11institutetext:  Department of Physics and Technology, UiT The Arctic
University of Norway, 9037, Tromsø, Norway 22institutetext: Department of
Mathematics and Statistics, UiT The Arctic University of Norway, 9037, Tromsø,
Norway 33institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, Université de Paris, Paris, France

# Modelling Solar Orbiter Dust Detection Rates
in Inner Heliosphere as a Poisson Process

S. Kočiščák<EMAIL_ADDRESS>A. Kvammen 11 I. Mann 11 S. H. Sørbye
22 A. Theodorsen 11 A. Zaslavsky 33

(Received Month DD, YYYY; accepted Month DD, YYYY)

###### Abstract

Context. Solar Orbiter provides dust detection capability in inner
heliosphere, but estimating physical properties of detected dust from the
collected data is far from straightforward.

Aims. First, a physical model for dust collection considering a Poisson
process is formulated. Second, it is shown that dust on hyperbolic orbits is
responsible for the majority of dust detections with Solar Orbiter’s Radio and
Plasma Waves (SolO/RPW). Third, the model for dust counts is fitted to
SolO/RPW data and parameters of the dust are inferred, namely: radial
velocity, hyperbolic meteoroids predominance, and solar radiation pressure to
gravity ratio as well as uncertainties of these.

Methods. Non-parametric model fitting is used to get the difference between
inbound and outbound detection rate and dust radial velocity is thus
estimated. A hierarchical Bayesian model is formulated and applied to
available SolO/RPW data. The model uses the methodology of Integrated Nested
Laplace Approximation, estimating parameters of dust and their uncertainties.

Results. SolO/RPW dust observations can be modelled as a Poisson process in a
Bayesian framework and observations up to this date are consistent with the
hyperbolic dust model with an additional background component. Analysis
suggests a radial velocity of the hyperbolic component around $63\pm
7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ with the predominance of
hyperbolic dust about $78\pm 4\text{\,}\%$. The results are consistent with
hyperbolic meteoroids originating between $0.02\text{\,}\mathrm{A}\mathrm{U}$
and $0.1\text{\,}\mathrm{A}\mathrm{U}$ and showing substantial deceleration,
which implies effective solar radiation pressure to gravity ratio $\gtrsim
0.5$. The flux of hyperbolic component at $1\text{\,}\mathrm{A}\mathrm{U}$ is
found to be $(1.1\pm
0.2)\text{\times}{10}^{-4}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$ and the
flux of background component at $1\text{\,}\mathrm{A}\mathrm{U}$ is found to
be $(5.4\pm
1.5)\text{\times}{10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$.

###### Key Words.:

interplanetary medium – interplanetary dust – Solar Orbiter – methods:
statistical – Bayesian inference

##  1 Introduction

###  1.1 Hyperbolic dust and $\beta$-meteoroids

Among dust detected with in-situ measurements within
$1\text{\,}\mathrm{A}\mathrm{U}$, particles on unbound hyperbolic trajectories
originating in the relative vicinity of the Sun play a major role, as was
already shown in case of measurements of Solar Orbiter (Zaslavsky et al.,
2021). Most of these hyperbolic particles of sub-micron size are believed to
be so-called $\beta$-meteoroids, generated by a high radiation pressure to
gravity ratio, denoted $\beta$:

$\qquad\qquad\qquad\qquad\beta=\frac{F_{radiation}}{F_{gravity}}.$ (1)

It is clear that in the region of dust sizes $s\gg\lambda$, where $s$ denotes
the dimension of a dust particle and $\lambda$ denotes the wavelength of
incident light, $F_{radiation}$ depends on dust particle’s cross section,
while $F_{gravity}$ depends on dust grain’s volume. Hence, the smaller the
particle, the higher the $\beta$ value. Maximum of $\beta$ is therefore
reached when $s\approx\lambda$ and usually $\beta_{max}\approx 1$. Notably,
both $F_{radiation}$ and $F_{gravity}$ depend on the inverse square of
heliocentric distance, hence $\beta$ remains constant for a given particle
throughout its trajectory. Note that $F_{radiation}$ and $F_{gravity}$ are the
predominant forces for the $\beta$-meteoroids, as electromagnetic forces
become relevant for dust grains of size $s<$100\text{\,}\mathrm{n}\mathrm{m}$$
(Mann et al., 2014). Dust of size $s<$100\text{\,}\mathrm{n}\mathrm{m}$$ can
also be on an unbound trajectory due to electromagnetic forces (Czechowski &
Mann, 2021; Mann & Czechowski, 2021).

For $\beta=1$, the grain neither accelerates nor decelerates due to Solar
influence. For $\beta=0.5$, effective Solar attraction is reduced to $1/2$,
which means that a sudden change in $\beta$ from $0$ to $0.5$ will cause an
originally circular orbit to become an unbound, parabolic orbit. Particles
with $\beta\approx 1$ are created mostly in collisions of larger dust
(Dohnanyi, 1972; Zook & Berg, 1975; Grün et al., 1985). Larger particles have
very low $\beta\ll 1$ and are therefore originally on Keplerian orbits
(referred to as initial orbits hereafter), hence $\beta=0.5$ could be
considered the minimal value needed for dust to become unbound.

The population of bound ($\beta\ll 0.5$) dust particles inside
$1\text{\,}\mathrm{A}\mathrm{U}$ is notably responsible for visual
observations of zodiacal light. Their spatial density has been observed to
depend on heliocentric distance approximately as $\sim r^{-1.3}$ (Leinert et
al., 1981), which holds well down to $20\text{\,}\mathrm{R}_{\odot}$, or
$0.1\text{\,}\mathrm{A}\mathrm{U}$. Inward of that distance, they show
shallower dependence, suggesting a maximum in density somewhere inward of
$0.05\text{\,}\mathrm{A}\mathrm{U}$, or $10\text{\,}\mathrm{R}_{\odot}$
(Stenborg et al., 2021). Regions with high density of bound dust is very
likely the region of origin of $\beta$-meteoroids, as the collision rate of
bound dust depends on the square of its spatial density (Mann & Czechowski,
2005).

As $\beta$-meteoroids likely make up most of the submicron hyperbolic dust,
the two terms are almost interchangeable for the purpose of the present
discussion. The term $\beta$-meteoroids is used where radiation pressure
ejection is important and the term hyperbolic dust is used where only
trajectories of the grains are relevant.

###  1.2 Impact ionization

Detection of bound dust particles is usually done remotely, both historically
(Van de Hulst, 1947; Leinert et al., 1981) and to current date (Howard et al.,
2019; Stenborg et al., 2021) taking advantage of light scattering properties
of these particles. Detection of sub-micron particles is done mostly in-situ
(at encounter with a particle), due to their insignificant light scattering
properties and low spatial density, often taking advantage of so-called impact
ionization effect (Friichtenicht, 1962; Alexander & Bohn, 1968). Impact
ionization dust detection is a passive data gathering process carried out by
either a specialized instrument (Dietzel et al., 1973; Srama et al., 2004) or
often as a byproduct of electric (Gurnett et al., 1997; Meyer-Vernet et al.,
1986; Kurth et al., 2006; Wang et al., 2006; Zaslavsky et al., 2012; Vaverka
et al., 2018; Malaspina et al., 2020; Mozer et al., 2020; Zaslavsky et al.,
2021; Nouzk et al., 2021) or magnetic (Malaspina et al., 2022; Gasque et al.,
2022) measurements. Due to high energy density present at the impact site,
free charge is generated upon hypervelocity dust impact. The charge generated
is partially picked up by the spacecraft body and/or antennas, which results
in specific signatures in fast electric measurements (Zaslavsky, 2015; Meyer-
Vernet et al., 2017; Vaverka et al., 2017; Mann et al., 2019; Shen et al.,
2021; Rackovic Babic et al., 2022). The amount of generated charge $Q$ has
been empirically found to approximately follow the equation

$Q=Am^{\gamma}v^{\alpha},$ (2)

where in the range of impact velocities $20<v<50$ achieved in laboratory
(Friichtenicht, 1962; Dietzel et al., 1972; Shu et al., 2012), $\gamma\approx
1$ and $3\lesssim\alpha\lesssim 5$. All three parameters $A$, $\gamma$, and
$\alpha$ are dependent on both the material of the grain and the target (Grün,
1984; Grün et al., 2007; Collette et al., 2014).

###  1.3 Dust mass distribution observations

For many decades now, it has been standard to express cumulative mass
distribution of dust near $1\text{\,}\mathrm{A}\mathrm{U}$ in terms of power-
law distribution over about $20$ decades of masses, from nanodust to comets
and asteroids and above. Clearly, the distribution is an approximation and the
distribution is described with different exponent in different intervals.
However, it is often the case that a single experiment is sensitive over
several orders of magnitude and finds that the mass distribution (number of
particles with mass at least $m$) follows a power-law

$F(m)=F(m_{0})\left(\frac{m}{m_{0}}\right)^{-\delta}$ (3)

over the observed range. For example, the work of Whipple (1967) reported
$\delta\approx 1.34$ for the mass range
$10^{-8}-10^{-1}$\mathrm{k}\mathrm{g}$$ and $\delta\approx 0.51$ for the mass
range $10^{-13}-10^{-8}$\mathrm{k}\mathrm{g}$$. Compiling previous estimates
and relying on the stationarity of dust cloud, Dohnanyi (1970) reported
$\delta\approx 7/6$ for sporadic meteoroids of masses from macroscopic down to
$10^{-11}$\mathrm{k}\mathrm{g}$$ and $\delta\approx 1/2$ between
$10^{-14}$\mathrm{k}\mathrm{g}$$ and $10^{-11}$\mathrm{k}\mathrm{g}$$. Grün et
al. (1985) suggested $\delta\approx 0.8$ in the range
$10^{-21}-10^{-17}$\mathrm{k}\mathrm{g}$$, that is $\beta$-meteoroids and
smaller. Recently, Zaslavsky et al. (2021) inferred $\delta\approx 0.34$ for
Solar Orbiter’s Radio and Plasma Waves (SolO/RPW for short) dust detections of
dust of $m\gtrsim 10^{-17}$\mathrm{k}\mathrm{g}$$. It is not clear, whether
eq. (3) represents a good approximation for $\beta$-meteoroids.

