= Solution
Work at $r\gg R_{\rm sch}$ 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 $r_c=R_{\rm sch}$,
$$
\frac{J_{\rm lc}^2}{2r_c^2}\simeq\frac{GM_H}{r_c},
\qquad J_{\rm lc}^2\simeq2GM_HR_{\rm sch}.
$$
At radius $r$, a velocity making angle $\vartheta$ with the inward radial direction has $J=rv\sin\vartheta$. Therefore the <loss cone> half-angle is
$$
\boxed{\vartheta_{\rm lc}(r,v)\simeq
\frac{\sqrt{2GM_HR_{\rm sch}}}{rv}.}
$$
Taking the characteristic speed to be the <circular speed> $v^2\sim GM_H/r$ gives
$$
\boxed{\vartheta_{\rm lc}\sim\sqrt{\frac{R_{\rm sch}}r},}
$$
with Newtonian circular-speed normalization $\vartheta_{\rm lc}^2\simeq2R_{\rm sch}/r$. The inward cap occupies solid angle $\Omega_{\rm lc}\simeq\pi\vartheta_{\rm lc}^2$, or a fraction $\vartheta_{\rm lc}^2/4$ 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 $\vartheta_{\rm lc}\propto r^{-1/2}$ for these characteristic Keplerian speeds.
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