How low can you go? The photoeccentric effect for planets of various sizes

It is well-known that the light curve of a transiting planet contains information about the planet's orbital period and size relative to the host star. More recently, it has been demonstrated that a tight constraint on an individual planet's eccentricity can sometimes be derived from the light curve via the "photoeccentric effect," the effect of a planet's eccentricity on the shape and duration of its light curve. This has only been studied for large planets and high signal-to-noise scenarios, raising the question of how well it can be measured for smaller planets or low signal-to-noise cases. We explore the limits of the photoeccentric effect over a wide range of planet parameters. The method hinges upon measuring g directly from the light curve, where g is the ratio of the planet's speed (projected on the plane of the sky) during transit to the speed expected for a circular orbit. We find that when the signal-to-noise in the measurement of g is <10, the ability to measure eccentricity with the photoeccentric effect decreases. We develop a "rule of thumb" that for per-point relative photometric uncertainties σ = {10^-3, 10^-4, 10^-5}, the critical values of the planet-star radius ratio are R_p /R∗ ≈ {0.1, 0.05, 0.03} for Kepler-like 30 minute integration times. We demonstrate how to predict the best-case uncertainty in eccentricity that can be found with the photoeccentric effect for any light curve. This clears the path to study eccentricities of individual planets of various sizes in the Kepler sample and future transit surveys.