Dips and Waves

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Dips and Waves
Number: 115
1st Author: Iain Hannah
2nd Author:
Published: 23 November 2009
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Introduction

There have been some heated discussion (for instance [1]) as to whether the "dip" seen in the mean electron spectrum <nFV> derived from RHESSI flare observations is a real feature since it can often be removed be correcting for albedo. But for flares with relatively low thermal mission the standard "thick-target" interpretation says that the "dip" must be there. This model only accounts for Coulomb collisions between the propagating beam of accelerated electrons and the background plasma. In this nugget we present simulation results from our recent paper [2] where we show what happens to the dip when you also include wave-particle interactions between the beam and background plasma.

Wave-particle Interactions

In the standard interpretation of RHESSI's hard X-ray emission, a negative powerlaw of acceelerated electrons (F0(E) ~E0) above a sharp low energy cutoff leaves the corona travelling down to the chromosphere. As they propagate they lose energy to the background plasma through Coloumb collisions, eventually losing their energy in the dense chromosphere, where they emit hard X-rays as observed by RHESSI, heating the local plasma which expands back upwards. It can be analytically shown the the resulting mean electron flux spectrum <nVF(E)> will also have a neagtive powerlaw above the cutoff (<nVF>~E1) but will have a positive one (<nVF(E)>~Eδ1) below the cutoff. The combination of this positive slope (increasing with increasing energy) and the falling thermal spectrum results in a local minima or "dip" in the total <nVF>.

We have simulated the propagation of such a powerlaw of acceleration electrons, Coulomb colision acting on the beam only, as shown in the left panel of Figure 1. But we have also ran a second set of numerical simulations in which we include the wave-particle interaction of the beam and background plasma. Namely we include beam-driven Langmuir wave turbulence. We want to include the waves-particle interactions as this non-collisional process is faster than collisions and the development of Langmuir waves from electron beams in solar flares is inferred from radio observations. These are self-consitently simulated using the quasi-linear approach describing the resonant interaction between the electrons and Langmuir waves. In these simulations we follow through time the electron distribution function f(v,x,t) and spectral energy density of the waves W(v,x,t). In Figure 1 we have snapshot from during the simulations showing f(v,x,t) for the coulomb collision only simulation (left) and f(v,x,t) and W(v,x,t) for the wave-particle simulation (middle and right panel). A movie of this can also be found here.

Figure 1: The electron distribution f and energy density of the waves W showing the simulation results from the 2 different simulations: Coulomb collisions only on the left, including wave-particle interactions on the right.) A movie of this can be found here

The immediate thing that happens is that the wave-particles very quickly flatten the low energy cutoff, producing a plateau in the electron distribution at low energies or velocities. The Coulomb collision alone are far slower at removing the low energy cutoff and produce the expected positive gradient in the electron distribution below the cutoff energy. To calculate the mean electron flux spectrum from our simulations we use the simulated f(v,x,t)/m, spatially integrating and averaging over time. The resulting spectra are shown in Figure 2.

Figure 2: The mean electron flux spectrum <nVF> for the simulation with Coulomb collisions actng on the beam only (left) and the inclusion of wave-particle interaction (right). The black line shows the simulation result, the orange dashed line an overplotted thermal model spectrum. The total spectrum is the dashed green line, indicating the presence of a dip in the Coulomb collision only case.

The positive slope increase in the Coulomb colision only case (left panel Figure 2) is clearly evident in the mean electron spectrum. As is the almost flat, though slighlty negative, spectrum at low energies when the wave-particle interactions are also considered. With the inclusion of a thermal model spectrum, using typical parameters for a small flare, we see the apearance of the local minima or "dip" in the beam only case but in the beam and waves case there is always a negative gradient.


Conclusions

The work shown here is a step towards a more complete treatment of electron transport in solar flares and highlights that the inclusion of wave-particle interactions flattens sharp low energy cutoffs in the inital accelerated electron distribution. There are still other processes that are not included here but it does demonstrate that the standard "thick-target" interpretation is insufficient to explain the RHESSI observations.

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