The Effects of Cosmic Expansion on Relaxation of Systems of Galaxies

The effects of cosmic expansion for the two-body relaxation time of galaxies in the Einstein-de Sitter universe are examined. We reformulate the scattering problem so that the time change of the cross section due to cosmic expansion can be taken into account. Next we calculate numerically the orbits of encounters to various sets of orbital parameters while specifying the initial states of the encounters: initial separation, impact parameter, and initial relative velocity between two galaxies. By combining the results of these studies, we obtain the two-body relaxation time as a function of the relative velocity v₀ at the initial epoch and the number density n of galaxies within proto-clusters of galaxies. The relaxation time is roughly the free-fall time for a small relative velocity, v₀ ≤ 300 km s^-1{(m / 10¹¹ M_solar)}^1/3 {[11/(1+z)]}^1/2, while it is proportional to v₀³ for a high relative velocity. The effects of the cosmic expansion make the relaxation time longer than that in non-expanding systems, especially for proto-clusters of galaxies with the number density n < 3 × 10³ Mpc^-3 (10¹¹}M_solar/m){[(1+z)/11]}³, where m is the mass of a galaxy and z is its redshift