ELECTRON DISTRIBUTION FUNCTIONS IN SOLAR FLARES FROM COMBINED X-RAY AND EXTREME-ULTRAVIOLET OBSERVATIONS

Solar flares accelerate large numbers of particles within seconds and maintain efficient acceleration over timescales of minutes to tens of minutes. Hard X-ray (HXR) observations of solar flares provide information on the electron acceleration and transport in solar flares and are crucial to better constrain various flare models (e.g., Holman et al. 2011, as a recent review). Observations with RHESSI (Lin et al. 2002) have significantly enriched our understanding of solar flare physics by providing accurate spatially resolved soft X-ray (SXR) and HXR spectra, which are used as the input to deduce the properties of the accelerated electrons (e.g., Kontar et al. 2011, as a recent review). RHESSI also makes it possible to analyze individual source spectra (coronal sources, footpoints), by means of imaging spectroscopy (e.g., Emslie et al. 2003; Battaglia & Benz 2006). The technique was recently used by Simões & Kontar (2013) to infer the characteristic electron acceleration rates at the looptop and at the footpoints of a number of flares, finding strong indication for particle trapping at the top of flaring loops. However, RHESSI detectors only provide X-ray observations above 3 keV, or even higher energies during large flares when attenuators are placed in front of the detectors to reduce dead-time, which results in much reduced flux at energies below ∼6–12 keV (Smith et al. 2002). While priceless to infer the properties of deka-keV electrons, it leaves an observational gap for electrons below 3 keV. Due to the typical shape of flare spectra, particles at these energies constitute the larger part of the total particle number in flares. It is important to extend the diagnostics to energies below 3 keV to answer questions such as whether the distribution at these energies is an iso-thermal Maxwellian. Unlike the higher X-ray energies, hot flaring plasma is normally studied using EUV and X-ray line emissions. Observations of the line intensities provide information on the temperature distribution of the solar plasma. With the recent launch of the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO; Lemen et al. 2012), it is now possible to deduce differential emission measures (DEMs) over a broad range of coronal temperatures with high spatial resolution. Complementing SXR and HXR observations with EUV observations allows the mean electron flux spectrum to be deduced over a wide range of energies from as low as 0.1 keV up to tens of keV.

In this paper we develop a method to infer the low-energy part of the electron flux spectrum from DEMs and apply the method to spatially resolved AIA observations. We first demonstrate the method on a set of synthetic DEMs for which the electron distribution function is known and then apply it to flare observations. Using simultaneous X-ray and EUV observations, we deduce, for the first time, the mean electron flux spectrum and estimate the local electron distribution function in various regions of the flaring atmosphere. The observations show that locally non-thermal electrons constitute a relatively modest fraction of the order of 10⁻² of the total electron numbers. Different flaring regions demonstrate distinct electron distributions: footpoint regions can be viewed consistent with a Maxwellian distribution and a non-thermal tail, while an above-the-looptop source deviates from a Maxwellian in the range above 0.5 keV.

Using line observations, the properties of thermal plasma are traditionally characterized by the DEM ξ(T) (cm⁻³ K⁻¹). Assuming a locally Maxwellian distribution with local temperature T(r) and density n(r) at spatial coordinate r, ξ(T) characterizes the amount of plasma at various plasma temperatures T(r), so that the integral over T gives the total EM EM = ∫ξ(T)dT. The Maxwellian distribution at temperature T(r) is given as

$$\begin{equation} F(E,r)=\frac{2^{3/2}}{(\pi m_e)^{1/2}}\frac{n(r)E}{(k_BT(r))^{3/2}}\exp {(-E/k_BT(r))}, \end{equation} \tag{ 1 }$$

where E is the electron kinetic energy, m_e is the electron mass, and k_b is the Boltzmann constant. This is related to the mean electron flux spectrum 〈nVF〉 in the emitting volume V and the DEM ξ(T) following, e.g., Brown & Emslie (1988):

$$\begin{eqnarray} \langle nVF\rangle &=&\int _Vn(r)F(E,r) \,dV \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &=&\int _Tn(r)\frac{2^{3/2}}{(\pi m_e)^{1/2}}\frac{n(r)E}{(k_BT(r))^{3/2}}\exp {(-E/k_BT(r))}\frac{dV}{dT}\,dT,\nonumber\\ \end{eqnarray} \tag{ 3 }$$

where n²dV/dT = ξ(T) is the DEM and thus

$$\begin{equation} \langle nVF\rangle =\frac{2^{3/2}E}{(\pi m_e)^{1/2}}\int _0^{\infty }\frac{\xi (T)}{(k_BT)^{3/2}}\exp {(-E/k_BT)}\,{d}T. \end{equation} \tag{ 4 }$$

