Evolution of the velocity dispersion of self-gravitating particles in disc potentials

The ratio of the vertical velocity dispersion to the radial one (σz /σR) of self-gravitating bodies in various disc potentials is investigated through two different numerical methods (statistical compilation of two-body encounters and N-body simulations). The velocity dispersion generated by two-body relaxation is considered. The ratio of dispersions is given as a function of a disc potential parameter, κ/Ω, where κ and Ω are the epicycle and circular frequencies (the parameters κ/Ω=1 and 2 correspond to Kepler rotation and solid-body rotation). For 1κ/Ω≲1.5, the velocity dispersion increases keeping some anisotropy (σzσR∼0.5–0.7) if the amplitude of radial excursion is larger than the tidal radius, while σzσR≪1 for smaller amplitude. On the other hand, for 1.5≲κ/Ω2.0, we found an isotropic state (σzσR≃1) in the intermediate-velocity regime, while an anisotropic state (σzσR<1) still exists for higher and lower velocity regimes. The range of the intermediate-velocity regime expands with κ/Ω. In the limit of solid-body rotation, the regime covers all of the velocity space. Thus, the velocity dispersion generally has two different anisotropic states for each disc potential (1κ/Ω<2) and one isotropic state for 1.5≲κ/Ω<2 where the individual states correspond to different amplitudes of velocity dispersion, while in the limit of solid-body rotation (κ/Ω=2.0), the entire velocity space is covered by the isotropic state.