A&A 506, 14371443 (2009)
Electron beam  plasma system with the return current and directivity of its Xray emission
M. Karlický  J. Kasparová
Astronomical Institute of the Academy of Sciences of the Czech Republic, 25165 Ondrejov, Czech Republic
Received 2 June 2009 / Accepted 29 July 2009
Abstract
Aims. An evolution of the electron distribution function in the
beamplasma system with the return current is computed numerically for
different parameters. The Xray bremsstrahlung corresponding to such an
electron distribution is calculated and the directivity of the Xray
emission is studied.
Methods. For computations of the electron distribution functions
we used a 3D particleincell electromagnetic code. The directivity of
the Xray emission was calculated using the angledependent
electronion bremsstrahlung crosssection.
Results. It was found that the resulting electron distribution
function depends on the magnetic field assumed along the electron beam
propagation direction. For small magnetic fields the electron
distribution function becomes broad in the direction perpendicular to
the beam propagation due to the Weibel instability and the return
current is formed by the electrons in a broad and shifted bulk of the
distribution. On the other hand, for stronger magnetic fields the
distribution is more extended in the beampropagation direction and the
return current is formed by the electrons in the extended distribution
tail. In all cases, the anisotropy of the electron distribution
decreases rapidly due to fast collisionless processes. However, the
magnetic field reduces this anisotropy decrease. The Xray directivity
shows the same trend and it is always closer to the isotropic case than
that in a simple beaming model.
Key words: Sun: flares  Sun: particle emission  Sun: Xrays, gamma rays
1 Introduction
It is commonly believed that the hard Xray emission in solar flares is produced by the bremsstrahlung process of energetic electrons in dense layers of the solar atmosphere (Brown 1971; TandbergHanssen & Emslie 1988).
It is also known that up to now this scenario has several unresolved drawbacks as summarized in the paper by Brown et al. (1990). For example, the bremsstrahlung mechanism generating the hard Xray bursts is of a very low efficiency and therefore huge electron beam fluxes 10^{12} erg s^{1} cm^{2} are required for an explanation of the observed Xray fluxes (Hoyng et al. 1978). It means that at the acceleration site in the low corona with a relatively low density ( cm^{3}), a substantial part of all plasma electrons needs to be accelerated. Furthermore, these electron beams represent huge electric currents that have to be neutralized by the return currents. The return current is a natural part of any beamplasma system (van den Oord 1990).
The beamplasma interaction has been studied for a long time, starting with the paper by Bohm & Gross (1949). While the first 1D models considered the electrostatic aspects of this interaction (twostream instability, generation of Langmuir waves, and quasilinear relaxation of the beam, see e.g. Melrose 1980; Birdsall & Langdon 1985; Benz 1993; Karlický 1997, and the references therein), new 3D studies include the return current and electromagnetic effects which lead to many further instabilities (Weibel, filamentation, oblique, Bell, Buneman, and so on, see Karlický 2009; Karlický & Bárta 2009; Bret 2009). (Remark: The Weibel instability in the sense used here and in the paper by Nishikawa et al. (2008) is also known as the filamentation instability (Bret 2009).) To cover all these processes, especially inductive processes neutralizing the total electric current, in the present study we use a general and fully selfconsistent (basic plasma physics) approach  a 3D electromagnetic particleincell (PIC) modelling.
All the abovementioned processes necessarily modify the electron distribution function in the flare Xray source. Moreover, contrary to simple models, which generally predict high anisotropy of electrons and Xrays, it was found that the observed hard Xray directivities are low (e.g. Kane 1983). Furthermore, Kontar & Brown (2006) found a low anisotropy of the electron distribution function in the Xray source by separating the reflected Xray emission from the direct one. They concluded that the conventional solar flare models with downward beaming are excluded.
