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Astron. Astrophys. 342, 257-270 (1999)

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3. The mapped core region

We describe in this section the results of our [FORMULA] mapping as well as our [FORMULA] measurements at selected positions.

3.1. Maps in [FORMULA] (1[FORMULA]0) and (2[FORMULA]1)

Our integrated intensity maps in [FORMULA] (2[FORMULA]1) and (1[FORMULA]0) are presented in Fig. 1a and 1b respectively. The main feature of the [FORMULA] maps Fig. 1 is a structure of dimension roughly [FORMULA] N-S (0.27 pc) which extends further in the E-W direction than the boundaries of our map.

While peaks are clearly observable in our integrated map (e.g. at offsets (0,0) and ([FORMULA],[FORMULA])), there is rather little contrast over the main structure. If one considers only that part of the cloud with integrated [FORMULA](1[FORMULA]0) intensity above 1 Kkms-1 (most of Fig. 1b), one sees that I([FORMULA](1[FORMULA]0)) varies only by a factor of 2.5. Thus, a priori, the [FORMULA] data give no strong evidence for high column density contrasts. Moreover, in spite of our relatively good linear resolution in [FORMULA](2[FORMULA]1), the peak line temperatures are only 3 K ([FORMULA]) which is considerably less than the temperature of 10 K typical in dense cores of this type (see Benson & Myers 1989). This suggests that the [FORMULA] lines are optically thin although there are other possibilities (low excitation temperature or a low filling factor). In contrast, the [FORMULA] lines, with typical temperatures of 10 K, observed by Dobashi et al. (1992), are clearly optically thick and presumably emitted in external layers of the cloud which are warmed up by external radiation fields.

Our data do show evidence of velocity structure within the cloud. This is evident when one examines Fig. 2 which shows observed spectra towards the central region of our map. There are clearly regions (e.g. around (0,[FORMULA])) where we observe two or more distinct velocity components. The line widths of individual components are relatively small and range down to 0.4 km s-1. This does not seem to be an artefact because the independently reduced (1[FORMULA]0) and (2[FORMULA]1) lines give very similar results.

[FIGURE] Fig. 2. [FORMULA] spectra of the central region on a [FORMULA] grid. The scale is [FORMULA] from -0.5 to 3 K, and [FORMULA]=0 to 8 kms-1. Thick lines show the (2[FORMULA]1) spectra, smoothed to the resolution of the (1[FORMULA]0) data, and thin lines the (1[FORMULA]0) lines.

More typical is a single Gaussian shaped line profile with a typical width of [FORMULA] kms-1 centered at [FORMULA] kms-1. We suspect that many such "simple Gaussians" are cases where it has not been possible to resolve blended components. Note however that all linewidths, even the smallest, are clearly larger than the thermal line width of [FORMULA], [FORMULA] in kms-1, but comparable to the speed of sound [FORMULA] which is a factor of 4.6 larger.

3.2. The velocity field

Channel maps (Fig. 3) and position-velocity plots of the [FORMULA] data reveal better than the maps of integrated intensities the fragmented spatial and kinematic structure of the observed region. At velocities [FORMULA] kms-1 we see a weak NW-SE orientated structure in [FORMULA](1[FORMULA]0) (Fig. 3) which vanishes at higher velocities. At velocities [FORMULA] kms-1 the emission is however mostly EW orientated and forms part of a ridge which continues further to the west, outside of the observed region. At [FORMULA] kms-1 we find a strong maximum of [FORMULA] K near offset ([FORMULA],[FORMULA]) and a NW-SE orientated ridge (position angle [FORMULA] counted counterclockwise from west) extending over [FORMULA] (0.27 pc). At [FORMULA] kms-1 another filament shows up, running nearly parallel to the first, but offset by [FORMULA] to the North. This filament peaks near (0,0) at [FORMULA] kms-1 and [FORMULA] K with [FORMULA]. While the first southern filament vanishes when going to higher velocities, the second filament continues, though weakening and rotating to become NS orientated at [FORMULA] kms-1.