Therefore, knowing the DEM, one can compute the electron flux spectrum in the emitting volume (electrons keV⁻¹ s⁻¹ cm⁻²). Although Equation (4), which is an equivalent of the Laplace transform of a function f(t)

$$\begin{equation} F(s)=\int _0^\infty \exp (-st)f(t)\,{d}t, \end{equation} \tag{ 5 }$$

is formally a straightforward integration over temperature, the numerical integration could be rather challenging due to the exponential kernel (e.g., Prato et al. 2006). Following Rossberg (2008), we rewrite the Laplace transform (Equation (5)) via the convolution integral, which will allow efficient numerical computations of 〈nVF〉 via ξ(T) and vice versa. Using the change of variables s = exp (y) and t = exp (− x), let us rewrite Equation (5) in the following form:

$$\begin{equation} F(e^y)=\int _{-\infty }^{\infty }K(y-x)h(x)\,{d}x, \end{equation} \tag{ 6 }$$

where K(y − x) = exp (y − x)exp [ − exp (y − x)] and h(x) = ϕ(e^−x) with $\phi (t)=\int _0^t f(t^{\prime }){d}t^{\prime }$. Equation (4) can be similarly brought into the form of Equation (5) using the variable change

$$\begin{equation} t=1/T \, ; \frac{dt}{dT}=-\frac{1}{T^2} \, ; dT=-\frac{1}{t^2}\,dt, \end{equation} \tag{ 7 }$$

which results in

$$\begin{equation} \langle nVF\rangle =\frac{2^{3/2}E}{(\pi m_e)^{1/2}k_B^{3/2}}\int _{0}^{\infty }\frac{\xi (T(t))}{t^{1/2}}\exp {(-Et/k_B)}\,\,{d}t, \end{equation} \tag{ 8 }$$

so that f(t) = (ξ(T(t))/t^1/2) and exp (− st) = exp (− Et/k_B) in Equation (5), which is then brought into the form of Equation (6) and solved.

2.1. Application on Synthetic DEM

We illustrate the method using two synthetic DEMs. The first is a single-temperature DEM, i.e., a δ-function in temperature space (Figure 1, top) at temperature T₀ = 5 MK. The mean electron flux spectrum corresponding to this DEM is calculated using Equations (4)–(8). The result is shown in the middle panel of Figure 1. This is compared to the Maxwellian distribution as defined in Equation (1). From Equation (2) one finds the mean electron flux spectrum for a uniform distribution over the whole volume as

$$\begin{equation} \langle nVF\rangle =n_e^2V\left(\frac{2}{k_BT}\right)^{3/2}\frac{E}{(\pi m_e)^{1/2}}\exp {(-E/k_BT)}, \end{equation} \tag{ 9 }$$

where $n_e^2V={\rm EM}$ is the total emission EM. As Figure 1 indicates, there is a very close agreement (better than 2% except for temperatures above ∼2 × 10⁷ K, where numerical effects become noticeable) between the analytical solution (Equation (9)) and the mean electron flux spectrum found by integrating the DEM (Equation (8)), validating the method. The second synthetic DEM for which the method is demonstrated is a Gaussian DEM function in log T:

$$\begin{equation} \xi (T)=\frac{\xi _0}{\sqrt{2\pi }\sigma ^2}\exp (-(\log T-\log T_0)^2/(2\sigma ^2)), \end{equation} \tag{ 10 }$$

with a peak temperature of T₀ = 5 MK, width σ = 0.1, and ξ₀ = 10³⁶ cm⁻³ K⁻¹, where ξ₀ was chosen such that the total EM is one order of magnitude larger than in the first case to simplify the presentation in Figure 1. We define the total EM as

