# Collisional Evolution of the Inner Zodiacal Cloud

J. R. Szalay Department of Astrophysical Sciences, Princeton University, 4 Ivy
Ln., Princeton, NJ 08540, USA  P. Pokorný Astrophysics Science Divison, NASA
Goddard Spaceflight Center, Greenbelt, MD, 20771, USA  Department of Physics,
The Catholic University of America, Washington, DC, 20064, USA  D. M.
Malaspina Department of Astrophysical and Planetary Sciences, University of
Colorado Boulder, Boulder, CO, USA  Laboratory for Atmospheric and Space
Physics, University of Colorado Boulder, Boulder, CO, USA  A. Pusack
Laboratory for Atmospheric and Space Physics, University of Colorado Boulder,
Boulder, CO, USA  S. D. Bale Space Sciences Laboratory, University of
California, Berkeley, CA, USA  Physics Department, University of California,
Berkeley, CA, USA  K. Battams US Naval Research Laboratory, 4555 Overlook
Avenue, SW, Washington, DC, USA  L. C. Gasque Space Sciences Laboratory,
University of California, Berkeley, CA, USA  Physics Department, University of
California, Berkeley, CA, USA  K. Goetz School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN, USA  H. Krüger Max-Planck-Institut
für Sonnensystemforschung, Göttingen, Germany  D. J. McComas Department of
Astrophysical Sciences, Princeton University, 4 Ivy Ln., Princeton, NJ 08540,
USA  N. A. Schwadron University of New Hampshire, Durham, NH, USA  P. Strub
Max-Planck-Institut für Sonnensystemforschung, Göttingen, Germany

(Received 21-Jan-2021; Revised 15-Apr-2021; Accepted 16-Apr-2021)

###### Abstract

The zodiacal cloud is one of the largest structures in the solar system and
strongly governed by meteoroid collisions near the Sun. Collisional erosion
occurs throughout the zodiacal cloud, yet it is historically difficult to
directly measure and has never been observed for discrete meteoroid streams.
After six orbits with Parker Solar Probe (PSP), its dust impact rates are
consistent with at least three distinct populations: bound zodiacal dust
grains on elliptic orbits ($\alpha$-meteoroids), unbound $\beta$-meteoroids on
hyperbolic orbits, and a third population of impactors that may either be
direct observations of discrete meteoroid streams, or their collisional
byproducts (“$\beta$-streams”). $\beta$-streams of varying intensities are
expected to be produced by all meteoroid streams, particularly in the inner
solar system, and are a universal phenomenon in all exozodiacal disks. We find
the majority of collisional erosion of the zodiacal cloud occurs in the range
of $10-20$ solar radii and expect this region to also produce the majority of
pick-up ions due to dust in the inner solar system. A zodiacal erosion rate of
at least $\sim$100 kg s-1 and flux of $\beta$-meteoroids at 1 au of
$0.4-0.8\times 10^{-4}$ m-2 s-1 is found to be consistent with the observed
impact rates. The $\beta$-meteoroids investigated here are not found to be
primarily responsible for the inner source of pick-up ions, suggesting
nanograins susceptible to electromagnetic forces with radii below $\sim$50 nm
are the inner source of pick-up ions. We expect the peak deposited energy flux
to PSP due to dust to increase in subsequent orbits, up to 7 times that
experienced during its sixth orbit.

Dust, Zodiacal Dust, Debris Disks, Collisions

††journal: PSJ

##  1 Introduction

The zodiacal dust distribution in the inner solar system is continuously
evolving. Grains orbiting the Sun slowly lose their angular momentum due to
Poynting-Robertson drag (Burns et al., 1979) and spiral in toward the Sun.
Additionally, cometary and asteroidal activity inputs dust into the zodiacal
cloud. Collisions occur between grains within the zodiacal cloud, which
fragment into numerous smaller grains (Kuchner & Stark, 2010). If these
collisional products are sufficiently small, they experience enough repulsive
force from radiation pressure to overcome the Sun’s attractive gravitational
force and leave the solar system on unbound, hyperbolic trajectories. Such
grains are called $\beta$-meteoroids, after the parameter $\beta$, the ratio
of radiation pressure to solar gravity (Zook & Berg, 1975).

Dust populations in the inner solar system have been observed both directly
via dedicated dust instruments, and indirectly, typically via signals produced
on spacecraft antenna. Dedicated dust measurements in the inner solar system
have been made with Pioneers 8 and 9 (Berg et al., 1973; Berg & Grün, 1973),
HEOS-2 (Hoffman et al., 1975; Hoffmann et al., 1975), Helios (Leinert et al.,
1978a, 1981; Grün et al., 1980; Altobelli et al., 2006; Krüger et al., 2020),
Ulysses (Wehry & Mann, 1999; Wehry et al., 2004; Landgraf et al., 2003;
Sterken et al., 2015; Strub et al., 2019), and Galileo (Grün et al., 1997).
These observations identified three populations of dust: larger grains on
bound orbits ($\alpha$-meteoroids), smaller grains on unbound orbits
($\beta$-meteoroids), and interstellar grains transiting the solar system.

The Parker Solar Probe (PSP) mission (Fox et al., 2016) transits the inner-
most portions of our solar system’s zodiacal dust cloud. During its first
orbit, PSP flew by the Sun with a perihelion distance of 0.17 au (36 solar
radii). Throughout the mission, PSP’s perihelion distance is incrementally
decreased via Venus flybys (Guo et al., 2021). The Solar Probe Science
Definition Team identified the exploration of “dusty plasma phenomena and
their influence on the solar wind and energetic particle formation” as one of
its four main science objectives (McComas et al., 2007). Many of the questions
to be resolved for this science objective rely on making measurements with a
dedicated dust analyzer with composition capability. While unequipped with a
dedicated dust detector, PSP registers dust impacts primarily via potential
measurements with the FIELDS instrument (Bale et al., 2016). In lieu of
measurements with a dedicated dust detector, these observations provide the
best in-situ characterization of the dust environment inside 0.3 au to date.
There has been an extensive history of measuring dust impacts with spacecraft
antenna measurements, for example: Voyager 2 (Gurnett et al., 1983), Vega
(Laakso et al., 1989), DS-1 (Tsurutani et al., 2003, 2004), Wind (Malaspina et
al., 2014; Kellogg et al., 2016; Malaspina et al., 2016), MAVEN (Andersson et
al., 2015), STEREO (Zaslavsky et al., 2012; Malaspina et al., 2015) and MMS
(Vaverka et al., 2018, 2019).

During its first three orbits, impact rates observed by PSP were consistent
with fluxes of high-speed, submicron-sized $\beta$-meteoroids leaving the
solar system on escaping orbits (Szalay et al., 2020b; Page et al., 2020;
Malaspina et al., 2020). Nano-sized grains which experience strong
electromagnetic forces were not found to be dominantly responsible for the
impact rates during PSP’s 2nd orbit (Mann & Czechowski, 2020). After six
orbits spanning three distinct orbital families (1-3, 4-5, 6), the PSP impact
rate data revealed a more complex picture than garnered from the first three
orbits. In this study, we present the first six orbits of PSP FIELDS impact
rate data in Section 2. We introduce a two-component dust model and compare
its results to the data in Section 3, showing how it reproduces the broad
features in the data. In Section 4, we present two possibilities for an
anomalous feature in the data not described by the two-component model. We
discuss implications for the upcoming orbits in Section 5 and conclude with a
discussion of these results in Section 6.

##  2 Impact Rates for Orbits 1-6

###  2.1 Dust identification and noise determination

The PSP Fields Experiment (FIELDS) (Bale et al., 2016) detects dust impacts on
the spacecraft surface. Dust grains traveling at high relative velocities ($>$
1 km/s) to the spacecraft vaporize and ionize upon impact, creating a
transient plasma cloud detected by the FIELDS electric field antennas as a
high amplitude (10’s to 1000’s of mV), impulsive (10’s of $\mu$s) voltage
spike. Since dust is a fundamental part of the ambient near-Sun environment,
and because FIELDS is sensitive to the voltage perturbations induced by dust
impacts, FIELDS unavoidably measures dust. Since the entire spacecraft surface
area acts as part of the ‘detector’, antenna-based dust detection has the
advantage of high count rates. PSP is bombarded by thousands of such impacts
each orbit, creating a statistically robust data set of dust detections (e.g.
Page et al., 2020; Malaspina et al., 2020; Szalay et al., 2020b).

