3. Simple 3-D free falls
3.1. The choice of the boundary conditions
A fractal structure such as observed in the ISM or in the galaxies distribution obeys a statistical translational invariance. If we run simulations with vacuum boundary condition, this invariance is grossly broken. On the other hand the quasi-periodic or fully-periodic conditions restore this invariance to some extent. They have been used in cosmological simulations. Hernquist et al. (1991) have compared the two models to analytical results and found that the fully-periodic model simulates more accurately the self-gravitating gas in an expanding universe. This is in agreement with expectations, since uniform expansion is automatically included in a fully-periodic model, while it is not in a quasi-periodic one. Klessen et al. (1998) and Klessen & Burkert (1999) applied Ewald method to simulations of the interstellar medium. This is not natural since expansion is necessary to validate Ewald method. Nevertheless one can argue that the dynamics is not altered by the use of Ewald method in regions with high density contrasts.
Hernquist's study was on the dynamics of the modes. We are more interested in the fractal properties. As far as we are concerned the question is wether the same initial conditions produce the same fractal properties in a cosmological framework (fully-periodic conditions) and in a ISM framework (quasi-periodic conditions). We will try to answer this question in Sect. 3.3.
3.2. The choice of the initial conditions
The power spectrum of the density fluctuations is most commonly described as a power law in cosmological simulations. Then, different exponents in the power law produce different fractal dimensions. The comoslogist can hope to find the fractal dimension from observations and go back to the initial power spectrum. In the case of the interstellar medium however, it is unlikely that the fractal dimension is the result of initial conditions since the life-time of the medium is much longer than the dynamical time of the structures. In this regard, all initial conditions should be erased and produce the same fractal dimension. We will check whether this is the case for power law density spectra with different exponents. Simple free falls with such initial conditions will allow the comparison between the two types of periodic boundary conditions and will allow us to compare between 3D and 2D models and different dissipation schemes.
3.3. Simulations and results
The four first simulations are free falls from power law initial conditions in either fully or quasi-periodic boundary conditions for two different power law exponents: and . About particles were used in the simulations. The time evolution of the particle distribution for and fully periodic condition is plotted in Fig. 1. If we compare the qualitative aspect of the matter distribution with those found in SPH simulations (Klessen et al. 1998), a difference appears. Filaments are less present in this N-body simulation than in SPH simulations. We believe that this is due to the strongly dissipative nature of SPH simulations. Indeed filaments form automaticaly, even without gravity , with initial conditions for the density field. Then, the strong dissipation of SPH codes is necessary to retain them; N-body codes seem to show that gravity by itself is not enough.
3.3.1. Fractal dimension
For each of the four simulations, the fractal dimension as a function of scale has been computed at different times of the free fall. Results are summarized in Fig. 2.
The fractal dimension of a mathematical fractal would appear either as an horizontal straight line or as a curve oscillating around the fractal dimension if the fractal has no randomization (like a Cantor set). For our system, it appears that at sufficiently early stages of the free fall, and for scales above the dissipative cutoff, the matter distribution is indeed fractal. However the dimension goes down with time to reach values of 2 for , and between 2 and 2.5 for . This is the fractal dimension for the total density, not for the fluctuations only. Stabilization of the fractal dimension marginally happens before the dissipation breaks the fractal.
One conclusion is however reasonable: as far as the fractal dimension is concerned, the difference between fully-periodic boundary conditions and quasi-periodic boundary conditions is not important. The deviation is a bit stronger in the case, which is understandable since produces weaker density contrasts in the initial conditions than and so the dynamics is less decoupled from the expansion. The effect of this conclusion is that we will not use the fully-periodic boundary conditions in further simulations since they require more CPU time and have no physical ground in the ISM framework. Clump mass spectra are given in appendix B.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000