1. 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

2. Soft Gamma-ray Repeater - neutron star (D ~ 10 km, M ≤ 1.5M Sun ) with magnetic field B ~ 10 14-15 G

3. Energy release in impulse phase (duration ≤ 1 s): up to 2×10 46 ergs High-quality, Q ≥ 10 5, high-frequency (18-2400 Hz) pulsations in ‘ringing tails’ of flares (~ 10 34-35 ergs). Background: ~ 0.1 Hz QPOs due to star rotation (dipole emission).

4. Ringing tail (Strohmayer & Watts ApJ 2006)

5. Starquakes: electric current generation driven by crust cracking ( Ruderman 1991) Neutron star (D ~ 10 km) with magnetic field B ~ 10 14-15 G

6. 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

7. 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 ~ 8×10 14 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 ≥ 10 4 (Levin 2006: Q ~30 ) - Broad discrete spectrum of fast QPO frequencies (20 -2400 Hz).

8. 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

9. RLC-circuit model SGR 1806-20 flare on Dec. 27, 2004. Total energy 5×10 39 J Circuit energy E = LI 2 /2, from loop geometry: For loop length l = 3×10 4 m, loop radius r = 3×10 3 m we obtain circuit inductance L ≈ 5×10 4 m ≈ 5×10 -3 H. From the energy of ‘ringing tail’ E = 0.5LI 2 = 10 37 J we derive loop electric current I ≈ 3×10 19 A. Using current magnitude we estimate the magnetic field minimum value: B φ ≈ I/cr ≈ 10 13 G < B q = m 2 c 3 /ħe = 4,4×10 13 G. Power released in ‘ringing tail’ W = R I 2 = 10 34 W → R = 2.3×10 -6 Ohm For anomalous (turbulent) conductivity σ eff = e 2 n/mν eff we get ν eff = (W p /nT)ω p ≈ 0.1 ω p

10. The origin of turbulent resistance ~ η I = 2.3×10 -6 Ohm, R eff ~ ν eff ~ W ~ I 2 Number density of e/p pair in trapped fireball: I = 2necS = 3 ×10 19 A → n = 2×10 16 см -3 → ω p = 8×10 12 s -1 (f p ≈ 1 THz ) ν eff = (W p /nk B T)ω p ≈ 10 -1 ω p Possible origin of small-scale turbulence: Beam instability in electron-positron plasma (Eichler et al 2002; Lyutikov 2002)

11. Self-excitation of current oscillations Equation for oscillations of electric current in a loop: Because R eff ~ ν eff ~ W~ I 2, R ~ αI 2 Current oscillations are excited if I < I max e.g. on the rising stage of a flare and on flare ‘tail’. δI → δB → SGR flux modulation

12. ν From minimal ( ν 1 = 18 Hz) and maximal ( ν 2 = 2384 Hz) frequencies of ‘ringing tail’ we can estimate capacities of loops in trapped fireball: C 1 ≈1,5×10 -2 F, C 2 ≈ 8×10 -7 F. From the other side, the loop capacity is (Zaitsev & Stepanov 2008): C ≈ ε A S/l, for S = πr 2 ≈ 3×10 11 cm 2, ε A = c 2 /V A 2 ≈ 1 и l = 3×10 6 cm, 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.

13. 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 = ν RLC ≈ 20-2500 Hz < ν Alfven = c/ l ≥ 10 4 Hz Because V A = c in magnetar coronae = c for ρ → 0 or B → ∞

14. Why we choose the SGR 1806-20 flare on Dec. 27, 2004 ? Flare start: 21:30:26,35 UT

15. Polar Geophysical Institute Tumanny Ionospheric Station

16. Flare start: 21:30:26,35 UT

17. Corona of SGR 1806-20: Diagnostics From loop geometry → L = 5×10 -3 Henry From ‘ringing tail’ energy LI 2 /2 → I = 3 ×10 19 A. From current value I ≈ B φ cr → B min ≈ 10 13 G < B q = 4.4×10 13 G From energy release rate W=RI 2 → R = 2.3×10 -6 Ohm From current and loop cross-section area → n = 2.5×10 16 cm -3 For R = 2.3×10 -6 Ohm collisional frequency ν eff = ( W p /nT)ω p = 6×10 -2 ω p For = 625 Hz → capacitance С = 1.3×10 -5 F Circuit quality Q = (R√C/L) -1 ≈ 8×10 5 From observations: Q = πν Δt ≥ 4×10 5 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.

18. 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 ×10 19 A, B min ≈ 10 13 G < B q = 4.4×10 13 G, n = 2.5×10 16 cm -3 Because B < B q = 4.4×10 13 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 = 10 21 A, B ≈ 4 ×10 14 G.