Draftversion January10,2017 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 NON-GAUSSIAN VELOCITY DISTRIBUTIONS IN SOLAR FLARES FROM EXTREME ULTRAVIOLET LINES: A POSSIBLE DIAGNOSTIC OF ION ACCELERATION Natasha L. S. Jeffrey, Lyndsay Fletcher & Nicolas Labrosse School ofPhysics&Astronomy,UniversityofGlasgow,G128QQ,Glasgow,UK Draft version January 10, 2017 ABSTRACT 7 In a solar flare, a large fraction of the magnetic energy released is converted rapidly to the kinetic 1 energy of non-thermal particles and bulk plasma motion. This will likely result in non-equilibrium 0 particle distributions and turbulent plasma conditions. We investigate this by analysing the profiles 2 of high-temperature extreme ultraviolet emission lines from a major flare (SOL2014-03-29T17:44) observedbythe EUV Imaging Spectrometer(EIS)onHinode. We findthatinmanylocationsthe line n profiles are non-Gaussian, consistent with a kappa-distribution of emitting ions with properties that a varyinspaceandtime. Attheflarefootpoints,closetositesofhardX-rayemissionfromnon-thermal J electrons,the κ-index for the Fe XVI 262.976˚A line at 3 MK takesvalues of3-5. In the corona,close 9 to a low-energy HXR source, the Fe XXIII 263.760 ˚A line at 15 MK shows κ values of typically 4-7. ] The observed trends in the κ parameter show that we are most likely detecting the properties of the R ion population rather than any instrumental effects. We calculate that a non-thermal ion population S could exist if locally accelerated on timescales 0.1 s. However, observations of net redshifts in the ≤ . linesalsoimplythe presenceofplasmadownflowswhichcouldleadtobulkturbulence,withincreased h non-Gaussianity in cooler regions. Both interpretations have important implications for theories of p solar flare particle acceleration. - o Subject headings: Sun: flares – Sun: UV radiation – Sun: X-rays, gamma rays – techniques: spectro- r scopic – line: profiles – atomic data t s a [ 1. INTRODUCTION almostimpossibletodetectwithmethodssuchasimpact 1 Solar flare extreme ultraviolet(EUV) spectral line ob- polarizationorcharge-exchange(e.g.Henoux et al.1990; Balanca & Feautrier 1998), that also require the pres- v servations with the Hinode (Kosugi et al. 2007) EUV ence of anisotropic ion beams, remaining inconclusive. 6 Imaging Spectrometer(EIS;Culhane et al.2007)provide Buttoassessthenon-thermalionenergycontentrequires 9 information on ion line emissions, plasma temperatures, knowledgeofthisacceleratedbutlow-energycomponent. 1 mass flows, ion abundances and electron densities (cf. Ion kappa velocity distributions (cf. Pierrard& Lazar 2 Milligan 2015). For most purposes, Gaussian fitting is 0 an excellent approximation for the low moments of the 2010; Livadiotis & McComas 2009) are routinely de- . spectralline: integratedintensity(zeromoment)andline tected in space physics e.g. Gloeckler & Geiss (1998), 01 centroid position (first moment), even if the line profile but the high density flare environment (ne >109 cm−3) 7 is non-Gaussian. But the shape of the EUV line pro- with thermalizing Coulomb collisions is very different to 1 file can be used to infer more about the velocity dis- thecollisionlesssolarwind. Ifsuchdistributionscanexist : tribution of the emitting ions. Jeffrey et al. (2016) ob- inflare conditions,they couldprovidea noveldiagnostic v servednon-GaussianspectrallinesinflareEUVemission, techniqueofsolarflareionaccelerationunavailableusing Xi showing that many unblended Fe XVI lines were consis- other methods. tent with a line shape produced by a kappa rather than The presence of plasma turbulence might be an ar a Maxwellian velocity distribution, in different flare re- alternative explanation of observed solar flare non- gions. Megakelvin flare temperatures produce spectral Gaussian spectral lines. Excess line broadening, or lines dominated by Doppler broadening, and physically, the presence of broadening larger than expected from such a line shape could be produced by (1) non-thermal isothermal ion motion, is often detected during a ions of . 