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A&A
Volume 588, April 2016
Article Number A115
Number of page(s) 14
Section The Sun
DOI https://doi.org/10.1051/0004-6361/201527474
Published online 28 March 2016

© ESO, 2016

1. Introduction

In solar flares, energy that is stored in nonpotential magnetic fields is released impulsively (presumably triggered by magnetic reconnection) and converted into kinetic energy of nonthermal particles and bulk mass motions, and into thermal energy of hot plasmas. In the standard flare model (the CSHKP model of eruptive solar flares; Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976), the energy is first transferred to nonthermal particles, which are guided by the magnetic field lines down to the chromospheric footpoints of the flare loops. There, they suffer collisional energy loss, which results in strong plasma heating that in turn leads to chromospheric evaporation, in which the heated plasma expands upward to fill the coronal flare loops. This scenario of beam-driven heating is strongly supported by the Neupert effect (cf. Neupert 1968; Veronig et al. 2005).

Both the nonthermal electrons, which carry a significant fraction of the energy released, and the hot thermal plasma (T> 10 MK) can be observed in the hard X-ray (HXR) regime due to nonthermal and thermal bremsstrahlung, respectively (the latter component can also be observed in soft X-rays). X-ray observations are thus crucial for understanding the physics of energy release and particle acceleration in solar flares – still two of the most important unsolved problems in solar physics.

Over the course of the past decade, the RHESSI instrument (Ramaty High Energy Solar Spectroscopic Imager; Lin et al. 2002) has dramatically increased our knowledge of the HXR signatures of solar flares. During the impulsive phase of a flare, HXR spectra typically consist of a steep thermal component at low energies and a nonthermal power law or broken power-law contribution at higher energies. With RHESSI, it is possible for the first time to clearly separate these two components, which allows deriving the parameters of the thermal plasma and of the accelerated nonthermal electrons through forward-fitting (cf. Holman et al. 2003) or inversion techniques (e.g., Kontar et al. 2005). A quantitative characterization of these parameters is essential for understanding energy release and particle acceleration in solar flares.

Flares range over several orders of magnitude with respect to energy, but even flares of the same size or importance can show quite different characteristics. Therefore it is important to study a larger number of events of different magnitudes before definite conclusions on flare physics can be made. While most RHESSI-related work can be characterized as case studies, there have been several relevant statistical studies focused on RHESSI data: Grigis & Benz (2004) have studied the spectral evolution of 24 flares, while Battaglia et al. (2005) have derived the dependence of spectral parameters on GOES size for 85 flares. Saint-Hilaire & Benz (2005) have studied thermal and nonthermal energies in 14 HXR peaks, while Hannah et al. (2008) have determined energies in a large number of microflares. More recently, Emslie et al. (2012) have extended the comprehensive approach of Emslie et al. (2004) to 38 solar eruptive events, and Caspi et al. (2014a) have studied the thermal properties of 37 events. However, many of these studies do not involve the same level of detailed analysis as the dedicated case studies do (for case studies, see, e.g., Saint-Hilaire & Benz 2002; Holman et al. 2003; Sui et al. 2005), they typically consider flares of relatively similar importance, and they usually study only a limited set of physical parameters over a narrow time range. Moreover, a comparison of the thermal energies derived is complicated by the fact that some studies derive thermal parameters from HXR spectra (e.g., Saint-Hilaire & Benz 2005) and others from GOES soft X-ray (SXR) fluxes (e.g., Emslie et al. 2012). The determination of the thermal source volume also varies between studies.

Our aim is to provide constraints on energy release and electron acceleration by bridging the gap between individual case studies and large statistical studies. Warmuth & Mann (2013a,b) have studied the HXR source sizes of 24 flares observed by RHESSI. Here, we continue this study by adding information gained from RHESSI and GOES spectroscopy. From this data set, we calculate parameters that characterize the properties of the thermal plasma and of the injected nonthermal electrons (see Sect. 2). We then analyze the parameter set in several different manners.

In this paper, we first quantify the range of values the parameters can take (Sect. 3). Most importantly, this provides upper limits for several key parameters, for instance, for the maximum injected electron flux that any electron acceleration mechanism must be able to reproduce. In Sect. 4 we study the dependence of the various physical parameters on the flare importance, while in Sect. 5 the relation between the thermal parameters as derived from RHESSI and GOES data is investigated in more detail. Section 6 studies the relation between thermal and nonthermal parameters. The conclusions of these studies are given in Sect. 7.

In a companion paper (Warmuth & Mann 2016, henceforth Paper II), we use the results derived here to study the energy partition (i.e., how the different energetics relate to one another) as a function of flare importance. In Paper II, we make general conclusions about the physics of energy release and include a discussion of probable systematic uncertainties.

2. Observations

2.1. Events

We have selected 24 flares from GOES class C3.4 to X17.2 to study how the characteristics of energy release and particle acceleration in flares change with flare importance. This is the same sample of events as used in Warmuth & Mann (2013a,b). Many of the events have been used previously for comparison with a shock-drift acceleration model (Mann et al. 2009; Warmuth et al. 2009b), and for the study of nonthermal energetics in the framework of magnetic reconnection (Mann & Warmuth 2011). The sample also contains some well-studied events, such as SOL2002-07-23T00:35 (e.g., Holman et al. 2003; Caspi & Lin 2010). All events are listed in Table 1 in ascending order of GOES importance. In addition to flare number, IAU event identifier (which gives the event date and the time of peak GOES flux), and GOES importance, the table gives the total time ranges (in minutes) that could be analyzed in each flare, Δt, and the time ranges of significant nonthermal HXR flux, Δtnth.

With seven C-class, eight M-class, and nine X-class flares (of which one is larger than X10), this sample covers nearly three magnitudes of the GOES importance scale. Weaker flares were not considered because we required good time-resolved imaging, which is not feasible for weaker events as they have limited count rates. A prerequisite for event selection was that the flares had to be well observed by RHESSI both spectroscopically and with respect to imaging and that they had to show a significant nonthermal component and two distinct nonthermal footpoints (cf. Warmuth & Mann 2013a). All events have RHESSI data coverage from the onset of the impulsive phase until after its end, with the exception of the X17.2 flare SOL2003-10-28T11:10, where RHESSI missed the most of the impulsive phase, but which was included as an example for an outstandingly energetic flare. We note that this event was omitted when studying correlations of maximum parameters, since the true maxima may have been missed. In tables the corresponding values are shown in brackets.

Table 1

Overview of the 24 analyzed flares.

2.2. Spectroscopy

We have obtained time series of RHESSI HXR count rate spectra for all 24 flares. In the majority of events, an integration time of 20 s was used, while in three events (nos. 3, 6, and 9), time bins of 12 s were necessary to adequately resolve the evolution during the impulsive phase. Only front detector 4 was used to obtain the count spectra because it had the best energy resolution (cf. Warmuth et al. 2009a). Using the OSPEX package1 of the RHESSI analysis tools, we folded spectral models of an isothermal component (employing the latest atomic data from the CHIANTI database; cf. Dere et al. 1997; Landi et al. 2013) plus a nonthermal thick-target bremsstrahlung component given by a power-law electron distribution (Brown 1971) through the full detector response matrix, and forward-fitted them to the background-subtracted count rate spectra (cf. Holman et al. 2003). The spectra were also corrected for decimation and photosperic albedo (Kontar et al. 2006), and using a single detector allowed us to best current correction methods for pulse pile-up and gain offset provided by OSPEX (cf. Warmuth et al. 2009a; Mann & Warmuth 2011). The energy bin width was 1/3 keV below 15 keV, 1 keV to 50 keV, 5 keV up to 100 keV, and 10 keV above that. The spectra were fitted up to 300 keV when possible, down to 6 keV for attenuator states A0 and A1, and down to 12 keV for attenuator state A3.

