# On the onset delays of solar energetic electrons and protons: Evidence for a
common accelerator

R.D. Strauss11affiliation: Center for Space Research, North-West University,
Potchefstroom, 2522, 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 , I.G. Richardson44affiliation: NASA/Goddard Space
Flight Center, Greenbelt, MD 20771, USA 55affiliation: Department of
Astronomy, University of Maryland, College Park, MD 20742, USA , J.P. van den
Berg11affiliation: Center for Space Research, North-West University,
Potchefstroom, 2522, South Africa , P.J. Steyn11affiliation: Center for Space
Research, North-West University, Potchefstroom, 2522, South Africa

###### Abstract

The processes responsible for the acceleration of solar energetic particles
(SEPs) are still not well understood, including whether SEP electrons and
protons are accelerated by common or separate processes. Using a numerical
particle transport model that includes both pitch-angle and perpendicular
spatial diffusion, we simulate, amongst other quantities, the onset delay for
MeV electrons and protons and compare the results to observations of SEPs from
widely-separated spacecraft. Such observations have previously been
interpreted, in a simple scenario assuming no perpendicular diffusion, as
evidence for different electron and proton sources. We show that, by assuming
a common particle source together with perpendicular diffusion, we are able to
simultaneously reproduce the onset delays for both electrons and protons. We
argue that this points towards a common accelerator for these particles.
Moreover, a relatively broad particle source is required in the model to
correctly describe the observations. This is suggestive of diffusive shock
acceleration occurring at large shock structures playing a significant role in
the acceleration of these SEPs.

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

=3 The Friends of AASTeX Collaboration

##  1 Introduction

The potential acceleration mechanisms for solar energetic particles (SEPs) are
still heavily debated. The most likely acceleration regions are magnetic
reconnection (or a closely related mechanism) in solar flares and shock
acceleration (either shock drift acceleration or diffusive shock acceleration)
related to shocks driven by coronal mass ejections (CMEs). SEP observations at
Earth have led to the (potentially oversimplified) classification of events
into impulsive (sometimes also referred to as electron-rich), that are
associated with short-duration solar flares, and gradual SEP events, that are
associated with CMEs and long-duration flares (e.g. Reames, 2013), though some
studies have suggested that there is a continuum of SEP event properties (e.g.
Cane et al., 2010). It is therefore not clear whether SEP electrons and
protons are accelerated by the same acceleration mechanism during the same
transient solar events.

Previous work has shown a clear empirical relationship between MeV electron
and proton measurements. For instance, the Posner (2007) Relativistic Electron
Alert System for Exploration (REleASE) algorithm uses an empirically
established relationship between relativistic electron intensities and
subsequent proton intensities to predict future proton levels. More recently
Dresing et al. (2022) found a similar dependence of electron and proton
intensities on shock parameters suggesting a common shock-related accelerator
for both species. Additionally, Richardson et al. (2014) found clear linear
(in logarithmic space) relationships between onset and peak delays at the
observing spacecraft relative to the onset of the related type III radio
emission for different levels of magnetic connection between the spacecraft
and the associated flare location. However, these linear relationships from
Richardson et al. (2014) are different for electrons and protons and cannot be
explained by simple ballistic motion between a single expanding source and the
observer, as also discussed by Kollhoff et al. (2021) in relation to the
widespread SEP event on November 29, 2020. If particle transport is neglected,
the different electron and proton dependencies can be explained by different
sources for these SEP electrons and protons, expanding in longitude at
different rates.

In this work we examine whether SEP electron and proton measurements at Earth
can in fact be explained by a common acceleration source when interplanetary
transport (i.e. particle scattering) is also considered. The focus is on
simulating the particle onset delays as presented by Richardson et al. (2014).
The modelling approach applied here includes perpendicular (cross-field)
diffusion. There is increasing evidence that perpendicular diffusion is an
essential transport process for SEPs. Kouloumvakos et al. (2022), for
instance, show that spacecraft that are magnetically unconnected to a CME-
driven shock can still observe a significant SEP increase. The low particle
anisotropy levels associated with such poorly connected observers (e.g.
Dresing et al., 2014) are also consistent with significant levels of cross-
field diffusion: As the perpendicular diffusion process is generally much
slower than parallel transport, the initial, highly anisotropic, beam of SEPs
predominantly propagates along the mean field without being scattered
significantly away from it. However, perpendicular diffusion becomes
increasingly effective at later times during the nearly-isotropic decay phase
of the SEP events. These nearly-isotropic particle distributions, being
scattered effectively perpendicular to the mean field later in the SEP event,
are usually associated with low levels of particle anisotropy. Additionally,
modelling by e.g. Dröge et al. (2016) has shown that significant levels of
perpendicular diffusion is needed to reproduce many observed SEP events.

