11institutetext: Swedish Institute of Space Physics, Uppsala, Sweden
11email<EMAIL_ADDRESS>22institutetext: Uppsala University, Department of
Astronomy and Space Physics, Uppsala, Sweden 33institutetext: Laboratoire de
Physique et Chimie de l’Environnement et de l’Espace, CNRS, Orléans, France
44institutetext: Laboratoire Lagrange, OCA, CNRS, UCA, Nice, France
55institutetext: Swedish Institute of Space Physics, Kiruna, Sweden
66institutetext: Umeå University, Department of Physics, Umeå, Sweden

We report on two independent mission-wide estimates of the ion flow speed
which are consistent with the bulk H2O+ ion velocities as measured by ICA. We
find the ion flow to consistently be much faster than the neutral gas over the
entire mission, lending further evidence that the ions are collisionally
decoupled from the neutrals in the coma. RPC measurements of ion speeds are
therefore not consistent with the assumptions made in previously published
plasma density models of the comet ionosphere at the start and end of the
mission. Also, the measured EUV flux is perfectly consistent with
independently derived values previously published from LAP and lends support
for the conclusions drawn regarding an attenuation of solar EUV from a distant
nanograin dust population, when the comet activity was high. The new density
dataset is consistent with the existing MIP density dataset, but facilitates
plasma analysis at much shorter timescales, and covers also long time periods
where densities were too low to be measured by MIP.

# Plasma densities, flow and Solar EUV flux at comet 67P

A cross-calibration approach

F. L. Johansson 1122 A. I. Eriksson 11 E. Vigren 11 L Bucciantini 33 P. Henri
3344 H. Nilsson 55
S. Bergman 5566 N. J. T. Edberg 11 G. Stenberg Wieser 55 E. Odelstad 11

(Received XX; accepted YY)

###### Abstract

Context.

Aims. During its two-year mission at comet 67P, Rosetta nearly continuously
monitored the inner coma plasma environment for gas production rates varying
over three orders of magnitude, at distances to the nucleus from a few to a
few hundred km. To achieve the best possible measurements, cross-calibration
of the plasma instruments is needed.

Methods. The goal is to provide a consistent plasma density data set for the
full mission, in the process providing a statistical characterisation of the
plasma in the inner coma and its evolution.

Results. We construct physical models for two different methods to cross-
calibrate the spacecraft potential and the ion current as measured by the
Rosetta Langmuir Probes (LAP) to the electron density as measured by the
Mutual Impedance Probe (MIP). We also describe the methods used to estimate
spacecraft potential, and validate the results with the Ion Composition
Analyser, (ICA).

Conclusions. We retrieve a continuous plasma density dataset for the entire
cometary mission with a much improved dynamical range compared to any plasma
instrument alone and, at times, improve the temporal resolution from 0.24-0.74
Hz to 57.8 Hz. The physical model also yields, at 3 hour time resolution, ion
flow speeds as well as a proxy for the solar EUV flux from the photoemission
from the Langmuir Probes.

###### Key Words.:

plasmas – comets:individual: 67P/Churyumov-Gerasimenko – space vehicles:
instruments, methods: data analysis, methods:statistical

##  1 Introduction

ESA’s comet chaser Rosetta studied the comet 67P/Churyumov-Gerasimenko in
unprecedented detail for more than two years from August 2014 to September
2016 (Taylor et al., 2017). In what amounted to a half-revolution around the
sun, the scientific package dedicated to the plasma environment, the _Rosetta
Plasma Consortium_ (RPC, Carr et al., 2007), studied the comet evolve through
different activity levels including perihelion at 1.24 AU and the dwindling
activity at the end of the mission, at 3.83 AU. The RPC included, among other
instruments, the _Langmuir probe_ (LAP, Eriksson et al., 2007, 2017), the _Ion
Composition Analyzer_ (ICA, Nilsson et al., 2007, 2017) and the _Mutual
Impedance Probe_ (MIP, Trotignon et al., 2007; Henri et al., 2017).

Figure 1: Sketch showing the effect on electron density at the Langmuir Probe
(LAP1) position from the electrostatic field from the (negative) spacecraft
potential that envelopes the sensor. Equipotentials of the electrostatic
potential field in the same plane as LAP1 and the spacecraft centre are
visualised by concentric blue circles.

As the spacecraft became significantly negatively charged in the cometary
environment (Odelstad et al., 2015, 2017; Johansson et al., 2020), the charged
particles that constitute the plasma environment were attracted to or repelled
from the spacecraft and instruments mounted on the spacecraft body or on short
booms protruding from the spacecraft body, as sketched in Figure 1. Low energy
particles, i.e. with energy in eV comparable to the spacecraft potential in
volts, are particularly affected: cometary electrons are significantly
repelled from the Langmuir Probe (Eriksson et al., 2017; Johansson et al.,
2020), and positive ions are perturbed (Bergman et al., 2019, 2020). As such,
this poses significant challenges for RPC’s ability to characterise the low
energy cometary plasma which dominates the comet environment (Edberg et al.,
2015; Odelstad et al., 2018; Gilet et al., 2019; Wattieaux et al., 2020), but
can be overcome through accurate modelling (Bergman et al., 2019; Johansson et
al., 2020) and cross-calibration (Heritier et al., 2017b; Breuillard et al.,
2019) to MIP measurements, which have been found to be almost unperturbed by a
negative spacecraft potential (Wattieaux et al., 2019).

We structure this paper as follows: After a brief description of the
instruments used in this study in Section 2, we describe the spacecraft
potential as measured by LAP, and how we can verify that measure using the
attracted ions observed by ICA in Section 3. We then show how the spacecraft
potential obstruct some of the LAP measurement modes, but also how we can use
the spacecraft potential to our advantage to recover electron densities via
cross-calibration in Section 4, and in a similar fashion cross-calibrate the
LAP ion currents to electron densities. We thereby obtain a dataset that
combines the dynamic range and high resolution of LAP with the accuracy of
MIP. Finally, we present a side-effect of the physical model of the cross-
calibration routine, which yields accurate estimates on the ion flow speed and
EUV flux and agree with independent measures and previously published results.

