The $^{12}$CO/$^{13}$CO ratio in turbulent molecular clouds

The $^{13}$CO molecule is often used as a column density tracer in regions where the $^{12}$CO emission saturates. The $^{13}$CO column density is then related to that of $^{12}$CO by a uniform isotopic ratio. A similar approximation is frequently used when deriving $^{13}$CO emission maps from numerical simulations of molecular clouds. To test this assumption we calculate the $^{12}$CO/$^{13}$CO ratio self-consistently, taking the isotope selective photodissociation and the chemical fractionation of CO into account. We model the coupled chemical, thermal and dynamical evolution and the emergent $^{13}$CO emission of isolated, starless molecular clouds in various environments. Selective photodissociation has a minimal effect on the ratio, while the chemical fractionation causes a factor of 2-3 decrease at intermediate cloud depths. The variation correlates with both the $^{12}$CO and the $^{13}$CO column densities. Neglecting the depth dependence results in $\leq$60 per cent error in $^{12}$CO column densities derived from $^{13}$CO. The same assumption causes $\leq$50 per cent disparity in the $^{13}$CO emission derived from simulated clouds. We show that the discrepancies can be corrected by a fitting formula. The formula is consistent with millimetre-wavelength isotopic ratio measurements of dense molecular clouds, but underestimates the ratios from the ultraviolet absorption of diffuse regions.