GLOBAL ENERGETICS OF SOLAR FLARES. III. NONTHERMAL ENERGIES

The nonthermal energy in flare electrons is generally calculated with the thick-target model (Brown 1971), which expresses the hard X-ray photon spectrum by a convolution of the electron injection spectrum with the Bethe–Heitler bremsstrahlung cross-section. According to this model, the observed hard X-ray photon spectrum $I(\varepsilon )$ observed at Earth can be approximated by a power-law function with a slope γ for the nonthermal energies, while the spectral index generally changes at the lower (thermal) energies. Thus, the nonthermal spectrum is defined as (e.g., see the textbook by Aschwanden 2004, chapter 13),

$$\begin{eqnarray}&&I(\varepsilon )=A\ {\varepsilon }^{-\gamma }\quad (\mathrm{photons}\ {\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}\ {\mathrm{keV}}^{-1}),\end{eqnarray} \tag{ 1 }$$

which yields a thick-target (nonthermal) electron injection spectrum f_e(e),

$$\begin{eqnarray}&&{f}_{e}(e)=2.68\times {10}^{33}\ b(\gamma ){{Ae}}^{-(\gamma +1)}\quad (\mathrm{electrons}\ {\mathrm{keV}}^{-1}\ {{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 2 }$$

which is a power-law function also, but with a slope $\delta =\gamma +1$ that is steeper by one, and $b(\gamma )$ is an auxiliary function related to the beta function. The detailed shape of a nonthermal electron spectrum that is affected by a low-energy cutoff is simulated in Holman (2003), showing a gradual flattening at lower energies. Note that we use the symbol ε for photon energies, while we use the symbol e for electron energies. The total power in nonthermal electrons above some cutoff energy e_c, i.e., $P(e\geqslant {e}_{c}$), is

$$\begin{eqnarray}&&P(e\geqslant {e}_{c})=4.3\times {10}^{24}\ \tfrac{b(\gamma )}{(\gamma -1)}\ A\ {({e}_{c})}^{-(\gamma -1)}\ \quad (\mathrm{erg}\,{{\rm{s}}}^{-1}).\end{eqnarray} \tag{ 3 }$$

Thus, the three observables of the photon flux A, the photon power-law slope γ, and the low-energy cutoff energy e_c are required to calculate the power during a selected flare time interval, which can be calculated with the OSPEX package of the SolarSoftWare library of the Interactive Data Language software (see RHESSI webpage http://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex_explanation.html).

In order to calculate the total nonthermal energy E_nt during an entire flare, we have to integrate the power as a function of time,

$$\begin{eqnarray}&&{E}_{\mathrm{nt}}={\int }_{{t}_{\mathrm{start}}}^{{t}_{\mathrm{end}}}P(e\gt {e}_{c}(t),t)\ {dt}\quad (\mathrm{erg}).\end{eqnarray} \tag{ 4 }$$

While the photon fluxes A(t) and the spectral slopes $\gamma (t)$ can readily be measured from a time series of hard X-ray photon spectra (Equation (1)), the largest uncertainty in the determination of the nonthermal energy is the low-energy cutoff energy e_c(t) between the thermal and nonthermal hard X-ray components, typically expected in the range of $\approx 10\mbox{--}30\,\mathrm{keV}$ (see Table 3 in Aschwanden 2007). In the following, we outline two different theoretical models of the low-energy cutoff that are applied in this study.

The bremsstrahlung spectrum $I(\varepsilon )$ of a thermal plasma with temperature T, as a function of the photon energy $\varepsilon =h\nu$, setting the coronal electron density equal to the ion density $(n={n}_{i}={n}_{e})$, and neglecting factors of the order of unity (such as the Gaunt factor $g(\nu ,T)$ in the approximation of the Bethe–Heitler bremsstrahlung cross-section), and the ion charge number, $Z\approx 1$, is (Brown 1974; Dulk & Dennis 1982),

$$\begin{eqnarray}&&I(\varepsilon )={I}_{0}\int \displaystyle \frac{\exp (-\varepsilon /{k}_{B}T)}{{T}^{1/2}}\displaystyle \frac{{dEM}(T)}{{dT}}\ {dT},\end{eqnarray} \tag{ 5 }$$

where ${I}_{0}\approx 8.1\times {10}^{-39}$ keV s⁻¹ cm⁻² keV⁻¹ and ${dEM}(T)/{dT}$ specifies the differential emission measure (DEM) ${n}^{2}{dV}$ in the element of volume dV corresponding to a temperature range of dT,

$$\begin{eqnarray}&&\left(\displaystyle \frac{{dEM}(T)}{{dT}}\right){dT}={n}^{2}(T)\ {dV}.\end{eqnarray} \tag{ 6 }$$

Regardless, whether we define this DEM distribution by an isothermal or by a multithermal plasma (Aschwanden 2007), the thermal spectrum $I(\varepsilon )$ falls off similarly to an exponential function at an energy of $\varepsilon \lesssim 20\,\mathrm{keV}$ (or up to ≲40 keV in extremal cases), while the nonthermal spectrum in the higher energy range of $\varepsilon \approx 20\mbox{--}100\,\mathrm{keV}$ can be approximated with a single (or broken) power-law function (Equation (1)).

