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Relaxation of N-body systems with additive r^-α interparticle forces

In Newtonian gravity the final states of cold dissipationless collapses are characterized by several structural and dynamical properties remarkably similar to those of observed elliptical galaxies. Are these properties a peculiarity of the Newtonian force or a more general feature of long-range forces? We study this problem by means of $N-$body simulations of dissipationless collapse of systems of particles interacting via additive $r^{-α}$ forces. We find that most of the results holding in Newtonian gravity are also valid for $α\neq2$. In particular the end products are triaxial and never flatter than an E7 system, their surface density profiles are well described by the Sérsic law, the global density slope-anisotropy inequality is obeyed, the differential energy distribution is an exponential over a large range of energies (for $α\geq1$), and the pseudo phase-space density is a power law of radius. In addition, we show that the process of virialization takes longer (in units of the system's dynamical time) for decreasing values of $α$, and becomes infinite for $α=-1$ (the harmonic oscillator). This is in agreement with the results of deep-MOND collapses (qualitatively corresponding to $α=1$) and it is due to the fact the force becomes more and more similar to the $α=-1$ case, where as well known no relaxation can happen and the system oscillates forever.