Epicyclic ellipse ($X/Y=\kappa /(2\Omega )$)

In a Cartesian frame rotating with an epicyclic guiding center, a small planar oscillation obeys $x=X\cos (\kappa t+\chi )$ and $y=-(2\Omega X/\kappa )\sin (\kappa t+\chi )$. The ellipse is elongated azimuthally when the radial epicyclic frequency satisfies $\kappa <2\Omega$. Conservation of angular momentum makes the motion around the guiding centre retrograde.