| In this thesis, I present results from the numerical study of the collapse of a self-gravitating collisionless stellar system. This investigation is divided into three parts. In the first stage, the system is taken to have a distribution function (DF) which is initially polytropic, with one spatial (r) and two velocity (vr, j 2) phase space directions. Although this system is stable to linear perturbations I introduce a non-linear disturbance by ‘cooling’ the system abruptly relative to virial equilibrium. Time evolution is followed using the coupled Collisionless Boltzmann (CBE) and Poisson's equations (or Vlasov equation [83]). By cooling clouds of various polytropic index n I am able to reproduce an instability observed by Henriksen & Widrow [31] in their investigation of self-similar collapse which may be the driving mechanism toward the complete relaxation of the system. In the second stage, the constraint of spherical symmetry is removed and a treecode is used to follow the evolution of the system. The initial distribution was taken to be a lowered Evans model (Kuijken & Dubinski [47]) which emulates a galactic halo, destabilized as above. Each particle is tagged with the value of the DF at its initial phase-space position. This allows us to observe the evolution of the velocity distribution directly as the system relaxes to a new equilibrium. I have shown in both the spherical and non-spherical collapse simulations evolution to new equilibrium configurations in which the velocity distribution approaches a Gaussian form. The evolution to this state has long been an open question, and in this work I am able to elucidate the process responsible and confirm predictions made from statistical considerations (Lynden-Bell [53]; Nakamura [63]). The third stage consists of a series of simulations of merging haloes. These simulations show a transition to a Gaussian velocity distribution which is increasingly suppressed as the initial separation is increased. Possible reasons for this are discussed. |
