3 Cloud parameters from the escape probability model

To interpret the small amount of information contained in observations of at most five transitions showing mainly Gaussian profiles and essentially unresolved, approximately circular symmetric intensity distributions, we need a simple cloud model that is both physically reasonable and characterised by few parameters. An obvious choice is a spherically symmetric model. This geometry reflects early phases and the large scale behaviour of several collapse simulations (e.g. Galli et al. 1999), whereas the inner parts of collapsing clouds are probably flattened structures (e.g. Li & Shu 1997).

Even in spherical geometry there is no simple way to solve the radiative transfer problem relating the cloud parameters to the emitted line intensities (see Appendix A.1). Thus we cannot compute the cloud properties directly from the observations.

3.1 Application of the escape probability approximation

A common approach is the escape probability approximation discussed in detail in Appendix A.2. Assuming that all cloud parameters, including the excitation temperatures, are constant within a spherical cloud volume one can derive a simple formalism relating the three parameters kinetic temperature $T_{\rm kin}$, gas density $n_{\rm H_2}$, and column density of radiating molecules on the scale of the global velocity variation $N_{\rm mol}/\Delta v$ to the line intensity at the cloud model surface. No assumption on molecular abundances is required.

Since a telescope does not provide a simple pencil beam we have to correct the model surface brightness temperature by the beam filling factor $\eta_{\rm f}$, given as the convolution integral of the normalised intensity distribution with the telescope beam pattern, to compute the observable beam temperature. Unfortunately, the brightness profile of the source is a non-analytic function where we can only give simple expressions for the central value observed in a beam much smaller than the source or for the integral value observed in a beam much larger than the source (Eqs. (A.9) and (A.11)). For intermediate situations we approximate the beam temperature by starting from both limits and using a beam filling factor given by the convolution integral of two Gaussians. The difference between the two values provides an estimate of the error made in the beam convolution.

To compute the integral we fitted the observed brightness distributions by Gaussians. Most cores are well approximated by slightly elongated Gaussians. W49A, S235B, Serpens, and $\rho$ Oph A show asymmetric scans so that the size determination is somewhat uncertain. The fit error is about 0.3' for these three sources. For the rest of the cores we obtain typical values of less than 0.2'. The true object size finally follows from the deconvolution of the measured intensity distribution with the telescope beam. The resulting source sizes in $\alpha$ and $\delta$ are given in Table 4. As the geometric mean is sufficient to compute the beam filling factor we do not expect any serious error from the fact that the cross-scans in $\alpha$ and $\delta$ do not necessarily trace the major axes of the brightness distribution. For sources which are considerably smaller than the beam widths of 53'', 107'', and 80'', respectively, only a rough size estimate is possible according to the nonlinearity of the deconvolution. This holds for W49A, W3, S235B, and partially S255. Most clouds, however, show an extent of the emission which is close to the beam size.

In general different values are obtained for the spatial FWHMs in the different lines. In Table 4 we find two classes of sources with respect to the variation of the source size depending on the transition observed. Most cores show a monotonic decrease of the visible size when going to higher transitions. This is expected from the picture that higher transitions are only excited in denser and smaller regions. Serpens, $\rho$ Oph A, and NGC 2024, however show the smallest width of the fit in the CS 2-1 transition. This is explained by eye inspecting the 2-1 maps and corresponding high-resolution observations from the literature where we see that the three sources break up into several clumps which are only separated in the 53'' beam but unresolved in the KOSMA beams. In these cases, we have restricted the analysis to the major core seen in the CS 2-1 maps using its size to compute the beam filling, although this approach introduces a small error in the data analysis by assigning the whole flux measured in the higher transitions to this central core.

 

 

Table 4: Source size corrected for beam convolution.

Source	FWHM in $\alpha$ [']			FWHM in $\delta$ [']
	2-1	5-4	7-6	2-1	5-4	7-6
W49A	1.0	1.1	0.3	1.5	1.7	1.3
W33	1.7	1.7	1.4	1.7	1.3	1.0
W51A	1.9	1.7	1.7	2.1	2.2	1.1
W3(OH)	2.1	1.1	1.3	1.7	1.4	1.4
W3	1.0	0.6	0.8	1.0	0.6	0.3
S255	1.9	1.6	1.1	1.2	0.6	1.1
S235B	1.0	2.2	1.1	0.9	0.2	0.3
S106	3.0	1.6		2.5	2.2
Serpens	2.0	2.2		2.1	2.3
DR21	3.0	2.6	1.8	1.5	1.7	1.8
Mon R2	3.2	3.0	1.1	3.6	3.2	1.6
NGC 2264	3.2	1.7	0.6	2.9	2.1
OMC-2	2.4	1.4	0.6	2.5	1.4	0.6
$\rho$ Oph A	1.3	3.0	2.6	1.9	2.9	2.5
NGC 2024	1.1	1.1	1.3	1.3	2.5	2.3

Applying the two limits for the beam size treatment (Eqs. (A.9) and (A.11)) using the sizes from Table 4 we find that the resulting gas and column densities are the same within 20% except for W49A, W3, S235B, and NGC 2264. The first three are small compared to the beams so that the results from Eq. (A.9) have to be rejected and only Eq. (A.11) can be used. For NGC 2264 we cannot provide a simple explanation for the difference so we give a relatively large error bar covering the results from both approximations.

