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This article was submitted to Space Physics, a section of the journal Frontiers in Astronomy and Space Sciences

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The electron velocity distributions measured _{⊥}/Τ_{//}). The analysis indicates that the parameters exhibit interdependence trends characterized by correlations between certain of these parameters and the kappa exponent (

Early measurements of solar wind electron distributions have revealed thermal cores well fitted by bi-Maxwellians, but non-Maxwellian high-velocity tails (

In the present analysis, we investigate the Kappa-distributed halo electrons by using the fits of solar wind electron velocity distributions established by _{c}, the core temperature parallel T_{c//} and perpendicular T_{c⊥} to the magnetic field_{
,
} and a bi-Kappa halo (subscript h) with four parameters, the halo density n_{h}, the halo parallel temperature T_{h//,} the halo perpendicular temperature T_{h⊥,} and the power law index _{B}T_{c} (where k_{B} = 1.38 10^{–23} J/K is the Boltzmann parameter) or about 2 times the thermal velocity (^{2}–10^{3} eV (

Basic information about solar wind electron observations in the present study.

Distance (AU) | Spacecraft | Number of data |
---|---|---|

0.35–0.45 | Helios | 14,427 |

0.9–1.1 | Cluster and Helios | 20,914 |

2.7–3.3 | Ulysses | 1,275 |

Our present study is structured as follows. In the next section, we start by analyzing the relationships between the kappa (

By the dual model itself, the electron velocity distribution is divided into two parts: the core is fitted by a bi-Maxwellian and the halo by a bi-Kappa (_{c}, T_{c//}, T_{c⊥}, n_{h}, T_{h//}, T_{h⊥},

Density of the halo electrons versus the kappa exponent κ at 0.35–0.45 AU

It is clear from

The results in

A direct correlation between n_{h} and _{h}, in agreement with the fact that lower densities means less important effects of collisions for the energetic electrons and thus stronger high-energy tails, more departed from a Maxwellian shape. Kappa distributions tend to standard Maxwellians when κ is very large (tending to infinity), as reviewed in

_{κ} = κ T_{M}/(κ-3/2) where T_{M} is the temperature of the Maxwellian distribution reproducing the core of a global Kappa function incorporating both the core and halo (^{^3}/2 (

Temperature of halo electrons versus

_{h⊥}/T_{h//}) of halo electrons as a function of the kappa parameter at 0.35–0.45 AU (left panel), 0.9–1.1 AU (middle panel) and 2.7–3.3 AU (right panel). The temperature anisotropy T_{h⊥}/T_{h//} of the halo is generally lower than 1, especially for

Temperature anisotropy of halo electrons versus κ at 0.35–0.45 AU

Density of the core electrons versus the exponent κ at 0.35–0.45 AU

Thus, low densities of the core corresponds to low values of

Note that the trend is inversed for the ratio of the core on halo density n_{c}/n_{h} as a function of _{c}/n_{h} (and on the contrary _{h}/n_{c} that is always <1). Since the slope of the halo density versus _{c}/n_{h} (that ranges between 5 and 300) is anti-correlated with _{
h
}

Temperature of core electrons versus κ at 0.35–0.45 AU

_{c⊥}/T_{c//}) of core electrons as a function of _{c⊥}/T_{c//} of the core is generally <1 and becomes closer to 1 for large

Temperature anisotropy of core electrons versus κ at 0.35–0.45 AU

The figures obtained for the total number density are not shown here because they are exactly similar to

The bulk velocity is not clearly related to

Bulk velocity of electrons versus κ of the halo at 0.35–0.45 AU

Note that on the contrary, a clear link between _{s} have a more important influence than the halo on the velocity to accelerate the wind in exospheric models (

_{c}T_{c} + n_{h}T_{h})/(n_{c} + n_{h}) of the electrons as a function of _{c}, T_{h}, n_{c} and n_{h}) to calculate the total temperature. Contrary to the halo temperature that was clearly anti-correlated to _{c}) does not show a clear correlation with

Total temperature of the electrons versus κ at 0.35–0.45 AU

The _{h}/n_{c}), even if _{h} and other electron characteristics, first considering the core density n_{c}.

Core density versus halo density for electrons observed by ULYSSES at 3 AU. Error-bars, average values and correlation coefficients as in

The total temperature illustrated in _{h} shows a modest increase with the density of the halo at low distances (0.4 AU, left). At larger distances, variation is less monotonic, i.e., high total temperatures are obtained for extremely low or extremely high halo densities. There is no clear correlation between the total temperature and the density of the halo. Plots are always very similar for parallel and perpendicular temperatures, and thus not shown because these details do not bring any new information.

