Torsional Oscillations of Relativistic Stars
with Dipole Magnetic Fields
Hajime Sotani (University of Thessaloniki)
withK. D. Kokkotas (University of Thessaloniki)
N. Stergioulas (University of Thessaloniki)
M. Vavoulidis (University of Thessaloniki)

QPOs in SGRs
Ref : Watts & Strohmayer (2006)

 SGRs• The most promising model :magnetar (Duncan & Thompson 1992)
Æ the magnetic stress in the solid crustÆ the gamma-ray burst

QPOs Giant flares in SGRs

• Up to now, three giant flares have been detected.
• SGR 0526–66 in 1979, SGR 1900+14 in 1998, SGR 1086–20 in 2004

• Peak luminosities : 1044 – 1046 erg/s• A decaying tail for several hundred seconds follows the flare.
 QPOs in decaying tail (Israel et al. 2005; Watts & Strohmayer 2005, 2006)• SGR 1900+14 : 28, 54, 84, and 155 Hz• SGR 1086–20 : 18, 26, 29, 92.5, 150, 626.5, and 1837 Hz(possible additional frequencies : 720 and 2384 Hz)
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How to explain QPOs QPOs is due to the torsional oscillation in crust ??

• In Newtonian; Hansen & Cioffi (1980), McDermott et al. (1998),
Carroll et al. (1986), Storhmayer (1991), …
• In GR; Schumaker & Thone (1983), Leins (1994),
Messions et al. (2001), Samuelsson & Andersson (2006)

Æ without magnetic field or non-realistic one !!
 QPOs can be explained by global Alfven modes ??
• Glampedakis et al. (2006)• simple toy model Æ the possibility to explain the observational data !!
 With dipole magnetic field & realistic EOS• Crust torsional modes• Global Alfven modes

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Models of Magnetar Ideal MHD approximation

Æ electric field is zero for comoving observer.
 Neglect the deformation due to the magnetic tension• Magnetic energy / gravitational energy~ 10-4 (B/1016[G])• Equilibrium configuratin :static and spherically symmetric one.
 Axisymmetric poloidal magnetic field In the crust : isotropic shear modulus μÆ Shear modulus depends on the EOS of crust.

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EOS
 Inner core : four EOS such as A, WFF3, APR, and L• EOS A is a very soft and EOS L is a very stiff.
 Crust : two EOS such as NV and DH• NV : Negele & Vautherin (1973)
ρ~ 2.4 x 1014 [g/cc] at the basis of crust
μ~ 1.267 x 1030ρ144/5 [erg/cc]• DH : Douchin & Haensel (2001)
ρ~ 1.28 x 1014 [g/cc] at the basis of crust
μ~ 1.74 x 1030ρ143/4 (Z/53)2 (616/A)4/3
x ((1-Xn)/0.41)4/3 [erg/cc]

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Perturbations
 Linearized the equation of motion and Maxwell’s equations• Axisymmetric axial perturbation• Cowling approximation () Perturbation equation to be solved

Æ eigenvalue equation of 1-dimensional form !
Æ the -th order perturbation is coupled with the ()-th order.

For simplicity, we neglect these coupling.
Æ We need to consider the case including coupling terms later…

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Boundary Conditions
 For pure crust torsional modes;• Zero traction condition at basis of crust and at stellar surface.
Æ at basis of crust, it is not true boundary condition with magnetic field !
Æ in the limit of non-magnetic field, it is correct one.
Æ thus for weak magnetic field this B.C. is not so bad.
 For global Alfven modes;•••

Regularity condition at stellar center
Zero traction condition at stellar surface
Traction is continuous at basis of crust
pure crust
torsional modes

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global Alfvenmodes

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Pure Crust Torsional Modes Non-magnetized case• For fundamental modes• Frequencies depend on the stellar parameters.
Æ it can vary by up to 30 ~ 50 %• The choice of the crust EOS does not affectsignificantly (1~5%)• For overtones• Frequencies are practically independent of the harmonic index .
• The variations in the frequencies due to different choices of both the highdensity and crust EOS are significant.
Æ ex) frequencies of first overtones vary the range of 500 – 1200 Hz.
 The effect of the magnetic fields• We derive the empirical formula such aswhere.
• For, the shift in the frequencies is significant.
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第19回理論懇シンポジウム
19回理論懇シンポジウム ＠ 立教大学,8Attempt to fit to Observational data Some of the stiff models fit quite well to the observational data

• SGR 1900+14 : L+NV25 (28, 54, 84, 155 Hz Æ 3t0, 6t0, 9t0, 17t0)• SGR 1806–20 : L+DH17 or L+NV20(18, 29, 92.5, 150, 626.5, 1837 Hz Æ 2t0, 3t0, 9t0, 15t0, lt1, lt4)
 However …

• it is difficult to explain the observational data of 26 Hz, because this data isvery close to the other data of 29 Hz.
• Similarly, the spacing between the 626.5 and 720 Hz in SGR 1806–20 may
be too small to be explained by consecutive overtones of crust torsionalmodes.
Æ it may be difficult to explain all frequencies by only crust torsional modes!!
Æ It may be necessary to consider the global Alfven modes !!

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Global Alfven Modes Realistic stellar models have very thin solid crust.

Æ the frequencies of global modes approaches that of the pure Alfven modes.
 For weak magnetic field,• The presence of solid crust has some effecton the global Alfven modes For strong magnetic field,• The effect diminishes !!
Æ The global modes for strong magnetic field
are almost identical to pure Alfven modes.
 The frequency of overtones depend onthe value of !!

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Comparison with Observational data All observational data can be explained by global Alfven modes !

Æ Owing to the non-degenerate of overtones
for different values of .
Ex) SGR 1806–20 :SGR 1900+14 :for A+DH14 stellar model.
Æ for other stellar models it is easy to fit !!
• Of course, this magnetic field is strong.
Æ it is easier to fit for the weaker magnetic
fields, because there exist more modes.
Ex) SGR 1806–20 : L+NV18 with
Æ 4a0, 8a0, 10a0, 2a6, and 2a11 for f < 150 Hz
 The maximum magnetic fields exist !!

• (0.8 ~ 1.2) x 1016 [G]
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11Conclusion
 QPOs in SGRs
 Some observational data can be explained by crust torsional modes.

Æ However, it is difficult to explain all observational data !!
 The overtones of global Alfven modes do not degenerate with .
Æ In order to explain the observational data, this is good feature!
 With global Alfven modes, it is possible to explain the allobservational data.
 The existence of maximum magnetic field.
• (0.8 ~ 1.2) x 1016 [G]

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Future Works
 Including the ()-th coupling terms
 Non-axisymmetric perturbation
 Different distribution of back-ground magnetic fields
 Polar perturbation
 Without Cowling approximation
 With the effect of deformation due to the magnetic pressure

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