Spectroscopic confirmation of the binary nature of the hybrid pulsator KIC 5709664 found with the frequency modulation method

We started a program to search for companions around hybrid $\delta$ Sct/$\gamma$ Dor stars with the frequency modulation method using Kepler data. Our best candidate was KIC 5709664, where we could identify Fourier peaks with sidelobes, suggesting binary orbital motion. We determined the orbital parameters with the phase modulation method and took spectroscopic measurements to confirm unambiguously the binary nature with radial velocities. The spectra show that the object is a double-lined spectroscopic binary, and we determined the orbital solutions from the radial velocity curve fit. The parameters from the phase modulation method and the radial velocity fits are in good agreement. We combined a radial velocity and phase modulation approach to determine the orbital parameters as accurately as possible. We determined that the pulsator is a hybrid $\delta$ Sct/$\gamma$ Dor star in an eccentric binary system with an orbital period of ~95 d and an eccentricity of 0.55. The measured mass ratio is 0.67. We analysed the pulsation content and extracted 38 frequencies with amplitudes greater than 20 $\mu$mag. At low frequencies, we found broad power excesses which are likely attributed to spots on the rotating surface of the lower-mass component. We inferred rotation periods of 0.56 and 2.53 d for the primary and secondary, respectively.