1. Diagnostics of Flare Plasma Using Pulsations in Microwave and X-Ray Emission
A.V.Stepanov1, Yu.G. Kopylova1, K.Shibasaki2,
Yu.T.Tsap3,V.F.Melnikov4, and T.B.Goldvarg5
1Pulkovo Observatory, St.Petersburg, Russia
2National Astronomical Observatory, Nobeyama, Japan
3Crimean Astrophysical Observatory, Nauchny, Crimea, Ukraine
4Research Institute of Radiophysics, Nizhny Novgorod, Russia
5Kalmyk State University, Elista, Russia
Solar Physics with the Nobeyama Radioheliograph
(Nobeyama Symposium 2004)
2004 October 26 – 29 Kiyosato , Japan

2. TRACE and VLBA/VLA → magnetic loops are fundamental structures of coronae of the Sun and late type stars (Bray et al. 1991; Benz et al.1998; Schrjiver et al. 1999) Moreover, the loops are typical magnetic elements of accretion discs and young stellar objects.
UV Ceti B (Feb 4,1996) VLBA/VLA at 3.6 cm. White circle ~ 0.8 milliarcsec

3. Parameters of coronal magnetic loops
Parameter the Sun red dwarf
Loop length, l cm (1-10)109 2 1011
Loop radius, r cm ~ – 109
Plasma density, n cm
Plasma temperature, T K 106 – 108
Magnetic field, B G 102 – 104
Magnetic mirror ratio,  – 100
Alfven velocity, VA cm/s ≥ 108
Emission measure, EM cm

4. Pulsing radio emission from flares

5. Radio pulsations from solar flares

6. MHD-oscillations of coronal loops
Eigen-modes of plasma cylinder
(Zaitsev & Stepanov 1975; Roberts 1983) .
Fast magneto sonic (radial) mode, m = 0:

7. Eigen-modes of plasma cylinder

8. (i) Radial oscillations
Fast magneto-sonic waves can undergo strong damping caused by radiation of FMS mode
to the environment plasma .The damping rate is (Meerson, Sasorov, Stepanov 1978)
(3)
There is no damping for e/i < k||2/ k┴2.  r2/l2, e.g. for comparatively thick loop. In this case
we have an ideal resonator for FMS-waves. Moreover, the plasma density inside flare loops can be
from two to three orders higher than outside (Doschek 1994). So, there is a jump of impedance for
FMS waves. Acoustic damping of the radial FMA oscillations of coronal loops caused by emission
of waves into the surrounding medium becomes insignificant (Zaitsev and Stepanov 1982;
Kopylova et al. 2002). Therefore, the analysis will be restricted to the dissipative processes only.
Under solar flare condition the most important damping mechanisms for fast sausage mode
are thermal conductivity and ion viscosity (Stepanov, Kopylova, Tsap et al. 2004):
Here θ is the angle between the magnetic field and wave vector .
Comparing the damping rates: for the plasma beta β ≈ 0.1 we find that ion viscosity less
important as compared to the electron thermal conductivity if θ < 78o.

9. MHD-approach: radial and ballooning modes (ii) Oscillation regime of ballooning mode
Curvature of magnetic field is important for ballooning mode.
Small deviations of plasma tongue having the scale L1= l/N where , l ≈ πR, R is the
curvature radius of a loop, N is the number of plasma tongues in a loop. Oscillations
occur as the result of acting of two forces: the force dealing with gas pressure gradient
and magnetic field line curvature F1 ~ p/R, and the back-ward force due to the
magnetic field tension F2 ~ B2/R . Dispersion equation for ballooning mode with taking
into account the scale of disturbances can be represented as (Mikhailovskii, 1974)
where is typical scale of plasma density inhomogeneity across the
magnetic field, is the transverse scale of plasma tongue. Period of the ballooning
oscillations →
Ideal MHD provide evidence for the analogy between the flute (ballooning) and
FMA modes (Burdo et al. 2000; Stepanov et al. 2004). So, to determine the damping
rate of ballooning mode we can use FMS expressions.

