  Exoplanet orbital eccentricities offer valuable clues about the history of
planetary systems. Eccentric, Jupiter-sized planets are particularly
interesting: they may link the "cold" Jupiters beyond the ice line to close-in
hot Jupiters, which are unlikely to have formed in situ. To date,
eccentricities of individual transiting planets primarily come from radial
velocity measurements. Kepler has discovered hundreds of transiting Jupiters
spanning a range of periods, but the faintness of the host stars precludes
radial velocity follow-up of most. Here we demonstrate a Bayesian method of
measuring an individual planet's eccentricity solely from its transit light
curve using prior knowledge of its host star's density. We show that eccentric
Jupiters are readily identified by their short ingress/egress/total transit
durations -- part of the "photoeccentric" light curve signature of a planet's
eccentricity --- even with long-cadence Kepler photometry and
loosely-constrained stellar parameters. A Markov Chain Monte Carlo exploration
of parameter posteriors naturally marginalizes over the periapse angle and
automatically accounts for the transit probability. To demonstrate, we use
three published transit light curves of HD 17156 b to measure an eccentricity
of e = 0.71 +0.16/-0.09, in good agreement with the discovery value e =
0.67+/-0.08 based on 33 radial-velocity measurements. We present two additional
tests using actual Kepler data. In each case the technique proves to be a
viable method of measuring exoplanet eccentricities and their confidence
intervals. Finally, we argue that this method is the most efficient, effective
means of identifying the extremely eccentric, proto hot Jupiters predicted by
Socrates et al. (2012).
