Issue 
A&A
Volume 603, July 2017



Article Number  A17  
Number of page(s)  8  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201630306  
Published online  03 July 2017 
Shockreflected electrons and Xray line spectra
Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 251 65 Ondřejov, Czech Republic
email: elena@asu.cas.cz
Received: 21 December 2016
Accepted: 11 May 2017
Aims. The aim of this paper is to try to explain the physical origin of the nonthermal electron distribution that is able to form the enhanced intensities of satellite lines in the Xray line spectra observed during the impulsive phases of some solar flares.
Methods. Synthetic Xray line spectra of the distributions composed of the distribution of shock reflected electrons and the background Maxwellian distribution are calculated in the approximation of nonMaxwellian ionization, recombination, excitation and deexcitation rates. The distribution of shock reflected electrons is determined analytically.
Results. We found that the distribution of electrons reflected at the nearlyperpendicular shock resembles, at its highenergy part, the so called ndistribution. Therefore it could be able to explain the enhanced intensities of Si xiid satellite lines. However, in the region immediately in front of the shock its effect is small because electrons in background Maxwellian plasma are much more numerous there. Therefore, we propose a model in which the shock reflected electrons propagate to regions with smaller densities and different temperatures. Combining the distribution of the shockreflected electrons with the Maxwellian distribution having different densities and temperatures we found that spectra with enhanced intensities of the satellite lines are formed at low densities and temperatures of the background plasma when the combined distribution is very similar to the ndistribution also in its lowenergy part. In these cases, the distribution of the shockreflected electrons controls the intensity ratio of the allowed Si xiii and Si xiv lines to the Si xiid satellite lines. The high electron densities of the background plasma reduce the effect of shockreflected electrons on the composed electron distribution function, which leads to the Maxwellian spectra.
Key words: Sun: flares / line: formation
© ESO, 2017
1. Introduction
It is commonly known that solarflare electron distributions can be strongly nonthermal. However, besides a highenergy tail, another kind of nonthermal distribution can also be present. Analysis of several softXrayflare spectra recorded by the SOLFLEX Bragg crystal spectrometer on the P781 spacecraft showed that the intensity ratios of Fe xxv resonance line to Fe xxivd satellite lines could be explained by the presence of nonthermal energy distribution with a “bump” during the early and peak phases of the flare (Seely et al. 1987). They also concluded that observed enhanced intensities of the satellite lines in comparison with Maxwellian synthetic spectra could not be assigned to the effect of a multithermal plasma.
High intensities of satellite lines are typical for the flare Xray line spectra. Phillips et al. (2006) used GOES flare temperatures to calculate intensities of satellite lines and compared them with those observed by the RESIK bentcrystal spectrometer aboard the CORONASF satellite (Sylwester et al. 2005). RESIK spectra covered 3.3–6.1 Å and included emission lines of Si , S , Cl , Ar , K , and many dielectronic satellite lines. They showed that the observed intensities of satellite lines are typically two times higher than theoretical ones.
Dzifčáková et al. (2008) adapted the analytic expression of peaked distribution proposed by Seely et al. (1987) as ndistribution (Fig. 1) to explain relative line intensities of silicon Xray flare spectra from the RESIK. They also considered that line spectra could be multithermal and calculated their differential emission measures (DEMs). These DEMs showed the presence of a dominant temperature component. Comparison of synthetic spectra for derived DEM’s with observations showed that they could not reproduce the intensity ratios of the allowed lines to the satellite ones (Dzifčáková et al. 2008, Figs. 11 and 12). However, synthetic spectra for the ndistribution were able to describe relative intensities of both the allowed and satellite lines. Spectra corresponding to ndistribution were observed during the impulsive phase of the flare. In decay phase and later, the spectra corresponded to an isothermal Maxwellian distribution (Dzifčáková et al. 2008, Fig. 13).
