ON THE ORIGIN OF HIGH-QUALITY

FAST QPOs FROM MAGNETARS

A.Stepanov (Pulkovo Observatory, St.Petersburg)V.Zaitsev (Institute of Applied Physics, N.Novgorod)

E.Valtaoja (Tuorla Observatory, Turku)

The XI Russian-Finnish Symposium on Radioastronomy

Pushchino, Russia, 18-22 Oct. 2010

Soft Gamma-ray Repeater -neutron star (D ~ 10 km, M ≤ 1.5MSun)with magnetic field B ~ 1014-15 G

Energy release in impulse phase (duration ≤ 1 s): up to 2×1046 ergs

High-quality, Q ≥ 105, high-frequency (18-2400

Hz)pulsations in ‘ringing tails’ of flares (~ 1034-35 ergs).

Background: ~ 0.1 Hz QPOs due to star rotation(dipole emission).

Ringing tail (Strohmayer & Watts ApJ 2006)

Starquakes: electric current generationdriven by crust cracking (Ruderman 1991)

Neutron star (D ~ 10 km) with magnetic field B ~ 1014-15 G

Flare scenario• Fireball (≥1 MeV electron-positron plasma + gamma-rays)

– the source of main pulse of flare.• Trapped fireball (≤ 1 MeV e/p plasma + γ) – a source of ‘ringing tail’

δB → SGR flux modulation

Existing models of magnetar fast QPOs

• Strohmayer & Watts (2006), Sotani et al (2008): Torsion Alfven oscillations of relativistic star with global dipole magnetic field.

• Levin (2006, 2007): Torsion oscillations of crust. Interaction between normal modes of magnetar’s crust and MHD-modes in its fluid core.

• Israel et al (2005): Coupling of toroidal seismic modes with Alfven waves propagating along magnetospheric field lines.

• Vietri et al (2007): Estimation of magnetic field from Cavallo-Fabian-Rees luminosity variability limit of ‘ringing tail’, B ~ 8×1014 G.

• Timokhin et al (2008) Variations of magnetospheric currents due to crust torsion oscillations.

• Glampedakis et al (2006): Interaction of global magneto-elastic vibrations of the star and fluid core.

• Bo Ma et al (2008): Standing slow magneto-sonic waves of flux tubes in magnetar coronae.

These models do not explain: - Excitation of oscillations in the ‘ringing tail’ and before impulse phase

- Very high Q-factor of fast QPOs, Q ≥ 104 (Levin 2006: Q ~30 )- Broad discrete spectrum of fast QPO frequencies (20 -2400 Hz).

Our approach: Coronal seismology (RLC-model)

Based on Beloborodov & Thompson model

(2007) For magnetar corona and on the model

of coronal loop as an equivalent electric

circuit (Zaitsev & Stepanov 1992).

Current in a loop is closed in metallic crust

Electric current appears due to crust cracking (Ruderman 1991)

Trapped fireball consists of ~ 5 -10 current-

currying loops (RLC-circuits).

Eigen-frequencies and Q-factors are

C

L

RQLC

1,2

1

RLC-circuit model

SGR 1806-20 flare on Dec. 27, 2004 . Total energy 5×1039J

Circuit energy E = LI2/2, from loop geometry:

For loop length l = 3×104 m, loop radius r = 3×103 m we obtaincircuit inductance L ≈ 5×104 m ≈ 5×10-3 H.

From the energy of ‘ringing tail’ E = 0.5LI 2 = 1037J we deriveloop electric current I ≈ 3×1019 A.

Using current magnitude we estimate the magnetic field minimum value:

Bφ ≈ I/cr ≈ 1013 G < Bq = m2c3/ħe = 4,4×1013 G.

