= Photon capture cross-section of a Reissner-Nordstrom black hole
{title2=$\sigma=2\pi r_{\mathrm{ph}}^3/(r_{\mathrm{ph}}-M)$}
For a <Reissner-Nordstrom black hole> with $|Q|\le M$, the outer unstable <circular null geodesic> has radius $r_{\mathrm{ph}}=(3M+\sqrt{9M^2-8Q^2})/2$. Maximizing $f/r^2$ for $f=1-2M/r+Q^2/r^2$ gives the critical <impact parameter> $b_c^2=r_{\mathrm{ph}}^2/f(r_{\mathrm{ph}})=2r_{\mathrm{ph}}^3/(r_{\mathrm{ph}}-M)$. The <geometrical optics> capture cross-section is $\pi b_c^2$, reducing to $27\pi M^2$ in <Schwarzschild spacetime> and $16\pi M^2$ in the extremal charged case. Finite-wavelength absorption need not equal this geometric limit.
// Target: general-relativity.bigb
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