Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. Perturbations on the time-scale of the orbital period of the perturber

Abstract

We study the long-term time-scale (i.e. period comaprable to the orbital period of the outer perturber object) transit timing variations in transiting exoplanetary systems which contain a further, more distant (a_2>>a_1) either planetary, or stellar companion. We give an analytical form of the O-C diagram (which describes such TTV-s) in trigonometric series, valid for arbitrary mutual inclinations, up to the sixth order in the inner eccentricity. We show that the dependence of the O-C on the orbital and physical parameters can be separated into three parts. Two of these are independent of the real physical parameters (i.e. masses, separations, periods) of a concrete system, and depend only on dimensionless orbital elements, and so, can be analyzed in general. We analyze these dimensionless amplitudes for different arbitrary initial parameters, as well as for two particular systems CoRoT-9b and HD 80606b. We find in general, that while the shape of the O-C strongly varies with the angular orbital elements, the net amplitude (departing from some specific configurations) depends only weakly on these elements, but strongly on the eccentricities. As an application, we illustrate how the formulae work for the weakly eccentric CoRoT-9b, and the highly eccentric HD 80606b. We consider also the question of detection, as well as the correct identification of such perturbations. Finally, we illustrate the operation and effectiveness of Kozai cycles with tidal friction (KCTF) in the case of HD 80606b.