# Jovian electrons in the inner heliosphere: Opportunities for Multi-
spacecraft Observations and Modeling

R. D. Strauss11affiliation: Center for Space Research, North-West University,
Potchefstroom, South Africa 22affiliation: National Institute for Theoretical
and Computational Sciences (NITheCS), South Africa , N. Dresing33affiliation:
Department of Physics and Astronomy, University of Turku, Turku, Finland , N.
E. Engelbrecht11affiliation: Center for Space Research, North-West University,
Potchefstroom, South Africa 22affiliation: National Institute for Theoretical
and Computational Sciences (NITheCS), South Africa , J. G.
Mitchell44affiliation: NASA Goddard Space Flight Center, Greenbelt, USA , P.
Kühl55affiliation: Institut für Experimentelle und Angewandte Physik,
Christian Albrechts-Universität zu Kiel, Germany , S. Jensen55affiliation:
Institut für Experimentelle und Angewandte Physik, Christian Albrechts-
Universität zu Kiel, Germany , S. Fleth55affiliation: Institut für
Experimentelle und Angewandte Physik, Christian Albrechts-Universität zu Kiel,
Germany , B. Sánchez-Cano66affiliation: School of Physics and Astronomy,
University of Leicester, Leicester, UK , A. Posner77affiliation: NASA/HQ,
Washington, DC, USA , J.S Rankin88affiliation: Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08540, USA , C. O.
Lee99affiliation: University of California at Berkeley, Berkeley, CA, USA ,
J.P. van den Berg11affiliation: Center for Space Research, North-West
University, Potchefstroom, South Africa , S.E.S. Ferreira11affiliation: Center
for Space Research, North-West University, Potchefstroom, South Africa , B.
Heber55affiliation: Institut für Experimentelle und Angewandte Physik,
Christian Albrechts-Universität zu Kiel, Germany

###### Abstract

In this paper we explore the idea of using multi-spacecraft observations of
Jovian electrons to measure the 3D distribution of these particles in the
inner heliosphere. We present simulations of Jovian electron intensities along
selected spacecraft trajectories for 2021 and compare these, admittedly
qualitatively, to these measurements. Using the data-model comparison we
emphasize how such a study can be used to constrain the transport parameters
in the inner heliosphere, and how this can lead to additional insight into
energetic particle transport. Model results are also shown along the expected
trajectories of selected spacecraft, including the off-ecliptic phase of the
Solar Orbiter mission from 2025 onward. Lastly, we revisit the use of
historical data and discuss upcoming missions that may contribute to Jovian
electron measurements.

cosmic rays — diffusion — Sun: heliosphere — solar wind — turbulence

=3 The Friends of AASTeX Collaboration

##  1 Introduction

Jupiter is the main source of a few MeV quiet-time electrons in the inner
heliosphere. At Earth, the intensities of these Jovian electrons are
characterized by a $\sim 13$ month periodicity due to the synodic period of
Jupiter (e.g. Chenette, 1980; Moses, 1987). Direct measurements by the Pioneer
10 spacecraft confirmed that these particles are indeed of Jovian origin (e.g.
Eraker, 1982). This $\sim 13$ month periodicity is due to changing levels of
magnetic connection between the observer (in this case near-Earth spacecraft)
and Jupiter, as illustrated in Fig. 1. When there is a good magnetic
connection, particles can travel predominantly along the Parker (1958) spiral.
During times of bad magnetic connection (the right panel of the figure),
particles need to be scattered perpendicular to the mean field to reach the
observer. This is usually a much slower process than field-aligned transport
(even with the addition of pitch-angle diffusion) so that particles spend much
more time being adiabatically cooled during their propagation, and
subsequently reach the observer with much lower intensities (see eg. Vogt et
al., 2022, and references therein). Vogt et al. (2020) also show that the
residence time (mean propagation time to Earth) of MeV Jovian electrons is
about 5 and 10 days for good and poor connection, respectively.

Figure 1: Examples of modelled pseudo-particle trajectories (three different
realizations of the numerical scheme are indicated by different colours) of 4
MeV Jovian electrons propagating between Jupiter and Earth. The Sun is located
at the origin, Jupiter at a distance of 5.2 au away from the Sun to the right
(the position indicated by a filled circle), and Earth at a radial distance of
1 au from the Sun (the position again indicated by a filled circle).
Simulations are performed for times of relatively good (left panel) and
relative bad magnetic connection (right panel) between the planets where the
spiral line indicated the Parker HMF line connected to Jupiter. The results
are taken from Strauss et al. (2011).

