Estimating the dense gas mass of molecular clouds using spatially unresolved 3,mm line observations

Emission lines such as are commonly used by extragalactic studies to trace high density molecular gas (nH₂ >,∼10^ cm^-3). Recent Milky Way studies have challenged their utility as unambiguous dense gas tracers, suggesting that a large fraction of their emission in nearby clouds is excited in low density gas. We aim to develop a new method to infer the sub-beam probability density function (PDF) of H_2 column densities and the dense gas mass within molecular clouds using spatially unresolved observations of molecular emission lines in the 3,mm band. We modelled spatially unresolved line integrated intensity measurements as the average of an emission function weighted by the sub-beam column density PDF. The emission function, which expresses the line integrated intensity as a function of the gas column density is an empirical fit to high resolution ($<0.05$,pc) multi-line observations of the Orion B molecular cloud. We assumed the column density PDF to be parametric, composed of a log-normal distribution at moderate column densities and a power-law distribution at higher column densities. To estimate the sub-beam column density PDF, we combined the emission model with a Bayesian inversion algorithm (implemented in the Beetroots code), which takes account of thermal noise and calibration errors. We validate our method by demonstrating that it recovers the true column density PDF of the Orion B cloud and reproduces the observed emission line integrated intensities within noise and calibration uncertainties. We applied the method to and observations of a $700 pc^2 field of view (FoV) in the nearby galaxy M51. On average, the model reproduces the observed intensities within 30%. The column density PDFs obtained for the spiral arm region within our test FoV are dominated by a power-law tail at high column densities, with slopes that are consistent with gravitational collapse. Outside the spiral arm, the column density PDFs are predominantly log-normal, consistent with supersonic isothermal turbulence setting the dynamical state of the molecular gas. We calculated the mass associated with the power-law tail of the column density PDFs and observe a strong, linear correlation between this mass and the 24μm surface brightness. Our method is a promising approach to infer the physical conditions within extragalactic molecular clouds using spectral line observations that are feasible with current millimetre facilities. Future work will extend the method to include additional physical parameters that are relevant for the dynamical state and star formation activity of molecular clouds.