5 [CI]/CO modelling

In a number of studies (e.g. Schilke et al. 1993; Tauber et al. 1995; Petitpas & Wilson 1998) column densities have been calculated assuming [CI] and CO emission to occur under optically thin LTE conditions in the high-temperature limit.

Figure 3: Model line ratio $I({\rm CI})/I(J=2$- $1 {\rm ^{13}CO}$) versus $T_{\rm kin}$ resulting for equal column densities $N({\rm CI})$/d $V = N({\rm CO})$/dV (solid lines). Curves for a range of total gas densities are marked in particles per cc. Dashed line: the ratio $I({\rm CI})/I(J=2$- $1 {\rm ^{13}CO}$) versus $T_{\rm ex}$, likewise requiring equal column densities $N({\rm CI})$/d $V = N({\rm CO})$/dV but assuming optically thin LTE conditions.

From Eqs. (1) and (2) by Tauber et al. (1995), it follows that:

$$\begin{eqnarray*}I([{\rm CI}])/I({\rm ^{13}CO}) = 0.0063\, A^{12}_{13}\, f(T_{\rm ex})\, N[{\rm CI}])/N({\rm CO}) \end{eqnarray*}$$

where

$$\begin{eqnarray*}f(T_{\rm ex}) = T_{\rm ex}/({\rm e}^{7/T_{\rm ex}}+3 {\rm e}^{-16.6/T_{\rm ex}}+5 {\rm e}^{-55.5/T_{\rm ex}}) \end{eqnarray*}$$

while we assume an isotopic abundance $A^{12}_{13} = [{\rm ^{12}CO}]/[{\rm ^{13}CO}]$ = 40 (cf. Mauersberger & Henkel 1993; Henkel et al. 1998). In Fig. 3 we have marked by a dashed line the expected line intensity ratios for the case $N[{\rm CI}])/N({\rm CO}) = 1$. In order to verify the correctness of the assumptions of low optical depth and LTE, we have used the Leiden radiative transfer models described by Jansen (1995) and Jansen et al. (1994) to calculate for gas volume densities ranging from $10^{2} \,{\rm cm^{-3}}$ to $10^{6} \,{\rm cm^{-3}}$ the [CI]/ ${\rm ^{13}CO}$ line intensity ratio corresponding to unit column density ratios. The calculation was performed for a representative value of the column density, $N[{\rm CI}]$/d $V = N({\rm CO})$/d $V = 1 \times 10^{17}$ cm $^{-2} \,{\rm {km\,s^{-1}}}^{-1}$ (cf. Israel & Baas 2001). Note that in this calculation, the temperature parameter is the kinetic temperature $T_{\rm kin}$ instead of the excitation temperature $T_{\rm ex}$. For neutral carbon, the two are not very different under the conditions considered, but for ${\rm ^{13}CO}$ the excitation temperature is generally much lower than the kinetic temperature over most of the relevant range. Under LTE conditions, the [CI]/ ${\rm ^{13}CO}$ ratio continuously increases with temperature $T_{\rm ex}$. In contrast, the radiative transfer calculation shows that this ratio is only weakly dependent on temperature above $T_{\rm kin} \approx 30$ K, and in fact decreases slowly with increasing temperature for densities up to $n(\,{\rm H_{2}}) \approx 10^{4}$. As Fig. 3 illustrates, the assumption of comparable excitation temperatures for ${\rm ^{13}CO}$ and [CI] is valid only for very high densities $n(\,{\rm H_{2}}) > 10^{6} \,{\rm cm^{-3}}$ which are unlikely to apply to our observed sample.