DISCRETE OSCILLATION MODES AND DAMPED STATIONARY DENSITY WAVES IN ONE-DIMENSIONAL COLLISIONLESS SYSTEMS

We consider small perturbations of equilibrium configurations of one-dimensional collisionless self-gravitating systems. We describe an approximate method based on a truncated set of moment equations for following the time evolution of the perturbations. In two special cases, namely constant density in real space and constant density in phase space, this fluid model describes the dynamics of the collisionless system correctly. In general, the predicted frequencies and the form of low-order oscillation modes agree remarkably well with the results of N-body simulations. The important role of the frequency of oscillation modes with regard to the finite band of particle frequencies and their harmonics is discussed, and the question of gaps between continuum segments is reinvestigated. With increasing central concentration of the equilibrium, discrete modes move closer to the adjacent continuum segment and, starting with higher order modes, eventually cross the border into the continuum. In a sequence of polytropes and King models, the lowest order breathing mode finally disappears for a polytropic index n congruent-to 2 and central potential depth W0 congruent-to 2, respectively. Within the continuum, branches of discrete oscillation modes are continued by special superpositions of singular modes which constitute slightly damped, stationary density waves (analogous to plasma oscillations). For finite perturbations, trapping of resonant particles may lead to oscillations in the wave amplitude. We conclude that stationary density waves may have very long lifetimes, even if their frequency lies within the continuum. This one-dimensional system is indeed sufficiently simple that it can be treated by a direct attack on the linearized Vlasov equation as in Weinberg, but the moment method developed here should be generalizable to three-dimensional systems in which exact solutions based on Vlasov methods are extremely complicated and prohibitively expensive.