Electron Acceleration and Transport in Microwave Flaring Loops

V. Melnikov

(Radiophysical Research Institute, Russia)

Nobeyama Symposium, 25-29 October 2004Kyosato, Japan

Nobeyama RadioheliographNobeyama Radioheliograph

Microwave flaring loops studies and their implication for the problem of

Electron Acceleration and Transport

The first works on microwave flaring loops using observations with high spatial resolution discovered two kinds of microwave sources:

- single compact loop-top sources and - double sources with their peaks located close to the conjugate magnetic footpoints.

(Marsh and Hurford(1980), Kundu et al.(1982), Kawabata et al. (1982), Nakajima (1983))

Several new studies were published during last years developing further understanding of the problem:

Hanaoka 1997; Nishio et al 1997 - footpoint and asymmetric sourcesBastian et al. 1998, - a review Lee et al 2000, - dynamics of spectral properties, electron anisotropyKundu et al 2001, - microwave brightness distribution in limb loopsAltyntsev et al. 2002, - subsecond sourcesMelnikov et al. 2002, - looptop source and inhomogeneity of electron distributionWhite et al. 2002 - modeling and derivation of microwave source parametersLee et al. 2002 – electron transport in two interacting loopsYokoyama et al. 2002, - spectral index distribution along a loopFleishman & Melnikov 2003, - influence of electron pitch angle anisotropyMelnikov et al 2004, - dynamics of brightness distribution along a loopKarlicky 2004 – looptop source on the postflare phaseSu & Huang 2004 – polarization of looptop and footpoint sources…

This talk will be restricted mainly to properties of

single well resolved loop-like microwave sources.

1) Microwave brightness distribution along a flaring loop;2) Spectral distribution along a flaring loop;3) Model simulations of high energy electron distributions along

a loop and their GS-emission; 4) Dynamics of the brightness and spectrum slope distribution

along a flaring loop;5) Gyrosynchrotron emission from anisotropic electron

distributions and its applications (intensity and spectral index distributions).

Loop top source in the event 12 Jan 2000 (a limb flare)

Red area – brightness distributionat 34 GHz.

Contours: - dark – hard X-ray emission (HXT/M2); -light – 17 GHz radio emission (NoRH)

Radio brightness and magnetic field distributions (a disk flare)

Microwave brightness distribution along flaring loops (f = 34 GHz).

Melnikov, Reznikova, Shibasaki 2004

Disagreement with the existing microwave loop models

The brightness peaks of optically thin GS emission have to be near thefootpoints of extended loops with a nonuniform magnetic field as shown by Alissandrakis and Preka-Papadema (1984), Klein et al (1984) due to strong dependence of GS intensity on the magnetic field strength.

For example, if the electron power law spectral index =4, then

The possibility to have a hump in brightness near the loop top due to the effect of optical thick emission (Preka-Papadema & Alissandrakis 1992, Bastian et al 1998) is ruled out in our case since for all the events under study the frequency spectral index between 17 and 34 GHz is negative and, therefore, the microwave emission from the loops is optically thin at least at 34 GHz.

Bastian, Benz & Gary (1998)

Attempts to reconcile existing models with NoRH observations

Kundu, Nindos, White & Grechnev, 2001

12 Jan 2000 (a limb flare)

Middle panel : Spatialprofiles of the constant field best-fit models as afunction of distance along the loop.

Bottom panel: for inhomogeneous magnetic field and different transverse dimensions of the model loop at 17 and 34 GHz(to get them both optically thick).

Time Delays of Emission From the Loop Top

1) The burst of emission from the loop top is delayed against the burst from the footpoint source by several seconds. This delay is more pronounced at 34 GHz than at 17 GHz.

2) Time profiles of emission from the loop top are wider and their decay is slower than those from the region near the footpoint.

17 GHz, I 17 GHz, V 34 GHz, I

13 March 2000

Time delays of emission from the loop top

3) There is almost no delay between the time profiles from the regions near two conjugate footpoints.

4) For all the events the flux peaks from footpoint sources are coincident in time with the flux peaks of hard X-ray emission (in Fig. the vertical line indicates the moment of the hard X-ray emission peak)

Time delays indicate the trapping and accumulation of high energy electrons in the upper part of an extended loop and lack of these processes near footpoints.

(Melnikov, Reznikova, Yokoyama, Shibasaki, 2002).

28 August 1999

Time delays at higher frequencies

1) The peak of the emission at higher frequency, 34 GHz, is delayed against that at 17 GHz.

2) This is well pronounced for the loop-top part of the sources. However, there are no such well seen delays for the footpoint parts.

3) The footpoint emission at both frequencies peaks almost simultaneously with the peak of the corresponding hard X-ray burst.

13 March 2000

Interpretation: Delays at higher frequencies from the looptop can be explained by longer lifetime of the trapped higher energy electrons responsible for the higher frequency emission (Melnikov & Magun, 1998).