Total temperature versus density of the halo at 0.35–0.45 AU

_{h}/T_{c} typically ranges from 2 to 10 and seems to slightly increase with the distance. The density ratio n_{c}/n_{h} ranges from 5 to 300 (which corresponds to n_{h}/n_{c} from 20% to less than 1%). Note that the temperature of the halo is directly related to the temperature of the core, with T_{h} larger when T_{c} is larger. The same is true for the density, as shown in

Ratio of the halo on core temperatures versus density ratio of the core on halo at 0.35–0.45 AU

In general, the temperature ratio is larger when the density ratio is larger, but the profiles of this dependency change much with heliospheric distance, as also found before for, e.g., _{h} decreases with increasing _{h}/T_{c} should correspond to lower heliospheric distances, as confirmed by our plots in _{h} and T_{c} reveals a natural correlation between a hotter core and a more thermalized and cooler halo. With the results in

It has been suggested that the electron halo observed in the solar wind may originate from nanoflare-accelerated electron beams below the solar surface (i.e., less than 1.1 R_{s}) through the nonlinear electron two-stream instability (ETSI) (_{c}/n_{h}>1.

_{c}/n_{h} (or a high n_{h}/n_{c}), which is expected because the halo temperature can contribute more to the total temperature. When the ratio n_{h}/n_{c} is small (thus n_{c}/n_{h} is high), this influence is minimal, and the temperature remains almost constant with the density ratio.

Total temperature versus density ratio of the core on halo at 0.35–0.45 AU

In the dataset used in the present paper, the entire electron velocity distribution is divided into two parts: f = f_{M} + f_{h}, i.e., the low-energy core fitted by a bi-Maxwellian distribution f_{M}, and the halo by a bi-kappa distribution f_{κ} (_{M}, T_{M//}, T_{M⊥}, n_{h}, T_{h//}, T_{h⊥}, κ_{h}) so that the characteristics of the suprathermal electrons depend not only on κ_{h}, but also on the number density and temperature ratio between the core and halo populations.

The fitting method of _{s}, isotropic or integrated over all pitch angles, thus using only 3 parameters to fit the VDF: n_{s}, T_{s} and κ_{s}. In _{s} = 2.7 in average for v < 550 km/s and _{s} = 2 in average for v > 550 km/s. A low κ_{s} in such a global Kappa distribution indicates then high suprathermal tails while κ_{s} tending to infinity corresponds to the Maxwellian case (_{s} is then the only parameter to give distinct information about the importance of the suprathermal electrons.

_{B}T/m)^{1/2} where m = 9.11 10^{–31} kg is the mass of the electrons and T their temperature in Kelvin. If the distribution is fitted on a sufficiently large energy range, the kappa index is the same for the single Kappa and for the sum of Maxwellian + Kappa, i.e., κ_{h} = κ_{s}. Indeed, the index kappa controls the slope of the high-energy tails. Note that halo electrons can be thermalized if κ_{h} tends to infinity, i.e., if the halo population is well described by a second Maxwellian with higher temperature. In a sum of two VDFs, the ratio of halo density on the core density indicates also the presence of the suprathermal particles and their importance, as illustrated in

_{kappa}= (1 + v^{2}/κ))^{(−κ−1)} in green with κ = 3) and sum of Maxwellian +0.03 Kappa (total in red): exp (-v^{2}) (in black) +0.03∗(1 + v^{2}/(2.5κ))^{(−κ−1)} (in blue). _{1} = 10^{10}, n_{2} = 10^{9}, T_{1} = 10^{5} K, T_{2} = 8 10^{5} K, and a single Kappa using here _{s} = 3, n_{s} = n_{1}+n_{2}κ_{s}
^{3/2} Γ(κ_{s}+1/2)/Γ(κ_{s}+1), T_{s} = T_{1}.

Links between a single Kappa and dual Maxwellian + Kappa can be established as follow:

1) _{s} = κ_{h} for isotropic distributions, so that the tails decrease with the same power law at high velocities_{.} This is true only if the fit is made on a sufficiently large range of the velocity, so that the tails are the same at large values (i.e., for normalized velocity v > 6 in

For a sum of two Maxwellians as illustrated in _{h}, but at very large velocities (here v > 8), the asymptotic behaviors differ. This shows that the quality of the fit and the values of the fit parameters will depend very much on the energy range that is taken into account. Note that the observations show in general tails decreasing more as power laws than exponentially. Anyway, in such an extreme case of a Maxwellian to represent the suprathermal tails, κ_{s} is not equal to κ_{h} (that would be very large for a Maxwellian) but should depend on n_{h} and T_{h}.

