On the Relation between Kappa Distribution Functions and the Plasma Beta Parameter in the Earth Magnetosphere: THEMIS observations
Adetayo V. Eyelade, Marina Stepanova, Cristobal M. Espinoza, Pablo S. Moya
aa r X i v : . [ phy s i c s . s p ace - ph ] J a n Draft version January 20, 2021
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On the Relation between Kappa Distribution Functions and the Plasma Beta Parameterin the Earth Magnetosphere: THEMIS observations.
Adetayo V. Eyelade, Marina Stepanova, Crist´obal M. Espinoza, and Pablo S. Moya Departmento de F´ısica, Universidad de Santiago de Chile (USACH), Santiago, Chile Departmento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile (Received January 20, 2021)
Submitted to ApJSSABSTRACTThe Earth’s magnetosphere represents a natural plasma laboratory that allows us to study thebehavior of particle distribution functions in the absence of Coulomb collisions, typically described bythe Kappa distributions. We have investigated the properties of these functions for ions and electronsin different magnetospheric regions, thereby making it possible to reveal the κ -parameters for a widerange of plasma beta ( β ) values (from 10 − to 10 ). This was done using simultaneous ion and electronmeasurements from the five Time History of Events and Macroscale Interactions during Substorms(THEMIS) spacecraft spanning the years 2008 to 2018. It was found that for a fixed plasma β , the κ -index and core energy ( E c ) of the distribution can be modeled by the power-law κ = AE γc for bothspecies, and the relation between β , κ , and E c is much more complex than earlier reported: both A and γ exhibit systematic dependencies with β . Our results indicate that β ∼ . − . β . For β >
1, both A and γ take nearly constant values, a featurethat is especially notable for the electrons and might be related to their demagnetization. The relationbetween β , κ , and E c that we present is an important result that can be used by theoretical models inthe future. Keywords:
Interplanetary medium (825), Planetary magnetosphere (997), Plasma astrophysics (1261),Plasma physics (2089), Space plasmas (1544) INTRODUCTIONUnderstanding the dynamics of charged energetic par-ticle interactions in space and astrophysical plasmas hasbeen one of the main challenges in the space physicscommunity over several decades. Owing to the lackof adequate collisions, these plasmas are usually ob-served in quasi-equilibrium stationary states differentfrom thermodynamic equilibrium. The kinetics of therelaxation process needed to reach a stationary stateand the properties of such state in which the plasmaand electromagnetic turbulence coexist, are still notwell understood for a long list of space and astrophysi-
Corresponding author: Adetayo V. [email protected] cal objects (Marsch 2006; Bruno & Carbone 2013; Yoon2017). These unsolved problems border on a lack of suf-ficient understanding of the interaction of charged par-ticles in plasma environments, such as the solar windand Earth’s magnetosphere, which are essentially col-lisionless plasma systems in non-equilibrium stationarystates.Over half a century since the introduction ofthe Kappa distribution function by Montgomery et al.(1965), several studies have shown that the prop-erties of collisionless plasmas can be well mod-eled by distributions with enhanced suprathermalpower-law tails, rather than Maxwellian distribu-tions. The Kappa distributions play a crucialrole in the description of plasma objects such asthe solar wind (Collier et al. 1996; Pierrard et al.1999; Mann et al. 2002; Livadiotis & McComas 2011;
Eyelade et al
Yoon 2014; Pierrard & Pieters 2014), the Earth’smagnetosheath (Vasyliunas 1968; Ogasawara et al.2013, 2015), and several regions in the mag-netosphere like the magnetotail (Grabbe 2000),the ring current (Pisarenko et al. 2002), and theplasma sheet (Christon et al. 1988; Kletzing et al. 2003;Stepanova & Antonova 2015; Espinoza et al. 2018;Kirpichev & Antonova 2020).The general form of the Kappa distribution is denotedby f : f ( E ; n α , κ α , E c α ) = n α (cid:16) m α πE cα (cid:17) Γ( κ α )Γ( κ α − ) √ κ α × h Eκ α E cα i − κ α − , (1)where f is the phase space density, E is the kinetic en-ergy, the sub-index α corresponds to the particle index,which can be electron (e) or ion (i); n α is the particledensity, m α is the particle mass, Γ is the Euler gammafunction, and κ α and E c α are the κ -parameter and char-acteristic or core energy, respectively. Kappa distribu-tions given by Equation (1) exhibit a thermal core withcharacteristic energy E c α and suprathermal tails, suchthat the total characteristic particle kinetic energy E total is given by E total = E cα κ α κ α − / , (2)which enables a straightforward comparison betweenKappa and Maxwellian distributions, and to outline theeffects of suprathermals as shown by Lazar et al. (2015,2016). The spectral index κ α is a measure of the slopeof the energy spectrum of the suprathermal particlesthat form the tail of the velocity distribution function.Hence κ α primarily provides a measure of the depar-ture of the stationary states from thermal equilibrium(Burlaga & F.