Schwarzschild Effective Potential and the ISCO

For a massive particle outside a Schwarzschild black hole the radial motion follows an effective potential $V_{\text{eff}}(r)$ that adds a relativistic $-1/r^3$ term to the usual Newtonian centrifugal-plus-gravity well. Circular orbits sit at its extrema: a minimum is stable, a maximum unstable. Lower the angular momentum and the minimum and maximum slide together and merge at $r = 6M$, the innermost stable circular orbit (ISCO). Inside the ISCO no stable circular orbit exists and matter spirals in, which is why accretion disks have a sharp inner edge and a fixed maximum efficiency. The playground plots $V_{\text{eff}}$ with the energy level, turning points, and the ISCO marked as you vary $L$.