) vs. luminosity-scaled black hole mass at emission wavelength $\log (\lambda /\textup{\AA})=3.5$ for a range of accretion discs (GR approximation with spin $a=0$). For brevity, we use $L_{3000,43} = L_{3000}/(10^{43} \mathrm{erg\,s^{-1}\,}$Å$^{-1})$. A warped 2D surface is seen in projection as a 1D line. Every point in this figure represents a 1-parameter family of discs with different $L_{3000}$ but identical $R_{\rm mean}/\sqrt{L_{3000}}$ and identical $t_{\rm mean}/\sqrt{L_{3000}}$. The variation of points seen in the projected plane is caused by variations in $R_{\rm ISCO}/R_{\rm mean}$.
• Figure 5: Illustration of the analytic approximation to size and time scales in simple thin-disc models. Top: Grid points are displayed as points and the analytic fits are plotted as lines. The colour scale represents the range of Eddington ratios and lines in a single panel are differentiated by wavelength. Bottom: Residuals as a function of wavelength and mass, showing that while the analytic solution is typically within 0.01 dex of the numerical calculation, the deviation can reach 0.05 dex at high masses and low luminosities.