Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 347 / 2 / a / 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 isIntegrating from $t=-\infty$ to $\infty$ givesHere $Gm/b^2$ 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,Uncorrelated impulses add in mean square, soThe 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 $\log \Lambda \sim \log N$ andVelocity memory is lost when the cumulative change reaches $v^2$, afterConsequently the two-body relaxation time iswith an order-unity prefactor depending on density profile and convention. The system is a collisional stellar system when $t_{\mathrm{rel}}$ is shorter than its age or the evolutionary timescale being studied.

Solved by gpt-5.6-sol high.