Spin-Orbit Angles of Kepler-13Ab and HAT-P-7b from Gravity-Darkened Transit Light Curves

Abstract

Analysis of the transit light curve deformed by the stellar gravity darkening allows us to photometrically measure both components of the spin-orbit angle $\psi$, its sky projection $\lambda$ and inclination of the stellar spin axis $i_{\star }$. In this paper, we apply the method to two transiting hot Jupiter systems monitored with the Kepler spacecraft, Kepler-13A and HAT-P-7. For Kepler-13A, we find $i_{\star }=81^{\circ }\pm 5^{\circ }$ and $\psi =60^{\circ }\pm 2^{\circ }$ adopting the spectroscopic constraint $\lambda =58.6^{\circ }\pm 2.0^{\circ }$ by Johnson et al. (2014). In our solution, the discrepancy between the above $\lambda$ and that previously reported by Barnes et al. (2011) is solved by fitting both of the two parameters in the quadratic limb-darkening law. We also report the temporal variation in the orbital inclination of Kepler-13Ab, $\mathrm{d}|\cos i_{\mathrm{orb}}|/\mathrm{d}t=(-7.0\pm 0.4)\times 10^{-6}{\mathrm{day}}^{-1}$, providing further evidence for the spin-orbit precession in this system. By fitting the precession model to the time series of $i_{\mathrm{orb}}$, $\lambda$, and $i_{\star }$ obtained with the gravity-darkened model, we constrain the stellar quadrupole moment $J_2=(6.1\pm 0.3)\times 10^{-5}$ for our new solution, which is several times smaller than $J_2=(1.66\pm 0.08)\times 10^{-4}$ obtained for the previous one. We show that the difference can be observable in the future evolution of $\lambda$, thus providing a possibility to test our solution with follow-up observations. The second target, HAT-P-7, is the first F-dwarf star analyzed with the gravity-darkening method. Our analysis points to a nearly pole-on configuration with $\psi =101^{\circ }\pm 2^{\circ }$ or $87^{\circ }\pm 2^{\circ }$ and the gravity-darkening exponent $\beta$ consistent with $0.25$. Such an observational constraint on $\beta$ can be useful for testing the theory of gravity darkening.

15 pages, 8 figures, accepted for publication in ApJ, typo in the abstract fixed

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