Nonthermal components of the electron distribution during two flares using both RESIK and RHESSI spectra were analyzed by Kulinová et al. (2011). They found that apart from a component corresponding to the electron beam, RHESSI spectra showed the other nonthermal component in keV energy range during the impulsive phase. This component was explained by the ndistribution with similar parameters to the ndistribution diagnosed independently from the silicon lines in the RESIK spectra. This nonthermal ndistribution was associated with the presence of the electron beam indicated by a radio burst.
Figure 2 demonstrates the effect of the ndistribution on the relative line intensities. The synthetic spectra for the Maxwellian distribution and for the ndistribution have the same intensity ratio of allowed lines, Si xiv 5.22 Å to Si xiii 5.68 Å. Similarly to observations, the intensities of the satellite lines Si xiid 5.82 Å and 5.56 Å are more than two times higher for the ndistribution than for the Maxwellian one.
Fig. 1
Comparison of the Maxwellian distribution (black line) with ndistributions with n = 3 (green), 5 (blue), and 11 (red). The mean energy of distributions is the same. 
Fig. 2
Xray line spectra for the Maxwellian distribution (black line) and the ndistribution with n = 3 (red). The intensities of the satellite lines Si xiid are enhanced for the ndistribution in the comparison with the Maxwellian spectrum. The line ratio of Si xiv 5.22 Å to Si xiii 5.68 Å line is the same in both spectra. 
The nonMaxwellian ndistribution is usually expressed as (e.g., Dzifcakova 1998) where E is the electron energy, k_{B} the Boltzmann constant, and n ∈ ⟨1,∞) the parameter of the distribution. The distribution is normalized to unity. For n = 1, the distribution is Maxwellian and then T is its temperature. The mean energy of the ndistribution depends on n(3)where τ is the pseudotemperature and T is just a parameter of the distribution. Contrary to the Maxwellian distribution, the ndistribution has narrower peak and is steeper at its highenergy part when their mean energies are the same (Fig. 1).
Up to now this electron distribution was used only as the parametric one and its physical origin remains unclear. There have been several attempts to explain processes forming the ndistribution in solar flare conditions. Some of its aspects, especially its highenergy part, were explained considering the return current in the beamplasma system (Dzifčáková & Karlický 2008; Karlický et al. 2012) or as the distribution of electrons accelerated in the doublelayer (Karlický 2012). However, both explanations have some drawbacks. While in the returncurrent explanation the drift associated with the return current needs to be greater that the thermal velocity of the background plasma, in the case of the doublelayer explanation, an existence of the double layer is not still confirmed.
Recently, Vandas & Karlický (2016) studied the distributions of electrons reflected at the nearly perpendicular shock. They found that these distributions exhibit some aspects of the ndistribution. However, in the close vicinity of the shock, the electrons in this distribution are much less numerous than that of the background plasma (which serves electrons for reflection in the shock). Thus, an effect of the ndistribution on the Xray line spectra from regions close to the shock is nearly negligible.
Therefore, in the present paper we assume that the shockreflected electrons propagate some distance from the shock where the background (Maxwellian) plasma electrons could have smaller density and different temperature compared with those at the shock vicinity, and thus produce some effects on the resulting Xray line spectra.
This paper is organized as follows: in Sect. 2 we present distributions of electrons reflected at the shocks having different parameters. Then in Sect. 3 we describe details of calculations of the Xray line spectra. The resulting spectra for various distributions of the shockreflected electrons as well as for various distributions of the background plasma are presented in Sect. 4. Finally, a discussion and conclusions are presented in Sects. 5 and 6, respectively.
2. Distribution of electrons reflected at a nearly perpendicular shock
Since the 1970s, through insitu spacecraft observations, it has been known that nearly perpendicular, fast, collisionless shocks are capable of accelerating electrons. Wu (1984) and Leroy & Mangeney (1984) suggested that the nearly perpendicular shock acts as a fast moving magnetic mirror. This kinematic point of view was subsequently supplemented by a dynamic one (KraussVarban & Wu 1989; Vandas 1989). Electrons drift in the shock layer due to gradient and curvature drifts; this drift is against the induced electric field (V × B), so electrons gain energy. Some of them are reflected at the magnetic field increase and form a beam traveling upstream away from the shock along magnetic field lines.
