Alfvén QPOs in magnetars in the anelastic approximation

We perform two-dimensional simulations of Alfvén oscillations in magnetars, modeled as relativistic stars with a dipolar magnetic field. We use the anelastic approximation to general relativistic magnetohydrodynamics, which allows for an effective suppression of fluid modes and an accurate description of Alfvén waves. In addition, we compute Alfvén oscillation frequencies along individual magnetic field lines with a semi-analytic approach, employing a short-wavelength approximation. Our main findings are as follows: a) we confirm the existence of two families of quasi-periodic oscillations (QPOs), with harmonics at integer multiples of the fundamental frequency, as was found in the linear study of Sotani, Kokkotas & Stergioulas (2008); b) the QPOs appearing near the magnetic axis are split into two groups, depending on their symmetry across the equatorial plane. The antisymmetric QPOs have only odd integer-multiple harmonics; c) the continuum obtained with our semi-analytic approach agrees remarkably well with QPOs obtained via the two-dimensional simulations, allowing for a clear interpretation of the QPOs as corresponding to turning points of the continuum. This agreement will allow for a comprehensive study of Alfvén QPOs for a larger number of different models, without the need for time-consuming simulations. Finally, we construct empirical relations for the QPO frequencies and compare them to observations of known Soft Gamma Repeaters. We find that, if the magnetic field of magnetars is characterized by a strong dipolar component, and QPOs are produced near the magnetic pole, then one can place an upper limit to the mean surface strength of the magnetic field of about 3 − 8 × 10G.