  Accretion onto central massive black holes in galaxies is often modelled with
the Bondi solution. In this paper we study a generalization of the classical
Bondi accretion theory, considering the additional effects of the gravitational
potential of the host galaxy, and of electron scattering in the optically thin
limit. We provide a general analysis of the bias in the estimates of the Bondi
radius and mass accretion rate, when adopting as fiducial values for the
density and temperature at infinity the values of these quantities measured at
finite distance from the central black hole. We also give general formulae to
compute the correction terms of the critical accretion parameter in relevant
asymptotic regimes. A full analytical discussion is presented in the case of an
Hernquist galaxy, when the problem reduces to the discussion of a cubic
equation, therefore allowing for more than one critical point in the accretion
structure. The results are useful for observational works (especially in the
case of low-luminosity systems), as well as for numerical simulations, where
accretion rates are usually defined in terms of the gas properties near the
