Synthetic CO emission and the $X_{\rm CO}$ factor of young molecular clouds: a convergence study

The properties of synthetic CO emission from 3D simulations of forming molecular clouds are studied within the SILCC-Zoom project. Since the time scales of cloud evolution and molecule formation are comparable, the simulations include a live chemical network. Two sets of simulations with an increasing spatial resolution (d$x=3.9$ pc to d$x=0.06$ pc) are used to investigate the convergence of the synthetic CO emission, which is computed by post-processing the simulation data with the RADMC-3D radiative transfer code. To determine the excitation conditions, it is necessary to include atomic hydrogen and helium alongside H$_2$, which increases the resulting CO emission by ~7-26 per cent. Combining the brightness temperature of $^{12}$CO and $^{13}$CO, we compare different methods to estimate the excitation temperature, the optical depth of the CO line and hence, the CO column density. An intensity-weighted average excitation temperature results in the most accurate estimate of the total CO mass. When the pixel-based excitation temperature is used to calculate the CO mass, it is over-/underestimated at low/high CO column densities where the assumption that $^{12}$CO is optically thick while $^{13}$CO is optically thin is not valid. Further, in order to obtain a converged total CO luminosity and hence <$X_{\rm CO}$> factor, the 3D simulation must have d$x\lesssim0.1$ pc. The <$X_{\rm CO}$> evolves over time and differs for the two clouds; yet pronounced differences with numerical resolution are found. Since high column density regions with a visual extinction larger than 3~mag are not resolved for d$x\gtrsim 1$~pc, in this case the H$_2$ mass and CO luminosity both differ significantly from the higher resolution results and the local $X_{\rm CO}$ is subject to strong noise. Our calculations suggest that synthetic CO emission maps are only converged for simulations with d$x\lesssim 0.1$ pc.