What Transit Timing Variations Ought to Look Like for Exoplanets in Near Eccentric Orbital Resonances

Transit timing variations for planets near eccentricity-type mean motion resonances

Authors:

Deck et al

Abstract:

We present a derivation of the transit timing variations (TTVs) of a pair of planets near the j:j-2 second order resonance on nearly circular and nearly coplanar orbits. We show that the TTVs of each planet are given by sinusoids with a frequency of j n_2-(j-2)n_1, where n_2 and n_1 are the mean motions of the outer and inner planets, respectively. The amplitude of the TTV depends on the mass of the perturbing planet, relative to the mass of the star, and on a function of the eccentricities and longitudes of pericenter of each planet. The phase of each sinusoid is approximately phi and phi+pi, where the phase phi also depends on the eccentricities and longitudes of pericenter. Therefore, the situation for second order resonances is analogous to the case of TTVs induced by two planets near a first order mean motion resonance. Degeneracies between planet masses and eccentricities/longitudes of pericenter occur when the small phase offset from pi cannot be resolved, and even when it can be, degeneracies persist between the two planet's eccentricities and longitudes of pericenter. In order to break degeneracies one must measure another, independent signal of the TTVs such as the short period "chopping" TTV. Alternatively, we show how the second order terms can be used to break degeneracies near first order resonances (e.g. 4:2 resonant terms near the 2:1 resonance). Lastly, we derive an approximate formulae for the TTVs of a pair of planets near any order eccentricity-type mean motion resonance, this shows that the same basic TTV structure holds for all eccentricity-type resonances. Our general formula reduces to previously derived results (Lithwick et al. 2012) near first order mean motion resonances.