The Ulysses spacecraft, with it’s highly inclined trajectory (e.g. Heber et
al., 1997), measured Jovian electrons propagating to high heliospheric
latitudes. Onboard Ulysses were a number of scientific instruments of which
the Kiel Electron Telescope (KET) covers a wide range of electron fluxes from
about 2.5 MeV to 6000 MeV (Simpson et al., 1992; Heber et al., 2001, 2002).
Assuming a quasi-constant Jovian source (e.g. Simpson et al., 1974; Eraker,
1982) located in the inner heliosphere, Jovian electrons may serve as a very
good probe of interplanetary transport conditions and the underlying processes
influencing their transport (e.g. Ferreira et al., 2001b). Subsequent
modelling of these particles by e.g. Ferreira et al. (2001a) and Moeketsi et
al. (2005) showed that these particles reach high latitudes due to efficient
perpendicular scattering in the meridional plane. Such studies, where Jovian
electron measurements are compared to simulation results, have led to a number
of similar insights regarding particle transport processes through turbulent
plasmas (see also Kissmann et al., 2003; Ferreira et al., 2004). Furthermore,
Jovian electron observations, due to the larger parallel mean free paths and
relatively drift-free transport of these particles, have been employed to
investigate the structure of the heliospheric magnetic field. Sternal et al.
(2011), for instance, combined electron transport modelling with careful
analyses of observations to find evidence for a Fisk (1996)-type heliospheric
magnetic field in Ulysses count rates of 2.5–7 MeV electrons.

The decentral location of the Jovian source, in combination with the Parker
heliospheric magnetic field geometry and the essentially drift-free
propagation of these particles, provides an almost ideal opportunity for the
investigation of low-energy electron diffusion coefficients in particular,
through careful modelling of observed Jovian intensities. Essentially,
perpendicular diffusion would be the dominant mechanism in their transport
during periods of poor magnetic connectivity of the observer with the Jovian
source, while parallel diffusion dominates during magnetically well-connected
intervals. To date, relatively few studies have taken advantage of this
phenomenon, as the majority of studies reporting on particle mean free paths
focus on modelling the transport of solar energetic particles (SEPs), or
galactic cosmic rays (for a brief review of these, see Engelbrecht et al.,
2022). Chenette et al. (1977) do take advantage of this to provide one of the
earliest observational estimates for the electron perpendicular mean free
path, with subsequent studies providing refined estimates including values for
the electron parallel mean free path (Vogt et al., 2020, 2022), and estimates
for the potential rigidity dependence of these particle’s perpendicular mean
free paths (Zhang et al., 2007). Engelbrecht et al. (2022) report on radial
and rigidity dependences, as well as values at $1$ au, for electron parallel
and perpendicular mean free paths which yield model results of spacecraft
observations of both good and poor magnetic connectivity with Earth. These
could then be compared with predictions for these quantities calculated from
various scattering theories. However, it should be noted that those authors
report a degeneracy in the parameter set required to fit single spacecraft
observations, which may be disentangled via comparisons with data sets
originating from multiple spacecraft, as described here.

Recently, Mitchell et al. (2022) reported measurements inside of $\sim 0.5$ au
made by the Parker Solar Probe (PSP) spacecraft. Although Jovian electron
measurements have been reported at such small radial distances before (e.g.
Eraker & Simpson (1979) reports Mariner 10 observations just inside 0.5 au
during the Mercury encounter), the clear signal from PSP and the peculiar time
profile have led to renewed interest in the propagation of Jovian electrons to
the very inner heliosphere. Similar to the earlier Ulysses results, Jovian
electrons from such a unique observational position, compared to extensive
modelling results, can again lead to insight and potential breakthroughs
regarding particle transport. Moreover, there is currently a fleet of
spacecraft in the inner heliosphere that have the ability to measure Jovian
electron intensities, including, PSP, STEREO A (STA), SOHO, BepiColombo
(hereinafter Bepi), Solar Orbiter, and MAVEN. Such multi-spacecraft Jovian
electron measurements have the potential to provide precisely the multiple
sets of observations required to provide new insights into MeV electron
transport.

In this paper we simulate the intensity of Jovian electrons along a number of
these spacecraft trajectories for 2021. This time interval was still in the
minimum of solar activity with only a couple of SEP events observed, mainly
towards the second part of the year. These simulations are qualitatively
compared to various in-situ measurements and we evaluate the ability of each
spacecraft to accurately measure Jovian electrons, and also the ability of the
model to reproduce the correct temporal variations. Thereafter, we perform
long-term simulations of the Jovian electron intensity along a few selected
spacecraft trajectories to show what type of time profiles can be expected in
the future. We do the same for historical spacecraft (Helios 1 and 2, and
near-Earth spacecraft) as motivation to re-analyze these older data. We close
the paper with a section devoted to a discussion of upcoming missions that can
contribute future Jovian electron measurements, and various avenues for future
work.

##  2 Spacecraft measurements

In this section we briefly discuss the different spacecraft missions and
instruments used to compile Jovian electron intensities for 2021. We have not
filtered out contributions from SEP events, which leads to significant
contamination in the second half of 2021. Additionally, the different
instruments provide electron intensities at different energies, making a
detailed quantitative study impractical at the moment. However, for the
purposes of this initial qualitative study, these issues will only effect the
results in a quantitative fashion and will therefore not affect the main
conclusions of the paper.