FIELDS has four 2 m whip antennas in the plane of PSP’s heat shield (V1, V2,
V3, V4), and a fifth, 21 cm antenna situated along the magnetometer boom (V5).
FIELDS dust counts are most readily determined in peak detection data from the
Time Domain Sampler (TDS) receiver. The TDS records signals from each antenna
in a bandpass from a few kHz to 1 MHz. It produces a data product called
TDSmax, which reports the largest (signed) amplitude value on each configured
channel recorded by the TDS over a short time interval. The TDSmax data
product also records the number of zero-crossings associated within $\sim$13
ms surrounding the time of the largest-amplitude signal. This information is
used to distinguish dust impact voltage spikes (few zero crossings) from
plasma waves (many zero crossings).

Voltage spikes produced by impulsive dust impact voltage signals are clearly
distinguished from plasma wave-induced signals in TDSmax data by amplitude
$>$50 mV and $<$5 zero crossings (Page et al., 2020). Since PSP’s first solar
encounter in 2018, TDSmax data for the V2 monopole channel have been
continuously produced whenever FIELDS is powered on, producing a vast database
of dust impacts. In this work, we analyze impacts registered by FIELDS above
50 mV identified in the V2 TDSmax dataset. In addition to count rates, FIELDS
can determine basic directionality information for each dust impact by
comparing voltage spike signal strength across its multiple antennas
(Malaspina et al., 2020). The time domain waveform shapes of these voltage
spikes, determined via high-resolution waveform captures, are consistent with
previous observations of dust impact voltage spikes (Mann et al., 2019; Bale
et al., 2020).

Changes in solar wind plasma density and temperature cause the spacecraft
floating potential to vary and could affect dust impact observations. Because
the amplitude of measured dust voltage spikes is partially determined by the
amount of impact-liberated charge recollected on spacecraft surfaces,
variations in the spacecraft floating potential will produce some variation in
the minimum detectable dust impact voltage spike amplitude, and therefore
variation in the dust count rate. However, the spacecraft floating potential
has varied weakly over orbits 1-6, with most excursions less than 5 Volts in
magnitude (Bale et al., 2020). Therefore, we do not expect the impact count
rate to vary significantly over any given orbit due to spacecraft charging
effects when a fixed amplitude threshold is used.

The identified dust impact rates have been consistent with a “memory-less”
processes, such that the probability of receiving an impact is independent of
the probability of when the next impact will occur (Poissonian). While there
may be additional non-Poissonian sources of similar waveforms (Mozer et al.,
2020), the finding that PSP’s identified dust impact rates are highly
consistent with a Poisson process, along with their waveform similarity to
known dust impact signals, reinforces that the vast majority of observed
abrupt voltage spikes are due to dust impacts (Page et al., 2020).

###  2.2 Impact rates

Figure 1a shows the impact rates observed by the FIELDS instrument on PSP for
each orbit in high resolution for a voltage threshold of 50 mV. Errors are
given for $\sqrt{N}$ counting statistics, where $N$ is the total number of
impacts in each measurement. To discern the dust populations responsible for
these impact rates, we seek to understand the large-scale, broad features in
the data. Hence, we apply a 1 day time binning to the data, shown in Figure
1b; this work focuses on these time-binned data. Figure 2a,b shows the rates
as a function of the distance from the center of the Sun. Figure 2c shows
color-coded trajectories according to the impact rates, where orbits 1-3, and
4-5 have been averaged together, and orbit 6 is shown individually.

There are multiple features of note across each of these six orbit in Figure
2. All orbits exhibit a peak on the inbound, pre-perihelion portion of the
orbital arc. Outbound, post-perihelion, orbits 4 and 5 show a clear second
peak in their impact rate profiles, while the remaining have an extended
“shoulder” that generally decreases monotonically from the pre-perihelion
peak. The location of the first peak occurs at similar longitudes/true
anomalies across the six orbits. We seek to explain these various features and
compare the observed impact rates to a dust model described in the next
section.

Figure 1: Impact rates as a function of days from perihelion for orbits 1-6
(color coded in the legend) with an impact threshold of 50 mV for high
temporal resolution (a) and 1 day bins (b). Error bars show $\sqrt{N}$
counting statistics.  Figure 2: Daily averaged impact rates as a function of
radial distance for orbits 1-6, separated by inbound (a) and outbound (b).
Error bars show $\sqrt{N}$ counting statistics, where due to the large number
of counts the error bars are often smaller than the plotting symbols
themselves. (c) Impact rates overlaid on the PSP trajectory in the ecliptic
J2000 frame, averaged over orbits 1-3, 4-5, and individually shown for orbit
6. Color and width of the color strip represents the impact rate.

##  3 Two-component Dust Model

###  3.1 Dust populations and terminology

There have been two distinct dust populations observed by in-situ detection
methods that are directly related to our zodiacal cloud. The bulk of the
material in the zodiacal cloud is comprised of meteoroids on bound, elliptic
orbits slowly spiraling into the Sun under the influence of Poynting-Robertson
drag (e.g. Grün et al., 1985). Grains with these orbital characteristics have
been termed $\alpha$-meteoroids (Grun & Zook, 1980). These grains collide and
fragment into smaller grains, which can experience sufficient force from
radiation pressure to be gravitationally unbound from the Sun. Once unbound,
grains move along hyperbolic orbits, with trajectories that are governed by
$\beta$, the ratio of solar radiation and gravitational forces (e.g. Zook &
Berg, 1975; Burns et al., 1979). Hence, these grains are termed
$\beta$-meteoroids. We incorporate both $\alpha$-meteoroids and
$\beta$-meteoroids into a two-component model to compare with PSP impact
rates. While there may also be a minor contribution of intersteller grains
(e.g. Grün et al., 1993; Sterken et al., 2012; Strub et al., 2019), we do not
model this population as their fluxes are expected to be particularly small
given their exclusion from the inner solar system due to radiation pressure.

###  3.2 Impact Charge Generation

Dust impacts into a target (PSP) generate an impact charge according to the
relation,

$Q_{\mathrm{imp}}=Cm_{\mathrm{imp}}^{a}v_{\mathrm{imp}}^{b},$ (1)

where $m_{\mathrm{imp}}$ is the impactor mass, $v_{\mathrm{imp}}$ is the
impactor speed, and $C$, $a$, and $b$ are empirically determined constants
based on the impactor and target properties (Auer, 2001). These parameters can
vary for different surfaces, and the constant $C$ has been experimentally
determined to span a large range (Collette et al., 2014). We will perform our
fits, described in the next sections, on $\tilde{Q}_{\mathrm{imp}}\equiv
Q_{\mathrm{imp}}/C$ to remove any dependence on this coefficient. Without a
specific calibration for the various surfaces and target materials on PSP, we
assume typical values of $a=1$, and $b=3.5$ (Auer, 2001) for the calculations
and analysis below, where $m_{\mathrm{imp}}$ is in kg and $v_{\mathrm{imp}}$
in km s-1. Any detection method based on the amount of impact charge
generated, in this case via the capacitively coupled transient voltage changes
measured by FIELDS, will have a critical value of impact charge below which an
impact creates too little charge to detect, $Q_{c}$. For any given impact
speed, $Q_{c}$ sets the minimum detectable radius via

$s_{min}=\left(\frac{3Q_{c}}{2\pi\rho
v_{\mathrm{imp}}^{b}}\right)^{1/3}=C_{a}v_{\mathrm{imp}}^{-b/3},$ (2)

where $\rho$ is the bulk mass density of the impactors and
$C_{a}=(3Q_{c}/2\pi\rho)^{1/3}$. We have left $b=3.5$ as a parameter to show
the dependence on this exponent in subsequent equations. To roughly estimate
the expected impact charge threshold, we assume PSP is spherical with a
capacitance of $C_{\mathrm{PSP}}=200$ pF. The voltage amplitude generated by a
dust impact is given by $Q_{\mathrm{imp}}=C_{\mathrm{PSP}}\Delta V/\Gamma_{q}$
where $\Delta V$ is the potential change and $\Gamma_{q}$ is the efficiency
factor, typically $\sim$0.2 (Collette et al., 2015, 2016). With a detection
threshold of $\Delta V_{c}=50$ mV and $\Gamma_{q}=0.2$, the threshold impact
charge is estimated to be $Q_{c}\approx 0.3\times 10^{9}$ electrons.

