Orbital solutions derived from radial velocities and time delays for four {\it Kepler} systems with A/F-type (candidate) hybrid pulsators.

The presence of A/F-type {\it Kepler} hybrid stars extending across the entire $\delta$ Sct-$\gamma$ Dor instability strips and beyond remains largely unexplained. In order to better understand these particular stars, we performed a multi-epoch spectroscopic study of 49 candidate A/F-type hybrid stars and one cool(er) hybrid object detected by the {\it Kepler} mission. We determined a lower limit of 27 % for the multiplicity fraction. For six spectroscopic systems, we also reported long-term variations of the time delays. For four systems, the time delay variations are fully coherent with those of the radial velocities and can be attributed to orbital motion. We aim to improve the orbital solutions for those systems with long orbital periods (order of 4-6 years) among the {\it Kepler} hybrid stars. The orbits are computed based on a simultaneous modelling of the RVs obtained with high-resolution spectrographs and the photometric time delays derived from time-dependent frequency analyses of the {\it Kepler} light curves. We refined the orbital solutions of four spectroscopic systems with A/F-type {\it Kepler} hybrid component stars: KIC 4480321, 5219533, 8975515 and KIC 9775454. Simultaneous modelling of both data types analysed together enabled us to improve the orbital solutions, obtain more robust and accurate information on the mass ratio, and identify the component with the short-period $\delta$ Sct-type pulsations. In several cases, we were also able to derive new constraints for the minimum component masses. From a search for regular frequency patterns in the high-frequency regime of the Fourier transforms of each system, we found no evidence of tidal splitting among the triple systems with close (inner) companions. However, some systems exhibit frequency spacings which can be explained by the mechanism of rotational splitting.