These fits yield the emission measure EM and temperature T of the isothermal plasma and the injected nonthermal electron flux as a function of electron energy E, F0(E), parameterized by the total injected flux Fnth, spectral indices δL and δH below and above the break energy EB, and low-energy cutoff EC of the injected nonthermal electron flux. In the majority of nonthermal spectra (82%), a single power law was sufficient to fit the observations. We note that the high-energy cutoff EH was not fitted since it is masked by the background. For power-law electron spectra with δ> 1 (as is the case for solar flares), this high cutoff is not relevant for the total flux or energy. In contrast, EC is a crucial parameter for deriving total fluxes and thus energetics. Unfortunately, the low-energy cutoff (visible in the photon spectra as a flattening toward lower energies) is usually masked by the thermal emission, with the rare exceptions of early impulsive flares (e.g., Sui et al. 2007) and late impulsive peaks (Warmuth et al. 2009a). We therefore always obtained the highest low-energy cutoff that was still consistent with the data. Therefore, the nonthermal fluxes, powers, and energies deduced in the following are lower limits.

In total, 1984 individual HXR spectra were fitted, of which 60% had a distinct nonthermal power-law component. Six percent of the spectra were obtained in attenuator state A0, 55% in A1, and 39% in A3. The systematic uncertainties were set to zero during spectral fitting. This resulted in a median reduced χ2 value of 1.2, meaning that the majority of the spectra was fitted very well.

With respect to the thermal component, we also obtained EM and T from the two SXR channels of the GOES X-ray Sensor (XRS), using the code developed by White et al. (2005), which is also based on the CHIANTI package. The pre-flare background was subtracted from the signal. The GOES response is more weighted to lower temperatures than RHESSI, therefore different values will be obtained for EM and T for a nonisothermal plasma. RHESSI- and GOES-derived quantities are identified throughout by the suffixes R and G (e.g., TR and TG for the temperatures derived from RHESSI and GOES data).

2.3. Imaging

Both the thermal and nonthermal source sizes were measured as a function of time for all 24 flares. This is described in detail in Warmuth & Mann (2013a,b), therfore here we give only a brief overview. In all events, a coronal thermal source and a pair of nonthermal chromospheric footpoints were detected. The thermal component was imaged in the 6−12 keV range, the footpoints either in the 25−50, 50−100, or 100−300 keV bands. We used four different imaging methods: CLEAN with uniform and natural weighting (Hurford et al. 2002), Visibiliy Forward Fit (Hurford et al. 2005), and MEM_NJIT (Schmahl et al. 2007).

We then derived various geometric parameters from the images (or, in the case of Visibiliy Forward Fit, directly from the fit). The parameters most relevant for the present study are the total footpoint area, Anth,tot, footpoint separation dFP, and coronal source volume, Vth. The latter parameter can either be derived directly from the thermal HXR source (Vdir) or indirectly from the footpoint areas, and their separation when a semi-circular loop is assumed (Vind). The former method seems to be more reliable (see Warmuth & Mann 2013b), therefore we used Vth = Vdir here.

After rejecting inappropriate imaging methods on a case-by-case basis, we adopted the mean of the values given by the remaining methods as the definite value for the geometric parameters, while the standard deviation provided an estimate of the uncertainties (see Warmuth & Mann 2013a). Typical relative uncertainties are 30% for thermal volumes and 40% for nonthermal footpoint areas.

2.4. Derived parameters

The quantities obtained from the spectral fits and the imaging observations were then used to obtain some of the crucial parameters with respect to energy release and particle acceleration in solar flares. Because of the count statistics, time ranges with good imaging are usually shorter than periods with good spectral fits, and time bins of images are often larger than for spectra. To combine them with spectral parameters, we therefore used a spline interpolation of the geometric parameters. For times earlier than the first good HXR image, we assumed the same geometric parameter values as determined from this first image, and for times later than the last useful HXR image, we adopted the values derived from this last image.

thumbnail Fig. 1

Distribution of the physical parameters of the hot thermal plasma. Plotted are histograms of emission measure EM, temperature T, electron number density n, thermal energy content Eth, radiative energy loss rate Prad, and conductive loss rate Pcond. Dashed lines indicate the median of the distributions. RHESSI-derived parameters are shown in black, while parameters based on GOES are indicated in gray.

With respect to the thermal component, one limitation of this study (which also applies to most other relevant publications) is the assumption of an isothermal plasma when fitting the RHESSI spectra. However, we can obtain some information about any non-isothermality by computing all the thermal parameters defined below from two independent data sets, namely from RHESSI HXR data and GOES SXR data (see Sect. 2.2). The problem of nonisothermality will be further discussed in Sect. 5.

The thermal energy as a function of time t is given by Eth(t)=3kT(t)EM(t) Vth(t)f,\begin{equation} E_\mathrm{th}(t)=3 k T(t) \sqrt{EM(t) \, V_\mathrm{th}(t) f} , \end{equation}(1)with k as Boltzmann’s constant and f the filling factor2. The number density of the thermal electrons is obtained by n(t)=EM(t)/(Vth(t)f).\begin{equation} n(t)=\sqrt{EM(t) / (V_\mathrm{th}(t) f)}. \end{equation}(2)In the following, we assume f = 1 , which implies that the thermal energies are upper limits, while the densities are lower limits. The consequences of smaller filling factors will be addressed in Paper II.

As a result of loss processes, the amount of energy that has to be supplied to the thermal plasma will always be higher than the maximum thermal energy reached during an event, which is therefore only a lower limit for the required energy input. We have to consider radiative and conductive losses. For the radiative energy loss rate, Prad(t), we used the radiative loss rates given by CHIANTI, assuming coronal abundances and Mazzotta et al. (1998) ionization equilibria. In first approximation, the radiative losses are reproduced by the approach of Cox & Tucker (1969), where Prad ~ EM × T− 1 / 2.

The conductive loss rates of the hot plasma through the two footpoints are approximated by Pcond(t)=Anth,tot(t)4dFP(t)π κ0 T(t)7/2,\begin{equation} P_\mathrm{cond}(t)= A_\mathrm{nth,tot}(t) \frac{4}{d_\mathrm{FP}(t) \pi} \, \kappa_\mathrm{0} \,T(t)^{7/2}, \end{equation}(3)with the classical Spitzer coeffcient for thermal conduction, κ0 = 10-6 erg cm-1 s-1 K−7/2, Anth,tot as the total footpoint area, and (dFPπ) / 4 as the loop half-length, which we adopted as the temperature scale length (cf. Phillips et al. 1996; Veronig et al. 2005). However, under typical solar flare conditions, this conductive flux usually saturates (e.g., Gray & Kilkenny 1980), with a local (Campbell 1984) and a nonlocal saturation regime (Karpen & DeVore 1987). We thus adopted the approach of Battaglia et al. (2009) to reduce the conductive fluxes appropriately. The total radiative and conductive losses, Erad and Econd, were finally obtained by integrating over the event duration. These event-integrated energetics will be discussed in Paper II.

In addition to the thermal energy, hot plasma also has potential energy (due to dense plasma that has filled coronal loops with heights of tens of Mm) and kinetic energy (due to plasma flows). In Paper II, we will show that these two components can be neglected for the discussion of energetics.

