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Second-order mean-motion resonances lead to an interesting phenomenon in thesculpting of the period ratio distribution due to their shape and width inperiod-ratio/eccentricity space. As the osculating periods librate inresonance, the time-averaged period ratio approaches the exactcommensurability. The width of second-order resonances increases withincreasing eccentricity, and thus more eccentric systems have a stronger peakat commensurability when averaged over sufficient time. The libration period isshort enough that this time-averaging behavior is expected to appear on thetimescale of the Kepler mission. Using N-body integrations of simulated planetpairs near the 5:3 and 3:1 mean-motion resonances, we investigate theeccentricity distribution consistent with the planet pairs observed by Kepler.This analysis, an approach independent from previous studies, shows nostatistically significant peak at the 3:1 resonance and a small peak at the 5:3resonance, placing an upper limit on the Rayleigh scale parameter, $\sigma$, ofthe eccentricity of the observed Kepler planets at $\sigma=0.245$ (3:1) and$\sigma=0.095$ (5:3) at 95% confidence, consistent with previous results fromother methods.

Period Ratio Sculpting Near Second-Order Mean-Motion Resonances