Thermal-nonthermal energy partition in solar flares derived from X-ray, EUV, and bolometric observations
aa r X i v : . [ a s t r o - ph . S R ] N ov Astronomy & Astrophysicsmanuscript no. 39529corr_mod © ESO 2020November 10, 2020
Thermal-nonthermal energy partition in solar flares derived fromX-ray, EUV, and bolometric observations
Discussion of recent studies
A. Warmuth and G. Mann Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, GermanyReceived < date > / Accepted < date > ABSTRACT
Context.
In solar flares, energy is released impulsively and is partly converted into thermal energy of hot plasmas and kinetic energyof accelerated nonthermal particles. It is crucial to constrain the partition of these two energy components to understand energy releaseand transport as well as particle acceleration in solar flares. Despite numerous e ff orts, no consensus on quantifying this energy balancehas yet been reached. Aims.
We aim to understand the reasons for the contradicting results on energy partition obtained by various recent studies. The over-arching question we address is whether there is su ffi cient energy in nonthermal particles to account for the thermal flare component. Methods.
We considered five recent studies that address the thermal-nonthermal energy partition in solar flares. Their results arereviewed, and their methods are compared and discussed in detail.
Results.
The main uncertainties in deriving the energy partition are identified as (a) the derivation of the di ff erential emission measure(DEM) distribution and (b) the role of the conductive energy loss for the thermal component, as well as (c) the determination of thelow-energy cuto ff for the injected electrons. The bolometric radiated energy, as a proxy for the total energy released in the flare, is auseful independent constraint on both thermal and nonthermal energetics. In most of the cases, the derived energetics are consistentwith this constraint. There are indications that the thermal-nonthermal energy partition changes with flare strength: in weak flares,there appears to be a deficit of energetic electrons, while the injected nonthermal energy is su ffi cient to account for the thermalcomponent in strong flares. This behavior is identified as the main cause of the dissimilar results in the studies we considered. Thechanging partition has two important consequences: (a) an additional direct (i.e. non-beam) heating mechanism has to be present, and(b) considering that the bolometric emission originates mainly from deeper atmospheric layers, conduction or waves are required asadditional energy transport mechanisms. Key words.
Sun: flares – Sun: X-rays, gamma rays – acceleration of particles
1. Introduction
In solar eruptive events such as flares (e.g. Fletcher et al.2011) and coronal mass ejections (CMEs; e.g., Chen 2011;Webb & Howard 2012), a large amount of energy ( ≤ erg)that is originally stored in nonpotential coronal magnetic fieldsis released impulsively and converted into other forms of en-ergy, presumably triggered by magnetic reconnection (e.g.,Priest 1982). Newly reconnected magnetic field lines rapidlymove away from the reconnection site, taking the plasma withthem. This forms two outflow jets. The outflow may also beheated, for example, by standing slow-mode shocks that sep-arate the inflow from the outflow in Petschek-style reconnec-tion (cf. Cargill & Priest 1982), as well as by fast-mode ter-mination shocks. Somewhere in this geometry, e ffi cient parti-cle acceleration due to an as yet poorly understood mecha-nism (cf. Zharkova et al. 2011; Mann 2015), is taking place. Thedownward-moving field lines form flaring loops that becomefilled with dense plasma that is evaporated from the chromo-sphere by strong heating. In major eruptive events, the unstablemagnetic structure forms a flux rope (in which a filament may beembedded) that is expelled from the corona. The upward-movingreconnected field lines become part of the flux rope, which is Send o ff print requests to : A. Warmuth, e-mail: [email protected] subsequently observed as a CME. This general scenario is sup-ported by many observations and represents our standard modelof a solar eruption. In its 2D form, it is known as the CSHKPmodel (cf. Carmichael 1964; Sturrock 1966; Hirayama 1974;Kopp & Pneuman 1976). Recently, the standard model has beenextended to 3D (see Aulanier et al. 2012, 2013; Janvier et al.2013).It is quite evident that a solar eruptive event is characterizedby a complex scenario of energy release, transport, and conver-sion. This starts with the free magnetic energy in the flaring ac-tive region and continues with the amount of energy that is actu-ally released, the kinetic and thermal energy of the reconnectionoutflow jets, the kinetic energy in accelerated particles and inevaporation flows, the thermal energy of evaporated plasma, andradiative and conductive energy losses of various plasmas. In thecase of eruptive flares, the kinetic and potential energy of CMEs,the energy of CME-driven shocks, and finally, the energy contentin solar energetic particles (SEPs) is added to this.A quantitative characterization of the di ff erent forms of en-ergy therefore represents a crucial observational constraint formodels of solar eruptions in general, as well as for magnetic re-connection, heating, and particle-acceleration processes in par-ticular. Several studies have tried to characterize the partition be-tween subsets of these energies in solar flares or eruptive events. Article number, page 1 of 15 & Aproofs: manuscript no. 39529corr_mod
In this context, three questions have attracted particular inter-est: (i) whether there is enough free magnetic energy in an ac-tive region to account for the total energy released in a flare orCME, (ii) what the energy partition is between flare and CME,and (iii) whether nonthermal particles can power the thermalcomponent in flares. It is generally found that enough free mag-netic energy is available to drive flares and CMEs (Emslie et al.2012; Aschwanden et al. 2017). With regard to the partition be-tween energy of the flare and the associated CME, the situationis less clear. Emslie et al. (2012) found energies on the same or-der of magnitude, and Aschwanden et al. (2017) concluded thatthe flare component dominates the energetics.In this study, we focus on the last question, whether thereis enough energy in nonthermal particles to heat the thermalplasma that is observed in solar flares. Hard X-ray (HXR) andgamma-ray observations clearly demonstrate that electrons andions are e ffi ciently accelerated to high energies during solarflares (cf. Holman et al. 2011; Vilmer et al. 2011). The mostwidely accepted mechanism for the generation of the thermalflare plasma is chromospheric evaporation by electron beams.This scenario is supported by the Neupert e ff ect (Neupert 1968),which refers to the observation that the time profile of nonther-mal HXR or microwave emission tends to closely match the timederivative of the (thermal) soft X-ray (SXR) flux. This impliesthat the energy is first released in the form of nonthermal elec-trons, which then follow the reconnected magnetic field linesdown to denser layers of the atmosphere where they thermal-ize and initiate chromospheric evaporation that then fills up theflaring loops with SXR-emitting plasma. The electron beam sce-nario is further supported by spatial and temporal correlationsof the nonthermal HXR emission and evaporation signaturessuch as hot upflows (e.g., Milligan et al. 2006; Tian et al. 2014;Li et al. 2015).There is no consensus so far about the answer to the simplequestion whether nonthermal electrons can provide su ffi cient en-ergy to power the thermal flare component because even the mostrecent studies give contradicting results. In Sect. 2 we brieflydescribe five relevant studies, their method, and the main con-clusions. In Sect. 3 we assess the treatment of various specificissues in the di ff erent studies, which allows us to set the conflict-ing results into perspective and derive a possible way forward.The conclusions are given in Sect. 4
2. Overview of studies
In the following, we describe five studies from the past 15 yearsthat meet two criteria: (a) they simultaneously determined thethermal energy of the hot plasma and the energy input by non-thermal electrons in the same flare events, and (b) they consid-ered a larger sample of flares in order to obtain statistically validresults and investigate correlations between di ff erent flare pa-rameters. While most of the selected studies have addressed sev-eral di ff erent topics and several of them have derived additionalflare and / or CME energetics, we focus on the issue of the parti-tion between thermal and nonthermal flare energy here.As an overview, Table 1 shows some basic characteristics ofthese studies. This includes the number of events studied, therange of SXR flare importance of the selected flares as measuredby the Geostationary Orbiting Environmental Satellites (GOES),the model used to characterize the thermal component (isother-mal vs. multithermal), the data source for the spectral and geo-metric parameters of the thermal component, and the energy-lossprocesses of the thermal plasma that were considered (radiationand conduction). Stoiser et al. (2007, henceforth referred to as S +
