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We present the results of a series of numerical simulations of gravitational collisionless N-body systems in equilibrium after a violent relaxation from cosmological initial conditions. The distribution function f of such systems has a complicated form due to the complex structure of the phase space of stellar orbits. This complexity makes hardly tractable the old problem of writing a simple model for f .H owever, we show that it is possible to benet from various statistical regularities of the phase space in order to compose a heuristic approximation for f. Such regularities are revealed if we decompose a system in a number of spherical shells. For each shell we dene thermodynamical quantities (e.g., temperatures) which, as we nd, vary smoothly with the radius r of the shell. Using these quantities, we nd a model that ts the number density function (E;L 2 ;r )i n each shell. For the greatest range of energies, this function tends to the form of the Stiavelli-Bertin (1987) model. By adding the contributions of all the spherical shells, we then nd a global model for f. While our method is based on a spherical approximation, we show that it reproduces very accurately the global proles of our triaxial N-body systems.

A "spherical shell number density" model for violently relaxed $\vec{N}$-body systems