For a plane shock wave and thermal plasma the distribution function of reflected electrons can be expressed analytically (Vandas & Karlický 2016) where e is the elementary charge, ΔΦ is the electrostatic crossshock potential, , B_{1} and B_{2} are the upstream and downstream magnetic fields, respectively, , V_{B} = V_{1n}/cosθ_{Bn}, V_{1n} is the normal component of the upstream plasma bulk velocity with respect to the shock front, m_{e} is the electron mass, and θ_{Bn} is the angle between the shock normal and the upstream magnetic field. The energy because the upstream plasma is assumed to be thermal with electron density n_{s}, temperature T_{s}, and thermal velocity v_{s} (“s” for “seed” because these electrons serve as seed particles for the acceleration process), so the initial distribution function of electrons is n_{s}f_{1}(E) where f_{1} is from Eq. (1) with n = 1 and T = T_{s}. We note that n with a subscript means a number density throughout the paper, while the sole n is the parameter of the ndistribution function. The distribution function (4) is not normalized to unity but is related to the number density n_{s} of seed particles. The velocity V_{B} is the velocity of the magnetic mirror; it must be comparable to electron velocities for an efficient electron acceleration. Because the upstream plasma velocity V_{1} is much smaller than the thermal speed v_{s}, θ_{Bn} must be close to 90° (i.e., a nearly perpendicular shock).
The number density n_{r} and mean energy ⟨ E_{r} ⟩ of reflected electrons are where erf is the error function.
For the numerical results presented in the following, we started with the same shock parameters to those used in Vandas & Karlický (2016): V_{1n} = 1000 km s^{1}, T_{s} = 5 MK, B_{2}/B_{1} = 1.6, θ_{Bn} = 84°, and ΔΦ = 80 V. The shock has Mach number 1.5, which is consistent with the value recently reported from observations (Chen et al. 2015). The given parameters correspond to case 3 listed in Table 1. For other considered cases, only one parameter was varied as shown in the last column of the table. Cases 1–5 were used for Xray spectra computations; they represent a variation in T_{s}, the seed plasma temperature (the temperature upstream of the shock). Reflected electrons from these cases were combined with background Maxwellian plasma with the temperature T_{B} (from 2 to 6 MK) and density n_{B} (n_{B}/n_{s} from 0.01 to 1). Construction of the resulting distribution functions is shown in Fig. 3.
Table 1 lists the relative number density n_{r}/n_{s} of reflected electrons, their mean energy ⟨ E_{r} ⟩ and the energy E_{max} where the distribution function reaches a maximum. The fifth column contains n, the parameter of an ndistribution function that best fits the distribution function of reflected electrons in its highenergy part. The result of the fit for case 3 is shown in Fig. 4.
The fitting procedure was the following: a value of n is determined that minimizes the difference between the distribution functions (of reflected electrons and the ndistribution) in logarithmic scale within the energy interval from E_{max} to an energy where the distribution function of reflected electrons fell by four orders of magnitude. E_{max} is the energy where the distribution function of reflected electrons has maximum (it is given in Table 1). The ndistribution function also depends on the parameter T. It is calculated for a given n from the formula so that the maxima of both distribution functions coincided (see Fig. 4)
List of model parameters, characterization of the corresponding distribution function of reflected electrons and its relationship to the ndistribution function.
