A Phase‐Space Approach to Collisionless Stellar Systems Using a Particle Method

A particle method for reproducing the phase space of collisionless stellarsystems is described. The key idea originates in Liouville's theorem whichstates that the distribution function (DF) at time t can be derived fromtracing necessary orbits back to t=0. To make this procedure feasible, aself-consistent field (SCF) method for solving Poisson's equation is adopted tocompute the orbits of arbitrary stars. As an example, for the violentrelaxation of a uniform-density sphere, the phase-space evolution which thecurrent method generates is compared to that obtained with a phase-space methodfor integrating the collisionless Boltzmann equation, on the assumption ofspherical symmetry. Then, excellent agreement is found between the two methodsif an optimal basis set for the SCF technique is chosen. Since thisreproduction method requires only the functional form of initial DFs but needsno assumptions about symmetry of the system, the success in reproducing thephase-space evolution implies that there would be no need of directly solvingthe collisionless Boltzmann equation in order to access phase space even forsystems without any special symmetries. The effects of basis sets used in SCFsimulations on the reproduced phase space are also discussed.