KIC 4142768: An Evolved Gamma Doradus/Delta Scuti Hybrid Pulsating Eclipsing Binary with Tidally Excited Oscillations

We present the characterization of KIC 4142768, an eclipsing binary with two evolved A-type stars in an eccentric orbit with a period of 14 days. We measure the fundamental parameters of the two components ($M_1=2.05M_{\odot}, R_1=2.96R_{\odot}$ and $M_2=2.05M_{\odot}, R_2=2.51R_{\odot}$) by combining {\it Kepler} photometry and spectra from {\it Keck} HIRES. The measured surface rotation rates are only one-fifth of the pseudo-synchronous rate of the eccentric orbit. Fourier spectrum of the light curve reveals hybrid pulsations of $\delta$ Scuti and $\gamma$ Doradus type, with pulsation frequencies at about $15-18$ day$^{-1}$ for p modes and about $0.2-1.2$ day$^{-1}$ for low-frequency g modes. Some of the g modes are exact orbital harmonics and are likely tidally excited. Their pulsation amplitudes and phases both agree with predictions from the linear tidal theory for $l=2, m=2$ prograde modes. We examine the period spacing patterns in the free oscillating g modes and identify them mostly as prograde sectoral dipole modes. The unstable frequency range and frequency spacing of p modes and the inferred asymptotic g-mode period spacings both agree with the stellar model for the primary star evolved to a late stage of the main sequence. The inferred rotation rate of the convective core boundary is very slow, similar to the small surface rotation rate inferred from the spectroscopy. The measured surface and near-core rotation rates provide constraints for testing the mechanism of angular momentum transfer and tidal synchronization in evolved eccentric binary star systems.