Fig. 3
Electron distributions (red lines) composed from the shock reflected electrons (green lines) with different background Maxwellian distribution (blue lines). Top: the distributions composed from the distribution of reflected electrons with ⟨ E_{r} ⟩ = 1.68 keV and T_{s} = 6 MK (case 4, Table 1) with Maxwellian distributions having different electron densities n_{B} = 0.01, 0.1, and 1.0 and fixed temperature T_{B} = 5 MK. Middle: the distributions composed from the distribution of reflected electrons (case 4) with the Maxwellian distributions having n_{B}/n_{s} = 0.1 and temperature T_{B} = 2–6 MK. Bottom: the distributions composed from different distributions of reflected electrons (⟨ E_{r} ⟩ = 1.19, 1.52, and 1.84 keV) (case 1, 3, and 5) with the Maxwellian distribution with T_{B} = 5 MK and n_{B}/n_{s} = 0.1. 
Fig. 4
Result of the fitting procedure for case 3. The distribution function of reflected electrons is given in red, and a corresponding ndistribution function in green. The bullet points to the common maximum of the distribution functions. 
From cases 1–5 we see that the value of n decreases with increasing T_{s}. In order to gain insight into how n changes with variations of shock parameters, additional cases are given in Table 1. Case 6 was treated in Vandas & Karlický (2016) where we tried to find shock parameters to fit the ndistribution function with n = 3 in its highenergy part. This was done by trial and error and visual inspection. Here a more objective procedure yields n = 3.3, not far from the desired value of 3. We see that the value of n decreases with increasing jump in magnetic field at the shock wave, or with decreasing electrostatic potential. It increases with θ_{Bn} rise, and this increase is very rapid above 87°.
3. Calculation of synthetic Xray line spectra
We calculated Xray line synthetic spectra for silicon in the same spectral interval as in Dzifčáková et al. (2008), that is, the RESIK spectral region 5.2–6.0 Å, in order to be able to compare them with observations. RESIK was a highresolution crystal Xray spectrometer (Sylwester et al. 2005) on board the Russian CORONASF mission. The RESIK spectral region is very useful for the diagnostics of electron distribution because it contains both allowed and satellite lines. In contrast to allowed lines, whose excitation rate is an integral of the product of the crosssection with distribution function from the excitation energy E_{x} to infinity (Fig. 5, crossway hatched areas), the intensities of the satellite lines depend on the number of electrons in the distribution with the energy corresponding to the excitation energy of the doubleexcited state of ion E_{d} (Fig. 5). This means that satellite lines are able to sample the electron distribution and their atypical nonMaxwellian intensities can be very sensitive indicators of the presence of a nonthermal electron distribution (Gabriel & Phillips 1979). The intensities of Si xiid satellite lines also depend on the number of the recombining Si xiii ions, therefore their ratios to Si xiii allowed lines do not depend on the ionization state of plasma.
Fig. 5
Two composed electron distributions for T_{B} = 5 ×10^{6} K, n_{B}/n_{s} = 0.10, and ⟨ E_{r} ⟩ = 1.19 keV and 1.84 keV with marked excitation energy E_{d} of the satellite line Si xiid 5.82 Å, excitation energy E_{x} of the allowed line Si xiii 5.68 Å, and the relative number of electrons over distribution that contributed to the excitation rate (crossway hatched areas), the ionization energy E_{I} of Si xiii to Si xiv and relative number of electrons over distribution that contributed to the ionization rate (horizontally and vertically hatched areas). 
The synthetic silicon Xray line spectra for the ndistributions were calculated by Dzifčáková et al. (2008). The enhanced intensities of the satellite lines Si xiid for ndistribution in comparison with Maxwellian intensities mean that the electron distribution must have an enhanced number of electrons with the energy corresponding to their excitation energies in comparison with the number of electrons in the Maxwellian distribution. However, the behavior of the whole spectrum with changes in temperature and distribution function is not so simple. The ratio of Si xiv 5.22 Å to Si xiii 5.68 Å also depends on the electron distribution through both the ionization and excitation state of the plasma. This ratio increases with temperature or pseudotemperature; differently, however, for different distributions. On the contrary, the relative intensity of Si xiiid 5.82 Å to Si xiii 5.68 Å decreases with the temperature.
