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12.6 Giant flares from Soft Gamma Repeaters

The discovery of quasi-periodic oscillations (QPO) in X-ray flux following giant flares from Soft Gamma Repeaters (SGR) has recently triggered a burst of intense theoretical research. Oscillations were detected at frequencies 18, 26, 29, 92.5, 150, 626.5 and 1837 Hz during the spectacular December 27, 2004 giant flare (the most intense ever observed in our Galaxy) from SGR 1806–20 [214421397] and at 28, 54, 84 and 155 Hz during the August 27, 1998 giant flare from SGR 1900+14 [396]. Evidence has also been reported for oscillations at 43.5 Hz during the March 5, 1979 event in SGR 0526–66 [34]. Soft Gamma Repeaters (SGR) are believed to be strongly-magnetized neutron stars or magnetars endowed with magnetic fields as high as 1014 – 1015 G (for a recent review, see, for instance, [426] and references therein; also see the home page of Robert C. Duncan [128]). Giant flares are interpreted as crustquakes induced by magnetic stresses. Such catastrophic events are likely to be accompanied by global seismic vibrations, as observed by terrestrial seismologists after large earthquakes. Among the large variety of possible oscillation modes, torsional shear modes in the crust are the most likely [129Jump To The Next Citation Point]. Shear flow in the crust is illustrated in Figure 58View Image. If confirmed, this would be the first direct detection of oscillations in a neutron star crust.

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 ℓ = 2 (the case ℓ = 1 corresponding to the crust uniformly rotating around the static core is ignored), is approximately given by [359Jump To The Next Citation Point]

∘ ---------- 1 − rg∕R ∘ -------------- fn=0,ℓ=2 ≃ 2πvt ---------, fn=0,ℓ ≃ fn=0,ℓ=2 (ℓ − 1)(ℓ + 2 ), (300 ) Rrcc
where rg is the Schwarzschild radius of the star, rcc the radius of the crust and vt 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 (n = 0) modes scale like
∘ -------------- fn=0,ℓ ≃ fn=0,ℓ=2 (ℓ − 1)(ℓ + 2) . (301 )
The frequencies of overtones n > 0 are independent of ℓ to a good approximation (provided ℓ is not too large compared to n), and can be roughly estimated as
rg 2π2nvr fn>0 ≃ (1 − -- )-------, (302 ) R ΔR
where vr is the speed of shear waves propagating radially with polarization in an angular direction. For a reasonable crustal equation of state, the crust thickness ΔR ≡ R − rcc can be estimated as [359Jump To The Next Citation Point]
ΔR [ r r ]− 1 ----≃ 1 + 21.5-g(1 − -g) . (303 ) R R R
If the crust is isotropic, the velocities v r and v t 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 n = 0, ℓ = 2 toroidal mode and the first overtone n = 1, ℓ = 1, respectively, puts stringent constraints on the mass and radius of the star, as shown in Figure 79View Image. 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 ℓ = 2 and ℓ = 4 toroidal modes, respectively. In this case, the mass and radius are much less constrained, as can be seen in Figure 80View Image. The identification of higher frequency QPOs is more controversial.

View Image

Figure 79: Constraints on the mass and radius of SGR 1806–20 obtained from the seismic analysis of quasi-periodic oscillations in X-ray emission during the December 27, 2004 giant flare. For comparison, the mass-radius relation for several equations of state is shown (see [359Jump To The Next Citation Point] for further details).

The effect of rotation on oscillation modes is to split the frequency of each mode with a given ℓ into 2ℓ + 1 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 [129Jump To The Next Citation Point290Jump To The Next Citation Point328257385Jump To The Next Citation Point]. Simple Newtonian estimates lead to an increase of the frequencies by a factor ∘ ------------2 1 + (B ∕B μ), where √ ---- B μ ≡ 4πμ is expressed in terms of the shear modulus μ [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 ---------------- ∘ 1 + α (B ∕B )2 n,ℓ μ, where α n,ℓ 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].

View Image

Figure 80: Constraints on the mass and radius of SGR 1900+14 obtained from the seismic analysis of the quasi-periodic oscillations in the X-ray emission during the August 27, 1998 giant flare. For comparison, the mass-radius relation for several equations of state is shown (see [359Jump To The Next Citation Point] for further details). From [359].

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 [7372], 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 [9091].

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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