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2 B68 as Bonnor-Ebert cloud

Throughout this paper, we make extensive use of the dust column density profile measured by Alves et al. (2001) and in parts also take up their proposal of B68 being a Bonnor-Ebert sphere (BES). In order to make transparent which assumptions are involved in deriving various quantities, we summarise in this section the concept of Bonnor-Ebert spheres and discuss its applicability to B68.

Assuming an isothermal, spherically symmetric distribution of gas in hydrostatic equilibrium, a cloud's density profile is governed by Eqs. (374) and (375) of Chandrasekhar (1939, p.156). As Bonnor (1956) and Ebert (1955) pointed out, these equations have a family of solutions characterised by the nondimensional radial parameter  \ensuremath{\xi_{\rm max}}, if the sphere is bound by a fixed external pressure \ensuremath{P_{\rm R}}. They also discovered that such a gaseous configuration is unstable to gravitational collapse if $\ensuremath{\xi_{\rm max}} >6.5$. With R being the radius where the pressure P(r) has dropped to \ensuremath{P_{\rm R}}, the physical parameters are determined by the scaling relation $\ensuremath{\xi_{\rm max}} =\xi(r=R)=\sqrt{4\pi{}G\ensuremath{\rho_{\rm c}} }R/a$, where \ensuremath{\rho_{\rm c}} is the central density, a the isothermal sound speed and G the gravitational constant.

In practice, the radius of a cloud cannot be directly observed, but only its angular diameter \ensuremath{\theta_{\rm R}} and in certain circumstances its distance D. Using the mean molecular weight m and the central number density $n_{\rm c}$ to write $\ensuremath{\rho_{\rm c}} = mn_{\rm c}$ and $a = \sqrt{kT/m}$, where k is the Boltzmann constant and $T\equiv\ensuremath{T_{\rm kin}} $ the kinetic temperature, we can express the relation between the physical parameters of a BES as

 \begin{displaymath}\frac{\ensuremath{\xi_{\rm max}} }{\ensuremath{\theta_{\rm R}} } = D\sqrt{\frac{4\pi{}G\ensuremath{n_{\rm c}} }{kT}}~m~.
\end{displaymath} (1)

The column density profile of a BES is determined by numerical integration. The column density towards the centre is $\ensuremath{N_{\rm c}} =K~\ensuremath{n_{\rm c}} ~D~\ensuremath{\theta_{\rm R}} $, where the constant K depends only on \ensuremath{\xi_{\rm max}}. The shape of the density profile as well as the shape of the column density profile depend only on \ensuremath{\xi_{\rm max}}, i.e. the normalised profiles $n/\ensuremath{n_{\rm c}} $ vs. $\theta/\ensuremath{\theta_{\rm R}} $ and $N/\ensuremath{N_{\rm c}} $ vs. $\theta/\ensuremath{\theta_{\rm R}} $ are fully determined by \ensuremath{\xi_{\rm max}}. Vice versa, if the normalised column density profile of a cloud is measured, \ensuremath{\xi_{\rm max}} can be determined without any knowledge of any of the parameters on the right hand side of Eq. (1). Not even \ensuremath{\theta_{\rm R}} has to be known for this, but naturally \ensuremath{\theta_{\rm R}} comes with the astrometric calibration of the CCD image.

Alves et al. (2001) determined the extinction profile of B68 and found it to have the same shape as the column density profile of a BES with $\ensuremath{\xi_{\rm max}} =6.9$. Does this mean that B68 actually is a BES? First of all, one has to assume that the shape of the gas column density profile has been measured, i.e. $N_{\rm gas}\propto\mbox{$E(H-K)$ }$. Except for Sect. 4.1.2, all results of this paper implicitly fall back on this assumption. But despite having the column (and hence number) density profile of a BES, the globule may still not be a BES in the sense that it may not be isothermal, it may not be in hydrostatic equilibrium and it may not be in equilibrium altogether. It is safe to say that B68 cannot be a perfect BES for a number of reasons: 1) Its shape is not perfectly circulary symmetric. 2) The molecular line-widths show that a small microturbulent velocity field is present. 3) The measured \ensuremath{\xi_{\rm max}} is larger than the critical value, i.e. B68 is unstable to gravitational collapse unless some additional support mechanism, e.g. a magnetic field, plays a role. Nevertheless, in the framework of inevitable idealisations in astronomy, the BES model fits B68 certainly to some degree (as the density profile fits near-perfectly). We will resume the BES model and discuss the conclusion from applying it to B68 in Sect. 6.2. Therefore we calculate the numerical constants involved in the following paragraphs. Note that up to and including Sect. 6.1 we refer with "cloud model'' and "BES profile'' to the fitted (column) density profile, which is based on the raw observations of Alves et al. (2001) and does not require the hydrostatic equilibrium assumption.

