Near-resonant transit-timing superperiod ($P_t=2\pi /|(j+1)n_2-j{n}_1|$)

For two nearly circular Kepler orbits near a first-order $(j+1):j$ period ratio, the slow resonant argument circulates on the displayed superperiod when longitude of periapsis precession is negligible. If $P_2/P_1=((j+1)/j)(1+\Delta )$, then $P_t=P_1(1+\Delta )/(j|\Delta |)$. Within resonant-argument libration, this circulation formula is replaced by an amplitude-dependent libration period.

Near a first-order $(j+1):j$ orbital resonance, a weak periodic resonant orbital perturbation can accumulate a large transit-timing variation. Let subscripts 1 and 2 denote inner and outer planets, define ${\mu }_2=M_2/M_{\star }\ll 1$, $\alpha =a_1/a_2$, and writeHere $n_i=2\pi /P_i$ are mean motions. Approximate the inner planet's disturbing potential per unit mass bywhere $f$ is the dimensionless coefficient of this chosen resonant harmonic, $e$ is its effectively constant free orbital eccentricity, ${\lambda }_i$ are mean longitudes and ${\varpi }_1$ is longitude of periapsis. Neglect precession and suppose the resonant argument circulates approximately uniformly.

The leading semimajor-axis equation gives $|{\dot{a}}_1|/a_1\sim 2j{\mu }_2\alpha n_1|f|e$. Since $\delta n_1/n_1=-3\delta a_1/(2a_1)$, one integration supplies a factor $1/|\omega |$ in the mean motion perturbation, and a second supplies it in the mean longitude. Converting the phase perturbation to a timing amplitude givesDropping $\Delta$ against one in the numerator gives the displayed near-resonant estimate. Its $e/{\Delta }^2$ enhancement assumes this term dominates direct phase and forced-eccentricity effects. Close enough to resonance, resonant-argument libration and changes of the free orbital eccentricity invalidate the expansion; the divergence does not predict arbitrarily large transit times.