Weak-field photon-orbit validity ($u=r_S/r\ll 1$)

The first-order null-orbit equation $(u^{\prime }{)}^2=r_S^2{E}^2/h^2-u^2(1-u)$ formally has a circular solution at $u=2/3$, with $|h/E|=3\sqrt{3}{r}_S/2$. Its second-order equation gives a perturbation $\delta u^{\prime \prime }=\delta u$, hence instability. However, $u=2/3$ is outside the weak-field expansion. The exact Schwarzschild spacetime has a corresponding unstable photon sphere, but that physical result needs the exact geometry; extrapolating the linearized metric is not a controlled proof.