A systematic study of the condensation of the corona and the application for $\Gamma_{\rm 2-10keV}-L_{\rm bol}/L_{\rm Edd}$ correlation in luminous active galactic nuclei

In this paper, we explained the observed $\Gamma_{\rm 2-10keV}-L_{\rm bol}/L_{\rm Edd}$ correlation in luminous active galactic nuclei within the framework of the condensation of the corona around a supermassive black hole (Liu et al. 2015; Qiao & Liu 2017). Specifically, we systemically test the effects of black hole mass $M$, the viscosity parameter $\alpha$, and the magnetic parameter $\beta$ (with magnetic pressure $p_{\rm m}=B^2/{8\pi}=(1-\beta)p_{\rm tot}$, $p_{\rm tot}=p_{\rm gas}+p_{\rm m}$) on the structure of the accretion disc and the corona, as well as the corresponding emergent spectra. It is found that the hard X-ray photon index $\Gamma_{\rm 2-10keV}$ nearly does not change with changing black hole mass $M$, or changing magnetic parameter $\beta$. Meanwhile, it is found that the geometry of the accretion flow, i.e., the relative configuration of the disc and corona, as well as the emergent spectra can be strongly affected by changing the value of $\alpha$. By comparing with a sample composed of 29 luminous active galactic nuclei with well constrained X-ray spectra and Eddington ratios, it is found that the observed $\Gamma_{\rm 2-10keV}-L_{\rm bol}/L_{\rm Edd}$ correlation can be well matched with a relatively bigger value of $\alpha$, i.e., $\alpha \sim 1$, as previously also suggested by Narayan (1996) for luminous accreting black holes.