Gravitational radiation from a particle in circular orbit around a black hole. III. Stability of circular orbits under radiation reaction

We use the Teukolsky perturbation formalism to show that (i) a particle in circular motion around a nonrotating black hole remains in a circular orbit under the influence of radiation reaction, and (ii) circular orbits are stable only if the orbital radius is greater than a critical radius ${\mathit{r}}_{\mathit{c}}$≃6.6792M, where M is the mass of the black hole. A circular orbit is stable if, when slightly perturbed so that it acquires a small eccentricity, the radiation reaction decreases the eccentricity; a circular orbit is unstable if the radiation reaction increases the eccentricity. Our analysis is restricted by four major assumptions: (i) the black hole is nonrotating, (ii) the eccentricity is always small, (iii) the gravitational perturbations are linear, and (iv) the adiabatic approximation (that the radiation reaction takes place over a time scale much larger than the orbital period) is valid. On the other hand, our analysis is not limited to weak-field, slow-motion situations; it is valid for particle motion in strong gravitational fields.