07) have de-rived thermal and nonthermal energies for 18 microflares (withbackground-subtracted GOES classes ranging from A3 to B7)that occurred within a single active region on 2003 Septem-ber 26. The thermal parameters were derived from isothermalfits to Reuven Ramaty High Energy Solar Spectroscopic Im-ager (RHESSI; Lin et al. 2002) HXR spectra at the flare peaks.The corresponding source volume was estimated from the foot-point brightenings observed at 1600 Å by the Transition Regionand Coronal Explorer (TRACE; Handy et al. 1999) under theassumption of a semicircular loop. The nonthermal energy inelectrons was calculated by a power-law fit to the nonthermalpart of the photon spectrum, conversion to electron beam powerby assuming thick-target emission and a fixed low-energy cuto ff of 10 keV, and integration over the time of detectable emissionabove 10 keV under the assumption of a triangular time profile.S +
07 reported that nonthermal dominates thermal energy, witha median ratio of ≈ The study of Emslie et al. (2012, henceforth E +
12) focused onderiving a broad range of flare, CME, and magnetic field en-ergetics for a sample of 38 larger flares (ranging from C6 toX28 in GOES class). This was achieved by applying the methodpreviously used on two large flares (Emslie et al. 2004, 2005).The peak thermal energy was deduced from isothermal fits toRHESSI spectra in combination with source areas (and thus vol-umes) obtained from RHESSI imaging. In addition, the radiativeloss of the hot plasma was derived from GOES observations byapplying an isothermal fit to the GOES fluxes and using the re-sulting temperature and emission measure to compute the radia-tive loss as given by the radiative loss function provided by theCHIANTI atomic database (Dere et al. 1997).The energy in nonthermal electrons was derived from thick-target fits to the RHESSI spectra and integration in time over thewhole event. In this process, the highest low-energy cuto ff s thatwere consistent with the data were used. Therefore the derivednonthermal energies are actually a lower estimate. With respectto the thermal-nonthermal energy partition, the study concludedthat the energy in injected electrons is su ffi cient to heat the hotflare plasma and to account for its radiative losses. The energetics of ten microflares was studied by Inglis & Christe(2014, henceforth IC14). Thermal energies were calculated froma multithermal model constrained by both RHESSI HXR spec-tra and extreme ultraviolet (EUV) fluxes from the Solar Dynam-ics Observatory (SDO; Pesnell et al. 2012) Atmospheric Imag-ing Assembly (AIA; Lemen et al. 2012). The thermal plasmavolume was obtained from RHESSI thermal source areas. Theradiative energy loss was derived from the di ff erential emis-sion measure (DEM) profiles using CHIANTI. The energy innonthermal electrons was constrained by fitting the high-energyHXR emission not accounted for by the multithermal model witha thick-target bremsstrahlung model (e ff ectively giving an upperestimate for the low-energy cuto ff and a lower estimate for thenonthermal power). IC14 concluded that the minimum nonther-mal energy content averages approximately 30% of the thermalenergy deduced from the multithermal model. Article number, page 2 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares
Table 1.
Overview of the flare energetics studies discussed. For details, see the main text. no. GOES class thermal data source data source thermalstudy flares range model thermal spectrum thermal volume lossesStoiser et al. 2007 (S +
07) 18 A3–B7 isoth. RHESSI TRACE –Emslie et al. 2012 (E +
12) 38 C5–X28 isotherm. RHESSI RHESSI rad.Inglis & Christe 2014 (IC14) 10 B3–B9 multitherm. RHESSI + AIA RHESSI rad.Warmuth & Mann 2016 (WM16) 24 C3–X17 isotherm. RHESSI + GOES RHESSI rad., cond.Aschwanden et al. 2017 (A +
17) 188 M1–X7 multitherm. AIA AIA rad.
In a series of studies, Warmuth & Mann performed a detailedcharacterization of the geometric (Warmuth & Mann 2013a,b)and spectral parameters (Warmuth & Mann 2016a) of 24 solarflares ranging from small C-class to large X-class flares usingHXR imaging and spectroscopy from RHESSI. Based on theseparameters, energy partition was studied by Warmuth & Mann(2016b, henceforth WM16). Thermal energy as a function oftime was derived from isothermal fits to the RHESSI spectracombined with RHESSI thermal source sizes. This was supple-mented by isothermal fits to the GOES fluxes. Radiative losseswere computed from the isothermal parameters provided byRHESSI and GOES using CHIANTI. The thermal parameterswere also used to derive the conductive energy loss, assumingSpitzer conductivity with the appropriate saturation limits (cf.Battaglia et al. 2009, and references therein). Finally, the totalheating requirement was obtained from the various thermal en-ergetics. Similar to E +
12 and IC14, a lower limit to the energyinput by nonthermal electrons was derived from thick-target fitsto the RHESSI spectra.WM16 found that conductive losses are energetically veryimportant. The total heating requirements can only be fulfilledby energetic electrons in stronger flares. In weak flares, the ther-mal requirements are higher by up to an order of magnitude thanthe nonthermal input.
In an extensive series of studies on energetics, Aschwan-den et al. (2017) quantified the free magnetic energy in ac-tive regions (Aschwanden et al. 2014), the energy of CMEs(Aschwanden 2016), and the thermal energy of the plasma(Aschwanden et al. 2015) as well as the energy in nonthermalelectrons (Aschwanden et al. 2016) in solar flares. The partitionbetween these energies in solar eruptive events is discussed inAschwanden et al. (2017). In the following, we collectively re-fer to the results of these studies as A + ff that was obtained from an analytical approximation to thewarm-target model introduced by Kontar et al. (2015). In thisstudy, we only consider the 188 flares for which both thermaland nonthermal energies are available. A +
17 concluded that the energy in nonthermal electrons isgenerally about an order of magnitude higher than the peakthermal energy. Recently, Aschwanden et al. (2019, henceforthA +
19) have constrained the low-energy cuto ff and nonthermalelectron energetics for the same event sample with four di ff er-ent methods, which has led to some modifications of the originalconclusions that we discuss in Sect. 3.4. We summarize the conclusions on thermal-nonthermal energypartition given by the five studies. E +
12 find that nonthermalelectrons can account for the thermal plasma, and both S + +
17 conclude that the electrons actually dominate flare en-ergetics. In contrast, IC14 find that there is not enough energy inthe nonthermal electrons to power the thermal plasma. Finally,WM16 conclude that the electrons can account for the thermalplasma only in stronger flares. This clearly shows that no con-sensus on the thermal-nonthermal energy partition in solar flareshas been reached yet.
3. Discussion
The thermal and nonthermal energetics discussed were derivedfrom X-ray and / or EUV observations, while it is well known thatsolar flares emit copiously at longer wavelengths. A meaningfuldiscussion of energy partition in flares is thus only possible whenthe total energy released in a solar flare is constrained first.Based on total solar irradiance (TSI) observations obtainedfrom the Total Irradiance Monitor (TIM) on the Solar Radiationand Climate Experiment (SORCE) spacecraft, the total (bolo-metric) energy radiated by flares has been measured individuallyfor a few large X-class flares (Woods et al. 2006, E + α = . ± . , which is alsoconsistent with the individual bolometric energies derived fromSORCE / TIM. The agreement between the two di ff erent instru- Article number, page 3 of 15 & Aproofs: manuscript no. 39529corr_mod ments and analysis techniques gives us some confidence in thevalidity of the derived energies.The bolometric energy is a proxy for the total energy releasedin solar flares. Regardless of the way in which energy is releasedin a solar flare (whether by direct heating of plasma, fast bulkflows, or nonthermal particles), in the end, everything is ther-malized and radiated away. Because the bolometric energy cov-ers the whole spectrum, we expect it to correspond to the totalenergy that has been released originally. This refers only to theenergy released in the flare and does not account for the energyof an associated CME or filament eruption.In the following, we therefore frequently compare the resultsof the five studies to the bolometric energies in order to (i) havean independent consistency check and (ii) assess the fraction ofthe various energies with respect to the total energy released.An additional important aspect is that the bolometric emis-sion is dominated by near-UV, white-light, and near-IR radi-ation that originates from comparatively cool and dense plas-mas in the chromosphere and photosphere (Woods et al. 2006;Kretzschmar 2011). Assuming that the primary energy releasetakes place in the corona, this places some stringent require-ments on the processes that transport the energy down to loweratmospheric layers (electrons, ions, conduction, and waves).
