We study how the estimation of the sonic Mach number ( \mathrm { 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 \mathrm { M _ { s } } to be overestimated by a factor of \approx 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-Alfvénic turbulence shows increased line broadening as compared with sub-Alfvénic 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 ^ { 2 } \mathrm { M _ { s } ^ { 2 } } , and the related column density standard deviation ( \sigma _ { N / \langle N \rangle } ) 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 / \langle N \rangle } – \mathrm { 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 / \langle N \rangle } 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 .