Because of the two different functional shapes, a cross-over energy ${\varepsilon }_{c}$ can often be defined from the change in the spectral slope between the thermal and the nonthermal spectral component. The electron energy spectrum, however, can have a substantially lower or higher cutoff energy (e.g., Holman 2003).

We represent the combined spectrum with the sum of the (exponential-like) thermal and the (power-law-like) nonthermal component, i.e.,

$$\begin{eqnarray}I(\varepsilon ) & = & {I}_{\mathrm{th}}(\varepsilon )+{I}_{\mathrm{nt}}(\varepsilon )\\ & = & {I}_{0}\displaystyle \int \displaystyle \frac{\exp (-\varepsilon /{k}_{B}T)}{{T}^{1/2}}\displaystyle \frac{{dEM}(T)}{{dT}}\ {dT}\ +\ A\ {\varepsilon }^{-\gamma },\end{eqnarray} \tag{ 7 }$$

where the cross-over energy ${\varepsilon }_{\mathrm{co}}$ can be determined in the (best-fit) model spectrum $I(\varepsilon )$ from the energy where the logarithmic slope is steepest, i.e., from the maximum of $\partial \mathrm{log}I(\varepsilon )/\partial \mathrm{log}\varepsilon$.

A new theoretical model has recently been developed that allows us to calculate the low-energy cutoff energy in the thick-target model directly, by including the "warming" of the cold thick-target plasma during the electron precipitation phase, when chromospheric heating and evaporation sets in (Kontar et al. 2015). Previous applications of the thick-target model generally assume cold (chromospheric) temperatures in the electron precipitation site (e.g., Holman et al. 2011, for a review). The theoretical derivation of the warm-target model has been analytically derived and tested with numerical simulations that include the effects of collisional energy diffusion and thermalization of fast electrons (Galloway et al. 2005; Goncharov et al. 2010; Jeffrey et al. 2014). According to this model, the effective low-energy cutoff e_c is a function of the temperature ${e}_{\mathrm{th}}={k}_{B}{T}_{e}$ of the warm-target plasma and the power-law slope $\delta =\gamma +1$ of the (nonthermal) electron flux,

$$\begin{eqnarray}&&{e}_{c}\approx (\xi +2)\ {k}_{B}{T}_{e}=\delta \ {k}_{B}{T}_{e}.\end{eqnarray} \tag{ 8 }$$

where $\xi =\gamma -1$ is the power-law slope of the source-integrated mean electron flux spectrum (see Equations (8)–(10) in Kontar et al. 2015), and T_e is the temperature of the warm target, which is a mixture or the cold preflare plasma and the heated evaporating plasma. Thus, for the temperature range of a medium-sized to a large X-class flare, which spans ${T}_{e}\approx 10\mbox{--}25$ MK, the temperature in energy units is ${E}_{\mathrm{th}}={k}_{B}{T}_{e}\approx 0.9\mbox{--}2.1\,\mathrm{keV}$, and for a range of power-law slopes of $\delta =3\mbox{--}6$ (Dennis 1985; Kontar et al. 2011), a range of ${e}_{c}\approx 3\mbox{--}13\,\mathrm{keV}$ is predicted for the low-energy cutoffs by this model.

Besides collisional heating of the warm chromospheric target, electron beams and beam-driven Langmuir wave turbulence may affect the low-energy cutoff additionally (Hannah et al. 2009). Alternative analytical models on the low-energy cutoff can be derived from a collisional time-of-flight model (Appendix A), from the Rosner–Tucker–Vaiana heating/cooling balance model (Appendix B), and from the runaway acceleration model (Appendix C).

For the measurement of the nonthermal energy (E_nt) of electrons during solar flares, we use the OSPEX (Object Spectral Executive) software, which is an object-oriented interface for X-ray spectral analysis of solar data, written by Richard Schwartz and others (see the RHESSI website at http://hesperia.gsfc.nasa.gov/ for documentation). The OSPEX software allows the user to read RHESSI data, to select and subtract a background, to select time intervals of interest, to select a combination of photon flux model components, and to fit those components to the spectrum in each selected time interval. During the fitting process, the response matrix is used to convert the photon model to the model counts that are fitted to the observed counts. The OSPEX software deals also with changes of attenuator states, decimation, pulse pile-up effects, and albedo effects, and provides procedures to calculate the nonthermal energy $({E}_{\mathrm{nt}})$ (according to the thick-target model) and the thermal energy (E_th) down to energies of $\gtrsim 3\,\mathrm{keV}$.