-Vi˜nas 2005). For κ α → ∞ , equation (1)becomes identical to the Maxwell distribution and ap-proaches the quasi-thermal core of the observed distri-bution. f ( E ; n α , E c α ) = n α (cid:18) m α πE c α (cid:19) / exp (cid:18) − EE c α (cid:19) (3)In the solar wind, several mechanisms have been re-ported in the literature that lead to the generation ofKappa distributions (Livadiotis et al. 2018). For in-stance, Lazar et al. (2017) observed the presence ofsuprathermal electron fluxes, which can be well mod-eled by Kappa distributions. They argued that the useof Kappa and bi-Kappa distributions allows a realisticinterpretation of non-thermal electrons and their effectson the electron firehose instability, where growth rates are observed to increase while the instability thresholdsand electron kappa ( κ e ) are seen to decrease.The study by Maksimovic et al. (1997) describes thefirst exospheric model of the solar wind based on KappaVelocity Distribution Functions for protons and elec-trons escaping from the corona. Their model provides apossible hint regarding key features of the solar windflow. It indicates that the fastest solar wind flowsthat originate from the corona holes are high-speedstreams with an enhanced high-velocity tail simulatedby a Kappa function with a small electron kappa κ e value. Whereas, for hot equatorial regions where theslow solar wind originates, the electron velocity distribu-tion functions are closer to the Maxwellian equilibrium,corresponding to κ e = ∞ .More recently, a significant relationship has beenfound between the κ -index and other plasma param-eters. For instance, it was established that κ -indexcorrelates with the solar wind density and tempera-ture (Livadiotis et al. 2018). Besides, the κ -index wasalso found to be connected with the polytropic in-dex and magnetic field (Livadiotis 2017). In addition,Livadiotis et al. (2018) remarked that the κ -index de-creases when the magnetic field’s long-range interac-tions induce correlations among particles, as the systemis turned away from thermal equilibrium. They founda strong correlation between the plasma β parameter,(defined as (cid:0) p/B / µ (cid:1) , where p is the plasma pres-sure, B is the magnetic field, and µ is the magneticpermeability) and kappa. This correlation takes placewhen β increases, as thermal pressure becomes domi-nant. Similarly, as the κ -index increases, the long-rangeinteractions due to the magnetic field become weaker.This observation further revealed that kappa regulationin the low beta regime is due to the magnetic field, whichinduces the correlation between particles.Furthermore, there are many studies in some specificregions of the Earth magnetosphere that utilized Kappadistributions. For instance, Christon et al. (1989, 1991)obtained ion and electron Kappa distributions in theplasma sheet using the particle instruments onboardthe International Sun-Earth Explorer 1 (ISEE 1). Theyfound that the κ -index ranges between 4 and 8 for bothions and electrons, with a most probable value between5 and 6, which shows that the spectral shape is dis-tinctly non-Maxwellian. Later, Haaland et al. (2010)found that the κ -index ranges between 3 and 6, usingdata of the Cluster satellites. Stepanova & Antonova(2015) utilized Kappa distributions to fit ion and elec-tron flux spectra for five events in which the THEMISsatellites were aligned along the plasma sheet. Theyobtained snapshots of kappa properties that show a ten- n the Relation between κ and β κ -index to increase in the tailward direc-tion. Espinoza et al. (2018) also used the Kappa distri-bution to model ions and electrons flux spectra alongthe plasma sheet. Their results reveal that κ i > κ e ,which suggests that non-thermal properties of the elec-trons are stronger than ions. Besides, their results showa persistent dawn-dusk asymmetry in the relative num-bers of energetic ions, which increases during substorms.This is consistent with the previous study of Wing et al.(2005).Recently, Kirpichev & Antonova (2020) measured the κ -parameters for ions in different magnetospheric re-gions and during quiet magnetospheric conditions. Theyfound that kappa depends on the core energy ( E c ) for awide energy range and a broad range of the plasma beta( β ) parameter. Their results support earlier findings,which showed that κ -index increases with E c in the mag-netosphere of the Earth (Christon et al. 1989) and thesolar wind (Collier 1999). However, despite the afore-mentioned studies, there is no systematic experimentalanalysis that focuses on the coupling between Kappa dis-tribution parameters (density, core energy, and κ -index)and the plasma beta β parameter in the Earth’s mag-netosphere. This study, for the first time, considers si-multaneous electron and ion measurements, we will ex-plore more precisely the relationship between κ -index,core energy E c , and plasma beta β , which will providea better understanding of plasma thermalization, therelation between ion and electron properties, and theimportance of the level of magnetization of each specieswithin the Earth’s magnetosphere, regardless of the de-tails of a given magnetospheric region.The paper is organized as follows: In section 2 wedescribe the data and methodology for obtaining the ionand electron Kappa distribution parameters and plasmabeta; In section 3 we present the results of the analysesand explore the relationship between kappa, beta, andcore energy E c . In section 4 we discuss the observedresults, and in section 5 we summarize and concludeour findings. INSTRUMENTATION AND DATA ANALYSISThe present study combines data sets of the multi-satellite mission Time History of Events and MacroscaleInteractions during Substorms (THEMIS), using all itssatellites (TH-A, TH-B, TH-C, TH-D, and TH-E), andspanning the years 2008 to 2018. The data was down-loaded via the THEMIS ftp website . All measurementswere constrained to the following Geocentric Solar Mag- http://themis.ssl.berkeley.edu/index.shtml -40-30-20-10010 X GSM (R E ) -30-20-100102030 Y G S M ( R E ) -2.5-2-1.5-1-0.500.511.5 -40-30-20-10010 X GSM (R E ) -30-20-100102030 Z G S M ( R E ) -2.5-2-1.5-1-0.500.511.5 Figure 1.
