11institutetext: Southwest Research Institute, San Antonio, TX 78238, USA
11email<EMAIL_ADDRESS>22institutetext: Department of Space and
Climate Physics, Mullard Space Science Laboratory, University College London,
Dorking, Surrey RH5 6NT, UK 33institutetext: Department of Mathematics,
Physics and Electrical Engineering, Northumbria University, Newcastle, UK
44institutetext: Space Science Center, University of New Hampshire, Durham NH
03824, USA 55institutetext: Planetek, Via Massaua, 12, 70132 Bari BA, Italy
66institutetext: Leonardo, Taranto, Italy 77institutetext: National Institute
for Astrophysics, Institute for Space Astrophysics and Planetology, Via del
Fosso del Cavaliere 100, I-00133 Roma, Italy

# Deriving the bulk properties of solar wind electrons observed by Solar
Orbiter

A preliminary study of electron plasma thermodynamics

G. Nicolaou 1122 R. T. Wicks 2233 C. J. Owen 22 D. O. Kataria 22 A.
Chandrasekhar 22 G. R. Lewis 22 D. Verscharen 2244 V. Fortunato 55 G. Mele 66
R. DeMarco 77 R. Bruno 77

(Received day month year; accepted day month year)

###### Abstract

Context. We demonstrate the calculation of solar wind electron bulk parameters
from recent observations by Solar Wind Analyser Electron Analyser System on
board Solar Orbiter. We use our methods to derive the electron bulk parameters
in a time interval of a few hours. We attempt a preliminary examination of the
polytropic behavior of the electrons by analyzing the derived electron density
and temperature. Moreover, we discuss the challenges in analyzing the
observations due to the spacecraft charging and photo-electron contamination
in the energy range ¡ 10 eV.

Aims. We derive bulk parameters of thermal solar wind electrons by analyzing
Solar Orbiter observations and we investigate if there is any typical
polytropic model that applies to the electron density and temperature
fluctuations.

Methods. We use the appropriate transformations to convert the observations to
velocity distribution functions in the instrument frame. We then derive the
electron bulk parameters by a) calculating the statistical moments of the
constructed velocity distribution functions and b) by fitting the constructed
distributions with analytical expressions. We firstly test our methods by
applying them to an artificial data-set, which we produce by using the forward
modeling technique.

Results. The forward model validates the analysis techniques which we use to
derive the electron bulk parameters. The calculation of the statistical
moments and the fitting method determines bulk parameters that are identical
within uncertainty to the input parameters we use to simulate the plasma
electrons in the first place. An application of our analysis technique to the
data reveals a nearly isothermal electron ”core”. The results are affected by
the spacecraft potential and the photo-electron contamination, which we need
to characterize in detail in future analyses.

###### Key Words.:

Instrumentation: miscellaneous – Methods: data analysis – Sun: heliosphere –
solar wind – Plasmas

##  1 Introduction

The ESA Solar Orbiter mission investigates the solar wind dynamics within the
inner heliosphere. The scientific payload consists of four in-situ and six
remote sensing instruments. The Solar Wind Analyser instrument suite (SWA,
Owen et al., 2020) is one of Solar Orbiter’s in-situ instruments and comprises
three sensors which share a common data processing unit. SWA measurements
resolve the velocity distribution functions (VDFs) of solar wind protons,
$\alpha$-particles, electrons and heavier ions in every few seconds. The orbit
of the spacecraft and the ability of SWA to measure the VDFs of the solar-wind
particles with such a high time resolution, gives us the great opportunity to
study physical mechanisms within a broad range of time-scales (even at kinetic
scales) in the inner heliosphere (e.g., Verscharen et al., 2019; Zouganelis,
I. et al., 2020).

SWA sensors are designed to operate in several modes. In normal mode
operations for example, the instruments resolve the three-dimensional (3D)
VDFs of the solar wind particles, while in burst mode the measurements
construct 2D pitch angle distributions that capture the shortest time-scale
processes. However, the scientific interpretation of the observations, often
requires sophisticated analysis tools and methods. Sometimes, a model of the
instrument’s response (i.e. a forward model) is the main tool to derive the
plasma bulk parameters, such as density, bulk flow velocity and temperature
(e.g. Nicolaou et al., 2014b, 2015b, 2019, 2020b; Wilson et al., 2008, 2017;
Cara et al., 2017; Elliott et al., 2016). For instance, Nicolaou et al.
(2014b) use a forward model of the Solar Wind Around Pluto instrument (SWAP,
McComas et al., 2008) on board New Horizons to fit the count distributions
observed by the instrument in the deep Jovian magnetosheath. The results
quantify the dynamical motions of the Jovian magnetotail and basic
thermodynamic properties of the plasma ions. Elliott et al. (2016, 2019) use a
forward model of the same instrument and derive the plasma properties in the
outer heliosphere. The derived properties reveal how the solar wind evolves as
it propagates through the heliosphere.

The development of such models is not just extremely useful for the analysis
of the observations, but also for the quantification of the errors in the
derived parameters which define the confidence level of the scientific results
(Nicolaou & Livadiotis, 2020; Criton et al., 2020). The basic principle is to
simulate the observations of a specific instrument in given plasma conditions.
The modeling requires the knowledge of the instrument’s characteristics which
are determined from the final hardware design and its detailed calibration. We
can then analyze the simulated measurements in a way we plan to analyze the
actual observations. Finally, we compare the analysis results with the input
plasma parameters we use to simulate the observations in the first place. The
comparison quantifies the capabilities of the instrument and the analysis
methods in resolving specific features of the plasma VDFs. For example, the
study by Nicolaou et al. (2018) demonstrates and quantifies the accuracy of a
novel method deriving the kappa indices of proton VDFs constructed from
measurements by SWA’s Proton Alpha Sensor (SWA-PAS). With the use of a forward
model, the authors show that the accuracy depends on the plasma parameters in
a complicated way.

