Short-term variability and mass loss in Be stars V. Space photometry and ground-based spectroscopy of $\gamma$ Cas

Context. Be stars are physically complex systems that continue to challenge theory to understand their rapid rotation, complex variability and decretion disks. $\gamma$ Cassiopeiae ($\gamma$ Cas) is one such star but is even more curious because of its unexplained hard thermal X-ray emission. Aims. We aim to examine the optical variability of $\gamma$ Cas and thereby to shed more light on its puzzling behaviour. Methods. Three hundred twenty-one archival H$\alpha$ spectra from 2006 to 2017 are analysed to search for frequencies corresponding to the 203.5 day orbit of the companion. Space photometry from the SMEI satellite from 2003 to 2011 and the BRITE-Constellation of nano-satellites between 2015 and 2019 is investigated in the period range from a couple of hours to a few days. Results. The orbital period of the companion of 203.5 days is confirmed with independent measurements from the structure of the H$\alpha$ line emission. A strong blue/red asymmetry in the amplitude distribution across the H$\alpha$ emission line could hint at a spiral structure in the decretion disk. With the space photometry, the known frequency of 0.82 d$^{-1}$ is confirmed in data from the early 2000s. A higher frequency of 2.48 d$^{-1}$ is present in the data from 2015 to 2019 and possibly also in the early 2000s. A third frequency at 1.25 d$^{-1}$ is proposed to exist in both SMEI and BRITE data. The only explanation covering all three rapid variations seems to be nonradial pulsation. The two higher frequencies are incompatible with rotation.