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.