##  2 Instrument description

The Rosetta plasma consortium included, among other instruments, the _Rosetta
dual Langmuir probe instrument_ (LAP), which consists of two 5 cm diameter
spherical electrostatic probes LAP1 and LAP2 and their associated electronics.
The primary parameter measured are the currents flowing to (or the voltage of)
the probes when some bias voltages (or bias currents) are applied to it. From
these measurements we can derive plasma characteristics such as plasma
density, electron temperature, ion flow speed, as well as spacecraft
potential, extreme ultraviolet (EUV) flux and electric field fluctuations. Not
all parameters are monitored simultaneously or uniformly over the mission, as
it is contingent on the bias voltage or current (i.e. the operational mode)
selected for each probe, and, ultimately, the science objective of each
operational period. LAP1 and LAP2 are situated at the ends of separate booms
of length 2.24 and 1.62 m, respectively, and described in further detail in
Eriksson et al. (2007).

The Mutual Impedance Probe, (MIP), is an active electric sensor that allows us
to derive the plasma density via the identification of the plasma resonance
frequency. MIP consists of two pairs of transmitting and receiving electrodes
(as well as their associated electronics), and measures the electric coupling
of the electrodes to the plasma by fluctuating a charge on the transmitters.
MIP is mounted on a 1 m long bar on the same boom as LAP1, but can also make
use of the LAP2 sensor for transmission, situated $\sim 4$ m away on the LAP2
boom. MIP is described in further detail by Trotignon et al. (2007).

The Ion Composition Analyzer, ICA, is a mass-resolving ion spectrometer
mounted on the Rosetta spacecraft body, and included in the RPC package. ICA
has a maximum energy range of a few eV to 40 keV, which it typically sweeps
every 12 s with a typical energy resolution of $dE/E=0.07$, as well as an
angular field-of-view of 360 $\times$ 90 $\deg$ (Odelstad et al., 2017;
Nilsson et al., 2017). As positive ions are accelerated into the ICA
instrument when the spacecraft potential is negative, we can estimate the
spacecraft potential also from the measured energy of the lowest energy ions,
as described in Odelstad et al. (2017).

Figure 2: Two example LAP sweeps from 2016-09-01 (blue and red stars) showing
significant spacecraft charging, as well as a visualisation of a selection of
parameters from the analysis: the spacecraft potential estimate from the zero-
crossing potential, $V_{\rm z}$ from each sweep (blue and red dots,
respectively), the estimated photoemission current for the first sweep (yellow
line), which has a discontinuity at $V_{\rm ph}$ (black triangle), the ion
current slope fits (green and red dashed lines) and the total current fit to
the first sweep (purple solid line). As the spacecraft charging is
significant, the electron current is poorly constrained by the bias range, but
is here modelled using a double Maxwellian distribution with characteristic
energies of 0.1 eV and 1.5 eV.

##  3 Spacecraft potential

The spacecraft potential, $V_{\rm S}$, is fundamental to the interpretation of
plasma data. As documented in detail by Odelstad et al. (2015, 2017), LAP uses
two complementary techniques of estimating the spacecraft potential. One
involves identifying the photoelectron knee potential $V_{\rm ph}$ from a
probe sweep, which is the negative of the potential where the probe is in
equilibrium with its immediate plasma surroundings and is neither attracting
nor repelling photoelectrons emitted from the probe surface. The other method
involves measuring the floating potential $V_{\rm f}$, which is the negative
of the potential for which the sum of all currents to the probe is zero. When
LAP is operating in floating potential mode, we measure $V_{\rm f}$ directly,
but we can also estimate it from a sweep (Figure 2), in which case we denote
this parameter $V_{\rm z}$ .

In an ideal scenario, $V_{\rm z}$ and $V_{\rm f}$ would be fully equivalent.
However, the sweep has discrete step size, 0.25 or 0.5 V in most modes, and
$V_{\rm z}$ has to be found by interpolation or fitting. Therefore, $V_{\rm
z}$ usually exhibits more noise than does $V_{\rm f}$, which is also immune to
a displacement current added by any capacitive effects when varying the bias
voltage. Nevertheless, the method of fitting $V_{\rm z}$ can increase the
range of the estimate by extrapolating beyond the sweep bias voltage window,
as shown in Figure 2. In one of the two sweeps in this example, $V_{\rm z}$
falls just outside the bias range of the sweep, but as we show below, the
extrapolation can reach much further. For sweeps with disturbances (noise or
otherwise) there can be several zero-crossings of current. In these cases,
each zero-crossing is ranked in descending order of longevity, i.e. the
distance to the next zero-crossing in either direction, as we expect
disturbances to be short-lasting over sweep time scales. Only the best ranked
zero-crossing is chosen for the $V_{\rm z}$ estimate.

In Figure 3 we show a validation of the automatic identification of $V_{\rm
z}$, beyond the sweep range, and in comparison to $V_{\rm f}$ and ICA
spacecraft potential estimates. No obvious discontinuities are seen between
the two methods in Figure 3. Statistically, as LAP alternated between
continuous floating potential mode and sweeps (yielding $V_{\rm f}$ and
$V_{\rm z}$ estimates, respectively) over the entire mission, we find a slight
shift with a median of 0.4 V and a median absolute deviation of 1.7 V, when we
move from one mode to the other.

Figure 3: Spacecraft potential estimates from LAP and ICA. Coloured circles
are the spacecraft potential estimates from LAP, coloured by data source. Red
circles are derived from $V_{\rm f}$. Yellow, blue and green circles are all
derived from $V_{\rm z}$, but green indicates values that are extrapolated
beyond the sweep range and blue indicates sweeps with several zero-crossings
(i.e. disturbances). The blue line is an -3.8 V offset corrected ICA estimate
derived from the negative of the lowest energy bin with at least five ion
detections, and filtered using a 50 point moving average. Note that there are
no obvious discontinuities in moving from $V_{\rm z}$ to $V_{\rm f}$.