###  3.3 Bound $\alpha$-meteoroids on elliptic orbits

We use the following functional form for the number density $n_{\alpha}$ of
bound $\alpha$-meteoroids on elliptic orbits,

$n_{\alpha}(r,a)=n_{0\alpha}f_{\alpha}(r)r^{-1.3}\left(\frac{s}{s_{0}}\right)^{-3\alpha}$
(3)

where $n_{0\alpha}$ is the cumulative number density at 1 au for grains with
radii greater than the reference grain size $s_{0}=1$ $\mu$m, approximating
the size distribution using a single power-law, and $r$ is the heliocentric
distance in au. Mass indices varying from 0.4 to 1.3 were observed over a
large mass range at 1 au (Grün et al., 1985) and modeling efforts suggest the
size distribution is expected to differ from this inside 1 au (Ishimoto &
Mann, 1998). To reduce model parameters, we assume a single power-law
dependence with $\alpha=0.9$, consistent with the size index for $\sim$100
$\mu$m grains at 1 au (Grün et al., 1985; Pokorný & Brown, 2016) and with the
collisionally produced size distributions assumed for the meteoroid stream
analysis discussed in this work (Appendix A). $f_{\alpha}(r)$ is given by the
following empirical relation (Stenborg et al., 2020) determined via fits to
the Wide-field Imager for Parker Solar Probe (WISPR) remote sensing data
(Vourlidas et al., 2016),

$f_{\alpha}(r)=\left\\{\begin{array}[]{rl}0&\ \ \text{if }r<r_{0},\\\
\frac{r-r_{0}}{r_{1}-r_{0}}&\ \ \text{if }r_{0}\leq r\leq r_{1},\\\ 1&\ \
\text{if }r>r_{1}.\end{array}\right.$ (4)

Equation 4 matches the Helios remote observations outside $r>r_{1}$ (Leinert
et al., 1978b) and provides an estimated representation for how the density
would trend inside $r_{1}$ towards the dust free zone (e.g. Leinert et al.,
1978a; Stenborg et al., 2020) approximated here to occur at
$r_{0}=5R_{\odot}$. We use a value of $r_{1}=19R_{\odot}$ as empirically
determined by comparison with WISPR data (Stenborg et al., 2020), which is
inside the perihelion distance for orbits 1-6 discussed in this work. All
grains are assumed to be on perfectly circular orbits with orbital speeds of
$v_{orb}=\sqrt{\mu/r}$, where $\mu=1.327\times 10^{20}$ m3 s-2 is the Sun’s
gravitational parameter. At any given location along PSP’s orbit, the observed
zodiacal impact flux is,

$\Phi_{\alpha}(r)=n_{0\alpha}f_{\alpha}(r)(s_{0}/C_{a})^{3\alpha}v_{\mathrm{imp}}^{1+\alpha
b}r^{-1.3}.$ (5)

The impact speed between $\alpha$-meteoroids and PSP is given by (e.g. Szalay
et al., 2020b),

$v_{\mathrm{imp}}^{2}=\mu\left(\frac{3}{r}-\frac{1}{a}-2\sqrt{\frac{a(1-e^{2})}{r^{3}}}\right),$
(6)

where $a$ and $e$ are Parker Solar Probe’s semi-major axis and eccentricity,
respectively, and PSP and all grains are assumed to have inclination $i=0$.
For bound grains on circular orbits, we assume they have the density and
$\beta$ characteristics of nominal old cometary grains (Wilck & Mann, 1996),
such that grains with radii above 0.5 $\mu$m have $\beta<0.5$ and are bound.

###  3.4 Unbound $\beta$-meteoroids on hyperbolic orbits

Initial analyses assumed all $\beta$-meteoroids were produced at a single
location and had the same $\beta$ value (Szalay et al., 2020b; Page et al.,
2020; Malaspina et al., 2020). Here, we utilize a forward-model to track
$\beta$-meteoroid trajectories from a dispersed source region with a
distribution of masses and $\beta$ values. We model the dynamics of
$\beta$-meteoroids under the attractive solar gravitational force and
repulsive solar radiation force: $F=\mu(1-\beta)/r^{2}$. Grains collisionally
produced at perihelion of their parent body’s orbit will be unbound from the
Sun’s gravity for $\beta\geq 0.5(1-e)$ where $e$ is the eccentricity of the
parent body (Grün et al., 1985). For example, $\beta$-meteoroids produced from
a parent grain on a perfectly circular orbit are unbound for $\beta\geq 0.5$.
The value of $\beta$ is dependent on the grain composition, albedo, porosity,
and size. We assume collisional products are similar to canonical “young”
cometary grains (Wilck & Mann, 1996) to determine the relation between mass
and $\beta$. Assuming collisions in the nominal zodiacal cloud occur between
grains on perfectly circular orbits, collisional products with radii less than
1.2 $\mu$m are unbound and become $\beta$-meteoroids.

We estimate PSP impact rates using the following numerical scheme. In a
uniform grid of 100 bins in radial position from 0 to 1 au, we release
$\beta$-meteoroids and track their location and speed as they travel away from
the Sun. The simulation is performed in 1D, assuming PSP and all grain
trajectories are in the same plane. From their radial speed, we accumulate
residence times for grains released at all initial positions. We assume the
mass distribution follows a power-law with differential slope $\alpha=0.9$
(Appendix A) and follow grain trajectories within 50 logarithmically spaced
mass bins corresponding to a radius range of 50 nm to 1.2 $\mu$m. We normalize
residence times from grain trajectories according to the size and collisional
mass production distribution to account for which initial starting locations
and masses contribute to the density at any location.

To approximate the functional form of collisional mass production, we follow
previous conventions (Zook & Berg, 1975; Steel & Elford, 1986), by assuming
the collision rate per unit distance is proportional to
$n_{\alpha}(r)^{2}v(r)\sigma(r)$, where $n_{\alpha}$ is the spatial zodiacal
density given in Equation 3, $v$ is the average impact speed, and $\sigma$ is
the cross sectional area. The average impact speed is assumed to follow the
Keplerian trend proportional to $r^{-0.5}$. Taking into account the threshold
energy to catastrophically destroy grains (Appendix A, Eq. A4), we approximate
the cross-sectional area to be proportional to $r^{-\alpha+2/3}$ . Therefore,
we represent the differential collisional mass production as,

$M^{+}(r)dV\propto n_{\alpha}(r)^{2}v(r)\sigma(r)4\pi r^{2}dr\propto
f^{2}_{z}r^{-1.33},$ (7)

where the integral over $\int M^{+}(r)dV=M_{\mathrm{tot}}$ is a fit parameter.
The density of $\beta$-meteoroids detectable by PSP is then
$n_{\beta}(r,a)\propto n_{\beta}(r)s_{min}^{-3\alpha}\propto
n_{\beta}v^{\alpha b}$. The incident impact flux of $\beta$-meteoroids
observed by PSP is

$\Phi_{\beta}=n_{\beta}(r,a)v(r)\propto n_{\beta}v_{imp}^{1+\alpha b}.$ (8)

###  3.5 Model fits

For any combination of the three fit parameters ($n_{0\alpha}$,
$\tilde{Q}_{c}$, and $M^{+}_{\mathrm{tot}}$), the model can produce synthetic
impact rates to the PSP spacecraft. The fits are performed using a $\chi^{2}$
minimization (Markwardt, 2009) between synthetic forward-modeled impact rates
and the rates observed by PSP. We individually fit each orbit, with fit values
given in Table 1.

Orbit | $n_{0\alpha}$ | $\tilde{Q}_{c}$ | $M^{+}_{\mathrm{tot}}$ |  $\Phi_{\beta}$(1 au)
---|---|---|---|---
|  (km${}^{-3})$ |  $(10^{9}$ e-) |  (kg s-1) |  $10^{-4}$ m-2 s-1
1 | 0.3 | 1.4 | 60 | 0.4
2 | 0.1 | 1.0 | 110 | 0.6
3 | 0.3 | 1.8 | 100 | 0.6
4 | 0.6 | 3.4 | 85 | 0.5
5 | 0.6 | 3.4 | 110 | 0.6
6 | 0.7 | 6.6 | 100 | 0.8
1-6 | 0.5 | 3.4 | 100 | 0.6
Table 1: Two-component fit values for individual orbits and all orbits
together. Columns 2-4 give the model fits directly from the two-component
model, while the last column ($\Phi_{\beta}$) gives the expected flux of
$\beta$-meteoroids at 1 au.

Figures 3 and 4 show the results of the model fits, where each orbit arc is
shown separately in Figure 3 along with model curves for all orbits up to and
including 24 to illustrate the expected profiles to be encountered in future
orbits. In Figure 4, the top sub-panels show the expected impact speeds. The
2nd sub-panels show the minimum detectable size threshold corresponding to the
fitted $\tilde{Q}_{c}\equiv Q_{c}/C$. These values of charge threshold, given
in Table 1, are similar to our back of the envelope estimate of $Q_{c}\approx
0.3\times 10^{9}$ e- for $C=0.04-0.2$, which is well within the range of
expected values for relevant spacecraft surface materials (Collette et al.,
2014). The 3rd sub-panels show the PSP data along with model rates for
$\alpha$-meteoroids, $\beta$-meteoroids, and combined total impact rate. The
last subpanels show the difference between the data and modeled rates.