For the nonthermal component, the total flux of nonthermal electrons as a function of time, Fnth(t) is obtained from the injected electron flux distribution F0(E,t) through Fnth(t)=∫EC(t)EHF0(E,t) dE.\begin{equation} F_\mathrm{nth}(t)= \int_{E_\mathrm{C}(t)}^{E_\mathrm{H}} F_\mathrm{0}(E,t) \, \mathrm{d}E. \end{equation}(4)EH was fixed at 32 MeV, which for all practical purposes is equivalent to having no high-energy cutoff at all. The kinetic power of the electrons, Pnth(t), was obtained by multiplying F0(E,t) with E before integrating. The total injected nonthermal energy in an event, Enth is then given by integrating Pnth(t) over time. We note again that all nonthermal parameters are lower estimates. The consequences of the low-energy cutoff will be further discussed in Paper II.

3. Parameter ranges, distributions, and uncertainties

thumbnail Fig. 2

As in Fig. 1, but showing the distribution of the nonthermal electron parameters. Plotted are histograms of the injected electron flux Fnth and kinetic and power Pnth, low-energy cutoff EC, spectral index of the injected electrons δ (for single and double power law; black histogram for δL and gray for δH), and break energy EB (for a double power law).

3.1. Thermal parameter distributions

Figure 1 shows the distribution of the basic physical parameters characterizing the hot thermal plasma component. The histograms were computed from all fitted time intervals and thus reflect the distribution of parameters over the evolution of all events. Included are the emission measure, temperature, electron density, thermal energy, and the radiative and conductive loss rate (the distribution of thermal volumes is shown in Fig. 11 in Warmuth & Mann 2013a). The histograms show both the RHESSI-derived (shown in black) and GOES-derived (gray) quantities and indicate the medians of the distributions. It is clearly evident that RHESSI yields systematically lower EM and higher T than GOES: EMR = 1045−1.3 × 1050 cm-3 (with the maximum near 1049 cm-3), while EMG = 1047 cm-3−9 × 1050 cm-3 (maximum around 5 × 1049 cm-3). Conversely, TR = 10−45 MK (maximum at 20 MK), while TG = 6−28 MK (maximum at 14 MK). This behavior is a well-known phenomenon (see, e.g., Battaglia et al. 2005; Veronig et al. 2005; Sui et al. 2005; Caspi et al. 2014a), which is generally interpreted as the consequence of a multithermal plasma combined with RHESSI’s higher sensitivity to high-temperature plasmas as compared to GOES. We discuss this problem in detail in Sect. 5.

We now compare RHESSI- and GOES-derived parameter distributions that have not yet been studied in detail. For the thermal electron densities, we obtain nR = 109−4 × 1011 cm-3 (maximum around 1011 cm-3) and nG = 9 × 109−9 × 1011 cm-3 (maximum around 2 × 1011 cm-3). The RHESSI-derived densities are lower because of the lower EM. Most densities are significantly higher than the typical densities of 109−1010 cm-3 derived for microflares by Hannah et al. (2008). The high-density end of the distribution is consistent with the densities of several 1011 cm-3 recently derived by Guo et al. (2012) using HXR imaging spectroscopy in conjunction with a collisional energy loss model.

The instantaneous thermal energies derived from RHESSI and GOES show a quite similar distribution, covering the range 1028−1031 erg (peaking near 1030 erg). In contrast, the energy losses derived from RHESSI and GOES show much larger discrepancies. The RHESSI-derived radiative loss rates (3 × 1022−3 × 1027 erg s-1) are significantly lower than the GOES-derived ones (5 × 1024−4 × 1028 erg s-1), which reflects the differences found for the emission measures. Conversely, the RHESSI-derived conductive loss rates (2 × 1026−9 × 1028 erg s-1) are higher than the corresponding values for GOES (5 × 1025−2 × 1028 erg s-1). This is an effect of the higher temperatures obtained from RHESSI. The distributions of Prad and Pcond thus show that the “RHESSI plasma” loses energy predominantly through conduction (which is on average higher than the radiative loss rate by an order of magnitude), while the “GOES plasma” loses energy equally as a result of radiation and conduction.

The relation between RHESSI- and GOES-derived thermal parameters is discussed in more detail in Sect. 5.

3.2. Nonthermal parameter distributions

Figure 2 shows the distribution of the parameters characterizing the injected electrons as derived from the RHESSI spectra. The injection rate Fnth ranges over nearly four orders of magnitude, from a few 1032 s-1 up to 2.6 × 1036 s-1, with a median of 4.1 × 1034 s-1. The corresponding kinetic power of the accelerated electrons, Pnth, lies in the range of 4 × 1025−3 × 1029 erg s-1, with a median of 2.5 × 1027 erg s-1. These distributions are consistent with those presented by Mann & Warmuth (2011) for a smaller event sample.

The energy flux densities can be obtained by dividing Pnth by the nonthermal footpoint area, Anth,tot. The resulting energy flux density distribution (not shown here) ranges from 2 × 108 to 8 × 1011 erg cm-2 s-1, with a maximum near 1010 erg cm-2 s-1, which incidentally is the threshold for explosive chromospheric evaporation as given by Fisher et al. (1985). We note that the maximum flux densities are almost by an order of magnitude higher than the 1011 erg cm-2 s-1 adopted by Allred et al. (2005) in their radiative hydrodynamic modeling of the atmosphere response to the electron beam. We note that the flux densities are almost certainly significantly underestimated, since the electron flux is a lower estimate, whereas the footpoint area is probably an upper estimate (cf. Warmuth & Mann 2013a).

The low-energy cutoff EC ranges from 11 to 135 keV, with the bulk of the distribution lying between 15 and 40 keV. We stress again that this represents the highest low-energy cutoff that is consistent with the data. For most of the energies below 40 keV, the true cutoff energy may well be lower because of masking by the thermal component. Conversely, the 7% of spectra with cutoff energies above 40 keV can be interpreted as cases where EC is indeed unusually high. For example, the X-class flare of 2005 Jan. 19 showed a late impulsive peak with EC> 100 keV, which either suggests two physically distinct acceleration processes acting in the same event, or alternatively a sudden shift in the characteristic parameters of the accelerator (cf. Warmuth et al. 2009a; Ireland et al. 2013). We note that the median of EC is 24.9 keV, which is consistent with the value found by Sui et al. (2005), who used a combination of spatial, spectral, and temporal analysis to determine the EC for an M-class flares.

We distinguish here between single and broken power-law spectra for electron spectral indices. For the former cases, the spectral slope δ ranges from 2.4 to 11.6, with a pronounced peak between 3 and 4. For the broken power laws, the indices below the break are δL = 1.4−6, and the indices above the break are δH = 2.5−9.3. The distributions have prominent peaks near 3 and 4.5, respectively. This reflects the well-established fact that the spectra become steeper above the break energy (for a counter-example, see Warmuth et al. 2009a). Finally, the break energy shows a broad distribution from 25 to 490 keV, with a large peak around 50 keV. We note that especially the higher break energies may not reflect an actual softening of the electron spectrum above a few hundred keV, but could also be caused by a high-energy cutoff that is significantly below the 32 MeV assumed here. While this issue is relevant for constraining acceleration processes, it is unimportant in terms of terms of energetics, since the overwhelming part of the energy released resides in electrons with several tens of keV.

Table 2

Maximum thermal plasma parameters.

3.3. Parameter uncertainties

How accurately can we determine the various physical parameters? For the spectral fit parameters derived from the RHESSI spectra, the uncertainties provided by OSPEX based on the curvature matrix in parameter space were used (for a detailed discussion of various other methods of error estimation, see Ireland et al. 2013). For EM and T derived from GOES, we adopted the uncertainties that are given by the Temperature and Emission measure-Based Background Subtraction method (TEBBS; see Ryan et al. 2012). The uncertainties of the parameters derived from both spectra and images were computed taking into account both the spectral fit parameter uncertainties and the uncertainties of the source sizes as determined in Warmuth & Mann (2013a). The error bars provided in this paper and in Paper II are all based on this approach.

