Radiative efficiency of black-hole accretion ($\eta$)

The radiative efficiency is $\eta =L/(\dot{M}{c}^2)$. For a zero-torque thin disk it is $1-E_{\mathrm{ISCO}}$, the binding energy per unit rest mass released before matter crosses the innermost stable circular orbit.

The maximal radiative efficiency is the available binding energy per unit accreted rest-mass energy when all energy released outside the capture boundary escapes as radiation. A relativistic zero-torque disk has ${\eta }_{\max }=1-E_{\mathrm{ISCO}}$.

The innermost stable circular orbit is the smallest stable circular timelike orbit around a black hole. Its radius and orbital energy depend on black-hole spin and on whether the orbit is prograde or retrograde.

For equatorial test-particle circular orbits in the extremal-spin limit, the prograde Kerr black hole innermost stable circular orbit has Boyer-Lindquist radius tending to $GM/c^2$ and specific energy tending to $1/\sqrt{3}$. The retrograde limit has radius $9GM/c^2$ and specific energy $5/(3\sqrt{3})$. The prograde radius approaching the horizon coordinate does not make the massive limiting orbit identical to the null horizon generator.^[ref]

The Schwarzschild coordinate rotation frequency at the marginally stable circular orbit, measured per unit time of a stationary clock at infinity. It scales inversely with black-hole mass. A finite-radius static clock adds a gravitational time-dilation factor; a quasi-periodic brightness signal need not directly equal this orbital frequency.

For ideal zero-torque equatorial thin discs, efficiency is $1-E_{\mathrm{ISCO}}$. The formal extremal retrograde, Schwarzschild and extremal prograde limiting radii are respectively $9,6,1$ in units $GM/c^2$, giving efficiencies $1-\sqrt{25/27}$, $1-\sqrt{8/9}$ and $1-1/\sqrt{3}$. Photon capture and non-ideal accretion can change the realized efficiency.