Follow-up observations of the binary system $\gamma$ Cep

We present new imaging and spectroscopic observations of the exoplanet host star $\gamma$ Cep A, and of its low-mass stellar companion $\gamma$ Cep B. We used AstraLux/CAHA to follow the orbital motion of the companion around its primary, whose radial velocity was determined with spectra of the star, taken with the spectrograph FLECHAS at the University Observatory Jena. We measured the astrometry of $\gamma$ Cep B relative to its primary in all AstraLux images and determined its apparent SDSS i'-band photometry, for which we obtained i'=9.84$\pm$0.17 mag. Using stellar evolutionary models and the Gaia parallax of the exoplanet host star, we derived the mass of $\gamma$ Cep B to be 0.39$\pm$0.03 M${_\odot}$. This is in good agreement with the mass of the companion, derived from its NIR photometry, given in the literature. With the detection limit, reached in our AstraLux images, we explored the detection space of potential additional companions in the $\gamma$ Cep binary system. In the background limited region at angular separations larger then 5 '' (or 69 au of proj. separation) companions down to 0.11 M$_\odot$ are detectable around $\gamma$ Cep A. The radial FoV, fully covered in our AstraLux images, exhibits a radius of 11.2 ''. This allows the detection of companions with proj. separations up to 155 au. However, except for $\gamma$ Cep B no additional companions could be imaged around the exoplanet host star. We redetermined the orbital solution of the $\gamma$ Cep binary system with the new AstraLux astrometry of $\gamma$ Cep B and the additional radial velocities of $\gamma$ Cep A, obtained from our FLECHAS spectroscopy of the star, combined with astrometric and radial velocity data from the literature. The determined orbital elements were used to derive the system parameters and to calculate specific future ephemeris for this intriguing exoplanet host binary star system.