Here, we develop a model of SWA’s Electron Analyser System (SWA-EAS) which is
designed to resolve the VDFs of solar wind plasma electrons (Owen et al.,
2020). Typical solar wind electron VDFs consist of three electron populations:
i) the thermal ”core”, which comprises the majority of the electron density,
ii) the nearly isotropic ”halo” which extends to higher energies and ii) the
field aligned beams, known as the ”strahl” electrons (e.g., Feldman et al.,
1975; Pilipp et al., 1987). In this paper, we use our model to produce the
expected normal mode observations for typical ”core” electrons, taking into
account the initial calibration of the instrument’s energy range and
resolution, angular range and resolution, and detection efficiency. In section
2, we describe briefly the SWA-EAS instrument and in section 3, we describe
how we model the VDFs in each of the two SWA-EAS sensor heads, how we convert
the observations onto the instrument frame and analyze them. In section 4, we
show modeled observations of typical thermal solar wind electrons, with their
velocities characterized by idealized VDFs, which we analyze in order to
validate the necessary coordinate transformations that construct the 3D VDFs
in the instrument frame. In section 5, we apply our analysis tools to a time
interval of SWA-EAS observations and we derive the plasma bulk properties that
allow us to study the thermodynamic behavior of the thermal plasma electrons.
More specifically, we investigate the relationship between the electron
density and temperature in order to derive the polytropic index, a useful
parameter for energy transfer studies. In section 6, we discuss the results of
this work and identify several challenges regarding the spacecraft potential
and photo-electron contamination, which we need to overcome in future
analyses. Finally, section 7 summarizes our conclusions.

##  2 SWA-EAS instrument

SWA-EAS is designed to measure the 3D VDFs of solar wind electrons at
heliocentric distances between $\sim$0.3 and $\sim$1 au. In order to achieve
full-sky observations, the instrument comprises two top-hat electrostatic
analyzer (ESA) heads; SWA-EAS 1 and SWA-EAS 2. The two heads are mounted
orthogonally on an electronics box installed on the main spacecraft boom in
the shadow of the spacecraft (see Figure 1). The reference frame of each SWA-
EAS head has its z-axis perpendicular to the top-hat plane, pointing towards
the box. The x-axis of SWA-EAS 1 points southwards while the x-axis of SWA-EAS
2 points northwards. The y-axis of each SWA-EAS head frame completes a right
handed orthogonal frame. The instrument frame has its x-axis along the anti-
sunward direction and its z-axis northwards. Each analyzer scans the elevation
angle $\Theta$ of the incoming electrons in its own frame, which is defined as
the angle between the particle velocity vector and the top-hat plane,
increasing towards negative z-axis. The measured elevations and the elevation
range are slightly different for each SWA-EAS head, but roughly, each head
covers the range between $-45^{\circ}$ and $+45^{\circ}$, in 16 steps of the
aperture deflectors. The elevation bandwidth $\Delta\Theta$ varies with the
elevation angle and it ranges from $\sim 3^{\circ}$ to $\sim 12^{\circ}$ for
both sensors (see left panel of Figure 2). The azimuth direction of the
incoming particles, for each SWA-EAS head, is defined as the angle between the
particle velocity projection on the x-y plane (top-hat plane) and the x-axis
in the corresponding SWA-EAS frame. Each SWA-EAS resolves the azimuth
direction $\Phi$ in each energy-elevation scan simultaneously, by using 32
azimuth sectors mounted on the position sensitive Micro Channel Plate (MCP)
detector which is parallel to the top-hat plane. The azimuth sectors in each
SWA-EAS head cover the full 2$\pi$ range. Each azimuth sector has a bandwidth
$\Delta\Phi\sim$11.25∘. Finally, each SWA-EAS measures the energy $E$ of the
plasma electrons in the range between $\sim$0.7 eV and 5 keV in 64 log-spaced
energy steps. The energy bandwidth of each energy step is $\Delta E\,/\,E$
$\sim 12.5\%$. The left panel of Figure 2 shows the elevation coverage of the
instrument ($\Theta\pm\Delta\Theta/2$), the middle panel shows the azimuth
angle coverage ($\Phi\pm\Delta\Phi/2$), while the right panel shows the energy
coverage ($E\pm\Delta E/2$).

Figure 1: Diagram of the instrument’s reference frame and the reference frame
of the two SWA-EAS heads. The anti-sunward direction is along the x-axis of
the instrument frame. The two SWA-EAS heads are mounted orthogonally on the
electronics box which is installed on the spacecraft boom in the shadow of the
spacecraft. Figure 2: (Left) the elevation angle, (middle) azimuth angle, and
(right) energy coverage for (black) SWA-EAS 1 and (red) SWA-EAS 2 heads. In
each panel, the circles indicate the center of each bin, while the error bars
correspond to their bandwidth.

##  3 Model and Methods

###  3.1 VDF model

We first model the solar wind electrons with their velocities following the
isotropic Maxwellian distribution functions:

$f(\@vec{u})=n(\pi
v_{\mathrm{th}}^{2})^{-\frac{3}{2}}\mathrm{exp}\left[-\frac{(\@vec{u}-\@vec{v})^{2}}{v_{\mathrm{th}}^{2}}\right],$
(1)

where $\@vec{u}$ is the particle velocity vector, $n$ is the electron density,
$v_{\mathrm{th}}$ is the thermal speed, and $\@vec{v}$ is the bulk flow
velocity vector of the plasma electrons. We define our input distribution
function, including the bulk velocity vector in the instrument reference
frame.

###  3.2 VDF in SWA-EAS heads

In order to model the distributions constructed from the observations of each
EAS head, we calculate the velocity vector in each EAS head frame by applying
rotation matrices. The velocity vector components in the SWA-EAS 1 frame are

$\begin{bmatrix}v_{\mathrm{x,EAS1}}\\\ v_{\mathrm{y,EAS1}}\\\
v_{\mathrm{z,EAS1}}\end{bmatrix}=\begin{bmatrix}0&0&-1\\\
\mathrm{sin}(45^{\circ})&-\mathrm{cos}(45^{\circ})&0\\\
-\mathrm{cos}(45^{\circ})&-\mathrm{sin}(45^{\circ})&0\end{bmatrix}\begin{bmatrix}v_{\mathrm{x}}\\\
v_{\mathrm{y}}\\\ v_{\mathrm{z}}\end{bmatrix}$ (2)

Similarly, the velocity components in the SWA-EAS 2 are

$\begin{bmatrix}v_{\mathrm{x,EAS2}}\\\ v_{\mathrm{y,EAS2}}\\\
v_{\mathrm{z,EAS2}}\end{bmatrix}=\begin{bmatrix}0&0&1\\\
\mathrm{sin}(45^{\circ})&\mathrm{cos}(45^{\circ})&0\\\
-\mathrm{cos}(45^{\circ})&\mathrm{sin}(45^{\circ})&0\end{bmatrix}\begin{bmatrix}v_{\mathrm{x}}\\\
v_{\mathrm{y}}\\\ v_{\mathrm{z}}\end{bmatrix}$ (3)