###  2.1 Parker Solar Probe

As mentioned above, Mitchell et al. (2022) recently reported observations of
Jovian electrons observed by the High-Energy Telescope (HET) instrument in
EPI-Hi within the Integrated Science Investigation of the Sun (IS$\odot$IS)
instrument suite on board PSP. The HET instrument utilizes a standard dE/dx
versus residual energy measurement technique to measure electrons in the
energy range $\sim$0.5 - 6 MeV. Further details on the IS$\odot$IS/EPI-Hi
instruments are provided by McComas et al. (2016) and Wiedenbeck et al.
(2017). Due to the several-month orbital period, as well as the high angular
(and radial) velocity and eccentricity of the PSP orbit, the spacecraft is
nominally magnetically connected to Jupiter for a brief ($\sim$10-15 day) time
period during each orbit. For roughly the first half of the PSP mission, these
brief periods of nominal connectivity occurred as the spacecraft was leaving
its closest approaches with the Sun as determined by the positions of the
spacecraft and Jupiter, as well as the average solar wind velocity during that
time.

###  2.2 Solar Orbiter

The High Energy Telescope (HET) of Solar Orbiter is part of the Energetic
Particle Detector (EPD; Rodríguez-Pacheco et al., 2020) suite. It consists of
two near-identical units located on the spacecraft with orthogonal viewing
directions. Each HET unit is a double-sided telescope stack consisting of a
central bismuth germanium oxide (BGO) scintillator (C) and two solid state
detectors (SSDs) on each side. Therefore, HET is capable of covering four
viewing directions in total. The four SSDs are segmented, with the outer
segments acting as anticoincidence. HET utilizes the dE/dx-E method and the
dE/dx-dE/dx method for stopping and penetrating particles, respectively. The
HET instrument is designed to cover the energy range from a few MeV/nuc up to
a few hundred MeV/nuc for ions (see Fig. 3 in Rodríguez-Pacheco et al., 2020)
and relativistic electrons to above 10 MeV. Additional information and updates
are also given in Wimmer-Schweingruber et al. (2021).

###  2.3 STEREO A

The High Energy Telescope (HET; von Rosenvinge et al., 2008) carried by both
spacecraft of the STEREO mission (Kaiser et al., 2008) measures protons in the
energy range of 13 to 100 MeV, and electrons from 0.7 to 4 MeV. We use STEREO
A/HET measurements of the highest electron energy channel E3, providing
2.8–4.0 MeV electron intensities. The instrument provides only one viewing
direction, which was aligned with the nominal direction of the Parker spiral
magnetic field at the start of the STEREO mission. However, since the
spacecraft was put upside down after its solar superior conjunction in
2014–2015, HET points perpendicular to the nominal Parker spiral.

###  2.4 BepiColombo

The BepiColombo Environment Radiation Monitor (BERM) on board the BepiColombo
mission (Benkhoff et al., 2010) is an instrument that is part of the
housekeeping suite of the European Mercury Planetary Orbiter (MPO). BERM is
designed to measure the radiation environment encountered by BepiColombo
during its cruise phase and after orbit insertion at Mercury, being one of the
few instruments that is operated continuously since its launch in 2018 (Pinto
et al., 2022). BERM measures electrons with energies from ${\sim}$170 keV to
$\sim$10 MeV, protons from $\sim$1.35 MeV to $\sim$160 MeV, and heavy ions
with Linear Energy Transfer from 1 to 50 MeV·mg-1·cm2. In this study, we use
the electron channel E4 that is a integral channel sensitive to electrons with
energies 1.1-$\infty$ MeV (Pinto et al., 2022).

###  2.5 SOHO

The Electron Proton Helium INstrument (EPHIN) is part of the Comprehensive
Suprathermal and Energetic Particle Analyzer (COSTEP; Müller-Mellin et al.,
1995) experiment aboard the SOHO spacecraft. The SOHO mission was launched in
December 1995 and is in orbit around the Lagrangian point L1. EPHIN consists
of 6 solid-state silicon detectors, labeled A-F from top to bottom, surrounded
by a scintillator tube that serves as anti-coincidence. The energy range for
stopping electrons extends from 0.2 MeV to about 10 MeV. The data is based on
the E1300 coincidence channel. Since 2017, the instrument has been operating
in failure modes D and E which removes the two lowest detectors from any
coincidence logic. This results in a wide energy range for the E1300. We
performed a _bow-tie_ analysis on this channel for a more sophisticated
calculation of the response factor. Assuming a power-law spectrum with
spectral indices between -2.5 and -1 the effective energy of this channel is
$E_{eff}=1.8$ MeV. Also the data was filtered for SEP events as discussed in
Sec. A of the appendix.

###  2.6 MAVEN

The Solar Energetic Particle (SEP) instrument on aboard the Mars Atmosphere
and Volatile EvolutioN (MAVEN), which provides the energy-flux spectra of high
energy particles (ions and electrons), is used (Larson et al., 2015). In
particular, data from the SEP1 sensor’s forward looking direction, which is
nominally pointing at the Parker spiral. Here we use the integrated SEP1
electron flux in the energy range 100 – 200 keV.

