5. Clumpy clouds
Using the models described in Sect. 2it is possible to create a large variety of clumpy model clouds for the radiative transfer simulations. In this section we shall present some results of the calculations made with clouds created either with the structure tree model or with the fractal model.
The model clouds consist of 36 36 36 cubic cells. All radiative transfer computations were made using the simulation method B and the number of rotational levels included into the calculations was 11. The lines were divided into 0.25 km s-1 wide channels both during the simulation and when calculating the spectrum. The spectra were calculated by convolving the emitted intensity with a gaussian beam. The FWHM of this beam was 6 times the length of the side of an individual cell. In the following the terms cloud diameter and cloud radius will refer to the cloud size along the edge of the cubic cloud.
5.1. Fractal clouds
The method used to create the fractal model clouds (see Sect. 2) has only one parameter, . The fractal dimension determined from the column density maps was found to depend only weakly on this. The density distribution is more affected with the proportion of denser cells increasing with larger parameter values. It should be noted that the shape of the density spectrum is completely unrelated to the fractal dimension. The fractal dimension is usually determined from the slope of the (log(A),log(P))- relation i.e. from the relation between areas and perimeters of the structures seen in the column density map at different column density levels. The relation is approximately linear and any monotonic transformation of the cell densities will preserve the fractal dimension. Also, for the observed line profiles the mass spectrum is probably much more important.
We present here calculations performed with three fractal clouds, F1, F2 and F3. The clouds have been created using different values of the parameter : =0.1 for F3, =0.4 for F1 and =0.7 for F2. All clouds have fractal dimensions of about 1.3 and are in this respect similar to real clouds. The fractal dimension was determined from the (log(A),log(P))- relation, but since the column density maps are relatively small, 36 36 pixels, the value of the dimension is consequently somewhat uncertain. We feel, however, that here the fractal dimension is not very important by itself.
The mass spectra of the three clouds are shown in Fig. 2 a. In Fig. 2 b the column density map of the cloud F1 is shown as an example. During each run the densities of the cloud cells were scaled to desired density ranges with a linear transformation. This scaling does not change the shape of the mass spectrum nor the fractal dimension.
5.1.1. Clouds with macroturbulence
In the model clouds the turbulence consists of two components: the turbulence within the cells and the random motions of the cells themselves. The first of these is reflected only in the intrinsic linewidths which were taken to be the same in all cells. This microturbulent velocity dispersion was set to =0.2 km/s. The macroturbulence was created by assigning to each cell a random velocity component generated from a normal distribution with =1.2 km/s. The linewidth is therefore mainly due to these cell velocities with smaller contributions from the microturbulence and the thermal line broadening. A small infall velocity v(r)=v0 was also added to the clouds with =0.5 and the velocity at the cloud surface v0 =0.15 km s-1. The results will therefore also indicate under what conditions such a velocity field can be deduced from the observed line profiles.
The results from the initial runs using these model clouds are shown in Fig. 3. The kinetic temperature of the clouds was set to a constant value of =40 K and the calculations were performed with the maximum densities scaled to 1.0 105 cm-3, 5.0 105 cm-3 and 1.0 106 cm-3. The cloud size was 0.23 pc 0.23 pc, which would correspond to , if the cloud was at the distance of 1000 pc. The spectra at the central positions of the clouds as well as at distance of 1/3 and 2/3 times the cloud radius are shown.
The differences between the model clouds are mainly due to different column densities which in turn result from the differences in the mass spectra. Using a beam with FWHM twice the value used in calculating the spectra the beam averaged column densities were 8.04 1013, 1.51 1014, and 3.06 1013 at the centre positions of F1, F2, and F3, respectively when the maximum densities were scaled to 1.0 105 cm-3. The differences between the models can be seen also by comparing the optical depths. The optical depths in the line centre are listed in Table 1.