RHESSI complements spectral information of the DEM distribution at the high-temperature side (${T}_{e}\gtrsim 16$ MK) (Caspi 2010; Caspi & Lin 2010; Caspi et al. 2014), while AIA/SDO provides DEM information at the low-temperature side (${T}_{e}\lesssim 16$ MK), as we determined in Paper II. For spectral modeling, we are using the two-component model vth+thick2_vnorm, which includes a thermal component at low energies and a (broken) power-law function at higher (nonthermal) energies. In our spectral fits, we are only interested in the transition from the thermal to the nonthermal spectrum, which can be expressed by an exponential-like plus a single-power-law function (Equation (7)), and thus we use only the lower power-law part of the two-component model vth+thick2_vnorm, while the spectral slope in the upper part was set to a constant (${\delta }_{2}=4$). In addition, we use calc_nontherm_electron_energy_flux of the OSPEX package to calculate the nonthermal energy flux in the thick-target model.

RHESSI Spectral Fitting Range Selection: in order to obtain a self-consistent measure of the nonthermal energy, which varies considerably during the duration of a flare or among different flares, we have to choose a spectral fitting range that covers a sufficient part of both the thermal and nonthermal components. We choose the maximum energy range $[{\varepsilon }_{1},{\varepsilon }_{2}]$, bound by ${\varepsilon }_{1}=6...10\,\mathrm{keV}$ and ${\varepsilon }_{2}=20...50\,\mathrm{keV}$, in which an acceptable (reduced) ${\chi }^{2}$-value ($\chi \lt 2.0$) is obtained for the spectral fit. The upper bound of the fitting range is mostly constrained by the photon count statistics, which is often too noisy for energies at ${\varepsilon }_{2}\gtrsim 30\,\mathrm{keV}$ during small flares (M-class here), given the time steps of ${\rm{\Delta }}t=20\,{\rm{s}}$ chosen throughout. The fitted energy ranges also cover the range of cross-over energies (10–28 keV) found in multithermal fitting of energy-dependent time delays (Aschwanden 2007).

As a general criticism, we have to be aware that the nonthermal spectral component could in addition also be confused with a multithermal component in the fitted spectral range of $\varepsilon \approx 10\mbox{--}30\,\mathrm{keV}$ (Aschwanden 2007), or with non-uniform ionization effects (Su et al. 2011), or with return-current losses (Holman 2012).

RHESSI Detector Selection: we used the standard option of OSPEX, where a spectral fit is calculated from the combined counts of a selectable set of RHESSI subcollimaters. RHESSI has nine (subcollimator) detectors that originially had near-identical sensitivities, but progressively deviated from each other as a result of steady degradation over time due to radiation damage from charged particles. Heating up the germanium restores the lost sensitivity and resolution, and thus five annealing procedures have been applied to RHESSI so far (second anneal at 2010 March 16–May 1; third at 2012 January 17–February 22; forth at 2014 June 26–August 13; and fifth at 2016 February 23–April 23). No science data are collected during the annealing periods. Based on the performance of the individual detector sensitivities, it is general practice to exclude detectors 2 and 7 in spectral fits. Furthermore, detectors 4 and 5 are considered to be unreliable after 2012 January (R. Schwartz 2016, private communication). Therefore, we select the set of detectors [1, 3, 4, 5, 6, 8, 9] in spectral fits up to the third anneal in 2012 January (events # 1–126 in Table 1), and the set of [1, 3, 6, 8, 9] after 2012 February (events # 154-395 in Table 1). Omitting detectors 4 and 5 in the latter set of 71 events yields a total nonthermal energy that is by a factor of ${q}_{\det }={E}_{\mathrm{nt}}[1,3,6,8,9]/{E}_{\mathrm{nt}}[1,3,4,5,6,8,9]=1.3\pm 0.5$ higher.

GOES Time Range and RHESSI Time Resolution: we download the GOES 1–8 Å light curves ${F}_{{GOES}}(t)$ and calculate the time derivative as a proxy for the hard X-ray time profile ${F}_{\mathrm{HXR}}\approx {{dF}}_{{GOES}}/{dt}$, as shown in Figures 1(a), 2(a), and 3(a). The start time ${t}_{\mathrm{start}}$, peak time ${t}_{\mathrm{peak}}$, and end time ${t}_{\mathrm{end}}$ are defined from the NOAA/GOES catalog. We compute consecutive spectra in time steps of ${\rm{\Delta }}t=20\,{\rm{s}}$. Note that RHESSI is a spinning spacecraft with a period of 4 s, which does not cause any modulation effects for 20 s time integrations.

