The origin of anisotropic velocity dispersion of particles in a disc potential

The velocity dispersion of particles in a disc potential is anisotropic. N-body simulations and observations show that the ratio between the radial component of the dispersion, σ_R, and the vertical one, σ_z, is σ_z/σ_R ≃ 0.6 for stars in a galactic disc in the solar neighbourhood, and σ_z/σ_R ≃ 0.5 for planetesimals in a Kepler potential. These ratios are smaller than the ‘isotropic’ ratio, σ_z/σ_R = 1. The velocity dispersion evolves through gravitational scattering between particles. To explain the anisotropic ratio, we performed analytical calculations using the two-body approximation which is similar to that of Lacey, although we calculate the logarithmic term ln Λ in the two-body approximation more exactly, since we found that the equilibrium ratio of σ_z/σ_R depends sensitively on the choice of ln Λ. We determined the effective ln Λ for each component of velocity distribution, while Lacey simply took ln Λ as a constant. The numerical results of orbital integrations show that our treatment is correct, whereas Lacey's overestimated $$d\sigma _{z}^{2}/dt$$ and underestimated $$d\sigma _{R}^{2}/dt$$ considerably, so that he overestimated the equilibrium ratio of σ_z/σ_R. We find that the ratio σ_z/σ_R approaches a value that is determined mainly by κ/Ω (where κ and Ω are the epicyclic frequency and the angular velocity of a local circular orbit). The equilibrium ratios are predicted to be about 0.5 for the Kepler potential (κ/Ω = 1) and about 0.6 for the galactic potential in the solar neighbourhood (κ/Ω ≃ 1.4). Therefore the analytical calculation here explains well the ratios σ_z/σ_R found by N-body simulations and observations.