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We study how the estimation of the sonic Mach number ($M_s$) from $^{13}$COlinewidths relates to the actual 3D sonic Mach number. For this purpose weanalyze MHD simulations which include post-processing to take radiativetransfer effects into account. As expected, we find very good agreement betweenthe linewidth estimated sonic Mach number and the actual sonic Mach number ofthe simulations for optically thin tracers. However, we find that opacitybroadening causes $M_s$ to be overestimated by a factor of ~ 1.16 - 1.3 whencalculated from optically thick $^{13}$CO lines. We also find that there is adependency on the magnetic field: super-Alfv\'enic turbulence shows increasedline broadening as compared with sub-Alfv\'enic turbulence for all values ofoptical depth for supersonic turbulence. Our results have implications for theobservationally derived sonic Mach number--density standard deviation($\sigma_{\rho/<\rho>}$) relationship, $\sigma^2_{\rho/<\rho>}=b^2M_s^2$, andthe related column density standard deviation ($\sigma_{N/<N>}$) sonic Machnumber relationship. In particular, we find that the parameter b, as anindicator of solenoidal vs. compressive driving, will be underestimated as aresult of opacity broadening. We compare the $\sigma_{N/<N>}-M_s$ relationderived from synthetic dust extinction maps and $^{13}$CO linewidths withrecent observational studies and find that solenoidally driven MHD turbulencesimulations have values of $\sigma_{N/<N>}$ which are lower than real molecularclouds. This may be due to the influence of self-gravity which should beincluded in simulations of molecular cloud dynamics.

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