1 MeV or (2) non-Gaussian turbulent velocity flare e.g. Antonucci & Dodero (1995); Dere & Mason (1993); Doschek et al. (1979, 1980); Alexander (1990); fluctuations, providing a new EUV diagnostic tool. Non-thermal flare particles are usually detected by X- Antonucci et al. (1986), and likely produced by either rayandgammarayobservations. MostflareshaveX-ray turbulent magnetic fluctuations (magnetohydrodyamic bremsstrahlung emission from keV electrons, currently (MHD) turbulence) or possibly by the superposition of detected with the Ramaty High Energy Solar Spectro- unresolved flows. Although, recent EIS studies in active scopic Imager (RHESSI; Lin et al. 2002). Only a small regionsandcoolerlinesinflaresshowedsomecorrelation between excess line width and directed Doppler shifts minority of (typically) large flares, e.g. SOL2002-07-23 (Krucker et al. 2003), have detectable gamma-ray line (e.g. Milligan 2011), other notable observations: larger emission,producedbyinteractionsbetweenMeVprotons broadening of hotter lines and isotropy (line broadening and heavier ions (cf. Vilmer et al. 2011), and hence the isseenforflareslocatedatallheliocentricangles),might propertiesandoccurrenceofsuchionsremainuncertain. beconsistentwithmagneticfluctuations. Otherindepen- Accelerated ions with energies less than a few MeV are dentobservationsusingX-rayimaginge.g. Kontar et al. 2 Jeffrey, Fletcher & Labrosse Figure 1. Left: Acartoonoftheflareandtheobservations. Right: Ifaninstrumentalcausecanbeeliminated,thentheEUVkappaline profilescouldbeproducedbythreephysicalscenarios: 1. anon-thermalionvelocitydistributionfromisotropicnon-thermalionmotions, 2. turbulentmotionsduetomagneticfluctuationsorpossiblyasuperpositionofunresolvedflowsor3. amulti-thermalplasmadistribution (notdiscussedinthispaper). (2011)alsoshowadditionalandindependentevidencefor Solar Dynamics Observatory (Pesnell et al. 2012) and MHD turbulence in the corona. Further, a recent study theDunnSolarTelescope(DST).Hence,ithasgenerated in preparation (Kontar et al., submitted PRL) shows a number of papers studying flare energy (Aschwanden that MHD turbulence can act as a crucial intermediary 2015), chromospheric evaporation and white light flare in the transfer of large amounts of energy from stressed emission (e.g. Heinzel & Kleint 2014; Battaglia et al. magnetic fields to accelerated particles. However, irre- 2015; Kleint et al. 2015; Li et al. 2015; Liu et al. 2015; spectiveofthecause,thisexcessturbulentmotionisusu- Young et al.2015;Heinzel et al.2016;Kleint et al.2016; ally assumed to produce a Gaussianline profile. Indeed, Kowalskiet al. 2016; Rubio da Costa et al. 2016), spec- plasmamotionsinastochasticturbulentsystemandde- tropolarimetric data (Judge et al. 2015), sunquakes scribedbyBrownianmotionwillproduceavelocityprob- (Judge et al. 2014; Matthews et al. 2015), Moreton ability distribution function (PDF) that is normally dis- waves (Francile et al. 2016) and soft X-ray pulsations tributed. However,large,sporadicmotionsfarexceeding (Sim˜oes et al. 2015). Here, we show that many Fe XVI the mean, may lead to a velocity PDF with larger and andFeXXIIIlines,producedatelectrontemperaturesof heavier tails than that of a Gaussian, and lead to EUV 3MKand 15MKrespectively,havealineshapecon- ∼ ∼ line profiles better described by a kappa or Lorentzian sistent with a k,appa velocity distribution. We discuss profile. For example, non-Gaussian magnetic fluctua- whether the observed non-Gaussian line profiles could tionsaremeasuredinspaceplasmas(Sorriso-Valvo et al. be produced by the EIS instrumental profile. We create 1999; Hnat et al. 2002; Pucci et al. 2016), with this in- mapsshowingthe spatialdistributionoffittedlineprop- termittency likely to exist on smaller scales in particu- erties such as the κ index and characteristic width that lar. Therefore, any evidence of non-Gaussian line pro- describe the velocity distribution at each location and files connectedto solarflare turbulence couldprovidean time. Finally, we weigh the evidence for the line shapes important observational constraint regarding the nature being due to non-Maxwellian flare-accelerated ions or of the turbulence, vital for MHD and kinetic modelling, to non-Gaussian turbulent velocity fluctuations, which which is not available via other techniques. Some possi- would be an observational first. ble causes of non-Gaussian spectral line profiles, includ- ing turbulence and accelerated ions are shown in Figure 2. CHOSEN FLARE AND METHOD 1. SOL2014-03-29T17:44isanX1.0flarewithcoordinates In this paper, we analyse flare SOL2014-03-29T17:44, [X=510”,Y=265”]. The X-ray emission starts around that shows the presence of non-Gaussian EUV spec- 17:44 UT and peaks in soft X-rays (SXR) at 17:48 tral lines. To date, SOL2014-03-29T17:44 is one of the UT (in Geostationary Operational Environmental∼Satel- best