Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements

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Article Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements

Georgios Nicolaou 1,* , Robert Wicks 1, George Livadiotis 2, Daniel Verscharen 1,3, Christopher Owen 1 and Dhiren Kataria 1

1 Department of Space and Climate Physics, Mullard Space Science Laboratory, University College London, Dorking, Surrey RH5 6NT, UK; [email protected] (R.W.); [email protected] (D.V.); [email protected] (C.O.); [email protected] (D.K.) 2 Southwest Research Institute, San Antonio, TX 78238, USA; [email protected] 3 Space Science Center, University of New Hampshire, Durham, NH 03824, USA * Correspondence: [email protected]

Received: 30 December 2019; Accepted: 11 January 2020; Published: 16 January 2020

Abstract: Electrostatic analysers measure the flux of plasma particles in velocity space and determine their velocity distribution function. There are occasions when science objectives require high time-resolution measurements, and the instrument operates in short measurement cycles, sampling only a portion of the velocity distribution function. One such high-resolution measurement strategy consists of sampling the two-dimensional pitch-angle distributions of the plasma particles, which describes the velocities of the particles with respect to the local magnetic field direction. Here, we investigate the accuracy of plasma bulk parameters from such high-resolution measurements. We simulate electron observations from the Solar Wind Analyser’s (SWA) Electron Analyser System (EAS) on board Solar Orbiter. We show that fitting analysis of the synthetic datasets determines the plasma temperature and kappa index of the distribution within 10% of their actual values, even at large heliocentric distances where the expected solar wind flux is very low. Interestingly, we show that although measurement points with zero counts are not statistically significant, they provide information about the particle distribution function which becomes important when the particle flux is low. We also examine the convergence of the fitting algorithm for expected plasma conditions and discuss the sources of statistical and systematic uncertainties.

Keywords: space plasma; kappa distributions; pitch-angle distributions; plasma instruments

1. Introduction The environment of interplanetary space is filled with a very low-density plasma, primarily consisting of electrons and protons with a small component of heavier ions. Typically, investigations of this plasma measure the flux of particles in energy and angle to determine the velocity distribution function (VDF) of different particle species. The VDF indicates how energy is distributed between particles of the same type, and can be analysed accordingly to provide bulk properties of the plasma such as density, velocity, and temperature. The accuracy of the derived plasma bulk parameters depends on the accuracy of the measurements and on the techniques we use to analyse the data. The velocities of space plasma particles often follow kappa distribution functions (see, e.g., in [1–6] and references therein). The kappa index labels and governs these distributions and it is another fundamental parameter that describes the thermodynamic state of the plasma. Theoretical works derived the kappa distribution from the foundations of Tsalis non-extensive statistical mechanics (see, e.g., in [7–9]). In addition, a large number of studies have determined kappa distribution functions in space plasma environments, such as the solar wind (see, e.g., in [10–15]), planetary magnetospheres (see,