We model the response of each SWA-EAS head, considering their energy and field
of view range and resolution. We take into account that each SWA-EAS head
resolves the distribution in an $E$, $\Theta$, $\Phi$ grid. We further assume
that the constructed VDF in each $E$, $\Theta$, $\Phi$ pixel has the value
corresponding to the exact center of the pixel. This simplification ignores
the shape of the VDF and a possibly non-uniform response of the instrument
within the bandwidth of each pixel. In Figure 3, we show an example of a
distribution modeled on (top panel) SWA-EAS 1 and (bottom panel) SWA-EAS 2.
For this example we use the input parameters $n$ = 100 cm-3, $\@vec{v}$=
440$\hat{x}$ km$\,\mathrm{s}^{-1}$, and $v_{\mathrm{th}}$ = 2000
km$\,\mathrm{s}^{-1}$.

Figure 3: Model distribution function constructed in the (top) SWA-EAS 1 and
(bottom) SWA-EAS 2 head. For this model we use input density $n$ = 100 cm-3,
bulk velocity $\@vec{v}$ = 440 km$\,\mathrm{s}^{-1}$ and direction along the
x-axis of the instrument frame, and $v_{\mathrm{th}}$ = 2000
km$\,\mathrm{s}^{-1}$. In the left panel of each row, we show the model
distribution function as a function of energy and elevation angle, summed over
azimuth. The right panels show the model distribution as a function of energy
and azimuth, summed over elevation.

###  3.3 Conversion to instrument frame

Each SWA-EAS head constructs part of the electron VDF from the observations.
The first step of the analysis combines the observations by both SWA-EAS heads
and determines the full 3D VDF in the instrument frame. We do that by defining
a grid on the instrument frame and applying the rotation matrices presented in
Equations 2 and 3 in order to project each pixel of the instrument frame to
the SWA-EAS 1 and SWA-EAS 2 frames respectively. Then, each projected pixel
assumes the VDF value constructed for the specific $E$, $\Theta$, $\Phi$ pixel
of an individual head that includes the projected point within the bandwidth
$\Delta E$, $\Delta\Theta$, $\Delta\Phi$. For pixels that are covered by both
heads, we apply the average of the two VDF values corresponding to each head’s
value in that bin. In Figure 4, we show 2D cuts of the model distribution
function shown in Figure 3, as constructed in the instrument frame with
resolution $\Delta u_{\mathrm{x}}\times\Delta u_{\mathrm{y}}\times\Delta
u_{\mathrm{z}}$ = $500\,\times\,500\,\times\,500\,\mathrm{km^{3}\,s^{-3}}$. In
principle, we can define the grid of the instrument frame as we wish in order
to either reduce the computation steps of our analysis or to improve the
accuracy of our analysis. The accuracy however, is bounded by the instrument
resolution and cannot increase indefinitely. Ideally, the grid definition
should be optimized for specific applications, depending on the shape of the
constructed VDFs.

After the construction of the 3D VDFs in the instrument frame, we calculate
the bulk parameters of the electrons, using two typical analysis methods. We
firstly calculate the plasma bulk parameters by numerically calculating the
statistical moments of the constructed velocity distribution functions. The
second method fits the constructed distribution functions with analytical
expressions by optimizing the electron bulk parameters.

Figure 4: 2D cuts of the model distribution function shown in Figure 3,
constructed in the instrument frame. (Left) the distribution function f at
$u_{\mathrm{z}}$ = 0 km$\,\mathrm{s}^{-1}$ as a function of $u_{\mathrm{x}}$
and $u_{\mathrm{y}}$, (middle) $f$ at $u_{\mathrm{y}}$ = 0
km$\,\mathrm{s}^{-1}$ as a function of $u_{\mathrm{x}}$ and $u_{\mathrm{z}}$,
and (right) $f$ at $u_{\mathrm{x}}$ = 0 km$\,\mathrm{s}^{-1}$ and as a
function of $u_{\mathrm{y}}$ and $u_{\mathrm{z}}$.

###  3.4 Velocity Moments

We firstly calculate the velocity moments of the electrons by numerically
integrating (summing discrete pixels) the constructed velocity distribution
functions. For the constructed
$f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})$ determined in the instrument
frame, the plasma density is given by

$n_{\mathrm{mom}}=\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})\mathrm{\Delta}u_{\mathrm{x}}\,\mathrm{\Delta}u_{\mathrm{y}}\,\mathrm{\Delta}u_{\mathrm{z}}.$
(4)

The bulk velocity components are

$v_{\mathrm{x,mom}}=\frac{\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}u_{\mathrm{x}}f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})\mathrm{\Delta}u_{\mathrm{x}}\,\mathrm{\Delta}u_{\mathrm{y}}\,\mathrm{\Delta}u_{\mathrm{z}}}{n_{\mathrm{mom}}},$
(5)

$v_{\mathrm{y,mom}}=\frac{\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}u_{\mathrm{y}}f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})\mathrm{\Delta}u_{\mathrm{x}}\,\mathrm{\Delta}u_{\mathrm{y}}\,\mathrm{\Delta}u_{\mathrm{z}}}{n_{\mathrm{mom}}},$
(6)

and

$v_{\mathrm{z,mom}}=\frac{\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}u_{\mathrm{z}}f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})\mathrm{\Delta}u_{\mathrm{x}}\,\mathrm{\Delta}u_{\mathrm{y}}\,\mathrm{\Delta}u_{\mathrm{z}}}{n_{\mathrm{mom}}}.$
(7)

The thermal speed is calculated as

$v_{\mathrm{th,mom}}=\sqrt{\frac{\frac{2}{3}\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}[(\@vec{u}-\@vec{v_{\mathrm{mom}}})^{2}]f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})\mathrm{\Delta}u_{\mathrm{x}}\,\mathrm{\Delta}u_{\mathrm{y}}\,\mathrm{\Delta}u_{\mathrm{z}}}{n_{\mathrm{mom}}}}.$
(8)

###  3.5 Fitting

The fitting analysis is widely used to determine the analytical form of
observed distribution functions. The fitting method optimizes the parameters
of an analytical expression for
$f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})$, which we define as
$f_{\mathrm{fit}}(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})$. Typically, a
fitting routine minimizes the chi-square parameter $\chi^{2}$, which in its
simplest form is