The spatial coverage of the measured spectrafor which both electron and ion Kappa fits were successful.The color code is used to represent the value of the electronplasma β e . Upper panel: X and Y gsm plane. Lower panel: X and Z gsm plane. netospheric (GSM) coordinate system: − ≤ X ≤ E , − ≤ Y ≤
30 R E , − ≤ Z ≤
10 R E , and atdistances larger than 5 R E from the center of the Earth.This region is depicted in Figure 1, where panels (a)and (b) show the spatial coverage in the X − Y GSM and X − Z GSM planes, respectively. Each position iscolor-coded with the measured electron plasma beta( β e ). All measurements used in this study were aver-aged over 12 minute long intervals, which is long enoughto make stable measurements of the particle fluxes, andat the same time, short enough to ensure that the dis-tributions do not change significantly over this time(Stepanova & Antonova 2015; Espinoza et al. 2018).The magnetic field data used in this study were ob-tained from the Flux Gate Magnetometer (FGM) on-board the THEMIS satellites (Auster et al. 2008). Theplasma particle data were obtained from the Electro-static Analyzers (ESA McFadden et al. 2008), with anenergy range from a few eV up to 30 keV for electronsand 25 keV for ions, and the Solid State Telescopes (SSTAngelopoulos 2008) with an energy range from 25 keVto 6 MeV. Further, we used level 2 full mode particleenergy fluxes, which are averaged over the particle in- Eyelade et al struments satellite rotation thereby significantly increas-ing the statistical measurements for each energy chan-nel. Magnetospheric plasmas are composed mainly ofprotons and electrons but we should also expect a lowfraction of heavy ions. Unfortunately, THEMIS particleinstruments do not distinguish protons from other ionspecies, hence, we refer to all of them as ions.In our study, only the central energies of the parti-cle instruments combined range were considered for fit-ting. This was done to avoid contamination from thespacecraft potential and photoelectrons at low energies( <
40 eV) and cosmic rays and low statistics at high en-ergies. The analyses were limited to the ranges 1 .
75 to210 keV for ions, and 0 .
36 and 203 . F , as shown below: F α ( E ) = n α √ π m α E E / c α Γ( κ α )Γ( κ α − ) √ κ α (cid:20) EκE c α (cid:21) − κ α − (4)Figure 2 illustrates examples of ion (upper panels) andelectron (bottom panels) energy flux spectra measuredby combining both particle instruments (ESA and SST)onboard the THEMIS satellites (solid lines). The circleson the plots are the average of the spectra obtained forthe 12 minutes time windows. The open circles repre-sent measurements from the ESA, while the filled circlesrepresent the SST. The error bars for the averaged fluxdata represent the spread between the maximum andminimum observed values. They were found to vary sig-nificantly between the ESA and SST data, so they werenormalized in the same way as Espinoza et al. (2018).The inverse squared of the error bars is used to de-fine weights for the fits, which were performed usinga non-linear least-squares method combined with theLevenberg-Mardquart algorithm.We visually inspected hundreds of spectra and decidedto work only with the fits that give a reduced chi-squared χ < Table 1.
Criteria applied to select magnetosphericplasmas. Parameter Symbol ConditionPlasma ion density (cm − ) n i > . B x − ) v T be well fitted by a Kappa distribution function for bothions and electrons. The final data set, analyzed in thenext section, includes the three kappa parameters fromthe fits and the plasma beta for ions and electrons.Tables 2 and 3 give some statistical properties of thedistributions of all the beta and kappa parameters ob-tained from the fits for ions and electrons, respectively.The data set was also divided into two groups, depend-ing on beta, and the same quantities are given for eachof these groups ( β < β > β e versus β i , κ e versus κ i , and E c e versus E c i , respectively. As expected, β i and β e are correlated since the plasma on large scalesis quasi-neutral, and the majority of the data we haveare from the plasma sheet and from the region that sur-rounds the Earth, which is filled with plasma sheet-likeplasma. In this case we expect ion temperatures to behigher than electron temperatures, with typical ion-to-electron temperature ratios T i /T e between 4 and 6 ac-cording to Baumjohann (1993); Borovsky et al. (1997);Espinoza et al. (2018).Further, the relation between κ e and κ i that weobserve is uncorrelated, which is consistent with low-statistic results by Christon et al. (1989). The correla-tion we find between E ce and E ci is rather low, and lessthan what was obtained by Christon et al. (1989). Thiscould be related to the fact that we did not limit ourstudy to the plasma sheet only. Nevertheless, these re-sults suggest that a more detailed analysis is needed toestablish whether there is a relation between β , κ , and E c , which is done in the next section. RELATIONSHIP BETWEEN BETA ( β ), KAPPA( κ ) AND CORE ENERGY ( E C )To assess the relation between beta, kappa, and coreenergy, we define a grid in the logarithmic ( β, κ ) spaceusing a cell size of ∆ log β = ∆ log κ i = 0 .
