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

Abstract

We develop a detailed methodology of determining three-dimensionally the angle between the stellar spin and the planetary orbit axis vectors, $\psi$, for transiting planetary systems. The determination of $\psi$ 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 $i_{\mathrm{orb}}$, and the projection of the spin--orbit angle onto the plane of the sky, $\lambda$. 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 $\psi$. 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 $\lambda \approx 180^{\circ }$. Our joint analysis indicates $i_{\star }\approx 30D$ and $\psi \approx 120^{\circ }$, 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 $\lambda$, 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 $\lambda \approx 0^{\circ }$, implying that the system is well-aligned. With the help of the tight constraint from asteroseismology, however, we obtain $i_{\star }={65.4}_{-6.4}^{+10.6}$ and $\psi ={26.9}_{-9.2}^{+7.0}$, and thus find that the system is actually mildly misaligned.

20 pages, 13 Figures, 5 Tables

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