 Relaxation time conventions in stellar dynamics

Weak equal-mass small-angle gravitational encounters give a single-star transverse kick $2Gm/(bu)$ and a relative-velocity kick twice as large. With encounter rate $2\pi bdbnu$, the single-star squared-velocity diffusion rate is the displayed $D_v$, while the relative rate is $4D_v$. A chosen reference squared velocity divided by either rate defines a relaxation estimate; specifying which rate and reference are used is essential to a numerical prefactor. If $u=\sqrt{2}\sigma$, the convention $3{\sigma }^2/(4D_v)$ gives $3{\sigma }^3/(16\pi \sqrt{2}n{G}^2{m}^2\ln \Lambda )$. The single-star convention $3{\sigma }^2/D_v$ is four times larger. Velocity-distribution averaging further changes numerical coefficients, while the scaling and Coulomb logarithm in stellar dynamics persist. The factor $n$ must be number density, not a dimensionless population count.