1. Thegrid is defined for the range -3 < log β < < log κ < N ), where emptybins contain less than ten measurements. n the Relation between κ and β -4 -2 E i , keV I on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THAk i =9.3 2.9E ci =10.7 5.3 keVn i =0.2 0.1 cm -3 (a) -4 -2 E i , keV I on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THDk i =10.9 7.9E ci =9.2 11.1 keVn i =0.1 0.3 cm -3 (b) -4 -2 E i , keV I on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THEk i =9.9 3.9E ci =10.1 6.2 keVn i =0.1 0.1 cm -3 (c) -3 -2 -1 E e , keV E l e c t r on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THAk e =3.0 0.1E ce =1.5 0.2 keVn e =0.1 0.0 cm -3 (d) -3 -2 -1 E e , keV E l e c t r on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THDk e =3.1 0.1E ce =1.4 0.2 keVn e =0.2 0.1 cm -3 (e) -3 -2 -1 E e , keV E l e c t r on E ne r g y F l u x , k e V / ( s c m s t e r k e V ) THEk e =3.7 0.2E ce =1.1 0.2 keVn e =0.1 0.0 cm -3 (f) Figure 2.
Examples of fits of Kappa distributions to the flux spectra of ions (a-c) and electrons (d-f), which are simultaneouslymeasured on 29th March 2008 between 10.4 UT and 10.5 UT by THA (first column), THD (second column), and THE (thirdcolumn). In each panel, the black solid lines are sub-datasets taken over a total of 12 minutes. Open circles represent averagesof the subsets measurements of low energy particles (ESA) and filled circles represent averages of the subsets of measurementsof high energy particles (SST). The red dash-line is the kappa function curve fitted to the open and filled circles.
The observed ion and electron kappa indices vary from1.5 (lowest possible state) to a little above 40, and theobserved beta values are from 10 − to nearly 10 (Tables2 and 3). The distribution of the observations in thisspace, however, presents a clear maximum, for both ionsand electrons. While for the ions, most cases are in therange 5 ≤ κ i ≤ . ≤ β ≤ .
7, the electronsexhibit slightly smaller kappa and beta values in general:4 ≤ κ e ≤ . ≤ β e ≤ . β i < .
01 and κ i >
10, or plasmas with β e >
30 and κ e > β ∼
1. For β >
1, the mean (and median) kappais almost constant, with a slight decrease towards largerbeta values. This feature is similar to the result obtainedby Kirpichev & Antonova (2020) (see their Figure 3) inthe case of ions, but with smaller kappa values. The core energies obtained from the considered fitscover the range 0 .
25 keV to 15 .
25 keV in the case of ions,and 0 .
25 keV to 9 .
25 keV for the electrons. In order tostudy the relation between κ and E c for a fixed β , itis necessary to obtain the most representative E c foreach ( β, κ ) bin. A popular candidate would be the meanvalue, which works well in the case of normal distribu-tions. In order to determine the degree to which theydepart from a normal distribution, we analyze the distri-bution of core energies in each bin. As observed in Fig-ures 5(b) and 6(b), the obtained distributions are notalways Gaussian. To characterize this, we determinedthe mean, median, and skewness for the E c distributionin each ( β, κ ) bin. Figures 5 and 6 show the mean E c values and the skewness for ions and electrons. As seenin both species, the hot plasma is negatively skewed;meanwhile, the cold plasma is positively skewed. There-fore, to ensure the robustness of our study, the analyseswere performed twice: once for the mean E c values andonce for the median E c values. We can confirm that theresults are qualitatively very similar for both cases. Me-dian and also mode E c values as a function of κ and β , Eyelade et al
Table 2.