[FIGURE] Fig. 3. [FORMULA](1[FORMULA]0) channel maps of IC 5146. The central LSR velocity of each velocity interval is indicated at the upper left corner of each panel. The velocity interval is [FORMULA] kms-1. Contours are at 0.3 (=3[FORMULA]) to 2.4 by 0.3 K ([FORMULA]). The western part of the emission appears more smoothly distributed because it was not sampled at half the HPBW (cf. Fig. 1).

3.3. The distribution of clumps

In order to quantify the structure better we decomposed the observed [FORMULA](1[FORMULA]0) dataset iteratively into 26 Gaussian shaped clumps using an automated algorithm developed by Stutzki & Güsten (1990) and discussed recently by Kramer et al. (1998b). The residual emission not decomposed into clumps is 5 [FORMULA] above the noise level. From the decomposed clumps we only selected clumps with deconvolved widths of at least 50% of the resolutions in all three dimensions ([FORMULA],[FORMULA],v). The spatial dynamic range of these (1[FORMULA]0) data, i.e. the ratio of observed spatial area to beam area, is only 40, and hence the number of clumps identified is small. We use here the (1[FORMULA]0) data set because it is spatially more homogeneously sampled than the (2[FORMULA]1) data, the noise is lower, and calibration accuracy is higher.

Most clumps show deconvolved radii close to the limit set by the resolution, though the size number distribution extends to radii of 0.1 pc ([FORMULA]). The average beam deconvolved aspect ratio of the larger clumps is 2.8. Clumps are generally aligned with the filamentary structure already found in the channel maps (Fig. 3). The number distribution of position angles of the clump elongations shows two maxima, a strong at [FORMULA] and a weaker at [FORMULA], which correspond to the position angles of the two filaments described above.

The number distribution of deconvolved clump line widths shows a maximum at [FORMULA] kms-1 clearly above the resolution, indicating that this width might be a property inherent to the clump ensemble. The largest clump linewidth found is 0.7 kms-1. These results confirm our earlier suspicion that line profiles consist of one or several subsonic or sonic components.

In Fig. 4 we plot the dependence of the velocity FWHM [FORMULA] and mass M versus the radius R of the individual clumps. The dashed lines correspond to the empirical relations [FORMULA], [FORMULA] found in a variety of clouds in the galaxy (e.g. Larson 1981, Myers 1983, Dame et al. 1986). The IC 5146 data set agrees roughly with these relations, although the scatter is large.

[FIGURE] Fig. 4. The dependence of velocity dispersion and mass versus radius for the individual clumps. The vertical dotted line corresponds to the resolution of [FORMULA] pc (HPBW[FORMULA]). The dashed lines correspond to [FORMULA] and [FORMULA] respectively.

To calculate clump masses [FORMULA] from intensities and column densities we assume optically thin emission, an excitation temperature of 10 K, LTE, and a standard abundance ratio [[FORMULA]]/[H2] (Frerking et al. 1982). In Sect. 3.6 we will show that these [FORMULA] column densities are overestimating the true column densities because [FORMULA] is somewhat subthermally excited (cf. Fig. 8c). Clump masses are in the range [FORMULA]. The average clump volume density derived from masses and radii is [FORMULA] cm-3. The total mass of all clumps is 32 [FORMULA], 73% of the total mass.

The dynamical timescale of the clumps should be roughly [FORMULA] where l is the clump size and thus it is more than about [FORMULA] years, given the range of clump sizes, 0.02-0.11 pc, and clump velocity widths, 0.3-0.7 kms-1.

For comparison with LTE masses we also calculated virial masses [FORMULA] from linewidths assuming gravitational boundedness and neglecting e.g. external pressure. All clumps are near, but slightly above, virial equilibrium: [FORMULA] (Fig. 5). The average ratio is [FORMULA].  2 A similar result was obtained by Dobashi et al. (1992, 1993) who observed the large scale distribution of [FORMULA] and [FORMULA](1[FORMULA]0) in IC 5146 with [FORMULA] resolution. They also find that clumps are nearly in virial equilibrium.

