  Solar system planets move on almost circular orbits. In strong contrast, many
massive gas giant exoplanets travel on highly elliptical orbits, whereas the
shape of the orbits of smaller, more terrestrial, exoplanets remained largely
elusive. Knowing the eccentricity distribution in systems of small planets
would be important as it holds information about the planet's formation and
evolution, and influences its habitability. We make these measurements using
photometry from the Kepler satellite and utilizing a method relying on Kepler's
second law, which relates the duration of a planetary transit to its orbital
eccentricity, if the stellar density is known. Our sample consists of 28 bright
stars with precise asteroseismic density measurements. These stars host 74
planets with an average radius of 2.6 $R_\oplus$. We find that the eccentricity
of planets in Kepler multi-planet systems is low and can be described by a
Rayleigh distribution with $\sigma$ = 0.049 $\pm$ 0.013. This is in full
agreement with solar system eccentricities, but in contrast to the eccentricity
distributions previously derived for exoplanets from radial velocity studies.
Our findings are helpful in identifying which planets are habitable because the
location of the habitable zone depends on eccentricity, and to determine
occurrence rates inferred for these planets because planets on circular orbits
are less likely to transit. For measuring eccentricity it is crucial to detect
and remove Transit Timing Variations (TTVs), and we present some previously
unreported TTVs. Finally transit durations help distinguish between false
positives and true planets and we use our measurements to confirm six new
exoplanets.
