Diffusive Transport of Energetic Electrons in the Solar Corona: X-ray and Radio Diagnostics
AAstronomy & Astrophysics manuscript no. paper_arxiv2 c (cid:13)
ESO 2018November 6, 2018
Diffusive Transport of Energetic Electrons in the Solar Corona:X-ray and Radio Diagnostics
S. Musset , , E. P. Kontar , and N. Vilmer LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ.Paris Diderot, Sorbonne Paris Cité School of Physics and Astronomy, University of Minnesota School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UKReceived ... / Accepted ...
ABSTRACT
Context.
Imaging spectroscopy in X-rays with RHESSI provide the possibility to investigate the spatial evolution ofthe X-ray emitting electron distribution and therefore to study the transport effects on energetic electrons during solarflares.
Aims.
We study the energy dependence of the energetic electron scattering mean free path in the solar corona.
Methods.
We use the imaging spectroscopy technique with RHESSI to study the evolution of energetic electrons dis-tribution in different part of the magnetic loop during the 2004 May 21 flare. These observations are compared withthe radio observations of the gyrosynchrotron radiation of the same flare by Kuznetsov & Kontar (2015), and with thepredictions of the diffusive transport model described by Kontar et al. (2014).
Results.
The X-ray analysis shows a trapping of energetic electrons in the corona and a spectral hardening of theenergetic electron distribution between the top of the loop and the footpoints. Coronal trapping of electrons is strongerfor the radio-emitting electrons than for the X-ray-emitting electrons. These observations can be explained by thediffusive transport model derived by Kontar et al. (2014).
Conclusions.
We show that the combination of X-ray and radio diagnostics is a powerful tool to study electron transportin the solar corona in different energy domains. We show that the diffusive transport model can explain our observations;and in the range 25-500 keV, the electron scattering mean free path decreases with electron energy. We can estimatefor the first time the scattering mean free path dependence on energy in the corona.
Key words.
Sun: flares - Sun: particle emission - Sun: X-rays - Transport of particles - RHESSI
1. Introduction
Particle transport between the acceleration site and the X-ray and radio emission sites is a key process that mustbe studied and understood in order to use X-ray and ra-dio diagnostics to study particle acceleration during solarflares. Indeed, transport mechanisms can modify the spatialand spectral distributions of energetic particles produced bythe acceleration process. The spatial and spectral distribu-tions of X-ray emitting electrons can be studied during solarflares using imaging spectroscopy in X-rays. This techniqueis therefore a useful tool to study the transport of energeticelectrons in magnetic loops.In addition to imaging and spectroscopy of solar flaresin X-ray and gamma-ray ranges (Lin et al. 2002), theReuven Ramaty High Energy Solar Spectroscopic Imager(RHESSI) provides the possibility to use imaging spec-troscopy in hard X-rays (HXR). This technique has beenused to study events which exhibit both footpoint and coro-nal HXR sources (e.g. Krucker & Lin 2002; Emslie et al.2003; Battaglia & Benz 2006; Piana et al. 2007; Simões &Kontar 2013). These studies show in particular that in someevents, X-ray emission in the coronal source is a combina-tion of both thermal and non-thermal emissions. Battaglia& Benz (2006) showed that the difference between the pho-ton spectral indexes in the coronal source and the footpoint sources was between 1.2 and 0.6 (in three flares) and be-tween 2.4 and 3.7 (in two flares), but not 2, the expectedvalue in the standard model. This discrepancy betweenexpected and observed differences implies that addition-nal transport effects are needed to explain these observa-tions. Battaglia & Benz (2006) interpreted the hardeningof the spectrum as the result of a filter mechanism causinglow-energetic electrons to preferentially loose energy beforereaching the chromosphere; candidates for this mechanismsbeing collisions and the electric field of the return current.More recently, Simões & Kontar (2013) compared the elec-tronic spectral indexes in coronal and footpoint sourcesand went to similar conclusions: on the four events stud-ied, the difference of electronic spectral index between thefootpoints and the coronal source lies between 0.2 and 1.0,while it is expected to be nul in the case of limited electroninteraction with the ambiant medium during the transportin the loop. This study also shows that the rate of non-thermal electrons in the coronal source is larger than inthe footpoints, by a factor ranging from ≈ ≈
8. Theseobservations suggest that a mechanism is responsible for en-ergetic electron trapping in the coronal source. Such mech-anism could be for instance magnetic mirroring or turbu-lent pitch-angle scattering. These observations carried withimaging spectroscopy in X-rays provide new constraints to
Article number, page 1 of 15 a r X i v : . [ a s t r o - ph . S R ] O c t &A proofs: manuscript no. paper_arxiv2 electron propagation models and are not compatible withthe predictions of the standard model, described in the fol-lowing.In the standard model of solar flares (see e.g. Sturrock1968; Arnoldy et al. 1968; Sweet 1969; Syrovatskii & Shmel-eva 1972), energetic electrons are accelerated in the coronaand then propagate along the magnetic field lines of coro-nal loops, losing a relatively small amount of their energyvia collisions with the particles of the ambiant plasma, un-til they reach the chromosphere, a denser medium wherethey loose instantaneously the bulk of their energy and arethermalized. During their propagation, energetic electronsradiate a bremsstrahlung emission which is detected in theX-ray range, both in the coronal loop and in the footpoints(see e.g. Holman et al. 2011; Kontar et al. 2011a, for recentreviews). In this standard model for the electron transport,we expect to see as many electrons leaving the looptopsource than arriving in the footpoint, since the propaga-tion time is much smaller than the collision time in thecorona, and than the time cadence of X-ray observations.For that reason, it is also expected to see the same spectraldistribution of energetic electrons in the looptop and in thefootpoints. Therefore, in this standard model, we expect tofind the same electron rate and the same electronic spec-tral index in the looptop and in the footpoints However,as it has been described in this introduction, recent anal-ysis of X-ray emission during solar flares (e.g. Battaglia &Benz 2006; Simões & Kontar 2013) shown that this stan-dard model for electron propagation could not explain theirobservations.Trapping of energetic electrons in the coronal part ofthe loop can be explained by the effect of a converging mag-netic field. The simpliest way to modelize magnetic mirrorsis to consider a magnetic loss cone for the pitch angle dis-tribution. The value of the loss-cone angle depends on themagnetic ratio σ = B F P /B LT . Aschwanden et al. (1999a);Tomczak & Ciborski (2007); Simões & Kontar (2013) cal-culated the magnetic ratios needed to explain X-ray ob-servations, assuming an isotropic pitch angle distribution,and found values lying between 1.1 and 5.0. However, mag-netic loss cones are an approximation for magnetic mirror-ing only valid for rapid variations of density and magneticfield amplitude near the footpoints of the magnetic loop.More realistic models of magnetic convergence have beendeveloped and the evolution of energetic electron popula-tions in the case of a converging magnetic field have beenstudied analytically (see e.g. Kennel & Petschek 1966; Ko-valev & Korolev 1981; Leach & Petrosian 1981; MacKinnon1991; Melrose & Brown 1976; Vilmer et al. 1986) or nu-merically (see e.g. Bai 1982; McClements 1992; Siversky &Zharkova 2009; Takakura 1986). These studies showed thatthe convergence of magnetic field causes energetic electrontrapping in the corona, but the value of the ratio of electronrates in the corona and in the footpoints depends on numer-ous parameters such as the density, the form of the magneticfield convergence or the electronic pitch angle distribution.We note that Takakura (1986) calculated in particular thedifference of spectral index between the coronal source andthe footpoints, lying between 0 and 0.8.Energetic electron trapping can also be explained byan alternative scenario, the diffusive transport of electronsdue to strong pitch angle scattering. Turbulent pitch anglescattering is the result of small scale magnetic fluctuationsaffecting the parallel transport of energetic electrons in flar- ing loops. The presence of such magnetic fluctuations issuggested by the increase of loop width which has been ob-served with RHESSI (Kontar et al. 2011b; Bian et al. 2011).Kontar et al. (2014) studied the effect of strong turbulentpitch angle scattering, leading