# The source of electrons at comet 67P

P. Stephenson1, A. Beth1,2, J. Deca3, M. Galand1, C. Goetz4, P. Henri5,6, K.
Heritier1, Z. Lewis,1 A. Moeslinger,2 H. Nilsson7 and M. Rubin8
1Department of Physics, Imperial College London, London, UK
2Department of Physics, Umeå University, Umeå, Sweden
3Laboratory for Atmospheric and Space Physics, University of Colorado,
Boulder, Colorado
4Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle Upon Tyne, UK
5Lagrange, OCA, CNRS, UCA, Nice, France
6LPC2E, CNRS, Orleans, France
7 Swedish Institute of Space Physics, Kiruna, Sweden
8Physikalisches Institut, University of Bern, Bern, Switzerland<EMAIL_ADDRESS>
(Jun 2023)

###### Abstract

We examine the origin of electrons in a weakly outgassing comet, using Rosetta
mission data and a 3D collisional model of electrons at a comet. We have
calculated a new dataset of electron-impact ionization (EII) frequency
throughout the Rosetta escort phase, with measurements of the Rosetta Plasma
Consortium’s Ion and Electron Sensor (RPC/IES). The EII frequency is evaluated
in 15-minute intervals and compared to other Rosetta datasets.

Electron-impact ionization is the dominant source of electrons at 67P away
from perihelion and is highly variable (by up to three orders of magnitude).
Around perihelion, EII is much less variable and less efficient than
photoionization at Rosetta. Several drivers of the EII frequency are
identified, including magnetic field strength and the outgassing rate.
Energetic electrons are correlated to the Rosetta-upstream solar wind
potential difference, confirming that the ionizing electrons are solar wind
electrons accelerated by an ambipolar field.

The collisional test particle model incorporates a spherically symmetric, pure
water coma and all the relevant electron-neutral collision processes. Electric
and magnetic fields are stationary model inputs, and are computed using a
fully-kinetic, collisionless Particle-in-Cell simulation. Collisional
electrons are modelled at outgassing rates of $Q=10^{26}$ s-1 and $Q=1.5\times
10^{27}$ s-1. Secondary electrons are the dominant population within a weakly
outgassing comet. These are produced by collisions of solar wind electrons
with the neutral coma.

The implications of large ion flow speed estimates at Rosetta, away from
perihelion, are discussed in relation to multi-instrument studies and the new
results of the EII frequency obtained in the present study.

###### keywords:

Comets:general – Comets: individual: 67P/CG

##  1 Introduction

As a comet approaches the Sun, near-surface ices are heated and sublimate,
producing an envelope of expanding neutral gas known as the coma. The coma can
be ionized by several processes, generating a population of ions and electrons
within it. These form a cometary ionosphere, which interacts with the solar
wind. Ionization can be driven by absorption of high energy, extreme
ultraviolet (EUV) photons or by collisions of energetic electrons. Solar wind
ions can collide with cometary neutrals, causing ionization or a charge
exchange process to produce a different ion species. The Rosetta mission
escorted comet 67P/Churyumov-Gerasimenko (67P) for two years along its orbit
from early Aug 2014 to 30 Sep 2016. During the escort phase, Rosetta gathered
an extensive and unique dataset of an evolving cometary ionosphere (e.g.,
Goetz et al., 2016; Heritier et al., 2017b). The cometary plasma environment
at Rosetta was probed by instruments of the Rosetta Plasma Consortium (RPC;
Carr et al., 2007) and the Rosetta Orbiter Spectrometer for Ion and Neutral
Analysis (ROSINA; Balsiger et al., 2007). At the end of mission, Rosetta made
a final, controlled descent to the surface of 67P, measuring the vertical
profile of a cometary ionosphere for the first time (Heritier et al., 2017a).

The electrons within the coma of 67P cannot be described as a single
population. Cold ($<1$ eV), warm ($\sim 10$ eV) and hot ($>15$ eV) populations
have been identified within the coma (Clark et al., 2015; Broiles et al.,
2016b, a; Eriksson et al., 2017; Engelhardt et al., 2018; Myllys et al., 2019;
Gilet et al., 2020; Wattieaux et al., 2020; Myllys et al., 2021). The warm
population is made up by newly born cometary electrons, produced by ionization
within the coma. The cold electrons are produced by cooling the warm
population through collisions with the neutral coma (Stephenson et al., 2022).
The hot population is likely made up of solar wind electrons that have been
energised by an ambipolar field that forms around the comet (Myllys et al.,
2019; Stephenson et al., 2022). The cold electron population was observed
frequently throughout the Rosetta mission, when the neutral gas should not
have been dense enough to substantially cool the warm electrons (Engelhardt et
al., 2018; Gilet et al., 2020; Wattieaux et al., 2020). These estimates
assumed that cometary electrons travelled approximately radially away from the
nucleus, like the neutral gas in which they are embedded. To evaluate the
energy degradation of electrons at a comet, Stephenson et al. (2022) applied a
3D collisional electron model. With trapping in the ambipolar potential well
and the gyration around the magnetic field, electron cooling becomes
substantially more efficient and a weakly outgassing comet can generate and
sustain a cold electron population.

At a comet with a low outgassing rate, $Q<2\times 10^{27}$ s-1, the ambipolar
field forms a potential well around the nucleus (Deca et al., 2017, 2019;
Divin et al., 2020). The potential well, which exceeds 60 V in depth at
$Q=10^{25}$ s-1, traps cometary electrons in the inner coma (Sishtla et al.,
2019; Stephenson et al., 2022). As well as causing electron cooling, the
ambipolar field accelerates and funnels solar wind electrons towards the
nucleus (Deca et al., 2017; Stephenson et al., 2022). The acceleration of
solar wind electrons by the ambipolar field is a likely source of ionizing
electrons within the coma. Measurements of the solar wind ions by the Ion
Composition Analyser (RPC/ICA; Nilsson et al., 2007) have been used to
estimate the potential difference between Rosetta and the upstream solar wind.
This potential fluctuates rapidly, but often exceeds several hundred electron
volts (Nilsson et al., 2022).

The electron density at Rosetta was measured by the Langmuir Probes (LAP;
Eriksson et al., 2007) and Mutual Impedance Probes (MIP; Trotignon et al.,
2007) of the Rosetta Plasma Consortium (RPC; Carr et al., 2007). Multi-
instrument models have been used to explain the electron density during
periods at large heliocentric distances (Galand et al., 2016; Heritier et al.,
2017a, 2018). These models include photoionization and electron-impact
ionization (EII) as a source of electrons, with EII frequencies derived from
measurements of energetic electrons by the Ion and Electron Sensor (RPC/IES;
Burch et al., 2007). EII was found to be a substantial but variable source of
cometary electrons over the considered intervals, sometimes dominating over
photoionization (Galand et al., 2016; Heritier et al., 2017a, 2018). The EII
frequency has not been assessed across the whole escort phase. Additionally,
the source and drivers of the ionizing electrons are not well known.

The variation of the energetic electron population throughout the coma is also
not well established. At large heliocentric distances, multi-instrument
analysis of far ultraviolet (FUV) emissions is consistent with a constant
energetic electron population with cometocentric distance (Chaufray et al.,
2017). FUV emissions from the coma of 67P were largely driven by dissociative
excitation by electron impact (Galand et al., 2020; Stephenson et al., 2021).
The models, which assume a constant energetic electron flux along the line-of-
sight and an ion bulk speed, $u_{i}$, equal to the neutral bulk speed,
$u_{gas}$, accurately reproduce the electron density and emission brightness
through a multi-instrument approach (Galand et al., 2016, 2020; Heritier et
al., 2017a, 2018; Stephenson et al., 2022).  The strong agreement between
modelled and observed FUV emissions during limb observation demonstrates that
emissions are driven by electrons accelerated on large scales (Galand et al.,
2020; Stephenson et al., 2021).

RPC/LAP and RPC/ICA measurements have been used to derive ion flow speeds in
excess of 10 km s-1 within 30 km of the nucleus at large heliocentric
distances (Nilsson et al., 2020; Johansson et al., 2021). The cometary ions
are born at the neutral outflow speed (0.4-1 km s-1; Marshall et al., 2017;
Biver et al., 2019). For such high speeds to be achieved at Rosetta, the
cometary ions must undergo substantial acceleration in the inner coma, which
conflicts with the findings of the studies combining multiple instruments and
models, where the ions are tied to the neutral gas ($u_{i}\approx u_{n}$; Beth
et al., 2022).

In addition to the cases at large heliocentric distances, there have been a
number of estimates of the ion velocity close to perihelion, particularly near
the diamagnetic cavity (Vigren et al., 2017; Odelstad et al., 2018; Bergman et
al., 2021a, b). These show accelerated ions at speeds of 7 km s-1 inside the
cavity, as well as ions streaming back towards the cavity. The multi-
instrument models of plasma density and FUV emissions are only applicable to
cases at large heliocentric distances and the weakly collisional regime
(Galand et al., 2016; Heritier et al., 2018), and cannot be compared with the
perihelion ion speed measurements. To extend the multi-instrument models to
higher outgassing rates, additional processes would need to be included,
particularly consideration of ion-neutral chemistry and electron-ion
recombination (Beth et al., 2022).

In this paper, we examine the source of the cometary electrons through two
methods (see Section 2). In Section 2.1, we calculate the ionization frequency
throughout the escort phase of Rosetta, through photoionization and electron-
impact ionization, based on in situ measurements. Section 2.3 outlines the
collisional test particle model used to model electrons in the cometary
environment and the simulation parameters.

The new dataset of electron-impact ionization frequency throughout the Rosetta
mission is compared to other measurements of the cometary environment, from
RPC and ROSINA, and properties of the spacecraft trajectory in Section 3. In
Section 4, the collisional test particle model is used to examine the origin
of electrons in the cometary environment, distinguishing between
photoelectrons, solar wind electrons and secondary electrons. Section 5
discusses the measurements of large ion flow speeds (>5 km s-1) by the RPC/LAP
and ICA instruments (Johansson et al., 2021; Nilsson et al., 2020) and the
difficulty of reconciling these measurements with multi-instrument studies of
the electron density and FUV emissions. The results are summarized in Section
6.

##  2 Methods

We address the source of electrons at comet 67P through two methods: analysis
of Rosetta data (see Section 2.1) and the application of a collisional test
particle model (Section 2.3; Stephenson et al., 2022).

###  2.1 Data from Rosetta mission

We calculate the ionization frequency from photoionization and electron-impact
ionization (EII) throughout the Rosetta mission. Section 2.1.2 outlines the
calculation of the EII frequency from in-situ measurements.

####  2.1.1 Ionization Frequencies

The photoionization frequency of water is given by

$\nu_{h\nu,\text{H}_{2}\text{O}}^{ioni}=\int\limits_{\lambda_{min}}^{\lambda_{th}}\sigma_{h\nu,\text{H}_{2}\text{O}}^{ioni}(\lambda)I(\lambda)\mathrm{d}\lambda.$
(1)

where $I(\lambda)$ is the photon flux at 67P.
$\sigma_{h\nu,\text{H}_{2}\text{O}}^{ioni}(\lambda)$ is the photoionization
cross section of water (Vigren & Galand, 2013).

The photon flux, $I(\lambda)$, is measured at 1 au by TIMED/SEE (1 day
resolution Woods et al., 2005), and then extrapolated to the position of comet
67P. The extrapolation assumes the flux follows $1/r_{h}^{2}$, with
heliocentric distance $r_{h}$, and includes a time-shift to correct for the
solar phase angle (Galand et al., 2016; Heritier et al., 2018). We assume
there is no attenuation of the solar flux in the coma, which is valid for
$Q<2\times 10^{27}$ s-1. We do not have measurements of the neutral gas column
between Rosetta and the Sun, but this is not significant at Rosetta during the
mission, as the cometocentric distance of the spacecraft varied with
outgassing (Heritier et al., 2018; Beth et al., 2019).

Solar wind ion impact ionization and solar wind charge exchange frequencies in
the coma were derived from in situ RPC/ICA measurements by Simon Wedlund et
al. (2019). These have a resolution up to 192 s. Close to perihelion, Rosetta
was inside the solar wind ion cavity (Behar et al., 2017; Nilsson et al.,
2017), so there are no estimates available for the solar wind ion impact or
charge exchange frequencies.