One potential reason for discrepancies between the various stud-ies are obviously di ff erent event selection criteria. In particular,the samples used di ff er significantly in terms of flare importance,for instance, as measured by the GOES peak flux. The minimumand maximum GOES class of the various flare samples are givenin Table 1. While E +
12 and A +
17 mostly considered M- and X-class flares and IC14 restricted their study to microflares, WM16covered the range from C- to X-class flares.If thermal and nonthermal energetics scale di ff erently withflare importance, then it is natural that di ff erent partitions arefound for these strongly dissimilar event samples. We addressthis issue in Sect. 3.6 after considering the dependence of ther-mal and nonthermal energetics on flare class in Sects. 3.3 and3.4. For an overview of the thermal energetics, we plot the peak ther-mal energies E th as a function of the peak GOES SXR flux asdetermined by the five studies in Fig. 1. We used the background-subtracted GOES peak fluxes for all studies except for E + +
17, for which only the unsubtracted fluxes are available(the same approach is used throughout this paper). For weakerevents, the subtracted fluxes are more meaningful because thebackground can be quite significant. However, the e ff ect of back-ground subtraction is negligible for the M- and X-class flares thatconstitute the event samples used by E +
12 and A +
17, so that wedo not introduce a bias by using the raw values for these twostudies.All studies show a good to excellent correlation with GOESclass. E +
12 and WM16 are very consistent and are about an or-der of magnitude lower than the bolometric energies, which areplotted in green for comparison. IC14 appears to be elevated byabout half an order of magnitude as compared to extrapolationsof E +
12 and WM16 to weaker flares (indicated by dotted linesin Fig. 1), while S +
07 is very consistent with the extrapolations. -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 pea k t he r m a l ene r g y E t h [ e r g ] C: 0.66 Aschwanden+ 2017C: 0.95 Warmuth & Mann 2016C: 0.89 Inglis & Christe 2014C: 0.56 Emslie+ 2012C: 0.75 Stoiser+ 2007bolometric energy (Emslie+ 2012)bolometric energy (Kretzschmar 2011)
Fig. 1.
Peak thermal energy E th vs. peak GOES flux for all fivestudies. For comparison, the total radiated energies E bol derived fromSORCE / TIM (green diamonds) and SOHO / VIRGO (green crosses; thegreen line is a power-law fit to these data points) are shown. C indicatesthe linear correlation coe ffi cient of the logarithms of the value pairs. Thedotted lines indicate power-law fits to the values of E +
12 and WM16.
Finally, the thermal energies of A +
17 are about an order of mag-nitude higher than those of the other studies, and they are roughlyequal to the bolometric energies. The question now is how thesedi ff erences can be explained. The five studies used quite diverse approaches to determine thethermal energy of the flare plasma from observations and alsoused di ff erent data analyses, which may lead to systematic dif-ferences that might account for the di ff erent results on energypartition.E +
12 used isothermal fits of the RHESSI HXR spectra to ob-tain emission measure and temperature as input for the derivationof thermal energies. WM16 combined this method with isother-mal fits of the GOES fluxes and derived three thermal energies:isothermal GOES, isothermal RHESSI, and a combined valuethat was obtained by assuming that half the emitting volume wasfilled by the GOES and the RHESSI plasma, respectively (thesecombined values are shown in Fig. 1).It is well known that RHESSI always yields highertemperatures and lower emission measures than GOES(cf. Battaglia et al. 2005; Ryan et al. 2014; Warmuth & Mann2016a). These di ff erences arise because flares are not trulyisothermal and the RHESSI temperature response is weightedtoward higher temperatures than GOES. The net e ff ect on ther-mal energies is shown in Fig. 2(a), where we show the GOES-derived peak thermal energies normalized by those derived basedon RHESSI as a function of GOES class for the events of WM16.On average, the GOES-derived energies are higher by a factor of1.4. This ratio does not depend on flare class. The correspondingratio of the combined thermal energy, shown in Fig. 2(b), showsthe corresponding ratios for the combined thermal energy, which Article number, page 4 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]0.00.51.01.52.02.5 E t h ( GO ES ) / E t h ( RH ESS I ) Warmuth & Mann 2016 (a) -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]0.00.51.01.52.02.5 E t h ( c o m b i ned ) / E t h ( RH ESS I ) Warmuth & Mann 2016 (b) -7 -6 -5 peak GOES flux [W m -2 ]0.00.51.01.52.02.5 E t h ( m u l t i t h . ) / E t h ( i s o t h . ) Inglis & Christe 2014 (c)
Fig. 2.
Ratios of peak thermal energies E th derived with di ff erent methods for the same flare samples plotted vs. peak GOES flux. (a) Ratio ofGOES- and RHESSI-derived energy in WM16. (b) Ratio of combined (using RHESSI and GOES) and RHESSI-derived energy in WM16. (c)Ratio of multithermal and isothermal energy in IC14. is higher on average than the RHESSI-derived energy by a factorof 1.7.IC +
14 went beyond the isothermal approximation and com-puted thermal energies from a DEM constrained both from EUVimaging with AIA and RHESSI HXR spectra. In principle, thisshould provide more realistic estimates of the thermal energy.The authors also computed the thermal energies correspondingto an isothermal fit, and the multithermal to isothermal energyratio is shown in Fig. 2(c). The energies are comparable, whichcan be explained by the fact that most of the thermal energy iscontributed by hot plasma that is constrained by RHESSI data(cf. the discussion in IC14). The di ff erent data source and ther-mal model employed by IC14 therefore cannot explain the ex-cess over extrapolations of the energies of E +
12 and WM16.Finally, A +
17 derived the multithermal energy at flare peaktime using a spatial synthesis method that fits a Gaussian DEMto each spatial pixel of a set of AIA images in the six coronalwavelengths. Figure 1 demonstrates that the resulting energiesare about an order of magnitude higher than those derived fromRHESSI and GOES. A potential explanation for this could be asevere underestimation of thermal energies due to the isothermalassumption of E +
12 and WM16. However, IC14 showed thatisothermal and multithermal energies can be quite comparable.While this was compared only for microflares, larger flares tendto be hotter, so that the thermal energy will be even more dom-inated by material that is seen in X-rays, and thus no increasedmismatch is to be expected.An alternative explanation is that the method of A +
17 over-estimated the thermal energies. IC14 showed that a single-Gaussian DEM profile cannot decrease su ffi ciently steeply athigh temperatures in order to be compatible with RHESSI X-rayobservations. While the multiple Gaussians used by A +
17 willmitigate this issue to some extent, we note that in fact most DEMreconstruction methods tend to derive too much plasma at hightemperatures, resulting in excessive X-ray emission (cf. Su et al.2018). This notion is supported by the fact that many thermal en-ergies of A +
17 are on the same order as the bolometric energy,or even exceed it.
The volume that is required for the computation of the thermalenergy is usually decomposed into an apparent volume V derivedfrom EUV or X-ray imaging of the thermal source and a volume -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 v o l u m e [ c m ] C: 0.55 Aschwanden+ 2017C: 0.70 Warmuth & Mann 2016C: 0.42 Inglis & Christe 2014C: 0.24 Emslie+ 2012C: 0.23 Stoiser+ 2007
Fig. 3.
