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

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2. Possibility of the generation of plasma waves by power-law electrons

2.1. Energy distribution function

We consider the generation of plasma waves in a system consisting of a background plasma with an electron density n and a minor part of fast electrons with a density [FORMULA] and a distribution function [FORMULA], where [FORMULA] and [FORMULA] denote the vector components parallel and perpendicular to the direction of the magnetic field [FORMULA], respectively. We suppose that the plasma is sufficiently dense ([FORMULA], where [FORMULA] and [FORMULA] are the plasma frequency and the electron gyrofrequency, respectively), and the plasma waves propagate nearly perpendicular to the magnetic field ([FORMULA], where k is the wave number of the plasma waves at [FORMULA]). The growth rate [FORMULA] of the plasma waves is given by the following formula (Mikhailovskij 1974):

[EQUATION]

It is seen from Eq. (1) that for instability ([FORMULA]) it is necessary that the derivative [FORMULA] is positive at least at some part of the integration path. If we take the distribution function in the form used by Benz & Kuijpers (1976)

[EQUATION]

where

[EQUATION]

is the step function and [FORMULA] ("mirror ratio") characterizes the magnetic loop trapping the particles, we find that the derivative of the distribution function

[EQUATION]

is always negative except at the boundary of the loss cone where it tends to infinity, since [FORMULA] is the delta function. In particular this part of the derivative leads to instability and must be considered in the evaluation of the integral in Eq. (1).

In our analysis we will apply a power-law distribution with a boundary of the loss cone that depends on energy:

[EQUATION]

with

[EQUATION]

and the condition [FORMULA], where [FORMULA] is the minimum value of the velocity of the power-law distribution. The distribution function (5) leads to a finite derivative [FORMULA] at the boundary of the loss cone. In the opposite case we would leave the frame of the kinetic approximation for the description of the instability and Eq. (1) for the growth rate would no longer be valid (Mikhailovskij 1974).

The normalization coefficient [FORMULA] in Eq. (5) follows from the condition

[EQUATION]

where [FORMULA] is the density of the power-law electrons. In the case that the mirror ratio is sufficiently large ([FORMULA]), the value of [FORMULA] is given by the following equation

[EQUATION]

The distribution functions (2) and also (5) are restricted to certain minimum values [FORMULA] characterizing the hardness of the power-law spectrum. In order to keep [FORMULA] finite, [FORMULA] is required, and in order to have a finite average velocity [FORMULA], it is necessary to take [FORMULA].

2.2. Growth rate of instability

Inserting the distribution function (5) into Eq. (1) and evaluating the integral, we obtain the following expression for the growth rate of the plasma waves at [FORMULA]:

[EQUATION]

where [FORMULA] and x are defined as

[EQUATION]

[EQUATION]

[FORMULA] is the gamma function and [FORMULA] is given by

[EQUATION]

Eq. (9) is obtained under the conditions [FORMULA] and [FORMULA].

The function [FORMULA] can be written in form of an expansion (Abramovitz & Stegun 1964)

[EQUATION]

which is strongly decreasing for values [FORMULA] we are interested in. Therefore, we can restrict our analysis to the first term of the expansion. Then, Eq. (9) simplifies to

[EQUATION]

From Eq. (14) follows that the instability is maximal for [FORMULA], i. e., for plasma waves with a sufficiently small phase velocity. For [FORMULA] the first term in the brackets of Eq. (14) is exponentially small and the instability vanishes. The instability condition for plasma waves with small phase velocities ([FORMULA]) can be written as follows

[EQUATION]

From Eq. (15) it can easily be seen that instability occurs for all relevant mirror ratios [FORMULA] and exponents [FORMULA] of the power-law spectrum. However, the maximum growth rate

[EQUATION]

decreases with increasing mirror ratio [FORMULA] which is related to a diminishing of the number of particles outside the loss cone contributing to the instability.

2.3. Influence of the background plasma

In the ambient plasma the excited plasma waves are subject to Landau damping. The damping rate [FORMULA] for the most unstable waves with [FORMULA] is given by the formula

[EQUATION]

where [FORMULA] is the electron thermal velocity of the background plasma (T - temperature, [FORMULA] - Boltzmann's constant). The condition [FORMULA] determines the values of the density of fast electrons for which the excited plasma waves are not seriously damped:

[EQUATION]

The dependence of the minimum density of the power-law electrons, beginning with that being able to generate plasma waves, on the mirror ratio [FORMULA] is shown in Fig. 1 for [FORMULA] using the ratio [FORMULA] as a parameter. The curves were derived from Eq. (18) under the assumption that Landau damping acts as the main attenuation mechanism of plasma waves in the background plasma. They were found to depend only weakly on [FORMULA] in the range [FORMULA].

[FIGURE] Fig. 1. Dependence of the minimum density of energetic power-law electrons allowing for the generation of plasma waves in a sufficiently hot plasma on the mirror ratio [FORMULA]. The numbers on the curves are the values of the ratio [FORMULA] according to the temperature of the background plasma and the minimum energy [FORMULA] of the power-law energy distribution. Assuming [FORMULA] keV, the corresponding temperatures would be [FORMULA] K (4), [FORMULA] K (6), [FORMULA] K (8), and [FORMULA] K (10). The curves were calculated on the basis of inequality (18) under the assumption that the main damping mechanism of the plasma waves is Landau damping.

It can be concluded from Fig. 1 that Landau damping suppresses the generation of plasma waves only in the immediate source region of the microwave bursts, i. e., in those parts of the flaring loops where a sufficiently hot plasma with temperatures of the order [FORMULA] K can be assumed. Hence for the excitation of plasma waves either an extremely high concentration of energetic electrons ([FORMULA]) or a sufficiently large minimum energy value [FORMULA] keV of the power-law electron energy distribution is necessary. The possibility of generation of plasma waves within the source region of the microwave bursts must be considered as a rather extreme case, although this possibility cannot be fully excluded.

On the other hand, the plasma in the immediate vicinity of a flaring magnetic loop can be sufficiently cold with a temperature [FORMULA] K (Benz et al. 1992). In this case even a relatively small number of power-law electrons escaping from the flare volume can generate plasma emission and the threshold of instability is determined by the damping of plasma waves due to electron-ion collisions in the plasma. Then the damping rate is given by

[EQUATION]

where

[EQUATION]

is the effective collision frequency of electron-ion encounters in the plasma (Zheleznyakov 1996, p. 261).

From Eqs. (17) and (19) follows that for [FORMULA] cm-3 and [FORMULA] cm s-1 ([FORMULA] keV) the collisional damping of the plasma waves begins to dominate over the Landau damping if the temperature of the background plasma satisfies the condition

[EQUATION]

In this case the maximum growth rate of the instability given by Eq. (16) exceeds the collisional damping if

[EQUATION]

Inequality (22) corresponds to the condition [FORMULA] under the assumption that [FORMULA], where [FORMULA] is determined by Eq. (21). Fig. 2 shows the dependence of the ratio [FORMULA] on the mirror ratio [FORMULA] for [FORMULA] and values of the temperature determined by condition (21). We see from Fig. 2 that in the given case the threshold of the excitation of plasma waves is sufficiently small ([FORMULA], depending on the mirror ratio).

[FIGURE] Fig. 2. Dependence of the threshold of the excitation of plasma waves on the mirror ratio [FORMULA] for the case that the damping of the plasma waves is due to collisions of electrons with ions in the background plasma, i. e., if the temperature of the plasma satisfies condition (21). Parameter values are [FORMULA] cm [FORMULA], and [FORMULA] cm s-1 ([FORMULA] keV).
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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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