6 Discussion

  
C${^{18}}\!$O abundance distribution

In pure gas-phase chemistry models the CO abundance is very stable. At gas densities over 10³  it is practically constant at all times. Therefore, any variation in the fractional CO abundance observed in B68 is probably due to accretion and desorption processes on dust grains. The adopted CO depletion law, as described in Sect. 4.1.3, is consistent with a steady state, i.e. with the situation where accretion and desorption are in equilibrium. In the time-dependend depletion model of Caselli et al. (2001) this situation corresponds to very late stages of chemical evolution. One plausible theory of the origin of globules is that they are remnants of dense cores of dark clouds or cometary globules (Reipurth 1983). This scenario supports the possibility that B68, being an aged object, could indeed have reached chemical equilibrium.

The assumption that A/B is constant includes the following assumptions: 1) The gas kinetic temperature is roughly constant, 2) the dust temperature remains everywhere below the critical temperature of CO desorption, which lies in the range 20-30 K (Léger et al. 1985; Takahashi & Williams 2000), and 3) the same desorption mechanisms are operating throughout the cloud. The constancy of the gas temperature is already built in the adopted physical model (first assumption). The average dust temperature of B68 is $\approx$13 K, which we have derived using ISOPHOT Serendipity Survey data (for calibration see Hotzel et al. 2001). Langer & Willacy (2001) claim the detection of a dust core of 8-9 K. Both observational results support the validity of the second assumption.

As discussed in detail by Watson & Salpeter (1972), Léger et al. (1985), Willacy & Millar (1998) and Takahashi & Williams (2000), the desorption mechanisms operating in dense dark clouds with no star formation are connected with cosmic rays, X-rays or H₂ formation on grains. The main process in all three cases is impulsive whole grain or spot heating, resulting in classical evaporation of adsorbed molecules. Cosmic rays can also contribute to desorption via chemical explosions, and via photo-desorption by UV photons resulting from excitation of H₂ molecules, but both processes are believed to be effective in the low extinction regions only (Léger et al. 1985). These processes are neglected hereafter, as is heating due to H₂ formation on grains because of its relative inefficiency compared with cosmic ray heating (Takahashi & Williams 2000). According to Léger et al. (1985) X-rays can heat small grains with radii in the range 200-400 Å more efficiently than cosmic rays, while the heating of larger grains is assumed to be dominated by cosmic rays.

The accretion and desorption constants are actually integrals over the grain cross section distribution and the velocity or energy distributions of the colliding particles. The accretion constant can be written as $A=2\epsilon\ensuremath{{\scriptstyle\langle}} {}v\ensuremath{{\scriptstyle\rangle}} {}S$, where $\epsilon$ is the total dust grain surface area per H nucleon, $\ensuremath{{\scriptstyle\langle}} {}v\ensuremath{{\scriptstyle\rangle}}$ is the average speed of the molecules in question (at 8 K: $\langle{}v(\mbox{C$\ensuremath{\:\!} ^{18}\ensuremath{\!\;\!} $ O})\rangle=\mbox{${7.5}\times10^{3}$ }~{\rm cm~s^{-1}}$) and S is the sticking probability, which is generally assumed to be unity in cold clouds (e.g. Sandford & Allamandola 1990). The factor of 2 comes from the assumption that all hydrogen is in molecular form. The value of $\epsilon$ depends on the assumed grain size distribution, and especially on the lower cut off of the grain radius, a_-. For example, the distribution adopted by Léger (1983), with a_-=50 Å, gives $\epsilon=\mbox{${1.4}\times10^{-21}$ }~\ensuremath{{\rm cm}^2}$. On the other hand, if one assumes that CO on grains with radii below 400 Å is efficiently desorbed by X-rays or by other processes (i.e. effectively no absorption on small grains), the effective $\epsilon$ becomes 3.5 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-22  . The accretion constants Acorresponding to these $\epsilon$-values are 2.1 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-17    and 5.3 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-18    .