2) n_{s} depends on n_{M} and n_{h} but it is not simply their sum. To have the VDFs normalized at the same value at v = 0: N_{s} = N_{M} + N_{h} ∼ N_{M} where N_{i} = n_{i} (m/(κ_{i} 2 π k T_{i}))^{3/2} Γ(κ_{i}+1)/Γ(κ_{i}-1/2) for Kappa functions as defined by _{M} = n_{M} (m/(2πkT_{M}))^{3/2} for Maxwellians. This gives a relation between the densities n of the different fit functions, but note that the kappa index κ_{s} of the single Kappa is necessary to determine the exact value of n_{s}. This relation is valid for the Olbertian form of the Kappa VDFs (

3) T_{s} depends in a non-trivial way on n_{M}, T_{M}, n_{h}, T_{h} and κ_{h} where T_{s} can be anisotropic, with different components in the parallel and perpendicular directions. T_{s} ∼ T_{M} gives generally good fits at low velocities, when the ratio n_{M}/n_{h} is large enough, as illustrated in

Suprathermal electrons are observed in the solar wind from 0.3 to 4 AU with a _{h}/n_{c,} (

The presence of suprathermal electrons at low altitude in the solar corona has important consequences for the characteristics of the solar wind. In exospheric models, the kappa parameter of the strahl suprathermal population has a direct influence on the acceleration of the wind (

Moments of Kappa VDFs obtained with the exospheric model from _{//}, dashed: T_{⊥}),

The proton temperature perpendicular to the direction of the magnetic field (dashed line) is higher than the parallel temperature (dotted line) at low distance, but this trend is inverted at large distance (see panel E), as indeed observed. For the electrons, the parallel temperature is larger than the perpendicular one, and the anisotropy increases with the radial distance (panel F). It becomes too large as compared to the observations at large distances (

Panel G shows that

The relations found in the previous sections are especially useful to determine links between the parameters that can be used to estimate boundary conditions in such solar wind models and provide predictions of solar wind characteristics at all radial distances in the Solar System.

To our knowledge, the present work provides the first study where the kappa exponent _{c}, n_{h}, T_{c}, T_{h}, T_{c⊥}/T_{c//,} T_{h⊥}/T_{h//}, n_{,} T, u) allowing us to explore the potential correlations, but also the influence of suprathermal halo electrons on the other properties of the solar wind plasma. We unveil correlations, more than apparent, between the observed kappa exponent, and the core, halo and total density of the electrons at all radial distances. We find strong links between

The population of energetic (suprathermal) electrons is already present at low distances. Even if κ is higher at low distance, the ratio of halo on core electrons remains high. This is important since the presence of suprathermal electrons tends to accelerate the wind, whatever the VDF is represented by a Kappa distribution or a sum of two Maxwellians, due to the higher escaping flux in exospheric models (

Low _{//} slightly larger than T_{⊥}. This can be explained by the lower effects of Coulomb collisions when the density is low, as simulated with solar wind models based on the solution of the Fokker-Planck equation (

It was shown recently that non-equilibrium systems described by linear Fokker-Planck equations for the VDFs yield steady state Kappa distributions for particular choices of the system parameters (

By fitting WIND electron measurements at energies up to 1.5 keV at 1 AU,

Nevertheless, the present results may contribute to understanding the source of the suprathermal electrons and determine their evolution with the radial distance. The low parameter

Publicly available datasets were analyzed in this study. This data can be found here:

VP has written this article and made the analyses of the data. SS made the fits of the velocity distribution functions to determine the parameters used in the study. ML contributed to the writing of the article.

This research was supported by the Solar-Terrestrial Center of Excellence. The SafeSpace project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 870437.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

VP thanks the editors for their invitation to contribute to this special issue about Solar Wind Evolution: Interplay of Large and Small Scale Physics. The authors thank the International Space Sciences Institute (ISSI) and the participants in a 2021 ISSI workshop for the project “Heliospheric energy budget: from kinetic scales to global solar wind dynamics”. ML acknowledges support in the framework of the project SIDC Data Exploitation (ESA Prodex-12).