= Solution
For constant <mean motions>, consecutive mean <conjunctions> are separated by $P_{\rm syn}=P_bP_c/(P_c-P_b)$ and advance in longitude by $\Delta\Lambda=2\pi/(P_c/P_b-1)$. To follow one of the $q$ branches, compare <conjunctions> $q$ steps apart. At exact resonance that advance is $2\pi p$; subtract it to measure the slow drift. Hence <conjunction-pattern precession near a mean-motion resonance> is
$$
\dot\Lambda_{bc}=\frac{q\Delta\Lambda-2\pi p}{qP_{\rm syn}}
=\frac{(p+q)n_c-pn_b}{q}
=-\frac{2\pi p\,\delta}{qP_b(1+\delta)}.
$$
The printed expression is its leading small-offset term:
$$
\boxed{\dot\Lambda_{bc}=-\frac{2\pi\delta}{P_b}\frac pq+O(\delta^2)}.
$$
Positive $\delta$ gives retrograde drift and negative $\delta$ prograde drift. The tracked-branch rotation time is about $qP_b/(p|\delta|)$; the full set of $q$ interchangeable directions repeats after a rotation $2\pi/q$, giving the usual mean-pattern super-period $P_b(1+\delta)/(p|\delta|)$. The slow, continuously tracked branch interpretation assumes the resonance neighbourhood $|\delta|(p+q)\ll q$; merely $|\delta|\ll1$ need not give a slow pattern for an arbitrarily large integer ratio $p/q$. At $\delta=0$ there is no offset-driven drift.
The allowed directions can consequently move through the observer's line of sight. Triple <exoplanet transits> require the $bc$ and $cd$ patterns, together with their timing phases, to overlap that line of sight within the finite <exoplanet transit> windows; differing drift rates can open and close such windows. A small offset neither guarantees a triple event nor prohibits it permanently.
Moreover, in a true <mean-motion resonance>, bounded $\phi_{bc}$ can coexist with a period offset because $(p+q)n_c-pn_b=q\dot\varpi_c+\dot\phi_{bc}$. The mean <conjunction> pattern then follows apsidal precession plus the <resonant-argument libration> correction. For appreciable <orbital eccentricity>, the true-conjunction geometry has the periodic corrections identified above; the drift calculation is a mean-phase result.
Back to article page
