Tumbling endstates for unstable rotating spheres - Implications for rotation in elliptical galaxies

We present 10^5^ particle numerical simulations for the evolution of unstable, radially anisotropic spherical systems which have a modest net rotation about the z axis. Reliability of these experiments is demonstrated by continuity of the results with small changes in the parameters as well as qualitative agreement with the analytic theory presented. As a result of the existence of a spectrum of non-axisymmetric modes unique to rotating systems, all our experiments produced endstates in which a triaxial bar tumbled over its major axis. Roughly half of the angular momentum of the system was in the pattern. The endstates were significantly more triaxial than those presented for the corresponding non-rotating systems in previous work. Long lived transient behaviour, lasting for several hundred crossing times, was exhibited by systems with weak non-axisymmetric instability, such as those with weaker rotation and those with counter streaming subsystems. A linear normal mode analysis for a general rotating spherical system is presented. We show that models of the type considered here can have a new set of non-axisymmetric overstabilities not associated solely with radial orbits. These occur in addition to the purely growing axisymmetric radial orbit instability. Instability to the non-axisymmetric modes can persist when the anisotropy has been reduced so that there is stability to axisymmetric modes. Simulations of models with identifiable dynamical subsystems, such as counter rotating cores, show separate spectra of normal modes, expected from the normal mode analysis, for each subsystem which can then interact. Angular momentum is redistributed by the interaction of stars with the various growing modes. We present a simple model for the slow down in the pattern speed of the tumbling bar due to transport of angular momentum on a long time-scale by dynamical friction. These results are directly applicable to the shapes of rotating elliptical galaxies. In simulations of weakly unstable systems isophotal twists associated with interacting subsystems developed. We constructed systems with counter rotating cores which had a similar ratio of streaming to random velocity to that observed in some elliptical galaxies. This feature was long lived, thus demonstrating that these systems do not need to have formed recently.