Entropy 2020, 22, 103; doi:10.3390/e22010103 www.mdpi.com/journal/entropy Entropy 2020, 22, 103 2 of 14 e.g., in [16–18]), cometary electrons (see, e.g., in [19]), and the outer heliosphere and inner heliosheath (see, e.g., in [20–26]). The high energy “tails” that characterize kappa distributions are associated with relatively low particle fluxes, and are thus not always clearly resolved in plasma measurements. In these cases, the analysis becomes challenging, because a smaller number of particles are measured leading to a higher statistical uncertainty. Nevertheless, the accurate description of the plasma requires the determination of the kappa index [2,3,27]. Therefore, it is essential to obtain high-quality measurements that resolve the high-energy tails of the plasma distribution functions, and to use the appropriate analysis methods to determine the plasma bulk parameters from the observations. Typical electrostatic analysers measure the flux of plasma particles over a finite range of energy and flow direction, at a given time. An almost complete scan through energies and directions is achieved by changing the voltages of the instrument components in a consecutive order. Space plasma is permeated by magnetic field which provides an important direction for anisotropy in the plasma. Charged particles gyrate around the magnetic field and particles can move relative to the magnetic field due to drift motions. The magnetic field is also an important direction for the propagation of plasma waves, wave-particle interactions, damping, heating and the generation of waves via instabilities from free-energy in the particle VDFs. Thus, in some operation modes, the measurement cycle of a plasma instrument is modified, and the instrument samples the particle energies in a limited range of directions only, providing the measurements to construct the two-dimensional (2D) pitch-angle distribution function of the plasma. Although the reduction by one dimension allows faster scans which provide high time-resolution measurements, it reduces the statistical significance of the data because the distribution is resolved in fewer points in velocity space. For example, the Plasma Electron and Current Experiment (PEACE) instrument [28] on board Cluster is designed to measure the plasma electrons within the Earth’s magnetosphere. PEACE consists of two top-hat electrostatic analysers, with look directions onto a plane perpendicular to the spin axis of the spacecraft. During the nominal operation mode, PEACE constructs the three dimensional (3D) VDF of the plasma electrons over the energy range from 0.59 eV to 26.4 keV. The 3D VDF can be constructed over half a spin period of the spacecraft, which corresponds to 2 seconds. In cases when the magnetic field direction lies within the instrument’s azimuthal field of view, we can construct the pitch angle distribution of the plasma electrons over a time period of just 62.5 milliseconds—the time it takes for the instrument to scan in energy. As another example, the Solar Wind Analyser’s Electron Analyser System (SWA-EAS [29]) on board Solar Orbiter, will measure the solar wind electrons in the energy range from 1 eV to 5 keV, within heliocentric distances from ~0.3 to 1 au. In its nominal operation mode, SWA-EAS completes an energy-direction scan constructing the entire 3D VDFs of the plasma electrons in ~1 s. In burst-mode, the instrument will measure the 2D pitch-angle distribution of the electrons over a period of 0.125 s, using a single deflection state. The accuracy of the derived plasma bulk parameters is a function of the electron flux, which depends on the solar wind density, and speed. Since Solar Orbiter will observe plasma within a wide range of heliocentric distances and heliographic latitudes, we expect a wide range of plasma fluxes to be sampled. In this study, we investigate the accuracy of our derivation of the plasma bulk parameters, such as the plasma density, temperature tensor and kappa index, from the analysis of the expected 2D pitch-angle measurements by SWA-EAS on board Solar Orbiter. We model the expected measurements of solar wind plasma electrons, considering the instrument’s ideal response, based on the initial instrument calibration. We then fit the synthetic dataset with an analytical model in order to derive the bulk parameters of the electrons. We compare the derived parameters with those from our input. Through this comparison, we quantify the accuracy of the derived parameters as a function of the recorded counts. In Section2, we describe in detail our instrument model and how we model and analyse the expected observations. In Section3, we present our results, which we discuss in detail in Section4. Section5 summarises our conclusions. Entropy 2020, 21, x 3 of 15 Entropy 2020, 22, 103 3 of 14 2. Methods 2. Methods 2.1. Instrument Model 2.1. InstrumentSWA-EAS Modelconsists of a pair of top-hat electrostatic analyser heads, orthogonally mounted on the spacecraftSWA-EAS boom. consists Each analyser of a pair head of top-hat includes electrostatic an aperture analyser deflector heads, system orthogonally and a position-sensitive mounted on the multichannelspacecraft boom. plate Each (MCP) analyser detector. head The includes instrument an aperture measures deflector the plasma system electron and afluxes position-sensitive as a function ofmultichannel kinetic energy plate and (MCP) angle detector. of incidence The instrumentfrom which measures the electron the plasmaVDF is calculated. electron fluxes SWA-EAS as a function scans theof kinetic energy energy of the andelectrons angle by of electrostatic incidence from selectio whichn in the 64 electron steps, spread VDF is ex calculated.ponentially SWA-EAS over the range scans betweenthe energy 1 eV of to the 5 electronskeV. Each by specific electrostatic kinetic selectionenergy step in 64E has steps, a width spread of exponentiallyΔΕ/Ε ~ 12.5%. In over the the nominal range operationbetween 1 mode, eV to 5the keV. aperture Each specific deflector kinetic system energy of each step analyserE has a width head ofuses∆E /voltagesE ~ 12.5%. applied In the to nominal plates atoperation the aperture mode, entrance the aperture to deflect deflector incoming system elec of eachtrons analyser and scans head through uses voltages the elevation applied direction to plates of at thethe plasma aperture particles, entrance defined to deflect as incomingthe complement electrons of the and angle scans within through the the particle elevation velocity direction vector of and the theplasma z-axis particles, perpendicular defined to as the the top-hat complement aperture of plane the angle (left within panel of the Figure particle 1). velocityEach of vectorthe two and SWA- the EASz-axis analyser perpendicular heads measure to the top-hat specific aperture elevation plane directions (left panel Θ within of Figure the1 range). Each from of the −45° two to SWA-EAS+45° in 16 uniformlyanalyser heads spaced measure electrostatic specific steps. elevation We model directions an idealΘ responsewithin the in which range each from Θ 45is measuredto +45 inwith 16 − ◦ ◦ auniformly resolution spaced of ΔΘ electrostatic= 6°. The azimuth steps. direction We model is an the ideal angle response within inthe which projection each Θof isthe measured velocity vector with a onresolution the top-hat of ∆Θ plane= 6 (◦x.- They plane) azimuth and the direction x-axis (right is the anglepanel withinof Figure the 1). projection Each SWA-EAS of the velocity analyser vector head resolveson the top-hat 32 discrete plane azimuth (x-y plane) directions and the Φx -axiscovering (right the panel full range of Figure from1). 0° Each to 360° SWA-EAS by 32 azimuth analyser sectors head mounted on the MCP. Each azimuth direction Φ has a resolution of ΔΦ = 11.25°. Combining two of resolves 32 discrete azimuth directions Φ covering the full range from 0◦ to 360◦ by 32 azimuth sectors themounted analyser on units the MCP. offset Each by 90° azimuth allows direction the full skyΦ has to be a resolutionmeasured. ofThus,∆Φ =one11.25 analyser. Combining can always two see of  ◦ thethe direction analyser unitstowards off setthe bymagnetic 90◦ allows field the vector full skyB. In to burst-mode be measured. operations, Thus, one the analyser analyser can head always that  → cansee thesee directionB performs towards two consecutive the magnetic energy field vector scans;B one. In with burst-mode the aperture operations, deflectors the analyserset to Θ head= θB, → followedthat can see by Ba scanperforms with twothe deflectors consecutive set energy to Θ = scans;-θB, where one with θB is the the aperture elevation deflectors angle of setthe tomagneticΘ = θB, fieldfollowed vector, by awhich scan withhas been the deflectors recorded setby totheΘ magnet= θ ,ometer where θ onis board the elevation Solar Orbiter angle of[30] the just magnetic before − B B thefield EAS vector, scan. which Thus, has the been flux recorded of electrons by the will magnetometer be measured onin boardtwo cones, Solar Orbiterone with [30 an] justedge before aligned the  alongEAS scan. and Thus,one anti-parallel the flux of electronswith B. will From be measuredthese measurements in two cones, a pitch-angle one with an distribution edge aligned can along be recovered.and one anti-parallel with →B. From these measurements a pitch-angle distribution can be recovered.

Figure 1. Schematic of a Solar Wind Analyser’s Electron Analyser System (SWA-EAS) top-hat analyser Figurehead and 1. itsSchematic angular ﬁeldof a of Solar view. Wind (Left) Analyser’s The elevation Elec angletron is Analyser deﬁned as System the complement (SWA-EAS) of the top-hat angle analyserbetween head the particle and its velocity angular vector field of and view. the (zLeft-axis,) The perpendicular elevation angle to the is top-hat defined plane. as the The complement elevation ofangle the ofangle the electronsbetween the is resolved particle invelocity 16 electrostatic vector and uniform the z-axis, steps. perpendicular (Right) The to azimuth the top-hat angle plane. is the Theangle elevation within theangle projection of the electrons of the velocity is resolved vector in on16 theelectrostatic top-hat plane uniform and steps. the x-axis. (Right Both) The SWA-EAS azimuth angleanalyser is the heads angle resolve within the the azimuth projection direction of the velocity on MCP vector detectors on the using top-hat 32 sectors. plane and the x-axis. Both SWA-EAS analyser heads resolve the azimuth direction on MCP detectors using 32 sectors. 2.2. Synthetic Dataset 2.2. SyntheticWe use theDataset well-established forward-modelling technique (see, e.g., in [31–39]) to simulate the expectedWe use SWA-EAS the well-established observations inforward-modelling the solar wind. We tech focusnique on the(see, characterization e.g., in [31–39]) of to supra-thermal simulate the expected SWA-EAS observations in the solar wind. We focus on the characterization of supra-thermal electrons, which we consider more challenging due to the high energy tails of their VDFs. We model

