# The Stability of the Electron Strahl against the Oblique Fast-
magnetosonic/Whistler Instability in the Inner Heliosphere

Seong-Yeop Jeong  Mullard Space Science Laboratory, University College London,
Dorking, RH5 6NT, UK<EMAIL_ADDRESS>Joel B. Abraham  Mullard Space
Science Laboratory, University College London, Dorking, RH5 6NT, UK;
<EMAIL_ADDRESS>Daniel Verscharen  Mullard Space Science Laboratory,
University College London, Dorking, RH5 6NT, UK<EMAIL_ADDRESS>Laura
Berčič  Mullard Space Science Laboratory, University College London, Dorking,
RH5 6NT, UK<EMAIL_ADDRESS>David Stansby  Mullard Space Science
Laboratory, University College London, Dorking, RH5 6NT, UK;
<EMAIL_ADDRESS>Georgios Nicolaou  Mullard Space Science Laboratory,
University College London, Dorking, RH5 6NT, UK<EMAIL_ADDRESS>Christopher J. Owen  Mullard Space Science Laboratory, University College
London, Dorking, RH5 6NT, UK<EMAIL_ADDRESS>Robert T. Wicks
Northumbria University, Newcastle, NE1 8ST, UK  Andrew N. Fazakerley  Mullard
Space Science Laboratory, University College London, Dorking, RH5 6NT, UK;
<EMAIL_ADDRESS>Jeffersson A. Agudelo Rueda  Mullard Space Science
Laboratory, University College London, Dorking, RH5 6NT, UK;
<EMAIL_ADDRESS>Mayur Bakrania  Mullard Space Science Laboratory,
University College London, Dorking, RH5 6NT, UK<EMAIL_ADDRESS>
###### Abstract

We analyze the micro-kinetic stability of the electron strahl in the solar
wind depending on heliocentric distance. The oblique fast-
magnetosonic/whistler (FM/W) instability has emerged in the literature as a
key candidate mechanism for the effective scattering of the electron strahl
into the electron halo population. Using data from Parker Solar Probe (PSP)
and Helios, we compare the measured strahl properties with the analytical
thresholds for the oblique FM/W instability in the low- and
high-$\beta_{\parallel c}$ regimes, where $\beta_{\parallel c}$ is the ratio
of the core parallel thermal pressure to the magnetic pressure. Our PSP and
Helios data show that the electron strahl is on average stable against the
oblique FM/W instability in the inner heliosphere. Our analysis suggests that
the instability, if at all, can only be excited sporadically and on short
timescales. We discuss the caveats of our analysis and potential alternative
explanations for the observed scattering of the electron strahl in the solar
wind. Furthermore, we recommend the numerical evaluation of the stability of
individual distributions in the future to account for any uncertainties in the
validity of the analytical expressions for the instability thresholds.

Unified Astronomy Thesaurus concepts: Solar wind (1534); Space plasma (1544);
Heliosphere (711);

##  1 Introduction

The solar-wind plasma is continuously emitted from the Sun into interplanetary
space (Parker, 1958; Neugebauer & Snyder, 1962). It expands quasi-spherically
and forms the heliosphere. The electrons are the most abundant particle
species and play an important role in the solar wind. They guarantee quasi-
neutrality and provide the solar wind with a significant heat flux through the
non-thermal properties of the electron velocity distribution functions (VDFs;
Marsch, 2006). In addition, the electrons generate a global ambipolar electric
field through their thermal pressure gradient (Jockers, 1970; Lemaire &
Scherer, 1970, 1971; Pierrard et al., 1999; Scudder, 2019; Berčič et al.,
2021).