Figure 3: Model fits for each orbit for inbound (pre-perihelion) and outbound
(post-perihelion) arcs. Individual PSP data shown as the black dots. The
individual $\alpha$-meteoroid and $\beta$-meteoroid fits are shown in the
green and purple lines respectively, where their sum is given by the black
overall impact rate fit. Fits for all PSP orbits up to orbit 24 are duplicated
in each panel, shown with grey lines. Figure 4: Model fits for each orbit.
Sub-panels show: modeled impact speed, minimum detectable size, data and model
impact rates, and the difference between data and model rates for orbits 1-6.

As perihelion is reduced, each successive orbital group shows an increasing
prominence in $\alpha$-meteoroids vs. $\beta$-meteoroids. For orbits 1-3, the
large majority of the impact rate profile can be reproduced with
$\beta$-meteoroids , notably for orbit 2 where almost no $\alpha$-meteoroids
are found to contribute to the impact rates. For orbits 4 and 5, which have a
lower perihelion distance, the post-perihelion portion has a region that is
dominanted by $\alpha$-meteoroids . Finally, for orbit 6, both the pre-
perihelion and post-perihelion arcs are dominanted by $\alpha$-meteoroids near
close approach.

Our model is able to reproduce the overall structure of the PSP measured
impact rates for each orbit with three notable exceptions. The first is
outside $\sim$0.7 au, where the model predicts larger counts than observed by
PSP. This feature is likely due to the non-standard orientation of the
spacecraft due to communications attitude changes via yaw maneuvers at the
outermost portions of the orbit. For example, the lower counts in Orbit 2’s
inbound arc before and after $\sim$0.75 au exhibit a step-like change. This
change exactly coincides with when PSP’s attitude transitioned from a non-
standard orientation with the heat shield off-pointed from the solar direction
to one in which PSP’s heat shield was directly pointed to the solar direction.
We use an effective collecting area assuming the PSP +X vector is always
aligned with the ram direction and that the +Y vector is aligned with PSP’s
orbital plane normal vector, which is often not the case outside $\sim$0.7 au
(see Malaspina et al. (2016) for spacecraft coordinate definitions). Outside
$\sim$0.7 au, PSP is often oriented with rotations about multiple axes
compared to its nominal solar pointed ram configuration. Estimating the
effective area more precisely for these configurations would require time-
dependent 3D modeling of the spacecraft geometry and could be investigated in
future studies.

The second feature not predicted by the model is the magnitude of the pre-
perihelion peak impact rates. A pre-perihelion peak in impact rate is expected
for $\beta$-meteoroid impactors due to the enhanced impact speed on the
inbound orbit arc (Szalay et al., 2020b). Since the impact rate from
$\alpha$-meteoroids is symmetric about perihelion, this leads to modeled rates
that always peak before perihelion. While the model predicts such a feature
and is able to reasonably reproduce the location of this peak, it is not able
to fully reproduce the large impact rates observed by PSP. Specifically,
orbits 2, 4, and 5 exhibit pre-perihelion enhancements over the model fits of
approximately 10 hr-1, 20 hr-1, and 35 hr-1 respectively.

The final aspect for which the model deviates from the data is in the post-
perihelion rates within $\sim$5 days from perihelion. Across all six orbits
discussed here, the model is deficient in reproducing larger impact rates in
this region. During orbits 1-3, and 6, this enhancement manifests itself as an
extended “shoulder” on the impact rate profile, which still monotonically
decreases from the expected pre-perihelion peak. During orbits 4-5, this
enhancement forms a second peak in the total observed impact rates, with the
data-model difference also clearly showing this unexpected feature. The
difference between data and model rates for the post-perihelion enhancement
maximizes during orbits 4-5. No such feature is predicted by the two-component
model and we present possible explanations for this post-perihelion
enhancement in Section 4.

The model also allows for a comparison with the fitted impact threshold across
the orbits. Figure 5 shows the impact speed and size ranges for the two dust
populations in this model. Gray contours show iso-impact-charge features,
where any impact on a specific contour is expected to generate the same impact
charge and therefore the same impact voltage spike magnitude. The fitted
$\tilde{Q}_{c}$ values are shown in black and the modeled $\alpha$-meteoroid
and $\beta$-meteoroid populations are marked with the two boxes. The shaded
portions of each box shows the range of detectable impactors for each
population according to the model fits. As shown here, for the similar values
of $\tilde{Q}_{c}$, PSP is expected to be able to detect a large portion of
available impactor populations. It also shows that even with the difference in
both speed and mass of the two populations, the impact charge relation
suggests both populations are able to be detected each orbit
contemporaneously.

Figure 5: Impact speed and size ranges for the two-component model. Grey lines
show iso-impact charges following $Q_{imp}\propto mv^{3.5}$, with model fitted
detection thresholds ($\tilde{Q}_{c}$) shown with the solid line in each sub-
panel. Green and purple boxes show the simulated ranges of $\alpha$-meteoroids
and $\beta$-meteoroids respectively, with the shaded portion indicating the
detectable range.

###  3.6 Collisional Production Rate in the Zodiacal Cloud

Figure 6: Collisional production rate in the zodiacal cloud. a) Forward-
modeled production rate per radial distance of $\beta$-meteoroids consistent
with the observed impact rates to PSP. The majority of zodiacal collisions
producing $\beta$-meteoroids occur within approximately 10 to 20 solar radii.
b) $\beta$-meteoroid density derived from the PSP impact fluxes and forward
model. Quantities are displayed as a function of distance from the center of
the Sun in au (bottom) and solar radii (top).

Given the model fits to the zodiacal dust cloud, we can provide estimates on
the total collisional production rate of grains in the inner solar system.
Volumetric mass loss rates are initially calculated in the model assuming
$\beta$-meteoroid fluxes are uniformly distributed throughout all inclinations
as the model is inherently one dimensional. The conversion from observed flux
back to $M^{+}$ assumes a full-sky averaged flux as an input. In reality, the
zodiacal cloud has an inclination distribution that consists primarily of low-
inclination orbits (e.g. Nesvorný et al., 2010). Since PSP observed the flux
near the ecliptic plane, which is larger than the latitudinally averaged flux,
we must apply a correction factor. Using a latitudinal dependence (Leinert et
al., 1978b) proportional to $f(\lambda)=1-\sin{|\lambda|}$, we scale the model
mass production values by $\langle f(\lambda)\rangle=0.36$. Table 1 shows the
results of the fitted total mass production rates for fits of each orbit
separately, and combined across all of the first six orbits in the last row.

The model predicts that the majority of collisions producing
$\beta$-meteoroids that stream throughout the solar system occur within 10 to
20 solar radii (Figure 6a). Inside of $\sim$10 solar radii, the modeled
$\beta$-meteoroid density monotonically decreases toward the Sun (Figure 6b),
indicating a precipitous drop in collisional production near the Sun and
suggestive that the size distribution of zodiacal material changes as a
function of radial distance (Ishimoto & Mann, 1998). The location of peak
production is consistent with a collisional source mechanism eroding the
zodiacal cloud, as opposed to sublimation driven erosion where grains are
expected to sublimate inside $\sim$10 solar radii (Mann et al., 2004). Across
all orbits, we find a zodiacal collisional production rate of at least
$M^{+}=100$ kg s-1 is consistent with the fluxes of $\beta$-meteoroids
observed by PSP. We have assumed that all collisional products are distributed
into $\beta$-meteoroids, hence this number is likely an underestimate, which
may explain why it is approximately 5 to 10 times lower than previous
collisional estimates (Grün et al., 1985).

If sub-micron sized grains are responsible for the abundance of pickup ions
near the Sun (Schwadron et al., 2000), we expect pickup ion production would
be enhanced near the peak $\beta$-meteoroid density location around 10 to 20
solar radii. Previous estimates found a lower limit for the dust geometric
cross-section of $\Gamma_{\sigma}\geq 1.3\times 10^{-17}$ cm-1 is needed to
produce inner source pickup ions (Schwadron et al., 2000), almost 2 decades
larger than the value used previously (Holzer, 1977; Fahr et al., 1981;
Gruntman, 1996), $\Gamma_{\sigma}=2\times 10^{-19}$ cm-1. A geometric cross
section for inner source pickup ions of $\Gamma_{\sigma}\geq 1.3\times
10^{-17}$ cm-1 is 4 decades larger than the $\beta$-meteoroid geometric cross
section found here based on collisional production in the zodiacal cloud; this
discrepancy raises the question about whether dust from zodiacal can explain
the inner source.

