An alternative approach to the relation between accretion luminosity and mass accretion rate

Abstract In this paper I introduce and discuss an alternative approach to the relation between accretion luminosity, Lacc, and mass accretion rate, ˙M : instead of the universally adopted Lacc = GM ˙M/R, I propose the dynamical definition Lacc = v2f˙M/2 where vf is the final velocity of the infalling matter at the surface of the accreting object of mass M and radius R. Both definitions are based on the energy conservation, but while the former assumes that matter is in free fall, the latter is valid always. By adopting the alternative form for Lacc, I show that the Eddington luminosity Led, when the outward radiation pressure wins on gravity, is never produced with a finite ˙M. Instead, Led is a limit asymptotically reached when ˙M → ¥. My argument is very simple, so I felt the need to find a possible explanation to why no one arrived to this conclusion before. To this aim, I give a brief presentation of the history of accretion, from the pioneer work of Hoyle and collaborators until the ’60s of last century, to show how the perception of the role of the radiation pressure in accretion evolved. I give also some practical applications of the formulae I derived, in the case of high-mass star formation and of the growth of super massive black holes. The study of these two processes, already complex per se, becomes more difficult to solve because of the existence of a limiting ˙M, named Eddington mass accretion rate or ˙Med, that it is supposed to generate a luminosity equal to Led, making it impossible to accrete at rate ˙M > ˙Med. Accretion rates higher than ˙Med are however necessary, as theory and observations show. My definition of Lacc takes naturally into account the work done by radiation pressure to slow down the infalling matter: as a consequence, Lacc does not increase linearly with ˙M and Led is only an asymptotic value.