11institutetext: Institut für Theoretische Physik, Lehrstuhl IV: Plasma-
Astroteilchenphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany 11email:
<EMAIL_ADDRESS>22institutetext: Research Department, Plasmas with Complex
Interactions, Ruhr-Universität Bochum, 44780 Bochum, Germany 33institutetext:
Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU
Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium 44institutetext: Department
of Physics and Astronomy, University of Turku, 20014 Turku, Finland

# Revisiting the Ulysses electron data with a triple fit of velocity
distributions

K. Scherer Revisiting the Ulysses electron data with a triple fit of velocity
distributionsRevisiting the Ulysses electron data with a triple fit of
velocity distributionsRevisiting the Ulysses electron data with a triple fit
of velocity distributionsRevisiting the Ulysses electron data with a triple
fit of velocity distributions E. Husidic Revisiting the Ulysses electron data
with a triple fit of velocity distributionsRevisiting the Ulysses electron
data with a triple fit of velocity distributionsRevisiting the Ulysses
electron data with a triple fit of velocity distributionsRevisiting the
Ulysses electron data with a triple fit of velocity distributions M. Lazar
Revisiting the Ulysses electron data with a triple fit of velocity
distributionsRevisiting the Ulysses electron data with a triple fit of
velocity distributions H. Fichtner Revisiting the Ulysses electron data with a
triple fit of velocity distributionsRevisiting the Ulysses electron data with
a triple fit of velocity distributionsRevisiting the Ulysses electron data
with a triple fit of velocity distributionsRevisiting the Ulysses electron
data with a triple fit of velocity distributions

###### Abstract

Context. Given their uniqueness, the Ulysses data can still provide us with
valuable new clues about the properties of plasma populations in the solar
wind, and, especially, about their variations with heliographic coordinates.
In the context of kinetic waves and instabilities in the solar wind plasma,
the electron temperature anisotropy plays a crucial role. To date, mainly two
electron populations (i.e., the core and the halo) have been surveyed using
anisotropic fitting models, and limiting in general to the ecliptic
observations.

Aims. We revisit the electron data reported by by the SWOOPS instrument on-
board of the Ulysses spacecraft between 1990 to early 2008. These observations
reveal velocity distributions out of thermal equilibrium, with anisotropies
(e.g., parallel drifts or/and different temperatures, parallel and
perpendicular to the background magnetic field), and quasi-thermal and
suprathermal populations with different properties.

Methods. We apply a 2D nonlinear least square fitting procedure, using the
Levenberg-Marquardt algorithm, to simultaneously fit the velocity electron
data (up to a few keV) with a triple model combining three distinct
populations: the more central quasi-thermal core and suprathermal halo, and a
second suprathermal population consisting mainly of the electron strahl (or
beaming population with a major field-aligned drift). The recently introduced
$\kappa$-cookbook is used to describe each component with the following
anisotropic distribution functions (recipes): Maxwellian, regularized
$\kappa$-, and generalized $\kappa$-distributions. Most relevant are triple
combinations selected as best fits (BFs) with minimum relative errors and
standard deviations.

Results. The number of BFs obtained for each fitting combination sum up to
80.6 % (70.7 % in the absence of coronal mass ejections) of the total number
of events. Showing the distribution of the BFs for the entire data set, during
the whole interval of time, enables us to identify the most representative
fitting combinations associated with either fast or slow winds, and different
phases of solar activity. The temperature anisotropy quantified by the best
fits is considered as a case study of the main parameters characterizing
electron populations. By comparison to the core, both suprathermal populations
exhibit higher temperature anisotropies, which slightly increase with the
energy of electrons. Moreover, these anisotropies manifest different
dependencies on the solar wind speed and heliographic coordinates, and are
highly conditioned by the fitting model.

Conclusions. These results demonstrate that the characterization of plasma
particles is highly dependent on the fitting models and their combinations,
and this method must be considered with caution. The multi-distribution
function fitting of velocity distributions has however a significant potential
to advance understanding of the solar wind kinetics and deserves further
quantitative analyses.

###### Key Words.:

plasmas, Sun: heliosphere, solar wind, methods: data analysis

##  1 Introduction

The solar wind is a hot and dilute plasma that constantly streams from the Sun
and fills interplanetary space (Marsch, 2006; Lazar, 2012). Its collisionpoor
nature allows for departures from thermal (Maxwellian) equilibrium (Kasper et
al., 2006; Štverák et al., 2008; Wilson et al., 2019b, 2020), which persist,
being most probably maintained by the the resonant interaction with wave
turbulence and fluctuations (Bale et al., 2009; Yoon, 2011; Alexandrova et
al., 2013). In-situ observations regularly reveal typical non-thermal
characteristics in the particles’ velocity distributions including the
following: (i) enhanced supra-thermal tails caused by an increased number of
particles in the high-energy regime of the distribution (Maksimovic et al.,
1997; Štverák et al., 2008; Mason & Gloeckler, 2012), (ii) temperature
anisotropies, that is, different temperatures parallel and perpendicular to
the ambient magnetic field (Kasper et al., 2006; Marsch, 2006; Štverák et al.,
2008), and (iii) anti-sunward field-aligned beams (also called strahls)
(Pilipp et al., 1987c; Pierrard et al., 2001; Wilson et al., 2019a).

