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The factor called $N$ in the formula must be the stellar number density $n$, not a dimensionless count. Indeed ${\sigma }^3/(G^2{m}^2)$ has dimensions time per volume, so a density in the denominator is essential. If $N_{\ast }$ is the population in a representative volume $V$, then $n=N_{\ast }/V$.

For a weak encounter between equal-mass stars, let $u$ be the asymptotic relative speed and $b$ the impact parameter. In the straight-line impulse approximation, one star's transverse kick isThe other receives the opposite kick, so the relative-velocity change is $\Delta u_{\perp }=4Gm/(bu)$. Encounters in $(b,b+db)$ occur at rate $2\pi bdbnu$. Independent random kicks add in squared magnitude, givingThis is the random-walk derivation of stellar relaxation. The Coulomb logarithm in stellar dynamics is $\ln \Lambda =\ln (b_{\max }/b_{\min })$. The upper cutoff is the size or inhomogeneity scale beyond which independent local encounters are inappropriate. The lower cutoff is of order the strong-deflection scale $G(2m)/u^2$, or a larger stellar-size/softening scale if applicable. A self-gravitating system commonly has $\Lambda$ of order its total number of stars; that population count is distinct from $n$ in the diffusion rate.

To state the numerical convention explicitly, use representative encounter speed $u=\sqrt{2}\sigma$ and define the relaxation estimate from the relative-velocity diffusion rate with reference squared velocity $3{\sigma }^2$. This givesThis recovers the printed coefficient under a specified convention. There is no convention-independent exact numerical prefactor determined by a representative one-dimensional velocity alone. Using the single-star diffusion rate with the same $3{\sigma }^2$ reference instead gives four times this time; averaging over a full velocity distribution also changes the prefactor. The relaxation time conventions in stellar dynamics must therefore distinguish single-star and relative kicks. The physical scaling is $T_R\propto {\sigma }^3/(n{G}^2{m}^2\ln \Lambda )$, and the printed $N$ must be interpreted as number density.