Statistics of ion β and kappa parameters.Parameter Minimum Maximum Mean Median Q Q σ For all β i : 47,058 spectra κ i E c i E Ti n i S k i -1.60 4.09 0.80 0.73 0.01 1.50 0.96 β i β i regime, β i
1: 38,708 spectra κ i E c i E Ti n i S k i -1.59 4.09 0.82 0.80 0 1.65 1.06 β i β i regime, β i >
1: 8,350 spectra κ i E c i E Ti n i S k i -0.42 3.42 0.78 0.71 0.26 1.16 0.72 β i Note — Energies are given in keV, and densities in cm − . E Ti is the iontotal energy, S k i is the skewness of the ion E c distribution. Q , Q , and σ represent the lower quartile, upper quartile, and standard deviation of eachquantity, respectively. for both species, can be seen in the Appendix (Figures12 and 13).Previous studies of Kappa distributions for ions havereported that kappa increases with core energy in a lin-ear fashion (Christon et al. 1989; Collier 1999). Mean-while, Kirpichev & Antonova (2020) has recently estab-lished that a power-law function of the form κ = AE γ can be used to describe the relationship between kappaand core energy in the case of ions. Here we use thesame function as Kirpichev & Antonova (2020) for bothspecies.Examples of kappa versus core energy, for some se-lected beta values, are shown in Figure 7, where theleft column (Panels (a),(c),(e), and (g)) show the re-sults for ions, and the right column (Panels (b), (d), (f),and (h)) for electrons. The horizontal error bars corre-spond to the standard deviation of the core energy E c in each ( κ, β ) bin. The fits to log-log data have linearcorrelation coefficients R > .
7. Thus we conclude thatunder fixed β conditions, the κ -index increases with E c for both ions and electrons, and that they relate via a power-law. Table 4 details the results of these fits for afew selected beta values. In order to establish whetherthere is a dependence of A and γ on β we plot all ob-tained fitting coefficients as shown in Figure 8. It wasfound that for both species, the relation between A andbeta has a clear minimum near β = 0 .
1, and that issymmetric with respect to it up to at least β = 1. Asimilar relation is observed for γ , which exhibits a max-imum at approximately the same value of β . To char-acterise this behaviour we use an empirical relation be-tween the power-law coefficients A or γ and β of theform a | log ( β/β ) | + b , where β is the location of theextremum. This function was fitted to A and γ dataaround the minimum or maximum (as examples, thedata used for the fits in the cases shown in Figure 8 areplotted with filled circles). The results of these fits canbe found in Table 5. In both cases, we find that β ∼ . A and γ , for ionsand electrons.As we have mentioned in section 1, the total charac-teristic particle kinetic energy given by Equation (2), n the Relation between κ and β Table 3.
Statistics for electron β and kappa parameters.Parameter Minimum Maximum Mean Median Q Q σ For all β e : 47,058 spectra κ e E c e E Te n e S k e -0.98 5.07 1.24 1.07 0.38 1.97 1.09 β e β e regime, β e
1: 44,774 spectra κ e E c e E Te n e S k e -0.98 5.07 1.41 1.35 0.53 2.10 1.13 β e β e regime, β e >
1: 2284 spectra κ e E c e E Te n e S k e -0.47 2.64 0.71 0.59 0 1.18 0.71 β e Note —Energies are given in keV, and densities in cm − . E Te is the electrontotal energy, S k e is the skewness of the electron E c distribution. Q , Q ,and σ represent the lower quartile, upper quartile, and standard deviationof each quantity, respectively. enables a straightforward comparison between Kappaand Maxwellian distributions, as it also considers theeffect of the suprathermal within the Kappa distribu-tion model. In order to see how the inclusion of thiseffect may alter our conclusions, we have repeated ouranalysis using E total instead of E c . Comparison of Fig.9 with Figs. 5 and 6 shows that for every combina-tion between β and κ , h E total i is larger than h E c i (seealso Tables 2 and 3). Nevertheless the statistical resultsare similar and, as expected, noticeable differences ap-pear only for very small κ values close to 3/2, where E total ≫ E c . In addition, we have corroborated that forboth species κ and E total also follow a power-law rela-tion κ = A T E γ T total , with different A T and γ T parametersfor different plasma beta (see Figure 10). Further, com-puting the value of our A T and γ T as function of beta,our results show that the relation between E total and κ is qualitatively the same as the in the case of E c (seeFigure 11). DISCUSSION
Table 4.
Power-law fitting coefficients A (Ampli-tude) and γ ( power-law index) obtained from thedependence of κ on E c , and their respective correla-tion coefficient ( R ). Values are given for both ionand electron and for some selected values of β . β A i γ i R A e γ e R The most evident result to emerge from our ob-servation is the clear positive correlation between β e and β i (see Figure 3(a)), which would be impossi- Eyelade et al -3 -2 -1 i -4 -3 -2 -1 e Scatter plots of ion and electron beta R =0.73 =0.88 0.002 (a) i e Scatter plots of ion and electron kappa R =0.05 (b) -2 -1 E ci -3 -2 -1 E c e Scatter plots of ion and electron Energy R =0.26 (c) Figure 3.
Interrelationship of ion and electron parameters β (a), κ (b), and E c (c). Table 5.
