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

Abstract

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 $\le$60 per cent error in ${}^{12}$CO column densities derived from ${}^{13}$CO. The same assumption causes $\le$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.

20 pages, 16 figures. Accepted for publication in MNRAS. Treatment of CO shielding was improved after referee feedback. No qualitative changes in results and conclusions, but the numerical values of the fitting formulae were updated

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