According to previous in-situ measurements of the electron VDF in the solar
wind, multiple deviations from a Maxwellian equilibrium state often occur. The
electron VDFs typically consist of three different electron populations: the
core, the halo, and the strahl. The core, which accounts for over 90% of the
electrons in the solar wind, has a relatively low energy ($\lesssim
50\,\mathrm{eV}$) compared to the other populations and is quasi-isotropic.
The halo has a greater energy ($\gtrsim 50\,\mathrm{eV}$) than the core and is
nearly isotropic as well (Štverák et al., 2008; Salem et al., 2021). Lastly,
the electron VDF often shows an energetic and highly field-aligned beam
population known as the strahl. The strahl carries most of the heat flux
(Feldman et al., 1975; Pilipp et al., 1987c) and becomes faint at large
heliocentric distances (Pilipp et al., 1987b; Štverák et al., 2009; Graham et
al., 2017). The continuing transfer of electrons from the strahl population
into the halo population with increasing distance suggests that a local
scattering mechanism modifies the shape of the VDF. This mechanism is very
important for the definition of the electron heat flux; however, its nature is
still unknown.

Electron measurements in the solar wind provide evidence that whistler waves
exchange energy with the electron strahl, which makes them a candidate to
explain the scattering of the strahl into the halo population (Pagel et al.,
2007; Lacombe et al., 2014; Graham et al., 2017). Linear theory suggests that
the FM/W wave propagating in the anti-Sunward direction with an oblique
wavevector with respect to the background magnetic field can be driven
unstable by the strahl through the anomalous cyclotron resonance and scatter
strahl electrons in pitch angle through velocity space (Vasko et al., 2019;
Verscharen et al., 2019). This strahl-driven instability has recently received
a significant amount of attention in the literature as a local strahl
scattering mechanism. Linear and quasi-linear theories as well as numerical
particle-in-cell (PIC) simulations support this picture (López et al., 2020;
Jeong et al., 2020; Micera et al., 2020, 2021; Sun et al., 2021).

As the solar wind begins its journey in the corona where Coulomb collisions
are more frequent than in the heliosphere, we anticipate that the
core–halo–strahl configuration develops outside the corona. It is unknown at
which heliocentric distance the local strahl scattering sets in. Based on a
linear stability analysis, Horaites et al. (2018a) and Schroeder et al. (2021)
argue against the action of a strahl-driven instability in the inner
heliosphere at all. Parker Solar Probe (PSP) data from encounters 4 and 5
suggest that the strahl, at times, drives the oblique FM/W wave unstable at
heliocentric distances below $54r_{s}$ (Halekas et al., 2021), where $r_{s}$
is the solar radius. A kinetic expansion model for the solar-wind electrons
suggests, however, that the oblique FM/W instability is unlikely to occur
regularly and scatter the electron strahl at heliocentric distances below
$20r_{s}$ (Jeong et al., 2022). Moreover, whistler waves are barely detected
by PSP within $28r_{s}$ (Cattell et al., 2022), supporting the notion that
strahl scattering by the oblique FM/W instability is not a universal – or even
common – process in the inner heliosphere. We note, however, that the required
whistler-wave amplitude for effective strahl scattering is small (Jeong et
al., 2020; Vasko et al., 2020) and thus could be hidden in the fluctuation
spectrum of the background turbulence in the solar wind (Zank et al., 1996;
Chen et al., 2013; Tong et al., 2019).

In the present paper, we use the analytical thresholds for the oblique FM/W
instability derived by Verscharen et al. (2019) to investigate the stability
of the electron strahl in the inner heliosphere. By using PSP and Helios data,
we quantify the analytical thresholds as functions of heliocentric distance.
Our results show that, on average, the electron strahl is not scattered by the
oblique FM/W instability in the inner heliosphere.

##  2 Methods

###  2.1 Thresholds for the Oblique FM/W Instability

To analyze the strahl scattering through the oblique FM/W instability, we
evaluate the analytical thresholds in the low- and high-$\beta_{\parallel c}$
regimes provided by Verscharen et al. (2019). These analytical equations have
been tested against numerical solutions for the thresholds of the oblique FM/W
instability from linear hot-plasma theory over a range of representative
solar-wind parameters. The transition between the low-$\beta_{\parallel c}$
and high-$\beta_{\parallel c}$ regimes occurs at