The absence of the similar delay in the footpoint source excludes the origin of the delay in the looptop due to electron spectral hardening during the acceleration process itself.

We suggest that the probable explanation of the loop-top source is trapping, accumulation and, as a

result,

an enhanced density of mildly relativistic electrons in the upper part of a flaring loop..

Kinetics of Nonthermal Electrons in Magnetic Loops

In a magnetic loop part of injected electrons are trapped due to magnetic mirroring and the other part precipitates into the loss-cone.

A real distribution strongly depends on the pitch-angle dependence of the injection function S(E,,s,t).

Fokker-Plank equation:

f

E

cf

ds

Bdc

s

fc

t

f

0

2

2

1ln

),,,(1 223

0

tsESfc

Kinetics of Nonthermal Electrons in Magnetic Loops If the energy of electrons E > 200 keV, their lifetime, , in a magnetic trap with n0=1010cm-3 is more than 50s. So, if their injection is impulsive, t < , the collisional terms in the Fokker-Plank equation can be omitted and the exact solution can be represented as follows.

Model of a microwave source

Magnetic field distribution: B(s) = B0(1+s2/s02)

Nonthermal electrons:- power law energy dependence: g(E) = g0 E-

- pitch-angle distributions:1. Beam-like distribution:

(0) = exp[-(1-02)/ 1

2]

2. Isotropic pitch-angle distribution 0 = C = const

3. Pancake distribution: (0) = exp[0

2/ 12]

Calculations of gyrosynchrotron emission from the model source

For calculations of gyrosynchrotron emission we use exact expressions for absorption and emission coefficients in the magnetized plasma (Eidman 1958, 1959; Ramaty 1969; Fleishman & Melnikov, 2003).

Melnikov V.F., Shibasaki K., Reznikova V.E. ApJ, 2002, 580, L185

Beam-like injection

Isotropic injection

Pancake injection

Results of simulations

These results were obtained only for the loops close to the limb, where the effect of magnetic field inhomogeneity is most strong. For calculations we choosed the angle between the magnetic field vector and line of sight =80o.

For this case two strong peaks of radio emission are observed even for theisotropic injection while the well defined peak of the number density existsin the loop cente.

Only in the case of the pancake injection we can get a pronounced radio brightness peak at the center of a limb loop. A degree of brightness concentration to the loop center (the ratio ILT/IFP can vary in a wide range depending on the parameter 0 of the gaussian distribution function.

What is the physical reason for the existence of microwave loop-top sources?

We consider that the most probable explanation of the loop-top source is a strong concentration of mildly relativistic electrons in the upper part of a flaring loop..

It follows from our simulations that this strong concentration can occur due to the perpendicular to magnetic field pitch-angle anisotropy of high energy

electrons.

These findings put important new constraints on - the particle acceleration/ injection mechanisms and - the kinetics of high energy electrons in flaring magnetic loops.

What acceleration+injection mechanisms can produce anisotropic distributions perpendicular to magnetic field?

Indeed:This conclusion is in disagreement with the current theoretical ideas on particle acceleration process in solar flares

- DC-electric field (along the loop axes) acceleration, - stochastic MHD-turbulent cascade acceleration

in which the acceleration occurs along the magnetic field lines.

On the other hand, the mechanisms like the acceleration in current sheets (e.g. Syrovatskii, Litvinenko and Somov, Somov and Kosugi, 1997) look quite favorable to produce highly anisotropic electron distributions near the looptop.I would also mention the model proposed by Vlahos et al., 2004. As well, all types of the betatron mechanism, for instance, considered by Karlicky and Kosugi, 2004, also agree with our finding.

A New approach to an old problem(L. Vlahos, 2004)

• From one current sheet to millions

Betatron Acceleration in Collapsing Magnetic Trap(Karlicky, Kosugi, 2004)

What’s about transport effects?

The second possibility to get anisotropic distribution of electrons is some transport effects:

Enhanced pitch-angle scattering and energy losses near footpoints due toa) higher plasma density in the lower parts of a loop;b) enhanced level of whistler (or some other) turbulence.

This is under consideration now (based on the F-PEq)In this case we expect some dynamics or redistribution of the microwave

brightness along the flaring loop: In the beginning of a burst the regions which are closer to footpoints should be relatively brighter than the looptop.

Some new constraints on the acceleration/transport models following

from spatial distribution dynamics

Let’s consider temporal variations of 1) the radio brightness and 2) frequency spectral slope distributions along a

microwave flaring loop

High frequency spectral index variations along a loop.

Yokoyama et al. 2002, ApJ, 576, L87.

High frequency spectral index variation along a loop

(Melnikov, Reznikova, Yokoyama, Shibasaki, 2002).

