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

The ratio of the vertical velocity dispersion to the radial one ( σ z  / σ R ) 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 1 κ /Ω≲1.5, the velocity dispersion increases keeping some anisotropy ( σ z σ R ∼0.5–0.7) if the amplitude of radial excursion is larger than the tidal radius, while σ z σ R ≪1 for smaller amplitude. On the other hand, for 1.5≲ κ /Ω2.0, we found an isotropic state ( σ z σ R ≃1) in the intermediate‐velocity regime, while an anisotropic state ( σ z σ R <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 (1 κ /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.