$$\begin{equation} {\rm EM}=\int _{T_{{\rm min}}}^{T_{{\rm max}}} \xi (T)dT, \end{equation} \tag{ 11 }$$

where T_min = 0.01 MK and T_max = 100 MK in the synthetic case. We again compare the electron flux spectrum inferred from the DEM with a single-temperature Maxwellian. Since a Gaussian DEM function does not represent a single-temperature distribution, some discrepancy, especially at higher energies, can be expected. As Figure 1 demonstrates, the inferred mean electron flux spectrum is still in good agreement with the single-temperature model at low energies, but diverges at higher energies, due to the contribution of non-zero DEM at temperatures higher than the peak temperature that was used for the Maxwellian model. This interpretation is supported by the fact that adding an additional Maxwellian temperature model with temperature 9 MK and emission measure 0.09 times the total EM of the first DEM model adequately describes the high-energy component and leads to a better agreement except for the highest temperatures (see Figure 1, blue dashed lines).

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We now apply the method to two events for which there were good simultaneous RHESSI and AIA observations. The first event was a GOES class C4.1 event on 2010 August 14 (henceforth SOL2010-08-14T10:05) with HXR peak time around 09:46 UT. It displays an EUV loop that is co-spatial with an SXR loop seen in RHESSI images. Because the event was relatively weak, the regions of interest were not saturated in all AIA wavelength channels (see Battaglia & Kontar 2012). The second event was a GOES M7.7 limb event on 2012 July 19 (henceforth SOL2012-07-19T05:58) with the first HXR peak at around 05:22 UT, showing two footpoints, an SXR coronal source, and an HXR above-the-looptop source. In this event the exposure times in all AIA wavelengths were short enough to provide unsaturated images even during the flare peak. For both events we chose the time interval with highest observed X-ray energies within the limitations given by both instruments such as saturation of AIA images and pileup in RHESSI spectra. For both events the mean electron distribution function at energies below ∼1 keV is calculated using AIA DEM measurements. RHESSI observations are used to find the distribution function above 3 keV up to the highest observed X-ray energies (∼25 keV for SOL2010-08-14T10:05 and 80 keV for SOL2012-07-19T05:58).

3.1. SOL2010-08-14T10:05

For this flare we chose a time interval during the rise phase of the flare when AIA images were not saturated, and a small non-thermal tail was observed with RHESSI. Panel (a) in Figure 2 shows a 131 Å AIA image overlaid with 20%, 30%, and 50% contours from a RHESSI 8–10 keV CLEAN image, calculated using grids 3–8 and natural weighting with clean-beam-width-factor 1.7 (corresponding to an effective beam FWHM of 9.1 arcsec). The RHESSI source is co-spatial with the top of the EUV loop and extending downward along the legs of the EUV loop. The bulk of the EUV emission originates from ribbons at the bottom of the loop, and AIA images in these regions consequently are saturated in most wavelength bands. Here we focus on the part of the loop that is co-spatial with the RHESSI source where no saturation occurred and the regularized inversion to find the DEM (Kontar et al. 2005) could be made with high confidence.

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3.1.1. RHESSI Spectrum and Electron Flux Distribution

The RHESSI full Sun spectrum suggests the presence of a thermal component and a small non-thermal tail extending to about 30 keV. The spectrum was fitted between 09:42 and 09:43 UT (attenuator state 0), during the rise phase of the flare but before pileup started to dominate energies above 10 keV (Smith et al. 2002), with a single-temperature component of 10.5 MK temperature and emission measure of 3 × 10⁴⁷ cm⁻³, and a thin-target power-law component with spectral index δ = 3.9 and low-energy cutoff of 10 keV. In addition to the RHESSI temperature, the temperature and emission measure from GOES were determined for the same time interval, giving T_GOES = 9.5 MK and EM_GOES = 5 × 10⁴⁷ cm⁻³. RHESSI images suggest that there is faint footpoint emission above ∼25 keV, but the bulk of the non-thermal emission below this energy seems to originate from the flaring loop. The mean electron flux spectrum for the thermal component is then just the Maxwellian distribution (Equation (9)). The electron flux spectrum for the non-thermal power-law component is easily found under the thin-target assumption (e.g., Tandberg-Hanssen & Emslie 1988).

