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

TL;DR

This paper develops a beam-averaged framework to infer the sub-beam $N_{H2}$ distribution in molecular clouds using unresolved 3 mm line observations. It combines a parametric LN+PL $N$-PDF with an empirically calibrated emission function derived from high-resolution Orion B data and retrieves the parameters via Bayesian inversion (Beetroots), validated against Orion B and applied to a 700×700 pc region in M51. The approach reproduces line intensities within ~30% and reveals a gravity-dominated PL tail in spiral-arm regions, while inter-arm regions are LN-dominated; the dense-gas mass correlates linearly with 24 μm emission in gravity-dominated zones, supporting a link between dense gas and star formation. This method offers a path to characterize extragalactic molecular gas conditions from current millimeter surveys, with planned extensions to include additional physical parameters and higher-J lines to improve universality and accuracy.

Abstract

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$μ$m surface brightness.

Figures (17)

• Figure 1: Binned trends of line integrated intensity as a function of column density. The data is binned in 30 equally sized bins of column density. Black circles correspond to the bin average, while the grey shading indicates the standard deviation in each bin. The black solid line is a smoothly varying double PL fit to the trends, specific to each emission line. The red dashed line shows for comparison the empirical fit to the Perseus cloud by Tafalla2021, assuming a kinetic temperature of 11 K. Each panel shows a different emission line: $^{12}$CO($J$=1$\rightarrow$0), $^{13}$CO($J$=1$\rightarrow$0) and HCO$^+$($J$=1$\rightarrow$0) (left to right, top row); C$^{18}$O($J$=1$\rightarrow$0), HCN($J$=1$\rightarrow$0) and $^{12}$CS($J$=2$\rightarrow$1) (middle); HNC($J$=1$\rightarrow$0) SO($J_K$=3$_2$$\rightarrow$2$_1$) and N$_2$H$^+$($J$=1$\rightarrow$0) (bottom). The standard Milky Way CO-to-H$_2$ conversion factor and its typical uncertainty Bolatto2013 is indicated in the top left panel. An HCN($J$=1$\rightarrow$0) dense gas conversion factor of 60 M$_\odot$ (K km s$^{-1}$)$^{-1}$ is indicated in the central panel as the green curve.
• Figure 2: A comparison of the reference and estimated $N$-PDFs when inverting the $N$-PDF on the spatially and spectrally averaged ORION-B data. The thick red line indicates the $N$-PDF as a histogram constructed directly from the dust-derived Orion B column densities, and the green line represents a $\chi^2$ fit to the red histogram. The estimated $N$-PDFs from the 10 000 MCMC iterations to sample the Bayesian posterior are shown with blue circles. The dashed orange line is the MAP estimation for the $N$-PDF. The vertical dotted black line indicates the limit below which the line intensities predicted by the emission function fall below the typical noise level of the data, that is 0.1 K km s$^{-1}$.
• Figure 3: Two-dimensional projections of the posterior PDF in the form of a scatter plot matrix. The matrix's diagonal shows the posterior PDF of each estimated parameter. The MAP estimation is represented as a vertical cyan line on the histograms and as a cyan square in the scatter plot. The true $N$-PDF parameters obtained by fitting the dust derived Orion B $N$-PDF is shown are shown as red crosses. The black dashed line show the range in PL index $\alpha$ of the $N$-PDF expected for gravitational collapse. The estimations closely match the reference values, although clear degeneracies are present in the posterior PDF.
• Figure 4: MAP estimations of the sub-beam $N$-PDF parameters across our M51 test region. Top: from left to right the panels show the mean column density of the LN part of $N$-PDF ($N_0$), the pixel area filling factor ($\eta$), and the average gas density (including blank sky contributions). Bottom: from left to right are displayed the width of the log-normal ($\sigma$), the column density of transition between the log-normal and power-law parts of the $N$-PDF ($N_{\rm{thresh}}$) and the power-law index ($\alpha$). Red contours in each panel indicate $^{13}$CO($J$=1$\rightarrow$0) integrated intensities of 4 and 12 K km s$^{-1}$ (dashed and solid contours, respectively). The white arrow indicates the direction to the galactic centre. To first order, the gas is denser and more gravitationally unstable inside the spiral arm than outside the arm.
• Figure 5: The spatial distribution of the mass of dense gas (left), the gas mass in the power-law part of the $N$-PDF (middle), and the 24$\,\mu$m surface brightness in our M51 target region. We use the 24$\,\mu$m emission as a proxy for star formation. The masses are derived from the MAP estimate of the $N$-PDF, using equations \ref{['eq:sigma_dense']} and \ref{['eq:sigma_pl']}. The red contours are the samed as in Figure \ref{['fig:m51:param-map']}. The masses of dense and PL gas appear highly correlated, with a similar spatial distribution as the 24$\mu$m emission.