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

TL;DR

This study measures the spin-orbit angles of Kepler-13Ab and HAT-P-7b using gravity-darkened transit light curves, revealing spin-orbit precession and constraining stellar quadrupole moments and gravity-darkening exponents.

Contribution

It applies the gravity-darkening method to two Kepler systems, providing new measurements of spin-orbit angles, stellar quadrupole moments, and testing gravity darkening theory for an F-dwarf star.

Findings

Kepler-13A has a spin-orbit angle of 60°±2° and stellar inclination of 81°±5°.

Detected temporal variation in Kepler-13Ab's orbital inclination indicating spin-orbit precession.

HAT-P-7 analyzed as a nearly pole-on system with gravity-darkening exponent consistent with 0.25.

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 $i_\star=81^\circ\pm5^\circ$ and $\psi=60^\circ\pm2^\circ$ adopting the spectroscopic constraint $\lambda=58.6^\circ\pm2.0^\circ$ 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, $\mathrm{d} |\cos i_{\rm orb}|/\mathrm{d}t=(-7.0\pm0.4)\times10^{-6}\,\mathrm{day}^{-1}$, providing further evidence for the spin-orbit precession in this system. By fitting the precession model to the time series of $i_{\rm orb}$, $\lambda$, and $i_\star$ obtained with the gravity-darkened model, we constrain the stellar quadrupole moment $J_2=(6.1\pm0.3)\times10^{-5}$ for our new solution, which is several times smaller than $J_2=(1.66\pm0.08)\times10^{-4}$ 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 $\psi=101^\circ\pm2^\circ$ or $87^\circ\pm2^\circ$ 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.