Spectral Evolution in Stochastic Acceleration Models

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Number: 31
1st Author: Paolo Grigis
2nd Author:
Published: 17 January 2006
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RHESSI observes hard X-rays and γ-rays from highly energetic particles that suddenly appear in and around solar flares. One should probably think of these particles as somewhat analogous to cosmic rays, and they have a very different distribution in energy from the Maxwellian distribution of the medium in which they appear. What accelerates these particles to such high energies, and why does a solar flare devote so much energy to this acceleration process?

There are several candidate theories but no consensus choice yet. Stochastic acceleration is a mechanism by which particles exchange energy with moving inhomogeneities of the electromagnetic fields in a plasma, such as waves. During this process, electrons can either lose or gain energy and therefore they follow a random walk in energy. This can be mathematically described by a diffusion equation.

Different scenarios for stochastic acceleration in solar flares have been presented. They differ in the choice of the properties of the waves responsible for the acceleration and the characterization of the particle energy distribution (spatially homogeneous, isotropic waves and particles being simplest). However, they all produce qualitatively similar results. In particular stochastic acceleration models can accelerate both electrons and ions, and the acceleration can be appropriately fast.

Particle spectra

In a stochastic accelerator, the acceleration proceeds as long as the wave energy is present, and the particle spectrum can become very hard. However the escape ot the particle from this voume may help to control the spectrum. Therefore we have added an escape term to the basic theory in order to see what effects would result. We compare the model results with observation as described by M. Battaglia earlier RHESSI nugget, focusing on the behavior of coronal sources, which in this model would correspond to the acceleration region.

The photon spectrum emitted by the accelerated electrons depends on the physical parameters of the accelerator (density, temperature, etc.) and on two main model parameters: IACC and τ. The acceleration parameter IACC controls the strength of the acceleration (and actually depends on the energy density in the turbulent waves and on their average wave vector). The escape time τ controls the escape rate of the accelerated particle: large τ correspond to better trapping (less escape). In first approximation, the spectra only depend on the product IACC • τ.

The acceleration of electrons out of the thermal distribution is shown in Movie 1 (where the Maxwellian distribution is shown in pink and the accelerated electrons in blue). The equilibrium spectra are plotted for different values of IACC • τ in Figure 1.

Figure 1: Equilibrium electron distribution function for different values of IACC • τ. The harder spectra correspond to a regime of stronger acceleration and better trapping, the softer spectra are the result of weak acceleration and strong escape losses. The soft-hard-soft behaviour is evident: harder spectra have a larger flux ([1], [2]). Observationally, this corresponds to a "pivot point" in the spectrum, which we now suggest an explanation for.

Pivot-point basics

Figure 1 explains qualitatively the presence of the soft-hard-soft effect, but can the model be compared to the observation in a more quantitative way? This can be done by checking whether the model predict the presence of a pivot point in the photon spectra. A pivot point is a common intersection point of the nonthermal part of the spectra measured at different times. Figure 2 (left) shows 7 power-law spectra crossing in a pivot point at 15 keV. The plot (right) of the spectral index vs. the logarithm of the flux measured at a fixed energy (say, 35 keV) is linear.

Figure 2: spectra crossing at a pivot point located at energy 15 keV (left panel). The plot of the spectral index vs. the logarithm of the flux at 35 keV is linear (right panel). Therefore it is possible to check for the presence of a pivot point in both observations and model results by looking at a plot showing the spectral indices vs. the logarithm of the flux measured at some fixed energy (like the right panel in Fig. 2): if a linear relation between the two parameters is found, then a pivot point is present, and its coordinates can be easily found from the slope and intercept of the straight line.

Model results

The model results for electrons and photons are shown in Fig. 3.

Figure 3: Model results showing the spectral index versus flux for electrons and photons. The deviation from the (dashed) best-fit straight line is small. The pivot-point energy for the photon spectra shown in Fig. 3 is 5 keV. For other values of the ambient parameters, the pivot-point energies lie in the 5-15 keV range. These are about a factor 2 lower than the values inferred from the RHESSI observations. Is there some way to modify the model to increase the energy of the pivot point? One possibility is to strongly increase trapping below 30 keV. Physically, this can be accomplished with a potential barrier around the acceleration region which does not let the low-energy particles escape. In this case the model pivot-point energy rises up to the observed values, but at the cost of an unobserved hardening of the photon spectrum below 25 keV.


Our stochastic acceleration model, endowed with a suitable escape term, reproduces the observed soft-hard-soft behaviour of flare looptop sources. Furthermore, the model predicts the presence of a pivot-point in the spectral evolution. This successes are however temperated by the fact that for reasonable values of the ambient plasma physical parameters, the predicted pivot-point energy is too low by a factor of about 2. Considering the strong simplifications that went into the model, this does not seem too bad. Nevertheless it is clear that more work is needed to unravel the intricacies of particle acceleration in solar flares.

Biographical note: Paolo Grigis is working on his PhD thesis at ETH Zurich in the group of Arnold Benz.

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