Entropy 2020, 22, 103 4 of 14 electrons, which we consider more challenging due to the high energy tails of their VDFs. We model solar wind electrons with their velocities following the bi-kappa distribution function (see, e.g., in [3–6])

  κ 1 " #3/2 2 2 − −  ( ) (→ → )  n me Γ(κ + 1)  me u u0, me u u 0,  f (→u) = p 1 + k − k + ⊥ − ⊥  , (1) T T 2π(κ 3/2)kB Γ(κ 1/2)  2kB(κ 3/2)T 2kB(κ 3/2)T  k ⊥ − − − k − ⊥ where n is the electron number density, u , →u are the particle velocity parallel and perpendicular to k ⊥ the magnetic ﬁeld respectively and u0, and →u 0, are the corresponding bulk velocities. T , and T are k ⊥ k ⊥ the parallel and the perpendicular temperature, respectively, and κ is the kappa index. Finally, Γ is the gamma function, kB is the Boltzmann constant and me is the electron mass. We consider a simple set-up with →B in the top-hat plane of an individual SWA-EAS analyser head (magnetic ﬁeld elevation angle θB = 0◦) and the analyser head performing a single scan through kinetic energy E and azimuth Φ, for elevation Θ = θB = 0◦, with acquisition time ∆τ. The instrument records an expected number of counts,

2 C (E, Θ = θ = 0, Φ) = G f (E, Θ = θ , Φ)E2∆τ, (2) exp B 2 B me where ∆E G = A ∆Θ∆Φ, (3) eff E is the geometric factor of the instrument and Aeff is the effective area, which is a function of the instrument’s aperture and the detector’s efficiency. For this study, we assume a constant Aeff which results in a constant G, although in reality, both could be functions of energy, elevation and azimuth. For further simplification, we only consider cases in which the bulk velocity vector points towards the x-axis, so that the elevation and the azimuth angle of the bulk velocity vector θu,0 = ϕu,0 = 0◦. We also set the azimuth angle of the magnetic field to ϕB = 45◦. We note that in this specific set-up, the bulk speed components have the same magnitude →u 0, = →u 0, , and the azimuth sectors of the instrument k ⊥ measure the pitch-angles of the particles. We assume that the registered counts C follow the Poisson distribution: C Cexp Cexp P(C) = e− . (4) C! In Figure2, we show two examples of the synthetic datasets, based on electron plasma densities 3 3 of n = 5 cm− (left panel), and n = 50 cm− (right panel). For both examples, the bulk velocity of 1 magnitude u0 = 500 kms− points along the x-axis, and we set κ = 3, T = 10 eV, and T = 20 eV. k ⊥ In our approach we neglect effects related to the spacecraft potential. In reality, the spacecraft floats at a potential caused by interaction with the space environment and depending on the composition of materials that cover the surface of the spacecraft. This potential can be of order 10 V or more. Furthermore, UV photons striking the spacecraft produce photoelectrons that form a cloud around the spacecraft. Thus, the incoming solar wind electron population can be accelerated by a few eV and obscured by a high-density population of electrons at all energies below the spacecraft potential. Furthermore, it is possible for the VDF of the electrons to have an irregular shape in 3D. We ignore these effects here as we are interested in the high-energy tail of the electron distribution which should not be affected by these phenomena, and we assume gyrotropy, i.e., that the VDF is symmetrical in rotation around the magnetic field direction. Entropy 22 Entropy 20202020,, 21,, 103x 55 of of 1415

Figure 2. Modelled counts as a function energy and azimuth direction on the analyser’s head frame Figure 2. Modelled counts as a function3 energy and azimuth3 direction on the analyser’s head frame for (left) plasma density n = 5 cm− and (right) n = 50 cm− . For both examples, the magnetic field for (left) plasma density n = 5 cm−3 and (right) n = 50 cm−3. For both examples, the magnetic field vector vector (magenta) is in the top-hat plane (Θ = θB = 0◦) in azimuth direction Φ = 45◦. The bulk flow (magenta) is in the top-hat plane1 (Θ = θB = 0°) in azimuth direction Φ = 45°. The bulk flow of the of the electrons u0 = 500 kms− along the x-axis (Θ = Φ = 0◦). The parallel temperature T = 10 eV, electrons u0 = 500 kms−1 along the x-axis (Θ = Φ = 0°). The parallel temperature T = 10k eV, the the perpendicular temperature T = 20 eV, and the kappa index κ = 3.  ⊥ perpendicular temperature T⊥ = 20 eV, and the kappa index κ = 3. We derive the plasma properties by fitting our analytical model of the expected counts in Equation (2) toWe the derive observations. the plasma During properties the actual by fitting operations our analytical of SWA-EAS, model→B ofwill the be expected determined counts from in Solar Orbiter’s magnetometer instrument [30]. Therefore, in our fitting analysis, the parallel and Equation (2) to the observations. During the actual operations of SWA-EAS, B will be determined perpendicularfrom Solar Orbiter’s directions magnetometer are known instrument parameters. [30]. Moreover, Therefore, we reducein our fitting the unknown analysis, parameters the parallel of and the fittedperpendicular model by directions obtaining are the known plasma parameters. bulk velocity Mo vectorreover, from we protonreduce measurementsthe unknown parameters by the Proton of Alphathe fitted Sensor model (SWA-PAS by obtaining [29]) on the board plasma the samebulk spacecraft.velocity vector This from approach proton assumes measurements that protons by andthe electronsProton Alpha have theSensor same (SWA-PAS bulk velocity, [29]) which on board is generally the same true spacecraft. for plasma This with approach zero net chargeassumes and that no significantprotons and drifts electrons among have the the ion species.same bulk We velocity, note, however, which is that generally a more true sophisticated for plasma approach with zero could net leavecharge the and electron no significant bulk velocity drifts asamong a free the parameter ion spec toies. be We derived note, however, by the fitting that analysis. a more sophisticated We then use aapproach chi-squared could minimization leave the electron algorithm bulk which velocity defines as a the free optimal parameter combination to be derived of the undeterminedby the fitting plasma parameters (n, κ, T , and T ) that minimize the chi-squared value analysis. We then use a chi-squaredk ⊥ minimization algorithm which defines the optimal combination

