= Critical photon orbits of Schwarzschild spacetime
{title2=$u-\tfrac13=-4Ae^\phi/(1+2Ae^\phi)^2$}
At the critical <impact parameter> $b=3\sqrt3M$, the null orbit equation factors as $(u')^2=(u-1/3)^2(2u+1/3)$ for $u=M/r$. One oriented family has the displayed solution. As $\phi\to-\infty$, positive $A$ approaches the <photon sphere> from larger radii and negative $A$ from smaller radii; $A=0$ is the circular orbit. The physical domain must retain $r>0$ and the positive square root in its implicit parametrization.
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