Thermal source volume plotted vs. peak GOES flux for all fivestudies. filling factor f that accounts for the possibility that the emittingplasma may only occupy a fraction of the apparent volume.With the exception of S +
07 and A +
17, all studies usedthe area of the thermal X-ray source as imaged by RHESSI toconstrain the apparent volume, and adopted f =
1. S +
07 de-rived the volume by assuming a semicircular loop with a cross-sectional area and loop length given by areas and separations ofthe footpoint brightenings observed at 1600 Å by TRACE, whileA +
17 estimated the volume from the flare area that was mea-sured above some appropriate threshold in the emission mea-sure per (macro)pixel that results from the spatial synthesis DEMmethod.In Fig. 3 we plot the volumes used to derive the peak thermalenergies as a function of GOES peak flux. All volumes showconsiderable scatter, for example, flare volumes span a range ofone order of magnitude for the same GOES class for most of thestudies, and even two orders of magnitude for A +
17. Despite this
Article number, page 5 of 15 & Aproofs: manuscript no. 39529corr_mod scatter, all studies show at least a moderate correlation betweensource volume and GOES class.It is noteworthy that the volumes of E +
12 and WM16 onthe one hand and A +
17 on the other hand are generally consis-tent, although they were derived with two completely di ff erentmethods. Conversely, the microflare volumes of IC14 are one totwo orders of magnitude larger than would be expected from theother four studies. Interestingly, this corresponds to the amountthat is required to explain the elevated thermal energies of IC14discussed in Sect. 3.3.1. While these sources could in principlebe intrinsically large, their disagreement with the other studiessuggests that an overestimation of source size is more likely.This may have been caused by an insu ffi cient accounting forthe tendency of the CLEAN imaging algorithm to provide sys-tematically larger source sizes (e.g. Warmuth & Mann 2013a).Another possibility might be that the di ffi culty RHESSI has inresolving small sources, which has been demonstrated for non-thermal sources (cf. Dennis & Pernak 2009; Warmuth & Mann2013b), might also apply to the potentially small thermal sourcesin microflares. While all studies assumed a filling factor of unity, we neverthe-less have to consider the validity of this assumption, and the con-sequences that result if this does not hold. A filling factor belowunity can a ff ect energy partition because it decreases the thermalenergies by a factor of f / , while it increases the radiative lossesby f − / . In the literature, very diverse results are reported on f .While studies in the EUV have tended to yield very low val-ues, that is, 0 . < f < . . < f < f can alsobe constrained by the requirement that the flare plasma has to bemagnetically contained, that is, the plasma beta has to be smallerthan unity. Thus the required coronal magnetic field strength isdependent on the filling factor according to B cor ∼ f / . Mea-suring magnetic field strengths in solar flares is challenging, butdi ff erent techniques consistently demonstrate that B cor is on theorder of a few 100 G in strong flares. For example, in the X8.2flare of 2017 September 10 field strengths of 520 G and 148 Gat heights (above the limb) of ≈
20 and 30 Mm were derivedby Gary et al. (2018) from observation of gyrosynchrotron emis-sion. In the same event, Kuridze et al. (2019) obtained 420 G and350 G at heights of 15 and 25 Mm from spectropolarimetry. Wecan now compare the required field strengths to these rather firmconstraints. Already for f =
1, the flares of Caspi et al. (2014a)require B cor of up to 160 G, while WM16 derive values of up to370 G. This would rule out filling factors below 0.1. This result issupported by spectroscopic observations using density-sensitivelines (e.g. Milligan et al. 2012). We conclude that while the fill-ing factor f remains a poorly constrained parameter, it is unlikelythat it a ff ects the results on energy partition in a substantial way. The peak thermal energy is only a lower limit of the energy thatis required to generate and sustain the thermal plasma due toenergy-loss processes. Deriving the true thermal-nonthermal en-ergy partition thus requires quantifying these losses and assess-ing their importance. A convenient measure for this is the ratio -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 r ad i a t ed ene r g y E r ad [ e r g ] C: 0.96 Warmuth & Mann 2016C: 0.80 Inglis & Christe 2014C: 0.71 Emslie+ 2012bolometric energy (Emslie+ 2012)bolometric energy (Kretzschmar 2011)
Fig. 4.
As in Fig. 1, but showing the energy radiated by the hot plasma, E rad , as a function of peak GOES flux for three studies. between the energy loss (integrated over the event) and the peakthermal energy, and we focus on this property in the followingdiscussions.The radiative losses of the hot plasma, E rad , have been con-sidered in all studies, with the exception of S +
07. Figure 4 shows E rad as a function of GOES peak flux for the di ff erent studies. Inall cases, E rad correlates well with GOES flux. Generally, E rad is clearly below the bolometric energy. This is consistent withthe understanding that the bolometric emission is dominated bynear-UV, white-light, and near-IR emission that originates fromcomparatively cool material located at lower heights in the solaratmosphere (Woods et al. 2006; Kretzschmar 2011).Figure 5 shows E rad / E th for the di ff erent studies. In A + E rad / E th = . ± . +
17. The corre-sponding mean ratios for the other studies and methods are listedin Table 2. Studies using multithermal DEM reconstructions(IC14, A +
17) obtain significantly lower ratios ( E rad / E th < . + E rad / E th increases for higher flare importance. While the radiative lossescan be neglected for B-class flares (this even holds when we as-sume that the thermal energies of IC14 are overestimated; cf.the discussion in Sect. 3.3.3), they generally dominate the peakthermal energy for X-class flares.This result can be understood as follows. A flare is heatedimpulsively and then cools down. In this case, we would expect E rad = E th (neglecting conductive losses). However, when heat-ing is more gradual, E rad > E th is observed because more energyis lost before the thermal peak is reached. This is consistent withthe results shown above: larger flares usually have longer du-rations and more extended impulsive phases as well, therefore Article number, page 6 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 -2 -1 r ad i a t i v e ene r g y l o ss / pea k t he r m a l ene r g y ( E r ad / E t h ) C: 0.73 Warmuth & Mann 2016C: 0.41 Inglis & Christe 2014C: 0.55 Emslie+ 2012
Fig. 5.
Energy radiated by the hot plasma normalized by the peak ther-mal energy, E rad / E th , vs. peak GOES flux for three studies. The dottedline denotes energy equipartition. Table 2.
Relation between radiative energy loss E rad and peak thermalenergy E th as derived by di ff erent studies and methods. Shown are thelogarithmic mean ratios, E rad / E th , and the correlation coe ffi cient of thelogarithms of the two quantities, C . study method E rad / E th C E +
12 RHESSI, isothermal 3 . ± .
12 0.71IC14 RHESSI + AIA, isothermal 0 . ± .
17 0.88IC14 RHESSI + AIA, multithermal 0 . ± .
16 0.89WM16 RHESSI, isothermal 0 . ± .
19 0.94WM16 GOES, isothermal 1 . ± .
17 0.96WM16 RHESSI + GOES, bithermal 1 . ± .
18 0.96A +
17 AIA, multithermal 0 . ± .
06 n / a E rad dominates. The question now is how we can understand E rad < E th in weak flares. IC14 noted this point and proposedthat in addition to strong conductive losses (see Sect. 3.3.6), afilling factor of about f ≈ × − would give an equipartition ofpeak thermal and radiated energy. As we showed in Sect. 3.3.4,the latter explanation may be inconsistent with the magnetic fieldstrength required to contain the plasma. A final possibility is thatthe thermal energy is overstimated (cf. Sect. 3.3.3).Finally, we address the low value of E rad / E th derived byA +
17. This could have resulted from the thermal energies thatare systematically higher than those of the other studies by aboutan order of magnitude. Moreover, A +
17 found that E rad / E th is anticorrelated with E th , which is in contrast to the trendseen in the other studies. A +
17 argued that this is qualita-tively consistent with cooling models that predict that radia-tive and conductive losses are anticorrelated at higher temper-atures (e.g. Cargill et al. 1995). However, the radiative loss rateis not strongly dependent on temperature, and the higher emis-sion measure in large flares usually overcompensates for this. Itappears that A +
17 derived systematically lower radiative lossesthan E +
12 and WM16 for large events: the maximum in A + -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 -1 c ondu c t i v e ene r g y l o ss / pea k t he r m a l ene r g y ( E c ond / E t h ) C: -0.57 Warmuth & Mann 2016
Fig. 6.
As in Fig. 5, but showing conductive energy loss of the hot flareplasma normalized by the peak thermal energy, E cond / E th , as a functionof peak GOES flux as derived by WM16. Table 3.
As in Table 2, but showing the relation between conductive en-ergy loss E cond and peak thermal energy E th as derived by WM16 usingthree di ff erent methods for determining the thermal plasma parameters. study method E cond / E th C WM16 RHESSI 9 . ± .
20 0.85WM16 GOES 2 . ± .
21 0.83WM16 RHESSI + GOES 4 . ± .
