Energetic Electron Distribution of the Coronal Acceleration Region: First results from Joint Microwave and Hard X-ray Imaging Spectroscopy
Bin Chen, Marina Battaglia, Säm Krucker, Katharine K. Reeves, Lindsay Glesener
DDraft version February 11, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Energetic Electron Distribution of the Coronal Acceleration Region: First results from JointMicrowave and Hard X-ray Imaging Spectroscopy
Bin Chen ( 陈 彬 ) , Marina Battaglia , S¨am Krucker , Katharine K. Reeves , and Lindsay Glesener Center for Solar-Terrestrial Research, New Jersey Institute of Technology, 323 M L King Jr. Blvd., Newark, NJ 07102-1982, USA University of Applied Sciences and Arts Northwestern Switzerland, 5210 Windisch, Switzerland Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA School of Physics & Astronomy, University of Minnesota Twin Cities, Minneapolis, MN 55455, USA (Received 2021 January 20; Revised 2021 February 8; Accepted 2021 February 9)
Submitted to ApJLABSTRACTNonthermal sources located above bright flare arcades, referred to as the “above-the-loop-top”sources, have been often suggested as the primary electron acceleration site in major solar flares.The X8.2 limb flare on 2017 September 10 features such an above-the-loop-top source, which wasobserved in both microwaves and hard X-rays (HXRs) by the Expanded Owens Valley Solar Array(EOVSA) and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), respectively.By combining the microwave and HXR imaging spectroscopy observations with multi-filter extremeultraviolet and soft X-ray imaging data, we derive the energetic electron distribution of this sourceover a broad energy range from <
10 keV up to ∼ MeV during the early impulsive phase of the flare.The best-fit electron distribution consists of a thermal “core” from ∼
25 MK plasma. Meanwhile, anonthermal power-law “tail” joins the thermal core at ∼
16 keV with a spectral index of ∼ ∼
160 keV to > >
20 to ∼ ∼
10 Alfv´en crossing times in the source. These resultsprovide strong support for the above-the-loop-top source as the primary site where an on-going bulkacceleration of energetic electrons is taking place very early in the flare energy release.
Keywords:
Solar flares (1496), Solar coronal mass ejections (310), Non-thermal radiation sources(1119), Solar x-ray emission (1536), Solar radio emission (1522), Solar magnetic recon-nection (1504), Solar energetic particles (1491) INTRODUCTIONHard X-ray (HXR) sources located above bright flarearcades, often referred to as the “above-the-loop-top”HXR sources, have been often considered as the pri-mary site for electron acceleration in major solar flares(Masuda et al. 1994; Krucker & Lin 2008; Krucker et al.2010; Chen & Petrosian 2012; Liu et al. 2013; Krucker &Battaglia 2014; Oka et al. 2015; Petrosian 2018). Theseabove-the-loop-top HXR sources are mainly due to
Corresponding author: Bin [email protected] bremsstrahlung radiation, which sometimes also showsa microwave counterpart (Melnikov et al. 2002; Kruckeret al. 2010; Gary et al. 2018; Krucker et al. 2020). Thelatter is due to gyrosynchrotron radiation from pre-sumably the same population of accelerated nonther-mal electrons gyrating in the coronal magnetic field, anargument corroborated in a recent statistical study byKrucker et al. (2020) which found a tight correlation be-tween the >
50 keV HXR peak flux and the 17 and 34GHz microwave peak flux from 40 large flares ( > M7) insolar cycles 23 and 24.However, previous studies often found that spectralindices of the electron distribution derived from HXRand microwave data are very different, deviating from a r X i v : . [ a s t r o - ph . S R ] F e b Chen et al. the simple assumption of an energetic electron distribu-tion with a common power-law shape (e.g., Kundu et al.1994; Silva et al. 2000; Asai et al. 2013). It has beenargued that such a discrepancy may be attributed tospectral breaks in the electron distribution since HXRand microwave emissions are sensitive to different en-ergy regimes: the typical energy ε of an electron emit-ting an HXR photon of energy (cid:15) is ε ≈ . (cid:15) –3 (cid:15) , whereasthe microwave emission is typically dominated by elec-trons with energies above ∼
