The relationship between black hole accretion rate and gas properties at the Bondi radius

The mass accretion rate determines the black hole accretion mode and the corresponding efficiency of active galactic nuclei (AGNs) feedback. In large-scale simulations studying galaxy formation and evolution, the Bondi radius can be at most marginally resolved. In these simulations, the Bondi accretion formula is always used to estimate the black hole accretion rate. The Bondi solution can not represent the real accretion process. We perform 77 simulations with varying density and temperature at Bondi radius. We find a formula to calculate the black hole accretion rate based on gas density and temperature at Bondi radius. We find that the formula can accurately predict the luminosity of observed low-luminosity AGNs. This formula can be used in sub-grid models in large-scale simulations with AGNs feedback.