1. RADIAL OSCILLATIONS OF CORONAL LOOPS AND FLARE PLASMA DIAGNOSTICS Yu.G.Kopylova, A.V.Stepanov, Yu.T.Tsap, A.V.Melnikov Pulkovo Observatory, St.Petersburg

2. The main structural elements of the Sun and late type stars coronae are magnetic loops TRACE, UV: direct observation of the MHD loops oscillations

3. 1. MHD waves in coronal loops; 2. Pulsating regime of magnetic reconnection; 3. Non-linear wave-wave or wave-particle interaction; 4. Modulation of the electric current in flare loops. Modulation of Flare Emission Coronal seismologyLoop plasma diagnostic Rosenberg suggested to associating pulsations of the radio emission with loop oscillations

4. The eigenmodes of coronal loops The emission in many wavelength ranges is effectively modulated by radial oscillations (RADIAL)

5. The coronal magnetic tube model Solutions inside the tubeoutside Axisymmetricmagnetic flux tube index Inside the tube outside Perturbed quantities

6. First analytical solution was obtained by Zaitsev and Stepanov (1975) Edwin and Roberts (1983) numerical calculations ??? Nakariakov et al. (2003) About the oscillation period estimation ? Trapped modes, no emission of MHD waves Solution outside the tube

7. Solution of dispersion equation for complex argument  a includes both leaky and trapped modes  a in general case is complex quantity

8. Trapped modes coincide with curves obtained by Edwin and Roberts (1983) Leaky modes Dispersion curves of radial FMA oscillations Zeros of

9. The period of the modes accompanied by the emission of MHD waves into the surrounding medium is determined by the radius of the tube a, not by its length L..s

10. THE MODULATION OF FLARE EMISSION BY THE RADIAL OSCILLATIONS OF CORONAL LOOPS The modulation of nonthermal gyrosynchrotron emission The magnetic field В and spectral index  estimation from ratio of modulation depths for optically thin and thick sources. From the Dulk formulae for emission coefficient of trapped electrons in optically thin 1 and thick 2 sources:   Pulsation are out of phase F f1 increases with decreasing F f2

11. The Flare of May 30, 1990 out of phase Pulsation of the microwave emission with period P =1.5 s on the time profiles at 15 and 9.375 GHz vary out of phase, M 1 = 2.5%, M 2 = 5%,. Assumptions: 1) Radial oscillations of the flare loop caused the emission modulation 2) The emission source at 15 GHz was optically thin but at 9.375 GHz optically thick Spectral index of electrons  = 4.4 Magnetic field B ~200 G

12. Plasma diagnostic using of the observable characteristics of the pulsations (the modulation depth M, the Q-factor, and the period P) Zaitsev and Stepanov, 1982 (X-ray pulsations) Q=  /  ? ξ = 0.9δ − 1.22 For microwave emission of solar flares nonthermal gyrosynchrotron mechanism is responsible

13. 3. Numerical solution of the dispersion equation 3. Numerical solution of the dispersion equation Comparison analysis of three methods have shown that for rarefied loops this mechanism defines oscillation damping Comparison analysis of three methods have shown that for rarefied loops this mechanism defines oscillation damping Analytical solution Z-S Energy method Numerical calculations Dependences of the Q factor on ratio of the Alfven speeds inside and outside the magnetic loop The damping of radial FMA oscillations I. Acoustic damping mechanism 1.Analytical solution of the dispersion equation 1.Analytical solution of the dispersion equation. 2.Energy method of the acoustic decrement calculation. 2. Energy method of the acoustic decrement calculation.

14. McLean and Sheridan (1973) have detected pulsations with P=4 s and rapid amplitude decrease. The solar flare of May 16, 1973 We’ll assume that density in the external region varies with height h in accordance to the Baumbach–Allen formula for electron density distribution Acoustic damping mechanism of loop radial oscillations Upper limit for electron density

15. The damping of radial FMA oscillations Total decrement. So the ion viscosity and thermal electron conductivity make a major contribution to the damping The comparison analysis of the dissipative processes decrements Joule losses Electron conductivity radiative losses Ion viscosity I. Dissipative processes

16. χ = 10ε/3 + 2, T [K], n [cm-3], B [G] The expression for determining the flare plasma parameters Taking into account expression for total decrement we modified the diagnostic method on a case of pulsations of the gyrosyncrotron emission

17. The Flare of August 28, 1999 Observations: NoRH (17 ГГц) АО NOAA № 8674 (Yokoyama et al.,2002) Flare region consisted of 2 emission sources The results of wavelet analysis for the emission intensity: 3 oscillation branches with 14, 7 and 2.4 s