100 keV (White et al. 2011;Asai et al. 2013). The discrepancy could also arise fromelectron transport between spatially separated HXR andmicrowave sources: the HXRs are usually dominated byprecipitated electrons at the footpoints of the flare ar-cade, while the microwaves are mainly due to trappedelectrons within the flare arcade or loop-top region (Mi-noshima et al. 2008; Kawate et al. 2012; Asai et al. 2013).Important progress has been made by studying par-tially occulted flares in which the bright footpoint HXRsources are hidden behind the solar limb. These eventsoffer the opportunity to study HXR sources in thecorona without the “contamination” from the footpointsources (e.g., Krucker & Lin 2008; Krucker et al. 2010;Effenberger et al. 2017; Petrosian 2018). However, thelatest microwave spectral imaging observations of theSOL2017-09-10 X8.2-class flare by the Expanded OwensValley Solar Array (EOVSA; Gary et al. 2018) revealedthat the microwave emission is present throughout thecoronal flaring region: from the loop-top and loop-legsof the flare arcade (Gary et al. 2018; Yu et al. 2020),to the entire reconnection current sheet and outflow re-gion (Chen et al. 2020b), and even to the footpoints ofthe erupting flux rope (Chen et al. 2020a). Therefore,imaging spectroscopy in both HXRs and microwaves isrequired to obtain the spatially resolved spectra fromthe above-the-loop-top source in isolation.Here we report the first spatially resolved HXR andmicrowave imaging spectroscopy of an above-the-loop-top source, based on the Reuven Ramaty High EnergySolar Spectroscopic Imager (RHESSI; Lin et al. 2002)and EOVSA observations of the SOL2017-09-10 X8.2limb flare during its early impulsive phase. Many as-pects of this flare event have already been studied by nu-merous works. The early impulsive phase is loosely de-fined as the period around the first HXR and microwavepeak at 15:54:20 UT (Figure 1(c) and (f)). It features anerupting cavity and an elongated plasma sheet in the lowcorona, which is interpreted as, respectively, an eruptingmagnetic flux rope viewed along its axis and a reconnec-tion current sheet viewed edge on (Yan et al. 2018; Chenet al. 2020a,b). An above-the-loop-top source, observedin both microwaves and HXR, is present above the SXR flare arcade (Gary et al. 2018; see also Figure 1(a)).As shown by Chen et al. (2020b), this source coincideswith a local minimum in the magnetic field and the loca-tion where most microwave-emitting electrons are con-centrated, serving as a “magnetic bottle” to confine andaccelerate electrons to high energies (see, e.g., recentmodeling results in Kong et al. 2019). Here we combinemicrowave, EUV, and HXR imaging and spectroscopy toconstruct the electron spectrum of the above-the-loop-top source over a wide energy range from <
10 keV upto MeV. We present the imaging spectroscopy resultsin Section 2. We then present the multi-wavelengthanalyses of the thermal and nonthermal properties ofthis source in Sections 3 and 4. In Section 5, we dis-cuss their implications in electron acceleration duringthe early phase of the flare energy release. SPATIALLY RESOLVED HXR ANDMICROWAVE SPECTRARHESSI had four detectors (1, 3, 6, 8) operating dur-ing the event. HXR imaging and spectroscopy resultsduring the early impulsive phase of the flare have beenalready reported in Gary et al. (2018). Briefly, we selecta 48-s time interval 15:53:56–15:54:44 UT (vertical bluestrip in Fig. 1(f)) for analysis as it is least impacted bypulse pileup (which occurs when two lower-energy pho-tons arrive at the detector within a short time and arecounted by the detector as a single higher-energy pho-ton). X-ray imaging using detectors 1 and 3 in 30–100keV reveals one compact HXR source located at onefootpoint of the bright flare arcade with a full-width-at-half-maximum (FWHM) size of about 3.3 (cid:48)(cid:48) . This foot-point source coincides with a compact white-light con-tinuum source observed by the Helioseismic and Mag-netic Imager aboard the Solar Dynamics Observatory(SDO/HMI; Scherrer et al. 2012) at this time (Garyet al. 2018; Jejˇciˇc et al. 2018).A two-step CLEAN procedure (Krucker et al. 2011)is used to reveal an extended coronal HXR above-the-loop-top source with a FWHM size of ∼ (cid:48)(cid:48) × (cid:48)(cid:48) , shownas purple contours in Figures 1(a) and (d). The coro-nal source is located above the bright flare arcade seenin both EUV (SDO/AIA 131 ˚A shown in Figure 1(a),sensitive to ∼
10 MK plasma) and SXR (Hinode/XRTBe-med image in Figure 1(d) and RHESSI 12–18 keVsource shown as red contours). Note the coronal sourceis completely absent (i.e., “resolved out”) from the im-age made with detector 3 alone (with a 6.8 (cid:48)(cid:48) resolution),confirming the extended nature of the coronal source . RHESSI detector modulation is not sensitive to angular scalesmuch larger than the angular resolution of the detector.
