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Appendix A: The radiative transfer problem in the escape probability approximation

A.1 The general radiative transfer problem

The physical parameters of a cloud model and the emerging line profiles and intensities are linked by the radiative transfer problem. It relates the molecular emission and absorption coefficient at one point to the radiation field determined by the emission and transfer of radiation at other locations in the cloud. The quantity entering the balance equations for the level populations at a point $\vec{r}$ is the local radiative energy density u within the frequency range for each transition:

$$u(\vec{r})=\int_{4\pi} {\rm d}\Omega\; u(\vec{r},\vec{n})$$

 

$$u(\vec{r},\vec{n})= {1 \over c} \int_{-\infty}^{\infty} {\rm d}\nu\; I_\nu(\vec{r},\vec{n}) \Phi_\nu (\vec{r},\vec{n})\;.$$ (A.1)

Here, $u(\vec{r},\vec{n})$ is the absorbable radiative energy coming from direction $\vec{n}$ and $I_\nu(\vec{r},\vec{n})$ is the intensity at a given frequency within this direction. It is determined by the radiative transfer equation

$$\vec{n} \nabla I_\nu(\vec{r},\vec{n}) = - \kappa_\nu(\vec{r},... ...}) I_\nu(\vec{r},\vec{n}) + \epsilon_\nu(\vec{r},\vec{n})\cdot$$ (A.2)

Assuming complete redistribution the profile for the absorption and the emission coefficients, $\kappa_\nu(\vec{n},s)$ and $\epsilon_\nu(\vec{n},s)$is given by the same local line profile $\Phi_\nu$. For Maxwellian velocity distributions it is a Gaussian:

$$\Phi_\nu(\vec{n},\vec{r})={1 \over \sqrt{\pi} \sigma} \exp\left(-{(\nu-\vec{n}\vec{v}(\vec{r}))^2\over \sigma^2}\right)\cdot$$ (A.3)

Here, $\vec{v}$ is the velocity of the local volume element written in units of the frequency $\nu$. The frequently used FWHM of the distribution is related to $\sigma$ by FWHM $=2\sqrt{\rm ln\,2}\,\sigma$.

For a molecular cloud this results in a huge system of integral equations interconnecting the level populations and intensities at all points within a cloud.

A.2 The escape probability model

A simple way to avoid the nonlinear equation system is the escape probability approximation that is widely applied to interpret molecular line data. It is based on the assumption that the excitation, and thus the absorption and emission coefficients, are constant within those parts of a cloud which are radiatively coupled. Then the radiative transfer equation (Eq. (B.13)) can be integrated analytically. We obtain for the integrated radiative energy density:

$$u(\vec{n})={1 \over c} \left [\beta(\vec{n}) I_{\rm bg}(\vec{... ...ta(\vec{n}) \right) {\epsilon\rm _l \over \kappa\rm _l}\right]$$ (A.4)

where the subscript l denotes line integrated quantities. The term $\beta(\vec{n})$ is the probability that a photon can escape or penetrate along the line of sight $\vec{n}$ from the considered point to the boundary of the interaction region or vice versa.

$$\beta(\vec{n}) \int_{-\infty}^{\infty} {\rm d}\nu\; \Phi_\nu(\vec{r},\vec{n}) \times \exp (-\tau_\nu(\vec{r},\vec{n}))$$ = (A.5)

$${\rm with} \tau_\nu(\vec{r},\vec{n})=\int_{-\infty}^{\vec{r}} {\rm d}s_{\vec{n}}\; \kappa\rm _l(\vec{r}) \Phi_\nu(\vec{r},\vec{n})$$ (A.6)

where the integration path d $s_{\vec{n}}$ follows the direction $\vec{n}$.

There are two main concepts to define the interaction region and thus to compute the escape probability. The first one is the large velocity gradient approximation introduced by Sobolev (1957). Here, the interaction region is determined by a velocity gradient in the cloud that displaces the line profiles along the line of sight so that distant regions are radiatively decoupled. When the resulting interaction region is sufficiently small it is justified to assume constant parameters. The escape probability then follows

 

$$\beta(\vec{n}) \left. \left[1-\exp \left( -\tau_{\rm LVG} \right)\right]\right/ \tau_{\rm LVG}$$ = (A.7)

$${\rm with} \tau_{\rm LVG}={\kappa\rm _l \over \vert(\vec{n}\nabla) (\vec{n}\vec{v})\vert}$$ (A.8)

(cf. Ossenkopf 1997). The radiative energy density in each direction is determined by two local quantities only: the source function $S=\epsilon\rm _l/\kappa\rm _l$and the optical depth of the interaction region $\tau_{\rm LVG}$. The observable brightness temperature at the line centre is constant over all regions with the same velocity gradient

$$T_{\rm B} \approx {c^2 \over 2 k \nu^2} \left( S - I_{\rm bg}\right) \left[1-\exp(-2\tau_{\rm LVG})\right]\cdot$$ (A.9)

In molecular clouds, the local velocity gradients are unknown, however. It is generally assumed that the total observed line width, which is composed from turbulent, thermal, and systematic contributions, can be used as the measure of the velocity gradient over the cloud size. This approach was applied by Plume et al. (1997) for the massive cores discussed in the text.

We used another method, the static escape probability model. It does not depend on the velocity structure but assumes a special geometry of the interaction region. Stutzki & Winnewisser (1985) solved the problem for a homogenous spherical cloud with constant excitation parameters. The resulting escape probability is taken to be constant

$$\beta=\int_{-\infty}^{\infty} {\rm d}\nu\; \Phi_\nu \times \exp (-\tau_\nu)$$ (A.10)

where $\tau_\nu$ is the optical depth at the cloud centre.

The surface brightness temperature towards the centre of the cloud is given by the same expression as Eq. (A.9) when we use the line integrated optical depth at the cloud centre $\tau\rm _l$ instead of $\tau_{\rm LVG}$. It decays with growing distance from the cloud centre. Averaged over the whole cloud, the brightness temperature at the line centre is given by

   

$$T_{\rm B} \approx {c^2 \over 2 k \nu^2} \left( S - I_{\rm bg}\right) \left(1-{e}(-2\tau\rm _l)\right)$$ (A.11)

$$\mbox{with} ~~~~{e}(x)={2 \over x^2} \left( 1 - \exp(-x)(1+x)\right).$$

This value would be observed with a beam much larger than the cloud.

When the velocity gradient in the LVG approximation is computed from the total line width and the cloud size, it turns out that both methods agree when applied to observations with a small beam towards the cloud centre. Only for large-beam observations, they differ in the functions in Eqs. (A.9) and (B.22), which are either $\exp(-2\tau)$ or $e(-2\tau)$, but result in similar values.

By setting up a table of beam temperatures from Eqs. (A.9) and (B.22) and comparing the observed line intensities with the tabulated values we can derive three parameters from the observations: the kinetic temperature $T_{\rm kin}$ and the gas density $n_{\rm H_2}$ providing mainly the source function, and the column density of the considered molecules relative to the line width $N/\Delta v$ providing the photon escape probability.