How $V_{\rm ph}$ and $V_{\rm f}$ relate to the spacecraft potential has been
studied in detail in Odelstad et al. (2017) whenever we can compute
simultaneous and good quality estimates from the offset of low energy ions in
ICA for a spacecraft that is sufficiently negatively charged. Over the entire
mission, $V_{\rm ph}$ and $V_{\rm f}$ diverge slightly as the spacecraft
potential approaches zero or becomes slightly positive, with $V_{\rm f}$
becoming non-linearly less sensitive to spacecraft potential changes, as shown
in Figure 4. Also, the method of detection for $V_{\rm ph}$ relies on
identifying the peak of the second derivative, i.e. the discontinuity of the
photoelectron current and, as such, relies on a very good signal-to-noise
ratio for an accurate detection. When the signal-to-noise ratio is poor, a
Blackman filter (Magnus & Gudmundsson, 2008) has been applied on the signal
before identifying the peak of the second derivative, which can introduce
errors and artefacts. However, as $V_{\rm ph}$ relates to the spacecraft
potential only by some factor (Odelstad et al., 2017), we created an empirical
model to map the less noise-sensitive $V_{\rm f}$ and $V_{\rm z}$ dataset to
equivalent $V_{\rm ph}$ values by use of the fit in Figure 4. We call this new
variable $U_{1}$ according to

$U_{1}=V_{\rm f}+5.5\ \exp\left(\frac{V_{\rm f}}{8.0}\right),$ (1)

where we note that the numerical factors have units of volts and that $U_{1}$
should be some factor $\alpha$ of the spacecraft potential, $V_{\rm S}$,
between 0.7 and 1 according to Odelstad et al. (2017). For the LAP $V_{\rm S}$
estimates plotted in Figure 3, this factor $\alpha$ is taken to be 0.95. With
this empirical model, 68 percent of all $U_{1}$ estimates fall within 1 V of
$V_{\rm ph}$ in Figure 4.

Figure 4: 2D histogram of 390 000 simultaneous spacecraft potential estimates
$V_{\rm ph}$ vs $V_{\rm z}$ or $V_{\rm f}$ in 200x200 bins, corresponding to
bin widths of $\sim$ 0.3 and 0.5 V, coloured by log10 counts for the entire
cometary mission. As $V_{\rm ph}$ is sensitive to noise, there are several
artefacts at e.g. +22 V and -25 V, but there is otherwise a clear agreement,
especially for large negative values as far as $V_{\rm ph}$ can measure. As
the spacecraft potential approaches zero, $V_{\rm f}$ and $V_{\rm z}$ diverges
non-linearly from $V_{\rm ph}$, the latter being considered the better
estimate for the spacecraft potential at these ranges. An empirical model was
fitted to move from $V_{\rm f}$ and $V_{\rm z}$ to $V_{\rm ph}$ and is plotted
in red.

##  4 Method

###  4.1 MIP-LAP cross-calibration

As described in Section 2, the RPC included two instruments targeting the bulk
plasma properties, LAP and MIP. The Langmuir probe technique used by LAP is
very flexible, enabling access to a broad number of plasma parameters like
electron density, electron temperature and ion temperature or flow speed, and
also provided integrated solar EUV flux, spacecraft potential and electric
field measurements. Other advantages are the broad dynamic range in plasma
density (see Section 6) and the possibility of very high time resolution (down
to 53 $\mu$s for LAP). However, perturbations from the spacecraft on the
plasma at the probe can sometimes be very large, and disentangling variations
of density and temperature is not always possible. The mutual impedance
technique used by MIP is entirely different, resting on observing the
transmission properties of the plasma for artificially injected oscillating
electric fields in the kHz to MHz range when in active mode, or observing
natural plasma oscillations when in passive mode. Compared to LAP, dynamic
range, maximum time resolution and resolution in density are all lower, but
MIP also has two major advantages. First, the main resonance frequency
identified by MIP effectively only depends on the plasma density, with very
little complication from temperature variations. Second, for the negative
spacecraft potential attained by Rosetta in the inner coma, the MIP density
value turns out to be insensitive to the spacecraft potential (Wattieaux et
al., 2019).

In order to combine the advantages of the two instruments and measurement
principles as discussed above, cross-calibrated datasets have been made
available on the ESA Planetary Systems Archive111http://psa.esa.int/(Besse et
al., 2018). In the following sections we will show some results of these
cross-calibrations as well as some by-products achieved during the process. In
principle, the cross-calibration uses the current or voltage measured by LAP
and finds a fit of this to available MIP density data points over some time
period. This fit can then be used either to obtain a plasma density with the
high resolution and/or dynamic range of LAP measurements and the robustness of
MIP measurements.

The upper panel in Figure 5 shows a mission-wide 2D histogram of the (32 s
average) MIP electron density vs. the (32 s average) current measured by one
of the two LAP probes, LAP1, at fixed negative bias voltage. For a non-
illuminated probe, this is mainly the current due to ion collection,
proportional to the plasma density as seen by the points extending down to
about 1 nA and extending to the top right corner. However, when the probe is
exposed to sunlight the resulting current cannot be lower than the
contribution from photoelectron emission, as seen by the large number of
points around to 10 to 50 nA. Section 4.3 will show how this contribution can
be identified and used for solar EUV monitoring. As described in (Johansson et
al., 2017), LAP2 has exhibited clear contamination signatures from a
capacitive and resistive layer, and as such, is rarely used for current cross-
calibration. Nevertheless, when in floating potential mode, there is no
electrical response from the contamination layer, and we have excellent
agreement between the two probes.

The lower panel of Figure 5 illustrates why only currents at negative bias
voltage (ion currents) are used for cross-calibration. This 2D histogram shows
currents obtained at positive bias voltage with respect to the spacecraft,
intended to attract electrons providing a current proportional to plasma
density. While some points indeed show the expected correlation, others
clearly do not, even exhibiting anti-correlation, and the spread of points is
very large. The basic reason for this is the highly negative potential
obtained by the spacecraft in high density plasmas (Johansson et al., 2020),
so high that the resulting voltage of the probe can even become negative with
respect to the plasma. Such charging is not a problem for the ion current,
which maintains a good correlation to the MIP plasma density. This also lends
observational support to the insensitivity of the MIP electron density to the
spacecraft potential as found by Wattieaux et al. (2019).

