Growth of Velocity Dispersions for Collapsing Spherical Stellar Systems

First, we have ensured that spherical nonrotating collisionless systemscollapse with almost retaining spherical configurations during initialcontraction phases even if they are allowed to collapse three-dimensionally.Next, on the assumption of spherical symmetry, we examine the evolution ofvelocity dispersions with collapse for the systems which have uniform orpower-law density profiles with Maxwellian velocity distributions byintegrating the collisionless Boltzmann equation directly. The results showthat as far as the initial contraction phases are concerned, the radialvelocity dispersion never grows faster than the tangential velocity dispersionexcept at small radii where the nearly isothermal nature remains, irrespectiveof the density profiles and virial ratios. This implies that velocityanisotropy as an initial condition should be a poor indicator for the radialorbit instability. The growing behavior of the velocity dispersions is brieflydiscussed from the viewpoint that phase space density is conserved incollisionless systems.