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

A typical globular cluster has undergone appreciable two-body relaxation, since its stellar relaxation time is $t_{\mathrm{rel}}\sim 0.1(N/\ln N)t_{\mathrm{cross}}$: for $N\sim 10^5$ and stellar crossing time $t_{\mathrm{cross}}\sim 10^6\mathrm{yr}$ this is about $10^9\mathrm{yr}$, shorter than its age, allowing a relatively relaxed, tidally limited structure. A galaxy with $N\sim 10^{11}$ and $t_{\mathrm{cross}}\sim 10^8\mathrm{yr}$ instead has $t_{\mathrm{rel}}\sim 4\times 10^{16}\mathrm{yr}$ and behaves as a collisionless stellar system, retaining differences from formation, galaxy mergers, and star formation. Relaxation erases more memory in clusters than in galaxies, although real globular clusters also differ in concentration and evolutionary state, so structural similarity is approximate.