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Astron. Astrophys. 320, L13-L16 (1997)

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3. Simulational results and discussion

3.1. Electron beam distributions

For an explanation of the observational discrepancies in hard X-ray and microwave emission we used the results of Paper II. An electron beam was assumed to be injected within 6 s with a power law in energy and Gaussian profiles in pitch angle cosines and time. The loop was considered to have a closed configuration with converging magnetic field, the beam electrons are assumed to be scattered anisotropically into both negative and positive pitch angles, inducing the electric field of return current. A temperature, density and macrovelocity of the ambient plasma were considered to result from a hydrodynamic response to the beam input, and ionization was governed by the hydrogen ionization.

3.1.1. Energy and angular distributions

Some of the beam electron distribution functions calculated for different depths, energy and pitch-angle cosines are plotted in Figure 1 (Paper II), where the dimensionless energy z is equal to 1 at energy of 15 keV, to 2 - at 30 keV etc.

Most important simulated features, obtained from the kinetic solutions for beam electrons, are the following:

  • A return current effect appears mostly at the transition region where a strong beam dissipation is caused by Ohmic heating of the ambient plasma (see Figure 1 (b,c) and Figure 2 in Paper II).
  • Powerful and hard electron beams with [FORMULA] and [FORMULA] produce higher induced electric field of return current and, therefore, more low energy electrons, moving backward to the source in the corona.
  • Softer and less powerful beams which have smaller upper energy limit ([FORMULA]) and wider initial angular distributions (Figure 2 in Paper II), can be thermalized completely at the transition region, transforming into secondary beams moving backwards with nearly normal distribution in energy.
  • A residual part of the initial beam electrons with higher energy ([FORMULA]) still can precipitate down to the photosphere as a beam with a power law in energy, but with narrower energy range and wider angular distributions.
  • The secondary beam of return current, returning to the source, causes a split in energy of initial beam. At the transition region, for harder beams, it results in a dip at 20-25 keV in energy distributions followed by maximum at 30-35 keV. After maximum the electron distributions have a single power law in energy with the initial spectral indices. For softer beams there is even a single maximum at 30-35 keV in the energy distributions with a fall of electron abundances in both lower and higher energy tails.
  • If electron beams are powerful enough to precipitate down to the photosphere, the dip disappears completely at the chromospheric level. But the abundances of low energy electrons increase in depth, and electron energy distributions become steeper than the initial ones, or their spectral indices become higher. It results in softening of initial beams during their precipitation into the deeper chromosphere.

3.1.2. X-ray bremsstrahlung emission

The hard X-ray bremsstrahlung flux was calculated from the beam electron distribution functions [FORMULA], [FORMULA] at the viewing angle [FORMULA], as the following:


where [FORMULA], [FORMULA] is an intensity unit from paper by Nocera et al. (1985), µ - pitch angle cosine, n - ambient plasma density, S is the area of flare, and R is the distance from an observer. The scattering cross-sections in the directions being parallel and perpendicular to magnetic field, [FORMULA] and [FORMULA], as well as the variables z, s, [FORMULA] and [FORMULA] were taken from Paper II.

In Figure 1 there are plotted hard X-ray fluxes calculated for electron beam distributions with viewing angle [FORMULA] from Paper II (Figures 1 and 2), (a) - for harder and more intensive beam with [FORMULA], [FORMULA] and (b) - for softer and weeker beam with [FORMULA], [FORMULA]. The energy range of hard X-ray fluxes of 15-300 keV (z=1-20) covered both the HXIS and HXRBS energy bands.

[FIGURE] Fig. 1a and b. The hard X-ray fluxes caused by electron beams with the initial parameters [FORMULA], [FORMULA] (a) and [FORMULA], [FORMULA] (b), C denotes a pure collisional model with anisotropic scattering, CEM denotes a model with collisions, electric and converging magnetic fields; 1 refers to upper energy limit of 150 keV, 2 - 300 keV.

The hard X-ray energy distributions reflect rather complicated energy distributions of beam electrons caused by the kinetic effects of beam precipitation. The main features, found from the hard X-ray energy distributions, are:

  • Despite the integration over pitch angles and electron energies, similarly to electron energy distributions, harder and more intensive X-ray fluxes (Figure 1a) have a dip at 20-25 keV in flux energy distributions followed by maximum at 25-30 keV. The dip is smaller than those in the electron distributions but still well pronounced.
  • Energy distributions after the dip retain initial spectral indices, whereas at lower energies, before the dip, the spectral indices are higher than after it, or the fluxes are much softer than those at higher energies.
  • X-ray flux in Figure 1(b), caused by softer and weaker beams with upper energy limit of 300 keV, shows flattening in the energy distributions at the slightly higher energies of 24-27 keV. The index before this flattening is higher than the initial one, whereas after flattening the distribution is similar to the initial one.
  • If the upper energy limit of electron energy is smaller (150 keV), the flux distribution becomes more flattened, or harder, than initial electron distribution, but at 27-28 keV the flux again reaches the initial spectral index.
  • X-ray fluxes, calculated in Figure 1(a,b) from 30 keV to lower energies for pure collisional electron distributions with the same spectral indices as after the dip, are much higher, than the one, calculated for more sophisticated distributions with the dip. Consequently, for these distributions the total X-ray fluxes, calculated as an integral in energy range of 15-300 keV, are lower by 60-70 [FORMULA] than those obtained for a pure collisional model with single power law.