Measured parameters for the empiri-cal relation a | log ( β/β ) | + b that describes thedependency between A and β or between γ and β .Fitting coefficient a β b R A i A Ti γ i -0.53 0.09 1.28 0.68 γ Ti -0.40 0.11 1.10 0.31 A e A Te γ e -0.48 0.09 1.17 0.82 γ Te -0.68 0.10 1.33 0.59 Note —The last column gives the correlation co-efficient. A T and γ T are amplitude and powerindex obtained from kappa and total energy re-lation. ble in the absence of an ion and electron tempera-ture correlation, hence a pressure correlation. In thiscase, we expect the ion temperature to be higher thanthe electron temperature, with a ratio of T i /T e be-tween 4 and 6 (Baumjohann 1993; Borovsky et al. 1997;Espinoza et al. 2018; Wang et al. 2012). Also, there is alack of correlation between κ e and κ i , which implies thatthe processes that regulate the macroscopic propertiesof each species are not related. Perhaps the time scalesfor the energy relaxation of electrons and ions are dif-ferent. Figure 3(c) clearly emphasizes this as E c e versus E c i are only slightly correlated.Despite the strong scattering observed for the κ val-ues, on average, κ increases with β up to β = 1, andthen it slowly decreases as β increases beyond 1. Thistrend is observed for both species, although the effect ismore pronounced for the electrons. This type of anal-ysis was carried out in the solar wind, and a similartrend was noted. Our results are consistent with theprevious studies of ion distributions by Livadiotis et al.(2018) (see their Figure 10(b)). The fact that the samerelation between kappa and beta is observed in both sys-tems, and at similar beta values, suggests this behavioris controlled by some universal property present in col-lisionless plasma systems.In the case of the magnetosphere, comparing withsimilar previous studies, we note that in our case,ion κ values are lower than those reported byKirpichev & Antonova (2020) (see their Figure 3), whichmay be related to the fact that they studied only someparticular regions of the magnetosphere. However, for n the Relation between κ and β -3 -2 -1 i i (a) -3 -2 -1 e e (b) Figure 4. . β − κ plane, for (a) ions, and (b)electrons. The colorbar indicates the number of observations h N i in logarithmic (base 10) color scale. The black and white solidlines represent Mean and Median kappa values in each beta bin, respectively. both ions and electrons, the κ values we found across dif-ferent regions of the magnetosphere, are similar to thoseobtained by Espinoza et al. (2018), even though thatstudy considered only the plasma sheet and restrictedthe dataset to β >
1. This good agreement also sug-gests that the relation between core energy, plasma betaand the kappa power-index may be explained by basicplasma physics processes rather than phenomenologicalor specific properties of the Earth’s magnetosphere.Our analysis of the relationship between β , κ , and E c (section 3) shows that some combinations of κ and β are not present in the analysed plasmas. For low β regimes this may be due to plasma dynamics in whichthe magnetic field dominates and drives the system to-wards specific configurations. On the other hand, forlarge β regimes, temperature fluctuations or kinetic in-stabilities are expected to dominate, which might makecertain plasma configurations more stable than oth-ers. A similar phenomenon was previously reportedby Livadiotis et al. (2018) for solar wind observations.Thus, in low and high plasma β regimes, the degrees offreedom decrease and make the plasma remain in just afew possible states.In Figure 5(a) we observe that on average, ion core en-ergies ( E c i ) are more enhanced as the plasma becomescloser to a Maxwellian: κ i > h E c i i > κ i ≤ h E c i i ≤ β i regimes. In the case of elec-trons, as shown in Figure 6(a), electron core energies E c e > κ values ( κ e ≥ κ e ≤
8) are typicalfor smaller energies E c e ≤ β e regimes. This correlation found between high energiesand high values of kappa may suggest that plasma heat-ing and the thermalization of the distribution may be inagreement with Collier (1999), who suggested that theincrement in kappa depends on ion core temperature,indicating that particle distributions of hotter plasmastend to Maxwellian functions.Previous studies have also shown that the κ -index usu-ally increases with core energy. For instance, Collier(1999) proposed that these two parameters have a lin-ear relation; meanwhile, Kirpichev & Antonova (2020)opted for a power-law dependency but using only ionmeasurements. Our study strongly agrees with the lat-ter, as power-law functions fit well the relation between κ and E c for all values of β parameter, and for bothspecies, as illustrated in Figure 7. Similar results areobtained for the relation between κ and E total (see Fig-ure 10). Moreover, we have found that the coefficientsof power-law fitting A (amplitude) and γ (power-law in-dex) depend strongly on β , for both species, such thatdifferent β values yield different values for A and γ (seeFigures 8 and 11). These results are in contrast withKirpichev & Antonova (2020), who obtained practicallyconstant values for the power index ∼ .
5. The dis-crepancy can be attributed to the fact that they applieddifferent, stronger restrictions to select plasmas for theiranalyses.Further, the increment in kappa value with energyis attributed to positive values of the power-index ( γ ),consistent with the result showing that larger values of0 Eyelade et al -3 -2 -1 i i (a) -3 -2 -1 i i -101234 (b) ion Energy distribution in black marked cell i = 10.00, i = 0.06S ki = -1.11 c oun t s Mean Eci = 7.74 keV (c) ion Energy distribution in red marked cell i = 3.98, i = 0.06S ki = 1.70 c oun t s Mean Eci = 2.67 keV (d)
Figure 5.
