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Astron. Astrophys. 322, 943-961 (1997)

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6. Discussion

The number of rotational levels included into the calculations was 11. This is sufficient since the number of molecules at the first omitted excitation level was found to be insignificant for all density and temperature combinations used. This was confirmed by scaling the densities in the clouds F2 and T3 to the highest density intervals used in the previous calculations and by repeating these calculations with 13 excitation levels. Towards the centre of the fractal cloud F2 the number of molecules on the excitation level [FORMULA] 11 (averaged over the line of sight) was already about 1.0 [FORMULA] 10-4 times less than the number of molecules on the level [FORMULA] 3. Similarly for cloud T3, with the average density close to 1.0 [FORMULA] 107 cm-3, the population of the level [FORMULA] 11 was about 500 times less than that of the most populated levels.

The collisional coefficients were taken from Green & Chapman (1978). There exists, however, also a more recent set of coefficients given in Turner et al. (1992). We repeated the calculations of the models F2 and T3 with the new coefficients using again the highest density ranges employed in the previous calculations. The differences in the line intensities were at least below 0.3 K i.e. no clear differences could be seen in the figures. Also the line profiles remained unchanged. Therefore, the choice of the collision coefficients does not affect the results of this paper.

6.1. Effects of discretization

Before we can draw any definite conclusions from e.g. the comparison of one-dimensional and three-dimensional models we must first consider the effects that the discretization has on the computed spectra. In one-dimensional clouds we can easily use a very fine grid size and thus make sure that the results are not affected by the discretization. For a three-dimensional cloud this is not always possible for practical reasons.

Each cell should be so small that the excitation conditions are approximately constant within the cell. In Fig. 12 we show a series of figures of the excitation temperature distributions in one-dimensional clouds with different sizes and densities. Compared with the physical parameters of a single cell these can give us an idea in which conditions the discretization may affect the outcome of the radiative transfer calculations. In fact this is the worst case scenario since in the three-dimensional cloud each cell is surrounded by similar cells and therefore the excitation conditions are more homogeneous. In the following we shall call the one-dimensional model clouds `cells' since the results are compared with individual cells of the three-dimensional clouds.

[FIGURE] Fig. 12. The radial dependence of the excitation temperature in spherical clumps subjected to background radiation only. In each frame the lines from top to bottom show the excitation temperatures of the transitions 2-1, 3-2, 5-4 and 7-6. The diameter of the cells is 0.0019 pc for the upper frames and 0.0075 pc for the frames on the lower row. The constant densities of the cells are 1.0 [FORMULA] 106 cm-3 (leftmost frames), 5.0 [FORMULA] 106 cm-3 and 1.0 [FORMULA] 107 cm-3 (rightmost frames)

The upper frames of Fig. 12 correspond to a cell size that is [FORMULA] 10% larger than the cell sizes in model clouds T1, T2 and T3. At the density of 5.0 [FORMULA] 106 cm-3 there are already noticeable changes in the excitation temperatures. For the transitions 5-4 [FORMULA] drops at the surface of the cell about 1 K from the value at the centre. However, since the effect is much smaller for cells embedded in a cloud we may still conclude that the chosen cell size does not cause significant errors to the line profiles calculated from the three-dimensional clouds. Also the optical depths of all cells in the clouds T1, T2 and T3 were less than 1.0.

The cell size used in connection with the fractal clouds was much larger. For example, in cloud F2 the optical depth of the transition CS(5-4) was in some cells over 2.0 when the maximum density of the cloud was scaled to 1.0 [FORMULA] 106 cm-3. The lower frames of Fig. 12 correspond to a cell size [FORMULA] 13% larger than in the fractal clouds. From this figure we can see that the drop in the excitation temperature of the 5-4 transition is already about 4 K when the density of the cell is 5.0 [FORMULA] 106 cm-3. In a cloud with the volume filling factor [FORMULA] this change in the excitation conditions is distributed over several cells and the effects of the discretization are again insignificant (see e.g. Fig. 1> in Park & Hong 1995). On the other hand, in e.g. Fig. 6 a the cloud has both a high density and a small volume filling factor. This means that many cells are subjected only to the background radiation and the excitation temperatures of the upper transitions should drop at the cell surface. Since this is not possible there cannot be any self absorption within the cell either, i.e. the large cell size reduces the amount of self absorption in the calculated spectra. The volume filling factor 0.05 would mean, on the average, only a few non-empty cells in each line of sight through the cloud composed of 36 [FORMULA] 36 [FORMULA] 36 cells. Because of the way the fractal clouds were created most of the mass is concentrated near the cloud centre and the volume filling factor is also significantly higher there. On the other hand, near the cloud surface the cells generally have very low optical depths. These factors may partly reduce the effects of the large cell size. However, these results should already be treated with some caution.