$\chi^{2}=\sum_{u_{\mathrm{x}}}\sum_{u_{\mathrm{y}}}\sum_{u_{\mathrm{z}}}[f(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})-f_{\mathrm{fit}}(u_{\mathrm{x}},u_{\mathrm{y}},u_{\mathrm{z}})]^{2},$
(9)

where we use Equation 1 as $f_{\mathrm{fit}}$ 111Occasionally, it is
preferable to fit the logarithm of the constructed VDFs, as this approach
captures better the tails of the distribution functions (e.g., Nicolaou et
al., 2018, 2020b). However, through this paper we use Equation 9 as we focus
on the analysis of the ”core” electrons., parameterized by the bulk parameters
we want to estimate:

$f_{\mathrm{fit}}(\@vec{u})=n_{\mathrm{fit}}(\pi
v_{\mathrm{th,fit}}^{2})^{-\frac{3}{2}}\mathrm{exp}\left[-\frac{(\@vec{u}-\@vec{v_{\mathrm{fit}}})^{2}}{v_{\mathrm{th,fit}}^{2}}\right].$
(10)

##  4 Model Results

We first test our analysis tools by using the forward model to simulate ten
VDFs of solar wind electrons with different bulk parameters. We consider
plasma electrons with their density increasing from 100 to 190 cm-3 (see
Figure 5). We set random fluctuations of the velocity vector components with
an amplitude $\delta v_{\mathrm{x}}$ = $\delta v_{\mathrm{y}}$ = $\delta
v_{\mathrm{z}}$ = 100 km$\,\mathrm{s}^{-1}$ and average values
$v_{\mathrm{x}}$ = 440 km$\,\mathrm{s}^{-1}$, $v_{\mathrm{y}}$ =
$v_{\mathrm{z}}$= 0 km$\,\mathrm{s}^{-1}$. Finally, the thermal speed
increases with the plasma density according to the adiabatic relationship
(i.e. $v_{\mathrm{th}}\propto n^{1/3}$), which is widely used to describe
space plasmas within the heliosphere (e.g., Parker, 1961; Zhang et al., 2016).
We then construct the VDFs in the Cartesian instrument frame with resolution
$\Delta u_{\mathrm{x}}\times\Delta u_{\mathrm{y}}\times\Delta u_{\mathrm{z}}$
= $500\,\times\,500\,\times\,500\,\mathrm{km^{3}\,s^{-3}}$ and analyze them as
explained in Sections 3.4 and 3.5 to derive the electron bulk parameters.

In Figure 5, we show the time series of the input and the derived parameters.
The derived parameters are almost identical to the corresponding input
parameters, indicating that the VDF transformation to the instrument frame and
the analysis tools we are using, are appropriate.

Figure 5: Time series of (black) input, (red) derived by moments and (blue)
derived by fitting plasma bulk parameters of ”core” electrons with their
velocities following isotropic Maxwell distribution functions. There is not an
apparent difference between the derived and input parameters and thus, our
analysis is appropriate for this plasma parameter range.

In order to examine the accuracy of our analysis tools for a wider range of
plasma parameters, we perform an additional test. We use the forward model to
simulate the observations of 500 Maxwellian VDFs, with their bulk parameters
randomly selected from normal distributions. Specifically, we simulate 500
observations with an average plasma density $\bar{n}$ = 100 cm-3 and standard
deviation $\sigma{\bar{n}}$ = 10 cm-3, average velocity components
$\bar{v}_{\mathrm{x}}$ = 440 km$\,\mathrm{s}^{-1}$, $\bar{v}_{\mathrm{y}}$ =
$\bar{v}_{\mathrm{z}}$ = 0 km$\,\mathrm{s}^{-1}$ with standard deviations
$\sigma{v_{\mathrm{x}}}$ = $\sigma{v_{\mathrm{y}}}$ = $\sigma{v_{\mathrm{z}}}$
= 50 km$\,\mathrm{s}^{-1}$ respectively, and an average thermal speed
$\bar{v}_{\mathrm{th}}$ = 2000 km$\,\mathrm{s}^{-1}$ with standard deviation
$\sigma{v_{\mathrm{th}}}$ = 200 km$\,\mathrm{s}^{-1}$. We then analyze the 500
modeled VDFs and compare the results with the input plasma parameters. The top
panels of Figure 6 show histograms of the 500 input parameters $n$,
—$\@vec{v}$—, and $v_{\mathrm{th}}$ and the corresponding parameters derived
by the moments and fitting analysis. The input bulk parameters and the
corresponding parameters derived by both analysis methods have similar
histograms. We quantify the agreement between the analysis results and the
actual parameters by studying histograms of the ratios of the derived
parameters over the corresponding input parameters (see bottom panels of
Figure 6). The moments analysis of our idealized input distributions derives
more accurately the parameters than the fitting analysis for the examined
range of input parameters. Nevertheless, the average values of the ratios in
both analysis methods are remarkably close to unity (within 0.4$\%$) and the
standard deviations are less than 0.2$\%$. Our results show that we can have
high confidence in our methods we use to re-sample and analyze EAS
measurements.

Figure 6: (Top left) Histogram of (gray) input, (red) derived by moments, and
(blue) derived by fit plasma density. (Top middle) Histogram of the input and
derived plasma speed and (top right) the input and derived thermal speed in
the same format. (Bottom left) Histogram of (red) derived by moments and
(blue) derived by fitting density divided by the input density. (Bottom
middle) Histogram of the derived over input speed and (bottom right) the
derived over input thermal speed in the same format.

##  5 Application to flight data

###  5.1 Time series of bulk parameters

We apply our analysis methods to a set of SWA-EAS measurements obtained on
15-06-2020 from 14:50:19UT to 17:35:19UT, when the spacecraft was at
$\sim$0.52 au from the Sun. We first convert the electron counts
$C(E,\Theta,\Phi)$ to electron VDFs in each SWA-EAS head. For the conversion
we use the simplified formula (e.g., Fränz et al., 2006; Lavraud & Larson,
2016)

$f(E,\Theta,\Phi)=\frac{m_{\mathrm{e}}^{2}C(E,\Theta,\Phi)}{2E^{2}G_{\mathrm{f}}(\Theta,\Phi)Q_{\mathrm{e}}(E)\Delta\tau},$
(11)

where $m_{\mathrm{e}}$ is the electron mass, $G_{\mathrm{f}}(\Theta,\Phi)$ is
the geometric factor of the electrostatic analyzer head as determined in
ground calibration, $Q_{\mathrm{e}}(E)$ is the quantum efficiency of the
detector, also as determined in ground calibration, and $\Delta\tau$ is the
acquisition time for each $E,\,\Theta$ scan. The $G_{\mathrm{f}}$ is
independent of energy, while the quantum efficiency depends only on the energy
of the detected particles. For nominal mode operations, $\Delta\tau\sim$ 0.85
milliseconds.