Although the distribution functions of shockreflected electrons are similar to ndistribution, a background Maxwellian distribution is also present and the true distribution is a combination of both distributions. Therefore, we performed new calculations of the ionization and excitation state for particular composed distributions to determine the plasma line emission. We suppose an equilibrium state in our calculation because high electron densities of flaring plasma lead to equilibrium times for the ionization around or below 1 s. Equilibrium times for the electron excitation are even shorter than for the ionization (e.g., Bradshaw 2009). The line emissivity, ε_{ij}, can be written (8)where h is the Planck constant, c is the speed of light, λ_{ij} is the wavelength of the transition between atomic level i and j, A_{ij} is the Einstein coefficient for spontaneous emission, is the number of ions with the excited level i, is the abundance of + ktimes ionized ions relative to the total number of ions for element X, A_{X} is the abundance of the element X relative to hydrogen, N_{H} is the total number of hydrogen ions, and N_{e} is the electron density. The line intensity is simply an integral of the emissivity along the line of sight in the optically thin coronal plasma.
In equilibrium state, the excitation equilibrium determines the ratio and the ionization equilibrium gives .
For the ionization, the direct electron ionization and autoionization are important in the coronal conditions. The dominant recombination processes are the radiative recombination and dielectronic recombination. In the equilibrium state, the total ionization is compensated by the total recombination.
The rate R of any elementary process can be written: (9)where σ is the crosssection for the specific elementary process, v is the electron velocity, E is the electron energy, and f(E) is the electron distribution as a function of the energy. The ionization rates for any electron distribution can be directly calculated from the ionization crosssections based on the atomic data of Dere (2007) and available in the CHIANTI 8.0 database (Del Zanna et al. 2015).
The method of Dzifčáková (1992), Wannawichian et al. (2003), and Dzifčáková & Dudík (2013) has been used to determine radiative recombination rates. We assumed that the crosssection for the radiative recombination, σ_{RR}, can be expressed (e.g., Osterbrock 1974) (10)where C_{RR} is a constant and η + 0.5 is a powerlaw index. The parameters C_{RR} and η were taken from Aldrovandi & Pequignot (1973), Landini & Monsignori Fossi (1990), Shull & van Steenberg (1982), Mazzotta et al. (1998), and Badnell (2006b).
For the dielectronic recombination rate, the following expression, valid for any distribution function, has been used (Dzifčáková 1992; Dzifčáková & Dudík 2013) (11)where A_{m} and E_{m} are the parameters from the Maxwellian approximations of the dielectronic recombination rates. The parameters have been taken from AbdelNaby et al. (2012), Altun et al. (2006, 2007, 2004), Zatsarinny et al. (2004, 2006, 2005a,b), Mitnik & Badnell (2004), Colgan et al. (2003, 2004), Bautista & Badnell (2007), and Badnell (2006a) and can be found in the CHIANTI database (Dere et al. 2009).
For the excitation equilibrium, the electron collisional excitation and deexcitation together with the spontaneous radiative decay transitions are important. The collision strengths Ω_{ij} instead of the crosssections σ_{ij} are commonly used for calculation of the electron excitation rates: (12)where ω_{i} is the statistical weight of the level i, E_{i} is the incident electron energy, and a_{0} is the Bohr radius.
The spline approximations of the Maxwellianaveraged collision strengths for the majority of the astronomically interesting ions of elements from H to Zn can be found in the CHIANTI database (Dere et al. 1997; Del Zanna et al. 2015). The original Ωs are usually inaccessible because of their huge data volumes. Therefore, Dzifčáková et al. (2015) developed method to calculate Ω for approximation of the nonMaxwellian excitation and deexcitation rates. The tests of this method on the κdistributions for the KAPPA package (Dzifčáková et al. 2015) showed precision within 5–10% in comparison with direct numerical calculations. The Ω approximations for Si xii–Si xiv based on the atomic data of Zhang et al. (1990), Phillips et al. (2006), Vainshtein & Safronova (1978, 1980), Sampson et al. (1983), Zhang & Sampson (1987), Aggarwal & Kingston (1992) and contained in the CHIANTI database were used to calculate the Xray spectra in the RESIK 5.2–6.0 Å spectral window.
