The origin and formation of cuspy density profiles through violent relaxation of stellar systems

It is shown that the cuspy density distributions observed in the cores of e lliptical galaxies can be realized by dissipationless gravitational collaps e. The initial models consist of power-law density spheres such as rho prop ortional to r(-1) with anisotropic velocity dispersions. Collapse simulatio ns are carried out by integrating the collisionless Boltzmann equation dire ctly, on the assumption of spherical symmetry. From the results obtained, t he extent of constant density cores, formed through violent relaxation, dec reases as the velocity anisotropy increases radially, and practically disap pears for extremely radially anisotropic models. As a result, the relaxed d ensity distributions become more cuspy with increasing radial velocity anis otropy. It is thus concluded that the velocity anisotropy could be a key in gredient for the formation of density cusps in a dissipationless collapse p icture. The velocity dispersions increase with radius in the cores accordin g to the nearly power-law density distributions. The power-law index, n, of the density profiles, defined as rho proportional to r(-n), changes from n approximate to 2.1 at intermediate radii to a shallower power than n appro ximate to 2.1 toward the centre. This density bend can be explained from ou r postulated local phase-space constraint that the phase-space density acce ssible to the relaxed state is determined at each radius by the maximum pha se-space density of the initial state.