Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 347 / 2

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.

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Equating stellar surface gravity to the differential black-hole acceleration givesso the tidal disruption radius isA nonrotating hole swallows the star without a visible disruption when this lies inside its capture scale, here approximated by the Schwarzschild radius $r_h=2G{M}_{\mathrm{BH}}/c^2$. Equating the two radii yields the Hills massFor a solar-type star this is of order $10^8{M}_{\odot }$.

The threshold does depend on spin. A Kerr black hole has spin- and inclination-dependent horizon, marginally bound, and capture radii. Prograde orbits around a rapidly rotating hole can approach more closely, allowing disruption by masses above the Schwarzschild Hills mass, whereas retrograde capture occurs farther out.

An intermediate-mass black hole lies well below this threshold for ordinary stars, so stars entering its loss cone are disrupted outside the horizon. The returning debris can grow the hole and produces a tidal disruption event that may reveal an otherwise quiescent cluster black hole through a flare. Dense clusters can supply repeated disruptions, although the rate depends on two-body relaxation, stellar collisions, binary interactions, and whether gravitational recoil or cluster dynamics ejects the hole.

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Insert the self-similar ansatz into the height-integrated equations. Mass conservation is already satisfied because $H\sim c_s/{\Omega }_K\propto R$ and $\rho RH{u}_R$ is constant. The angular-momentum and energy equations reduce towhile radial momentum givesDefineSolving the quadratic gives the exact advection-dominated accretion flow coefficientsFor ${\alpha }^2\ll 1$,and therefore

Efficient cooling means $f\to 0$ and hence ${ϵ}^{\prime }=ϵ/f\to \infty$ for fixed $\gamma <5/3$. ThenThe flow is therefore cold, nearly Keplerian, slowly accreting, and geometrically thin: the standard thin-disk limit.

For significant advection, $f=O(1)$ and all three deviations are explicit:The gas is hot and thick, pressure supplies part of the radial support, rotation is sub-Keplerian, and dissipated entropy is carried inward. As $\gamma \to 5/3$, ${ϵ}^{\prime }\to 0$, so $\Omega \to 0$, $c_s^2\to (2/5)v_K^2$, and $u_R\to -(3\alpha /5)v_K$: the self-similar rotating solution approaches a hot Bondi-like inflow. Sagittarius A* is the standard supermassive example: its luminosity is tiny compared with its Eddington luminosity despite an available gas supply, and its hot optically thin spectrum and low radiative efficiency are described by an ADAF or the broader radiatively inefficient accretion-flow family.

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