Fig. 6
Synthetic spectra of the composed distributions (red lines) in dependence on the density n_{B} of the background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 
Fig. 7
Synthetic spectra of the composed distributions (red lines) in dependence on the background temperature T_{B} for low electron densities of background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 
Fig. 8
Synthetic spectra of the composed distributions (red lines) in dependence on the mean energy ⟨ E_{r} ⟩ of reflected electrons (or the temperature T_{s} of the seed plasma; from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 
Fig. 9
Synthetic spectra of the composed distributions (red lines) in dependence on the background temperature T_{B} for high electron densities of background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 
4. Xray line spectra for the shockreflected electrons together with different background electron distributions
Considering different mean energies of the reflected electrons (cases 1–5, Table 1) and the background plasma with different parameters we calculated the Xray line spectra and compared them with Maxwellian ones.
We found that the spectra with the enhanced intensities of the Si xiid satellite lines similar to the ndistribution spectra are formed for the low mean energies of the shockreflected electrons. The enhancement is higher for the lower electron densities of the background plasma (Fig. 6). The top spectrum in Fig. 6 is formed by the composed distribution, which is more similar to the ndistribution not only in its highenergy part but also at low energies, as can be seen by the comparison of the distribution for n = 3 in Fig. 1 with the composed distribution for ⟨ E_{r} ⟩ = 1.68 keV and n_{B}/n_{s} = 0.01 in Fig. 3 (top). The low number of the lowenergy electrons results in a decrease of the recombination rate and increase of the ionization degree, which causes higher relative intensity of Si xiv 5.22 Å in comparison with Si xiii 5.68 Å (Fig. 6, top).
The temperature of the background plasma has only a small effect on the spectra at the low background electron densities (Fig. 7) because the changes of the recombination rates with temperature T_{B} are smaller than the changes with n_{B}/n_{s}. For the high densities of the background electrons, the Maxwellian part of the composed distribution becomes more important and spectra look more or less Maxwellian (Fig. 6, bottom).
The most distinct effect of the shockreflected electrons on the spectra was found in the case with the low background plasma densities (Fig. 8). With increase of the mean energy of reflected electrons, the relative intensity of Si xiv line increases and the enhancement of the satellite line intensities decreases. The shift of the energy maximum of the shockreflected electrons reflects an increase in the number of electrons capable of exciting and ionizing the Si xiii ions (Fig. 5, hatched areas are much larger for distribution with ⟨ E_{r} ⟩ = 1.84 keV). This rise results in a strong increase of the ionization and excitation rates. However, the increase in the number of electrons with energy E_{d} is not so strong (Fig. 5), therefore the intensity of the satellite lines decreases with regard to the allowed lines.
The Si xiid satellite line enhancement disappears for the high mean energies of the shockreflected electrons and high electron densities of the background plasma (Fig. 9, top). On the contrary, the opposite effect, that is, a decrease in the Si xiid satellite line intensities in comparison with the Maxwellian case, can be observed (Fig. 9, bottom). This decrease is higher for the low temperatures of the background plasma. In this case the shock reflected electrons strongly influence total energy of the composed distribution and form the highenergy tail of the distribution. The shape of these distributions is similar to the Maxwellian distribution with a power law tail and also to the κdistributions, and naturally forms similar spectra. However, the increase in the background temperature leads to an increase in the number of highenergy electrons in the Maxwellian distribution and inhibits the effect of the shockreflected electrons on the composed distribution, and spectra become Maxwellian (Fig. 9, top).