Based on the work of Léger et al. (1985) and Hasegawa & Herbst (1993), Caselli et al. (2001) assumed that the desorption in L1544 is dominated by thermal evaporation due to heating by relatively heavy cosmic rays. The cosmic ray desorption rate for CO derived by Hasegawa & Herbst (1993), $k_{\rm CRD}(\mbox{CO})=\mbox{${9.8}\times10^{-15}$ }~{\rm s}^{-1}$, is based on the Fe nuclei flux derived by Léger et al. (1985), which is consistent with a total H₂ ionization rate of 10^-17 s^-1. Other assumptions used for the indicated value of $k_{\rm CRD}$ are that 1) the average grain radius is about 1000 Å (needed for the calculation of the fraction of the time spent by grains in the vicinity of 70 K), 2) the adsorption energy of CO is 1210 K and 3) that the characteristic adsorbate vibrational frequency of CO is 10¹² s^-1. The uncertainty of the cosmic ray desorption rate is a good order of magnitude (Caselli et al. 2001).

Assuming that the mentioned process dominates the replenishment of gas-phase CO, we set $B=k_{\rm CRD}(\mbox{CO})$ and use the accretion constant for a_-=400 Å. Then we get $A/B=\mbox{${5.4}\times10^{-4}$ }~\ensuremath{{\rm cm}^{-3}}$, and the corresponding value of the parameter $Z=\frac{5}{6}\ensuremath{n_{\rm c}} ~A/B$ is about 100. In view of the large uncertainty of the desorption constant, all Z values from $\sim$10 to 10³ fit the model of Hasegawa & Herbst (1993).

The modelling of the parameter Zto the observed column density profile in Sect. 4.1.3 gave a best fit for Z=180 and the fit became noticeably worse for $Z\sim50$. We must consider values down to Z=20 however, as the Monte Carlo analysis in Sect. 4.2 still produced good results for this number. The latter modelling also set the upper bound to $Z\sim300$, but an even tighter limit is set from the cosmic abundances of H, C and ¹⁸O: With the solar abundance [C]/[H $]=\mbox{${4.7}\times10^{-4}$ }$ (Lambert 1978) and the terrestrial isotopic ratio [¹⁶O]/[¹⁸O]=489 (Duley & Williams 1984, p.175), we get $\ensuremath{\widehat{Y}_{\rm C^{18}O}} ^{\rm max}=\mbox{${1.9}\times10^{-6}$ }$. For B68, with $\ensuremath{N_{\rm c}} (\mbox{H$_2$ })=\mbox{${2.5}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}}$ (Eq. (12)), this means $\ensuremath{N_{\rm c}} ^{\rm max}(\ensuremath{\mbox{C$\ensuremath{\:\!}^{18}\en... ...$ O},{\rm gas+dust}} )=\mbox{${4.8}\times10^{16}$ }~\ensuremath{{\rm cm}^{-2}}$. This number is reached for $Z\approx200$ (see Fig. 4b). For our lower limit Z=20 we find $\ensuremath{N_{\rm c}} (\ensuremath{\mbox{C$\ensuremath{\:\!}^{18}\ensuremath{\... ...$ O},{\rm gas+dust}} )=\mbox{${0.6}\times10^{16}$ }~\ensuremath{{\rm cm}^{-2}}$, corresponding to 13% of carbon nuclei being bound in CO molecules. More recent determinations of solar and stellar carbon abundances suggest that the average galactic value is a factor of 2 lower than the value of Lambert (1978) used above (see discussion in Snow & Witt 1995). Therefore, even though a relatively high [C]/[H] ratio may be present in B68 (as in the sun), it is save to exclude Z values exceeding 200.

The reasonable Z range derived from our observations is thus 20<Z<200, corresponding to only 5% to 0.5% of all CO molecules in the centre of B68 being in the gas phase. The large overlap with the above prediction deduced from the model of Hasegawa & Herbst (1993) suggests that the degree of CO depletion can indeed be understood in terms of accretion and cosmic ray induced desorption.

The derived range for is considerably higher than the commonly quoted fractional abundance value $\ensuremath{Y_{\rm C^{18}O}} =\mbox{${1.7}\times10^{-7}$ }$ of Frerking et al. (1982), which is based on a CO vs.  comparison in the $\rho$ Ophiuchus and Taurus molecular cloud complexes. This is to be expected because they ultimately measured the gas phase CO. To compare our results with their value, we have calculated the gas-phase fractional abundance at the outer boundary: $n_{\rm R}$(C¹⁸O, gas)/$n_{\rm R}$(H $_2)={1.6}\times10^{-3}$/ ${1.5}\times10^{4}={1.1}\times10^{-7}$. This value lies between the values of Frerking et al. (1982) for Taurus "envelopes'' (0.7 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-7) and "dense cores'' (1.7 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-7). Considering that drops towards the centre of B68, we would have observed a lower fractional abundance if we had directly used column densities, as has been done in most other studies. Therefore, our value is comparable to the value for the Taurus envelopes. Harjunpää & Mattila (1996) investigated the molecular clouds Chamaeleon I, R Coronae Australis and the Coalsack and determined the $N(\mbox{C$\ensuremath{\:\!} ^{18}\ensuremath{\!\;\!} $ O})$ vs. E(J-K) relations, corresponding to fractional abundances between 0.7 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-7 (Coalsack) and 2 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-7 (Chamaeleon I). is clearly lower in B68 than it is in the active star forming regions R Coronae Australis, Chamaeleon I and $\rho$ Ophiuchus. This points towards a possible relation between the depletion degree and the star formation activity.

