On the Maximum Luminosity of Galaxies and Their Central Black Holes: Feedback From Momentum-Driven Winds

Abstract

We investigate large-scale galactic winds driven by momentum deposition. Momentum injection is provided by (1) radiation pressure produced by the continuum absorption and scattering of UV photons on dust grains and (2) supernovae. UV radiation can be produced by a starburst or AGN activity. We argue that momentum-driven winds are an efficient mechanism for feedback during the formation of galaxies. We show that above a limiting luminosity, momentum deposition from star formation can expel a significant fraction of the gas in a galaxy. The limiting, Eddington-like luminosity is $L_{\rm M}\simeq(4f_g c/G) σ^4$, where $σ$ is the galaxy velocity dispersion and $f_g$ is the gas fraction. A starburst that attains $L_{\rm M}$ moderates its star formation rate and its luminosity does not increase significantly further. We argue that ellipticals attain this limit during their growth at $z \gtrsim 1$ and that this is the origin of the Faber-Jackson relation. We show that Lyman break galaxies and ultra-luminous infrared galaxies have luminosities near $L_{\rm M}$. Star formation is unlikely to efficiently remove gas from very small scales in galactic nuclei, i.e., scales much smaller than that of a nuclear starburst. This gas is available to fuel a central black hole (BH). We argue that a BH clears gas out of its galactic nucleus when the luminosity of the BH itself reaches $\approx L_{\rm M}$. This shuts off the fuel supply to the BH and may also terminate star formation in the surrounding galaxy. As a result, the BH mass is fixed to be $M_{\rm BH}\simeq (f_g κ_{\rm es}/πG^2)σ^4$, where $κ_{\rm es}$ is the electron scattering opacity. This limit is in accord with the observed $M_{\rm BH}-σ$ relation. (Abridged)