The Column Density Variance-Sonic Mach Number Relationship

Although there are a wealth of column density tracers for both the molecular and diffuse interstellar medium, there are very few observational studies investigating the relationship between the density variance ($\sigma^2$) and the sonic Mach number (${\cal M}_s$). This is in part due to the fact that the $\sigma^2$-${\cal M}_s$ relationship is derived, via MHD simulations, for the 3D density variance only, which is not a direct observable. We investigate the utility of a 2D column density $\sigma_{\Sigma/\Sigma_0}^2$-${\cal M}_s$ relationship using solenoidally driven isothermal MHD simulations and find that the best fit follows closely the form of the 3D density $\sigma_{\rho/\rho_0}^2$-${\cal M}_s$ trend but includes a scaling parameter $A$ such that: $\sigma_{ln(\Sigma/\Sigma_0)}^2=A\times ln(1+b^2{\cal M}_s^2)$, where $A=0.11$ and $b=1/3$. This relation is consistent with the observational data reported for the Taurus and IC 5146 molecular clouds with $b=0.5$ and $A=0.16$ and $b=0.5$ and $A=0.12$, respectively. These results open up the possibility of using the 2D column density values of $\sigma^2$ for investigations of the relation between the sonic Mach number and the PDF variance in addition to existing PDF sonic Mach number relations.