We want to characterize the part of the VDF that contains the thermal ”core”
electrons. Therefore, we exclude data-points in energy channels $E\,>\,$ 68
eV, which is about the typical energy range of supra-thermal electrons and
electron beams (e.g., Feldman et al., 1975; Maksimovic et al., 2005).
Moreover, the lower energy range of the constructed distribution is
contaminated by photo-electrons which are produced on the spacecraft body,
mainly by UV radiation. These photo-electrons are accelerated by the
spacecraft potential. Our preliminary investigation of the data-set, shows
that the photo-electron count distributions have their peaks below 10 eV (see
Appendix A). Therefore, in this first attempt to analyze the data, we exclude
the measurements in energy channels $E\,<\,$ 10 eV. In the left panel of
Figure 7, we show an example of a measured electron VDF as a function of
log${}_{10}(E)$ and $\Theta$, summed over $\Phi$, constructed in the EAS 1
head. In the right panel of Figure 7, we show the VDF part with energies
between $\sim$ 10 and $\sim$ 68 eV which is the VDF part we analyze.

We also find a systematic difference between the VDFs constructed by the two
EAS heads. We apply a preliminary correction in order to eliminate the
differences in the measurements by EAS 1 and EAS 2 while observing the same
VDF. The details of the sensitivity cross-calibration are given in Appendix B.
We note however, that the two sensors will be cross calibrated in the near
future. The detailed in-flight calibration will be based on the analysis of
the entire data-set obtained up to this time and will provide detailed
efficiency corrections.

Figure 7: (Left) The electron VDF as a function of log${}_{10}(E)$ and
$\Theta$, summed over $\Phi$, constructed on the SWA-EAS 1 head from
observations obtained at 14:50:19UT on 15-06-2020. (Right) the same
distribution with all electrons with energies below $\sim$10 eV and above
$\sim$68 eV removed.

After the energy range selection and the scaling of $f_{\mathrm{EAS2}}$, we
construct the VDF in the instrument frame and apply the moments and the
fitting analysis to derive the plasma electron bulk parameters. Figure 8 shows
the results of this analysis. We also calculate and show the typical
statistical uncertainty of the derived parameters. Our calculations assume
that the observed counts follow the Poisson distribution (see Appendix C). The
electron density derived by moments $n_{\mathrm{mom}}$ is significantly
smaller than the density derived by fitting $n_{\mathrm{fit}}$ within the
entire interval. This is an expected result since we exclude a significant
portion (i.e., the part outside the energy range between 10 and 68 eV) of the
analyzed VDF and the derived $n_{\mathrm{mom}}$ is actually a partial density.
The electron velocity components derived by both methods follow a similar
pattern. However, the absolute values are different, especially for the
$v_{\mathrm{y}}$ component. Finally, the thermal speeds derived by both
methods follow a very similar pattern, with $v_{\mathrm{th,mom}}$ being $\sim
20\%$ larger than $v_{\mathrm{th,fit}}$. This difference is also expected
considering that the formula deriving $v_{\mathrm{th,mom}}$ has the partial
(underestimated) density in the denominator (Equation 8). In Section 6, we
discuss further the expected differences between the derived parameters.

We quantify the linear relationships and the linear correlation coefficients
between the parameters derived by moments and the corresponding parameters
derived by fitting. In Table 1, we show the results of this analysis. The two
densities $n_{\mathrm{mom}}$ and $n_{\mathrm{fit}}$ are almost perfectly
correlated (Pearson coefficient $\sim$ 0.99). All plasma velocity components
have a strong correlation (Pearson coefficient ¿ 0.9). The correlation between
the derived thermal speeds is smaller, but still significant ($\sim$0.78).

Figure 8: Bulk parameters of solar wind electrons derived by (black) moments analysis and (blue) fitting analysis of the velocity distribution functions constructed from observations obtained between 14:50:19UT and 17:35:19UT on 15-06-2020. From top to bottom, we show the electron density, velocity along x-direction, velocity along y-direction, velocity along z-direction, and thermal speed. In each panel, we show the typical statistical uncertainty for the derived parameters. Table 1: Linear relationships and correlation coefficients between the bulk parameters derived by moments and derived by fitting analysis.  Parameter | Linear Relationship | Correlation
---|---|---
$n$ |  $n_{\mathrm{fit}}=1.72n_{\mathrm{mom}}-9.73$ (cm-3) | 0.99
$v_{\mathrm{x}}$ |  $v_{\mathrm{x,fit}}=0.80v_{\mathrm{x,mom}}-42.17$ (km$\,\mathrm{s^{-1}}$) | 0.93
$v_{\mathrm{y}}$ |  $v_{\mathrm{y,fit}}=1.02v_{\mathrm{y,mom}}+46.52$ (km$\,\mathrm{s^{-1}}$) | 0.93
$v_{\mathrm{z}}$ |  $v_{\mathrm{z,fit}}=1.06v_{\mathrm{z,mom}}+3.40$ (km$\,\mathrm{s^{-1}}$) | 0.94
$v_{\mathrm{th}}$ |  $v_{\mathrm{th,fit}}=1.70v_{\mathrm{th,mom}}-2135.85$ (km$\,\mathrm{s^{-1}}$) | 0.78

###  5.2 Comparison with RPW densities

We compare the electron density values we derive from EAS-SWA observations
with the electron density values derived by the Solar Orbiter Radio and Plasma
Waves (RPW) instrument (Maksimovic, M. et al., 2020) on board Solar Orbiter,
for the same time interval (Figure 9). Although there is a systematical offset
between the derived values, the two instruments observe a similar pattern of
the density variations for the most of the time interval we examine. This is
of course an encouraging result, showing the analysis of data from two
different instruments with a completely different measurement principle,
determine similar dynamic variations of the electron density.