$\beta_{\parallel c}\approx 0.25,$ (1)

where $\beta_{\parallel j}\equiv 8\pi n_{j}k_{B}T_{\parallel j}/B_{0}^{2}$,
$n_{j}$ is the density of population $j$, $T_{\parallel j}$ is the temperature
of population $j$ parallel to the background magnetic field,
$\boldsymbol{B}_{0}$ is the background magnetic field, and $k_{B}$ is the
Boltzmann constant. The subscripts $e$, $c$ and $s$ indicate quantities
related to the total electrons, the core and the strahl, respectively. The
plasma crosses the analytical threshold of the oblique FM/W instability in the
low-$\beta_{\parallel c}$ regime if

$U_{s}\gtrsim 3v_{\parallel th,c},$ (2)

where $v_{\parallel th,c}=\sqrt{2k_{B}T_{\parallel c}/m_{e}}$ is the core
parallel thermal speed, $U_{s}$ is the strahl bulk speed, and $m_{e}$ is the
mass of an electron. The plasma crosses the analytical threshold of the
oblique FM/W instability in the high-$\beta_{\parallel c}$ regime if

$U_{s}\gtrsim\left[2\frac{n_{s}}{n_{c}}\sqrt{\frac{T_{\parallel
s}}{T_{\parallel c}}}v_{Ae}^{2}v_{\parallel
th,c}^{2}\frac{(1+\cos\theta)}{(1-\cos\theta)\cos\theta}\right]^{1/4},$ (3)

where $v_{Ae}=B_{0}/\sqrt{4\pi n_{e}m_{e}}$, and $\theta$ is the angle between
wavevector $\boldsymbol{k}$ and $\boldsymbol{B}_{0}$111The original derivation
of the instability thresholds by Verscharen et al. (2019) assumes isotropic
electrons. However, it is straight-forward to extend this framework to
anisotropic electrons, in which case we retrieve Eqs. (2) and (3)..

###  2.2 Data Analysis of PSP and Helios Data

We base our data analysis on the PSP dataset created by Abraham et al. (2021).
Our PSP data were recorded at heliocentric distances between $25r_{s}$ and
$90r_{s}$. This dataset uses fits to the observed level-3 electron VDFs
provided by the Solar Wind Electron Alphas and Protons (SWEAP) instrument
suite (Kasper et al., 2016; Whittlesey et al., 2020) on board PSP. SWEAP
consists of two electrostatic analysers, SPAN A and B, which combined measure
electrons arriving from across almost the full sky using orthogonally
positioned $120^{\circ}\times 240^{\circ}$ fields of view, over an energy
range from 2 eV to 1793 eV during our measurement intervals. During the
encounter, the sampling cadence of PSP’s SPAN detector is $13.98s$, and its
integration time is $6.965s$. We fit the VDFs in the magnetic-field-aligned
velocity frame and ignore any data points below 30 eV to avoid the inclusion
of secondary electrons. The analysis method fits a bi-Maxwellian distribution
to the core, a bi-$\kappa$-distribution to the halo and a drifting bi-
Maxwellian distribution to the strahl. We only allow the strahl to drift in
the parallel direction to the magnetic field.

On Helios 1 and 2 (Rosenbauer et al., 1981; Pilipp et al., 1987a), electrons
were measured by the electron particle instrument I2, part of the E1 Plasma
Experiment. Our Helios data were recorded at heliocentric distances between
$62r_{s}$ and $210r_{s}$. A narrow instrument aperture ($19^{\circ}\times
2^{\circ}$) spins with the spacecraft and covers a full range of $360^{\circ}$
in azimuth, providing two-dimensional electron VDFs. Helios took
$16\,\mathrm{s}$ to measure a full two-dimensional electron VDF with a cadence
of $40\,\mathrm{s}$. For context, we also use proton density and velocity as
well as magnetic-field measurements as described by Berčič et al. (2019,
Section 3). We analyze the electron VDFs in the magnetic-field-aligned frame,
in which the solar-wind protons have zero bulk speed. We model the
distributions with a sum of a bi-Maxwellian distribution representing the
core, a bi-$\kappa$-distribution representing the halo, and a bi-Maxwellian
distribution representing the strahl. We only allow the core and the strahl to
have drift velocities in the direction parallel to the magnetic field.

In both datasets, we discard all VDFs for which the fit results for the
density of the halo or the strahl is greater than the density of the core.