We have assumed a constant kinetic temperature for all clouds. If the cloud were subjected to some external heating mechanism, the kinetic temperatures of the clumps should increase towards the surface and the excitation temperatures of most lines would be more constant. At lower densities, however, the excitation temperature of the 2-1 transition increases towards the surface. Therefore the discretization will produce some errors in any case.

6.2. Effects of clumpiness: line profiles and intensities

The comparison of the three-dimensional clouds of Fig. 6 (T2 in Fig. 10 a) and the one-dimensional clouds of Fig. 8 (Fig. 11) shows very clearly the effect that the density distribution has on the observed line profiles. The self-absorption is stronger in the 1D-clouds and it is also enforced by the density structure which causes a drop in the excitation temperatures towards the cloud surface. However, both the 1D- and the 3D-clouds have approximately similar large scale density distributions. Clearly the clumpy cloud structure and the macroturbulence are the key factors causing the observed differences.

The spectra of the 1D-clouds have somewhat lower intensities than the spectra of the corresponding 3D-clouds even if the effect of the self-absorption is taken into account. This effect which is particularly clear at higher densities illustrates some of the difficulties in making comparisons between different kinds of models. The range of densities and the column densities averaged over all four beam sizes were about the same in both 1D- and 3D-clouds. As the 3D-clouds effectively have [FORMULA] 1.0 the 1D-clouds with the same column densities must have lower densities in the cells. Therefore lower line intensities are observed in 1D-clouds as compared to 3D-clouds with the same column density.

Clouds F1, F2 and F2 have different distributions of cell densities (see Fig. 2). In F3 the majority of cells have a relatively low density while in F2 the proportion of dense cells is much higher. In general, these differences cause similar changes in the observed spectra as the volume filling factor. The effect of lowering the cell densities is not necessarily the same as that of lowering the volume filling factor, however. On the contrary, low density can also mean low excitation temperatures and this may in turn lead to increased self-absorption i.e. the effect might be quite the opposite to that of a low volume filling factor. In Fig. 13 we show the spectra from three clouds with cells of two densities. In each cloud half of the cells have a density of 5.0 [FORMULA] 106 cm-3 and the other half a density of 5.0 [FORMULA] 104 cm-3 (Fig. 13, frame a), 2.5 [FORMULA] 105 cm-3 (Frame b) or 1.0 [FORMULA] 106 cm-3 (Frame c). The figure shows that the denser cells are responsible for the major part of emission in all cases since the differences between the models are relatively small. It can also be seen that lowering the density of part of the cells has indeed lead to decreased self-absorption e.g. in CS(5-4) and CS(7-6). Therefore, the clumpy structure seems to be able to reduce the self-absorption when the density contrast between the clumps and the interclump material is modest. Note also that the line intensities have remained about the same in all three cases. The intensity of e.g. the 5-4 transition decreases, however, with increasing column density while the intensity of the optically thinner 2-1 transition increases.

[FIGURE] Fig. 13. The spectra from clouds with the cells having only two possible density values. Half of the cells have a density of 5.0 [FORMULA] 106 cm-3. The density in the other half is shown in frame a 5.0 [FORMULA] 104 cm-3, in frame b 2.5 [FORMULA] 105 cm-3 and in frame c 1.0 [FORMULA] 106 cm-3. The kinetic temperature is 40 K and the cloud size is the same as for e.g. T1

If the maximum density of the clouds is scaled to the same value and the differences in the cell density distributions are allowed to modify the total column density, the amount of self absorption increases with the column density (see Figs. 3 and 4). On the other hand, in Fig. 5 the column densities are identical but the cloud sizes are correspondingly different. The effect of the cell density distribution is quite clear, i.e. very different line intensities are observed in e.g. clouds F1 and F2. This shows, once again, the uncertainty in determining the column density without a detailed knowledge of the actual density structure.

6.3. Effects of clumpiness: line ratios

The line ratios depend mostly on the temperature and the average density, and the differences between different clumpy density distributions are small. The density distribution is still important since it affects the self-absorption and in this respect the differences between e.g. the clumpy models and the spherically symmetric models were very large.

The line ratios depend therefore also on the actual distribution of the gas, the turbulence and the velocity field. As the lines become optically thick, reducing the volume filling factor affects the observed maximum antenna temperature only slightly. On the other hand, an increase in the column density may even reduce the peak line intensity because of the self-absorption. Since the intensity of the optically thin lines is directly proportional to f, the clumpy structure has a strong influence on the line ratios. Fig. 6 a is a good example of this and the similar effect can be seen in Fig. 13.

There are some differences between the different fractal models in Fig. 3. The differences are not very large, however, and can be explained by differences in the mean densities and the self-absorption effects caused by the different column densities. Compared with the fractal clouds the line ratios observed from model clouds created with the structure tree statistics are more similar to each other. This is natural since all clumps in the tree models have the same density distribution, although at different size scales, and therefore the distributions of the cell densities are similar. The column densities of the tree models fall into a narrower range than those of the fractal models.

