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

We derive the transit timing variations (TTVs) of two planets near a second order mean motion resonance on nearly circular 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 $j \ge 3$ is an integer characterizing the resonance and $n_2$ 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 both the eccentricities and longitudes of pericenter of each planet. The TTVs of the two planets are approximated anti-correlated, with phases of $ϕ$ and $\approx ϕ+π$, where the phase $ϕ$ also depends on the eccentricities and longitudes of pericenter. Therefore, the TTVs caused by proximity to a second order mean motion resonance do not in general uniquely determine both planet masses, eccentricities, and pericenters. This is completely analogous to the case of TTVs induced by two planets near a first order mean motion resonance. We explore how other TTV signals, such as the short-period synodic TTV or a first order resonant TTV, in combination with the second order resonant TTV, can break degeneracies. Lastly, we derive approximate formulae for the TTVs of planets near any order eccentricity-type mean motion resonance; this shows that the same basic sinusoidal TTV structure holds for all eccentricity-type resonances. Our general formula reduces to previously derived results near first order mean motion resonances.