3.1.2. Mean Electron Flux Spectrum from AIA Differential Emission Measure

The AIA DEM per area (cm⁻⁵ K⁻¹) was calculated using the regularized inversion method developed for RHESSI by, e.g., Kontar et al. (2005) and adapted for SDO/AIA by Hannah & Kontar (2012). Such a DEM can be calculated pixel by pixel or for any finite area (e.g., Hannah & Kontar 2013; Battaglia & Kontar 2012). For direct comparison with RHESSI results the EUV region that is co-spatial with the RHESSI emission was analyzed under the assumption that the same emitting plasma is observed in all wavelengths. RHESSI full-Sun spectra are dominated by the flaring emission, and it is often assumed that the bulk of the emission originates from a region that corresponds to the size of the 50% contours in a RHESSI image. However, AIA images clearly outline the whole loop, as do the RHESSI contours down to 20%. We therefore calculate the total AIA DEM from several areas, namely, the ones corresponding to 50%, 30%, and 20% contours in the RHESSI 8–10 keV CLEAN image, and use the result from the 50% and the 20% contours as a confidence interval. The DEM from within the 50% contour is shown in panel (b) of Figure 2. The error bars represent the uncertainties of both the DEM and the temperature, i.e., the effective temperature resolution, obtained from the regularized inversion (see Hannah & Kontar 2012, for full details). The DEM suggests the presence of two main temperature components, a weak one at 2 MK and one at 10 MK. The low-temperature component can most likely be attributed to background emission, while the high-temperature component is dominated by flaring emission (see also Battaglia & Kontar 2012). Note that in this case we did not impose a positivity constraint on the reconstructed DEM (see the Appendix) because the assumption of a positive DEM is quite strong and only correct in the case of purely thermal plasma. From the DEM, the mean electron flux spectrum is calculated using the method described in Section 2. Figure 2 shows the mean electron flux spectrum in units of (electrons cm⁻² keV⁻¹ s⁻¹) as a function of energy from the combined AIA and RHESSI observations, where we use the result from the 50% contours and 20% contours as a confidence interval. Dividing by energy and multiplying with $m_e^2$, we can also display the spectrum as a velocity distribution function 〈nVf(v)〉 (Figure 2, panel (b)). The distribution found from AIA is consistent with a Maxwellian of temperature T = 6 MK and emission measure EM = 4.5 × 10⁴⁶ cm⁻³, but deviates from the Maxwellian distribution at energies greater than 1 keV. The extrapolation of the RHESSI thermal distribution into the AIA regime is a factor ∼3 larger than the distribution from AIA. We discuss several reasons for this discrepancy in Section 4. The overall distribution over all energies resembles particle distributions often found in the solar wind with a core-halo-strahl structure (see Marsch 2006, for a review).

3.2. SOL2012-07-19T05:58

For this limb event three distinct sources were observed with RHESSI (SXR coronal source, HXR above-the-looptop source, HXR footpoints; see Figure 3). The event has been analyzed in detail by Liu et al. (2013) with respect to several aspects of its time evolution and with a focus on the coronal densities by Krucker & Battaglia (2013). AIA exposure times where as short as 0.2 s during the course of the flare. Thus, there are unsaturated images in all wavelength channels even at the flare peak time. Here we focus on the same time interval (05:20:30 to 05:23:02 UT, attenuator state 1) used by Krucker & Battaglia (2013), who analyzed the first HXR peak using imaging spectroscopy, and we present mean electron distribution functions for three different sources observed by RHESSI: the SXR coronal source, the HXR above-the-looptop source, and the northern footpoint. A weak second footpoint that was likely occulted was also observed. STEREO images of the region suggest the presence of loops or a loop arcade for which, as seen from Earth, the southern footpoint would be occulted. Note that the northern footpoint was also likely partly occulted. Footpoint sources at higher energies are formed deeper down in the chromosphere and have a vertical extent of at least 1 Mm (Battaglia & Kontar 2011; Kontar et al. 2010). Thus, the higher energies will be more occulted relative to lower energies, resulting in a softer HXR spectrum and lower observed flux. Therefore, the inferred electron flux spectrum from an occulted footpoint will represent a lower limit.