##  3 Results

###  3.1 Fit Parameters from PSP and Helios

We bin the fitted data from both PSP and Helios into 50 radial-distance bins
of equal width. We first separate our PSP and Helios data into the
low-$\beta_{\parallel c}$ regime and high-$\beta_{\parallel c}$ regime
according to Eq. (1). We show the profiles of $T_{\parallel c}$, $T_{\parallel
s}$, $B_{0}$, $n_{c}/n_{e}$, $n_{s}/n_{e}$ and $v_{Ae}$, which all appear on
the right-hand sides of Eqs. (2) and (3), as functions of heliocentric
distance in Fig. 1.

Figure 1: Radial profiles of $T_{\parallel c}$, $T_{\parallel s}$, $B_{0}$,
$n_{c}/n_{e}$, $n_{s}/n_{e}$ and $v_{Ae}$ in our PSP and Helios datasets,
separated between the low-$\beta_{\parallel c}$ and the high-$\beta_{\parallel
c}$ regimes according to Eq. (1).

The red and blue lines correspond to the parameters in the
low-$\beta_{\parallel c}$ regime while the yellow and green lines correspond
to the parameters in the high-$\beta_{\parallel c}$ regime from PSP and
Helios.

PSP predominately samples streams of slow solar wind while the Helios dataset
includes a broader variety of solar wind. Therefore, the PSP and Helios
datasets do not connect exactly, especially in terms of $T_{\parallel s}$,
$n_{c}/n_{e}$ and $n_{s}/n_{e}$ in the low-$\beta_{\parallel c}$ cases. Fig.
1(c) shows the magnetic field averaged over the respective particle sampling
times of the electron instruments for PSP and Helios.

For our evaluation of the instability thresholds as functions of heliocentric
distance, we apply the binned mean profiles of the parameters shown in Fig. 1
to Eqs. (2) and (3).

###  3.2 Threshold in the Low-$\beta_{\parallel c}$ Regime

In Fig. 2, the blue (PSP) and red (Helios) lines show the profiles of the
right-hand side of Eq. (2), and the yellow (PSP) and green (Helios) lines show
the measured profiles of $U_{s}$.

Figure 2:  Profiles of the low-$\beta_{\parallel c}$ threshold of the oblique
FM/W instability from Eq. (2) and $U_{s}$ as functions of heliocentric
distance. All PSP and Helios data underlying this figure have
$\beta_{\parallel c}<0.25$.

The strahl bulk speed does not cross the low-$\beta_{\parallel c}$ threshold
in the shown range of heliocentric distances on average and, thus, does not
fulfill Eq. (2). The low-$\beta_{\parallel c}$ threshold and $U_{s}$ decrease
quasi-monotonously. Near the Sun, the difference between the
low-$\beta_{\parallel c}$ threshold and $U_{s}$ decreases with heliocentric
distance. However, beyond $\sim 80r_{s}$ from the Sun, the difference is
nearly constant. The error bars of the low-$\beta_{\parallel c}$ threshold and
$U_{s}$ do not overlap in the shown range of heliocentric distances.

###  3.3 Threshold in the High-$\beta_{\parallel c}$ Regime

Previous studies of the oblique FM/W instability by Vasko et al. (2019),
Verscharen et al. (2019), López et al. (2020), Jeong et al. (2020), Micera et
al. (2020), Micera et al. (2021), and Sun et al. (2021) suggest that the angle
$\theta$ in Eq. (3) typically ranges from $\theta=55^{\circ}$ to $65^{\circ}$.
Eq. (3) does not significantly depend on $\theta$ in this range compared to
its much stronger dependence on $T_{\parallel c}$, $T_{\parallel s}$, $n_{c}$,
$n_{s}$ and $v_{Ae}$. Therefore, we use a representative value of
$\theta=60^{\circ}$.

In Fig. 3, the blue (PSP) and red (Helios) lines show the profiles of the
right-hand side of Eq. (3), and the yellow (PSP) and green (Helios) lines show
the measured profile of $U_{s}$.

Figure 3: Profiles of the high-$\beta_{\parallel c}$ threshold of the oblique
FM/W instability from Eq. (3) and $U_{s}$ as functions of heliocentric
distance. All PSP and Helios data underlying this figure have
$\beta_{\parallel c}>0.25$.