Table 1. The parameters of the fractal cloud models. The columns are: the model cloud (column 1), the maximum density in the cloud (column 2), kinetic temperature (column 3), column density towards the cloud centre (column 4), the optical depths in the line centre towards the cloud centre (columns 5-8) and the figure in which the spectra are shown (column 9)
Cloud F3 has the lowest column densities and no self-absorption can be seen in in its spectra. In clouds F1 and F2 some line profiles are flat-topped due to self-absorption. The effect is strongest for the transitions 3-2 and 5-4 which, in F2, are clearly self-absorbed when the maximum density is 5.0 105 cm-3. At higher densities these lines have clearly non-Gaussian profiles also in F1. Because of the large macroturbulence the line profiles are not very smooth and it is difficult to deduce the existence of the infall motion from the small asymmetry in the lines. The line ratios, for a given model and maximum density, remain nearly constant in all positions.
In Fig. 4 we show the centre point spectra with the kinetic temperature of the clouds set to 20 K. As expected the relative intensity of the transitions CS(2-1) and CS(3-2) has increased. These lines show also stronger self-absorption features at this lower temperature. The change in the kinetic temperature has, however, somewhat different effects in the three model clouds. The line intensities are least affected by the drop in the kinetic temperature in the cloud F3 which has the lowest average density. As the mean density of the model increases the changes in the spectra get more noticeable. Apparently the cloud F3 is optically thin to such an extent that the excitation temperatures are not strongly affected by the kinetic temperature while e.g. cloud F2 is close to thermalization due to its higher average density. While the drop in temperature has generally decreased the intensity of the transitions 5-4 and 7-6, the line CS(5-4) is still flat topped in both high density models, F1 and F2.
It is difficult to say to what extent the differences in the observed spectra are due to the different density distributions in the clouds and to what extent to the different column densities. To study this question we scaled the physical sizes of clouds F1 and F3 in such a way that the column densities at the centre positions of the clouds (averaged over a beam with FWHM twice that used for spectrum calculations) were approximately the same as for F2 in Fig. 3b when the maximum density was scaled to 5.0 105 cm-3. The linear size of F1 was therefore increased by a factor of 1.8 and the size of F3 by a factor of 4.7. Since the beam size is much smaller than the size of the clouds, the differences between spectra from different clouds should reflect only the effects of the density distribution along the line of sight. The spectra are shown in Fig. 5.
Since the model F2 has the highest average density it has also the strongest emission in all transitions. The enlarged column densities of clouds F1 and F3 do increase the intensities of all transitions but increased self-absorption and increased line-widths are also evident.
Real clouds may have very low volume filling factors f and in many regions of massive star formation the volume filling factor has been estimated to lie in the range 0.1-0.2 (e.g. Zhou et al. 1994; Mundy et al. 1986). Although the fractal model clouds have a large fraction of low density cells, the effective volume filling factor is still rather high. To study the effect of smaller f we modified the clouds by removing a number of randomly selected cells from the clouds. This procedure may also change the fractal dimension of the clouds. We selected only cloud F1 for this study. In Fig. 6 we show the spectra from F1 with the maximum density scaled to 5.0 106 cm-3. The three frames show the spectra from the central position when the volume filling factor was set to 0.05, 0.15 and 0.4.
The volume filling factor has the expected effects. Although the density has been increased from Fig. 3a the lines are generally less self-absorbed. The lines become wider with increasing volume filling factor. Note that in the last frame all lines are much stronger than in Fig. 3a and the line profiles are still almost Gaussian. The low volume filling factor combined with the existence of macroturbulent motions seems therefore to be effective in reducing self-absorption. The changes in the volume filling factor have also affected the line ratios. The intensity of CS(2-1) clearly increases with increasing f while the relative intensity of CS(5-4) has dropped slightly.
The small-scale structure in the line profiles is entirely due to the macroturbulence. As the volume filling factor gets lower the spectra from individual cells are averaged with fewer other cells and the line profile appears `clumpy'. This sub-structure is of course entirely dependent on the cell size used in the model cloud. Since the lines are emitted in different fashion from cells with different densities, the profiles of different transitions may also deviate significantly from each other (see e.g. lines CS(2-1) and CS(7-6) in Fig. 6).
In Fig. 6 we show similar spectra from the cloud F1 with lower maximum density, 1.0 106 cm-3. The `noise' in the profiles of e.g. CS(5-4) seem a little higher than in the case of higher gas density. This probably reflects the fact that fewer cells have the densities required to emit these transitions.
5.1.2. Microturbulent clouds
It is also interesting to compare the calculations with the results from microturbulent models. In the previous calculations the microturbulence was only =0.2 km s-1 while the macroscopic random motions of the cells were mainly responsible for the observed linewidths.
