Period Ratio Sculpting near Second-order Mean-motion Resonances

Abstract Second-order mean-motion resonances lead to an interesting phenomenon in the sculpting of the period-ratio distribution, due to their shape and width in period-ratio/eccentricity space. As the osculating periods librate in resonance, the time-averaged period ratio approaches the exact commensurability. The width of second-order resonances increases with increasing eccentricity, and thus more eccentric systems have a stronger peak at commensurability when averaged over sufficient time. The libration period is short enough that this time-averaging behavior is expected to appear on the timescale of the Kepler mission. Using N-body integrations of simulated planet pairs near the 5:3 and 3:1 mean-motion resonances, we investigate the eccentricity distribution consistent with the planet pairs observed by Kepler. This analysis, an approach independent from previous studies, shows no statistically significant peak at the 3:1 resonance and a small peak at the 5:3 resonance, placing an upper limit on the Rayleigh scale parameter, σ, of the eccentricity of the observed Kepler planets at σ = 0.245 (3:1) and σ = 0.095 (5:3) at 95% confidence, consistent with previous results from other methods.