The large columns of dusty gas enshrouding and fuelling star-formation in young, massive stellar clusters may render such systems optically thick to radiation well into the infrared. This raises the prospect that both direct radiation pressure produced by absorption of photons leaving stellar surfaces and indirect radiation pressure from photons absorbed and then re-emitted by dust grains may be important sources of feedback in such systems. Here we evaluate this possibility by deriving the conditions under which a spheroidal, self-gravitating, mixed gas-star cloud can avoid catastrophic disruption by the combined effects of direct and indirect radiation pressure. We show that radiation pressure sets a maximum star cluster formation efficiency of $epsilon_{rm max} sim 0.9$ at a (very large) gas surface density of $sim 10^5 M_odot$ pc$^{-2} (Z_odot/Z) simeq 20$ g cm$^{-2} (Z_odot/Z)$, but that gas clouds above this limit undergo significant radiation-driven expansion during star formation, leading to a maximum stellar surface density very near this value for all star clusters. Data on the central surface mass density of compact stellar systems, while sparse and partly confused by dynamical effects, are broadly consistent with the existence of a metallicity-dependent upper-limit comparable to this value. Our results imply that this limit may preclude the formation of the progenitors of intermediate-mass black holes for systems with $Z gtrsim 0.2 Z_odot$.
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