 Solution

A photon arriving from infinity crosses the barrier precisely when $c^2{k}^2/2$ exceeds the maximum in part (c):At infinity its spatial tangent has speed in affine-parameter units $ck$, while $h$ is its angular momentum per unit affine normalization. A straight asymptotic ray with impact parameter $b$ therefore has $|h|=bck$. ThusAt equality, the incoming ray approaches the unstable photon sphere asymptotically and is not captured. A larger impact parameter gives an outer turning point and scattering back to infinity.