In the present paper we want to demonstrate the importance of the abovementioned processes on the evolution of the beamplasma system with the return current. Our aim is to show their effects on the anisotropy of the electron distribution function in this system and thus on the directivity of the corresponding Xray emission. Using the 3D electromagnetic PIC model, for the first time in the study of Xray directivity, we compute the evolution of the beamplasma system with the return current depending on the magnetic field in the beam propagation direction. Then, assuming that the resulting electron distribution functions generate Xray bremsstrahlung, we calculate the directivity of the associated Xray emission. (For a detailed analysis of the instabilities and waves produced in the studied beamplasma system, see Karlický et al. 2008; Karlický 2009; Karlický & Bárta 2009.)
The layout of the paper is as follows: in Sect. 2 we outline our model. The results of computations of the electron distribution functions with the return current are shown in Sect. 3. In Sect. 4 we present the corresponding Xray directivities. Finally, in Section 5 the results are discussed and conclusions given.
2 Model
Table 1: Model parameters.
For our study we used a 3D (3 spatial and 3 velocity components) relativistic electromagnetic PIC code (Buneman 1993). The system sizes are , , and (where is the grid size).
For a basic set of models we initiated a spatially homogeneous electronproton plasma with the protonelectron mass ratio (Models AF, and KO in Table 1). This is unrealistic and it was chosen to shorten the proton skin depth and computations. Nevertheless, the ratio is still sufficient to well separate the dynamics of electrons and protons. For comparison we added models with the mass ratio and 100 (Models GJ in Table 1). The electron thermal velocity is (the corresponding temperature is MK), where c is the speed of light. In all models, 160 electrons and 160 protons per cube grid were used. The plasma frequency is and the electron Debye length is . In the models with the protonelectron mass ratio , the electron and proton skin depths are and , respectively.
Then, we included one monoenergetic beam homogeneous throughout the numerical box (see Models AM). Note that due to the physical and numerical simplicity and the propagation effect in which faster electrons escape from the slower ones, in most cases we consider monoenergetic electron beams, although in the interpretation of solar flare hard Xrays, the powerlaw distributions are used. The powerlaw distributions are derived as mean distributions over the whole Xray source for much longer timescales than those considered in the present study. In much smaller flare volumes and on much shorter timescales, the monoenergetic beam is a reasonable choice. Nevertheless, in Models N and O we added computations with the beam having a powerlaw distribution function. To show effects of instabilities distinctly we chose its powerlaw index (in the velocity space) as 1.5, and the lowvelocity cutoff of 0.09 c.
Table 2: The real spatial and time scales as a function of the chosen plasma density .
To keep the total current zero in these models in the initial states, we
shifted the background plasma electrons in the velocity space (i.e. we
initiated the return current) according to the relation
,
where
is the velocity
of the electron beam,
and
are the beam and
background plasma densities (for this type of initiation see
Niemiec et al. 2008). The beam velocity was chosen to be
or 0.333
(in the z direction), see Table 1. The ratio of the beam and plasma densities
was taken as
(Models AL and NO), and
(Model M).
Figure 1: The electron distribution functions in Model B at four different times: at the initial state a), at b), at c), and d). Crosses correspond to f(v_{z}), dotted and dashed lines display f(v_{x}) and f(v_{y}), respectively. Note that f(v_{x}) and f(v_{y}) overlap. The single cross in the part a) at v/c = 0.666 denotes the monoenergetic electron beam. 