In the electron distributions up to a few keV, three prominent components can
be identified (Pierrard et al., 2001; Wilson et al., 2019a). First, the core
of the distribution is represented by a quasi-thermal component, with up to 80
to 90 % of the total particle number density, and well described by a
Maxwellian distribution. Second, with about 5 to 10% of the total number
density, a suprathermal component, commonly referred to as halo, contributes
to an enhancement of the distribution tails and can be modeled by an Olbertian
Kappa (or $\kappa$–) power-law distribution function. Third, a further
constituent termed beam or strahl can be found and has a noticeable field-
aligned drift (or relative beaming speed) (Pilipp et al., 1987c; Marsch,
2006). The number density of the strahl population is even less than that of
the halo, and can also be described by a (drifting) $\kappa$-distribution
(Wilson et al., 2019a). While these three components can individually be
described by the mentioned distribution functions, different combinations are
employed for an overall fit. If the beam has a very low density, a dual model
can be used to fit the core and halo (Lazar et al., 2017). Otherwise, if the
beam appears more clearly, a triple model that includes the strahl can be
invoked (Wilson et al., 2019a). In closed magnetic field topologies, such as
coronal loops, where double strahls, that is, two counterbeaming strahls have
been observed, see Lazar et al. (2014) and references therein, a quadruple
model can be applied (Macneil et al., 2020). Sometimes a superhalo component
is mentioned (Yoon et al., 2013), but these populations may enhance the higher
energy tails above 10 keV (Lin, 1998). Thus, with the electron data up to a
few keV, and excluding those associated with closed magnetic field lines of
coronal mass ejections, the present study is limited to a triple model (see
Sec. 2).

Particles in heliospheric plasmas such as the solar wind are subject to
processes involving non-thermal acceleration. Their distribution tails then no
longer exhibit a Maxwellian (i.e., exponential) cutoff, but often a decreasing
power law. Such non-equilibrium distributions are well parameterized by the
Kappa distribution, introduced empirically by Olbert (1968), and published for
the first time by Vasyliunas (1968) as a global fitting model that does not
distinguish between core and halo. More rigorous analyses involve a
combination of multiple (anisotropic) distribution functions, including
Maxwellian and Kappa distributions (Pilipp et al., 1987a, b, c; Maksimovic et
al., 2005; Štverák et al., 2008). The Kappa distribution did not only prove to
be a powerful tool for modeling non-thermal distributions, but has also become
notorious for its critical limitation in defining macroscopic physical
properties by the velocity moments, for example of order $l$, which diverge
for low power exponents $\kappa<(l+1)/2$ (Lazar & Fichtner, 2021). For this
reason, a generalization of the (isotropic) standard Kappa distribution has
recently been introduced by Scherer et al. (2017), termed the regularized
Kappa distribution, for which all velocity moments converge. An extension to
the anisotropic regularized Kappa distribution was then presented in Scherer
et al. (2019). The mathematical definitions of these distribution functions
are given in Sec. 2.

The present paper aims at a re-evaluation of the Ulysses electron data
obtained between 1990 and 2008, and is building on the work in Scherer et al.
(2021). For a realistic analysis, we incorporated the anisotropic nature of
the distributions by applying a 2D fitting method. In order to take potential
single components of the distributions into account, we use a triple model
including a quasi-thermal core, a suprathermal halo, and a suprathermal strahl
component. For the model distributions, we chose the anisotropic bi-
Maxwellian, the regularized bi-Kappa, and the generalized anisotropic
regularized Kappa distribution, which was introduced in an attempt to unify
the various commonly used Kappa distributions (Scherer et al., 2021). By
establishing conditions defining good fits and best fits, in section 2 we
describe their distributions for the entire data set, and for each year in
part. Section 3 contains a breakdown of the Ulysses data according to latitude
and solar wind speed, and establish a connection to the point in the solar
cycle at that time. After introducing formulas used to compute electron
parameter (e.g., temperature anisotropies), in section 4 we consider
temperature anisotropy as a case-study, and identify correlations between
temperature anisotropy and other quantities such as the solar wind speed,
parallel plasma beta and other parameters more specific to distribution
models. The paper ends with conclusions in Sec. 5.

##  2 The models

We fit the Ulysses electron data from the SWOOPS instrument (Bame et al.,
1992, 111 http://ufa.esac.esa.int/ufa/\\#data) (in ”Additional datasets”) by
assuming that there exist up to three electron populations: a core component
indicated by the subscript $c$, a halo component by the subscript $h$, and a
strahl component by the subscript $s$. The total combined distribution
function and its moments are indicated by the subscript $t$:

$\displaystyle f_{t}=f_{c}+f_{h}+f_{s}\,,$ (1)

where the distribution functions $f_{i}$, $i\in\\{c,h,s\\}$ are described
below. We seek to satisfy the following condition:

$\displaystyle n_{c}>n_{h}>n_{s}$ (2)

with $n_{i}$, $i\in\\{c,h,s\\}$ denoting corresponding number densities.

The distribution function $f_{i}$ with $i\in\\{c,h,s\\}$ can be an anisotropic
Maxwellian distribution (AMD), a regularized anisotropic Kappa-distribution
(RAK) or a generalized anisotropic Kappa-distribution (GAK). These types of
distributions can be described by the recipes introduced in Scherer et al.
(2020), where the general recipe
$(\eta_{\parallel},\eta_{\perp},\zeta,\xi_{\parallel},\xi_{\perp})$,
abbreviated already with GAK, is given by

$\displaystyle
f_{GAK}(\eta_{\parallel},\eta_{\perp},\zeta,\xi_{\parallel},\xi_{\perp})=n_{0}\,N_{{GAK}}$
(3)
$\displaystyle\times\left(1+\frac{(v_{\parallel}-u)^{2}}{\eta_{\parallel}\Theta_{\parallel}^{2}}+\frac{v_{\perp}^{2}}{\eta_{\perp}\Theta_{\perp}^{2}}\right)^{-\zeta}\exp\left({-\xi_{\parallel}\frac{(v_{\parallel}-u)^{2}}{\Theta_{\parallel}^{2}}-\xi_{\perp}\frac{v_{\perp}^{2}}{\Theta_{\perp}^{2}}}\right)\,,$

where $v_{\parallel}$ and $v_{\perp}$ are the parallel and perpendicular
velocity components, respectively, with respect to the magnetic field, and $u$
is a parallel drift speed. The parameters
$\eta_{\parallel},\eta_{\perp},\zeta,\xi_{\parallel},\xi_{\perp}$ are
constants with respect to velocity, space and time, and $\Theta_{\parallel}$
and $\Theta_{\perp}$ normalize the velocity components and often are termed
thermal speeds. The normalization constant of the distribution function in Eq.
(3) reads

