Alfven quasi-periodic oscillations in magnetars

We investigate torsional Alfven oscillations of relativistic stars with a global dipole magnetic field, via two-dimensional numerical simulations. We find that (i) there exist two families of quasi-periodic oscillations (QPOs) with harmonics at integer multiples of the fundamental frequency, (ii) 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, (iii) the QPOs are long-lived, (iv) for the chosen form of dipolar magnetic field, the frequency ratio of the lower to upper fundamental QPOs is approximate to 0.6, independent of the equilibrium model or of the strength of the magnetic field, and (v) within a representative sample of equations of state 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. The lower and upper QPOs can be interpreted as corresponding to the edges or turning points of an Alfven continuum, according to the model proposed by Levin (2007). Several of the low-frequency QPOs observed in the X-ray tail of SGR 1806-20 can readily be identified with the Alfven QPOs we compute. In particular, one could identify the 18- and 30-Hz observed frequencies with the fundamental lower and upper QPOs, correspondingly, while the observed frequencies of 92 and 150 Hz are then integer multiples of the fundamental upper QPO frequency (three and five times, correspondingly).With this identification, we obtain an upper limit on the strength of the magnetic field of SGR 1806-20 (if is dominated by a dipolar component) between approximate to 3 and 7x10(15) G. Furthermore, we show that an identification of the observed frequency of 26 Hz with the frequency of the fundamental torsional l = 2 oscillation of the magnetar's crust is compatible with a magnetar mass of about from 1.4 to 1.6M circle dot and an equation of state (EOS) that is very stiff (if the magnetic field strength is near its upper limit) or moderately stiff (for lower values of the magnetic field).