= Conjunction-pattern precession near a mean-motion resonance
{title2=$\dot\Lambda$}
With constant <mean motions> and $P_c/P_b=(1+\delta)(p+q)/p$, mean <conjunctions> advance by $\Delta\Lambda=2\pi/(P_c/P_b-1)$ each <synodic period>. Tracking one branch every $q$ <conjunctions> and subtracting $2\pi p$ gives
$$
\dot\Lambda=\frac{q\Delta\Lambda-2\pi p}{qP_{\rm syn}}
=-\frac{2\pi p\delta}{qP_b(1+\delta)}.
$$
The usual formula $-2\pi p\delta/(qP_b)$ keeps only first order in $\delta$. A labelled branch takes $qP_b(1+\delta)/(p|\delta|)$ to turn once, whereas the full set of $q$ interchangeable directions repeats after $P_b(1+\delta)/(p|\delta|)$. Finite <orbital eccentricity> introduces true-conjunction corrections.
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