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Astron. Astrophys. 328, 390-398 (1997)

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3. The energy density of the plasma waves

The plasma waves generated by fast electrons lead to a diffusion of these electrons inside the loss cone and to their escape from the magnetic trap. The characteristic diffusion time [FORMULA] depends on the size of the source region of the fast particles. In the case of a sufficiently large loop size the diffusion time is smaller than the mean life time of the fast particles in the magnetic trap (Bespalov et al. 1991):

[EQUATION]

Here [FORMULA] denotes the length of the magnetic trap and [FORMULA] the average velocity of the fast particles. For characteristic values of [FORMULA] and [FORMULA] the diffusion time is of the order of some fraction of a second to a few seconds which is much less than the flaring time of the generation of fast particles (which is about 1 minute for an impulsive flare). Thus, for an estimation of the energy density of the plasma waves we can apply the quasi-linear theory under stationary conditions since the quasi-linear diffusion of the particles into the loss cone is compensated by the source of fast particles, and the growth rate of plasma waves under quasi-linear conditions compensates its damping by electron-ion collisions or Landau damping.

A stationary model of the generation of plasma waves at the upper hybrid frequency [FORMULA] was considered by Shaposhnikov (1988). He found that under the assumption of a one-dimensional diffusion along the line

[EQUATION]

in velocity space with conservation of the pitch angle of the fast electrons the energy density of plasma waves can be expressed by the formula

[EQUATION]

where [FORMULA] cm-3 s-1 ] is the source function which determines the amount of fast particles released in the flare source per volume unit and time unit, [FORMULA] denotes the dissipation rate of the energy of the plasma waves which is, in our case, determined either by Landau damping or by electron-ion collisions:

[EQUATION]

In the stationary state the source function J can be expressed in terms of the fast-particle density [FORMULA] and the mean lifetime [FORMULA] of particles in the magnetic trap:

[EQUATION]

[FORMULA] can be estimated in the following way (Bespalov et al. 1991):

[EQUATION]

where

[EQUATION]

is the number of energetic particles in the magnetic trap with unit cross-section at the footpoint where [FORMULA], and

[EQUATION]

is the particle flux at the footpoint of the magnetic tube.

For moderate diffusion, where [FORMULA], the velocity distribution function is quasi-isotropic and the loss cone is practically filled with fast particles, although there is no turbulent mirror at the footpoints of the flux tube yet which could impede the escape of particles from the loss cone. Hence one obtains the following relations for N, S, and [FORMULA] (Bespalov & Trakhtengerts 1986):

[EQUATION]

Then Eqs. (25), (27), and (31) allow to calculate the energy density of the plasma waves from the density of the fast electrons [FORMULA] and the main parameters of the magnetic flux tube:

[EQUATION]

For strong quasi-linear diffusion, i. e., if [FORMULA], at the footpoints of the magnetic traps turbulent mirrors develop which reduce the escape of particles from the loss cone. This fact diminishes the stream of particles at the footpoints of the trap ([FORMULA]). In this case Eq. (32) yields, for a given particle density [FORMULA], an upper limit of the energy density of the plasma waves.

Usually, under conditions of the lower solar atmosphere, the parameter [FORMULA] is small; therefore the energy density of the plasma waves reaches only a fraction of the energy density of the fast particles. Let, e. g., [FORMULA], [FORMULA] K, [FORMULA] cm, [FORMULA], and [FORMULA] cm s-1, we obtain from Eq. (32) [FORMULA]. Consequently, the ratio of the energy density of the plasma waves to the thermal energy density of the background plasma, which determines the efficiency of the conversion of the plasma waves into electromagnetic waves, is of order

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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