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, and a fraction of this goes into 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 themselves strongly affected by density inhomogeneity, or by wave-wave interactions, and it is exactly these processes which can lead to modification of 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 (Melrose, 1980), and add collisional terms, and terms describing plasma density inhomogeneity and wave-wave interactions (Kontar et al, 2012). The former are treated using a diffusive treatment, derived in (Ratcliffe et al, 2012).

The primary quantity of interest is the time-averaged electron flux as a function of electron energy, as this directly related to the observed HXR emission. In Figure 1 we show this distribution 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 no effect on this distribution, which confirms a well known previous result (McClements, 1986). 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 (above 200keV the simulations cannot be trusted as relativistic effects become very important). Moreover, we can expect an enhancement of the HXR emission of several times, or perhaps an orer of magnitude within this range.

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.

RatcliffeKontarKarlickyFig2.png


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-dimensional 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-dimensional and self-consistent.

The fact that these two very different approaches return similar effects in terms of electron acceleration is perhaps surprising, and is a strong argument in favour of such an acceleration effect occuring.

Conclusions

The effects of Langmuir waves on HXR emission from an electron beam were considered a long time ago, but until now a full self consistent treatement of the Langmuir wave evolution was not performed. In the former case, the Langmuir waves were found to have no effect on the time-averaged electron spectrum. However in our simulations, Langmuir wave evolution is seen to lead to significant electron acceleration, and thus a modified HXR spectrum.

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.

To conclude, by answering the question posed in our abstract- yes, Langmuir wave generation and evolution can affect the electron spectrum in such a way as to be visible in the HXR emission from loop footpoints.

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