The ${\sigma }_{\mathrm{H}\beta }$-based dimensionless accretion rate and its connection with the corona for AGN

Abstract

With respect to the $\mathrm{H}\beta$ full width at half-maximum ($\mathrm{FWH}{\mathrm{M}}_{\mathrm{H}\beta }$), the broad $\mathrm{H}\beta$ line dispersion (${\sigma }_{\mathrm{H}\beta }$) was preferred as a velocity tracer to calculate the single-epoch supermassive black hole mass ($M_{\mathrm{BH}}$) suggested by \cite{Yu2020b}. For a compiled sample of 311 broad-line active galactic nuclei (AGN) with measured hard X-ray photon index ($z<0.7$), ${\sigma }_{\mathrm{H}\beta }$ and the optical Fe II relative strength ($R_{\mathrm{Fe}}$) are measured from their optical spectra, which are used to calculate ${\sigma }_{\mathrm{H}\beta }$-based virial $M_{\mathrm{BH}}$ and dimensionless accretion rate ($\dot{\mathcal{M}}$). With respect to $\mathrm{FWH}{\mathrm{M}}_{\mathrm{H}\beta }$, it is found that the mean value of ${\sigma }_{\mathrm{H}\beta }$-based $M_{\mathrm{BH}}$ is on average larger by 0.26 dex, and the mean value of ${\sigma }_{\mathrm{H}\beta }$-based $\dot{\mathcal{M}}$ is on average smaller by 0.51 dex. It is found that there exists a non-linear relationship between the Eddington ratio ($L_{\mathrm{Bol}}/L_{\mathrm{Edd}}$) and $\dot{\mathcal{M}}$, i.e., $L_{\mathrm{Bol}}/L_{\mathrm{Edd}}\propto {\dot{\mathcal{M}}}^{0.56\pm 0.01}$. This non-linear relationship comes from the accretion efficiency $\eta$, which is smaller for AGN with higher $\dot{\mathcal{M}}$. We find a strong bivariate correlation of the fraction of energy released in the corona $F_{\mathrm{X}}$ with $\dot{\mathcal{M}}$ and \mbh, $F_{\mathrm{X}}\propto {\dot{\mathcal{M}}}^{-0.57\pm 0.05}{M}_{\mathrm{BH}}^{-0.54\pm 0.06}$. The flat slope of $-0.57\pm 0.05$ favours the shear stress tensor of the accretion disk being proportional to the geometric mean of gas pressure and total pressure. We find a strong bivariate relation of $\Gamma$ with $\dot{\mathcal{M}}$ and $F_{\mathrm{X}}$, $\Gamma \propto {\dot{\mathcal{M}}}^{-0.21\pm 0.02}{F}_{\mathrm{X}}^{0.02\pm 0.04}$. The hard X-ray spectrum becomes softer with increasing of $F_{\mathrm{X}}$, although the scatter is large.

18 pages, 9 figures, ApJ, in press

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