Figure 9: Electron plasma density (black) as derived here from the SWA-EAS
observations analysis and (red) as observed by RPW instrument on-board.

###  5.3 Electron thermodynamics

One of Solar Orbiter’s goals is the understanding of physical mechanisms which
involve energy transfer between the plasma species and electromagnetic fields
(Zouganelis, I. et al., 2020). Such processes may occur on several time-scales
giving characteristic signatures in the VDFs of some or all the plasma
species. The investigation of plasma processes involving energy transfer is
sometimes achieved through the determination of the polytropic behavior of the
plasma. The polytropic process is the transition of a plasma from one
thermodynamic state to another following a specific relationship between the
density $n$ and temperature $T$ (e.g., Parker, 1961):

$T\propto n^{\gamma-1},$ (12)

where $\gamma$ is the polytropic index which governs the process. For the
special case with $\gamma$ = 1, the plasma has a constant temperature and the
process is called isothermal. An isobaric process (constant pressure $P\propto
nT$) has $\gamma$ =0, while an isochoric process (constant $n$) has $\gamma$ =
$\infty$. Finally, in an adiabatic process, there is no energy exchange and
the polytropic index is equal to the ratio of the specific heats
$\gamma=c_{\mathrm{p}}/c_{\mathrm{v}}$. The polytropic equation achieves a
closure to the VDF moments hierarchy through the relationship between higher
order moments (e.g., temperature, pressure) and the zeroth order moment; the
plasma density. Polytropic closures are basic ingredients for fluid
descriptions of plasmas. However, recent studies show that the fluid
description also applies to short-scale fluctuations in the solar wind (e.g.,
Wu et al., 2019; Verscharen et al., 2019). Typical analyses determine $\gamma$
from the linear relationship between the logarithms of $n$ and $T$. From
Equation 12 we get

$\mathrm{log_{10}}(T)=(\gamma-1)\mathrm{log_{10}}(n)+\mathrm{const}.$ (13)

For example, several studies fit Equation 13 to observations in order to
derive the polytropic behavior of solar wind and its structures in the inner
heliosphere (e.g., Totten et al., 1995; Bavassano et al., 1996; Newbury et
al., 1997; Nicolaou et al., 2020a), at $\sim$ 1 au (e.g., Osherovich et al.,
1993; Kartalev et al., 2006; Nicolaou et al., 2014a; Livadiotis & Desai, 2016;
Livadiotis, 2018a, b) and in the outer heliosphere (e.g., Livadiotis &
McComas, 2013; Elliott et al., 2019). Some other studies use the same method
to derive the polytropic behavior of the plasma within Earth’s magnetosphere
(e.g., Pang et al., 2016; Park et al., 2019), Jovian magnetosheath and
boundary layer (e.g., Nicolaou et al., 2014b, 2015a), Saturnian magnetosphere
(e.g., Arridge et al., 2009; Dialynas et al., 2018), and more.

We examine if the polytropic model applies to the electron bulk parameters we
derive here from SWA-EAS observations. In Figure 10, we show the log10($T$)
vs. log10($n$) for the time interval shown in Figure 8. The left panel shows
the parameters derived by the moments calculation, and the right panel the
parameters derived by fitting. The density and the temperature are anti-
correlated. We calculate a Pearson correlation coefficient $\sim$ -0.87 for
the moment parameters and $\sim$ -0.69 for the fitting parameters. In both
panels, the magenta line is the linear model fitted to the data-set. The
calculated slopes indicate $\gamma\sim$ 0.82 when the moments are used and
$\gamma\sim$ 0.66 when the fitting parameters are used. These values show a
polytropic behavior within the isothermal ($\gamma$ = 1) and isobaric
($\gamma$ = 0) range. According to this result, if the electrons have three
effective kinetic degrees of freedom, then there is an energy transfer
associated with their bulk parameter fluctuations. It is important to note
that proper polytropic behavior investigations should be carefully applied to
identified streamlines (e.g., Kartalev et al., 2006; Nicolaou et al., 2014a;
Pang et al., 2016; Livadiotis, 2018a, b; Nicolaou & Livadiotis, 2019;
Livadiotis & Nicolaou, 2021; Nicolaou et al., 2021). The identification of
streamlines demands analysis of plasma and magnetic field data. We plan such a
detailed analysis in the near future, when the cross calibrated data from SWA-
EAS are available. In the next section we discuss further our results.

Figure 10: log${}_{10}T$ vs. $log_{10}n$ of solar wind electrons, as derived
by (left) statistical moments analysis and (right) fitting analysis of the
first 100 3D-VDFs obtained in 15-06-2020. The magenta dashed lines in each
plot are isobaric lines. There is a clear anti-correlation between
log${}_{10}(T)$ and log${}_{10}(n)$. However, the small value of the slope
describing the relationship between the observables, indicates a nearly
isothermal electron plasma. The statistical error bar in the left panel is not
shown as it is comparable to the data symbol size.

##  6 Discussion

Our SWA-EAS response model validates the VDF transformation to the instrument
frame and confirms the accurate derivation of the electron plasma bulk
parameters comparing both a moments calculation and fitting analysis. The
application of our methods to flight data shows that the statistical velocity
moments and the Maxwellian fits to the analyzed VDFs derive parameters with
similar patterns, but different absolute values.

Firstly, the zeroth-order velocity moments derive smaller densities than the
densities derived by the Maxwellian fits to the same VDFs. This is an expected
result, considering that we exclude a big portion of the VDFs from our
analysis. By excluding measurements in energies ¡ 10 eV which are contaminated
by photo-electrons, we also exclude the actual thermal solar-wind electrons at
these energies. Therefore, the derived moments are actually partial moments.
On the other hand, the Maxwellian fitting interpolates the missing VDFs in the
low and high energy range, accounting for the missing parts assuming a
Maxwellian VDF. Importantly, besides a systematical offset, the densities we
derive compare fairly well with the densities derived by the RPW instrument.