Table 1: Time periods considered during the mission (see Figure 1b). Period | Start Date | End Date | Heliocentric Distance [au]
---|---|---|---
Early Mission | 08-Sep-2014 | 05-Feb-2015 | 3.4 - 2.4
Around perihelion | 10-May-2015 | 02-Dec-2015 | 1.24 - 1.8
Late Mission | 01-Apr-2016 | 30-Sep-2016 | 2.7 - 3.8
Pre-perihelion | 08-Sep-2014 | 11-Aug-2015 | 3.4 - 1.24
Post-perihelion | 11-Aug-2015 | 30-Sep-2016 | 1.24 - 3.8
Pre-excursion | 08-Sep-2014 | 21-Sep-2015 | 3.4 - 1.24
Post-excursion | 23-Oct-2015 | 30-Sep-2016 | 1.51 - 3.8
Table 2: Datasets from the Rosetta mission, retrieved from the European Space Agency’s Planetary Science Archive (PSA), which are compared to the electron-impact ionization frequency. The potential difference between Rosetta and the solar wind, $\Delta U$, is calculated by Nilsson et al. (2022). 1 Balsiger et al. (2007). 2 Trotignon et al. (2007). 3 Eriksson et al. (2007). 4 Glassmeier et al. (2007). 5 Nilsson et al. (2007). Name | Equation/Symbol | Units | Instrument
---|---|---|---
Heliocentric distance | $r_{h}$ | au | -
Cometocentric Distance | $r_{Ros}$ | km | -
Spacecraft Latitude | - | ∘ | -
Spacecraft Longitude | - | ∘ | -
Neutral Density | $n_{\ce{H2O}}$ | cm-3 | ROSINA 1
Local Outgassing Proxy | $n_{\ce{H2O}}\times r_{Ros}^{2}$ | cm-1 | ROSINA 1
Electron Density | $n_{e}$ | cm-3 | RPC/MIP 2 & LAP 3
Spacecraft Potential | $V_{S/C}$ | V | RPC/LAP 3
Magnetic Field Strength | $B$ | nT | RPC/MAG 4
Clock Angle | $\arctan(B_{y}/B_{z})$ | ∘ | RPC/MAG 4
Cone Angle | $\arctan(\sqrt{B_{y}^{2}+B_{z}^{2}}/B_{x})$ | ∘ | RPC/MAG 4
H2O^+ Density ($<60$ eV) | $n_{\ce{H2O+}}$ | cm-3 | RPC/ICA 5
SW-Rosetta Potential Diff. | $\Delta U$ | V | RPC/ICA 5

####  2.1.2 Electron impact ionization frequency

We calculate the electron-impact ionization frequency throughout the Rosetta
mission, from measurements of the energetic electron population RPC/IES (Burch
et al., 2007). RPC/IES measured the electron differential particle flux,
$J(E)$, from 4.32 eV up to 17 keV at the detector, throughout the mission with
scans typically taking $\sim 2$ mins. In April 2015, half of the anodes
degraded, so these have been excluded from the analysis throughout the
mission. The electron differential particle flux, $J(E)$ [cm-2s-1eV-1], is
computed as described in Stephenson et al. (2021), assuming that the electron
flux is isotropic. The electron fluxes are calculated using half the anodes of
RPC/IES as the other half degraded in April 2015 (Broiles et al., 2016a).
While there could be up to a maximum factor of 2 uncertainty arising from the
assumption of isotropy, the thermal speed of the energetic electron population
is large compared to the bulk solar wind speed throughout the cometary
environment, so no significant anisotropy in the electron distribution is
expected. The flux is also corrected for the persistently negative substantial
spacecraft potential ($V_{S/C}<-10$ V) measured by the Langmuir probe, using
Liouville’s theorem (in which the phase space density is conserved along the
particle trajectories):

$\frac{J(E)}{E}=\frac{J_{IES}}{E_{IES}}\text{, with
}E\,\text{[$\mathrm{eV}$]}=E_{IES}\,\text{[$\mathrm{eV}$]}-qV_{S/C}\,\text{[$\mathrm{V}$]}.$
(2)

The $IES$ subscript indicates the quantities measured in situ by the
instrument, while those unlabelled are the quantities in the local cometary
environment. The spacecraft potential repelled low energy electrons away from
the detector and limited the threshold electron energy for detection
(electrons in the coma must have at least
$E_{\text{min}}\text{[eV]}=4.32\,\text{[eV]}-qV_{S/C}$ [V] to be detected for
$V_{S/C}<4.32$ V).

The electron particle flux is calculated in 15-minute intervals throughout the
mission, typically containing three or four scans of RPC/IES. After July 2016,
38% of the 15-minute intervals contain two RPC/IES scans, 12% contain one
scan, with the other 50% containing three or more. The electron flux in each
interval is extended to energies below the detection limit, assuming a
constant electron flux. The extrapolation to low energies does not
substantially affect the ionization frequency, as the electron-impact
ionization cross sections are small near the threshold ($E_{Th}\approx
12\leavevmode\nobreak\ eV$) (Itikawa & Mason, 2005).

The electron-impact ionization frequency for collisions with water
($\nu^{\text{ioni}}_{e,\ce{H2O}}$ [s-1]) is given by

$\nu^{\text{ioni}}_{e,\ce{H2O}}=\sum\limits_{X}\int\limits_{E_{Th,X}}^{E_{\text{Max}}}\sigma^{\text{ioni},X}_{e,\ce{H2O}}(E)\>J(E)\mathrm{d}E,$
(3)

where the summation is over the different ion products, $X$ (e.g., H2O+, OH+,
O+, H+). $\sigma^{\text{ioni},X}_{e,\ce{H2O}}(E)$ is the electron-impact
ionization cross section for electron collisions with water. The electron-
impact ionization cross-sections for water are based on measurements by Straub
et al. (1998), with a minor update to the instrument calibration applied by
Lindsay & Mangan (2003), as recommended by Itikawa & Mason (2005) and Song et
al. (2021).

The upper limit, $E_{\text{Max}}=300$ eV, is where the energetic electron flux
measured by RPC/IES becomes small (orders of magnitude less than those
measured at $E<100$ eV), reaching the 1 count per scan level. The retrieved
ionization frequencies are not dependent on the upper energy limit of the
integration. An increased upper energy limit from 300 eV to 450 eV only
results in an increase in $\nu_{e}^{\text{ioni}}$ of $<0.1$%.

###  2.2 Comparison with other Rosetta datasets

The ionization frequencies (particularly from EII) are compared to properties
of the coma (e.g. electron density) and spacecraft trajectory (e.g. spacecraft
latitude) to confirm the source of electrons and identify drivers of EII.
These are listed in Table 2, with the relevant instruments for each dataset.
The comparison datasets are time-averaged over the same 15-minute intervals as
the EII frequency.

The EII frequency is generally compared to other datasets using 2D histogram.
The counts in each column along the $x$-axis are normalised to 1, to attempt
to minimize sampling biases in the spacecraft trajectory. Mean, median and
quartiles are given for each column.

###  2.3 Test particle modelling

Table 3: Parameters of the test particle simulations. $Q$ is the cometary outgassing rate, while $u_{gas}$ is the neutral outflow velocity. The photoionization frequency ($\nu_{h\nu}^{ioni}$) and photoelectron temperature($T_{h\nu,e}$) are also given. $n_{SW},T_{SW,e},u_{SW,x},B_{SW}$ are the density, electron temperature, $x$ component of the bulk velocity and magnetic field strength of the undisturbed upstream solar wind, respectively. $\mathrm{d}x_{EB}$ gives the resolution of the electric and magnetic fields used as an input to the simulation (see Section 2.3). $\mathrm{d}t$ is the timestep used to generate the test particle trajectories. $N_{h\nu,pls}$ and $N_{SW,pls}$ are the number of macroparticles used in the photoelectron and solar wind electron simulations, respectively. Parameter |  | Simulation 1 | Simulation 2
---|---|---|---
$Q$ | [s-1] | $10^{26}$ | $1.5\times 10^{27}$
$u_{gas}$ | [km s-1] | 1 | 1
$\nu_{h\nu}^{ioni}$ | [s-1] | $1.32\times 10^{-7}$ | $1.32\times 10^{-7}$
$T_{h\nu,e}$ | [eV] | 10 | 10
$n_{SW}$ | [cm-3] | 1 | 1
$T_{e,SW}$ | [eV] | 10 | 10
$u_{SW,x}$ | [km s-1] | 400 | 400
$B_{SW,y}$ | [nT] | 6 | 6
$\mathrm{d}x_{EB}$ | [km] | 7.66 | 10
$\mathrm{d}t$ | [s] | $10^{-6}$ | $10^{-6}$
$N_{SW,pls}$ | - | $15\times 10^{6}$ | $3.48\times 10^{6}$
$N_{h\nu,pls}$ | - | $4.41\times 10^{6}$ | $1.43\times 10^{6}$

In addition to analysis of the Rosetta dataset, we have modelled electron
impact ionization throughout the coma using a 3D collisional electron test
particle model (Stephenson et al., 2022). The model includes photoelectrons,
solar wind electrons and secondary electrons, generated within the coma and at
the simulation boundaries. The $x$-axis of the domain points in the Sun-comet
direction, the $y$-axis is aligned with the interplanetary magnetic field, and
the $z$-axis completes the set. The extent of the domain is given by
$-770\text{\leavevmode\nobreak\ km}<x<1430\text{\leavevmode\nobreak\ km}$ and
$-1100\text{\leavevmode\nobreak\ km}<y,z<1100\text{\leavevmode\nobreak\ km}$,
with the comet nucleus at the origin.

The electrons move through the domain, pushed by stationary electric and
magnetic fields. The input electric and magnetic fields are taken from a
fully-kinetic but collisionless iPiC3D simulation (Deca et al., 2017, 2019),
run using the same simulation parameters as these authors (see Table 3). The
fields in the PiC model are evaluated on a $\mathrm{d}x_{EB}=7.66$ km grid for
the lower outgassing rate and 10 km grid for the higher outgassing rate. The
PiC model uses a reduced ion-electron mass ratio ($m_{e}=m_{p}/100$) to reduce
computation time, whereas the physical electron mass is used in the test
particle model (see discussion of the impact of the electron mass in
Stephenson et al., 2022).

In our model, the electrons can undergo collisions with molecules in the
neutral coma. The neutral coma is pure water and spherically symmetric, with a
constant outflow velocity for the neutral gas (Haser, 1957). The relevant
electron-water collisions across the energy ranges of interest (Itikawa &
Mason, 2005) are included in the model. This includes elastic scattering (Cho
et al., 2004; Faure et al., 2004) and inelastic collisions such as excitations
and electron-impact ionization. The collision processes are treated as
stochastic processes.

The model outputs are the energy distribution function (edf) of electrons,
$f(\bm{x},E)$ [cm-3eV-1], for each electron population (photoelectrons, solar
wind and secondaries of each). The density of each electron population is
given by:

$n_{e}(\bm{x})=\int f(\bm{x},E)\;\mathrm{d}E$ (4)

The EII frequency of each population is given by

$\nu^{\text{ioni}}_{e}(\bm{x})=\sum\limits_{X}\int_{E_{Th,X}}^{E_{\text{Max}}}\sigma_{e}^{\text{ioni},X}(E)\times
f(\bm{x},E)\times\sqrt{\frac{2E}{m_{e}}}\>\mathrm{d}E$ (5)

The speed factor $\sqrt{\frac{2E}{m_{e}}}$ converts the edf to an electron
flux. As for the measured electron fluxes, there are few electrons with
energies exceeding the integration limit and the derived ionization
frequencies are not sensitive to $E_{\text{Max}}$ ($<0.1$% change beyond 300
eV).

The secondary electrons produced within the simulation are generated through
stochastic collisions, rather than using Eq. 5 (Stephenson et al., 2022).
However, far from the nucleus the frequency of ionization collisions is quite
low, so the statistics are better when directly calculating the EII frequency
from the edf. Closer to the nucleus, the EII frequency calculated from the edf
is the same as produced through stochastic collisions within 10%. Up to seven
generations of electrons are simulated, with each impact ionization collision
produced 10 macroparticles with their total weight equal to that of the
ionizing electron (Stephenson et al., 2022). Beyond seven generations, the
weight of each macroparticle is reduced by $10^{7}$ compared to the particles
of the first generation. The total number of macroparticles also falls with
generation, so any subsequent generations have negligible impact on the
simulation outputs.

##  3 Ionization frequency throughout the Rosetta mission

We first analyse the new dataset of electron-impact ionization frequency
throughout the Rosetta mission to determine the source of electrons within the
coma and some key drivers of the e-impact ionization frequency. Section 3.1
examines the source of the electrons within the coma, while Section 3.2
identifies the origin of the ionizing electrons.

(a)(b) Figure 1: (a) Ionization processes throughout the Rosetta escort phase.
The electron-impact ionization frequency (blue) is calculated in 15-minute
intervals, from RPC/IES measurements (Burch et al., 2007). Median, 25th and
75th percentiles for the EII freqeuncy are given for 15 day periods across the
escort phase (pink). Photoionization frequency (green) is calculated from
daily measurements of solar flux at 1au by TIMED/SEE (Woods et al., 2005) and
then extrapolated to 67P. The electron-impact ionization frequency has been
fitted with respect to (orange) local outgassing rate, (yellow) heliocentric
distance and (purple) $(T-T_{PH})$. The coefficients and goodness of the fits
are listed in Table 4 and the time periods for the fits are outlined in Table
1. The photoionization frequency is fitted using Eq. 6 (black). (b)
Heliocentric distance (blue) and cometocentric distance (orange) throughout
the Rosetta mission. The time intervals used in the analysis and listed in
Table 1 are shown.

###  3.1 Source of electrons in the coma

####  3.1.1 Comparison of ionization sources

Figure 1 shows the frequency of the two major ionization processes at comet
67P throughout the Rosetta mission, photoionization and electron-impact
ionization.

