Trace the Self-Gravitating Gas Using CO Isotopologues

Recent studies have shown that the star formation rate (SFR) correlates tightly and linearly with the mass of gravitationally bound gas, which can be delineated from the power-law tail of the column-density probability distribution function ($N$-PDF) derived from dust emission observations. This relationship holds across four orders of magnitude within the Milky Way–spanning low-mass to high-mass star-forming regions and encompassing the extreme environment of the Central Molecular Zone. Building on this framework, we present a new approach for estimating the mass of gravitationally bound gas in molecular clouds using multi-line CO isotopologue observations. Our sample includes 16 molecular clouds with robust detections in ${}^{12}$CO, ${}^{13}$CO, and C${}^{18}$O $J$ = 1-0, spanning both massive inner Galaxy clouds and nearby star-forming regions. We find that the $N$-PDFs derived from combined CO isotopologue data recover the characteristic log-normal plus power-law profiles seen in dust-based studies. The mass and spatial distribution of the self-gravitating structures estimated from both dust-based and CO-based methods agree well throughout the sample. This indicates that the CO isotopologue combination can robustly trace the self-gravitating component via the $N$-PDF method and provides a reliable, scalable, and velocity-resolved alternative to dust emission for identifying the star-forming gas in molecular clouds.