##  2 The numerical transport model

Figure 1: The left panel shows the assumed slab turbulence spectrum at Earth,
as a function of the parallel wavenumber, while the shaded regions indicate
where electrons (red shading) and protons (blue shading) in the assumed energy
ranges will resonate. The right panel shows the resulting electron and proton
parallel and perpendicular mean-free-paths as used in this study. Also shown
is the magnetic focusing length.

In this work we simulate the transport of SEPs through the turbulent
interplanetary medium by solving the following focused transport equation for
the distribution function, $f$,

$\displaystyle\frac{\partial f}{\partial t}+\nabla\cdot\left(\mu
v\hat{b}f\right)+\frac{\partial}{\partial\mu}\left(\frac{1-\mu^{2}}{2L}vf\right)$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\mu}\left(D_{\mu\mu}\frac{\partial
f}{\partial\mu}\right)$ (1) $\displaystyle+$
$\displaystyle\nabla\cdot\left(\mathbf{D}^{(x)}_{\perp}\cdot\nabla f\right)$

using the approach outlined by Strauss & Fichtner (2015). Here, $\mu$ is the
particle pitch-angle cosine with respect to the mean magnetic field, directed
in the $\hat{b}$ direction (see e.g. van den Berg et al., 2020). This model
was previously applied to near-relativistic electron transport by Strauss et
al. (2017) and Strauss et al. (2020). In these previous applications, the
focus was on calculating the pitch-angle diffusion coefficient, $D_{\mu\mu}$,
and perpendicular diffusion coefficient, $D_{\perp}$, from first principles
using observed solar wind turbulence values. We continue with that approach
here, but apply the model to both SEP electron and proton transport. Although
we have tried, as far as possible, to constrain all the turbulence quantities
from measurements, we accept that there are still some uncertainties related
to these quantities and the associated diffusion coefficients calculated from
them. The left panel of Fig. 1 shows the assumed slab turbulence spectrum, as
a function of the parallel wavenumber $k_{||}$, at Earth. The red and blue
shaded regions indicate where electrons and protons in the assumed energy
ranges will, respectively, resonante. Here we use the energy ranges of
Richardson et al. (2014), namely 0.7 – 4 MeV for electrons and 14 – 24 MeV for
protons. For this illustrative example we assume that the particles will
resonante at $k_{||}\sim r_{L}^{-1}$, where $r_{L}$ is the particles’ Larmor
radius (Strauss et al., 2020). This calculation illustrates that protons and
electrons, in the assumed energy ranges, will resonate in the inertial range
of the slab turbulence, with resonate wavenumbers very close to each other. As
usual, a Parker (1958) heliospheric magnetic field (HMF) geometry is assumed
with an associated focusing length, $L$. This value, along with the resulting
parallel and perpendicular mean-free-paths for electrons and protons, using
the calculations of Strauss et al. (2017), are shown, as a function of radial
distance, in the right panel of Fig. 1. The perpendicular diffusion
coefficient is discussed in more detail in the next section. We do not include
drift effects as these are generally considered to be negligible for low
energy (i.e. $\sim$ MeV) particles (Engelbrecht et al., 2017; van den Berg et
al., 2021).

