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Acceleration of electrons in a large-scale electric eld of coronal magnetic loops
V.V. Zaitsev
Institute of Applied Physics, Nizhny Novgorod, Russia

Introduction
The main part of the energy, released during solar and stellar ares, goes on the acceleration of energetic particles. By this the bulk of electrons and ions is accelerated up to the energies 100 KeV and 100 MeV respectively during the impulsive solar ares Miller, et al., 1997. These values of energy correspond to the observed hard X-ray and - ray emissions in lines. Besides, - ray emission in continuum and the emission observed sometimes during neutral pions dissociation indicate that the energy of electrons and ions in a are can reach 10 MeV and 1 GeV respectively. If one will suppose that hard X-ray emission of a are occurs as a result of bremsstrahlung emission of fast electrons entering the chromosphere a thick-target non-thermal model Emslie, et al., 1981; McClymont and Can eld, 1986; Can eld and Gayley, 1987; Mariska, et al., 1989, then it follows that the impulsive are should produce energetic electrons with the energies " 20 KeV dur_ ing the time 10 ,,100 s with the production rate of about N =1037 s,1 . Therefore, the _e" 20 KeV 3 1029 erg s,1 during rate of the energy release in this case is about E 100 s. This corresponds to a total energy of electrons Ee" 20 KeV 3 1031 erg and a total number of electrons Ne" 20 KeV 1039. Necessary value of the electrons acceleration rate decreases if one will suppose that the spectrum of hard X-ray emission with the energy " 30 KeV is generated by a hot plasma with the corresponding temperature T 3 107 K, whereas the emission of higher energies is produced by fast electrons with a power-law spectrum. This is a hybrid thermal nonthermal model by Holman and Benka 1992. In this case necessary energetic electrons _ " 20 KeV production rate decreases up to N = 2 1035 s,1 for the injection time about 100 s. This gives Ne" 20 KeV 2 1037, E_e" 20 KeV 6 1027 erg s,1 or Ee" 20 KeV 6 1029 erg. Taking account of a well correlation of impulsive ares with coronal magnetic loops, let us consider acceleration of electrons by a large-scale electric eld generated during a convective motion of a photospheric plasma in a coronal magnetic loop's foot-points. A coronal magnetic loop can accumulate an amount of energy up to 5 1032 erg, su cient even for the explanation of the energy release of a large are. This energy is accumulated in a non-potential part of a magnetic eld, appearing as a result of a high electric current up to 3 1012 A, running along a loop from one its foot-point to an other through the coronal part of the loop and upper layers of the photosphere Zaitsev, et al., 1998. On the red dwarf stars free energy of current-carrying magnetic loops can reach values of 3 1036 erg.

Acceleration region: Chromosphere or Corona
In order to provide the uxes of fast electrons observed during the ares a large amount of particles should be involved into a regime of acceleration. What could appear to be a


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source of these particles in the case of the acceleration taking place in a coronal magnetic loop? The total amount of particles in a aring magnetic loop in the case of its length 1 5 109 cm, cross-section 1018 cm2 and plasma density1010 cm,3 inside is 1 5 1037. If we take into account the fact that any realistic mechanism of acceleration in plasma accelerates only a small part of a total amount of particles, then the number of accelerated electrons appears to be insu cient for the explanation of observed uxes of energetic particles even in the most suitable case when a hybrid thermal non-thermal model of hard X-ray emission generation Ne 2 1037 is realized. Total amount of electrons in a coronal part of a magnetic loop is insu cient for providing necessary acceleration regime. For a magnetic loop there exist in principle two important possible sources, providing a su cient amount of particles for the acceleration process. Firstly, this is a chromospheric part of the loop, where in the heights interval from the temperature minimum up to the transition chromosphere-corona region about 5 1040 particles is contained. This estimation is correct for the case of the loop's cross-section near a foot-point about 1018 cm2 or even less, if one will take into account the increase of a magnetic loop's cross-section with a height. For the case of particle acceleration taking place in a chromospheric part of a magnetic loop the total amount of particles is quite su cient for providing the acceleration rates discussed above. The second possibility of enriching of a magnetic loop with the particles during a are appears in the case when the said loop interacts with a prominence. In this situation the are is initiated by a ute instability causing a penetration of a dense plasma of a prominence into the current channel of the loop. The number of particles provided by the prominence during the are tf 100 s can be estimated as N = 2R0 rtnpVptf , where rt 5 108 cm is a thickness of the tongue of prominence plasma penetrating into the current channel, np plasma number density in the prominence, and Vt VTi 2 106cm s,1 characteristic velocity of plasma penetration through the loop's surface into the current channel this velocity is taken to be equal to the ions thermal velocity in the case of T = 5 104 K. For the above parameters we obtain N 1038. This exceeds about an order the value of N necessary for the hybrid thermal non-thermal model, but a few times less then N in the thick-target non-thermal model of hard X-ray emission. Therefore, as a conclusion it follows that to provide particle acceleration mechanisms in the most powerful solar ares with a su cient total number of particles the most appropriate location of the acceleration region is in a chromospheric part of a coronal magnetic loop. Whereas for the ares of lower energy release the acceleration region can be localized in the vicinity of the top of a loop, and necessary amount of particles can be provided by a plasma of a prominence, penetrating the loop.

