Mass and Orbit Determination from Transit Timing Variations of Exoplanets

The timing variations of transits of an exoplanet provide means of detecting additional planets in the system. The short-period and resonant variations of the transit signal are probably the most diagnostic of the perturbing planet's mass and orbit. The method can be sensitive to small perturbing masses near the transiting planet and for orbits at mean motion resonances. It is not evident, however, how the mass and orbit of the perturbing planet can be determined from the observed variations of transit times. This is a difficult inverse problem. Direct N-body integrations are computationally too expensive to provide an adequate sampling of parameter space. Here we develop an alternative method based on analytic perturbation theory. We find that this new method is typically ~10⁴ times faster than direct N-body integrations. The perturbation theory that we use here has an adequate precision to predict timing variations for most planetary orbits except those with very large eccentricities where the expansion of the disturbing function is divergent. By applying the perturbation method to the inverse problem we determine the number and precision of the measured transit times that are required for the unique and correct characterization of the perturbing planet. We find that the required precision is typically a small fraction (~15%-30%) of the full transit timing variation amplitude. Very high precision observations of transits will therefore be needed. We discuss the optimal observation strategy to characterize a planetary system from the transit timing variations. We find that the timing of secondary transits, if measured with adequate precision, can help to alleviate the problem with the degeneracy of solutions of the inverse problem.