Instabilities and pulsations in models of the B-type supergiant $\kappa$ Cassiopeiae (HD 2905)

For the B-type supergiant $\kappa$ Cassiopeiae (HD 2905) variabilities with periods between several hours and a few days have been observed both photometrically and spectroscopically. A recent study of this star by Sim\'on-D\'{\i}az et al. (2018) has revealed variability with a dominant period of 2.7 days. To understand this variability, we present a linear non-adiabatic stability analysis with respect to radial perturbations for models of $\kappa$ Cassiopeiae. Instabilities associated with the fundamental mode and the first overtone are identified for models with masses between 27 M$_{\odot}$ and 44 M$_{\odot}$. For selected models, the instabilities are followed into the non-linear regime by numerical simulations. As a result, finite amplitude pulsations with periods between 3 and 1.8 days are found. The model with a mass of 34.5 M$_{\odot}$ exhibits a pulsation period of 2.7 days consistent with the observations. In the non-linear regime, the instabilities may cause a substantial inflation of the envelope.