Luminosity of black hole accretion disc

Three times the Schwarzschild radius corresponds to the closest stable circular orbit around a black hole. The general idea is that as matters moves in towards the black hole it gets stuck in an accretion disc where angular momentum has to be moved outwards in order to allow the matter to move inwards. However, once the matter gets inside $3r_s$, that problem disappears and the material is able to flow straight into the black hole.

Thus when we observe black hole accretion discs we expect them to be truncated at $3r_s$.

So I think the argument then is along the lines of - the gravitational potential energy of unit mass falling to $3r_s$ is converted into an orbital kinetic energy of $0.5v^2 = GM/6r_s$ per unit mass and the rest is converted to radiation. Thus
$$L = left[frac{GM}{3r_s} - frac{GM}{6r_s}right] frac{dM}{dt}$$
$$ L = frac{GM}{6r_s} frac{dM}{dt} = frac{1}{12}c^2 frac{dM}{dt} .$$