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3.2.1. RHESSI Spectrum and Electron Flux Distribution

Using full-Sun spectroscopy for the SXR coronal source and fitting a single-temperature thermal model, Krucker & Battaglia (2013) found T = 23 MK, EM = 4 × 10⁴⁸ cm⁻³. The GOES temperature and emission measure during the corresponding time interval were T_GOES = 15.5 MK, EM_GOES = 10⁴⁹ cm⁻³. Imaging spectroscopy of the above-the-looptop HXR source resulted in an electron spectral index δ = 3.2 ± 0.2 between 28 and 50 keV, and the spectral index of the northern footpoint was δ = 2.0 ± 0.2 between 20 and 70 keV, where a thin-target assumption was used in both cases. In the intermediate energy range between ∼14 and 20 keV the coronal source and the footpoints are seen in the images. Such a complex source structure makes image reconstruction with RHESSI rather difficult, and it is often problematic to distinguish reconstruction noise from actual source emission (Hurford et al. 2002; Battaglia & Benz 2006). We therefore only estimated an upper limit of the electron flux in this energy range by fitting a thermal (Maxwellian) component to the X-ray spectrum (compare Figure 3).

3.2.2. Mean Electron Flux Spectrum from AIA Differential Emission Measure

The AIA DEM and subsequently the mean electron flux spectrum were calculated within the three regions as defined by contours from a RHESSI CLEAN image at 5–7 keV using grids 3, 5, 6 with uniform weighting (giving an effective CLEAN beam FWHM of 8.7 arcsec) in the case of the coronal source. For the HXR sources two-step CLEAN (Krucker et al. 2011) images at 30–70 keV were made using grids 1–5 (uniform weighting, effective beam FWHM = 3 arcsec) to image the footpoints and grids 4–9 (uniform weighting, effective beam FWHM = 17 arcsec) to image the above-the-looptop source. For the HXR footpoints and SXR coronal source the 30%, 50%, and 70% contours were used, and we use the 30% and 70% contours as a confidence interval. The coronal HXR source showed extended emission over a large area, and we use the 50% and 70% contours to give a confidence interval, since the 30% contours overlap with the coronal SXR source.

We find that the mean electron flux spectrum derived from the AIA data for the coronal SXR source at energies between 0.1 and 2 keV can be approximated with a Maxwellian distribution of temperature 18 MK and emission measure 10⁴⁸ cm⁻³. As in the previous event, the extrapolation of the RHESSI spectrum to energies below 3 keV is a factor of ∼2 larger than the AIA values. The electron flux spectrum derived from AIA for the HXR footpoint is consistent with a Maxwellian at temperature T = 10 MK and emission measure EM = 10⁴⁷ cm⁻³. The spectrum of the HXR above-the-looptop source is consistent with a Maxwellian at temperature 5 MK and emission measure 5.5 × 10⁴⁶ cm⁻³ only up to about 0.6 keV and deviates from the Maxwellian distribution at higher energies. For a direct comparison of the number of electrons between the distinct regions it is useful to compare the mean electron spectra per unit volume. We define the volume as V = A^3/2, where A is the area of the 50% RHESSI contours of the SXR coronal source and the HXR above-the-looptop source. For the footpoint the effective area over which EUV emission is observed is smaller than the 50% contour by about a factor of three and the EUV emission originates from a height above the centroid of the HXR emission (Figure 3, panel (a)). Thus, we define the area of EUV emission from the footpoint region as the area of the 50% HXR contour divided by 3. Figure 4 shows a comparison between the mean electron flux spectra per unit volume from each of the sources. The figure indicates that the number of electrons per unit volume is highest in the footpoint (a factor of six larger than the SXR coronal source) and is lowest in the HXR above-the-looptop source (one order of magnitude smaller than in the SXR coronal source).