The distribution of inner source pickup ions indicates that the grain
population producing the inner source peaks at $\sim$10 $R_{\odot}$, which is
within the region of maximum $\beta$-meteoroid collisional production between
10 and 20 $R_{\odot}$ found here. While not modeled here, collisional
fragmentation can also produce grains with radii $\lesssim 50$ nm. This
population of nanograins, which are highly susceptible to electromagnetic
forces, can become trapped in the inner-most regions of the solar system
inside $\sim$30 $R_{\odot}$(Czechowski & Mann, 2010). Impacts from nanograins
in the range of $30-40$ nm have been estimated to have impact speeds in the
range of $20-45$ km s-1 during PSP’s first three orbits (Mann & Czechowski,
2020). Such impacts would produce an impact charge of approximately
$10^{5}-10^{6}$ electrons, orders of magnitude lower than the estimated
detection threshold of $\tilde{Q}_{c}\sim 10^{9}$ electrons even accounting
for a few orders of magnitude variation in $C$ (Collette et al., 2014). Hence,
this population of nanograins is not expected to be directly detectable in the
FIELDS impact data.

Dust grains can serve as the neutralizing agent for solar wind ions that
penetrate these grains, and could potentially explain the large fluxes of
inner source pickup ions observed (Schwadron et al., 2000). Wimmer-
Schweingruber & Bochsler (2003) suggest that the size of such dust particles
is less than or comparable to the penetration range of solar wind ions in dust
material, $<$100 nm. Thus, the collisional production of zodiacal grains
discussed here also suggests the generation of extremely small dust particles
that may explain the origin of inner source pickup observed in the inner
heliosphere (Schwadron et al., 2000). We have ruled out larger
$\beta$-meteoroids for the inner source of pickup ions, given their low
geometric cross section due in part to their relatively short lifetimes in the
inner solar system. If dust is responsible for generating the inner source,
then it must be from nanograins with radii below $\sim$50 nm. While this
nanograin population is likely not detectable directly with FIELDS
measurements, PSP may be able to observe pickup ions in the inner solar system
directly with PSP’s SWEAP instrument (Kasper et al., 2016). These pickup ions
can also seed energetic particles accelerated via shocks in the inner solar
system that could be detected by the IS$\odot$IS instrument (McComas et al.,
2016, 2019). Hence, the inner-most PSP orbits may yield critical information
on the inner-source of pickup ions across its in-situ instrument suites.

##  4 Post-perihelion enhancement

A post-perihelion impact rate enhancement is not predicted by the two-
component model and is observed in every orbit in varying magnitudes. Since
PSP cannot directly measure dust density and speed distributions, we use a
model to estimate these quantities from count rates. The enhancement in the
peak rates above the two-component model rates in orbits 4-5 amounts to a
measured flux of approximately $3\times 10^{-3}$ m-2 s-1, which correspond to
densities in the range $30-300$ km-3 for $v_{imp}=10-100$ km s-1. We
investigate two possibilities to explain these post-perihelion enhancements,
both of which are related to meteoroid streams: direct meteoroid stream
encounters or $\beta$-meteoroids created by collisions between meteoroid
streams and the nominal zodiacal cloud.

###  4.1 Direct meteoroid stream encounter

The estimated densities necessary to account for the post-perihelion
enhancement are similar to densities of meteoroid streams inferred by Helios
in-situ dust measurements of $10-100$ km-3 (Krüger et al., 2020). Applying a
similar analysis, we utilize the Interplanetary Meteoroid Environment for
eXploration (IMEX) model (e.g. Soja et al., 2019) to estimate the number flux
PSP encounters near transits with cometary orbits. This model tracks the
trajectories of grains with radii greater than 100 $\mu$m that are released
due to cometary activity to build up a density distribution along the comets
orbit. It accounts for gravitational and solar radiation forces acting on dust
grains, as well as the collisional lifetimes. Grains with radii smaller than
100 $\mu$m are also likely to exist in a more extended cross-sectional area,
however, they dynamically decouple from their parent comets’ orbits much more
rapidly for smaller sizes.

Figure 7: Orbits of meteoroid stream parent bodies that transit near PSP’s orbit 4-5 post-perihelion impact rate enhancement in the ECLIPJ2000 frame for side (a) and top-down (b) views.  Body | $d_{\mathrm{min}}$ | $v_{\mathrm{imp}}$ | $n_{\mathrm{lim}}$
---|---|---|---
| (au) ($R_{\odot}$) | (km s-1) | (km-3)
(3200) Phaethon | 0.056 (12) | 62 | 48
342P/SOHO | 0.0069 (1.5) | 54 | 55
323P/SOHO | 0.017 (3.7) | 131 | 23
96P/Machholz | 0.12 (25) | 87 | 34
Table 2: Meteoroid stream candidates. At the time of closest approach of each
orbit to PSP’s orbit, $d_{\mathrm{min}}$ is the minimum distance,
$v_{\mathrm{imp}}$ is the impact speed, and $n_{\mathrm{lim}}$ is the density
limit that would be needed to generate an impact flux of $3\times 10^{-3}$ m-2
s-1 at PSP.

We performed a comprehensive search across all known comets in the inner solar
system to determine which orbits have the closest approach points near the
region of peak impact rate enhancement in PSP’s orbits $4-5$. From this
search, we identified three cometary candidates: 342P/SOHO, 323P/SOHO, and
96P/Machholz (Table 2). We also investigate the Geminids meteoroid stream,
whose parent body is likely asteroid (3200) Phaethon. Figure 7 shows these
four orbits along with the PSP orbits and region of enhanced post-perihelion
impact rates for orbits 4-5. Dots mark the location on each orbit closest to
the Orbit $4-5$ enhancement. As shown in this figure, the stream parent bodies
identified have close transits with PSP’s orbit, with 342P transiting the
nearest to the region of interest. For the cometary candidates, only one had
sufficient brightness data to constrain the stream number densities: 96P. For
this stream, we find its impact rates would peak a few days before the
observed enhancement with an upper limit of $\sim$10-3 hr-1, making it an
unlikely explanation for the post-perihelion enhancement.

Estimating the peak rates for 342P and 323P requires comprehensive knowledge
of the absolute brightness of these comets not available at the time of this
analysis. Of these two candidates, 323P/SOHO does not exhibit activity like a
normal comet (Battams & Knight, 2017); its activity is similar to 322P/SOHO
which exhibits activity more similar to an asteroid than a comet (Knight et
al., 2016). Thus, dust production from 323P is likely minimal, and a
substantial debris trail is unlikely. While we cannot directly estimate these
meteoroid stream densities, with knowledge of the impact speeds from each
stream, we can estimate the required densities necessary to produce the orbit
4-5 enhancement, $n_{\mathrm{lim}}$ given in Table 2. These density limits are
at the position of PSP. Given that PSP flies within a few solar radii of 342P
and 323P, these density limits are applicable to core of the stream. Note,
$n_{\mathrm{lim}}$ is the limiting density needed to produce the observed
impact rates; it does not signify that any of these meteoroid streams actually
have these densities, as that would require an exact determination of the
source of the PSP impact rate enhancement.

Phaethon, the parent body of the Geminids meteoroid stream, does not
continually replenish this stream in sufficient quantities via the relatively
low activity level of active asteroid Phaethon (e.g. Jewitt & Li, 2010; Jewitt
et al., 2013), nor is meteoroid bombardment of Phaethon sufficient to source
the stream (Szalay et al., 2019). However, unlike the other meteoroid streams
discussed here, we have direct observations of its flux. At 1 au, the number
flux of Geminids with radii $\geq$100 $\mu$m is $4\times 10^{-11}$ m-2 s-1,
which corresponds to a number density of $1.2\times 10^{-6}$ km-3 for their
heliocentric speed of 33 km s-1 (Blaauw, 2017). Its closest encounter with the
PSP trajectory is at $\sim$0.14 au, where the Geminids have a heliocentric
speed of $\sim$109 km s-1. Scaling the density as $r^{-1.5}$ (Appendix A) at
0.14 au gives an expected density of $2\times 10^{-5}$ km-3. With a relative
impact speed between PSP and the Geminids of 62 km s-1, an effective area of 6
m2, and assuming PSP flew directly through the core of this meteoroid stream,
PSP could expect impact rates of $3\times 10^{-5}$ hr-1 for grains with radii
$\geq 100$ $\mu$m, well below the rates necessary to explain the impact rate
feature. The cumulative mass index of the Geminids was found to be
$\alpha_{g}=0.68$. Even if we assume a power-law size distribution holds to
radii as small as 1 $\mu$m, the expected impact rate from grains $\geq$1
$\mu$m would be 0.4 hr-1, still significantly lower than the $\sim$40 hr-1
observed in the enhancement. PSP also does not transit directly through the
core of the stream, its close approach point during the enhancement period is
approximately 0.06 au from the core of the stream. Therefore, we do not
attribute the enhancement to direct Geminids observations, however, we explore
another alternative related to the Geminids in the next section.

