Kepler-50: Two Planets in Close Resonance Perturbed by a Non-transiting Planet
Kepler-50 has two transiting planets close to the 6:5 commensurability. We analyze transit times measured from Kepler data with dynamical fits. We attribute the residual signal to a near-resonance non-transiting third planet in the system. The parameters of the third planet are degenerate. We identified 55 (15) regions of interest for an exterior (interior) perturber, and characterize system parameters in each region of interest with posterior sampling. We find that removing samples that are not long-term stable provides few additional constraints on posteriors, and that the transit times do not confirm the likely 2-body resonant state of the transiting planets as $\approx$ 68\% of posterior samples that are likely stable are also likely in libration. We identify a small fraction of samples that are in 3-body libration. Despite the degeneracies, we recover robust, tightly constrained masses for Kepler-50 b and c ($2.35^{+0.48}_{-0.43}$ $M_{\oplus}$ and $4.43^{+0.45}_{-0.49}$ $M_{\oplus}$, respectively) and strongly detected non-zero relative eccentricity for Kepler-50 b and c with ($e_c\cos\varpi_c$ - $e_b\cos\varpi_b$) = -0.045 $\pm 0.003$, and ($e_c\sin\varpi_c$ - $e_b\sin\varpi_b$) = -0.030 $\pm 0.004$. These close neighbors are in the radius valley and sub-Neptune by size, respectively ($1.750 \pm 0.038$ $R_{\oplus}$ and $2.079 \pm 0.060$ $R_{\oplus}$). With one of the closest planet pairs known and one of the closest to first-order commensurability, the Kepler-50 system is an especially valuable benchmark for planet formation theory.