Determination of three-dimensional spin-orbit angle with joint analysis of asteroseismology, transit lightcurve, and the Rossiter-McLaughlin effect: Cases of HAT-P-7 and Kepler-25

We develop a detailed methodology of determining three-dimensionally the angle between the stellar spin and the planetary orbit axis vectors, ψ, for transiting planetary systems. The determination of ψ requires the independent estimates of the inclination angles of the stellar spin axis and of the planetary orbital axis with respect to the line of sight, iBlack Star and i_orb, and the projection of the spin-orbit angle on to the plane of the sky, λ. These are mainly derived from asteroseismology, transit lightcurve, and the Rossiter-McLaughlin effect, respectively. The detailed joint analysis of those three datasets enables an accurate and precise determination of the numerous parameters characterizing the planetary system, in addition to ψ. We demonstrate the power of the joint analysis for the two specific systems HAT-P-7 and Kepler-25. HAT-P-7b is the first exoplanet suspected to be a retrograde (or polar) planet because of the significant misalignment λ ≈ 180°. Our joint analysis indicates iBlack Star ≈ 30° and ψ ≈ 120°, suggesting that the planetary orbit is closer to polar rather than retrograde. Kepler-25 is one of the few multi-transiting planetary systems with measured λ, and hosts two short-period transiting planets and one outer non-transiting planet. The projected spin-orbit angle of the larger transiting planet, Kepler-25c, has been measured to be λ ≈ 0°, implying that the system is well aligned. With the help of the tight constraint from asteroseismology, however, we obtain iBlack Star = 65°.4_-6°.4^+10°.6 and ψ = 26.°9_-9°.2^+7°.0, and thus find that the system is actually mildly misaligned. This is the first detection of the spin-orbit misalignment for the multiple planetary system with a main-sequence host star, and points to mechanisms that tilt a stellar spin axis relative to its protoplanetary disk.