Photoionization (green, Figure 1a) is the steadiest process throughout the
mission. The photoionization frequency is evaluated at Rosetta, assuming there
is no absorption of the flux between the Sun and the spacecraft. The frequency
varies roughly with $1/r_{h}^{2}$ from $10^{-7}$ s-1 at the start of mission
to $5.2\times 10^{-7}$ s-1 on 17 Jul 2015, due to the variation in the photon
flux. Shorter timescale oscillations are driven by the solar rotation with a
period of $\sim 28$ days. Additionally, the solar activity decreased over the
Rosetta mission leading to smaller photon fluxes towards the end of mission.

The electron impact ionization frequency (blue) fluctuated much more than
photoionization. Far from perihelion, the EII frequency varied by three orders
of magnitude and was often much higher than the photoionization frequency.
Near the end of mission, the EII frequency reached up to $1.4\times 10^{-6}$
s-1, whereas photoionization frequency was only $5\times 10^{-8}$ s-1. In the
late mission period (see Table 1), the photoionization frequency was
$(4-8)\times 10^{-8}$ s-1 while the EII frequency varied between $10^{-9}$ s-1
and $4\times 10^{-6}$ s-1 with a median of $1.68\times 10^{-7}$ s-1. At times,
photoionization was still the main source of cometary electrons, as the EII
frequency was so variable. This is consistent with previous studies of the
electron-impact ionization frequency over shorter time intervals (Galand et
al., 2016; Heritier et al., 2017a, 2018). Towards the start of the escort
phase, the minima in EII frequency were comparable to the photoionization
frequency at $1.2\times 10^{-7}$ s-1.

Close to perihelion, the variability in EII frequency was greatly diminished
to only a factor of 10 over short intervals. As a result, it was smaller than
the photoionization frequency throughout this period (up to $5\times 10^{-7}$
s-1). This may not be true throughout the coma, as photoabsorption is
significant in the inner coma at large outgassing rates ($r<70$ km for
$Q=3\times 10^{28}$ s-1 at the terminator; Heritier et al., 2018; Beth et al.,
2019).

In addition to photoionization and electron-impact, charge exchange (CX) and
solar wind ion impact (SWII) are sources of cometary ions. Throughout the
Rosetta mission, the frequencies of CX and SWII were typically both very small
compared to the electron impact ionization frequency ($10^{-8}$ s-1 for CX and
$10^{-10}$ s-1 for SWII; Simon Wedlund et al., 2019). On short timescales,
solar wind charge exchange was a substantial source of cometary ions, but this
was infrequent. Charge exchange does not have a net contribution to the plasma
production, as no free electrons are generated. However, it can modify the
plasma density through changes in transport and the fields due to mass loading
of the solar wind.

At large heliocentric distances, EII is often the major source of electrons
within the coma, although it is highly variable. When the EII is at a minimum,
photoionization is also an important source of electrons away from perihelion.
Around perihelion, the EII frequency is much less variable than at large
heliocentric distances and was consistently lower than the photoionization
frequency at Rosetta. The median EII frequencies (pink, Figure 1a) are similar
between perihelion and towards the end of mission, but the peaks in the EII
frequency were much larger towards the end of mission. At large heliocentric
distances, EII is the major source of electrons within the coma, although it
is highly variable. When the EII is at a minimum, photoionization is also an
important source of electrons away from perihelion. Around perihelion, the EII
frequency is smaller than found at large heliocentric distances and is weaker
than photoionization.

Fits to the median values of the photoionization frequency and electron-impact
ionization frequency are shown in Figure 1a. The photoionization frequency at
1au decreased approximately linearly over the Rosetta mission, due to the
decrease in solar activity (Heritier et al., 2018). The approximate
photoionization frequency (in s-1) at Rosetta (black, Figure 1a) is given by:

$\nu_{h\nu}^{\text{ioni}}=\frac{1}{r_{h}^{2}}(p_{0}+p_{1}(T-T_{PH}))$ (6)

where $T-T_{PH}$ is the number of days since perihelion, $r_{h}$ is in au,
$p_{0}=7.18\times 10^{-7}$ s-1 au2 and $p_{1}=-3.23\times 10^{-10}$ s-1au2
days-1.

The electron impact ionization frequency has been fitted using several
variables: heliocentric distance ($r_{h}$ [au]), local outgassing rate
($nr^{2}$ [cm-1]) and days since perihelion. The fits with heliocentric
distance and outgassing are split into two periods around the excursion in
September 2015 (see Table 1 and red, Fig. 1a), as the EII frequency behaved
differently through these periods (see Sections 3.2.3). After the excursion,
heliocentric distance was not a good predictor of the EII frequency, so the
fit over this period is not shown in Figure 1a. The median EII frequency is
much more variable after the excursion, which may partially be driven by the
arrival of corotating interaction regions at 67P from Jun-Sep 2016 (Edberg et
al., 2016a; Hajra et al., 2018; Stephenson et al., 2021).

The dependence between the variables $x=r_{h}$ (yellow, Fig. 1a), $nr^{2}$
(orange), $(T-T_{PH})$ (purple) and the EII frequency are given by:

$\log_{10}(\nu_{e}^{\text{ioni}})=\sum\limits_{n}p_{n}x^{n},$ (7)

where the polynomial coefficients and adjusted-$R^{2}$ values are listed in
Table 4. The plotted fit for $x=nr^{2}$ has been calculated using the median
local outgassing rates throughout the mission, with the pre and post-excursion
periods (see Table 1) split into 100 equal parts of 90 and 82 hours
respectively.

The dependence of the EII frequency on the local outgassing rate (orange,
Figure 1) is discussed in Section 3.2.3 and seen in Figure 6.

Table 4: Coefficients, $p_{n}$, of fits to the median electron-impact ionization frequencies throughout the Rosetta mission (see Eq. 7). The independent variable, $x$, for each fit is given in addition to the adjusted-R-squared value for each fit. The time periods for each fit are outlined in Table 1. Each of the fits is shown in Figure 1a. The fit to the local outgassing rate is shown in Figure 6 and the pre-excursion fit to the heliocentirc distance is shown in Figure 16a. Variable | Period | $p_{0}$ | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$ | Adj-R-sq
---|---|---|---|---|---|---|---
$r_{h}$ [au] | Pre-Exc | -2.4918 | -7.8105 | 4.16 | -0.6599 | - | 0.65
$\log_{10}$($nr^{2}$ [cm-1]) | Pre-Exc | -2127 | 744.6 | -97.62 | 5.665 | -0.1228 | 0.922
$\log_{10}$($nr^{2}$ [cm-1]) | Post-Exc | 540.3 | -225.3 | 34.28 | -2.29 | 0.05674 | 0.6399
$T-T_{PH}$ [days] | All | -7.1690 | $-3.34\times 10^{-3}$ | $1.36\times 10^{-5}$ | $3.14\times 10^{-8}$ | $-1.14\times 10^{-10}$ | 0.56

####  3.1.2 EII vs electron density

In order to confirm that EII is the major source of electrons away from
perihelion, we compare the EII frequency with the electron density at Rosetta,
derived from the cross-calibration between RPC/MIP and LAP (Johansson et al.,
2021). The electron and ion density are dependent on cometocentric distance,
with $n_{e}\propto\frac{r-r_{67P}}{r^{2}}$ when transport is significant
(Galand et al., 2016; Heritier et al., 2017a, 2018; Beth et al., 2022). For
the sake of simplicity, we assume a spherical comet with an equivalent radius
of $r_{67P}=1.7$ km (Jorda et al., 2016). In order to remove the dependence of
Rosetta’s distance to the nucleus, where $n_{e}$ is measured, the electron
density is corrected. For an ionization frequency (and gas outflow speed,
$u_{gas}$) independent of cometocentric distance the corrected electron
density is given by:

$n_{e,corr}=n_{e}\times\frac{r^{2}}{r-r_{67P}}=\frac{Q\nu^{ioni}}{4\pi
u_{gas}u_{i}},$ (8)

where $u_{i}$ is the bulk radial ion speed (Galand et al., 2016; Heritier et
al., 2018; Beth et al., 2022). This has the same unit as a column density. The
correction to the electron density does not significantly change the
correlation early and late in the mission as the bulk of the measurements is
taken over a narrow range of cometocentric distances (see Figure 1b).

Figure 2 shows the relationship between the corrected electron density and
electron impact ionization frequency. The corrected electron density is
positively correlated with EII frequency in the early and late mission
periods, when the EII frequency is not small compared to the photoionization
frequency (median values have $R^{2}=0.978$ for
$\nu_{e}^{\text{ioni}}>10^{-7}$ s-1 for the early period) When the EII
frequency is relatively small, the electron density does not vary
significantly with EII frequency. This occurs when the photoionization
frequency is comparable to or larger than the EII frequency, so most cometary
electrons are photoelectrons. This occurs at $\nu_{e}^{\text{ioni}}<1.5\times
10^{-7}$ s-1 in the early mission period, although the statistics are smaller
over this range (only 550 intervals with $\nu_{e}^{\text{ioni}}<10^{-7}$ s-1
out of 8926 in Figure 2a).

The corrected electron density is well correlated to the total ionization
frequency (photoionization and EII) across the whole range of frequencies,
with the lower limit being the photoionization frequency. Around perihelion,
the median corrected electron density shows no significant dependence on the
EII frequency. Electron-impact ionization is significantly weaker than
photoionization around perihelion, so it is not the key source of cometary
electrons over the period (see Figure 1). The comparison to the total
ionization frequency is not useful around perihelion, as it is dominated by
photoionization and does not vary substantially. Other sources of variation in
the coma are more important for the electron density around perihelion, such
as the outgassing rate.

The EII frequency is also strongly correlated to the spacecraft potential
early and late in the mission, with more EII leading to more negative
potentials (see Appendix A.2). A hotter and denser electron population drives
more negative spacecraft potentials (Odelstad et al., 2015; Johansson et al.,
2021), while containing more energetic electrons that can drive impact
ionization (e.g. Cravens et al., 1987). Around perihelion, the spacecraft
potential was not strongly dependent on EII frequency, supporting the
conclusion that EII was not the major source of electrons close to the Sun.

Early MissionAround PHLate Mission(a)(b)(c)$\begin{aligned}
\log_{10}\big{(}\frac{y}{\mathrm{m}^{-2}}\big{)}&=\\\
0.555&\log_{10}\big{(}\frac{x}{\mathrm{s}^{-1}}\big{)}+15.65\end{aligned}$$R^{2}=0.978$$\begin{aligned}
\log_{10}\big{(}\frac{y}{\mathrm{m}^{-2}}\big{)}=\\\
0.254\log_{10}\big{(}\frac{x}{\mathrm{s}^{-1}}\big{)}+14.19\end{aligned}$$R^{2}=0.936$
Figure 2: (a-c) Histogram of the cross-calibrated electron density measured at
Rosetta by RPC/MIP and LAP (Trotignon et al., 2007; Eriksson et al., 2007;
Johansson et al., 2021), corrected for cometocentric distance (assuming
$n_{e}\propto 1/r$, Heritier et al., 2017a), against electron-impact
ionization frequency. Median (black circles) and quartile values (error bars)
are plotted for each bin along $x$. Linear fits are given for the early and
late periods (blue crosses). The corrected electron density has the same units
as a column density.

####  3.1.3 EII vs water ion density

Figure 3 shows the correlation between the EII frequency and the low energy
water ion density, derived from moments of the RPC/ICA data (Nilsson et al.,
2007). The density estimate includes contributions from all ions near 18 amu,
such as H2O+, H3O+ and NH4+. The RPC/ICA measurements of the low energy
cometary ions were less frequently available than the RPC/LAP-MIP electron
density or RPC/IES electron flux measurements, with 4113, 9598 and 8065
intervals included in Figures 3a, b and c, respectively (compared to 8926,
17746 and 15000 intervals in the corresponding plots of Figure 2).The moments
of the RPC/ICA data are 1 to 1.5 orders of magnitude smaller than the electron
densities measured concurrently by the RPC/LAP and RPC/MIP instruments.
Therefore, RPC/ICA must be missing a large part of the cometary ion
population. This may be a result of the limited field of view of the
instrument, or a consequence of the strong deflection of low energy cometary
ions by the spacecraft potential (Bergman et al., 2020). However, we are
interested in the variation rather than the magnitude of the ion population.
The density calculation also neglects the deflection of the cometary ions near
the detector (Bergman et al., 2020).

As with the electron density, the low energy cometary ion density, $n_{i}$, is
normalized for the cometocentric distance by using
$n_{i,corr}=n_{i}\times\frac{r^{2}}{r-r_{67P}}$. The corrected low-energy
cometary ion density is well correlated to the EII frequency, both early and
late in the mission. Far from perihelion, an increasing EII frequency is
associated with higher cometary ion densities. Late in the mission, the
cometary water ion density is fairly constant with EII frequency when the EII
frequency is small ($<3\times 10^{-8}$ s-1). At larger ionization frequencies,
there is a clear positive correlation (median values have $R=0.9645$, and
$R=0.9760$ early and late in the mission, respectively). Around perihelion,
the density of cometary ions is not strongly dependent on the EII frequency
($R=-0.41$).

This supports the conclusions in Section 3.1.2, that ionization within the
coma is primarily driven by EII when 67P was far from perihelion. The bulk of
ions and electrons in the coma are produced by EII, as expected from Figure 1.
Around perihelion, EII is a weaker process and less ionizing than
photoionization, so the plasma density in the coma is not strongly dependent
on the EII frequency.