$\displaystyle N_{\mathrm{GAK}}^{-1}=$
$\displaystyle\sqrt{\pi^{3}}\Theta_{\parallel}\Theta_{\perp}^{2}\eta_{\parallel}^{\frac{1}{2}}\eta_{\perp}$
(4)
$\displaystyle\times\int\limits_{0}^{1}U\left(\frac{3}{2},\frac{5}{2}-\zeta,\xi_{\perp}\eta_{\perp}+(\xi_{\parallel}\eta_{\parallel}-\xi_{\perp}\eta_{\perp})t^{2}\right)\mathrm{d}t$

with $U(a,b,x)$ being the Kummer-U function. The AMD is then given by the
recipe $(1,1,0,1,1)$, while the RAK is represented by
$(\kappa,\kappa,\kappa+1,\xi_{\parallel},\xi_{\perp})$ To these dependent
variables come in addition the dependent variables
$n_{0},\Theta_{{}_{\parallel}},\Theta_{{}_{\perp}}$, and $u$. Thus, for the
AMD we have to fit 4 parameters, for the GAK 9 and for the RAK 7. We allowed
that the number of data points equals the number of free parameters, which is
rarely the case. Usually there are more than 60 data points to be fitted. If
there where less, the mean error and the standard deviation are larger than
0.3 (see below).

Figure 1: Histogram of the GFs of the entire data set for each fitted (total)
distribution $f_{ijk}$. Blue boxes show the additional GFs for the events
during CMEs. See Table 1 for the corresponding values.

For the total combined distribution function we introduce the generic notation
$f_{t}=f_{ijk}$, where this time indices $i,j$ and $k$ indicate the fitting
models. Thus, to avoid further clumsy notation, we use the index $1$ for the
AMD, $2$ for the GAK, and $3$ for the RAK, while $0$ indicates that no model
is used for the respective component (i.e., for singular or dual fitting
models). We write, for example, for a distribution function

$\displaystyle f_{123}=f_{c,1}+f_{h,2}+f_{s,3}\,,$ (5)

which means that $f_{123}$ has an AMD core, a GAK halo and an RAK
superhalo/strahl. We used the fitting method described in Scherer et al.
(2021), which is similar to the one applied by Wilson et al. (2020). The data
were fitted with the following combinations of distribution functions:
$f_{100}$, $f_{200}$, $f_{300}$, $f_{110}$, $f_{120}$, $f_{130}$, $f_{111}$,
$f_{112}$, $f_{113}$, $f_{121}$, $f_{122}$, $f_{123}$, $f_{131}$, $f_{132}$,
$f_{133}$. For dual or triple combinations, we always assume that the core
distribution is Maxwellian.

To check the quality of the fits we define the relative error

$\displaystyle E_{i}=\frac{|f_{fit,i}-f_{obs,i}|}{f_{obs,i}}$ (6)

for each data point $i$, and the mean relative error $<E>$ and its standard
deviation $\sigma$ as

$\displaystyle<E>$ $\displaystyle=\frac{1}{N}\sum\limits_{i=1}^{N}E_{i}\,,$
(7) $\displaystyle\sigma$
$\displaystyle=\sqrt{\frac{1}{N-1}\sum\limits_{i=1}^{N}(<E>-E_{i})^{2}}\,.$
(8)

For a good fit (GF), see Fig. 1, we require that $<E>\;\leq 0.3$ and
$\sigma\leq 0.3$ (Scherer et al., 2021). The events which do not obey this
condition or condition (2) for the number densities are rejected. Furthermore,
we define the best fit (BF), see Fig. 2, as the minimum of $<E>$ and $\sigma$
of all GFs.

Our analysis covers the Ulysses data from the launch in late 1990 to early
2008. There are in total 324,450 events, including 30,558 events during
coronal mass ejections (CMEs), which are taken from Richardson (2014). During
CMEs (reduced in number, below 10 $\%$ of the number of events of a certain
relevance) the electrons may exhibit a double strahl or two beaming
components, sunward and anti-sunward, moving along the closed magnetic field
topology. Thus, a double strahl (more or less symmetric) is not reproduced by
our models, but it can mimic a suprathermal component with an excess of
temperature anisotropy in the direction parallel to the magnetic field. To
avoid such a confusing interpretation, the events during CMEs are counted
separately in our present analysis. In addition, we have rejected 2,139
events, which violated condition (2). The total number of bad fits is about
20%. Ideally, a data set consists of 400 data points with finite values, but
most of the time there are much fewer such points, due to the missing data. If
the number of these points is too low, the fits become unreliable, as
indicated by the mean error and the standard deviation in the case of rejected
fits. The data set is given in keV without any error estimates, thus we were
not able to weight the data according to their observational errors, and
therefore the weight is always unity.

In Fig. 1 we show the distribution of GFs for all individual distributions
functions $f_{ijk}$. The most reduced relevance can be attributed to singular
fits, like $f_{100}$, $f_{200}$ or $f_{300}$, with less than
$4\leavevmode\nobreak\ \%$ of total events, and those reproducing the halo
with a Maxwellian and the strahl with GAK ($f_{112}$ with GFs
$<7\leavevmode\nobreak\ \%$) or RAK ($f_{113}$, with GFs
$<3\leavevmode\nobreak\ \%$); see Table 1. The highest peak is given by a
standard combination of three AMDs, that is, the $f_{111}$ combination, with
the ability to provide GFs for about 70 $\%$ of the total data (in the absence
of CMEs, with green color). However, GFs are also obtained with all the other
combinations involving GAK or RAK for describing the suprathermal populations.
These combinations dominate the histogram with representations between 20 %
and more than 50 $\%$ of the total number of events: $f_{130}$ with $\sim
22.5\leavevmode\nobreak\ \%$, $f_{133}$ with $\sim 29\leavevmode\nobreak\ \%$,
$f_{120}$, $f_{122}$ $f_{123}$ and $f_{132}$, each of them approaching
$40\leavevmode\nobreak\ \%$, $f_{131}$ with $\sim 45\leavevmode\nobreak\ \%$,
and $f_{121}$ with $\sim 54\leavevmode\nobreak\ \%$; see again Table 1.