  
6.2 Temperature and distance

As discussed in Sect. 2, the concept of Bonnor-Ebert spheres imposes Eq. (1) on the relation between certain observable parameters. Assuming that the state of isothermal hydrostatic equilibrium does apply to B68, we can derive its distance and mass from Eq. (2) if the kinetic temperature and the central column density are known.

There are several reasons to doubt the high kinetic temperature of 16 K derived by Bourke et al. (1995). The first comes from their own results. By using this kinetic temperature and the excitation temperature of the (J,K)=(1,1) inversion transition of NH₃, they derive a hydrogen number density of $n(\mbox{H$_2$ })=\mbox{${9.1}\times10^{3}$ }~\ensuremath{{\rm cm}^{-3}}$. This is a good order of magnitude lower than the value from the BES model. According to Eq. (2) of Ho & Townes (1983), an overestimate of the kinetic temperature leads to an underestimate of the H₂ number density, which suggests that Bourke et al. (1995) used a too high value for . Secondly, our observations and Monte Carlo modelling results are in agreement with the assumption of a nearly homogenous excitation temperature of $\ensuremath{T_{\rm ex}}\mbox{$(J=1\mbox{--}0)$ }=8$ K for $\ensuremath{\:\!} ^{13}\ensuremath{\!\;\!}$CO and C $\ensuremath{\:\!} ^{18}\ensuremath{\!\;\!}$O, which in turn is roughly equal to . These results agree with the earlier observations of Avery et al. (1987). Moreover, temperatures derived from ammonia in other globules without internal heating sources lie at around 10 K (Lemme et al. 1996). Finally, the modelling results of Zucconi et al. (2001) for a BES with similar characteristics to B68 suggest that the dust temperature decreases well below 10 K in the dense inner parts, which is consistent with a low gas temperature in such an object.

After taking T=8 K as the most likely kinetic temperature, the distance can be checked by using the formula for the central column density ( $=1.2N(\mbox{H$_2$ })$) in Eq. (2) and the canonical $N(\mbox{H$_2$ })/\mbox{$E(H-K)$ }$ ratio given in Eq. (11). As the NIR reddening at the cloud center is $\mbox{$E(H-K)$ }=2.07~\mbox{mag}$, we find that the distance to the cloud is about 70 pc. The adoption of the non-standard ratio given in Eq. (13) would bring the cloud still nearer, which would however be unlikely on the basis of the Monte Carlo results (see Sect. 4.2). Therefore, it seems reasonable to assume that the cloud is located on the near side of the Ophiuchus complex, i.e. at a distance of 80 pc (de Geus et al. 1989).

Summarizing, T=8 K and D=80 pc are the most likely values that are consistent with B68 being a Bonnor-Ebert sphere and our own observations. This is the first distance estimate ever for this globule, which is not based on the ad hoc assumption that B68 is at the same distance as the centre of the Ophiuchus giant molecular cloud complex. The small distance and temperature values imply a significantly lower mass than previously estimated: The parameters of B68 as calculated by Eq. (2) are $\ensuremath{n_{\rm c}} =\mbox{${2.61}\times10^{5}$ }~\ensuremath{{\rm cm}^{-3}}$, $\ensuremath{N_{\rm c}} =\mbox{${2.59}\times10^{22}$ }~\ensuremath{{\rm cm}^{-2}}$, $M=0.67~\ensuremath{{M}_\odot}$ and $\ensuremath{P_{\rm R}} =\mbox{${1.69}\times10^{-12}$ }$ Pa. The derived value for the external pressure, which is needed to contain the BES, is not too far away from the pressure of the Loop 1 superbubble (0.9-1.2 $\ensuremath{\;\!}\times\ensuremath{\;\!}$10^-12 Pa, Breitschwerdt et al. 2000).