Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 4 / vii

Let $u={\lambda }_2-{\Omega }_2$ be the outer body's argument of latitude in the nearly circular limit and let $\Delta ={\lambda }_2-{\lambda }_1$ be its azimuth in the frame rotating with $M_1$. At the $q=2$ libration centre,Viewed along the star--$M_1$ line, the projected coordinates therefore have the schematic formThis relation is the general sketch prescription. For the simplest $p=1$ resonance it is a figure-eight-like curve, with conjunctions on the vertical axis at maximum positive or negative height and node crossings in the reference plane. More generally the curve repeats with the $p$ vertical oscillations and two relative azimuthal circuits required by the $(p+2):p$ commensurability.