1. University of Tuebingen K.D.Kokkotas & N.Stergioulas
Alfven Oscillations in Magnetars H.S., K.D.Kokkotas, & N.Stergioulas, MNRAS 375, 261 (2007) H.S., K.D.Kokkotas, & N.Stergioulas, MNRAS 385, L5 (2008) H.S. & K.D.Kokkotas, arXiv: [astro-ph]
Hajime SOTANI
University of Tuebingen
with
K.D.Kokkotas & N.Stergioulas

2. Introduction ~SGRs & QPOs~
Soft gamma Repeaters (SGRs) : objects that arise the gamma-ray flare activity.
Source…Magnetar ?? (neutron stars with strong magnetic fields > 1014 G)
Scenario for outbreak of gamma-ray burst (Duncan & Thompson (1992))
 During the evolution of magnetars, the magnetic stress accumulates in the crust.
 When this stress is released, the gamma-ray could emit !
Giant Flare from SGRs ( ergs/s)
SGR 0526–66 in March
SGR in August
SGR 1806–20 in December
In the decaying tail after the flare, QPOs are found !!  Barat et.al. (1983); Israel et.al. 2005; Watts & Strohmayer 2005, 2006
SGR : 23ms (43Hz), B ~ 4 x 1014G, L ~ 1044 ergs/s
SGR : B > 4 x 1014G, 28, 54, 84, and 155 Hz
SGR 1806–20 : B ~ 8 x 1014G, L ~ 1046 ergs/s 18, 26, 30, 92.5, 150, 626.5, and 1837 Hz (also 720Hz ?? and 2384 Hz ??)
Ref : Watts & Strohmayer (2006)
QPOs exist !!
Within this model the short bursts are produced by the propagation of Alfven waves in the magnetosphere, driven by magnetic field diffusion across small cracks in the neutron star crust.
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Strohmayer & Watts (2006)

3. Introduction ~How to explain QPOs~
QPOs of SGRs are due to the crust torsional oscillations ??
In Newtonian; Hansen & Cioffi (1980), McDermott et al. (1998), Carroll et al. (1986), Storhmayer (1991), …
 the case without magnetic fields 
In GR; Schumaker & Thone (1983), Leins (1994), Messions et al. (2001), Samuelsson & Andersson (2006)
Alternative possibility : Alfven oscillations in the core region ??
Glampedakis et al. (2006), Levin(2006)
Levin(2007) … the QPO frequencies are enhanced at their edges or turning point.
 How do the Alfven Oscillations behave in the realistic stellar models ??
We examine the Alfven Oscillations in the realistic stellar models to know the feature of these oscillations and
to explain the observational evidences.
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4. Models of Magnetar Perturbations
Ideal MHD approximation
 Electric fields are zero for comoving observer.
The stellar deformation due to the magnetic fields are neglect.
Magnetic energy / gravitational energy ~ 10-4 (B/1016[G])2
Equilibrium configuration : static spherically symmetric
Axisymmetric poloidal magnetic fields
Linearizing the equation of motion and Maxwell equations
Axisymmetric perturbation ( m = 0 )  axial perturbation is independent from polar perturbation
Cowling approximation ( )
Perturbations
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5. Crust torsional oscillations
H.S., K.D.Kokkotas, & N.Stergioulas (2007)
We examine the torsional oscillations restricted in the crust region.  we omit the coupling with the core oscillations.
We can see the magnetic effect for
Frequencies for fundamental mode depend strongly on the stellar properties, where they change in 30 ~ 50 %
Frequencies for overtones are independent from the harmonic index .
Frequencies depend strongly on the both EOS of crust and core.  ex) first overtones are in the range of 500 – 1200 Hz.
We find the empirical formula such as
depend on the stellar models
The attempt to explain the observational data
is partially successful, but it may be difficult
to explain all data.
especially 18, 26, 30 Hz
for SGR
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6. Calculations for Alfven Oscillations
H.S., K.D.Kokkotas, & N.Stergioulas (2008)
We calculate the Alfven oscillations
On the whole star without the crust regions
Axisymmetric poloidal magnetic fields
2D evolutions for axisymmetric torsional oscillations with numerical viscosity (Kreiss-Oliger dissipation)
Perturbed variable :
Two dimensional evolutionary equations
Boundary conditions
(regularity condition)
(vanishing traction)
(axisymmetry)
(equatorial symmetry of even )
4th-order Kreiss-Oliger dissipation
* more detail studies : Colaiuda, et al. (arXiv: ) Cerda-Duran, et al. (arXiv: )
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7. Feature of Oscillations
FFT on the different position in the magnetar
The both edges of continuum are enhanced, which are corresponding to 25 and 41 Hz for the fundamental QPOs.
 two families; upper and lower QPOs
The oscillations on the axis are same phase, but those on the other stellar location have different phase  not normal-modes but continuum
we can find that the feature of continuum arise near the equatorial plane.
