Transit-timing variation (TTV)

A departure of transit arrival times from a constant-period ephemeris can reveal gravitational interactions with other bodies. Dynamical fits may constrain an exoplanet mass independently of transit depth, providing a mass to combine with radius for exoplanet interior models.

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.

A first-order resonant orbital perturbation changes the semi-major axis and therefore the mean motion. Integrating the sinusoidal mean-motion variation over the near-resonant transit-timing superperiod gives a mean-longitude amplitude proportional to ${\mu }_{2}e/{\Delta }^2$. Converting phase to transit time gives the displayed amplitude ratio. It assumes small perturbations, circulating resonant argument, effectively fixed free orbital eccentricity, and dominance of this enhanced term over direct phase and forced-eccentricity contributions. The formal divergence as $\Delta \to 0$ signals failure of this expansion, not infinitely large transit times.