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, $i_\star$ and $i_{\rm orb}$, and the projection of the spin--orbit angle onto 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 $λ\approx 180^\circ$. Our joint analysis indicates $i_\star \approx {30D}$ and $ψ\approx 120^\circ$, 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 $λ\approx 0^\circ$, implying that the system is well-aligned. With the help of the tight constraint from asteroseismology, however, we obtain $i_\star={65.4}^{+{10.6}}_{-{6.4}}$ and $ψ={26.9}^{+{7.0}}_{-{9.2}}$, and thus find that the system is actually mildly misaligned.