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

Work at $r\gg R_{\mathrm{sch}}$ in the region where the central black hole dominates the gravitational potential. Let $J$ be a star's specific angular momentum. Near its very small pericentre, the centrifugal term balances the deep gravitational potential. For an approximately parabolic passage with capture pericentre $r_c=R_{\mathrm{sch}}$,At radius $r$, a velocity making angle $\vartheta$ with the inward radial direction has $J=rv\sin \vartheta$. Therefore the loss cone half-angle isTaking the characteristic speed to be the circular speed $v^2\sim G{M}_H/r$ giveswith Newtonian circular-speed normalization ${\vartheta }_{\mathrm{lc}}^2\simeq 2R_{\mathrm{sch}}/r$. The inward cap occupies solid angle ${\Omega }_{\mathrm{lc}}\simeq \pi {\vartheta }_{\mathrm{lc}}^2$, or a fraction ${\vartheta }_{\mathrm{lc}}^2/4$ of an isotropic velocity sphere. Counting low-angular-momentum trajectories in both radial directions changes only an order-one factor; each such orbit is removed at its next pericentre.

The capture occurs in the strong-field region, so this Newtonian pericentre estimate is not an exact relativistic capture coefficient. The Schwarzschild capture calculation alters an order-one constant, while retaining the small-angle scaling ${\vartheta }_{\mathrm{lc}}\propto r^{-1/2}$ for these characteristic Keplerian speeds.

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.

The sign of cluster evolution depends on the orbital energy of the captured stars, not just on their small angular momentum. Let the instantaneous capture rate be $\Gamma >0$, so the remaining stellar mass $M_s=Nm$ has ${\dot{M}}_s=-m\Gamma$, while ${\dot{M}}_H\simeq m\Gamma$. A captured star has negative specific orbital energy ${ϵ}_{\mathrm{cap}}$ measured outside the absorbing region. Removing it increases the energy of the surviving stars by $-m{ϵ}_{\mathrm{cap}}$; growth of the central black hole instead deepens their potential.

To turn this balance into a radius estimate, assume the surviving cluster evolves slowly through homologous virial equilibria of scale $R$. Write its stellar self-energy and black-hole interaction energy aswhere positive $a,b$ depend on the density shape. The virial theorem givesAt the instant of capture, the stellar-system energy budget, excluding feedback and energy carried by other escaping stars, isThe second term is the work of the changing central potential; it must not be confused with the change in the number of stars. Differentiating the virial expression and using ${\dot{M}}_H=m\Gamma$ givesThis estimates the radius response of a general homologous cluster once its structure and capture energies are specified.

For a simpler locally Keplerian subsystem, neglect its stellar self-gravity, so $E=-bG{M}_H{M}_s/(2R)$. Put ${ϵ}_{\mathrm{cap}}=\xi E/M_s$, with $\xi >0$ measuring how tightly bound a captured star is relative to the average orbital energy. The preceding balance reduces toThis is Keplerian cluster radius evolution under stellar capture. Removing representative-energy stars, $\xi =1$, leaves unchanged the characteristic orbits if $M_H$ is fixed; with black-hole growth it contracts them at rate $\dot{R}/R\simeq -{\dot{M}}_H/M_H$. Preferential removal of more tightly bound stars drives expansion if $\xi -1>M_s/M_H$. In that heating-dominated case,so inserting the empty-cone flux gives $t_R\sim t_{\mathrm{rel}}\log (1/{\mathcal{R}}_{\mathrm{lc}})/|\xi -1|$ for the one-zone estimate. Growth-dominated contraction instead has timescale $M_H/(m\Gamma )$.

For a Kepler orbit of semimajor axis $a_{\mathrm{cap}}$, ${ϵ}_{\mathrm{cap}}=-G{M}_H/(2a_{\mathrm{cap}})$ and $\xi =R/(b{a}_{\mathrm{cap}})$. Reaching $R_{\mathrm{sch}}$ does not imply that the orbit had binding energy of order $c^2$: a nearly parabolic star can reach that pericentre with almost zero orbital energy. Therefore expansion is expected when strongly bound central stars are preferentially swallowed, but the supplied information does not uniquely fix the sign or rate for the entire remaining cluster. Captured-energy selection, the density profile and any energy feedback must be specified. The balances above give both the conventional expansion estimate and the contraction case without assuming missing data.