##  3 Jovian electron transport modelling

For the work presented here, we use the model of Strauss et al. (2011),
rewritten by Dunzlaff et al. (2015), to solve the Parker (1965) transport
equation and simulate the intensities of Jovian electrons in the inner
heliosphere. This model was also used previously to calculate Jovian electrons
propagation times (Strauss et al., 2013), a calculation later refined by Vogt
et al. (2020). For this model we adopt a Parker (1958) heliospheric magnetic
field with a nominal solar wind speed of 400 km/s. The Jovian electron source
is introduced in the model by specifying the Jovian source function of Vogt et
al. (2018) at the position of Jupiter.

Here we use a relatively ad hoc expressions, consistent with those used by
e.g. Vogt et al. (2022), to model the transport coefficients of low-energy
electrons, given by

$\lambda_{||}(r)=\frac{\lambda_{0}}{2}\left(1+\frac{r}{r_{0}}\right),$ (1)

where $\lambda_{0}=0.1$ au is a reference value at $r_{0}=1$ au. For the
perpendicular mean free path it is assumed that
$\lambda_{\perp}=\chi\lambda_{||}$ with $\chi=0.015$, in an approach similar
to that of Ferreira et al. (2001a). As there is some uncertainty in these
values, we also include a $25\%$ error bar, so that $\lambda_{0}=0.1\pm 0.025$
au and $\chi=0.015\pm 0.00375$. Note that the diffusion coefficient $\kappa$
is related to the above mean free paths by $\kappa=v\lambda/3$ in the standard
way (see, e.g., Shalchi, 2009). Although these expressions are considerably
simpler than those yielded by various scattering theories (see, e.g.,
Engelbrecht et al., 2022, and references therein), there are several
advantages implicit to their use. Primarily, they yield values at $1$ au, as
well as energy and radial dependencies, consistent with what is expected from
prior modelling endeavours and theory, such that the results for free
parameters can, in principle, be compared to theory to draw conclusions as to
the behaviour required of these coefficients so as to fit various sets of
observations (e.g. Palmer, 1982; Bieber et al., 1994; Dröge, 2000; Engelbrecht
et al., 2022). A secondary advantage lies in the tractability of these
expressions relative to those derived from first principles. It should be
noted that, as a first approach, no solar cycle dependence is assumed in the
transport coefficients employed here. Such a dependence has been inferred in
the past (Kanekal et al., 2003), and is expected from theory, given the
dependence of particle mean free paths on observed solar cycle-dependent
turbulence quantities (e.g. Zhao et al., 2018; Burger et al., 2022), which in
turn has been demonstrated to influence the modulation of galactic cosmic ray
protons (e.g. Moloto & Engelbrecht, 2020). Modelling these effects, however,
would require modelling electron mean free paths and the turbulence quantities
they depend on from first principles (e.g. Engelbrecht & Strauss, 2018;
Engelbrecht, 2019), which remains beyond the scope of the present study.
However, as shown by Kanekal et al. (2003), any such time-dependent modulation
effects are generally small for Jovian electrons and the changing magnetic
connectivity between the Jovian source and the spacecraft remains the dominant
effect. This is supported by the results of prior simulations investigating
the residence times of Jovian electrons versus those of higher energy galactic
cosmic ray electrons: Strauss et al. (2011) report galactic cosmic ray
electron residence times of several hundred days, as opposed to residence
times of only a few days for Jovian electrons (see also Vogt et al., 2020,
2022). As the Jovian residence times are very small relative to the typical
solar cycle-related variation times of heliospheric plasma parameters, it
would be reasonable to neglect temporal variations on these time scales, and
the more significant influence of the relatively shorter 13-month periodicity
associated with varying magnetic connectivity could also be expected to be
more significant. It should also be noted, however, that caution needs be
taken when drawing conclusions as to low-energy ($\sim$ MeV) electron
modulation by comparing modulation effects observed for high energy galactic
cosmic ray electrons and Jovian electrons. This is due to the fact that these
particles are influenced by different transport mechanisms. For instance, the
transport of high energy galactic cosmic ray electrons is significantly
influenced by drift effects, while that of Jovians is not (see, e.g., Ferreira
et al., 2001a; Engelbrecht, 2019). Furthermore, the diffusion coefficients of
these particles also differ significantly (e.g. Engelbrecht et al., 2022, and
references therein). These latter points, when considered in combination with
the fact that Jovian electrons are exposed to a modulation volume considerably
smaller than that experienced by galactic electrons, further support the
observational evidence presented by Kanekal et al. (2003) for the predominant
influence of magnetic connectivity on time scales short relative to solar
cycle time scales on the transport of Jovian electrons.