5. Discussion
Now a question arises concerning whether or not, in the impulsive phase of some solar flares, where the enhanced intensities of the Si xiid satellite lines were observed, there are regions in some distance from shocks with much lower densities where the reflected electrons propagate and thus produce these intense satellite lines.
Besides shocks that are believed to be generated in connection with the magnetic field reconnection and plasmoid formation and their interactions (Bárta et al. 2008; Vlahos et al. 2016), the so called termination shock is expected to be formed near the upper part of the flare arcade due to fast outflowing plasma from the reconnection site lying directly above (see the flare scenario, e.g., Fig. 1 in the paper by Mann et al. 2009). This termination shock could be that where the electrons are accelerated and reflected. To fulfill conditions for the enhanced intensities of Si xiid satellite lines, the reflected electrons need to propagate to regions with lower densities. Are there such regions in the flare cusp structure at some distance from the termination shock?
During propagation of the shockreflected electrons the bumpontail instability can generate Langmuir waves and then the radio emission. This process by itself is important in detection of the flare termination shock (Chen et al. 2015). However, we expect that the plasma in the flare cusp structure in front of the termination shock is in the state of a strong turbulence with some lowdensity regions. We note that the presence of the turbulence can reduce an effect of the bumpontail instability (Melrose 1980, p. 210). Nevertheless, at the present state of observations and modelling we cannot confirm these lowdensity regions. Another possibility is that the shockreflected electrons propagate out of the flare cusp structure, where there are certainly such lowdensity regions.
Our results have some general implications. They are valid for any nearly perpendicular shocks generated in solar flares (standing or propagating) with an appropriate lowdensity region aside from these shocks.
In order for the distribution of reflected electrons to be similar to an ndistribution function, their source distribution must be Maxwellian or very close to it. Moreover, the resulting distribution is formed by single reflections inside the shock layer in our calculations, so the role of random processes, such as pitchangle scattering, is neglected. If some of these assumptions were not in fact valid, the distribution of reflected electrons would resemble a κdistribution function rather than an ndistribution function (Vandas 1991).
Our model implicitly assumes processes in the collisionless regime, which gives an upper limit for plasma densities considered immediately in front of the shock of about 10^{11} cm^{3}. Furthermore, the waveparticle interactions can also modify the distribution function of the reflected electrons propagating some distance from the shock to lowdensity regions. Nevertheless we assume that the energy distribution of these electrons is still appropriate to generate the enhanced intensities Si xiid satellite lines. It can be supported by insitu observations of reflected electrons from planetary bow shocks, which are detectable as beams very far from their source regions. Anderson (1981) reports observations of reflected electrons by Earth’s bow shock from a distance of 1.5 Gm (nearly two solar radii).
6. Conclusions
We have found that the highenergy part of the ndistribution function can be matched by a distribution function of reflected electrons at a nearly perpendicular shock. The effective n depends on shockwave parameters; it increases with decreasing magnetic field jump at the shock wave, decreasing crossshock electrostatic potential, increasing of shock velocity, or with increasing θ_{Bn}. Furthermore, it increases above 10 very rapidly with θ_{Bn} over 87°. Temporal variations of n following from observations (Kulinová et al. 2011) could be attributed to temporal changes of shockwave parameters, or changes of a shock region where reflected electrons come from.
We have shown that the distribution of the shockreflected electrons influences the relation between the excitation of the allowed and satellite lines and their relative intensities. Spectra similar to ndistribution spectra can be formed by the distribution composed from the shockreflected electrons with low mean energy and background Maxwellian plasma at lower electron densities and temperatures. This combination of parameters enables the shockreflected electrons to dominate the electron distribution shape and to enhance intensities of the Si xiid satellite lines. Conversely, the high electron densities of the background plasma inhibit the effect of the reflected electrons on the shape of the distribution function and spectra come to be Maxwellian. In an extreme case, for the low temperature and high density background plasma, the reflected electrons form the highenergy tail of the distribution, which has an opposite effect on the satellite line intensities compared to the ndistribution and resembles an effect of the κdistribution function. It is interesting that the same mechanism, but under different conditions, can produce spectral effects attributed to the n or κdistribution functions.