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RHESSI Quick-look Data: in a next step, we inspect the RHESSI quick-look time profiles (Figures 1(b), 2(b), and 3(b)), which show photon counts in different energy channels in the range of 6–300 keV. Based on these RHESSI time profiles, we select time intervals for background subtraction. Generally, we select a time interval at flare start as the background interval (in 90%), and subtract this preflare spectrum for the entire flare time interval. Only in a few cases (10%) where the preflare flux is higher than the postflare flux, we choose a time interval at flare end for background subtraction. The RHESSI quick-look data show changes in the attenuator state (e.g., Figures 2(b), 3(b)), which are automatically handled in most time intervals with the OSPEX software, unless there is a change in the attenuator state during a selected time interval itself, in which case this time interval is removed from the spectral analysis. The quick-look data occasionally show data gaps that are caused when RHESSI enters spacecraft night in its near-Earth orbit. If the data gap does not occur during the flare peak of hard X-ray emission, we still include the event in the analysis, as long as the time interval of dominant nonthermal HXR emission is covered (such as in event #219 in Figure 2(b)).

OSPEX Spectral Fitting: for spectral fitting, we perform first a semi-calibration and store the detector response matrix (DRM), and then run a spectral fit with the fit function vth+thick2_vnorm using the OSPEX software, optimizing the following model fit parameters (for each time interval t):

EM(t)	= emission measure in units of 1049 cm−3
Te(t)	= plasma temperature in units of keV (1 keV = 11.6 MK)
A(t)	= photon flux at $\varepsilon =50\,\mathrm{keV}$
$\delta (t)$	= negative power-law index of electron spectrum
ec(t)	= low-energy cutoff

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Examples of spectral fits are shown in Figures 1(c), 2(c), and 3(c), fitted at the time of the peak power ${P}_{\mathrm{co}}(t)$ (indicated with red vertical lines in Figures 1, 2, and 3). The best-fit spectrum yields a cross-over energy e_co between the thermal and nonthermal spectral component. Alternatively, the warm-target model of Kontar et al. (2015) yields a low-energy cutoff value e_wt. The fitted energy ranges are listed in Table 1 and are indicated with dotted vertical lines in Figures 1(g), 2(g), and 3(g). The goodness-of-fit is quantified with the ${\chi }^{2}$-value criterion. In the case of bad fits of the ${\chi }^{2}$-values ($\chi \gt 2$), we changed either the fitted energy range (in 13%), the selected interval for background subtraction (10%), or the fitted time range (5%).

Three examples of analyzed flare events are shown in Figures 1, 2, and 3, including one of the smallest events (Figure 1: #387, GOES M1.0 class), an event with multi-peak characteristics (Figure 2; #219, GOES M2.0 class), and one of the largest events (Figure 3; #12, GOES X2.2 class). In all three cases, we show the time evolution of the most important fit parameters in the various panels ((d) through (j)) of Figures 1–3: (d) the thermal emission measure ${EM}(t);$ (e) the temperature evolution ${T}_{e}(t);$ (f) the nonthermal photon flux ${I}_{\mathrm{nt}}(t)$ at 50 keV; (g) the power-law slope $\delta (t);$ (h) the goodness-of-fit $\chi (t);$ (i) the nonthermal power ${P}_{\mathrm{wt}}(t)$ using the low cutoff energy based on the warm-target model (Section 2.3); and (j) the low-energy cutoff ${e}_{\mathrm{wt}}(t)$ of the warm-target model. In the examples shown in Figures 1, 2 and 3, we see that the thermal emission measure EM(t) increases during the rise time of the GOES flux, while the temperature T_e(t) decreases, which indicates both, namely density and temperature increases due to chromospheric evaporation, as well as subsequent plasma cooling, during the impulsive flare phase. Since multiple heating and cooling cycles overlap during a flare, we see both effects simultaneously. The cases shown in Figures 1, 2 and 3 show also that the nonthermal flux ${I}_{\mathrm{nt}}(t)$ (Figures 1(f), 2(f), and 3(f)) and the power ${P}_{\mathrm{wt}}(t)$ (Figures 1(i), 2(i), 3(i)) are correlated with the GOES time derivative (Figures 1(a), 2(a), and 3(a)).

The goodness of the spectral fits computed with the OSPEX code is specified with the ${\chi }^{2}$-criterion, based on the least-square difference between the theoretical spectral model (isothermal plus power-law nonthermal function) and the observed counts in the fitted energy range [${\varepsilon }_{1}-{\varepsilon }_{2}$]. The fitted energy time interval (with a resolution of 1 keV) has about ${n}_{\mathrm{bin}}\approx 30-10=20$ energy bins, while the model has four (${n}_{\mathrm{par}}=4)$ free parameters $({EM},{T}_{e},{A}_{50},\delta )$, yielding a degree of freedom ${n}_{\mathrm{free}}={n}_{\mathrm{bin}}-{n}_{\mathrm{par}}\approx 20-4=16$. In our spectral analysis of 191 flare events, we performed spectral fits, with an average of ${n}_{t}\approx 27$ time steps per event, amounting to a total of ${N}_{\mathrm{spec}}\approx 191\times 27=5157$ spectral fits. The values $\chi (t)$ of three events are shown in Figures 1(h), 2(h), and 3(h). The median values of these three events are $\chi =1.4,1.0$, and 1.3. We obtained in all 191 events a median goodness-of-fit value of $\chi \lt 2$, after adjustment of the fitted energy range if necessary. The mean and standard deviations of the median ${\chi }^{2}$-values of all 191 events is $\chi =1.2\pm 0.4$, which indicates that the fitted spectral model is adequate in the chosen fitted energy range. Of course, if one particular model, such as the two-component thermal–nonthermal model chosen here (Equation (7)), is found to be consistent with the data according to an acceptable goodness-of-fit criterion, it does not rule out alternative models. For instance, the thermal component is often modeled with an isothermal (single-temperature) spectrum, while a multithermal power-law function was found to fit the thermal flare component in most flares equally well (Aschwanden 2007).