900 925 950 975 1000 1025 1050 1075 1100Solar X (arcsec)2502252001751501251007550 S o l a r Y ( a r c s e c ) a Plasma Sheet Erupting Flux Rope
SDO/AIA 131 Å T o t a l F l u x ( s f u ) b Microwave Flux Spectra F l u x D e n s i t y ( s f u ) c Early Impulsive Main Impulsive
Microwave Light Curves S o l a r Y ( a r c s e c ) d Above-the-Looptop 50-100 keVFlare Arcade 12-18 keVFootpoint 30-100 keV
XRT Be Med-Open 15:54:36
20 40 60 80 100 140Energy (keV)10 X - r a y Sp e c t r u m ( p h o t o n s s c m k e V ) e X-ray Photon Flux Spectra
Thermal ArcadeFlare IntegratedBackgroundCoronalFootpoint 15:50 15:54 15:58 16:02UT Time0.00.20.40.60.81.01.21.4 N o r m a li z e d I n t e n s i t y f Early Impulsive Main Impulsive
X-ray Light Curves
GOES 1-8 Å GOES 1-8 Å derivativeRHESSI 50-100 keV Figure 1.
Microwave and HXR imaging spectroscopy of the above-the-loop-top source. (a) EOVSA microwave sources (filledcontours; 40% of maximum), RHESSI 12–18 keV flare arcade SXR source (red contours; 30%, 60%, 90% of maximum), RHESSI30–100 keV footpoint HXR source (blue contours; 50% and 90% of maximum), and RHESSI 50–100 keV above-the-loop-topHXR source (purple contours; 50% and 90% of maximum) overlaid on SDO/AIA 131 ˚A image at 15:53:48 UT. (b) Blue tored solid curves show total-power (full-Sun-integrated) microwave spectra from 15:53:56 UT to 15:54:40 UT. Circle symbolsshow the microwave spectra from the above-the-loop-top source region (integrated within the white box in (a)). Black symbolsindicates the average spectrum within this time interval. (c) Total-power microwave light curves at selected frequencies. (d)More detailed view of the same RHESSI X-ray sources overlaid on Hinode/XRT Be-med image at 15:54:36 UT. The field ofview is shown as a gray box in (a). (e) RHESSI X-ray spectroscopy results showing the flare-integrated spectrum (black) andthe background spectrum (gray), as well as the spectral component from the thermal arcade (dark red). Blue symbols arethe spectrum of the footpoint source obtained by imaging spectroscopy. Short purple dashes are the differences between theflare-integrated spectrum and 1,000 trials of power-law fits for the footpoint source (blue lines), representing the emission fromthe extended coronal source. Purple plus symbols are the average photon flux of the coronal source over each energy band usedfor imaging spectroscopy. (f) GOES 1–8 ˚A SXR light curve (black), SXR derivative (gray), and RHESSI 50–100 keV HXR lightcurve (blue). The blue strip in (c) and (f) indicates the time interval shown in (b) during the early impulsive phase.
HXR imaging spectroscopy is first performed on thefootpoint source using detector 3 by imaging at sevenHXR energy bands centered at 37.5, 42.5, 47.5, 55, 65,80, and 105 keV. Figure 1(e) shows the best fit of thefootpoint photon spectrum (blue crosses) using a power-law function F ( (cid:15) ) = A (cid:15) − γ , yielding a photon spectralindex of γ = 3 . ± . ( A ) = 5 . ± .
