= Finite-azimuth Schwarzschild photon plunge
{c}
{title2=$r\sim\tfrac\mu2(\phi_*-\phi)^2$}
For a captured nonradial <null geodesic> with $u=1/r$, the large-$u$ orbit equation is $(du/d\phi)^2\sim2\mu u^3$. Integration on the inward branch gives
$$
\phi_*-\phi\sim\sqrt{2/(\mu u)},\qquad
r\sim\frac\mu2(\phi_*-\phi)^2.
$$
The <curvature singularity> is reached at finite azimuth. This does not prevent a near-critical ray from winding repeatedly near the <photon sphere> before its final plunge. A radial ray has constant azimuth, so this asymptotic coordinate parametrization is inapplicable to it.
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