Conjunction-pattern precession near a mean-motion resonance ($\dot{\Lambda }$)

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 $2\pi p$ givesThe usual formula $-2\pi p\delta /(q{P}_b)$ keeps only first order in $\delta$. A labelled branch takes $q{P}_b(1+\delta )/(p|\delta |)$ 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.

For an exterior circular grain with radiation-pressure coefficient $\beta$, set $q=\sqrt{1-\beta }$ and $ϵ=a/a_p-1$. Its next conjunction occurs after the synodic period, during which the planet's unwrapped mean longitude advances by $\Lambda =2\pi /[1-q(1+ϵ{)}^{-3/2}]$. A weak planetary scattering encounter changes this by $\delta \Lambda \simeq -3\pi q(1+ϵ{)}^{-5/2}\delta ϵ/[1-q(1+ϵ{)}^{-3/2}{]}^2$. Unwrapped longitude counts turns; reducing it modulo $2\pi$ before comparison would erase this measure of timing sensitivity.