of the undetermined plasma parameters (Nn, κ, T , and T⊥ ) that minimize2 the chi-squared value X Ci  Cexp,i(n0, κ0, T0, T0 ) 2 1  −  χ =  k ⊥  , (5) N R  σi  2 − i=1N − ′′κ ′′ 2 1 CCiiexp, (, n , T ,T)⊥ χ =  , (5) where N is the total number of data-pointsNR− we=  consider forσ fitting, R is the number of the model’s free i 1 i parameters, and σi is the standard deviation of each count Ci. The model we fit to the observations is a functionwhere N ofis then, κ total, T , number and T , of so data-pointsR = 4. In plasma we consider applications, for fitting, we R typically is the number use σ iof= the√C model’si, as within free k ⊥ aparameters, complete scan,and σ wei is obtainthe standard only one deviation measurement of each count for each Ci. specificThe model point we infit velocityto the observations space, and is we a assumefunction thatof n,it κ, represents T , and T⊥ the, so average R = 4. In value plasma and applications, the variance we of typically the expected use σ number= C , of asparticles. within a  ii For each measurement sample, we perform two fittings; one which excludes all points with C = 0, complete scan, we obtain only one measurement for each specific point in velocity space, andi we and one which includes all points with C = 0 in the analysis. We note that, in the second type of fitting, assume that it represents the average valuei and the variance of the expected number of particles. For we assign an uncertainty of σi = 1 to each Ci = 0 measurement (see, e.g., in [40]). In Figure3, we show each measurement sample, we perform two fittings; one which excludes all points with Ci = 0, and one an example of our synthetic dataset and the corresponding model fitted to it. (The software used in which includes all points with Ci = 0 in the analysis. We note that, in the second type of fitting, we assign this study is available for download at www.github.com/gnicolaou85/Elfit) an uncertainty of σi = 1 to each Ci = 0 measurement (see, e.g., in [40]). In Figure 3, we show an example of our synthetic dataset and the corresponding model fitted to it.

Entropy 20202020,, 2221,, 103x 66 of of 1415

Figure 3. (Left) Modelled counts as a function of energy and azimuth direction (instrument frame), Figure 3. (Left) Modelled3 counts1 as a function of energy and azimuth direction (instrument frame), using n = 20 cm− , u0 = 500 kms− towards the x-axis (Θ = Φ = 0◦), κ = 3, T = 10 eV, T = 20 eV, and a using n = 20 cm−3, u0 = 500 kms−1 towards the x-axis (Θ = Φ = 0°), κ = 3, T k= 10 eV, T⊥ = 20 eV, and a magnetic-field direction (magenta) in the top-hat plane (Θ = 0◦ and Φ = 45◦). (Right) Result of our fitmagnetic-field to the modelled direction observations. (magenta) The in the model top-hat finds plane the optimal(Θ = 0° and combination Φ = 45°). ( ofRightn, κ), ResultT , and ofT ourthat fit 2 k ⊥ minimizesto the modelled the χ observations.value (see text The for more).model finds the optimal combination of n, κ, T , and T⊥ that  2 3. Resultsminimizes the χ value (see text for more).

3. ResultsFor each set of input plasma conditions, we simulate 200 measurement samples which we fit to derive the electron density, temperature and kappa, as explained in Section2. By investigating the For each set of input plasma conditions, we simulate 200 measurement samples which we fit to histograms of the derived parameters, we verify that 200 samples are sufficient to derive statistically derive the electron density, temperature and kappa, as explained in Section 2. By investigating the significant results, especially within the low density (low particle flux) range. The histograms of the histograms of the derived parameters, we verify that 200 samples are sufficient to derive statistically derived parameters from the analysis of 200 samples with input parameters n = 7 cm 3, u = 500 kms 1 significant results, especially within the low density (low particle flux) range. The −histograms0 of the− pointing along the x-axis, κ = 3, T = 10 eV, and T = 20 eV, are shown in Figure4. Table1 shows derived parameters from the analysisk of 200 samples⊥ with input parameters n = 7 cm−3, u0 = 500 kms−1 their average values and standard deviations. For these input plasma conditions, the average derived pointing along the x-axis, κ = 3, T = 10 eV, and T⊥ = 20 eV, are shown in Figure 4. Table 1 shows plasma density is by ~23% lower than the input plasma density if the fitting includes Ci = 0, and by ~13%their average lower than values the and input standard density deviations. when Ci = For0 are these not includedinput plasma in the conditions, fit. The standard the average deviation derived of 3 i theplasma derived density densities is by is~23%σn ~0.2 lower cm than− for the both input fits. plasma On average, density the if fitting the fitting analysis includes that includes C = 0, andCi = by0 i overestimates~13% lower than the the kappa input index density by ~7%, when while C = the0 are fitting not included that excludes in theC ifit.= 0The underestimates standard deviation kappa byof σ −3 ~23the %.derived The standard densities deviations is n ~0.2 of cm the derived for both kappa fits. On indices average, are ~ the 0.2 andfitting 0.1 analysis respectively. that includes The average Ci = derived0 overestimatesT is by the ~6% kappa lower index than by the ~7 actual%, while value the when fittingC ithat= 0 excludes are included Ci = 0 in underestimates the fit, and by kappa ~25% k largerby ~23 than%. The the standard actual value deviations when C ofi = the0 are derived not included kappa indices in the fit.are The ~ 0.2 standard and 0.1 deviationrespectively. of T Theis k 0.3average eV when derivedCi = 0T are is fittedby ~6% and lower 0.7eV than when the Cactuali = 0 arevalue not when fitted. C Thei = 0 fitare that included includes in theCi = fit,0 derivesand by  accurately T within the standard deviation σT = 0.6 eV. The fit that excludes Ci = 0 from the fit ~25% larger ⊥than the actual value when Ci = 0 are⊥ not included in the fit. The standard deviation of overestimates T by ~19%, with standard deviation σT = 1 eV. T is 0.3 eV when Ci = 0 are fitted and 0.7 eV when Ci = 0 are not fitted. The fit that includes Ci = 0  ⊥ ⊥ 3 3 The solar orbiter expects densities between ~5 cm− and ~500 cm− . In Figure5, we show the T σT i derivedderives accurately parameters as⊥ functions within the of standard the plasma deviation density over⊥ = the 0.6 expected eV. The range.fit that Theexcludes red points C = 0 showfrom the averagefit (overoverestimates 200 samples) valuesT⊥ of the derived by parameters~19%, aswith determined standard by our fits excludingdeviation pointsσ T⊥ with = 1 eV.Ci = 0, and the blue points show our results for the fit analysis including points with Ci = 0. The shadowed regions represent the standard deviations of the corresponding values. The horizontal axis on the top shows the maximum number of the expected counts (peak value at an individual E, Φ) for the specific input parameters. On the lower side of the density (and counts) range, the plasma

Entropy 2020, 22, 103 7 of 14 density is underestimated in both ﬁtting strategies; however, it is more accurately determined by both Entropy 2020, 21, x 3 7 of 15 ﬁtting strategies when n > 50 cm− (Cmax > 100).

