Nonuniqueness and structural stability of self-consistent models of elliptical galaxies

Abstract

(Abridged:) Schwarzschild's method was used to construct equilibrium solutions to the collisionless Boltzmann equation corresponding to a Plummer sphere, which were compared with analytical results to test the robustness of the numerical method and its efficiency in probing the degeneracy of the solution space. The method was then used to construct triaxial equilibria with no known analytical solutions and to study their nonuniqueness. The model that was studied, the triaxial Dehnen potential, contains a central density cusp and admits both regular and chaotic orbits. It was found that, for a weak density cusp, self-consistent models do not exist if the chaotic orbits are assumed to be completely mixed so as to yield a time-independent building-block: only the innermost 65% of the mass can be mixed. In these inner regions, it is possible to obtain alternative solutions that contain significantly different numbers of chaotic orbits, yet yield (at least approximately) the same mass density distribution. However, it is not clear whether these solutions are truly time-independent, since the unmixed chaotic orbits in the outer regions, which do not sample an invariant measure, can cause secular evolution. When using Schwarzschild's method, one must be very careful to sample accessible phase space as comprehensively and as densely as possible, while ensuring that each orbit is a truly time-independent building block. Some of the numerical equilibria were sampled to generate initial conditions for N-body simulations to test the structural stability of the models. Preliminary work showed that no catastrophic evolution takes place, but there is a weak tendency for the configuration to become more nearly axisymmetric over several dynamical times. It is not yet clear whether this tendency is real or a numerical artifact.