5 Deriving H₂ column densities from $\vec {E(H-K)}$

In order to estimate the column density profile of B68, Alves et al. (2001) derived the colour excesses E(H-K) of more than a thousand background stars behind the cloud (the underlying assumption is that $E(H-K) \propto N_{\rm gas}$). Referring to the standard, i.e. $\ensuremath{R_{\rm V}} =3.1$, interstellar reddening law of Mathis (1990), they used the relationship $\ensuremath{A_{\rm V}} =14.67~E(H-K)$ to plot the visual extinction profile. By convention, is used to present extinction or reddening data. is however not the best parameter to be converted to hydrogen column density, because the conversion factor depends on grain properties, which can be different in different environments (Kim & Martin 1996). We therefore use E(H-K) for converting the extinction data to gas column density. The best fitting BES extinction profile reaches $\ensuremath{A_{\rm V}} =30.3$ mag (Sect. 2), hence follows $\mbox{$E(H-K)$ }=2.07$ mag in the centre of the globule.

From UV observations in the direction of diffuse clouds Bohlin et al. (1978) determined the relation between reddening in the optical and hydrogen column density:

$$\frac{N(H {\sc i})+2N(\mbox{H$_2$ })}{E(B-V)} = \mbox{${5.8}\times10^{21}$ }~\ensuremath{{\rm cm}^{-2}} ~\mbox{mag}^{-1}.$$ (9)

This relationship (Eq. (9)), which is frequently used in "scaling'' the dust column densities to $N(\mbox{H$_2$ })$, may be seriously in error in the case of dense clouds. Practically no dense-cloud lines of sight were included in the sample of Bohlin et al. (1978). Diplas & Savage (1994) re-examined the ratio $N(H {\sc i})/\mbox{$E(B-V)$ }$ for a larger sample of sight lines. They found that the ratio increases to a value of $\mbox{${7.8}\times10^{21}$ }~\ensuremath{{\rm cm}^{-2}} ~\ensuremath{{\rm mag^{-1}}}\pm 12\%$, for sight lines involving target stars located in dusty regions where $n(H {\sc i}) > 1.5~\ensuremath{{\rm cm}^{-3}}$. This increase is also expected for theoretical reasons, since a depletion of small particles in dense clouds leads to a reduced reddening efficiency (see e.g. Kim & Martin 1996). A counter-example are two obscured lines of sight, studied recently by the Far Ultraviolet Spectroscopic Explorer (FUSE), i.e. HD 73882 with $\ensuremath{A_{\rm V}} =2.44~\mbox{mag}$ (Snow et al. 2000) and HD 110432 behind the Coalsack dark nebula with $\ensuremath{A_{\rm V}} =1.32~\mbox{mag}$ (Rachford et al. 2001). For these two lines of sight, the numerical factor in Eq. (9) is 5.1 (HD 73882) and 4.2 (HD 110432) instead of 5.8. The conclusion is that there is no evidence for a substantial increase in the H₂-to-extinction ratio towards those "translucent lines of sight'' in denser dust clouds which can still be studied by means of ultraviolet observations.

Cardelli et al. (1989) derived a family of extinction laws for both diffuse and dense regions, parameterised by the total-to-selective extinction ratio $\ensuremath{R_{\rm V}} = \ensuremath{A_{\rm V}} /\mbox{$E(B-V)$ }$. Their extinction curve for the canonical value $\ensuremath{R_{\rm V}} =3.1$, which is an average for lines of sight penetrating the diffuse interstellar medium, results in the colour excess ratio

$$\frac{E(H-K)}{E(B-V)} = 0.236.$$ (10)

In the case of dense dark clouds, where all hydrogen is in molecular form, Eqs. (9) and (10) lead to

$$\frac{N(\mbox{H$_2$ })}{E(H-K)} = \mbox{${1.23}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}} ~\ensuremath{{\rm mag^{-1}}}\;.$$ (11)

We apply this to B68 and obtain

$$\ensuremath{N_{\rm c}} (\mbox{H$_2$ })=\mbox{${2.5}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}}$$ (12)

in the centre of the globule, keeping in mind that Eq. (11) is ultimately derived from the diffuse dust sample of Bohlin et al. (1978).

Even though there is reason to assume the extinction law at near-infrared (NIR) wavelengths to be independent of environment (Mathis 1990), the normalisation of the extinction curve (with respect to hydrogen column density) may still show a dependency. Observational evidence either favouring or opposing the latter dependency would help to assess the applicability of Eq. (11) to B68, but is still scarce. The $(N(H {\sc i})+2N(\mbox{H$_2$ }))/E(H-K)$ ratio determined towards $\rho$ Oph A(HD 147933), which is seen through an extinction layer of $E(B-V)=0.47~\mbox{mag}$ on the outskirts of the dense $\rho$ Oph cloud core, is close to the value indicated in Eq. (11) (de Boer et al. 1986; Clayton & Mathis 1988).

Using NIR spectroscopy, H₂ column densities can be probed in much denser clouds. Lacy et al. (1994) have detected the $\mbox{H$_2$ }(v=1\mbox{--}0)\;S(0)$ line (4498 cm^-1) in absorption towards NGC2024 IRS2, and they derive $N(\mbox{H$_2$ })=\mbox{${3.5 \pm 1.4}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}}$. By modelling the observed spectral energy distribution of IRS2 at 1.65, 2.2, and 4.64 Ueurmnm, Jiang et al. (1984) have derived a colour excess of $E(H-K)=1.5\pm0.3~\mbox{mag}$ in this direction (using extinction curve No. 15 of van de Hulst 1957). Furthermore, Maihara et al. (1990) have observed the Br$\alpha$/Br$\gamma$ line ratio towards the compact H II region surrounding IRS2. The resulting colour excess is $E({\rm Br}\gamma-{\rm Br}\alpha)=1.31\pm0.13~\mbox{mag}$, which corresponds to $E(H-K) = 1.35\pm0.14~\mbox{mag}$ (Cardelli et al. 1989). Adopting the mean of these two colour excesses, we end up with the ratio

$$\frac{N(\mbox{H$_2$ })}{E(H-K)} = \mbox{${2.5\pm1.1}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}} ~\ensuremath{{\rm mag^{-1}}} .$$ (13)

This ratio is $\sim$2 times larger than the diffuse dust value (Eq. (11)). However, the uncertainties are large, and the lower limit of Eq. (13) is close to the value in Eq. (11). Moreover, because the value in Eq. (13) is still based on a single source we cannot adopt it as a proven observational result. It does suggest, however, that the H₂-to-extinction ratio may increase in dense cloud cores (even for NIR extinction), and the use of Eq. (11) for B68 may result in an underestimate of $N(\mbox{H$_2$ })$.