[FIGURE] Fig. 5. The ratio of virial mass and the mass calculated from [FORMULA] column densities, [FORMULA]/[FORMULA], as a function of clump mass [FORMULA] for all 26 individual clumps.

3.4. [FORMULA] spectra

The simplest check on whether the [FORMULA] lines are optically thick is to measure the spectra of the rarer CO isotopomer [FORMULA]. We thus took [FORMULA] spectra at 24 positions (see Fig. 1b) mainly along a diagonal cut through the mapped core region, including the position of the peak in visual extinction of 28 mag (Fig. 1a), and at 7 positions in the direction of background stars. To estimate upper limits for, or if possible to determine, the [FORMULA] optical depth [FORMULA], we then applied two techniques. First, we obtained a limit on the [FORMULA] (1[FORMULA]0) optical depth using the measured relative intensities of the three hyperfine satellites of this transition (see Frerking & Langer 1981 for the 1[FORMULA]0 transition, and Lovas & Krupenie 1974, Townes & Schawlow 1975 for 2[FORMULA]1, cf. Ladd et al. 1998). Deviations from the intrinsic relative line strengths are an indication of high optical depth. We transform this into a limit on the [FORMULA](1[FORMULA]0) optical depth [FORMULA] using an assumed relative abundance [FORMULA] for [[FORMULA]]/[[FORMULA]] of 3.65 (Wilson & Rood 1994, Penzias 1981). At all positions observed in [FORMULA] we find [FORMULA].

A second technique is simply to use the measured ratio of integrated intensities of the [FORMULA] and corresponding [FORMULA] lines. With the plausible assumptions of equal excitation temperatures and beam filling factors for both isotopomers one then has for the ratio of integrated [FORMULA] and [FORMULA] line intensities [FORMULA] that:

[EQUATION]

This may be solved to determine the optical depth [FORMULA], averaged over the spectral line, in the 1[FORMULA]0 or 2[FORMULA]1 transitions. This latter technique yields more critical limits since it measures directly saturation effects in the more abundant isotope. The limits obtained are given in Table 1. We find [FORMULA][FORMULA]3.0 at all positions observed in [FORMULA]. In other words, no evidence is found in our data for compact clumps with [FORMULA](1[FORMULA]0) optical depths higher than 0.5.

We believe that 0.5 is a conservative upper limit and our data set is consistent with optically thin [FORMULA](1[FORMULA]0) emission at all positions. The mean and scatter of the line ratios [FORMULA], [FORMULA], correspond to optically thin emission and the scatter is consistent with the calibration error of [FORMULA]%. This conclusion is also supported by the lack of correlation between optical extinctions and optical depths in Table 1. Even when taking into account only those 10 positions with optical extinctions higher than 20 mag, the average line ratio and scatter, [FORMULA], are consistent with [FORMULA].

To make this clearer, we used integrated intensities [FORMULA] (Table 1) of [FORMULA], which is certainly optically thin, to calculate the corresponding [FORMULA] intensities, [FORMULA], and to compare the ratio [FORMULA]/[FORMULA] with the corresponding ratio [FORMULA]/[FORMULA] (Table 1) calculated from the measured [FORMULA] intensities. For extinctions in the range 15 mag[FORMULA][FORMULA][FORMULA] mag, both ratios are almost exactly the same, [FORMULA] for [FORMULA]/[FORMULA] and [FORMULA] for [FORMULA]/[FORMULA], indicating that in fact [FORMULA] is optically thin even at these high optical extinctions.

One could in principle apply the same techniques as above to derive upper limits on the optical depth in the [FORMULA](2[FORMULA]1) line. However, we conclude that the quality of our (1[FORMULA]0) data is such that it is more accurate to use the fact that (see next section) the excitation temperature [FORMULA] between the J=2 and 1 levels is in the range 6-10 K. The ratio of 2[FORMULA]1 and 1[FORMULA]0 optical depths [FORMULA], assuming constant [FORMULA] for both transitions, is given by:

[EQUATION]

For [FORMULA] in the range 6-10 K, [FORMULA] varies between 1.2 and 1.9. Thus, we expect the [FORMULA](2[FORMULA]1) optical depth to be relatively well constrained by the limits obtained for [FORMULA](1[FORMULA]0), and that at all positions observed in [FORMULA], the [FORMULA](2[FORMULA]1) optical depth is less than 1.