Figure 2: Histogram for the BFs of the entire data set. The green boxes show the BFs without CMEs, the blue boxes show the additional BFs with CMEs. See Table 1 for the corresponding values. Table 1: Number of events (in $\%$) corresponding to the BFs and to the GFs of all combinations of distribution functions in Figs. 1 and 2. $f_{ijk}$ | BF w/o CMEs | BF | GF w/o CMEs | GF
---|---|---|---|---
$f_{100}$ | $0.0$ | $0.0$ | $0.0$ | $0.0$
$f_{200}$ | $0.3$ | $0.3$ | $3.2$ | $0.3$
$f_{300}$ | $0.1$ | $0.1$ | $3.9$ | $4.2$
$f_{110}$ | $0.0$ | $0.0$ | $1.5$ | $1.7$
$f_{120}$ | $7.1$ | $8.2$ | $37.7$ | $42.7$
$f_{130}$ | $1.7$ | $1.9$ | $22.4$ | $25.2$
$f_{111}$ | $21.2$ | $23.9$ | $69.9$ | $78.9$
$f_{112}$ | $0.2$ | $0.2$ | $5.8$ | $6.5$
$f_{113}$ | $0.1$ | $0.1$ | $2.6$ | $2.9$
$f_{121}$ | $19.1$ | $21.9$ | $53.9$ | $61.4$
$f_{122}$ | $3.2$ | $3.8$ | $37.7$ | $43.2$
$f_{123}$ | $1.1$ | $1.3$ | $35.6$ | $40.4$
$f_{131}$ | $12.1$ | $13.6$ | $44.3$ | $49.8$
$f_{132}$ | $3.8$ | $4.5$ | $39.5$ | $44.9$
$f_{133}$ | $0.7$ | $0.8$ | $28.8$ | $32.7$

These results are refined in Fig. 2, which shows the distribution of the BFs
for the entire data set (with green for the events without CMEs). The number
of BFs (in $\%$) obtained for each combination of distribution functions are
given in Table 1, and together sum up to 80.6 % (70.7 % without CMEs) of the
total number of events. Thus, each valid event is assigned one of these
fitting combinations $f_{ijk}$. It can be seen that 21.2 % of the events
without CMEs are best fitted by the $f_{111}$ combination. This is followed at
short difference by the $f_{121}$ with 19.1 %, and then by $f_{131}$ with
12.1%, $f_{120}$ with 7.1 %, $f_{132}$ with 3.8 %, $f_{122}$ with 3.2 %, and
$f_{130}$ with 1.7 %. Remarkable is the existence of dual core-halo
distributions (in the absence of strahl), i.e., $f_{120}$ and $f_{130}$, with
almost 9 % of the relevant events. BFs around 1 % are obtained for
combinations like $f_{123}$ and $f_{133}$, while singular distributions using
GAK (i.e., $f_{200}$) or RAK (i.e., $f_{300}$) have a very reduced presence,
with 0.3 % and 0.1 %, respectively. Fitting models involving generalized Kappa
distributions, like GAK or RAK, sum up together to 56.7 % of the events (49.5
% without CMEs). It is noteworthy that 59.4 % of the events (52.4 % without
CMEs) have the strahl component best reproduced by an AMD, while halos are
described by a GAK in about 35 % of the events (30.5 % without CMEs), and by
an RKD in about 21 % of the events (18.3 % without CMEs). We must also outline
that the most prominent fit combines three AMDs, for 24% of all events (21.2 %
without CMEs).

In order to check if the more complicated GAK-halo distribution ($f_{121}$)
could be replaced by a simpler RAK-halo distribution ($f_{131}$), we compare
the BFs of the $f_{121}$ distribution with the GFs of the $f_{131}$
distribution. Figure 3 displays the mean error of the BFs of the $f_{121}$
distribution against the mean error of GFs of the corresponding $f_{131}$
distribution. It can be seen that about half of the $f_{121}$ distributions
can be replaced by the $f_{131}$ distribution and obtaining still a GF. In
total we have 63,821 $f_{121}$ BF events, which can be replaced by 37,263 GF
events of the $f_{131}$ distribution. Thus, about 58% of the $f_{121}$
distribution could be replaced by the $f_{131}$ distribution. This can be
helpful in analytic studies of dispersion relations. Nevertheless, we still
have about 42% (26,558) events of the $f_{121}$ distribution for which the
fits of the $f_{131}$ distribution give only bad results.

Figure 3: Correlating the mean error of GFs of the $f_{121}$ distribution and
the mean error of GFs of the $f_{131}$ distribution. The red line indicates
the threshold for a GF.

The total number of bad fits is about 19.4% or 62,843 events, together with
2,139 rejected events (0.66%). These data shall be handled individually (about
20%), because the fit procedure needs to start with a different initial guess,
the data sets are too spare to be fitted, or the data cannot be fitted with
the above combinations of distribution functions. In the present analysis,
from the total number of events we consider the remaining majority of 80 %, or
259,365 events. In the following we discuss the correlation between
macroscopic parameters, and concentrate only on the BFs, which are unique. We
mainly refer to the most representative combinations like $f_{111},f_{121}$,
and $f_{131}$, although the analysis may also take into account the less
prominent examples like $f_{120},f_{122}$, or $f_{132}$.

Figure 4: The Ulysses solar wind speeds during time. Adapted from McComas
(2008). The three upper panels show the solar wind proton speed as a function
of heliographic latitude in a polar coordinate system, during solar minimum
(right upper panel), solar maximum (middle upper panel) and again solar
minimum (left upper panels).