Typically, the relative uncertainties (defined as the one-sigma uncertainty divided by the parameter value) of the various spectral parameters range from below 1% up to 100%. The median relative uncertainties are 8% for EM and 2% for T as derived from RHESSI (the GOES values are broadly similar). In comparison, the geometric source parameters have larger median uncertainties, e.g., 30% for the thermal volume (Warmuth & Mann 2013a). This means that the errors of parameters derived from spectra and images are dominated by the geometric uncertainties. Hence, the resulting median relative uncertainties are 20% for electron density, 30% for thermal energy, 10% for radiative loss rate, and 30% for conductive loss rate.

For the injected nonthermal electrons, we refrain from providing an error estimate since it is clear that in most cases the true low-energy cutoff is masked by the thermal component. This results in a large fit uncertainty of EC and consequently also for the electron flux and power. Instead, the fitted electron fluxes and corresponding power have to be interpreted as lower limits to the true values.

4. Scaling of physical parameters with flare importance

4.1. Scaling of thermal plasma parameters

After determining the ranges and distributions of the physical parameters, we now proceed to study how the parameters vary with flare importance as characterized by the peak GOES long-channel SXR flux, FG (for example, FG = 10-6 W m-2 corresponds to a C1 flare, and FG = 10-4 W m-2 to an X1 flare). In this work, FG refers to the background-subtracted peak fluxes. For all events, Table 2 gives FG and the maximum values of the emission measures EMR and EMG, temperatures TR and TG, and thermal electron densities nR and nG, as derived from RHESSI and GOES data. The correlation of these maximum thermal parameters with peak GOES flux are shown in Fig. 3. We also show power-law fits obtained with the Bivariate Correlated Errors and intrinsic Scatter (BCES) bisector estimator (Akritas & Bershady 1996), a method based on performing two weighted linear least-squares regression fits on the data, or on the logs of the data – log x on log y and log y on log x. The bisector is obtained as log y = b + αlog x, and the corresponding power law is y=10bxα,\begin{equation} y=10^b x^\alpha, \end{equation}(5)with α as the slope and b as the intercept (for a more detailed discussion, see Warmuth & Mann 2013b).

There is an excellent correlation (rank correlation coefficient R = 0.96) between maximum RHESSI emission measure and GOES peak flux (top left panel in Fig. 3), which can be parametrized as EMR=1052.65×FG0.87\hbox{$EM_\mathrm{R}=10^{52.65} \times F_\mathrm{G}^{0.87}$}. Slightly steeper power laws were found by Battaglia et al. (2005) and Hannah et al. (2008), which are shown in the figure for comparison. We find a reasonable agreement with the values of Battaglia et al. (2005), which at first sight may be surprising because these EMs were derived for the nonthermal HXR peak times, which typically occur earlier than the maximum of the EM. However, this is counterbalanced by the fact that Battaglia et al. (2005) did not use the recent CHIANTI abundances, which has resulted in EMs up to four times larger than the novel method (cf. White et al. 2005). In contrast, the fit of microflares by Hannah et al. (2008) gives emission measures that are systematically higher than ours by a factor of ≈5, although they were derived with a similar method. This may indicate that the power law found for microflares cannot be simply extended to stronger events (cf. the discussion in Ryan et al. 2012).

As for the GOES-derived emission measures (top right in Fig. 3), it is not surprising to find an excellent correlation (R = 1) with peak GOES flux, and a relation that is close to linear, that is, EMG=1053.47×FG0.94\hbox{$EM_\mathrm{G}=10^{53.47} \times F_\mathrm{G}^{0.94}$}. A slightly flatter relation (α = 0.86) has been found by Ryan et al. (2012) for a large sample of GOES flares.

As seen in Fig. 3 (left middle), the RHESSI-derived maximum temperatures show a good correlation with peak GOES flux (R = 0.72) and can be fitted with a power law according to TR=102.04×FG0.14\hbox{$T_\mathrm{R}=10^{2.04} \times F_\mathrm{G}^{0.14}$}. Correlations with temperature are usually fitted with exponential functions in the form of y=γlog10x+d,\begin{equation} y = \gamma \log_{10} x + d, \end{equation}(6)and the corresponding fit for our sample is \begin{lxirformule}$T_\mathrm{R} = 9.46 \log_{10} F_\mathrm{G} + 70.11$\end{lxirformule} (indicated by the dashed gray line). This can be compared to two previous studies that found quite distinct results (see also Fig. 3). Battaglia et al. (2005) found a comparatively shallow increase of TR with flare importance (γ = 3, albeit with a large uncertainty), while Caspi et al. (2014a) derived a much steeper increase (γ = 14). Our relation lies between these two results, but agrees more closely with Caspi et al. (2014a). Our log 10FG − TR relation becomes steeper with increasing GOES class, so that the slope is roughly consistent with the one derived by Battaglia et al. (2005) for weaker flares, while X class flares agree more closely with the slope derived by Caspi et al. (2014a). This indicates that the very different slopes of the two previous studies have resulted form a selection effect that is due to the different event samples, namely flares from class B1 to M6 in Battaglia et al. (2005), and predominantly stronger events from C9.2 to X10 in Caspi et al. (2014a). Our study thus shows that there are not two distinct flare populations for the relation between maximum TR and GOES class. The relation is better reproduced by a power law than by an exponential function.

We find a well-defined (R = 0.88) but significantly shallower relation for the GOES maximum temperatures, that is, TG=101.79 × FG0.12\hbox{$T_\mathrm{G}=10^{1.79} \,\times\, F_\mathrm{G}^{0.12}$} and TG = 5.37log 10FG + 42.26. This relation agrees well with previous results of Feldman et al. (1996), Ryan et al. (2012), and Caspi et al. (2014a), who found slopes of γ = 5.4, 3.85, and 4.6, respectively.

The bottom panels in Fig. 3 show the maximum electron densities as derived from RHESSI and GOES. In both cases, a moderate correlation with peak GOES flux is found (R = 0.53 and 0.66, respectively), and the fitted power laws are quite similar with α = 0.37.

In summary, we find that the maximum emission measures (both RHESSI and GOES derived) correlate very well with the GOES peak flux, while maximum temperatures also show a good correlation. In the latter case, RHESSI shows a much steeper rise of T with flare importance than GOES. The maximum thermal electron densities show a moderate correlation, with very similar slopes for RHESSI and GOES. Warmuth & Mann (2013b) found the thermal volumes at the time of the GOES peak flux to be reasonably well correlated (R = 0.74) with the peak flux according to Vth=1029.71×FG0.74\hbox{$V_\mathrm{th}=10^{29.71} \times F_\mathrm{G}^{0.74}$}. Thus flares of a larger importance are characterized by a higher temperature and emission measure, where the latter quantity results from both a higher density and a larger volume.

thumbnail Fig. 3

Thermal parameters as a function of background-subtracted GOES peak flux. Shown are the maxima of the emission measure EM (top), temperature T (middle), and electron density n (bottom) derived from RHESSI (left column) and GOES data (right column). Also shown are power-law fits obtained with the BCES bisector method (full black line). The slope α and intercept b of the obtained power law are shown together with the rank correlation coefficient R . Additionally, an exponential fit is shown for the temperatures (dashed gray lines), and the corresponding fit parameters γ and d are indicated (see main text). Where appropriate, correlations found by the following studies are indicated by dash-dotted lines: Feldman et al. (1996), Battaglia et al. (2005), Hannah et al. (2008), Ryan et al. (2012), Caspi et al. (2014a).

