 Resonant conjunction geometry

For an eccentric outer body and circular inner perturber, the exterior resonant argument is $\varphi =(p+q){\lambda }_{\mathrm{out}}-p{\lambda }_{\mathrm{in}}-q\varpi$. At equal mean longitudes $\Lambda$, conjunction phases areThe synodic period and coprime $p,q$ determine the visitation order, with advance $2\pi p/q$. For actual true alignment, write $x=f$ and $y=M(f)$ for the eccentric body's true anomaly and mean anomaly. Then $\varphi =(p+q)y-px$ instead of $qx$. The evenly spaced directions are a leading-small-eccentricity approximation.