## 5. Clumpy cloudsUsing 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 ## 5.1. Fractal cloudsThe 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 macroturbulenceIn 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)=v 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
10
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
10
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 10 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
10
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
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 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 10 ## 5.1.2. Microturbulent cloudsIt 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 In the following the clouds have no macroturbulence but the
microturbulence has been increased to
= 1.5 km s
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
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 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 modelsIn 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 treesThe 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
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 ## 5.2.1. Clouds with macroturbulenceThe amount of microturbulence was first set to
=0.2 km s 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 10
One can again see large differences in the observed line ratios.
The beam-averaged column densities towards the cloud centres were 7.59
10 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
10 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.
## 5.2.2. Microturbulent cloudsWe show in Fig. 10 b spectra calculated with microturbulent
models where =1.5 km s 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
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 treesIn 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 |