to a diffusive transport ofenergetic electrons in the loop, during solar flares. Theycompared the predictions of the model with observations offour flares and estimated for these events that the charac-teristic mean free path for this diffusive transport was of − cm, which is smaller than the typical size of aloop ( ≈ × cm) and comparable to the size of coronalsources ( ≈ × cm). Therefore, the authors concludedthat pitch-angle scattering du to magnetic fluctuations ina collisional plasma is likely to be present in flaring loops.The diffusive transport of electrons and ions is alsostudied since several decades in the interplanetary mediumwhere in-situ measurements of particles are made. Jokipii(1966) developed the first description of particle scatter-ing in varying magnetic field. In this analysis, the magneticfield is considered as the superposition of a constant fieldand a smaller fluctuating component which is an homo-geneous random function of position with zero mean. Thiswork was improved in later approachs (see e.g. Dröge 2000a,for a review). Some studies focussed on the possible rigid-ity dependence of the particle mean free path. Palmer(1982) studied the values of the mean free path measuredfor solar particle events near the Earth and found that al-though the values could vary of two orders of magnitude,no dependence in rigidity was found: the values of the meanfree path at different rigidities were found mostly between0.08 and 0.3 AU, in the so-called ’consensus range’. How-ever, later studies revisited this consensus (see e.g. Bieberet al. 1994; Dröge 2000b) and showed in particular that theelectron scattering mean free path is rigidity dependant. Inparticular, Dröge (2000b) showed that the electron meanfree path varies as a power law with rigidity, in the range0.1-1 MV, with a slope of -0.2. More recently, Agueda et al.(2014) found the same kind of rigidity dependence for sixsolar particle events (over seven studied), in the 0.3-0.5 MVrange, with slopes varying between -0.3 and -1.2.In this paper, we present X-ray observations of one flarewhich exhibit a non-thermal loop-top X-ray source. TheM2.6 flare on 2004 May 21 flare is located near the solarlimb and was well observed by RHESSI, the Nobeyama Ra-dio Heliograph (NoRH) and the Nobeyama Radio Polarime-ters (NoRP). Kuznetsov & Kontar (2015) showed that thegyrosynchrotron emissions observed at 17 and 34 GHz withthe NoRH were cospatial with the X-ray emission (evenif the centroid of the X-ray emission is shifted of about 6arcsec under the position of the centroid of the 34 GHzemission), where a looptop source and two footpoints arevisible. They also deduced from the NoRP spectra of themicrowave emission that the absolute value of the electronicspectral index was about 2.7. The authors simulated thegyrosynchrotron emission with the recently developed IDLtool GX Simulator, using a linear force-free extrapolation ofthe magnetic field of the loop. The results of their simula-tion was compared with the microwave data to deduce thespatial and spectral properties of the radio-emitting ener- The rigidity R of a charged particle is defined by R = pc/q where p is the momentum of the particle, q its charge and c thespeed of light. For relativistic particle, R = (cid:112) E ( E + 2 mc ) /q where E is the kinetic energy and m the mass of the particle.Article number, page 2 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona getic electrons. They found that the microwave emission ismostly produced by electrons of a few hundreds of keV hav-ing a hard spectrum (with an absolute value of the spectralindex around 2). They also showed that the spatial distri-bution of energetic electrons with energy above 60 keV isstrongly peaked near the top of the flaring loop, implyingthat there is a coronal trapping of energetic electrons duringthis event. The peak of the spatial distribution of energeticelectrons is shifted of 3.2 Mm in regards to the top of theloop where the magnetic field is minimal. According to theauthors, this spatial distribution of energetic electrons isdue to a combination of the processes of particle acceler-ation, trapping, and scattering. However, the authors didnot calculate the scattering rate but focussed on the distri-bution of electrons in the loop.The aim of this paper is the study of the trapping of en-ergetic electrons in two distinct energy domains. For thatpurpose, we used the analysis of Kuznetsov & Kontar (2015)of the radio-emitting energetic electrons above a few hun-dred of keV, and we analysed the X-ray emission of ener-getic electrons with energies below 100 keV. We thereforeshow in this paper that the electron scattering mean freepath decreases with increasing electron energy. In section2 is presented the imaging spectroscopy of the 2007, May21 flare in X-rays. The spatial and spectral distributionsderived from the X-ray observations are presented in sec-tion 3. The interpretation of the X-ray observations, thecomparison between X-ray and radio observations, and thecomparison with the predictions of the diffusive transportmodel of Kontar et al. (2014) are discussed in section 4,along with the energy dependence of the energetic electronscattering mean free path in the frame of that model. Alter-native mechanisms and improvement of the diffusive trans-port models are discussed in section 5. The main results aresummarized in section 6.
2. X-ray imaging spectroscopy at the peak of theflare
The M2.6 flare on 2004 May 21 flare, in active region10618, was detected by RHESSI in the 3-100 keV range. TheRHESSI corrected count rates at relevant energies bins arepresented on figure 1, together with the X-ray flux fromGOES. In this figure the count rates are corrected fromthe changes of attenuator state and decimation state. Thepeak of the RHESSI count rates is around 23:50 UT, whichis about 2 minutes before the GOES X-ray peak. On figure1, the vertical dashed-dotted lines show the time interval(23:49:30 to 23:50:30 UT) chosen to image the X-ray emit-ting sources. We chose the time interval with the highestsignal above 25 keV. Note that a consequent peak in the 25-100 keV lightcurve is visible between 23:56:00 and 23:58:30UT; but it was not possible to reconstruct a reliable imageabove 25 keV during this time interval. The photon statis-tics is also too low to enable reliable imaging spectroscopyon other one-minute intervals after the X-ray peak of theflare, due to the high level of noise in the images, and there-fore the time evolution of X-ray emission is not discussedin this paper. the lower limit adopted in Kuznetsov & Kontar (2015) is 60keV even if the radio emissivity is maximum for electrons of afew hundred of keV (Kuznetsov, private communication). Fig. 1.
RHESSI corrected count rates between 23:35 and 00:05UT, in different energy ranges (green: 12-25 keV, cyan: 25-50keV, orange: 50-100 keV) and GOES flux between 1.0 and 8.0Å (dashed line). The vertical dashed-dotted lines at 23:49:30and 23:50:30 UT mark the time interval used for imaging spec-troscopy. a r cs e c Fig. 2.
CLEAN image (beam factor 1.7) between 23:49:30 and23:50:30 UT, at 25-50 keV. Contours are 30%, 50%, 70% and90% of CLEAN images at 10-25 keV (blue), 25-50 keV (green)and 50-100 keV (orange). Boxes 0, 1, 2 are used for imagingspectroscopy of the looptop source, the first footpoint and thesecond footpoint respectively.
Image and contours at 12-25, 25-50 and 50-100 keV arepresented in figure 2. The geometry of the source can beinterpreted as a single loop structure with two footpoints.A coronal hard-X-ray source is visible on the top of the
Article number, page 3 of 15 &A proofs: manuscript no. paper_arxiv2 loop structure at 12-25 and 25-50 keV, and the two foot-points are visible in the 25-50 and 50-100 keV ranges. Theloop was divided in three regions (see figure 2) in order todo imaging spectroscopy on the looptop source and on thetwo footpoints. The image reconstruction was done over a60-second time interval during the main hard X-ray peak,between 23:49:30 and 23:50:30 UT, using the CLEAN algo-rithm (Hurford et al. 2002) with a beam factor value of 1.7.The beam factor was carefully chosen as it has an impor-tant impact on the determination of the source sizes (seesection 2.3 and appendix A). All collimators except the firstone (with the smallest pitch) were used, achieving a spatialresolution of 3.9 arcsec.To do imaging spectroscopy, we reconstructed CLEANimages in 20 narrow energy bins between 10 and 100 keV,with increasing width of the bins with energy (2 keV widthbetween 10 and 30 keV, 3 keV width between 30 and 45 keV,5 keV width between 45 and 60 keV, 15 keV bin between60 and 75 keV and 25 keV bin between 75 and 100 keV).Three images over the 20 images produced are presented infigure 3, with the 50 % contours in red. On these images,the looptop source is visible between 10 and 36 keV, andthe footpoints are visible above 28 keV. The visibility oflooptop and footpoint sources in the images is of courselimited by the dynamic range of the images.
