Opacity Broadening of $^{13}$CO Linewidths and its Effect on the Variance-Sonic Mach Number Relation

We study how the estimation of the sonic Mach number ($M_s$) from $^{13}$CO linewidths relates to the actual 3D sonic Mach number. For this purpose we analyze MHD simulations which include post-processing to take radiative transfer effects into account. As expected, we find very good agreement between the linewidth estimated sonic Mach number and the actual sonic Mach number of the simulations for optically thin tracers. However, we find that opacity broadening causes $M_s$ to be overestimated by a factor of ~ 1.16 - 1.3 when calculated from optically thick $^{13}$CO lines. We also find that there is a dependency on the magnetic field: super-Alfv\'enic turbulence shows increased line broadening as compared with sub-Alfv\'enic turbulence for all values of optical depth for supersonic turbulence. Our results have implications for the observationally derived sonic Mach number--density standard deviation ($\sigma_{\rho/<\rho>}$) relationship, $\sigma^2_{\rho/<\rho>}=b^2M_s^2$, and the related column density standard deviation ($\sigma_{N/<N>}$) sonic Mach number relationship. In particular, we find that the parameter b, as an indicator of solenoidal vs. compressive driving, will be underestimated as a result of opacity broadening. We compare the $\sigma_{N/<N>}-M_s$ relation derived from synthetic dust extinction maps and $^{13}$CO linewidths with recent observational studies and find that solenoidally driven MHD turbulence simulations have values of $\sigma_{N/<N>}$ which are lower than real molecular clouds. This may be due to the influence of self-gravity which should be included in simulations of molecular cloud dynamics.