The $^{12}\textrm{CO}/^{13}\textrm{CO}$ ratio in turbulent molecular clouds
Carbon monoxide (CO) isotopes are commonly used as tracers of the column density. The most abundant CO isotope, 12CO, is usually optically thick and so provides only a lower limit for the column density. In these regions, less abundant isotopes, such as 13CO are observed. When these 13CO observations are then used to infer 12CO column densities, a uniform 12CO/13CO isotope ratio is often adopted. A similar approximation is frequently made when comparing results from cloud simulations including chemistry to observations. In this work, we examine the impact of selective photodissociation and chemical fractionation on the 12CO/13CO isotopic ratio, with the aim of testing the validity of a uniform scaling between 12CO and 13CO. We follow the coupled chemical, thermal and dynamical evolution of isolated molecular clouds with several different metallicities, radiation fields and initial densities, and then post-process them with line radiative transfer in order to obtain maps of the emergent 13CO emission. We find that the CO isotope ratio shows a close correlation to both the 12CO and the 13CO column densities. The ratio varies within a factor of 2-3 with the CO isotope column density. When a uniform ratio is assumed, this results in up to 60 per cent error in 12CO column density estimates derived from observed 13CO column density. The same assumption results in up to 50 per cent difference in the intensity of 13CO emission derived from 12CO number density distributions. We show that this effect can be corrected by the use of a fitting formula derived from our simulations. The proposed formula is consistent with millimetre-wavelength measurements of the CO isotope ratio, but underestimates the ratios from ultraviolet absorption data. The fitting formula nevertheless allows us to obtain more accurate 13CO emission maps from simulations and to estimate the isotope ratio from 13CO observations.