We may here note that the spacecraft potential, MIP electron density estimates
and LAP currents are all measures of plasma parameters on slightly different
spatial scales. The currents to the 5 cm diameter Langmuir probe mainly
depends on the plasma within about a meter222The electrostatic field from the
Langmuir probe does not reach much farther than the Debye length, but in
practice its reach is limited to about a meter even when the Debye length is
longer as the distance at which the vacuum field from a sphere at potential
$\sim$30 V has decayed to the typical energy of the plasma particles collected
($\sim$5 eV) is on the order of 10 probe radii. from the probe and therefore
is the most local estimate, while the spacecraft potential (derived from
currents to a largely conductive spacecraft with a wingspan of 32 m) is the
least local. When there are plasma variations on scales below tens of meters,
even ideal sets of measurements would only agree in an average sense. As the
Debye length in the inner coma typically is $\lesssim 1$ m (Gilet, N. et al.,
2020), such small scale plasma variations are fully possible, meaning that the
scatter observed when comparing e.g. LAP probe current to MIP plasma density
is not necessarily due to measurement uncertainties.

Figure 5: LAP currents in 32-second averages vs simultaneous 32-second average
MIP densities over the mission. Top: MIP density vs LAP1 current at negative
bias voltage ($V_{\rm b}<-15$ V, referred to as ”ion current”), coloured by
log10 counts in each bin. Bottom: MIP density vs LAP1 electron currents
($V_{\rm b}$ ¿20 V), coloured by log10 counts in each bin. Figure 6: Mission-
wide 2-D histogram of MIP density vs simultaneous LAP spacecraft potential
proxy $U_{1}$ in 100x200 bins coloured by log10 counts. The upper sensitivity
limit of MIP in LDL mode is just below 300 cm-3, and is visible as a partial
cut-off.

###  4.2 Cross-calibration of spacecraft potential to electron density

MIP identifies the electron density via detection of the plasma frequency.
Typically, every 1.4-4.3 seconds, an oscillating electric field is injected in
the plasma at stepped frequencies through different electric transmitters. The
electric potential that propagates in the plasma is simultaneously measured on
the two MIP receivers, from which the complex (amplitude and phase) mutual
impedance spectra is built. Eventually, the resonance observed in the mutual
impedance spectra at the plasma frequency enables us to retrieve the electron
density at a cadence of 0.24-0.74 Hz.

There are two fundamental operational modes of MIP, namely the SDL and LDL
modes, used intermittently for smaller and larger Debye lengths, respectively.
Therefore, the density targeted by the MIP instrument is limited to two
different ranges, and as such, on the selection of a suitable operational mode
based on the predicted density days/weeks in advance. It is therefore not
surprising that there are intervals in the cometary mission where the electron
density falls outside of MIP sensitivity range (see Section 5, Figure 9).
Thus, special care has to be taken when the average density is outside the
measurement range but the dynamic variations of the plasma allow MIP to
sporadically detect the lower or upper edge of the density range. If a MIP
sample (i.e. a positive MIP plasma resonance identification) is not adjacent
to multiple positive identifications (at least five within the ten closest
identifications), the MIP sample is disregarded for cross-calibration.

The spacecraft potential is the potential for which the sum of all currents to
the spacecraft is zero. Defining a positive current as the flow of positive
charges from the spacecraft, the dominating positive and negative current
contributions for Rosetta are the cometary electron current and the
photoemission current, respectively. In such a plasma, for a negative and
conductive spacecraft, $V_{\rm S}$ becomes (Odelstad et al., 2017)

$V_{\rm S}\approx-T_{\rm e}\log\left(C\,n\sqrt{T_{\rm e}}\ \right),$ (2)

where $n$ is the number density of the electrons, which are assumed to be a
Maxwellian population of characteristic temperature $T_{\rm e}$, given in eV,
and $C$ is a constant not depending on plasma properties.

We therefore expected the spacecraft potential to be much more sensitive to
the temperature of these electrons than to the electron density. In a recent
study (Johansson et al., 2020), we have shown that this is not the case, and
also presented a detailed model explaining why the Rosetta spacecraft
potential is sensitive to the density of electrons, regardless of temperature,
allowing even cold (0.1 eV) electrons to reach a spacecraft that is often
charged -10 to -20 V. The primary reason is the presence of exposed positively
biased conductors on the front-side edges of the solar panels.

As several studies (Heritier et al., 2017b; Breuillard et al., 2019; Johansson
et al., 2020) show, and as is visible in Figure 6, there is a strong relation
between the logarithm of the electron density and the spacecraft potential
proxy $U_{1}$. This relation is even more evident over shorter time windows,
shown in Figure 7, as the photoemission current, which is included in $C$ in
Equation 2, can be assumed to be constant. As the orbital parameters are not
uniform over the entire mission, some branching is evident in the mission
overview in Figure 6 at the lowest densities, which Rosetta only visited on a
selection of heliocentric distances, and thus, specific ranges of
photoemission current.

The spacecraft potential is, as such, clearly a good proxy for the electron
density, and we relate the two in a similar fashion as Breuillard et al.
(2019) by rearranging Equation 2:

$n=P_{2}\exp\left(-\frac{U_{1}}{P_{1}}\right),$ (3)

where $P_{1}$ and $P_{2}$ are both constants over the time interval. $P_{1}$
is defined as $P_{1}=\alpha k_{B}T_{\rm e}/e$, with the definition of $\alpha$
from Section 3.

The cross-calibration of electron density to simultaneous $U_{1}$ estimates
was performed with a 3-day window that is stepped in 1-day steps over the
entire mission according to Equation 3, with some outlier removal, as
specified in RO-IRFU-LAP-
XCAL333http://psa.esac.esa.int/pub/mirror/INTERNATIONAL-ROSETTA-
MISSION/RPCLAP/RO-C-RPCLAP-5-PRL-DERIV2-V1.0/DOCUMENT/RO-IRFU-LAP-XCAL.PDF.
This rather long calibration window ensures a physical interpretation of each
fit, allows us to bridge longer gaps in MIP data, and generate a continuous
and mission-wide electron density estimate, at the time resolution of the LAP
$U_{1}$ estimate. The dynamical range is also much improved both above and
below the sensitivity range of MIP regardless of its operational mode.
Temporal solar EUV events, or rapid and significant fluctuations in electron
temperature will not be correctly captured by the the model, but the general
trend in the electron density data will still be valid.