Apparently, the formation of the spectrum with high intensities of the Si xiid satellite lines by the shockreflected electrons needs specific conditions. To decipher whether or not these conditions are really present in the impulsive phase of some solar flares, further and more detailed observations are required.
Acknowledgments
We acknowledge support from Grants P209/12/0103, 1716447S, 1419376S, 1706065S, 1517490S, and 1613277S by the Grant Agency of the Czech Republic and from the AV ČR grant RVO:67985815.
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All Tables
List of model parameters, characterization of the corresponding distribution function of reflected electrons and its relationship to the ndistribution function.
All Figures
Fig. 1
Comparison of the Maxwellian distribution (black line) with ndistributions with n = 3 (green), 5 (blue), and 11 (red). The mean energy of distributions is the same. 

In the text 
Fig. 2
Xray line spectra for the Maxwellian distribution (black line) and the ndistribution with n = 3 (red). The intensities of the satellite lines Si xiid are enhanced for the ndistribution in the comparison with the Maxwellian spectrum. The line ratio of Si xiv 5.22 Å to Si xiii 5.68 Å line is the same in both spectra. 

In the text 
Fig. 3
Electron distributions (red lines) composed from the shock reflected electrons (green lines) with different background Maxwellian distribution (blue lines). Top: the distributions composed from the distribution of reflected electrons with ⟨ E_{r} ⟩ = 1.68 keV and T_{s} = 6 MK (case 4, Table 1) with Maxwellian distributions having different electron densities n_{B} = 0.01, 0.1, and 1.0 and fixed temperature T_{B} = 5 MK. Middle: the distributions composed from the distribution of reflected electrons (case 4) with the Maxwellian distributions having n_{B}/n_{s} = 0.1 and temperature T_{B} = 2–6 MK. Bottom: the distributions composed from different distributions of reflected electrons (⟨ E_{r} ⟩ = 1.19, 1.52, and 1.84 keV) (case 1, 3, and 5) with the Maxwellian distribution with T_{B} = 5 MK and n_{B}/n_{s} = 0.1. 

In the text 
Fig. 4
Result of the fitting procedure for case 3. The distribution function of reflected electrons is given in red, and a corresponding ndistribution function in green. The bullet points to the common maximum of the distribution functions. 

In the text 
Fig. 5
Two composed electron distributions for T_{B} = 5 ×10^{6} K, n_{B}/n_{s} = 0.10, and ⟨ E_{r} ⟩ = 1.19 keV and 1.84 keV with marked excitation energy E_{d} of the satellite line Si xiid 5.82 Å, excitation energy E_{x} of the allowed line Si xiii 5.68 Å, and the relative number of electrons over distribution that contributed to the excitation rate (crossway hatched areas), the ionization energy E_{I} of Si xiii to Si xiv and relative number of electrons over distribution that contributed to the ionization rate (horizontally and vertically hatched areas). 

In the text 
Fig. 6
Synthetic spectra of the composed distributions (red lines) in dependence on the density n_{B} of the background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 

In the text 
Fig. 7
Synthetic spectra of the composed distributions (red lines) in dependence on the background temperature T_{B} for low electron densities of background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 

In the text 
Fig. 8
Synthetic spectra of the composed distributions (red lines) in dependence on the mean energy ⟨ E_{r} ⟩ of reflected electrons (or the temperature T_{s} of the seed plasma; from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 

In the text 
Fig. 9
Synthetic spectra of the composed distributions (red lines) in dependence on the background temperature T_{B} for high electron densities of background plasma (from top to bottom). For comparison, the spectra of Maxwellian distributions are added (black lines). 

In the text 
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