25Hz(L0)
41Hz(U0) L1 U1 U2 U3 U1 U2 U3 U1 U2 U3
41Hz(U0)
41Hz(U0)
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8. Empirical Formula
The lower and upper Alfven QPO frequencies with realistic stellar models
the ratio of are almost 0.6 independently of the stellar models.
the frequencies of overtones are nearly integer multiplies of the fundamental one;
the frequencies are proportional to the magnetic field strength.
Empirical formula for the frequencies of lower and upper Alfven QPOs
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9. Attempt to fit to Observational data
Now we have empirical formula for the frequencies of Alfven QPOs and of crust torsional modes.
On the other hand the observed frequencies of QPOs in SGRs
SGR : 28, 54, 84, 155 Hz
SGR : 18, 26, 30, 92.5, 150 Hz
crust torsional modes :
lower Alfven QPOs :
upper Alfven QPOs : x2 x3
crust torsional oscillation ? or polar oscillation ? x3 x5
0.6
crust torsional oscillation ?
EOS L :
EOS APR :
Observed QPOs restrict on the stellar model and/or magnetic field strength !
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10. How about Polar Oscillations ?
H.S. & K.D.Kokkotas, arXiv: [astro-ph]
No one knows whether the polar oscillations could be continuum or discrete modes.
We can check this feature of oscillation spectrum by seeing
FFT at the various points inside star
phase of the specific frequencies in FFT
If oscillations are continuum,
the peaks in FFT depends on the observation points.
the phase of each peaks in FFT also depends on the observation positions.
If oscillations are discrete modes,
the peaks in FFT are independent from the observation points.
the phase of each peaks in FFT is constant inside the star.
How dose the polar oscillations in magnetar depend on the magnetic fields ??
By changing the magnetic field strength, we should check the dependence.
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11. code test
Without magnetic field, the stellar oscillations can be solved as an eigenvalue problem.
fluid mode such as f and p modes
In order to know the accuracy of numerical code, we compare the frequencies determined with frequency domain and with time domain.
Polytropic star with 1.4 solar mass
2D region of (r,theta), where theta is in the range between 0 and pi/2.
50 x 40 grid points
Evolution time is 200 msec.
Deviation : 2.6% for 2 f mode and 3.7% for 4 f mode l
f (E)
f (T)
1 p(E)
1 p(T)
2 p(E)
2 p(T) 2
2.68
2.75
6.71
6.87
9.97
10.21 3
3.26
3.41
7.65
7.89
11.07
11.38 4
3.75
3.89
8.49
8.73
12.08
12.38
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12. FFT FFT at various points inside the star B = 1 x 1016 Gauss
The frequencies of peak in FFT are independent from the observation positions.
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13. phase The phase of each peak
Phase with the frequency of peak in FFT is constant.
With two results such as FFT and phase, the polar Alfven oscillations are not continuum but discrete modes.
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14. polar Alfven oscillations
Dependence of polar Alfven modes on magnetic strength
Frequency ~ a few hundred Hz
These can be used to explain the observed higher QPOs frequencies.
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15. fluid modes
The dependence of fluid modes frequencies on the magnetic field strength
We can expect that these frequencies would be almost independent from the magnetic field strength unless the magnetic field are quite strong.
We can see the effect of the magnetic field lager than 1x1016Gauss.
That deviation from the frequencies without magnetic field are only less than a few %.
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16. Conclusion Future Works
The attempt to explain the observed QPO frequencies in SGRs by using only crust torsional modes are partially successful, but it may be impossible to explain all evidences.
The oscillations of magnetars could be continuum, where two families exist in the Alfven oscillations (upper and lower QPOs).
With crust torsional oscillations, we can explain the observed QPO frequencies.
With two results such as FFT and phase, the polar Alfven oscillations are not continuum but discrete modes.
Polar Alfven modes can be used to explain the higher QPO frequencies.
Fluid modes in Magnetar are almost independent from the magnetic field strength.
With the effect of crust on the Alfven torsional oscillations, how do the spectrum become ?? Discrete or Continuum ??
Non-axisymmetric oscillations
Future Works
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17. END

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19. Future works
How to connect between the Alfven and crust oscillations ??
The polar oscillations.
The non-axisymmetric oscillations.
The effect of the stellar deformation.
Without Cowling approximation ??
With rotation ???
