Suppose we have a star of mass orbiting a black hole of mass . The semimajor of is . Now suppose we introduce a dipole perturbation to the force , where is a scalar, but not necessarily a constant, while is a constant vector. Under the influence of the perturbed force, the total angular momentum may change, but the component parallel to doesn't. This state can be easily justified
We denote the angle between and by . Since , where is the eccentricity, the conservation of yields
Hence, under the influence of such a perturbation, the inclination can grow at the expense of the eccentricity, and vice versa.
Now suppose that the perturbation is due to another star with mass and semimajor axis . We are now interested in the time scale on which the inclination - eccentricity exchange occurs. We will assume that the eccentricity are not very close to 1. The magnitude of the angular momentum of is . The torque exerts on is . However, the contribution of this term cancels out when goes round. The next leading order term is the different between the torques exerted on at opposite phases of its orbit . The time scale is therefore given by
where denotes the period of each star around .