###  4.2 $\beta$-stream encounter

As an alternative to a direct meteoroid stream encounter, the post-perihelion
enhancement could be due to $\beta$-meteoroids produced by collisions between
meteoroid streams and the nominal zodiacal dust distribution. The timing and
location of the post-perihelion enhancement coincide with PSP’s intersection
of $\beta$-meteoroid trajectories produced by collisions between the Geminids
meteoroid stream and the nominal zodiacal cloud (Fig. 8), which we term a
“$\beta$-stream”. From previous estimates of the Geminids collisional lifetime
and total stream mass, the stream losses $\sim$6 to 40 kg s-1 from collisions
and would register impact rates on PSP in the range of $\sim$0.1 to 30 hr-1
during the post-perihelion enhancement period for orbits $4-5$ (Appendix A).
The enhanced impact rate features post-perihelion above the modeled rates are
therefore qualitatively consistent with Geminids $\beta$-stream fluxes. Figure
8 shows the possible detection geometry for a Geminids $\beta$-stream. The
blue contours show the approximate density profile of larger meteoroids in the
Geminids stream, integrated from $>$14,000 orbits of visual meteors
(Jenniskens et al., 2018). The gray fan-like shape shows the example path
taken by $\beta$-meteoroids for $\beta=0.7$ produced in the region in the
Geminids (grey tube) that would account for 95% of potentially detectable
fluxes during PSP’s orbits $4-5$.

A $\beta$-stream detection could also explain why PSP observed a post-
perihelion enhancement for each orbit, as the $\beta$-stream would be expected
to be dispersed and could intersect multiple orbits. However, to reach PSP
from the core of the Geminids stream, the collisional process must generate
fragments with transverse spread $\sim$10-20∘ out of plane above their
original trajectories (Figure 11). If the enhancement is due to the Geminids
$\beta$-stream, it suggests the Geminids stream could be more spatially
extended than inferred from the core of larger grains which cause visual
meteors at Earth (Appendix B).

There are many additional meteoroid streams which could produce
$\beta$-streams detectable by PSP, listed in the Appendix in Table 4. The
Geminid $\beta$-stream is preliminarily a favorable candidate and initial
analyses on the directionality inferred by FIELDS observations also support
this source mechanism (Pusack et al., 2021). A comprehensive comparison
between multiple meteoroid stream origins along with directionality and
amplitude analysis would be crucial in determining the source of the post-
perihelion enhancement.

Figure 8: $\beta$-stream geometry. Impact rates overlaid on the PSP trajectory
in the ecliptic J2000 frame, averaged over orbits 1-3 for the outer-most orbit
and individually shown for orbits 4-6. Color and width of the color strip
encode rate. The gray fan-like shape shows the example path taken by
$\beta$-meteoroids for $\beta=0.7$ produced in region in the Geminids that
accounts for 95% of detectable fluxes during PSP’s orbits 4-5. Blue contours
show the approximate density profile of larger meteoroids in the Geminids
stream.

##  5 Implications for PSP’s Inner Orbits

###  5.1 Predicted impact rates

After orbit 7, PSP’s perihelion is inside $r_{1}=19R_{\odot}$ (Table 3), such
that the density of bound $\alpha$-meteoroids on elliptic orbits is no longer
expected to increase with a power-law relation. Therefore, inside this
critical distance, the PSP impact rates will be more governed by the relative
impact speed to dust. Figure 9 shows the expected impactor parameters for the
two-component model in the upcoming orbital groupings, along with the peak
impact speeds and rates for $\alpha$-meteoroids and $\beta$-meteoroids. Figure
3 also highlights the expected impact rates for subsequent orbits in the four
grey curves with the largest impact rates.

Figure 9: Model predictions for PSP orbits 8-24, with sub-panels described in
Figure 4.

###  5.2 Deposited Impact Energy to PSP

Dust presents a hazard to the PSP spacecraft, both via direct impacts to
instruments and to the spacecraft subsystems. On orbit 2, the IS$\odot$IS
instrument registered a permanent increase of noise in one of its 80
aperatures which has been attributed to a dust impact (Hill et al., 2020). The
timing and orientation of the affected aperture during the noise event
suggested the impact was a $\beta$-meteoroid (Szalay et al., 2020b), as it
occurred exactly when the flux of $\beta$-meteoroids to that aperture was
expected to peak and the orientation at that time would have prevented
$\alpha$-meteoroids on perfectly circular orbits from impacting the aperture
foil. Initial models suggested a total of 1-10 more impacts of this type could
occur between orbits 1-6 (Szalay et al., 2020b), however, no more apertures on
IS$\odot$IS have registered a similar permanent noise-inducing event.
Additionally, the WISPR instrument often records images which are dominated by
features attributed to impact ejecta spray from the spacecraft due to dust
impacts.

Orbit | $r_{\textrm{ph}}$ | $v^{\textrm{max}}_{\alpha}$ | $v^{\textrm{max}}_{\beta}$ | $R_{\alpha}$ | $R_{\beta}$ | $R_{tot}$ | $\frac{E_{\alpha}}{E_{\alpha}(6)}$ | $\frac{E_{\beta}}{E_{\beta}(6)}$ | $\frac{EF^{\textrm{max}}_{\alpha}}{EF^{\textrm{max}}_{\alpha}(6)}$ | $\frac{EF^{\textrm{max}}_{\beta}}{EF^{\textrm{max}}_{\beta}(6)}$
---|---|---|---|---|---|---|---|---|---|---
|  (au) ($R_{\odot}$) |  (km s-1) |  (km s-1) |  (hr-1) |  (hr-1) |  (hr-1) |  |  |  |
1-3 | 0.16 (36) | 39 | 160 | 6 | 31 | 33 | 0.26 | 0.62 | 0.16 | 0.37
4-5 | 0.13 (28) | 46 | 163 | 20 | 49 | 59 | 0.45 | 0.71 | 0.36 | 0.58
6-7 | 0.095 (20) | 57 | 167 | 51 | 83 | 130 | 1 | 1 | 1 | 1
8-9 | 0.074 (16) | 66 | 171 | 79 | 110 | 187 | 1.7 | 1.2 | 2.2 | 1.5
10-16 | 0.062 (13) | 74 | 174 | 88 | 120 | 212 | 2.2 | 1.3 | 3.7 | 1.9
17-21 | 0.053 (11) | 81 | 176 | 97 | 140 | 228 | 3.0 | 1.5 | 5.3 | 2.3
22-24 | 0.046 (9.9) | 89 | 179 | 110 | 150 | 253 | 3.7 | 1.6 | 7.3 | 2.7
Table 3:  Predicted quantities during each of PSP’s orbital groups.
$r_{\textrm{ph}}$ $-$ perihelion distance, $v^{\textrm{max}}$ $-$ peak impact
speed, $R$ $-$ peak detectable impact rate, $E$ $-$ total deposited energy
throughout the orbit, and $EF^{\textrm{max}}$ $-$ peak energy flux. Ratios are
given with respect to values during orbit 6.

Table 3 gives an indication of the expected maximum impact speed and rate or
all orbit groups. By orbit 6, the $\beta$-meteoroid impactor population has a
peak impact speed of 167 km s-1, not far from its overall mission peak of 179
km s-1. Between orbit 6 and 24, the impact rate of $\beta$-meteoroids is
expected to approximately double. Therefore, the $\beta$-meteoroid population
is expected to minimally increase in overall impact energy for the remainder
of the mission. On the other hand, the $\alpha$-meteoroid population increases
significantly in both impact speed and total impact flux. We estimate the peak
and total deposited energy for each orbit. The peak deposited energy flux is
determined by multiplying the total peak impact rate (independent of detection
threshold) by the average energy per impact. The total deposited energy is the
integral of the impact rate multiplied by average energy per impact along the
entire orbit. Table 3 lists these quantities as ratios to their values during
orbit 6 to give an indication of how much these quantities increase in
subsequent orbits. As given in this table, the peak energy flux is expected to
be as much as $\sim$7 times larger during orbits $22-24$ compared to orbit 6.
Additionally, the possibility exists that PSP could directly encounter
meteoroid streams from cometary activity, posing a hazard in addition to those
discussed here.

##  6 Discussion and Conclusions

PSP impact rates carry critical information on the collisional environment in
the inner solar system. Initial analyses of the first three PSP orbits
concluded the impact rates were consistent with $\beta$-meteoroids (Szalay et
al., 2020b; Page et al., 2020; Malaspina et al., 2020), with a subsequent
analysis finding electromagnetic forces were mostly negligible for the
dynamics of the observed impactor population (Mann & Czechowski, 2020). With
six orbits of data spanning three separate orbital families, the impact rate
profiles have revealed additional substructure that is not explainable by
$\beta$-meteoroids alone, particularly after orbit 3. Given that PSP does not
have a dedicated dust detector capable of measuring the mass and/or velocity
distributions of impacting grains, a number of assumptions must be made to
interpret this dataset, simplify the data-model comparison framework, and
minimize the number of free modeling parameters. We find that the broad,
overall impact rate profiles are consistent with two dust components comprised
of bound $\alpha$-meteoroids on elliptical orbits and unbound
$\beta$-meteoroids on hyperbolic orbits; additional more localized sources
might account for smaller-scale features.