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20. Numerical Results 2 Distribution of effective amplitude for upper QPOs
Distribution of effective amplitude for lower QPOs
The maximum exists near the axis.
There are nodal lines along certain magnetic field lines.
For the overtone, the additional “horizontal” nodal line exits.
The effective amplitude for lower QPOs is partially limited to a region around the closed magnetic field lines.
For the overtone, the nodal line divides the region of maximum amplitude.
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21. Numerical Results 3 Magnetic lines of interior region
For the L0 QPOs, an oscillating region is restricted to roughly 0.5<x<1.0, while it appears to be "touching" the surface for 0<z<0.3.
We can see that the last closed field line inside stellar surface originates at about x=0.58.
The field lines that originate from the region 0.5<x<0.58 are open and cross the surface at 0<z<0.3
 While the maximum oscillational amplitude is inside the region of closed lines, there is still a broader region, including open field lines.
 The information of oscillations with lower QPOs could leak out of the star.
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22. Numerical Results 4
At the location inside the star where the effective amplitude becomes maximum, time evolution of are …
We can see the long-lived QPOs
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23. Perturbations Linearizing the equation of motion and Maxwell equations
Axisymmetric axial perturbation ( m=0 )  axial perturbation is independent from polar perturbation
Cowling approximation ( )
Perturbation equations to solve … function of
2D wave equations with some boundary conditions
Alternative method
expanding by the spherical harmonics function Ylm
1D eigen-value problem
 : function of　　　
 the -th order perturbation : coupled with the ( )-th order
For simplicity, the coupling terms neglect.  “truncated method”
 we find that for weak magnetic fields this is good approximation at least for the case of crust torsional oscillation!!
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24. The thickness of crust : 3 – 12 %
Adopted EOS
Core : four EOS ( A, WFF3, APR, and L )
EOS A : a very soft; EOS L : a very stiff.
crust : two EOS ( NV and DH )
NV : Negele & Vautherin (1973) [g/cc] at the basis of crust [erg/cc]
DH : Douchin & Haensel (2001) [g/cc] at the basis of crust [erg/cc]
The thickness of crust : 3 – 12 %
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25. Attempt to fit to Observational data
Some of stellar models with stiff EOS are good agreement with the observational data.
SGR : L+NV25 (28, 54, 84, 155 Hz  3t0, 6t0, 9t0, 17t0)
SGR 1806–20 : L+DH17 or L+NV20 (18, 30, 92.5, 150, 626.5, 1837 Hz  2t0, 3t0, 9t0, 15t0, lt1, lt4)
However …
It is quite difficult to explain the both data of 26 & 30 Hz.  there is no modes between 2t0 and 3t0.  26Hz is very close to the different data of 30 Hz !
Similarly, there is same problem in the observational data of SGR 1806–20 such as and 720 Hz by only using the crust overtones.
This attempt is partially successful, but it may be impossible to explain by using only crust torsional modes.
26Hz Hz ??
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26. Models of Magnetar Ideal MHD approximation
 Electric fields are zero for comoving observer.
The stellar deformation due to the magnetic fields are neglect.
Magnetic energy / gravitational energy ~ 10-4 (B/1016[G])2
Equilibrium configuration : static spherically symmetric
Axisymmetric poloidal magnetic fields
In the crust : isotropic shear modulus μ
 Shear modulus … depend on EOS of crust
: number density of ions
+Ze： average charge of ions
: average space between ions
Strohmayer, Ogata, Iyetomi, Ichimaru, Van Horn (ApJ 375,679 (1991))
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27. Boundary Conditions for crust oscillations
Zero traction basis of crust ＆ the stellar surface.
Notice
Strictly basis of crust, this boundary condition is not acceptable, because of the existence of magnetic lines
 Probably, we should adopt continuous traction the basis of crust
 crust region is coupled with the core fluid
However, in the limit of non-magnetic field, “zero traction condition” is correct.
Thus for the weak magnetic field, zero traction condition is not so strange 　　 B < 4 x 1015 G？
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28. Torsional Oscillations of Relativistic Stars with Dipole Magnetic Fields
Case without magnetic fields
For fundamental modes
Frequencies depend strongly on the stellar properties.  change in 30 ~ 50 %
Frequencies are almost independent from the crust EOS  change in 1~5%
For overtones
Frequencies are independent from the harmonic index .
Depend strongly on the both EOS of crust and core.  ex) first overtones are in the range of 500 – 1200 Hz.
Effect of the magnetic fields
We found the empirical formula, such as
For , the frequencies are dramatically changing.
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29. Equilibrium models
The magnetic field configurations for distribution (i)
The magnetic field configurations for distribution (ii)
The component of magnetic field is not smooth at stellar surface.
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