The extinction profile of B68 only deviates noticeably from a BES profile in the outermost parts (for $r>100\ensuremath{^{\prime\prime}} $, see Fig. 2 of Alves et al. 2001). As the fitting parameters, particularly  \ensuremath{\theta_{\rm R}}, change considerably if one tries to scale the BES profile to also those data points, we have done a \ensuremath {\chi ^2}-fit to the data of Alves et al. (2001, who kindly provided us with the data), deliberately ignoring data points at $r>100\ensuremath{^{\prime\prime}} $ at this stage. The lowest \ensuremath {\chi ^2} was found for $\ensuremath{\xi_{\rm max}} =6.99$, which corresponds to a centre-to-edge density contrast of 17.1. The scaling parameters for this particular BES extinction profile are $\ensuremath{\theta_{\rm R}} =106.35\ensuremath{^{\prime\prime}} $ and $\ensuremath{A_{\rm V}} =30.3$ mag (see also Fig. 4a). The actual peak extinction value is not needed in Eq. (1), but will be used in Sects. 5 and 6 to derive the H2-to-extinction ratio in B68. Varying \ensuremath{\xi_{\rm max}} by one decimal increases \ensuremath {\chi ^2} by 2%, while the scaling parameters \ensuremath{\theta_{\rm R}} and \ensuremath{A_{\rm V}} are modified by 0.6% and 0.3% respectively. The density contrast, being an increasing function of \ensuremath{\xi_{\rm max}}, is the least precise parameter, changing between 16.5 and 17.7 for the given variation of \ensuremath{\xi_{\rm max}}.

For a BES with $\ensuremath{\xi_{\rm max}} =6.99$ and $\ensuremath{\theta_{\rm R}} =106.35\ensuremath{^{\prime\prime}} $, we have calculated the following values for the radius R, the central gas particle number density \ensuremath{n_{\rm c}}, the peak column density \ensuremath{N_{\rm c}}, the total mass M and the external pressure \ensuremath{P_{\rm R}}:

\begin{displaymath}R = 10635~{\rm AU} ~ \quad \quad \quad \quad \quad\! \mathcal{D}
\end{displaymath}


\begin{displaymath}n_{\rm c} = 2.090 \times10^{5}~{\rm cm}^{-3} \quad \quad\! \mathcal{D}^{-2} \quad\!\!\mathcal{T}
\end{displaymath}


 \begin{displaymath}{N_{\rm c}} = 2.590 \times 10^{22}~{\rm cm}^{-2} ~~ \quad \mathcal{D}^{-1} \quad\!\!\mathcal{T}
\end{displaymath} (2)


\begin{displaymath}M = 1.049~{M}_\odot \quad \quad \quad \quad \quad \mathcal{D}~~~~ \quad\!\!\!\mathcal{T}
\end{displaymath}


\begin{displaymath}P_{\rm R} = 1.687 \times10^{12}~{\rm Pa} ~~ \quad \quad \mathcal{D}^{-2} \quad\!\!\mathcal{T}^2,
\end{displaymath}

abbreviating $\mathcal{D} = \frac{D}{100~\ensuremath{{\rm pc}} }$ and $\mathcal{T} = \frac{T}{10~\ensuremath{{\rm K}} }$. The external pressure follows from \ensuremath{n_{\rm c}}, as the centre-to-edge density contrast only depends on \ensuremath{\xi_{\rm max}}.

We assume H2 and He are the only species contributing substantially to mass and pressure, with a fractional helium abundance of $\ensuremath{n_{\rm He}} /\ensuremath{n_{\rm H_2}} =0.2$ . Under this assumption, the mean molecular weight is m = 2.329  \ensuremath{{m_{\rm H}}} ( \ensuremath{{m_{\rm H}}} being the atomic hydrogen mass). Throughout this paper, we use the commonly used rounded value m=2.33  \ensuremath{{m_{\rm H}}}. For other values of m, the quantities in Eq. (2) have to be scaled as $\ensuremath{n_{\rm c}}\propto m^{-2}$, $\ensuremath{N_{\rm c}}\propto m^{-2}$, $M\propto m^{-1}$ and $\ensuremath{P_{\rm R}}\propto m^{-2}$.

If two of the three remaining unknown variables (D, T and  \ensuremath{n_{\rm c}}) in Eq. (1) are known, the third one follows from Eq. (1) and the cloud model is fixed. This way we first calculated \ensuremath{n_{\rm c}} in Eq. (2), then from the fixed cloud model we determined \ensuremath{N_{\rm c}} and M. If all three variables D, T and \ensuremath{n_{\rm c}} can be determined from observations, Eq. (1) allows a consistency check of the involved parameters. In practice, for a small globule like B68 both T and \ensuremath{n_{\rm c}} can be determined to some accuracy e.g. by ammonia observations, while a distance estimate remains difficult due to the lack of foreground stars. In fact, Eq. (1) can be used to determine the distance under these circumstances.


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