The bulk velocity components derived by moments are strongly correlated with
the corresponding velocity components derived by fitting (Pearson coefficient
¿0.9). However, occasionally, the absolute values exhibit large differences (¿
50 km$\,\mathrm{s}^{-1}$). The $v_{\mathrm{x}}$ velocity component is almost
always negative within the analyzed data-set, reaching a minimum value -150
km$\,\mathrm{s}^{-1}$. Both the $v_{\mathrm{y}}$ and $v_{\mathrm{z}}$
components fluctuate roughly between -100 and +100 km$\,\mathrm{s}^{-1}$. A
negative $v_{\mathrm{x}}$ component indicates a bulk flow towards the Sun,
which is the opposite direction of the expected solar wind bulk flow. However,
we acknowledge that our instrument does not resolve energies below $\sim$ 0.7
eV, which correspond to electron speeds ¡ 500 km$\,\mathrm{s}^{-1}$.
Additionally, here we analyze only the measurements obtained at energies ¿ 10
eV, in order to exclude photo electrons. The energy resolution of the
instrument $\Delta E\,/\,E\sim$ 12.5 $\%$, corresponds to a speed resolution
$\Delta u\,/\,u=\Delta E\,/\,2E\,\sim$ 6.25 $\%$. This means that even in the
lowest energy bin we analyze here (10eV), we measure electrons with $u$ =
$\sqrt{2E/m}$ $\sim$1800 km$\,\mathrm{s}^{-1}$ and the speed bandwidth is as
large as $\Delta u\,/\,u$ = 118 km$\,\mathrm{s}^{-1}$, which is a significant
fraction of the expected solar wind speed. In fact, the peaks of the core
electron VDFs which describe best the electron bulk speed are well beyond the
analyzed energy range. In order to get an estimation of how the erroneous flow
direction affects the derivation of the other bulk parameters, we simulate a
VDF with bulk parameters that are typical within the time interval we examine;
$n$ = 75 cm-3, a pure sunward flow $V_{\mathrm{x}}$ = -150
km$\,\mathrm{s}^{-1}$ and $V_{\mathrm{th}}$ = 2050 km$\,\mathrm{s}^{-1}$. We
fit the simulated VDF with a drifting Maxwellian function with a pure
antisunward flow $V_{\mathrm{x}}$ = 440 km$\,\mathrm{s}^{-}1$, which is the
expected solar wind flow. The fit derives the plasma density and temperature
within 5$\%$ and 10$\%$ of their values, respectively.

The thermal speed derived by the moments analysis is linearly correlated but
systematically larger than the thermal speed derived by the fitting analysis.
The difference between the two is of the order of 400 km$\,\mathrm{s}^{-1}$.
The main reason for this difference is again the fact that the derived moments
are derived from the analysis of partial VDFs (not the entire energy range).
According to Equation 8, the derived thermal speed is proportional to
$n_{\mathrm{mom}}^{-1/2}$. Therefore, the underestimation of
$n_{\mathrm{mom}}$ due to the excluded lower energy range, results in an
overestimation of $v_{\mathrm{th,mom}}$. Other differences between
$v_{\mathrm{th,mom}}$ and $v_{\mathrm{th,fit}}$ are possibly due to deviations
of the constructed VDFs from the Maxwell distribution function. For instance,
if the observed VDFs exhibit non-Maxwellian high energy tails, the fitting
analysis underestimates the derived temperature (e.g., Nicolaou & Livadiotis,
2016; Livadiotis, 2018b). In general, the data we analyze here do not exhibit
large deviations from the Maxwellian model, as we show in the typical
1D-fitting example in Figure 11.

Figure 11: The VDF of plasma electrons as a function of energy, observed by
SWA-EAS at 14:50:19UT on 15-06-2020, averaged over elevation and azimuth
directions. The dashed-magenta shows the corresponding Maxwellian fit to the
observations.

By using either the derived moments or the fitting parameters, we find a
similar polytropic model for the plasma electrons (Figure 10). This result
verifies that the moment and the fitting parameters capture fluctuations in
$n$ and $T$ in a comparable way, besides the significant differences of their
absolute-value offsets for the reasons discussed earlier in this section.
According to our analysis, the polytropic model that describes our results has
$\gamma\sim$ 0.82 when we use the derived moments or $\gamma\sim$ 0.66 when we
use the fitting parameters. These values are close to the isothermal model
which is consistent with energy transfer during the bulk fluctuations of
plasma particles with three effective kinetic degrees of freedom. A large heat
flux in the electrons may lead to the quasi-isothermal behavior of thermal
solar wind electrons (e.g., Hollweg, 1976).

We remind the reader that this study characterizes only the electrons in the
core of the VDF without applying sophisticated corrections for the photo-
electron contamination and the spacecraft potential which are expected to
affect the determination of the VDFs (e.g., Song et al., 1997; Salem et al.,
2001). Although, we exclude low energy electrons (¡10 eV) from our analysis,
we expect that a few volts of spacecraft potential can affect the accuracy of
our derivations. The spacecraft potential modifies the energies and the
directions of the particles (e.g., Lewis et al., 2008, 2010; Lavraud & Larson,
2016; Voshchepynets et al., 2018; Bergman et al., 2020, 2021). Detailed
studies of the spacecraft potential and its geometry should be prioritized in
the future to improve the accuracy of the electron bulk properties.

##  7 Summary and Conclusions

We analyze the first measurements by the Solar Wind Analyser’s Electron
Analyser System (SWA-EAS) on board Solar Orbiter. For the evaluation of the
analysis method we developed a forward model of the instrument’s response. We
derive the electron plasma bulk parameters by using two classic analysis
methods; calculation of the statistical velocity moments of the constructed
velocity distribution functions and fitting the velocity distribution
functions with analytical expressions. We finally investigate if there is a
general polytropic behavior supported by the analyzed data-set. In summary, we
conclude that

  1. 1.

Our method successfully converts the observations by both SWA-EAS heads to
velocity distribution functions in the instrument frame;

  2. 2.

Our analysis derives accurate electron plasma bulk parameters for a typical
solar wind plasma;

  3. 3.

Low energy photo-electrons contaminate the observations within the low energy
range ¡10 eV. By excluding the contaminated parts of the analyzed
distributions we underestimated the density moment and overestimate the
thermal speed moment;

  4. 4.

The spacecraft potential is comparable to the energy of the electrons in the
”core” of the velocity distribution function. Therefore the spacecraft
potential is expected to manipulate the energies and the directions of the
solar wind electrons just before they get detected by SWA-EAS. A thorough
study of the spacecraft potential is required in order to correct the energies
and the directions of the observed electrons.

  5. 5.

The fitting process described here is able to improve some of the calculated
moment inaccuracies, i.e., the fitted density is almost two times higher
because the fit can compensate for the lost counts in the low energy range and
so calculates the temperature more accurately also. There is a remaining
issue, likely caused by the spacecraft photo-electrons accelerated by the
spacecraft potential, that is making the velocity calculated from both moments
and fits to be significantly modified from the expected solar wind, to the
extend that the radial velocity is pointing in the wrong direction.

