Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 44 / 4 / b / Solution

A small loss cone does not by itself specify its supply rate. Introduce a characteristic orbital radius $r$, an orbital period $P\sim r/v$, the number $N_r$ of stars on the relevant energy range, and their local number density $n$. Weak encounters of impact parameter $b$ give velocity kicks of order $Gm/(bv)$. Summing their squared kicks over encounter rate $2\pi bdbnv$ givesHere $\log \Lambda$ is the Coulomb logarithm in stellar dynamics. This derives the stellar relaxation time relevant to angular-momentum scattering, up to the velocity-averaging constants.

Let $J_c^2\sim G{M}_{H}r$ and define the dimensionless cone size ${\mathcal{R}}_{\mathrm{lc}}=J_{\mathrm{lc}}^2/J_c^2\sim R_{\mathrm{sch}}/r$. In one period, two-body relaxation changes $J^2$ by order $J_c^{2}P/t_{\mathrm{rel}}$, so the refilling parameter isIf refilling is sufficiently rapid to keep the cone populated, an isotropic fraction of order ${\mathcal{R}}_{\mathrm{lc}}$ reaches its absorbing pericentre each orbit. The full-cone rate isIf $q\ll 1$, the cone is depleted and capture is diffusion-limited. At fixed orbital energy, the two transverse angular-momentum coordinates have an approximately two-dimensional diffusion equation. Its steady radial solution between the absorbing boundary $J_{\mathrm{lc}}$ and the reservoir $J_c$ satisfiesThe gradient supplies a constant inward flux. With diffusion coefficient of order $J_c^2/t_{\mathrm{rel}}$ and reservoir population $N_r$, this givesConstants of two depend on using $J$ or $J^2$ and on the definition of $t_{\mathrm{rel}}$. A rough interpolation uses the slower of the orbital draining and diffusion-supply rates; an accurate transition requires an orbit-averaged diffusion calculation. These are the empty and full loss-cone capture rates. In particular, multiplying $N_r/t_{\mathrm{rel}}$ by the cone's tiny solid-angle fraction would incorrectly treat diffusive feeding as independent isotropic redraws.

For a one-zone black-hole-dominated cluster with $N_r\sim N$, $n\sim N/r^3$ and $v^2\sim G{M}_H/r$,The mass ingestion rate is ${\dot{M}}_H\simeq m\Gamma$. A radially extended cluster requires summing the energy-dependent fluxes, not replacing every radius by the same $r$. No numerical capture rate follows from $N,m,M_H$ alone: its radius/density, orbital distribution and refilling regime are required. The displayed relations are the requested estimates with those physical scales explicit.