3.2. Discussion

Let us compare the observations of hard X-ray fluxes in different energy bands, discussed in Section 2, with the simulations which can explain the discrepancies in interpretation surprisingly well.

In the present kinetic model electron beams are found to be thermalised completely if their upper energy limit is [FORMULA] for softer beams and even higher for harder beams. For beams to be able precipitate down to the chromosphere, a higher upper energy limit ([FORMULA] 180 keV) is required. This is very close to the energy limit of 200 keV, deduced for X-rays and microwaves by McKinnon et al. (1985). The abundances of electrons with energies higher than 200 keV, deduced from the electron distribution functions in Paper II, are dramatically decreased in depth by the return current effect. But even at the injection site the ratio of electron abundances with energies below and higher 200 keV is about 4 orders of magnitude. An integration in depth, used in the observational interpretation, results in the ratio of about 3 orders of magnitude, which matches those deduced by Koshugi (1986) and Nishio et al. (1995) from hard X-ray and microwave observations.

The energy limit in simulations is assosiated with the effect of induced electric field and can be even higher, if the electrons are injected not along magnetic field lines but at bigger pitch angles. Less energetic electrons lose their energy mainly in the corona and transition region, producing the observed hard X-ray emission. More energetic electrons can precipitate down to the chromosphere, losing their energy in Ohmic and collisional losses with ions and neutrals with a production of the observed microwave emission. Owing to wide pitch angle distributions in depth, the electrons, responsible for this emission, can be taken to be isotropic (McKinnon et al. (1985)).

Let's evaluate the parameters under which a two stream instability, destroying such distributions by a quasi-linear relaxation and, in turn, causing Langmuire waves, denoted by Emslie and Smith (1984)), will take place in our models. There are two kinds of the beam pairs which can produce the instabily: a direct electron beam and ambient plasma electrons as well as the direct beam and the beam of return current.

For the first pair to comply with the beams instability criteria (formula (14) in the paper by Emslie and Smith (1984)), very powerful beams with initial energy fluxes [FORMULA] are required, as in our models the dips followed by bump-in-tail appear at the depths [FORMULA], where an electron temperature is about [FORMULA] K and ambient plasma density is about [FORMULA].

For the second pair of beams: direct and return current beams, the situation is a little more complicated, as their abundances and energies are related each to other. Our preliminary estimations show that the instability criteria above is satisfied: at chromospheric levels - for any beam parameters and at coronal levels - for the beams with the initial flux [FORMULA]. We did not consider either of such beams in the present simulations, but we are planning to investig this problem in our future papers, as it is likely to explain some puzzles in microwave observations.

Therefore, we can conclude that in the presented models a particle-particle approximation can be used without a significant effect on the beam electron distributions.

The most difficult problem for a pure collisional model was to explain the difference in spectral indices, deduced from softer and harder ranges of X-ray and microwave observations. In the present kinetic model this difference is explained by a variable in depth action of the induced electric field of return current. This field causes a dip in electron energy distributions at 20-25 keV at the corona and transition region, which disappears at chromospheric levels, where, however, the electron distributions become softer.

The integration in depth of electron distribution functions leads to X-ray fluxes having depressions at the energies where electron distributions have dips. X-ray fluxes are also divided by these depressions in two parts: lower energy and higher energy ones. A lower energy part, with [FORMULA], is shifted to lower energies by width of the dip (5-6 keV). Owing to integration of electron distributions in column depth, the lower energy fluxes have much softer energy distributions than the initial ones, which results from softening of electron energy distributions at chromospheric levels described above. Since a higher energy flux, [FORMULA] keV, is not affected by the induced electric field, so it has the same spectral index as in the initial distributions. The difference in spectral indices between lower and higher energy parts is about 1-1.5 which is comparable with those deduced from the observations.

From the explanations above, the difference in total X-ray fluxes, deduced from the HXIS (15-30keV) and from the extrapolated HXRBS (25-300 keV) observations are fairly understandable. Since the extrapolation of the HXRBS fluxes from 30 keV to lower energy range has been done in a pure collisional model, so the fluxes do not show the depression caused by Ohmic losses. Thus they are not shifted to lower energy and produce higher total X-ray fluxes, than those deduced from distributions with the depression. The difference between these total fluxes is about [FORMULA] which is close to the ratios found from the observations.

Therefore, we can conclude, that the observational discrepancies arise from their interpretation by a pure collisional electron beam precipitation into the atmosphere. The more sophisticated kinetic approach considering anisotropic scattering of electron beams in the ambient plasma with partial ionisation in a presence of induced electric field of return current and converging magnetic field can give a reasonable explanation of such observations. And from the differences in total fluxes and spectral indices it should be possible to extract information about the magnitude of the induced electric field. This reinforces the point that the interpretation of hard X-ray and microwave emission requires more realistic kinetic models, which can provide more realistic pictures of electron beam dynamics.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998