Power released in ‘ringing tail’ W = R I 2 = 1034 W → R = 2.3×10-6 Ohm

For anomalous (turbulent) conductivity σeff = e2n/mνeff

we get νeff = (Wp /nT)ωp ≈ 0.1 ωp

474

ln2rl

lL

The origin of turbulent resistance

~ ηI

= 2.3×10-6 Ohm , Reff ~ νeff ~ W ~ I2

Number density of e/p pair in trapped fireball:

I = 2necS = 3×1019 A → n = 2×1016 см-3 → ωp= 8×1012 s-1 (fp ≈1 THz )

νeff = (Wp /nkBT)ωp ≈ 10-1 ωp

Possible origin of small-scale turbulence: Beam instability in electron-positron

plasma (Eichler et al 2002; Lyutikov 2002)

2

4

p

eff

effeff S

lS

lRR

emfIRdtdI

cL

eff 2

Self-excitation of current oscillations

Equation for oscillations of electric current in a loop:

Because Reff ~ νeff ~ W~ I2, R ~ αI2

Current oscillations are excited if I < Imax e.g. on the rising stage of

a flare and on flare ‘tail’.

δI → δB → SGR flux modulation

0~~~

2max

22

2

2

CI

tI

IItI

cL

From minimal (ν1 = 18 Hz) and maximal (νν2 = 2384 Hz) frequencies of ‘ringing tail’ we can estimate capacities of loops in trapped fireball:

C1 ≈1,5×10-2 F, C2 ≈ 8×10-7 F.

From the other side, the loop capacity is (Zaitsev & Stepanov 2008):

C ≈ εAS/l, for S = πr2 ≈ 3×1011cm2,

εA = c2/VA2 ≈ 1 и l = 3×106cm, C ≈ 10-7 F.

We can get various loop capacities C = 10-2 - 10-7 F for the loops with

different lengths l and cross-sectional areas S.

Magnetar coronal loop – a system with compact parameters?

Oscillations of electric current should be in-phase in all points of a loop. On the other hand, variations of the current propagate along the loop with the Alfven velocity. Therefore, for the condition of phase coincidence, the Alfven time should be substantially smaller than the period of oscillations .

QPO-frequency ν = νRLC ≈ 20-2500 Hz < νAlfven = c/l ≥ 104 Hz

Because VA = c in magnetar coronae

= c for ρ → 0 or B → ∞

2

241

B

c

c

k

Why we choose the SGR 1806-20 flare on Dec. 27, 2004?

Flare start: 21:30:26,35 UT

Polar Geophysical InstituteTumanny Ionospheric Station

Flare start: 21:30:26,35 UT

Corona of SGR 1806-20: Diagnostics

From loop geometry → L = 5×10-3 Henry From ‘ringing tail’ energy LI2/2 → I = 3×1019 A.

From current value I ≈ Bφ cr → Bmin ≈ 1013 G < Bq = 4.4×1013 GFrom energy release rate W=RI2 → R = 2.3×10-6 OhmFrom current and loop cross-section area → n = 2.5×1016 cm-3

For R = 2.3×10-6 Ohm collisional frequency νeff = (Wp /nT)ωp = 6×10-2 ωp For = 625 Hz → capacitance С = 1.3×10-5 FCircuit quality Q = (R√C/L)-1≈ 8×105 From observations: Q = πνΔt ≥ 4×105 for train duration Δt ≥ 200 с.

Various high-quality QPO’s detected in giant flare of SGR 1806-20 (ν =18, 30, 92, 150, 625, 1480 Hz) are due to persistence of loops with various geometry, plasma density, and magnetic field in a fireball.

12

LC

Summary

• Phenomenological approach: “ringing tile” - a trapped fireball - as a set of current-carrying coronal loops is quite effective diagnostic tool for magnetar corona.

I = 3×1019 A, Bmin ≈ 1013 G < Bq = 4.4×1013 G, n = 2.5×1016 cm-3

• Because B < Bq = 4.4×1013 G, the physical processes in ‘trapped fireball’ can be studied in non-quantum plasma approach.

• Estimations from energetic reasons give us real physical parameters of magnetars. For impulse phase (fireball)

I = 1021 A, B ≈ 4×1014 G.