Return currents and soft-hard-soft spectral evolution
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|1st Author:||Valentina Zharkova|
|2nd Author:||Mykola Gordovskyy|
|Published:||24 January 2006|
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Solar hard X-ray spectra normally have a "broken power law," of the form A(h&nu)-&delta where &delta changes abruptly at a specific "break energy." RHESSI now gives us high-resolution views of such spectra, showing spectral indices at lower energies (<70 KeV) being smaller by 1.5-2.5 powers than those at higher energies. The "break energy" at the elbow separating these spectral parts varies in a range of 30-60 keV; it is lower for harder beams and higher for softer ones.
Another striking property of these flare hard X-ray spectra in the impulsive phase is the "soft-hard-soft" spectral evolution noted in previous nuggets   . In this pattern the softer spectra (larger &delta) systematically occur at lower flux levels, so that the hard X-ray maxima have the hardest spectra (smallest &delta) while the harder ones correspond to maxima in the hard X-ray emission. Such a pattern even appears in "loop-top" sources .
The theory we describe here explains these properties in a natural manner by considering the "return current" resulting from the electric field created by the charge imbalance of the original particle beam. We illustrate this theory with several movies as linked below.
Electron beam kinetics in Coulomb collisions and Ohmic losses
These variations of hard X-ray photon spectra can be explained by the kinetics of an electron beam originating in the corona, i.e. the "thick target" model. We do this by taking into account the effects of Coulomb collisions and Ohmic losses in the electric field induced by precipitating electron beams. This theory makes use of the time-dependent Fokker-Plank equation in a flaring atmosphere with the exponential variations of density and temperature.
Let us assume a model in which electrons are injected at the top of a loop (see the cartoon and Movie 1) and "precipitate" downwards in the solar atmosphere. The number of precipitating electrons can vary from 107 to 109 cm-3. While precipitating these electrons constitute a current, and thus must create a significant self-induced electric field that creates a return current for compensation. The return current consists of ambient electrons, plus whatever primary electrons that have scattered back into the upward direction (see Movie 2). By this means we have a full electric circuit of precipitating and returning electrons that keeps the whole system neutral as Ampere's law requires.
Self-induced electric field
The self-induced electric field E is associated with the number of precipitating electrons at a given depth that is dependent on the injected beam parameters as shown in the Figure 1, which shows the electric field variation with depth. Precipitating electrons with energies below about 100 keV strongly feel this electric field. In fact they lose much more energy to it than to the ordinary Coulomb collisions, the basic mechanism for stopping a fast electron in a plasma. In Table 1 we have tabulated the full stopping depths for electrons, with energies from 10 to 300 KeV losing their energy completely in collisions (first column) and various electric fields (the next 3 columns). It can be seen that electric field losses can dominate collisional ones at very high levels in the corona at column depths of 1018 cm-2 if the electric field is high enough (the last column in Table 1). Note that the electric field also can actually cause the reflection of the lowest-energy electrons, which can thus contribute to the return current themselves.
Figure 1: Variation of electric field with depth Figure 2/3: Differences of electron power-law indices as a function of X-ray spectral index &delta (upper), and beam flux (lower). The lower-energy electrons lose much more energy in the self-induced field and collisions than in the case of collisions only, and this results in their numbers being significantly reduced. This, in turn, leads to a harder X-ray photon spectrum, i.e. a flattening at lower energies. This could be the origin of the broken power law observed in the photon spectra, especially well by RHESSI. Movie 3 and Movie 4 make this clear. The higher the beam energy flux F and the higher its electron spectral index &gamma, the bigger is the difference between the spectral indices in the broken power law (see Figure 2 and Figure 3). Table 2 compares the photon spectral indices for different initial fluxes F and single spectral indices &gamma (not &delta, which refers to the X-ray photons here) of the primary electron beam.
Explaining the Soft-Hard-Soft pattern
Armed with this physics, we can explain the dependence of the photon spectral index on the electron initial fluxes and thus the so-called the "soft-hard-soft" pattern described in the earlier RHESSI science nuggets   . Movies 3 and 4 illustrate these variations.
At flare onset, the initial fluxes are smaller, so are the self-induced electric fields induced by the injected electrons. As a result, the photon spectra have the indices close to what would be expected for a very weak beam, ie dominated by ordinary collisional interactions. At flare maximum, the energy fluxes of electron beams are much stronger and induce much higher electric fields. This leads to a strong flattening of their photon spectra as discussed above. After the flare maximum the beam fluxes decrease again, restoring the low-energy spectral slope; see Movie 3.
We note also that because of the strong effect of deceleration of beam electrons by the self-induced electric field as seen from comparing the stopping depths in Table 1, this can explain the SHS pattern not only in the footpoints but also in the loop-top sources (). where particles are accelerated; it is an automatic consequence if the loop-top region has sufficient density to emit bremsstrahlung hard X-rays.
Biographical note: Valentina Zharkova and Mykola Gordovskyy are theoretical physicists at the University of Bradford (UK).