observed flares in history. As well as observa- lite (GOES) 1-8 ˚A). The hard X-ray (HXR, >25 keV) tions with RHESSI and Hinode EIS, the flare was also emissionpeaksaround17:46UT.TheflareRHESSI and observed by the Interface Region Imaging Spectrograph GOES X-ray light curves are shown in Figure 2. The (IRIS; De Pontieu et al.2014),instrumentsonboardthe start and end times of six EIS rasters covering the rise, Non-Gaussian Velocity Distributions in Solar Flares from Extreme Ultraviolet Lines 3 non-Gaussian line profiles to determine the underlying velocity distribution.1 For the case of an accelerated ion population and fol- lowing Bian et al. (2014), a 3-D kappa ion velocity dis- tribution f(v) of the first kind can be written as n Γ(κ) v2 −κ f(v)= 1+ π3/2v3 κ3/2Γ(κ 3/2) κv2 th − (cid:18) th(cid:19) (1) v2 −κ =A 1+ v κv2 (cid:18) th(cid:19) where n = f(v)d3v is the number density associ- ated with an acccelerated ion distribution and v = th R 2k T/M is a Maxwellian thermal velocity at temper- B ature T (for k the Boltzmann constant and M the ion B mpass),andΓ(z)= ∞tz−1e−tdtistheGammafunction. 0 To make the link to observed line profiles, we need to convert EquatioRn 1 to a 1-D line-of-sight velocity v . k The1-Dionvelocitydistributionis givenbythe integral over all perpendicular velocities v , so that, assuming Figure 2. RHESSI (top) and GOES (bottom) light curves for ⊥ isotropy, flare SOL2014-03-29T17:44. The grey dashed lines indicate the start and end times of six EIS rasters covering the flare and the timesofstudy. ∞ f(v )= f(v)2πv dv k ⊥ ⊥ peak and decay times of SOL2014-03-29T17:44 are in- Z0 dicated by grey dotted lines in Figure 2, denoting the ∞ v2+v2 −κ k ⊥ time intervals under study. EIS observes SOL2014-03- =Av 1+ κv2 2πv⊥dv⊥ 29T17:44infast-rasteringmode. Eachrasteristwomin- Z0 th ! utes and fourteen seconds long, with slit movements ev- πκv2 v2 −κ+1 ery 12 seconds. The slit scans in the X direction from =A th 1+ k solar∼westtoeast. The1′′ slitisusedduringtheobserva- vκ−1 κvt2h! tions,moving3′′.99everyslitjump. Thenaturalbinning in the Y direction is 1′′. n Γ(κ 1) vk2 −κ+1 f(v )= − 1+ FiTguhreem3o.rpThhoelogtwyoofimSOagLe2s01in4-0th3-e293T0417˚A:44piasssshboawndn ionf → k π1/2κ1/2vth Γ(κ−3/2) κvt2h! (2) SDOAtmosphericImagingAssembly(AIA; Lemen et al. 2012) at 17:46:58 UT and 17:49:22 UT show the mainly where n= ∞ f(v )dv . unsaturated flare ribbons. RHESSI X-ray contours at −∞ k k As κ , Equation 22 tends to a 1-D isothermal 10-25 keV and either 25-50 keV or 50-100 keV are over- → ∞R Maxwellian distribution. In this form, we can think of laid. During raster 17:46:14UT, two HXR footpoints at v as the thermal speed of a Maxwellian ion popula- 50-100 keV are present, at either side of a lower energy th tion before acceleration, or a characteristic speed of the 10-25 keV coronal source. At the later time, the 50- distribution. For low κ and large v , we can approxi- 100 keV HXR footpoints disappear but we still observe k mate the ion velocity distribution as a power law with X-rays up to 50 keV. The EIS intensity contours from the Fe XVI and Fe XXIII EIS rasters are also displayed. f(vk)≈vk−2(κ−1) =vk−β. The EIS data is aligned with AIA using the procedure The line-of-sight velocity distribution is related to the eis aia offsets.pro, with a 5′′ error in Y. We assume that emitted line profile by f(v ) I(λ) dλ = I(λ)λ0, for AIA and RHESSI are well-aligned for the purposes of k ∝ dvk c wavelength λ, rest wavelength λ and speed of light c, our analysis. 0 giving The EISdata in the Y directionis binned into 2′′ bins (from 1′′) improving the signal-to-noise ratio and line (λ λ )2 −κ+1 fitting goodness-of-fit. The EIS instrumental broaden- I(λ)=A 1+ − 0 (3) ing Winst using the 1′′ slit is Winst = 0.059 ˚A (the full λ(cid:18) κ2σκ2 (cid:19) width at half maximum, FWHM) assuming a Gaussian where v2 = 2k T/M = 2σ2c2/λ2 and A A c/λ . instrumental profile. th B κ 0 λ ∝ v 0 As κ in Equation 3, the line shape becomes Gaus- →∞ 2.1. Non-Gaussian ion and plasma velocity 1 It isalsopossiblethat amulti-thermalplasmaalongthe line- distributions of-sightcouldberesponsible,particularlyiftheionsandelectrons have different temperature distributions, but this isnot discussed The EIS data for SOL2014-03-29T17:44 includes two here. suitably strong, unblended spectral lines formed at dif- 2 Equation 2 is slightly different to the kappa function used in ferenttemperatures: FeXVI( 2.5 4MK,logT =6.4) Jeffreyetal. (2016), where