Each of the 20 images reconstructed between 10 and 100keV contributes to a single point in the spectrum. The spec-tra of each region defined in figure 2 were fitted using acombination of a thermal and a non-thermal componentsin OSPEX (Schwartz et al. 2002). The three spectra result-ing from the fits are displayed in figure 4 and the values ofthe free parameters are described in table 1.The thermal model has two free parameters which areadjusted during the fit: the temperature and emission mea-sure of the X-ray emitting plasma. The non-thermal partof the spectra was fitted with two different models com-puting the X-ray flux from a single power-law distribu-tion of energetic electrons. In the looptop source (region0 in figure 2), we assume for simplicity that energetic elec-trons lose only a small portion of their energy through col-lisions and that the region can be considered as a thin tar-get. The free parameters of the thin target model are theelectronic spectral index δ LT and a normalisation factor (cid:10) ¯ nV ¯ F (cid:11) = (cid:16)(cid:82) ∞ E (cid:10) ¯ nV ¯ F ( E ) (cid:11) dE (cid:17) (electrons / s / cm ), where ¯ n is the mean density of the thin target, V is its vol-ume, and ¯ F ( E ) is the energetic electron mean spectrumin electrons / s / cm / keV (see equation B.6 in appendix). Inthe footpoints (regions 1 and 2 in figure 2), the densityis much higher and the energetic electrons lose instanta-neously all their energy in the target, considered as a thicktarget. The free parameters of the thick target model are theelectronic spectral index δ F P and the electron rate above E , ˙ N (electrons / s), entering the target (see equation B.15in appendix). In each case, a minimum correction for albedowas used (assuming an isotropic beam of electrons), andthe low energy cutoff E of the non-thermal model (thickor thin target models) was fixed to 25 keV, since when thisparameter was set free in the spectral analysis, it reached23 keV. Table 1.
Values of the free spectral parameters obtained forthe looptop source, and for the first and second footpoints. EMand T are the emission measure and the temperature (thermalcomponent). (cid:10) ¯ nV ¯ F (cid:11) is the normalisation factor derived for thelooptop source in the thin target approximation, ˙ N is the elec-tron rate above 25 keV derived for the footpoints in the thicktarget approximation, δ is the electron spectral index derived inboth thin and thick target approximations. Note that the non-thermal parameters refers to the electron distribution directlyand not to the photon spectrum. First SecondLooptop Footpoint FootpointEM ( × cm − ) . ± . . ± .
08 0 . ± . T (keV) . ± . . ± . . ± . T ( × K) ± . ± . ± . (cid:10) ¯ nV ¯ F (cid:11) > keV( × e − cm − s − ) . ± . N ( × e − s − ) . ± .
03 0 . ± . δ . ± . . ± . . ± . In the further calculation of the electron rate for the dif-ferent X-ray sources (see section 3.1), we need to estimatethe sizes of the different X-ray emitting regions: the coro-nal source and the footpoints. Moreover, we distinguish thethermal X-ray emitting region from the non-thermal X-ray emitting region in the coronal source. We use the 50%CLEAN contours from the images to estimate the length,width or area of the X-ray sources. The CLEAN imageswere produced with a beam factor of 1.7. The determina-tion and the influence of this parameter are discussed inappendix A.The size of the thermal coronal source was measured at10-12 keV, to ensure the X-ray emission to be entirely ther-mal (see the looptop spectrum, on the left panel in figure4). The size of the non-thermal X-ray source at the looptopis measured at 26-28 keV, since at this energy, the looptopsource is still visible in the image, and the X-ray spectra ispredominantly non-thermal (see the looptop spectrum, onthe left panel in figure 4). Finally, the area of the footpointsis taken at 60-75 keV. The measurements of the length andwidth of the coronal sources are represented by green andblue arrows respectively in figure 3 and schematically ex-plained in figure 5. The measured sizes and areas are sum-marized in table 2.The emission measure EM given by the spectral anal-ysis (see table 1) of the thermal part of the coronal source,and the estimation of the size of the thermal source (seetable 2), leads to the following estimation of the density: ¯ n = (cid:114) EMV th = (cid:114) EML th A th (1)where ¯ n is the density (in cm − ), V th is the volume of thethermal source (in cm ). L th , W th and A th are respectivelythe length, the width and the cross-section of the thermal Article number, page 4 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona
Fig. 3.
RHESSI images in 3 of the 20 energy bins used for imaging spectroscopy, integrated between 23:49:30 and 23:50:30 UT,with the 50 % of the maximum value enlightened in red. Source length and width are shown with green and blue arrows respectively.At 10-12 keV, the X-ray emission is thermal and we therefore show the length and width of the thermal source L th and W th . At26-28 keV, the emission is non-thermal and we therefore show the lenght and width of the non-thermal looptop source L LT and W LT . At 60-75 keV, the area of the footpoint sources is calculated with the 50 % contour in red. Fig. 4.
Count flux spectra (data and fit) with residuals, for the looptop source (left), the first footpoint (middle) and the secondfoopoints (right), as defined by the black boxes in figure 2. The spectra derived from the data is shown in black. The blue curverepresent the thermal component of the fit, the green curve represents the non-thermal component. The pink dashed-dotted linerepresents the component due to albedo correction. The red curve is the total fitted spectrum. source, with A th = π ( W th / . The assumed geometry ofthe loop is described in figure 5.The mean plasma density obtained is ¯ n = (1 . ± . × cm − . Note that this value is in the range of densitiescalculated by Simões & Kontar (2013) for events where anon-thermal looptop source is visible.
3. Determination of the spatial and spectraldistributions of X-ray emitting energeticelectrons
As explained below, the electron rate above E = 25 keVof electrons leaving the looptop source is found to be about ˙ N LT = (0 . ± . × electrons s − . Article number, page 5 of 15 &A proofs: manuscript no. paper_arxiv2
Table 2.
Measured sizes of the thermal and non-thermal (loop-top and footpoints) sources, using the 50% contours of theCLEAN images as shown in figure 3.
Width Length Area(Mm) (Mm) (Mm )Thermal source (10-12 keV) 5.3 7.0Looptoop source (26-28 keV) 5.8 9.61st Footpoint (60-75 keV) 17.62nd Footpoint (60-75 keV) 19.2 Electrons escapinglooptop to footpoints
Looptop source 𝑊 𝑙𝑡 Footpoint
Footpoint 𝐿 𝑙𝑡 𝐴 𝑙𝑡 𝐴 𝑙𝑡 𝑧 Acceleration region 𝐴 𝐹𝑃 Fig. 5.
Sketch of a symmetrical magnetic loop. The limits of thelooptop sources are the cross-sections of the loop with area A LT shown in grey. The length L LT and the width W LT of the loop-top X-ray source are used to determine the size of the looptopsource, which is approximated to a cylinder of diameter W LT .Blue arrows represent the electron rate for electrons leaving thelooptop source of cross-section A LT . Indeed, the electron rate ˙ N LT (in electrons s − ) in thelooptop source is given by: ˙ N LT = A LT (cid:90) ∞ E ¯ F ( E ) dE (2)Where E is the low energy cutoff (in keV), A LT (cm )is the cross-section of the looptop source (assuming a sym-metrical source) as shown in figure 5, and ¯ F ( E ) is the meanenergetic electron spectrum (in electrons s − cm − keV − ).We assume energetic electrons propagating in both direc-tions along the loop axis (see the blue arrows in figure 5).For a source with an homogeneous ambiant plasma, wecan express the looptop electron rate as follows: ˙ N LT = A LT (cid:90) ∞ E (cid:10) ¯ nV ¯ F (cid:11) ¯ nV dE = 1¯ nL LT (cid:90) ∞ E (cid:10) ¯ nV ¯ F (cid:11) dE (3) (cid:10) ¯ nV ¯ F (cid:11) = (cid:82) ∞ E (cid:10) ¯ nV ¯ F (cid:11) dE is given by the spectral analy-sis of the looptop source (see table 1) and L LT has beenmeasured on the 26-28 keV CLEAN image (see table 2).The electron rate obtained for the looptop source, ˙ N LT = (0 . ± . × electrons s − , is compared tothe electrons rates obtained by the spectral analysis ofthe two footpoints, which are (0 . ± . × and Fig. 6.
