The maximum accretion rate of hot gas in dark matter haloes

 We revisit the question of ‘hot mode’ versus ‘cold mode’ accretion on to galaxies using steady-state cooling flow solutions and idealized 3D hydrodynamic simulations. We demonstrate that for the hot accretion mode to exist, the cooling time is required to be longer than the free-fall time near the radius where the gas is rotationally supported, Rcirc, i.e. the existence of the hot mode depends on physical conditions at the galaxy scale rather than on physical conditions at the halo scale. When allowing for the depletion of the halo baryon fraction relative to the cosmic mean, the longer cooling times imply that a virialized gaseous halo may form in halo masses below the threshold of $\sim 10^{12}\, {\rm M_{\odot }}$ derived for baryon-complete haloes. We show that for any halo mass there is a maximum accretion rate for which the gas is virialized throughout the halo and can accrete via the hot mode of ${\dot{M}}_{\rm crit}\approx 0.7(v_{\rm c}/100\, \rm km\ s^{-1})^{5.4}(R_{\rm circ}/10\, {\rm kpc})(Z/\, {\rm Z_{\odot }})^{-0.9}\, {\rm M_{\odot }}\, {\rm yr}^{-1}$, where Z and vc are the metallicity and circular velocity measured at Rcirc. For accretion rates $\gtrsim {\dot{M}}_{\rm crit}$ the volume-filling gas phase can in principle be ‘transonic’ – virialized in the outer halo but cool and free-falling near the galaxy. We compare ${\dot{M}}_{\rm crit}$ to the average star formation rate (SFR) in haloes at 0 < z < 10 implied by the stellar-mass–halo-mass relation. For a plausible metallicity evolution with redshift, we find that ${\rm SFR}\lesssim {\dot{M}}_{\rm crit}$ at most masses and redshifts, suggesting that the SFR of galaxies could be primarily sustained by the hot mode in halo masses well below the classic threshold of $\sim 10^{12}\, {\rm M_{\odot }}$.