###  3.1 Initial modelling results

Figure 2: Selected spacecraft trajectories for 2021. In the bottom panels,
these trajectories are overlaid onto simulation results for 4 MeV Jovian
electrons with the Parker spiral field connected to Jupiter also shown. The
colorbar shows the intensity, normalized to 100%, of Jovian electrons on a
logarithmic scale. Figure 3: The lines and shaded uncertainty regions are
modelling results for 4 MeV Jovian electrons along the selected spacecraft
trajectories. These modelled intensities are in units of /m2/s/sr/MeV.
Spacecraft measurements are included as the scatter points with varying units:
PSP/HET: counts/s, SolO/HET: /cm2/s/sr/MeV, STA/HET: /cm2/s/sr/MeV, Bepi/BERM:
counts/s, SOHO/EPHIN: /cm2/s/sr/MeV, MAVEN/SEP1: /cm2/s/sr/keV. The red
vertical lines on the SolO panel indicate the occurrence of multi-spacecraft
SEP events from the SERPENTINE SEP catalog.

The top panel of Fig. 2 shows the trajectories, in terms of radial distance
and longitude, of the selected spacecraft in 2021 as taken from
https://omniweb.gsfc.nasa.gov/coho/helios/heli.html. The shaded regions
indicate times when the different spacecraft are magnetically well-connected
to the Jovian source; see also Fig. 3. The color of the shading corresponds to
that of the figure legend. In the bottom panel we again show the trajectories
(with a zoomed in image on the right), overlaid onto the simulated 4 MeV
Jovian electron intensity as taken from Strauss et al. (2011). The motion of
the Jovian source is also indicated on the figures and also included into the
model calculations.

We now read the selected spacecraft trajectories, using a daily cadence, into
the model of Strauss et al. (2011) and compute the intensity of 4 MeV
electrons. For the time being, we neglect any temporal changes in both the
heliosphere (e.g. differing solar wind speeds) and in the transport
coefficients. These results are shown in the different panels of Fig. 3 for
each spacecraft trajectory separately. The results using the nominal value of
$\lambda=0.1$ au are shown as the thick lines, while the shaded ranges
indicate the uncertainty introduced by varying the transport coefficients by
25%. Again, the vertical shaded regions indicate times when the spacecraft are
magnetically well connected to Jupiter.

The measurement from the different spacecraft, summarized in the previous
section, are included as the scatter points. As some of these measurements are
not converted to a differential intensity (e.g. the counting rate from PSP)
and as a thorough inter-calibration has not yet been performed (the energy
bins of the spacecraft are also different), these intensity values are
included on a separate axis on the right and scaled by eye to compare with the
modelling results. While a more detailed quantitative comparison definitely
needs to be done in the future, we are still able to qualitatively identify
the general trends in both the simulation results and the measurements using
the simplified approach outlined here. For most of these detectors typical
uncertainties are not routinely calculated and/or given. Statistical errors
are usually quoted for some instruments but these are, at best, a lower limit
for the actual uncertainty. As such we did not consider any data uncertainties
in this work.

The modelled profile at the position of SOHO (green curves) and STA (purple
curves) show the expected $\sim 13$ month periodicity. The measurements at STA
(green symbols) also reflect this profile, albeit with some scatter in the
second half of the year due to multiple SEP events. SOHO (purple symbols),
however, does not see the expected trend. It has to be noted though, that the
electron flux observed by SOHO is dominated by SEPs except for the first third
of the year. The red vertical lines in the SolO panel indicate the occurrence
of 25–40 MeV proton SEP events that were observed by at least two spacecraft,
taken from the SERPENTINE SEP catalog
111https://data.serpentine-h2020.eu/catalogs/sep-sc25/. The absence of a clear
variation in the first months could have been caused by unfavorable solar wind
speeds and/or interaction regions preventing a magnetic connection to Jupiter.
Due to Mars’ slow orbital motion, the MAVEN simulations (blue curves) show a
very weak Jovian electron signal. The corresponding measurements (red symbols)
do not show any clear trend and are mostly consistent with constant fluxes.
The faster orbiting spacecraft show quasi-periodically varying intensities.
The SolO simulation (yellow curves) shows a periodicity of $\sim 250$ days,
BepiColombo (black curves) $\sim 200$ days, and PSP (red curves) a period of
$\sim 100$ days. The BepiColombo measurements (yellow symbols) show a hint of
a Jovian signal, which may be enhanced by means of a more careful removal of
SEP events. A very clear signal is seen in both SolO (gray symbols) and PSP
(blue symbols) measurements. The simulated (and measured) PSP time profile is
particularly interesting. Due to PSP’s highly eccentric orbit, the time
profile does not show sinusoidal variations, but rather a pulse-like profile.
This can be understood by referring to the bottom right panel of Fig. 2: The
Jovian intensity profile at a spacecraft, in the ecliptic plane, is determined
by both its radial and azimuthal position. While the radial changes are
usually negligible for spacecraft at e.g. Earth’s orbit, PSP’s measurements
are highly sensitive to these changes. In 2021, just before perihelion is
reached, PSP measures a minimum Jovian intensity. This is because it is not
magnetically connected to Jupiter, but it is also very near to the Sun and
hence far from Jupiter. Just past perihelion, PSP is magnetically well-
connected to Jupiter and measures a peak in the time profile, even though it
is still near the Sun. The maximum Jovian intensity along PSP’s orbit will
only be measured during a magnetically well-connected aphelion. Such times are
considered in the next section.