As an inner boundary condition ($r_{0}=0.05$ AU) to the model, the following
function is specified

$\displaystyle
f(r=r_{0},\phi,t)=\frac{C}{t}\exp\left[-\frac{\tau_{a}}{t}-\frac{t}{\tau_{e}}\right]\exp\left[-\frac{\left(\phi-\phi_{0}\right)}{2\sigma^{2}}\right].$
(2)

The time dependence of this function is determined by the so-called
acceleration ($\tau_{a}=1$ hr) and escape ($\tau_{e}=1$ hr) timescales, while
a Gaussian source (in terms of longitude) is specified where the broadness,
$\sigma$, can be varied. This function quantifies the assumed accelerated SEP
distribution released into the interplanetary medium from the particle
accelerator. With the uncertainty surrounding the acceleration process and the
fact that these timescales cannot be directly measured, these values are
subject to change in future simulations. While $\tau_{e}$ seemingly has only a
small effect on the simulated intensities, the calculated peak delay
(discussed later) is sensitive to the choice of $\tau_{a}$. Work is under way
to further constrain these model inputs.

###  2.1 Efficiency of perpendicular diffusion

As with previous work we assume that perpendicular diffusion is governed by
the so-called field line random walk (FLRW) process (Jokipii, 1966), leading
to a perpendicular diffusion coefficient of the form

$D_{\perp}=a|\mu|v\kappa_{FL},$ (3)

where $\kappa_{FL}$ is the so-called FLRW diffusion coefficient which
describes the diffusion of the magnetic field lines and depends on the
underlying turbulence quantities (see also Strauss et al., 2016). Note that
$D_{\perp}$ is proportional to particle speed and that, in general, faster
particles diffuse faster perpendicular to the mean HMF. The factor $a$ is
introduced to account for the fact that particles have a finite gyro-radius
and cannot perfectly follow diffusing field lines (e.g. Shalchi, 2009). The
efficiency of perpendicular diffusion in this model therefore strongly depends
on the seemingly ad hoc factor $a$ which can thus be interpreted as the
probability that particles are tied to fluctuating magnetic field lines (e.g.
Qin & Shalchi, 2014). While some progress has been made to derive $a$ from
first principles (Shalchi, 2015, 2020), in previous work, Strauss et al.
(2017) treated $a$ as a free parameter and obtained a good comparison between
SEP measurements, for 100 keV electrons, and modelling results when $a=0.2$.
Although this value is based on a comparison with a single particle species at
a single energy, we adopt $a=0.2$ for all simulations performed in this work.

##  3 Analytical Estimates and Model Results

###  3.1 Analytical Estimates

Figure 2: The left and middle panels are analytical estimates of the onset
delay, for a SEP point source, as a function of magnetic connectivity for
protons and electrons, respectively. The right panel shows the relationship
between the electron and proton onset delays which is independent of
$\lambda_{\perp}$. Observations are taken from Richardson et al. (2014), while
analytical estimates are shown for different assumptions of $\lambda_{\perp}$.

In this section we start by estimating the onset delay as a function of
magnetic connection using simplistic analytical arguments. For an observer
perfectly magnetically connected to the SEP source, the SEPs simply have to
propagate along the mean HMF. Assuming this happens in a ballistic (scatter
free) fashion we can estimate the onset delay as $\tau_{0}=\Delta s_{||}/v$
with $\Delta s_{||}=1.2$ AU the distance the SEPs would cover along a nominal
spiral magnetic field line to 1 AU and $v$ the particle speed. For 1.7 MeV
electrons this is $v_{e}\sim 7$ AU/hr and for 18 MeV protons, $v_{p}\sim 1.4$
AU/hr. If we now assume that particles propagate diffusively across HMF lines
to reach magnetically unconnected observers, and that this process takes an
additional time $\tau_{d}$, the onset delay becomes $\tau=\tau_{0}+\tau_{d}$.
We can estimate $\tau_{d}$ by assuming diffusive motion perpendicular to the
mean HMF is due to perpendicular diffusion. For an isotropic distribution
(implying significant scattering), the diffusive propagation time can be
evaluated as $\tau_{d}\sim\left(\Delta
s_{\perp}\right)^{2}/\left(6\kappa_{\perp}\right)$ (see e.g. Strauss et al.,
2011). The isotropic form of the perpendicular diffusion coefficient is

