Resolved Maps of the CO-to-H$_2$ Conversion Factor in 41 Nearby Galaxies

We measure the CO-to-H$_2$ conversion factor ($\alpha_\mathrm{CO}$) in 41 galaxies at 2 kpc resolution, using dust surface density inferred from far-IR emission as a tracer of the gas surface density and assuming a constant dust-to-metals ratio. We have $\sim940$ and $\sim660$ independent measurements of $\alpha_\mathrm{CO}$ for CO (2-1) and (1-0), respectively. The mean values for $\alpha_\mathrm{CO~(2-1)}$ and $\alpha_\mathrm{CO~(1-0)}$ are $6.9^{+3.9}_{-4.5}$ and $3.1^{+1.6}_{-1.7}$, respectively. The CO-intensity-weighted mean for $\alpha_\mathrm{CO~(2-1)}$ is 4.37, and 2.58 for $\alpha_\mathrm{CO~(1-0)}$. We examine how $\alpha_\mathrm{CO}$ scales with several physical quantities, e.g.\ star-formation rate (SFR), stellar mass, and interstellar radiation field (ISRF). Among them, the strength of the ISRF and the resolved specific SFR have the strongest anti-correlation with spatially resolved $\alpha_\mathrm{CO}$. No strong correlations are found between galaxy-averaged $\alpha_\mathrm{CO}$ and galaxy properties. We also find that $\alpha_\mathrm{CO}$ decreases with stellar mass surface density ($\Sigma_\star$) in the dense regions ($\Sigma_\star\geq200~{\rm M_\odot~pc^{-2}}$), following the power-law relations $\alpha_\mathrm{CO~(2-1)}\propto\Sigma_\star^{-0.5}$ and $\alpha_\mathrm{CO~(1-0)}\propto\Sigma_\star^{-0.3}$. The power-law index is insensitive to the assumed D/M. Meanwhile, the discrepancy between the power-law index for CO~(2-1) and (1-0) data indicates a variation in the CO line ratio traced by $\Sigma_\star$. We interpret the decrease in $\alpha_\mathrm{CO}$ as a result of higher velocity dispersion compared to isolated, self-gravitating clouds. The decrease in $\alpha_\mathrm{CO}$ at high $\Sigma_\star$ is important for assessing molecular gas content and star-formation efficiency in the centers of galaxies, bridging MW-like to starburst-like conversion factors.