Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 12 January 2011 (MN L A T E X style file v2.2) On the modelling of accretion onto SMBHs Alexander Hobbs * , Chris Power, Sergei Nayakshin & Andrew R. King Dept. of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK 12 January 2011 ABSTRACT Accretion onto supermassive black holes (SMBHs) embedded in the potential of a host galaxy is frequently modelled by the Bondi-Hoyle formalism. Here we examine the validity of this approach analytically and numerically. We argue that only for hot virialised gas with zero angular momentum is the Bondi-Hoyle solution adequate. In the opposite extreme, when cooling is efficient, the gas is in a state of free-fall long before it reaches the black hole. In this paper we demonstrate that in this regime the Bondi-Hoyle formalism can be erroneous by orders of magnitude in either direction. It may also impose unphysical trends in the results by being wrong by different factors for different halo masses. We propose that future numerical simulations of SMBH growth should either modify the prescription to include the free-fall regime or use the “accretion disc particle” method advocated by Power et al. (2010). Key words: 1 INTRODUCTION Over the last decade, compelling observational evidence has revealed that many galaxies in the local Universe harbour supermassive black holes (SMBHs) with masses 10 6 < ∼ M bh /M⊙ < ∼ 10 9 in their centres. During the same pe- riod, surveys of the distant Universe uncovered the existence of quasars at z ∼ 6, when the Universe < ∼ 1/10 th of its cur- rent age; this implies that many SMBHs had already assem- bled their mass by this time (cf. Fan et al., 2006). Our understanding of the physics that dictates the growth of SMBHs is incomplete. Black holes grow by accret- ing low angular momentum material from their surround- ings, yet the character of the accretion flow onto an SMBH is governed by physical processes as diverse as galaxy merg- ers (e.g., Hopkins & Quataert, 2009), turbulence induced by stellar feedback (cf. Hobbs et al., 2010) and black hole accretion-driven outflows (e.g., Nayakshin & Power, 2010). Black hole growth is routinely modelled in galaxy for- mation simulations (cf. Springel et al., 2005) and the impor- tance of SMBHs in shaping the properties of galaxies is now well established (cf. Croton et al., 2006; Bower et al., 2006). The majority of galaxy formation simulations published in the literature incorporate what we shall term the “Bondi- Hoyle model” for black hole growth (cf. Springel et al., 2005), which derives from the work of Bondi & Hoyle (1944) and Bondi (1952) - hereafter B&H. This model assumes the sim- plest possible accretion flow, where the gas is at rest at in- finity and accretes steadily onto a black hole, subject only ⋆ E-mail: alexander.hobbs@astro.le.ac.uk to the (Newtonian) gravity of the latter, which is modelled as a point mass. Simulations that model the idealised physical problem as it is set out in B&H produce results that are in good agreement with the analytical solution (Ruffert, 1994). In galaxy formation simulations, however, gas is expected to have non-zero angular momentum that provides a natural barrier to accretion (Power et al., 2010). Gas settles into a disc whose dimensions are set by the angular momentum of the accretion flow, and only the material with the very low- est specific angular momentum can accrete, as the viscous timescale for material to be transported through the disc may already be comparable to the age of the Universe at R ∼ 1 pc (King, 2010). In this short paper we suspend, for the moment, our disbelief that gaseous infall can proceed entirely radially from large scales. We consider spherically-symmetric accre- tion flows (i.e., zero angular momentum) onto an SMBH embedded in the potential of a massive dark matter halo. This is an example of a situation where we might expect the Bondi (1952) formula to provide a reasonable estimate of the accretion rate onto the SMBH. We argue that the Bondi (1952) formula, designed for accretion onto “naked black holes”, can only be applied to accretion onto astrophysical SMBHs, i.e., those embedded in massive dark matter halos, if the gas in the latter is at or near hydrostatic equilibrium. It is only in this case that one of the key assumptions of Bondi (1952) – the “gas being at rest at infinity” – is satisfied (infinity meaning outside the Bondi radius; see below). This is not what is expected in gas-rich epochs when c  0000 RAS