4.2. Scaling of nonthermal electron parameters

Table 3

Parameters of the injected electrons at the time of the nonthermal HXR peak.

thumbnail Fig. 4

As in Fig. 3, but showing the parameters of the injected electrons at the peak time of the nonthermal HXR flux as a function of background-subtracted GOES peak flux. Shown is the injected electron flux Fnth and corresponding kinetic power Pnth, the low-energy cutoff EC, and the spectral index below the spectral break δL.

Battaglia et al. (2005) have established correlations between GOES peak flux and nonthermal HXR spectral parameters. Here, we advance from this and determine correlations with the injected nonthermal electrons. We considered the electron parameters at the time of maximum nonthermal HXR flux (defined as the 25−50 or 50−100 keV band, according to flare importance). Table 3 lists the electron fluxes, powers, low-energy cutoffs, and spectral indices below the spectral break (only two cases showed an actual break). The correlation of these nonthermal parameters with peak GOES flux are shown in Fig. 4.

Both the electron fluxes and corresponding kinetic powers at the time of the nonthermal HXR peak show a good correlation with the peak GOES flux (R = 0.81 and 0.88, respectively). The power scales according to Pnth=1031.64×FG0.84\hbox{$P_\mathrm{nth}=10^{31.64} \times F_\mathrm{G}^{0.84}$}. The correlation is weaker for the low-energy cutoff (R = 0.59), where a slight increase with GOES flux is found (α = 0.14). This increase mainly reflects the more pronounced thermal HXR component in flares with greater importance, and not necessarily truly higher cutoff energies in stronger flares. Finally, the electron spectral index δL shows a moderate tendency (R = −0.47) to decrease with peak GOES flux (α = −0.16). In contrast to this, Battaglia et al. (2005) did not find a statistically significant correlation between GOES peak flux and photon spectral index.

We conclude that the nonthermal electron fluxes and corresponding powers at the time of the HXR flux peak3 are highly correlated with the GOES importance, which is consistent with the standard flare scenario where the SXR-emitting thermal component is generated by beam-driven chromospheric evaportion (see Sect. 1).

5. Thermal plasma component derived from RHESSI and GOES

5.1. Comparison of maximum values

We now come back to the thermal plasma component, where we have confirmed that RHESSI and GOES data yield different thermal parameters. In this section, we will investigate how this observational fact can be interpreted.

Let us first quantify the difference between RHESSI- and GOES-derived plasma parameters in more detail. In Fig. 5, the maximum emission measures and maximum temperatures given by RHESSI are plotted versus the corresponding maximum parameters as derived from GOES. It should be noted that RHESSI and GOES give different peak times for the thermal parameters. EMR tends to peak before EMG with a median time difference of 0.8 min, while TR peaks even more clearly before TG with a median time difference of 2 min. In Sect. 5.1, we will therefore continue with a comparison of cotemporal values.

The maximum EMs are highly correlated (R = 0.96) and follow a power law according to EMR=103.49×EMG0.92\hbox{$EM_\mathrm{R}=10^{3.49} \times EM_\mathrm{G}^{0.92}$}. Conversely, the maximum temperatures are also well-correlated (R = 0.72). Within the uncertainties, the relation is consistent with a linear one (α ≈ 1), and a linear fit in the form of y=ϵx+g\begin{equation} y= \epsilon x + g \end{equation}(7)yields TR = 1.78TG − 4.61. This relation lies between TR = 1.12TG − 3.12 found by Battaglia et al. (2005) and TR = 3.2TG − 28.6 derived by Caspi et al. (2014a). Caspi et al. (2014a) have suggested that their steeper relation may be appropriate for strong flares, while the shallower relations of Battaglia et al. (2005) and also Hannah et al. (2008) may reflect the weaker flares of those event samples (see also the discussion in Sect. 4.1). While we do indeed find a tendency for an increase of steepness for stronger flares, this is not significant in the maximum temperature data set (α = 1.11 ± 0.16).

thumbnail Fig. 5

Comparison of maximum thermal parameters derived from RHESSI to parameters as given by GOES. Shown are the maximum emission measures EM (left) and temperatures T (right). We also show power-law fits (full lines), the slope and intercept of the obtained power law, α and b, and the rank correlation coefficient, R. For the temperatures, a linear fit (dashed gray line, with slope ϵ and intercept g) is also shown. The dotted black line denotes x = y. At the bottom of the plots, the median ratio (M [ .../... ]) of the parameters derived from RHESSI and GOES is shown. The temperature correlations found by Battaglia et al. (2005) and Caspi et al. (2014a) are indicated by dash-dotted lines.

5.2. Comparison of cotemporal values

thumbnail Fig. 6

As in Fig. 5, but comparing thermal parameters derived from RHESSI to cotemporal parameters as given by GOES for all analyzed time steps (giving 1 951 value pairs). Shown is the emission measure EM, temperature T, electron density n, thermal energy Eth, radiative energy loss rate Prad, and conductive loss rate Pcond. For clarity, no error bars are plotted here.

After comparing maximum values, we now investigate the relation between various RHESSI- and GOES-derived parameters obtained at the same times. This is shown in Fig. 6 for all events and time steps. For all parameters, the values given by RHESSI and GOES are very well correlated (R ≥ 0.87).

Compared to the maximum emission measures, the cotemporal EMR rises more steeply with EMG (α> 1), which is mainly caused by the steepening of the relation at lower EMs. While the median ratio is EMR/EMG = 0.21, below EMG ≈ 1048 cm-3, EMR is smaller than EMG by one to two orders of magnitude. Conversely, the relation for the cotemporal temperatures is very similar to the one found for the maximum temperatures. However, the much larger number of value pairs now clearly shows the nonlinearity of the relation (α = 1.16 ± 0.01). We note that for TG ≤ 20 MK and TR ≤ 25 MK, the bulk of the data points is consistent with the more shallow slope found by Battaglia et al. (2005) for the maximum temperatures, while the slope increases significantly for higher temperatures. This again supports the notion that the vastly different slopes found by Battaglia et al. (2005) and Caspi et al. (2014a) have been caused by a selection effect.

The electron densities show a very similar behavior as the emission measures, with α = 1.2. On average, nR is half as large as nG, but for low densities nR ≤ 1010 cm-3 this ratio decreases to 0.1.

Continuing with the energetics of the thermal component, we find that the thermal energies are extremely well correlated (R = 0.99) and show a nearly linear relation (α = 1.04). On average, Eth,R amounts to two-thirds of Eth,G. The radiative energy loss rate is mainly dependent on the emission measure, thus we find a similar nonlinear relation (α = 1.25). The radiative losses derived from RHESSI are significantly lower than from GOES, with a median ratio Prad,R/Prad,G = 0.15. The relation becomes much steeper at low loss rates, where Prad,R/Prad,G can be as low as 0.01. The situation is different for the conductive loss rate. There, the relation is almost linear (α = 1.03), and the losses are higher for RHESSI, with a median ratio of Pcond,R/Pcond,G = 2.4. This is clearly an effect of the higher temperatures given by RHESSI.

