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

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1. Introduction

From many studies over the last two decades, it has been established that the interstellar medium (ISM) has a clumpy hierarchical structure, approaching a fractal structure independent of scale over 4 to 6 orders of magnitude in sizes (Larson 1981, Scalo 1985, Falgarone et al. 1992, Heithausen et al. 1998). The structure extends up to giant molecular clouds (GMC) of 100 pc scale, and possibly down to 10 AU scale, as revealed by HI absorption VLBI (Diamond et al. 1989, Faison et al. 1998) or extreme scattering events in quasar monitoring (Fiedler et al. 1987, Fiedler et al. 1994). It is not yet clear which mechanism is the main responsible for this structure; it could be driven by turbulence, since the Reynolds number is very high, or self-gravity, since clouds appear to be virialized on most of the scales, with the help of magnetic fields, differential rotation, etc...

One problem of the ISM turbulence is that the relative velocities of clumps are supersonic, leading to high dissipation, and short lifetime of the structures. An energy source should then be provided to maintain the turbulence. It could be provided by star formation (stellar winds, bipolar flows, supernovae, etc... Norman & Silk 1980). However, the power-law relations observed between size and line-width, for example, are the same in regions of star-formation or quiescent regions, either in the galactic disk, or even outside of the optical disk, in the large HI extensions, where very little star formation occurs. In these last regions alternative processes of energy input must be available. A first possibilty is the injection by the galactic shear. Additionally, since there is no heating sources in the gas, the dense cold clumps should be bathing at least in the cosmological background radiation, at a temperature of 3 K (Pfenniger & Combes 1994). In any case, in a nearly isothermal regime, the ISM should fragment recursively (e.g. Hoyle 1953), and be Jeans unstable at every scale, down to the smallest fragments, where the cooling time becomes of the same order as the collapse time, i.e. when a quasi-adiabatic regime is reached (Rees 1976).

In this work, we try to investigate the effect of self-gravity through N-body simulations. We are not interested in star formation, but essentially in the fractal structure formation, that could be driven essentially by gravity (e.g. de Vega et al. 1996). Our aim is to reach a quasi-stationary state, where there is statistical equilibrium between coalescence and fragmentation of the clouds. This is possible when the cooling (energy dissipated through cloud collisions, and subsequent radiation) is compensated by an energy flux due to external sources: cosmic rays, star formation, differential shear... Previous simulations of ISM fragmentation have been performed to study the formation of condensed cores, some with isolated boundary conditions (where the cloud globally collapses and forms stars, e.g. Boss 1997, Burkert et al. 1997), or in periodic boundary conditions (Klessen 1997, Klessen et al. 1998). The latter authors assume that the cloud at large scale is stable, supported by turbulence, or other processes. They follow the over-densities in a given range of scales, and schematically stop the condensed cores as sink particles when they should form structures below their resolution. We also adopt periodic boundary conditions, since numerical simulations are very limited in their scale dynamics, and we can only consider scales much smaller than the large-scale cut-off of the fractal structure. We are also limited by our spatial resolution: the smallest scale considered is far larger than the physical small-scale cut-off of the structures. Our dynamics is computed within this range of scales.

Our goal is to achieve long enough integration time for the system to reach a stationary state, stationary in a statistical sense. This state should have its energy confined in a narrow domain, thus presenting the density contrasts of a fragmented medium while avoiding gravitational collapse after a few dynamical times. Only when those conditions are fulfilled, can we try to build a meaningful model of the gas. Let us consider for example the turbulence concepts.

Theoretical attempts have been made to describe the interstellar medium as a turbulent system. While dealing with a system both self-gravitating and compressible, the standard approach has been to adapt Kolmogorov picture of incompressible turbulence. The first classical assumption is that the rate of energy transfer between scales is constant within the so-called inertial range. This inertial range is delimited by a dissipative scale range at small scales and a large scale where energy is fed into the system. In the case of the interstellar medium, the energy source can be the galactic shear, or the galactic magnetic field, or, on smaller scales, stellar winds and such. In this classical picture of turbulence, one derives the relation [FORMULA], where [FORMULA] is the velocity of structures on scale L. If we consider now that the structures are virialized at all scales, we get the relation [FORMULA], where [FORMULA] is the density of structures on scale L. This produces a fractal dimension [FORMULA]. A more consistent version is to take compressibility into account in Kolmogorov cascade; then [FORMULA]. Adding virialization, we get the relations [FORMULA] and [FORMULA]. This produces the fractal dimension [FORMULA]. It should be emphasized again that all these scenarios assume a quasi-stationary regime.

However, numerical simulations of molecular cloud fragmentation have been, so far, carried out in dissipative schemes which allow for efficient clumping (e.g Klessen et al. 1998, Monaghan & Lattanzio 1991). As such, they do not reach stationary states. Our approach is to add an energy input making up for the dissipative loss. After relaxation from initial conditions we attempt to reach a fragmented but non-collapsing state of the medium that does not decay into homogeneous or centrally condensed states. It is then meaningful to compute the velocity and density fields power spectra and test the standard theoretical assumptions. It is also an opportunity to investigate the possible fractal structure of the gas. Indeed, starting from a weakly perturbed homogeneous density field, the formation time of a fractal density field independent of the initial conditions is at the very least of the order of the free-fall time, and more likely many times longer. A long integration time should permit full apparition of a fractal mass distribution, independent of the initial conditions.

This program encounters both standard and specific difficulties. Galaxies clustering as well as ISM clumping require large density contrasts, that is to say high spatial resolution in numerical simulations. This point is tackled, be it uneasily, by adaptative-mesh algorithms or tree algorithms, or multi-scale schemes ([FORMULA]). Another problem is the need for higher time resolution in collapsed region. This is CPU-time consuming or is dealt with by multiple time steps. This point is of primary sensitivity in our simulation since we need to follow accurately the internal dynamics of the clumps and filaments to avoid total energy divergence. Finally we need a long integration time to reach the stationary state. This is directly in competition for CPU-time with spatial resolution for given computing resources.

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Online publication: August 23, 2000