20 0.85is E rad ≈ erg, while it approaches 10 erg in E +
12. Thecause for this is unclear because both E +
12 and A +
17 used thesame method to derive E rad . While conductive losses have been considered by manyauthors to investigate flare thermal evolution (e.g.Aschwanden & Alexander 2001; Ryan et al. 2013) and byseveral case studies of chromospheric evaporation (e.g.Battaglia et al. 2009), WM16 provided the first systematic treat-ment of conductive losses in terms of energetics. Conversely,conduction has been neglected by the other three studies ofenergetics discussed here.Applying Spitzer heat conduction (Spitzer 1962), WM16found very large conductive losses, and the corresponding log-arithmic mean ratios of E cond / E th are listed in Table 3 for thethree di ff erent methods used by WM16. In particular, the rela-tive importance of the conductive losses is dependent on GOESclass, which is illustrated in Figure 6 (the energies plotted referto the combination of RHESSI- and GOES-derived plasma pa-rameters). For C-class flares, the conductive losses exceed thepeak thermal energies by up to one order of magnitude, and thisratio decreases for larger flares. This is the opposite of what wasfound for the radiative losses (cf. Fig. 5). Article number, page 7 of 15 & Aproofs: manuscript no. 39529corr_mod
The large conductive losses were the main reason for theconclusion in WM16 that the nonthermal energy is only su ffi -cient to power the thermal flare component in the larger events.We need to determine whether such large losses are realistic.The estimation of the conductive loss involves several un-certainties. To arrive at the conductive energy loss rate, theconductive flux density has to be integrated over the cross-sectional area of the coronal loop footpoints. WM16 took thisas the HXR footpoint area. However, RHESSI may have is-sues with properly measuring footpoint sizes (Dennis & Pernak2009; Warmuth & Mann 2013b), which could lead to an overes-timate of the conductive losses. Conversely, the HXR footpointsmap the area where nonthermal electrons are currently precipi-tating, which may only comprise a fraction of a flaring loop filledwith hot plasma. If this e ff ect dominates, then conductive losseswould be underestimated.Another source of uncertainty is the thermal gradient length,which was taken as the half-loop length by WM16. It can be ar-gued that significantly shorter lengths (i.e. corresponding to thetransition region height; cf. Fletcher et al. 2013) could be appro-priate, which again would result in even larger losses.A more fundamental issue is the validity of using Spitzerheat conductivity. Under typical solar flare conditions, the heatflux is usually saturated (e.g. Campbell 1984; Karpen & DeVore1987), and this limitation has been accounted for by WM16.However, an additional e ff ect was not considered by WM16that has been pointed out by Brown et al. (1979): turbulencegenerated in a flaring loop can scatter electrons. Several re-cent studies have demonstrated that this can drastically suppressthe parallel heat flux (cf. Bian et al. 2016; Emslie & Bian 2018;Roberg-Clark et al. 2018). A high level of turbulence could thuse ff ectively switch o ff conduction as a loss term. As a conse-quence, the heating requirements for the thermal flare compo-nent in WM16 would drop, in particular for the smaller flares, forwhich conduction was found to be more important. This meansthat one of the main conclusions of WM16 would have to bemodified: if conduction is suppressed, the nonthermal energy in-put can account for the thermal requirements even in smallerflares.While there are arguments for a strong suppression of theconductive heat flux, there is also observational evidence forconduction-driven evaporation (e.g. Antiochos & Sturrock1978; Zarro & Lemen 1988; Czaykowska et al. 2001;Battaglia et al. 2009; Battaglia et al. 2015), which suggeststhat the suppression may not be as severe as predicted, or at leastnot in all cases. This opens yet another question about the roleof conduction in energy partition. In assuming that the energytransported to the deeper atmospheric layers is completelyradiated away, WM16 have treated conduction solely as aloss term. However, as the evidence for conduction-drivenevaporation shows, at least part of the energy conducted tothe chromosphere might be spent in generating newly heatedand subsequently evaporated plasma. This would amount to a“reprocessing” of the conductive losses, and would in e ff ectlower the total heating requirements of the hot plasma. Weconclude that while conductive losses may potentially be veryimportant for the energy partition and transport, several openquestions remain that will have to be properly addressed beforeany reliable quantitative assessment can be made. -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 non t he r m a l ene r g y E n t h [ e r g ] C: 0.64 Aschwanden+ 2017C: 0.95 Warmuth & Mann 2016C: 0.52 Inglis & Christe 2014C: 0.38 Emslie+ 2012C: 0.79 Stoiser+ 2007bolometric energy (Emslie+ 2012)bolometric energy (Kretzschmar 2011)
Fig. 7.
As in Fig. 1, but showing the energy in nonthermal electrons, E nth , plotted vs. peak GOES flux for all five studies. The nonthermal component was derived from a much more ho-mogeneous data set than the thermal component in the observa-tions: all studies exclusively used HXR spectroscopic data ob-tained with RHESSI to constrain the energy in accelerated elec-trons. All studies used the collisional thick-target model to de-rive the spectrum of the injected electrons. While E +
12, IC14,and WM16 followed the standard practice of using the highestlow-energy cuto ff consistent with the data, A +
17 constrainedthe cuto ff with an analytical approximation of the warm-targetmodel developed by Kontar et al. (2015). S +
07 adopted a cuto ff of 10 keV for all events, a value which was found to be consistentwith the HXR spectra.In Figure 7 we plot the energy in nonthermal electrons, E nth ,derived by the five studies as a function of GOES peak flux. Allstudies show a correlation with GOES class, but at a slightlylower level than the thermal energies (cf. Fig. 1). The energies ofE +
12 and WM16 are consistent, and the energies of IC14 alsoagree with extrapolations of E +
12 and WM16 to lower GOESclasses. The values derived by these three studies are consistentwith the bolometric energy, and there is a trend for the nonther-mal energy to decrease with respect to the bolometric energy insmaller flares (IC14 and WM16). In contrast, the nonthermal en-ergies of S +
07 and A +
17 are significantly higher. Particularlyin the latter case, E nth is more than an order of magnitude higherthan E bol in the majority of the events. This strongly suggests thatthe nonthermal energies have been overestimated by A +
17 (andto a lesser degree by S +
07) because it is impossible that sucha huge amount of energy is injected into the solar atmospherewithout being thermalized and subsequently radiated away, thusbeing detected in the bolometric emission. This clearly demon-strates the benefit of using the bolometric energy as an indepen-dent constraint on energetics.
Article number, page 8 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 non t he r m a l ene r g y E n t h [ e r g ] C: 0.54 cross-over -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 non t he r m a l ene r g y E n t h [ e r g ] C: 0.27 warm-target approximation -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 non t he r m a l ene r g y E n t h [ e r g ] C: 0.12 time-of-flight -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 non t he r m a l ene r g y E n t h [ e r g ] C: 0.41 total electron numberbolometric energy (Emslie+ 2012)bolometric energy (Kretzschmar 2011)
Fig. 8.
As in Fig. 7, but showing the energy in nonthermal electrons, E nth , as a function of peak GOES flux as derived with four di ff erent methodsused by Aschwanden et al. (2019). Following up on A +
17 and using the same event sample,A +
19 have determined the low-energy cuto ff and the nonther-mal energy using four di ff erent methods, which are discussed inSect. 3.4.3. The relation of the corresponding nonthermal en-ergies with GOES class is plotted in Fig. 8. With the excep-tion of the spectral cross-over method (also applied by E + +
17. Moreover, the di ff er-ent methods yield quite consistent nonthermal energies, with theexception of the cross-over method, which gives substantiallylower energies. All five studies considered here have used the collisional thick-target bremsstrahlung model (Brown 1971) to derive the flux(and hence the kinetic power) of the injected electrons from ob-served HXR photon spectra. The main di ff erence was the wayin which the low-energy cuto ff of the nonthermal electron dis-tribution was constrained, which is addressed in the followingSect. 3.4.3. S +
07 fit a photon power law to the spectra that wasthen converted into a thick-target electron flux. In the classical thick-target model, the background plasma isconsidered to be “cold” in the sense that the thermal speed ofthe particles is much slower than the speed of the injected elec-trons. However, this assumption becomes invalid when the targetis heated during the flaring process. Thus, Kontar et al. (2015)have recently developed the warm-target model, and extensionof the cold-target model that takes into account the physics ofcollisional energy di ff usion and thermalization of fast electronsin the background plasma. As additional input, the warm-targetmodel requires the target temperature, density, and length, whichmeans that imaging observations are required in addition HXRspectra.The warm-target model can be employed to obtain an up-per estimate for the injected nonthermal energy because in con-trast to the cold-target model, the low-energy cuto ff cannot bemade arbitrarily small: the thermalized electrons leave their sig-nature on the photon spectrum. This approach was taken byKontar et al. (2019) in a detailed study of the peak of a singleflare, but has not been applied to a larger event sample so far. The low-energy cuto ff E C is the crucial parameter for constrain-ing the energy content of the injected electrons. It is also the mostelusive parameter because its spectral signature (a flattening ofthe photon spectrum below the break) is usually masked by the Article number, page 9 of 15 & Aproofs: manuscript no. 39529corr_mod -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]110100 l o w - ene r g y c u t o ff E C [ k e V ] C: 0.26 Aschwanden+ 2019 (WTA)C: 0.06 Aschwanden+ 2017C: 0.72 Warmuth & Mann 2016C: 0.17 Inglis & Christe 2014Stoiser+ 2007
Fig. 9.
