Evolving pulsation of the slowly rotating magnetic $\beta$ Cep star $\xi^1$ CMa

Recent BRITE-Constellation space photometry of the slowly rotating, magnetic $\beta$ Cep pulsator $\xi^1$ CMa permits a new analysis of its pulsation properties. Analysis of the two-colour BRITE data reveals the well-known single pulsation period of $0.209$ d, along with its first and second harmonics. A similar analysis of SMEI and TESS observations yields compatible results, with the higher precision TESS observations also revealing several low-amplitude modes with frequencies below 5 d$^{-1}$; some of these are likely $g$ modes. The phase lag between photometric and radial velocity maxima - equal to 0.334 cycles - is significantly larger than the typical value of $1/4$ observed in other large-amplitude $\beta$ Cep stars. The phase lag, as well as the strong dependence of phase of maximum light on wavelength, can be reconciled with seismic models only if the dominant mode is the fundamental radial mode. We employ all published photometric and radial velocity measurements, spanning over a century, to evaluate the stability of the pulsation period. The $O-C$ diagram exhibits a clear parabolic shape consistent with a mean rate of period change $\dot P=0.34\pm 0.02$ s/cen. The residuals from the best-fit parabola exhibit scatter that is substantially larger than the uncertainties. In particular, dense sampling obtained during the past $\sim$20 years suggests more complex and rapid period variations. Those data cannot be coherently phased with the mean rate of period change, and instead require $\dot P\sim0.9$ s/cen. We examine the potential contributions of binarity, stellar evolution, and stellar rotation and magnetism to understand the apparent period evolution.