The Discovery of the Long-Period, Eccentric Planet Kepler-88 d and System Characterization with Radial Velocities and Photodynamical Analysis

We present the discovery of Kepler-88 d (P_d = 1403±14 , M sin i_d = 970±50M⊕ = 3.05±0.16M_J, e_d = 0.42±0.04) based on six years of radial velocity (RV) follow-up from the W. M. Keck Observatory High Resolution Echelle Spectrometer spectrograph. Kepler-88 has two previously identified planets. Kepler-88 b (KOI-142.01) transits in the NASA Kepler photometry and has very large transit timing variations (TTVs). Nesvorný et al. performed a dynamical analysis of the TTVs to uniquely identify the orbital period and mass of the perturbing planet (Kepler-88 c), which was later was confirmed with RVs from the Observatoire de Haute-Provence (OHP). To fully explore the architecture of this system, we performed photodynamical modeling on the Kepler photometry combined with the RVs from Keck and OHP and stellar parameters from spectroscopy and Gaia. Planet d is not detectable in the photometry, and long-baseline RVs are needed to ascertain its presence. A photodynamical model simultaneously optimized to fit the RVs and Kepler photometry yields the most precise planet masses and orbital properties yet for b and c: P_b = 10.91647±0.00014days, M_b = 9.5±1.2M⊕, P_c = 22.2649±0.0007 days, and M_c = 214.1±5.3M⊕. The photodynamical solution also finds that planets b and c have low eccentricites and low mutual inclination, are apsidally anti-aligned, and have conjunctions on the same hemisphere of the star. Continued RV follow-up of systems with small planets will improve our understanding of the link between inner planetary system architectures and giant planets