Spatially integrated and density weighted mean fluxspectra for the looptop (dashed line) and the footpoint sources(plain lines) deduced from X-ray observations (black his-tograms), and computed with the diffusive transport of Kontaret al. (2014) model with n = 9 . × cm − , d = 5 . Mmand λ = 1 . × cm (red). The dotted vertical line marks theenergy at which the coronal and the second footpoint spectracross. (0 . ± . × electrons s − . The sum of the ratesfrom the footpoints is therefore significantly lower than therate needed to explain the nonthermal emission in the coro-nal source: the ratio ˙ N LT ˙ N FP is about 2.2 for this event, forelectrons above E = 25 keV. Using equations B.6 and B.15 (in appendix B) and theresults of the spectral analysis displayed in table 1, thespatially integrated density weighted mean flux spectra (cid:104) nV F ( E ) (cid:105) of the looptop source and footpoints are plot-ted on figure 6, in black. On this figure, the energies forwhich the footpoint spectra are crossing the looptop spec-trum are 50 keV and 60 keV for the first and the secondfootpoints respectively.The spatial distribution of energetic electrons (cid:104) nV F (cid:105) at25 keV is also known in three locations in the loop (looptopand footpoints). The distance between the footpoints andthe looptop source is estimated by taking the distance be-tween the centers of the boxes defined in figure 2. Kuznetsov& Kontar (2015) showed that the maximum of the spatialdistribution of energetic electrons was shifted of 3.2 Mm inregards with the top of the magnetic loop. This three-pointdistribution is shown in figure 7.The number density of energetic electrons with energy E > E min , n E min b (in electrons cm − ), is defined as: n E min b ≡ (cid:90) ∞ E min F ( E ) v ( E ) dE (4)where v is the velocity of the electrons. In the following, wedistinguish the estimation of n E min b in the thin and in thethick target models. The details of the calculations are inappendix B. Article number, page 6 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona
Fig. 7.
Spatial distribution of energetic electrons at 25 keVdeduced from X-ray observations (black crosses) and computedwith the diffusive transport model of Kontar et al. (2014) with n = 9 . × cm − , d = 5 . Mm and λ = 1 . × cm (redlines). The looptop source is shifted of 3.2 Mm in regards tothe top of the loop, as it has been described in Kuznetsov &Kontar (2015). For the model, the dashed or dotted lines marka confidence interval around the computed value marked by theplain line. The detailed description is in section 4.1. Using equations B.7 and B.16 with E min = E = 25 keV, we can evaluate the electron density of energetic elec-trons with energy E > keV, n b in the thin and thicktarget models respectively. We found n b = (15 ± × electrons cm − in the corona and n b = (9 ± × and (4 ± × electrons cm − in each footpoint. Note that tocalculate the number density of energetic electrons from theobservations, we need an estimation of the area of the cross-section of the loop A LT . From the size estimation displayedin table 2, we found A LT = π ( W LT / = 26 Mm . Thespatial distribution of the energetic electron density above25 keV electrons in the flaring loop is plotted on figure 8.
4. Interpretation of the observations in the contextof diffusive transport of energetic electrons
The spectral indexes of the electron distribution from theX-ray emission, using the thin and thick target models aresummarized in table 1. While electron spectral indexes inboth footpoints are very close and will be considered assimilar, the electron spectral index in the loop top sourceis softer by ≈ . Furthermore the ratio ˙ N LT ˙ N FP of the electronrate in the looptop source and in the footpoints is found tobe around 2.2. These values are similar to values found forother events (see e.g. Simões & Kontar 2013). These resultssuggest that a significant number of high energy electronsare confined in the coronal region. Such a confinement ofhigh energy electrons can result from magnetic mirroring orturbulent pitch angle scattering as demonstrated in Kontaret al. (2014); Bian et al. (2011). In the following we will Fig. 8.
Spatial distribution of the density of energetic electronswith energy
E > keV, deduced from observations ( n b , blackcrosses). The spatial distributions of n b calculated with thediffusive transport model of Kontar et al. (2014), with A LT = 26 Mm , d = 5 . Mm, n = 9 . × cm − and λ = 1 . × cm(red lines) is also plotted. The looptop source is shifted of 3.2Mm in regards to the top of the loop, as it has been describedin Kuznetsov & Kontar (2015). The detailed description is insection 4.1. Fig. 9.
Spatial distribution of the density of energetic electronswith energy
E > keV, at looptop, estimated from X-ray ob-servations ( n b , black cross) and radio observations (black plainline, see Kuznetsov & Kontar 2015). The spatial distributionsof n b calculated with the diffusive transport model of Kontaret al. (2014), with A LT = 26 Mm , d = 5 . Mm, n = 9 . × cm − and λ = 1 . × cm (red lines) and λ = 10 cm (greenlines), are also plotted. The looptop source is shifted of 3.2 Mmin regards to the top of the loop, as it has been described inKuznetsov & Kontar (2015). The detailed description is in sec-tion 4.1. Article number, page 7 of 15 &A proofs: manuscript no. paper_arxiv2 investigate whether the confinement observed in this flarecan be explained in the context of the diffusive transportmodel of Kontar et al. (2014). In this model, strong turbu-lent pitch angle scattering, due to small scale fluctuationsof the magnetic field, is responsible for a diffusive paral-lel transport of energetic particules and finally results as aconfinement mechanism. In section 4.1.2 we will compare the observed spatial andspectral distributions of (cid:104) nV F (cid:105) with the ones calculatedin the diffusive transport model described in Kontar et al.(2014). The distribution F D ( E, z ) (electrons / cm / s / keV)of energetic electrons of energy E at a position z along themagnetic loop is indeed described by the following equation(Kontar et al. 2014): F D ( E, z ) =
EKn (cid:90) ∞ E dE (cid:48) F ( E (cid:48) ) (cid:112) aπ ( E (cid:48) − E ) + 2 d × exp (cid:18) − z a ( E (cid:48) − E ) + 2 d (cid:19) (5)Where F ( E ) is the initial distribution of energetic electronsin the source (acceleration) region supposed to be spatiallyextended, d is the size of the acceleration region (Gaussianform), n is the plasma ambiant density, K = 2 πe Λ is thecollisional parameter, and a = λ/ (6 Kn ) where λ is thepitch-angle scattering mean free path of the electrons. Inthe work of Kontar et al. (2014), this mean free path λ isconsidered to be independent of energy.In equation 5 the plasma density n , the size of the accel-eration region d and the electron scattering mean free path λ are parameters to the model. The effects of these param-eters on both spatial and spectral distributions of energeticelectrons is shown in figure 10 and summarized in table3. The spectral indexes obtained in the corona and in thefootpoints for each set of parameters are described in table4. As shown on figure 10, the different parameters do notinfluence the two distributions in the same way. For in-stance, when d or λ increases, the spatial distribution be-comes broader (note that the shape of the distribution re-mains different), but the effects on the spectra are not thesame: increasing d will have almost no impact on the coro-nal spectrum whereas increasing λ leads to a softening ofthe coronal spectrum.It can be seen that the increase of density leads to anenhanced trapping of energetic electrons, even if the scat-tering mean free path remains unchanged.In addition, we note that there is a limit value of thesize of the acceleration region below which the effect of thisparameter is negligible: this is the case when d (cid:28) aπ ( E (cid:48) − E ) (see equation 5). In our conditions, the influence of d onthe energetic electron distributions is negligible for d (cid:46) cm. In the following, the electron distribution in the loop topand in the footpoints are computed through integration of equation 5 on z (respectively from -7 Mm to 5 Mm for looptop sources and from -9 Mm to - ∞ and from +13 Mm to+ ∞ for footpoint sources). In order to fit the distribu-tions derived from these equations to the observations, weneed to determine the injected distribution of electrons F ,and make the parameters n , d and λ vary. As discussed inKontar et al. (2014) and also shown in section 4.1.1, theincrease of diffusion due to pitch-angle scattering resultsin enhanced coronal emission and weaker footpoint emis-sion than in the standard non diffusive case, due to theincrease of the time spent by the electrons in the corona.In the diffusive case, the spectrum of the electrons in thecorona becomes progressively flatter, with decreasing scat-tering mean free path. However, as shown in section 4.1.1,the mean electron spectrum in the footpoint is less affectedby the increase of diffusion that the electron spectrum inthe corona. This is why in the following of the paper, weassume that the injection spectrum of the energetic elec-trons is given by the spectral index of the population of theelectrons entering the footpoints. The injected electron ratecannot be directly inferred from the results of the spectralanalysis of the observations: indeed, the value found in thefootpoints is too small because there are trapped electronswhile the electron rate computed in the corona (see sec-tion 3.1) is too large since there is some trapping effects.Therefore, the injected electron rate is considered as a freeparameter to fit the model to the observations, with how-ever the constraint that its value ˙ N i must be between thetwo boundaries ˙ N F P and ˙ N LT . This parameter has no ef-fect on the spatial distribution of energetic electrons butimpacts the normalisation of the spatial distribution of thedensity of energetic electrons.Once the initial distribution of electrons is determined,the spatial and spectral distributions of energetic electronsin the coronal source and in the footpoints depends on thedensity of the ambiant medium n , the size of the accelera-tion region d and the electron scattering mean free path λ (see equation 5). We search for the best set of parameterswhich can reproduce at the same time the spatial distri-bution of energetic electrons at 25 keV and the spectraldistribution of energetic electrons in the footpoints (figures6 and 7). As described in section 4.1.1, each parameter af-fects both spatial and spectral distributions. The major ef-fect is found for the width of the spatial distribution andthe slope of the electron spectrum in the corona. As seen intable 4, although the slope of the energetic electrons in thefootpoints never very far to the slope of the electron spec-trum in the footpoints that will be produced in the stan-dard case. The space of four parameters ( ˙ N , n , λ , d ) wasexplored by producing predicted spatial and spectral dis-tributions of energetic electrons that could be compared tothe ones deduced from X-ray observations. A χ was com-puted for each set of parameter, as described in appendix C.The minimal χ was found for the following set of param-eters: ˙ N = (cid:0) +0 . − . (cid:1) × s − , n = (cid:0) . +6 . − . (cid:1) × cm − , λ = (cid:0) . +0 . − . (cid:1) × cm, d = (cid:0) . +0 . − . (cid:1) × cm. The uncer-tainties on the parameters represent the values for whichthe χ exceed the minimal χ by at least 5% (see appendixC). The modeled distributions are plotted in figures 6 and7. We can note that the slope of the looptop spectra is notwell recovered by the model. On the other hand, the mod-eled spatial distribution does not seem to be as peaked thatexpected from the data. This is a consequence of a trade- Article number, page 8 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona 𝑛 = 5 × 10 cm −3 𝑛 = 4 × 10 cm −3 𝑛 = 10 cm −3 𝑛 = 4 × 10 cm −3 𝑛 = 4 × 10 cm −3 𝑑 = 6 × 10 cmλ = 3 × 10 cm𝑑 = 2 × 10 cmλ = 3 × 10 cm 𝑑 = 2 × 10 cm𝑑 = 6 × 10 cm𝑑 = 12 × 10 cmλ = 10 cmλ = 3 × 10 cmλ = 10 cm Fig. 10.
Influence of the free parameters in equation 5 ( n the plasma density, d the size of the acceleration region, and λ thescattering mean free path) on the spatial (left panels) and spectral (right panels) distributions of energetic electrons. In the toppanels, d and λ are constant and n = 5 × cm − (blue), n = 4 × cm − (green) and n = 10 cm − (orange). In the middlepanels, n and λ are constant and d = 2 × cm (blue), d = 6 × cm (green) and d = 12 × cm (orange). In the bottompanels, n and d are constant and λ = 10 cm (blue), λ = 3 × cm (green) and λ = 10 cm (orange). The dotted vertical linesmark the energies at which the coronal spectrum crosses the footpoint spectrum. off that happen during the fit to both spatial and spectraldistributions. Nevertheless, the models fit the data with adensity close to the density deduced from the observations,and a electron injection rate close to the electron rate de-duced in the looptop source. The spatial distribution of the energetic electron densityis also computed with the set of parameters found aboveand compared with the values derived from the fit of theobservations (see figure 8). The cross-section of the loop A LT has been fixed to 26 Mm (as this area has been used Article number, page 9 of 15 &A proofs: manuscript no. paper_arxiv2
Table 3.
Summary of the influence of the density n , the size of the acceleration region d and the scattering mean free path λ onthe spatial and spectral distributions of energetic electrons in the frame of the diffusive transport model. Parameter Effect on spatial distribution Effect on spectraWhen n increases The spatial distribution gets narrower and The spectra gets harder and thethe peak of the distribution decreases energy at which coronal and footpointspectra cross increasesWhen d increases The spatial distribution gets broader and The footpoint spectrum gets softer andthe peak of the distribution decreases the energy at which coronal and footpointspectra cross decreasesWhen λ increases The spatial distribution gets broader and The spectra get softer andthe peak of the distribution decreases the energy at which coronal and foortpointspectra cross decreases Table 4.
Influence of the density n , the size of the accelerationregion d and the scattering mean free path λ on the spectralindex of non-thermal electron distributions in the frame of thediffusive transport model. The spectral indexes δ C of the coronalspectra is measured with a linear regression between 25 and 65keV; the spectral indexes δ FP are measured in the same waybetween 50 keV and 100 keV. When n varies, d = 2 × cmand λ = 3 × cm; when d varies, n = 4 × cm − and λ = 3 × cm; and when λ varies, n = 4 × cm − and d = 6 × cm. Density n × × δ C . ± .
01 2 . ± .
01 2 . ± . δ F P , E > keV . ± .
01 1 . ± .
01 1 . ± . Size d (cm) × × × δ C . ± .
01 2 . ± .
01 2 . ± . δ F P , E > keV . ± .
01 1 . ± .
01 2 . ± . Lambda λ (cm) × δ C . ± .
01 2 . ± .
01 2 . ± . δ F P , E > keV . ± .
01 1 . ± .
01 1 . ± . to calculate the electron density from the observations). Inthe different plots, the estimation of the error on the valuesof δ thick and ˙ N LT is taken into account and is responsiblefor the error intervals around the distributions derived fromthe model and visible in figures 7, 8 and 9. The 2004 May 21 flare gyrosynchrotron emission has beenstudied by Kuznetsov & Kontar (2015). The gyrosyn-chrotron emission is produced mostly by electrons of en-ergies around 400 keV, and therefore the radio observationsof the flare allows to study energetic electrons in a differ-ent energy domain than the X-ray analysis, X-ray emittingelectrons being mostly in the 25-100 keV energy range.