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Because computations in the PIC models are dimensionless, the results are valid for a broad range of plasma densities. The real time and spatial scales are given by specifying the plasma density. Table 2 summarizes temporal and spatial scales (the interval of computations and the Debye length) for the plasma densities in the 10^{8}10^{11} cm^{3} range. The processes under study are very fast. The collisional processes are much longer, see the collisional free time (1/) in Table 2. The numerical system size is small (45 45 600 , i.e. for the plasma density e.g. cm^{3} it gives 76 cm 76 cm 1010 cm). Since the periodic boundary conditions are used, in reality the studied problem is infinite in space.
Figure 2: The electron distribution functions at as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. Notation is the same as in Fig. 1. 

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Figure 3: The electron distribution functions in electron energies (thick lines) at as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. For comparison in each panel the initial electron plasma distribution is added (thinner lines). 

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The beam density and the corresponding beam energy flux is given by the chosen plasma density , = (1/8 and 1/40), and the beam velocities (see Table 1). For example, for cm^{3}, / , and c, the beam density cm^{3} and the beam energy flux erg s^{1} cm^{3}.
Because we want to study the influence of the magnetic field, in the models we consider several values of the ratio of the electroncyclotron and electronplasma frequencies ( , 0.1, 0.5, 0.7, 1.0, and 1.3  see Table 1). Note that in the space close to the flare acceleration site in the low corona there is plasma of relatively low density. Thus, for the huge electron beam fluxes required for an explanation of the observed Xray bursts, such high ratios of are needed. In all models, the periodic boundary conditions were used.
3 Results of 3D PIC simulations
As an illustration of the time evolution of the electron distribution function
in the beamplasma system with the return current, Fig. 1 shows this evolution
for Model B. As can be seen, due to the twostream instability (Michailovskij
1975), a plateau of the distribution function f(v_{z}) (in the beam propagation
direction) on the beam side is formed. Moreover, some small part of the
electrons even increased their energy due to their interaction with generated
Langmuir waves. Simultaneously, the distribution functions f(v_{x}) and
f(v_{y}), i.e. the distribution functions in the directions perpendicular to
that of the beam propagation, are strongly heated. This is due to the
Weibel instability (1959, see also Nishikawa et al. 2006).
Figure 4: Time evolution of the ratio of the electron kinetic parallel and perpendicular energies as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. 

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Figure 5: The electron distribution functions at as a function of the mass ratio:  two upper plots,  two middle plots, and  two bottom plots for two values of ( left column) and 1.3 ( right column). Notation is the same as in Fig. 1. 

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To demonstrate how the magnetic field influences the resulting electron
distribution function, Fig. 2 presents the distribution functions for six
values of the ratio of the electroncyclotron and electronplasma frequencies
(
,
0.1,0.5, 0.7, 1.0, and 1.3  Models AF, Table 1). It is evident that with the increase of the ratio
,
the role of the Weibel instability is
more and more reduced, the distribution functions in the direction
perpendicular to the beam propagation f(v_{x}) and f(v_{y}) are less heated. On
the other hand, the problem of the return current formation becomes more and
more onedimensional and a more extended tail on the return current side is
formed (compare Model A and F in Fig. 2, see also Karlický et al. 2008;
Karlický 2009; Karlický & Bárta 2009). In Fig. 3 the same results are expressed in terms of the
electron distribution functions depending on the electron energies. Although
this type of description is more common in flare research, the distribution
functions in
velocity space presented in Fig. 2 carry more information than those in Fig. 3
and thus they are more physically relevant in describing the studied processes.
The ratio of the electron kinetic energies in the direction parallel and
perpendicular to that of beam propagation, which expresses the ``anisotropy'' of
the system, is shown in Fig. 4. The ratio of energies is defined as:
(1) 
where n is the number of electrons in the whole numerical box. As can be seen in Fig. 4, the collisionless (waveparticle) processes very rapidly decrease the ``anisotropy'' on time scales shorter than . This process is faster and more efficient for lower magnetic fields. While the ending ratio is for Model F ( ), in Model A ( ) this ratio is only .
In Fig. 5 a comparison of models with three different mass ratios ( , 16, 100) and two values of the ratio (0.0 and 1.3) is made. While in the cases with (the electronpositron plasma) the strong heating of the distribution functions f(v_{x}) and f(v_{y}) can be seen even for the strong magnetic field ( ), for the protonelectron plasma the resulting f(v_{x}) and f(v_{y}) for and 100 do not differ significantly. Note that in the model with the proton skin depth is greater than the system sizes L_{x} and L_{y}.
We also compared the evolution of the electron distribution functions in Models A and F with Models K and L, i.e. the models with a lower initial beam velocity ( ). We found that only the extent of the returncurrent tail in Model L is shorter than that in Model F. It is a natural consequence of the greater beam velocity in Model F than in Model L. Furthermore, it was found that Model M gave qualitatively the same results as Model A.
In Figs. 6 and 7 the electron distribution functions in Models N and O, i.e. in the models with the powerlaw beam and with two different ratio of electroncyclotron and electronplasma frequencies ( and 1.3) are shown. Because these models are not subject to the bumpontail instability there are no significant changes in the distribution f(v_{z}) on the beam distribution side. On the other hand, the Weibel instability plays its role, especially in the case without the magnetic field (Model N). Once again, in Model N the plasma is heated in the direction perpendicular to that of beam propagation, whereas in Model O, the return current is formed by the extended distribution tail.
Figure 6: The electron distribution functions in Model N with the powerlaw beam at four different times: at the initial state a), at b), at c), and d). Notation is the same as in Fig. 1. 