A representative value for the electron temperature during a flare can be defined in various ways. In Paper II, we measured the peak temperature ${T}_{\mathrm{AIA}}$ of the DEM distribution at the peak time of the flare, as well as the emission measure-weighted temperature T_w (Equation (13) in Paper II), which approximately characterizes the "centroid" of the (logarithmic) DEM function. The mean ratio of these two temperature values was found to be ${q}_{T}={T}_{\mathrm{AIA}}/{T}_{w}=0.31$ within a standard deviation by a factor of 2.0 (Figure 4, left panel). The emission measure-weighted temperature T_w is generally found to be higher, because near-symmetric DEM functions as a function of the logarithmic temperature are highly asymmetric on a linear temperature scale, with a centroid that is substantially higher than the logarithmic centroid.

On the other hand, spectral fits of RHESSI data with an isothermal component are known to have a strong bias toward the highest temperatures occurring in a flare, because the fitted energy range covers only the high-temperature tail of the DEM distribution function (Battaglia et al. 2005; Caspi et al. 2014; Ryan et al. 2014). A statistical study demonstrated that the high-temperature bias of RHESSI by fitting in the photon energy range of $\varepsilon \approx 6\mbox{--}12\,\mathrm{keV}$ amounts to a factor of ${T}_{R}/{T}_{\mathrm{AIA}}=1.9\pm 1.0$ (Ryan et al. 2014). Here we find that all RHESSI temperatures averaged during each flare are found in a range of ${T}_{R}=16\mbox{--}40\,\mathrm{MK}$, which is about equal to the emission measure-weighted temperature, i.e., ${T}_{R}/{T}_{w}=0.90$ within a factor of 1.4 (Figure 4, right panel). The 1σ ranges (containing 67% of the values) of the various temperature definitions are ${T}_{\mathrm{AIA}}\approx 3\mbox{--}14\,\mathrm{MK}$, ${T}_{w}\approx 20\mbox{--}30\,\mathrm{MK}$, and ${T}_{R}\approx 19\mbox{--}28\,\mathrm{MK}$. Thus, we should keep these different temperature definitions in mind when we calculate the low-energy cutoff e_c(t) as a function of the RHESSI temperature T_R(t) (Equation (8) for the warm-target model).

The most decisive parameter in the determination of the nonthermal energy E_nt is the low-energy cutoff e_c (Equation (4)), which is directly proportional to the temperature T_e in the warm target (Equation (8)). The relevant temperature is a mixture of preflare plasma temperatures and upflowing evaporating flare plasma. In the absence of a sound model, we resort to the mean value of the DEM peak temperatures determined in flaring active regions, as determined with AIA in Paper II, yielding a mean value of ${T}_{\mathrm{AIA}}=8.8\pm 6.0\,\mathrm{MK}$ (Figure 4 left panel), averaged over N = 380 M and X-class flare events. For the subset of 191 flare events observed with RHESSI, this mean value is T_e = 8.6 MK, or ${k}_{B}{T}_{e}=0.74\,\mathrm{keV}$. Note that a deviation of the plasma temperature by a factor of two will result into a deviation in the determination of the nonthermal energy E_nt by about an order of magnitude (using a power law with a typical slope of $\gamma \approx 4$ in Equations (3) and (4).

The nonthermal energy in electrons, calculated as a time integral E_nt (Equation (4)), using the low-energy cutoff according to the warm thick-target model ${e}_{\mathrm{wt}}(t)$ (Section 2.3; Equation (8)), or alternatively the thermal/nonthermal cross-over energy ${e}_{\mathrm{co}}(t)$ (Section 2.2), is the main objective of this study. Examples of the time evolution of the nonthermal parameters [$A(t),\delta (t)$, ${e}_{\mathrm{co}}(t)$, ${e}_{\mathrm{wt}}(t)$] and the resulting nonthermal energies ${{dE}}_{\mathrm{nt}}(t)$ are shown in Figures 1–3. In Figure 5, we show statistical results of these parameters. Investigating the dependence of these parameters on the flare temperature T_R we find that both the low-energy cutoff energy e_wt (Figure 5(a)) as well as the nonthermal (warm-target) energy E_nt (Figure 5(b)) are uncorrelated with the RHESSI temperature.