Early MissionAround PHLate Mission(a)(b)(c)$R^{2}=0.930$$\begin{aligned}
\log_{10}\big{(}\frac{y}{\mathrm{m}^{-2}}\big{)}=\\\
1.057\log_{10}\big{(}\frac{x}{\mathrm{s}^{-1}}\big{)}+17.63\end{aligned}$$\begin{aligned}
\log_{10}\big{(}\frac{y}{\mathrm{m}^{-2}}\big{)}&=0.797\log_{10}\big{(}\frac{x}{\mathrm{s}^{-1}}\big{)}\\\
+&15.78\end{aligned}$$R^{2}=0.953$ Figure 3: Electron-impact ionization
frequency against low-energy cometary water ion density (a) early in the
mission, (b) around perihelion and (c) late in the mission. Median (black
circles) and quartile values (error bars) are plotted for each bin along $x$.
Linear fits are given for the early and late periods (blue crosses).

###  3.2 Drivers of electron-impact ionization

Having confirmed that EII is the major source of cometary electrons at large
heliocentric distances, we next identify some key drivers of the ionizing
population of electrons.

####  3.2.1 EII vs magnetic field strength

Early MissionAround PHLate
Mission(a)(b)(c)$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.0238\frac{x}{\mathrm{nT}}-6.62$$R^{2}=0.825$$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.0281\frac{x}{\mathrm{nT}}-7.39$$R^{2}=0.844$
Figure 4: Electron-impact ionization frequency vs. magnetic field magnitude
(a) early in the mission, (b) around perihelion and (c) late in the mission.
Median (black circles) and quartile values (error bars) are plotted for each
bin along the $x$-axis. Linear fits are given for the early and late periods
(blue crosses).

Figure 4 shows the relation between the EII frequency and magnetic field
strength. Early and late in the mission, the ionization frequency increases
with increasing magnetic field strength, with a steeper gradient late in the
mission (0.0238 in the early period vs. 0.0281 in the late period). The
correlation holds well for weak magnetic fields ($<17$ nT in the early period,
and $<12$ nT for the late period), but plateaus at
$\nu^{\text{ioni}}_{e,\text{H}_{2}\text{O}}=10^{-7}$ s-1 in the late mission
period. However, at the higher field strengths, there are far fewer data
points with less than 100 per bin beyond $B=20$ nT, where the median EII
frequencies show more variability.

Around perihelion, there is no significant variation of the EII frequency with
magnetic field strength (Figure 4b). The median EII frequency only varied
within a factor of 2 between $B=0$ nT and $B=50$ nT, whereas the EII frequency
increased by a factor of 5 between $B=0$ nT and $B=25$ nT at large
heliocentric distances. The EII frequency decreases slightly with increasing
magnetic field strength up to 25 nT, and then increases.

The EII frequency varies non-linearly with the clock and cone angles of the
magnetic field, both early and late in the mission. However, the non-linear
relationship with the orientation of the field likely reflects the correlation
between the angles and magnetic field strength (see Appendix A.3).

The relationship with magnetic field strength early and late in the mission
may indicate that EII is strongest in regions where pile up is substantial.
The pile-up of the solar wind magnetic field is also associated with an
increase in the solar wind electron density. It is found deeper in the coma,
where the ambipolar potential well is more likely to be deep. This indicates
that solar wind electrons, accelerated by the ambipolar field, are piling up
with the magnetic field and driving ionization. Additionally, magnetic
mirroring can lead to an increase in the perpendicular electron energy in
regions of high magnetic field. A large perpendicular energy, compared to
parallel energy, leads to more efficient electron trapping within the
potential well and therefore the trapped electrons spend more time in a region
where ionization can occur.

####  3.2.2 EII vs. potential well

Figure 5 shows the correlation between the Rosetta-upstream solar wind
potential difference ($\Delta U$) and the EII frequency. The electric
potential difference between the observation point and the upstream solar wind
is derived from in situ measurements of the solar wind ions (H+, He^++) by
RPC/ICA (Nilsson et al., 2007, 2022).

Both early and late in the Rosetta mission, the EII is very well correlated to
the solar wind potential difference at low outgassing rates. Early in the
mission, the median EII frequency increased from $2.5\times 10^{-7}$ s-1 at 50
V to $8\times 10^{-7}$ s-1 at 300 V. Late in the mission, the EII frequency
increased from $3\times 10^{-8}$ s-1 to $2\times 10^{-7}$ s-1 between 40 and
250 V. At large values of $\Delta U$ ($>500$ V in the early mission and $>400$
V in the late period) the median values show much more variability and the
positive correlation seems to break down, but the statistics here are weaker
with fewer than 30 points per bin.

It was not possible to estimate the potential difference to the solar wind
around perihelion, as Rosetta was in the solar wind ion cavity (Nilsson et
al., 2017).

The solar wind potential difference is a local measurement of the ambipolar
potential well that formed around 67P. As the EII frequency is well correlated
to the potential difference, it is very likely that the ionizing electron
population are solar wind electrons that have been accelerated by the
ambipolar field.

Early MissionLate
Mission(a)(b)$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=1.25\times
10^{-3}\frac{x}{\mathrm{V}}-6.69$$R^{2}=0.847$$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=2.08\times
10^{-3}\frac{x}{\mathrm{V}}-7.43$$R^{2}=0.814$ Figure 5: EII frequency against
SW-Rosetta potential difference (a) early in the mission and (b) late in the
mission. Median (black circles) and quartile values (error bars) are plotted
for each bin along the $x$-axis. Linear fits are given for the early and late
periods (blue crosses).

####  3.2.3 EII vs outgassing rate

Pre-PHPost-PH(a)(b) Figure 6: Electron-impact ionization frequency against a
proxy measurement of the outgassing rate ($n\times r^{2}$, where $n$ is the
neutral density measured by ROSINA Balsiger et al., 2007) for the (a) pre-
perihelion and (b) post-perihelion phases. Median (black circles) and quartile
values (error bars) are plotted for each bin along the $x$-axis. The fit to
the median EII frequency as a function of local outgassing rate is shown in
each period in blue crosses (see Table 4).

Figure 6 shows the variation of the EII frequency with a proxy of the local
outgassing rate, $n\times r^{2}$. The neutral density, $n$, is measured
locally by ROSINA/COPS, with a correction for the composition using
ROSINA/DFMS (Balsiger et al., 2007; Gasc et al., 2017). This neglects
variation of the neutral gas outflow velocity, which varied between 400 m s-1
and 900 m s-1 throughout the escort phase (Biver et al., 2019). Both pre- and
post-perihelion, the EII frequency has a non-linear relationship (see Figure
6) with the proxy for the local outgassing rate and shows little variability
around the wider trend (especially in comparison to the relationship with
heliocentric distance, Section A.1). At low outgassing rates ($n\times
r^{2}<3\times 10^{10}$ cm-1), the ionization frequency increases with
outgassing rate. As the outgassing increases further, the ionization frequency
remains fairly constant before beginning to decrease. The EII frequency then
falls until $n\times r^{2}=3\times 10^{11}$ cm-1, after which the ionization
frequency remains constant or shows a marginal increase.

Post perihelion, the magnitude of the ionization frequency is smaller than
early in the mission, particularly at large heliocentric distances (see Figure
1). This may be driven by the decreasing solar activity throughout the
mission, which results in a reduction in the EUV flux for a given heliocentric
distance (Lean et al., 2003; Woods & Rottman, 2002; Woods et al., 2012). The
EUV flux drives photoionization in the coma which is critical for generating
the ambipolar potential well, into which solar wind electrons can be
accelerated.

The variation of the ionization frequency with outgassing may be indicative of
the evolution of the ambipolar potential well throughout the mission. Far from
perihelion, the coma is very weakly collisional and increased outgassing
produces more electrons locally and strengthens the electron pressure gradient
in the coma ($\bm{E}_{Ambi}=-\frac{1}{en_{e}}\nabla(n_{e}k_{B}T_{e})$). This
increases the ambipolar field strength and causes increased acceleration of
solar wind electrons into the coma. As the coma becomes denser, it also
becomes increasingly collisional. The electron-neutral collisions, in
conjunction with trapping in the ambipolar field, become strong enough to cool
electrons within the inner coma (Stephenson et al., 2022). This weakens the
pressure gradient in the inner coma and the ambipolar field strength. In this
way, negative feedback from the cooling on the potential well is established
(Stephenson et al., 2022). This also results in less acceleration and
funnelling of the solar wind electrons into the inner coma. The feedback of
collisions onto the ambipolar field is discussed further in Section 5.

At higher outgassing rates, the electron cooling and feedback become stronger,
further damping the ambipolar potential well. Close to perihelion, the
ambipolar field is effectively quenched in the inner coma by the electron-
neutral collisional cooling.

A similar relationship is also observed between the EII frequency and
heliocentric distance (see Section A.1), but with much more variability,
especially post-perihelion. This may be caused by the variation with
cometocentric distance which is unaccounted for or by the outgassing variation
across the surface of the comet, especially between hemispheres.

In the early and late mission periods (see Table 1), the correlation with
outgassing rate is also observed in the relation with spacecraft latitude. In
the early period, the EII frequency increases with latitude from the southern
to the northern hemisphere, where the outgassing was stronger (Hansen et al.,
2016; Fougere et al., 2016; Läuter et al., 2018). Late in the mission, the
strongly outgassing southern latitudes were associated with the highest EII
frequencies. Both the early and late periods have outgassing rates in the
lower range of those seen in Figure 6 ($n\times r^{2}<3\times 10^{10}$ cm-1).
In the two periods, outgassing was positively correlated with EII frequency,
so the more outgassing hemisphere typically saw larger ionization rates.
Closer to perihelion (i.e., $n\times r^{2}>3\times 10^{10}$ cm-1), outgassing
and EII frequency are negatively correlated, so the less outgassing latitudes
may see higher levels of EII.

###  3.3 Electron impact ionization vs. cometocentric distance

Figure 7 shows the correlation between EII frequency and the cometocentric
distance during three periods of the mission. Both early and late in the
mission, there is no obvious correlation between the EII frequency and
cometocentric distance. Early in the mission, the cometocentric distance did
not vary substantially and the bulk of the measurements were taken at 20 km or
30 km from the nucleus. At other distances, there is little data over the
early mission period and no clear correlation with cometocentric distance.

Late in the mission, Rosetta collected data more evenly between cometocentric
distances of 5 and 30 km. This period shows a slight increase in the electron-
impact ionization frequency with increasing cometocentric distance. There are
measurements up to 500 km in the late period, but not enough datapoints beyond
32 km (only 131 points compared to 14,500 for $r<32$ km) to draw any
significant conclusions.

The trajectory of Rosetta was tuned throughout the mission to approximately
maintain the neutral density at the spacecraft. Therefore, larger
cometocentric distances are linked to higher outgassing rates. Furthermore, in
the late mission period, the coma was weakly collisional and higher outgassing
rates led to increased EII frequencies. Hence, these two factors could be a
driver of the increase in EII frequency with cometocentric distance late in
the mission (see Figure 7c).

Around perihelion, the EII frequency does exhibit a dependence on
cometocentric distance. Between 200 and 600 km, the EII frequency decreases
with increasing cometocentric distance from $\nu_{e}^{\text{ioni}}=10^{-7}$
s-1 to $4\times 10^{-8}$ s-1 at 600 km. This could be driven by a process
accelerating electrons towards the nucleus. However, beyond 400 km there is
limited data and this largely originates from the excursion around perihelion
(see Figure 1b), so could be driven by temporal changes. For example, a CME
was observed during the inbound leg of the excursion at $\sim 800$ km (Edberg
et al., 2016b), which drives the peak in ionization frequency at that
cometocentric distance.

Closer to the nucleus ($<200$ km), the EII frequency decreases with decreasing
cometocentric distances. This may indicate the region in which electron-
neutral collisions can efficiently cool the suprathermal electrons.

During the mission, there were a number of periods where Rosetta rapidly
changed cometocentric distance, such as the excursion at the end of March
2016. The EII frequency did not behave in the same way between manoeuvres or
between the outbound and inbound motion of the spacecraft. At times, the EII
frequency would decrease as the spacecraft moved away from the nucleus, such
as the increase from 28 km to 260 km from 15-17 Feb 2015. However, as the
spacecraft returned to 100 km on 25 Feb 2015, the EII frequency fluctuates but
shows no overall change. The fluctuations of the EII frequency throughout the
manoeuvres greatly exceeded the overall change in magnitude with cometocentric
distance.

Early MissionAround PHLate
Mission(a)(b)(c)$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.0014\frac{x}{\mathrm{km}}-6.42$$R^{2}=0.055$$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.0177\frac{x}{\mathrm{km}}-7.40$$R^{2}=0.660$
Figure 7: Electron-impact ionization frequency vs. cometocentric distance (a)
early in the mission, (b) around perihelion and (c) late in the mission.
Median (black circles) and quartile values (error bars) are plotted for each
bin along $x$, in addition to the mean values (black crosses). Linear fits to
the median values (blue crosses) are shown for the early and late mission
periods below in the ranges 10 to 32 km and 5 to 32 km, respectively.

###  3.4 Summary of ionization frequency throughout the Rosetta mission

(a)(b)(c)(d)(e) Figure 8: Summary of the median, 25th and 75th percentiles for
Figures 2 to 6, for each time period (see Table 1 and Figure 1).

