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
]
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.
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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].
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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 magneto-elastic 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 two-fluid description of the crust reviewed in Section 10.2. Following the same arguments as for two-fluid 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 co-moving 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 neutron-star–oscillation problem deserves further theoretical study. The prospect of probing neutron star crusts by analyzing the X-ray emission of giant magnetar flares is very promising.
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