From this analysis, we can only put limits on the [FORMULA] optical depth over the region covered by our [FORMULA] measurements. We cannot exclude the possibility that the [FORMULA] optical depth becomes high in regions of our map for which [FORMULA] data are unavailable. This possibility however seems unlikely. There is no evidence for example (see next section) that the molecular excitation is different outside the region observed in [FORMULA]. There is also no evidence for larger linewidths. We conclude therefore that the [FORMULA] optical depth is low, i.e. [FORMULA], throughout the region mapped and that [FORMULA] can be used to obtain reliable estimates of CO column density.

3.5. [FORMULA] excitation

In Fig. 6, we display our map of the integrated main beam intensity ratios [FORMULA] defined by:

[EQUATION]

[FIGURE] Fig. 6. The ratio of integrated main beam [FORMULA](2[FORMULA]1) and (1[FORMULA]0) temperatures [FORMULA]. The ratio is blanked at those positions where the intensities of one of the transitions are less than [FORMULA]. The [FORMULA](2[FORMULA]1) data were smoothed to the spatial resolution of the (1[FORMULA]0) data. The contours delineate ratios of 0.6 and 1.0 Kkms-1 ([FORMULA]) which bracket the ratios found at most mapped positions.

The mean and scatter are [FORMULA]=0.83[FORMULA]0.22. Note that the observational error on the line ratio is very similar, [FORMULA]%. Another view is given by Fig. 8b which shows that the (2[FORMULA]1)/(1[FORMULA]0) ratio varies between 0.4 and 1.3 along a NW-SE cut. Assuming optically thin emission in both lines, and equal excitation temperature and beam filling factor for both transitions, we find for the ratio of integrated intensities:

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA](2[FORMULA]1), and [FORMULA](1[FORMULA]0). The quantity [FORMULA] is a slowly varying function of [FORMULA] of order unity. From Eq. 4, we find that [FORMULA] varies between 4 K and 9 K along the cut. The average ratio of 0.8 corresponds to an excitation temperature of 6 K. We conclude that either the kinetic temperatures in the shielded cloud interiors are small ([FORMULA] K) or the excitation is sub-thermal, i.e. densities are low.

We can investigate the latter point using an LVG analysis for [FORMULA] using collision rates from McKee et al. (1982). This radiative transfer model assumes a constant kinetic temperature, column- and volume density to calculate the emergent line temperatures and optical depths as well as the level populations and excitation temperatures. Since the observed lines are essentially optically thin, the magnitude of [FORMULA] column densities has only little influence on the ratios, i.e. trapping plays a negligible role. In Fig. 7 we show computed values of the line ratio [FORMULA] (Eq. 3) as a function of local densities [FORMULA](H2) for kinetic temperatures of 4, 6, 8, and 10 K, which bracket the range of likely values. In general, high volume densities lead to a higher population of the [FORMULA] level relative to the [FORMULA] level and hence a higher [FORMULA] ratio, for a constant column density. One sees that the kinetic temperatures must be greater than [FORMULA] K, to be consistent with the measured [FORMULA] ratios, found over most of the map (Fig. 6). Local densities in the range [FORMULA] to [FORMULA] cm-3 are sufficient to account for the observed [FORMULA] ratio, if the kinetic temperature is 10 K.

[FIGURE] Fig. 7. Results of LVG calculations showing the ratio [FORMULA] as a function of local volume density [FORMULA](H2) for different kinetic temperatures. The thick and thin lines correspond to N([FORMULA])/[FORMULA]cm-2/kms-1 and [FORMULA]cm-2/kms-1 respectively, which bracket the range of possible column densities assuming linewidths of 1 kms-1. The dashed horizontal lines bracket the range of measured line ratios [FORMULA] over nearly the full map and also indicate the mean ratio [FORMULA][FORMULA][FORMULA].