10. MHD-approach: radial and ballooning modes (i) Diagnostics of coronal loop plasma
Nobeyama flare observations (≥17 GHz) → optically thin, τ < 1
Flux F = ηΩd ~ B0.90δ Modulation depth M = (Fmax – Fmin)/ Fmax
M = 2ξ ΔB/B, ξ = 0.90δ (a)
Origin of loop MHD oscillations: flare energy release, chromosphere evaporation, …
Thus plasma beta β ≈ 2ΔB/B = M/ξ (b)
For θ < 78o electron thermal conductivity play the main contribution to the damping of
both FMS and ballooning modes.
Quality (c)
Oscillation period PFMS = r (cs2 + VA2) -1/ Pball = 2l/VAN (d)
From Eqs. (a) – (d) → Formulas for calculations of loop plasma parameters using
characteristics of radio pulsations caused by ballooning and FMS oscillations

11. Modulation of gyrosynchrotron emission
Consider the effect of radial coronal-loop oscillations B = B0 + ΔB sin ωt on the microwave
radiation from solar flares for which the nonthermal gyrosynchrotron mechanism is generally
accepted (Bastian 1998). The spectral fluxes for optically thin and optically thick sources:
F1 = η1Ωd ~ B0.90δ-1.22, τ1 <1; F2 = η2 Ω/ k2 ~ B – 1.02 – 0.08δ, τ2 >1 (1)
where Ω is the solid angle of the source and d is its characteristic thickness. The emission (η)
and absorption (k) coefficients are expressed by Dulk (1985). In Eqs. (1) the law of magnetic
flux conservation (d ~ B -1/2, Ω ~ B -1/2) and moderate diffusion regime of energetic electrons
(Stepanov & Tsap, 2002) were took into account. From Eqs.(1) one can see that
radio oscillations in optically thin and optically thick sources should be in anti-phase.
Modulation depth: M = (Fmax – Fmin)/ Fmax for optically thin and thick sources are
M1= 2(0.90δ – 1.22)ΔB/B, M2= 2(0.08δ )ΔB/B. For the same population of
electrons we determined the power-law spectral index:

13. Modulation of gyrosynchrotron emission: Event of May 23,1990 (Qin et al. 1996 Solar Phys.163, 383)
Flux Fmax ~ 103 sfu, Pulsation period ≈ 1.5 s, Anti-phase at and 15.0 GHz Modulation depth: M1(15 GHz, τ ≈ 0.1 ) = 5%, M2(9.4 GHz, τ ≈ 2 ) = 2.5%. Electron power-law index: ≈ 4.4 Magnetic field B(δ,θ,τ,νpeak) ≈ 190 G (Qin et al: 130 G), νpeak ≈ 10.6 GHz

14. MHD-approach: radial and ballooning modes (ii) Diagnostics of coronal loop plasma
Ballooning oscillations Radial (FMS) oscillations
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χ = 10ε/3 + 2, ε = M/ξ
Temperature T is in K, the density n is in cm-3, and the magnetic field B is in G

16. Modulation of gyrosynchrotron emission: Event of May 8,1998
M3.1 X-class flare NOAA S15W82
Pulsation period P ≈ 16 s
Q-factor ≈ 25, Modulation M ≈ 0.3
Ballooning mode oscillations:
Number of plasma tongues N = 4
Loop length ≈ 8109 cm
The angle θ ≈ 66o < 78o (!) → Thermal
conductivity is most important for damping
of ballooning oscillations.
For δ = 4.5 (Lee et al. 2002) one can obtain
the loop plasma parameters:
Plasma temperature T ≈ 4107 K
Electron number density n ≈ 3.71010cm-3
Magnetic field B ≈ 220 G
Plasma beta β ≈ 0.11
Nobeyama 17 GHz
Yohkoh

17. Modulation of gyrosynchrotron emission: Event of August 28,1999
Model: Two-loop configuration
Ballooning oscillations: P ≈ 14 s (compact loop),
a gap (00:55:45-56:30 UT) : ballooning instability
and hot plasma injection into extended loop
Radial FMS oscillation: P ≈ 2.4 s (extended loop),
started after plasma injection (00:56 UT)
Parameter Extended loop Compact loop
T , K × × 107
n, cm− × × 1010
B, G β
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Fit well to soft X-ray diagnostics

19. Loop-loop interaction