###  1.4 Poisson point process

A Poisson point process is a stochastic process defined by the following
properties:

  1. 1.

Poisson distribution of counts within arbitrarily chosen bounded region (for
example a temporal interval),

  2. 2.

Statistical independence of counts within disjoint regions (temporal
intervals), and

  3. 3.

No two events can happen at the exact same location (time).

In case of dust detection in the solar system, we assume the third condition
met, as the detection rate is much lower than the detection process duration.
The first two conditions demand that a detection of a dust particle does not
influence the probability of detection of any other particle in any other time
point, for example particles do not interact, do not come in swarms, and their
reservoir cannot be depleted. All these can be assumed, as particles are
likely formed far away from the spacecraft, are sparsely distributed, and
their trajectories are uncorrelated. In that case, a Poisson process is the
simplest conceivable model and it is natural to consider dust counts as an
inhomogeneous Poisson point process, that is a Poisson process with non-
constant rate. This means that the rate depends on other parameters, in our
case distance from the Sun and spacecraft velocity. In fact, the observed
number of detections within a naturally considered temporal interval, for
example an hour or a day, is usually a low number. This implies a considerable
probability of $0$ detections, which makes the random variable of detections
per temporal interval a poor fit to often considered normal distribution,
which allows for negative numbers. Hence, Poisson distribution of counts
should be considered.

###  1.5 Bayesian inference

Inference of variable detection rate could be done by least-squares fitting of
a model onto a time-series of detections per unit time, as is often done. A
least squares fit produces the maximum likelihood estimate when the error of
the data (residuals) are normally distributed. However, detections per unit
time have a Poisson distribution, as discussed in sec. 1.4. It is possible to
obtain a maximum likelihood estimate with more careful analysis, but
uncertainty is not directly accessible and must be estimated by other means
(for example bootstrap). Adapting a procedure designed specifically to fit a
Poisson process to Poisson observations grants the resolution needed to fit a
complicated model precisely. Moreover, given we meet model assumptions, we can
make more meaningful error estimates and potentially compare competing models
in a meaningful way.

In the present work we take advantage of Bayesian inference, which is a
general procedure that works with models for observations with unknown
parameters and meets both of the above-mentioned criteria: it handles the
Poisson distribution and provides an uncertainty estimate. In this approach,
unknown parameters are regarded as random variables coming from an unknown
distribution, about which some prior information is available (in the form of
a prior belief, or prior distribution). The procedure infers the posterior
(improved) distribution of unknown parameters based on the prior distribution
and observed data. This distribution automatically carries information about
uncertainty.

###  1.6 Integrated Nested Laplace Approximation

Integrated nested Laplace approximation (Rue et al. (2009, 2017), INLA for
short) implements approximate Bayesian inference for a wide class of three-
stage hierarchical models. This class of models contains multilevel (nested)
models, spatio-temporal models, survival models, and others (Gómez-Rubio,
2020). A decisive advantage of INLA above other Bayesian methods (for example
sampling-based methods) is its computational efficiency allowing for fitting
more complicated models to more observations within available time, making it
the method of choice for the course of this work. The inference is carried out
using the R-INLA package (Martins et al., 2013; Rue et al., 2017) for the
Bayesian inference.

###  1.7 Paper structure

In section 2 we briefly introduce the Solar Orbiter mission, its dust
measurement results and the data product that we used throughout the work.
Section 3 is a discussion and analysis of observed hyperbolic dust velocity.
The fitting of dust detection rate using INLA is presented in section 4 and we
conclude our findings in section 5. Finally, an outlook for Solar Orbiter and
other missions is briefly discussed in section 6.

##  2 Solar Orbiter’s dust observations and data products

Solar Orbiter (SolO) is a spacecraft that orbits the Sun on an elliptical
trajectory. SolO underwent several gravity assists and its orbital parameters
have therefore changed several times since its launch in early 2020. As of
summer 2022, SolO has low inclination, effectively making measurements in the
ecliptic plane. Its aphelion is close to $1\text{\,}\mathrm{A}\mathrm{U}$ and
perihelion $0.3\text{\,}\mathrm{A}\mathrm{U}$, but for the majority of its
mission so far, its perihelion was close to
$0.5\text{\,}\mathrm{A}\mathrm{U}$.

Radio and Plasma Waves (RPW) is an experiment onboard SolO designed to measure
both the electric and the magnetic field in three components in a wide
frequency band, from near-DC to $16\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ in
case of the electric field (Maksimovic et al., 2020). The measurements of
electric fields, crucially for the present work, allow for detection of cosmic
dust impacts, which is one of the auxiliary scientific objectives of SolO. A
part of the data analyzed here was accessed at Solar Orbiter/RPW Investigation
(2022), specifically TDS waveform electrical data (Level 2).

In their recent work, Zaslavsky et al. (2021) described properties of SolO/RPW
as a dust detector. It has the capacity
$C\approx$250\text{\,}\mathrm{p}\mathrm{F}$$, sensitivity to pulses
$V\gtrsim$5\text{\,}\mathrm{m}\mathrm{V}$$, a collection area of
$S_{col}\approx$8\text{\,}\mathrm{m}^{2}$$, and a duty cycle of
$D\approx$6.2\text{\,}\%$$. They showed that SolO/RPW is indeed capable of
dust detections and that these could be modelled as hyperbolic dust. The
authors mostly discuss $\beta$-meteoroids, as they are likely the observed
population, but in principle the model fits to any hyperbolic dust population.
The model for the detection rate $R$ presented in aforementioned work,

$\displaystyle\begin{split}R=&\
F_{1AU}S_{col}\left(\frac{r}{$1\text{\,}\mathrm{A}\mathrm{U}$}\right)^{-2}\frac{v_{impact}}{v_{dust}}\left(\frac{v_{impact}}{v_{impact}($1\text{\,}\mathrm{A}\mathrm{U}$)}\right)^{\alpha\delta},\end{split}$
(4a) $\displaystyle\begin{split}\qquad v_{impact}=&\
|\@vec{v_{dust}}-\@vec{v_{SolO}}|,\\\ =&\
\sqrt{\left(v_{dust}^{radial}-v_{SolO}^{radial}\right)^{2}+\left(v_{dust}^{azimuthal}-v_{SolO}^{azimuthal}\right)^{2}}\end{split}$
(4b)

has three parameters: $\alpha\delta$, $F_{1AU}S_{col}$, and $v_{dust}$. Note
that, both $\alpha\delta$ and $F_{1AU}S_{col}$ are products of two quantities.
All three parameters could have a spatio-temporal dependence.

The model shows a good fit to the data with the parameters taken constant:
outward radial heliocentric speed of
$v_{dust}\approx$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ and
exponent $\alpha\delta\approx 1.3$. The value for $v_{\beta}$ was inferred by
relating the difference in detection rate in the inbound and the outbound leg
of an orbit (Zaslavsky et al., 2021) and plugged into equation (4a). A value
of $\delta\approx 0.34$, which is a dimensionless parameter in mass
distribution of detected dust grains (see eq. (3)), was inferred from the
distribution of impact pulse amplitudes. The value of $\alpha$, which stands
for power of velocity in charge-yield equation (2) is deduced from the
knowledge of $\alpha\delta$ and $\delta$. $\alpha$ is often measured in
laboratory setup and its inferred value is compatible with ground-based
measurements (Collette et al., 2014). The arameter $F_{1AU}$ stands for the
flux at $1\text{\,}\mathrm{A}\mathrm{U}$,
$F_{1AU}\approx$8\text{\times}{10}^{-5}\text{\,}\mathrm{m}^{-}\mathrm{2}\mathrm{s}^{-}\mathrm{1}$$,
see Zaslavsky et al. (2021) for details.

In addition to L2 SolO/RPW data, this work makes use of the data product
provided by Kvammen et al. (2022), which is a result of a convolutional neural
network classified time domain sampled data. It builds on a supervised
classification algorithm trained using a randomly chosen sub-sample of
manually labelled data. Its main advantage over (in itself time consuming)
visual inspection of all data automatically classified as dust is that it is
fully automatic and reasonably time consuming. Therefore it allows for not
only type 1 error correction (detection confirmation), but also for type 2
error correction implying search for dust in the vast data that was not
classified as dust by an on-board algorithm a priori. Although no supervised
classifier could get rid of human bias and error completely, this data
provides the most reliable SolO dust detection data available to this date, as
was shown in Kvammen et al. (2022). The data set consists of $4606$ dust
detections aquired over approx. $669$ hours within $457$ days. We refer to
this data as TDS/TSWF-E/CNN and it is publicly accessible, see Kvammen et al.
(2022).

##  3 Impact rates and velocities of hyperbolic dust

###  3.1 Single-particle velocities

$\beta$-meteoroids are moving mostly radially outward from their region of
origin, which is located well within $0.5\text{\,}\mathrm{A}\mathrm{U}$. Figs.
1, 2 display possible single-particle velocity profiles, see appendix. A for
underlying equations. As $\beta<0.5$ leads to finite aphelion, while
$\beta\approx 1$ requires a rather specific set of parameters, values of
$\beta\gtrsim 0.5$ are shown. Note that this choice is inconsequential and for
illustration purposes only, as we do not presuppose a $\beta$ value in further
analysis. In fact, we do not presuppose that the observed population is
$\beta$-meteoroids, though that is likely the case. The effective initial
orbit of $\beta$ grain’s parent body must lie outside of near-solar dust-free
zone, but in the region with high bound dust concentration, which restrains
the $r_{0}$ values shown. As shown in Fig. 1, radial $\beta$-meteoroid
velocities expected between $0.5\text{\,}\mathrm{A}\mathrm{U}$ and
$1\text{\,}\mathrm{A}\mathrm{U}$ are between
$30\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ and
$90\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ for the given
combinations of parameters and nearly independent of heliocentric distance
(nearly constant). Solar gravity and radiation pressure forces are central
forces, therefore the $\beta$ value does not influence azimuthal velocity as a
function of heliocentric distance, which is governed by angular momentum
conservation and initial orbit only. Azimuthal velocities of
$\beta$-meteoroids of chosen parameters between
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and $1\text{\,}\mathrm{A}\mathrm{U}$ are
therefore between $7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ and
$30\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ and decreasing $\propto
r^{-1}$, as shown in Fig. 2.

