# On the Determination of Kappa Distribution Functions from Space Plasma Observations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{B}is the Boltzmann constant. Finally, κ is the kappa index that labels and governs the VDF. In order to describe accurately space plasmas, we need high-quality measurements which allow the accurate determination of the VDF. State-of-the-art instruments, such as top-hat electrostatic analyzers, are capable of measuring plasma particle fluxes in velocity space, constructing the 3D VDFs of the plasma particles. Due to technological limitations associated with the instrument’s resolution, range, and efficiency, the 3D VDFs are not always perfectly resolved. Inaccuracies in the measurements can lead to inaccurate description of the plasma. Furthermore, the total error of the derived plasma parameters also depends on the method we use to analyze the observations [34,35,36,37].

_{K}denotes the kinetic degrees of freedom. In Equation (2) we use the notation of the invariant kappa index ${\kappa}_{0}\equiv \kappa -\frac{{d}_{k}}{2}$ (for more details see [48,49]). As explicitly shown by [6], only the 0th (α = 0) and the 2nd (α = 2) order ε

_{Κ}moments do not depend on κ, while any other order is a combination of κ and T. In this study we examine 3D VDFs, therefore d

_{K}= 3, and $\kappa ={\kappa}_{0}+\frac{3}{2}$. In this consideration, κ ranges between 3/2 and ∞. According to Equation (2) and as discussed in [6], only moments of order α ≤ 2 converge for all possible κ values. For instance, the first order moment (α = 1) for a 3D VDF is

^{1}as a function of κ, for five different temperatures. For all the temperatures we show, there is a sharp increase of ${M}^{1}$ as a function of κ within the range 1.5 < κ < 4, and a plateau for κ > 4. The numerical calculation of Equation (3) leads to the determination of κ. Such a novel calculation could be useful for future analyses and/or could be applied on-board in future operations for fast estimations. However, we firstly need to validate this method considering plasma measurements with realistic uncertainties, obtained by an instrument with realistic detection efficiency, field of view, and resolution.

## 2. Methods

#### 2.1. Synthetic Data Set

_{0}the effective aperture, which is a function of the geometric aperture and the detection efficiency (for more see [35,36,37]). For our study, we use ${A}_{0}=4.4\times {10}^{-6}$ m

^{2}, and combined with the energy resolution and the solid angle covered by the instrument’s angular resolution results to $\mathrm{G}=2\times {10}^{-9}$ m

^{2}∙eV/eV∙sr. Although Equation (4) gives the expected average number of detected particles for each E, Θ, Φ pixel, in reality, the registered counts C

_{out}follow the Poisson distribution function of average C(E,Θ,Φ), with measurement probability

^{−3}, ${u}_{0}$ = 500 kms

^{−1}with direction towards Θ = 0° and Φ = 0°, T = 20 eV and κ = 3.

#### 2.2. Statistical Moments

_{out}(E,Θ,Φ) ~ C(Ε,Θ,Φ) and the kinetic energy distribution function is constructed from the observations, using the inverse of Equation (4):

_{out}, i.e., the plasma density

_{out}is determined by solving

_{out}by numerically solving Equation (12). The accurate derivation of κ

_{out}, depends on the accuracy of T

_{out}and ${M}_{\mathrm{out}}^{\alpha}$, which we examine through this paper.

## 3. Results

^{−3}, ${u}_{0}$ = 500 kms

^{−1}towards Θ = Φ = 0°, T = 20 eV, and κ = 3, which are typical solar wind proton parameters within the heliocentric distance range from 0.3 to 1au (e.g., [57,58]). We model 1000 observation samples for the specific input parameters. We analyze each sample as explained in Section 2.2 in order to determine n

_{out}, u

_{0,out}, T

_{out}, and κ

_{out}. In Figure 3, we show the histograms of the derived plasma parameters for the 1000 modeled observation samples. In the example shown in Figure 3, we calculate κ

_{out}from the first order energy moment ${M}_{\mathrm{out}}^{1}$ (α = 1). On average, the analysis of the specific plasma underestimates the plasma density and temperature and overestimates the kappa index. More specifically, the average n

_{out}is ~19.8 cm

^{−3}, which is by ~1% smaller than the actual n. The average T

_{out}is 19.4 eV, which is by ~3% smaller than the actual T. The average κ

_{out}is about 3.5, while the actual κ = 3. Finally, we find that the average plasma speed does not deviate from the actual value.