The column density maps of the fractal clouds and the clouds created with structure trees are very different. The spectra of e.g. F3 in Fig. 3c and T3 in Fig. 10 have, however, comparable line intensities. In order to see the effects of the model according to which the clumpy structure is generated we scaled the cloud F1 down by a factor of 4. The physical size of the cloud, 0.058 pc in each dimension, is then the same as the size of the clouds generated using structure trees. The spectra are shown in Fig. 15 a. Comparing these spectra with e.g. the corresponding spectra from the cloud T3 one can see that the differences are very small. This seems to indicate that the actual method used to generate the the clumpy structure is only of secondary importance and the line ratios are mainly determined by the overall distribution of the cell densities. The result is encouraging since it indicates that real clouds can be modeled with clumpy density distributions even though the exact nature of the small scale clumpiness is still unknown.

[FIGURE] Fig. 14. The peak antenna temperature of CS(2-1) (dashed lines) and the ratio between CS(7-6) and CS(2-1) lines (solid lines) as the function of maximum density in the cloud and the beam averaged column density. The calculations have been made with the fractal cloud F1 using a beam with FWHM equal to one sixth of the cloud diameter
[FIGURE] Fig. 15a and b. a The spectra from the cloud F1 after scaling the size down to the size of the clouds generated with a structure-tree. The maximum densities of the clouds are 1.0 [FORMULA] 106 cm-3, 5.0 [FORMULA] 106 cm-3 and 1.0 [FORMULA] 107 cm-3 for the subframes from left to right. The kinetic temperature is 40 K. b The spectra from cloud F1 with maximum density scaled to 1.0 [FORMULA] 107 cm-3 after the kinetic temperature was modified. The radial dependence of the kinetic temperature was first set according to [FORMULA] (leftmost subframes) or [FORMULA] (rightmost subframes). The temperatures were then scaled to range from 30 K to 60 K. Upper subframes show the spectra from the macroturbulent clouds and the lower frames show spectra from the corresponding microturbulent clouds. The maximum density was 1.0 [FORMULA] 107 cm-3 for all clouds

The line ratios in clumpy clouds should be different from the line ratios in microturbulent clouds or LVG models and e.g. the column densities predicted by clumpy models should be higher. In order to study these differences we scaled the density and size of cloud F1 and produced spectra from a large density and column density range. The results for [FORMULA] (2-1) and the ratio [FORMULA] (7-6)/ [FORMULA] (2-1) are shown Fig. 14 as the function of the maximum cloud density and the beam averaged column density.

This graph can be compared with the LVG calculations of Zinchenko et al. 1994 (their Fig. 22). Our y-axis is the total column density integrated over the line and not N/ [FORMULA] as in the LVG graphs. In the optically thin case the FWHM of the CS line was in our calculations about 3.5km s-1. Therefore e.g. the column density value N=1.0 [FORMULA] 1014 cm-3 corresponds to N/ [FORMULA] =0.27 [FORMULA] 1014 cm-3 /km s-1 in the line centre. When this difference is taken into account our model predicts only slightly higher column densities than the LVG model. This may indicate that the cloud F1 is not clumpy enough (see Fig. 2 b) for the effects of clumpiness to show more clearly.

From Fig. 2 a we can see that the mean density of the cloud F1 is around 20% of the maximum value. The predicted mean densities are therefore lower than the densities predicted by the LVG model. The differences are larger at higher densities where the lines become saturated. At low column densities the value [FORMULA] (7-6)/ [FORMULA] (2-1)=1.2 is reached in the LVG model at density 3 [FORMULA] 107 cm-3 while the corresponding mean density in our model is lower by a factor of three.

6.4. Effects of kinetic temperature distribution

We have so far assumed in all our calculations a constant kinetic temperature. This is of course not true for real interstellar molecular clouds. The kinetic temperature may either increase or decrease with distance from the cloud centre depending on strengths of the internal and external heating mechanisms. The temperature structure can change both the line ratios and the line profiles and e.g. the self-absorption features are highly dependent on the temperature distribution.

At this stage, we have not carried out detailed investigations of these effects. Instead, we shall present only one example that shows some of the qualitative effects involved. The kinetic temperatures of the cloud in Fig. 15 a were modified with the temperatures either increasing or decreasing towards the cloud centre according to exponential laws [FORMULA] and [FORMULA]. The temperatures were then scaled to the ranges 30-60 K. The spectra towards these two model clouds are shown in Fig. 15 b together with spectra from the corresponding microturbulent clouds. These spectra can be compared also with the constant temperature model in Fig. 15 a with [FORMULA] =40.0 K.

The intensities in Fig. 15 b are higher in rightmost frames since in these cases the temperature in the dense centre region is higher. For this reason the line ratios are somewhat different. As expected, the self-absorption is stronger in clouds where the kinetic temperature drops towards the cloud surface. For the line profiles the differences between the microturbulent and macroturbulent models are larger, however.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998