If dust detections on SolO/RPW correspond to hyperbolic dust, a difference in
detection rate $R_{in}$ vs. $R_{out}$ due to spacecraft radial velocity should
be present, as indeed is the case. It was shown by Zaslavsky et al. (2021)
that this approach allows for order of magnitude estimation of the radial
component of dust velocity
$v_{dust;rad}\approx$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$,
which is in line with expectations. In the present work, we extend the
approach to estimate continuous heliocentric distance dependent dust radial
velocities, using the data product of Kvammen et al. (2022) and taking into
account more unknown variables that influence our estimates.

Figure 1: Radial velocity profiles of $\beta$-meteoroids released by a sudden
parameter change (for example due to a collision) from an initially circular
orbit. A selection of $\beta$ values and origins ($r_{0}$) is shown. Figure 2:
Azimuthal velocity profiles of $\beta$-meteoroids released by a sudden
parameter change (for example due to a collision) from an initially circular
orbit. A selection of origins ($r_{0}$) is shown, the $\beta$ value is not
relevant.

###  3.2 Velocity estimation

A first estimate of $v_{dust;rad}$ is obtained if a model for dust collection
rate with linear dependence on relative SolO and dust velocity $v_{impact}$ is
assumed ($R\propto v_{impact}$). This corresponds to linear dependence on
volume of space scanned per unit of time only:

$v_{dust;rad}\approx\frac{R_{in}+R_{out}}{R_{in}-R_{out}}|v_{sc;rad}|,$ (5)

where $|v_{sc;rad}|$ is the absolute value of the spacecraft’s radial velocity
at a given heliocentric distance. $R_{in}$ and $R_{out}$ are obtained at the
same heliocentric distance but in inbound and outbound legs of the orbit
respectively. If, however, a different dependence of $R(v_{impact})$ is
assumed, eq. (5) changes. Assuming $R\propto v_{impact}^{q}$, a second
estimate of $v_{dust;rad}$ is obtained by

$\displaystyle\begin{split}v_{dust;rad}&=|v_{sc;rad}|\frac{\left(R_{in}^{2/q}+R_{out}^{2/q}\right)}{\left(R_{in}^{2/q}-R_{out}^{2/q}\right)}+\frac{\sqrt{\tilde{D}}}{\left(R_{in}^{2/q}-R_{out}^{2/q}\right)},\end{split}$
(6a)
$\displaystyle\begin{split}\tilde{D}&=v_{sc;rad}^{2}\left(R_{in}^{2/q}+R_{out}^{2/q}\right)^{2}-v_{sc}^{2}\left(R_{in}^{2/q}-R_{out}^{2/q}\right)^{2},\end{split}$
(6b)

where $q$ is equivalent to $1+\alpha\delta$ in eq. (4a) and $\tilde{D}$ has no
direct physical interpretation. We took the spacecraft’s azimuthal velocity
into account, but not the dust’s azimuthal velocity, as that would be a second
unknown component, for which we do not have enough information. It is
nonetheless possible to correct for assumed dust azimuthal velocity by
subtracting it from $v_{sc}$, see appendix B for derivation of eqs. (5) –
(6b).

It follows from eq. (6a) that having $R_{in}$, $R_{out}$ observed, the value
of $q>1$ will lead to higher velocity estimate than in the case of $q=1$, an
estimate that is higher by a factor of $q$ in first order approximation.
Zaslavsky et al. (2021) reported inferred velocities
$v_{dust;rad}\approx$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$
assuming $q=1$ and shown compatibility of detection rates with the model
assuming $q=1+\alpha\delta\approx 2.3$ according to eq. (4a). Assumptions met,
$v_{dust;rad}\approx$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ is
likely an underestimate. The most important assumption here is that the dust
indeed comes from a hyperbolic population.

Note that the assumption that all detected dust grains are hyperbolic is
difficult to verify or falsify. The most prominent trend in detections is that
the counts diminish with increasing heliocentric distance, which could easily
hide a plethora of other components, like bound dust or interstellar dust.
First correction to the assumption that all detections come from hyperbolic
dust stream is the assumption of having a two component field: hyperbolic dust
and sporadic (background) detections, the latter having no dependence on
spacecraft location nor velocity.

###  3.3 Velocity inference

Figure 3: SolO heliocentric distance and absolute value of its radial
velocity. Colors separate individual branches of the orbit that come with
changes in orbital parameters at gravity assists. Dashed lines correspond to
all the combinations of radial velocity and location, while solid lines denote
that SolO passed through both inbound and outbound arms for the combination.
The horizontal dashed line denotes SolO radial velocity of
$5\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$.

Assuming the dust flux is not explicitly dependent on time, the dust detection
rate is a function of orbital phase as long as the orbital parameters don’t
change. Conversely, gravity assists change orbital parameters, such as
perihelion, aphelion, and eccentricity. In the present analysis, we therefore
treat sets of orbits, delimited by gravity assists, as separate data sets. In
this way, data is aggregated for several orbits with the same orbital
parameters, but we do not aggregate incompatible measurements. For instance:
dust detection counts recorded near $0.6\text{\,}\mathrm{A}\mathrm{U}$ on
branches 2 and 3 are expected to be very different due to vastly different
SolO radial velocity, see SolO radial velocity and its heliocentric location
throughout its trajectory in Fig. 3. Minor orbital alterations between gravity
assists are neglected.

Since 29th June 2020, SolO has undergone 3 gravity assists, producing 4
distinct sets of data. The chronologically last of these so far did not
accumulate data sufficient for analysis, and crucially did not produce any
detections in the inbound part of the orbit at the time of analysis, hence the
first 3 branches were used. The difference between detection rate in inbound
and outbound leg of an orbit could be used for dust radial velocity inference,
provided that radial spacecraft velocity is not negligible, compared to dust
radial speed. Hence data represented by dashed lines in Fig. 3 are not used
for this analysis. Data with radial spacecraft velocity
$>$5\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ are not used as they
carry little information (see horizontal dashed line in Fig. 3).

Figure 4: Non-parametric fitting of detection rate observed between 29th JUN
2020 and 27th NOV 2021 in inbound and outbound part of the trajectory,
branches separated by gravity assists on 26th DEC 2020 and 8th AUG 2021. The
lines are the results of non-parametric fitting and only grayed intervals are
used for further analysis, compare with Fig. 3.

In order to estimate radially dependent velocity, we produced smooth estimates
of radially dependent detection rates, as defined in TDS/TSWF-E/CNN data set.
The fitting was done separately for inbound and outbound legs, separately for
each gravity assist delimited data set. In order to not rely on assumptions,
we decided to use non-parametric fitting, specifically Nadaraya–Watson kernel
regression (Watson, 1964; Nadaraya, 1964) with a Gaussian kernel ($FWHM\approx
2.355\sigma=$0.15\text{\,}\mathrm{A}\mathrm{U}$$). This is a simple and robust
local-averaging fitting procedure, producing $C^{\infty}$ estimates. Dust
detection counts are Poisson random variables, therefore they have variance
equal to their mean value. To evaluate uncertainty, we constructed confidence
intervals for the non-parametric fit by bootstrapping on daily dust counts:
new samples were generated with original counts as rates for new Poisson-
distributed random variables. For illustration of all three data sets and
fitted rates, see Fig. 4. For every branch, we only used the heliocentric
distance interval where both inbound and outbound legs are available, bounded
by the innermost and the outermost detection on the leg, see the grayed areas
in Fig. 4. We did not use the $r<$0.62\text{\,}\mathrm{A}\mathrm{U}$$ of
branch 2, as there are no outbound detections near
$0.6\text{\,}\mathrm{A}\mathrm{U}$ and the detections near
$0.5\text{\,}\mathrm{A}\mathrm{U}$ show little difference between the inbound
and the outbound leg. This may be due to spatial limitation of the given
model, an unlikely combination due to scarce data, a truly higher radial
velocity, or a combination of more effects.

Having smooth detection rate estimates, we produced velocity estimates using
eq. (6a), see Fig. 5. Bootstrap samples of detection rates are used to
calculate the shown percentile confidence intervals. Notably, not all the
bootstrap samples for $\lambda_{bg}=4$ allowed for a solution, which is
apparent from the jitters of the blue curve at
$r>$0.75\text{\,}\mathrm{A}\mathrm{U}$$. Confidence intervals are constructed
from the solutions that were obtained. This issue is to be expected, as
$\lambda_{bg}=4$ implies very little hyperbolic dust at
$r>$0.8\text{\,}\mathrm{A}\mathrm{U}$$ (see Fig. 4) and therefore uncertainty
in the inferred velocity. The estimate shown in Fig. 5 assumes
$\alpha\delta=1.3$ and an initial heliocentric distance of
$0.1\text{\,}\mathrm{A}\mathrm{U}$, the latter in the form of correction for
dust azimuthal velocity.

Figure 5: Velocity estimated from TDS/TSWF-E/CNN. Different colors correspond
to different assumed background rates. Shades correspond to $50\text{\,}\%$
confidence intervals and the solid lines correspond to median values for a
given heliocentric distance. Different branches of
$\lambda_{bg}=$0\text{\,}\mathrm{h}^{-1}$$ are labeled. Only parts of the
branches where $v_{SolO}>$5\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$
are shown.

To further estimate uncertainty, we included three relevant parameters (in
total):

  * •

a background (non-hyperbolic) rate $\lambda_{bg}$, corrected for by
subtraction from the estimated detection rate,

  * •

the product $\alpha\delta$ included in eq. (6a),

  * •

azimuthal dust velocity corresponding to different initial circular orbits, as
shown in Fig. 2, giving a straight-forward generalization of eq. (6a).

