Finite self-gravitating systems

We have studied in detail numerical simulations, and some simple analytical results, of what appears at first sight perhaps a very simple initial condition which one might consider naturally in the study of the virialization of self-gravitating systems: a large number N of point particles are randomly distributed in a sphere, and evolved from a cold (i.e. zero velocity) start. The “simplicity” of the family of initial conditions is that they are characterized by a single parameter, the particle number N (as the system is open the units may be defined by the system size, its total mass and Newton’s constant G). Further as N→∞ the initial conditions tends to that considered as a starting point for analyses of non-linear structure formation in cosmology — the “cold spheric al collapse” model, in which a perfectly uniform spherical overdensity embedded in an expanding universe is considered.

A random sampling with particles of such a flat density profile is the simplest discrete realisation of this theoretical model one can envisage, and might thus be expected perhaps to be the subject of much study. In practice this initial condition has been but little studied  because of the intrinsic difficulty in its numerical integration which is related to a property of the uniform limit: as N→∞ the evolution of the system leads to a singularity after a finite time,as all the mass arrives at the origin after a time t coll ∼ 1/√Gρ where ρ is the initial mass density. While for any finite N system the singularity does not occur, the typical size of th e region the system contracts to before “turn-around” decreases as N increases, and the typical particle velocities grow.

This makes the numerical integration very costly, and limits greatly the accessible particle N compared to other warm or less homogeneous initial conditions. One motivation for this study comes thus from the “uniform spherical collapse model”: when calculating predictions for the masses and abundances of halos in the framework, one of the critical assumptions is that all the mass and energy in the initially collapsing region is ultimately virialized in the collapsed structure. The question arises as to whether this is generically true independently of the initial conditions. In the present case — which we have noted is, in a simple sense, the “closest” initial condition to the exactly uniform case — it turns out, interestingly, that this is not a good approximation: the violence of the collapse leads to an ejection of energy from the system (as kinetic energy of particles which escape with positive energy). Our numerical analysis, coupled to an analytical scaling argument, lead us to conclude that the ejected energy is in fact unbounded above as N increases, so that these collapses can be characterized as causing “explosions”, with purely Newtonian gravitational physics.

The characterization of the collapse of an isolated cloud of self-gravitating particles represents an important theoretical problem that can be possibly relevant for the comprehension of the formation astrophysical relaxed objects, as for instance globular clusters, galaxies etc. It is known, since the first numerical simulations, that when an isolated self-gravitating particles system is initially cold, it collapses violently and then it relaxes to produce a virialized quasi-equilibrium state in a relatively short time scale. While the collapse of an uniform spherical cloud was the case mostly studied in the literature many authors have focused on spherical models with non-trivial density profiles and/or with significant non-zero velocities.

Beyond the case of a spherical cloud of self-gravitating particles, we have studied the dynamics of the cold collapse by using initially cold and slightly ellipsoidal clouds or of clouds of an irregular shape. This system was chosen because it represents a relatively simple configuration for determining the effects of the deviations from spherical symmetry during the cold collapse. While the main characteristics of the virialized object are very similar to the spherical cloud case, we found that ejected particles are characterized by highly asymmetrical shapes, whose features can be traced in the initial deviations from spherical symmetry that are amplified during the cold collapse phase. Ejected particles can indeed form flattened configurations even though the initial cloud was very close to spherical.

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