Low-energy cuto ff s E C plotted vs. GOES peak flux as derived bywith the spectral cross-over method by S +
07, IC14, and WM16 (here,the mean and standard deviation of the cuto ff is shown for each flare),and with the warm-target approximation by A +
17 and A + much stronger thermal emission. There are a few exceptions tothis: in early impulsive events (Sui et al. 2007) and in flares withlate impulsive peaks (Warmuth et al. 2009a; Ireland et al. 2013),the low-energy cuto ff can be observed directly in the spectra.The standard practice to cope with the masking of the cuto ff isto use the highest low-energy cuto ff that is still consistent withthe data, and thus obtain a lower estimate for the flux and powerof the energetic electrons (see e.g. Holman et al. 2011). This wasdone by E +
12, IC14, and WM16. In addition, it was one of themethods used by A +
19, where it is referred to as the spectralcross-over method. S +
07 used a modification of this approach:they obtained the spectral cross-over in the range of 9–12 keV,but adopted a fixed cuto ff of 10 keV for all their events.The lower limit on nonthermal energy provided by the cross-over method is to some degree dependent on the way in whichthe thermal part of the X-ray spectrum is fit. While all studiesused an isothermal model, IC14 additionally considered the mul-tithermal case. This resulted in lower nonthermal energies, with alogarithmic mean of E nth , multi / E nth , iso = . ± .
34. A multither-mal fit thus gives the absolute lower limit on the electron energycontent. Although the median di ff erence of the cuto ff energies inIC14 was only 2.5 keV, this nevertheless resulted in a substan-tial e ff ect on the total energies. This is generally the case for thesteep spectral indices that are characteristic for microflares (i.e.electron spectral indices ≈
8; cf. Christe et al. 2008). This strongsensitivity might be one reasons for the apparent overestimationof nonthermal energies by S +
07, where the universally adoptedcuto ff of 10 keV is a few keV lower than in most of the events ofIC14 (cf. Fig. 9).A +
17 applied a di ff erent method to constrain E C . They usedan approximation from the warm-target model of Kontar et al.(2015) that provides an e ff ective low-energy cuto ff that is lin-early dependent on the electron spectral index and the warm-target temperature. As temperature, 8.6 MK was adopted for all flares and time intervals, which corresponded to the mean EM-weighted temperature derived from the DEM reconstructions.This approach has resulted in significantly lower cuto ff ener-gies (with a mean of E C = . E C >
10 keV), and thereforeproduced higher nonthermal energies. To illustrate this, we showthe low-energy cuto ff s determined by S +
07, IC14, WM16, andA +
17 as a function of GOES peak flux in Fig. 9.A +
19 expanded on the issue of the low-energy cuto ff by ap-plying four di ff erent constraints on E C to the event sample ofA +
17. These were (a) the familiar cross-over method (CO), (b)the warm-target approximation used by A +
17 (WTA), but nowusing the geometric mean of RHESSI- and AIA-derived tem-peratures measured in each individual event, (c) a model basedon the equivalence of the time-of-flight (TOF) of electrons andthe collisional deflection time, and (d) the total electron numbermodel (TEN), which relies on the number of electrons that isavailable for acceleration in the flaring region.A +
19 found that generally the WTA, TOF, and TEN modelsyield consistent results on the cuto ff energies, which are found tobe around 10 keV on average. For comparison, we have includedin Fig. 9 the WTA values of A +
19. The e ff ect of the higher tem-peratures used by A +
19 is evident when it is compared to theresults of A +
17. While these cuto ff s are significantly higher thanthe average value of A +
17, they correspond to the lowest valuesfor E C derived by IC14 and WM16 from the cross-over method.We note that the nonthermal energies derived by A +
17 andA +
19 generally show a significantly stronger scatter for flares ofcomparable importance than the other studies. A possible causefor this might be the automatic fitting procedures that had to ap-plied because of the large sample size.
While a single HXR spectrum is su ffi cient to derive a thermalenergy, a thick-target fit only yields the power of the injectedelectrons. Obtaining the nonthermal energy thus requires a timeintegration over the event (similar to the radiative and conductivelosses), or at least an assumption about the duration of the non-thermal energy input. E +
12, WM16 and A +
17 have all split eachindividual flare into time bins (typically of 20 s duration), per-formed spectral fits for each bin, and integrated over the wholeevent.In contrast, low count rates and short durations of the mi-croflares studied by S +
07 and IC14 demanded a di ff erent ap-proach, and in both cases only a single spectrum was fit for eachflare. S +
07 fit a spectrum with 12 s integration time obtainedaround the HXR peak and integrated the nonthermal power overthe time of detectable emission above 10 keV under the assump-tion of a triangular time profile. IC14 used spectra with 60 s inte-gration time and integrated the nonthermal power over the sameduration. When a single spectral fit is used, it results in a lessrealistic characterization of the nonthermal emission. This mayhave contributed to the large discrepancy between the nonther-mal energies derived by S +
07 and IC14.
When we discussed nonthermal energetics, we only consid-ered the energetics of electrons so far. To quantify the thermal-nonthermal energy partition and the acceleration e ffi ciency, weneed to consider the contribution of energetic ions as well. Un-fortunately, the energy content of ions is even less well con- Article number, page 10 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares strained than that of electrons. In strong flares, observation ofgamma-ray lines allows the determination of the energetics of > +
12 con-cluded that the energy content of electron and ions is generallycomparable within an order of magnitude (with a logarithmicmean of E nth , i / E nth , e = . ± . Our discussion of flare energetics has focused on thermal ener-gies and energy losses of the hot plasma, and on the energy innonthermal particles. However, several additional energy com-ponents may be important for understanding energy partition ingeneral. We briefly address them.The hot thermal plasma not only contains thermal energy, butalso gravitational energy (because the bulk of the flare plasmahas to be transported from the chromosphere to larger coronalheights), and kinetic energy of the evaporation flows. WM16demonstrated that these energies are far lower than the thermalenergy. The kinetic flow energy is more than an order of mag-nitude lower than the peak thermal energy, and the gravitationalenergy is a full two orders of magnitude lower.A di ff erent type of flow is directly associated with the re-connection process. The reconnection outflow jets contain thekinetic energy of the bulk flow, and additionally, the kinetic en-ergy associated with turbulence. Several models assume that theenergy required for particle acceleration is either taken fromthe bulk outflow (through acceleration at shock waves, e.g.Mann et al. 2009) or from turbulent flows within the outflow(through stochastic acceleration, e.g. Petrosian 2012, and refer-ences therein). In this scenario, the flow energy should not beadded to the thermal-nonthermal energy budget. It rather rep-resents the reservoir from which the energy in nonthermal par-ticles is supplied and is thus an intermediate step between thedissipated magnetic energy and the accelerated particles.For this model to be feasible, the flow energy should at leastbe comparable to the total nonthermal energy. This is indeedsupported by observations. Warmuth et al. (2009b) constrainedthe density and speed of the outflow with radio observationsof termination shocks and found the kinetic energy su ffi cient toaccount for the nonthermal particles. Conversely, Kontar et al.(2017) derived the kinetic energy associated with small-scale tur-bulent mass motions from the nonthermal velocity broadening ofEUV lines and found it to be very consistent with the energy inaccelerated particles.Another component we did not consider is the warm coronalplasma (i.e., plasma at temperatures of below 5 MK that does notproduce significant X-ray emission) that produces a prolongedsecondary peak in EUV emission during the gradual phase ofa small fraction of flares (13%), the so-called EUV late phase(cf. Woods et al. 2011). This component, located in distinct loopsystems, implies additional heating requirements. Just consider-ing radiative losses in EUV, it has been shown that the EUV latephase can be up to four times more energetic than the main phase(Liu et al. 2015), and numerical modeling suggests that the peak Table 4.
