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

The angleobeys the D'Alembert characteristic. Because reflection in the reference plane makes the disturbing function even in inclination, the nodal coefficient must be even, and thereforeAt astronomical conjunction, ${\lambda }_1={\lambda }_2={\lambda }_c$, soThus $\varphi /q$ is the angular distance from the ascending node of $M_2$ to the conjunction. When $M_2$ is at that node, ${\lambda }_2={\Omega }_2$, andUp to the chosen sign convention, $\varphi /p$ is the angular separation between the inner body and the node when the outer body crosses the reference plane. These two views explain geometrically why libration confines both conjunction and node-crossing phases.