## 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,

2,

3,

4,

5,

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,

8,

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,

11,

12,

13,

14,

15]), planetary magnetospheres (see, e.g., in [

16,

17,

18]), cometary electrons (see, e.g., in [

19]), and the outer heliosphere and inner heliosheath (see, e.g., in [

20,

21,

22,

23,

24,

25,

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

Section 2, we describe in detail our instrument model and how we model and analyse the expected observations. In

Section 3, we present our results, which we discuss in detail in

Section 4.

Section 5 summarises our conclusions.

## 3. Results

For 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

Section 2. By investigating the histograms of the derived parameters, we verify that 200 samples are sufficient to derive statistically significant results, especially within the low density (low particle flux) range. The histograms of the derived parameters from the analysis of 200 samples with input parameters

n = 7 cm

^{−3},

u_{0} = 500 kms

^{−1} pointing along the x-axis,

κ = 3,

${T}_{\parallel}$ = 10 eV, and

${T}_{\perp}$ = 20 eV, are shown in

Figure 4.

Table 1 shows their average values and standard deviations. For these input plasma conditions, the average derived plasma density is by ~23% lower than the input plasma density if the fitting includes

C_{i} = 0, and by ~13% lower than the input density when

C_{i} = 0 are not included in the fit. The standard deviation of the derived densities is

${\sigma}_{n}$ ~0.2 cm

^{−3} for both fits. On average, the fitting analysis that includes

C_{i} = 0 overestimates the kappa index by ~7%, while the fitting that excludes

C_{i} = 0 underestimates kappa by ~23 %. The standard deviations of the derived kappa indices are ~ 0.2 and 0.1 respectively. The average derived

${T}_{\parallel}$ is by ~6% lower than the actual value when

C_{i} = 0 are included in the fit, and by ~25% larger than the actual value when

C_{i} = 0 are not included in the fit. The standard deviation of

${T}_{\parallel}$ is 0.3 eV when

C_{i} = 0 are fitted and 0.7 eV when

C_{i} = 0 are not fitted. The fit that includes

C_{i} = 0 derives accurately

${T}_{\perp}$ within the standard deviation

$\sigma {T}_{\perp}$ = 0.6 eV. The fit that excludes

C_{i} = 0 from the fit overestimates

${T}_{\perp}$ by ~19%, with standard deviation

$\sigma {T}_{\perp}$ = 1 eV.

The solar orbiter expects densities between ~5 cm

^{−3} and ~500 cm

^{−3}. In

Figure 5, we show the derived parameters as functions of the plasma density over the expected range. The red points show the average (over 200 samples) values of the derived parameters as determined by our fits excluding points with

C_{i} = 0, and the blue points show our results for the fit analysis including points with

C_{i} = 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} (

C_{max} > 100).

The kappa index is significantly misestimated (up to 35%) over the low input density range, if the analysis excludes points with C_{i} = 0. For instance, when n = 5 cm^{−3}, the derived index κ_{out} ~ 2. Interestingly, for the same plasma, κ_{out} is accurately determined when C_{i} = 0 measurements are included in our fit. However, as the density increases to n > 20 cm^{−3}, the kappa index is more accurately determined when derived by fits excluding points with C_{i} = 0.

For the input parameters we examine, the fitting analysis that excludes points with C_{i} = 0 significantly overestimates the plasma temperature in the low-density range (n < 10 cm^{−3}). For instance, when n = 5 cm^{−3} (C_{max} ~ 10), the derived ${T}_{\parallel}$ ~ 17 eV, which is 1.7 times greater than its actual input value, and the derived ${T}_{\perp}$ ~ 30 eV, which is 1.5 times greater than its actual input value. For the same plasma conditions, when the analysis includes points with C_{i} = 0, the derived temperatures do not deviate from their actual values by more than 4 %. Nevertheless, the two fitting analyses determine similar temperatures for n > 10 cm^{−3}.

Finally, we examine the fit convergence as a function of the plasma parameters; the minimum

χ^{2} defines the best fit parameters. Previous studies [

27,

33] show that the accurate estimation of the plasma temperature depends on the accurate determination of the kappa index. In

Figure 6, we show 2D plots of the

χ^{2} value as a function of the model parameters

κ,

${T}_{\parallel}$, and

${T}_{\perp}$. For each panel, we show

χ^{2} as a function of two parameters at a time, and we keep the remaining model parameters to their values as determined by the best fit. We perform our calculations for input plasmas with two different densities:

n = 10 cm

^{−3}, and

n = 50 cm

^{−3}. In both examples, we use

κ = 3,

u_{0} = 500 kms

^{−1} pointing along the x-axis,

${T}_{\parallel}$ = 10 eV, and

${T}_{\perp}$ = 20 eV as the input parameters. Red areas on the plots in

Figure 6 indicate

χ^{2} > 1. For a fixed acquisition time Δ

τ and fixed geometric factor

G, higher densities result in higher counts and a smaller area of low

χ^{2}, which indicates that the derived parameters possess a smaller uncertainty. The non-axisymmetric shape of the area of low

χ^{2}, indicates an interdependency of the derived kappa index and the temperatures. More specifically, the area of low

χ^{2} shifts towards higher temperatures and gets broader along the vertical axis for smaller

κ.