$\kappa_{\perp}=\frac{1}{2}\int_{-1}^{+1}D_{\perp}(\mu)d\mu=\frac{av}{2}\kappa_{FL},$
(4)

and, in terms of the perpendicular mean-free-path,
$\lambda_{\perp}=3\kappa_{\perp}/v=(3/2)a\kappa_{FL}$. Note that
$\lambda_{\perp}$ for the FLRW has no speed (energy) dependence and is
determined completely by the underlying magnetic turbulence. If we assume the
distance covered perpendicular to the field, $\Delta s_{\perp}$, is only
directed in the azimuthal direction (i.e. neglecting the non-radial geometry
of the HMF), we can estimate $\Delta s_{\perp}\sim\Delta s_{||}\Delta\phi$
where $\Delta\phi$ is the azimuthal angle away from the best magnetic
connection to the source. The resulting onset delay then becomes

$\tau=\frac{\Delta s_{||}}{v}\left[1+\frac{\Delta
s_{||}}{2\lambda_{\perp}}\left(\Delta\phi\right)^{2}\right].$ (5)

These estimates are shown for electrons and protons in Fig. 2 for different
values of $\lambda_{\perp}$. The left and middle panels show the calculated
onset delay as a function of magnetic connectivity for protons (left) and
electrons (middle). The right panel shows the resulting relationship between
the electron and proton onset delays, $\tau_{p}=(v_{e}/v_{p})\tau_{e}$ that
only depends on the ratio of the particles’ speeds in this approach.

In this simplified derivation we assume ballistic motion parallel to the
field, but diffusive motion of an isotropic distribution perpendicular to the
mean HMF. The effect of pitch-angle scattering and potentially anisotropic
distributions are therefore not included consistently and must be included in
a modelling approach to obtain a correct estimate of the onset delay. This
will be done in the following section where a more realistic Parkerian
geometry, that is not simply a radially directed magnetic field, is also
included. However, even with this very simplified approach we note that a
combination of motion along the mean HMF and diffusion perpendicular to the
mean field is sufficient to explain the observed dependence of the onset delay
on magnetic connection. Additionally, by assuming a FLRW-type perpendicular
diffusion coefficient (which is itself proportional to particle speed) we can
explain, to first order, the linear relationship between the observed electron
and proton onset delays. A more detailed calculation is performed in the next
section.

###  3.2 On the Geometry of Perpendicular Diffusion

Figure 3: Contour plots of the SEP proton (left) and electron (right)
intensity 5 hrs after particle injection.

Assuming a narrow source with $\sigma=5^{\circ}$, Fig. 3 shows contour plots
of the normalized intensity of 14 MeV protons (left panel) and 0.7 MeV
electrons (right panel) 5 hrs after the initial particle release. The orbit of
Earth (circular at a radius of 1 AU) is shown, along with the HMF line
connected to the center of the SEP injection, as thick white lines. Also
included, as thin dotted white lines, are line elements perpendicular to the
HMF at different radial positions. Perpendicular diffusion acts to transport
particles along these line elements. It is important to keep in mind that
perpendicular diffusion does not imply motion purely in the longitudinal
direction, but owing to the Parker HMF geometry (the HMF spiral angle is $\sim
45^{\circ}$ at Earth’s orbit), particles are also transported in the radial
direction. For a fixed source this usually implies more efficient particle
transport towards the west of the best-connected HMF line, as these particles
are transported away from the Sun, leading to asymmetrical distributions in
terms of longitude. In this context, see also the recent simulations of
Laitinen et al. (2023).

###  3.3 Longitudinal Dependence of SEP Intensity

Figure 4: The histograms show the observed distributions of the connection
angle (left panels) and full-width half maximum (FWHM; right panels) of 14 –
24 MeV protons (top panels; in blue) and 0.7 – 4 MeV electrons (bottom panels;
in red) from Fig. 24 of Richardson et al. (2014). Observational averages are
indicated by the filled circles, while the open green symbols show the results
from the best-fit numerical model.

Figure 5: The left panel shows the maximum omni-directional intensity as a
function of the magnetic connection angle for best-fit model assuming a source
width of $\sigma=25^{\circ}$ indicated by the black dashed curve. The right
panel shows the resulting intensities for different assumptions of the source
size (indicated in the legend).