The spatial distributions of electrons at 25 keV (figure 7)and of the density of energetic electrons in the loop n b (fig-ure 8), show that most of the energetic electrons with energy E > keV are located in the looptop source. Moreover, anasymmetry is seen between the two footpoints. Both resultsare in agreement with the results obtained by Kuznetsov &Kontar (2015) who calculated the spatial distribution of thedensity of energetic electrons with energy E > keV ( n b )from observations of the gyrosynchrotron emission (see fig-ure 7 in Kuznetsov & Kontar 2015, and figure 9 in thepresent paper). To compare with the results of Kuznetsov& Kontar (2015), we estimate the number density of ener-getic electrons above 60 keV from the X-ray observations,using the relation n b ≈ n b (60 / − δ +1 / with δ = 4 . : n b ≈ . × electrons cm − in the corona. This estima-tion of n b from X-ray producing electrons at the looptopis plotted as a cross in figure 9. It is about 2 times smallerthan the value found by Kuznetsov & Kontar (2015) fromradio observations (see figure 9). This difference could beexplained if there is a break in the power-law spectrumof the energetic electrons with a smaller spectral index athigher energy or if the thin target approximation in thecoronal source must be relaxed as suggested by the flatterspectrum of electrons in the coronal source derived in themodel.Independent of this quantitative comparison of relativenumbers of electrons producing X-rays and radio emissions,we compare the relative spatial distributions of both X-rayemitting-electrons and radio emitting-electrons by compar-ing the ratio between the maximum electron density in theloop and the number density in the footpoints. The ratiobetween the number densities of energetic electrons at thelooptop and in the footpoints n b,LT /n b,F P , from the X-raymeasurements, are 1.6 and 3.8 for the first and the secondfootpoint respectively. Taking the values at the same dis-tance in figure 7 in Kuznetsov & Kontar (2015), the ratios n b,LT /n b,F P , averaged over the three times, are 7.7 and 9.We note that the ratio is much higher for the distributiondeduced at energies above 60 keV from the gyrosynchrotronemission than for the one deduced at 25 keV from HXR ob-servations. This implies that the X-ray emitting energeticelectron spatial distribution is less strongly peaked than Article number, page 10 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona the spatial distribution deduced from microwave emissions,as seen in figure 9 and that the high energy electrons re-sponsible for gyrosynchrotron emissions are more confinedin the corona than the lower energy ones. Based on thediscussion of section 4.1.1 and assuming that X-ray and ra-dio emissions are produced by the same electrons injectedand confined in the same loop (the values of d and n re-mains unchanged), the only way to produce a more spa-tially peaked distribution is to vary the scattering mean freepath λ . Therefore, a second fit was performed on the dis-tribution of the density of energetic electrons deduced fromradio observations, the only free parameter being the scat-tering mean free path λ . The ambiant density and size of theacceleration region are kept as they were found by fittingthe distributions deduced from X-ray observations. Figure9 shows the modeled distribution with λ = (cid:0) +4 − . (cid:1) × cm, which produced the smallest χ . Details of the fit aredescribed in appendix C.
5. Discussion
In this paper, we focused on the interpretation of our ob-servations in the frame of the diffusive transport model de-scribed in Kontar et al. (2014). We showed that the diffu-sive transport model of Kontar et al. (2014) can explain theobserved trapping of energetic electrons in the corona. Inparticular, the model explains the electron spectrum in thefootpoint, the hardening of the footpoint spectrum com-pared to the coronal spectrum, and the spatial distribu-tion of energetic electrons along the loop. However, diffusivetransport of energetic electrons in not the only mechanismthat can explain electron trapping in the corona and weinclude a discussion about trapping with magnetic mirrorsat the end of this section.
The spatial distributions of X-ray emitting and radio emit-ting electrons can be reproduced in the context of the dif-fusive transport model of Kontar et al. (2014) only by as-suming that the scattering mean free path of energetic elec-trons decreases with increasing electron energy, which ex-plains why the trapping of energetic electrons in the coronais stronger at higher energies. This conclusion is of coursecontradictory to the assumption of the model in which thescattering mean free path is constant with energy and showsthat to completely study the behaviour of X-ray and radioemissions, a new model should be developed in which thescattering mean free path depends on energy. This is how-ever out of the scope of the present paper. We shall howeverdiscuss the result on the energy dependance of the scatter-ing mean free path with respect with what is observed inthe interplanetary medium. Several studies (see e.g. Dröge2000b; Agueda et al. 2014) have found for interplanetaryelectrons in range 0.1 - 1 MV a power law dependence ofthe electron mean free path on rigidity, with a negativepower law index. We therefore also assume a power lawdependence of the electron scattering mean free path withenergy in the present study. We have only two data points,the first one derived from X-ray radiation above 25 keV,and the second one derived from radio observation, whichis produced mostly by electrons at ± keV (private Fig. 11.
Energy dependence of the scattering mean free pathcalculated with the diffusive transport model of Kontar et al.(2014), with A LT = 26 Mm , d = 5 . Mm, n = 9 . × cm − .The uncertainties on the values of the mean free path derivedfrom the fit (see appendix C) and on the energy of the radio-emitting electrons are taken into account. The two most extremeslopes for the power law dependence of the mean free path areplotted. communication from A. Kuznetsov). The mean free pathscalculated in this paper are plotted as a function of elec-tron energy in figure 11. Given the uncertainty about theenergy of radio-emitting electrons, and the uncertainty onthe mean free path, we can calculate the slope of the powerlaw in two limit cases: -1.9 and -0.3. The correspondingslopes for the mean free path dependence in rigidity are be-tween -3.4 and -0.5. Although it is clear that the scatteringmean free path is decreasing with increasing electron energyand rigidity, a large range of slopes are consistent with ourdata. It should be pointed out that, 5 out of 7 events studiedby Agueda et al. (2014) have shown slopes for the rigiditydependence of the scattering mean free path of electrons inthe interplanetary medium that could be consistent withour observations. We note that the diffusive transport model predictions didnot perfectly reproduce the observations in the details. Inparticular, the predicted looptop source spectrum is flatterthan the spectrum deduced from the X-ray analysis of theflare. This difference could be due to the fact that someeffects are not taken into account in the diffusive trans-port equation that has been used in this paper, such asthe effect of a converging magnetic field. This discrepancycan also be explained by the fact that the thin target ap-proximation might not be valid for the coronal source inthis context. Indeed, the density calculated in the coronalsource ( . ± cm − ) is quite high and the source couldbe considered as a thick target for low energy electrons.Moreover, the diffusion of energetic electrons in the corona Article number, page 11 of 15 &A proofs: manuscript no. paper_arxiv2 leads to enhanced time spent by the electrons in the tar-get, where they loose more energy than assumed in the thintarget approximation. However, assuming that the coronalX-ray source is a thick target in the spectral analysis donot improve the agreement between the data and the pre-dictions of the diffusive transport model. The X-ray coronalsource is most probably neither a thick nor a thin target,but is in between, with a density where none of these twoapproximations are completely valid. Finally, the normal-isation of the modeled distribution of the density of en-ergetic electrons above 60 keV is not well recovered. Thismight indicate that fewer energetic electrons are acceleratedat energies above 60 keV than expected (e.g. the injectionelectron rate decreases with energy), but it most probablydue to the fact that the model produces a too-flat coronalspectrum and therefore overestimate the number of high-energy electrons in the loop.We show that the diffusive transport can explain the ob-served spatial and spectral distributions in the X-ray andradio ranges, if the mean free path is energy dependent. Themean free path has been assumed to be constant in Kontaret al. (2014): a further development of this model shouldinclude the energy dependance of the mean free path, aswell as the relativistic effects, to allow a more precise com-parison of the model prediction with combined X-ray andradio observations.
