Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 316 / 3 / ii

In the mean-conjunction approximation set ${\lambda }_b={\lambda }_c=\Lambda$. The resonant conjunction geometry is thenThere are $q$ phase directions spaced by $2\pi /q$. The integer $p$ determines their temporal order: at exact resonance the synodic period is $P_{\mathrm{syn}}=(p+q)P_b/q=p{P}_c/q$, and the conjunction longitude advances by $2\pi p/q$ per conjunction. For coprime $p,q$ this visits all $q$ branches; resonant-argument libration broadens each direction by about its libration amplitude of a resonant argument divided by $q$.

Mean conjunction and actual alignment are different for an eccentric orbit. If $\Lambda$ is the true common longitude, write $x=\Lambda -{\varpi }_c=f_c$ and $y=M_c(f_c)$, with $M_c$ the mean anomaly. Because ${\lambda }_c={\varpi }_c+y$ while the circular planet has ${\lambda }_b=\Lambda$, the exact geometric relation isThe displayed equally spaced directions use $y=x+O(e_c)$, hence are leading-small-eccentricity geometry. With appreciable $e_c$, use Kepler's equation to locate true conjunctions; equating true and mean longitudes silently is not exact.