Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 4 / ii / Solution

At small separations the extended dark-matter haloes overlap, and Chandrasekhar dynamical friction transfers orbital energy to gravitational wakes while tidal interactions also redistribute orbital energy. A characteristic decay time scales as $t_{\mathrm{df}}\sim v{r}^2/(Gm\ln \Lambda )$, so sufficiently massive close pairs evolve rapidly towards a galaxy merger and spend relatively little time at small separation. Short residence time depletes the close-pair population; the quoted separation scale is characteristic of the population, not a universal exclusion radius.