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Quasi-periodic oscillations in superfluid magnetars

We study the time-evolution of axisymmetric oscillations of superfluid magnetars with a poloidal magnetic field and an elastic crust, working in Newtonian gravity. Extending earlier models, we study the effects of composition gradients and entrainment on the magneto-elastic wave spectrum and on the potential identification of the observed quasi periodic oscillations (QPOs). We use two-fluid polytropic equations of state to construct our stellar models, which mimic realistic composition gradient configurations. The basic features of the axial axisymmetric spectrum of normal fluid stars are reproduced by our results and in addition we find several magneto-elastic waves with a mixed character. In the core, these oscillations mimic the shear mode pattern of the crust as a result of the strong dynamical coupling between these two regions. Incorporating the most recent entrainment configurations in our models, we find that they have a double effect on the spectrum: the magnetic oscillations of the core have a frequency enhancement, while the mixed magneto-elastic waves originating in the crust are moved towards the frequencies of the single-fluid case. The distribution of lower-frequency magneto-elastic oscillations for our models is qualitatively similar to the observed magnetar QPOs with $ν< 155 $Hz. \emph{In particular, some of these QPOs could represent mixed magneto-elastic oscillations with frequencies not greatly different from the crustal modes of an unmagnetised star.} We find that many QPOs could even be accounted for using a model with a relatively weak polar field of $B_{p} \simeq 3\cdot 10^{14}$G, because of the superfluid enhancement of magnetic oscillations. Finally, we discuss the possible identification of 625 and 1837~Hz QPOs either with non-axisymmetric modes or with high-frequency axisymmetric QPOs excited by crustal mode overtones.