The strahl bulk speed does not cross the high-$\beta_{\parallel c}$ threshold
in the shown range of heliocentric distances on average and, thus, does not
fulfill Eq. (3). The high-$\beta_{\parallel c}$ threshold and $U_{s}$ quasi-
monotonously decrease at all explored heliocentric distances. Like in the
low-$\beta_{\parallel c}$ case, near the Sun, the difference between the
high-$\beta_{\parallel c}$ threshold and $U_{s}$ decreases with heliocentric
distance. However, beyond $\sim 80r_{s}$ from the Sun, the difference is
approximately constant. The error bars of the high-$\beta_{\parallel c}$
threshold and $U_{s}$ do not overlap at all explored heliocentric distances.

##  4 Discussion and Conclusions

We analyze the stability of the solar-wind electron strahl against the oblique
FM/W instability for self-induced strahl scattering in two different
$\beta_{\parallel c}$ regimes. By quantifying the plasma parameters based on
binned and averaged electron data from PSP and Helios, as shown in Fig. 1, we
evaluate the analytical instability thresholds as functions of heliocentric
distance.

Our results show that Eqs. (2) and (3) are, for plasma with the observed
average radial profiles, not satisfied at heliocentric distances between
$25r_{s}$ and $210r_{s}$. In Figs. 2 and 3, the low- and
high-$\beta_{\parallel c}$ thresholds are always greater than $U_{s}$. For
both the low-$\beta_{\parallel c}$ and for the high-$\beta_{\parallel c}$
cases, the difference between the thresholds and $U_{s}$ decreases with
heliocentric distance near the Sun as predicted by Jeong et al. (2022).
However, beyond a heliocentric distance of $\sim 80r_{s}$, the difference
between the thresholds and $U_{s}$ is nearly constant. Assuming that our PSP
and Helios data are representative and that our Eqs. (2) and (3) reflect the
thresholds accurately, our findings suggest that the strahl is not scattered
by the oblique FM/W instability in the inner heliosphere, in agreement with
the arguments presented by Horaites et al. (2018a) and Schroeder et al.
(2021).

According to quasi-linear theory and particle-in-cell simulations (Jeong et
al., 2020; Micera et al., 2020, 2021), the oblique FM/W instability, if
excited, scatters the electron strahl efficiently on a timescale shorter than
$1\,\mathrm{s}$ in the inner heliosphere. This rapid action of the instability
and the radial decrease of the thresholds suggest that the measured $U_{s}$
would closely follow the threshold of marginal stability at all distances
beyond the critical distance $r_{\mathrm{crit}}$, at which the threshold is
crossed for the first time. Neither Fig. 2 nor Fig. 3 suggest the existence of
$r_{\mathrm{crit}}$ at heliocentric distances less than 1 au. However, we
propose some uncertainties to our results.

Our analysis is based on electron measurements with a finite sampling cadence
($\sim 14\,\mathrm{s}$ by PSP and $\sim 40\,\mathrm{s}$ by Helios). Even if
the instability scatters the electron strahl, PSP and Helios would be unlikely
to detect the moment of strahl scattering because it occurs so quickly. If the
magnetic field changes significantly during such a sampling interval, the
corresponding electron measurement would be deteriorated. Since the averaged
amplitude of magnetic-field variations at the timescale of the particle
measurements is small, we assume that this effect has a small impact on our
results. However, we recommend repeating our study with even faster plasma
instrumentation (e.g., with Solar Orbiter’s EAS instrument in burst mode,
Owen, C. J. et al., 2020) for comparison in the future.

