= Energy-dependent Schwarzschild material capture threshold
{c}
{title2=$h_c^2=\mu c^2r_c^2/(r_c-3\mu)$}
For an incoming timelike Schwarzschild <geodesic> with asymptotic <Lorentz factor> $k\ge1$, the critical unstable circular orbit obeys
$$
k^2=\frac{(r_c-2\mu)^2}{r_c(r_c-3\mu)},\qquad
h_c^2=\frac{\mu c^2r_c^2}{r_c-3\mu}.
$$
At $k=1$, $r_c=4\mu$ and $|h_c|=4\mu c$. For $k>1$, the critical radius lies between $3\mu$ and $4\mu$ and $|h_c|>4\mu c$. The marginally bound angular-momentum bound remains sufficient at larger energies but is no longer the exact threshold. Origin at infinity and an inward initial branch are essential; an orbit already inside a barrier is a different <initial value problem>.
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