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

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.