Figure 4. Histograms of (top left) density nout,(top right) kappa index κout,(bottom left) parallel Figure 4. Histograms of (top left) density nout, (top right) kappa index κout, (bottom left) parallel temperature T ,out and (bottom right) perpendicular temperature T ,out, as determined from the k ⊥ temperature T ,out and (bottom right) perpendicular temperature3 T⊥,out , as determined1 from the analysis of 200 measurement samples of plasma with n = 7 cm− , u0 = 500 kms− pointing along the xanalysis-axis, κ = of3, 200T =measurement10 eV and T samples= 20 eV. of The plasma blue histogramswith n = 7 correspondcm−3, u0 = 500 to kms values−1 pointing derived along by a ﬁt the that x- k ⊥ includesaxis, κ = points3, T = with 10 eVCi =and0, whileT⊥ =the 20 redeV. histogramsThe blue histograms represent corr valuesespond derived to values by a ﬁt derived that excludes by a fit  points with C = 0. that includesi points with Ci = 0, while the red histograms represent values derived by a fit that excludesTable points 1. withThe C inputi = 0. and the derived bulk parameters for the histograms in Figure4.

Table 1. The input and the derived bulk parameteFit rs for the histogramsFit in Figure 4. Parameter Input Including Excluding Ci = 0Fit Points Ci = 0 PointsFit Parametern (cm 3) Input7 Including 5.4 0.2 6.1 Excluding0.2 − ± ± κ 3Ci 3.2 = 0 points0.2 2.3 Ci0.1 = 0 points ± ± n (cmT-3) (eV) 7 10 9.4 5.4 ±0.3 0.2 12.5 0.7 6.1 ± 0.2 k ± ± T (eV) 20 19.8 0.6 23.7 1.0 κ ⊥ 3 3.2± ± 0.2 ± 2.3 ± 0.1 T (eV)  10 9.4 ± 0.3 12.5 ± 0.7 The kappa index is significantly misestimated (up to 35%) over the low input density range, T⊥ (eV) 20 19.8 ± 0.6 23.7 ± 1.0 3 if the analysis excludes points with Ci = 0. For instance, when n = 5 cm− , the derived index κout ~ 2. −3 −3 Interestingly,The solar for orbiter the same expects plasma, densitiesκout betweenis accurately ~5 cm determined and ~500 when cm . CIni =Figure0 measurements 5, we show arethe 3 includedderived parameters in our fit. However, as functions as the of density the plasma increases dens toityn over> 20 the cm −expected, the kappa range. index The is red more points accurately show determinedthe average when(over derived200 samples) by fits values excluding of the points derive withd parametersCi = 0. as determined by our fits excluding points with Ci = 0, and the blue points show our results for the fit analysis including points with Ci = 0. The shadowed regions represent the standard deviations of the corresponding values. The horizontal axis on the top shows the maximum number of the expected counts (peak value at an individual E, Φ) for the specific input parameters. On the lower side of the density (and counts) range, the plasma density is underestimated in both fitting strategies; however, it is more accurately determined by both fitting strategies when n > 50 cm−3 (Cmax > 100).

Entropy 2020, 21, x 8 of 15

The kappa index is significantly misestimated (up to 35%) over the low input density range, if the analysis excludes points with Ci = 0. For instance, when n = 5 cm−3, the derived index κout ~ 2. Interestingly, for the same plasma, κout is accurately determined when Ci = 0 measurements are Entropy 2020, 22, 103 8 of 14 included in our fit. However, as the density increases to n > 20cm−3, the kappa index is more accurately determined when derived by fits excluding points with Ci = 0. For thethe inputinput parametersparameters wewe examine,examine, thethe fittingfitting analysisanalysis thatthat excludesexcludes pointspoints withwith CCii == 00 3 significantlysignificantly overestimatesoverestimates the the plasma plasma temperature temperature in the in low-density the low-density range (nrange< 10 cm(n −< ).10 For cm instance,-3). For 3 wheninstance,n = when5 cm− n =(C 5max cm−~3 ( 10),Cmax the ~ 10), derived the derivedT ~ 17 T eV, ~which 17 eV, iswhich 1.7 times is 1.7 greater times greater than its than actual its actual input k  value, and the derived T ~ 30 eV, which is 1.5 times greater than its actual input value. For the same input value, and the derived⊥ T⊥ ~ 30 eV, which is 1.5 times greater than its actual input value. For plasma conditions, when the analysis includes points with Ci = 0, the derived temperatures do not the same plasma conditions, when the analysis includes points with Ci = 0, the derived temperatures deviate from their actual values by more than 4 %. Nevertheless, the two fitting analyses determine do not deviate from their actual values3 by more than 4 %. Nevertheless, the two fitting analyses similar temperatures for n > 10 cm− . determine similar temperatures for n > 10 cm−3.