3.6. Column density of [FORMULA]

One can slightly improve the accuracy of the [FORMULA] column density determination relative to the simple LTE method, because we have two transitions and can thus estimate the molecular excitation. We assume a kinetic temperature of 10 K and use the LVG model to determine the fraction of molecules in rotational states [FORMULA] from the observed intensity ratio [FORMULA]. Then the "LVG column density" is determined on the basis of the observed intensity in [FORMULA](1[FORMULA]0) assuming it is optically thin.

The results are shown in Fig. 8c along a NW-SE cut (see Fig. 1). A lower limit [FORMULA] was computed assuming only the two lowest levels to be populated and an "upper limit" [FORMULA] computed on the basis of the (1[FORMULA]0) intensity assuming LTE with [FORMULA] = 6 K, the lowest temperature consistent with the average ratio [FORMULA] of 0.8 (Fig. 7). One sees from this that the [FORMULA] column density is certainly determined by our measurements to better than a factor of two.

[FIGURE] Fig. 8a-c. Diagonal cut in NW-SE direction a showing the variation of integrated [FORMULA](1[FORMULA]0) and [FORMULA](2[FORMULA]1) main beam temperatures smoothed to the same spatial resolution. The estimated observational errors due to the calibration uncertainty and the uncertainty of the applied telescope efficiencies are 13% and 20% respectively. b  The line ratio [FORMULA] along the cut. The average value is 0.8 and the estimated error is [FORMULA]%. c  Variation of [FORMULA] column densities along the cut. The best estimate of the column density [FORMULA] is bracketed by the LTE column density and the column density [FORMULA] assuming that only the lowest two levels are populated.

Next, we used the same method to derive LVG column densities [FORMULA] over the mapped region for all positions with integrated intensities a factor 6 above the noise level (Fig. 9). To allow a comparison with the NIR data, we smoothed the [FORMULA](1[FORMULA]0) and (2[FORMULA]1) maps with a Gaussian to a resolution of [FORMULA] sampled on a [FORMULA] grid.

[FIGURE] Fig. 9. Map of the [FORMULA] LVG column density in contours overlayed on the map of visual extinction in greyscales, both with a resolution of [FORMULA] on a [FORMULA] fully sampled grid. Grey levels are 3 to 27 by 2 mag. Contour levels are (0.8 to 2.0 by 0.2) [FORMULA] cm-2.

The [FORMULA] LVG column density varies by less than a factor of 3 between [FORMULA] and [FORMULA] cm-2. Additionally our data are consistent with volume densities in the range [FORMULA] to [FORMULA] cm-3 and a kinetic temperature of 10 K. For this range of parameters, the LVG model gives [FORMULA](1[FORMULA]0) optical depths of less than 0.55 and [FORMULA](2[FORMULA]1)[FORMULA], consistent with the results of Sect. 3.4. The resultant (1[FORMULA]0) excitation temperatures range between 7 and 10 K, and are slightly lower for the (2[FORMULA]1) line. If we take the abundance ratio, [[FORMULA]]/[H2] = 1.7 10- 7, e.g. Frerking et al. (1982), then the molecular hydrogen column density varies between [FORMULA] and [FORMULA] cm-2. Extinctions may be converted to hydrogen column densities using the ratio obtained by Bohlin et al. (1978): H2/AV = 9.36 1020 cm-2mag- 1. In the following, we will speak of the "canonical" abundance ratio

[EQUATION]

obtained with the above ratios. The visual extinction thus derived from the above H2 column density varies between 5 and 14 magnitudes.

Using the LVG model, our limits on the [FORMULA] opacity show that a [FORMULA] column density of [FORMULA] cm-2 is an upper limit for kinetic temperatures [FORMULA] 10 K and volume densities [FORMULA] cm-3. This would correspond to an optical extinction of 20 mag using the canonical ratios.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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