Figure 10: Summary showing the various identified and potentially detected
dust populations along with data from orbit 6 (inbound) and orbit 4 (outbound)
to highlight characteristic features observed across orbits 1-6.

Figure 10 highlights the various dust populations discussed in this work, as
well as the three notable features in the data that the model is not able to
fully reproduce. First, outside $\sim$0.7 au, the model predicts larger impact
rates than observed due to non-standard and rapidly changing spacecraft
orientations which reduce PSP’s effective area. Second, while the model
predicts a pre-perihelion impact rate peak for each orbit due to the
asymmetric impact speeds for $\beta$-meteoroids during the inbound and
outbound portions of each PSP orbit, it is not able to fully reproduce the
magnitude of this enhancement for a subset of the first six orbits. A
comprehensive dynamical model (e.g. Pokorný & Kuchner, 2019) could potentially
reveal additional substructure not captured due to the assumptions involved in
the idealized two-component model discussed here and may better predict this
pre-perihelion enhancement. We note the impact rate due to grains with radii
$\sim$10’s nm that experience non-negligible electromagnetic forces are
expected to be even more asymmetric than those considered here (Mann &
Czechowski, 2020). Such a population, which was not modeled here, is expected
to sharply peak in impact rate before perihelion. However, such a population
is not expected to be directly detectable, as it would generate impact charges
orders of magnitude lower than our estimated impact threshold.

The largest discrepancy between the data and model occurs a few days after
perihelion. All six orbits’ impact rate profiles exhibit an enhancement above
the model predictions during this period, most evident in the sharp second
peaks during orbits 4 & 5\. We suggest two possible explanations originating
from a meteoroid stream: 1) direct encounters with meteoroid streams or 2)
$\beta$-streams, the collisional byproducts of meteoroid streams. A
$\beta$-stream from the Geminids meteoroid stream is a favorable candidate and
estimates suggest such a process could provide the necessary fluxes to explain
this feature. It is also favorable as such a population could potentially be
observable across multiple of PSP’s orbital families. However, determining the
origin for this impactor population presents difficulties, particularly given
that FIELDS is not a dedicated dust instrument. Additional information on the
directionality and amplitude distribution has provided additional critical
insight specifically on the orbit 4 enhancement and preliminarily favors the
Geminids $\beta$-stream origin (Pusack et al., 2021); a comprehensive
comparison of directionality and amplitude distributions across multiple
orbits would help further characterize the origin of this third impactor
population.

If the post-perihelion enhancement is due to a Geminids $\beta$-stream, it
would be the first direct observation that asteroidal and cometary debris
trails collisionally erode as they transit the zodiacal cloud. Hence, PSP may
be an efficient laboratory to investigate collisional processes and could
monitor the activity and catastrophic breakup of asteroids and comets in the
inner solar system. It could also reveal additional meteoroid streams, either
directly or through their $\beta$-streams, that would be difficult to detect
via other means; such a detection would be similar to the observation of an
intense meteoroid shower at Mercury attributed to Comet Encke, which does not
intersect Earth’s orbit (Christou et al., 2015). Many of the candidate parent
bodies for the enhancement have low activity. If the enhancement is due to one
these streams, it could indicate additional streams similar to the Geminids
exist, where the total stream mass is comparable to the mass of the parent
body likely as the result of a catastrophic event in the past few thousand
years.

By fitting to the observed PSP impact rates, we estimate the total zodiacal
collisional mass production rate to be at least $M^{+}=100$ kg s-1,
approximately an order of magnitude lower than previous collisional estimates
(Grün et al., 1985). The discrepancy between these two mass estimates is
likely related to our assumption that all collisional products become
$\beta$-meteoroids, suggesting our estimate is a lower limit. Propagating the
model results to 1 au, the flux of $\beta$-meteoroids (Table 1) is estimated
to be $0.4-0.8\times 10^{-4}$ m-2. This is similar to previous PSP estimates
of $0.3-0.7\times 10^{-4}$ m-2 based on a simpler $\beta$-meteoroid model
(Szalay et al., 2020b) and $0.4-0.5\times 10^{-4}$ m-2 from a distributed
$\beta$-meteoroid source model for grains with radii between 100 to 140 nm
(Mann & Czechowski, 2020). These values are notably similar to fluxes of
$\beta$-meteoroids derived from STEREO antenna measurements of $0.2-1\times
10^{-4}$ m-2 (Zaslavsky et al., 2012). All antenna-based estimates are lower
than fluxes derived from dedicated dust detectors onboard Pioneers 8 and 9 of
$6\times 10^{-4}$ m-2 (Berg et al., 1973) and Ulysses of $2\times 10^{-4}$ m-2
(Wehry & Mann, 1999; Wehry et al., 2004). Discrepancies could be due to the
different detection mechanisms and relative sensitivies of the various
detection mechanisms, or might indicate an intrinsic variability in
$\beta$-meteoroid flux as suggested from Ulysses measurements (Wehry et al.,
2004).

We expect the majority of collisions in the zodiacal cloud occur in the region
spanning approximately 10 to 20$R_{\odot}$ and if the inner source of pickup
ions is of dust origin (Schwadron et al., 2000), it must be from grains with
radii less than $\sim$50 nm. These findings underscore the prevalence of
collisions and $\beta$-meteoroids within our solar system, which are a crucial
part of the lifecycle of the zodiacal cloud. $\beta$-meteoroids could also be
important in space-weathering processes on airless bodies, particularly in the
inner solar system where impact speeds and fluxes are high. An airless body
with an eccentric orbit like PSP’s would experience orders of magnitude higher
fluxes and impact speeds for both $\alpha$-meteoroids and $\beta$-meteoroids
compared the Moon (e.g. Szalay et al., 2019), where $\beta$-meteoroids have
been suggested to be an important driver in producing impact ejecta (Szalay et
al., 2020a). Based on the updated fluxes derived in this work, $\sim$109
$\beta$-meteoroids impact the Moon’s surface each second. If even a small
fraction of those impactors are able to liberate lunar surface material, it
could be an important space weathering driver.

PSP’s subsequent orbits will allow us to further probe the diverse dust
populations in the innermost regions of our solar system and enable us to
directly compare impact rate measurements with the remote sensing observations
(Stenborg et al., 2020) that have suggested a departure from the nominal
power-law scaling of the zodiacal dust density inside 19$R_{\odot}$. We
summarize our results below,

  * •

The broad PSP impact rate features in orbits $1-6$ can be explained by two
dust sources: $\alpha$-meteoroids on bound, elliptic orbits and
$\beta$-meteoroids on unbound, hyperbolic orbits;

  * •

A prominent excursion from model predictions after each perihelion suggests
meteoroid streams may be observed, either directly or by their collisional
byproducts through a “$\beta$-stream”;

  * •

$\beta$-streams are expected to be a universal phenomenon in all exozodiacal
disks;

  * •

PSP is likely unable to detect nanograins with radii less than $\sim$50 nm;

  * •

Future PSP orbits are expected to experience increasingly intense fluxes from
$\alpha$-meteoroids;

  * •

A zodiacal erosion rate of at least $\sim$100 kg s-1 is consistent with the
observed impact rates;

  * •

The flux of $\beta$-meteoroids at 1 au is estimated to be $0.4-0.8\times
10^{-4}$ m-2 s-1;

  * •

The majority of zodiacal collisions producing $\beta$-meteoroids occur in a
region from $\sim$10$-20$ $R_{\odot}$;

  * •

If the inner source of pick-up ions is due to dust, it must be from nanograins
with radii below $\sim$50 nm.

We thank the many Parker Solar Probe team members that enabled these
observations. We acknowledge NASA Contract NNN06AA01C. P.P. was supported by
NASA Solar System Workings award number 80NSSC21K0153. The IMEX Dust Streams
in Space model was developed under ESA funding (contract 4000106316/12/NL/AF -
IMEX).