5.3. Evolutionary effects and multithermality

thumbnail Fig. 7

As in Fig. 5, but showing the relation between thermal parameters derived from cotemporal RHESSI and GOES data for three different evolutionary stages of the flares. Compared are the emission measures (left column) and temperatures (middle column). Additionally, the corresponding thermal volumes are compared to the volume at the GOES SXR flux peak, Vth,peak (right column). The three evolutionary stages correspond to the times of the maximum RHESSI temperature (top row), the SXR peak (middle row), and the latest analyzed time interval (t = tend; bottom row). RHESSI gives lower emission measures and higher temperatures than GOES in the early flare phase, when the thermal volumes are comparatively large.

The comparisons above have shown that RHESSI gives lower emission measures and higher temperatures than GOES for all time intervals. For the emission measures, we have noted deviations from the fitted power law at low EMs, which are typically observed in the early flare phase. This suggests that there may be an evolutionary aspect to the relation of RHESSI- and GOES-derived parameters. To investigate this possibility further, Fig. 7 shows the emission measures (left column) and temperatures (middle column) as given by RHESSI and GOES for three distinct evolutionary phases. The top row shows the relations for the time of the maximum temperature given by RHESSI, which occurs in the early impulsive phase. The middle row corresponds to the time of peak GOES flux, while the bottom row shows the last time interval that could be analyzed in each flare (late phase).

In the early impulsive phase, the emission measure ratio is low (median EMR/EMG = 0.11), and the EM relation is clearly nonlinear (α = 1.31). Both at flare peak and in the late phase, the EM ratio is higher (with medians of EMR/EMG = 0.25 and 0.22, respectively), and the EM relation is closer to linear. Conversely, the temperature ratio in the early impulsive phase is very high (with a median of TR/TG = 1.89), while it decreases to a median of ≈1.35 at flare peak and end (the latter ratio closely agrees with the results of Ryan et al. 2014). In summary, while the parameter relations at flare peak and in the late phase are consistent with what we found for the whole flare durations (cf. Fig. 6), the early impulsive phase is characterized by a larger difference between RHESSI- and GOES-derived thermal parameters. Taking the first analyzed time interval instead of the time of maximum RHESSI temperature gives comparable results, which confirms that the RHESSI plasma has a significantly lower EM and is much hotter than the GOES plasma during the entire early phase.

Commonly, the fact that RHESSI gives lower EMs and higher temperatures than GOES is interpreted as the signature of a multithermal plasma, which can be characterized by a differential emission measure (DEM). Evidence for multithermal plasmas has been found by EUV and SXR spectroscopy (e.g., Dere & Cook 1979; McTiernan et al. 1999; Chifor et al. 2007) and by RHESSI HXR observations (Aschwanden 2007; Caspi & Lin 2010; Jain et al. 2011). Recent studies using multiband EUV images from SDO/AIA (Fletcher et al. 2013; Ryan et al. 2014; Aschwanden et al. 2015) and EUV spectra by SDO/EVE (Warren et al. 2013; Kennedy et al. 2013; Caspi et al. 2014b) and by Hinode/EIS (Graham et al. 2013) have finally provided well-constrained DEM distributions and have shown conclusively that the thermal component in solar flares is indeed multithermal in nature. The DEM distributions derived in these studies tend to be broad, and often a bimodal distribution is found, with a low-temperature peak at 1–2 MK (which is probably due to non-flaring foreground and background coronal material; cf. Battaglia & Kontar 2012; Krucker & Battaglia 2014), and a broad high-temperature peak ranging from ≈10 MK up to some ≤30 MK. In a few events, there is evidence for an additional super-hot (T ≥ 30 MK) component (Caspi & Lin 2010; Caspi et al. 2014a) from RHESSI, but this has not yet been corroborated by EUV spectroscopy.

Since RHESSI’s response is more weighted towards higher temperatures (T ≥ 10 MK, with sensitivity increasing with temperatures) than GOES, this implies that it will detect a smaller high-temperature component (or high-temperature tail) of the DEM distribution in addition to a more dominant bulk of the distribution at lower temperatures that is primarily detected in the GOES channels. An isothermal fit to the RHESSI data will therefore yield higher temperatures (and lower EMs) than for GOES data, which are not sensitive to plasmas hotter than ≈25 MK.

During most of the flare’s evolution, the shape of the DEM seems to change only moderately. For instance, this can be deduced by comparing the TR/TG ratios at flare peak and in the late phase in Fig. 7, which are almost equal. However, the particularly pronounced difference in the early impulsive phase implies a significantly different shape of the DEM distribution, namely a broader distribution. This could result from an additional super-hot component (cf. Caspi & Lin 2010) or an enhanced high-temperature tail of the distribution. Indeed, EUV spectroscopy has confirmed that there is an evolution from broader DEM distributions in the early phase to narrower ones in the later phase (cf. Caspi et al. 2014b) .

The question now arises why the strong heating that is required to generate the particularly hot RHESSI plasma should be present only early in the event, even well before the nonthermal energy input has reached its maximum. Considering HXR imaging data may offer a clue to the problem. Warmuth & Mann (2013b) showed that in 87% of the events the early flare phase is characterized by comparatively large thermal sources that are located rather high in the corona and show an initial downward motion. For comparison, the right column in Fig. 7 shows the thermal source volumes normalized by the source volume at GOES flare peak for the three evolutionary stages. This clearly shows that most of the early source volumes are comparatively large.

We have suggested that this behavior might be due to the initial contraction of newly reconnected magnetic loops, but we also hinted at a second possibility that is connected with directly heated coronal plasma versus evaporated thermal plasma. Early in the event, nonthermal energy input by particle beams is low, and not much chromospheric plasma will be evaporated. Instead, the thermal X-ray emission will be dominated by coronal plasma that is heated directly in situ (for spectroscopic evidence of direct heating, see Brosius 2012) by various mechanisms, including slow-mode shocks in a Petschek-type reconnection scenario (cf. Petschek 1964; Cargill & Priest 1982; Longcope & Guidoni 2011), fast-mode shocks in the reconnection outflow (cf. Mann et al. 2009; Warmuth et al. 2009a), betatron heating in a collapsing magnetic trap (e.g., Karlický & Kosugi 2004), and turbulence (e.g., Miller et al. 1996; Oreshina & Oreshina 2013). This directly heated plasma will be observed as a hot and comparatively tenuous source by RHESSI, which is exactly what we observe: initially, TR> 20 MK, and the densities are a third of the densities at flare peak. Emission from this directly heated component does not contribute significantly to the GOES bands, which are already dominated by the evaporated plasma early in the event. This explains the much lower densities and higher temperatures measured by RHESSI in the early phase.

As the flare progresses, more chromospheric plasma is evaporated. Since the temperature responses of GOES and RHESSI overlap, the evaporated material contributes more and more to the RHESSI signal. RHESSI now observes both the directly heated coronal plasma and the cooler and denser evaporated material. As a result, the temperatures and emission measures derived from isothermal fits of the RHESSI spectra are closer to those given by GOES at flare peak and in the later phase.

With the additional spectral information, we can now confirm that the scenario proposed in Warmuth & Mann (2013b) is indeed consistent with the observations. We can thus provide a physical interpretation for the well-known differences in thermal parameters derived from RHESSI and GOES, namely the combination of a cooler (T ≈ 10−25 MK) evaporated component and a hotter (T> 25 MK) directly heated coronal component. In all likelihood, the directly heated component is present at least during the whole impulsive phase, but it is observable by RHESSI only as long as the evaporated plasma does not dominate the emission. Otherwise, both the RHESSI images and spectra will result from a combination of both plasma components that will be increasingly dominated by the evaporated plasma. The two components are not necessarily isothermal.

