= Solution
During a weak encounter, approximate one star's trajectory by a straight line with speed $v$ and <impact parameter> $b$. At longitudinal coordinate $z=vt$, the transverse acceleration is
$$
a_\perp=\frac{Gm b}{(b^2+v^2t^2)^{3/2}}.
$$
Integrating from $t=-\infty$ to $\infty$ gives
$$
\boxed{\delta v_\perp=\frac{2Gm}{bv}
=\frac{2Gm b}{b^2v}}.
$$
Here $Gm/b^2$ sets the gravitational acceleration, $b/v$ is the encounter duration, and the geometric projection produces the displayed transverse impulse.
In one crossing, the number of encounters with impact parameters in $(b,b+db)$ is, up to the system-geometry convention,
$$
dn\simeq\frac{2Nb\,db}{R^2}.
$$
Uncorrelated impulses add in mean square, so
$$
\left\langle(\Delta v)^2\right\rangle_{\rm cross}
=\int_{b_{\min}}^{b_{\max}}(\delta v)^2dn
\simeq\frac{8NG^2m^2}{R^2v^2}
\log\!\frac{b_{\max}}{b_{\min}}.
$$
The <virial theorem> gives $v^2\sim GNm/R$. Taking $b_{\max}\sim R$ and the strong-deflection scale $b_{\min}\sim Gm/v^2\sim R/N$ gives the <Coulomb logarithm in stellar dynamics> $\log\Lambda\sim\log N$ and
$$
\frac{\langle(\Delta v)^2\rangle_{\rm cross}}{v^2}
\simeq\frac{8\log\Lambda}{N}.
$$
Velocity memory is lost when the cumulative change reaches $v^2$, after
$$
\boxed{n_{\rm cross}\simeq\frac{N}{8\log\Lambda}
\simeq\frac{N}{8\log N}}.
$$
Consequently the <two-body relaxation> time is
$$
\boxed{t_{\rm rel}\simeq\frac{N}{8\log N}\frac{R}{v}},
$$
with an order-unity prefactor depending on density profile and convention. The system is a <collisional stellar system> when $t_{\rm rel}$ is shorter than its age or the evolutionary timescale being studied.
Solved by gpt-5.6-sol high.
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