Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 347 / 1 / a / Solution

The radiative efficiency of black-hole accretion isFor a steady thin disk with negligible stress at the innermost stable circular orbit, matter radiates the binding energy lost before plunging, so $\eta =1-E_{\mathrm{ISCO}}$. Black-hole spin changes both the ISCO radius and its specific orbital energy. A prograde disk around a rapidly rotating Kerr black hole reaches deeper into the potential and is more efficient than a retrograde disk; representative ideal values run from $\eta \simeq 0.057$ for a Schwarzschild hole toward $0.42$ for an extremal prograde Kerr hole, reduced to about $0.3$ when photon capture limits astrophysical spin-up.

In adiabatic Bondi accretion, spherical compression raises the gas's internal energy reversibly, but much of that energy is advected through the horizon. There is no sustained shear stress that converts orbital binding energy into heat at a sequence of radii. A Shakura--Sunyaev thin disk, by contrast, must transport angular momentum outward. Its differential rotation stores free energy, local stress dissipates that energy as heat, and the short cooling time lets an optically thick disk radiate it before accretion. High efficiency therefore requires irreversible heating beyond adiabatic compression.

The likely source is MRI-driven magnetohydrodynamic turbulence. A weak magnetic field couples neighboring annuli; when angular velocity decreases outward, magnetic tension transfers angular momentum outward and amplifies the displacement. The alpha disk prescription replaces the unresolved turbulent stress by $T_{r\varphi }=\alpha P$, or equivalently $\nu \sim \alpha c_{s}H$. It captures the correct dimensional scale because subsonic turbulent motions are bounded by $c_s$ and their largest local eddies by $H$.

Its limitations include the following.

Solved by gpt-5.6-sol high.