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 $L$.
WHAT TO TRY
- Watch the orbit precess: unlike a closed Newtonian ellipse, the relativistic orbit traces a rosette because the perihelion advances every revolution, and the energy line and moving radius marker on the V_eff curve below show exactly where the particle is in its potential well.
- Lower L toward L_ISCO = 2 sqrt(3): the well shallows and the perihelion presses inward toward the ISCO at r = 6M where stable orbits end. Push the orbit-energy slider past 1 and the particle clears the barrier and plunges through the horizon, the way accretion-disk matter does at the inner edge.
- Switch to photon mode: a light ray with impact parameter b is deflected when b exceeds 3 sqrt(3) M and captured below it, circling the photon sphere at r = 3M that sets the black-hole shadow.