the index ( κ) was used instead of and Fe XXIII ( 15 16 MK,≈logT−= 7.2). We use the ( κ+1). − ≈ − − 4 Jeffrey, Fletcher & Labrosse Figure 3. Top row: Two SDO AIA images of SOL2014-03-29T17:44 using the 304 ˚A passband (green background image) at times of 17:46:59UT(left)and17:49:22UT (right),times withintwodifferent EISrasters. RHESSIcontours at10-25keV (red) andeither 25-50 keV or 50-100 keV (navy blue) are displayed at levels of 50 % and 70 % of the maximum. Fe XXIII (left) and Fe XVI (right) intensity contoursaredisplayedinpurple,at30%,50%and70%ofthemaximum. Bottom row: SpectrallinesofeitherFeXXIII(left)orFeXVI (right) observed at the location of the rectangular box shown inthe top images. Each lineis fitted withthe KG1, KG2 and SG fits (see text for details). The small panels display the line peak and right wings in detail so that the fits can be clearly seen. The reduced χ2, residualsandfitparametersofκandσ arealsodisplayed. sian. Also, if κ = 2, the line profile is the same as a volution of F(u ) with the ion velocity distribution, but k Lorentzian. Hence, a kappa line profile can be used as a theoveralllineprofileanditsnon-Gaussianitycouldstill generalline fitting form that can coverthe specific cases beapproximatedbyakappalinedistribution. Hence,re- of both Gaussian and Lorentzian line profiles. A kappa gardlessofthephysicalprocess,akappalineprofileisan distributionmightalsobeusedtodescribeaspectrumof excellent starting point for the detection and analysis of velocities F(u ) produced by plasma turbulence. In this non-Gaussianionorplasmavelocities. Evenifthekappa k case, the plasma velocity distribution (excluding the ion distributiondoesnotdescribealltheunderlyingphysics, thermal motions) could be described by, it provides a mathematically convenient line profile for the determination of non-thermal/non-Gaussian veloci- F(u )= F0 Γ(κ−1) 1+ (uk−u1)2 −κ+1 (4) ntioetspfroosmsibHlein),opderoEvIidSindgataafi(wtthinegrefumnocrteiodnettahialetdcafinttrinanggies k κ1/2Γ(κ−3/2)(cid:18) κu20 (cid:19) from a Gaussian to a Lorentzian. where u is the plasma velocity, F is a function depen- k 0 2.2. EIS line fitting of FeXVI and FeXXIII dent on plasma properties, u is a characteristic speed 0 of the turbulence and u is a bulk flow plasma velocity. The Fe XVI and Fe XXIII lines are fitted with a sin- 1 The overall velocity distribution would then be a con- gleGaussiantoestimatetheGaussianintensity,centroid Non-Gaussian Velocity Distributions in Solar Flares from Extreme Ultraviolet Lines 5 and line width. Lines with skewness S > 0.08 indicat- where o are the observedintensity values, ǫ arethe ob- i i | | ing lackofsymmetry,andprobablemovingcomponents, served intensity error values, m are the model values i are removed from the study. Many of the Fe XVI and and degree of freedom DOF=numberof datapoints Fe XXIII lines fitted with a Gaussian have high reduced numberof fittedparameters, and by examining the fi−t χ2 values, greater than 6. From the Gaussian fitting, residuals R= o−m. ǫ evenaftertheremovalofaGaussianinstrumentalprofile The bottom row of Figure 3 displays two examples: with FWHM W = 0.059 ˚A, the Doppler broadening one Fe XVI and one Fe XXIII profile and fit. Here the inst in most regions is larger than expected from an isother- lines are fitted with: (1.) a physical kappa - instrumen- mal plasma. The Gaussianline widths after the removal tal Gaussian fit (KG1), (2.) an instrumental kappa - of W for Fe XXIII can be as large as 0.12 ˚A and physical Gaussian fit (KG2) and (3.) a single Gaussian inst for Fe XVI as large as 0.08 ˚A. The expected isothermal (SG). The corresponding spatial locations are indicated widths for Fe XXIII and Fe XVI are W 0.1 ˚A (for in the images shown in the top row of Figure 3 by the th ∼ rectangularboxesandslitpositions(dashedlines). Each logT =7.2) and W 0.04 ˚A (for logT =6.4). th ∼ imagedisplaystheAIA304˚Apassbandwheretwonorth Next, as in Jeffrey et al. (2016), we re-fit the lines and south ribbons can be clearly seen. RHESSI X-ray withaconvolvedkappa-Gaussiandistribution,account- contours at 10-25keV and 25-50 keV or 50-100 keV and ing for (1.) a Gaussian EIS instrumental profile with Fe XVI or Fe XXIII contours are displayed. Figure 3 W = 0.059 ˚A and (2.) the possibility of a non- inst shows how the kappa part of the KG1 fit is able to ac- Maxwellianvelocitydistributionresultinginline profiles count for the higher peaks and broader wings of the ob- with higher peaks and ‘heavier’ wings than a Gaussian. served spectral lines. For both profiles in Figure 3, the The convolved kappa ( )- Gaussian ( ) line profile is K G single Gaussian fits produce the large reduced χ2 values given by of χ2 = 7.6 (Fe XXIII) and χ2 = 5.1 (Fe XVI). The G G KG1 fits give the lowestreduced χ2 values of χ2 =1.3 KG (λ)= (λ) (λ)=A[0]+A[1] (Fe XXIII) and χ2 = 1.7 (Fe XVI). The KG2 fits pro- W G ∗K × KG −A[4]+1 duce higher reduced χ2 values than the KG1 fits with exp (λ′ −A[2])2 1+ (λ−λ′ −A[2])2 χ2KG2 = 4.4 (Fe XXIII) and χ2KG2 = 6.8 (Fe XVI). The Xλ′ − 2σI2 ! 