Understanding biases in measurements of molecular cloud kinematics using line emission

ABSTRACT Molecular line observations using a variety of tracers are often used to investigate the kinematic structure of molecular clouds. However, measurements of cloud velocity dispersions with different lines, even in the same region, often yield inconsistent results. The reasons for this disagreement are not entirely clear, since molecular line observations are subject to a number of biases. In this paper, we untangle and investigate various factors that drive linewidth measurement biases by constructing synthetic position–position–velocity cubes for a variety of tracers from a suite of self-gravitating magnetohydrodynamic simulations of molecular clouds. We compare linewidths derived from synthetic observations of these data cubes to the true values in the simulations. We find that differences in linewidth as measured by different tracers are driven by a combination of density-dependent excitation, whereby tracers that are sensitive to higher densities sample smaller regions with smaller velocity dispersions, opacity broadening, especially for highly optically thick tracers such as CO, and finite resolution and sensitivity, which suppress the wings of emission lines. We find that, at fixed signal-to-noise ratio, three commonly used tracers, the J = 4 → 3 line of CO, the J = 1 → 0 line of C18O, and the (1,1) inversion transition of NH3, generally offer the best compromise between these competing biases, and produce estimates of the velocity dispersion that reflect the true kinematics of a molecular cloud to an accuracy of $\approx 10{{\ \rm per\ cent}}$ regardless of the cloud magnetic field strengths, evolutionary state, or orientations of the line of sight relative to the magnetic field. Tracers excited primarily in gas denser than that traced by NH3 tend to underestimate the true velocity dispersion by $\approx 20{{\ \rm per\ cent}}$ on average, while low-density tracers that are highly optically thick tend to have biases of comparable size in the opposite direction.

The relation between the turbulent Mach number and observed fractal dimensions of turbulent clouds

Supersonic turbulence is a key player in controlling the structure and star formation potential of molecular clouds (MCs). The three-dimensional (3D) turbulent Mach number, $\operatorname{\mathcal {M}}$, allows us to predict the rate of star formation. However, determining Mach numbers in observations is challenging because it requires accurate measurements of the velocity dispersion. Moreover, observations are limited to two-dimensional (2D) projections of the MCs and velocity information can usually only be obtained for the line-of-sight component. Here we present a new method that allows us to estimate $\operatorname{\mathcal {M}}$ from the 2D column density, Σ, by analysing the fractal dimension, $\mathcal {D}$. We do this by computing $\mathcal {D}$ for six simulations, ranging between 1 and 100 in $\operatorname{\mathcal {M}}$. From this data we are able to construct an empirical relation, $\log \operatorname{\mathcal {M}}(\mathcal {D}) = \xi _1(\operatorname{erfc}^{-1} [(\mathcal {D}-\operatorname{\mathcal {D}_\text{min}})/\Omega ] + \xi _2),$ where $\operatorname{erfc}^{-1}$ is the inverse complimentary error function, $\operatorname{\mathcal {D}_\text{min}}= 1.55 \pm 0.13$ is the minimum fractal dimension of Σ, Ω = 0.22 ± 0.07, ξ1 = 0.9 ± 0.1, and ξ2 = 0.2 ± 0.2. We test the accuracy of this new relation on column density maps from Herschel observations of two quiescent subregions in the Polaris Flare MC, ‘saxophone’ and ‘quiet’. We measure $\operatorname{\mathcal {M}}\sim 10$ and $\operatorname{\mathcal {M}}\sim 2$ for the subregions, respectively, which are similar to previous estimates based on measuring the velocity dispersion from molecular line data. These results show that this new empirical relation can provide useful estimates of the cloud kinematics, solely based upon the geometry from the column density of the cloud.