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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Instrument Model

_{B}, followed by a scan with the deflectors set to Θ = −θ

_{B}, where θ

_{B}is the elevation angle of the magnetic field vector, which has been recorded by the magnetometer on board Solar Orbiter [30] just before the EAS scan. Thus, the flux of electrons will be measured in two cones, one with an edge aligned along and one anti-parallel with $\overrightarrow{B}$. From these measurements a pitch-angle distribution can be recovered.

#### 2.2. Synthetic Dataset

_{B}is the Boltzmann constant and m

_{e}is the electron mass.

_{B}= 0°) and the analyser head performing a single scan through kinetic energy E and azimuth Φ, for elevation Θ = θ

_{B}= 0°, with acquisition time Δτ. The instrument records an expected number of counts,

_{eff}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 A

_{eff}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 θ

_{u,0}= φ

_{u,0}= 0°. We also set the azimuth angle of the magnetic field to φ

_{B}= 45°. We note that in this specific set-up, the bulk speed components have the same magnitude $\left|{\overrightarrow{u}}_{0,\parallel}\right|=\left|{\overrightarrow{u}}_{0,\perp}\right|$, and the azimuth sectors of the instrument measure the pitch-angles of the particles. We assume that the registered counts C follow the Poisson distribution:

^{−3}(left panel), and n = 50 cm

^{−3}(right panel). For both examples, the bulk velocity of magnitude u

_{0}= 500 kms

^{−1}points along the x-axis, and we set κ = 3, ${T}_{\parallel}$ = 10 eV, and ${T}_{\perp}$ = 20 eV.

_{i}is the standard deviation of each count C

_{i}. The model we fit to the observations is a function of n, κ, ${T}_{\parallel}$, and ${T}_{\perp}$, so R = 4. In plasma applications, we typically use ${\sigma}_{i}=\sqrt{{C}_{i}}$, as within a complete scan, we obtain only one measurement for each specific point in velocity space, and we assume that it represents the average value and the variance of the expected number of particles. For each measurement sample, we perform two fittings; one which excludes all points with C

_{i}= 0, and one which includes all points with C

_{i}= 0 in the analysis. We note that, in the second type of fitting, we assign an uncertainty of σ

_{i}= 1 to each C

_{i}= 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. (The software used in this study is available for download at www.github.com/gnicolaou85/Elfit)

## 3. Results

^{−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.

^{−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).

_{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.

_{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}.

^{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 Δτ 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

_{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.

_{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.

_{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.

^{−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.

## 5. Conclusions

^{2}minimization method.

- The fit analysis of plasma measurements with relatively high flux (C
_{max}> 30) estimates the plasma temperature and kappa index more accurately if it excludes measurement points with C_{i}= 0. The corresponding analysis of measurements with low particle flux (C_{max}< 30) estimates the temperature and kappa index more accurately if it includes measurement points with C_{i}= 0. Although C_{i}= 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 (C
_{max}< 100). We show that the misestimation is due to the asymmetry of the Poisson distribution and the assigned uncertainties to the data points.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic of a Solar Wind Analyser’s Electron Analyser System (SWA-EAS) top-hat analyser head and its angular field of view. (

**Left**) The elevation angle is defined as the complement of the angle between the particle velocity vector and the z-axis, perpendicular to the top-hat plane. The elevation angle of the electrons is resolved in 16 electrostatic uniform steps. (

**Right**) The azimuth angle is the angle within the projection of the velocity vector on the top-hat plane and the x-axis. Both SWA-EAS analyser heads resolve the azimuth direction on MCP detectors using 32 sectors.

**Figure 2.**Modelled counts as a function energy and azimuth direction on the analyser’s head frame for (

**left**) plasma density n = 5 cm

^{−3}and (

**right**) n = 50 cm

^{−3}. For both examples, the magnetic field vector (magenta) is in the top-hat plane (Θ = θ

_{B}= 0°) in azimuth direction Φ = 45°. The bulk flow of the electrons u

_{0}= 500 kms

^{−1}along the x-axis (Θ = Φ = 0°). The parallel temperature ${T}_{\parallel}$ = 10 eV, the perpendicular temperature ${T}_{\perp}$ = 20 eV, and the kappa index κ = 3.

**Figure 3.**(

**Left**) Modelled counts as a function of energy and azimuth direction (instrument frame), using n = 20 cm

^{−3}, u

_{0}= 500 kms

^{−1}towards the x-axis (Θ = Φ = 0°), κ = 3, ${T}_{\parallel}$ = 10 eV, ${T}_{\perp}$ = 20 eV, and a magnetic-field direction (magenta) in the top-hat plane (Θ = 0° and Φ = 45°). (

**Right**) Result of our fit to the modelled observations. The model finds the optimal combination of n, κ, ${T}_{\parallel}$, and ${T}_{\perp}$ that minimizes the χ

^{2}value (see text for more).