We have an estimate of the region of likely velocities (see Fig. 6), given
reasonable variation of free parameters. The analysis shows the velocity to be
mostly between ${40}{km/s}$ and ${100}{km/s}$ and the rate of constant, non-
hyperbolic dust between $0-4${\mathrm{h}}^{-1}$$ (according to Fig. 6). The
background detection rate $\lambda_{bg}$ could clearly not be lower than $0$
and the result suggests it could not be much higher than
$\approx$4\text{\,}\mathrm{h}^{-1}$$, because no solutions of (6a) in that
case, as missing solutions in Fig. 5 show. The reason is, that the difference
between inbound and outbound rates are observed to be too high to be explained
in the case of $\lambda_{bg}=$4\text{\,}\mathrm{h}^{-1}$$. A rather low amount
($\lesssim$1\text{\,}\mathrm{h}^{-1}$$) of non-hyperbolic dust would imply
higher velocity, in the range of
$\approx$100\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$. The
conclusion is that the higher the background detection rate $\lambda_{bg}$ is,
the lower is the underlying dust velocity. Similarly, a higher $\alpha\delta$
product implies higher velocity and larger initial radius (in case of
$\beta$-meteoroids) implies higher underlying radial velocity. Furthermore,
assuming $\beta$-meteoroids, low velocities
$\gtrsim$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ imply a low
$\beta$ factor (see Fig. 1). While bearing many uncertainties, this inference
is very robust as it does not depend on a specific model for dust, in
particular it is independent on dust spatial density as a function of
heliocentric distance, because we only compare observations on the same
heliocentric distance. The background component is among the biggest unknowns,
see appendix C for full velocity profiles that produce the data in Fig. 6.

Figure 6: Velocity estimated from TDS/TSWF-E/CNN. Single velocities are
obtained from profiles for better readability. The points are constructed as
averages of velocities at $0.65,0.75,0.85$\mathrm{A}\mathrm{U}$$ for all the
branches, where a velocity is the median solution for all the bootstrap
replications. For full profiles see appendix C.

###  3.4 Spatial density

If the hyperbolic grains do not accelerate (for example $\beta$-meteoroids
with $\beta\approx 1$ are assumed), a radial dependence of spatial density of
$\sim r^{-2}$ is the result. This is not the case if acceleration or
deceleration is present. Particularly, it makes sense to assume slowing dust
($\beta<1$), as figure 5 suggests slowing rather than accelerating dust. Also,
$\beta\approx 1$ or even $\beta>1$ needs a rather specific set of conditions
(combination of material and specific size, see Mann (2010)), while
$0.5<\beta<1$ is possible for a broad range of dust parameters. The observed
effective $\beta$ is then determined by aggregation of all components. The
equation for detection rate (4a) contains $r^{-2}$, but it remains the correct
expression for dust flux even if $v_{dust}$ is not constant. In that case,
$r^{-2}$ should not be interpreted as a spatial density of dust, but as a
geometric factor. The spatial density is then expressed through the non-
constant $v_{dust}$. However, this makes it very difficult to fit model (eq.
(4a)) to the data, as $v_{dust}$ is no longer a numeric parameter, but a
function of $r$.

Let us stick to spatial density view and examine the effective exponent of $r$
given $\beta$-meteoroids with some $0.5\leq\beta<1$. Fig. 7 shows an example
of how the $\beta$ value influences the spatial dust density (for spatial dust
density calculation, see appendix A). The analysis of spatial density as a
result of deceleration does not require the dust to be $\beta$-meteoroids, but
the relation to $\beta$ value clearly does. In Fig. 7, an initial orbit of
$0.1\text{\,}\mathrm{A}\mathrm{U}$ is assumed, see appendix E for plots of
dust spatial density variation similar to Fig. 7 for different initial orbits.
The particular exponents depend on the initial orbit, but the general trend of
lower $\beta$ value implying deceleration remains.

Figure 7: Modelled dust spatial densities for different $\beta$ values
assuming a circular initial orbit of $0.1\text{\,}\mathrm{A}\mathrm{U}$. Solid
lines show the spatial density and are normalized to the density at
$0.5\text{\,}\mathrm{A}\mathrm{U}$. Dashed lines are approximations to the
solid lines, assuming power dependence on $r$.

##  4 Daily count inference

###  4.1 Model formulation

We decided to model the number of dust detections within a day as a Poisson
distributed random variable, as dust detection itself is a prime example of a
Poisson point process in time, as discussed in section 1.4. The rate $\lambda$
of the process is considered dependent on multiple parameters $\@vec{\theta}$.
Notably, we consider $\lambda$ to not be explicitly dependent on time, but to
be temporarily dependent indirectly, through the orbital parameters and the
orbital phase of the spacecraft. Importantly, the rate is also considered
dependent on the parameters of the dust cloud. Therefore, we formulate a
hierarchical Bayesian model with $5$ parameters:
$\epsilon_{v},\epsilon_{r},\lambda_{\beta},\lambda_{bg},\nu_{r}$, that we for
simplicity denote
$\@vec{\theta}=(\epsilon_{v},\epsilon_{r},\lambda_{\beta},\lambda_{bg},\nu_{r})$.
These parameters are used to model the rate $\lambda$ and by extension
detected counts, see equations (7a) to (7d). Note that the rate $\lambda$ (see
eq. (7b)) is a generalisation of eq. (4a) with an additional constant
(background) term and a variable exponent of heliocentric distance. In order
to keep the present model simple and yet allow for non-constant velocity, we
use the parameters $\nu_{r}$ and $\epsilon_{r}$ as mean radial dust velocity
and density exponent respectively, see appendix D for further interpretation.
This combination allows for fitting of a single constant mean velocity and
effective acceleration (through the exponent $\epsilon_{r}$) at the same time
with just two constant scalar parameters. Mind though, that the parameter
$\nu_{r}$ has the exact meaning of mean velocity only in the case of constant
velocity. Given our generalization, it is not necessarily the mean velocity
over the orbit, but an effective velocity parameter. This is acceptable, as
the velocity is reasonably approximated by a constant between
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and $1\text{\,}\mathrm{A}\mathrm{U}$, even
in the case of $\beta=0.5$ (see Fig. 1).

The Poisson likelihood (eq. (7a)) includes exposure time $E$ (in hours) and
the rate $\lambda$ (in detections per hour). Then eq. (7c) is a straight-
forward definition of relative velocity between the spacecraft and the dust
particle, while eq. (7d) describes the decomposition of dust velocity into
radial and azimuthal components. In practice, the model defines the parameter
$\nu_{r}$ as the radial velocity of a dust particle and the variable
$\nu_{azimuthal}$ as the azimuthal velocity of the same particle, but only
$\nu_{r}$ is regarded as a random variable. The variable $v_{a}$ is directly
related to heliocentric distance $|\@vec{r}|$ according to eq. (7e), which is
approximately equivalent to the $r_{0}=$0.1\text{\,}\mathrm{A}\mathrm{U}$$
line in Fig. 2. This is due to the simpler and less important dependence of
$v_{impact}$ on $v_{a}$ and as a compromise in order to keep the number of
free parameters reasonable with respect to available data (the attempts to fit
$6$ parameters were not fruitful). The main goal of the fitting procedure is
to get the marginal posterior distributions of each of the parameters
$\@vec{\theta}$ of the model

$\displaystyle\begin{split}\qquad\qquad N|\lambda,\@vec{\theta}&\sim
Poiss(E\cdot\lambda(\theta)),\end{split}$ (7a)
$\displaystyle\begin{split}\lambda(\@vec{\theta})&=\lambda_{\beta}\cdot
v_{impact}^{\epsilon_{v}}\cdot r^{\epsilon_{r}}+\lambda_{bg},\end{split}$ (7b)
$\displaystyle\begin{split}v_{impact}&=\frac{|\@vec{v_{SolO}}-\@vec{v_{dust}}|}{$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$},\end{split}$
(7c)
$\displaystyle\begin{split}\@vec{v_{dust}}&=\nu_{r}\cdot\@vec{e_{r}}+v_{a}\cdot\@vec{e_{\phi}},\end{split}$
(7d)
<EMAIL_ADDRESS>(7e)

There are $N$ detections observed in a given day, the exposure $E$ is known,
and so are the location and velocity of SolO. In equation (7c), a
dimensionless parameter is constructed — it has computational advantages if
$v_{impact}\approx 1$, as rather high power of the variable is computed in the
process. Eqs. (7c) to (7e) explain the role of the parameter $v_{r}$ in eq.
(7b) and are only separated from (7b) for better readability. Note that purely
2D motion of dust particles, within the ecliptic plane, is assumed in eq.
(7d).

The parameter $\epsilon_{v}$ is the exponent of $v_{impact}$ in the mean rate
formula and incorporates the dependence on rate of volume scanning
($V/t\propto S\cdot v_{impact}$), hence $v_{impact}^{1}$, and the dependence
on charge yield $\alpha$ and dust mass power-law exponent $\delta$ in the form
of $v^{\alpha\delta}$. The dependence is then
$v_{impact}^{1+\alpha\delta}=v_{impact}^{\epsilon_{v}}$.

The parameter $\epsilon_{r}$ is the exponent of heliocentric distance $r$,
notably influenced by acceleration / deceleration of dust, as discussed in
sec. 3.4. See appendix D for further interpretation.

The parameter $\lambda_{\beta}$ plays the role of a normalization constant,
accounting for an absolute dust spatial density and spacecraft detection area
and holds the physical unit of $hour^{-1}$. It is uninteresting to study this
parameter in itself, in the sense that it merely normalizes the model so that
the detection rate correspond to the observed mean rate and has no consequence
on the physical characteristics of any given particle.

The parameter $\lambda_{bg}$ has the meaning of detections per hour as well,
but has a clearer interpretation as the background detection rate, that is the
rate of detections that are not attributable to hyperbolic dust.

The parameter $\nu_{r}$ also has a very direct meaning, which is the mean
outward radial velocity of the hyperbolic dust in our experimental range. Note
that variation of impact velocity is still allowed by variation in spacecraft
velocity $\@vec{v_{SolO}}$. Acceleration is accounted for in $\epsilon_{r}$.

