# Planar Precession Frequency of Orbit

What is the general relation between orbital precession $\Phi$, orbital frequency $\Omega$ and a radial perturbation frequency $\omega$?

For certain cases the answer is "clear", for example:

1) If $\Omega = \omega$ the perturbation performs one oscillation in the same time as one orbit is completed, hence there is no precession and $\Phi = 0$. (Or is $\Phi = \infty$?). I believe this suggests a form $\Phi \propto (\omega - \Omega)$

2) If $\omega = 2\Omega$ it takes the perturbation half an orbit to perform an oscillation. Hence $\Phi = 2\Omega$. This is perhaps suggestive of some type of fractional form $\Phi \propto \frac{\omega}{\Omega}$

I have now been drawing several diagrams with precessing orbits but don't seem able to work out the general relationship between $\Omega, \omega$ and $\Phi$. Anybody who knows?

I can see that the angle swept out during one oscillation of the perturbation is $$\Delta\theta = \Omega \frac{2\pi}{\omega}$$ and the precession should be something on form: $$\Phi = \frac{\Delta\theta-2\pi n}{2 \pi/\omega}$$ where n is an integer. However, the value of the integer depends on the ratio of $\omega$ and $\Omega$ and from here on I am stuck...