In the following the clouds have no macroturbulence but the microturbulence has been increased to = 1.5 km s-1. The line-widths should be comparable to the earlier calculations with macroturbulent clouds. In Fig. 7 a we present calculations with model clouds that are, apart from the turbulence, exactly the same as in Fig. 3. The infall velocity still exists in the clouds so that the model is not purely microturbulent. The infall velocity is, however, much smaller than the linewidths.
The most important difference between Fig. 7 a and the macroturbulent case is the marked increase in the self-absorption. In the microturbulent clouds the self-absorption features set in at lower densities, and at higher densities the spectra have clear absorption dips instead of being just flat-topped. With the absence of the noise from the macroturbulence and due to the increased self-absorption, the line asymmetry caused by the infall motion can also be clearly seen. All line intensities have remained approximately the same as in Fig. 3 and the differences are mainly linked to the increased absorption (e.g. cloud F2, lines CS(3-2) and CS(2-1)).
The results of microturbulent models corresponding to the macroturbulent ones presented in Fig. 4 are shown in Fig. 7 b. These clouds have a kinetic temperature of =20 K. Here also the differences from the macroturbulent case are visible even at the lowest densities.
Comparison between microturbulent models with different kinetic temperatures has similarities with the previous comparison between macroturbulent models with different temperatures. The lowering of the temperature does not radically change the line ratios but e.g. the self-absorption of CS(2-1) has increased.
To see the effect of the volume filling factor more clearly we made also comparisons with some clouds with f =1.0. For this purpose we calculated the spectra for spherically symmetric, one-dimensional models. The clouds are divided into 30 shells, all with equal thickness. The density was set according to law n and scaled approximately to the same range as in the corresponding fractal cloud. The beam averaged column densities were calculated at the cloud centre as well as at distances corresponding to 1/3 and 2/3 of the radius of the three-dimensional cloud. The cloud parameters were modified in such a way that the column densities agreed to within 10% with the corresponding values from the three-dimensional fractal cloud. The clouds should therefore have similar large-scale density distributions as their three-dimensional counterparts. In Fig. 8 we show the spectra from the one-dimensional cloud corresponding to the three-dimensional model calculations shown in Fig. 6 (upper frames).
The correspondence between the column densities of the one-dimensional and the three-dimensional cloud cannot be exact since the density structure of a fractal cloud does not obey exponential laws. Also, the mass spectra of 1D and 3D clouds differ and therefore the differences in the calculated spectra are not just caused by differences in f. These complications must be kept in mind when comparisons are made between different models.
The self-absorption of the spectra in Fig. 8 is very strong even though the line intensities are similar to or lower than in the spectra from the three-dimensional clouds with similar column and volume densities. This can be seen as further evidence for the fact that clumpy clouds produce spectra with less self-absorption. However, direct comparison between the spectra is difficult as can be seen from e.g. the different line ratios. The differences in the mass spectra and the cloud sizes are clearly responsible for some of the differences.
5.1.3. Summary of the calculations with fractal cloud models
In Table 1 we list the parameters of all the fractal models for which the calculations were described in the previous chapters. The table lists also column densities and the optical depths through the cloud in the direction of the cloud centre. The optical depths are calculated for the line centre. Note also that the values are calculated along a single line of sight and e.g. the column densities might vary significantly even between close positions.
5.2. Clouds created with structure trees
The column density maps of three clouds created with structure trees are shown in Fig. 9. The basic density distribution of the clouds and the added clumps is n r-0.5. Since the density distribution was also used as the probability distribution for the position of the sub-clumps the column density clearly peaks close to the centre of the maps. In all created clouds the density of each sub-clump is five times the density of parent clump.
Table 2. Parameters of the structure trees used to create the model clouds T1, T2 and T3. The branching factor is the number of sub-clumps within a clump, N is the number of levels in the structure tree with the root cloud included as one of the levels and is the ratio of the linear sizes between a sub-clump and a clump
Cloud T1 has a branching factor of =5 and each clump has a size 0.5 times that of its parent. Due to the relatively coarse discretization there are only four levels in the structure tree and the smallest clumps have a diameter of 4.5 cells. Cloud T2 is similar to T1 except that it has a branching factor of 10 and the size ratio between sub-clumps and clumps is 0.3.