Figure 8 summarises Figures 2 to 6. Analysis of the ionization frequencies
demonstrate that, at large heliocentric distances, electron-impact ionization
is the main source of ionization, as the EII frequency is positively
correlated with the electron and ion densities (yellow and blue, Figures 8a,
b). Around perihelion, the electron and ion densities show little dependence
on the EII frequency (red, Figures 8a, b), as photoionization is more ionizing
than EII during this period (green vs blue, Figure 1). The magnetic field
strength and the Rosetta-upstream solar wind potential difference are key
drivers of the EII frequency (Figures 8c, d), when comet 67P was far from
perihelion. These indicate that the ionizing electrons originate in the solar
wind and are accelerated into the inner coma by an ambipolar electric field.
The EII frequency behaves non-linearly with respect to changes in the local
outgassing rate (Figure 8e), which may reflect the changes in the ambipolar
potential well. At low outgassing rates, a denser coma results in a deeper
potential well and an increased EII frequency. However, at higher outgassing
rates, electron-neutral collisions are more significant and lead to electron
cooling and damping of the ambipolar field in the inner coma. This limits
acceleration of solar wind electrons into the inner coma and leads to a
reduction in the EII frequency.

##  4 Origin of cometary electrons with test particle simulations

When measuring electrons, it is not possible to distinguish the origin of the
electron, whether from the solar wind or produced in the cometary environment
through photoionization or EII. Additionally, for those electrons produced by
e-impact, it is also not possible to determine the origin of the ionizing
electron. With the test particle model, electrons are distinguished by their
origin and, in the case of secondary electrons, by the origin of the ionizing
electron. In this way, we can disentangle the most substantial electron
populations within the coma and the populations which drive ionization.

###  4.1 Comparison of electron sources

Figure 9 shows the electron density calculated using the collisional test
particle simulation at outgassing rates of (a-c) $Q=10^{26}$ s-1 and (d-f)
$Q=1.5\times 10^{27}$ s-1 (see Table 3 for other simulation parameters). For
67P, the outgassing rates correspond to heliocentric distances of 2.5-2.8 au
and 1.8-2 au, respectively (Läuter et al., 2018; Biver et al., 2019). The
density of photoelectrons (a, d), solar wind electrons (b, e) and secondary
electrons (c, f) are separated. Total electron densities for each outgassing
rate are shown in Figure 22, and fractional density of each population are
seen in Figure 23.

The simulation axes are aligned to the upstream solar wind conditions with the
bulk solar wind along $\hat{\bm{x}}$, the solar wind magnetic field aligned
with $\hat{\bm{y}}$ and the convective electric field pointing in the
$-\hat{\bm{z}}$ direction.

In the low outgassing case, the photoelectrons (Figure 9a) are confined to a
small region in the inner coma, with the densest region ($n_{e}>10$ cm-3) only
extending 50 km from the nucleus. There is also a narrow region of high
density that extends tailwards from the inner coma in the $+x$ and $+z$
direction. These electrons have been transported away from the dense innermost
region by the $\bm{E}\times\bm{B}$ drift.

The photoelectrons are primarily produced in the inner coma, where the neutral
density (following $1/r^{2}$) is at its highest. There is no substantial
absorption of the solar photons within the weakly outgassing comet ($<10\%$
for $Q<2\times 10^{27}$ s-1), so a constant photoionization frequency
($\nu_{e}^{h\nu}=1.32\times 10^{-7}$ s-1, Table 3) is used throughout the
coma.

Therefore, the production of photoelectrons also scales with $1/r^{2}$.
Produced locally at 10 eV, the photoelectrons are effectively trapped in the
coma by the ambipolar field.

The solar wind electrons are found throughout the coma with a substantial
density across the simulation domain (Figure 9b). At the simulation
boundaries, the solar wind electron density is 1 cm-3 and forms the dominant
source of electrons at large cometocentric distances. The solar wind electron
density does increase to 8 cm-3 in the inner coma, as the electrons are
funnelled towards the nucleus by the ambipolar electric field.

Close to the nucleus ($<40$ km), the solar wind and photoelectrons have
comparable densities, with solar wind electrons larger by $\sim 50$%. The
solar wind electrons are typically more energetic than the photoelectrons
within the dense region. Consequently, it is more likely that solar wind
electrons, rather than photoelectrons, will ionize neutral species, producing
more secondary electrons while degrading their own energy.

Secondary electrons form a dense population in the inner coma ($<30$ km,
Figure 9c), with a density similar to the total of both the photoelectrons and
solar wind electrons. Newly-born secondary electrons are energetic enough to
themselves cause impact ionization but are trapped in the dense, inner region
of the coma by the ambipolar electric field. The confinement makes the
secondary electrons more likely to ionize than the primary solar wind
electrons, which are unlikely to remain in the inner coma for long. With
further generations, the secondary electrons become less dense and cooler as
the energy is distributed between more electrons, in addition to the inelastic
collisions that dissipate energy. As electrons become cold, it also becomes
more likely that they will recombine with cometary ions.

The total secondary electron density decreases rapidly with increasing
cometocentric distance, and at a much faster rate than the photoelectron
population. Beyond 50 km, the secondary electron density drops below the
photoelectrons and beyond 100 km the solar wind electrons dominate the
density. The faster decline, compared to photoelectrons, results from the fall
in electron-impact ionization frequency with distance from the nucleus (see
Figures 10 and 12).

At $Q=1.5\times 10^{27}$ s-1, the photoelectron population (Figure 9d) is
denser and extends to larger cometocentric distances than in the low
outgassing case. The same photoionization frequency is used as in the lower
outgassing case (Table 3), so the production of photoelectrons is 15 times
larger due to the increased outgassing. A similar increase (between 10 and 20)
is seen in the photoelectron density in the inner coma of the low outgassing
case ($<50$ km). The region of high photoelectron density extends up to 300 km
from the nucleus in the higher outgassing case, extending towards $+z$. The
denser ionosphere at $Q=1.5\times 10^{27}$ s-1 produces a larger region of
interaction with the solar wind plasma.

The solar wind electrons are initiated at the boundary with the same
parameters as used at $Q=10^{26}$ s-1. However, the solar wind density is
enhanced ($n_{e,SW}>10$ cm-3) over a larger region due to the greater extent
of the ambipolar field for $Q=1.5\times 10^{27}$ s-1. The solar wind electrons
in the inner coma closely resemble the photoelectron population in the region.
The solar wind electrons, having been accelerated by the ambipolar field,
undergo collisions with the neutral coma. They are then efficiently trapped by
the ambipolar potential well, leading to further cooling, followed by
transport by particle drifts.

The electrons further from the coma are not confined by the ambipolar field,
although they still are accelerated and deflected by it. In the dense region
of photoelectrons ($n_{e,PE}>5$ cm-3), the solar wind electrons are a factor 3
denser than the photoelectrons (see Figures 9d and 9e), although this drops to
1 in the densest photoelectron region.

The solar wind electrons are dense over a much larger region than the
photoelectrons, but there is a depleted region extending from the inner coma
towards $+z$. The low density region is associated with a peak in the
ambipolar potential, which deflects electrons away from the region.

The secondary electron cloud has a similar shape to the photoelectrons, but in
the inner coma the secondary electron density reaches 1400 cm-3. This is
substantially larger than the photoelectron and solar wind electron
populations, which collectively peak at 700 cm-3. In a non-collisional
simulation at the higher outgassing rate ($Q=1.5\times 10^{27}$ s-1), the
photoelectron density peaks at 900 cm-3 while the solar wind is a minor
contribution in the inner coma ($<16$ cm-3). Despite the additional production
(EII) and cooling (inelastic excitations) in the collisional simulation, the
cometary electrons may escape more easily from the coma in the collisional
case due to elastic scattering collisions. These can redistribute the parallel
and perpendicular energies and allow particles to escape the potential well.
Conversely, the solar wind electrons are more efficiently confined when
collisions are included. They undergo inelastic collisions in the inner coma,
lose energy, and become trapped by the potential well.

The total electron density in the coma is larger in the collisional case, due
to the additional substantial contribution from electron-impact ionization.
The third and fourth generations of electrons are significant at $Q=1.5\times
10^{27}$ s-1, unlike in the lower outgassing case, as the accelerated solar
wind electrons produce a population of energetic secondaries in the inner
coma. The ambipolar potential well is deeper at the higher outgassing rate,
meaning the secondary electrons are born at higher energies and more capable
of causing ionization. Additionally, the higher outgassing rate increases the
production rate of secondaries and boosts the energetic electron flux ($J(E)$,
see Eq. 3).

The impact of the additional electrons on the electric fields in the coma is
assessed using the Generalised Ohm’s law and assuming the magnetic field is
unchanged from the collisionless case (Stephenson et al., 2022). This is a
reasonable assumption, as magnetic field saturates during pile up and is
resistant to changes in electron density (Goetz et al., 2017). The electron
component of the Hall field ($\bm{E}_{Hall,e}=-\bm{u}_{e}\times\bm{B}$) and
the ambipolar field ($\bm{E}_{Ambi}=-\frac{1}{e\,n_{e}}\nabla p_{e}$) are
unaffected outside the inner coma ($r>40$ km), by the additional electrons
(see Figure 24). Closer to the nucleus, electron-neutral collisions quench the
Hall and ambipolar fields, as the bulk velocity and electron temperatures are
small in the region. However, the ambipolar potential well is constructed over
a much larger spatial scale ($>200$ km), so the potential in the inner coma is
reduced by at most 40% (see Appendix C).

In both outgassing cases, the secondary electrons are more abundant in the
inner coma than photoelectrons. Solar wind electrons also have a substantial
contribution to the total density. Electron-impact ionization is the major
source of cometary electrons within a weakly outgassing comet. As the
outgassing increases, it is likely that photoionization will become dominant,
as the cooling of the ionizing electrons becomes more efficient. This was
observed at 67P during the Rosetta mission around perihelion (see Figure 1a),
but the high outgassing cases cannot be addressed with the test particle
model.

PhotoelectronsSolar Wind ElectronsSecondary
Electrons(a)(b)(c)(d)(e)(f)$\bm{Q=10^{26}}$ s-1$\bm{Q=1.5\cdot 10^{27}}$
s-1$\bm{u_{SW}}$$\bm{E}_{SW}$$\bm{B}_{SW}$ Figure 9: Electron density of (a &
d) photoelectrons, (b & e) solar wind electrons and (c & f) secondary
electrons for outgassing rates of (a-c) $Q=10^{26}$ s-1 and (d-f) $Q=1.5\times
10^{27}$ s-1. The densities are plotted in the $xz$ plane. The colour bars are
over different ranges for the two outgassing cases. Streamlines are shown for
electrons following the $\bm{E}\times\bm{B}$ drift velocity in the $x$-$z$
plane.

###  4.2 Source of ionizing electrons

####  4.2.1 EII frequency from different electron populations

PhotoelectronsSolar Wind ElectronsSecondary
Electrons(a)(b)(c)(d)(e)(f)$\bm{Q=10^{26}}$ s-1$\bm{Q=1.5\cdot 10^{27}}$
s-1$\bm{u_{SW}}$$\bm{E}_{SW}$$\bm{B}_{SW}$ Figure 10: Electron-impact
ionization frequency from collisions by (a,d) photoelectrons, (b,e) solar wind
electrons and (c,f) secondary electrons for outgassing rates of (a-c)
$Q=10^{26}$ s-1 and (d-f) $Q=1.5\times 10^{27}$ s-1. Streamlines are shown for
electrons following the $\bm{E}\times\bm{B}$ drift velocity in the $x$-$z$
plane.

Having determined that the bulk of cometary electrons are produced by
electron-impact ionization at medium-to-large heliocentric distances, we now
examine the origin of the ionizing electrons. Figure 10 shows the electron-
impact ionization frequency from each of the electron populations, calculated
from the collisional test particle model using Eq. 5.

In the lowest outgassing case, the EII frequency from photoelectrons peaks at
$6\times 10^{-8}$ s-1 in the inner coma (see Figure 10a). This decreases
quickly with cometocentric distance to $10^{-8}$ s-1 at 50 km. The number of
photoelectrons reduces further from the nucleus (see Figure 9a), and the
typical electron energy also falls in a shallower region of the potential
well. The EII frequency from photoelectrons is smaller than from
photoionization ($\nu_{e}^{h\nu}=1.32\times 10^{-7}$ s-1) throughout the coma,
which is typical for an optically thin coma.

The solar wind electrons are much more ionizing than photoelectrons throughout
the coma (Figure 10b), with a peak at $6\times 10^{-7}$ s-1. The solar wind
electrons are ionizing over a large region of the coma, with the increased
frequency close to the nucleus driven by the acceleration in the ambipolar
potential well. The EII frequency from solar wind electrons exceeds the
photoionization frequency up to 200 km from the nucleus, and dominates the
contribution to the EII frequency within the coma (compared with
photoelectrons).

Secondary electrons generate additional electrons within the coma through
electron-impact collisions. This process is only substantial over a small
region in the inner coma (see Figure 10c), as secondary electrons are well
confined by the ambipolar field. Very close to the nucleus ($<70$ km), the EII
frequency from secondary electrons exceeds that from photoelectrons. The
secondary electrons, produced by collisions of solar wind electrons, are much
more energetic than the newly born photoelectrons and therefore more likely to
cause ionization.

