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Suppose we have a star of mass  m_1 orbiting a black hole of mass  M . The semimajor of  m_1 is  a_1 . Now suppose we introduce a dipole perturbation to the force \alpha \vec{r}_0, where \alpha is a scalar, but not necessarily a constant, while \vec{r}_0 is a constant vector. Under the influence of the perturbed force, the total angular momentum may change, but the component parallel to \vec{r}_0 doesn't. This state can be easily justified

 \frac{d}{dt} \left(\vec{r}_0 \cdot \vec{L} \right ) = \frac{d}{dt} \left(\vec{r}_0 \cdot \vec{r} \times \vec{v} \right ) = \vec{r}_0 \cdot \vec{r} \times \frac{d \vec{v}}{dt} \propto \vec{r}_0 \cdot \vec{r} \times \vec{r_0} = 0

We denote the angle between  \vec{r}_0 and  \vec{L} by  i . Since  L \propto \sqrt{1-e^2}, where e is the eccentricity, the conservation of  \vec{L} \cdot \vec{r}_0 yields

 \cos i \sqrt{1-e^2} = \rm const

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  m_2 and semimajor axis a_2 \gg a_1. 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 m_1 is L_1 \approx \sqrt{G M a_1}. The torque m_2 exerts on m_1 is G m_2/a_2^2 a_1. However, the contribution of this term cancels out when m_1 goes round. The next leading order term is the different between the torques exerted on </math>m_1</math> at opposite phases of its orbit  \dot{L} \approx G m_2 a_1^2/a_2^3 . The time scale is therefore given by

 T_{lk} \approx \frac{L}{\dot{L}} \approx \frac{\sqrt{G M/a_1^3}}{G m2/a_2^3} \approx \frac{P_2^2}{P_1} \frac{M}{m_2}

where  P denotes the period of each star around M.