Core energy distributions and skewness examples. (a) 2-D plot of average ion core energy h E ci i (color bar) in the β i − κ i plane. (b) The skewness S ki of the distributions of E ci in each cell of the β i − κ i plane. (c) Histogram of one E ci distribution to illustrate a negative skewness. This particular distribution corresponds to the bin marked with a black cellin panels (a) and (b). (d) Histogram of one E ci distribution to illustrate a positive skewness. This particular distributioncorresponds to the bin marked with a red cell in panels (a) and (b). the core energy correspond to larger κ . However, wefound the relation is non-linear. In low beta plasma( β < . γ value for ions and electrons, and thesame happens in the high beta regime ( β > .
1) as seenin Figure 8. Meanwhile, for a specific range of betavalue ( β ∼ . − .
3) the relation between kappa andenergy is stronger showing a maximum value of γ forboth species. As previously mentioned, this is consis-tent with Livadiotis et al. (2018). Thus, the presence ofextremums near β ∼ .
1, as shown in Figure 8, indi-cates the existence of two regimes. Regardless the de- tails of the solar wind or the magnetosphere, in poorlycollisional space plasma systems it is expected plasmamagnetization to dominate in the low beta regime, andtherefore the relation between κ and core energy shouldbe stronger. On the other hand, in the case of a largebeta plasma correlations are mainly due to temperaturefluctuations such that the magnetic field becomes lessrelevant for the determination of κ .Between these two separated regimes there is an inter-mediate range in which other plasma effects should berelevant. For instance, the difference between cold andhot plasma, or the effectiveness of instabilities that was n the Relation between κ and β -3 -2 -1 e e (a) -3 -2 -1 e e (b) Electron Energy distribution in black marked cell e = 7.94, e = 0.63S ke = -0.24 c oun t s Mean Ece = 3.42 keV (c)
Electrons Energy distribution in red marked cell e = 3.16, e = 0.63S ke = 1.94 c oun t s Mean Ece = 1.05 keV (d)
Figure 6.
Core energy distributions and skewness examples. (a) 2-D plot of average electron core energy h E ce i (color bar)in the β e − κ e plane. (b) The skewness S ke of the distributions of E ce in each cell of the β e − κ e plane. (c) Histogram ofone E ce distribution to illustrate a negative skewness. This particular distribution corresponds to the bin marked with a blackcell in panels (a) and (b). (d) Histogram of one E ce distribution to illustrate a positive skewness. This particular distributioncorresponds to the bin marked with a red cell in panels (a) and (b). responsible for the power-law coefficients trend in Figure8. This is consistent with theoretical linear analysis doneby Mace & Sydora (2010) in which deviations from thecold plasma approximation of whistler waves propagat-ing through a bi-Kappa distributed plasma were foundto be relevant for β > .
1, such that β ∼ . β < . β > .
1) in which temperatureeffects become relevant.Recently a similar behavior was described byL´opez et al. (2020), in which different regimes of plasma waves and instabilities were found to be strongly depen-dent on plasma beta. For instance, at β < . β > . β < . β ∼ . Eyelade et al -1 (E i (keV)) -1 ( k i ) A i =2.60 i =0.56R =0.94 i =0.01 (a) -1 (E e (keV)) -1 ( k e ) A e =4.88 e =0.82R =0.93 e =0.01 (b) -1 (E i (keV)) -1 ( k i ) A i =1.02 i =1.22R =0.79 i =0.10 (c) -1 (E e (keV)) -1 ( k e ) A e =2.72 e =1.15R =0.74 e =0.10 (d) -1 (E i (keV)) -1 ( k i ) A i =2.55 i =0.81R =0.88 i =1.00 (e) -1 (E e (keV)) -1 ( k e ) A e =4.25 e =0.61R =0.99 e =1.00 (f) -1 (E i (keV)) -1 ( k i ) A i =3.91 i =0.49R =0.92 i =10.00 (g) -1 (E e (keV)) -1 ( k e ) A e =3.91 e =0.49R =0.92 e =10.00 (h) Figure 7.
Plots of the dependence of ion (left) and electron (right) core energies with κ , for different constant β values.Correlation coefficients and power-law fitting coefficients A (Amplitude), and γ (power-law index) are shown on the plots. n the Relation between κ and β -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 log ( i ) A i R =0.80 a*abs(log ( i / A ))+b (a) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 log ( e ) A e R =0.95 a*abs(log ( e / e0 A ))+b (b) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 log ( i ) i R =0.68 a*abs(log ( i / ))+b (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 log ( e ) e R =0.82 a*abs(log ( e / e0 ))+b (d) Figure 8.
Dependency of the fitted power-law coefficient A (top) and γ (bottom) with β . The left panels are for ions and theright panels for electrons. Eyelade et al a larger variation of κ , while in the high plasma betaregime ( β > .
