RADIAL OSCILLATIONS OF CORONAL LOOPS RADIAL OSCILLATIONS OF CORONAL LOOPS AND FLARE PLASMA DIAGNOSTICSAND FLARE PLASMA DIAGNOSTICS

Yu.G.Kopylova, A.V.Stepanov,

Yu.T.Tsap, A.V.Melnikov

Pulkovo Observatory, St.Petersburg

The main structural elements of the Sun and late type stars coronae are magnetic loops

TRACE, UV: direct observation of the MHD loops oscillations

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 seismology Loop plasma diagnostic

Rosenberg suggested to associating pulsations of the radio emission with loop oscillations

The eigenmodes of coronal loops

The emission in many wavelength ranges is effectively modulated by radial oscillations

(RADIAL)

The coronal magnetic tube modelThe coronal magnetic tube model

Solutions inside the tube outside

Axisymmetric magnetic flux tube

indexindex Inside the tube

outside

Perturbed quantities

First analytical solution was obtained by Zaitsev and Stepanov (1975)

)()(

)()()(

)( )1(0

)1(1222

0

1222

aHaH

VkaJaJ

Vke

eeAii

i

iiAee

Edwin and Roberts (1983) numerical calculations

???

0

0 0 0

2 / ,

2 2.62

GSM P A P Ae

A A

P L C C C C

a aP

j C C

Nakariakov et al. (2003)

About the oscillation period estimation

??

Trapped modes, no emission of MHD waves

Solution outside the tube

22

222

22222

222222

))(())((

sA

sAT

TsA

As

CVCV

cCkCV

VkCk

Solution of dispersion equation for complex argument a includes both leaky and trapped modes

a in general case is complex quantity

Trapped modes coincide with curves obtained by Edwin and Roberts (1983)

Leaky modes

Dispersion curves of radial FMA oscillations

Zeros of

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

THE MODULATION OF FLARE EMISSION BY THE RADIAL OSCILLATIONS OF CORONAL LOOPSThe modulation of nonthermal gyrosynchrotron emission

BB

M

BB

M

02.1sin08.0

,22.1sin90.0

2

1

,~

1 ,~02.108.0

2

22.190.01

BF

BF

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 thin1 and thick2 sources:

Pulsation are out of phase

Ff1 increases with decreasing Ff2

The Flare of May 30, The Flare of May 30, 19901990

Pulsation of the microwave emission with period P =1.5 s on the time

profiles at 15 and 9.375 GHz vary out of phaseout of phase, M1 = 2.5%, M2 = 5%, .AssumptionsAssumptions:: 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

Plasma diagnostic using of the observable characteristics of the pulsations (the modulation depth M, the Q-factor, and the

period P)

//2 MBB

Zaitsev and StepanovZaitsev and Stepanov, 1982, 1982 ( (X-ray pulsations)X-ray pulsations)

Q=/?

,/2~0ar

,2)( maxminmax BB

FFFM

,~ 22.19.0 BFfξ = 0.9δ − 1.22

For microwave emission of solar flares nonthermal gyrosynchrotron mechanism is responsible

3.3. Numerical solution of the dispersion equationNumerical solution of the dispersion equation

Comparison analysis of three methods have shown Comparison analysis of three methods have shown that for rarefied loops this mechanism defines that for rarefied loops this mechanism defines oscillation damping oscillation damping

Analytical solution Z-SEnergy methodNumerical calculations

e

i

nn

Q2

Dependences of the Q factor on ratio of the Alfven speeds inside and outside the magnetic loop

TThe damping of he damping of radial FMAradial FMA oscillations oscillations

I. Acoustic damping mechanism1.1. Analytical solution of the dispersion equationAnalytical solution of the dispersion equation.

2.2. Energy method Energy method of the acoustic decrement calculationof the acoustic decrement calculation..

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

TThe damping of he damping of radial FMAradial FMA oscillations oscillations

Total decrement. So the ion viscosity and thermal electron conductivity make a major contribution to the damping

TTR

20105)(

The comparison analysis of the dissipative processes decrements Joule losses

Electron conductivity

radiative losses

Ion viscosity

I. Dissipative processes

2

28

~102.1

Pr

T

2

28

~102.1

Pr

T

2/34

22/7311 2sin~

102

P

rQn 2/34

22/7311 2sin~

102

P

rQn

4/53

4/72/52/117 2sin~

109.2

P

rQB 4/53

4/72/52/117 2sin~

109.2

P

rQB

χχ = 10= 10εε/3 + 2, /3 + 2,

T T [K], [K], n n [cm-3], [cm-3], B B [G][G]

The expression for determining the flare plasma parameters

,/2~0ar

Taking into account expression for total decrement we modified the diagnostic method on a case of pulsations of the gyrosyncrotron emission

The Flare of August 28, 1999Observations: 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

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 × 107 5.2 × 107

n, cm-3 1.5 × 1010 4 × 1010

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

Modulation of nonthermal Modulation of nonthermal bremsstralungbremsstralung from loop footpoints from loop footpoints((optical, hard X-ray emission)optical, hard X-ray emission)

FFFM /)( minmax

The emission flux determined by the variations of the fast electrons flux .

,cv ,/Q

2320 M

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.

Oscillations of Optical Emission on the star EV Lac

Assumptions1. 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

K107.3 7T

Гс320B

311ñì106.1 n

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)

P =10 s, Q = 30, ΔF = 0.1

The FlareThe Flare on November 4,on November 4, 2003 2003 on EQ Peg B (M5E)on EQ Peg B (M5E) ((ULTRACAMULTRACAM))

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.Mathioudakis et al. (2006(2006)) non-leakynon-leaky (trapped) radial oscillations (trapped) radial oscillations

We assume that leaky radial We assume that leaky radial oscillations were excited. oscillations were excited.

K105 7T

312ñm104 n

G1100B

ñm108.1 9L

1

La ~?

P1 8 sP2 11 sP3 12 s

The period drifts to longer values during the flare

P1 P2 P3

L ~ 1010 сm

P P1 P2 P3

T, K 9.6×107 8.1×107 7.7×107

n, сm-3 3.7×101

1

3.5×101

1

3.4×1011

B, G 780 700 670

Change of oscillation period in timeChange of oscillation period in time

22

2

SiAij CV

aP

Parameters decreased

during the flare

ConclusionsConclusions::

The radial oscillations of solar and stellar coronal loops in most The radial oscillations of solar and stellar coronal loops in most casescases are leaky. The period of the leaky modes is determined by are leaky. The period of the leaky modes is determined by the radius of the tube, not by its length.the radius of the tube, not by its length.

For dense flare loops dissipation of radial oscillations is For dense flare loops dissipation of radial oscillations is determined by ion viscosity and the electron thermal conductivity. determined by ion viscosity and the electron thermal conductivity. For rarefied loops acoustic damping mechanism plays the main For rarefied loops acoustic damping mechanism plays the main role. role.

Methods of diagnostics for the flare loop parameters based on the Methods of diagnostics for the flare loop parameters based on the observed period, quality-factor, and modulation depth of the observed period, quality-factor, and modulation depth of the nonthermal emission pulsations are suggested and applied to the nonthermal emission pulsations are suggested and applied to the analysis of several solar and stellar flareanalysis of several solar and stellar flare events.events.

Kopylova Yu.G., Stepanov A.V., Tsap Yu.T.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 alStepanov 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 alKopylova Yu.G., A.V. Melnikov, Stepanov A.V. et al., Ast.Lett., V.33, 2007, №10, p.706–713.

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