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4 Directivity of Xray emission
Knowing the electron distribution function
,
an instantaneous Xray
bremsstrahlung, i.e. the socalled thintarget emission (e.g. Brown et al.
2003) can be calculated. To account for the anisotropy of
,
we
considered the angledependent electronion bremsstrahlung crosssection
differential in the electron energy E and the solid
angle of the incoming electron, where
is the photon energy
and
is the angle between the electron precollision velocity and direction
of the photon emission (Gluckstern & Hull 1953). We used the expression for
given in Appendix of Massone et al. (2004), which
includes the Elwert (1939) Coulomb correction. The crosssection was evaluated
using hsi_reg_ge_angle_cross.pro
available in the Solar Software.
Figure 8 shows the Xray directivity, i.e. the ratio of the angledependent to integral photon spectrum , where and is the polar and azimuthal angle, respectively, is the solid angle. The zaxis of the coordinate system is chosen to be along the beam propagation direction. Note that due to axial symmetry of the problem around the zaxis, the photon spectrum is also independent of , so . Assuming that the beam propagates along the local normal line towards the photosphere, Fig. 8 displays a variation of the Xray directivity observed from different viewing angles: the cases with and correspond to the forward (the direction to the photosphere) and backward (the direction to the Earth's observer when the Xray source is at the disc centre) emissions, while the case with denotes the emission in the perpendicular direction (the Xray source placed on the solar limb).
The behaviour of the Xray directivity is closely related to the corresponding electron distribution. Comparing Model A and F at the time with Models AF in the initial state (i.e. the case with a simple beaming) in Fig. 8, it can be seen that values of the directivity, especially in the backward direction, become closer to the value 1 (the isotropic case). Therefore, the global directivity decreased during the evolution of the electron distribution. Furthermore, we can see that the directivity values for in Model A are closer to the isotropic case than those in Model F. This is due to the strong heating of the plasma in the direction perpendicular to the beam propagation and it is caused by the Weibel instability in Model A (the case with zero magnetic field).
We also defined the electron directivity , similarly to the Xray one. Models A and F at time are presented in Fig. 9 and show in another way the electron distribution characteristics discussed above in Sect. 3, Fig. 2. Comparing these electron directivities, we can see that they differ more distinctly than the corresponding Xray directivities (Fig. 8). Such a difference is caused by the strong smoothing effect of the bremsstrahlung crosssection.
We also calculated the Xray directivities for Models KL and NO. They show the same changes as follows from the comparison of plots in Fig. 8, but these changes are less pronounced due to smaller changes of the anisotropy in these models.
Figure 7: The electron distribution functions in Model O with the powerlaw beam at four different times: at the initial state a), at b), at c), and d). Notation is the same as in Fig. 1. 

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5 Discussion and conclusions
Varying the ratio of electroncyclotron and electronplasma frequencies
,
it was found that the magnetic field
influences the evolution of the electron distribution function in electron beam
 plasma system with a return current. While for small magnetic fields
(
)
the electron distribution
function becomes broad in the direction perpendicular to the beam propagation
due to the Weibel instability and the return current is formed by the electrons
in a broad and shifted bulk of the distribution, for stronger magnetic fields
(
)
the distribution is more
extended in the beampropagation direction and the return current is formed by
the electrons in an extended distribution tail. Assuming the magnetic field and
electron density as B = 100 G and
cm^{3}relevant to solar flares, the ratio of the electroncyclotron and
electronplasma frequencies is
.
In such conditions the Weibel instability plays a role, but it is reduced for a
higher magnetic field. The evolution is influenced also by the twostream
instability. Besides the formation of the plateau of the electron distribution
on the electron beam side, the simultaneously generated Langmuir waves even
accelerate a small part of the electrons.
Figure 8: The Xray directivity in several energies for corresponding to Models A and F at and the Xray directivity in the initial state for Models AF (the case of simple beaming). The horizontal solid line represents the isotropic case, the dashed vertical line denotes the viewing angle for a limb source. 

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Figure 9: The electron directivity in several energies for Models A and F at . The horizontal solid line represents the isotropic case, the dashed vertical line denotes the viewing angle for a limb source. The corresponding Xray directivities are shown in Fig. 8. 