5) kappa remains more or less the same. SUMMARY AND CONCLUSIONThe observations of ion and electron energy fluxes,made by the THEMIS mission instruments (section 2),have provided the opportunity to study the behavior ofthe Kappa distribution parameters for 47,058 cases, inwhich the spectra of both species were successfully mod-elled with Kappa functions. The plasma measurementswere made in the geomagnetic tail and in the plasmathat surrounds the Earth beyond 7 R E . More specifi-cally, the studied region corresponds to − ≤ X ≤ E , − ≤ Y ≤
30 R E , and − ≤ Z ≤
10 R E andonly plasmas that satisfy the restrictions summarised inTable 1 were considered. We studied the relationshipbetween κ and E c for a wide range of the plasma β pa-rameter, from 10 − to 10 .This is the first time this type of research is carriedout using THEMIS data covering different regions ofthe magnetosphere, and one important finding of ourresearch is the presence of Kappa distributions in manyregions of the magnetosphere. On average, the κ indicesof ions and electrons were found to increase with β , up to β ∼ β > κ indices tendto decrease slowly. In addition, certain combinationsof κ and β are absent in the studied plasmas (for in-stance β i < .
01 and κ i >
10 for ions, or β e >
30 and κ e > κ ≥ E c i ≥ κ e > E c e ≥ κ = AE γc , showing that kappaincreases with energy, and the relation between κ andcore energy is stronger for a specific range of beta value( β ∼ . − . κ (Moya et al. 2020). Moreover, we found that both A and γ depend on β ,the values of A are found to exhibit a minimum near β ∼ .
1; while the power-law indices γ exhibit a maximumat around the same value of β . The observed trend forboth species suggests a universal plasma transition ataround β ∼ . κ in-dices depending on the value of β . When β is small(i.e. in a strongly magnetized plasma), κ is more de-pendant on the magnetic field than on instabilities orfluctuations; and therefore plasma waves and instabili-ties have little effects on kappa. There exist a transitionnear β ∼ . − . κ ,which is consistent with the results presented in Figure8. Finally, for β > .
5, kinetic instability thresholdsprevent the plasma parameters to deviate from a quasi-equilibrium state, and the value of κ is expected to bedetermined mostly by temperature fluctuations. Con-sistently, Figure 8 shows that for β > .
0, both A and γ deviate from the observed trend and tend to take nearlyconstant values. This effect is particularly noticeablefor A e , which may be evidence of electron demagne-tization, which often takes place in highly fluctuatingelectric and magnetic fields (Antonova et al. 1999), andthat are commonly observed at high plasma beta ( β ≥ n the Relation between κ and β -3 -2 -1 i i (a) -3 -2 -1 i i -101234 (b) -3 -2 -1 e e (c) -3 -2 -1 e e (d) Figure 9. (a) 2-D plot of ion total energy E Ti (color bar) in the β i − κ i plane. (b) The skewness S ki of the distributions of E Ti in each cell of the β i − κ i plane. (c) 2-D plot of electron total energy E Te (color bar) in the β e − κ e plane. (d) The skewness S ki of the distributions of E Te in each cell of the β e − κ e plane. Eyelade et al -1 E Ti (keV) -1 i A Ti =1.17 Ti =0.85R =0.89 i =0.01 (a) -1 E Te (keV) -1 e A Te =3.98 Te =0.69R =0.89 e =0.01 (b) -1 E Ti (keV) -1 i A Ti =2.23 Ti =0.69R =0.99 i =0.50 (c) -1 E Te (keV) -1 e A Te =2.57 Te =0.89R =0.98 e =0.50 (d) -1 E Ti (keV) -1 i A Ti =1.88 Ti =0.91R =0.97 i =1.00 (e) -1 E Te (keV) -1 e A Te =3.40 Te =0.66R =0.98 e =1.00 (f) -1 E Ti (keV) -1 i A Ti =1.79 Ti =1.12R =0.70 i =6.31 (g) -1 E Te (keV) -1 e A Te =4.05 Te =0.31R =0.87 e =6.31 (h) Figure 10.
Plots of the dependence of ion (left) and electron (right) Total energies with κ , for different constant β values.Correlation coefficients and power-law fitting coefficients A T (Amplitude), and γ T (power-law index) are shown on the plots. n the Relation between κ and β -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 log ( i ) -0.500.511.522.533.54 A T i R =0.47 a*abs(log ( i / A T ))+b (a) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 log ( e ) A T e R =0.78 a*abs(log ( e / e0 A T ))+b (b) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 log ( i ) -0.200.20.40.60.811.21.41.61.8 T i R =0.31 a*abs(log ( i / T ))+b (c) -2.5 -2 -1.5 -1 -0.5 0 0.5 1 log ( e ) -1.5-1-0.500.511.522.5 T e R =0.59 a*abs(log ( e / e0 T ))+b (d) Figure 11.
Dependency of the fitted power-law coefficient A T (top) and γ T (bottom) with β . The left panels are for ions andthe right panels for electrons. Eyelade et al