The corresponding sun spot number (solar activity) is shown in the lower
panel, where the black line gives the sunspot number and the red line the tilt
angle. Indicated in the lower inlet are the dominant distribution functions
during that period to illustrate the solar cycle dependence of the electron
distribution functions as shown in Fig. 6.

Figure 5: Histograms of the number of events depending on the solar wind speed
during the Ulysses mission: from low latitudes $<$$$ (upper panel) and from
high latitudes $>$$$ (lower panel).

##  3 Time and speed variations

We remind the reader of the time and latitude dependence of the solar wind
speed along the Ulysses trajectory (McComas et al., 2008), and reprint in Fig.
4 the angular distribution from McComas et al. (2008). It can be seen that
during the first and third latitude scan, there are low solar wind speeds
below $\approx$$$ latitude and high speeds above $\approx$$$ latitude, and
almost no intermediate speeds. This is different during a more active Sun in
the second scan, when the speeds scatter over all latitudes.

The difference in solar wind speeds depending on the latitude becomes also
evident in Fig. 5, which shows the histograms of the number of events (in %)
as a function of the solar wind speed. We split into a part for low latitudes
$<$$$ (upper panel) and a part for high latitudes $>$$$ (lower panel). This
time the events (and the corresponding values) without CMEs are given in
black, while those during CMEs in blue. The gap at intermediate speeds, with
about 2.5 % of events, is obvious around $\approx 650$
$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$. From these histograms one can also
observe that the low speeds cluster around 450 km/s at low latitudes near the
ecliptic (upper panel), while the high speeds cluster around 750 km/s at high
latitudes towards the poles and coronal holes (lower panel). Thus, high speeds
are mainly observed during solar minimum (see also McGregor et al., 2011).

|
---|---

year | $f_{120}$ | $f_{111}$ | $f_{121}$ | $f_{122}$ | $f_{131}$ | Sum
---|---|---|---|---|---|---
1990 | $0.00$ | $0.32$ | $0.06$ | $0.01$ | $0.05$ | $0.39$
1991 | $0.38$ | $1.90$ | $1.13$ | $0.15$ | $0.93$ | $3.72$
1992 | $1.16$ | $0.63$ | $1.53$ | $0.31$ | $0.53$ | $3.94$
1993 | $0.55$ | $0.74$ | $1.26$ | $0.20$ | $0.68$ | $2.96$
1994 | $0.05$ | $2.08$ | $0.96$ | $0.07$ | $1.55$ | $3.24$
1995 | $0.24$ | $2.97$ | $0.95$ | $0.09$ | $1.11$ | $4.33$
1996 | $0.76$ | $0.90$ | $1.76$ | $0.26$ | $1.08$ | $3.95$
$\sum^{1996}_{1990}$ | $3.14$ | $9.54$ | $7.65$ | $1.87$ | $5.94$ | $22.44$
1997 | $0.83$ | $0.38$ | $2.48$ | $0.64$ | $0.45$ | $4.98$
1998 | $0.96$ | $0.67$ | $2.68$ | $0.55$ | $0.49$ | $5.41$
1999 | $0.89$ | $1.00$ | $1.83$ | $0.37$ | $0.70$ | $4.46$
2000 | $0.52$ | $2.21$ | $1.38$ | $0.19$ | $1.13$ | $4.48$
$\sum^{2000}_{1997}$ | $3.20$ | $2.64$ | $8.37$ | $1.75$ | $2.77$ | $19.33$
2001 | $0.12$ | $3.52$ | $0.67$ | $0.10$ | $1.06$ | $4.50$
2002 | $0.52$ | $1.75$ | $1.28$ | $0.24$ | $1.02$ | $4.01$
$\sum^{2002}_{2001}$ | $0.64$ | $5.27$ | $1.95$ | $0.34$ | $2.08$ | $8.51$
2003 | $0.73$ | $1.34$ | $1.76$ | $0.30$ | $0.88$ | $4.43$
2004 | $0.35$ | $0.56$ | $1.10$ | $0.25$ | $0.26$ | $2.52$
2005 | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$
2006 | $0.11$ | $0.24$ | $0.30$ | $0.03$ | $0.22$ | $0.70$
2007 | $0.05$ | $2.63$ | $0.68$ | $0.07$ | $1.37$ | $3.51$
2008 | $0.00$ | $0.09$ | $0.03$ | $0.00$ | $0.05$ | $0.13$
$\sum^{1008}_{2003}$ | $1.24$ | $4.86$ | $3.87$ | $0.65$ | $2.78$ | $11.29$

Figure 6: The histogram (left panel) and table of relative numbers (left
panel) splitted into a yearly basis (right panel). Left panel: Histogram for
the BFs of individual years. Inside a distribution function $f_{ijk}$ each bar
represents the events per year, from 1990 to 2008. The data for 2005 are
missing. The data for 1990 at the beginning of the mission are very sparse,
which is also the case toward the end of the mission (after 2005). Right
panel: The relative number of events by the BFs of the five most
representative distribution functions. For each intermediary row,
”$\sum^{yy}_{yy}$” gives the sum of the years above, and the last column ”Sum”
gives the sums of each row. The sum of the last column is 61.57%. Thus, with
these five distribution functions we cover approximately 62% of all events
(70.7% events without CME).

