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Appendix A: Is there a way to quantify the degree of departure of plasma from the Maxwellian distribution?

The diagnostic technique described in Sect. 4.2 provides a method to measure the κ of the distribution that governs the electrons in the plasma. In this appendix, we consider the case where the observed plasma is mostly Maxwellian. We demonstrate how the degree of departure from the purely Maxwellian plasma can then be quantified. To do this, we apply the diagnostic technique from Fig. 7 to the following case study.

We assume that the observed emission along the line-of-sight is originating in two emitting volumes. One of these volumes is characterized by a Maxwellian distribution, temperature T and the emission measure EM(Maxw). The other volume is characterized by a κ-distribution with a given κ, the same T, and a different EM(κ). The emerging continuum signal is then given by the sum of the contributions of the free-free and free-bound continua from these two volumes.

Fig. A.1 Determination of the apparent κ from the combination of a Maxwellian plasma and a plasma with a κ-distribution with known κ. The known κ is labeled in the title of each image in the panel. Black lines and their respective coding are the same as in Fig. 7. Colored lines denote the dependence of the relative height of the ionization edges on the emission measure ratio EM(Maxw)/EM(κ) = 10-2 (red), 10-1 (orange), 1 (green) and 10 (blue). Asterisks denote the points corresponding to log(T/K) = 6.8, 7.0, and 7.2. Except for the top left panel for κ = 2, these points are always close to the respective isotherms (gray lines).

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Then, using the diagnostic plots in Fig. 7 to determine κ leads, except at very low T, to higher apparent values of κ depending on the ratio EM(Maxw)/EM(κ) (Fig. A.1). For a plasma that is mostly Maxwellian, i.e., EM(Maxw)/EM(κ) ≪  1, the diagnostics leads to almost Maxwellian plasma. That is, using the Si xiii and Si xiv edges for a combination of Maxwellian plasma and plasma with κ = 2, the height of the edges is consistent with Maxwellian plasma if EM(Maxw)/EM(κ) = 10^-2 (red line in Fig. A.1top left). However, the apparent value of κ can be equal to  ≈10 if the EM(Maxw)/EM(κ) is ten times higher, i.e., equal to 10^-1 (orange line in Fig. A.1top left).

On the other hand, if the emission measure of the plasma with the κ-distribution is much greater than the EM(Maxw), then the diagnosed value of κ approaches the true value (Fig. A.1).

We can thus conclude that if the plasma has the same T, or at most a narrow DEM, any increase in the height of the ionization edge must be the consequence of the presence of κ-distributions.

We note here that the presented case study is fairly simple, because we assumed that the value of T is the same for the Maxwellian plasma and the plasma with a κ-distribution. Of course, the scenario can be complicated in many ways (e.g., diffferent T, or a multi-strand plasma characterized with a DEM and varying κ). In those cases, the diagnostics using only two ionization edges is clearly inappropriate and the observer would have to carefully check for the signatures of plasma multithermality, e.g., appearance of ionization edges or lines formed at higher or lower T.

We also note that if the Maxwellian and κ-distributed plasmas can be spatially resolved, the plot in Fig. 7 can be simply used for each spatial pixel separately and should lead to the correct results.

The same analysis as presented here could be also repeated for a combination of Maxwellian plasma and a plasma characterized with an n-distribution. However, then the ionization edges would be decreased with respect to the Maxwellian distribution, and therefore could prove to be much more difficult to be observed. For this reason, we chose to demonstrate the diagnostics using the κ-distributions.

Finally, we note that using RHESSI observations of the flare plasma, Kulinová et al. (2011) were able to resolve the contribution of the bresstrahlung produced by plasma with an n-distribution from the thermal component. This was possible because the bremsstrahlung from n-distributions manifests itself at energies of  ≳4   keV, while the thermal component peaks at  ≈7 keV. However, the closeness of these two components leads to large errors in determining n from RHESSI observations (Table A.7 in Kulinová et al. 2011).

© ESO, 2012