Why is the Star Formation Rate Proportional to Dense Gas Mass?

One of the most profound empirical laws of star formation is the Gao-Solomon relation, a linear correlation between the star formation rate (SFR) and the dense molecular gas mass. It is puzzling how the complicated physics in star-formation results in this surprisingly simple proportionality. Using archival Herschel and Atacama Large Millimeter/submillimeter Array Observations, we derived the masses of the most massive cores ($M^{\rm max}_{\rm core}$) and masses of the gravitationally bound gas ($M_{\rm gas}^{\rm bound}$) in the parent molecular clouds for a sample of low-mass and high-mass star-forming regions. We discovered a significant correlation $\log(M^{\rm max}_{\rm core}/M_{\odot}) = 0.506 \log(M_{\rm gas}^{\rm bound}/M_{\odot})-0.32$. Our discovered $M^{\rm max}_{\rm core}$-$M_{\rm gas}^{\rm bound}$ correlation can be approximately converted to the Gao-Solomon relation if there is (1) a constant 30% efficiency of converting $M^{\rm max}_{\rm core}$ to the mass of the most massive star ($m^{\rm max}_{\rm star}$), and (2) if SFR and $m^{\rm max}_{\rm star}$ are tightly related through $\log({\rm SFR}/(M_{\odot} {\rm yr}^{-1})) = 2.04 \log(m^{\rm max}_{\rm star}/M_{\odot})-5.80$. Intriguingly, both requirements have been suggested by previous theoretical studies (c.f. Yan et al. 2017). Based on this result, we hypothesize that the Gao-Solomon relation is a consequence of combining the following three non-trivial relations (i) SFR vs. $m^{\rm max}_{\rm star}$, (ii) $m^{\rm max}_{\rm star}$ vs. $M^{\rm max}_{\rm core}$, and (iii) $M^{\rm max}_{\rm core}$ vs. $M_{\rm gas}^{\rm bound}$. This finding may open a new possibility to understand the Gao-Solomon relation in an analytic sense.