At $Q=1.5\times 10^{27}$ s-1, the EII frequency from photoelectron collisions
(Figure 10d) is substantial over a large region extending 100 km upstream of
the nucleus and up to 380 km downstream. The largest EII frequencies are not
found in the innermost coma, despite the large electron density in the region
(see Figure 9d-f). The peak EII frequencies in all three populations are found
at $(50,0,-50)$ km (Figure 10). Here, the ambipolar potential well is deep
($>150$ V) and the neutral coma is not dense enough to efficiently cool the
electrons in the region.

As seen at lower outgassing, solar wind electrons are more ionizing than
photoelectrons throughout the simulation domain. However, the secondary
electrons exceed the EII frequency of the solar wind electrons in the inner
coma. The secondary electrons, particularly those produced by collisions of
solar wind electrons in the inner coma, are very energetic and can drive
electron-impact ionizations. The secondaries are also produced within the
ambipolar potential well, so are more efficiently trapped than the solar wind
electrons. Therefore, they are more likely to collide with the neutral gas and
produce further generations of electrons. The further generations of electrons
are less ionizing as the input energy is distributed between more electrons
and some energy is dissipated through supplying the threshold energy for
inelastic excitations. The secondary electrons that drive ionization are
almost entirely descendents of the accelerated solar wind population, rather
than the lower energy photoelectrons.

####  4.2.2 EII frequency vs magnetic field strength

(a)(b)$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.229\frac{x}{\mathrm{nT}}-9.349$$R^{2}=0.9788$$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=0.0877\frac{x}{\mathrm{nT}}-8.2795$$R^{2}=0.9442$
Figure 11: Electron-impact ionization frequency against magnetic field
strength throughout the coma for outgassing rates of (a) $Q=10^{26}$ s-1 and
(b) $Q=1.5\times 10^{27}$ s-1. Median (black circles) and quartile values
(error bars) are plotted for each. Linear fits are given for both outgassing
rates (blue crosses).

The analysis applied to the data from Rosetta in Section 3 can also be applied
to the outputs of the test particle simulation. The values calculated for each
cell of the test particle simulation are organized into a 2D histogram in
Figures 11 and 12, with the median and the quartiles for each bin of the
magnetic field strength also shown. The results from the test particle model
represent a single 3D snapshot under a single set of conditions, whereas the
Rosetta data provided a single measurement point that moved through an
evolving coma.

Figure 11 shows the relation between the EII frequency and the magnetic field
strength in the test particle simulations. Under both outgassing rates, the
EII frequency increased with magnetic field strength. At the lower outgassing
rate, the EII frequency increases from $3.2\times 10^{-8}$ s-1 at 8 nT to
$10^{-6}$ s-1 at 13 nT (Figure 11a).

At $Q=1.5\times 10^{27}$ s-1, the EII frequency increases from $10^{-8}$ s-1
at 6 nT to $1.6\times 10^{-6}$ s-1 at 25 nT (Figure 11b). The EII frequency
decreases at very large magnetic field strengths ($B>25$ nT, Figure 11b).
These cells are found close to the nucleus, where the electron-neutral
collisions efficiently degrade the high energy tail of the solar wind
electrons, quenching the ionization processes.

At both outgassing rates, there is an apparent discontinuity in the EII
frequency between 6 and 6.5 nT, which reflects that very few regions in the
simulation domain have magnetic fields below 6.5 nT (see top of Figure 11a).
This is more a reflection of the sampled domain rather than any physical
process. Field strengths of 6.5 nT correspond to the region of upstream solar
wind which has been slightly compressed near the upstream boundary. The
electrons are unaccelerated solar wind electrons ($T_{e,SW}=10$ eV), which are
not energetic enough to cause substantial ionization. The smallest magnetic
field strengths ($<6$ nT) are found $\sim 300$ km downstream of the nucleus
between $y=\pm 30$ km.

At $Q=1.5\times 10^{27}$ s-1, magnetic field strengths below 6 nT are confined
to a small region found $\sim 300$ km downstream of the nucleus between $y=\pm
30$ km. The lowest EII frequencies at $10^{-9}$ s-1 are found at (-50, 0, 200)
km in the region depleted of electrons (see Figure 9d-f). This region is a
maximum in the parallel electric potential at 50 V above the undisturbed solar
wind. The potential barrier decelerates and deflects electrons, reducing the
probability of ionization collisions occurring in the region.

The positive correlation with magnetic field strength agrees qualitatively
with the results from the Rosetta mission (see Section 3.2.1), although the
gradients of the correlations are much larger in the fits to the test particle
model (0.23 and 0.088 vs. 0.024 and 0.028). The ionizing electrons are
accelerated solar wind electrons, which are funnelled into the inner coma. The
electrons are then trapped and pile up in a dense region where they can
produce many secondary electrons.

####  4.2.3 EII frequency vs cometocentric distance

Figure 12 shows the variation of the EII frequency with cometocentric distance
throughout the coma. Under both outgassing conditions, the EII frequency
increases with decreasing distance to the nucleus.

In the low outgassing case, the EII frequency is fairly constant at large
distances from the coma (Figure 12a), as these points are in the undisturbed
solar wind. Closer to the nucleus, the electrons are accelerated by the
ambipolar potential well and the EII frequency increases. The ionization
frequency is dominated by collisions of solar wind electrons or their
secondary electrons. The EII frequency increases from $10^{-7}$ s-1 to
$10^{-6}$ s-1 between $r=100$ km and the nucleus. The cooling processes in the
inner coma do cause some degradation at $Q=10^{26}$ s-1, but only a small
fraction of the total electrons become cold. Consequently, the EII frequency
continues to increase all the way to the surface. There is some uncertainty in
the behaviour of the electrons in the inner coma, as the field resolution
(7.66 km, Table 3) is larger than the comet nucleus ($1.7$ km, Jorda et al.,
2016).

At $Q=1.5\times 10^{27}$ s-1, the behaviour is very similar to the case at
$Q=10^{26}$ s-1 away from the inner coma. Far from the nucleus ($r>400$ km) in
the (somewhat) pristine solar wind, the EII frequency is steady at $10^{-8}$
s-1. All three electron populations are more ionizing closer to the nucleus
(50-350 km), with the total EII frequency increasing by a factor 10. Even
closer to the nucleus ($r<50$ km), the EII frequency decreases as the neutral
gas is dense enough to substantially cool the high energy electrons.

The median EII frequencies at the low outgassing rate are very close to a
$\nu_{e}^{\text{ioni}}\propto 1/r$ profile (blue dashed line, Figure 12)
throughout the wider coma. Similar behaviour is seen between 100 and 400 km at
$Q=1.5\times 10^{27}$ s-1, but the collisional cooling in the inner coma
causes deviation from this trend. At small cometocentric distances ($r<50$
km), the EII frequency increases with cometocentric distance at $Q=1.5\times
10^{27}$ s-1, with a gradient of
$\Delta(\log_{10}\nu_{e}^{\text{ioni}})/\Delta r=0.0119$.

The behaviour of the EII frequency in the inner coma is subject to several
caveats, particularly in the higher outgassing case. Firstly, the resolution
of the fields is coarse (7.66 km and 10 km, Table 3) compared to the size of
the nucleus, which may have impacts on the inner coma. Additionally, there is
no feedback from collisional processes onto the fields in the test particle
simulations. Cooling of electrons in the inner coma reduces the electron
pressure gradient and consequently the ambipolar field in the inner coma
(Stephenson et al., 2022). The quenched ambipolar field in the inner coma
results in a potential well that is flatter and reduced in depth by 20% close
to the nucleus (grey region, Figure 14), compared to when collisions are
neglected. However, the ambipolar field is still efficient at trapping of
cometary electrons near the nucleus and accelerating solar wind electrons into
the inner coma, as the potential well is built up over a large range of
cometocentric distances, including regions without substantial electron
cooling.

The flattening of the potential well in the inner coma would result in a
region of constant ionization frequency, as energetic solar wind electrons
would not be funnelled closer to the nucleus or undergo further acceleration.
In Figure 12, this would lead to a plateau in the ionization frequency below
$\approx 50$ km, and a deviation away from the $1/r$ trend, which occurs
without any collisional feedback.

(a)(b)$\frac{y}{\mathrm{s}^{-1}}=9.36\times
10^{-6}\mathrm{km}/x$$R^{2}=0.980$$\frac{y}{\mathrm{s}^{-1}}=4.162\times
10^{-5}\mathrm{km}/x$$R^{2}=0.736$ Figure 12: Total electron-impact ionization
frequency against cometocentric distance throughout the coma from test
particle simulations at outgassing rates of (a) $Q=10^{26}$ s-1 and (b)
$Q=1.5\times 10^{27}$ s-1. Median (black circles) and quartile values (error
bars) are plotted for each bin of cometocentric distance, $r$. The range of
cometocentric distances probed by Rosetta im the early and late periods (see
Table 1) are show in the top plots for each period. Fits of a $1/r$ profile
are given for both outgassing rates (blue crosses) for $r>100$ km. At
$Q=1.5\times 10^{27}$ s-1, the function is fit between 100 and 400 km. The
spatial resolution of the fields (see Table 3) and lack of feedback of
collisional processes on the fields limits the validity of fits below $r=100$
km.

##  5 Ion acceleration and electron impact ionization away from perihelion

Table 5: Set of test cases to examine ion acceleration in the coma. Rosetta has been set at a cometocentric distance of 30 km.  Case | Ion bulk velocity | Electron-impact freq
---|---|---
1 | Constant at 0.6 km s-1 | Constant at $3\times 10^{-7}$ s-1
2 | Constant at 5 km s-1 | Constant at $3\times 10^{-7}$ s-1
3 |  Linear increase. 0.6 km s-1 at surface. 5 km s-1 at Rosetta. |  Constant at $3\times 10^{-7}$ s-1
4 |  Linear increase. 0.6 km s-1 at surface. 5 km s-1 at Rosetta. |  $\propto 1/r$. $3\times 10^{-7}$ s-1 at Rosetta
(a)(b) Figure 13: (a) Electron density and (b) electron-impact ionization
frequency as a function of cometocentric distance for the four test cases (1)
- (4) outlined in Table 5. The black crosses give the electron density and
ionization frequency measured at Rosetta (at 30 km). The ionization
frequencies for cases (1) - (3) are identical in (b). The error bars allow for
10% error in the outgassing rate and outflow velocity, as well as 30% error on
the electron impact ionization frequency. Figure 14: Ambipolar potential well
along $y$ at $x,z=7.66$ km for a collisionless (blue) and collisional (red)
electron test particle simulation at $Q=10^{26}$ s-1 (see Table 3). The
collisional region of the coma is highlighted in grey and is associated with a
flattened potential well.

Multi-instrument models have accurately reproduced the measured electron
density (and ion density through quasineutrality) at Rosetta, when far from
perihelion (Galand et al., 2016; Heritier et al., 2017a, 2018). These are
based on the continuity equation for the total ion density, which is given by
(Galand et al., 2016):

$\frac{1}{r^{2}}\frac{\mathrm{d}}{\mathrm{d}r}\bigg{(}n_{i}(r)r^{2}u_{i}(r)\bigg{)}=P_{i}(r)-L_{i}(r),$
(9)

where $n_{i}$ is the total ion density, $P_{i}$ is the production rate of ions
through ionization and $L_{i}$ is the chemical loss rate through electron-ion
recombination. Far from perihelion, transport dominated over recombination at
67P, so the last term can be neglected (Galand et al., 2016). In Eq. 9, we
have assumed a spherically symmetric cometary plasma so $u_{i}$ is only the
radial ion velocity.

The multi-instrument approaches assume ions flow radially outwards at the
neutral gas velocity and that the ionization frequency is constant throughout
the coma (in the optically thin regime). The neutral coma and electron-
ionization frequency are derived from measurements at Rosetta taken at similar
times.

The production $P_{i}$ of cometary ions and electrons are dominated by
electron-impact ionization at large heliocentric distances (see Section 3).
Therefore, the electron density measured at Rosetta $r_{Ros}$ is subject to
the outgassing rate and to two main, competing drivers: (1) the electron
impact ionization frequency at $r<r_{Ros}$ (through
$P_{i}=(\nu_{h\nu}^{ioni}+\nu_{e}^{\text{ioni}})n_{n}$, where $n_{n}=Q/(4\pi
u_{gas}r^{2})$ is the neutral density) and (2) the ion velocity, $u_{i}(r)$
(contained in the left hand side of Eq. 9) at Rosetta. The outgassing rate and
EII frequency at Rosetta varied by orders of magnitude within several hours,
so the multi-instrument modelled densities must be driven by almost-concurrent
measurements at Rosetta and cannot be used to explain the electron densities
over long periods with a single set of inputs.

Given that the electron density magnitude and variation is well captured with
the multi-instrument model (Galand et al., 2016; Heritier et al., 2017a,
2018), an increase from the modelled ion velocity would have to be compensated
by increasing the modelled EII frequency in the coma. However, the EII
frequency is also constrained by measurements at Rosetta, so would have to be
varied between Rosetta and the nucleus’ surface.