Relation between energy in nonthermal electrons E nth and peakthermal energy E th as derived by di ff erent studies and methods. Shownare the logarithmic mean ratios, E nth / E th , and the correlation coe ffi cientof the logarithms of the two quantities, C . study method E nth / E th C S +
07 CO 28 . ± .
29 0.76E +
12 CO 6 . ± .
21 0.32IC14 CO, isotherm. 0 . ± .
24 0.75IC14 CO, multitherm. 0 . ± .
21 0.84WM16 CO, RHESSI 4 . ± .
17 0.93WM16 CO, GOES 3 . ± .
18 0.93WM16 CO, combined 2 . ± .
18 0.93A +
17 WTA 6 . ± .
15 0.42A +
19 CO 0 . ± .
17 0.34A +
19 WTA 0 . ± .
26 0.05A +
19 TOF 0 . ± .
32 0.19A +
19 TEN 0 . ± .
21 0.34
Notes.
The methods used to constrain the low-energy cuto ff are spec-tral cross-over (CO), warm-target approximation (WTA), time-of-flight(TOF), and total electron number (TEN). heating rate for the late-phase loops may be at least as high asfor the main flaring loops (Dai et al. 2018).It has been proposed that the heating required by the EUVlate phase may be supplied by the thermalization of the en-ergy contained in a flux rope after a failed eruption (Wang et al.2016). If this is the case, then the existence of the EUV late phasedoes not directly a ff ect the considerations on flare energy parti-tion we made here because it rather relates to the CME energet-ics. However, this does demonstrate the close interplay betweenthe processes we tend to divide into flare and CME. A full under-standing of solar eruptive events will require taking into accountboth manifestations of the phenomenon. After discussing in detail the issues that a ff ect the derivation ofthermal and nonthermal energetics in solar flares, we return tothe basic question of energy partition. Because all thermal en-ergetics and most of the nonthermal energies scale rather wellwith GOES peak flux, we now assess the dependence of the en-ergy balance on flare importance.We first consider the relation between peak thermal energy E th and energy in nonthermal electrons, E nth . Table 4 shows themean (logarithmic) ratios of E nth / E th as derived by all studiesand methods considered here, and the correlation between thetwo parameters. While S +
07, E +
12, WM16, and A +
17 reportedthat the energy in electrons is more than su ffi cient to accountfor the thermal energy, both IC14 and A +
19 reported a deficitof electrons. The correlation between the two quantities is lowerthan the correlation of each individual energy component withpeak GOES flux (cf. Figs. 1 and 7), which is not surprising con-sidering the uncertainties involved in the derivation of both ther-mal and nonthermal components.Going beyond mean values, we plot the ratio E nth / E th as afunction of peak GOES flux in Fig. 10. While the individualstudies show only low correlations between the energy ratio andGOES class (with the exception of S +
07 and WM16), the dif-ferent results considered together might indicate a trend: the mi-croflares of IC14 all show a deficit of nonthermal energy, the
Article number, page 11 of 15 & Aproofs: manuscript no. 39529corr_mod -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 -3 -2 -1 non t he r m a l ene r g y / pea k t he r m a l ene r g y ( E n t h / E t h ) C: -0.03 Aschwanden+ 2019 (WTA)C: 0.27 Aschwanden+ 2017C: 0.47 Warmuth & Mann 2016C: -0.04 Inglis & Christe 2014C: -0.02 Emslie+ 2012C: 0.59 Stoiser+ 2007
Fig. 10.
Energy in nonthermal electrons normalized by peak thermalenergy, E nth / E th , vs. peak GOES flux as derived by the five studies. Inaddition, the ratio is shown for the nonthermal energies derived with thewarm-target approximation in A + thermal component in larger events of E +
12 can all easily be ac-counted for by the injected electrons, and the flares of WM16tie these two regimes together. For A +
17 and A +
19, the scatteris too severe to discern any trend. However, the microflare en-ergetics derived by S +
07 are evidently not consistent with thisscenario because they suggest a clear dominance of the nonther-mal energy.We thus have two studies of microflares that yield contra-dicting results on energy partition. As an independent check,we now consider the large statistical study on microflares byHannah et al. (2008). We did not discussed it here in detail be-cause it lacks the quantification of nonthermal energies (onlypeak powers of injected electrons were provided). However,we note that the median thermal energy of 9 161 microflareswas 10 erg, while the median peak power in electrons for4 236 microflares was 10 erg s − . The typical time duration ofnonthermal HXR emission in microflares was given as ≈
10 s,which would correspond to a median electron energy contentof 10 erg in the microflare sample. This means that on aver-age, the energetic electron input can account for only ≈
10% ofthe peak thermal energy in microflares. This is consistent withthe results of IC14 and extrapolated ratios of WM16. We thusconclude that the nonthermal energies of IC14 are more realis-tic than those of S +
07. This is supported by the disagreementbetween the nonthermal and the bolometric energies in S + E nth . Still, thediscrepancies between the microflare studies clearly illustratethe di ffi culty in accurately measuring the nonthermal energy insmall events, which is caused by poor statistics (low number ofnonthermal counts), issues with background subtraction, and thesteep spectra that lead to a very high sensitivity of the derivedelectron flux with respect to the low-energy cuto ff .The overall trend of a decreasing nonthermal-to-thermal ra-tio with decreasing flare importance is confirmed when the ther- -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 -3 -2 -1 non t he r m a l ene r g y / hea t i ng r equ i r e m en t s ( E n t h / E h ) C: 0.60 Warmuth & Mann 2016C: -0.06 Inglis & Christe 2014C: -0.26 Emslie+ 2012
Fig. 11.
As in Fig. 10, but showing the energy in nonthermal electronsnormalized by the total heating requirements, E nth / E h , vs. peak GOESflux as derived by E +
12, IC14, and WM16. The energy required to gen-erate and sustain the hot flare component is approximated by addingthe peak thermal energy and the radiative losses, E th + E rad , for E + mal losses are considered as well. In Fig. 11 we plot the ratio ofnonthermal electron energy over the total heating requirementsof the hot plasma (A +
17 and A +
19 are omitted here because oftheir large scatter). The latter quantity is approximated by addingthe peak thermal energy and the radiative losses, E th + E rad , forE +
12 and IC14. For WM16, we used the total heating require-ments derived from a time integration of the thermal energychange rate plus the radiative and conductive energy-loss rates.
The decreasing nonthermal to thermal energy ratio for weakerflares established in the previous section might be a spurioustrend due to two issues connected to the thermal energetics.First, the low nonthermal ratio in IC14 might primarily be dueto an overestimate of the thermal energy caused by an oversizedsource volume (cf. Sect. 3.3.3). Second, the decreasing ratio inWM16 is partly due to the conductive losses that are more im-portant for weaker flares. When conduction is suppressed or thelosses are reprocessed (cf. Sect.3.3.6), this trend is significantlyweaker.We can avoid these issues when we compare the nonthermalenergy to the bolometric radiated energy. We stress once againthat E bol is a proxy for the total energy that is released in a flare.We thus define E nth / E bol as the nonthermal fraction, which is thefraction of energy dissipated that is converted into nonthermalparticles (electrons, in our case).Table 5 shows the logarithmic averages for all studies andmethods E nth / E bol and the correlation of the two quantities,while in Fig. 12 we plot this nonthermal ratio for the individ- Article number, page 12 of 15. Warmuth, G. Mann: Thermal-nonthermal energy partition in solar flares
Table 5.
As in Table 4, but showing the relation between energy in non-thermal electrons E nth and bolometric radiated energy E bol as derived bydi ff erent studies and methods. E bol is derived from a power-law fit to thevalues obtained by Kretzschmar (2011). study method E nth / E bol C S +
07 CO 2 . ± .
29 0.44E +
12 CO 0 . ± .
21 -0.43IC14 CO, isotherm. 0 . ± .