We emphasize that the hotter (presumably directly heated) component is apparently present in all flares, not just in strong X-class flares. The temperature difference increases steadily with GOES class, but we cannot assign a temperature threshold (e.g., TR = 30 MK as adopted by Caspi et al. 2014a) above which a flare could be called “super-hot”. Likewise, emission measure or density do not show any apparent jumps as a function of temperature. Based on back-extrapolating the temperature relations to lower GOES classes and supported by the shallower relations derived by Battaglia et al. (2005) and Hannah et al. (2008), Caspi et al. (2014a) have suggested that flares weaker than C4 may lack a directly heated component. However, in our data we do not see any drastic change in the temperature relation around GOES class C4. Instead, there is a smooth flattening of the TR − FG relation toward weaker flares, so that even in microflares TR will be higher than TG. We conclude that X-class flares that have particularly high RHESSI temperatures are not physically distinct from weaker flares. Instead, all studied flares are consistent with a “super-hot" (used in the relative sense) directly heated thermal plasma component in addition to a hot evaporated component.

6. Relation between thermal plasma and injected nonthermal electrons

thumbnail Fig. 8

Thermal flare parameters plotted versus cotemporal energy input rate by nonthermal electrons, Pnth. Shown are temperatures T (top), emission measure increase rates dEM/ dt (middle), and thermal energy increase rate dEth/ dt (bottom), derived from RHESSI (left column) and GOES data (right column). Also shown are power-law fits (full lines), the slope and intercept of the obtained power law, α and b, and the rank correlation coefficient, R. For T, an exponential fit (dashed gray line) is also shown together with the fit parameters γ and d. For dEth/ dt, the dotted line denotes x = y, and the median ratio M of the thermal energy increase and the nonthermal input is indicated.

In Sect. 4.1 we have established that flares of higher GOES importance are characterized by larger maximum electron fluxes, which is consistent with beam-driven evaporation. If this is really the case, we expect to see a good correlation between thermal flare parameters and the nonthermal energy input. Indeed, Battaglia et al. (2005) have found that a higher photon flux at 35 keV is associated with higher plasma temperatures and higher emission measures and GOES flux. Here, we proceed by considering the actual energy input by the injected electrons instead of just the photon flux at a fixed energy.

In the first row of Fig. 8, the RHESSI and GOES temperatures are plotted versus the cotemporal injected power Pnth. Both temperatures show a reasonable correlation with nonthermal energy input (R = 0.56 and 0.61, respectively), but in contrast to TG, the relation between TR and Pnth bifurcates into two branches at higher temperatures and energy input rates. The split occurs above Pnth ≈ 1027 erg s-1 and TR ≈ 25 MK. The high-temperature branch primarily results from X-class flares, in particular from the impulsive phase before the GOES flux peak.

In contrast to temperature, the emission measures do not show a correlation with energy input rate. This is expected because the amount of plasma that is contributing to the observed EM accumulates over time and is therefore not related to the instantaneous energy input. However, we can consider the time derivative of the EM: assuming that beam-driven evaporation is taking place, and neglecting losses, the increase of emission measure per time interval should be proportional to the nonthermal energy input rate. The middle row in Fig. 8 shows dEM/ dt versus Pnth. Here, we include only positive values, that is, actual increases of EM, and only the time ranges up to the GOES flux peak, since later on loss terms become more important. Both RHESSI- and GOES-derived emission measure gains show reasonable correlation with nonthermal energy input. The GOES parameters show a significantly tighter correlation (R = 0.66, as opposed to R = 0.5 for RHESSI).

We now consider an even more direct way of studying the relation between thermal plasma and nonthermal electrons, namely by comparing the time derivative of the thermal energy dEth/ dt to the cotemporal nonthermal energy input Pnth. This is shown in the bottom row of Fig. 8 (again, only positive values are included). For both RHESSI- and GOES derived thermal energies, we find a moderate correlation (R = 0.52) with the energy input by nonthermal electrons. Both relations are close to linear (α ≈ 1), and the thermal energy increase rate is lower than the input rate by electrons for most time intervals (the median ratio is about a third). Both these characteristics are consistent with the notion of chromospheric evaporation by nonthermal electron beams. The fact that the correlation is only moderately strong could result from several effects (see also the discussion of the Neupert effect in Veronig et al. 2005). With regard to the input, the nonthermal electron power values are lower estimates because of the unknown low-energy cutoff, and an additional input by ions is not considered here. The uncertainty of thermal energies is at the 30% level. Another effect is that the injected energy will not be exclusively converted to thermal energy of the hot plasma, but will partially go into plasma flows (see discussion in Paper II). Additionally, the hot plasma suffers both radiative and conductive losses. The magnitude of these losses varies with time, which will lead to a weakening of the correlation between dEth/ dt and Pnth. Finally, various forms of non-beam mechanisms could contribute to plasma heating. In these cases, conduction will not only act as a loss term, but also as a gain term that is due to conduction-induced chromospheric evaporation. This matter will be discussed in more detail in Paper II.

We conclude that the correlations of temperatures as well as emission measure and thermal energy increase rates with nonthermal electron power support the notion of beam-driven chromospheric evaporation and heating. However, how can we interpret the bifurcation seen in the relation between RHESSI-derived temperatures and nonthermal input? We note that the high-temperature branch mainly results from the impulsive phase of X-class flares, for which we have derived a particularly large difference between TR and TG. In Sect. 5.3 we have attributed this to a directly heated coronal plasma component that contributes to the RHESSI response, but is not detected in the GOES data. The high-temperature branch could thus be due to the contribution of the directly heated component. This branch shows a significantly steeper increase with nonthermal power than the GOES temperatures do, whereas the slope of the low-temperature branch is consistent with GOES. This implies that the temperature of the evaporated plasma rises comparatively slowly with nonthermal electron power, whereas the rise is significantly more steep for the directly heated component.

In addition, we note that the scatter of the points in the high-temperature branch is larger than in the low-temperature one, and that the correlation of the emission measure increase rate is weaker for RHESSI than for GOES. This implies a loser relation with the nonthermal electron input and thus suggest the contribution of an additional plasma component that is not (primarily) being heated and/or evaporated by an electron beam, but is instead heated in situ by the processes mentioned in Sect. 5.3.

7. Summary and conclusions

Constraining the physics of solar flares – with respect to both energy release and particle acceleration – requires a comprehensive characterization of the physical parameters of the thermal plasma and the accelerated nonthermal particles, which we have done in this paper by using HXR imaging and spectroscopy from RHESSI. In contrast to most previous studies, we have (a) derived these parameters for sample of flares with a wide range of GOES importance (24 flares from C3.4 to X17.2); (b) considered whole time series of parameters for each flare instead of using maximum parameters alone; (c) conducted a detailed geometric characterization of HXR source sizes (see Warmuth & Mann 2013a,b); and (d) used both RHESSI and GOES data to derive the thermal flare parameters. We derived the ranges and distributions of the physical parameters, their scaling with flare importance (i.e., peak GOES flux), the relation between RHESSI- and GOES-derived thermal parameters, and the relation between thermal and nonthermal parameters. The main results from this analysis are summarized below.

Maximum emission measure and temperature are strongly correlated with GOES peak X-ray flux for both RHESSI- and GOES-derived parameters. RHESSI data show a significantly steeper rise of temperature with flare importance than GOES data. While previous studies have derived very dissimilar slopes for the RHESSI temperature increase, we showed that this has most probably resulted from selection effects. The higher maximum emission measures of stronger flares arise from a combination of higher densities and larger thermal source volumes, with the latter contribution being more important.