##  4 Long term simulations

Figure 4: Long-term simulations of 4 MeV Jovian electrons at the position of
Earth. Times of best magnetic connection between the spacecraft and Jupiter
are also indicated by vertical shading.

Figure 5: Long-term simulations of 4 MeV Jovian electrons at the position of
PSP (left, in red) and BepiColombo (right, in black). Times of best magnetic
connection between the spacecraft and Jupiter are also indicated by vertical
shading. Figure 6: Long-term simulations of 4 MeV Jovian electrons at the
position of SolO. Times of best magnetic connection between the spacecraft and
Jupiter are also indicated by vertical shading.

Here, we repeat the calculations shown in the previous section, but use the
position of selected spacecraft from 2021 to 2025 (for PSP and BepiColombo)
and, for Earth and SolO, from 2021 to 2030. These trajectories and the
resulting model calculations are shown in Figs. 4, 5, and 6.

The simulated intensity at Earth shows, again, the expected $\sim 13$ month
periodicity, but interestingly also shows the effect of a changing radial
position: The absolute maximum Jovian intensity for this time period, is
predicted in 2023 when Earth is at aphelion during the best magnetic
connection with Jupiter, while the maximum during late 2028, when Earth is at
perihelion during best magnetic connection, is 25% lower. This is, of course,
much smaller than the factor $\sim 3$ change that results from the changing
magnetic connection (i.e. the difference from minimum to maximum during a
$\sim 13$ month period).

The simulated profiles along PSP’s future trajectory show the effect of a
highly eccentric trajectory, with the Jovian intensity profile changing
significantly from 2021 to e.g. 2024, where the best magnetic connection will
be measured at aphelion. Similar changes are also predicted along the
BepiColombo profile, albeit not as accentuated as for PSP.

For the simulations along SolO’s trajectory, as shown in Fig. 6, we also
include the effect of SolO’s inclined orbit later in the mission. Up to
$\sim$2028, expected Jovian intensity profiles display sharply peaked profiles
similar to what is seen for PSP (Fig. 5), for similar reasons. Beyond these
times, the model yields more gradual, less pulse-like profiles more
reminiscent of what is expected for, say, BepiColombo, with a periodicity of
$\sim 135$ days. Care should, however, be taken in the interpretation of these
latter profiles. During the period of interest, the solar activity cycle would
be descending towards solar minimum, and as such the assumption made here of a
latitudinally-constant solar wind speed would be less valid (see, e.g.,
McComas et al., 2000). For instance, higher solar wind speeds at higher
latitudes would influence the heliospheric magnetic field winding angle,
thereby altering the level (and period) of magnetic connectivity to the
source. This latter phenomenon would be further complicated due to the
possible presence of a Fisk-type heliospheric magnetic field (see Steyn &
Burger, 2020, and references therein), which could significantly affect the
transport of Jovian electrons (e.g. Sternal et al., 2011). Lastly, it should
be noted that particle transport coefficients, due to both the solar cycle
dependence of turbulence parameters and the differing nature of turbulence in
the fast solar wind versus the slow solar wind (e.g. Forsyth et al., 1996;
Bavassano et al., 2000a, b), would be expected to be considerably different to
those applicable to slow solar wind conditions (e.g. Engelbrecht & Burger,
2013).

##  5 Historical, Upcoming, and Anticipated measurements

Figure 7: Historical simulations of 4 MeV Jovian electrons at the position of
Earth (green), Helios 1 (red), and Helios 2 (blue) from 1975 until 1985

###  5.1 IMP, Helios 1, and Helios 2

As a last model application we solve the model to obtain the 4 MeV Jovian
electron intensity along the trajectories of Earth (the position of the
Interplanetary Monitoring Platform, IMP, spacecraft), Helios 1, and Helios 2,
and present the results of these simulations in Fig. 7.

###  5.2 HEPD-02

The second High Energy Particle Detector (HEPD-02), planned to be aboard the
second China Seismo-Electromagnetic Satellite (CSES-02), is specifically
designed to detect cosmic ray protons with energies ranging from 30-200 MeV
and electrons with energies from 3-100 MeV, improving on the capabilities of
HEP-01 aboard CSES-01 (see, e.g., Bartocci et al., 2020; Scotti & CSES-Limadou
Collaboration, 2023). Furthermore, information as to detected particle’s pitch
angles, albeit relative to the Earth’s magnetic field, will also be available
(De Santis et al., 2022). This mission is expected to launch in 2023.

