  The importance of radiation pressure feedback in galaxy formation has been
extensively debated over the last decade. The regime of greatest uncertainty is
in the most actively star-forming galaxies, where large dust columns can
potentially produce a dust-reprocessed infrared radiation field with enough
pressure to drive turbulence or eject material. Here we derive the conditions
under which a self-gravitating, mixed gas-star disc can remain hydrostatic
despite trapped radiation pressure. Consistently taking into account the
self-gravity of the medium, the star- and dust-to-gas ratios, and the effects
of turbulent motions not driven by radiation, we show that galaxies can achieve
a maximum Eddington-limited star formation rate per unit area
$\dot{\Sigma}_{\rm *,crit} \sim 10^3 M_{\odot}$ pc$^{-2}$ Myr$^{-1}$,
corresponding to a critical flux of $F_{\rm *,crit} \sim 10^{13} L_{\odot}$
kpc$^{-2}$ similar to previous estimates; higher fluxes eject mass in bulk,
halting further star formation. Conversely, we show that in galaxies below this
limit, our one-dimensional models imply simple vertical hydrostatic equilibrium
and that radiation pressure is ineffective at driving turbulence or ejecting
matter. Because the vast majority of star-forming galaxies lie below the
maximum limit for typical dust-to-gas ratios, we conclude that infrared
radiation pressure is likely unimportant for all but the most extreme systems
on galaxy-wide scales. Thus, while radiation pressure does not explain the
Kennicutt-Schmidt relation, it does impose an upper truncation on it. Our
predicted truncation is in good agreement with the highest observed gas and
star formation rate surface densities found both locally and at high redshift.