Figure 7: An example of the sliding cross-calibration window. 2D histogram of
MIP density vs $U_{1}$, 50x50 bins coloured by counts. The orthogonal least
squares fit (black line) yields an electron temperature of 5.85 eV for
$1.053U_{1}=V_{\rm S}$, as per Figure 3. Figure 8: An example of the sliding
cross-calibration window for ion current: MIP density vs $I_{\rm-}$ for three
hours around 2016-05-23T23:30:00. A linear orthogonal least squares fit yields
an effective ion speed of 9.85 km/s and an estimate for $I_{\rm ph0}$ of -6.92
nA.

###  4.3 Cross-calibration of ion current to electron density

In a similar, but more straight-forward fashion, one can relate the ion
current to the ion density, and by argument of quasi-neutrality, the electron
density (under the assumption that charged cometary dust does not contribute
much to the overall current balance). At the bias potentials considered
($V_{\rm b}\leq-15$ V), the Langmuir probe is repelling electrons, such that
the current contribution from plasma electrons can be assumed to be
negligible, including secondary emission from electron impact. Whenever
cometary cold ions are present in the ICA data, they appear supersonic, i.e.
the thermal speed of ions is much lower than their flow speed (Bergman et al.,
2021). This is supported also by the identification of a cometary ion wake
behind the spacecraft as discussed in Odelstad et al. (2018). It can be shown
(Sagalyn et al., 1963; Fahleson, 1967) that for a supersonic flow of ions in a
plasma (a cold ion beam), the current to a sunlit probe attracting ions is
given by

$I_{\rm-}=I_{\rm ph0}-n\pi r_{\rm p}^{2}q_{\rm i}\sqrt{\frac{2E_{\rm
i}}{m_{i}}}\left(1-\frac{q_{\rm i}V_{\rm p}}{E_{\rm i}}\right),$ (4)

where $I_{\rm ph0}$ is the photosaturation current of the probe, defined to be
negative, $r_{\rm p}$ is the radius of the probe, $E_{\rm i}$ is the energy of
ions of charge $q_{\rm i}$ and mass $m_{\rm i}$, and $V_{p}$ is the absolute
potential of the probe according to $V_{\rm p}=V_{\rm b}+V_{\rm S}$. In this
cold ion limit we have neglected the thermal velocity component, and we define
the effective ion speed $u_{\rm eff,i}$ such that $E_{\rm i}=0.5m_{\rm
i}u_{\rm eff,i}^{2}$. For a warm flowing ion plasma, $E_{\rm i}$ may instead
be interpreted as a typical ion energy, combining the thermal and ram energy.
A factor of $4/\pi$ then appears inside the square root term in Equation (4)
(Lindqvist et al., 1994), but the linearity of the current-voltage relation
remains for all attractive probe potentials (Mott-Smith & Langmuir, 1926;
Fahleson, 1967).

If we assume that the plasma density varies much faster than the ion energy
and the EUV flux (and therefore, $I_{\rm ph0}$), and we ignore small
perturbations arising from probe potential variations, we can expect a linear
relation between $I_{\rm-}$ and the electron density.

This is shown in an example cross-calibration window in Figure 8, where the
x-intersect yields the photosaturation current. In a similar fashion as in
Section 4.2, we perform a linear orthogonal least square fit of coinciding
$I_{\rm-}$ measurements and MIP electron density estimates over a window
spanning three hours, which is stepped with one hour over the entire cometary
mission. If there are several $I_{\rm-}$ measurements, during a MIP density
sampling interval, an average is taken. Measurements of $I_{\rm-}$ for a
shadowed probe are analysed separately, assuming $I_{\rm-}(n=0)=0$. Some
outliers are removed according to RO-IRFU-LAP-XCALfootnote 3.

During periods with low electron density and few coinciding MIP and LAP data
points (before 2015-01-01 and during the so-called night side excursion around
2016-04-01), where no electron density estimates would otherwise be produced,
the calibration instead considers a combined dataset of MIP and LAP sweep
densities (obtained from fits as in the example in Figure 2) for the cross-
calibration. In this case, the cross-calibration is applied over a larger
calibration window (15-day window, 5-day step size) to improve the fitting
performance. As the spacecraft potential is low or positive during these
periods, the reliability of the LAP sweep electron density estimates is
considered to be the best available, although still sensitive to electron
temperature variations.

##  5 Results

Figure 9: Two examples that illustrates the performance of the cross-
calibration. Top: New Years Eve 2014, when the MIP instrument (blue dots) was
occasionally in a measurement mode that is not sensitive to low densities, but
LAP was continuously measuring spacecraft potential, and recovers the density
via cross-calibration to spacecraft potential (red line). The errors in the
cross-calibration are dominated by uncertainties in MIP, and are typically 25
percent in both datasets. Bottom: The time resolution of the electron density
dataset is also improved, here via a $\sim$60 Hz ion current cross-calibration
as described in Section 4.3. Figure 10: Mission-wide histograms of electron
density from the cross-calibration of LAP ion current to MIP in blue bars, of
spacecraft potential to MIP (purple) and the MIP densities in yellow bars.
Also plotted on the axis to the right is the fraction of MIP counts to MIP-LAP
$I_{-}$ counts in each density bin. As the LAP current measurement has finite
resolution, fractional errors exceed unity below 5 cm-3 even for a perfect
calibration of the ion current to density, and are therefore not included.
Figure 11: Photosaturation current estimates during the cometary mission from
the cross-calibration of ion current to MIP (purple dots) with an associated
error bar from the fit of less than five percent, three different
photosaturation current estimates are as described in Johansson et al. (2017),
where blue and yellow are two out of three independent estimates from LAP with
associated error bars, and the black line is a(flare-removed) photosaturation
estimate as measured from an EUV monitor at Mars orbit. Also plotted, a low-
pass ($1/f<9.5$ days) filtered profile of all photosaturation estimates from
LAP (red line).

