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One of the most active fields of research of modern-day astrophysics is thatof massive black hole formation and co-evolution with the host galaxy. In theseinvestigations, ranging from cosmological simulations, to semi-analyticalmodeling, to observational studies, the Bondi solution for accretion on acentral point mass is widely adopted. In this work we generalize the classicalBondi accretion theory to take into account the effects of the gravitationalpotential of the host galaxy, and of radiation pressure in the optically thinlimit. Then, we present the fully analytical solution, in terms of theLambert-Euler $W$-function, for isothermal accretion in Jaffe and Hernquistgalaxies with a central black hole. The flow structure is found to be sensitiveto the shape of the mass profile of the host galaxy. These results and theformulae that are provided, mostly important the one for the critical accretionparameter, allow for a direct evaluation of all flow properties, and are thenuseful for the above mentioned studies. As an application, we examine thedeparture from the true mass accretion rate of estimates obtained using the gasproperties at various distances from the black hole, under the hypothesis ofclassical Bondi accretion. An overestimate is obtained from regions close tothe black hole, and an underestimate outside a few Bondi radii; the exactposition of the transition between the two kinds of departure depends on thegalaxy model.

Isothermal Bondi accretion in Jaffe and Hernquist galaxies with a central black hole: fully analytical solutions