In Fig. 6 the histogram of Fig. 2 is divided in individual years. This helps
us to find the combinations of distribution functions relevant for each orbit
of the Ulysses missions, and implicitly for different solar activities. These
combinations are also indicated in Fig. 4, in order of their relevance as
follows: [$f_{111}$, $f_{121}$, $f_{131}$, $f_{120}$] for the first and third
orbits, and [$f_{121}$, $f_{111}$, $f_{131}$, $f_{120}$] for the second orbit.
On the other hand, the $f_{111}$ and $f_{131}$ distributions have a dip around
the years 1996 to 2000, which is the ascending phase of solar cycle 23 (see
Fig. 4), while the $f_{121}$ and $f_{122}$ distributions show a maximum.
Unfortunately, that is the only ascending phase which is covered by the
Ulysses mission. Therefore, we can only guess that during the ascending phases
of a solar cycle, the $f_{121}$ ($f_{122}$) distributions are more
appropriate. In two declining phases (that of solar cycle 22 and 23) the
distribution functions $f_{111}$ and $f_{131}$ give the most relevant best
fits, although $f_{121}$ is also well represented in this case. Thus, in a
declining phase of a solar cycle, the electron distributions are better
reproduced by three Maxwellians, meaning that they are, individually, closer
to thermal equilibrium. Contrarily, in the rising phase, when the solar
activity increases, the halo distribution is not well fitted by Maxwellians,
meaning that the particles are no longer in thermal equilibrium. The above
holds also true for the period around 2002, where the $f_{111}$ and $f_{131}$
distributions have a minimum, and later at 2007 show a maximum (and vice versa
for the $f_{121}$ and $f_{122}$ distributions). However, because this is at
the end of the misson and no further data are available, we cannot safely
conclude that this time dependence is verified by observations. The reason is
that the time series only covers parts of a solar Hale cycle, but needed were
at least a few such cycles. From a theoretical point of view one may explain
the time dependence by the changing magnetic field in the rising phase and the
beginning of the solar activity. Thus, the non-equilibrium distributions
$f_{121}$ ($f_{122}$) can be caused by an enhanced flare activity. This
proposed context needs a more detailed analysis including all spacecraft and
solar data, which is far beyond the topic of this work.

##  4 Case study: temperature anisotropy

Figure 7: Correlating the solar wind speed (x-axis) given in units of 100 km/s
and the temperature anisotropy (y-axis) for all events. The number of events
$N_{vA}^{i}$ is given in a logarithmic scale. Here we show the BFs of all
fits, while in the following we also show the fits for $f_{111},f_{121}$, and
$f_{131}$ distributions. The upper-left panel shows the total anisotropy, the
upper-right panel shows that of the core, while the lower-left and lower-right
panels that of the halo and strahl, respectively.

The four temperature anisotropies we define as

$\displaystyle
A_{i}=\frac{T_{{}_{\perp},i}}{T_{{}_{\parallel},i}}=\frac{P_{{}_{\perp},1}}{P_{{}_{\parallel},i}}\,,\qquad
i\in\\{t,c,h,s\\}$ (9)

and additionally, we define the parallel plasma beta as

$\displaystyle\beta_{\parallel,i}=\frac{8\pi P_{\parallel,i}}{B^{2}},\qquad
i\in\\{t,c,h,s\\}\,.$ (10)

with magnetic field magnitude $B$. The $\beta_{\perp}$ can be defined, but is
not discussed here. See appendix A for further information including the
representation method in Fig. 7 to Fig. 10. Furthermore, we only discuss the
events without CMEs, and the distributions in the CMEs will be left for future
work. We display the total number of events on a rainbow colour-coded scale,
while the number of core data is displayed in a reddish, of the halo in a
blueish, and of the strahl in a greenish color scale.

Figure 8: Correlating the solar wind speed and the temperature anisotropy
without CMEs. The results shown in the columns from left to right are for
$f_{all}$, $f_{111}$, $f_{121}$ and $f_{131}$, respectively. Rows from top to
bottom display the results for the total, core, halo and strahl distributions,
respectively. See text for more details.

###  4.1 Temperature anisotropy and solar wind speed.

Figures 7 and 8 display the colour-coded binned number of events $N_{vA}^{i},\
i\in\\{t,c,h,s\\}$ (see Sec. A.1), in a plot with the solar wind speed versus
the temperature anisotropy (Eq. 9). In Fig. 7 we correlate temperature
anisotropy and solar wind speeds, for the total distribution function of all
events, (left-upper panel), and only for the core (right-upper panel), for the
halo (left-lower panel), and for the strahl (right-lower panel). In the upper-
left panel, four maxima in the number of all events $N_{vA}^{t}$ can be
identified: the first one between $v_{sw}=300-500$ km/s and an anisotropy of
$A\approx 1$, the second one between $v_{sw}\approx 700$ km/s and
$v_{sw}\approx 800$ km/s and $A\approx 1$, and the third and fourth at about
the same solar wind speeds, but at lower temperature anisotropies $A\approx
0.5$. The core distribution contributes mainly to the first and second maxima
(right-upper panel), while the halo distribution determines the third and
fourth maxima, and contributes also to the first and second maximum (left-
lower panel). The strahl distribution (right-lower panel) contributes
primarily to the first and second maxima. This implies that the temperature
anisotropy is mainly caused by the halo and strahl components, while the core
distribution is well fitted with an isotropic ($A=1$ temperature) distribution
function. The dip at solar wind speeds about $v_{sw}\approx 650$
$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ is due to the fact that these
speeds are rare (see Figs. 4 and 5).

Figure 8 is structured in a similar way as Fig. 7 and shows a comparison of
$f_{all}$ with $f_{111},f_{121}$, and $f_{131}$ (the columns from left to
right) by plotting $N_{vA}^{i}$ of the corresponding total, core, halo, and
strahl distributions. It can be seen that the $f_{111}$ distributions have
maxima at temperature anisotropies around $A=1$ for all four plots: total,
core, halo, and strahl. The $f_{121}$ distribution has two maxima along
$A\approx 1$ and $A\approx 0.5$, where the former is mainly the contribution
from the core, while the latter are contributions from the halo and the
strahl. The core of the $f_{121}$ distribution has a similar behavior as that
of $f_{111}$ distribution, that is, it scatters mainly around $A=1$. A similar
behavior shows the $f_{131}$ distribution, except that the halo does not
scatter as much as the halo of the $f_{121}$ distribution. Also, the total
anisotropy is smoother for the $f_{131}$ distribution. The total anisotropy
for the $f_{111}$ distribution scatters only around $A=1$ (temperature
isotropy), while the total of the $f_{121}$ distribution has two maxima around
$A=1$ and $A=0.5$, similar to the $f_{131}$ distributions, except that the
second maximum around $A=0.5$ is not very pronounced. The dips are again
explained by the absence of speeds about $v_{sw}\approx 650$
$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ (see above).