**Figure 4.**Histograms of (

**top left**) density n

_{out}, (

**top right**) kappa index κ

_{out}, (

**bottom left**) parallel temperature ${T}_{\parallel ,\mathrm{out}}$ and (

**bottom right**) perpendicular temperature ${T}_{\perp ,\mathrm{out}}$, as determined from the analysis of 200 measurement samples of plasma with 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. The blue histograms correspond to values derived by a fit that includes points with C

_{i}= 0, while the red histograms represent values derived by a fit that excludes points with C

_{i}= 0.

**Figure 5.**(

**From top to bottom**) The derived electron density over input density, kappa index, parallel and perpendicular temperature as functions of the input plasma density. The red points represent the mean values (over 200 samples) of the parameters derived by fitting only the measurements with C

_{i}≥ 1. The blue points represent the mean values of the parameters derived by fitting to all measurements including those with C

_{i}= 0. The shadowed regions represent the standard deviations of the derived parameters.

**Figure 6.**2D histograms of the χ

^{2}value as a function of (

**top**) the modelled κ and ${T}_{\parallel}$ and (

**bottom**) the modelled κ and ${T}_{\perp}$, as derived for plasma with two different input densities; (

**left**) n = 10 cm

^{−3}, and (

**right**) n = 50 cm

^{−3}. In both examples, we use u

_{0}= 500 kms

^{−1}pointing along the x-axis, κ = 3, ${T}_{\parallel}$ = 10 eV, and ${T}_{\perp}$ = 20 eV as input parameters.

**Figure 7.**Number of counts as a function of energy for the pitch-angle with the maximum flux assuming a plasma with n = 5 cm

^{−3}, κ = 3, ${T}_{\parallel}$ = 10 eV and ${T}_{\perp}$= 20 eV. The blue line is the fitted model to the observations by (

**left**) excluding points with C

_{i}= 0 which are shown with red colour, and (

**right**) including points with C

_{i}= 0. The magenta line is the expected counts C

_{exp}, given by Equation (2). The labels in each panel show the parameters as derived by the corresponding fit.

**Figure 8.**Poisson distribution with average value (

**top**) C

_{exp}= 1, (

**middle**) C

_{exp}= 3, and (

**bottom**) C

_{exp}= 5. The vertical lines indicate the two modes of the distribution, C

_{exp}(blue) and C

_{exp}− 1 (orange) respectively. For small average values, the Poisson distribution is asymmetric, and the probability to measure number of counts lower than the average value is significant. This can bias the results to lower densities.

**Figure 9.**Number of the expected average counts C

_{exp}as a function of energy in the pitch-angle bin with the maximum particle flux, considering the same plasma conditions as in the example shown in Figure 7. The blue curve is the model fitted to the observations by (

**left**) excluding points with C

_{i}= 0 which are shown with red colour, and (

**right**) including points with C

_{i}=0. The orange curve is the mode C

_{exp}− 1. In each panel, we show the parameters as derived by the corresponding fit. In the absence of statistical fluctuations, both fitting strategies derive identical bulk parameters.

**Table 1.**The input and the derived bulk parameters for the histograms in Figure 4.

Parameter | Input | Fit Including C _{i} = 0 Points | Fit Excluding C _{i} = 0 Points |
---|---|---|---|

n (cm^{−3}) | 7 | 5.4 ± 0.2 | 6.1 ± 0.2 |

κ | 3 | 3.2 ± 0.2 | 2.3 ± 0.1 |

${T}_{\parallel}$ (eV) | 10 | 9.4 ± 0.3 | 12.5 ± 0.7 |

${T}_{\perp}$ (eV) | 20 | 19.8 ± 0.6 | 23.7 ± 1.0 |

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## Share and Cite

**MDPI and ACS Style**

Nicolaou, G.; Wicks, R.; Livadiotis, G.; Verscharen, D.; Owen, C.; Kataria, D.
Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements. *Entropy* **2020**, *22*, 103.
https://doi.org/10.3390/e22010103

**AMA Style**

Nicolaou G, Wicks R, Livadiotis G, Verscharen D, Owen C, Kataria D.
Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements. *Entropy*. 2020; 22(1):103.
https://doi.org/10.3390/e22010103

**Chicago/Turabian Style**

Nicolaou, Georgios, Robert Wicks, George Livadiotis, Daniel Verscharen, Christopher Owen, and Dhiren Kataria.
2020. "Determining the Bulk Parameters of Plasma Electrons from Pitch-Angle Distribution Measurements" *Entropy* 22, no. 1: 103.
https://doi.org/10.3390/e22010103