###  5.3 IMAP

The upcoming Interstellar Mapping and Acceleration Probe (IMAP) mission
(McComas et al., 2018) will continue relativistic electron measurements in the
near-Earth solar wind with its High-energy Ion Telescope (HIT). Measurements
from IMAP can serve as a future baseline value at Earth, and provide the
linkage between ageing detectors on SOHO and ACE. IMAP is scheduled to be
launched in 2025 into an orbit around the Sun–Earth L1 Lagrange point.

###  5.4 ARTEMIS and Space Weather Needs

It has been shown that MeV electron measurements can be exploited for space
weather purposes (Posner, 2007). The premise of the Relativistic Electron
Alert System for Exploration (REleASE) algorithm is that all major SEP events
release relativistic electrons at the onset, which arrive at 1 au tens of
minutes before the more hazardous ions. This property of relativistic
electrons is particularly relevant for the upcoming ARTEMIS era, during which
the human presence on the Moon and in transit to and from Mars requires
advance-warning of radiation hazards. The current implementation of REleASE,
HESPERIA (High Energy Solar Particle Events forecasting and Analysis) –
REleASE (Malandraki & Crosby, 2018) uses ACE/EPAM observations from the Earth-
Sun Lagrange point 1 (L1) that are available in real time. The system is part
of the SEP Scoreboard effort and as such is undergoing near-real-time testing.
Current studies show that additional needs for relativistic electron
observations from other vantage points exist. For example, human exploration
of Mars would require a Mars upstream monitor (e.g. Posner & Strauss, 2020;
Lee et al., 2023). There are considerations to equip the transfer vehicle
humans would use on their outward journey to Mars and return to Earth with
relativistic electron detectors. Other space weather missions under
consideration would lead to the Sun–Earth Lagrange points L4 (Posner et al.,
2021) and L5 (Akioka et al., 2005) and could be equipped with MeV electron
instrumentation, although their use is not directly driven by space weather
needs. Reliable, high-quality MeV electron measurements at multiple, well
thought-out locations in the heliosphere would not only benefit human
exploration, but would support in-depth Jovian electron studies. From a
physics point-of-view, Jovian electrons and SEP electrons propagate in exactly
the same fashion through the turbulent interplanetary medium. One can
therefore make the argument that dedicated space weather missions, and their
users, will also benefit from better understanding electron transport
conditions in the heliosphere, and their potential variability, that we reach
through studying Jovian electron transport.

###  5.5 Jupiter Icy Moons Explorer

JUpiter ICy moons Explorer (Juice) is an ESA-mission, launched on 14 April
2023, to study the Jovian system. On board is the PEP (Particle Environment
Package), a particle spectrometer to measure neutral and charged particles in
the Jupiter system, including low energy Jovian electrons and other energetic
particle populations (Pinto et al., ????; Barabash et al., 2022). Measurements
taken during this mission could be used to revisit the form of the Jovian
electron source spectrum employed in transport studies, providing an
invaluable additional constraint for future modelling endeavours.

##  6 Discussion

In this paper we discuss how multi-spacecraft Jovian electron measurements can
be used to study the fundamental transport processes that ultimately determine
their intensities throughout the heliosphere. We show an example of such a
study using a simplified model that assumes relatively ad hoc transport
coefficients in a time-stationary heliosphere. Such a study would need
multiple sets of observations at various locations, taken during overlapping
periods of time so as to disentangle solar-cycle and other temporally-
dependent influences on transport coefficients (see, e.g., Engelbrecht et al.,
2022). Careful comparisons of the results of such studies with theory would
not only improve our understanding of the theoretical underpinnings of
electron diffusion, but could also give indirect measures of hard-to-measure
turbulence quantities pertaining to the dissipation range of solar wind
turbulence, as for example it is known that low-energy electron parallel mean
free paths are strongly dependent on both the dissipation range onset
wavenumber and spectral index (see, e.g., Engelbrecht & Burger, 2013). This
would also have the potential to provide a novel, albeit indirect measure of
the solar-cycle dependence of these quantities.

We present a comparison between different spacecraft measurements of Jovian
electrons and simulation results specifically for 2021. The model is able to
capture the observed Jovian electron features very well, especially the
somewhat peculiar time profiles that were observed by PSP. The PSP results
confirm the ability of Jovian electrons to reach small heliocentric distances
without being completely impeded by the outward moving solar wind. However,
for the majority of the time periods during which PSP was nominally
magnetically connected to Jupiter, including the second perihelion of 2021,
IS$\odot$IS did not observe a coincident Jovian electron enhancement as
predicted by the model. Several potential reasons for the absence of Jovian
electron enhancements during these periods are explored in Mitchell et al.
(2022), including impediments from stream interaction regions (SIRs),
modulation of the Jovian source by the solar wind dynamic pressure, and the
obstruction of the flow of Jovian electrons by the heliospheric current sheet
(HCS) when the spacecraft and Jupiter lie on opposite sides of the HCS. The
later time profile along PSP’s trajectory is predicted to differ substantially
from that measured in 2021 due to PSP’s highly eccentric orbit. The maximum
intensities measured during each orbit are predicted to occur near aphelion
later in the mission, resulting in a more constant profile with a deep
depression near perihelion. The predicted intensity profile from 2022 to $\sim
2025$ at SolO’s position is particularly interesting to study due to a sharply
peaked profile near perihelion.

