CO isotopologue-derived molecular gas conditions and CO-to-H$_2$ conversion factors in M51

Over the past decade, several millimeter interferometer programs have mapped the nearby star-forming galaxy M51 at a spatial resolution of ${\le}170$ pc. This study combines observations from three major programs: the PdBI Arcsecond Whirlpool Survey (PAWS), the SMA M51 large program (SMA-PAWS), and the Surveying the Whirlpool at Arcseconds with NOEMA (SWAN). The dataset includes the (1-0) and (2-1) rotational transitions of $^{12}$CO, $^{13}$CO, and C$^{18}$O isotopologues. The observations cover the $r{<}\rm 3\,kpc$ region including center and part of the disk, thereby ensuring strong detections of the weaker $^{13}$CO and C$^{18}$O lines. All observations are convolved in this analysis to an angular resolution of 4$''$, corresponding to a physical scale of ${\sim}$170 pc. We investigate empirical line ratio relations and quantitatively evaluate molecular gas conditions such as temperature, density, and the CO-to-H$_2$ conversion factor ($\alpha_{\rm CO}$). We employ two approaches to study the molecular gas conditions: (i) assuming local thermal equilibrium (LTE) to analytically determine the CO column density and $\alpha_{\rm CO}$, and (ii) using non-LTE modeling with RADEX to fit physical conditions to observed CO isotopologue intensities. We find that the $\alpha_{\rm CO}$ values {in the center and along the inner spiral arm} are $\sim$0.5 dex (LTE) and ${\sim}$0.1 dex (non-LTE) below the Milky Way inner disk value. The average non-LTE $\alpha_{\rm CO}$ is $2.4{\pm}0.5$ M$_\odot$ pc$^{-2}$ (K km s$^{-1}$)$^{-1}$. While both methods show dispersion due to underlying assumptions, the scatter is larger for LTE-derived values. This study underscores the necessity for robust CO line modeling to accurately constrain the molecular ISM's physical and chemical conditions in nearby galaxies.