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We present and demonstrate a new method to infer the mean electron flux spectrum from DEM results derived from AIA images in different EUV wavelength channels. The method was applied to two well-observed flares. The first event displays an EUV loop that is co-spatial with an SXR loop observed by RHESSI, and the mean electron flux spectrum was derived for the loop as a whole. The second event can be divided into HXR footpoints, SXR coronal source, and HXR above-the-looptop source. Each of the three sources was analyzed separately. In all presented cases the result from AIA observations is a factor of 2.5–8 lower than the extrapolated thermal model from the RHESSI fit. There are several possible explanations for this discrepancy: the RHESSI fit, insensitivity of AIA to high temperatures, the method to infer the DEM, and two physically distinct particle populations. The RHESSI data-analysis tools are developed to a high standard, and it is believed that the instrument is well understood. In combination with good counting statistics in flares, this generally leads to rather small uncertainties of the fitted parameters (e.g., Ireland et al. 2013). However, for weak flares and when using imaging spectroscopy, it is often possible to fit different models (i.e., different temperature and emission measure) with equal χ² value. Further, the attenuator state affects the low-energy limit to which a confident fit can be made (Smith et al. 2002). This it not an issue for SOL2010-08-14T10:05, since no attenuator was in place, but with the thin attenuator in place during SOL2012-07-19T05:58 the spectral fit could only be performed down to 6 keV. In addition, it has recently been found that the thermal blanket thickness of the RHESSI instrument could be overestimated by up to 30% (B. Dennis 2013, private communication). This affects thermal fits in attenuator state 0 and in the case of SOL2010-08-14T10:05 could explain the discrepancy at least partly. All of these factors introduce an additional uncertainty not reflected in the purely statistical errors. We therefore assume an upper limit on the uncertainty of the fitted emission measure of a factor of two. This could bring the observed RHESSI emission measure down within the range of the AIA value. Comparison with GOES gives a slightly higher emission measure (5 × 10⁴⁷ cm⁻³ opposed to 3 × 10⁴⁷ cm⁻³) and slightly lower temperature (9.5 MK opposed to 10.5 MK) for SOL2010-08-14T10:05. The difference is more striking for SOL2012-07-19T05:58 (10⁴⁹ cm⁻³ opposed to 4 × 10⁴⁸ cm⁻³ and 15.5 MK opposed to 23 MK). The tendency of RHESSI emission measures to be smaller than those derived from GOES observations is a common pattern observed repeatedly in the past (e.g., Hannah et al. 2008; Battaglia et al. 2005). One explanation is that this reflects RHESSI's limited sensitivity to temperatures below ∼8 MK, temperatures to which GOES is sensitive. However, it has to be noted that GOES observations are full-Sun measurements with no way of knowing the exact position and extent of the source. In Figures 2 and 3 we assumed the same area as measured with RHESSI, but this is only an approximation. In summary, the fitted RHESSI emission measure in cooler flares is probably a lower limit of the true emission because of reduced sensitivity; on the other hand, the emission measure could be overestimated in the attenuator 0 state. Another explanation in certain flares is the main temperature sensitivity range of AIA. The response function of the high-temperature wavelength channels 131 Å and 193 Å peaks at 12 MK and 16 MK, respectively, and falls off sharply at higher temperatures. If the bulk of the plasma is at higher temperatures than 12–16 MK, AIA will not be sensitive to its signatures, resulting in an underestimated total emission measure. This could explain the discrepancy in the SXR coronal source of SOL2012-07-19T05:58, for which a RHESSI temperature of 23 MK was found. On the other hand, the RHESSI fit of SOL2010-08-14T10:05 suggests a temperature of 10.5 MK, which falls within the main AIA sensitivity range. To investigate the systematics of the DEM code and the influence of parameters such as the data error and use of a positivity constraint in the regularized inversion, we performed a systematic study using idealized Gaussian DEMs with different peak temperatures. From these, the expected AIA data number (DN) was calculated and used to find the DEM from regularized inversion (see the Appendix). In addition, a model with two Gaussian DEMs was used, emulating the often-found double-peak structure of the DEM. Figure 5 shows the model DEM and the reconstructed DEM for several values of peak temperature and several relative intensities in the case of two Gaussians. Generally there is a better agreement between model and reconstructed DEM without the imposed positivity constraint. In the case of purely Poisson error without any systematic terms the shape of the DEM is well reconstructed except for peak temperatures higher than log T ∼ 7, where some of the DEM seems to be "redistributed" to low temperatures at around log T = 6.2. The same but opposite effect occurs for very low peak temperatures of around log T ∼ 6. This is probably due to the 193 Å response, which has two peaks, one at log T = 6.2 and one at log T = 7.2. This effect is worsened when the positivity constraint is applied. In all cases, the total emission measure defined as ${\rm EM}=\int _{\log T=5.7}^{\log T=7.5} \xi (T) dT$ is reduced relative to the total model emission measure (Figure 6) by a factor of up to 30 when the positivity constraint is used and up to 2 without the positivity constraint. However, the effect is not systematic enough to allow for implementation of an empirical correction in the DEM reconstruction.