High frequency spectral index variation along a loop

We have confirmed the result by Yokoyama et al (2002) thatMicrowave spectrum near footpoints is considerably softer (by ~ 0.5-1) than near the loop top during the main peak of the bursts under our study

High frequency spectral index variation along a loop

1) || decreases in the rise and decay phases of the bursts both near footpoints and near the looptop (in agreement with the result by Melnikov & Magun 1998 obtained without spatial resolution);

2) A new finding: the decrease of || (flattening of the microwave spectrum) during the decay phase goes remarkably faster in the regions close to the footpoints than near the loop top.

Melnikov, Shibasaki, Reznikova, ApJ, 2002 Melnikov, Reznikova, Shibasaki, ApJ, 2004

Brightness distribution dynamics, 13 March 2000

Brightness distribution dynamics, 23 October 2000

Brightness distribution dynamics, 24 August 2002 (main peak, 34 GHz)

The microwave brightness along the flaring loop is not constant: - In the beginning of a burst the regions which are closer to footpoints are relatively brighter than the looptop. - In the burst maximum and on the decay phase the situation changes to the opposite.

The ratio of the intensities Ilt/Ifp is always higher for 34 GHz than for 17 GHz. Possibly it means that higher energy electrons are more concentrated to the looptop than less energetic ones. In the other words, they are more anisotropic perpendicular to magnetic field.

Some transport effects?

We expect some dynamics or redistribution of the microwave brightness and spectrum along the flaring loop because of enhanced pitch-angle scattering and energy losses near footpoints due to

a) higher plasma density in the lower parts of a loop;b) enhanced level of whistler (or some other) turbulence.

This is under consideration now (based on solving the Fokker-Plank Equation for the non-stationary case).

Model simulations of the GS-spectrum from electrons with pitch-angle anisotropy

(Fleishman & Melnikov, ApJ 2003)

Spectral slope at high frequencies

N=kE- - power law electron distribution

J ~ f- - frequency spectrum

Synchrotron emission (s>100, E/mc2>>1)

= (-1)/2 (Korchak and Terletskii 1952;

Getmantsev 1952)

Gyrosynchrotron emission (s<100, E/mc2<1)

= 0.90 -1.22 (Dulk and Marsh 1982)

Influence of the electron distribution anisotropy on the frequency spectrum

The angular width of the emission beam:

~ -1 = mc2/E

In ultra-relativistic limit ( -1 << 1), the anisotropy does not change the emission spectrum.

For solar flare conditions, the broad band microwave emission are mainly generated by mildly relativistic electrons

the anisotropy does influence the emission spectrum.

Model

Since our goal is to reveal the proper effect of the pitch-angle anisotropy on the gyrosynchrotron emission, we consider :

- a uniform source with - a constant magnetic field and - time-independent distribution of fast electrons.

For calculations we used exact expressions for absorption and emission coefficients in the magnetized plasma (Eidman 1958, 1959; Ramaty 1969).

(Fleishman & Melnikov, ApJ 2003)

2. Pitch-angle distribution of the sin-N type:

3. Pitch-angle distribution of the gaussian type:

1. Power-law distribution of electrons over momentum:

Results of simulations (for the case of rarefied plasma,

p/B = 0.3)

We see considerable change of microwave parameters with for the quasi-parallel propagation ( =0.8):

a) decrease of intensity;

b) increase of the degree of polarization and

c) Increase of the spectral index

Spectral index approaches the relativistic limit from above.

1. Electron pitch-angle distribution of the sin-N type.

f2() ~ exp{-2/ 02}

= cos(), =B^V.

In this case we see much stronger dependence of all emission parameters on the anisotropy compared with the sin-N pitch-angle distribution is clearly seen.

Spectral index approaches the relativistic limit from above.

Results of simulations:

2. Electron pitch-angle distribution of the loss-cone

type:

Implications for the observed spectral index and

intensity variations

The results presented show clearly that a pitch-angle anisotropy can make an important contribution to the observed spectral index and intensity variations in the optically thin part of the spectrum.

For the disc flares the foot-point source is observed at a quasiparallel direction, while the loop-top source is observed at a quasitransverse direction.

Since anisotropic distributions provide systematically softer spectra and lower intensity of gyrosynchrotron radiation (at quasiparallel directions, the case of the foot-point source) than the isotropic or weakly anisotropic distribution (at quasitransverse directions, the case of the loop-top source), this agrees well with the Nobeyama Radioheliograph data.

Conclusions We have shown that the microwave looptop sources indicate on strong concentration of mildly relativistic electrons in the upper part of a flaring loop.

The strong concentration is connected with the perpendicular to magnetic field pitch-angle anisotropy of high energy electrons.

The microwave brightness along the flaring loop is not constant: In the beginning of a burst the regions which are closer to footpoints are relatively brighter than the looptop. In the burst maximum and on the decay phase the situation changes to the opposite.

Pitch-angle anisotropy of high energy electrons can influence significantly the intensity, spectrum and polarization along a microwave flaring loops.

All these findings put important new constraints on the particle acceleration/injection mechanisms and on the kinetics of high energy electrons in flaring magnetic loops.