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

2. The model clouds

We have made calculations with clouds of three different geometries. The simplest are spherically symmetric and therefore essentially one-dimensional. The second type is cylindrically symmetric (i.e. two-dimensional) and the third, three-dimensional. In this paper we shall concentrate on the three-dimensional models with some results on the one-dimensional clouds shown for comparison.

For the radiative transfer calculations one must first set the density, velocity field, turbulence and kinetic temperature in each cell of the cloud. These quantities remain unchanged during the calculation and only the relative populations of the different energy levels are changed.

We have used two methods to generate the density distributions of the three-dimensional clouds. These are based on the structure-tree statistics of Houlahan & Scalo (1992) and the fractal model of Hetem Jr. & Lépine (1993). Both methods are based on the the observed properties of the molecular clouds and are not directly linked to theoretical hydrodynamic models.

Structure-tree statistics describes the cloud as a hierarchical structure of clumps. The making of the model cloud starts with one big clump and proceeds recursively as sub-clumps are generated within each clump. The density distribution of a clump is identical to that of the parent with the size scaled down with some factor, , and the centre density multiplied with another factor, . In our models the clumps have a density distribution , where r is the distance from the centre of the clump, R the size of the structure and a free parameter.

It is well known that molecular clouds show similar structure on many different scales (see e.g. Dickman et al. 1990; Falgarone et al. 1991). This fractal nature of the clouds seems to extend down to the smallest observable scales, i.e. at least down to 0.01 pc. The fractal dimension of most interstellar clouds is close to 1.3 (see e.g. Dickman et.al. 1990).

We calculate the density of fractal model clouds according to the Model 1 of Hetem & Lépine (1993). The algorithm has only one free parameter, , which determines the fractal dimension of the cloud. The algorithm starts by dividing a cubic cloud into eight sub-cubes with the mass of the cloud divided between the sub-cubes according to random numbers. The procedure is repeated recursively on all sub-cubes. In order to force the density to increase towards the centre of the cloud we have used the variation in which after each division the densest sub-cell is moved closest to the centre and the least dense cell furthest away (see Hetem & Lépine 1993).

The turbulence in the model clouds is divided into two parts. Microturbulence is the velocity dispersion within a cell and together with the kinetic temperature of the gas it determines the intrinsic line width. Macroturbulence is the random velocity component assigned to a cell. The assignment of turbulence to the model clouds is discussed in more detail in Sect.  5.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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