Neglecting the effects of rotation and magnetic fields, and ignoring the presence of neutron superfluid in the inner crust, but taking into account the elasticity of the crust in general relativity, the frequency of the fundamental toroidal crustal mode of multipolarity (the case corresponding to the crust uniformly rotating around the static core is ignored), is approximately given by [359]
where is the Schwarzschild radius of the star, the radius of the crust and is the speed of shear waves propagating in an angular direction and polarized in the mutually orthogonal angular direction. The frequencies of the higher fundamental () modes scale like The frequencies of overtones are independent of to a good approximation (provided is not too large compared to ), and can be roughly estimated as where is the speed of shear waves propagating radially with polarization in an angular direction. For a reasonable crustal equation of state, the crust thickness can be estimated as [359] If the crust is isotropic, the velocities and are equal.The analysis of QPOs in SGRs can potentially provide valuable information on the properties of the crust and, more generally, on the structure of neutron stars. The identification of both the 29 Hz and 626.5 Hz QPOs in the 2004 giant flare from SGR 1806–20 as the fundamental , toroidal mode and the first overtone , , respectively, puts stringent constraints on the mass and radius of the star, as shown in Figure 79. This constraint rules out some stiff equations of state based on the relativistic mean field theory proposed by Glendenning [166]. The 28 Hz and 54 Hz QPOs in the 1998 flare from SGR 1900+14 have been identified with the and toroidal modes, respectively. In this case, the mass and radius are much less constrained, as can be seen in Figure 80. The identification of higher frequency QPOs is more controversial.

The effect of rotation on oscillation modes is to split the frequency of each mode with a given into frequencies. It has recently been pointed out that some of the resulting modes might, thus, become secularly unstable, according to the Chandrasekhar–Friedman–Schutz (CFS) criterion [411]. The study of oscillation modes becomes even more difficult in the presence of a magnetic field. Roughly speaking, the effects of the magnetic field increase the mode frequencies [129, 290, 328, 257, 385]. Simple Newtonian estimates lead to an increase of the frequencies by a factor , where is expressed in terms of the shear modulus [129]. Sotani and et al. [385] recently carried out calculations in general relativity with a dipole magnetic field and found numerically that the frequencies are increased by a factor , where is a numerical coefficient. However, it has been emphasized by Messios et al. [290] that the effects of the magnetic field strongly depend on its configuration. The most important effect is to couple the crust to the core so that the whole stellar interior vibrates during a giant flare [165]. Low frequency QPOs could, thus, be associated with magnetohydrodynamic (MHD) modes in the core [262].

Another important aspect to be addressed is the presence of neutron superfluid, which permeates the inner crust. The formalism for treating a superfluid in a magnetoelastic medium has been recently developed both in general relativity [85] and in the Newtonian limit [73, 72], based on a variational principle. This formalism has not yet been applied to study oscillation modes in magnetars. However, we can anticipate the effects of the neutron superfluid using the twofluid description of the crust reviewed in Section 10.2. Following the same arguments as for twofluid models of neutron star cores [13], two classes of oscillations can be expected to exist in the inner crust, depending on whether neutron superfluid is comoving or countermoving with the crust. The countermoving modes are predicted to be very sensitive to entrainment effects, which are very strong in the crust [90, 91].
The neutronstar–oscillation problem deserves further theoretical study. The prospect of probing neutron star crusts by analyzing the Xray emission of giant magnetar flares is very promising.
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