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

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4. Simple 2-D free falls

We consider a 2-D system where particles have only two cartesian space coordinates but still obey a [FORMULA] interaction force.

4.1. Motivations for the two dimensional study

Turning to 2-D simulations allows access, for given computing resources, to a wider scale-range of the dynamics. Or, for given computing resources and a given scale-range, to larger integration times. The later is what we will need in the simulation with energy injection. In practice we will choose to use [FORMULA] particles in 2-D simulations to access both a larger integration time and a wider scale range. Indeed, between a 3-D simulation with [FORMULA] particles and a 2-D simulation with [FORMULA] particles one improves the scale range from [FORMULA] to [FORMULA]. At the same time one can increase the integration-time by a factor of [FORMULA] at equal CPU cost.

We can also add a physical reason to these computational justifications. The simulations with an energy injection aim to describe the behaviour of molecular clouds in the thin galatic disk. This medium has a strong anisotropy between the two dimensions in the galactic plane and the third one. As such it is a suitable candidate for a bidimensional modeling.

Moreover, studying a bidimensional system will allow us to check how the fractal dimension, dependent or not of the initial conditions, is affected by dimension of the space.

4.2. Simulations with 2 different dissipative schemes

Here again initial conditions with a power law density power spectrum are used. Exponent [FORMULA] is chosen to allow a comparison with the 3-D simulations. Boundary conditions are quasi-periodic. We have carried out two simulations with the two different dissipative schemes described in Sect. 2.4. The fractal dimensions at different stages of each of these two simulations are shown in Fig. 3.

[FIGURE] Fig. 3. Fractal dimension as function of scale at different stages of the time evolution. The simulations are two-dimensionnal with 10609 particles. The dissipative scheme is modified between the two simulations according to the description in Sect. 2.4.

In both simulations the fractal dimension of the homogeneous initial density field is 2 at scales above mean inter-particle distance and 0 below. Then the density fluctuations develop and the fractal dimension goes down to about [FORMULA] in both simulations. This value was not reached in 3-D simulations. This shows that the dimension of the space has an influence on the fractal dimension of a self-gravitating system evolved from initial density fields with a power law power spectrum. Indeed spherical configurations are favoured in 3 dimensions and may produce a different type of fractal than cylindrical configurations which are favored in two dimensions.

Another point is that the fractal dimension of the system at small scales is indeed sensitive to the dissipative scheme. The second scheme even alters the fractality of large scales. So even if we believe that the fractality of a self-gravitating system is controlled by gravity, we must keep in mind the fact that the dissipation has an influence on the result. In this regard, discrepancies should appear between N-body simulations including SPH or other codes.

The clump mass spectra are not conclusive due to insufficient number of clumps formed with 10609 particles.

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000