###  4.2 Prior distributions of parameters

For Bayesian inference, choosing reasonable priors is important. Ideally,
priors should be informative (narrow) enough to capture the prior knowledge
about parameters, but vague (wide) enough so that they still allow for
additional information to play a role. It is physically infeasible for the
parameters $\lambda_{\beta}$ and $\lambda_{bg}$ to be negative, as they have a
meaning of detection rate. Furthermore, positive radial velocity is also
required by the model to work. Therefore, we opted for gamma priors for these
three parameters. Although we are quite sure about the sign of the parameters
$\epsilon_{v}$, $\epsilon_{r}$, neither the model nor the physical unit
actually rules out the possibility of $\epsilon_{v}$, $\epsilon_{r}$ having
any sign. We therefore opted for normal priors for $\epsilon_{v}$,
$\epsilon_{r}$. The choice of prior family for $\epsilon_{v}$, $\epsilon_{r}$,
and $\nu_{r}$ is of little importance. Generally speaking, prior choice makes
less of a difference the more data are analyzed.

In order to incorporate our actual prior belief about the model, we chose what
we believe are moderately informative priors for the parameters. The following
paragraphs discuss our choice. For a graphical representation of the prior
distributions of the parameters, see Fig. 8.

The parameter $\epsilon_{v}$ stands for $1+\alpha\delta$. Since we have
indications from Zaslavsky et al. (2021) that $\delta\approx 0.3$ and most
laboratory experiments show (Collette et al., 2014) that
$3\lesssim\alpha\lesssim 5$, we expect
$1.9\lesssim\epsilon_{v}=1+\alpha\delta\lesssim 2.5$. We therefore choose the
prior $\epsilon_{v}\sim Norm(mean=2.2,stdev=0.2)$, which places emphasis on
the range $2.0<\epsilon_{v}<2.4$ and yet doesn’t prohibit any real
$\epsilon_{v}$.

Provided that there are no major sources of dust between
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and $1\text{\,}\mathrm{A}\mathrm{U}$ and
provided that dust does not either accelerate nor decelerate,
$\epsilon_{r}=-2$, which follows easily from mass conservation. If we relax
the latter assumption, then $\epsilon_{r}\neq-2$. In fact, the dependence will
no longer follow $r^{\epsilon_{r}}$ exactly, but as is shown in Fig. 7, for
$\beta$-meteoroids of $0.5\lesssim\beta\lesssim 1$ the dependence is very
similar to $r^{\epsilon_{r}}$ with $-2\lesssim\epsilon_{r}\lesssim-1.59$. We
therefore chose a prior $\epsilon_{r}\sim Norm(mean=-1.8,stdev=0.2)$ which
emphasizes the range $-2.0<\epsilon_{r}<-1.6$ but in principle allows for any
real $\epsilon_{r}$. As for the parameter $\lambda_{\beta}$, we know it is on
the order of the total rate, which is on average
$6.9\text{\,}\mathrm{h}^{-1}$. The interpretation of the parameter is made
more opaque by the normalization in eq. (7c). However, the factor of
$v_{impact}^{\epsilon_{v}}$ is on the order of $1$ and the factor of
$r^{\epsilon_{r}}$ is $>1$, hence we expect $1\lesssim\lambda_{\beta}\lesssim
10$. We chose a less informative prior of $\lambda_{\beta}\sim
Gamma(shape=3,scale=1)$.

Fig. 5 showed that for background detections,
$\lambda_{bg}<$4\text{\,}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}^{-1}$$ is
feasible. We chose a less informative prior $\lambda_{bg}\sim
Gamma(shape=3,scale=1)$, which is wide and allows for any positive
$\lambda_{bg}$.

Based on Fig. 6, we believe that values
$$40\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$\lesssim\nu_{r}\lesssim$80\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$
are mostly expected. We chose the prior $\nu_{r}\sim Gamma(shape=10,scale=5)$
that emphasizes that range, with the mean of
$50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$, which is the value that
Zaslavsky et al. (2021) reported. This prior still allows for any positive
value of $\nu_{r}$.

Figure 8: Prior and posterior distributions for the parameters $\theta$. The
prior distributions are described in text. Summary statistics for posterior
distributions are described in Tab. 1.

###  4.3 Posterior distributions

The analysis is performed using TDS/TSWF-E/CNN data. For an analogous analysis
performed on SolO on-board identified dust impacts, see appendix H. Posteriors
are inferred using R-INLA. The model (7b) is complicated by the steep
dependence of the rate $\lambda$, especially the dependence on exponential
parameters $\epsilon_{v},\epsilon_{r}$. Note that a radial velocity
$\gtrsim$60\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ is consistent
with detection rate $\lambda_{bg}\approx 1.5$ and $\alpha\delta\approx 1.0$,
according to Fig. 6. It is important to note that the exact choice of priors
and other parameters, such as the reference azimuthal velocity in eq. (7c) and
procedure starting point (as INLA works on a grid largely defined by the
initial point) influences the exact result, although no major difference is
encountered when parameters or priors are reasonably varied (see appendix I).
As for the initial point, the mode of the joint prior was used:
$\@vec{\theta}=(2.2,-1.8,2,2,45)$.

Several measures can be used to evaluate the appropriateness of a model to a
data set. We inspected the conditional predictive ordinates (CPO) and the
predictive integral transform (PIT), that indicated no issues, see appendix F
for details.

| $Mean$ | St. dev.
---|---|---
$\epsilon_{v}$ | 2.04 | 0.20
$\epsilon_{r}$ | -1.61 | 0.16
$\lambda_{\beta}$ | 1.96 | 0.38
$\lambda_{bg}$ | 1.54 | 0.25
$\nu_{r}$ | 63.4 | 6.7
Table 1: The marginal posterior mean and the standard deviation for all the
parameters, see Fig. 8 for visual representation of the posterior
distributions.

###  4.4 Discussion of the posterior distribution

The inferred posterior distribution of velocities shown in Fig. 8 is not to be
interpreted as a distribution of velocities within the dust cloud directly,
but rather as a distribution of the effective mean velocities encountered on
each day, or even better — the uncertainty in effective velocity. There could
indeed be dust grains with velocity well off the effective support of the
posterior distribution, as long as the mean of all velocities does not exceed
the region indicated by the posterior distribution.

The parameters $\@vec{\theta}$ are not independent. Fig. 9 shows the
covariance between the parameters $\nu_{r}$ (radial velocity) and
$\lambda_{bg}$ (background detection rate). Negative correlation suggests that
higher $\nu_{r}$ is likely in case of lower $\lambda_{bg}$. This offers a
sanity check: higher velocity would mean lower difference between inbound and
outbound flux, which has a similar effect to the higher background component
scenario — negative correlation between $\nu_{r}$ and $\lambda_{bg}$ is thus
expected. For covariances between all parameters, see appendix G.

Figure 9: The covariance between $\nu_{r}$ (radial dust velocity) and
$\lambda_{bg}$ (background detection rate), correlation is $-0.3$. The red
line is the mean $\nu_{r}$ conditioned on $\lambda_{bg}$. Produced by sampling
from the joint posterior distribution of $\@vec{\theta}$.

The TDS/TSWF-E/CNN data set contains $6.9\text{\,}\mathrm{h}^{-1}$ detections
on average. The inferred value of $\lambda_{bg}=$1.54\pm
0.25\text{\,}\mathrm{h}^{-1}$$ implies that in total $78\pm 4\text{\,}\%$ of
dust is attributed to hyperbolic dust within the model. The constant
background $\lambda_{bg}$ is the simplest available generalization and is
therefore likely an oversimplification. Hyperbolic dust detection rate shows
strong negative correlation with heliocentric distance. If, for instance, non-
hyperbolic dust shows similar anti-correlation, the actual non-hyperbolic
component is higher than inferred. Conversely, if non-hyperbolic prevalence
shows correlation with heliocentric distance, the actual non-hyperbolic
component is lower than inferred. Both cases would also imply changes to the
inferred parameters of hyperbolic dust, see appendix J for a visual
explanation. If non-hyperbolic is mostly non-dust (for example misattributed
electrical phenomena), independence on heliocentric distance is reasonable.
However, if most non-hyperbolic are bound (Keplerian) dust particles, then
anti-correlation is expected. If interstellar dust (ISD) streaming
predominantly from one direction approximately within the ecliptic plane is
present, positive correlation is also feasible due to velocity vector
orientation.

Indeed, ISD was observed (Baguhl et al., 1996; Zaslavsky et al., 2012;
Malaspina et al., 2014) to arrive mainly from $258\text{\,}{}^{\circ}{}$
ecliptic longitude. The highest flux is observed when a spacecraft has anti-
parallel velocity, which vaguely coincides with higher heliocentric distance
phase of SolO’s orbit so far. If ISD is an important contribution to
$\lambda_{bg}$, the actual background flux may be lower than the suggested
$\lambda_{bg}\approx$1.5\text{\,}\mathrm{h}^{-1}$$. For now, ISD is not
apparent in SolO data and the fact that models fit well without ISD suggests
it is not an important component of SolO detections. Its identification with
SolO is however beyond the scope of the present work, but remains worthy of
future investigation, especially since ISD may become more important during
the current solar cycle (Mann, 2010). For now, no bound dust particles are
apparent either, nor are the retrograde dust particles. If the constant
background is a crude oversimplification and the non-hyperbolic component has
a prominent dependence on heliocentric distance, the present interpretation of
the parameters $\@vec{\theta}$ is not correct, as the model is not on point.
Inclusion of more parameters in the model (for example a more sophisticated
non-hyperbolic term) may be feasible with more data in coming months.

The posterior mean of the detection rate is shown in Fig. 10 in units of:
$\mathrm{m}^{-2}\mathrm{h}^{-1}$, assuming a detection area of
$8\text{\,}\mathrm{m}^{2}$ (SolO thermal shield approx. area); and
$\mathrm{d}\mathrm{a}\mathrm{y}^{-1}$, taking into account the detection time
per day and extrapolating to $24\text{\,}\mathrm{h}$. Note that the credible
intervals reflect the uncertainty of the inferred mean detection rate (the
uncertainty of our knowledge, given the data), which is the same uncertainty
as visualised in Fig. 8. The spread of data points in Fig. 10 is much wider
and mostly defined by the variance of Poisson random variable, given the mean
rate, rather than the uncertainty in the mean rate. The prediction intervals
of the Poisson random variable are shown in Fig. 11 and there, data points
seem to be appropriately covered by the credible intervals.

Figure 10: The estimated posterior mean of the dust impact with
$90\text{\,}\%$ HPD credible intervals. The credible intervals are not
supposed to cover the data scatter (see text for interpretation of shown
credible intervals). Figure 11: The estimated posterior mean of the hyperbolic
dust detections and HPD prediction intervals. The Prediction intervals are
supposed to cover the data scatter (see text for interpretation of shown
prediction intervals).

The inferred value of the parameter $\epsilon_{r}\approx-1.6$ suggests that
dust grains are slowing substantially on their way out of the inner
heliosphere between $0.5\text{\,}\mathrm{A}\mathrm{U}$ and
$1\text{\,}\mathrm{A}\mathrm{U}$, resulting in a spatial distribution
different from trivial $\lambda\propto r^{-2}$ case. See Fig. 7 for
comparison. With inferred velocity of $63\pm
7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ between
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and $1\text{\,}\mathrm{A}\mathrm{U}$,
significant deceleration suggest much higher velocity closer to the Sun.
Assuming $\beta$-meteoroids with a circular initial orbit, the $\epsilon_{r}$
value implies a specific $\beta$ value needed for just the right level of
deceleration. Deceleration is a result of energy transfer from kinetic to
potential, and therefore, given initial heliocentric distance, the
deceleration rate depends on initial velocity. This makes the assumption of
circular initial orbit crucial when we are to infer the $\beta$ parameter. For
example, a $\beta$ value needed to explain an observed $\epsilon_{r}$ is
different if the $\beta$-meteoroid parent object has eccentricity $0.3$,
rather than $0$. For analysis of implied $\beta$ values in the case of
circular parent orbit, see Fig. 12. Various initial parent body orbit radii
are shown in Fig. 12 to demonstrate that the model is not very sensitive to
that parameter. For comparison, note that velocities
$\gtrsim$60\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ are consistent
with $\beta\approx 0.6$ and origin between $0.05\text{\,}\mathrm{A}\mathrm{U}$
and $0.1\text{\,}\mathrm{A}\mathrm{U}$, according to Fig. 1. Note that
$$0.05\text{\,}\mathrm{A}\mathrm{U}$\approx$10\text{\,}\mathrm{R}_{\odot}$$,
where $R_{\odot}$ is the Solar radius.

However, it is feasible to expect a parent body with eccentricity of $0.3$, as
the mean eccentricity in the inner asteroid belt is $e\approx 0.15$ (Malhotra
& Wang, 2016). If a dust grain is ejected from a given heliocentric distance
$r$, the eccentricity $e=0.3$ implies $+14\%$ ejection speed if $r$ is the
perihelion and $-16\%$ ejection speed if $r$ is the aphelion, compared to
ejection from circular orbit of radius $r$. In the case of $e\neq 0$,
$\beta>0.5$ is not the right condition for unbound $\beta$-meteoroid. In fact,
for $e=0.3$ the condition is approximately $\beta>0.35$ for perihelion, and
$\beta>0.65$ for aphelion ejection. Mind that the $+14\%$ could also be
$\Delta v$ transferred at collision, as collisions between larger dust objects
are likely a major source of $\beta$-meteoroids. Then the $14\%$ relative
speed would for instance correspond to collision of two asteroids on circular
orbits with relative inclination of $8\text{\,}{}^{\circ}$, which is also a
very feasible scenario. For instance, if the example of $+14\%$ of $\Delta v$
(or eccentricity of $0.3$) is a good representative of the process, the
resulting implied $\beta$ would be not $\beta\gtrsim 0.5$, but rather
$\beta\gtrsim 0.35$. Eccentricities and relative velocities in zodiacal cloud
remain uncertain. Note that even in the described case, we are still
considering a dust grain with purely azimuthal velocity at liberation, which
is yet another simplification.

Figure 12: $\beta$ parameter resulting from $\epsilon_{v}$ posterior
distribution under the assumption of circular parent body orbit orbit (see
text for discussion).

###  4.5 Comparison with previous results

As Solar Orbiter has been operated only since 2020 and will be operated at
least until 2027, the results presented in this paper are one of the earlier
for the mission. Based on similar data, though collected over shorter time
period, Zaslavsky et al. (2021) reported several physical parameters of the
$\beta$-meteoroid population. Interestingly, they reported radial velocity to
be about $50\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$, which is
within 2 standard deviations from $63\pm
7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ reported here, but bear in
mind that the number is inferred under substantially different assumptions.
The velocity is crucial for inference of $\beta$-meteoroid flux at
$1\text{\,}\mathrm{A}\mathrm{U}$ for example, which Zaslavsky et al. (2021)
reported to be
$8\text{\times}{10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$. Under our
model assumptions (constant radial dust velocity) and taking joint posterior
distribution of the parameters, we report $(1.6\pm
0.1)\text{\times}{10}^{-4}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$ for
hyperbolic dust and the residual component ($\lambda_{bg}$) together (measured
on a stationary spherical object, per $m^{2}$ of cross section), a value
higher by a factor of $\approx 2$. For the component consistent with
hyperbolic dust only, we report the flux of $(1.1\pm
0.2)\text{\times}{10}^{-4}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$ and for the
component attributed to the residual, background component $(5.4\pm
1.5)\text{\times}{10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$. As for
$\alpha\delta$ (mind that $\alpha\delta=\epsilon_{v}-1$ here), Zaslavsky et
al. (2021) reported consistency with $\alpha\delta=1.3$, while we report
$\epsilon_{v}-1=$1.04\pm 0.20\text{\,}$$.

As for comparison of present results with $\beta$-meteoroid flux near
$1\text{\,}\mathrm{A}\mathrm{U}$, Wehry & Mann (1999) reported the flux of
$\beta$-meteoroids in ecliptic plane detected by Ulysses between
$1.0-1.6$\mathrm{A}\mathrm{U}$$ to be $(1.5\pm
0.3)\text{\times}{10}^{-4}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$. Zaslavsky
et al. (2012) reported flux of $\beta$-meteoroids at
$1\text{\,}\mathrm{A}\mathrm{U}$ of size $100-300$\mathrm{n}\mathrm{m}$$ on
STEREO/Waves in the range of
$1-6\times${10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$$, a somewhat
lower value than reported here. SolO detections are likely of
$100\text{\,}\mathrm{n}\mathrm{m}$ and larger dust, but upper limit is
somewhat higher for SolO due to wider dynamic range
($3-150$\mathrm{m}\mathrm{V}$$ for STEREO and $3-700$\mathrm{m}\mathrm{V}$$
for SolO), which may account for some of the difference. Malaspina et al.
(2015) however reported the value for STEREO/Waves by about a factor of $2.5$
higher than Zaslavsky et al. (2012), which is on its upper bound virtually
identical with the value reported here. For the Wind/WAVES experiment,
Malaspina et al. (2014) reported $(2.7\pm
1.4)\text{\times}{10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$ for the sum
of $\beta$-meteoroids and interstellar dust of $0.1-11$\mu\mathrm{m}$$ size.
It remains open if and how much interstellar dust contributes to the
measurements of SolO/RPW analyzed in the present work. Recently, Szalay et al.
(2021) reported $4-8\cdot${10}^{-5}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$$
for $\beta$-meteoroid flux at $1\text{\,}\mathrm{A}\mathrm{U}$ measured with
Parker Solar Probe. The upper bound of this estimate is similar to the value
reported in the present work.

##  5 Conclusions

  1. 1.

We presented the analysis of the velocity of hyperbolic dust grains between
$$0.5\text{\,}\mathrm{A}\mathrm{U}$-$1\text{\,}\mathrm{A}\mathrm{U}$$ based on
highest-quality available data on daily dust detections by SolO/RPW, including
discussion of implications for the velocity in case of non-hyperbolic (be it
other dust population or false detections) component to the counts. Velocities
in the range $30-110$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$$ are compatible
with the data.

  2. 2.

We presented a Bayesian hierarchical model and demonstrated how it is used to
infer physical parameters of hyperbolic dust population in the studied region.
It is likely that $1.5\pm 0.3\text{\,}\mathrm{h}^{-1}$ are in fact not caused
by hyperbolic dust. Then observations are consistent with a mean radial
velocity of the hyperbolic component $63\pm
7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ between
$0.5\text{\,}\mathrm{A}\mathrm{U}$ and $1\text{\,}\mathrm{A}\mathrm{U}$.
Spatial dependence of the detection rate suggests substantial deceleration of
the observed hyperbolic dust particles. If they are $\beta$-meteoroids, the
value of $\beta$ is likely just above the liberation threshold, specifically
$\beta\gtrsim 0.5$ under the assumption of circular orbits of parent bodies.
Hence closer to their origin, they likely have velocities higher than the
inferred $63\pm 7\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$.

  3. 3.

As a result of our modelling, we provide estimates of hyperbolic dust flux at
$1\text{\,}\mathrm{A}\mathrm{U}$ of $(1.1\pm
0.2)\text{\times}{10}^{-4}\text{\,}\mathrm{m}^{-2}\mathrm{s}^{-1}$, a value
compatible with other relevant measurements.

##  6 Outlook

SolO will get significant inclination in 2025, which will require further
generalization of the model, to account for dust distribution out of the
ecliptic plane. Out of ecliptic parameters of hyperbolic dust will likely
provide more information on in-ecliptic hyperbolic dust, such as its parent
bodies’ mean eccentricity. Due to independence on $\beta$, knowledge of
azimuthal velocity would be a good indicator of $\beta$-meteoroids origin, but
it is hard to infer as the azimuthal component is much smaller than the radial
component. Out of ecliptic detections may help in this regard as well.

