Most astrophysical structures result from gravitational instabilities, from large-scale cosmological structures down to planets. Yet, among the least understood topics in astrophysics we find galaxy formation and star formation, which both involve fragmentation and the nonlinear growth of structures occurring during the non-linear phases of gravitational instability.
Perhaps one of the fundamental reasons why fragmentation and structure formation via gravitational instability appears so difficult is that we lack consistent theoretical tools allowing to combine gravity with gas physics. Indeed, it is too often ignored that classical thermodynamics does not hold for gravitating systems, because these are non-extensive in the thermodynamical sense (Landsberg 1972, 1984; Tsallis 1999; Plastino & Plastino 1999). Actually, many other natural systems do not respect the requisites of thermodynamics. Such systems often feature interesting phenomena such as growing long-range correlations or phase transitions. Among the symptoms of a fundamental deep problem in gravitating systems is the appearance of negative specific heat (Lynden-Bell & Lynden-Bell 1977; Lynden-Bell 1998), which was seen for a long time as a paradox in statistical mechanics, since negative specific heat was thought to be impossible.
Presently, the only available approach to follow the nonlinear phases of gravitational instabilities is to carry out numerical simulations. Among all the existing methods, N-body techniques are thought to be the most effective to simulate the continuous case as well as the granular phases of self-gravitating systems.
Yet, despite the considerable success of these methods in reproducing many observed features, many fundamental problems remain. As mentioned above, the fragmentation and structure formation are not clearly understood. Related to this, CDM simulations conflict with observations at galactic scales (Moore 1999; Bullock et al. 2001; Bolatto et al. 2002), and no theory of the ISM is presently able to predict the conditions of star formation. Most of the time the star formation process relies on recipes with few physical constraints.
In situations where N-body simulations are successful (e.g. hot stellar systems), gravitational dynamics is sufficient to account for their main global properties and additional microscopic physics can be neglected. But when gravitational instability via fragmentation involves small-scale physics, the outcome may be strongly dependent on the properties of the small-scale physics. In other words, in situations where the growth on singularities triggered by gravity is allowed, the chaotic nature of gravitating systems make them sensitive to the perturbations induced by non-gravitational physics.
Therefore it is important to understand the properties of N-body systems subjected to various perturbations. For these purposes, a numerical study of perturbed, self-gravitating N-body systems is carried out.
Among the relevant perturbations we expect that boundary conditions at small and large scales, as well as dissipative factors, can play a key role. In order to characterize the individual effects of perturbations, in the tradition of analytical models, one is advised to deliberately use simplified models.
A study of dissipative systems is important because such systems may develop long-range correlations. In the typical ISM, radiative cooling is very effective and induces a temporary energy flow leading the system far from equilibrium (Dyson & Williams 1997). From laboratory experiments it is well known that systems outside of equilibrium may spontaneously develop spatio-temporal structures (Glansdorff & Prigogine 1971; Nicolis & Prigogine 1977; Prigogine 1980; Melo 1994).
A permanent energy flow is induced when energy loss due to dissipation is replenished, that is, when the system is continuously driven, e.g., by time-dependent boundary conditions. Such systems may develop persistent long-range correlations. Astrophysical examples of this are the growth of structures in cosmological simulations or the long-term persistence of filamentary structures in shearing flows (Toomre & Kalnajs 1991; Huber & Pfenniger 2001a; Wisdom & Tremaine 1988; Salo 1995; Pfenniger 1998). Among other things, the effect of time-dependent potential perturbations on dissipative self-gravitating spheres is studied in this paper.
In the next section we briefly review some theoretical results of the thermodynamics of self-gravitating isothermal spheres. The model is presented in Sect. 3 and the applied methods to carry out the structure analysis for the simulated systems are explained in Sect. 4. The results are presented in Sects. 5-7. In Sect. 5 quasi-equilibrium states of N-body models are compared with analytical predictions. Section 6 is dedicated to a study of long-range correlations appearing in the interval of negative specific heat during the collapsing transition of gravitating systems. Finally, in Sect. 7 the evolution of systems subjected to an energy-flow is discussed.
Copyright ESO 2002