Zeeman splitting of torsional oscillation frequencies of magnetars

Magnetars form a special class of neutron stars possessing superstrong magnetic fields and demonstrating power flares triggered likely by these fields. Observations of such flares reveal the presence of quasi-periodic oscillations (QPOs) at certain frequencies; they are thought to be excited in the flares. QPOs carry potentially important information on magnetar structure, magnetic field, and mechanisms of magnetar activity. We calculate frequencies of torsional (magneto-elastic) oscillations of the magnetar crust treating the magnetic field effects in the first order of perturbation theory. The theory predicts splitting of non-magnetic oscillation frequencies into Zeeman components. Zeeman splitting of torsional oscillation spectrum of magnetars was suggested, clearly described and estimated by Shaisultanov and Eichler (2009) but their work has not been given considerable attention. To extend it we suggest the technique of calculating oscillation frequencies including Zeeman splitting at not too strong magnetic fields for arbitrary magnetic field configuration. Zeeman splitting enriches the oscillation spectrum and simplifies theoretical interpretation of observations. We calculate several low-frequency oscillations of magnetars with pure dipole magnetic field in the crust. The results qualitatively agree with low-frequency QPOs detected in the hyperflare of SGR 1806--20, and in the giant flare of SGR 1900+14.