The size of the resulting parameter range in $T_{\rm kin}$, $n_{\rm H_2}$, and $N_{\rm CS}/\Delta v$ is determined by the accuracy of the observations. For two cores it was only possible to set a lower limit to the gas density. Moreover, we were not able to provide any good constraint to the cloud temperature for all sources. Values between about 30 K and 150 K are possible. Hence, an independent determination of the cloud temperatures is required. Several different methods based on optically thick CO, NH₃ or dust observations are discussed in the literature and we used the values from the references given in Table 5. In addition to these values we also used 50 K as assumed by Plume et al. (1997) as "standard'' temperature in the parameter determination for massive cores.

3.2 Resulting core parameters

   

Table 5: Clump parameters derived from the escape probability model.

Source	$T_{\rm kin}$	$n_{\rm H_2}$	$\times/\div$	$N_{\rm CS}/\Delta v$	$\times/\div$
	[K]	[cm-3]		[cm-2/kms-1]
W49A(a)	20 $^{{\rm a}}$	> $7.3\times10^6$		$2.2\times10^{14}$	1.3
	50 $^{{\rm a}}$	$1.3\times10^6$	1.5	$2.1\times10^{14}$	1.2
W49A(b)	20 $^{{\rm a}}$	$4.5\times10^6$	1.6	$2.8\times10^{14}$	1.3
	50 $^{{\rm a}}$	$7.6\times10^5$	1.5	$2.7\times10^{14}$	1.2
W33	40 $^{{\rm b}}$	$1.7\times10^6$	1.7	$2.3\times10^{14}$	1.2
	50 $^{{\rm c}}$	$1.2\times10^6$	1.6	$2.3\times10^{14}$	1.2
W51A	20 $^{{\rm a}}$	$2.1\times10^7$	1.6	$4.8\times10^{14}$	1.7
	50	$1.1\times10^6$	1.5	$4.1\times10^{14}$	1.4
	57 $^{{\rm a}}$	$8.9\times10^5$	1.4	$3.7\times10^{14}$	1.4
W3(OH)	30 $^{{\rm d}}$	$1.9\times10^6$	1.5	$1.3\times10^{14}$	1.2
	50	$8.5\times10^5$	1.4	$1.3\times10^{14}$	1.2
W3	30 $^{{\rm e}}$	$7.1\times10^6$	2.9	$7.4\times10^{14}$	1.5
	50	$9.8\times10^5$	1.8	$8.1\times10^{14}$	1.4
	55 $^{{\rm f}}$	$7.4\times10^5$	1.6	$8.3\times10^{14}$	1.3
S255	40 $^{{\rm g}}$	$1.3\times10^6$	1.6	$1.3\times10^{14}$	1.3
	50	$9.3\times10^5$	1.5	$1.2\times10^{14}$	1.3
S235B	40 $^{{\rm h}}$	$1.2\times10^6$	1.5	$3.7\times10^{13}$	1.3
	50	$8.9\times10^5$	1.5	$3.6\times10^{13}$	1.3
S106	10 $^{{\rm i}}$	${>}6.3\times10^6$		$4.7\times10^{13}$	1.4
	25 $^{{\rm i}}$	$7.1\times10^5$	1.5	$3.8\times10^{13}$	1.4
	50	$2.5\times10^5$	1.6	$3.9\times10^{13}$	1.4
Serpens	25 $^{{\rm j}}$	$4.6\times10^5$	1.2	$2.8\times10^{13}$	1.5
	50	$1.9\times10^5$	1.6	$2.8\times10^{13}$	1.5
DR21	35 $^{{\rm k}}$	$1.5\times10^6$	1.7	$9.3\times10^{13}$	1.9
	50	$9.1\times10^5$	1.6	$8.7\times10^{13}$	1.4
Mon R2	25 $^{{\rm l}}$	$1.7\times10^6$	1.8	$4.5\times10^{13}$	1.5
	50 $^{{\rm m}}$	$6.2\times10^5$	1.7	$3.7\times10^{13}$	1.7
NGC 2264	25 $^{{\rm n}}$	$1.4\times10^6$	2.4	$7.9\times10^{13}$	2.6
	50	$4.7\times10^5$	2.3	$7.8\times10^{13}$	2.5
OMC-2	19 $^{{\rm q}}$	$1.9\times10^6$	1.6	$6.3\times10^{13}$	1.4
	24 $^{{\rm o,p}}$	$1.1\times10^6$	1.5	$6.2\times10^{13}$	1.4
	50	$3.4\times10^5$	1.7	$6.2\times10^{13}$	1.4
$\rho$ Oph A	25 $^{{\rm r}}$	$9.5\times10^5$	1.6	$2.0\times10^{13}$	1.4
	50	$3.7\times10^5$	1.8	$2.0\times10^{13}$	1.5
NGC 2024	25 $^{{\rm s}}$	$9.1\times10^6$	2.9	> $4.5\times10^{14}$
	40 $^{{\rm t}}$	$2.1\times10^6$	2.0	> $3.5\times10^{14}$
	50	$1.6\times10^6$	1.7	> $3.5\times10^{14}$