Acceleration Mechanism
To explain the appearance of a large amount of fast particles during solar ares a number of concrete mechanisms of particle acceleration was proposed. These mechanisms could be separated on three main classes: 1 stochastic acceleration by waves; 2 acceleration by shockwaves; 3 direct DC electric eld acceleration. The easiest way to accelerate a particle, probably, is to accelerate it directly in a region of energy release of a are. This way is called as a DC electric eld acceleration. As an accelerating electric eld here appear a large-scale electric eld E of a coronal magnetic loop. By this, in the case of a presence of a magnetic eld B j B j j E j in a plasma particles will be accelerated bya


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pro jection of electric eld onto a magnetic one Ek = jEBj . If the value of the longitudinal B e ! 2 component of the electric eld Ek is less then the Dreiser's eld: ED = V p , then in the Te acceleration process runaway there will be involved only the electrons with velocities v ED =Ek 1=2VTe, where VTe is a thermal velocity of electrons, Coulomb logarithm, !p Lengmure electron frequency, e the electron's charge. Kinetic theory yields the following formula for the runaway production rate Knoepfel and Strong, 1979:
p _ N s,1 = 0:35neVax3=8 exp , 2x , x ; 4

1
a

: where e = 5T53n2 is e ective frequency of electron-ion collisions, x = ED , V = Ek of the acceleration region.

volume

In a coronal part of a steady-state magnetic current-carrying loop with I = 1012 A, R0 = 5 108 cm, n = 1010 cm,3, and T = 106 107 K the electric eld Ek is too small to produce an observable acceleration of particles ED =Ek 200. The highest values of electric eld are generated in a dynamo-region in the foot-points of a magnetic loop, where charge separation, caused by a convective ow of a photospheric plasma and di erent interaction of electrons and ions with a magnetic eld of magnetic tube takes place. In this case a longitudinal component of the electric eld is Zaitsev and Khodachenko, 1997 1 , F Vr B 2  Br ; Ek = 2 , F enc2 1 + B 2 B 2

n where F = n m a manm is relative density of neutrals, n electrons number density, a a+ i e2n F2 = m + Coulomb conductivity, = 2 , F c2nm , ia e ective frequency e ei ea i ia of ion-neutral collisions, Br radial component of the magnetic eld, which is assumed to be small Br B . Therefore in a vertical magnetic tube with a converging radial ow of partially ionized photospheric plasma and Br = 0 there is no a DC electric eld acceleration. Acceleration of electrons appears during a ute instability developmentin a base of a magnetic tube when entering a currentchannel plasma tongue is inhomogeneous with height. In this case it is possible to show that a radial component of a magnetic eld is generated @ Zt 3 Br = B @z V t0dt0 0 and a longitudinal electric eld, causing acceleration of electrons, appears. The electric eld Ek increases during the heating of foot-points of a loop, because the heating increases and decreases . By this the relative density of neutrals F decreases too. In the case of a signi cant heating when an almost complete ionization takes place in the tube F 1 the value B 2 1, and the formula 2 simpli es:


62 1 Vr B  !e Br ; Ek = 2 c B ei In the opposite case, when the condition B
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B

2

1:

4

1 is realized, formula 2 yields B
2

, Ek = 1 F 2F miVr ia Br ; eB

1

5

Note that for Vr 0 the component Ek hasadownward direction and accelerates electrons towards the corona, whereas ions are accelerated towards the photosphere. In the case of a signi cant heating of the bases of a magnetic tube, when B ratio of the Dreiser's eld to the accelerating one is
! ED =7:7 10,5 n2  2=2:6 n 2 B ,2 3 104  T Ek B 2Vr T 5=2 20 1015 10,3 Vr 106
,5=2
2

