= Null effective potential in Schwarzschild spacetime
{title2=$V_{\rm eff}=\frac{h^2}{2r^2}(1-2\mu/r),\quad\mu=GM/c^2$}
For an equatorial <null geodesic> in <Schwarzschild spacetime>, let $k=(1-2\mu/r)\,dt/d\lambda$ and $h=r^2d\phi/d\lambda$ be the conserved quantities for an <affine parameter> $\lambda$. The null <metric> condition gives $\dot r^2/2+V_{\rm eff}=c^2k^2/2$. For $h\ne0$, the <effective potential> is zero at the <Schwarzschild event horizon>, positive outside, and tends to zero at infinity. Its unique exterior maximum is at $r=3\mu$, with value $h^2/(54\mu^2)$ and second <derivative> $-h^2/(81\mu^4)$. This maximum gives the unstable <photon sphere>.
The <impact parameter> of an incoming ray is $b=|h|/(ck)$ for $k>0$. Capture requires its radial energy to exceed the maximum, namely $b<3\sqrt3\mu$. At equality the incoming ray approaches the circular orbit asymptotically; it does not cross the horizon. For $h=0$, the potential is identically zero and the ingoing radial ray is captured.
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