32 -0.05IC14 CO, multitherm. 0 . ± .
33 0.04WM16 CO 0 . ± .
17 0.56A +
17 WTA 5 . ± .
13 0.39A +
19 CO 0 . ± .
16 0.28A +
19 WTA 0 . ± .
24 0.06A +
19 TOF 0 . ± .
32 -0.05A +
19 TEN 0 . ± .
21 0.20ual flares of S +
07, E +
12, IC14, and WM16 as a function ofpeak GOES flux (again, the results of A +
17 and A +
19 are notshown here due to their large scatter). Figure 12 demonstratesthat E nth / E bol also follows the overall trend of a decreasing non-thermal fraction for weaker flares (when S +
07 is not considered,where E nth has most likely been overestimated): while E nth / E bol is of the order of unity in X-class flares (but with significant scat-ter), it is an order of magnitude lower in C-class flares and mi-croflares. This behavior might be an artifact caused by a system-atic underestimation of the nonthermal energy in weaker events,as indeed all nonthermal energies plotted in Fig. 12 are lowerlimits. However, WM16 pointed out that this is an unlikely sce-nario because it would require either a systematically lower low-energy cuto ff or a higher contribution of ions in weaker flares.For neither scenario do we have observational evidence or a the-oretical justification. We therefore identified the changing non-thermal to thermal energy fraction as the main reason for thedissimilar results on energy partition provided by the di ff erentstudies.We therefore conclude that in strong flares, the nonthermalenergy input is su ffi cient to generate and sustain the hot thermalplasma, which is consistent with the well-established scenarioof chromospheric evaporation driven by electron beams (e.g.Milligan et al. 2006; Tian et al. 2014). Conversely, in weakerevents (microflares and C-class flares), the nonthermal energyinput appears to be insu ffi cient. This implies the presence of anadditional non-beam heating mechanism, and indeed, several re-cent studies have found evidence of direct heating of coronalplasma (Caspi & Lin 2010; Caspi et al. 2015, WM16, A + ffi ciency that increases with flare im-portance. According to this scenario, the acceleration mecha-nism works at a low e ffi ciency in weaker flares, which meansthat a comparatively small fraction of the dissipated magneticenergy is used to accelerate particles, and a larger fraction goesinto direct heating. We stress that any acceleration mechanismwill always generate an enhanced thermal particle distributionin addition to the nonthermal component. In stronger events, theacceleration process appears to operate at higher e ffi ciency, thusconverting a large fraction of the energy released into nonther-mal energy.Several acceleration mechanisms predict varying nonthermalfractions that are consistent with this understanding. For exam-ple, the e ffi ciency of shock-drift acceleration of electrons at atermination shock (cf. Mann et al. 2009; Warmuth et al. 2009b)is strongly dependent on the temperature and speed of the re- -8 -7 -6 -5 -4 -3 -2 peak GOES flux [W m -2 ]10 -3 -2 -1 non t he r m a l ene r g y / bo l o m e t r i c ene r g y ( E n t h / E bo l ) C: 0.56 Warmuth & Mann 2016C: 0.04 Inglis & Christe 2014C: -0.43 Emslie+ 2012C: 0.44 Stoiser+ 2007
Fig. 12.
As in Fig. 11, but showing the energy in nonthermal electronsnormalized by bolometric radiated energy, E nth / E bol , vs. peak GOESflux as derived by S +
07, E +
12, IC14, and WM16. E bol is derived froma power-law fit to the values obtained by Kretzschmar (2011). connection outflow jet, which will both be lower in a weakerevent. However, heating at the termination shock will be less sus-ceptible to these parameters, and thus a lower nonthermal frac-tion will result. Another example was presented by Dahlin et al.(2016), who performed kinetic simulations of reconnection andfound that the guide field strongly a ff ects the nonthermal frac-tion. In the presence of a strong guide field, plasma is predom-inantly heated, whereas a weak guide field (e.g. in a highlytwisted flux rope) allows more e ffi cient acceleration.The low nonthermal fraction E nth / E bol in weaker flares hasanother important consequence. It has been demonstrated thatthe bolometric emission is dominated by near-UV, white-light,and near-IR radiation, which mainly originates from compara-tively cool material located at deeper layers of the solar atmo-sphere (chromosphere and photosphere). When we accept thatthe primary energy release takes place in the corona, energyhas to be transported to these deeper layers in order to heatthe material. Based on energetics, we have shown that electronbeams are not su ffi cient for this in weaker flares. We need an ad-ditional energy transport mechanism. Proton beams have beenproposed (e.g. Emslie et al. 1998), but again it is not evidentwhy smaller events should be proton-dominated (cf. Sect. 3.4.5).WM16 have shown that the conductive losses are a viable addi-tional energy transport process that can quantitatively reproducethe bolometric energy. However, this will not work when con-duction is suppressed or when a significant fraction of the con-ductive flux is recycled in the form of conduction-driven evap-oration. In this case, the only energy transport mechanism leftare magnetohydrodynamic waves (see e.g. Fletcher & Hudson2008; Russell & Stackhouse 2013; Reep & Russell 2016).
4. Conclusions
We have reviewed in detail five recent studies that deter-mined both the thermal and nonthermal energy content in sam-
Article number, page 13 of 15 & Aproofs: manuscript no. 39529corr_mod ples of solar flares: Stoiser et al. (2007, S + + + + + ff . Thespectral cross-over method gives an lower estimate for theinjected energy, and as applied by E +
12, IC14, and WM16,this has yielded consistent results. Three alternative methodsapplied by A +
19 have given comparable average energies.4. The bolometric radiated energy, as a proxy for the total en-ergy released in a flare, is a useful independent constraint onboth thermal and nonthermal energetics. While the results ofmost studies are consistent with it, the nonthermal energiesA +
17 clearly violate this constraint.5. Generally, thermal and nonthermal energies have shown rea-sonable correlations with the peak SXR flux as measured byGOES.6. Considering all studies together, we note that the thermal-nonthermal energy partition changes with flare importance.In weak flares, there appears to be a deficit of energetic elec-trons, while the injected nonthermal energy is su ffi cient toaccount for the thermal component in strong flares. Thistendency is found for the ratio of nonthermal energy topeak thermal energy, to total heating requirements (includinglosses), and to the bolometric energy, which to some degreemitigates the uncertainties in the determination of the ther-mal energetics. Thus the changing energy partition is iden-tified as the main cause of the dissimilar results obtained bythe di ff erent studies.7. As a consequence, an additional direct heating process hasto be present, and considering that the bolometric emissionoriginates mainly from deeper atmospheric layers, conduc-tion or waves are required as additional energy transportmechanisms.An improvement in our understanding of energy partition insolar flares, and hence flare physics in general, will require sev-eral steps. On the thermal side, energetics have to be derivedfrom DEM distributions that are either constrained by both EUVand X-ray data (e.g. Battaglia & Kontar 2013; Inglis & Christe2014; Caspi et al. 2014b) or reconstructed with algorithms thathave been shown to produce results consistent with X-ray spec-troscopy (cf. Su et al. 2018). Perhaps even more importantly, theimportance of conductive losses has to be quantified with more confidence. This will require detailed time-resolved studies ofthe role of conduction, involving observations, theory, and nu-merical simulations.Improved constraints on the low-energy cuto ff will be crucialfor the nonthermal component. We propose to conduct system-atic studies using the full warm-target model (not just an ap-proximation), which will for the first time provide upper bound-aries on the energy in injected electrons (as opposed to thelower boundaries provided by the common spectral cross-overmethod). This should be combined with novel approaches, suchas using the time profile of X-ray emission at di ff erent ener-gies to constrain the cuto ff (cf. Dennis & Tolbert 2019). We sug-gest that all these methods should be applied to flares of variousGOES classes in order to ascertain whether a dependence of en-ergy partition on flare importance truly exists. Acknowledgements.
The work of A. W. was supported by DLR under grant No.50 QL 1701. We acknowledge support from the International Space Science In-stitute through the ISSI team on “Solar flare acceleration signatures and theirconnection to solar energetic particles”. We thank Andrew Inglis, Brian Dennis,Gordon Emslie, and Markus Aschwanden for the provision of supplementarydata and helpful discussions.
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