3. The uncertainties are obtainedfrom 1,000 trials of power-law fits after perturbing the data points of the footpoint source by adding randomnoises within their uncertainties. The differences be-tween the flare-integrated spectrum (black histogram)and the power-law fits of the footpoint spectrum (bluelines) are shown as the short purple dashes. Their aver-age values and standard deviations within each energyband used for footpoint imaging spectroscopy are shownas the purple cross symbols, representing the photonspectrum from the extended coronal HXR source. While
Chen et al. the photon spectrum of the coronal source is consider-ably softer than that of the footpoint source at above ∼
60 keV, the photon flux of the footpoint and coronalHXR source are comparable at lower energies (within afactor of two at ∼ < DIFFERENTIAL EMISSION MEASUREANALYSISIn order to constrain the thermal properties of theabove-the-loop-top source region, we perform differen-tial emission measure (DEM) analysis by combiningimages from six SDO/AIA passbands (94, 131, 171,193, 211, and 335 ˚A) at around 15:54:20 UT and theHinode/XRT Be-med image obtained at the closesttime (15:54:36 UT). These EUV and SXR bands com-bined provide a broad temperature sensitivity of coro-nal plasma from < (cid:46)
100 MK (Golub et al.2007; Lemen et al. 2012). The DEM analysis was car-ried out using a robustly tested DEM reconstructionroutine xrt dem iterative2 (Golub et al. 2004; Weberet al. 2004), which returns a distribution of the DEMas a function of temperature T , i.e., ξ ( T ) = d ( n d ) /dT (where n th is the thermal electron density and d is the column depth) from selected spatial regions in the im-age. Coronal abundances are assumed for the above-the-loop-top source, following the analysis made by Warrenet al. (2018) based on data from SDO/AIA and the EUVImaging Spectrometer (EIS) aboard Hinode.We divide the above-the-loop-top region into sixteen4 (cid:48)(cid:48) × (cid:48)(cid:48) small boxes (Figure 2) and take the averageintensities within each small box to perform the DEManalysis. A representative DEM distribution of a smallbox located near the centroid of the above-the-loop-topHXR source is shown in Figure 2(c), which displays aprominent peak at T ≈
25 MK (log T = 7 . ∼
25 MK albeit with larger un-certainties owing to the more substantial intensity vari-ations across the region. The average column emissionmeasure is ∼ × cm − . Taking a column depth d with the same value as the size of the above-the-loop-top source (20 (cid:48)(cid:48) –30 (cid:48)(cid:48) ), the average thermal electron den-sity of the above-the-loop-top source is about 3–4 × cm − . Our DEM analysis results are broadly consis-tent with those derived from EUV spectroscopy datamade by the Extreme-Ultraviolet Imaging Spectrometeraboard Hinode (Warren et al. 2018; Polito et al. 2018),although the latter used measurements made during themain peak of the flare (5–10 minutes after our time ofinterest). JOINT HXR AND MICROWAVE SPECTRALANALYSISThe HXR and microwave spectra of the above-the-loop-top source obtained from spatially resolvedRHESSI and EOVSA data offer a unique opportunityof reconstructing the underlying energetic electron dis-tribution over a wide energy range. As introduced inSection 1, the HXR spectrum provides excellent con-straints for the nonthermal electrons at relatively lowenergies ( (cid:46)
100 keV in this event), while the microwavedata complement the HXR diagnostics with added con-straints for the more energetic, (cid:38)
100 keV electrons. Inaddition, the optically-thick part of the microwave spec-trum also provides diagnostics for the thermal electrondensity, complementing the DEM analysis discussed inSection 3.For the nonthermal electron density distribution f ( ε ) = dn nth ( ε ) /dε , we adopt a broken power-law func-tional form with free parameters including the total elec-tron density n nth , low-energy cutoff ε min , break energy
940 960 980 1000Solar X (arcsec)180170160150140130120110 S o l a r Y ( a r c s e c ) a AIA 335 A 2017-09-10 15:54:26
940 960 980 1000Solar X (arcsec)180170160150140130120110 S o l a r Y ( a r c s e c ) b AIA 94 A 2017-09-10 15:54:24
940 960 980 1000Solar X (arcsec)180170160150140130120110 S o l a r Y ( a r c s e c ) d AIA 131 A 2017-09-10 15:54:20
940 960 980 1000Solar X (arcsec)180170160150140130120110 S o l a r Y ( a r c s e c ) e XRT Be med-Open 2017-09-10 15:54:36 T ( K )10 D E M ( c m K ) c DEM of Red Box T ( K )10 D E M ( c m K ) f DEMs of All Boxes
DEM of Average IntensityDEM Averaged Over Boxes
Figure 2.