1 the

2 B : 6 20 Br

An important point is a strong dependence of ED on the temperature and magnetic eld, Ek which can vary widely in a dynamo region. One can see that even for Br 0:1B avalue of the accelerating eld can reach the Dreiser's one if the bases are heated up to the temperature T =3:5 106 K. By this all the electrons appear in a runaway regime, and the electric eld reaches the value of 17 V cm,1. Such a eld on a scale h 108 cm can accelerate particles up to the highest energy of 1 GeV. Some features of electron acceleration in a super-Dreiser elds were considered by Litvinenko 1996. In fact such an extremely high electric eld appear in conditions of a maximal possible value of a magnetic eld about 103 G and a signi cant heating of the photospheric bases of the magnetic tube, which are not always realized. Therefore, the estimations, made above, demonstrate a possibility of an e ective particle acceleration in current-carrying magnetic loops. When acceleration of electrons takes place in chromospheric bases of magnetic loops the fast electrons production rate su cient for the hybrid thermal non-thermal model of generation of hard X-ray emission, being higher then 1035 s,1 , is realized for n =1011 cm,3, T =105 K, in the acceleration region, the radius of the tube R0 =108 cm, and a height of the acceleration region h = 1000 km. By this, Ek =2:15 10,3 V cm,1, ED =Ek = 26, and the energy of the bulk of accelerated electrons is 200 KeV.

Electric Current of Accelerated Electrons
One more important question is connected with an electric current, caused by the accel_ erated electrons. If fast electrons production rate during a are is N 1035 s,1 , then an _ 1:6 1015 A should electric current, caused by these electrons and having a value I = eN run. Assuming this current to run in a magnetic loop with a cross-section S =1018 cm2 one will obtain the corresponding magnetic eld B 6 106 G, which in fact is never observed in coronal magnetic structures.


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Usually two ways of solving this contradiction are considered. The rst way deals with a hypothesis of the accelerated electrons current lamentation, when a current channel is supposed to be splitted onto many thin current ropes with an opposite current directions in the neighboring ropes. As a result the integral magnetic eld of the current channel doesn't exceed an observed value. At the same time it is unclear how such a lamentation can appear. An other way to overcome the above paradox is connected with a formation of a backward current in plasma Hammer and Rostoker, 1970; Cox and Bennet, 1970; Lee and Sudan, 1971; Lovelace and Sudan, 1971. Let's consider for example a beam of fast electrons, having a radius r0 and injected into plasma along a z-axis of an external magnetic eld. The azimuthal component B' of the magnetic eld in every xed point will change during the leading front of the beam passing. The change of B' leads to appearance of electric eld Ez on the leading front of the beam of fast electrons. This eld acts on electrons of the background plasma in sucha way that a current, opposite to the current of injected electrons appears. Therefore the total current, and as a result, B' and Ez decrease. If the radius of the beam of fast electrons r0 exceeds a shielding scale = c=!p, then there is no a magnetic eld in regions with r r0. The current of the beam is compensated by the backward current of plasma, which whole runs almost inside the beam. A condition of a total compensation of currents look as the following: c=!p r0, eit 1, where t is a time of injection. For the times satisfying a condition eit 1 the backward current decays and the currents neutralization gradually disappears. A characteristic time of the backward current decay is determined however by the magnetic di usion time
2 tD = c2r0 ;

7

which for r0, being of the order of magnitude of a transverse scale of a magnetic loop, exceeds signi cantly all the characteristic times of aring processes. Therefore it is possible to assume that the injection of accelerated electrons doesn't change the external magnetic eld. The Lentz's law allows the beam of accelerated electrons to penetrate in plasma without losses of energy on a modi cation of the magnetic eld.

Conclusion
In this work the main attention is paid to the important role of the convective motions of plasma in the photosphere and lower chromosphere of the Sun and other stars of later spectral classes for the large scale electric eld generation and acceleration of particles. The optimal with respect to the fast electrons production rate place of location of the acceleration region is in chromospheric bases of magnetic loops where fast electrons pro_ duction rates N 1035 s,1 and average energies of accelerated electrons " 100 KeV could be provided. Under the special conditions the maximum value of accelerated electrons energy " 1 GeV can be reached. Injection of fast electrons into the coronal part of a magnetic loop doesn't lead to a generation of a strong magnetic eld. This is caused by the appearance of a backward current in plasma, which compensates the current of accelerated electrons.
This research was supp orted by Russian Foundation of Basic Researches grant 99-02-18244.


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References
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