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The collisionless processes cause a very fast decrease of the ratio of the electron kinetic parallel and perpendicular (with respect to the beam propagation direction) energies and lead to a decrease of the ``anisotropy'' of the system. Thus, the distribution function rapidly deviates from that with simple beaming. This can be also expressed by a decrease of the directivity of the associated Xray bremsstrahlung emission. This fact agrees with the statement of Kontar & Brown (2006) that conventional solar flare models with a simple downward beaming should be excluded.
An additional aspect of the present study is that the inclusion and physical necessity of the return current in the beam  plasma system resolves the problem of number of electrons needed for an acceleration of the dense electron beam in the corona where the density is relatively low. The return current simply carries the same amount of electrons as in the electron beam back to the acceleration site. However, the return current does not have the same distribution function as the initially injected beam.
Variations of the Xray directivity obtained in our models are of a level comparable to those in the electron beam propagation models by Langer & Petrosian (1977, Fig. 1) and Leach & Petrosian (1983, Fig. 4). However, there is an important difference between our model and the models by Langer & Petrosian (1977) and Leach & Petrosian (1983). We treat only collisionless processes which were neglected in the previous studies. Due to the very short time scales in our computations, no effects of longer beam propagation or collision scattering are included in the electron beam evolution.
Therefore, the similar level of Xray directivies suggests that a comparable level of isotropisation of the electron distribution function caused by the collisional processes can be produced by the studied waveparticle processes on much shorter time scales. Moreover, it means that these fast processes should not be neglected in Xray directivity studies.
Our study is not aimed at a direct comparison with observations, mainly due to the large difference between simulated and observationally available time scales. Nevertheless, the paper by Kontar & Brown (2006) allows us to compare our simulations with their derived ratio of downwardtoupward electron distributions, . The comparison reveals an agreement between inferred and Model F within the confidence interval up to 50 keV. At higher energies, our models predict a directivity higher than that obtained from observations.
The results presented here could be appropriate for lowdensity parts of flare loops where the collisionless processes are dominant. Furthermore, one may consider them as input into simulations (on much longer time scales) which treat a propagation of the beam in the environment where Coulomb collisions play a significant role, such as the transition region and the chromosphere. Since all these processes (collisionless on long time scales, collisional and even ionization processes in the background plasma) lead to further isotropisation of the particle distribution, we speculate that the resulting electron distribution and Xray directivity would be much closer to the isotropic case, as was recently found from Xray observations (Kontar & Brown 2006).
AcknowledgementsAll computations were performed on the parallel computer OCAS (Ondrejov Cluster for Astrophysical Simulations, see http://wave.asu.cas.cz/ocas). This research was supported by the grant IAA300030701 (GA CR) and the research project AV0Z10030501 (Astronomical Institute). The authors thank the referee for constructive comments that improved the paper.
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All Tables
Table 1: Model parameters.
Table 2: The real spatial and time scales as a function of the chosen plasma density .
All Figures
Figure 1: The electron distribution functions in Model B at four different times: at the initial state a), at b), at c), and d). Crosses correspond to f(v_{z}), dotted and dashed lines display f(v_{x}) and f(v_{y}), respectively. Note that f(v_{x}) and f(v_{y}) overlap. The single cross in the part a) at v/c = 0.666 denotes the monoenergetic electron beam. 

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In the text 
Figure 2: The electron distribution functions at as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. Notation is the same as in Fig. 1. 

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In the text 
Figure 3: The electron distribution functions in electron energies (thick lines) at as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. For comparison in each panel the initial electron plasma distribution is added (thinner lines). 

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In the text 
Figure 4: Time evolution of the ratio of the electron kinetic parallel and perpendicular energies as a function of the magnetic field in Models AF with , 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. 

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In the text 
Figure 5: The electron distribution functions at as a function of the mass ratio:  two upper plots,  two middle plots, and  two bottom plots for two values of ( left column) and 1.3 ( right column). Notation is the same as in Fig. 1. 

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In the text 
Figure 6: The electron distribution functions in Model N with the powerlaw beam at four different times: at the initial state a), at b), at c), and d). Notation is the same as in Fig. 1. 

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In the text 
Figure 7: The electron distribution functions in Model O with the powerlaw beam at four different times: at the initial state a), at b), at c), and d). Notation is the same as in Fig. 1. 

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In the text 
Figure 8: The Xray directivity in several energies for corresponding to Models A and F at and the Xray directivity in the initial state for Models AF (the case of simple beaming). The horizontal solid line represents the isotropic case, the dashed vertical line denotes the viewing angle for a limb source. 

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In the text 
Figure 9: The electron directivity in several energies for Models A and F at . The horizontal solid line represents the isotropic case, the dashed vertical line denotes the viewing angle for a limb source. The corresponding Xray directivities are shown in Fig. 8. 

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In the text 
Copyright ESO 2009