Constraining the H2 column densities in the diffuse interstellar medium using dust extinction and HI data

Carbon monoxide (CO) is a poor tracer of H$_{2}$ in the diffuse interstellar medium (ISM), where most of the carbon is not incorporated into CO molecules unlike the situation at higher extinctions. We present a novel, indirect method to constrain H$_{2}$ column densities ($N_{H_{2}}$) without employing CO observations. We show that previously-recognized nonlinearities in the relation between the extinction, $A_{V} ({H}_{2})$, derived from dust emission and the HI column density ($N_{HI}$) are due to the presence of molecular gas. We employ archival $N_{H_{2}}$ data, obtained from the UV spectra of stars, and calculate $A_{V} ({H}_{2})$ towards these sight lines using 3D extinction maps. We derive an empirical relation between $A_{V} ({H}_{2})$ and $N_{H_{2}}$, which we use to constrain $N_{H_{2}}$ in the diffuse ISM. We construct a $N_{H_{2}}$ map of our Galaxy and compare it to the CO integrated intensity ($W_{CO}$) distribution. We find that the average ratio ($X_{CO}$) between $N_{H_{2}}$ and $W_{CO}$ is approximately equal to $2 \times 10^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$, consistent with previous estimates. However, we find that the $X_{CO}$ factor varies by orders of magnitude on arcminute scales between the outer and the central portions of molecular clouds. For regions with $N_{H_{2}} \gtrsim 10^{20}$ cm$^{-2}$, we estimate that the average H$_{2}$ fractional abundance, $f_{{H}_{2}}$ = $2N_{H_{2}}$/(2$N_{H_{2}}$ + $N_{{HI}}$), is 0.25. Multiple (distinct) largely atomic clouds are likely found along high-extinction sightlines ($A_{V} \geq 1$ mag), hence limiting $f_{{H}_{2}}$ in these directions. More than $50 \%$ of the lines of sight with $N_{H_{2}} \geq 10^{20}$ cm$^{-2}$ are untraceable by CO with a $J$ = 1-0 sensitivity limit $W_{CO} = 1$ K km s$^{-1}$.