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, sigma(R), and the vertical one, sigma(z), is sigma(z)/sigma(R) congruent-to 0.6 for stars in a galactic disc in the solar neighbourhood, and sigma(z)/sigma(R) = 0.5 for planetesimals in a Kepler potential. These ratios are smaller than the 'isotropic' ratio, sigma(z)/sigma(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 In LAMBDA in the two-body approximation more exactly, since we found that the equilibrium ratio of sigma(z)/sigma(R) depends sensitively on the choice of In LAMBDA. We determined the effective ln LAMBDA for each component of velocity distribution, while Lacey simply took ln LAMBDA as a constant. The numerical results of orbital integrations show that our treatment is correct, whereas Lacey's overestimated dsigma(z)2/dt and underestimated dsigma(R)2/dt considerably, so that he overestimated the equilibrium ratio of sigma(z)/sigma(R). We find that the ratio sigma(z)/sigma(R) approaches a value that is determined mainly by kappa/OMEGA (where kappa and OMEGA 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 (kappa/OMEGA = 1) and about 0.6 for the galactic potential in the solar neighbourhood (kappa/OMEGA congruent-to 1.4). Therefore the analytical calculation here explains well the ratios sigma(z)/sigma(R) found by N-body simulations and observations.