The $f_{121}$ distribution shows the strongest scattering in the halo and
superhalo component. Nevertheless, because this is the BF, it indicates that
there might be another distribution function involved not covered by the
fitted AMD, RKD or GAK distributions. The scattering in the fitted halo by the
$f_{121}$ distribution might also occur due to the general nature of the GAK.
We do not make a closer inspection here, but state that the bulk part of the
data is in the maxima of the $f_{121}$ distribution, so that the GAK can be
used for fitting.

To conclude this section, we point out that the core has in this analysis a
bi-Maxwellian distribution, which can be replaced by an isotropic Maxwell
distribution, while the halo and superhalo/strahl are best fitted with
anisotropic distributions (here an RAK or GAK). In many cases we can also
replace the GAK by the RAK (see the discussion in Sec. 2 above).

Figure 9: Correlating the parallel plasma beta and the temperature anisotropy.
Panels are organized as in Fig. 8. See text for more details

###  4.2 Temperature anisotropy and parallel plasma beta

Figure 9 displays the temperature anisotropy as a function of the parallel
plasma beta. In the top panel of first column the total temperature anisotropy
of all events is shown. For all distributions, the core events (second row)
show small deviations from isotropy and distribute regularly, more or less
parallel to the $x$-axis. The halo events (first column, third row)
predominantly show an excess of parallel temperature
$T_{\parallel,h}>T_{\perp,h}$, revealing a mushroom-like shape, with the leg
at $\beta_{\parallel}\approx 0.1$, and the hat widely ranging from very low to
high values of $\beta_{\parallel,h}$, e.g., $10^{-4}<\beta_{\parallel,h}<50$.
The strahl events (first column, fourth row) are similar, but spread at
slightly lower values of $\beta_{\parallel,s}<1$. The distributions of these
data are very similar to those obtained by Štverák et al. (2008) with the
ecliptic electron data.

The second, third and fourth columns of Fig. 9 display the events fitted by
$f_{111}$, $f_{121}$, and $f_{131}$, respectively. It can be seen that the
$f_{111}$ distribution has a very weak scattering in all components and is
quite similar to the above mentioned plots by Štverák et al. (2008). This is
also true for the core events of all other distributions (second row).
However, the halo events of the $f_{121}$ and $f_{131}$ distributions show
much stronger scattering (third row). Although more restrained, the strahl
component (fourth row) show the same variation. For all distributions, the leg
is thus a feature mainly resulting from the other distributions not shown
here, e.g., $f_{120}$ and $f_{122}$ distributions.

###  4.3 Temperature anisotropy and
$\kappa,\eta_{{}_{\parallel}},\eta_{{}_{\perp}}$, and $\zeta$ parameters

Figure 10: The correlation between the temperature anisotropy and parameters
of the halo components of the distributions are displayed. The temperature
anisotropy is compared with the $\kappa$ parameter of the $f_{131}$
distribution (left upper panel), and with the $\eta_{{}_{\parallel}}$,
$\eta_{{}_{\perp}}$ and $\zeta$ parameters of the $f_{121}$ distribution
(right upper panel, left lower panel and right lower panel, respectively). For
more explanation see text.

In Fig. 10 we show the correlation between parameters of the halo components
of the $f_{131}$ and $f_{121}$ distributions with the temperature anisotropy.
The color coding is now given at the top of each panel. In the upper left
panel the scattering of the $\kappa$ values of the $f_{131}$ distribution is
shown. We can see that $\kappa$ has only values between 0 to about 2.5. It is
also evident that the lower the $\kappa$ values are, the lower is the halo
temperature anisotropy, that is, the perpendicular temperature is much higher
than the parallel one. The reason for $\kappa$ values only below 2.5 can be
due to the fact that for higher $\kappa$ values the distribution is comparable
to a Maxwellian, and thus the distribution $f_{121}$ approaches $f_{111}$.
However, this statement requires more research.

For the $f_{121}$ distribution, the scattering of $\eta_{{}_{\parallel}}$,
$\eta_{{}_{\perp}}$ and $\zeta$ parameters is similar to the scattering of
$\kappa$ values in the $f_{121}$ distribution, except that
$\eta_{{}_{\parallel}}$ has higher values (up to 5, see right upper panel),
while $\eta_{{}_{\perp}}$ values have a similar range as $\kappa$ (left lower
panel). The $\zeta$ parameter (right lower panel) is similar to the
$\eta_{{}_{\perp}}$ parameter. The $f_{131}$ halo component shows also an
increased scattering in the anisotropy around $\eta_{{}_{\parallel}}=0$,
$\eta_{{}_{\perp}}=0$ and $\zeta=0$, which can be caused by the fitting
procedure due to the small values of the corresponding parameters.

We computed a linear regression for the data shown in Fig. 10 via

$\displaystyle A(x)=\frac{T_{{}_{\perp}h}}{T_{{}_{\parallel}h}}=ax+b\,,$ (11)

where $x\in\\{\kappa,\eta_{{}_{\parallel}},\eta_{{}_{\perp}}.\zeta\\}$. The
values for the fits are listed in Table 2.

Table 2: The fit values for the linear regression (Eq. 11) as shown in Fig. 10. $A(x)$ | a | b
---|---|---
$A(\kappa)$ | 0.25 | 0.11
$A(\eta_{{}_{\parallel}})$ | 0.16 | 0.19
$A(\eta_{{}_{\perp}})$ | 0.15 | 0.27
$A(\zeta$) | 0.23 | -0.16

The linear regression is quite good for $A(\kappa)$, but for
$A(\eta_{{}_{\parallel}})$ and $A(\eta_{{}_{\perp}})$ the events form a curve
asymptotically approaching $A=1$. The fit for $A(\zeta)$ runs through a cloud,
which clusters around the linear regression and scatters toward higher
anisotropies. Nevertheless, the simplified regressions can help to study the
temperature anisotropy in the halo with increasing values of
$\kappa,\eta_{{}_{\parallel}},\eta_{{}_{\perp}}$ and $\zeta$.