2A[3]2A[4] ! lKinGe1s dfiitsp(plahyyesdicianl kFaipgupraep3roafirlee)twgioveesxaamlopwleesrwgohoedrenetshse- (5) of-fitthantheKG2fit(instrumentalkappaprofile). The where there are five free fit parameters A. For further example line fits in Figure 3 support a physical rather detailsseeJeffrey et al.(2016). FromEquation5,weare than an instrumental origin since the lines are best fit- interested in determining the values of the kappa index ted with different kappa parameters, and not the single, κ and characteristic width σκ (fit parameters A[4] and fixedκI andσI values of the constraint,as we mightex- A[3] respectively); parameters that provide information pect if the non-Gaussian part of the profile was wholly aboutthe velocitydistribution. We callthis fitKG1. As instrumental. We discuss this in greater detail in Ap- discussedinJeffrey et al.(2016),thisfunctionisagener- pendix A and later in subsection 3.2. alizedVoigtfunction,withthetraditionalVoigtfunction, InFigure3,theresidualsforeachspectrallinearealso a convolution of a Gaussian and a Lorentzian, being the shown. ForthechosenFeXXIIIline,theresidualsclearly limiting case when A[4] =κ=2. showthattheKG1modelisabetterfitfortheline,asin- It is possible that all or part of the non-Gaussian line dicated by the low χ2 value. This is particularly not- KG1 shape results from the instrumental profile. There is no icable around the peak and the wings of the line, where reason for the EIS instrumental profile to be Gaussian, the KG1 residuals are very close to zero (values within althoughitmaybeextremelywell-approximatedassuch. 2), compared to the fixed KG2 and SG residuals (val- Forexample,thespectrometermightbeexpectedtohave ±ues within 4). Again, for the chosen Fe XVI line, the ± an instrumental response closer to a sinc2λ function. To KG1 residuals show that this model is a better descrip- accountfor the possibility of a non-Gaussianinstrumen- tionofthelinethanaGaussian(SG),forallwavelengths tal response we fit another convolved kappa - Gaussian covering the line profile (again the KG1 residual values (see EquationA1 inAppendix A),where the kappa part are within 2). ± is fixed to represent an instrumental profile with chosen We perform two line profile studies. The initial study κ and σ and the Gaussianparametersare free to vary, fits, with KG1,KG2 and SG functions, lines that satisfy I I representing a physical line profile. The EIS instrumen- the following two criteria: tal profile can be approximated by a Gaussian profile 1. Lines must have an absolute value of skewnessless with FWHM W = 0.059 ˚A. Therefore the kappa in- inst than0.08(toremovelineswithmovingcomponents strumentalprofileisconstrainedbytherequirementthat as discussed. Also see Jeffrey et al. 2016). κ and σ produce W =0.059 ˚A when approximated I I inst by a Gaussian. To obtain this we choose κI = 3 and 2. The estimated noise level (calculated as the stan- σI =0.0395 ˚A. This parameter choice is not unique and darddeviationoftheratiooftheintensityerrorsto thechoiceofvaluesarediscussedfurtherinAppendix A. intensity for each line) for the line must be below We call this fit KG2. 9% and the ratio of the integrated intensity error The line goodness-of-fits are judged by a combi- to integrated intensity less than 0.9%. nation of “judgement by eye”, a reduced χ2 = (o m )2 Followingthe line fitting with KG1, KG2 and SG pro- 1 i− i from the weighted least squares fit, files, we identify those fits where we are confident that DOF ǫ2 i i X 6 Jeffrey, Fletcher & Labrosse Figure 4. Gaussian (SG) and kappa-Gaussian (KG1 and KG2) fits for Fe XVI. Left column: kappa-Gaussian fit using a Gaussian instrumental profile(KG1), middle column: kappa-Gaussian usingakappainstrumental profile(KG2)andright column: singleGaussian fit(SG). Row 1: κindex, row 2: 2√2ln2 σ ofeachfit(σκ orσG)androw 3: reduced χ2 values