  6. 6.

This first analysis of the derived bulk parameters, suggest that the observed
fluctuations of the thermal electrons have a large heat flux, leading to a
quasi-isothermal variability.

Our future plan is to fully characterize the resolved 3D VDFs from the fully
calibrated data and derive their bulk properties. We will then investigate the
polytropic behavior of the entire solar wind electron population (including
supra-thermal electrons and electron beams if possible) and compare with the
polytropic behavior of solar wind protons and heavier ions observed by Solar
Wind Analyser’s Proton Alpha Sensor (SWA-PAS) and Heavy Ion Sensor (SWA-HIS)
respectively (Owen et al., 2020).

###### Acknowledgements.

G.N., R.T.W., C.J.O. and D.V. are supported by the STFC Consolidated Grant to
UCL/MSSL, ST/S000240/1. D.V. is supported by STFC Ernest Rutherford Fellowship
ST/P003826/1.

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##  Appendix A Photo-electron Contamination

Photo-electrons are produced on the spacecraft body and are accelerated by the
spacecraft potential. If the paths of the accelerated photo-electrons are
within the range of the sampled directions, we expect to detect them at
energies that are comparable with the spacecraft potential. The distributions
of the observed photo-electrons overlap with the VDFs of solar wind electrons,
making it very difficult to resolve the actual solar wind VDFs within the
lower energy range. In this first analysis attempt, we detect the energy range
of the majority of the photo-electrons and we exclude all the measurements
obtained within this energy range.

In order to investigate the energy range of the photo-electrons, we study the
count distribution as a function of energy within the analyzed interval. The
top left panel of Figure 12 shows this distribution for the EAS 1 head, while
the top right panel shows the same distribution for the EAS 2. The black data-
points in both panels show the median values of counts as a function of
energy. The bottom panels show and the median value of the counts measured in
each energy bin, along with a photo-electron model and a thermal electron
model. Both models are based on Maxwellian VDFs. We clearly see two peaks in
the count distributions. The lower energy peak corresponds to the photo-
electrons, while the second peak corresponds to the thermal electrons. We set
10 eV (red dashed line in Figure 12) as the energy level that separates the
photo-electron and thermal electron distributions.

Figure 12: (Top left) Distribution of counts as a function of energy for the
observations obtained from 14:50:19UT to 17:35:19UT on 15-06-2020, by EAS 1
head and (top right) by EAS 2 head. Each bin of the histogram shows the sum of
the counts within the time interval, elevation bins and azimuth sectors. The
black data-points show the median value of counts in each energy bin. (Bottom
left) The median value of counts in each energy bin as measured by EAS 1, and
(bottom right) by EAS2. The red and magenta dashed lines show models of the
photo-electrons and thermal electrons respectively. Photo-electrons and
thermal electrons are well separated 10 eV.

##  Appendix B Sensitivity cross-calibration

By using the most recent laboratory calibration factors, our analysis reveals
systematically larger $f(E,\Theta,\Phi)$ values constructed by EAS 1 than the
corresponding values constructed by EAS 2 for the same distribution. Figure 13
shows this systematic difference. In the top panel, the black data points show
the VDF as a function of energy, constructed by EAS 1 ($f_{\mathrm{EAS1}}$) in
a single elevation bin and azimuth sector, which sample the +z direction in
the reference frame of the instrument. The red data-points in the same panel
show the corresponding values of the VDF constructed by EAS 2
($f_{\mathrm{EAS2}}$) for the same direction. The shadowed region corresponds
to the energy range which we analyze in this study. The top-right panel shows
a histogram of log10($f_{\mathrm{EAS1}}/f_{\mathrm{EAS2}}$). The peak of the
distribution is clearly $>$ 0\. The bottom panels of Figure 13 show the same
plots for $f_{\mathrm{EAS2}}$ multiplied by a factor of 1.65. The scaling
factor recovers the differences between the sensitivity of the two SWA-EAS
heads during the analyzed time interval. Therefore, we use this scaling factor
for our preliminary study. However, we plan a detailed cross calibration for
the entire data-set in the near future.

Figure 13: (Top left) The VDF of solar wind electrons as a function of $E$,
constructed from the (black) SWA-EAS 1 and (red) SWA-EAS 2 head, by including
only the elevation and azimuth bins closest to the +z direction in the
instrument frame. The purple shadowed region marks the energy range we analyze
in this study. (Top right) Histogram of the ratio values of the VDF
constructed from SWA-EAS 1 measurements, divided by the corresponding VDF
constructed by SWA-EAS 2 measurements. (Bottom left) The same as in the top
left panel, but with SWA-EAS 2 measurements multiplied by 1.65, and (bottom
right) shows the corresponding ratio after the multiplication.

##  Appendix C Typical statistical uncertainties

We estimate the typical statistical uncertainty (measurements error) of the
derived plasma parameters, assuming that the number of counts recorded by the
instrument follows the Poisson distribution. For our estimations, we simulate
1000 measurement samples of a non-drifting plasma with $n$ = 75 cm-3 and
$V_{\mathrm{th}}$ = 2050 km$\,\mathrm{s}^{-1}$. This is a typical set of
electron plasma parameters within the range of the time series in Figure 8. We
then analyze the simulated samples the same way we analyze the observations.
We show the histograms of the derived parameters in Figure 14. The standard
deviation of each histogram is then used as the characteristic error bar of
the corresponding parameter in Figures 8 and 10. Note however that our
preliminary analysis uses the initial calibration factors as determined from
laboratory testing instrument response models. These initial calibration
factors do not account for spacecraft charging effects, or the systematic
offset from the response of other instrument’s on-board. These corrections are
planned for the near future and may reveal additional systematical errors in
the derived parameters.

Figure 14: Histograms of the electrons plasma parameters derived by (top)
statistical moments and by (bottom) fitting analysis of 1000 simulated plasma
samples of plasma with $n$ = 75 cm-3 and $V_{\mathrm{th}}$ = 2050
km$\,\mathrm{s}^{-1}$. In our simulations we assume that the number of counts
follows the Poisson distribution. In each panel we show the standard deviation
of the histogram, which we consider as the typical statistical error for the
corresponding parameter. Those are used as error bars in Figures 8 and 10.