Figure 5. (From top to bottom) The derived electron density over input density, kappa index, Figure 5. (From top to bottom) The derived electron density over input density, kappa index, parallel parallel and perpendicular temperature as functions of the input plasma density. The red points and perpendicular temperature as functions of the input plasma density. The red points represent the represent the mean values (over 200 samples) of the parameters derived by fitting only the measurements mean values (over 200 samples) of the parameters derived by fitting only the measurements with Ci with C 1. The blue points represent the mean values of the parameters derived by fitting to all ≥ 1. Thei ≥ blue points represent the mean values of the parameters derived by fitting to all measurements including those with Ci = 0. The shadowed regions represent the standard deviations of measurements including those with Ci = 0. The shadowed regions represent the standard deviations the derived parameters. of the derived parameters. . Finally, we examine the fit convergence as a function of the plasma parameters; the minimum χ2 definesFinally, the we best examine fit parameters. the fit convergence Previous studies as a function [27,33] showof the thatplasma the accurateparameters; estimation the minimum of the 2 plasmaχ defines temperature the best fit depends parameters. on the Previous accurate studies determination [27,33] show of the that kappa the index. accurate In Figure estimation6, we showof the 2Dplasma plots temperature of the χ2 value depends as a function on the accurate of the model determination parameters ofκ the, T kappa, and T index.. For In each Figure panel, 6, we show 2 k ⊥ 2D2 plots of the χ value as a function of the model parameters κ, T , and T⊥ . For each panel, we χ as a function of two parameters at a time, and we keep the remaining model parameters to their valuesshow χ as2 as determined a function by of thetwo best parameters fit. We perform at a time, our and calculations we keep forthe inputremaining plasmas model with parameters two different to 3 3 1 densities:their valuesn =as10 determined cm− , and byn =the50 best cm− fit.. InWe both perform examples, our calculations we use κ = for3, u input0 = 500 plasmas kms− withpointing two alongdifferent the densities: x-axis, T =n =10 10 eV, cm and−3, andT =n =20 50 eV cm as−3 the. In inputboth examples, parameters. we Red use areasκ = 3, onu0 = the 500 plots kms in−1 pointing Figure6 k ⊥ indicate χ2 > 1. For a fixed acquisition time ∆τ and fixed geometric factor G, higher densities result in higher counts and a smaller area of low χ2, which indicates that the derived parameters possess a smaller uncertainty. The non-axisymmetric shape of the area of low χ2, indicates an interdependency Entropy 2020, 21, x 9 of 15

along the x-axis, T = 10 eV, and T⊥ = 20 eV as the input parameters. Red areas on the plots in  EntropyFigure2020 6 indicate, 22, 103 χ2 > 1. For a fixed acquisition time Δτ and fixed geometric factor G, higher densities9 of 14 result in higher counts and a smaller area of low χ2, which indicates that the derived parameters possess a smaller uncertainty. The non-axisymmetric shape of the area of low χ2, indicates an 2 ofinterdependency the derived kappa of the index derived and thekappa temperatures. index and the More temperatures. speciﬁcally, More the area specifically, of low χ theshifts area towards of low higherχ2 shifts temperatures towards higher and temperatures gets broader alongand gets the broader vertical axisalong for the smaller verticalκ. axis for smaller κ.

Figure 6. 2D histograms of the χ2 value as a function of (top) the modelled κ and T and (bottom) Figure 6. 2D histograms of the χ2 value as a function of (top) the modelled κ and T andk (bottom) the3 the modelled κ and T , as derived for plasma with two diﬀerent input densities; (left) n = 10 cm− , ⊥ 3 1 andmodelled (right κ) nand= 50T⊥ cm, as− .derived In both for examples, plasma with we use twuo0 different= 500 kms input− pointing densities; along (left) the n =x-axis, 10 cm−κ3, =and3, T = 10 eV, and T = 20 eV as input parameters. (right) n = 50 cm−3. In both examples, we use u0 = 500 kms−1 pointing along the x-axis, κ = 3, T = 10 k ⊥ 

4. DiscussioneV, and T⊥ = 20 eV as input parameters. We examine the accuracy of plasma electron bulk parameters as derived from the analysis of the expected4. Discussion observations by SWA-EAS operating in burst-mode. Generally, the accuracy of the derived parametersWe examine is a function the accuracy of the of flux plasma of plasma electron particles bulk parameters (number of as recorded derived from counts). the analysis Our analysis of the showsexpected that observations the fit analysis by of SWA-EAS samples with operating a significant in burst-mode. amount of Generally, counts (Cmax the> accuracy30), derives of the accurately derived theparameters plasma parametersis a function when of the measurement flux of plasma points particles with zero(number counts of recorded (Ci = 0) are counts). excluded Our from analysis the analysis.shows that We the show fit analysis that when of analysingsamples with observations a significant with amount a low amountof counts of counts(Cmax > (30),Cmax derives< 30), theaccurately accuracy the of plasma some parameters parameters is when improved measurement if measurement points pointswith zero with counts zero counts (Ci = 0) are are included excluded in thefrom fitting the analysis.analysis. We show that when analysing observations with a low amount of counts (CmaxThe < 30), uncertainty the accuracy of of each some measurement parameters Cisi improvedis approximated if measurement with σi =points√Ci ,with assuming zero counts that theare σi 1 recordedincluded signalin the followsfitting analysis. the Poisson distribution. This results to a relative uncertainty C = which i √Ci becomes considerably large for small C and propagates significant statisticalσ = and systematic errors The uncertainty of each measurementi Ci is approximated with iiC , assuming that the in the derivation of the plasma bulk properties. Although Poisson statistics does not reallyσ apply to i 1 Crecordedi = 0, these signal bins follows indicate the points Poisson of velocity distributi spaceon. withThis lowresults fluxes to a which relative do uncertainty not reach the detection= threshold. This information is still useful, especially when the overall signal is weak. Ci C i whichIn becomes Figure7, weconsiderably show the distributionlarge for small of counts Ci and aspropagates a function significant of energy, statistical recorded inand the systematic azimuth 3 anode which contains the maximum number of counts Cmax, assuming a plasma with n = 5 cm− , κ = 3, T = 10 eV and T = 20 eV. The blue curves in both panels show the analytical model fitted to the data k ⊥ Entropy 2020, 21, x 10 of 15

