Doppler planet searches have discovered that giant planets follow orbits with a wide range of orbital eccentricities, revolutionizing theories of planet formation. The discovery of hundreds of exoplanet candidates by NASAs Kepler mission enables astronomers to characterize the eccentricity distribution of small exoplanets. Measuring the eccentricity of individual planets is only practical in favorable cases that are amenable to complementary techniques (e.g., radial velocities, transit timing variations, occultation photometry). Yet even in the absence of individual eccentricities, it is possible to study the distribution of eccentricities based on the distribution of transit durations (relative to the maximum transit duration for a circular orbit). We analyze the transit duration distribution of Kepler planet candidates. We find that for host stars with T_eff > 5100 K we cannot invert this to infer the eccentricity distribution at this time due to uncertainties and possible systematics in the host star densities. With this limitation in mind, we compare the observed transit duration distribution with models to rule out extreme distributions. If we assume a Rayleigh eccentricity distribution for Kepler planet candidates, then we find best-fits with a mean eccentricity of 0.1-0.25 for host stars with T_eff < 5100 K. We compare the transit duration distribution for different subsets of Kepler planet candidates and discuss tentative trends with planetary radius and multiplicity. High-precision spectroscopic follow-up observations for a large sample of host stars will be required to confirm which trends are real and which are the results of systematic errors in stellar radii. Finally, we identify planet candidates that must be eccentric or have a significantly underestimated stellar radius.
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