Richardson et al. (2014) examined the longitudinal dependence of the intensity
of multi-spacecraft SEP proton and electron events in the energy ranges of 14
– 24 MeV and 0.7 – 4 MeV, respectively. This was done by fitting Gaussian
distributions to the peak intensities at the different observing spacecraft
(STEREO A/B and SOHO). The histograms in Fig. 4 show these results: The top
(blue) distributions are for protons and the bottom (red) distributions are
for electrons. The left panels show the connection angle (i.e. the angular
offset between the peak of the Gaussian function and the nominal Parker HMF
line connected to the parent active region) with positive values indicating a
shift towards the west (i.e. direction of solar rotation). Westward shifts are
also inferred in other studies of multi-spacecraft events e.g., Lario et al.
(2014), Cohen et al. (2017), and Bruno & Richardson (2021). The right panels
show the width of the Gaussian function in terms of the full-width half
maximum (FWHM). All the distributions show inter-event variation which is most
likely due to different interplanetary conditions during each event and the
difference in the accelerator of each event, e.g. larger or smaller CMEs.
While we accept that each SEP event will be different, we aim in this work to
rather reproduce the average characteristics of these events. These averages
are indicated above the histograms in Fig. 4 as solid circles with an
associated error bar.

The only remaining free parameter in our modelling approach is the size of the
particle source, $\sigma$. We now simulate SEP intensities for electrons with
energies of 0.7 and 4 MeV, and protons with energies of 14 and 25 MeV, the
limits of the energy ranges considered by Richardson et al. (2014), calculate
the maximum omni-directional intensity as a function of longitude, and, from
this, calculate both the resulting connection angle and FWHM. The size of the
assumed source is then adjusted, in increments of $5^{\circ}$, until a good
comparison is obtained with the average values of Richardson et al. (2014). A
best fit is obtained by assuming $\sigma=25^{\circ}$. These modelling results
are included in Fig. 4 as the green symbols with the associated error bar
indicating the deviation for the different energies (i.e. the upper and lower
boundaries of the energy channels). Interestingly, both the model and
observations indicate a systematic offset of the distribution towards the west
(a positive connection angle of $\sim 15^{\circ}$). In the modelling approach
this can be explained by the geometry of the perpendicular diffusion process,
as discussed in Sec. 3.2, leading to more effective (perpendicular) transport
in this direction. For a best fit value of $\sigma=25^{\circ}$, the left panel
of Fig. 5 shows the maximum omni-directional intensity as a function of
connection angle for the different particle species of different energies,
while the black dashed line shows the assumed source size. It is clear that
perpendicular diffusion broadens this initial source, while shifting the peak
of the distribution towards the west. The right panel shows the resulting
distribution for different source sizes.

In this section we adjusted the remaining free parameter in our model, the
size of the SEP source, until the measured distribution of the omni-
directional intensity could be reproduced. However, if just the particle
intensity is considered, this could lead to a degeneracy in the modelling
approach: Different combinations of particle sources and levels of
perpendicular diffusion can reproduce the same distribution at Earth’s orbit.
Additional observables are needed to uniquely determine all transport
parameters. This was done in Strauss et al. (2017) and we used their value of
$a=0.2$ to fix the level of perpendicular diffusion. However, as we will show
in the next section, the model set-up with $a=0.2$ and $\sigma=25^{\circ}$,
does also lead to a decent comparison with the onset delays of MeV electrons
and protons. We are therefore confident that the model gives a fair
representation of the observations.

###  3.4 Electron and Proton Onset Delays

Figure 6: The left panel shows the calculated maximum omni-directional
intensity as a function of longitude (i.e. magnetic connection), the middle
panel the onset delay in minutes, and the right panel the peak delay in hours.
The horizontal dashed line in the left panel indicates 50% of the peak value.
Simulations are performed for electrons and protons of different energies as
indicated on the legend. All measurements are taken from Richardson et al.
(2014).  Figure 7: The same as shown in Fig. 6, but model results are now
shown for different source sizes. Figure 8: The calculated proton and electron
onset delays (the different symbols) compared to the measurements of
Richardson et al. (2014). The left panel shows the model results for different
source sizes, while the right panel also assumes a lower background level in
the model.

In order to model particle onset delays, a relative background needs to be
chosen in the model. We follow the previous approach of Strauss et al. (2017)
of calculating the maximum omni-directional intensity, for all longitudes, at
1 AU, and then defining the background as 1/1000th of the peak intensity. The
onset delay is then simply calculated as the time from the start of the
simulation until this background level is crossed at each longitude. The
effects of different background levels are illustrated in Sec. 3.5.