Differential emission measure analysis of the above-the-loop-top source by combining SDO/AIA and Hinode/XRTmulti-band images. Panels in the left two columns show four examples of the seven EUV/SXR bands used for the DEManalysis. Contours are the RHESSI X-ray sources same as those in Figure 1(d). The sixteen 4 (cid:48)(cid:48) × (cid:48)(cid:48) regions used for derivingthe DEM results within the above-the-loop-top region are shown as white boxes. Panel (c) shows an example of the derivedDEM curve for a selected region (shown as the red box in left panels). The blue curve in panel (f) shows the DEM averagedover those derived from the 16 small boxes, while the black curve is the DEM derived from the average intensity within theentire above-the-loop-top region. Gray curves in panels (c) and (f) show the Monte Carlo runs. ε b , and spectral indices δ (cid:48) and δ (cid:48) below and above thebreak energy, respectively. Additional parameters usedfor the joint spectral analysis include the magnetic fieldstrength B , thermal electron density n th , and viewingangle θ (with regard to the magnetic field direction).The thermal temperature T of the above-the-loop-topsource is fixed to 25 MK according the DEM results andthe column depth d is set to 20 (cid:48)(cid:48) .To forward-fit the HXR emission, we assume the thin-target bremsstrahlung model and use the Python codesavailable in the SunPy package sunxspex . For themicrowave emission, we forward-calculate the gyrosyn-chrotron radiation from the broken power-law electrondistribution by adopting the fast gyrosynchrotron codesdeveloped by Fleishman & Kuznetsov (2010). We use a https://github.com/sunpy/sunxspex/blob/master/sunxspex/emission.py, which is a Python version of the standard routine brm2 thintarget.pro available in SolarSoft IDL. global minimization method differential evolution available in Scipy’s optimize package to fit the observedHXR and microwave spectra jointly. We note that thenumber of the independent measurements used for theminimization (35) is much greater than the degrees offreedom (8) adopted in the joint fit, therefore constitut-ing a well-determined minimization problem.To test the robustness of the fit results, we also employa Bayesian-statistics-based Markov chain Monte Carlo(MCMC) method (implemented in an open-source soft-ware emcee included in the Python package lmfit ), toevaluate the probability distributions of the fit parame-ters, which are in turn used to estimate the associateduncertainties (Figure 5; see Appendix for details). Thebest-fit results for both the microwave and HXR spectra,shown as thick black curves in Figure 3(a) and (b), re-spectively, agree very well with the distributions of the https://lmfit.github.io/lmfit-py Chen et al.
Frequency (GHz)10 F l u x ( s f u ) a Microwave Spectrum
Broken Powerlaw FitNonthermal ALT2.5 5.0 7.5 10.0 12.5 15.0Frequency (GHz)202 S i g m a P h o t o n F l u x ( p h o t o n s s c m k e V ) b X-ray Spectrum
Thermal ArcadeThermal ALT (Fit)Thermal ALT (DEM)Broken Powerlaw FitNonthermal ALT20 40 60 80 100 140Energy (keV)202 S i g m a Energy (keV)10 D i ff e r e n t i a l D e n s i t y ( e l e c t r o n s k e V c m ) Thermal Core (25 MK) Nonthermal Tail = 3.6 = 8.5 c Electron Density Distribution
Thermal ALT (Fit)Thermal ALT (DEM)Broken Powerlaw Fit
Figure 3.
Joint HXR and microwave spectral fit of the above-the-loop-top source (“ALT” in the figure legends). (a) Best fitfor the microwave spectrum (thick black curve) and the residuals. Microwave spectra from 200 of 1000 MCMC samples of theparameter space and the corresponding residuals are shown as thin purple curves. (b) Best fit for the above-the-loop-top X-raysource (thick black curve) and the residuals. Thin purple curves are the calculated HXR spectra from the same MCMC samples.Thin-target bremsstrahlung is assumed for the nonthermal HXR emission. The corresponding thermal bremsstrahlung X-rayspectra from the MCMC samples and DEM analysis are shown as the thin red curves and thick orange curve, respectively.(c) Corresponding best-fit and MCMC samples of the thermal component (red curves) and broken powerlaw electron densitydistribution (thick black curve and thin purple curves). The DEM-derived thermal electron density distribution is shown as thethick orange curve. Also shown are the calculated microwave and HXR spectra for two test cases of the nonthermal electrondistribution: dash-dot curves are for a power-law spectrum that does not break at ε b , and dashed curves are for a spectrumthat cuts off completely at ε b . See text for discussions. Frequency (GHz)10 F l u x ( s f u ) a Microwave Spectra S i g m a P h o t o n F l u x ( p h o t o n s s c m k e V ) b X-ray Spectra
30 40 60 80 100 140Energy (keV)202 S i g m a Energy (keV)10 D i ff e r e n t i a l D e n s i t y ( e l e c t r o n s k e V c m ) Thermal Core (25 MK) Nonthermal TailTime = 3.6 = 25.0 = 5.8 c Electron Density Distribution S e c o n d s s i n c e : : Figure 4.