foreachfit. TheRHESSI lightcurves arealsodisplayed,withgreydashedliness×howingthetimeofobservation(timet3). Theparametersfromlinesshowninthisfiguresatisfy criteria(1.) and(2.) only(initalstudy,seetextfordetails). KG1 is the best fit, according to the following extra cri- From the work in Jeffrey et al. (2016) and by testing teria: model lines with different levels of Gaussian noise, we found that lines with a noise value less than 10% were 3. The reducedχ2 values of the kappa - Gaussianfits ∼ usually suitable (i.e. small intensity error values) for a must be less than 5.0. linemodelcomparison. Theintegratedintensity errorto 4. Thereducedχ2 valuesoftheGaussianfitsmustbe integratedintensityratioof0.9%waschosenbytrialand greater than 3.0. error and by examining how this value changed for lines foundtobeeithersuitableorunsuitableforstudy. Crite- 5. The ratio χ2G/χ2KG must be greater than 2.0 (for rion2 allowsus to quicklyremovea large fractionof un- both KG1 and KG2). suitable lines in each raster without examining each line Criterion 2 is used as a “noise value”, which we de- in detail, since each map has a total of 660 lines. Fur- ∼ fine as 100% STD(ǫ/o) (for STD=standard deviation). ther, criteria 3-5 help to find non-Gaussian line shapes × Non-Gaussian Velocity Distributions in Solar Flares from Extreme Ultraviolet Lines 7 Figure 5. Gaussian (SG) and kappa-Gaussian (KG1 and KG2) fits for Fe XXIII. Left column: kappa-Gaussian fit using a Gaussian instrumental profile(KG1), middle column: kappa-Gaussian usingakappainstrumental profile(KG2)andright column: singleGaussian fit(SG). Row 1: κindex, row 2: 2√2ln2 σ ofeachfit(σκ orσG)androw 3: reduced χ2 values foreachfit. TheRHESSI lightcurves arealsodisplayed,withgreydashedliness×howingthetimeofobservation(timet3). Theparametersfromlinesshowninthisfiguresatisfy criteria(1.) and(2.) only(initalstudy,seetextfordetails). andremovelineswithlargererrorsthatcanbewell-fitted intensitiesusingthecodesofKlimchuk et al.(2016)that by all models (i.e. all producing low χ2 values), helping account for finite binning in wavelength before the lines to pinpoint and only examine lines that have a definite are fitted. non-Gaussian shape. In particular, criteria 4 and 5 are used to remove lines where the Gaussian model has low 3. RESULTS χ2 < 3 since we want to look at (a) non-Gaussian lines and (b) remove noisy lines well-fitted by any model. 3.1. Initial comparison of the KG1, KG2 and SG fits None of the Fe XVI regions contained warm pixels (as In Figures 4 and 5, maps of line fit parameters κ and discussedinJeffrey et al.2016)butfourFeXXIIIregions W = 2√2ln2 σ , and the goodness-of-fit χ2 are dis- κ didcontainwarmpixels. IntheinitialanalysisofSection played. These×are shown for a single EIS raster time of 3 warm pixels are included but they are removed in the 17:48:23 UT (start time, t ) and for each of the three 3 further analysis of KG1. We also varied the EIS line fits: KG1 (first column, three panels), KG2 (second col- 8 Jeffrey, Fletcher & Labrosse umn,twopanels)andSG(thirdcolumn,twopanels),for there is a trend between changes in κ index and W, for FeXVI (Figure 4) and FeXXIII (Figure 5). Line widths both Fe XVI and Fe XXIII lines. Only lines that satisfy W are displayed as a ‘Gaussian FWHM’ for easy com- all five criteria listed in Section 2 are shown in Figure parisonwithlinewidthsfoundfromGaussianlinefitting. 6. We look for common trends that might suggest that At this time, fits satisfying criteria 1 and 2 are located the kappa line profiles are due to an instrumental pro- withinthe FeXVI30%intensitycontourandalongthe cess instead of a physical one. Importantly, we compare n∼orthern ribbon, and for Fe XXIII, mainly within the the observed KG1 values in Figure 6 with Figure 13 in Fe XXIII 30% contour and close to the 10-20 keV and Appendix A. Figure 13 displays the results of two mod- 25-50 keV X-ray contours. elledlinescloselyrepresentingFeXVIandFeXXIII.Each FortheFeXVIKG1fit,thelowestvaluesofχ2 (<3) modelledline ischosento haveaninstrumentalresponse KG1 are located at the edges of the Fe XVI source and along either represented by: (1.) sinc2λ function (as discussed the northern ribbon. Closer inspection of the actual line inAppendixA)orby(2.) akappafunctionwiththecho- cfiltotsinegtofotrhealclefinttsreshofowthsetFheatXtVhIeshoiugrhceχa2KrGe1du(∼e1to0)thvealluinees sfietntinpgarfuamncettieornsKκGI 2=).3Tahnedn,σeIa=ch0m.0o3d9e5lle˚Ad(inthsterusammenetaasl