Monte Carlo (MC) simulation is a very flexible method for carrying out radiative transfer calculations. From early on it has been used both for spectral line calculations (e.g. Magnan 1970) and for the simulation of light scattering (e.g. Sandford 1973; Mattila 1970). It was quickly recognized that with MC one is not limited to simple models e.g. the effects of inhomogeneities in the cloud structure could be studied (van Blerkom 1971). However, most of the models remained still relatively simple. In 1979 Bernes published an important article on the use of MC for molecular emission line calculations. The model clouds were spherically symmetric and had a uniform density.
Over the years, as computing power has increased, the models used in MC calculations have become more detailed. MC simulation is a very common tool in studies of light scattering (e.g. Whitney & Hartmann 1992; Fischer et al. 1994; Lehtinen & Mattila 1996; Voshchinnikov et al. 1996; Fischer et al. 1996). Over the past few years MC has been used extensively also in the study of line emission in e.g. comets (Rousselot et al. 1994), accretion disks (Knigge et al. 1995), masers (Spaans & Langevelde 1992), supernovae (Mazzali & Lucy 1993; Mazzali et al. 1995; Zhang & Wang 1996) and molecular clouds (e.g. Gonzáles-Alfonso & Cernicharo 1993; Choi et al. 1995).
CS observations of interstellar clouds have been modeled lately with MC calculations by Zinchenko et al. (1994) and Choi et al. (1995). In these studies the model clouds have been spherically symmetric i.e. essentially one-dimensional. Molecular clouds have, however, a much more complicated structure and the interest in the study of these inhomogeneities has increased. The effects of inhomogeneity on the temperature structure and chemistry of interstellar molecular clouds have already been studied by Boisse (1990) and Spaans (1996) with the aid of MC simulations.
In order to study the effects of small scale density fluctuations on molecular emission lines, the Monte Carlo procedure has been extended to full three-dimensional cloud models (Park & Hong 1995, 1996). The advances in computer technology have made it possible to study more realistic models. In the future further computer development is likely to increase the importance of the Monte Carlo method.
Monte Carlo calculations of the radiative transfer problem are based on the simulation of the basic physical processes with the aid of computer-generated random numbers. Compared with other methods of solution MC simulation is extremely simple since one has to deal only with the basic formulae of the simulated physical processes.
Besides simplicity MC has also other significant advantages in radiative transfer calculations. MC simulation does not require simplifying assumptions as in micro-turbulent (e.g. Leung & Liszt 1976; Liszt & Leung 1977) or large velocity gradient (LVG) models (e.g. Goldreich & Kwan 1974; de Jong et al. 1975). The velocity field as well as the the density distribution of the cloud can be arbitrary. Therefore clumpy clouds can be studied with essentially the same program as simple, spherically symmetric clouds.
The long CPU-time needed for the simulation is almost the only negative side of the MC method. The random error in the results is approximately inversely proportional to the square root of the number of simulated photons and therefore a substantially increased accuracy can be obtained only at the cost of much longer computing times. The situation gets worse with higher dimensions. If a cloud is discretized in three dimensions each simulated photon interacts only with a small fraction of all the cells. The dependence between the number of cells and the required execution time is over-linear. For these reasons one is forced to use a low resolution discretization for the studied cloud.
A reference field was used by Bernes (1979) as a way to reduce the errors associated with the use of random numbers. In addition to this method we have also made some other changes and additions to the original simulation scheme and have found that they may, in some cases, reduce the execution times significantly.
The aim of the present study is two-fold. In the first part the methods of radiative transfer simulation are studied. We introduce a new simulation scheme that should prove effective especially in the study of clumpy clouds. We will also discuss some implementation details which can be used to reduce the execution times needed for the simulation.
These principles are embodied into a program that serves as the basis for the second part of this study in which the effects of clumpy density distribution on the observed CS lines are studied. We are mainly interested in qualitative effects, e.g. the existence or absence of the self-absorption features, line broadening etc. Similar studies have already been conducted using the CO molecule (Park & Hong 1995). It is necessary, however, to extend this work to other molecular species and especially those probing higher densities. CS is particularly suitable for this study because it has many easily observable rotational transitions in the mm and sub-mm range.
The density structures of the model clouds will be generated either with structure trees or with fractal models. These distributions are clearly more realistic than what has been used in model calculations so far. The comparison of different density distributions is needed so that the general characteristics of the clumpy cloud structure can be identified. Our knowledge about the exact nature of the small scale clumpiness is still very limited. Comparison of different clumpy models should proove useful in resolving these uncertainties.
This study serves also as a preparation for a more detailed analysis of the CS and C34 S observations from southern massive star forming cores (Juvela 1996). In this future work the observations will be interpreted with the aid of model calculations and the correspondence between different cloud models and the observations will be addressed.
After a short presentation of the model clouds in Sect. 2we shall discuss the implementation of the simulation procedures in Sect. 3. A few test problems as well as some comparisons with earlier simulation programs will be shown in Sect. 4. The results from the molecular line calculations made with clumpy cloud models are shown in Sect. 5and the results are discussed in Sect. 6. The final conclusions are presented in Sect 7.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998