Electron re-acceleration and HXR emission

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Introduction

During solar flares, vast amounts of energy are released from the Sun's magnetic field, part of which leads to particle acceleration. A fast electron beam thus produced can propagate along a coronal loop, and produce Hard X-ray (HXR) emission at the loop footpoints by collisional bremsstrahlung in the dense chromosphere. This HXR emission is one of the primary diagnostics of such electron beams, and is usually interpreted using the Cold Thick Target Model (CTTM, Brown 1971). This assumes, among other things, that the emitting electrons are modified only by collisions.

However, when such fast electron beams propagate in plasma, Langmuir (plasma) waves are generated. These waves are strongly affected by density inhomogeneities, or by wave-wave interactions, and this evolution can have significant effects on the electron distribution. These processes are complicated and non-linear, and are thus best addressed via simulations. Several simulation methods are applicable in this situation. In this nugget we show some quasilinear and PIC (particle-in-cell) simulations of the electron evolution and briefly discuss the possible effects on HXR emission.

Quasilinear Simulations

One model situation describing these electron beams is to consider the collisional relaxation of an electron beam. We may reasonably assume the electron and Langmuir waves dynamics are along a single direction, the direction of beam propagation, and so need only 1-dimensional simulations. We begin from the well established quasilinear equations (reviewed in e.g. Melrose, 1980), and add collisional terms. Plasma density inhomogeneities are included as diffusion of Langmuir waves in wavenumber and we also consider wave-wave interactions. For the details of this treatment see (Kontar et al, 2012; Ratcliffe et al, 2012).

Figure 1: Time-averaged electron fluxes. In each panel the black line shows the case without Langmuir wave generation, while the blue lines show, respectively: left panel-with Langmuir wave generation, middle panel-with plasma density fluctuations, and right panel-fluctuating density plasma plus wave-wave interactions.

The primary quantity of interest is the time-averaged electron flux as a function of electron energy, which is directly related to the observed HXR emission. In Figure 1 we show this for the simulation models with and without Langmuir wave generation (left panel), with Langmuir wave generation and evolution due to density inhomogeneities (middle panel), and finally with wave-wave processes also included (right panel).

We see from Figure 1 that the Langmuir wave generation alone has a very weak effect, which confirms a well known previous result (e.g. Hamilton and Petrosian, 1987; McClements, 1987). However, evolution of the Langmuir waves can produce significant changes in the HXR emission. For the model parameters chosen, this occurs primarily between 20 and 200 keV, and within this range we can expect an increase in HXR emission of a few times, or perhaps an order of magnitude.

PIC simulations

In Karlický and Kontar (2012) a 3-D particle-in-cell code is used to consider a similar problem, that of the monoenergetic beam injected into plasma, with the effects of wave-wave interactions included. We initiated a spatially homogeneous electron-proton plasma with the proton-electron mass ratio $m_p/m_e$=16. This ratio was chosen to shorten the computational times and keep the proton skin-depth shorter than the dimensions of the numerical box. This ratio is still sufficient to clearly separate the dynamics of electrons and protons. The electron thermal velocity was $v_{Te}$ = 0.06 $c$, where $c$ is the speed of light. The electron plasma frequency is $\omega_{pe}$ = 0.05 and the electron Debye length was $\lambda_\mathrm{D}$ = $v_{Te}$/$\omega_{pe}$ = 0.6 $\Delta$. Then we included a mono-energetic beam that was homogeneous throughout the numerical box. We introduced an appropriate return current to keep the total current in the system zero. The beam velocity was chosen to be $v_b/c = 0.666$. The ratio of the beam to the plasma densities was $n_{b}/n_{e}= 1/8$. The periodic boundary conditions were used.

Figure 2: The electron energy distributions (solid lines) at $\omega_\mathrm{pe} t$ = 200 as a function of the magnetic field in models A-F with $\omega_{ce}/\omega_{pe}$ = 0.0, 0.1, 0.5, 0.7, 1.0, and 1.3, respectively. For comparison in each panel we plot the initial electron plasma distribution together with the initial monoenergetic beam (dashed lines).

Using this model we made several computational runs, in which we also changed the magnetic field, oriented in the beam propagation direction, see Karlick\'y and Kontar (2012). As an illustration, in Fig. 2 we present the electron energy distributions at the time $\omega_{pe} t$ = 200, for the magnetic field expressed through the ratio $\omega_{ce}/\omega_{pe}$ = 0.0, 0.1, 0.5, 0.7, 1.0, and 1.3, respectively (models A-F) ($\omega_{ce}$ in the electron-cyclotron frequency). As can be see here, there are electrons accelerated above their initial energy. This result confirms those of the kinetic simulations. Furthermore, we found that the number of these electrons increases with the magnetic field increase. It is due to that in the 3-D beam-plasma system with low magnetic fields the Weibel instability reduces this acceleration process.

Complementary approaches

The two simulation methods presented here are very different, and each have their advantages and disadvantages. In brief:

Quasilinear simulations as used here consider only a 1-D model, using weak turbulence theory. The magnetic field is ignored, except as a guiding force for the electron beam. Computationally they are simple and fast, and the beam-plasma interaction is well treated by such a model. Moreover, we can argue in favour of almost 1-d electron dynamics.

PIC simulations are computationally demanding, and thus require such approximations as a small electron-proton mass ratio, and a small number of particles. However, the effects of magnetic field can be included, and the treatment is fully 3-D and self-consistent.

Thus the two methods offer very good independent confirmation, and together give a strong argument for such an acceleration effect occurring.

Conclusions

The effects of Langmuir waves on HXR emission from an electron beam were considered a long time ago, but only as an energy loss process for the beam, where they were found to have no effect on the time-averaged electron spectrum. However our simulations found significant electron acceleration. This can solve, at least in part, the electron number problem, which says that the total number of accelerated electrons is impossibly large. By redistributing electrons to energies above 20keV, the acceleration reduces the original number of electrons required.

We may still recover the electron spectrum generating the HXR emission using standard inversion techniques, but this will only return the electron distribution as injected into the dense region from which bremsstrahlung originates, and this will NOT be the same as the originally accelerated electron spectrum. Moreover, the complex nature of the beam-wave interactions means we cannot easily, if at all, recover this original spectrum.

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