e to the centrifugal barrier might collide with the one coming from the opposite side of the equatorial plane and form a shocked region. Fig. \[fig:acc\_whole\](a) depicts these structures, both of which can absorb (or scatter) the ionizing photons coming from the inner part, forming a shadowed region behind them. We assume that the shadowing effect considered here is caused by the outer disc structures, and thus the appearance of a slim disc is not essential in forming the shadowed region. The outer structure of the disc is highly uncertain and probably varies depending on the BH mass, accretion rate, angular momentum, metallicity of inflowing gas, etc.. In Section \[sec:direction\_dep\], we model the shadowing effect with a simple parametric fashion.

### Luminosity {#sec:efficiency}

In our simulation, we determine the luminosity of the BH radiation $L$ depending on $\dot{M}$ evaluated at the inner boundary at each time step. To model the luminosity, we adopt the fitting formula [@Watarai:2000aa], $$\begin{aligned}
 L = 
\begin{cases}
{\displaystyle}2\, L_{\mathrm{E}}\,\left[1+\ln\left(\frac{\dot{m}}{20}\right)\right] & \dot{m} > 20\\
{\displaystyle}0.1\, L_{\mathrm{E}}\, \dot{m}  & \dot{m} < 20
\end{cases}\,,
\label{eq:3}\end{aligned}$$ where $\dot{m}\equiv \dot{M}/\dot{M}_{\mathrm{E}}$. [@Watarai:2000aa] obtained this formula by fitting the dependence of the luminosity on the accretion rate in the 1D stationary disc model, taking into account the (dis-)appearance of the slim disc depending on $\dot{M}$ . When $\dot{M}$ is low ($\dot{m} < 20$), the radiative efficiency is fixed at 10%, which agrees with that of the standard disc. For rapid accretion with $\dot{m} > 20$, the second term $2 L_{\mathrm{E}}\, \ln(\dot{m}/20)$ represents the luminosity from the innermost slim disc, where the photon advection reduces the radiative efficiency. Note that the luminosity $L$ increases logarithmically with $\dot{M}$ and can even exceed $L_{\mathrm{E}}$ because a large fraction of the emitted photons escape from the di