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We develop and apply methods to extract planet masses and eccentricities fromobserved transit time variations (TTVs). First, we derive simple analyticexpressions for the TTV that include the effects of both first- andsecond-order resonances. Second, we use N-body Markov chain Monte Carlo (MCMC)simulations, as well as the analytic formulae, to measure the masses andeccentricities of ten planets discovered by Kepler that have not previouslybeen analyzed. Most of the ten planets have low densities. Using the analyticexpressions to partially circumvent degeneracies, we measure smalleccentricities of a few percent or less.

Numerical and Analytical Modelling of Transit Time Variations