We extract densities and eccentricities of 139 sub-Jovian planets by analyzing transit time variations (TTVs) obtained by the Kepler mission through Quarter 12. We partially circumvent the degeneracies that plague TTV inversion with the help of an analytical formula for the TTV. From the observed TTV phases, we find that most of these planets have eccentricities of order a few percent. More precisely, the r.m.s. eccentricity is 0.018 +0.005 −0.004 , and planets smaller than 2.5R ⊕ are around twice as eccentric as those bigger than 2.5 R ⊕ . We also find a best-fit density-radius relationship ρ ≈ 3 g/cm 3 × (R/3R ⊕ ) −2.3 for the 56 planets that likely have small eccentricity and hence small statistical correction to their masses. Many planets larger than 2.5R ⊕ are less dense than water, implying that their radii are largely set by a massive hydrogen atmosphere.

We conduct a uniform analysis of the transit timing variations (TTVs) of 145 planets from 55 Kepler multiplanet systems to infer planet masses and eccentricities. Eighty of these planets do not have previously reported mass and eccentricity measurements. We employ two complementary methods to fit TTVs: Markov chain Monte Carlo simulations based on N -body integration and an analytic fitting approach. Mass measurements of 49 planets, including 12 without previously reported masses, meet our criterion for classification as robust. Using mass and radius measurements, we infer the masses of planets' gaseous envelopes for both our TTV sample as well as transiting planets with radial velocity observations. Insight from analytic TTV formulae allows us to partially circumvent degeneracies inherent to inferring eccentricities from TTV observations. We find that planet eccentricities are generally small, typically a few percent, but in many instances are non-zero.

We combine analytical understanding of resonant dynamics in two-planet systems with machine-learning techniques to train a model capable of robustly classifying stability in compact multiplanet systems over long timescales of 109 orbits. Our Stability of Planetary Orbital Configurations Klassifier (SPOCK) predicts stability using physically motivated summary statistics measured in integrations of the first 104 orbits, thus achieving speed-ups of up to 105 over full simulations. This computationally opens up the stability-constrained characterization of multiplanet systems. Our model, trained on ∼100,000 three-planet systems sampled at discrete resonances, generalizes both to a sample spanning a continuous period-ratio range, as well as to a large five-planet sample with qualitatively different configurations to our training dataset. Our approach significantly outperforms previous methods based on systems’ angular momentum deficit, chaos indicators, and parametrized fits to numerical integrations. We use SPOCK to constrain the free eccentricities between the inner and outer pairs of planets in the Kepler-431 system of three approximately Earth-sized planets to both be below 0.05. Our stability analysis provides significantly stronger eccentricity constraints than currently achievable through either radial velocity or transit-duration measurements for small planets and within a factor of a few of systems that exhibit transit-timing variations (TTVs). Given that current exoplanet-detection strategies now rarely allow for strong TTV constraints [S. Hadden, T. Barclay, M. J. Payne, M. J. Holman, Astrophys. J. 158, 146 (2019)], SPOCK enables a powerful complementary method for precisely characterizing compact multiplanet systems. We publicly release SPOCK for community use.

We develop and apply methods to extract planet masses and eccentricities from observed transittimingvariations (TTVs). First, we derive simple analytic expressions for the TTVthat include the effects of both first-and secondorder resonances. Second, we use N-body Markov chain Monte Carlo simulations, as well as the analytic formulae, to measure the masses and eccentricities of 10planets discovered by Kepler that have not previously been analyzed. Most of the 10planets have low densities. Using the analytic expressions to partially circumvent degeneracies, we measure small eccentricities of a few percent or less.

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom (1980)'s pioneering work. But whereas Wisdom considered the overlap of firstorder resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet, and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian (Hadden in prep). We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.