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If we use the thermal–nonthermal cross-over method to estimate the low-energy cutoff, we find a systematically higher value, ${e}_{\mathrm{co}}\gtrsim {e}_{\mathrm{wt}}$ (Figure 5(c)). Consequently, the nonthermal energy estimated with the cross-over method is systematically lower than the nonthermal energy calculated with the warm-target model (Figure 5(d)). This result strongly depends on the assumption of the warm-target temperature. Based on a mean temperature of T_e = 8.6 MK found in the active regions analyzed here, we derive low-energy cutoff energies of ${e}_{\mathrm{wt}}=6.2\pm 1.6\,\mathrm{keV}$ for the warm-target model, which is significantly lower than the cross-over energies ${e}_{\mathrm{co}}=21\pm 6\,\mathrm{keV}$. If we adopt the warm-target model, we conclude that the cross-over method over-estimates the low-energy cutoff and underestimates the nonthermal energies.

In Figure 6, we show scatter plots of the nonthermal energy E_nt measured here with other forms of previously determined energies, such as the magnetic energy E_mag (Paper I) and the (total pre-impulsive and post-impulsive) thermal energies E_th (Paper II). The energy ratios are characterized with the means of the logarithmic energies in the following. The ratios between the three forms of energies are shown separately for the cross-over method in the left-hand panels of Figure 6, and for the warm-target model in the right-hand panels of Figure 6.

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The ratios between the nonthermal energies and the magnetically dissipated energy is ${E}_{\mathrm{co}}/{E}_{\mathrm{mag}}=0.01$ for the cross-over method, or ${E}_{\mathrm{wt}}/{E}_{\mathrm{mag}}=0.41$ for the warm-target model, respectively. Thus, the warm-target model yields ratios that are closer to unity, which is expected in terms of magnetic reconnection processes, where most of the magnetic energy is converted into particle acceleration. We find that the dissipated magnetic energy is sufficient to supply the energy in nonthermal particles in 71% for the warm-target model, or in 97% for the cross-over model (Figures 6(a) and (b)).

The ratios between the thermal energies and the magnetically dissipated energy is ${E}_{\mathrm{th}}/{E}_{\mathrm{mag}}=0.08$ for both the cross-over or the warm-target model (Figures 6(c) and (d)). We find that the dissipated magnetic energy is sufficient to supply the thermal energy in 95%.

Comparing the thermal with the nonthermal energies, we find a mean ratio of ${E}_{\mathrm{th}}/{E}_{\mathrm{wt}}=0.15$ for the warm-target model, or ${E}_{\mathrm{th}}/{E}_{\mathrm{co}}=6.46$ for the cross-over method. We find that the nonthermal energy is sufficient to supply the thermal energy in 85% for the warm-target model (Figure 6(f)), but only in 29% for the cross-over method. Thus, the warm-target model yields values that are closer to the expectations of the standard thick-target model, where the thermal energy is entirely produced by the nonthermal energy of precipitating (nonthermal) electrons.

We show the comparison of nonthermal and thermal energies also in the form of cumulative size distributions in Figure 7, for the subset of 75 flares for which all three forms of energy (magnetic, thermal, nonthermal) could be calculated. We find that the nonthermal energy is typically an order of magnitude larger than the thermal energy in the statistical average. The nonthermal energy is smaller than the magnetic energy, as expected for magnetic reconnection processes, for smaller flares with energies of ${E}_{\mathrm{nt}}\lt 3\times {10}^{32}$ erg. However, we find the opposite result for larger flares, with the nonthermal energy exceeding the magnetically dissipated energy, for large events with ${E}_{\mathrm{nt}}\gt 3\times {10}^{32}$ erg. Since the uncertainties in nonthermal energies are about an order of magnitude and the dissipated magnetic energy exceeds the nonthermal energy in 71% (Figure 6(b)), we suspect that the largest nonthermal energies are overestimated, which would indicate that a higher value of the low-energy cutoff or a higher flare plasma temperature (than the mean active region temperature T_e = 8.6 MK used here) could ameliorate the overestimated nonthermal energies.

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We compare the occurrence frequency distributions of magnetic, nonthermal, and thermal energies, as well as those of the direct RHESSI observables: the peak counts P, total counts C, and durations D (Figure 8). As a caveat, we have to be aware that these values for P and C are obtained from the online RHESSI flare catalog, and thus are not well-calibrated because they do not take attenuation or decimation into account. Nevertheless, taking these raw values, the magnetic and thermal energies have similar power-law slopes of $\alpha \approx 2.0$, while the nonthermal energies have a slightly flatter slope of ${\alpha }_{\mathrm{nt}}=1.41\pm 0.10$, which can be compared with a previous study, where a power-law slope of ${\alpha }_{\mathrm{nt}}=1.53\pm 0.02$ was found (Crosby et al. 1993). The latter study is actually based on larger statistics, containing 2878 flare events observed with HXRBS/SMM during 1980–1982 (Crosby et al. 1993), but with a higher assumed low-energy cutoff of ${e}_{c}\gt 25\,\mathrm{keV}$.

