What is the real accretion rate onto a black hole for low angular momentum accretion?

Mass accretion rate is a key parameter in accretion disk theory. It determines black hole accretion mode. In large scale cosmological simulations studying galaxy formation and evolution, Bondi radius can at most be marginally resolved. In those simulations, Bondi accretion formula is always used to estimate black hole accretion rate. Bondi solution is too simple, which cannot represent the real accretion process. We perform simulations of hot accretion flow at parsec scale irradiated by a low luminosity active galactic nucleus (AGN). We perform 77 simulations with varying density and temperature at outer boundary (10 parsec). Our purpose is to find a formula to calculate real black hole accretion rate based on the gas density and temperature at parsec scale. We define Eddington accretion rate to be $\dot M_{\rm Edd}=10L_{\rm Edd}/c^2$, with $L_{\rm Edd}$ and $c$ been Eddington luminosity and speed of light respectively. We set black hole mass to be $10^8M_{\odot}$, $M_{\odot}$ is solar mass. We find that black hole accretion rate can be expressed as $\frac{\dot M_{\rm BH} }{\dot M_{\rm Edd}}=10^{-3.11} (\rho_0/10^{-22}{\rm g cm}^{-3})^{1.36} (T_0/10^7 {\rm K})^{-1.9}$, with $\rho_0$ and $T_0$ being density and temperature at parsec scale, respectively. We find the formula can accurately predict the luminosity of observed low-luminosity AGNs (with black hole mass $\sim 10^8M_{\odot}$). This formula can be used in the sub-grid models in large scale cosmological simulations with a black hole mass of $\sim 10^8M_{\odot}$.