Two-body relaxation time

Remarks

An encounter with impact parameter $b$ gives $\Delta v_\perp \approx 2 G m/(b v)$, where $G$ is the gravitational constant. Summing over the encounters of one crossing, with $b$ between a smallest value $b_{\min}$ and $R$, gives $\sum (\Delta v_\perp)^2 \approx 8 N (G m/(R v))^2 \ln \Lambda$ with $\Lambda = R/b_{\min}$. If the system is in virial equilibrium, $v^2 \approx G N m/R$, and so $$t_{\text{relax}} \approx \frac{N}{8 \ln \Lambda} \, t_{\text{cross}}, \qquad \Lambda \approx N.$$ The factor $\ln \Lambda$ shows that all ranges of distance contribute equally, so distant weak encounters dominate over close strong ones.

Example: a globular cluster with $N = 10^5$ and $t_{\text{cross}} \approx 10^5$ years has $t_{\text{relax}} \approx 10^5/(8 \cdot 11.5) \cdot 10^5 \approx 10^8$ years, shorter than its age of about $10^{10}$ years.

Non-example: a galaxy with $N = 10^{11}$ and $t_{\text{cross}} \approx 10^8$ years has $t_{\text{relax}}$ of order $10^{17}$ years. So the galaxy does not relax within the age of the universe, and it is collisionless.