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Astron. Astrophys. 360, 1096-1106 (2000)

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Appendix A: computing the fractal dimension

By fractal dimension we mean more accurately the correlation fractal dimension. We will use the notation [FORMULA], defined as follows: if we consider a fractal set of points of equal weight located at [FORMULA], the fractal dimension of the ensemble obeys the relation:

[EQUATION]

The brackets stand for an ensemble average. In computational applications and in physical systems, the [FORMULA] limit is not reached. However the relation should hold at scales where the boundary effects created by the finite size of the system are negligible.

In practice, for a non-fractal ensemble of points, [FORMULA] usually depends on r (but not always). For a fractal it is independent of r, or at least it oscillates around a mean value. The latter case can happen for a set with a strong scale periodicity like the Cantor set. Examples are given in Fig. A.1. As the above remarks show, [FORMULA] is not a definitive criterion of fractality.

[FIGURE] Fig. A1. The fractal correlation dimension is plotted as a function of the scale for two different sets of points: the first (not fractal) set is an isothermal sphere and a second is a fractal cantor-like set build recursively on 8 levels with a small part of randomization. This shows that [FORMULA] independent of r is neither a sufficient nor a mecessary condition of fractality

According to the definition, the computation of the fractal dimension is based on the following method. [FORMULA] is the mean mass in a sphere of radius r centered on a particle. This mass is actually computed for each particle using the tree search, then it is averaged and the value [FORMULA] is derived. The average can be made on a subset of points to lower the computation cost, the result is then noisier.

Appendix B: clump mass spectrum

Although the results are not very conclusive, we give the mass distribution of the clumps in the 3-D simple freefall, to allow comparison with observational data and other numerical simulations. Plotting [FORMULA] against [FORMULA], the slope infered from observation data is -0.5. Thus the clump mass spectrum behaves as a power law: [FORMULA].

To produce the mass spectrum, the first step is to define individual clumps. We have used an algorithm very similar to the one proposed by Williams et al. (1994). We compute the density field from an interpolation of the distribution of the particles with a cloud-in-cell scheme. It should be mentione d that this procedure tends to produce an artificially high number of very small clumps associated to isolated particles or pair of particules which should not be considered as clumps at all.

On Fig. B.1 the clump mass spectrum is given at different times of the evolution and for the two types of initial conditions ([FORMULA]). Quasi-periodic boundary conditions are used. As time increases, as we can see, the clump mass spectrum slope decreases to reach [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. At later times, the power law is broken by the final dissipative collapse. According to this diagnosis, the case [FORMULA] is closer to the observed values. However, once again, this simulation provides only a transient state which is unlikely to be a good model of a GMC.

[FIGURE] Fig. B1. Clump mass spectra for two values of [FORMULA] at different evolution times. The given time-unit is about 1/10 of the free fall time. At [FORMULA] the spectra show a [FORMULA] slope for [FORMULA] and a [FORMULA] for [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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