The performance of the ion current cross-calibration procedure can be seen in
Figure 9. Here, we recover densities in periods where MIP does not produce
electron densities, and (bottom) the temporal resolution has increased
dramatically, but is well in agreement with MIP density estimates whenever
available.

For comparison and as a mission overview, we plot a histogram of the resulting
cross-calibrated plasma density datasets, as well as the MIP density in Figure
10. The cross-calibrated datasets cover a wider dynamic range of densities, at
least for the ion current, and for density ranges fully within the range of
the MIP sensitivity, we see a near constant ratio between the two datasets, as
expected for two comparable measures with unequal temporal resolution. Also as
expected, the ratio drops at the lower edge of the MIP sensitivity range. As
noted at the end of Section 4.2, the ion current, the MIP density and the
spacecraft potential are all measures at slightly different spatial scales.
And even though there is no obvious reason why the density estimates from the
spacecraft potential method has an upper limit, estimates above 3000 cm-3 on
this (32 m) scale seems rare with this method, and is instead most common on
Langmuir probe current scales. A feature indicative perhaps of the sizes of
the plasma structures that move past the Rosetta spacecraft.

The physical interpretation of the cross-calibration coefficients yields an
estimate for the photosaturation current $I_{\rm ph0}$ from the x-intersect of
the fit in Figure 8 as $I_{\rm-}(n=0)=I_{\rm ph0}$. This new estimate of
$I_{\rm ph0}$ agrees very well with other methods of obtaining the
photoemission current published in Johansson et al. (2017), as shown in Figure
11.

Figure 12: Effective ion speed estimates from the slope of the ion current,
${\rm d}I/{\rm d}V_{\rm b}$, from Langmuir Probe sweeps (median of 19 hour
bins plotted in red) and from ${\rm d}I/{\rm d}n$ using three hour long
$\sim$60 Hz ion current cross-calibration windows (blue). Both methods are
using coinciding plasma density estimates from MIP.

For the slope coefficient of the ion current cross-calibration fit (${\rm
d}I/{\rm d}n$), we can solve for the effective ion speed via Equation 4
assuming the spacecraft potential fluctuation is small compared to $V_{\rm p}$
and that ions are singularly charged with a mass of 19 amu (Heritier et al.,
2017a). The resulting speed is plotted in Figure 12 and compared with a sweep
derived ion speed estimate (${\rm d}I/{\rm d}V$), where we evaluate the slope
of the ion current ${\rm d}I/dV_{\rm b}$ using a linear orthogonal least
square fit (see Figure 2), as well as a coinciding MIP density measurement, as
described in detail in Vigren et al. (2017). Random errors are estimated from
the uncertainty in the least square fits, assuming the errors are normally
distributed, and is dominated by MIP sweep frequency discretisation. As the
number of MIP and LAP measurement points are much greater in the cross-
calibration window than in the sweep estimates, we try to reduce the random
errors in the sweep estimates by binning the data in 500 bins of equal length
(19 hour) over the entire mission.

##  6 Discussion

Before discussing the implications of the cross-calibration, where we focus on
the ion-current cross-calibration, we shall first discuss error sources.

###  6.1 Errors

In the cross-calibrated densities, the random errors associated with the
uncertainty of the associated LAP measurement is very small, typically 5 cm-3,
assuming an ion velocity of 5 kms-1. However, the standard error in the
calibration is dominated by the uncertainty of the associated MIP measurement
(typically 25 percent), but is also influenced on the fluctuation of the EUV
flux to the probe (if sunlit) or the spacecraft during the calibration
interval (typically less than 2 nA and much less than 10 percent). As the
final fractional error (assuming the errors are normally distributed) is the
sum of the squares of the fractional errors, the error margins of MIP and the
cross-calibrated densities is more or less identical, and for the comparison
in Figure 9 we plot all datasets without the $\approx$ 25 percent fractional
uncertainties.

Systematic errors arise from the limited validity of the underlying assumption
that the ion density at the probe position is equal to the electron density
MIP measures, and at least proportional to the average electron current to the
spacecraft, which is an error we cannot easily quantify. If this assumption is
not correct, the slope in the fit, and therefore the effective velocity of the
ions we estimate from the ion current cross-calibration should diverge from
the true value by some factor equal to the difference in ion and electron
densities during the calibration interval. Such effects include sheath effects
from the spacecraft potential for ions, and may require further investigation
with spacecraft-plasma interaction simulations. However, we can test the
assumption by comparing the effective velocity estimate to ICA ion
measurements, as we will do in subsection 6.3

If the ion velocity (for the ion current cross calibration) or the electron
temperature of the thermal electron population that dictates the spacecraft
potential dependence on electron density (Johansson et al., 2020) is not
constant during the cross-calibration interval, errors would grow, and the
resulting fit would be poor. Effective ion velocities and photoemission
currents resulting from fits with correlation coefficient less than 0.7 have
therefore been removed.

It should also be noted that the ion current to a Langmuir probe from ions
with a superposition of velocity distributions is the superposition of the
current from each velocity distribution. Although a Maxwellian simplifies the
interpretation of the LAP effective ion speed, it is not dependent on a
Maxwellian assumption. The equations that govern the cross-calibration and the
ion speed estimates rely on Orbital Motion Limited (OML) theory, i.e. on the
assumption that the Langmuir probe size (5 cm) is much smaller than the Debye
length, which is generally true for the entire mission (Gilet, N. et al.,
2020; Wattieaux et al., 2020). If the Debye length is at any point smaller
than the probe size, in very cold and dense plasmas, we note that OML theory
yields an upper limit on the collected current. In such cases, we would expect
to see more points above the linear trend in Figure 8 when the density is
high, which we generally do not. Similarly, it would yield a significant
underestimation of the photoemission when the density is high, but this is not
observed in Figure 11. Additionally, if the thermal velocity is the dominant
component of the velocity, in contrast to the findings in Odelstad et al.
(2018) and Bergman et al. (2021), our cold ion approximation in Equation 4
would lead to an overestimation of $u_{\rm eff,i}$ by 13 percent.

