Energy ejection in the collapse of a cold spherical self-gravitating cloud

When an open system of classical point particles interacting by Newtonian gravity collapses and relaxes violently, an arbitrary amount of energy may, in principle, be carried away by particles which escape to infinity. We investigate here, using numerical simulations, how this released energy and other related quantities (notably the binding energy and size of the virialized structure) depend on the initial conditions, for the one-parameter family of starting configurations given by randomly distributing N cold particles in a spherical volume. Previous studies have established that the minimal size reached by the system scales approximately as   N ^1/3  , a behaviour which follows trivially when the growth of perturbations (which regularize the singularity of the cold collapse in the   N  →∞  limit) is assumed to be unaffected by the boundaries. Our study shows that the energy ejected grows approximately in proportion to   N ^1/3  , while the fraction of the initial mass ejected grows only very slowly with N , approximately logarithmically, in the range of N simulated. We examine in detail the mechanism of this mass and energy ejection, showing explicitly that it arises from the interplay of the growth of perturbations with the finite size of the system. A net lag of particles compared to their uniform spherical collapse trajectories develops first at the boundaries and then propagates into the volume during the collapse. Particles in the outer shells are then ejected as they scatter through the time-dependent potential of an already re-expanding central core. Using modified initial configurations, we explore the importance of fluctuations at different scales and discreteness (i.e. non-Vlasov) effects in the dynamics.