Lidov Kozai Mechanism

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 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 $M$.