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

A maximum stellar surface density Σ_max∼ 3 × 10⁵M_⊙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.