##  5 Conclusions

In the present work we used a 2D fitting method, which accounts for different
temperatures parallel and perpendicular with respect to the background
magnetic field, to fit electron velocity distributions obtained during the
Ulysses mission. In doing so, we used a triple model to fit the core, halo and
superhalo/strahl populations within the total distribution. As model functions
we applied an anisotropic Maxwellian distribution (AMD), a generalized
anisotropic Kappa (GAK) and a regularized anisotropic Kappa (RAK)
distribution. Our findings indicate a time dependence of the electron
distributions on the solar cycle. Unfortunately, the data series is not long
enough to solidly confirm such a behavior. Nevertheless, the results suggest
that in a declining phase of a solar cycle, the individual electron
distributions are best described by three Maxwellians, meaning that they are
in thermal equilibrium. In the ascending phase, when the solar activity
increases, the halo distribution is not well fitted by a Maxwellian, which
means that the particles are out of thermal equilibrium. This behaviour
deserves more attention and should be be combined with other spacecraft data.
However, due to the unique solar polar orbit of Ulysses and the fact that most
of the other spacecraft are in the ecliptic (HELIOS) or close to the Sun
(Parker Space Probe, Solar Orbiter), one has in principle to subtract distance
and latitude effects.

While the core distribution can most likely be fitted with an isotropic
Maxwell distribution as the temperature anisotropy is close to 1 (see the
reddish color panels in the figures in Sec. 4), the halo and superhalo/strahl
(the blueish and greenish colo panels) are best fitted with anisotropic
distributions (here with an RAK or a GAK. In many cases we can also replace
the GAK by the simpler RAK, resulting in a slightly worse fit (see Fig. 3).

The parallel plasma beta correlation with the temperature anisotropy is
similar for the three distributions $f_{111},f_{121}$, and $f_{131}$, but
shows a stipe for the halo distribution in the sample of all distributions. We
also showed the correlation between the temperature anisotropy and the
$\kappa$ parameter of the $f_{131}$ distribution as well as the
$\eta_{{}_{\parallel}}$, $\eta_{{}_{\perp}}$ and $\zeta$ parameters of the
$f_{121}$ distribution. The temperature anisotropy shows almost a linear
dependence on $\kappa$ with increasing values of $\kappa$, and the temperature
anisotropy becomes more isotropic. For the GAK parameters
($\eta_{{}_{\parallel}},\eta_{{}_{\perp}},\zeta$) we found also a linear
dependence of the temperature anisotropy.

Finally, we want to direct the attention to Fig. 4 which shows that during a
solar cycle also the type of the distribution functions changes from mainly a
$f_{111}$ type near solar minimum to a more general $f_{121}$ type close to
solar maximum and back to $f_{111}$ for the next minimum phase. These results
demonstrate that the multi-distribution function fitting of velocity
distributions has a significant potential to advance our understanding of the
solar wind kinetics and, therefore, deserves further quantitative analyses.

The data sets were derived from sources in the public domain, available at
`http://ufa.esac.esa.int/ufa/#data/`.

###### Acknowledgements.

The authors acknowledge support from the Ruhr-University Bochum and the
Katholieke Universiteit Leuven. EH is grateful to the Space Weather Awareness
Training Network (SWATNet) funded by the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 955620.

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##  Appendix A The pressure and temperature moments

We repeat here shortly the definitions given in Paschmann et al. (1998) and
Scherer et al. (2021). The partial parallel and perpendicular thermal
pressures are given as follows:

$\displaystyle P_{th}=$
$\displaystyle\frac{1}{3}\left(\sum\limits_{i}P_{\parallel,i}+2\sum\limits_{i}P_{\perp,i}\right)\qquad
i\in\\{c,h,s\\}$ (12) $\displaystyle\Rightarrow$ (13)

The total pressure $P_{tot}$ is the sum of the thermal pressure plus (twice)
the ram pressures (see Scherer et al. (2021) for further explanation).

The respective temperatures are given by the ideal gas law as

$\displaystyle T_{\parallel,i}$
$\displaystyle\equiv\frac{P_{\parallel,i}}{k_{B}n_{i}}\,,\qquad\qquad
T_{\perp,i}\equiv\frac{P_{\perp,i}}{k_{B}n_{i}}\,,\qquad i\in\\{c,h,s\\}$
(14a) with $k_{B}$ denoting Boltzmann’s constant, and $\displaystyle
T_{\parallel,t}=\sum\limits_{i}\frac{n_{i}T_{\parallel,i}}{n_{t}}\,,\qquad\qquad
T_{\perp,t}=\sum\limits_{i}\frac{n_{i}T_{\perp,i}}{n_{t}}\,,$ (14b) where
number density $n_{t}$ is given by $\displaystyle
n_{t}=\sum\limits_{i}n_{i}\,,\qquad i\in\\{c,h,s\\}\,.$ (14c)

###  A.1 Representation method

In the graphical representations we divide the $x$\- and $y$-axis in 100 sub-
intervals $\Delta x_{i}$ and $\Delta y_{j}$, and compute the total number of
events (data points) $N_{all}$ in each interval $N_{i,j}/(\Delta x_{i}\Delta
y_{j})$ via

$\displaystyle N_{all}=\sum\limits_{i,j}N_{i,j}/(\Delta x_{i}\Delta y_{j})\,.$
(15)

As stated above, we only discuss the macroscopic parameters for the most
relevant distribution functions, according to their BFs, that is,
$f_{all},f_{111},f_{121}$, and $f_{131}$, where $f_{all}$ are all the BFs
including all distributions $f_{ijk}$. We always show first the total, core,
halo and superhalo/strahl moments for $f_{all}$, and then we show how the
moments for the single distributions $f_{111},f_{121}$ and $f_{131}$
contribute to $f_{all}$.