18. Loop-loop interaction model: Ballooning oscillations: P ≈ 14 and 7 s Sausage oscillations: P ≈ 2.4 s Parameters Extended loop Compact loop T, K 2.5 × 10 7 5.2 × 10 7 n, cm -3 1.5 × 10 10 4 × 10 10 B, G 150 230 β 0.04 0.11 ________________________________ 14 and 7 s pulsations have time gap: 1 and 2 harmonicas of ballooning modes Ballooning mode or plasma tongue oscillations excite in dense compact loop. Due to gas pressure rise the violation of oscillation conditions appears and ballooning instability develops. Development of ballooning instability results in the time gap. Injection of hot plasma from compact into extended loop occurs. Radial oscillation with 2.4 s of the large loop caused by the gas pressure rise are excited. As soon as the compact loop was liberated from excess pressure the oscillations of plasma tongues with 14 and 7 s resumed. 2.4 s 7 s 14 s FLARE SCINARIO

19. Modulation of nonthermal bremsstralung from loop footpoints (optical, hard X-ray emission) The emission flux determined by the variations of the fast electrons flux. Based on the model proposed by Zaitsev and Stepanov for radial modes excitation and taking into account total damping decrement we have derived expression for T,n,B estimation.

20. Oscillations of Optical Emission on the star EV Lac Assumptions 1.Optical emission occurs due to nonthermal bremsstalung mechanism. 2.Pulsations of flares emission are produced by the excitation of sausage loop oscillations P=13 c, Q=50, M=0.2 During simultaneous observations of three flares on EV Lac : Terskol Peak ( Northern Caucasus ), Stephanion Observatory (Greece), Crimean Observatory, Belogradchik (Bulgaria) in-phase oscillations with Р = 10-30 s were detected in the U and B bands Zhilyaev et al. (2000), U: ΔF  0.2, B: ΔF  0.05, (flare 11.09.98)

21. P =10 s, Q = 30, Δ F = 0.1 The Flare on November 4, 2003 on EQ Peg B (M5E) (ULTRACAM) Taking T, B, n, L from Haisch scaling laws (Mullan et al., 2006) Mathioudakis et al. have connected pulsation with trapped sausage mode. Mathioudakis et al. (2006) non-leaky (trapped) radial oscillations We assume that leaky radial oscillations were excited. ?

22. P 1  8 s P 2  11 s P 3  12 s The period drifts to longer values during the flare P 1 P 2 P 3 L ~ 10 10 сm PP1P1 P2P2 P3P3 T, K 9.6×10 7 8.1×10 7 7.7×10 7 n, сm -3 3.7×10 11 3.5×10 11 3.4×10 11 B, G 780780700670670 Change of oscillation period in time Parameters decreased during the flare

23. Conclusions: The radial oscillations of solar and stellar coronal loops in most casesare leaky. The period of the leaky modes is determined by the radius of the tube, not by its length. For dense flare loops dissipation of radial oscillations is determined by ion viscosity and the electron thermal conductivity. For rarefied loops acoustic damping mechanism plays the main role. The radial oscillations of solar and stellar coronal loops in most cases are leaky. The period of the leaky modes is determined by the radius of the tube, not by its length. For dense flare loops dissipation of radial oscillations is determined by ion viscosity and the electron thermal conductivity. For rarefied loops acoustic damping mechanism plays the main role. Methods of diagnostics for the flare loop parameters based on the observed period, quality-factor, and modulation depth of the nonthermal emission pulsations are suggested and applied to the analysis of several solar and stellar flare events. Kopylova Yu.G., Stepanov A.V., Tsap Yu.T. Stepanov A.V., Kopylova Yu.G., Tsap Yu.T., et al Kopylova Yu.G., Stepanov A.V., Tsap Yu.T., Ast. Lett., 2002, V.28, №11, p.783-879. Stepanov A.V., Kopylova Yu.G., Tsap Yu.T., et al., Ast.Lett., V.30, № 7, 2004, p.480-488. Stepanov A.V., Kopylova Yu.G., Tsap Yu.T, Kuprianova E.G., Stepanov A.V., Kopylova Yu.G., Tsap Yu.T, Kuprianova E.G., Ast.Lett., V.30, № 9, 2005, p.612-619. Kopylova Yu.G., A.V. Melnikov, Stepanov A.V. et al Kopylova Yu.G., A.V. Melnikov, Stepanov A.V. et al., Ast.Lett., V.33, 2007, №10, p.706– 713. Publications: