11institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Université de Paris, France
11email<EMAIL_ADDRESS>22institutetext: Arctic University of
Norway, Tromsø, Norway 33institutetext: Institute of Atmospheric Physics of
the Czech Academy of Sciences, Prague, Czechia 44institutetext: Space Research
Center, Polish Academy of Sciences, Warsaw, Poland 55institutetext: Faculty of
Mathematics and Physics, Charles University, Prague, Czech Republic
66institutetext: CNES, 18 Avenue Edouard Belin, 31400 Toulouse, France
77institutetext: Department of astronomy, Faculty of Mathematics, University
of Belgrade, Serbia 88institutetext: Space Sciences Laboratory, University of
California, Berkeley, CA, USA 99institutetext: Physics Department, University
of California, Berkeley, CA, USA 1010institutetext: Swedish Institute of Space
Physics, Uppsala, Sweden 1111institutetext: Radboud Radio Lab, Department of
Astrophysics, Radboud University, Nijmegen, The Netherlands 1212institutetext:
LPP, CNRS, Ecole Polytechnique, Sorbonne Université, Observatoire de Paris,
Université Paris-Saclay, Palaiseau, Paris, France 1313institutetext: LPC2E,
CNRS, 3A avenue de la Recherche Scientifique, Orléans, France
1414institutetext: Université d’Orléans, Orléans, France 1515institutetext:
Technische Universität Dresden, Würzburger Str. 35, D-01187 Dresden, Germany
1616institutetext: Space Research Institute, Austrian Academy of Sciences,
Graz, Austria 1717institutetext: Astronomical Institute of the Czech Academy
of Sciences, Prague, Czechia 1818institutetext: Department of Space and Plasma
Physics, School of Electrical Engineering and Computer Science, Royal
Institute of Technology, Stockholm, Sweden

# First dust measurements with the Solar Orbiter Radio and Plasma Wave
instrument

A. Zaslavsky 11 I. Mann 22 J. Soucek 33 A. Czechowski 44 D. Pisa 33 J. Vaverka
55 N. Meyer-Vernet 11 M. Maksimovic 11 E. Lorfèvre 66 K. Issautier 11 K.
Racković Babić 11 7 7 S. D. Bale 8899 M. Morooka 1010 A. Vecchio 1111 T. Chust
1212 Y. Khotyaintsev 1010 V. Krasnoselskikh 1313 M. Kretzschmar 1313 14 14 D.
Plettemeier 1515 M. Steller 1616 Š. Štverák 33 17 17 P. Trávníček 88 17 17 A.
Vaivads 1010 18 18

###### Abstract

Context. Impacts of dust grains on spacecraft are known to produce typical
impulsive signals in the voltage waveform recorded at the terminals of
electric antennas. Such signals are, as could be expected, routinely detected
by the Time Domain Sampler (TDS) system of the Radio and Plasma Waves (RPW)
instrument aboard Solar Orbiter.

Aims. We investigate the capabilities of RPW in terms of interplanetary dust
studies and present the first analysis of dust impacts recorded by this
instrument. Our purpose is to characterize the dust population observed in
terms of size, flux and velocity.

Methods. We briefly discuss previously developed models of voltage pulses
generation after a dust impact onto a spacecraft and present the relevant
technical parameters for Solar Orbiter RPW as a dust detector. Then we present
the statistical analysis of the dust impacts recorded by RPW/TDS from April
20th, 2020 to February 27th, 2021 between 0.5 AU and 1 AU.

Results. The study of the dust impact rate along Solar Orbiter’s orbit shows
that the dust population studied presents a radial velocity component directed
outward from the Sun, the order of magnitude of which can be roughly estimated
as $v_{r,dust}\simeq 50$ km.$s^{-1}$. This is consistent with the flux of
impactors being dominated by $\beta$-meteoroids. We estimate the cumulative
flux of these grains at 1 AU to be roughly $F_{\beta}\simeq 8\times 10^{-5}$
m-2s-1, for particles of radius $r\gtrsim 100$ nm. The power law index
$\delta$ of the cumulative mass flux of the impactors is evaluated by two
differents methods (direct observations of voltage pulses and indirect effect
on the impact rate dependency on the impact speed). Both methods give a result
$\delta\simeq 0.3-0.4$.

Conclusions. Solar Orbiter RPW proves to be a suitable instrument for
interplanetary dust studies, and the dust detection algorithm implemented in
the TDS subsystem an efficient tool for fluxes estimation. These first results
are promising for the continuation of the mission, in particular for the in-
situ study of the dust cloud outside the ecliptic plane, which Solar Orbiter
will be the first spacecraft to explore.

###### Key Words.:

Interplanetary dust ; Solar system

##  1 Introduction

For several decades, radio and plasma wave instruments have demonstrated their
ability to probe dust in different space environments. Voyager’s plasma wave
instrument (Gurnett et al., 1983) and planetary radio astronomy experiment
(Aubier et al., 1983) both observed broadband signals interpreted as produced
by dust impacts during the crossing of Saturn’s E ring by Voyager 1 and G ring
by Voyager 2 – the plasma wave instrument, operated in dipole mode with a
roughly symmetrical configuration, observing smaller amplitude signals that
the radio experiment, operated in monopole mode. The technique was used again
along Voyager 2 orbit, with dust measurements at Uranus (Meyer-Vernet et al.,
1986; Gurnett et al., 1987) and Neptune (Gurnett et al., 1991; Pedersen et
al., 1991). Voyager measurements at the outer solar system’s planets were
followed by others in space environments with an expected high dust flux, like
cometary trails with e.g. VEGA 2’s plasma wave instrument at comet Halley
(Oberc, 1990).

From the years 2000’s or so, radio analyzers aboard missions such as Wind
(Bougeret et al., 1995), Cassini (Gurnett et al., 2004) or STEREO (Bougeret et
al., 2008) recorded a large number of electric waveforms characteristic of
dust impacts. The improvement in the technical performance of these radio
detectors compared to the previous generation (the higher sampling frequency
of the waveform analyzers in particular) and the large number of examples
available to study has led to a better understanding of the mechanisms
involved in the voltage pulses generation after a dust impact. Several recent
studies detail this work of modelling and comparison to available data, such
as works by Zaslavsky (2015) on STEREO, Meyer-Vernet et al. (2017) or Ye et
al. (2019) on Cassini and Vaverka et al. (2019) on MMS. The paper by Mann et
al. (2019) provides a complete summary of the works performed on various
spacecraft, a prospect for missions Parker Solar Probe and Solar Orbiter and a
review of the voltage pulse mechanism in our present state of understanding.
The paper by Lhotka et al. (2020) also presents a detailed analysis of
spacecraft charging processes in various plasma environments and an
application to dust impacts on MMS.

To quickly summarize these models, voltage pulses result from the production
of free electric charges by impact ionization after a grain of solid matter
hits the spacecraft. These charges modify the spacecraft (and, depending on
the impact location, the antennas) potential through two main effects, one
being the perturbation of the electric current equilibrium between the
spacecraft and the surrounding plasma due to the collection by the spacecraft
of some of these free electric charges, the other being the perturbation of
the spacecraft potential by electrostatic influence from these free charges
that occurs when the impact ions/electrons cloud is not neutral on overall
(which happens after some of the charges have been collected or have escaped
away from the spacecraft).

Along the years, and thanks to these refinements in the pulses modeling, radio
analyzers have thus proven to be able to provide robust estimates of dust
fluxes in various mass ranges, varying from the nanometer to the micron.
Examples of successful use of this technique to derive dust fluxes include the
detection of nanometer sized dust with STEREO/WAVES (Meyer-Vernet et al.,
2009b) and Cassini/RPWS (Meyer-Vernet et al., 2009a), measurements of the
interstellar dust flux and direction at 1 AU by STEREO/WAVES (Zaslavsky et
al., 2012) and Wind/WAVES (Malaspina et al., 2014) or measurements of the
micron to ten microns sized dust density in the vicinity of Saturn by
Cassini/RPWS (Ye et al., 2014, 2018).

The present paper, in the continuation of these works, is devoted to the study
of the dust impact data recorded by the Radio and Plasma Waves instrument, and
to the derivation of the interplanetary dust fluxes along Solar Orbiter’s
orbit. This is of particular importance since in-situ measurements of
interplanetary dust in the inner heliosphere, which are necessary to constrain
and validate dust production models, are not numerous. Notably there are only
few data on the dust collision fragments that form in the inner solar system
and then are ejected outward. Flux estimates for these dust grains, denoted as
$\beta$-meteoroids, were made based on Helios observations (Zook & Berg, 1975)
and based on Ulysses observations (Wehry & Mann, 1999). The dust collision
evolution inside 1AU were studied with model calculations having only few
observational constrains (Mann et al., 2004).

Recently the Parker Solar Probe (PSP) operates the FIELDS instrument (Bale et
al., 2016) which provides observations of dust impacts. The PSP dust
observations have been presented in a number of recent works (Page et al.,
2020; Szalay et al., 2020; Malaspina et al., 2020). These observations provide
a number of interesting results on dust fluxes in the close vicinity of the
Sun. Interesting in the context of this paper are the works on the second
orbit of PSP with dust measurements between ca. 0.16 and 0.6 AU. Szalay et al.
(2020) showed that the observations during the second solar encounter could be
explained with particles that form as collision fragments near the Sun and
then are ejected by the radiation pressure force. Mann & Czechowski (2021)
showed that the same fluxes could be explained with a model that combines
collisional production of dust particles and their dynamics influenced by
gravity, radiation pressure and Lorentz force; the latter was found to have
only a small effect on the particles that were observed with PSP during the
second orbit.

In this paper we shall first present and discuss, in section 2, the waveform
data recorded by the instrument and the specificities of Solar Orbiter as a
dust detector. Then in section 3 we focus on the statistical study of the time
repartition and of the voltage amplitudes of the hits recorded. Finally, in a
last section, we build on this statistical study to determine the flux of the
dust population observed, and compare our results to the one obtained by other
missions and to theoretical predictions.

##  2 Dust measurements with RPW

The RPW instrument aboard Solar Orbiter, a complete description of which is
given by Maksimovic, M. et al. (2020), is designed to measure and analyze the
electric field fluctuations from near-DC to 16 MHz and magnetic fluctuations
from several Hz to 0.5 MHz. In this article we are mainly interested by the
electric waveforms provided by the time domain sampler (TDS) subsystem of RPW.
TDS provides digitized snapshots of the voltage measured between the terminals
of two of the spacecraft electric antennas (dipole mode) or between one the
spacecraft electric antennas and the spacecraft ground (monopole mode). The
waveforms used in this article were sampled at a rate of 262.1 kHz, after
being processed by an analog high-pass filter with a cutoff around 100 Hz.

Figure 1: Six examples of TDS snapshots showing impulsive signals interpreted
as due to dust impacts onto the spacecraft. Signals recorded on the three
different channels are represented in different colors. The left column shows
signals recorded in monopole mode (SE1 mode) whereas the right column shows
signals recorded in dipole on the two first channels and monopole on the third
one (XLD1 mode).

Figure 1 shows examples of impulsive signals recorded by TDS that we interpret
as due to dust impacts. The examples shown here were selected from the so-
called triggered snapshots which were flagged by the on-board detection
algorithm as dust probable impacts. See section 3.1 for the description of the
algorithm.

The left column shows events recorded in SE1 mode (for which TDS samples three
monopoles V1, V2 and V3) whereas the right column shows events recorded in
XLD1 mode (two channels measure dipole voltages V1-V3 and V2-V1, and the third
channel is linked to the monopole V2). Each channel (CH1, CH2, and CH3)
represents voltage difference measured between individual antennas (V1, V2,
and V3) for dipoles or between one antenna and the spacecraft body (SC) for
monopoles. It should be noted that presented data are not corrected for
transfer function (the existence of “overshoots” in these data is probably
artificial). Examples shown on the lower panels of left column are very
typical of signals detected at the terminal of a monopole antenna by electron
collection, similar to the one detected by STEREO/WAVES for instance.

###  2.1 Voltage pulses and their link to mass and velocity of impacting dust
grains

Before discussing these pictures, let us remind the mechanism through which
voltage pulses are thought to be produced. First, a dust grain impacts the
spacecraft body and expels from it some material, part of which is ionized.
The amount of free electric charge $Q>0$ in the (overall neutral) cloud of
expelled material is a function of the mass $m$ of the impacting grain and of
the relative velocity $v$ of the impactor with respect to the target. The
measurement of $Q$ after an impact in a dedicated collector, together with an
independent measurement of $v$ in order to deduce the mass $m$ of the
impactor, is the general operating principle of dust impact ionisation
detectors (Auer, 2001). Experiments show that the function $Q(m,v)$ follows
the general scaling

$Q(m,v)\simeq Amv^{\alpha}$ (1)

where $m$ is expressed in kg and $v$ in km.s-1. Since the parameters $A$ and
$\alpha$ quite strongly depends on the impacted material (Collette et al.,
2014), it is of course preferable for the charge yield of the collector
material as a function of $m$ and $v$ to be measured on ground. In the
absence, for the moment, of such measurements for Solar Orbiter’s surface
material, we will have to use approximate values for $A$ and $\alpha$. Dietzel
et al. (1973) or McBride & McDonnell (1999)) suggest the use of $A\sim 1$ and
$\alpha\sim 3.5$, which is a rather high charge yield – we shall discuss this
point when evaluating the size of the impactors in the last section of this
article.

In the absence of a dedicated and well calibrated collector, but in the
presence of electric antennas operated in monopole mode, one can still quite
reliably deduce the amount of charge $Q$ released during an impact, thanks to
the different dynamics of the electrons and the heavier positive charges in
the expelled cloud of ionized matter. The dynamics of the light electrons
quickly decouple from the one of the heavier positively charged matter
(Pantellini et al., 2012). The electrons are quickly collected by (or repelled
away from) the spacecraft, letting positive charges unscreened in the vicinity
of the spacecraft. For a positively charged spacecraft, it can be shown that
the combination of the effect of quick electron collection and ions getting
repelled away from the spacecraft will produce a maximal change in the
spacecraft ground potential equal, to a good approximation, to
$\delta\varphi_{sc}\simeq-Q/C_{sc}$ (here $C_{sc}$ is the spacecraft’s body
capacitance).

In monopole mode the signal recorded is
$V(t)=\Gamma\left(\varphi_{ant}(t)-\varphi_{sc}(t)\right)$, where
$\varphi_{ant}$ is the monopole antenna potential and $\Gamma$ a transfer
function accounting for the (mostly capacitive) coupling between the antenna
and the spacecraft body: $\Gamma=C_{ant}/(C_{ant}+C_{stray})$ (in this formula
$C_{ant}$ is the antenna’s capacitance and $C_{stray}$ accounts for the
capacitive coupling through the preamplifier, the mechanical base of antenna
and various wires). If one assumes that $\varphi_{ant}$ is roughly constant
during the process, then the charge $Q$ produced by impact ionisation is quite
simply linked to the peak of the voltage pulse measured in monopole mode by

$Q(m,v)\simeq\frac{C_{sc}V_{peak}}{\Gamma}.$ (2)

The use of formulas (1) and (2) therefore makes it possible to link the
properties of the impacting grain to those of the measured voltage signal.

We can see, in the light of these explanations, why monopole measurements are
favourable to dust detection. The main changes induced by the impact occurs in
the spacecraft ground potential, while antennas potentials stay roughly
constant. Dipole measurements, which measure the variation of an antenna’s
potential relative to another antenna, are therefore quite insensitive to this
process. Still, signals are quite frequently observed in dipole mode, as can
be seen on the right panels of Fig.1.

For a signal to be observed in dipole mode, it must produce a signal quite
stronger on a particular antenna than on the two others. An example of such a
signal recorded on monopole mode can be seen on the top left panel of Fig.1,
with a peak amplitude on monopole V3 much larger than on V1 and V2. An
interpretation for these signals is that the impact location may be close to a
particular antenna, the potential of which would in turn undergo a much
stronger variation under electrostatic influence from the positively charged
cloud than the other monopoles (Meyer-Vernet et al., 2014). The derivation of
the charge $Q$ from dipole measurements is therefore more complicated, since
the amplitude of the voltage pulse not only depends on $Q$ but also on the
position of the impact with respect to the monopoles. An order of magnitude of
such a signal is $V_{peak}\sim\Gamma Q/(4\pi\epsilon_{0}L_{ant})$ (assuming
only one arm of the dipole sees the whole unscreened charge Q), so that the
charge in the cloud can be linked to the peak voltage in dipole mode by

$Q(m,v)\sim\frac{4\pi\epsilon_{0}L_{ant}V_{peak,dipole}}{\Gamma},$ (3)

where $L_{ant}$ is the length of an antenna. Importantly, it should be noted
that, unlike the signal observed in monopole mode (for the at least two
monopoles showing similar peak voltages), which is quite accurately linked to
the released charge $Q$ by eq.(2), eq.(3) only provides a rough order of
magnitude, since the voltage produced will in fact be very dependent on the
location of the impact. An impact occurring at equidistance from two arms of
the dipole for instance, would produce a very small signal in dipole mode,
even for an important release of charge, whereas an impact cloud expanding in
the close vicinity of a particular dipole arm could produce a signal quite
stronger than the estimation above.

On the right column of figure 1 (XLD1 mode) are shown a few examples of events
where the signal is mainly registered by a single antenna - hence not mainly
produced by the variation of the spacecraft potential, but rather by
electrostatic influence on a particular antenna. The impact must be close to
antenna V2 on cases shown on top and bottom right panels (that show identical
pulses in channels CH2 and CH3) and to antenna V1 in the middle right panel
(inverted pulses in channels CH1 and CH2).

###  2.2 Parameters for Solar Orbiter as a dust detector

Now that we have detailed the main principles through which we interpret
voltage pulses after a dust impact, we describe in this part their application
to the specific case of Solar Orbiter.

Solar Orbiter orbits the Sun along a series of roughly elliptical orbits, with
a minimum perihelion of 0.28 AU and a maximum inclination with respect to the
ecliptic above $30^{\circ}$. A description of the mission and its science
objectives can be found in Müller, D. et al. (2020).

RPW’s electric sensors are three stacer monopoles of length $L=6.5$ m and
radius $a=0.015$ m, mounted on 1 m rigid booms to separate them from the
spacecraft body, electrically biased in order to reduce variations of their
potential with respect to the plasma at low frequencies (Maksimovic, M. et
al., 2020). They are in the same plane and make angles of roughly
$120^{\circ}$ with each other. The disposition of the antennas with respect to
the spacecraft body is shown on Fig.2. Let us note, in relation with the
previous discussion on the generation of electric pulses after a dust impact,
that the three monopoles are mounted on opposite sides of the spacecraft quite
far away from each other (similarly to the case of spacecraft like WIND, MMS
or Parker Solar Probe, but unlike the cases of Voyager, Cassini or STEREO),
which implies that the effect of electrostatic influence can be very different
from an antenna to the others, and explains why signals are frequently
observed in dipole mode.

Figure 2: Geometrical configuration of RPW’s electric sensors (labelled ANT1,
2 and 3), heat shield and deployed solar panels with respect to the spacecraft
body.

The capacitance associated to each monopole base has been measured on ground.
Numbers found are $C_{b}=76.3\pm 4$ pF for the monopole V1, $C_{b}=78.9\pm 3$
pF for V2 and $C_{b}=76.2\pm 2.7$ pF for V3. The stray capacitance is to a
good approximation the sum of this base capacitance and of the preamplifier
capacitance $C_{amp}$, which, when both preamplifiers are ON, is $C_{amp}=29$
pF. Since the base capacitances are equal within measurement uncertainties, we
shall use for the three monopoles the same value of the stray capacitance,
$C_{stray}\simeq 77+29\simeq 108$ pF.

Assuming the monopoles are in vacuum and considering the spacecraft as an
infinite ground plane, one finds the capacitance of a monopole
$C_{ant}=2\pi\epsilon_{0}L/(\ln{L/a}-1)\simeq 71$ pF. The presence of solar
wind’s plasma, however, at the frequencies we are interested in (smaller than
the local plasma frequency), decrease the capacitance to values $C_{ant}\sim
2\pi\epsilon_{0}L/(\ln{L_{D}/a})$ where $L_{D}$ is the local Debye length
(Meyer-Vernet & Perche, 1989). With the Debye length varying in a range $\sim
3-10$ m along the spacecraft’s orbit, one obtains $C_{ant}\sim 55-70$ pF (the
capacitance increasing when going closer to the Sun). Using these values, the
attenuation factor can be evaluated as
$\Gamma=C_{ant}/(C_{ant}+C_{stray})\simeq 0.34-0.39$.

The isopotential surface of the spacecraft’s body has been evaluated from the
Solar Orbiter numerical model. This value can be estimated from 25.11 m2,
taking into account only the surface of the 5 satellite walls plus the heat
shield, up to 31.43 m2 when we consider the envelope including the overall
spacecraft.. The solar panels (6 panels of $2.1\text{ m}\times 1.2$ m) are, at
least their backside, isopotential to the spacecraft body. The additional
surface is then $S_{panels}=15.12$ m2 (taking only one side into account).
Therefore we can evaluate the area of the interface between the plasma and the
isopotential parts of the spacecraft to be $S_{sc}\simeq
S_{body}+S_{panels}\simeq 43.4\pm 3.2\text{ m}^{2}$. The capacitance of a
conductor of such a complex shape is difficult to estimate. An order of
magnitude estimation is given by the vacuum capacitance of the sphere of
equivalent surface $C_{sc}\sim\epsilon_{0}\sqrt{4\pi S_{sc}}\sim 210$ pF. This
number is an underestimation that does not take into account sheath effects at
the surface of the satellite between other effects, and we shall, in this
paper use a value $C_{sc}=250$ pF when this number is needed. A more
appropriate determination of $C_{sc}$ could be made using a numerical model
for Solar Orbiter and its interaction with the surrounding plasma, a work that
will be undertaken in the future.

The surface $S_{sc}$ discussed above would approximately correspond to the
dust collecting area if the dust population velocity distribution was
isotropic in the frame of the spacecraft. As we will see in the following, it
is probably not the case for the majority of the dust observed by Solar
Orbiter, the velocity of which is mostly directed toward the heat shield.
Therefore the dust collecting area to consider is strongly reduced compared
$S_{sc}$, and is rather of the order of the heat shield’s surface
$S_{col}\simeq 2.5\text{ m}\times 3.1\text{ m}\simeq 8$ m2.

Finally let us briefly discuss the relaxation time of perturbations of Solar
Orbiter’s floating potential. Linear theory gives $\tau_{sc}\simeq
C_{sc}T_{ph}/I_{e}$, where $I_{e}=en_{e}v_{e}S_{sc}$ is the electron current
onto the spacecraft isopotential surface ($e$ is the electron charge, $n_{e}$
the local electron density and $v_{e}=\sqrt{kT_{e}/(2\pi m_{e})}$, with
$T_{e}$ the local electron temperature and $m_{e}$ the electron mass).
$T_{ph}\sim 3$ V is the temperature of the photoelectrons ejected from the
spacecraft body expressed in electric potential unit. The assumption
underlying this formula is that photoelectron current from the spacecraft is
dominating over solar wind’s electron current, the spacecraft potential being
as a consequence positive (assumption justified pretty much all along Solar
Orbiter’s trajectory, putting apart short periods in the shadow of Venus).
Assuming typical variation of electron parameters in the solar wind (see e.g.
Issautier et al. (1998)) one obtains for the relaxation time $\tau_{sc}\sim
40$ $\mu$s at 1 AU and $\sim 10$ $\mu$s at 0.5 AU. These numbers have their
importance in that the estimation of the peak amplitude of the pulses observed
in monopole mode $V_{peak}\sim-Q/C_{sc}$ is strictly valid only in the case
were the rise time of the pulse $\tau_{rise}$ (controlled by the positive
charges dynamics in the vicinity of the spacecraft) isn’t large compared to
the relaxation time $\tau_{sc}$ of spacecraft electric potential
perturbations. In the opposite case where $\tau_{sc}$ is small compared to
$\tau_{rise}$ the amplitude of the signal is reduced by a factor of the order
of $\tau_{sc}/\tau_{rise}\ll 1$ (Zaslavsky, 2015). This effect will not be
taken into account in this article. A more precise study of the waveforms –
which requires a very careful calibration work – will be the subject of
forthcoming studies, and will among other things make it possible to quantify
this effect.

##  3 Statistical analysis of dust impacts

###  3.1 Impact rate and estimation of the impactors radial velocity

In this section we present the results of the analysis of the impacts voltage
pulses recorded along Solar Orbiter’s orbit from April 20th, 2020 to February
27th, 2021. For this purpose, we shall mainly use the data provided by the on-
board analysis of TDS samples through an algorithm that flags a snapshot as
being produced by a dust impact if it contains a signal impulsive enough. This
dust detection algorithm has been working with constant settings from April
20th 2020, hence the date at which we start our analysis.

The detection algorithm described in detail in (Soucek & other authors, 2021)
works as follows: the instrument takes one waveform snapshot of 16384 samples
every second. Each snapshot exceeding a minimum amplitude threshold is
processed by the TDS on-board software to calculate the maximum and median
amplitude and calculate the Fourier spectrum from the snapshot. From this
spectrum the software determines the frequency corresponding to the largest
spectral peak and the FWHM (full width at half maximum) bandwidth of this
peak. The algorithm than classifies the observed snapshots based on comparing
the ratio $R$ between the maximum and median absolute value in the snapshot
and the spectral bandwidth to predefined thresholds. Specifically, snapshots
with large $R$ and large spectral bandwidth are identified as dust impacts.
This way, the algorithm allows to identify even relatively small amplitude
dust spikes.

The outcome of this detection is then used to select the “best” wave and dust
snapshots to be transmitted to the ground and also to build statistical data
products. The key data product used here is the number of detected dust
impacts in a 16 second interval which is transmitted in short statistical data
packet every 16 seconds. Due to the fact that the detection algorithm only
examines one snapshot of 62 ms every second, the reported dust counts are much
lower than the actual number of dust impacts, but the number of detected dust
impacts can be considered directly proportional to the actual number of dust
impacts.

Since some impulsive signals may be erroneously taken for dust (e.g. solitary
waves, (Vaverka et al., 2018) and various spacecraft generated effects), the
dataset has been cleaned by removing all periods of fast sampling at 524 kHz,
all measurements influenced by active sweeps performed by the BIAS subsystem
of RPW and several days (in particular during commissioning) where TDS was
configured to non-standard operation modes. We also removed the Venus flyby
interval on December 27th, 2020 when TDS detected numerous solitary waves and
counted them as dust impacts.

We considered this TDS dust data product on a timescale of 1h, and computed
the impact rate for each hour by dividing the number of snapshots flagged as
dust by the total number of snapshots recorded during this hour multiplied by
the duration of one snapshot ($\Delta t=62$ ms): impact rate
$R=N_{impact}/(N_{snapshots}\Delta t)$.

Figure 3: Upper panel: raw cumulative dust count as a function of time since
April, 1st, 2020. By raw, it is meant that it is the integer number of impact
provided by the detection algorithm, uncorrected for the instrument’s duty
cycle. Middle panel: Impact rate as a function of time since April, 1st, 2020.
Each point correspond to a 2 days time interval. The error bars are computed
from $\sqrt{N}/\Delta t$, where $\Delta t$ is the time covered by TDS
measurements on the 2 days considered, and $N$ the number of impacts during
these 2 days. The yellow areas correspond to time periods on which the number
of occurrence distributions of Fig.5 have been computed. The green areas
correspond to times periods on which the amplitude distributions shown on the
right panel of Fig.6 have been computed. Lower panel: Distance from the
spacecraft to the Sun in astronomical units, as a function of time since
April, 1st, 2020.

Fig.3 shows the evolution of the impact rates with time. The upper panel shows
the raw (i.e. not corrected for the duty cycle) cumulative impact number,
showing that the total number of impact detected by the algorithm during the
period of our study is 2758. One can also notice several small data gaps
corresponding to periods during which the instrument is switched off.

The middle panel shows the impact rate as a function of time, showing an
increase of the flux with decreasing distance to the Sun, a general behaviour
in agreement with remote and in-situ measurements from Helios (Leinert et al.,
1981) or Parker Solar Probe (Szalay et al., 2020; Page et al., 2020).

Figure 4: Left panel: impact rate as a function of radial distance. The black
points show fluxes recorded when the spacecraft is moving toward the Sun, the
red points when moving outward from the Sun. Each point is computed by
averaging the impact rate data on 50 distance intervals linearly spaced
between 0.5 and 1 AU. Error bars show one standard deviation error on the
computation of the mean. Middle panel : radial component of the dust velocity
$V_{r,dust}$ computed from eq.(4). Error bars are computed by propagating
errors on the impact rates shown on the left panel. Right panel : histogram of
the obtained values of $V_{r,dust}$.

Fig.4 illustrates the evolution of the impact rate with distance from the Sun.
On the left panel of this figure, the impact rates measured when the
spacecraft is going toward the Sun (spacecraft radial velocity $v_{r,sc}<0$)
are separated from the ones measured when the spacecraft is going outward
($v_{r,sc}>0$). It can be seen that the impact rate is in average slightly
larger when $v_{r,sc}<0$ than when $v_{r,sc}>0$. This is consistent with the
dust population measured having a mean velocity directed outward from the Sun.

One can use this difference in the impact rates to obtain a first estimation
of the radial velocity $v_{r,dust}$ of the dust population: assuming that the
velocity and fluxes are function of the distance to the Sun only (neglecting
all time variability) and neglecting – for an order of magnitude estimation –
the effect of tangential velocities, it can easily be shown that

$v_{r,dust}\sim\frac{R_{in}+R_{out}}{R_{in}-R_{out}}\left|v_{r,sc}\right|$ (4)

where $v_{r,sc}$ is the radial velocity of the spacecraft, $R_{in}$ the impact
rate when the spacecraft is going toward the Sun and $R_{out}$ when going
outward. The middle panel of Fig.4 shows the result of applying formula (4)
with the impact rates $R_{in}$ and $R_{out}$ shown on the left panel. The
error bars are very large but the mean value obtained is reasonably constant
with radial distance. The right panel shows a histogram of the values of
$v_{r,dust}$ obtained, with a peak value at $v_{r,dust}=30$ km.s-1 and an
averaged value around $50$ km.s-1. The large errorbars, the variations of the
measured fluxes and the use of a simple model let us only hope for an order of
magnitude estimation of course, but this value is consistent with expectations
for small particles on hyperbolic orbits, and to results from numerical
simulations of particles dynamics discussed in the last section of this paper.

Fig. 5 shows the distribution of the number of impacts recorded during time
intervals of 6 hours, for three different values of the impact rate. The three
intervals on which these number of occurrence distributions have been computed
are highlighted in yellow on Fig.3. They correspond to distances to the Sun
$\sim 0.7$ AU (left panel, mean impact rate $\sim 140$ day-1), $\sim 1$ AU
(middle panel, mean impact rate $\sim 50$ day-1) and $\sim 0.5$ AU (right
panel, mean impact rate $\sim 350$ day-1). The comparison with Poisson
distribution, over-plotted in red, shows a very good agreement consistent with
the data being due to independent events at roughly constant rates, as
expected for interplanetary dust impacts.

Figure 5: Distribution of number of impacts per time intervals of 6 hours, for
three different periods of time corresponding to different mean impact rates.
The red curve shows the Poisson distribution calculated from the mean impact
rate observed in each of these periods of time. Errorbars on the histograms
values are $\sqrt{N_{bin}}$, with $N_{bin}$ the number of counts in the
histogram bin.

###  3.2 Peak voltages distribution and power-law index of the impactor’s
cumulative mass flux

In order to further characterize the population of dust grains impacting Solar
Orbiter, we look at the distribution of voltages measured in monopole mode. To
this purpose we look at the snapshots dataset, which does not include all of
the dust impacts detected by the onboard algorithm on which the results of the
previous section are based. We shall assume that the subset of triggered
snapshots is a random sample from the ensemble of all the recorded dust impact
signals, and therefore that both share the same statistical properties – or
since this can’t be exactly true, we shall assume that selection bias are
small and do not impact importantly the voltage amplitude statistics.

Measurements in monopole mode, as discussed in section 2, are required in
order to properly (as unambiguously as possible) link the peak amplitude of
the pulse to the charge generated by impact ionisation. Unfortunately they
are, concerning monopoles V1 and V3, only active during a small fraction of
the mission time, i.e. mainly during two periods, from May 30th to June 8th
(185 dust snapshots telemetered) and from July 27th to August 13th (103 dust
snapshots). Things are different for monopole V2, which is quite routinely
operated with 934 dust snapshots telemetered from April 1st to November 1st of
2020. Hence the particular focus on monopole V2 in the following.

Figure 6: Normalized density (i.e. number of dust detected per unit voltage
interval) of voltage peaks in monopole mode. The left panel shows the
distribution of the voltage peaks on all monopoles from Apr. 1st to Nov. 30th,
2020. Different colors accounts for different monopoles: V1 (blue), V2/SE1
(red), V3 (green), V2/XLD1 (black). The middle panel shows the distribution of
peak amplitudes on monopole V2 during two different periods (highlighted in
green on fig.3). Red points correspond to the high impact rate period close to
the perihelion (May 13th – Jun 8th), black points to the low impact rate
period around the aphelion (Sep 17th – Nov 3nd). The right panel shows the
result of the least square fitting of all the data recorded on monopole V2
from Apr. 1st to Nov. 30th, 2020. The slope of the red curve is $-1.34\pm
0.07$. On all these figures, errorbars are computed from
$\sqrt{N_{bin}}/(\Delta V_{bin}N_{tot})$, where $N_{bin}$ is the number of
counts in the bin, $\Delta V_{bin}$ the bin width in mV and $N_{tot}$ the
total number of events from which the histogram is computed.

Fig.6 shows the normalized distributions of peak voltage associated to dust
impacts for different monopoles, and different time periods. We can see that
the peak voltage distribution has a power-law behaviour. The left panel shows
distributions computed on the whole period Apr. 1st - Nov. 31st, with
different colors corresponding to peak voltages at the terminals of different
monopoles. One can see that voltage distributions are similar on all monopoles
- which is to be expected since in average all monopole should react in the
same manner to dust impacts. The results of least-square fitting these
histograms with power laws is presented in table 1, where it can be seen that
power-law exponents are, within uncertainties, equal from a monopole to
another, around $-4/3$.

Table 1: Power law indices of the peak voltage distributions. The uncertainties show 95$\%$ confidence interval on the linear regression coefficient. Monopole | Time interval | Number of events | Power-law index
---|---|---|---
V1 (SE1 mode) | Apr. 1st – Nov. 31st | 328 | $-1.34\pm 0.14$
V2 (SE1 mode) | Apr. 1st – Nov. 31st | 328 | $-1.34\pm 0.14$
V2 (XLD1 mode) | Apr. 1st – Nov. 31st | 934 | $-1.37\pm 0.10$
V3 (SE1 mode) | Apr. 1st – Nov. 31st | 328 | $-1.36\pm 0.11$
V2 (SE1 mode) | May 30th – Jun. 8th (Perihelion) | 185 | $-1.37\pm 0.19$
V2 (XLD1 mode) | Sep. 17th – Nov. 2nd (Aphelion) | 161 | $-1.20\pm 0.17$
V2 (SE1 and XLD1 modes) | Apr. 1st – Nov. 31st | 1262 | $-1.34\pm 0.07$

The middle panel of Fig.6 compares the peak amplitude recorded on monopole V2
at different distances from the Sun. The limited number of impacts on which
such a comparison can be made imply quite important uncertainties on the
values of the distribution. The two last lines of table 1 show the results of
linear fitting for these distributions, showing a slope a bit steeper at the
perihelion than at the aphelion. This difference being within the
uncertainties, it is hard to conclude on this result; one should wait for more
statistics to see if this trend is confirmed.

The right panel of Fig.6 shows the distributions of all impacts recorded on
monopole V2 on the whole time of our voltage amplitude analysis, i.e. from
Apr. 1st to Nov. 31st, 2020, and the corresponding power-law least-square fit,
with index $-1.34\pm 0.07$.

From these observations it seems reasonable to approximate the rate of
observation of signals having peak voltages between $V_{peak}$ and
$V_{peak}+dV_{peak}$ by $dR=g(V_{peak})dV_{peak}$, with

$g(V_{peak})=g(V_{0})\left(\frac{V_{peak}}{V_{0}}\right)^{-a}$ (5)

where $a\simeq 1.34$ and $V_{0}$ an arbitrary voltage in the range where the
power-law behaviour applies.

An interesting result, of course, would consist in the derivation from these
data of an information on the mass distribution of the impacting dust
particles. We have seen in section 2.1 that the released charge $Q$, and hence
the peak voltage $V_{peak}$ is linked to the mass, but also to the velocity of
the impacting particle, and we do not have an independent measurement of the
latter for each of the impacts. Therefore one can only make inferences on the
mass distribution by assuming the existence of a relationship $v(m)$ between
the mass of the particle and its velocity with respect to the spacecraft. If
such a relationship exists, then the function $V_{peak}(m,v)$ becomes a
function of $m$ only and – under the assumption that two different values of
$m$ cannot produce a peak voltage of the same amplitude, i.e. that the
function $V_{peak}(m)$ is bijective on the observed mass interval – the mass
distribution of the impactors $f(m)$ can be directly linked to the measured
distribution $g(V_{peak})$ by

$f(m)=g(V_{peak})\left|\frac{dV_{peak}}{dm}\right|.$ (6)

For a first order estimation, one could assume that the impact velocity is
independent of the mass on the given mass range. Then, according eqs.(1) and
(2),

$V_{peak}\simeq\frac{\Gamma}{C_{sc}}Amv^{\alpha}$ (7)

is a linear function of $m$, and the mass distribution trivially has the same
power-law shape than the distribution of voltage peaks. The cumulative mass
flux of particles of mass larger than $m$ onto the spacecraft (defined as
$F(m)=\int_{m}^{+\infty}f(m^{\prime})dm^{\prime}$) would then be given by

$F(m)=F(m_{0})\left(\frac{m}{m_{0}}\right)^{-\delta}$ (8)

where $\delta=a-1\simeq 0.34$ and $F(m_{0})$ is the cumulative flux of
particles of mass larger than $m_{0}$ (that may depend on the distance to the
Sun).

Let us note that this estimation of the power-law index $\delta$ of the
cumulative mass flux in the observed mass range must likely be an
underestimation, since if the velocity is an increasing function of $m$ (which
is probably the case in the observed mass range, see next section), $f(m)$
will decrease with a steeper slope than $g(V_{peak})$. This can easily be seen
from eq.(6), taking for instance $V_{peak}\propto m^{1+\varepsilon}$ with
$\varepsilon>0$. One would then obtain for $f(m)$ a power law index
$a+(a-1)\varepsilon$ which is always larger than $a$ (if $a>1$ which is the
case here). Therefore, even without any precise knowledge of the function
$v(m)$, but under the assumption that this is an increasing function of $m$ in
the observed mass interval, it is possible to derive from these observations
of peak voltages a lower bound for the power law index of the cumulative mass
flux $\delta\gtrsim 0.34$.

To conclude this part let us note that a more detailed derivation of the mass
distribution of the particles could be obtained from these measurements by
computing the function $v(m)$ from numerical simulations, with proper
assumptions on initial conditions and dust $\beta$ parameter. Since this
function may depend on the distance from the Sun (which may explain the
possible change of the power law index of the voltage distributions from
perihelion to aphelion), this study may also require some time to accumulate
more statistics and be able to construct not too noisy distributions of
voltages at different distances from the Sun. Such a work is beyond the scope
of this first results paper, but it offers an interesting perspective for a
future study.

##  4 Estimation of the $\beta$-meteoroids flux and comparison to models and
results from other missions

We now compare the observed impact rates to a dust flux model that bases on a
number of assumptions. The existence of dust in the inner solar system can be
inferred from the brightness of the Zodiacal light and the F-corona which show
that dust in the approximate 1 to 100 micrometer size range forms a cloud with
overall cylindrical symmetry about an axis through the center of the Sun,
perpendicular to the ecliptic (cf. Mann et al. (2014)). The size distribution
at 1 AU is estimated from a number of different in-situ observations and
described in the interplanetary dust flux model (Gruen et al., 1985). The
majority of dust grains form by fragmentation from comets, asteroids and their
fragment grains. The larger grains are in bound orbit about the sun; as a
result of the Poynting – Robertson effect they lose orbital energy and angular
momentum and fall into the Sun after time scales of the order of $10^{5}$
years. However, the majority of the Zodiacal dust is within shorter time
fragmented by collision with other dust grains, the smaller fragments leave
the inner solar system and collision production from larger grains is needed
to replenish the dust cloud (Mann et al., 2004). The dust with sizes smaller
than roughly a micrometer experiences a larger radiation pressure force which
is directed outward. If the radiation pressure to gravity ratio, often denoted
as $\beta$ is sufficiently high, the dust can be ejected outward. Those grains
in hyperbolic orbits are often denoted as $\beta$-meteoroids those in in bound
orbits as $\alpha$-meteoroids (Zook & Berg, 1975). Assuming that the larger
dust grain that is fragmented moves in a circular orbit, its fragment attains
a hyperbolic orbit if the radiation pressure to gravity ratio $\beta$ exceeds
0.5. Based on light scattering models for different assumed dust compositions
(Wilck & Mann, 1996), this is the case for dust with sizes less than few 100
nm.

###  4.1 Mass of the impactors

The mass of the impactors can be estimated from the voltage observed, using
eq.(7) and spacecraft parameters from section 2.2. For this one needs an
estimation of the velocity of the impactors and of the charge yield of the
impacted material. For the particles velocity, we could have used the order of
magnitude obtained from the observations of impact rates in direction forward
and backward with respect to the Sun. But, as pointed out in section 3.1 this
is only a rough estimation, associated to large uncertainties. Therefore we
chose to rely in this section on estimations of dust particles velocities from
numerical simulations of dust dynamics in the interplanetary medium. The
simulation we use takes into account gravitational and radiation pressure
forces, but not the electromagnetic force (that should not be dominant for
particles of sizes $\gtrsim 40$ nm). Initial conditions and values of $\beta$
are chosen consistently with observations of dust from asteroidal origin.

Figure 7: Left panel : relative velocity with respect to Solar Orbiter as a
function of particle mass. The solid line shows the average along the
spacecraft trajectory, the upper dahsed line shows the impact velocity at the
perihelion and the lower dashed line the impact velocity at the aphelion.
Right panel : expected peak voltage as a function of the mass of the impactor,
calculated using values of impact velocities shown on the left panel. The
solid line shows the peak voltage for the averaged velocity (solid line from
left panel) and dashed lines for the perihelion and apehlion impact
velocities. The black curves assume a high charge yield with parameters
$A=0.7$ and $\alpha=3.5$ and the red curves a lower charge yield with
parameters $A=2.5\times 10^{-3}$ and $\alpha=4.5$. The points show the values
of the velocity/peak voltage obtained from 4 numerical simulations,
corresponding to size ranges $40-75$ nm, $75-100$ nm, $100-140$ nm and
$140-200$ nm. The points are placed at the middle of the mass interval for
each of the 4 simulated size ranges. The horizontal green line shows the
detection threshold $\sim 5$ mV.

The charge yield of the impacted material is another unknown of our study,
which of course plays an important role in determining the mass of the
impactors. Fig.7 shows the impact velocities from the numerical simulations
and the expected peak voltage for two examples of charge yield. The dark area
on the right panel shows the region of expected voltage peaks as a function of
the mass of the impactor for parameters $A=0.7$ and $\alpha=3.5$ from McBride
& McDonnell (1999). The red area is obtained for parameters $A=2.5\times
10^{-3}$ and $\alpha=4.5$, a quite lower charge yield, corresponding to
materials like germanium-coated Kapton, or solar cells and MLI (multi-layer
thermal insulation) from STEREO’s spacecraft, the charge yield of which were
measured on ground and can be found in table 1 of Collette et al. (2014).

Fig. 7 shows that particles in the size range $40-75$ nm, regardless of the
charge yield parameters used, are too small and not fast enough to produce
measurable signals. Grains with sizes $75-100$ nm should produce (for the
larger of them) signals above the detection threshold in the high charge yield
case, but not in the low charge yield one; grains with sizes $\gtrsim 100$ nm
finally, should produce measurable signals regardless of the precise charge
yield. An estimation of the mass of the smallest particles detected (we
consider a threshold voltage of 5 mV) from this figure gives $m\simeq
7.4\times 10^{-18}$ kg (high charge yield) and $m\simeq 1.2\times 10^{-17}$ kg
(low charge yield), corresponding to sizes (assuming, as in the whole of this
paper, a mass density $\rho=2.5$ g.cm-3 and grains of spherical shapes for
size-mass conversions) $r\simeq 89$ nm and $r\simeq 103$ nm respectively – one
can quite confidently affirm that the smallest grains detected have sizes
around 100 nm.

The curves for different charge yields diverge when masses are increased, and
for signals of amplitude e.g. $30$ mV, one have sizes $r\simeq 107$ nm (high
yield) and $r\simeq 150$ nm (low yield), a larger mismatch of course, showing
the necessity of ground measurement if one wants to reach a better mass
calibration on the whole voltage interval. That being said, the power-law
decrease of the peak voltages distribution shown in the previous section
implies that the fluxes are dominated by impact from small grains, so that the
determination of the mass of impactors producing high amplitudes peaks is less
critical for our study.

This discussion of particles masses is based on the measurement of the voltage
pulses in monopole mode, which, as was discussed previously, are the most
reliable when it comes to estimating the charge released by impact ionisation,
and therefore the dust particles masses. However, the dust detection algorithm
from which fluxes are computed works on a TDS channel that is operated in
dipole mode most of the time – when the instrument is on XLD1 mode. The
discussion above, and the curve presented on Fig.7 stays mainly relevant to
this case as long as the signal produced is proportional to the charge
released $Q$. According to the estimation given by eq.(3) this should be the
case for dipole measurements when considering a large enough number of hits.
Assuming $V_{dipole}=\Gamma Q/4\pi\epsilon_{0}L_{ant}$, the smallest mass
measured in dipole would be a factor $4\pi\epsilon_{0}L_{ant}/C_{sc}\sim 3$
larger (and therefore the sizes a factor $\sim 1.4$ larger). But the precise
factor is complicated to evaluate, and could be closer to 1 in average since
observations of waveforms shows that differences in peak voltage from a
monopole to another is often of the order of magnitude of the peak voltage
itself. This provides a clear motivation for for developing a quantitative
theoretical modeling of the generation of signals generation in dipole mode,
at least on a statistical basis.

###  4.2 Flux of $\beta$-meteoroids, comparison to predictions and
measurements from other spacecraft

Fig.8 presents the impact rates averaged over 2 days, already presented on
Fig.3 of this article, together with impact rates computed for three models of
the beta-meteoroid flux. The green line is the most simple model, with the
impact rate given by

$R=F_{1AU}S_{col}\left(\frac{r}{\text{1
AU}}\right)^{-2}\frac{v_{impact}}{v_{\beta}}$ (9)

where $S_{col}=8$ m2 is the collection area (taken equal to the heat shield
surface, cf. section 2.2),
$v_{impact}=\left|\left|v_{\beta}-v_{sc}\right|\right|$ is the relative speed
between the spacecraft and the $\beta$-meteoroids, the velocity of which is
taken radial and constant $v_{\beta}=50$ km.s-1. $F_{\beta,1AU}$ is the flux
of particles in the detection range, which is to a good approximation equal to
the cumulative flux of particles larger than $\sim 100$ nm. The $1/r^{2}$
scaling of the $\beta$-meteoroids flux is a consequence of mass conservation
(and of their production rate by fragmentation of larger particles being
negligible at the distances from the Sun at which the spacecraft orbits). A
fit of the data with eq.(9) gives a value of the cumulative flux
$F_{1AU}\simeq 8\times 10^{-5}$ m-2s-1.

Figure 8: Dust impact rate onto the spacecraft - the dots show the same data
as the one presented on fig.3. The green curve shows the expected impact rate
for beta meteoroids having a constant outward velocity $v_{\beta}=50$ km/s,
and a flux density $F_{1AU}=8\times 10^{-5}$ m-2s-1 at 1 AU. The red curve
shows the flux from eq.(10) with $F_{1AU}=8\times 10^{-5}$ m-2s-1 and
$\alpha\delta=1.3$. The black curve shows the prediction from numerical
simulation based on a model of production of dust by collisional
fragmentation. The collecting area is assumed to be equal to the heat shield’s
surface, $S_{col}=8$ m2.

This simple model provides, as can be seen on the figure, a fairly good
agreement with the data, although it seems to systematically underestimate the
high impact rate values around the perihelion. A reason for that can be seen
from the left panel of Fig.7: the mass of an impactor producing a given peak
voltage will be smaller at the perihelion than at the aphelion (because the
spacecraft’s velocity with in the Sun’s frame is larger at the perihelion that
at the aphelion). Since smaller particles have higher fluxes, an increase in
the rate of measurable signals is expected close to the Sun as compared to the
previous simple model.

This effect of decrease of the dust particles mass measured when approaching
the Sun can be quantified by assuming a cumulative mass flux varying in power
law with an exponent $\delta$, as written in previous section’s eq.(8). The
impact rate is then found to be

$R=F_{1AU}S_{col}\left(\frac{r}{\text{1
AU}}\right)^{-2}\frac{v_{impact}}{v_{\beta}}\left(\frac{v_{impact}}{v_{impact}(\text{1
AU})}\right)^{\alpha\delta}$ (10)

where $F_{1AU}$ is (as previously) the cumulative flux of particles above the
detection threshold at 1 AU, and the factor $(v_{impact}/v_{impact}(\text{1
AU}))^{\alpha\delta}$ accounts for the variation of the mass of the detected
impactors with the impact velocity.

One can fit eq.(10) to the data in order to obtain the product $\alpha\delta$.
The resulting curve is shown in red on Fig.8. It shows a better agreement with
data than the previous model, and fits quite well the high impact rates
observed at the perihelion. The value obtained is $\alpha\delta=1.3$, which,
considering values of $\alpha=3.5-4.5$, provides an estimation of the power-
law index of the cumulative mass distribution of the impactors
$\delta=0.29-0.37$. This value – obtained from a dataset including no voltage
measurements but only impact counts per unit time – is similar to the one
derived in the previous section by fitting peak voltage distributions,
indicating that the estimation of $\delta$ in RPW’s detection range seems
quite robust. Let us note finally that, assuming the measurement of $\delta$
from voltage distributions to be reliable enough, one could use this
estimation of $\alpha\delta$ to independently estimate the power-law index
$\alpha$ of the charge yield for Solar Orbiter material. We would then obtain
$\alpha\simeq 1.3/0.34\simeq 3.9$.

The black curve on Fig.8, finally, shows the prediction of impacts onto the
spacecraft from a model of production of small dust grains by collisional
fragmentation. This model assumes that the parent bodies move in Keplerian
orbits within the circumsolar dust disk. Their mass distribution is a modified
version of the interplanetary dust flux model (Gruen et al., 1985). The size
distribution of the collision fragments are described based on models by
Tielens et al. (1994) and Jones et al. (1996). It describes the fragmentation
and partial vaporization of a target and projectile composed of a certain dust
material. The vaporized and fragmented mass of the target are proportional to
the projectile mass and to the velocity – and material-dependent coefficients.
The mass distribution of fragments is of the form $m^{-0.76}$, with the
largest fragment mass specified as some (collision-velocity dependent)
fraction of the target mass. A brief description of the collision model is
given in Mann & Czechowski (2005). The derivation of dust fluxes is described
in Mann & Czechowski (2021). They are obtained from the dust trajectories
under the influence of gravity and radiation pressure (since the Lorentz force
does not have a strong influence for the considered dust sizes). The same
trajectories were used to produce Fig.7 of this article. The black curve shows
the prediction from this model of the number of particles comprised between
100 nm and 200 nm impacting a surface $S_{col}=8$ m2 per day. The curve
obtained from the model considers a constant minimal size of detected dust at
$100$ nm all along the trajectory. It does not include the effect of variation
of minimal mass detected discussed previously. It fits, without varying any
free parameter, well the high impact rates at perihelion but quite
overestimates the low fluxes period - a result that would tend to indicate
that the size of the particles detected is closer to 100 nm at the perihelion
and probably a bit larger at aphelion.

Some effects that can play a role on the derivation of the particle flux from
the observed impact rates have not been taken into account in this first
study. They include a better model for the collection surface and its possible
variation with distance to the Sun (the direction of dust’s velocity in the
spacecraft frame varying along the trajectory), although given the mostly
cubic shape of the spacecraft this effect is not expected to be very
important. A possible difference in charge yield for an impact on the heat
shield surface and one of the other 5 spacecraft walls could also have some
importance.

The result that we obtained for the $\beta$-meteoroid flux is similar
(although slightly higher) than the one derived using STEREO/WAVES by
Zaslavsky et al. (2012), of $F_{\beta}\simeq 1-6\times 10^{-5}$ m-2s-1 at 1
AU. Similar also to the measurement with Solar Probe Plus FIELD instrument
$F_{\beta}\simeq 3-7\times 10^{-5}$ m-2s-1 interpolated at 1 AU (Szalay et
al., 2020). Similar, finally, to theoretical expectations, as shown in this
paper.

To conclude, let us note that, if the presented data can be well described
with a flux of $\beta$-meteoroids, it lacks, in comparison with observations
from STEREO (Zaslavsky et al., 2012; Belheouane et al., 2012) or Wind
(Malaspina et al., 2014) an observed flux of interstellar dust. The latter
studies indeed showed a noticeable component of the impact rate modulated
along the solar apex direction, that is not observed with Solar Orbiter RPW.
This lack of an apparent interstellar dust component in the data is puzzling.
It could be explained by a deflection of the interstellar dust grains in the
solar magnetic field, and the consequent depletion of their flux inside 1 AU –
which is expected for grains of small size (Mann, 2010). This is a point which
deserves further studies.

##  5 Conclusions

  1. 1.

The analysis of the first data from Solar Orbiter Radio and Plasma Wave
instrument shows this instrument to be a quite reliable dust detector for dust
grains in the size range $\gtrsim$ 100 nm. Fluxes of particles derived are
consistent with previous observations in this size range and with theoretical
prediction from models of dust production by collisional fragmentation.

  2. 2.

The analysis of the difference in impact rates when the spacecraft’s velocity
vector is directed in the sunward or anti-sunward direction is shown to be
able to provide a direct measurement of the order of magnitude of dust grains
radial velocities $v_{r,dust}\sim 50$ km.s-1 consistent with theoretical
predictions in the observed size range.

  3. 3.

The analysis of voltage distribution in monopole mode, and the analysis of the
impact rates taking into account a variation of the mass of the smallest
grains detected as a function of the impact velocity provide two independent
methods for estimating the power law index $\delta$ of the cumulative mass
flux of particles in our detection range. These two methods consistently
provide a result $\delta\simeq 1/3$.

  4. 4.

These first results are on overall very promising. Still, numerous works
remain to be done, in particular a modeling of the signal generation mechanism
in monopole mode that would including the effect of variation of the floating
potential relaxation time with the local plasma parameters, but also models of
signal generation in dipole mode. Such works, together with a more precise
modeling of grain’s velocity and additional statistics could make it possible
to derive more information on the dust cumulative mass flux from the peak
voltage distributions.

  5. 5.

Solar Orbiter will reach perihelion close to 0.3 AU in spring 2022. The impact
rate should be $\sim$ 3 times (without taking the mass detection threshold
effect) larger at this point than at the perihelion studied in this article.
Solar Orbiter’s orbit will also reach increasingly higher latitudes in the
years to come, and will provide the first in-situ exploration of the dust
cloud out of the ecliptic. These perspectives are very promising, and these
first results show that RPW will have the capabilities to provide scientific
return from these opportunities.

###### Acknowledgements.

IM has been supported by the Research Council of Norway (grant no. 262941). JV
was supported by the Czech Science Foundation under the project 20–13616Y. MM
and YK are supported by the Swedish National Space Agency grant 20/136. AZ, IM
and JV acknowledge discussions during the ISSI team on dust impacts at the
International Space Science Institute in Bern, Switzerland.

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