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There is also the possibility of several distinct particle populations with different temperatures being present. The mean electron flux spectra derived from AIA at the footpoints and the HXR above-the-looptop source in SOL2012-07-19T05:58 are consistent with a Maxwellian distribution, but with a deviation near 1 keV, while RHESSI observes a power-law distribution at energies above 25 keV. Unfortunately, there were no reliable RHESSI observations possible between 3 and ∼25 keV for these two sources. We fitted a thermal component to the spectrum from imaging spectroscopy of the two sources in this energy range. While potentially not physically meaningful because the spectrum mainly consists of reconstruction noise from the different sources, this provides a reasonable upper limit of the flux (compare Figure 3) and suggests that the presence of a second Maxwellian distribution similar to the one found in the SXR coronal source and in the loop of SOL2010-08-14T10:05 is feasible.

Combining X-ray observations with EUV observations from RHESSI allows the mean electron flux spectrum from different sources in solar flares to be inferred from 0.1 keV up to several tens of keV, thus enabling diagnostics of the low-energy component of the spectrum inaccessible with RHESSI alone. There is still a gap in spectral coverage where uncertainties are rather large at the intermediate energies between about 1 and 3 keV where neither instrument is sensitive for reliable measurements. This is the most likely cause for the discrepancy between the total emission measures of RHESSI and AIA. The total spectrum can be described as the combination of a Maxwellian core, a secondary "halo" component, and a non-thermal tail similar to distributions often seen in the solar wind (e.g., Lin et al. 1997). An analytical description of the total solar flare spectrum and a quantitative comparison with solar wind spectra will be the subject of future work.

Financial support by the European Commission through the FP7 HESPE network (FP7-2010-SPACE-263086) is gratefully acknowledged. This work is supported by an STFC Consolidated Grant (E.P.K.) and by the Swiss National Science Foundation (M.B.). We thank the referee for valuable comments and suggestions.

To investigate the influence of parameters such as the data error and use of a positivity constraint in the regularized inversion method that was used for the analysis of the presented events, we performed a systematic study using idealized Gaussian DEMs. An idealized Gaussian DEM in logT of the form $({{\rm DEM}_0}/{\sqrt{2\pi \sigma }})\exp [-(\log T-\log T_0)^2/(2\sigma ^2)]\times 10^6/T_0$ was used where σ = 0.15 for six values of log T₀ between 5.8 and 7.5 and DEM₀ = 3.27 × 10²² cm⁻⁵ K⁻¹. From this, the expected AIA DN was calculated and used to find the DEM from inversion for an assumed data error of ${\rm DN}_{{\rm err}}=\sqrt{{\rm DN}}$ and including a "systematic error" ${\rm DN}_{{\rm err}}=\sqrt{{\rm DN}+(0.2\times {\rm DN})^2}$ to account for calibration uncertainties (e.g., Landi & Young 2010). In both cases, the DEM was found once without other constraints and once imposing a constraint for positive DEM at all temperatures. In a second step, a DEM with two Gaussian components (with peak temperatures log T_cold = 6.2 and log T_hot = 7.04, where DEM(log T_cold) = k × DEM(log T_hot) with k between 10⁻⁵ and 1) was used to emulate the often-observed two-peak structure. Figure 5 shows the input model along with the reconstructed DEMs for all temperatures. We can now define the total emission measure as the integral over the DEM ${\rm EM}=\int _{\log T=5.7}^{\log T=7.5} \xi (T) dT$ and compare this as a function of log T₀ and k (Figure 6).