dl$, with $F_{\rm D}(r)$ being the CO depletion factor [*within*]{} the core, thus function of the cloud radius $r$; see Section \[schemistry\]). We anticipate here that the $f_{\rm D}$ value at the cloud center is only a very small fraction ($\sim$ 1%) of $F_{\rm D}$ in the central few thousand AU of the core.

–  column densities, deuterium fractionation, and volume densities {#sdeut}
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The  and  column densities have been determined using the “constant–$T_{\rm ex}$” (CTEX) approximation, which reduces to simple analytic expressions (see Appendix A in Caselli et al. [@cas02b]), and the Large Velocity Gradient (LVG) approximation. Both approaches give reasonable column density estimates as long as optical depths are small. When, as for example for (1–0), the optical depths in the main hyperfine satellites are large, one is best (independent of method) to use the weakest of the satellites or alternatively the optical depth inferred from the intensity ratio of the weakest satellite to the strong main components. The errors in any case stem from the difficulties in estimating the optical depth of thick lines compounded with possible non-LTE effects for the hyperfine satellites (Caselli et al. [@cmt95]). Errors due to the estimate of the partition function (i.e. 