Near the solar minimum of 2020, a so-called de-focusing solar magnetic field
configuration was present in the solar system (Mann, 2010). This configuration
favours dust depletion of the near-ecliptic region due to the Lorentz force
acting on ISD particles entering the solar system. With solar cycle 25,
focusing field configuration will return at some time before the solar minimum
of 2031. It is possible that a significant ISD component will be observed in
years following the solar maximum of 2025, which will, if observed, provide
new opportunities for dust population discrimination and more comprehensive
dust cloud description thanks to SolO/RPW data.

###### Acknowledgements.

Author contributions: Concept: SK, AT, IM. Data analysis: SK, SHS, AK.
Interpretation: SK, IM, AT, SHS, AK, AZ. Manuscript preparation: SK. The code
and the data used in present work are publicly available at
github.com/SamuelKo1607/solo_dust_2022. This work made use of publicly
available data provided by A. Kvammen:
github.com/AndreasKvammen/ML_dust_detection. SK and AT are supported by the
Tromsø Research Foundation under the grant 19-SG-AT. This work is supported by
the Research Council of Norway (grant number 262941). Authors sincerely
appreciate the support of Solar Orbiter/RPW Investigation team. This work was
made possible by R-INLA package, authors thank to R-INLA team, see r-inla.org.

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##  Appendix A Single-particle velocity and spatial density

For the purpose of figs. 2 and 1, dust grains are assumed to move within the
ecliptics, liberated from initially circular orbit and with their motion
governed by the gravity and solar radiation pressure only, therefore

$\displaystyle\begin{split}|v|&=\sqrt{v_{0}^{2}+2GM(1-\beta)\left(\frac{1}{r}-\frac{1}{r_{0}}\right)},\end{split}$
(8) $\displaystyle\begin{split}v_{tan}=v_{0}\frac{r_{0}}{r},\end{split}$ (9)
$\displaystyle\begin{split}v_{rad}=\sqrt{v^{2}-v_{tan}^{2}},\end{split}$ (10)

where $v_{0}$ is the initial (purely radial) velocity and $r_{0}$ is the
initial heliocentric distance (radius of the circular orbit). Furthermore,
given a radial velocity profile of a radially escaping dust grain
$v_{rad}(r)$, dust spatial density $\rho$ at a heliocentric distance $r$ is

$\displaystyle\begin{split}\rho(r)&=\rho(r_{0})\left(\frac{r_{0}}{r}\right)^{2}\frac{v_{rad}(r_{0})}{v_{rad}(r)},\end{split}$
(11)

where $r_{0}$ is a reference heliocentric distance.

##  Appendix B Dust radial velocity estimation

Suppose the detection rate is proportional to $v_{relative}^{q}$:

$\displaystyle\begin{split}R&=R_{0}\cdot
v_{relative}^{q}=R_{0}\left(\@vec{v_{dust}}-\@vec{v_{sc}}\right)^{q}\end{split}$
(12)
$\displaystyle\begin{split}&=R_{0}\left[\sqrt{(v_{dust;rad}-v_{sc;rad})^{2}+v_{sc;azim}^{2}}\right]^{q},\end{split}$
(13)

where we assumed $v_{dust;azim}=0$. Then at any given heliocentric distance
$r$:

$\displaystyle\begin{split}R_{in}^{2/q}&=R_{0}^{2/q}\left((v_{dust;rad}+|v_{sc;rad}|)^{2}+v_{sc;azim}^{2}\right),\end{split}$
(14)
$\displaystyle\begin{split}R_{out}^{2/q}&=R_{0}^{2/q}\left((v_{dust;rad}-|v_{sc;rad}|)^{2}+v_{sc;azim}^{2}\right),\end{split}$
(15)

and therefore:

$\frac{R_{in}^{2/q}}{R_{out}^{2/q}}=\frac{(v_{dust;rad}+|v_{sc;rad}|)^{2}+v_{sc;azim}^{2}}{(v_{dust;rad}-|v_{sc;rad}|)^{2}+v_{sc;azim}^{2}},$
(16)

from which:

$\displaystyle\begin{split}0&=v_{dust;rad}^{2}\cdot\left(R_{in}^{2/q}-R_{out}^{2/q}\right)\end{split}$
(17)
$\displaystyle\begin{split}&+v_{dust;rad}\cdot\left(-2v_{sc;rad}\left(R_{in}^{2/q}+R_{out}^{2/q}\right)\right)\end{split}$
(18)
$\displaystyle\begin{split}&+\left(R_{in}^{2/q}-R_{out}^{2/q}\right)\cdot\left(v_{sc;rad}^{2}+v_{sc;azim}^{2}\right),\end{split}$
(19)

which leads to a quadratic root:

$\displaystyle\begin{split}v_{dust;rad}&=\frac{2|v_{sc;rad}|\left(R_{in}^{2/q}+R_{out}^{2/q}\right)\pm\sqrt{D}}{2\left(R_{in}^{2/q}-R_{out}^{2/q}\right)},\end{split}$
(21)
$\displaystyle\begin{split}D&=4v_{sc;rad}^{2}\left(R_{in}^{2/q}+R_{out}^{2/q}\right)^{2}-4v_{sc}^{2}\left(R_{in}^{2/q}-R_{out}^{2/q}\right)^{2},\end{split}$
(22)

where $(+)$ in eq. (21) leads to positive velocity $v_{sc;rad}$. It is easy to
see that in the special case of $q=1;v_{sc;azim}=0$:

$\displaystyle\begin{split}D&=4v_{sc;rad}^{2}\left[\left(R_{in}^{2}+R_{out}^{2}\right)^{2}-\left(R_{in}^{2}-R_{out}^{2}\right)^{2}\right],\end{split}$
(23)
$\displaystyle\begin{split}&=16v_{sc;rad}^{2}R_{in}^{2}R_{out}^{2},\end{split}$
(24)

and by extension:

$\displaystyle\begin{split}v_{dust;rad}&=\frac{|v_{sc;rad}|\left[\left(R_{in}^{2}+R_{out}^{2}\right)\pm
2R_{in}R_{out}\right]}{\left(R_{in}^{2}-R_{out}^{2}\right)},\end{split}$ (25)

which is

$\displaystyle\begin{split}v_{dust;rad}&=\frac{|v_{sc;rad}|\left(R_{in}+R_{out}\right)}{\left(R_{in}-R_{out}\right)}\end{split}$
(26)

for $(+)$ in numerator.

##  Appendix C Velocity inference — full velocity profiles

The velocity profiles inferred in section 3.2 are shown in Figs. 13 to 15
(compare to Figs. 5 and 6). Note that the missing solutions (jittery line) for
heliocentric distance $>$0.7\text{\,}\mathrm{A}\mathrm{U}$$ and
$\lambda_{bg}=4$ cause incomplete data shown in Fig. 6. These solutions only
exist for some combinations of the free parameters, in particular for
$\lambda_{bg}=4$.

Figure 13: Velocity estimated from TDS/TSWF-E/CNN under the assumption of
$\alpha\delta=1.0$. Panels correspond to different initial heliocentric
distances. Colors correspond to different assumptions of background detection
rate. Figure 14: Velocity estimated from TDS/TSWF-E/CNN under the assumption
of $\alpha\delta=1.3$. Panels correspond to different initial heliocentric
distances. Colors correspond to different assumptions of background detection
rate. Figure 15: Velocity estimated from TDS/TSWF-E/CNN under the assumption
of $\alpha\delta=1.6$. Panels correspond to different initial heliocentric
distances. Colors correspond to different assumptions of background detection
rate.

##  Appendix D Interpretation of the parameter $\epsilon_{r}$

Intuitive explanation of the parameter $\nu_{r}$ as the mean dust velocity and
of the factor $r^{\epsilon_{r}}$ as the spatial density can be clarified,
assuming $\nu_{r}\propto r^{\xi}$. With the model (7b), the non-constant
component $\tilde{R}$ of the rate $R$ is proportional to

$\displaystyle\begin{split}\tilde{R}&\propto r^{\epsilon_{r}}\cdot
v_{impact}^{\epsilon_{v}},\end{split}$ (27)

where $\epsilon_{r}=-2$ in case of no acceleration of the dust. $\epsilon_{v}$
is explained as $\epsilon_{v}=1+\alpha\delta$, therefore

$\displaystyle\begin{split}\tilde{R}&\propto r^{-2}\cdot v_{impact}^{1}\cdot
v_{impact}^{\alpha\delta}.\end{split}$ (28)

The factor of $v_{impact}^{1}$ actually comes from the proportionality

$\displaystyle\begin{split}\tilde{R}&\propto\frac{v_{impact}}{\nu_{r}},\end{split}$
(29)

if we assume only radial motion for simplicity, compare with eq. (4a). It is
also apparent from the following: Assuming stationary spacecraft
($v_{impact}=\nu_{r}$), the detection rate (in $\mathrm{s}^{-1}$) is a product
of the flux $F(r)$ (in $\mathrm{s}^{-1}$) and the detection area $S$ (in
$\mathrm{m}^{2}$), independently of $\nu_{r}$. We therefore have for non-
accelerating dust:

$\displaystyle\begin{split}\tilde{R}&\propto
r^{-2}\cdot\frac{v_{impact}}{\nu_{r}}\cdot
v_{impact}^{\alpha\delta},\end{split}$ (30)

and finally assuming $\nu_{r}\propto r^{\xi}$ we get

$\displaystyle\begin{split}\tilde{R}&\propto r^{-2}\cdot r^{-\xi}\cdot
v_{impact}\cdot v_{impact}^{\alpha\delta}=r^{-2-\xi}\cdot
v_{impact}^{\epsilon_{v}},\end{split}$ (31)

therefore

$\displaystyle\begin{split}\epsilon_{r}&=-2-\xi.\end{split}$ (32)

There is a dichotomy in eq. (31) in that the assumption of $v_{dust}\propto
r^{\xi}$ was used to expand the $v_{dust}$ but not the $v_{impact}$. This is
one way of interpreting the approximation described in section 4.1, we assume
a non-constant dust velocity in the factor for spatial dust density, but a
constant radial dust velocity in the expression for $v_{impact}$, see eq.
(7c). This is done because of a clear relation of $\epsilon_{r}\lessgtr-2$ to
acceleration and deceleration of the dust, which in our case
($\epsilon_{r}\approx-1.6\implies\xi\approx-0.4$) reveals that the dust is
decidedly decelerating.

