Let us now consider the determination of the cloud volume density, ntot. Here the idea is first to evaluate column densities along each line of sight. An optically-thin transition is clearly the optimal choice for this purpose. Although 13C16O often only marginally fulfills the requirement, it is much more easily detected than rarer isotopes, such as 12 C18O, that are unequivocally optically thin. Accordingly, many CO-based estimates of ntot start with the column density N13
We have already used the fact that the column density of any species is proportional to the optical thickness of an emitted spectral line. The actual proportionality constant between NCO and At013 follows in a straightforward manner from the transfer equation for spectral lines. From Equations (C.15) and (C.16) in Appendix C, we find
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