he inflow rate into the sink cell, according to the model described here. We first describe the structure of the BH accretion disc in Section \[sec:acc\_disc\], which motivates our subgrid model. Then we give the expressions for the luminosity and directional dependence in Sections \[sec:efficiency\] and \[sec:direction\_dep\], respectively.

### BH accretion disc with shadowing effect {#sec:acc_disc}

Fig. \[fig:acc\_whole\](a) shows the expected inner structure including a BH accretion disc that motivates our subgrid model. Below we explain the inner and outer parts of the structure presented in Fig. \[fig:acc\_whole\](a) in this order. We also describe resulting directional dependences of the BH irradiation.

In the inner part, we see that the circum-BH disc consists of the two different types of accretion discs. One is the innermost geometrically thick (aspect ratio $\sim 1$) slim disc appearing inside the photon trapping radius $r_{\mathrm{tr}} \equiv (\dot{M}/\dot{M}_{\mathrm{E}})\,r_{\mathrm{Sch}}$, where the cooling via radial advection balances with the viscous heating [e.g., @Begelman:1978aa; @Abramowicz:1988aa]. The other is the geometrically thin (aspect ratio $\ll 1$) standard accretion disc appearing outside $r_{\mathrm{tr}}$, where the radiative loss from the disc surfaces is the main cooling process [e.g., @Shakura:1973aa]. When $\dot{M}/\dot{M}_{\mathrm{E}} < 1$, the slim disc disappears and the standard accretion disc extends all the way to the inner disk edge. We model the luminosity based on this consideration in Section \[sec:efficiency\]. Since the surface temperature of the disc increases with decreasing the radius $r$, ionizing photons mostly come from the hot innermost part.

In the outer part, a disc wind might be launched from the disc surface photo-heated by the high-energy photons from the inner region [see @Proga:2000aa; @Proga:2004aa; @Nomura:2013aa; @Nomura:2016aa for line-driven disc wind of AGNs]. In addition, around the outer edge of the disc, the vertically falling flow du