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

We aim to develop a new method to infer the sub-beam probability density function (PDF) of H2 column densities and the dense gas mass within molecular clouds using spatially unresolved observations of molecular emission lines in the 3 mm band. We model 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. The column density PDF is assumed to be parametric, composed of a lognormal distribution at moderate column densities and a power law distribution at higher column densities. To estimate the sub-beam column density PDF, the emission model is combined with a Bayesian inversion algorithm (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, reproducing the observed emission line integrated intensities. We apply the method to 12CO(J=1-0), 13CO(J=1-0), C18O(J=1-0), HCN(J=1-0), HCO+(J=1-0) and N2H+(J=1-0) observations of a 700 x 700 pc2 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 lognormal, consistent with supersonic isothermal turbulence. We calculate the mass associated with the powerlaw tail of the column density PDFs and observe a strong, linear correlation between this mass and the 24$\mu$m surface brightness.