Temporal evolution of the above-the-loop-top source and the corresponding fit results. (a) Temporally resolvedEOVSA microwave spectra of the above-the-loop-top source from 15:53:56 UT to 15:54:28 UT when an increasing microwaveflux is present at all frequencies (circle symbols colored in time from blue to red) and their best fits (solid curves). (b) RHESSIHXR spectra based on the 48-s averaged spectrum of the above-the-loop-top source but scaled in time according to the flare-integrated 50–100 keV light curve. Best-fit results are shown as solid curves. (c) The corresponding time evolution of the best-fitnonthermal electron distribution shown in the same color scheme.
MCMC runs (thin purple curves). The correspondingbest-fit and MCMC runs of the broken power-law elec- tron distribution are shown in Figure 3(c). The ther-mal electron spectra returned from the MCMC runs ofthe joint fit are also displayed as thin red curves, inagreement with the DEM analysis results (thick orangecurve).As expected, the electron spectral index below thebreak energy δ (cid:48) = 3 . +0 . − . is quite nicely constrained,thanks to the HXR measurements with relatively smalluncertainties below ∼
100 keV. The break energy ε b isat 158 +14 − keV, above which the spectral index of thehigher energy electrons shows a significant break-down,with δ (cid:48) > . n th ≈ . +0 . − . × cm − (red curves in Fig-ure 3(c) show the corresponding electron distribution),consistent with that estimated from the DEM analysis( n th ≈ . × cm − with a column depth of d = 20 (cid:48)(cid:48) ;orange curve in Figure 3(c)). The X-ray emission ex-pected from the thermal component of the above-the-loop-top source, shown in Figure 3(b) as red and or-ange curves (calculated using the microwave- and DEM-constrained thermal electron density, respectively, and T = 25 MK), is “buried” under the X-ray emission fromthe thermal flare arcade (dark red curve in Figure 3(b))in the flare-integrated RHESSI X-ray spectrum. Such aweak thermal emission also renders it difficult to detectin RHESSI SXR images with the presence of the brightflare arcade (c.f., the 12–18 keV image shown in Fig-ure 1(d)). In the Hinode/XRT Be-med image (whichis sensitive to a broad temperature range of ∼ ∼
16 keVnearly seamlessly, implying a continuous electron popu-lation resulted from the bulk energization process. Wenote that while it is rather challenging to constrain thelow-energy cutoff ε min of the nonthermal electron spec-trum using the HXR data alone (Holman et al. 2011; seealso recent developments in Kontar et al. 2019 and ref-erences therein with the warm-target approach), the mi-crowave spectrum is sensitive to, among others, the to-tal nonthermal electron density n nth (see, e.g., Movie S2in Fleishman et al. 2020). The latter depends stronglyon ε min , particularly when a tight constraint for δ (cid:48) andthe nonthermal electron density above ∼
40 keV is al-ready available from the HXR diagnostics. Therefore,our joint microwave–HXR fit complements each otherto provide an optimal constraint for both the low-energycutoff ε min = 16 +3 − keV and the total nonthermal elec-tron density n nth = 1 . +0 . − . × cm − . The magneticfield strength returned from the fit is 845 +198 − G, consis- tent with earlier findings of a strong coronal magneticfield in this flare (and the associated active region) by us-ing direct spectroscopy measurements (Gary et al. 2018;Kuridze et al. 2019; Fleishman et al. 2020; Chen et al.2020b) and indirect extrapolation estimates (Longcopeet al. 2018; Anfinogentov et al. 2019). The correspond-ing plasma β = P gas /P B = 8 πn th kT /B is ∼ v A = B/ √ πn th µm H is around 10,000km s − .The steep break-down of the electron spectrum above ∼
160 keV resulting from the joint HXR/microwave fitis mainly determined by the observed microwave spec-trum. To demonstrate the necessity of such a spectralbreak down, we calculate the HXR and microwave spec-tra from two limiting cases of the electron distribution.Case A: A single power-law having the same best-fitspectral index of δ (cid:48) but extending beyond ε b to largeenergies (dash-dot curve in Figure 3(c)); Case B: Thesame power-law spectrum that cuts off completely at ε b (dashed curve in Figure 3(c)). Case A results in amicrowave spectrum that greatly exceeds the observedflux at above ∼ ε b also explainsthe relatively loose constraints on the upper bound of δ (cid:48) ,as demonstrated by the MCMC results. In either case,the corresponding HXR spectrum shows minor differ-ences at the highest measured energies that are nearlyindistinguishable within the uncertainties (dash-dot anddashed curves in Figure 3(b)). DISCUSSIONSIn the previous sections, we have combined microwave,EUV, and X-ray imaging and spectroscopy data of thesame coronal above-the-loop-top source observed duringthe early impulsive phase of the SOL2017-09-10 flare toderive a comprehensive, and nearly continuous energeticelectron spectrum from <
10 keV to ∼ ∼ δ (cid:48) ≈ . δ (cid:48) > .