###  6.2 EUV intensity

We are able to resolve the solar sidereal rotational period of $\sim$26 days
in our photosaturation current from the cross-calibration plotted in Figure
11, and also obtain a good agreement with other photosaturation estimates from
LAP (of which the shadow-crossing method is believed to be the most accurate).
Therefore, we find this result to be another confirmation of the results
published in Johansson et al. (2017) as well as of the physical model behind
the cross-calibration.

We also note that if there is a significant amount of higher energy ($>20$ eV)
electrons, such as the population giving rise to electron-impact ionisation in
the cometary plasma, this population would also be able to excite and emit
secondary electrons from the probe surface. For Cassini in the Saturn
magnetosphere, Garnier et al. (2012) found such secondary emission to be a
significant contributor to the Langmuir probe current at negative bias
voltage. The source population in this case was identified as electrons with
energy in the approximate range 250-450 eV. As significant electron fluxes in
this energy range have been detected also by Rosetta (e.g. Clark et al., 2015;
Myllys et al., 2019) one may ask if such currents appreciably contaminate our
photoemission estimates. If the secondary yield is higher than 1, i.e. more
than one electron emitted per incoming high energy electron, then this would
constitute an additive offset in all LAP measured $I_{\rm ph0}$ estimates.
However, the shadow crossing method is insensitive to this secondary emission,
except in the highly unlikely case of the electron flux being parallel to the
direction from the Sun. The good agreement between the shadow crossing method
with other LAP estimates, whenever available, suggests that secondary electron
emission due to impact of high energy electrons generally has little to no
effect on our results.

###  6.3 Effective ion speed

The ion speed estimate from the cross-calibration agrees generally very well
with the sweep derived ion speed in Figure 12, and again confirms the validity
of the physical model in the cross-calibration. They are also very much in
line with the average flow velocities as reported by ICA (Nilsson et al.,
2020), also during the events investigated in detail by Bergman et al. (2021)
as well as previously published LAP estimates (Vigren et al., 2017; Odelstad
et al., 2018). Since the two velocity estimates in Figure 12 arise from two
different methods (${\rm d}I/{\rm d}V_{\rm b}$ vs ${\rm d}I/{\rm d}n$), where
the internal instrumental offsets are very different (as one involves stepping
voltages with the LAP bias circuitry), we conclude that we have a very good
grasp of instrument calibration and (as all $I_{\rm ph0}$ estimates agree), a
near-zero contamination on LAP1.

Figure 13: Top: Median of 19 hour bins of ICA H2O+ion bulk velocities in terms
of magnitude (black line) and purely radial component (purple line) with
corresponding $\pm 1\sigma$ spread (shaded) from the standard deviation of the
mean, as well as the two LAP effective ion speed estimates from the cross-
calibration in Section 4.3 (blue line) and 19 hour binned medians from sweeps
(red line) vs time for a selected time period where the radial distance was
rapidly varied. Bottom: radial distance vs time for the same time period. At
$\sim$ 1000 km to 30 km the radial component of the ion velocity seems to
dominate, and we have good agreement for all estimates, when available. Even
closer to the comet nucleus the radial component decreases and perpendicular
components grow, and as expected, both LAP effective ion speed estimates
increase as the magnitude of the velocity increases.

However, for the entire cometary mission, the ion speed estimate is much
larger than the neutral velocity of 0.4-1 kms-1 (Hansen et al., 2016) and thus
speaks against the assumption made in previous ionospheric models (Heritier et
al., 2017b, 2018; Galand et al., 2016), at large heliocentric distances, that
ions are flowing with the neutral gas speed. This assumption is not strictly
necessary for such ionospheric model density result, and we note that to
reconcile our ion speed estimate with these models, either the radial velocity
of ions is not the dominant component of the ion flow, or there have to be
some mechanism that increase ionisation between Rosetta and the comet nucleus.

As LAP can not resolve ion density independently, we note that any spacecraft
sheath related effects which would alter the ion density at the probe location
would affect both LAP estimates equally. Validating the ion density with ICA
over the entire mission is not currently feasible , mainly due to the limited
and distorted field of view (Bergman et al., 2019, 2020), such that the ion
density estimate would differ from the expectation value at any time,
depending on geometry and ion flow direction. However, this has a very limited
effect on the ICA ion speed estimate (Bergman et al., 2021). Except for an ion
beam that is invariably outside of the ICA instrument field of view during the
entire Rosetta mission, we see no reason why the cometary bulk ion speed
should be systematically incorrect after a simple correction for the
spacecraft potential.

Therefore, to further cross-validate our result, we plot the two LAP estimates
together with the magnitude and the radial component of the ICA $<60$ eV H2O+
(or H3O+) bulk drift velocity ($u_{i}^{bulk}$ and ${u_{r,i}^{bulk}}$,
respectively) from Nilsson et al. (2020) in Figure 13 for an interval near the
tail-side excursion. The estimates are not perfectly equivalent but should
still correlate well with the magnitude of the ICA estimate, when available.

Here, we find (within error bars) that the radial velocity is dominant and
positive at large cometocentric distances (from $\sim$ 30 to 1000 km), and
increasing with radial distance, which was also noted in the beginning of this
interval in Behar et al. (2018). Also, as expected, both LAP effective ion
speed estimates correlate and agree within error bars with the magnitude of
the ICA H2O+ bulk velocity when available, pointing to the existence of a
radial electric field such as an ambipolar electric field (Madanian et al.,
2016; Vigren & Eriksson, 2017; Berčič et al., 2018; Deca et al., 2019). As the
ambipolar electric field is proportional to the electron pressure gradient, a
radial expansion is consistent with increasing radial ion velocities with
cometocentric distance. Moreover, an ambipolar electric field would accelerate
electrons falling inward and, as more electrons reach electron-impact
ionisation energies, provide increased ionisation between Rosetta and the
comet nucleus, as postulated earlier in this section.