The derivation of Eqs. (2) and (3) by Verscharen et al. (2019) is based on the
assumption that the oblique FM/W instability is most efficiently driven by
electrons with $v_{\parallel}=U_{s}$ where $v_{\parallel}$ is the velocity
coordinate in the direction parallel with respect to $B_{0}$ in the solar-wind
frame. However, according to the predictions from quasi-linear theory and PIC
simulations (Jeong et al., 2020; Micera et al., 2020, 2021), strahl scattering
by the oblique FM/W instability most efficiently occurs in the
$v_{\parallel}$-range $\gtrsim U_{s}$. For example, Jeong et al. (2020) show
that the oblique FM/W instability in the high-$\beta_{\parallel c}$ regime is
most efficiently driven by electrons with $v_{\parallel}\approx 2U_{s}$.
Applying this argument in the derivation of Eqs. (2) and (3), we find a
threshold for the high-$\beta_{\parallel c}$ case that is lower by a factor of
two than shown in Fig. 3. Such a modification would lead to a small difference
between the two curves shown in Fig. 3. If we adjusted the analytical
thresholds by a constant factor greater than 2, the discovered constant
difference between the thresholds and $U_{s}$ at $r\gtrsim 80r_{s}$ would be
consistent with the limitation of $U_{s}$ through the oblique FM/W instability
with $r_{\mathrm{crit}}\approx 80r_{s}$.

The uncertainties of the analytical instability thresholds can be circumvented
by calculating the stability of individually measured electron VDFs with
numerical tools like the ALPS (Verscharen & Chandran, 2018) or LEOPARD
(Astfalk & Jenko, 2017). These numerical codes are able to calculate the full
hot-plasma dispersion relation based on our PSP and Helios data without
assuming a bi-Maxwellian shape of the electron VDF. In addition, it would be
worthwhile to investigate the occurrence of oblique FM/W waves as a result of
the oblique FM/W instability. This method would also allow for an independent
determination of $r_{\mathrm{crit}}$ and thus an independent determination of
a potential correction factor to our analytical instability thresholds.

Our observation of the decreasing trends in $U_{s}$ (Figs. 2 and 3) as well as
earlier observations of the broadening of the strahl pitch-angle and the
scattering of the strahl into the halo (Hammond et al., 1996; Maksimovic et
al., 2005; Štverák et al., 2009; Graham et al., 2017) provide evidence that
strahl scattering occurs over a wide range of distances in the solar wind.
Therefore, if the oblique FM/W instability is _not_ responsible for this
evolution, other processes must explain the observed trends.

Other linear instabilities, including the quasi-parallel whistler heat-flux
instability (Tong et al., 2019; Kuzichev et al., 2019; Vasko et al., 2020) are
not efficient strahl-scattering mechanisms (Verscharen et al., 2019). However,
nonlinear processes involving combinations of primary and secondary
instabilities in connection with solar-wind expansion reproduce observed wave
and particle features in the solar wind (Micera et al., 2021). Coulomb
collisions have been suggested to contribute to the strahl evolution even
though the collisional mean free path of suprathermal electrons is large
(Landi et al., 2012; Horaites et al., 2018b, 2019). Another possible strahl-
scattering mechanism involves the dissipation of existing plasma fluctuations
(especially whistler waves, Vocks, 2012; Vocks et al., 2005). Although
whistler waves have been sporadically observed in the solar wind (Lacombe et
al., 2014; Agapitov et al., 2020; Kretzschmar, M. et al., 2021; Cattell, C. et
al., 2021; Jagarlamudi, V. K. et al., 2021), their origin and possible links
to solar-wind turbulence are still not well understood.

D.V. is supported by STFC Ernest Rutherford Fellowship ST/P003826/1. J. B. A.
is supported by STFC grant ST/T506485/1. D.V., C.J.O., A.N.F., D.S., and L.B.
are supported by STFC Consolidated Grant ST/S000240/1. R. T. W. is funded by
STFC consolidated Grant ST/V006320/1. J.A.A.R. is supported by the ESA
Networking/Partnering Initiative (NPI) contract 4000127929/19/NL/MH/mg and the
Colombian programme Pasaporte a la Ciencia, Foco Sociedad – Reto 3, ICETEX
grant 3933061. M. B. is supported by a UCL Impact Studentship, joint funded by
the ESA NPI contract 4000125082/18/NL/MH/ic. This work was discussed at the
“Joint Electron Project” at MSSL. The authors acknowledge insightful
discussions within the International Team “Heliospheric Energy Budget: From
Kinetic Scales to Global Solar Wind Dynamics” at the International Space
Science Institute (ISSI) in Bern led by M. E. Innocenti and A. Tenerani.

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