Innermost stable circular orbit

Remarks

For a timelike geodesic in the plane $\theta = \pi/2$ with proper time as parameter, $E = f \dot t$ and $L = r^2 \dot\phi$ are conserved. The normalisation $g(\dot\gamma, \dot\gamma) = -1$ gives $f h\, \dot r^2 = E^2 - V_L(r)$, so $V_L$ plays the role of an effective potential.

Example: for the Schwarzschild metric, $f(r) = 1 - 2M/r$ on $(2M, \infty)$ with $M > 0$. Then $V_L'(r) = 0$ gives $L^2 = M r^2 / (r - 3M)$, so circular orbits exist exactly for $r > 3M$. A computation shows that $V_L''(r_0) > 0$ at such an orbit exactly when $r_0 > 6M$. Hence $r_{\mathrm{ISCO}} = 6M$, where the orbit is marginally stable, with $L^2 = 12 M^2$.

Non-example: in Newtonian gravity the effective potential $-M/r + L^2/(2r^2)$ has a strict minimum at $r_0 = L^2/M$ for every $L > 0$. Every radius carries a stable circular orbit, and the infimum is $0$, which is not the radius of any orbit.

For the Kerr metric of mass $M$ and spin parameter $a$, the equatorial ISCO depends on the direction of motion. It moves inward towards $r = M$ for co-rotating orbits and outward towards $r = 9M$ for counter-rotating orbits as $\abs{a} \to M$.