Using the best-fit model of the previous section, Fig. 6 shows the resulting
onset delays. The left panel, once again, shows the maximum omni-directional
intensity as a function of longitude (different lines correspond to electrons
and protons of different energies as indicated in the legend). The middle
panel shows the measurements from Richardson et al. (2014) as the data points
(blue again corresponding to protons and red to electrons), while the lines
are model calculations. If the modelled intensity, at a given longitude, did
not cross the background levels, the onset delay is not defined and no model
values are shown. As expected, the minimum onset delay is seen at the best
magnetic connection between the observer and the source, where the first
arriving (usually very anisotropic) particles reach the observer in an almost
scatter-free fashion. The onset delay then increases for increasing magnetic
connection due to the relative slow (perpendicular) diffusion process that
transports the particles across magnetic field lines. Although the
measurements have significant event-to-event variations, the general trend and
ball-park values are consistent with the model results.

The right panel of Fig. 6 shows the calculated and observed peak delay as a
function of longitude. This is calculated as the time that elapsed between
particle injection (i.e. the start of the simulation) and the occurrence of
the maximum omni-directional intensity at each longitude. Although there is,
once again, a fair comparison between model results and observations, the
model is unable to reproduce the very long peak delays observed for electrons.
This is addressed in more detail in Sec. 3.6. Fig. 7 is similar to Fig. 6 but
shows the effect of a changing source size on the model calculations.
Generally, a larger source leads to shorter onset and peak delays.

The results shown in Figs. 6 and 7 are now combined and presented in Fig. 8.
Here we show, in the left panel, the calculated proton onset delay as a
function of the calculated electron onset delay. Calculations, shown here as
the different symbols, are performed for different source sizes and again
compared to measurements from Richardson et al. (2014) (grey data points and
green fit to the data). The model results compare well with the measurements
and confirm the linear relationship between the electron and proton onset
delays. Due to the east-west asymmetry present in the model, the modelled
relationship is not completely linear, but exhibits a closed loop. In the
right panel of Fig. 8, the model calculation is repeated, but for a different
level of the assumed model background as discussed in the next section. While
the best-fit model compares well with the measurements, we argue, again, that
the variation of e.g. the background level and accelerator sizes from event-
to-event can contribute to explaining the large inter-event variation observed
for the onset delays.

Figure 9: Onset delays for protons, similar to Fig. 6, but simulations are now
performed for different background levels in the model.

###  3.5 Caveats in the Simulation of the Onset Delay

The modelled onset delay is dependent on the assumed background level in the
model. The same is true for the observations (see also Laitinen et al., 2015;
Zhao et al., 2019). When the (modelled) differential intensity increases
slowly with time, as is the case for observers magnetically disconnected from
the source, a lower background will result in a shorter onset delay. This is
illustrated in Fig. 9 where the background level is changed from 1/10000th of
the peak intensity to 1/10th of the peak intensity. The default case is again
that of 1/1000th of the peak intensity. It is interesting to note that these
different onset delay calculations might also help to explain the observed
scatter (inter-event-variations) in the observations: In the model, the same
peak intensity is always obtained and the background level is adjusted to
simulate the effect of changing background levels whereas for observations of
different events, the background of the instrument may remain relatively
constant, but the magnitudes (i.e. peak intensity) of the events change. The
observed pre-event background from preceding events, if present, may also vary
from event to event. In each case, the relative background, i.e. the peak
intensity to background ratio, changes from event to event and can lead to the
variation presented in Fig. 9.

###  3.6 Electron and Proton Peak Delays

Figure 10: The modelled peak delay as a function of the onset delay for
protons (left panel) and electrons (right panel). The measurements are again
taken from Richardson et al. (2014). Note that, when no model values are
shown, the model intensity did not cross the background level at that
longitude.