As Rosetta moves below 30 km, the radial velocity component decreases, but
perpendicular components grow and start to dominate. We suspect that this
could again be an effect of an ambipolar electric field as the comet activity
is inhomogeneous and bursty by nature, which would create strong non-radial
electron-pressure gradients, and therefore, form an ambipolar electric field
with non-radial components. It seems reasonable to expect this field to be
strongest near the nucleus, where gas density gradients are great. Also, ions
accelerated by a non-radial field close to the comet would also appear radial
for a faraway observer with a limited viewing angle resolution. This
hypothesis is highly speculative, but could perhaps be tested in the future
using simulations similar to Deca et al. (2019) and Divin et al. (2020), but
resolving the nucleus and allowing for a more realistic outgassing.

Figure 14: 2-D histograms of ion speed vs distance to comet surface coloured
by counts, with a common x-axis. Top: Radial bulk speed of water ions from
ICA, $u^{bulk}_{r,i}$ in 90x40 bins. Bottom: The union of the two LAP
effective flow speed $u_{\rm eff,i}$ estimates in 90x40 bins and two least-
square regression fit (red solid or dashed line) for two selected intervals,
with errors in the coefficients estimated from the least-square regression.

This trend seems also generally true for the entire mission, as plotted in
Figure 14, where the combined dataset of the two LAP effective ion speeds,
$u_{\rm eff,i}$, shows large (presumably non-radial) velocities below 30 km
from the comet surface. We also find and an acceleration with radial distance
beyond 30 km distance, as reported by (Berčič et al., 2018) using ICA
measurements.

Nevertheless, all measured estimates suggest that the ions are moving much
faster than the neutrals over the entire mission. The discrepancy between
$u_{\rm eff,i}$ and ${u_{r,i}^{bulk}}$ outside 30 km from the comet surface,
although there is overlap, could perhaps be attributed to that the ion density
is enhanced at the probe position. If so, the assumption of quasineutrality
via $n_{\rm e}=n_{\rm i}$, is incorrect, perhaps by significant electron
depletion by dust (Morooka et al., 2011), and would need further
investigation. There might also be field of view effects on the (mostly nadir
pointing) ICA instrument, which warrants further investigation, and is why we,
in line with Nilsson et al. (2020), limit our comparison to the radial
velocity component in ICA.

If there was a strong bias against ions at lower energies in both ICA and LAP,
for instance if the spacecraft potential more easily deflects low energy ions
away from detection surfaces, we would expect there to be a strong dependence
on the drift energies on spacecraft potential shown in Figure 5 in Nilsson et
al. (2020), but we find none. Additionally, as shown in a recent case study by
Bergman et al. (2021), there were no indication that there is a significant
low-energy cometary ion population present, as detailed spacecraft-plasma
interaction simulations of such a environments were incompatible with ICA
measurements. Moreover, the instrument field of view increases for low energy
ions on a negatively charged spacecraft (Bergman et al., 2019), but some
uncertainties still remain regarding the geometric factor at low ion energies
(Bergman et al., 2021). Still, the two LAP estimates, as well as the ICA ion
velocity shows elevated ion velocities, for which at least the radial
component increases with radial distance. This is consistent with a radial
ambipolar electric field that has been predicted to be capable of accelerating
the cold new-born ions from the neutral speed to the speed we observe (Vigren
& Eriksson, 2017).

##  7 Conclusions

We have devised and verified two methods to recover a mission-wide plasma
density dataset from the LAP spacecraft potential estimates and the LAP ion
current by cross-calibrating the estimate to MIP density whenever available.
As a result, we improve the dynamic range as well as the temporal resolution
of the RPC plasma density dataset up to a factor of 240 and facilitate plasma
analysis at much shorter timescales. The dataset has been made available on
the PSA (Eriksson et al., 2020), and at the CDPP on
AMDA444http://amda.irap.omp.eu. The spacecraft potential and the effective ion
speeds resulting from the MIP-LAP ion current cross-calibration model have
been successfully cross-validated with ICA, an instrument with a fundamentally
different measurement principle.

The physical model that enables the cross-calibration allows for an almost
continuous (3 hour cadence) estimate of the effective ion speed and, when the
probe is sunlit, the photosaturation current. The latter we find to be well in
agreement with independent methods in published studies (Johansson et al.,
2017) and provides support for the conclusions drawn therein regarding
attenuation of the EUV.

The ion speed estimates are found to be large $\sim$ 5 kms-1 and mostly radial
in altitudes above $\sim$30 km, which is in line with previously published
LAP-derived ion speeds (Odelstad et al., 2018; Vigren et al., 2017), and with
recent ICA estimates of the ion bulk velocity (Nilsson et al., 2020; Bergman
et al., 2021) of H2O+, but in disagreement with the assumption that the ions
flow with the speed of the neutrals. This assumption has been made in several
modelling works that target plasma densities, and in an average sense
successfully reproduce observations at low activity (Vigren et al., 2016;
Galand et al., 2016; Heritier et al., 2017a, b, 2018). As a faster radial ion
flow would decrease these model density estimates at the Rosetta position,
some other error, such as a process which increases the rate of ionisation
must also be present. However, further investigation is needed to confirm that
the inferred bulk speeds are representative of the actual drift speed.

The elevated velocities in and of itself points to an electric field present
throughout the entire cometary mission, capable of accelerating ions and
increasing ionisation via electron impact ionisation between Rosetta and the
comet nucleus. Other candidates for the elevated ion speeds (above the neutral
speed the ions are born at) includes the wave processes already detected at
the comet (André et al., 2017; Ruhunusiri et al., 2020; Karlsson et al.,
2017).

###### Acknowledgements.

_Rosetta_ is an ESA mission with contributions from its member states and
NASA. This work would not have been possible without the collective efforts
over a quarter of a century of all involved in the project and the RPC. This
research was funded by the Swedish National Space Agency under grant Dnr
168/15. The cross-calibration of LAP and MIP data was supported by ESA as part
of the Rosetta Extended Archive activities, under contract
4000118957/16/ES/JD. Work at LPC2E is also supported by CNES. We acknowledge
the staff of Centre de Données de la Physique des Plasmas (CDPP) for the use
of Automated Multi-Dataset Analysis (AMDA). We also acknowledge the ESA
Planetary Science Archive for archiving and reviewing the LAP data.

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