FANDOM


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 $.