Extracting Planet Mass and Eccentricity From TTV data

TL;DR

This paper derives analytical formulas for TTV signals of Kepler planet pairs near resonances, enabling accurate mass estimation and insights into their eccentricities and evolutionary history.

Contribution

It introduces simple analytical formulas separating free and forced eccentricities, improving planet mass determination from TTV data.

Findings

Free eccentricities are small (<0.01), indicating substantial dissipation.

Planet masses are accurately determined within a factor of 2 from TTV amplitudes.

Some planets exhibit non-zero free eccentricity, suggesting post-resonance eccentricity excitation.

Abstract

Most planet pairs in the Kepler data that have measured transit time variations (TTV) are near first-order mean-motion resonances. We derive analytical formulae for their TTV signals. We separate planet eccentricity into free and forced parts, where the forced part is purely due to the planets' proximity to resonance. This separation yields simple analytical formulae. The phase of the TTV depends sensitively on the presence of free eccentricity: if the free eccentricity vanishes, the TTV will be in phase with the longitude of conjunctions. This effect is easily detectable in current TTV data. The amplitude of the TTV depends on planet mass and free eccentricity, and it determines planet mass uniquely only when the free eccentricity is sufficiently small. We proceed to analyze the TTV signals of six short period Kepler pairs. We find that three of these pairs (Kepler-18,24,25) have TTV phase consistent with zero. The other three (Kepler-23,28,32) have small TTV phases, but ones that are distinctly non-zero. We deduce that the free eccentricities of the planets are small, < 0.01, but not always vanishing. Furthermore, as a consequence of this, we deduce that the true masses of the planets are fairly accurately determined by the TTV amplitudes, within a factor < 2. The smallness of the free eccentricities suggests that the planets have experienced substantial dissipation. This is consistent with the hypothesis that the observed pile-up of Kepler pairs near mean-motion resonances is caused by resonant repulsion. But the fact that some of the planets have non-vanishing free eccentricity suggests that after resonant repulsion occurred there was a subsequent phase in the planets' evolution when their eccentricities were modestly excited, perhaps by interplanetary interactions.