## 4. Discussion

We examine the accuracy of plasma electron bulk parameters as derived from the analysis of the expected observations by SWA-EAS operating in burst-mode. Generally, the accuracy of the derived parameters is a function of the flux of plasma particles (number of recorded counts). Our analysis shows that the fit analysis of samples with a significant amount of counts (C_{max} > 30), derives accurately the plasma parameters when measurement points with zero counts (C_{i} = 0) are excluded from the analysis. We show that when analysing observations with a low amount of counts (C_{max} < 30), the accuracy of some parameters is improved if measurement points with zero counts are included in the fitting analysis.

The uncertainty of each measurement C_{i} is approximated with ${\sigma}_{i}=\sqrt{{C}_{i}}$, assuming that the recorded signal follows the Poisson distribution. This results to a relative uncertainty $\frac{{\sigma}_{i}}{{C}_{i}}=\frac{1}{{\sqrt{C}}_{i}}$ which becomes considerably large for small C_{i} and propagates significant statistical and systematic errors in the derivation of the plasma bulk properties. Although Poisson statistics does not really apply to C_{i} = 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 anode which contains the maximum number of counts

C_{max}, assuming a plasma with

n = 5 cm

^{−3},

κ = 3,

${T}_{\parallel}$ = 10 eV and

${T}_{\perp}$ = 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 of counts. In the left panel, the fitting analysis excludes measurements with

C_{i} = 0 (red points), whereas in the right panel, the fitting analysis includes measurements with

C_{i} = 0 with standard deviation

σ_{i} = 1. The kappa index of the fitted model in the left panel is ~2, and the distribution’s high-energy tail is prominent up to 2 keV, based on the inclusion of the non-zero data-points beyond ~400 eV, neglecting all points with

C_{i} = 0 in between. On the other hand, the fit result in the right panel does not have a prominent tail extending beyond 400 eV, as it fits all

C_{i} = 0 points within that energy range. In this case,

κ_{out} ~ 3 which is an accurate estimation of the actual kappa index of the modelled plasma.

As we show in

Figure 5 and

Figure 7, both fitting strategies underestimate the plasma density when the input density is

n < 50 cm

^{−3}. This is partially due to the asymmetry that characterizes the Poisson distributions of low counts. In

Figure 8, we show the Poisson distributions for

C_{exp} = 1,

C_{exp} = 3 and

C_{exp} = 5. Each distribution has two modes (most frequent values); the higher mode which is equal to the average value of the distribution

C_{exp}, and the lower mode =

C_{exp} − 1. The relative difference between the two modes increases with decreasing

C_{exp}. Samples drawn from a Poisson distribution with a small

C_{exp}, as is the case at low densities expected to be measured by SWA-EAS at ~1 au, will more likely undersample the distribution than oversample, and so densities will be underestimated. In addition, within a measurement cycle, each point of velocity space is sampled only once. In our analysis, we consider that an individual

C_{i} obtained at a specific point of velocity space is representative of the average value

C_{exp} with uncertainty

${\sigma}_{i}=\sqrt{{C}_{i}}$. This introduces an additional systematic error. We illustrate this error by considering an exemplar Poisson distribution with

C_{exp} = 5. If we obtain one measurement, the probabilities for observing

C_{i} = 1 and

C_{i} = 9 are almost the same (see

Figure 8). However, in our fitting analysis, the corresponding uncertainties for

C_{i} = 1 and

C_{i} = 9, are

σ_{i} = 1 and

σ_{i} = 3, respectively, and a

χ^{2} minimization model fitted to these two points will shift towards

C_{i} = 1, as the specific point has a bigger weight

${\sigma}_{i}^{-2}={C}_{i}^{-1}=1$. 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 (

Figure 9). For the specific example, we consider plasma with the same bulk properties as in the example shown in

Figure 7 and we show that in the absence of statistical fluctuations, both fits derive identical results. The orange curve in both panels is the lower mode of the Poisson distribution

C_{exp} − 1, which corresponds to a distribution with lower density. In reality, the errors associated with the statistical uncertainty of the measurements decrease with increasing number of counts and/or when analysing average counts over multiple samples of the same plasma.