Pitfalls when observationally characterizing the relative formation rates of stars and stellar clusters in galaxies

Stars generally form in aggregates, some of which are bound (clusters) while others are unbound and disperse on short ($sim10$ Myr) timescales (associations). The fraction of stars forming in bound clusters ($Gamma$) is a fundamental outcome of the star formation process. Recent observational and theoretical work has suggested that $Gamma$ increases with the gas surface density ($Sigma$) or star formation rate (SFR) surface density ($Sigma_{rm SFR}$), both within galaxies and between different ones. However, a recent paper by Chandar et al. has challenged these results, showing that the $total$ number of stellar aggregates per unit SFR does not vary systematically with the host galaxys absolute SFR. In this Letter, we show that no variations are expected when no distinction is made between bound and unbound aggregates, because the sum of these two fractions should be close to unity. We also demonstrate that any scaling of $Gamma$ with the absolute SFR is much weaker than with $Sigma_{rm SFR}$, due to the mass-radius-SFR relation of star-forming main sequence galaxies. The environmental variation of $Gamma$ should therefore be probed as a function of area-normalised quantities, such as $Sigma$ or $Sigma_{rm SFR}$. We present a set of guidelines for meaningful observational tests of cluster formation theories and show that these resolve the reported discrepancy.