In all events and time intervals, RHESSI gives systematically higher temperatures and lower emission measures than GOES, which is a signature of a multithermal plasma. As derived from RHESSI, strong X-class flares are characterized by emission measures of up to ≈1050 cm-3, temperatures of 45 MK, and electron densities of 4 × 1011 cm-3. The corresponding maximum GOES values are 9 × 1050 cm-3 for EM, 28 MK for temperature, and 9 × 1011 cm-3 for density. RHESSI yields significantly higher conductive energy losses than radiative losses, while the two loss terms are comparable when considering GOES data.

RHESSI yields higher temperatures and lower emission measures than GOES, which is particularly pronounced in the early impulsive phase, where the thermal HXR sources tend to be comparatively large and are located higher in the corona (see Warmuth & Mann 2013b). We interpret these characteristics in terms of a combination of a cooler (T ≈ 10−25 MK) evaporated plasma component that is detected by both RHESSI and GOES, and a hotter (T> 25 MK) directly heated coronal component that is only observed by RHESSI. The hotter component is present at least during the whole impulsive phase, but it is clearly observable by RHESSI only as long as the evaporated plasma does not dominate the emission. Generally, both the RHESSI images and spectra result from a combination of the two plasma components that will be increasingly dominated by the evaporated plasma as the flare progresses. The hotter component is present in both weak and strong flares, but is more pronounced in stronger flares.

The maximum nonthermal electron fluxes and corresponding energy input rates are highly correlated with the GOES importance, which is consistent with the standard flare scenario where the SXR-emitting thermal component is generated by beam-driven chromospheric evaporation. Higher maximum fluxes tend to be characterized by harder electron spectra. Strong X-class flares show electron fluxes of up to ≈2 × 1036 s-1, kinetic energy input rates of 3 × 1029 erg s-1, and energy flux densities of 8 × 1011 erg cm-2 s-1.

Temperature as well as the time derivative of emission measure and thermal energy correlates with the nonthermal electron energy input rate, which again supports the notion of beam-driven evaporation. However, the relation between the RHESSI temperature and nonthermal input rate bifurcates into two branches at higher temperatures. The high-temperature branch shows a steeper slope and primarily results from the impulsive phase of X-class flares. These characteristics again suggest that RHESSI observes a coronal plasma component in the impulsive flare phase that is not evaporated by an electron beam, but instead heated in situ by different mechanisms, such as slow-mode shocks, fast-mode shocks, and turbulence.

Using both RHESSI and GOES data, in combination with considering not only peak times, but the whole flare duration, has allowed us to better understand the nature of the thermal flare plasma and its heating mechanisms. We confirm the presence of a multithermal plasma, which has been invoked recently by EUV and X-ray spectroscopy. In particular, the comparison of thermal flare parameters with the energy input by nonthermal electrons has provided novel evidence for both chromospheric evaporation and direct coronal heating. In the companion Paper II, we will continue our investigation by focusing on energetics. In particular, we will study the energy partition in the flares, and how the partition changes with flare importance. This will allow us to better understand the energy release, transport, and conversion processes in solar flares.

2

We also used the RHESSI-derived thermal source volumes for the parameters obtained from GOES.

3

Very similar results are obtained when the peak time of the nonthermal power is chosen instead.

Acknowledgments

The work of A. W. was supported by DLR under grant No. 50 QL 0001. The authors are grateful to the RHESSI Team for the free access to the data and the development of the software.

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All Tables

Table 1

Overview of the 24 analyzed flares.

In the text
Table 2

Maximum thermal plasma parameters.

In the text
Table 3

Parameters of the injected electrons at the time of the nonthermal HXR peak.

In the text

All Figures

thumbnail Fig. 1

Distribution of the physical parameters of the hot thermal plasma. Plotted are histograms of emission measure EM, temperature T, electron number density n, thermal energy content Eth, radiative energy loss rate Prad, and conductive loss rate Pcond. Dashed lines indicate the median of the distributions. RHESSI-derived parameters are shown in black, while parameters based on GOES are indicated in gray.
In the text
thumbnail Fig. 2

As in Fig. 1, but showing the distribution of the nonthermal electron parameters. Plotted are histograms of the injected electron flux Fnth and kinetic and power Pnth, low-energy cutoff EC, spectral index of the injected electrons δ (for single and double power law; black histogram for δL and gray for δH), and break energy EB (for a double power law).
In the text
thumbnail Fig. 3

Thermal parameters as a function of background-subtracted GOES peak flux. Shown are the maxima of the emission measure EM (top), temperature T (middle), and electron density n (bottom) derived from RHESSI (left column) and GOES data (right column). Also shown are power-law fits obtained with the BCES bisector method (full black line). The slope α and intercept b of the obtained power law are shown together with the rank correlation coefficient R . Additionally, an exponential fit is shown for the temperatures (dashed gray lines), and the corresponding fit parameters γ and d are indicated (see main text). Where appropriate, correlations found by the following studies are indicated by dash-dotted lines: Feldman et al. (1996), Battaglia et al. (2005), Hannah et al. (2008), Ryan et al. (2012), Caspi et al. (2014a).
In the text
thumbnail Fig. 4

As in Fig. 3, but showing the parameters of the injected electrons at the peak time of the nonthermal HXR flux as a function of background-subtracted GOES peak flux. Shown is the injected electron flux Fnth and corresponding kinetic power Pnth, the low-energy cutoff EC, and the spectral index below the spectral break δL.
In the text
thumbnail Fig. 5

Comparison of maximum thermal parameters derived from RHESSI to parameters as given by GOES. Shown are the maximum emission measures EM (left) and temperatures T (right). We also show power-law fits (full lines), the slope and intercept of the obtained power law, α and b, and the rank correlation coefficient, R. For the temperatures, a linear fit (dashed gray line, with slope ϵ and intercept g) is also shown. The dotted black line denotes x = y. At the bottom of the plots, the median ratio (M [ .../... ]) of the parameters derived from RHESSI and GOES is shown. The temperature correlations found by Battaglia et al. (2005) and Caspi et al. (2014a) are indicated by dash-dotted lines.
In the text
thumbnail Fig. 6

As in Fig. 5, but comparing thermal parameters derived from RHESSI to cotemporal parameters as given by GOES for all analyzed time steps (giving 1 951 value pairs). Shown is the emission measure EM, temperature T, electron density n, thermal energy Eth, radiative energy loss rate Prad, and conductive loss rate Pcond. For clarity, no error bars are plotted here.
In the text
thumbnail Fig. 7

As in Fig. 5, but showing the relation between thermal parameters derived from cotemporal RHESSI and GOES data for three different evolutionary stages of the flares. Compared are the emission measures (left column) and temperatures (middle column). Additionally, the corresponding thermal volumes are compared to the volume at the GOES SXR flux peak, Vth,peak (right column). The three evolutionary stages correspond to the times of the maximum RHESSI temperature (top row), the SXR peak (middle row), and the latest analyzed time interval (t = tend; bottom row). RHESSI gives lower emission measures and higher temperatures than GOES in the early flare phase, when the thermal volumes are comparatively large.
In the text
thumbnail Fig. 8

Thermal flare parameters plotted versus cotemporal energy input rate by nonthermal electrons, Pnth. Shown are temperatures T (top), emission measure increase rates dEM/ dt (middle), and thermal energy increase rate dEth/ dt (bottom), derived from RHESSI (left column) and GOES data (right column). Also shown are power-law fits (full lines), the slope and intercept of the obtained power law, α and b, and the rank correlation coefficient, R. For T, an exponential fit (dashed gray line) is also shown together with the fit parameters γ and d. For dEth/ dt, the dotted line denotes x = y, and the median ratio M of the thermal energy increase and the nonthermal input is indicated.
In the text

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