We note that some spacecraft measurements, e.g. those of SOHO and BepiColombo,
do not show a clear Jovian signal. It is not yet entirely clear as to why this
is the case, and future studies, combining both numerical transport modelling
and further data analyses, will investigate this phenomenon in more depth. We
also note that for BepiColombo, the area of the entrance to the top detector
is very small, 0.5 mm2 (Pinto et al., 2022) and the field of view is
$40^{\circ}$, which may be contributing to a reduce rate of detection of
Jovian particles, and which statistic uncertainty will be considered in future
analysis.

While the proposed multi-spacecraft Jovian electron studies are compelling,
long-term historical data-sets can also yield novel insights into phenomena
observed in the newer observations. Here we show, as an example, the expected
profiles along selected spacecraft trajectories from 1975 – 1985. These
measurements show the same features as their modern-day counterparts. Similar
multi-spacecraft Jovian electron studies would therefore also be possible by
using historical data, and would accordingly require more careful modelling of
the temporal dependence of these particle’s transport coefficients and the
turbulence quantities they depend upon. Although this has been done to a
degree in various galactic cosmic ray proton studies, to date no such physics-
first studies pertaining to the time-dependent transport of MeV electrons
exist.

Several planned or proposed future missions will have the capability to
measure Jovian electrons. These valuable measurements can help to constrain
the transport processes of MeV electrons through the turbulent interplanetary
medium. We argue that such studies will also be beneficial to space weather
studies, e.g. forecasting SEP events, as well as other human space-borne
endeavours, and that dedicated space weather missions could, and should, also
be leveraged to measure Jovian electrons.

The modelling endeavours discussed in this paper can be refined and improved
in several ways. Firstly, the models and parameters employed for large-scale
plasma quantities, such as the solar wind speed and heliospheric magnetic
field, can be brought closer to spacecraft observations in terms of, say,
solar cycle dependencies, to further investigate the relative significance of
temporal effects. Jovian transport coefficients can also, for instance, be
modelled in a more fundamental way, incorporating the latest results from
various particle scattering theories (e.g. Shalchi, 2020; Engelbrecht et al.,
2022), so as to improve the ability of the model in terms of extrapolation as
well. This has already been done for galactic electrons (e.g. Dempers &
Engelbrecht, 2020; Engelbrecht, 2019), and the results of such studies can be
applied to the study of Jovians. Such an approach, however, would require
careful modelling of turbulence quantities that such transport coefficients
are functions of. For this, too, several successful turbulence transport
models exist (see, e.g., the reviews of Oughton & Engelbrecht, 2021; Adhikari
et al., 2021; Fraternale et al., 2022) that can readily be applied to such
studies.

This work is based on the research supported in part by the National Research
Foundation of South Africa (NRF grant numbers RA170929263913, SRUG220322419,
and 137793). Opinions expressed and conclusions arrived at are those of the
authors and are not necessarily to be attributed to the NRF. The
responsibility of the contents of this work is with the authors. N.D. is
grateful for support by the Academy of Finland (SHOCKSEE, grant No. 346902).
B.S.-C. acknowledges support through UK-STFC Ernest Rutherford Fellowship
ST/V004115/1 and STFC grant ST/Y000439/1. The EPD and this work was supported
by the German Space Agency (Deutsches Zentrum für Luft- und Raumfahrt, e.V.,
DLR) under grant number 50OT2002 to the Kiel University. Figures prepared with
Matplotlib (Hunter, 2007) and certain calculations done with NumPy (Harris et
al., 2020).

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##  Appendix A Filter method of EPHIN data

To better see the modulation of the electrons in the time profiles, it is
necessary to filter out the SEP events. In order to do this we used two
electron channels, the E150 ($E_{eff}=0.5$ MeV) and E1300 ($E_{eff}=1.8$ MeV)
channels, and determined the spectral slope for an assumed power-law spectrum
on hourly averages. In the top panel of fig. 8 we have plotted these spectral
indices against the ratios of the proton P4 and P25 count rates. The P4 / P25
ratio serves as a proxy of the spectral index of the protons. Since the energy
spectrum is generally softer during SEP events, we assume that electrons
outside of the red box shown in the figure are of solar origin. The data used
in this study consist only of the electrons inside the selection box, reducing
the contribution of SEP events. The bottom panel of fig. 8 shows the time
profile of the E1300 electron flux from 2019 to 2023 as a daily average. The
teal and grey dots show the filtered (i.e. red selection box) and unfiltered
data, respectively.

Figure 8: Electron spectral index plotted against the ratio of proton channels
P4 and P25 (top) and resulting time profile of E1300 flux from 2019 to 2023
(bottom).