On the equilibrium morphology of systems drawn from spherical collapse experiments

We present a purely theoretical study of the morphological evolution of self-gravitating systems formed through the dissipation-less collapse of N point sources. We explore the effects of resolution in mass and length on the growth of triaxial structures formed by an instability triggered by an excess of radial orbits. We point out that as resolution increases, the equilibria shift, from mildly prolate, to oblate. A number of particles N ≃ 100, 000 or larger is required for convergence of axial aspect ratios. An upper bound for the softening, ǫ ≈ 1/256, is also identified. We then study the properties of a set of equilibria formed from scale-free cold initial mass distributions, ρ ∝ r ; 0 6 γ 6 2. Oblateness is enhanced for initially more peaked structures (larger γ’s). We map the run of density in space and find no evidence for a power-law inner structure when γ 6 3/2 down to a mass fraction ∼< 0.1% of the total. However when 3/2 < γ 6 2 the mass profile in equilibrium is well matched by a power-law of index ≈ γ out to a mass fraction ≈ 10%. We interpret this in terms of less effective violent relaxation for more peaked profiles when more phase mixing takes place at the centre. We map out the velocity field of the equilibria and note that at small radii the velocity coarsegrained distribution function is Maxwellian to a very good approximation. We extend our study to non-scale-free initial conditions and finite but sub-virial kinetic energy. For cold collapses the equilibria are again oblate, as the scale-free models. With increasing kinetic energy the equilibria first shift to prolate morphology and then to spherical symmetry.