0. Such a spectral breakdown is an indication for the relatively small numberof microwave-emitting, mildly relativistic electrons veryearly in the flare energy release.
Chen et al.
However, several minutes later when the mi-crowave/HXR fluxes reach their peak at ∼ δ (cid:48) ≈ . ∼
50 keV with a photon spec-tral index of γ ≈ . δ (cid:48) ≈ . δ (cid:48) ≈ . (cid:38)
100 keV electrons and hardens the electron distribu-tion within a few minutes.It is beyond the scope of the current work to expandour joint microwave/HXR spectral analysis to theselater times, particularly because a significant pileup ef-fect exists for the HXR data. However, the temporallyresolved microwave imaging spectroscopy data does al-low us to gain some insights on the temporal evolu-tion of the high-energy electrons within this short in-terval during the early impulsive phase. Since RHESSIimaging spectroscopy does not provide temporally re-solved HXR spectra for the above-the-loop-top sourceduring this period, as a first-order approximation, wescale the absolute HXR photon flux of the 48-s-averagedspectrum (i.e., purple crosses in Figure 1(e)) as a func-tion of time according to the temporal variation of theflare-integrated 50–100 keV count rate in Figure 1(f),producing the pseudo temporally resolved HXR spec-tra shown in Figure 4(b). For simplicity, we further fixthe density of the thermal electrons n th , viewing an-gle θ , and the spectral index below the break δ (cid:48) to thebest-fit values from the time-averaged spectra, and setthe break energy to ε b = 120 keV. Figure 4 shows thetime-dependent fit results and the corresponding elec-tron density distribution, colored in time from blue tored for the interval from 15:53:56 to 15:54:30 UT whenthe microwave fluxes at all frequencies are growing. Inaccordance with the increasing microwave flux, the high-energy electron population above ε b experiences a rapidhardening: the electron spectral index above the break δ (cid:48) decreases from >
20 to ∼ (cid:46)
10 Alfv´encrossing times τ A in the above-the-loop-top source re-gion ( τ A = L/v A , where L = 20 (cid:48)(cid:48) –30 (cid:48)(cid:48) is the source size).Such a rapid spectral hardening of nonthermal electronswithin several Alfv´en crossing times has been found inrecent particle acceleration simulations for macroscalelow plasma β systems (Guidoni et al. 2016; Li et al. 2018; Arnold et al. 2020). We caution that, however,the results demonstrated here are subject to the valid-ity of the assumptions we adopt: both the fraction ofthe HXR flux in the above-the-loop-top source and thespectral shape of the HXR photon spectrum (which ismainly determined by δ (cid:48) ) remain unchanged during this34-s period, which cannot be examined in detail dueto the unavailability of HXR imaging spectroscopy at afiner time resolution.In our analysis, the thin-target bremsstrahlung sce-nario is assumed. The assumption is largely validsince the critical energy for stopping the electrons inthe above-the-loop-top source due to Coulomb collisions ε c ≈ . N / (cid:46)
20 keV (where N = n th L is the col-umn density in units of 10 cm − ), well below mostenergies in the nonthermal electron spectrum. However,under the strong diffusion limit for which electrons un-dergo a random-walk-type transport process, the HXRabove-the-loop-top source can become a thick target(Metcalf & Alexander 1999; Sim˜oes & Kontar 2013; Pet-rosian 2018). Nevertheless, such a coronal thick-targetscenario is deemed unlikely for our case because of thepresence of a bright 30–100 keV footpoint source, whichindicates abundant precipitated electrons at >
