A CONSERVATIVE NUMERICAL TECHNIQUE FOR COLLISIONLESS DYNAMIC-SYSTEMS - COMPARISON OF THE RADIAL AND CIRCULAR ORBIT INSTABILITIES

We have developed a smooth potential numerical technique for studying dynamical systems. It has a very low noise level, giving a very low level of numerical diffusion of orbits in phase space and consequently a high degree of orbital integrity. This allows us to follow a dynamical system down to the level of individual orbits for many orbital periods, and the global behaviour including resonant effects for hundreds of dynamical crossing times for the system. The memory usage of the computer code is scarcely more than that needed for the particle positions and velocities when 105 or more particles are used, so it can be run comfortably on currently available minicomputers. Since the code uses an analytic estimate of the potential it is possible to follow the detailed evolution of the potential, density or other integrals over the distribution function during a run from the behaviour of a few coefficients. It does not require frequent 'writes' of the position and velocity data set and consequent large disc memory usage; these only need be output at intervals when the distribution function itself is to be analysed. The code has been applied to the radial and circular orbit instabilities in spherical systems and gives good agreement with the linear theory. We can readily distinguish between the weak overstability of near circular orbits and the strong purely growing radial orbit instability. We can identify the resonant stars which drive these instabilities, and require no less than 105 particles to follow the precession of these orbits with sufficient accuracy to describe adequately all but the strongest instabilities. We follow the growth through to the final non-linear state. © Royal Astronomical Society.