Gravitational wave damping from a self-gravitating oscillating ring of matter around a black hole

We consider the space time structure of a black hole-ring system in which a non-rotating black hole is surrounded by a self-gravitating thick accretion disk, here simplified to be a thin ring. We consider the vertical oscillation of the ring about its equilibrium position caused by a perturbation along the vertical direction. We compute the gravity wave luminosity and loss rate of the angular momentum to study how the perturbation gradually dampens after a continuous emission of the gravitational wave. We present the results of a few such systems and place them on the LISA sensitivity curve to show that some such systems should be observable.