The last cloud, T3, has a branching factor of 100 and the structure tree has only two levels. The cloud consists therefore of 100 clumps with a radius of 3.6 cells. The clumps are generated preferentially close to the cloud centre with a probability r-0.5 when cells, and a constant probability inside this radius. This means that in T3 the central region is practically filled with high density gas and the volume filling factor drops only close to the the cloud surface. The clumps are allowed to intersect but the density of the intersection is updated only once and therefore the density in the central region should be almost constant.
5.2.1. Clouds with macroturbulence
The amount of microturbulence was first set to =0.2 km s-1 and the macroturbulence to =1.2 km s-1. A small infall velocity was again added as v(r)=v0 , where =0.5 and the infall velocity at the cloud surface v0 =0.15 km s-1.
We present the spectra calculated towards the central positions of the clouds in Fig. 10 a. The maximum densities of all clouds were scaled to 1.0 106, 5.0 106 and 1.0 107 cm-3. The clouds have a size of 0.058 pc in all three dimensions. These clouds are therefore both smaller and denser than the fractal clouds.
One can again see large differences in the observed line ratios. The beam-averaged column densities towards the cloud centres were 7.59 1013 cm-2 for T1, 8.73 1013 cm-2 for T2, and 1.55 1014 cm-2 for T3 when the maximum density of the clouds was scaled to 1.0 106 cm-3. When the structure tree has only two levels, as is the case for T3, the proportion of high density cells is rather high. As soon as the structure tree has more levels the density differences between the densest clumps and the ambient cloud become large and, since the densities in all clouds were scaled to the same range, clouds T1 and T2 have lower average densities. These differences are reflected in the relative intensity and the self-absorption of the line CS(7-6).
The highest density range used for the fractal clouds was the same as the lowest density range here i.e. from zero to 1.0 106 cm-3. Comparing the corresponding spectra here and in Fig. 3 we notice that the line ratios have indeed remained about the same. For example models F3 and T3 seem to be very close to each other.
Despite the higher densities the optical depths are somewhat lower than in the fractal clouds. The optical depths of the models are listed also Table 3.
Table 3. The parameters of model clouds in which the density distribution was generated using structure trees. The columns are: the model cloud (column 1), the maximum density in the cloud (column 2), kinetic temperature (column 3), column density towards the cloud centre (column 4), the optical depths in the line centre towards the cloud centre (columns 5-8) and the figure in which the spectra are shown (column 9)
5.2.2. Microturbulent clouds
We show in Fig. 10 b spectra calculated with microturbulent models where =1.5 km s-1. The infall velocity is the same as in the previous clouds so that the models are not purely microturbulent. The densities are scaled to the same ranges as in in Fig. 10 a.
The same effects can be seen here as in Fig. 7 a i.e. the spectra show increased self-absorption and larger line-widths compared with the macroturbulent case. Changes in the line ratios are also mainly correlated with the self-absorption.
In Fig. 11 we show again some spectra calculated with one-dimensional models. The model cloud consists of 30 spheres of equal thickness. The densities are scaled to the same ranges as in the three-dimensional cloud T2 in Fig. 10 a. The radial density dependence is n r-1.0 and the cloud sizes have been scaled so that the beam-averaged column densities are similar at the cloud centres as well as at distances of 1/3 and 2/3 of the radius of the three-dimensional cloud. The column densities of the 1D and 3D clouds agree within a few percent in all three positions.
The lines calculated from the 1D cloud are again very strongly self-absorbed compared even with the microturbulent spectra. The intensities of the lines are generally similar in the 1D and the 3D clouds. This seems to indicate that the correspondence between the 1D and 3D models is rather good. The strong self-absorption is, however, partly due to the existence of low density gas with the density systematically dropping towards the cloud surface. Together with microturbulent velocity structure this gives the best conditions for self-absorption.
5.2.3. Summary of the clouds created with structure trees
In Table 3 we list the parameters of the model clouds created using structure trees. The basic models T1, T2 and T3 have different density distributions and the actual model clouds were created by scaling these densities and by setting different values of turbulence etc. The table lists also the column densities and the optical depths through the clouds calculated along a single line of sight. The optical depths are the values for the line centre.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998