Away from perihelion, cometary ion radial bulk and effective total speeds have
been measured in excess of 10 km s-1 in the inner coma by RPC/LAP (Johansson
et al., 2021) and RPC/ICA (Nilsson et al., 2020), which have substantial
implications on the EII frequency throughout the coma. The key uncertainty in
estimates of the ion bulk velocity is the impact of the spacecraft potential,
which acclerates and deflects ions near the spacecraft, with low energy
cometary ions being most strongly deflected (Bergman et al., 2020). Low energy
water ions moving through a typical spacecraft potential of $V_{S/C}=-15$ V
could increase in speed by up to 12.5 km s-1.

For ions to achieve the large measured radial outflow velocities, they must
undergo substantial acceleration in the inner coma, as they are born at the
neutral flow speed ($0.4-1$ km s-1; Marshall et al., 2017; Biver et al.,
2019). The radial ion velocity measurements are substantially larger than
estimated by Vigren & Eriksson (2017) at $Q=2\times 10^{27}$ s-1. The 1D model
with a constant ambipolar field resulted in an ion velocity of 2.5 km s-1 at
100 km.

For consistency to be maintained between the multi-instrument models of
electron density and the high estimates of ion bulk velocity, higher EII
frequencies are required close to the nucleus. The effective bulk speeds
$\sim$10 km s-1 could have non-radial components, which would moderate the
high EII frequencies needed to maintain the observed plasma density.

There has also been a number of measurements of the ion bulk velocities close
to perihelion, particularly in the vicinity of the diamagnetic cavity (Bergman
et al., 2021a, b).These also find ions with large bulk speed ($>5$ km s-1), as
well as a population of ions streaming towards the comet. The presence of the
diamagnetic cavity substantially alters the plasma environment and the coma is
much more collisional than at large heliocentric distances. Therefore, this
discussion only applies to the case at large heliocentric distances and does
not comment on the conditions or measurements around perihelion.

Figure 13 illustrates the ion density profile at a point in time derived under
the prior assumptions of the multi-instrument model approach (blue, case 1),
using an outgassing of $Q=10^{26}$ s-1. The conditions for each test case are
outlined in Table 5. The electron density and EII frequency are both measured
at Rosetta, which we have chosen to be at $r=30$ km (black crosses, Figures
13a and b). This was a typical cometocentric distance of Rosetta when away
from perihelion.

We have neglected photoionization throughout the coma, as EII was found to be
the dominant source of electrons in the coma far from perihelion (see Figure
1), especially during the early phase of the mission. If the ions undergo
linear acceleration and the electron-impact ionization frequency varies as
$\nu^{\text{ioni}}_{e}(r)\propto\frac{1}{r^{n}}$, the plasma density from Eq.
9 is given by:

$\begin{split}n_{i}(r)=\frac{Q\nu^{\text{ioni}}_{e}(r_{Ros})r_{Ros}}{4\pi
u_{gas}\times u_{i}(r)\times
r^{2}}\times\quad\quad\quad\quad\quad\quad\quad\quad\quad\\\
\quad\quad\quad\quad\begin{cases}\log(\frac{r}{r_{67P}})&\text{if }n=1\\\
\frac{1}{n-1}\big{(}\frac{r_{Ros}}{r}\big{)}^{n-1}\bigg{[}\big{(}\frac{r}{r_{67P}}\big{)}^{n-1}-1\bigg{]}&\text{otherwise};\end{cases}\end{split}$
(10)

using the boundary condition $n_{i}=0$ at $r=r_{67P}$.

The relation $\nu^{\text{ioni}}_{e}(r)\propto r^{-1}$ is observed in the test
particle simulations (though this conclusion has limitations discussed
hereafter), particularly at $Q=10^{26}$ s-1 (see Figure 12a). It has
previously been used in studies of FUV emissions, when all electrons were
assumed to be energetic enough to excite (Feldman et al., 2015). This is
indeed the dependence expected in the case where the EII frequency follows the
total electron density.

From in-situ observations, the variation of EII frequency with cometocentric
distance at comet 67P remains uncertain at large heliocentric distances. The
data from the Rosetta mission do not show a clear correlation to or consistent
behaviour with cometocentric distance (see Section 3.3).

The test particle simulations suggest that the EII frequency could increase by
a factor of 10 within 100 km of the nucleus. However, these results do not
include any feedback of collisional processes onto the electric and magnetic
fields, which may damp the electron-impact ionization in the inner coma (see
Section 4.2.3), resulting in a more constant ionization frequency close to the
nucleus.

Cases 2 (red) and 3 (yellow; Figure 13a) show the ion density profiles with
higher ion velocities, but no change in the EII frequency (same as Case 1,
blue). The radial ion velocity is set to 5 km s-1 at Rosetta, an average
radial velocity according to the RPC/LAP and RPC/ICA measurements away from
perihelion. In both scenarios, the ion density cannot be reproduced at
Rosetta, regardless of whether the ions are born at 5 km s-1 (red) or are
accelerated from $u_{gas}=0.6$ km s-1 in the coma (yellow). This is
inconsistent with the constraints of the multi-instrument models.

In Case 4, we have increased the ionization frequency (purple, Figure 13b)
close to the nucleus to compensate for the high ion velocities, using a
$\nu_{e}^{\text{ioni}}\propto 1/r$ profile. This profile assumes the EII
frequency varies similarly to the bulk plasma density, as has previously been
used in FUV emission models (Feldman et al., 2015). This relation with
ionization frequency restores the electron density to within a factor of 3 of
those observed at Rosetta.

However, the energetic electron population (which drives EII) is greatly
boosted near the nucleus in Case 4, compared to the previously assumed case
(yellow, Figure 13b). The energetic electrons are increased by a factor of 10
in the inner coma, and would need to be further increased to fully reproduce
the plasma density measured at Rosetta. Using $\nu_{e}^{\text{ioni}}\propto
1/r^{1.89}$ would fully resolve the discrepancy in the electron density
(purple vs. black cross, Figure 13a). This profile would increase the
energetic electron population in the inner coma even further.

While this would reconcile the high ion velocity and plasma density
measurements, the EII frequency throughout the coma is not a free parameter.
Multi-instrument models of FUV emissions from the coma have shown that
emissions are driven by electron impact and are consistent with a constant EII
frequency along the line of sight (Galand et al., 2020; Stephenson et al.,
2021), as was concluded by Chaufray et al. (2017). FUV emissions are driven by
the same energetic electron population as the ionizing electrons, so the
emission frequency is expected to show the same spatial variation as the EII
frequency. The FUV emissions are also more sensitive to the EII frequency in
the inner coma than the electron density, which is accumulated over the radial
column. Each shell of width $\mathrm{d}r$ contributes
$\nu_{e}^{em}(r)\mathrm{d}r/r^{2}$ to the total brightness and
$\nu_{e}^{\text{ioni}}(r)\mathrm{d}r$ to the electron density, where
$\nu_{e}^{em}(r)$ is the emission frequency.

Consequently, an increase by a factor 10-15 in the energetic electron
population in the inner coma would result in an increase in FUV emissions by
an order of magnitude compared to the case of constant electron flux. While
there are uncertainties in the multi-instrument FUV studies, an increase in
the energetic electron population by a factor of 10 would produce a clear
discrepancy between the modelled and observed emission brightnesses.

Additionally, the direct measurements of the energetic electrons by RPC/IES do
not show any clear variation with cometocentric distance, when at large
heliocentric distance (see Figure 7). This is reinforced by the RPC/IES
measurements at the end of mission, which did not vary significantly down to
the surface (Heritier et al., 2017a).

In contrast, the test particle simulations do indicate that the ionization
frequency should increase closer to the nucleus (see Figure 12). However, the
EII frequency increases by only a factor 3 within 30 km of the nucleus. This
is much less than the required increase to reconcile the ion flow speed and
electron density measurements.

No Collisional CoolingCollisional
Cooling$\bm{n_{e}}$$\bm{n_{e}}$$\nu_{e}^{\text{ioni}}$$\nu_{e}^{\text{ioni}}$$\bm{V_{Pot}}$$\bm{V_{Pot}}$$\bm{u_{i}}$$\bm{u_{i}}$$\bm{r}$
Figure 15: Illustration of how electron density $n_{e}$, ion bulk speed,
$u_{i}$, ambipolar potential and EII frequency may vary with cometocentric
distance in a coma (top half) without collisional cooling and (bottom half)
with collisional cooling. The black line in the bottom half bounds the
collisional region of the coma, inside which the potential, EII frequency and
ion bulk speed are constant. Inside the collisional region, the ion speed,
ambipolar potential, and EII frequency are constant. In the non collisional
case, the ions accelerate throughout the coma, from the comet surface, and the
ambipolar potential decreases all the way to the surface. This also results in
a peak in the EII frequency very close to the nucleus.

The test particle simulations also lack any feedback from electron collisions
on the fields in the simulation. Using the outputs of the test particle
simulations, we can estimate the feedback of collisional process on the
ambipolar field using the Generalised Ohm’s Law (Stephenson et al., 2022):

$\bm{E}_{Ambi}=-\frac{1}{en_{e}}\nabla(n_{e}k_{B}T_{e}).$ (11)

Figure 14 shows the ambipolar potential well along the magnetic field line
that passes through (10, 0, 10) km at $Q=10^{26}$ s-1, for a collisionless
(blue) and collisional (red) electron simulation. In the collisionless case,
the potential well would lead to an increasing ionization frequency down to
the surface as solar wind electrons undergo acceleration into the potential
well. This would be consistent with Case 4 and the ion speed measurements,
although the increase in ionization frequency would still not be of the
required magnitude (see Figure 12a).

When collisions are included the electron temperatures are reduced in a
collisional region in the inner coma (Stephenson et al., 2022). This reduces
the ambipolar field strength and flattens the potential well near the nucleus.
In the grey highlighted region, electrons would undergo little acceleration
and the electron impact ionization frequency would be approximately constant.
This is consistent with the approach in Case 1 and used for the multi-
instrument observations. The consistent cold electron observations at Rosetta,
even at large heliocentric distances, indicate that the spacecraft was often
in this region of lower temperatures and the flattened ambipolar potential
(Eriksson et al., 2017; Engelhardt et al., 2018; Gilet et al., 2020; Wattieaux
et al., 2020). Therefore, the case of constant EII frequency between Rosetta
and the surface (as used in Case 1) is the more reasonable assumption when the
electric fields are accounted for. Beyond the collisional region (grey Figure
14), acceleration of the cometary ions is expected. The 80 V potential
difference translates to 28 km s-1 for water ions, although this acceleration,
which occurs over 300 km (potential well scale) is small compared to the
acceleration by pick up ($E_{conv}=-2.4\times 10^{-3}$ V m-1 results in 720 V
over 300 km).

While there may be some variation in the EII frequency with cometocentric
distance, a variation of factor 10 within 30 km of the nucleus is inconsistent
with the particle spectral measurements and the FUV emissions on the one hand,
and with the results of the test particle model on the other hand. When
allowing the EII frequency to increase by a factor of 3 close to the nucleus
(more than any discrepancy seen in the multi-instrument FUV studies), the
maximum radial ion velocity allowed is twice the outflow gas velocity,
significantly smaller than the estimated ion bulk speeds from Rosetta
observations. Direct measurements of the energetic electrons do not suggest
such substantial enhancement in the inner coma.

Figure 15 illustrates how the electron density, ion bulk speed, ambipolar
potential and EII frequency may vary in a weakly collisional coma (bottom
half) compared to the case without collisions (top half), with the collisional
region bounded by the black line in the bottom half. Within the collisional
region ion bulk speed, the potential and EII frequency are approximately
constant with cometocentric distance, consistent with the multi-instrument
models. Outside the collisional region, the ambipolar potential increases,
leading to ion acceleration and reduction of the EII frequency (possibly with
a $1/r$ profile as seen in Figure 12).

As the EII frequency does not substantially vary between Rosetta and the
nucleus at large heliocentric distances, it still remains unclear how the
large radial ion bulk speeds can be reconciled with the direct measurements
and multi-instrument models of the plasma density and FUV brightnesses.

##  6 Conclusion

We have calculated a new dataset of the electron-impact ionization frequency
throughout the Rosetta mission, using data from RPC/IES. Electron-impact
ionization is the dominant source of electrons in the coma at large
heliocentric distances (Figure 1) and varies by up to three orders of
magnitude over short periods. Far from perihelion, the electron density at
Rosetta is strongly dependent on the EII frequency (Figure 2). This
relationship is also seen in respect to the spacecraft potential and the
cometary ion density (Figure 17).

Around perihelion, the EII frequency is much less variable (one order of
magnitude) and is less ionizing than photoionization (see Figure 1). Hence, it
is not surprising that the electron and ion populations are not strongly
dependent on the EII frequency near perihelion (Figures 2 and 3).

Away from perihelion, the EII frequency is positively correlated with the
solar wind potential difference (estimated from RPC/ICA measurements; see
Figure 5), confirming that the ionizing electrons originate in the solar wind
and are accelerated in the cometary environment. The solar wind electrons are
accelerated by an ambipolar potential well.

