The maximum stellar surface density due to the failure of stellar feedback

A maximum stellar surface density Σ _max∼ 3 × 10^5{M_\odot pc^{-2}} is observed across all classes of dense stellar systems (e.g. star clusters, galactic nuclei, etc.), spanning ∼8 orders of magnitude in mass. It has been proposed that this characteristic scale is set by some dynamical feedback mechanism preventing collapse beyond a certain surface density. However, simple analytic models and detailed simulations of star formation moderated by feedback from massive stars argue that feedback becomes less efficient at higher surface densities (with the star formation efficiency increasing as ∼Σ/Σ_crit). We therefore propose an alternative model wherein stellar feedback becomes ineffective at moderating star formation above some Σ_crit, so the supply of star-forming gas is rapidly converted to stars before the system can contract to higher surface density. We show that such a model - with Σ_crit taken directly from the theory - naturally predicts the observed Σ_max. We find Σ_max ∼ 100Σ_crit because the gas consumption time is longer than the global free-fall time even when feedback is ineffective. Moreover, the predicted Σ_max is robust to spatial scale and metallicity, and is preserved even if multiple episodes of star formation/gas inflow occur. In this context, the observed Σ_max directly tells us where feedback fails.