The Rise and Fall of The Low Energy Cut Off

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Number: 89
1st Author: Ewan Dickson
2nd Author: Eduard Kontar
Published: 17 November 2008
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A typical spatially integrated electron spectrum of a solar flare is often viewed as a combination of thermal and non-thermal components: the former follows the Maxwell distribution and the latter is a power-law or broken power-law. Although power-law fits (N ~ E-a) are frequently used to describe particle spectra, we note that a pure power-law spectrum diverges for small energies. This can make the total number of solar flare electrons infinite. To combat this artifact a low energy cutoff is often imposed at some (often arbitrary) value. In the X-ray continuum spectrum of a solar flare, the soft (thermal) and hard (nonthermal) X-ray spectra apparently merge together seamlessly, so this addition has always been somewhat controversial. The ambient plasma has a finite temperature, typically of order 107 K in a flare, and the non-thermal spectrum should merge with this background Maxwellian at a few times kT. Because "non-thermal" electrons below a few kT would actually gain energy from the thermal background, a recent paper by Gordon Emslie has declared "death to the low energy cutoff!" (Note that the journal's editor rejected the use of mortality in the actual title of that paper).

While Emslie's suggested scenario declares the death of an arbitrary value of low energy cutoff, the nails into its coffin seemingly have been pulled by RHESSI's spectroscopic observations at high resolution. In practice, the RHESSI spectra often require the introduction of a low energy cutoff well above the value of a few kT that this theory expects. More sophisticated techniques based on regularization theory have led to the same puzzling conclusion: the observations appear to require that the electron spectrum have a dip, something like a cutoff. How can the observations of dips in the inverted electron spectra be explained?


The Compton scattering of X-rays in the lower solar atmosphere, below the emission source, will distort the X-ray spectrum of a flare. This Compton backscattering makes its strongest contributions in the range 30 - 100 keV and for flares near disk centre. The addition of a scattered component thus causes a flattening of the X-ray spectrum at low energies. This effect could then lead to electron spectra appearing to require a cutoff. To determine the electron flux regularised inversion was used, the advantage of this technique being that it is model-independent. The more traditional "forward fitting" method uses a previously specified parameterized function (usually containing a thermal component plus a power law) has the disadvantage that it can easily miss unexpected, but real, features such as a dip. It is thus explicitly model-dependent.

The technique of applying a correction for albedo and then using inversion to determine the electron spectrum of flares which appeared to require a low energy cut off was first performed in 2005 (see here and here for details). It was found that the correction did remove the dip from the electron spectrum for a few events.

How common is the low energy cutoff?

To determine whether the removal of the dip is just an effect which occurs in the particular flares examined or whether it is a more general systematic effect, we have now carried out a statistical survey of flares in the RHESSI catalogue. Although for most flares the thermal component completely dominates at low energies and so would hide any genuine cutoff, there are flares with weak thermal components. These are exactly the flares which suggest that a low energy cut off is necessary. Their power-law spectra ten to have large values of γ, the photon spectral index fitted only in the range 15 to 20 keV.

We selected a total of 177 flares (see here for details). For each of these flares the electron spectra was calculated and examined for unusual features such as the presence of a dip. This survey turned up 18 cases of clear dips. Figure 1 shows an example of such event. After application of a correction for the albedo, the physics of which is well-understood, none of the electron spectra showed a dip (as also shown for the example in Figure 1).

Figure 1: Plot of electron spectra for the 1 April 2004 23:00 UTC solar flare. The blue line denotes the uncorrected electron spectrum and the green line denotes the electron spectrum after correction for albedo. Both lines are plotted with 1σ error bars. The dip depth, d, is shown as the difference between the minimum and the following maximum.

The dips seem to occur in a fairly narrow energy range with minima of between 13 and 19 keV. Theoretically, the heliographic location of the flare affects the magnitude of the albedo correction, so we sorted our flare list to confirm the presence of this property. Figure 2a shows the dependence on heliocentric location quite clearly (note that cosine values of 0 correspond to the limb, and 1 to disk center). The flares showing dips are far more likely to occur near the solar centre (μ near 1).

Figure 2a:Histogram of the 18 events with a significant dip as a function of μ, the cosine of the heliocentric angle of the flare.
Figure 2b:Histogram showing the percentage of flares with a dip as a function of spectral index γ
Figure 2

This result is consistent with the theory that the flattened spectra, and hence the dips, are caused by the influence of the albedo. Of the flares with dips, over 70 % occurred at points with a μ>0.5.As justification for the choice of flares studied, the number of flares is calculated as a percentage of total flares studied for given ranges in γ0 (Figure 2b). As expected, dips are far more likely to occur for flares with a low value of γ0.


The small number of flares found with dips (18 out of 177) suggests that the vast majority of flare spectra are not so flat that they require a low energy cutoff. Of the flares found with flat spectra and spectral indices between 1.5 and 2, over 60% show a dip (Figure 2). In all of the flares studied the dip appears to be consistent with the albedo model. Observations which had previously required a low energy cutoff can be also understood in terms of the spectrum being the result of albedo distortion. Most of the flattened spectra are found in events where the flare occurs close to the disk centre, and therefore where the albedo has the greatest effect. Even for flares near the solar limb the correction for albedo still alters the spectrum and can remove the need for a dip in the spatially integrated electron spectrum. The results also suggest that if there is a low energy cutoff it is buried deeper in the thermal component and therefore has a value below ~12 keV.

As even isotropic emission correction for albedo can satisfactorily explain the flattening of spectra, and dips in the electron spectra, it seems possible that we have pounded the nails back into the coffin, and right now the low-energy cutoff seems deader than ever.

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