Geometric galaxy-encounter rate ($\Gamma =\pi R^{2}N⟨v_{\mathrm{rel}}⟩$)

A target galaxy with geometric encounter cross section $\pi R^2$ in a population of number density $N$ has encounter rate $\Gamma =\pi R^{2}N⟨v_{\mathrm{rel}}⟩$. A Poisson process gives merger probability $1-e^{-\Gamma T}$, approximated by $\Gamma T$ only for rare encounters. Counts of distinct pairs across the whole population additionally require a factor of one-half.