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 Keplerdata. 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 modulationmethod 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 velocitycurve fit. The parameters from the phase modulationmethod and the radial velocityfits are in good agreement. We combined a radial velocityand phase modulationapproach 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 periodof ~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 periodsof 0.56 and 2.53 d for the primary and secondary, respectively.