AGN jet power and feedback characterised by Bondi accretion in brightest cluster galaxies

(Abridged) We propose a new method to estimate the Bondi (hot gas) accretion rates, \dot M_B, onto the supermassive black holes (SMBHs) at the centres of elliptical galaxies. It can be applied even if the Bondi radius is not well-resolved in X-ray observations. This method is based on two simple assumptions: (1) hot gas outside the Bondi radius is in nearly a hydrostatic equilibrium, and (2) the gas temperature near the galaxy centre is close to the virial temperature. We apply this method to 28 bright elliptical galaxies in nearby galaxy clusters (27 of them are the brightest cluster galaxies; BCGs). We find a correlation between the Bondi accretion rates and the power of jets associated with the SMBHs over four orders of magnitude in \dot M_B. For most galaxies, the accretion rates are large enough to account for the jet powers. Our results indicate that the feedback from the active galactic nuclei (AGN) correlates with the properties of the hot gas surrounding the SMBHs. We also find that more massive SMBHs in BCGs tend to have larger specific growth rates. This may explain the hyper masses (~ 10^10 M_sun) of some of the SMBHs. Comparison between the accretion rates and the X-ray luminosities of the AGN suggests that the AGN in the BCGs are extremely radiatively inefficient compared with X-ray binaries in the Milky Way, even when their Eddington accretion ratio, \dot M_B/\dot M_Edd, exceeds 0.01. Lastly, we find a tight correlation between the Bondi accretion rates and the X-ray luminosities of cool cores. Their relation is linear and the power generated by the Bondi accretion is large enough to compensate the radiative cooling of the cool cores. Although the `classical' Bondi accretion model is a greatly oversimplified one, the correlations we find here demonstrate that the accretion onto the SMBHs reflects broadly the properties of the Bondi accretion.