Energy-dependent Schwarzschild material capture threshold ($h_c^2=\mu c^2{r}_c^2/(r_c-3\mu )$)

For an incoming timelike Schwarzschild geodesic with asymptotic Lorentz factor $k\ge 1$, the critical unstable circular orbit obeysAt $k=1$, $r_c=4\mu$ 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.