havingamovingcomponentnotremovedbytheskewness responseisconvolvedwithaGaussianlinerepresentative condition (the shape of some line profiles with a large ofa physicalline profile and the line width of this Gaus- moving component can lead to the line shape having a sianisvariedbetweensensiblevaluesforbothFeXVIand lowerskewnessthan0.08). TheKG1W valuesinregions FeXXIII(seeAppendix A).Eachresultingmodelledline oflowχ2 arebetween0.04˚Aand0.07˚A.FortheKG2 isfittedwiththeKG1fittingfunction,andtheKG1fitted KG1 fit,theχ2 valuesarelowinanumberoflocations,but valuesofκindexversusW valuesarethenplottedinFig- KG2 withhighervaluesthantheKG1fit. Theκindexandσ ure 13. Figure 13 shows that as the (physical) Gaussian κ values for the KG2 fit are kept constant at 3 and 0.0395 width of the modelled line increases, so do the resulting ˚A respectively,andtheKG2Gaussianwidths(2√2ln2 KG1 fit values of κ index and W, for all modelled lines. σ ) are found to be > 0.05 ˚A. The SG χ2 values ar×e The KG1 parameters found from actual fitting to the G SG observedlines and originally shown in Figure 6 are then higher (often greater than 3) and the SG widths are > re-plotted in Figure 13 for comparison with the model 0.06 ˚A. For the Fe XXIII KG1 fit, the majority of χ2 KG1 line results (lines only satisying criteria (1.) and (2.) in values are again very low (mainly <3), apart from two Section 2 are also shown). Both Figure 6 and Figure 13 pointsthathaveveryhighχ2 values(greaterthan16). KG1 show that the observed values show a range of different Again,oncloserinspection,theselinesappeartoinclude W values for a given κ value (and vice-versa), which is blue-shifted moving components (for all fits). For KG1, not suggested by the model line results. The results for the κ index values are found to be between 4 and 10 Fe XXIII do not match the expected curves at all, while and W between 0.08 ˚A and 0.10 ˚A. For KG2, the W valuesaregreater∼than 0.10˚A buttheχ2 valuesare there is a much better match for FeXVI,althoughagain ∼ KG2 we see different values of W for a given κ index. There- low ( 4). The χ2 for the Fe XXIII SG fits are again fore, this test is suggestive (but not conclusive) that the highe≤r, just as forStGhe Fe XVI fits, with values above 6. observed non-Gaussian line profiles are physical instead The uncertainties associated with σ inferred from each of instrumental and we interpret the KG1 fitting results of the KG1, KG2 and SG fits are small, of the order as such in the next subsection3. 10−3 ˚A or less. The errors for the KG1 κ values are of the order10−1 forbothFeXVIandFeXXIII.Theinitial 3.3. Further analysis of the KG1 lines analysis and Figures 4 and 5 show three main results: InFigure 7, the flare is shownat the six different (EIS start) times t = 17:44:00 UT, t = 17:46:14 UT, t = 1. Spatial patterns for κ index and characteristic 1 2 3 17:48:28 UT, t = 17:50:42 UT, t =17:52:55 UT and width σ (KG1) emerge and this is discussed fur- 4 5 ther in sκubsection 3.3. t6 =17:55:09UT. EachmapshowsAIA304˚A, RHESSI andEIScontours. MapsoftheFeXVIandFeXXIIIKG1 2. TheKG1andKG2W valuesaresmallerthanthose fit parameters, κ and W =2√2ln2σ , are displayed for κ found from the SG fit, which requires the presence regions that satisfy all the criteria listed in Section 2 in of larger excess line broadening to explain the ob- Figure 7. At times t andt , two HXR footpoints (ener- 1 2 served values. gies > 50 keV) and an X-ray coronalsource are present. At times t t , the HXR footpoints disappear but the 3. OKGve2raalnl,dthSeGKvGal1ueχs2K,Gfo1rvbaoluthesFaereXsVmIaalnledrFtheaXnXtIhIeI Xth-erraeyicsonroo3n−Fael s6XouXrIcIeI ecmanisssitoilnl bsueitoabbsleervfoerd.anAatlytsiimsedute1 (with most reduced χ2KG1 values less than two). toa lowsignal-to-noiseratioandhighskewness,likewise for Fe XVI at t and t . The Fe XVI and Fe XXIII KG1 1 2 fittingparameters(κandW)pluserrorsandreducedχ2 3.2. Further evidence against an instrumental origin values are also shown in Table 1. For comparision, the for the non-Gaussian property KG2 and SG W and χ2 values are also shown for each Before analysingthe KG1 fits in detail, we present ev- idencethatthe non-Gaussiancomponentofthe linepro- 3 Althoughwehaveprovidedevidenceofwhythenon-Gaussian files are more consistent with a physical rather than an lineprofilesaremorelikelytobephysical,itisdifficulttoruleout aninstrumentalcausecompletely. Therefore,ifwewishtoperform instrumental cause. Further details are provided in