= Extremal dilaton metric photon capture
{title2=$B_c=2M,\quad \sigma_{\rm capture}=4\pi M^2$}
For $f=1-2M/r$ and squared areal radius $C=r(r-b)$, an equatorial <null geodesic> obeys $\dot r^2=E^2[1-B^2f/C]$, with <impact parameter> $B=L/E$. At $b=2M$, $f/C=1/r^2$ in the exterior. Its supremum is $1/(4M^2)$ at the null singular boundary, rather than at a regular exterior <photon sphere>. Rays with $B>2M$ turn at $r=B$; rays with $B<2M$ reach the boundary. Assuming those rays are absorbed gives the displayed geometric-optics cross-section despite zero limiting horizon area.
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