_{n}

_{,out}~ 0.1 cm

^{−3}, σ

_{u}

_{0},

_{out}~ 0.2 kms

^{−1}, σ

_{T}

_{,out}~ 0.1 eV, and σ

_{κ}

_{,out}~ 0.05. The total error of the derived plasma parameters (statistical and systematic) depends on the plasma input and the accuracy with which the instrument measures the particle flux (e.g., [34,35,36,37]). In Section 4, we discuss further the sources of errors.

_{out}as calculated from the kinetic energy moments of different orders α. In Figure 4, we show the average κ

_{out}and its standard deviation, as functions of α for the same input plasma parameters as in the example in Figure 3. We investigate the results for α values within 0 and 2, which are the boundaries of the converging energy moment orders (see also [6] and references therein). For each α value, we analyze 1000 samples following the Poisson distribution in Equation (5). The derived kappa index κ

_{out}~ 3.95 for $\alpha \to 0$ and κ

_{out}~ 3.4 for $\alpha \to 2$. The standard deviation of the mean κ

_{out}values is σ

_{κ,out}~ 0.07 for $\alpha \to 0$ and reduces to σ

_{κ,out}~ 0.05 for $\alpha \to 2$. In the next section we discuss in detail our results.

## 4. Discussion

_{out}is based on the accuracy of T

_{out}and ${M}_{\mathrm{out}}^{\alpha}$. Any systematic error of T

_{out}and/or ${M}_{\mathrm{out}}^{\alpha}$ results in a systematic error of κ

_{out}. In Figure 5, we examine the values of κ

_{out}as a function of T

_{out}and the first order energy moment ${M}_{\mathrm{out}}^{1}$. The top left panel shows the histogram of ${M}_{\mathrm{out}}^{1}$ and the lower right panel the histogram of T

_{out}as derived from the analysis of 1000 samples considering plasma protons with n = 20 cm

^{−3}, ${u}_{0}$ = 500 kms

^{−1}towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3. In the top right panel, we show the solution matrix for κ

_{out}as a function of T

_{out}and ${M}_{\mathrm{out}}^{1}$, as calculated from Equation (12). On the same matrix, we indicate the input and the average derived parameters in our analysis. The derived κ

_{out}in our example is overestimated due to the misestimation of T

_{out}by ~3% and the overestimation of ${M}_{\mathrm{out}}^{1}$ by just 0.5%.

_{out}is under-sampled. Moreover, plasma instruments resolve the distribution function in finite ΔΕ, ΔΘ, and ΔΦ intervals. As a result, the shape of the actual distribution within individual ΔΕ, ΔΘ, and ΔΦ pixels and its contribution to the statistical moments cannot be quantified. Similarly, the distribution is sampled in discrete energy and angular steps, and the statistical moments are numerically calculated according to the specific limited sampling (binning).

_{out}and T

_{out}. On the other hand, colder plasmas have narrower VDFs, which are harder to sample with a limited angular resolution. In another example, plasmas with higher densities will increase the number of recorded counts, therefore will reduce the statistical (Poisson) error. The detailed characterization of the accuracy as a function of the plasma parameters is beyond the scope if this study but will be the subject of a future project.

