= Simultaneous transit in a resonant chain
= Simultaneous transits in a resonant chain
{synonym}
In leading mean-conjunction geometry a common observer phase $x$ for two <resonant arguments> requires $qx=\phi_1$ and $nx=\phi_2$ modulo $2\pi$. With $d=\gcd(q,n)$, the exact modular criterion is
$$
\frac{n\phi_1-q\phi_2}{d}\in2\pi\mathbb Z.
$$
Necessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb Z$. For exterior centres $\pi$ and interior centres $0,\pi,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed <exoplanet transit>: timing, observer orientation, <resonant-argument libration> widths and true-versus-mean longitude must also be considered.
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