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While we determined the dissipated magnetic energies E_mag (Paper I; called ${E}_{\mathrm{diss}}$ therein), thermal energies E_th (Paper II), and the nonthermal energies E_nt, we can ask now the question how the energy partition from primary to secondary energy dissipation works in solar flares. Many solar flare models are based on a magnetic reconnection process, where a stressed non-potential magnetic field becomes unstable and undergoes a reconfiguration toward a lower magnetic energy state, releasing during this process some amount ${E}_{\mathrm{mag}}={q}_{\mathrm{diss}}{E}_{\mathrm{free}}$ of the magnetic free energy ${E}_{\mathrm{free}}$ (defined by the difference between the non-potential and the potential energy, ${E}_{\mathrm{free}}={E}_{\mathrm{np}}-{E}_{p}$). Excluding alternative energy sources, we hypothesize that this dissipated magnetic energy E_mag is considered to be the entire available primary energy input, while other energy conversion processes represent secondary steps that need to add up in the energy budget,

$$\begin{eqnarray}&&{E}_{\mathrm{mag}}=({E}_{\mathrm{nt}}+{E}_{\mathrm{cme}}+...)\gt {E}_{\mathrm{nt}},\end{eqnarray} \tag{ 9 }$$

such as the nonthermal energy E_nt that goes into acceleration of particles, or the energy ${E}_{\mathrm{cme}}$ to accelerate an accompanying CME. The nonthermal energy E_nt may be further subdivided into energies in electrons ${E}_{\mathrm{nt},{\rm{e}}}$ and ions ${E}_{\mathrm{nt},{\rm{i}}}$,

$$\begin{eqnarray}&&{E}_{\mathrm{nt}}=({E}_{\mathrm{nt},{\rm{e}}}+{E}_{\mathrm{nt},{\rm{i}}}+...)\gt {E}_{\mathrm{nt},{\rm{e}}},\end{eqnarray} \tag{ 10 }$$

while the CME energy ${E}_{\mathrm{cme}}$ consists of the kinetic energy ${E}_{\mathrm{kin}}$ and the gravitational potential energy ${E}_{\mathrm{grav}}$, and part of it may be converted into acceleration of particles in the interplanetary CME shock (${E}_{\mathrm{nt},\mathrm{cme}}$), which are particularly present in solar energetic particle events,

$$\begin{eqnarray}&&{E}_{\mathrm{cme}}={E}_{\mathrm{kin}}+{E}_{\mathrm{grav}}+{E}_{\mathrm{nt},\mathrm{cme}}+....\end{eqnarray} \tag{ 11 }$$

We have to be careful to avoid double-counting secondary energies, because there may be some tertiary energy conversion processes, such as heating of chromospheric plasma according to the thick-target bremsstrahlung model, E_th, while upgoing nonthermal particles escape into interplanetary space, carrying an energy of ${E}_{\mathrm{nt},\mathrm{esc}}$,

$$\begin{eqnarray}&&{E}_{\mathrm{nt}}=({E}_{\mathrm{th}}+{E}_{\mathrm{nt},\mathrm{esc}}+...)\gt {E}_{\mathrm{th}}.\end{eqnarray} \tag{ 12 }$$

Since we have measured only three types of energies so far, ${E}_{\mathrm{mag}},{E}_{\mathrm{nt}}$, and E_nt, we can only test the inequalities given on the right-hand-side of Equations (9) and (12) at this point.

Based on the nonthermal energies in electrons determined in this work, we can answer the question whether the so far measured magnetic energy is sufficient to accelerate the electrons observed in hard X-rays, i.e., ${E}_{\mathrm{mag}}\gt {E}_{\mathrm{nt}}$, as expected for magnetic reconnection models. Relying on the warm-target model we found that 41% of the dissipated magnetic energy (with a standard deviation of about an order of magnitude) is converted into acceleration of nonthermal electrons, or a total amount of $\approx 82 \%$ for both electrons and ions in the case of equipartition, while the rest is available to accelerate CMEs. There are few statistical estimates of the flare energy budget in the literature (besides the work of Emslie et al. 2012; Warmuth & Mann 2016). One early study quoted that the nonthermal energy in electrons $\gt 20$ keV contains 10%–50% of the total energy output for the 1972 August flares (Lin & Hudson 1976; Hudson & Ryan 1995), which is consistent with our result of 41% within the measurement uncertainties.