Trapping of energetic electrons in the coronal part of theloop can be explained by the effect of a converging mag-netic field. In this event, the area of the section of the loopcalculated (26 Mm ) at the ends of the coronal source islarger than the area of the footpoints deduced from the X-ray observations (see table 2); this observation is in favorof a magnetic convergence of the loop. If we considere amagnetic loss cone for the electron pitch angle distribution,the loss-cone angle α depends on the magnetic ratio σ , asdescribed in the introduction. The trapped fraction of theenergetic electron distribution is deduced from X-ray ob-servations and is − ˙ N FP ˙ N LT (see Simões & Kontar 2013, formore details). Simões & Kontar (2013) showed that in thecase of an isotropic pitch-angle distribution, the trappedfraction of the energetic electron distribution is equal to µ , the cosine of the losscone angle α . We can thereforeretrieve the value of σ needed to explain the observed ˙ N LT ˙ N FP ratio in the case of an isotropic pitch-angle distribution :we found σ ≈ . , which is close to the values found bySimões & Kontar (2013); Aschwanden et al. (1999b); Tom-czak & Ciborski (2007) and explains the observed ratio ofcross-sections of the loop at the looptop and in the foot-points. This expected value σ of the magnetic ratio can becompared to the magnetic ratio measured in the loop σ r .To estimate the magnetic ratio σ r of the coronal loop, wecan use the magnetic extrapolation from Kuznetsov & Kon-tar (2015). Note that in doing so, we assume that the HXRand gyrosynchrotron emissions are produced in a same mag-netic loop, as mentioned in the introduction. As it can beseen in Figure 6 in Kuznetsov & Kontar (2015), at the loop-top of the reconstructed magnetic loop, the magnetic fieldstrength is B LT ≈ G.Using the estimation of the source length seen in X-ray,we determined the value of the magnetic field at the sup- posed position of the mirrors (at each end of the observedcoronal X-ray source) and found values of ± and ± G, leading to the following values of the mageticratio: σ r ≈ . and . . This is consistent with the ratio ofloop cross-sections at looptop and footpoints deduced fromthe X-ray images, ≈ . and . for the two footpoints.The magnetic ratio measured is therefore just enough toexplain the ratio of electron rate ˙ N LT ˙ N FP deduced from theX-ray observations. However, with this model, it is a pri-ori not possible to explain why the trapping of energeticelectrons is stronger at higher energies, and why a spectralhardening with a difference of 1 between electronic spec-tral slopes is observed between the looptop and footpointsources. We can also note that Kuznetsov & Kontar (2015)showed a shift between the centroid of the gyrosynchrotonsource and the top of the magnetic loop where the mag-netic field is minimal. In the case of electron trapping dueto magnetic mirroring, we expect to have a maximum emis-sion where the magnetic field is minimum.
6. Summary and conclusion
The summary of our observations is the following:1. The difference between the footpoint and looptop spec-tral indexes is about 1, which suggests that a mechanismis hardening the electron spectrum during the transport.This can be explained by trapping of energetic electronsin the corona.2. The ratio of the looptop and footpoint electron rateabove 25 keV, ˙ N LT ˙ N FP , has a value of 2.2, suggesting thatpart of the energetic electrons are trapped in the coronalpart of the loop.3. The spatial distribution of HXR-emitting electrons ispeaked near the looptop, but less peaked than the spa-tial distribution of microwave-emitting electrons sincethe ratio of energetic electron density between the loop-top and the footpoints is more than two time higherfor radio-emitting electrons above 60 keV than X-rayemitting electrons above 25 keV.4. The spectral and spatial distribution of energetic elec-trons, deduced from both X-ray and radio observations,can be explained by a diffusive transport model of Kon-tar et al. (2014), with a mean free path decreasing withincreasing electron energy.5. The mean free path for electron energies between 25 and100 keV is of the order of . × cm, which is smallerthan the length of the loop. These values are comparableto values found by Kontar et al. (2014). The mean freepath is also smaller than the size of the accelerationregion calculated with the model ( . × cm), whichsuggests that electrons can potentially be acceleratedfor a longer time.6. The scattering mean free path for electron energiesaround 400 keV ( cm) is significantly smaller thanthe mean free path estimated at lower energies. Similardependence of the scattering mean free path over elec-tron energies has been found in the case of interplan-etary electron transport, in the same range of electronenergy. We note that the potential slopes of the energydependence of the scattering mean free path in the so-lar corona are in agreement with some of the slopes ob-served for interplanetary electrons. Article number, page 12 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona
Trapping due to magnetic mirroring is not known to beenergy dependent and this mechanism cannot fully explainour observations.The diffusive transport model enable the reproductionof our different observations, such as the spectral slope inthe footpoints, some spectral hardening between the loop-top and the footpoints, and the spatial distributions of elec-tron density deduced from both X-ray and radio observa-tions.Imaging spectroscopy in HXR is a powerful tool to studyelectron transport during solar flares. This study shouldencourage the development of predictions on the spatialdistribution of electrons and on the evolution of the spec-tral index of non-thermal electron energy distribution bythe various transport models. The simultaneous observa-tion of non-thermal X-ray sources in both the coronal partof the loop and its footpoints is rare due to the facts thatfootpoints sources are usually brighter than coronal sourcesand that indirect imaging intruments such as RHESSI havea limited dynamic range. Instruments using focusing op-tics for hard X-ray imaging (such as a FOXSI spacecraft)would therefore provide very useful observations of faintnon-thermal coronal sources in the presence of bright foot-points and therefore add interesting cases in which to studyelectron transport during flares.Finally, this study shows that the combination of X-rayand radio diagnotics for energetic electrons in closed loopsduring flare enable to study the energy dependence of trans-port properties in the solar corona, such as the scatteringmean free path. Such observations could constrain in someextend some properties of the turbulence spectrum in thesolar corona.
Acknowledgements.
We thank the RHESSI team for producing freeaccess to data, Alexey Kuznetsov for his help with the radio data,and our referee for his useful comments. Sophie Musset acknowl-edges the CNES and the LABEX ESEP (N ◦ ∗ initiative (convention N ◦ ANR-10-IDEX-0001-02), as well as theProgramme National Soleil-Terre (PNST). EPK was supported by aSTFC consolidated grant ST/L000741/1.
Appendix A: CLEAN beam factor
The CLEAN algorithm is an iterative algorithm based onthe assumption that the X-ray image is well represented bya superposition of point sources convolved with the pointspread function (PSF) of the instrument (see e.g. Hur-ford et al. 2002). The CLEAN algorithm developed for theRHESSI image analysis has one parameter called ’beamfactor’, which represents the effective resolution of the sub-collimators used to reconstruct the image.In this paper, the value for the beam factor was chosento have CLEAN images as close as possible to images recon-structed with the visibility forward fit VISFF (see Schmahlet al. 2007 for the definition of visibilities and Xu et al.2008 for examples of application) and the PIXON (Metcalfet al. 1996; Hurford et al. 2002) algorithms. This is a stan-dard procedure to ensure that CLEAN agrees with otheralgorithms for the image reconstruction (see e.g. Dennis &Pernak 2009; Kontar et al. 2010).The determination of the best value of the beam factorfor the image reconstruction has an important impact onthe X-ray source size determination on CLEAN images. For example, when using the default value of the beam factor 1,the measured sizes are roughly 1.5 times greater than thesizes estimated on CLEAN images with a beam factor of1.7 or on a PIXON image.