$^{{\rm a}}$ Sievers et al. (1991), $^{{\rm b}}$ Goldsmith & Mao (1983), $^{{\rm c}}$ Haschick & Ho (1983), $^{{\rm d}}$ Wilson et al. (1991), $^{{\rm e}}$ Tieftrunk et al. (1998), $^{{\rm f}}$ Tieftrunk et al. (1995), $^{{\rm g}}$ Jaffe et al. (1984), $^{{\rm h}}$ Nakano & Yoshida (1986), $^{{\rm i}}$ Roberts et al. (1997), $^{{\rm j}}$ McMullin et al. (1994), $^{{\rm k}}$ Garden & Carlstrom (1992), $^{{\rm l}}$ Montalban et al. (1990), $^{{\rm m}}$ Giannakopoulou et al. (1997), $^{{\rm n}}$ Krügel et al. (1987), $^{{\rm o}}$ Castets & Langer (1995), $^{{\rm p}}$ Batrla et al. (1983), $^{{\rm q}}$ Cesaroni & Wilson (1994), $^{{\rm r}}$ Liseau et al. (1995), $^{{\rm s}}$ Liseau et al. (1995), $^{{\rm t}}$ Mezger et al. (1992).

Table 5 lists the parameters from the escape probability model for all cores. Whereas the column density is well constrained for most clouds, there is a considerable uncertainty in the gas density resulting from the unknown cloud temperature. At the temperature of 50 K we obtain average values and logarithmic standard deviation factors of

$$\langle n_{\rm H_{2}}\rangle = 7.9\times10^5 \qquad \times/\div 1.5$$ (1)

$$\langle N_{\rm CS}/\Delta v\rangle = 1.2\times10^{14} \qquad \times/\div 2.8.$$

From the 71 cores analysed by Plume et al. (1997) assuming this temperature they obtained

$$\langle n_{\rm H_{2}}\rangle = 8.5\times10^5 \qquad \times/\div 1.7$$ (2)

$$\langle N_{\rm CS}/\Delta v\rangle = 2.5\times10^{14} \qquad \times/\div 3.1.$$

This good agreement indicates first that both investigations study the same type of clouds visible in the CS transitions. Second, this shows that the static escape probability (Eq. (A.11)) used here and the LVG escape probability (Eq. (A.9)) applied by Plume et al. (1997) differ only marginally as discussed already by Stutzki & Winnewisser (1985). Third, the observational data from both telescopes give almost equivalent results, i.e. the reliability of the parameters hardly profits from using the 10 times smaller beam of the IRAM telescope when the escape probability approximation is used.

For NGC 2024 we are able to test the consistency of the results from the CS and the C³⁴S observations. The resulting hydrogen densities agree for both isotopes within 20% at all temperatures assumed. We obtain $9\times10^6$ cm^-3 at 25 K (Mezger et al. 1992) and $3.2\times10^6$ cm^-3 at 40 K (Ho et al. 1993). Unfortunately, this is a core where we can only give a lower limit to the column densities. The limits deviate by a factor 13, which is significantly different from the terrestrial isotopic ratio of 23 but close to the value 10 derived by Mundy et al. (1986) for the isotopic ratio in NGC 2024.

The escape probability model provides a first estimate to the physical parameters but its limitations are obvious. It is definitely not justified to assume constant parameters within the whole cloud. Moreover, several observations are in contradiction to the parameters from the escape probability models. Lada et al. (1997) detected the CS 10-9 transition in NGC 2024 and Plume et al. (1997) observed the 10-9 and 14-13 transitions in S255 and W3(OH). The critical densities for these transitions are about $6\times10^7$ cm^-3 and $2\times10^8$ cm^-3 respectively. From the densities in Table 5 one would conclude that these transitions are not excited. Hence, a more sophisticated model has to be applied to obtain a physically reasonable explanation of the measurements. Plume et al. (1997) suggested a two-component model or continuous density gradients to resolve this contradiction. We will discuss a self-consistent radiative transfer model including a radial density profile in the following.