##  Appendix E Spatial density profiles

Assumed initial orbital distance influences the relationship between $\beta$
values and spatial dust density profiles. In Fig. 7, initial orbit of
$0.1\text{\,}\mathrm{A}\mathrm{U}$ is assumed. See plots 16 and 17 for similar
plots with different assumptions of initial orbit. See Fig. 18 for similar
plot if eccentricity of $e=0.2$ and perihelion ejection with perihelion of
$r=$0.1\text{\,}\mathrm{A}\mathrm{U}$$ is assumed. Although the estimates of
the exponents vary, the general conclusion of lower $\beta$ implying lower
exponent holds. Note that the profile gets significantly influenced when
$\beta$ is close to threshold, and that it depends on the initial orbit and
eccentricity. Therefore, $\beta<$0.5\text{\,}$$ is shown where $0.5\text{\,}$
is much higher than the liberation threshold.

Figure 16: Modelled dust spatial densities for different $\beta$ values
assuming a circular initial orbit of $0.05\text{\,}\mathrm{A}\mathrm{U}$.
Solid lines show the spatial density and are normalized to the density at
$0.5\text{\,}\mathrm{A}\mathrm{U}$. Dashed lines are approximations to the
solid lines, assuming power dependence on $r$. Figure 17: Modelled dust
spatial densities for different $\beta$ values assuming a circular initial
orbit of $0.2\text{\,}\mathrm{A}\mathrm{U}$. Solid lines show the spatial
density and are normalized to the density at
$0.5\text{\,}\mathrm{A}\mathrm{U}$. Dashed lines are approximations to the
solid lines, assuming power dependence on $r$. Figure 18: Modelled dust
spatial densities for different $\beta$ values assuming elliptical initial
orbit with eccentricity of $e=0.2$ and perihelion ejection. A perihelion of
$0.1\text{\,}\mathrm{A}\mathrm{U}$ is assumed. Solid lines show the spatial
density and are normalized to the density at
$0.5\text{\,}\mathrm{A}\mathrm{U}$. Dashed lines are approximations to the
solid lines, assuming power dependence on $r$.

##  Appendix F Model assessment

There are several options for model evaluation implemented in R-INLA (Gómez-
Rubio 2020, Chapter 2.4). We briefly describe two measures of choice.

The conditional predictive ordinates (CPO, see Pettit (1990)) for a given
observation point gives the posterior probability of each observation when
this observation is ommited in the model fitting. CPO is used to detect
surprising observations or outliers. We examined the fit for failure flags for
all the points, that would suggest a contradiction between the model and a
data point. No failures were encountered.

The predictive integral transform (PIT, see Marshall & Spiegelhalter (2003))
measures the probability that a new observation will be lower than the
observed value for each observation point individually. Histogram of the PIT
values should therefore be similar to the uniform distribution between $0$ and
$1$ when the model explains data well, see Fig. 19.

Figure 19: Histogram of PIT values for the model described in section 4.3.
Mean and standard deviation are compared to the values for uniform
distribution.

##  Appendix G Covariance plots of posteriors

As is shown in figure 20, basically all parameter pairs show substantial
correlation. The pairs hold useful information, but are hardly surprising and
are easy to interpret, having the model eq. (7b) in mind. The correlation is
unimportant in the case of $\lambda_{\beta}$, which has a role of a
normalization constant.

Figure 20: Illustration of the covariance between all parameter pairs, constructed using sampling from the joint posterior distribution of all parameters. | $\epsilon_{v}$ | $\epsilon_{r}$ | $\lambda_{\beta}$ | $\lambda_{bg}$ | $\nu_{r}$
---|---|---|---|---|---
$\epsilon_{v}$ |  | 0.451 | -0.109 | 0.068 | -0.015
$\epsilon_{r}$ | 0.451 |  | 0.392 | -0.621 | 0.027
$\lambda_{\beta}$ | -0.109 | 0.392 |  | -0.244 | -0.799
$\lambda_{bg}$ | 0.068 | -0.621 | -0.244 |  | -0.303
$\nu_{r}$ | -0.015 | -0.027 | -0.799 | -0.303 |
Table 2: Covariance between all parameter pairs, constructed using sampling
from the joint posterior distribution of all parameters.

##  Appendix H Model fitting to original data

The procedure described in sec. 4 was also applied to the original TDS data,
meaning impacts identified onboard SolO, described by Maksimovic et al.
(2020), which are different from TDS/TSWF-E/CNN data (Kvammen et al. 2022)
used otherwise in sec. 4. We have every reason to believe that CNN-refined
data used in sec. 4 are more reliable, with fewer type 1 and type 2 errors,
but the original data was used previously (Zaslavsky et al. 2021) and the
inspection of the result of the procedure is instructive nonetheless. The
results are presented in Fig. 21.

The most important and intuitive difference is that a much lower
$\lambda_{bg}$ is inferred in this case (compare with Fig. 8). As described in
Kvammen et al. (2022), the CNN procedure identifies substantially more impacts
near-aphelion, which suggests more background dust, everything else being
equal. Importantly, the inferred velocity $\nu_{r}$ doesn’t change
substantially, even though $\epsilon_{v}$ and especially $\epsilon_{r}$ do
change consequentially. Importantly, $\epsilon_{r}<-2$ implies accelerating
dust, which implies $\beta>1$ and requires specific material and size of the
grains, hence this is unlikely — at least for $\beta$-meteoroids. Note that
$\epsilon_{r}$ is effectively far from our prior expectations, providing poor
fit to our prior knowledge. These results lend additional credence to the
improvement of the CNN data.

Figure 21: Prior and posterior distributions of parameters, making use of original TDS (onboard processed) data. Prior distributions are described in text, sec. 4. Posterior distributions are described in Tab. 3. Posteriors from Fig. 8 are shown as reference in dashed lines for comparison. | $Mean$ | St. dev.
---|---|---
$\epsilon_{v}$ | 1.84 | 0.10
$\epsilon_{r}$ | -2.33 | 0.09
$\lambda_{\beta}$ | 1.45 | 0.16
$\lambda_{bg}$ | 0.23 | 0.08
$\nu_{r}$ | 69.8 | 4.06
Table 3: Summary statistics for the parameters, making use of original TDS
(onboard processed) data. For visual representation see Fig. 21.

##  Appendix I Variation of priors and other parameters

It is true that the priors themselves express the uncertainty in prior
knowledge, but to demonstrate the robustness of the analysis, we here show
perturbed priors and resulting posterior combinations (Figs. 22 and 23), to
show that the result, although somewhat dependent on the prior selection, does
not change dramatically if priors are chosen arbitrarily slightly different.
Also the choice of the value of parameter $v_{a}$ (which is not a free
parameter in our modelling, see eq. (7e)) is examined here (see Figs. 24 and
25). Last but not least, we show the posteriors in case of change of the
initialization of the iterative procedure to estimate the parameters (Fig.
26). We do not claim that any of these results is as trustworthy as the main
result shown in Fig. 8, we had reasoning behind choosing the priors and
parameters that we chose. Note that the mean of the marginal posteriors shown
in Figs. 22 to 26 lie within high credibility regions of posteriors shown in
Fig. 8 and vice versa, which supports the claim that the analysis presented
here is robust. Observe that parameter values inferred with lower precision
(wider posterior distributions) are more susceptible to change due to change
in parameters, which is in line with expectations and with the meaning of
precision here. Good choice of priors is still important for getting the
highest quality estimate, but the result is not critically sensitive.

Figure 22: Prior and posterior distributions of the parameters, the priors
being substantially wider (less informative). Priors and posteriors from Fig.
8 are shown in dashed lines for comparison. Figure 23: Prior and posterior
distributions of parameters, priors shifted towards lower values. Priors and
posteriors from Fig. 8 are shown in dashed lines for comparison. Figure 24:
Prior and posterior distributions of parameters, fixed parameter of azimuthal
velocity is changed from $12\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$
at $0.75\text{\,}\mathrm{A}\mathrm{U}$ to constant
$0\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$. Posteriors from Fig. 8
are shown in dashed lines for comparison. Figure 25: Prior and posterior
distributions of parameters, the fixed parameter of azimuthal velocity at
$0.75\text{\,}\mathrm{A}\mathrm{U}$ is changed from
$12\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$ to
$24\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$, which is a value higher
by $100\text{\,}\%$. Posteriors from Fig. 8 are shown in dashed lines for
comparison. Figure 26: Prior and posterior distributions of parameters, with
starting point of
$\epsilon_{v}=3;\epsilon_{r}=-3;\lambda_{\beta}=3;\lambda_{bg}=3;\nu_{r}=30$.
Posteriors from Fig. 8 are shown in dashed lines for comparison.

##  Appendix J Possible background profiles

In the present analysis, the non-hyperbolic component was assumed to be
present and constant. See Fig. 27 for examples of possible non-hyperbolic
component profiles, as discussed in section 4.4. In the plot, mean rate of
non-hyperbolic component is the same in all three panels. Note that despite
that, the dynamic range (that is ratios of maximum over minimum values)
changes significantly as a result of the change in the temporal profile of the
non-hyperbolic component. Significant deviation from the constant case would
therefore likely change both the inferred prevalence of the non-hyperbolic
component and the parameters of the hyperbolic grains.

Figure 27: Detection rate: selection of different combinations of hyperbolic
and non-hyperbolic compounding to the same observed detection rate. In panel
a) the background component is independent on heliocentric distance. In panel
b) the non-hyperbolic component is negatively correlated with heliocentric
distance. In panel c) the non-hyperbolic component is positively correlated
with heliocentric distance.