Stellar relaxation time ($t_{\mathrm{rel}}$)

The stellar relaxation time is the time on which random gravitational encounters change a star's velocity by an amount comparable to its characteristic random velocity. A weak straight-line encounter at relative speed $u$ and impact parameter $b$ gives a kick $2Gm/(bu)$. Three-dimensional encounter counting gives $d⟨|\Delta \mathbf{v}{|}^2⟩/dt=8\pi G^2{m}^{2}n\log (b_{\max }/b_{\min })/u$. Definitions of relaxation, velocity averaging and system structure alter order-one coefficients; the robust spherical scaling is $t_{\mathrm{rel}}\sim v^3/(G^2{m}^{2}n\log \Lambda )\sim (N/\log N)t_{\mathrm{dyn}}$. Collisionless simulation requirements depend on the duration and permitted fractional relaxation, not a universal minimum particle number.

Small-angle gravitational encounters occur at rate $2\pi bdbnu$ and supply kicks $2Gm/(bu)$. Independent kick variances add to $d⟨\Delta v^2⟩/dt=8\pi G^2{m}^{2}n\ln \Lambda /u$. Setting the accumulated variance to $u^2$ and using virial scaling gives a stellar relaxation time of order $0.1N/\ln N$ stellar crossing times, with the coefficient dependent on profile and crossing convention.