The EII frequency is well correlated with magnetic field strength (Figure 4),
indicating that EII is more ionizing in regions where the solar wind slows
down and piles up. The outgassing rate (Figure 6) is also a key driver of the
EII frequency, although non-linearly. At low outgassing rates, the EII
frequency increases with outgassing rate, before plateauing, and subsequently
decreasing. This may reflect the variation in the ambipolar potential well
with outgassing. At low outgassing, there are few electron-neutral collisions
in the coma so increasing outgassing leads to a deeper potential well, as the
spatial scale of the cometary ionosphere increases. At intermediate rates,
collisions start to become important and cold electrons form in the inner
coma. This feedback reduces the ambipolar field in the inner coma and limits
the depth of the potential well. As the outgassing increases further, the
electron cooling increases and extends farther from the nucleus and the
ambipolar potential well is damped.

The origin of electrons within the coma is simulated using a 3D collisional
test particle model of electrons at a comet (see Section 4). At outgassing
rates of $Q=10^{26}$ s-1 and $Q=1.5\times 10^{27}$ s-1, electron-impact
ionization was found to be the major source of electrons within the coma
(Figure 9) and the ionizing electrons are solar wind electrons, or cometary
electrons that have been produced by solar wind electron-impacts. The test
particle model also shows a positive correlation between magnetic field
strength, as was seen in the Rosetta data.

There is not a clear variation in the EII frequency with cometocentric
distance away from perihelion, using the RPC/IES data. In the test particle
simulations, the EII frequency does increase closer to the nucleus, as the
electrons are accelerated into the inner coma by the ambipolar potential well.
However, there is no feedback of collisions onto the fields in the test
particle model, which damps the ambipolar field in the inner coma and limits
the EII frequency. When the feedback of collisions on the ambipolar field is
considered (see Figure 14), the potential well has a much flatter shape in the
collisional region of the coma. This would result in a roughly constant
ionization frequency in the collisional region, where cold electrons are
present and electron temperatures are low. Cold electrons were observed
throughout the Rosetta mission, so there should be little difference in the
ambipolar potential between Rosetta and 67P. For more thorough examination of
this feedback between collisional processes and the fields in the cometary
environment, a collisional PiC model would be necessary. This approach would
also allow application to higher outgassing rates, where the test particle
approach is limited by the lack of feedback between fields and collisions. We
consider recent measurements of the ion flow speeds at comet 67P far from
perihelion, which indicate ions undergo acceleration from the neutral outflow
speed to over 5 km s-1 between production and Rosetta. This acceleration would
have to be compensated by increased electron-impact ionization within the
inner coma, to restore the electron density to the observed levels. However,
the required factor 10-15 increase in EII frequency close to the nucleus is
inconsistent with the FUV emissions from the coma. It is also well beyond the
increase in EII frequency in the test particle model (factor 3 within 30 km),
which can be treated as an upper bound.

The discrepancy between the ion flow speed measurements and the multi-
instrument studies applied to large heliocentric distances remains an
outstanding question, but cannot reasonably be explained by an arbitrary
enhancement in the electron-impact ionization close to the nucleus.

## Acknowledgements

We gratefully acknowledge the work of the many engineers, technicians and
scientists involved in the Rosetta mission. In particular, we acknowldege the
work of the ROSINA team, with PI K. Altwegg, and the RPC instrument teams,
including the PIs: A. Eriksson of RPC/LAP, J. Burch of RPC/IES, KH. Glassmeier
of RPC/MAG and C. Carr of RPC/PIU. Without their contributions, ROSINA and RPC
would not have produced such outstanding results. Rosetta is an ESA mission
with contributions from its member states and NASA.

Work at Imperial College has been supported by the Science and Technology
Facilities Council (STFC) of the UK under studentships ST/S505432/1 and
ST/W507519/1 and under grants ST/5000364/1 and ST/W001071/1. Work at the
University of Bern was funded by the Canton of Bern and the Swiss National
Science Foundation (200020_207312). Work at LPC2E and Lagrange laboratories is
co-funded by CNES. Work at Umeå University was supported by the Swedish
National Space Agency (SNSA) grant 108/18. J. D. gratefully acknowledge
support from NASA’s Rosetta Data Analysis Program, Grant No. 80NSSC19K1305,
and the NASA High-End Computing (HEC) Program through the NASA Advanced
Supercomputing (NAS) Division at Ames Research Center.

## Data Availability

The data underlying this article are available on the PSA at
https://psa.esa.int. The electron-impact ionization frequencies are available
at 10.5281/zenodo.8048344.

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##  Appendix A Variation of electron impact ionization frequency during the
Rosetta mission

In this section, we show the correlation of the electron-impact ionization
frequency (EII) with additional properties of the cometary environment to
those discussed in Section 3.

###  A.1 EII vs. heliocentric distance

Firstly, Figure 16 shows the relationship between heliocentric distance and
EII frequency, in addition to the fit to the EII frequency outlined in Table
4. In the early mission period, the EII frequency varies non-linearly with
heliocentric distance, reflecting the same behaviour as observed in the
relationship with local outgassing rate (see Figure 6). However, post-
perihelion the correlation with heliocentric distance becomes much weaker and
shows significant variance around any trend. This may result from the more
asymmetric outgassing rate post perihelion, with strong CO2 outgassing from
the southern hemisphere (Läuter et al., 2018).

Early MissionLate Mission(a)(b) Figure 16: Electron-impact ionization
frequency vs. the heliocentric distance of comet 67P (a) pre-perihelion and
(b) post-perihelion. Median (black circles) and quartile values (error bars)
are plotted for each bin along the $x$-axis. The top panels show the number of
counts for each bin of heliocentric distance. The fit to the EII frequency
with heliocentric distance is shown in blue crosses in the left panel (see
Table 4).

###  A.2 EII vs. Spacecraft potential

The EII frequency is negatively correlated with the spacecraft potential
(Figure 17), as measured by RPC/LAP, during both the early and late mission
periods (see Table 1). Around perihelion, the EII frequency is much less
dependent on the spacecraft potential and is largely constant at
$\nu_{e}^{\text{ioni}}\approx 8\times 10^{-8}$ s-1.

The spacecraft potential is a function of the electron temperature and density
(Odelstad et al., 2015; Johansson et al., 2021), so the correlations in Figure
17 can be interpreted in the same way as Figure 2. Denser, hotter plasmas
result in higher EII frequencies and more negative spacecraft potentials. This
behaviour is seen at large heliocentric distances, as electron impact
ionization was the major source of electrons within the coma during these
periods.

Early MissionAround PHLate
Mission(a)(b)(c)$\log_{10}(\frac{y}{\mathrm{s}^{-1}})=-0.0418\frac{x}{\mathrm{V}}-6.73$$R^{2}=0.839$$R^{2}=0.916$$\begin{aligned}
\log_{10}(\frac{y}{\mathrm{s}^{-1}})&=\\\
-0.0573&\frac{x}{\mathrm{V}}-7.70\end{aligned}$ Figure 17: Electron-impact
ionization frequency vs. the spacecraft potential measured by RPC/LAP (a)
early in the mission, (b) around perihelion and (c) late in the mission.
Median (black circles) and quartile values (error bars) are plotted for each
bin along the $x$-axis. The top panels show the number of intervals for each
bin along $x$. Linear fits are given for the early and late periods (blue
crosses).

###  A.3 EII vs magnetic field angles

In Section 3.2.1 and Section 4.2.2, we have demonstrated the dependence of the
EII frequency on the magnetic field strength in the coma. In this section, we
examine the correlation betweent the EII frequency and the orientation of the
magnetic field. We consider the clock and cone angles which are given by:

$\displaystyle\theta_{\text{Clock}}$ $\displaystyle=\arctan(B_{y}/B_{z})$ (12)
$\displaystyle\theta_{\text{Cone}}$
$\displaystyle=\arctan(\sqrt{B^{2}_{y}+B^{2}_{z}}/B_{x})$ (13)

Figures 18 and 19 show the dependence of the EII frequency on the clock and
cone angles during the early, perihelion and late mission periods (see Table
1). The EII frequency does not depend substantially on either the clock or
cone angles in the early and perihelion periods. In the late mission period,
there is a non-linear relationship with the clock angle with a minimum near
0∘. Additionally, the EII frequency increases with cone angle during this time
(see Figure 19c).

The non-linear behaviour with clock angle is likely an effect of the sampling
with magnetic field strength, as the field strength shows similar behaviour
with clock angle as is exhbited by the EII frequency. The positive correlation
with cone angle could be interpreted as increased draping of magnetic field
lines (at high cone angles) being associated with higher EII frequencies.
However, the cone angle is also correlated with the magnetic field strength
(see Figure 21), which is shown to be a driver of the EII frequency in Section
3.2.1. The positive correlation with EII frequency between
$\theta_{\text{Cone}}=[0,100]^{\circ}$ and the subsequent decrease to
$\theta_{\text{Cone}}=125^{\circ}$ can be attributed to the varying magnetic
field strength. Beyond $\theta_{\text{Cone}}=150^{\circ}$, the EII frequency
increases while the median magnetic field strength decreases, in contrast to
expectations. However, the number of datapoints is small at large cone angles
and the variability in the median and quartile EII frequencies is large.

Early MissionAround PHLate Mission(a)(b)(c) Figure 18: Electron-impact
ionization frequency vs. the clock angle of the magentic field (see Eq. 12)
(a) early in the mission, (b) around perihelion and (c) late in the mission.
Median (black circles) and quartile values (error bars) are plotted for each
bin along the $x$-axis. The top panels show the number of intervals for each
bin along $x$. Early MissionAround PHLate Mission(a)(b)(c) Figure 19:
Electron-impact ionization frequency vs. the cone angle of the magnetic field
(see Eq. 13) (a) early in the mission, (b) around perihelion and (c) late in
the mission. Median (black circles) and quartile values (error bars) are
plotted for each bin along the $x$-axis. The top panels show the number of
intervals for each bin along $x$. Early MissionAround PHLate Mission(a)(b)(c)
Figure 20: The clock angle of the magnetic field (see Eq. 12) vs. the magnetic
field strength (a) early in the mission, (b) around perihelion and (c) late in
the mission. Median (black circles) and quartile values (error bars) are
plotted for each bin along the $x$-axis. The top panels show the number of
intervals for each bin along $x$. Early MissionAround PHLate Mission(a)(b)(c)
Figure 21: The cone angle of the magentic field (see Eq. 13) vs. the magnetic
field strength (a) early in the mission, (b) around perihelion and (c) late in
the mission. Median (black circles) and quartile values (error bars) are
plotted for each bin along the $x$-axis. The top panels show the number of
intervals for each bin along $x$.

##  Appendix B Total electron density and population fractions throughout the
coma

Figure 22 shows the total electron density from the test particle simulation
for the two outgassing rates (see Table 3). The fractional contribution from
each electron population (photoelectron, solar wind and secondary electrons)
is separated in Figure 23. Away from the inner coma, the electron density is
dominated by the solar wind population. Closer to the nucleus, the
photoelectron and secondary electrons make up an increasingly large fraction
of the total density. For both outgassing rates, electron impact ionization is
stronger than photoionization close to the nucleus, resulting in large
fractions of secondary electrons ($>0.5$).

(a)(b)$\bm{u_{SW}}$$\bm{E}_{SW}$$\bm{B}_{SW}$ Figure 22: Total electron
density obtained for simulations at outgassing rates of (a) $Q=10^{26}$ s-1
and (b) $Q=1.5\times 10^{27}$ s-1 in the $x$-$z$ plane. Streamlines are shown
for electrons following the $\bm{E}\times\bm{B}$ drift velocity in the $x$-$z$
plane. PhotoelectronsSolar Wind ElectronsSecondary
Electrons(a)(b)(c)(d)(e)(f)$\bm{Q=10^{26}}$ s-1$\bm{Q=1.5\cdot 10^{27}}$
s-1$\bm{u_{SW}}$$\bm{E}_{SW}$$\bm{B}_{SW}$ Figure 23: Fraction of total
electron density made up by (a,d) photoelectrons, (b,e) solar wind electrons
and (c,f) secondary electrons for outgassing rates of (a-c) $Q=10^{26}$ s-1
and (d-f) $Q=1.5\times 10^{27}$ s-1 in the $x$-$z$ plane. Streamlines are
shown for electrons following the $\bm{E}\times\bm{B}$ drift velocity in the
$x$-$z$ plane.

##  Appendix C Ambipolar and Hall fields in the cometary coma

Figure 24 shows the hall and ambipolar fields from non-collisional and
collisional test particle simulations of a comet with outgassing $Q=10^{26}$
s-1. The Hall field only includes the electron component, as the ions are not
modelled in the particle simulations.

$\displaystyle\bm{E}_{Hall,e}$ $\displaystyle=-\bm{u}_{e}\times\bm{B}$ (14)

While this does not capture the total Hall field, the difference between the
collisional and non-collisional cases shows the role of collisions in
mitigating the Hall field in the inner coma.

(a)(b)(c)(d)(e)(f)CollisionlessCollisional$\bm{u_{SW}}$$\bm{E}_{SW}$$\bm{B}_{SW}$
Figure 24: (a,d) Electron component of the Hall field (see Eq. 14), (b,e) the
ambipolar electric field (Eq. 11) and (c,f) ambipolar potential well from
(a-c) collisionless and (d-f) collisional test particle simulations at
$Q=10^{26}$ s-1.