Comparing the energy ranges determined in this global flare energetics project with those obtained from 38 events in Emslie et al. (2012), we find higher amounts of nonthermal flare electron energies in the statistical average, covering the range of ${E}_{\mathrm{nt}}\approx (20\mbox{--}2000)\times {10}^{30}\,\mathrm{erg}$ (Figure 9), which is mostly accounted for by a lower value of the low-energy cutoff predicted by the warm-target model (Kontar et al. 2015) for some events, while cutoff energies with the highest acceptable value of the ${\chi }^{2}$ were used in Emslie et al. (2012). The magnetically dissipated energies appear to be overestimated by an order of magnitude (Figure 9) in Emslie et al. (2012), based on the ad hoc assumption that the dissipated energy amounts to 30% of the potential field energy therein (Paper I). On the other hand, the thermal energies appear to be underestimated by at least an order of magnitude (Figure 9) in Emslie et al. (2012) due to the isothermal approximation, as discussed in Paper II.

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A second question we can answer is whether the nonthermal energy in electrons is sufficient to heat the flare plasma by the chromospheric evaporation process, as expected in the thick-target model according to the Neupert effect (Dennis & Zarro 1993), which requires ${E}_{\mathrm{nt}}\gt {E}_{\mathrm{th}}$. Based on the warm-target model, we found a mean (logarithmic) ratio of ${E}_{\mathrm{th}}=0.15\ {E}_{\mathrm{nt}}$ (Figure 6(f)). The fraction of flares that have a thermal energy less than the nonthermal energy, as expected in the standard thick-target model, amounts in our analysis to $\approx 85 \%$ for the warm-target method, or $\approx 29 \%$ for the cross-over model.

This means that the thick-target model could be insufficient to supply enough energy to explain the thermal energy produced by the chromospheric evaporation process in about 15% of the flares for the warm-target model, or in 71% for the cross-over model. Thus, the cross-over model would pose a series problem for the thick-target model. The insufficiency of the thick-target model has been addressed as a failure of the theoretical Neupert effect (Veronig et al. 2005; Warmuth & Mann 2016), which invokes testing of the correlation between the electron beam power (from RHESSI) and the time derivative of the thermal energy heating rate (from GOES). From such studies, it was concluded that (1) fast electrons are not the main source of soft X-ray plasma supply and heating, (2) the beam low cutoff energy varies with time, or (3) the theoretical Neupert effect is strongly affected by the source geometry (Veronig et al. 2005). If the thermally dominated flares cannot be fully explained by the thick-target model, additional heating sources besides precipitating electrons would be required. The most popular alternative to the thick-target model is heating by thermal conduction fronts (Brown et al. 1979; Emslie & Brown 1980; Smith & Brown 1980; Smith & Harmony 1982; Batchelor et al. 1985; Reep et al. 2016). Other forms of direct heating (for an overview see chapter 16 in Aschwanden 2004) occur via (1) resistive or Joule heating processes, such as anomalous resistivity heating (Duijveman et al. 1981; Holman 1985; Tsuneta 1985), ion-acoustic waves (Rosner et al. 1978a), electron ion-cyclotron waves (Hinata 1980), (2) slow-shock heating (Cargill & Priest 1983; Hick & Priest 1989), (3) electron beam heating by Coulomb collisional loss in the corona (Fletcher 1995, 1996; Fletcher & Martens 1998), (4) proton beam heating by kinetic Alfvén waves (Voitenko 1995, 1996), or (5) inductive current heating (Melrose 1995, 1997).

The thick-target model fails to explain the observed amount of thermal energy only in a small number of flares for the warm-target model, while it is a larger number of events for the cross-over method. However, it is more likely that the cross-over method overestimates the low-energy cutoff, which underestimates the nonthermal energies, while the physics-based warm-target model leads to higher nonthermal energies, in which case the problem with the insufficiency of the thick-target model goes away.

We outlined two different methods to infer a low-energy cutoff. The first method consists of measuring the cross-over between the fitted thermal and nonthermal spectral components, which yields an upper limit on the low-energy cutoff, but a statistical test demonstrates that the obtained values (${e}_{\mathrm{co}}=21\pm 6\,\mathrm{keV}$) are significantly higher than those obtained from the warm-target model (${e}_{\mathrm{wt}}=6.2\pm 1.6\,\mathrm{keV}$). There are pros and cons for each method. The cross-over method requires a dominant thermal component, which is not always detectable in the spectrum, in which case the cross-over energy has a large uncertainty. The warm-target model requires the measurement of the (warm) flare temperature, which is measured at lower values from DEMs at EUV wavelengths than from hard X-ray spectra observed with RHESSI. Moreover, the spatial temperature distribution is very inhomogeneous and the location with the dominant temperature component relevant for the warm-target collisional energy loss may be a mixture of colder preflare plasma in active regions and heated evaporating flare plasma at the location of instantaneous electron precipitation. In summary, the value of the low-energy cutoff is strongly dependent on the assumed warm-target temperature, for which no physical model is established yet.

In this study, we also investigated the temporal evolution of the low-energy cutoff e_c(t), for instance, as shown in Figures 1(j), 2(j), and 3(j), but we do not recognize a systematic pattern indicating how the evolution of this low-energy cutoff is related to other flare parameters.