Electron re-acceleration and HXR emission

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Revision as of 10:49, 15 February 2013

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 magnetic loop, and produce Hard X-ray (HXR) emission at its footpoints by collisional http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/xrayc.html bremsstrahlung] in the dense chromosphere. This HXR emission is one of the primary diagnostics of such electron beams, and is usually interpreted using the "thick target" model. This assumes, among other things, that the distribution function of the emitting electrons is modified only by collisions.

However, when such fast electron beams propagate in a 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 (ref. [1]), and add collisional terms. Plasma density inhomogeneities are included as diffusion of Langmuir waves in wavenumber and we also consider wave-wave interactions (ref. [2]).

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 closely 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. However, the time 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

It is also possible to use a 3D "particle-in-cell" (PIC) simulation used to consider a similar problem, that of a monoenergetic beam injected into a plasma, with the effects of wave-wave interactions included. In our PIC simulations the proton-electron mass ratio was taken to be 16 for computational reasons, but that is sufficient for these simulations. The initial electron beam was homogeneous throughout the numerical box of the simulation, and an appropriate return current introduced to keep the total current in the system zero. Periodic boundary conditions were used.

Figure 2: The electron energy distributions (solid lines) at (normalized) time t = 200. The magnetic field is zero in model A and increases through models B to F. 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. In Fig. 2 we present the the resulting electron energy distributions at the (normalized) time t = 200, for the magnetic field expressed through the ratio of the electron-cyclotron frequency to the plasma frequency, equal to 0.0, 0.1, 0.5, 0.7, 1.0, and 1.3, respectively (models A-F). As can be see here, there are electrons accelerated above their initial energy, and the number of these increases with the magnetic field strength. This is due to the Weibel instability, which in the 3-D beam-plasma system with low magnetic fields 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 such simulations are simple and fast, and the beam-plasma interaction is well treated by such a model. Moreover, we can argue in favour of an 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 in terms of the coronal electron content. By redistributing electrons to energies above 20keV, the re-acceleration described in these simulations reduces the original number of electrons required.

We may still use the HXR spectrum to deduce the electron spectrum generating the HXR emission, via use of 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.

References

[1] Plasma Astrophysics

[2] Density Fluctuations and the Acceleration of Electrons by Beam-generated Langmuir Waves in the Solar Corona