A family of models of partially relaxed stellar systems

N-body simulations of collisionless collapse have offered important clues for the construction of realistic stellar dynamical models of elliptical galaxies. Understanding this idealized and relatively simple process, by which stellar systems can reach partially relaxed equilibrium configurations (characterized by isotropic central regions and radially anisotropic envelopes), is a prerequisite to more ambitious attempts at constructing physically justified models of elliptical galaxies in which the problem of galaxy formation is set in the generally accepted cosmological context of hierarchical clustering. In a previous paper we have discussed the dynamical properties of a family of models of partially relaxed stellar systems (the $f^{(\nu)}$ models), designed to incorporate the qualitative properties of the products of collisionless collapse at small and at large radii. Here we revisit the problem of incomplete violent relaxation, by making a direct comparison between the detailed properties of such family of models and those of the products of collisionless collapse found in N-body simulations that we have run for the purpose. Surprisingly, the models thus identified are able to match the simulated density distributions over nine orders of magnitude and also to provide an excellent fit to the anisotropy profiles and a good representation of the overall structure in phase space. The end-products of the simulations and the best-fitting models turn out to be characterized by a level of pressure anisotropy close to the threshold for the onset of the radial-orbit instability. The conservation of Q, a third quantity that is argued to be approximately conserved in addition to total energy and total number of particles as a basis for the construction of the $f^{(\nu)}$ family, is discussed and tested numerically