139 Kepler planets transit time variations

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 the order of a few percent.

More precisely, the rms eccentricity is 0.018_-0.004_^+0.005^, and planets smaller than 2.5 R_{earth}_ are around twice as eccentric as those bigger than 2.5 R_{earth}_. We also find a best-fit density-radius relationship {rho}~3 g/cm^3^x(R/3 R_{earth}_)^-2.3^ for the 56 planets that likely have small eccentricity and hence small statistical correction to their masses. Many planets larger than 2.5 R_{earth}_are less dense than water, implying that their radii are largely set by a massive hydrogen atmosphere.