Ap- more detailed flare spectroscopy studies and use line shape as a pendix A. In Figure 6, we plot the KG1 κ values versus reliablediagnostic tool inthe future, then the exact instrumental the characteristic widths W =2 2ln(2)σ to observe if profilemustbelaboratorytestedbeforelaunch. κ p Non-Gaussian Velocity Distributions in Solar Flares from Extreme Ultraviolet Lines 9 Figure 6. TheKG1fitvaluesofκversusW =2√2ln2 σκ ateachtimeforFeXVI(toprow)andFeXXIII(bottom row). Byplottingκ × versus2√2ln2 σκ welookfortrendsthatmightindicatethatnon-Gaussianlineprofilesareduetotheinstrumentalresponseinsteadof aphysicalproce×ss. ThisfigureiscomparedwithFigure13inAppendixA.Themultiplevalues ofκforagivenW (andvice-versa)isone observationthatsupportsaphysicalcause. line. Overall, over 60 % of the lines shown in Figure 7 Att ,t andt ,theFeXVIW valuesarebetween0.03 4 5 6 and Table 1 have KG1 χ2 2.0. ˚A and 0.05 ˚A, slightly lower than the Fe XVI values at KG1 ≤ t with the majority between 0.05-0.06 ˚A. Overall, the 3.3.1. FeXXIII la2rgest KG1 W values occur at early times for Fe XXIII During the interval starting at t (covering the HXR and FeXVI. 2 peak), five Fe XXIII regions satisfy the five criteria in Section 2.2. These cover part of the coronal 10-25 keV 4. PHYSICAL INTERPRETATION AND X-ray source and lie within the Fe XXIII 50% contour DISCUSSION line. Wecanseethattheκindexincreasesfromnorthto Our analysis can be summarised as follows: south, with the lowest values of κ close to the centre of the coronal X-ray source increasing from κ 3.8 to κ 1. Non-Gaussian line profiles consistent with kappa ∼ ∼ 6.5. Similarly,thelargestvaluesofW occurclosertothe distributionsofemittingionswerefoundduringthe centre of the Fe XXIII source, with values ranging from flare, close to the flare loop-top, HXR footpoints W = 0.09 ˚A to W = 0.11 ˚A. At t3 the HXR footpoints andribbons(similartoSOL2013-05-15T01:45anal- disappear and there is an X-ray coronal source located ysed in Jeffrey et al. 2016). closetoX =520′′,Y =275′′. WefitwithKG1FeXXIII, linesfromelevenlocationsalongthesouthernedgeofthe 2. FeXVIlinesexhibitingkappaprofilesweresituated coronal X-ray source, finding κ between 4 and 9. The further from the coronal source than the Fe XXIII W values are lower than t , ranging between W 0.08 lines,andofteninregionswhereHXRsourceswere 2 ˚AandW 0.095˚A.Att thereareeightregionssu∼itable previously observed. 4 ∼ for study with κ ranging between 4.5 and 6. W ranges between0.07and0.095˚A, withallvalues lowerthanthe 3. Fe XXIII lines exhibiting kappa profiles were situ- expected Gaussianthermalwidth of 0.1 ˚A. At t and t , ated close to the coronal source and appeared to 5 6 move with the coronal source over time. the FeXXIIIκ values are 6 7 with W 0.08 0.095 ˚A. ≈ − ∼ − 4. TheκindexvaluesoftheFeXVIlinesweresmaller than those of Fe XXIII and not so systematic in 3.3.2. FeXVI terms of position and value. Overall,theκvaluesfoundforFeXVIaresmallerthan thoseforFeXXIII.Attimet therearesixlocationswith 5. FeXXIIIshowedinterestingspatialvariationsclose 3 Fe XVI lines satisfying our criteria. Close to the centre to the coronalX-raysourceswith smaller values of of the Fe XVI source and overlapping slightly with the κ index locatedcloserto the X-raycoronalsources edge of the coronal X-ray source we find two locations early in the flare. with κ values of 3 and 4. At X = 495′′, Y = 265′′, close to the eastern footpoint, the κ values are between We considered the possibility that the observed non- 3.5 and 4. Overall, the κ values for Fe XVI are lower Gaussian line profiles result from the EIS instrumental than for Fe XXIII at this time. At time t there are response. Although we cannot rule this out completely, 4 nine suitable Fe XVI pixels, with κ between 2.5 and 4. we find that parameter trends for κ index and σ (par- κ These are scattered, mostly located at the periphery of ticularlyforFeXXIII)donotbehaveaswewouldexpect the main Fe XVI source and at some distance from the iftheinstrumentalresponsewerenon-Gaussian. We will coronal X-ray source. now discuss the possible origins of the results. 10 Jeffrey, Fletcher & Labrosse Figure 7. Maps ofSOL2014-03-29T17:44 duringsixdifferent EISraster times (timet1 t6 increases fromtopto bottom), showingthe − resultsof KG1fits satisyingallfive criteria(Section 2). Columns one and two: FeXVI κ(1) and W =2√2ln2 σκ (2). Columns three × and four: FeXXIIIκ(3)andW =2√2ln2 σκ (4). ThevaluesarealsoshowninTable1. ×