errors in the derivation of the plasma bulk properties. Although Poisson statistics does not really apply to Ci = 0, these bins indicate points of velocity space with low fluxes which do not reach the detection threshold. This information is still useful, especially when the overall signal is weak. In Figure 7, we show the distribution of counts as a function of energy, recorded in the azimuth −3 Entropyanode2020 which, 22, 103 contains the maximum number of counts Cmax, assuming a plasma with n = 105 ofcm 14 , κ = 3, T = 10 eV and T⊥ = 20 eV. The blue curves in both panels show the analytical model fitted to  the data (fitted to the 2D distribution of counts) and the magenta curve indicates the expected number (fitted to the 2D distribution of counts) and the magenta curve indicates the expected number of counts. of counts. In the left panel, the fitting analysis excludes measurements with Ci = 0 (red points), In the left panel, the fitting analysis excludes measurements with Ci = 0 (red points), whereas in whereas in the right panel, the fitting analysis includes measurements with Ci = 0 with standard the right panel, the fitting analysis includes measurements with Ci = 0 with standard deviation σi deviation σi = 1. The kappa index of the fitted model in the left panel is ~2, and the distribution’s high- = 1. The kappa index of the fitted model in the left panel is ~2, and the distribution’s high-energy energy tail is prominent up to 2 keV, based on the inclusion of the non-zero data-points beyond ~400 tail is prominent up to 2 keV, based on the inclusion of the non-zero data-points beyond ~400 eV, eV, neglecting all points with Ci = 0 in between. On the other hand, the fit result in the right panel neglecting all points with Ci = 0 in between. On the other hand, the fit result in the right panel does does not have a prominent tail extending beyond 400 eV, as it fits all Ci = 0 points within that energy not have a prominent tail extending beyond 400 eV, as it fits all Ci = 0 points within that energy range. range. In this case, κout ~ 3 which is an accurate estimation of the actual kappa index of the modelled In this case, κ ~ 3 which is an accurate estimation of the actual kappa index of the modelled plasma. plasma. out

FigureFigure 7. 7. NumberNumber ofof countscounts asas aa functionfunction ofof energyenergy forfor the the pitch-angle pitch-angle with with the the maximum maximum flux flux 3 assuming a plasma with n = 5 cm −3, κ = 3, T = 10 eV and T = 20 eV. The blue line is the fitted model assuming a plasma with n = 5 cm− , κ = 3, T = 10 eV and T⊥ = 20 eV. The blue line is the fitted model k ⊥ to the observations by (left) excluding points with Ci = 0 which are shown with red colour, and (right) to the observations by (left) excluding points with Ci = 0 which are shown with red colour, and (right) including points with Ci = 0. The magenta line is the expected counts Cexp, given by Equation (2). The including points with Ci = 0. The magenta line is the expected counts Cexp, given by Equation (2). The labels in each panel show the parameters as derived by the corresponding fit. labels in each panel show the parameters as derived by the corresponding fit. As we show in Figures5 and7, both fitting strategies underestimate the plasma density when the As we show in Figures3 5 and 7, both fitting strategies underestimate the plasma density when input density is n < 50 cm− . This is partially due to the asymmetry that characterizes the Poisson the input density is n < 50 cm−3. This is partially due to the asymmetry that characterizes the Poisson distributions of low counts. In Figure8, we show the Poisson distributions for Cexp = 1, Cexp = 3 distributions of low counts. In Figure 8, we show the Poisson distributions for Cexp = 1, Cexp = 3 and and Cexp = 5. Each distribution has two modes (most frequent values); the higher mode which is Cexp = 5. Each distribution has two modes (most frequent values); the higher mode which is equal to equal to the average value of the distribution Cexp, and the lower mode = Cexp 1. The relative the average value of the distribution Cexp, and the lower mode = Cexp−1. The −relative difference difference between the two modes increases with decreasing Cexp. Samples drawn from a Poisson between the two modes increases with decreasing Cexp. Samples drawn from a Poisson distribution distribution with a small Cexp, as is the case at low densities expected to be measured by SWA-EAS atwith ~1 au,a small will C moreexp, as likely is the undersamplecase at low densities the distribution expected thanto be oversample, measured by and SWA-EAS so densities at ~1 willau, will be underestimated.more likely undersample In addition, the within distribution a measurement than over cycle,sample, each and point so ofdensities velocity will space be isunderestimated. sampled only In addition, within a measurement cycle, each point of velocity space is sampled only once. In our once. In our analysis, we consider that an individual Ci obtained at a specific point of velocity space is analysis, we consider that an individual Ci obtained at a specific point of velocity space is representative of the average value Cexp with uncertainty σi = √Ci. This introduces an additional σ = systematicrepresentative error. of We the illustrate average this value error C byexp consideringwith uncertainty an exemplariiC Poisson. This distribution introduces withan additionalCexp = 5. Ifsystematic we obtain oneerror. measurement, We illustrate the this probabilities error by for considering observing Cani = exemplar1 and Ci = Poisson9 are almost distribution the same (seewith FigureCexp =8 5.). If However, we obtain in one our measuremen fitting analysis,t, the the probabilities corresponding for uncertainties observing Ci for= 1 andCi = C1i and= 9 areCi =almost9, are theσi 2 = 1 and σi = 3, respectively, and a χ minimization model fitted to these two points will shift towards 2 = 1 = Ci = 1, as the specific point has a bigger weight σi− Ci− 1. This specific systematic error purely depends on the statistical uncertainty of the measurements. We prove this by fitting the mean values of the Poisson distribution (or higher mode value) with both fitting strategies (Figure9). For the specific example, we consider plasma with the same bulk properties as in the example shown in Figure7 and Entropy 2020, 21, x 11 of 15

same (see Figure 8). However, in our fitting analysis, the corresponding uncertainties for Ci = 1 and Ci = 9, are σi = 1 and σi = 3, respectively, and a χ2 minimization model fitted to these two points will Entropy 2020, 22, 103 11 of 14 σ −−21== shift towards Ci = 1, as the specific point has a bigger weight iiC 1. This specific systematic error purely depends on the statistical uncertainty of the measurements. We prove this by fitting the we show thatmean in values the absence of the Poisson of statistical distribution ﬂuctuations, (or higher bothmode ﬁtsvalue) derive with identicalboth fitting results. strategies The orange (Figure 9). For the specific example, we consider plasma with the same bulk properties as in the curve in both panels is the lower mode of the Poisson distribution Cexp 1, which corresponds to a example shown in Figure 7 and we show that in the absence of statistical fluctuations,− both fits derive distributionidentical with lowerresults. The density. orange In curve reality, in both the panels errors is the associated lower modewith of the thePoisson statistical distribution uncertainty Cexp−1, of the measurementswhich decrease corresponds with to a increasing distribution number with lower of countsdensity. andIn reality,/or when the errors analysing associated average with the counts over multiple samplesstatistical ofuncertainty the same of plasma.the measurements decrease with increasing number of counts and/or when analysing average counts over multiple samples of the same plasma.

