Alfven QPOs in Magnetars

We investigate torsional Alfven oscillations of relativistic stars with a global dipole magnetic field, via 2D numerical simulations. We find that a) there exist two families of quasi-periodic oscillations (QPOs) with harmonics at integer multiples of the fundamental frequency, b) the lower-frequency QPO is related to the region of closed field lines, near the equator, while the higher-frequency QPO is generated near the magnetic axis, c) the QPOs are long-lived, d) for the chosen form of dipolar magnetic field, the frequency ratio of the lower to upper fundamental QPOs is about 0.6, independent of the equilibrium model or of the strength of the magnetic field, and e) within a representative sample of EOSs and of various magnetar masses, the Alfven QPO frequencies are given by accurate empirical relations that depend only on the compactness of the star and on the magnetic field strength. Compared to the observational frequencies, we also obtain an upper limit on the strength of magnetic field of SGR 1806-20 (if it is dominated by a dipolar component) between ∼ 3 and 7x1015 Gauss.