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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. 5R_. We also find a best-fit density-radius relationship ~3 g/cm³ (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.
Hadden et al. (Tue,) studied this question.