Appendix B: X-ray production in thin- andthick-targets
The bremsstrahlung photon flux at energy (cid:15) , I ( (cid:15) ) , producedby an energetic electron flux density distribution F ( E, r ) (electrons/cm /s/keV) in an emitting source (a target) ofplasma density n and volume V is expressed as: I ( (cid:15) ) = 14 πR (cid:90) V (cid:90) ∞ (cid:15) n ( r ) F ( E, r ) Q ( (cid:15), E ) dEdV (B.1)Where Q ( (cid:15), E ) is the differential bremsstrahlung cross-section, and the integration is done over the target volumeand all contributing electron energies, which are all electronenergies above the photon energy (cid:15) .We can see that the X-ray spectrum I ( (cid:15) ) is linked toboth the energetic electron distribution and the ambiantplasma properties (density and volume of the target).For spectral observations, we deal with a spatially-integrated form of equation B.1: I ( (cid:15) ) = 14 πR (cid:90) ∞ (cid:15) (cid:10) ¯ nV ¯ F ( E ) (cid:11) Q ( (cid:15), E ) dE (B.2)where ¯ n = (1 /V ) (cid:82) V n ( r ) dV and ¯ F ( E ) (electrons/cm /s/keV) is the mean electron flux distri-bution, i.e. the plasma-density-weighted, target-averagedelectron flux density distribution (Brown et al. 2003;Kontar et al. 2011a; Holman et al. 2011), defined as: ¯ F ( E ) = 1¯ nV (cid:90) V n ( r ) F ( E, r ) dV (B.3)Since the quantity ¯ nV is dimensionless, the units of (cid:10) ¯ nV ¯ F ( E ) (cid:11) are the same as those of the electron flux(electrons/cm /s/keV). (cid:10) ¯ nV ¯ F ( E ) (cid:11) is a quantity which canbe retrieved from the X-ray spectrum I ( (cid:15) ) without anymodel assumption and therefore, is the quantity derivedduring spectroscopic diagnosics of the X-ray emission. Toretrieve the product (cid:10) ¯ nV ¯ F (cid:11) , in principle, we only need toknow the bremsstrahlung cross-section Q ( (cid:15), E ) .In our study, we were particularly interested by thenumber density of energetic electrons with energy E >E min , n E min b (in electrons cm − ), which is defined as: n E min b ≡ (cid:90) ∞ E min F ( E ) v dE (B.4)where v is the velocity of the electrons. It can also be ex-pressed as: n E min b ≡ (cid:90) ∞ E min (cid:10) ¯ nV ¯ F ( E ) (cid:11) ¯ nV v dE (B.5)We distinguish two approximations, the thin-target andthe thick-target models. In the thin target model, energeticelectrons lose only a small fraction of their energy while theypass through the target, whereas in the thick target model,energetic electrons lose all their supra-thermal energy in thetarget.In the following, we describe how the product (cid:10) ¯ nV ¯ F (cid:11) isexpressed in the thin- and thick-target models in OSPEXand how we estimate the energetic electron number density n b (cm − ). Article number, page 13 of 15 &A proofs: manuscript no. paper_arxiv2
Appendix B.1: Thin target model
We assume a power-law distribution for the electron meanspectrum: ¯ F ( E ) ∝ E − δ thin . In OSPEX, the proportionalityconstant is defined such as we can write the spatially inte-grated density weighted mean flux spectrum (cid:10) ¯ nV ¯ F ( E ) (cid:11) (inelectrons s − cm − keV − ) as: (cid:10) ¯ nV ¯ F ( E ) (cid:11) = (cid:10) ¯ nV ¯ F (cid:11) δ thin − E (cid:18) EE (cid:19) − δ thin , E > E (B.6)where δ thin and (cid:10) ¯ nV ¯ F (cid:11) = (cid:16)(cid:82) ∞ E (cid:10) ¯ nV ¯ F ( E ) (cid:11) dE (cid:17) are thespectral index and the normalisation factor given by thespectral analysis (see table 1).Equation B.5 and can be integrated over E, using equa-tion B.6 to obtain: n E min b = (cid:10) ¯ nV ¯ F (cid:11) ¯ nV δ thin − δ thin − / E − / min (cid:112) m/ (cid:18) E E min (cid:19) δ thin − (B.7)where m is the electron mass (in keV / c ). Appendix B.2: Thick target model
In the thick target model, energetic electrons lose all theirsupra-thermal energy through efficient collisions. Therefore,the energetic electron spectrum ¯ F is different from the in-jected electrons spectrum F . In fact, we need to integratethe injection spectrum over all energies in the X-ray emit-ting source.Therefore, the number of photons of energy between (cid:15) and (cid:15) + δ(cid:15) produced by an electron of initial energy E is: ν ( (cid:15), E ) = (cid:90) t F t =0 n ( r ) Q ( (cid:15), E ( t )) v ( t ) dt (B.8)where t F is the time at which all energetic electrons havebeen thermalized. Since energetic electrons are losing en-ergy at a rate dE/dt , the time integration can be replacedby an integration over energy: ν ( (cid:15), E ) = (cid:90) E (cid:15) n ( r ) Q ( (cid:15), E ) v ( E ) | dE/dt | dE (B.9)Energetic electrons lose their energy by Coulomb colli-sions with the electrons of the ambient plasma, and in thatcase the energy loss rate is expressed as: dE/dt = − ( K/E ) n ( r ) v ( E ) (B.10)where K = 2 πe Λ , with Λ is the Coulomb logarithm, e isthe electron charge, n the density of the plasma, and v isthe speed of the energetic electron.If we consider the injected electron spectrum F ( E ) ,the X-ray spectrum can be express as: I ( (cid:15) ) = A πR (cid:90) ∞ E = (cid:15) F ( E ) ν ( (cid:15), E ) dE (B.11)where A is the area of the thick target source. Using equation B.10 in equation B.9, we can rewriteequation B.11 in the following way: I ( (cid:15) ) = A πR K (cid:90) ∞ E = (cid:15) F ( E ) (cid:90) ∞ E = (cid:15) EQ ( (cid:15), E ) dEdE (B.12)and by changing the integration order, and comparing withequation B.2: (cid:10) ¯ nV ¯ F ( E ) (cid:11) = A EK (cid:90) ∞ E = E F ( E ) dE (B.13)Once again, we assume the injection spectrum to have apower-law dependence in energy, F ∝ E − δ thick . In OSPEX,the injection spectrum F ( E ) (electrons / sec / cm / keV) hasthe following form: F ( E ) = ˙ NA δ thick − E (cid:18) EE (cid:19) − δ thick , E > E (B.14)where ˙ N is the injection electron rate (in electrons s − ),and δ F P is the spectral index.After integration of equation B.13, the spatially inte-grated density weigthed mean flux spectrum is: (cid:10) ¯ nV ¯ F ( E ) (cid:11) = ˙ NK E (cid:18) EE (cid:19) − δ thick +2 (B.15)Equation B.5 is also valid for the thick target model.Using equation B.15 and after integration, the density ofenergetic electrons, in electrons / cm , in the thick target,is: n E min b = ˙ NK (cid:112) m/ nV E / min δ thick − / (cid:18) E E min (cid:19) δ thick − (B.16) Appendix C: Model fitting
The first fit of the model to the data was performed usingthe X-ray observations. The electron mean spectra for thecoronal source and one footpoint, as well as the spatial dis-tribution of electrons at 25 keV, are modeled and comparedto the same distributions deduced from the X-ray observa-tions. These distributions are visible in figure 6 and 7. The χ is calculated by comparing the looptop spectra between22 keV and 39 keV, where the observed spectra are mostlynon-thermal; by comparing the footpoint spectra between24 and 100 keV ; and by comparing the spatial distributiona the three data points deduced from the observations. Theerrors on the observations are derived from the errors foundon the free parameters in the spectral analysis (see table 1).The evolution of the χ in regards to the free parameters( n , λ , d ) is displayed in figure C.1. To provide uncertaintieson the values of those parameters, we looked at the valuesfor which the χ was 5% larger than its minimum. Theresulting density is between × and . × cm − ,the size of the acceleration region is between and . Mmand the scattering mean free path is between × and . × cm.We note that the final value of the χ is quite big, whichis due in particular to the fact that the spectral slope of thecoronal spectrum is not well recovered.The fit of the model to the spatial distribution of thedensity of energetic electrons above 60 keV deduced from Article number, page 14 of 15. Musset et al.: Diffusive transport of energetic electrons in the solar corona
Fig. C.1.
Evolution of the values of the χ with the free parameters in the model. Left: evolution of χ with ambiant density n ,with λ = 1 . × cm and d = 5 . Mm. Middle: evolution of χ with the size of the acceleration region d , with λ = 1 . × cmand n = 9 . × cm − . Right: evolution of χ with the scattering mean free path, with d = 5 . Mm and n = 9 . × cm − .The horizontal line marks the limit of 5% of the minimal χ value that as been used to determine uncertainties on the best valuesfor the model free parameters. radio observations was performed by comparing the mod-eled distributions on artificially created data points between-17 and +17 Mm, spaced of 0.5 Mm each. The error on thedistribution deduced from observations was set to 10% ofthe value, since this is the maximum error on that distribu-tion according to Kuznetsov & Kontar (2015). The rangeof values of the scattering mean free path that lead to thebest fit within 5% of the minimum χ is × to × cm. References