= Solution
The <stellar relaxation time> is the <time> on which accumulated discrete gravitational encounters change a typical <star>'s <velocity> by an amount comparable to its original <velocity>. It concerns <two-body relaxation>, rather than the much faster orbital evolution in a smooth <gravitational potential>. A <collisionless stellar system> requires this <time> to greatly exceed the duration being studied.
In the straight-line <impulse approximation>, an encounter at relative <speed> $u$ has transverse <acceleration> $Gmb/(b^2+u^2t^2)^{3/2}$. Thus
$$
\Delta v_\perp=\int_{-\infty}^{\infty}\frac{Gmb\,dt}{(b^2+u^2t^2)^{3/2}}=\frac{2Gm}{bu}.
$$
A <small-angle gravitational encounter> needs $\Delta v_\perp\ll u$. The transition to order-one deflections is therefore $b\sim Gm/u^2$, ignoring factors such as the equal-mass relative deflection. For the spherical estimate $u\sim v$,
$$
\boxed{b_{\min}\sim\frac{Gm}{v^2}.}
$$
This is the lower cutoff of the weak-scattering estimate; closer encounters actually occur and require strong-scattering treatment.
The number of encounters in <time> $T$ with <impact parameters> in $[b,b+db]$ is $2\pi b\,db\,nuT$. Independent random transverse directions make mean kicks cancel while their <variances> add. Consequently
$$
\frac{\langle|\Delta\mathbf v|^2\rangle}{T}=\int_{b_{\min}}^R\left(\frac{2Gm}{bu}\right)^2 2\pi bnu\,db=\frac{8\pi G^2m^2n}{u}\log\Lambda,\qquad\Lambda=\frac R{b_{\min}}.
$$
Each logarithmic interval contributes equally, giving the <Coulomb logarithm in stellar dynamics>. Defining a deflection <time> by <variance> $v^2$ and setting $u=v$ gives $v^3/(8\pi G^2m^2n\log\Lambda)$. The paper instead uses an order-one normalization three quarters of this estimate:
$$
\boxed{t_{\rm rel}\simeq\frac3{32\pi}\frac{v^3}{G^2m^2n\log\Lambda}.}
$$
Both have the same physical scaling. The specified characteristic <speed>, the word “comparable”, the velocity-distribution average and strong-encounter cutoff do not fix that numerical coefficient uniquely. The crude equal-speed impulse calculation must not be claimed to determine $3/(32\pi)$ exactly.
For a self-gravitating, approximately virialized system, the <virial theorem> gives $v^2\sim GNm/R$. With $n=3N/(4\pi R^3)$, $\Lambda\sim N$ and the <stellar crossing time> $t_{\rm dyn}=R/v$, substitution into the paper's convention gives
$$
\boxed{t_{\rm rel}\simeq\frac{N}{8\log N}\,t_{\rm dyn}.}
$$
An externally dominated <gravitational potential> or a different structural constant changes this substitution. For an <N-body simulation> lasting $T=K t_{\rm dyn}$, a tolerable fractional velocity-squared diffusion $\varepsilon\ll1$ requires $T/t_{\rm rel}\lesssim\varepsilon$, hence
$$
\boxed{\frac{N}{8\log N}\gtrsim\frac K\varepsilon.}
$$
There is no unique smallest $N$ without $K$, the error tolerance and the <force> prescription. For scale, equality at $t_{\rm rel}/t_{\rm dyn}=100$ occurs near $N=7.1\times10^3$; $N=10^5$ gives about $1.1\times10^3$ crossing times. A calculation lasting 100 crossing times therefore needs substantially more than the first threshold to have negligible relaxation. <Gravitational softening> can raise the effective cutoff and reduce artificial scattering, but it also sets the spatial <force> resolution.
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