The pulsating magnetosphere of the extremely slowly rotating magnetic $β$ Cep star $ξ^1$ CMa

$ξ^1$ CMa is a monoperiodically pulsating, magnetic $β$ Cep star with magnetospheric X-ray emission which, uniquely amongst magnetic stars, is clearly modulated with the star's pulsation period. The rotational period $P_{\rm rot}$ has yet to be identified, with multiple competing claims in the literature. We present an analysis of a large ESPaDOnS dataset with a 9-year baseline. The longitudinal magnetic field $\langle B_{\rm Z}\rangle$ shows a significant annual variation, suggesting that $P_{\rm rot}$ is at least on the order of decades. The possibility that the star's H$α$ emission originates around a classical Be companion star is explored and rejected based upon VLTI AMBER and PIONIER interferometry, indicating that the emission must instead originate in the star's magnetosphere and should therefore also be modulated with $P_{\rm rot}$. Period analysis of H$α$ equivalent widths measured from ESPaDOnS and CORALIE spectra indicates $P_{\rm rot} > 30$ yr. All evidence thus supports that $ξ^1$ CMa is a very slowly rotating magnetic star hosting a dynamical magnetosphere. H$α$ also shows evidence for modulation with the pulsation period, a phenomenon which we show cannot be explained by variability of the underlying photospheric line profile, i.e. it may reflect changes in the quantity and distribution of magnetically confined plasma in the circumstellar environment. In comparison to other magnetic stars with similar stellar properties, $ξ^1$ CMa is by far the most slowly rotating magnetic B-type star, is the only slowly rotating B-type star with a magnetosphere detectable in H$α$ (and thus, the coolest star with an optically detectable dynamical magnetosphere), and is the only known early-type magnetic star with H$α$ emission modulated by both pulsation and rotation.