## 5. Conclusions

- The velocity moments of the observed distribution underestimate the plasma density and temperature, but they provide an accurate estimation of the plasma bulk speed.
- The calculation of the kinetic energy moments of order between 0 and 2 leads to the estimation of the kappa index value. The accuracy of the derived index value is slightly improved as the order of the used energy moment increases. Nevertheless, due to instrument limitations, the analysis systematically overestimates the kappa index of the plasma.
- The misestimations of the plasma parameters are due to the instrument’s limited efficiency, energy and angular range, resolution, and limited sampling of the actual plasma distribution. Our analysis quantifies the error of the derived parameters for a specific instrument design and plasma conditions. Similarly, future applications could quantify the expected errors by adjusting the instrument and plasma parameters. Moreover, our results could drive future instrument designs in order to achieve the desired accuracy in specific applications.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The first order kinetic energy moment ${M}^{1}$ as a function of the kappa index κ, for five different plasma temperatures T.

**Figure 2.**Measurement sample for plasma with n = 20 cm

^{−3}, ${u}_{0}$ = 500 kms

^{−1}towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3, recorded by the top-hat electrostatic analyzer design we consider in this study. The left panel shows the registered number of counts C

_{out}as a function of log

_{10}(E) and Θ integrated over Φ, while the right panel shows C

_{out}as a function of log

_{10}(E) and Φ, integrated over Θ.

**Figure 3.**Histograms of the derived (

**top left**) n

_{out}, (

**top right**) u

_{0,out}, (

**bottom left**) T

_{out}, and (

**bottom right**) κ

_{out}, by the analysis of 1000 measurement samples considering plasma with n = 20 cm

^{−3}, ${u}_{0}$ = 500 kms

^{−1}towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3. In each panel, we show the mean value μ and the standard deviation σ of the derived moments, while the vertical blue dashed line indicates the corresponding input value.

**Figure 4.**The mean kappa index κ

_{out}and its standard deviation σ

_{κ,out}as functions of the energy moment order we use to analyze the data-set.

**Figure 5.**(

**Top left**) The occurrence of ${M}_{\mathrm{out}}^{1}$ and (

**lower right**) T

_{out}, as derived from the analysis of 1000 samples of plasma with n = 20 cm

^{−3}, ${u}_{0}$ = 500 kms

^{−1}towards Θ = 0° and Φ = 0°, T = 20 eV, and κ = 3. (

**Top right**) Solutions of κ

_{out}as a function of T

_{out}and ${M}_{\mathrm{out}}^{1}$ according to Equation (12). On each panel, the blue lines indicate the input parameters and the black lines the derived parameters in our example.

**Figure 6.**Solutions of κ

_{out}as a function of T

_{out}and ${M}_{\mathrm{out}}^{1}$ according to Equation (12). The black circle indicates the average parameters as derived from the analysis of 1000 observation samples by our standard instrument model with field of view −22.5° < Θ < +22.5°, −45° < Φ < +45°, and angular resolution ΔΘ = 5° and ΔΦ = 6° respectively. The green circle indicates the average parameters as derived from the analysis of 1000 observation samples by an instrument with field of view −45° < Θ < +45°, −45° < Φ < +45°, and angular resolution ΔΘ = ΔΦ = 2.5°. The input plasma parameters are the same as in Section 3 and are indicated by the blue circle.

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Nicolaou, G.; Livadiotis, G.; Wicks, R.T.
On the Determination of Kappa Distribution Functions from Space Plasma Observations. *Entropy* **2020**, *22*, 212.
https://doi.org/10.3390/e22020212

**AMA Style**

Nicolaou G, Livadiotis G, Wicks RT.
On the Determination of Kappa Distribution Functions from Space Plasma Observations. *Entropy*. 2020; 22(2):212.
https://doi.org/10.3390/e22020212

**Chicago/Turabian Style**

Nicolaou, Georgios, George Livadiotis, and Robert T. Wicks.
2020. "On the Determination of Kappa Distribution Functions from Space Plasma Observations" *Entropy* 22, no. 2: 212.
https://doi.org/10.3390/e22020212