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, Russia2National Astronomical Observatory, Nobeyama, Japan

3Crimean Astrophysical Observatory, Nauchny, Crimea, Ukraine4Research 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

TRACE and VLBA/VLA → magnetic loops are fundamental structures of coronaeof 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)UV Ceti B (Feb 4,1996)VLBA/VLA at 3.6 cm.VLBA/VLA at 3.6 cm.White circle ~ 0.8 milliarcsecWhite circle ~ 0.8 milliarcsec

Parameters of coronal magnetic loops

Parameter the Sun red dwarf----------------------------------------------------------------Loop length, l cm (1-10)109 2109 - 31011

Loop radius, r cm ~108 108 – 109

Plasma density, n cm-3 1010 - 1011 1010 - 1012

Plasma temperature, T K 106 - 107 3106 – 108

Magnetic field, B G 102 - 103 3102 – 104

Magnetic mirror ratio, 2 – 100Alfven velocity, VA cm/s 108 ≥ 108 Emission measure, EM cm-3 1047 - 1050 1050 - 1053

Pulsing radio emission from flares

Radio pulsations from solar flares

MHD-oscillations of coronal loops

Eigen-modes of plasma cylinder

(Zaitsev & Stepanov 1975; Roberts 1983)

.

Fast magneto sonic (radial) mode, m = 0:

)(

)(

)(

)()1(

)1(

rH

rH

rJ

rJ

em

em

im

im

2||222

||222

42

)(k

cckcc AsAs

22

||2

22||

2

Ae

Ai

ee

ii

ck

ck

2/1222/12||

2 )()( AisiFMS cckk

Eigen-modes of plasma cylinder

(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.

2

2||

2

2|| ,

2

k

k

k

k

i

e

i

eFMSFMS

222

2

sincos3

1

eic m

M 2

2

sin212

1

eiv m

M

2

3

cos

104

c

v

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.

R

pVk A

22||

2

a

aa

,

,

1 xnna

)2(

12 llNV

lT

Ap

NV

lT

Ap

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-1.22 Modulation depth M = (Fmax – Fmin)/ Fmax

M = 2ξ ΔB/B, ξ = 0.90δ - 1.22 (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 + VA

2) -1/2 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

2sin105.6

22

2/32

PnTQ

2sin105.6

22

2/32

PnTQ

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 magneticflux 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 + 1.02)ΔB/B. For the same population of electrons we determined the power-law spectral index:

21

21

/08.090.0

/02.122.1

MM

MM

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 9.375 and 15.0 GHzModulation 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

21

21

/08.090.0

/02.122.1

MM

MM

MHD-approach: radial and ballooning modes (ii) Diagnostics of coronal loop plasma

Ballooning oscillations Radial (FMS) oscillations

________________________________________________________________χ = 10ε/3 + 2, ε = M/ξTemperature T is in K, the density n is in cm-3, and the magnetic field B is in G

22

28104.2

PN

lT

22

28104.2

PN

lT

2

28102.1

P

rT

2

28102.1

P

rT

43

22/7311 2sin

107.5PN

Qln

43

22/7311 2sin

107.5PN

Qln

2/34

22/7311 2sin

102

P

Qrn 2/34

22/7311 2sin

102

P

Qrn

32/5

4/72/52/117 2sin

108.6PN

lQB

32/5

4/72/52/117 2sin

108.6PN

lQB

4/53

4/72/52/117 2sin

109.2

P

rQB 4/53

4/72/52/117 2sin

109.2

P

rQB

Modulation of gyrosynchrotron emission: Event of May 8,1998

M3.1 X-class flare NOAA 8210 S15W82

Pulsation period P ≈ 16 s

Q-factor ≈ 25, Modulation M ≈ 0.3

Ballooning mode oscillations:

Number of plasma tongues N = 4

Loop length ≈ 8109 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 ≈ 4107 K

Electron number density n ≈ 3.71010cm-3

Magnetic field B ≈ 220 G

Plasma beta β ≈ 0.11

Nobeyama 17 GHz

Yohkoh

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 2.5 × 107 5.2 × 107

n, cm−3 1.5 × 1010 4 × 1010

B, G 150 280

β 0.04 0.11__________________________________________________________________

Fit well to soft X-ray diagnostics

Loop-loop interaction