Figure 8. PoissonFigure 8. distributionPoisson distribution with averagewith average value value (top (top) C)exp Cexp= =1, 1, (middle(middle) )CCexpexp = 3,= and3, and (bottom (bottom) ) Cexp = 5. The verticalCexp = 5. The lines vertical indicate lines theindicate two the modes two modes of the of the distribution, distribution, CCexpexp (blue)(blue) and andCexp-1C (orange)exp 1 (orange) − respectively.respectively. For small For average small average values, values, the the Poisson Poisson distributiondistribution is asymmetric, is asymmetric, and the and probability the probability to to measure numbermeasure number of counts of counts lower lower than than the the average average valuevalue isis significant. signiﬁcant. This can This bias can thebias results the to results to lower densities. lowerEntropy densities. 2020, 21, x 12 of 15

Figure 9. NumberFigure 9. Number of the expectedof the expected average average counts counts CCexpexp asas a function a function of energy of energy in the pitch-angle in the pitch-angle bin bin with the maximumwith the maximum particle particle ﬂux, flux, considering considering thethe same same plasma plasma conditions conditions as in the as example in the shown example in shown Figure 7. The blue curve is the model fitted to the observations by (left) excluding points with Ci = 0 in Figure7. The blue curve is the model ﬁtted to the observations by ( left) excluding points with Ci which are shown with red colour, and (right) including points with Ci =0. The orange curve is the

= 0 whichmode are shown Cexp-1. Inwith each panel, red colour,we show andthe parameters (right) including as derived by points the corresponding with Ci = fit.0. In The the absence orange curve is the mode Cofexp statistical1. In fluctuations, each panel, both we fitting show strategies the parameters derive identical as bulk derived parameters. by the corresponding fit. In the − absence of statistical fluctuations, both fitting strategies derive identical bulk parameters. 5. Conclusions We investigate the accuracy of the electron plasma bulk parameters as derived by our analysis of SWA-EAS high-time resolution measurements. We assume electrons with velocity distribution function following the anisotropic bi-kappa distribution function and simulate the expected counts in the instrument’s frame based on a realistic instrument model. We derive the plasma bulk parameters by fitting the observations with an analytical model, using the χ2 minimization method. Our analysis shows the following.

• The fit analysis of plasma measurements with relatively high flux (Cmax > 30) estimates the plasma temperature and kappa index more accurately if it excludes measurement points with Ci = 0. The corresponding analysis of measurements with low particle flux (Cmax < 30) estimates the temperature and kappa index more accurately if it includes measurement points with Ci = 0. Although Ci = 0 is a measurement with a large uncertainty, it contains information that becomes useful when the overall signal is weak. • Examination of the fit convergence indicates that the determination of the plasma temperature and the determination of the kappa index are interdependent. As expected, the uncertainty of the derived parameters decreases with increasing particle flux. • The plasma density is underestimated when the particle flux is low (Cmax < 100). We show that the misestimation is due to the asymmetry of the Poisson distribution and the assigned uncertainties to the data points. Our study predicts the accuracy of the future SWA-EAS measurements. However, our methods can be applied to investigate and/or improve the accuracy of similar fitting analyses which determine kappa distribution functions. Although this paper uses the kappa distribution to describe the VDFs of plasma particles, the kappa distribution function is used to describe several other mechanisms as well (e.g., the angular scattering of particles passing through graphene and carbon foils [41,42]).

Author Contributions: Conceptualisation, G.N., R.T.W, G.L., D.V., C.J.O. and D.O.K; methodology, G.N., R.T.W., G.L. and D.V.; software, G.N.; validation, G.N., R.T.W., G.L. and D.V.; formal analysis, G.N.;

Entropy 2020, 22, 103 12 of 14

5. Conclusions We investigate the accuracy of the electron plasma bulk parameters as derived by our analysis of SWA-EAS high-time resolution measurements. We assume electrons with velocity distribution function following the anisotropic bi-kappa distribution function and simulate the expected counts in the instrument’s frame based on a realistic instrument model. We derive the plasma bulk parameters by ﬁtting the observations with an analytical model, using the χ2 minimization method. Our analysis shows the following.

The fit analysis of plasma measurements with relatively high flux (Cmax > 30) estimates the plasma • temperature and kappa index more accurately if it excludes measurement points with Ci = 0. The corresponding analysis of measurements with low particle flux (Cmax < 30) estimates the temperature and kappa index more accurately if it includes measurement points with Ci = 0. Although Ci = 0 is a measurement with a large uncertainty, it contains information that becomes useful when the overall signal is weak. Examination of the fit convergence indicates that the determination of the plasma temperature • and the determination of the kappa index are interdependent. As expected, the uncertainty of the derived parameters decreases with increasing particle flux.

The plasma density is underestimated when the particle ﬂux is low (Cmax < 100). We show that the • misestimation is due to the asymmetry of the Poisson distribution and the assigned uncertainties to the data points. Our study predicts the accuracy of the future SWA-EAS measurements. However, our methods can be applied to investigate and/or improve the accuracy of similar ﬁtting analyses which determine kappa distribution functions. Although this paper uses the kappa distribution to describe the VDFs of plasma particles, the kappa distribution function is used to describe several other mechanisms as well (e.g., the angular scattering of particles passing through graphene and carbon foils [41,42]).

Author Contributions: Conceptualisation, G.N., R.W., G.L., D.V., C.O. and D.K.; methodology, G.N., R.W., G.L. and D.V.; software, G.N.; validation, G.N., R.W., G.L. and D.V.; formal analysis, G.N.; investigation, G.N., R.W., G.L., D.V., C.O. and D.K.; resources, G.N., R.W., G.L., D.V., C.O. and D.K.; data curation, G.N.; writing—original draft preparation, G.N.; writing—review and editing, G.N., R.W., G.L., D.V., C.O. and D.K.; visualisation, G.N.; supervision, R.W., D.V. and C.O.; project administration, R.W. and C.O.; funding acquisition, R.W. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: G.N., R.W. and D.V. are supported by the STFC Consolidated Grant to UCL/MSSL, ST/S000240/1. G.L. is supported by Nasa’s projects NNX17AB74G and 80NSSC18K0520. D.V. is supported by the STFC Ernest Rutherford Fellowship ST/P003826/1. Conﬂicts of Interest: The authors declare no conﬂicts of interest.

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