30 keV.An intermediate thin-thick target scenario may still bepossible in the case of a transitional weak-to-strong dif-fusion. A detailed investigation is beyond the scope ofthis Letter.Although a broken power-law function for the non-thermal electron distribution is found to agree very wellwith the observed microwave and HXR spectra, ourresults do not necessarily exclude the possibility of adifferent type of source electron distribution. Distri-butions that are intense and flat at low energies, butweaker and steeper at high energies, may also work.One such example is the kappa distribution (Oka et al.2015; Battaglia et al. 2015, 2019). In addition, despitethat our data provide direct diagnostics for the above-the-loop-top source in isolation, transport effects withinthe above-the-loop-top source (e.g., trapping, scattering,collisional loss) may alter the “pristine” flare-acceleratedelectron spectrum, leading to a spectral break-downat higher energies (Melrose & Brown 1976; Wheatland& Melrose 1995; Fletcher & Martens 1998; Metcalf &Alexander 1999; Petrosian 2018). While care mustbe taken when interpreting the observations, the rapidhardening of the electron spectrum does suggest a likelyongoing-acceleration. Last but not least, in our analy-sis, an isotropic electron angular distribution is assumed.Amendments need to be made if a significant anisotropyof the electron distribution is present (Fleishman & Mel-nikov 2003; Massone et al. 2004; Kaˇsparov´a et al. 2007;Chen & Bastian 2012), although observational evidencefor the anisotropy in the above-the-looptop sources hasbeen elusive. Our data do not provide adequate con-straints for distinguishing these above scenarios. Fur-ther progress calls for next-generation microwave andHXR instrumentation that can provide high dynamicrange imaging spectroscopy observations along with si-multaneously high spatial, temporal, and spectral reso-lution, such as the Frequency Agile Solar Radiotelescope(FASR; Bastian et al. 2019) and a spacecraft version ofthe Focusing Optics X-ray Solar Imager sounding rocket(FOXSI; Krucker et al. 2013). ACKNOWLEDGMENTSWe are grateful to Lyndsay Fletcher, Eduard Kon-tar, Dale Gary, Gregory Fleishman, James Drake, HarryArnold, Fan Guo, and Xiaocan Li for helpful discussionswithin the SolFER DRIVE Science Center collabora-tion. The work is supported partly by NASA DRIVEScience Center grant 80NSSC20K0627. EOVSA oper-ation is supported by NSF grant AST-1910354. B.C.is supported by NSF grant AGS-1654382 and NASAgrant 80NSSC20K1318 to NJIT. K.R. is supported byNSF grant AGS-1923365 to SAO. L.G. is supported byNASA grant 80NSSC20K1277. Hinode is a Japanesemission developed and launched by ISAS/JAXA, withNAOJ as domestic partner and NASA and STFC (UK)as international partners. It is operated by these agen-cies in cooperation with ESA and NSC (Norway).
Facilities:
OVRO:SA, RHESSI, SDO, Hinode
Software:
Astropy (Astropy Collaboration et al.2018), CASA (McMullin et al. 2007), LMFIT (Newvilleet al. 2014), NumPy (Harris et al. 2020), SciPy (Virtanenet al. 2020), SunPy (SunPy Community et al. 2020),0
Chen et al.
APPENDIX A. MARKOV CHAIN MONTE CARLO ANALYSIS OF THE JOINT SPECTRAL FITWe employ a Markov chain Monte Carlo (MCMC) method to evaluate the fit results. The technique and software usedfor the MCMC analysis are identical to those described in the Methods section of Chen et al. (2020b). Figure 5 shows theMCMC analysis results in the form of a corner plot, in which the diagonal panels show the one-dimensional projectionsof the probability distributions of the respective fit parameters. The two-dimensional histograms of the probabilitydistributions between pairs of the fit parameters are shown as the non-diagonal panels. Solid horizontal/vertical linesin each panel indicate the best-fit values from the minimization. The widths of the distributions provide optimalestimates for the uncertainties of the respective fit parameters. The uncertainties shown for each fit parameter areestimated using the 16% and 84% quantiles of the respective 1-D histograms, which correspond to approximately the1- σ level (1 − e − . ≈ . B (G) = 845 +19847 ( d e g ) (deg) = 26 +103 . . . . n t h ( × c m ) n th (×10 cm ) = 2.4 +0.90.6 . . . . . n n t h ( × c m ) n nth (×10 cm ) = 1.1 +0.50.4 . . . . . = 3.6 +0.10.7
481 21 62 0 = 8.5 +8.20.4 b ( k e V ) b (keV) = 158 +1446 B (G)
81 62 43 2 m i n ( k e V ) (deg) . . . . n th (×10 cm ) . . . . . n nth (×10 cm ) . . . . . b (keV) min (keV) min (keV) = 16 +36 Figure 5.
Markov chain Monte Carlo analysis of the joint microwave and HXR fit results. See Appendix for details.