As the last part of our work we investigate the modelled relationship between
the peak and onset delays. Fig. 10 shows these simulation results for protons
(blue in left panel) and electrons (red in right panel) for different particle
energies (different lines). These calculations are again compared to the
observations of Richardson et al. (2014). For protons, the model reproduces
the observed linear dependence very well, although these curves are non-
linear: Due to the small asymmetry introduced by the Parker HMF, the onset
time is slightly shorter at western longitudes and slightly longer at eastern
longitudes. On these figures this manifests of this “figure of eight” shape.
These results suggest that both the onset delay and peak delay is at a minimum
at the best magnetic connection between the source and observer while both
increase for increasingly poor connectivity. For electrons, however, the
calculated peak delays are much shorter than the values derived from
observations. At present it is not clear why this is the case, although we
speculate on possible reasons for this discrepancy in the next section.

##  4 Discussion

Using transport coefficients calculated from first principles, we are able to
simultaneously reproduce the observed longitudinal intensity distribution of
14 - 24 MeV protons and 0.7 - 4 MeV electrons. This is done by assuming an
average source size of $\sigma=25^{\circ}$. Of course, the measurements show a
lot of inter-event variation which can be explained by different
interplanetary transport conditions, varying source sizes and, of course,
varying levels of peak intensity relative to the background.

When perpendicular diffusion is included in the model, we note a systematic
shift of the peak intensity to the west of the best magnetic connection due to
the geometry of the Parker HMF. Interestingly, a similar shift is seen in both
the measurements, but also in previous simulation work by other authors, e.g.
He & Wan (2015).

Our modelling approach can simultaneously reproduce SEP electron and proton
onset delays by assuming the same particle source. When interplanetary
transport is included, we do not need to invoke different particle sources
expanding at different rates to explain the Richardson et al. (2014)
observations. The shortest onset delays are noted at the best magnetic
connection, while the delay increases away from best connection due to the
slow perpendicular transport process that transports particles across field
lines. In fact, we are also able to reproduce the linear relationship observed
between electron and proton onset delays by assuming the FLRW process of
perpendicular diffusion, where the diffusion coefficient scales linearly with
the particle speed. Electrons, being much more mobile than protons at these
energies, therefore move faster both along and perpendicular to the mean
field.

Our results therefore point towards MeV electrons and protons having a common
accelerator or source. The transport model itself cannot distinguish between a
shock-related or flare-related process. However, it is clear that the model
requires both an extended source ($\sigma\sim 25^{\circ}$) and some level of
perpendicular diffusion to explain the observed broadness of the SEP events.
Such an extended source could point towards a CME-related source. On the other
hand, Dresing et al. (2016) find no evidence of interplanetary shock
acceleration for $\sim 100$ keV electrons, while a comparison with remote
sensing observations seem to suggest a flare association for these low energy
electrons (e.g. Dresing et al., 2021). In previous modelling work we found
that 100 keV electron observations can be reproduced, on average, by assuming
a compact source with $\sigma\sim 5^{\circ}$. This likely points to a compact
flaring source being the main accelerator of electrons at these low energies.
Our present and previous simulation results are therefore consistent with the
findings of Dresing et al. (2022): Low energy ($\sim$100s of keV) electrons
are most likely related to acceleration in compact flaring regions, while
$\sim$MeV electrons are most likely produced in more extended CME-driven shock
regions. Additionally, our results suggest that $\sim$MeV electrons and $\sim
10$ MeV protons are produced at the same source.

One possible deficiency of our present modelling approach is the inability to
reproduce the observed peak delays for electrons. The reason for this
discrepancy is presently not known. While this could be due to an incorrect
level of pitch-angle diffusion included in the model (more diffusion generally
leads to longer peak delays), this is unlikely as the amount of pitch-angle
scattering needed to increase the peak delay by an order of magnitude would
also increase the onset delay with a similar amount. This would be
inconsistent with the observations. These longer peak delays are more likely
related to an extended injection of SEP electrons into the interplanetary
medium due to either particle trapping or particle re-acceleration at the
expanding CME shock; an effect not captured by our current injection profile.
This discrepancy continues to be a topic of further investigation.

This work is based on the research supported in part by the National Research
Foundation of South Africa (NRF grant numbers SRUG220322419 and
RA170929263913). 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).
IGR acknowledges support from NASA programs NNH19ZDA001N-LWS,
NNH19ZDA001N-HSR, and from the STEREO mission. Figures prepared with
Matplotlib (Hunter, 2007) and certain calculations done with NumPy (Harris et
al., 2020).

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