##  Appendix A Geminids $\beta$-stream Collisional Production Rate

The collisional lifetime of the Geminids meteoroid stream has been previously
estimated as $\tau_{c}$ = $20-40\times 10^{3}$ yr (Steel & Elford, 1986). The
total mass in the stream is estimated to be $M_{gem}=2-7\times 10^{13}$ kg
(Blaauw, 2017). The average mass loss (or collisional production rate) can
then be approximated to be $M^{+}=\Delta M_{gem}/\Delta t\approx
e^{-1}M_{gem}/\tau_{c}=6-40$ kg s-1. The number production rate is related to
the mass production rate as

$N^{+}=M^{+}\frac{1-\alpha}{\alpha}\frac{m^{-\alpha}_{min}-m^{-\alpha}_{max}}{m^{1-\alpha}_{max}-m^{1-\alpha}_{min}},$
(A1)

where $\alpha$ is the cumulative mass index. We use a value of $\alpha=0.9$,
measured for collisional ejecta above the Moon (Horányi et al., 2015), which
is comprised of silicate material that is collisionally ground to sub-micron
sized via impacts with speeds of $\sim$$10-60$ km s-1 (Pokorný et al., 2019).
The lunar ejecta environment is a reasonable proxy for the Geminids
$\beta$-stream collisional products, as they are also expected to be silicate
and have relative impact speeds with the zodiacal grains $\sim$$30-50$ km s-1
near perihelion. The location of the post-perihelion enhancement was found to
be consistent with trajectories of grains released from the Geminids with
$\beta\gtrsim 0.5$. For a nominal asteroidal grain (Wilck & Mann, 1996), this
corresponds to a mass range of $m_{min}=4.7\times 10^{-18}$ kg and
$m_{max}=1.7\times 10^{-15}$ kg, or grain radius range of 70 to 500 nm.
Assuming the entirety of Geminids collisional byproducts are created in this
mass range, we estimate $N^{+}=1.7-12\times 10^{17}$ s-1 grains are
collisionally produced.

The collisional probability between the Geminids and nominal zodiacal cloud is
expected follow $P=n_{g}n_{\alpha}v\sigma_{gz}\Delta V$ where $n$ is number
density, $v$ is the average impact speed, $\sigma$ is the collisional cross
section, $\Delta V$ is the volume the collisions occur in, and subscript ‘g’
corresponds to the Geminids (Steel & Elford, 1986). Assuming the cross-
sectional area of the Geminids tube is proportional to $r^{2}$ and the orbital
speed is proportional to $r^{-0.5}$, the Geminids density is estimated to
follow $n_{g}(r)=n_{0g}r^{-1.5}$, where $r$ is radial distance. To calculate
the impact speed, we assume the Geminids stream always impacts zodiacal grains
on circular orbits with the same inclination. The impact speed is

$v^{2}=\mu\left(\frac{3}{r}-\frac{1}{a}-2\sqrt{\frac{a(1-e^{2})}{r^{3}}}\right),$
(A2)

where we use the Geminids parent body Phaethon’s orbital elements of semi-
major axis $a=1.27$ au and eccentricity $e=0.89$ as a proxy for the Geminids.
The collisional cross sectional area is dominated by the much larger Geminids
grains, therefore $\sigma=\pi s_{g}^{2}$.

Incorporating the latitudinal dependence (Leinert et al., 1978b) into the
zodiacal density distribution given in Equation 3, we assume the cumulative
density for grains with radii $>s$ is

$n_{\alpha}(r,\lambda,a)=n_{0\alpha}f_{\alpha}(r)(1-\sin{|\lambda|})r^{-1.3}\left(\frac{s_{\alpha}}{s_{0}}\right)^{-3\alpha},$
(A3)

where $\lambda$ is the ecliptic latitude. For simplicity, we assume the
cumulative mass index $\alpha$ of the zodiacal cloud is the same as the
Geminids. The value of $s_{\alpha}$ is the minimum size zodiacal grain that
can catastrophically destroy a Geminids grain upon impact. A key parameter to
estimate is the ratio of mass in an impact required to catastrophically
destroy the larger mass, $\Gamma=m_{g}/m_{\alpha}$, where $m_{g}$ is the
larger mass as the Geminids is comprised of grains in the size range of 100
$\mu$m to 1 mm (Blaauw, 2017). $\Gamma$ is related to the threshold energy
(Grün et al., 1985) $Q_{s}=v^{2}/2\Gamma$. Hence, the size of a bound zodiacal
grain that can catastrophically destroy a Geminids grain is

$s_{\alpha}=\left(\frac{3Q_{s}m_{g}}{2\pi\rho_{\alpha}u^{2}}\right)^{1/3},$
(A4)

where $\rho_{\alpha}$ is the average grain mass density. The last term in $P$,
the collision volume, is calculated per constant unit length along the
Geminids tube to be $\Delta V\propto r^{2}$. To determine the region where the
majority of collisions occur, we express collision probability as a
proportionality relation for all varying terms,

$P\propto(1-\sin{|\lambda|})v(r)^{2\alpha+1}r^{-0.8}$ (A5)

PSP would be impacted by collision products a distance $d$ from the source
region. Assuming the $\beta$-stream is produced in a cone of solid angle
$\Omega$, the flux to PSP is

$F=\frac{N^{+}}{\Omega
d^{2}}\left(\frac{v_{\mathrm{imp}}}{v_{\mathrm{g}}}\right).$ (A6)

The normalized product $FP$ gives the amount of detectable flux at PSP. Using
PSP’s location during its peak anomalous post-perihelion enhancement for
orbits 4 and 5 as a reference position, 95% of the flux it could detect from
the Geminids $\beta$-stream originates from a portion in the Geminids tube
within 0.15 to 0.27 au pre-perihelion (grey tube along Geminids orbit, Figure
8, when PSP is within 0.09 to 0.29 au from the core of the stream. We estimate
this region to account for $\sim$12% of the total collisional mass loss in the
Geminids. A speed ratio of $v_{\mathrm{imp}}/v_{\mathrm{g}}\approx 0.1$ is
used based on the relative speeds between the Geminids $\beta$-stream grains
and PSP. Assumping PSP’s effective area (Page et al., 2020) during this time
is 6 m2 and a $10-20^{\circ}$ cone half-angle for $\Omega$, we estimate PSP
should register count rates from the Geminids in the ranges of $0.1-30$ hr-1.

##  Appendix B Geminids $\beta$-stream ejection geometry

Figure 11: Side view of Geminids $\beta$-stream geometry. All panels show the
same quantities as Figure 8 in the ecliptic J2000 y-z plane for $\beta=0.7$,
with impact rates for solely orbit 4. a) Nominal case of $\beta$-meteoroid
trajectories released from the core of the Geminids stream and inheriting the
initial speeds from the parent meteoroids. b) The same trajectories as panel
(a), offset by $z=+0.03$ au, $1\sigma$ of the width at this location predicted
by radar meteor observations (Hajduková Jr et al., 2017). c) $\beta$-meteoroid
trajectories tilted by 15∘ in the $+z$ direction. d) $\beta$-meteoroid
trajectories combining both panels (b) and (c), offset by $z=+0.03$ au and
with a 15∘ $+z$ cone angle.

While the trajectories of the Geminids $\beta$-stream intersect the PSP orbit
exactly where the post-perihelion enhancement is observed in the ecliptic x-y
plane, these trajectories transit under the PSP orbit (Figure 11). We
investigate two effects that could enable PSP to experience fluxes from the
$\beta$-stream: angular spread and a larger collisional source region.

The blue contours in Figure 11 show the spatial extent of the tube by
integrating $\sim$14,000 orbits of visual Geminids meteors observed at Earth
(Jenniskens et al., 2018). The orbits we use to map the tube structure form
the more concentrated “core” of the stream, as they are from visual
observations of larger meteoroids that would maintain orbits very similar to
Phaethon. This core is also likely responsible for the narrow Geminids feature
observed by PSP’s WISPR imager (Battams et al., 2020). Radar observations of
smaller meteoroids (Hajduková Jr et al., 2017) more subject to additional non-
gravitational perturbations indicate that 1$\sigma$ of the reconstructed width
at the $\beta$-stream source region discussed here is $\sim$0.03 au.

Additionally, Earth might not be representatively sampling the entirety of the
Geminids tube, potentially missing a significant fraction of meteoroids that
comprise a larger structure than the one we recreate here. A survey of the
Geminids shower activity over the last few decades in combination with
dynamical modeling shows Earth might be transiting near the edge of the stream
structure (Ryabova & Rendtel, 2018). For these reasons, we expect the spatial
extent to be more dispersed throughout the stream than portrayed here. We
therefore offset the initial coordinates of the $\beta$-stream by 0.03 au and
find the shower does intersect a portion of the enhanced impact rate region
along PSP’s orbit (Figure 11b).

Alternatively, assuming the collisional process produces fragments within a
cone, we rotate the velocity vectors vertically about a radial direction by
15∘ (Figure 11c) and find this also provides an intersection between the
$\beta$-stream and PSP enhancement region. Since we find a combination of both
corrections (Figure 11d) to best match the location of the enhancement, if a
Geminids $\beta$-stream is responsible for the post-perihelion enhancements in
orbits 4 and 5, the tube could be larger than modeled here and collisional
fragments could spray into a cone of $\sim$15∘.

##  Appendix C Additional $\beta$-streams which intersect PSP’s orbit

In addition to investigating the Geminids, which we favor as a likely
$\beta$-stream candidate, we performed a comprehensive search to test if any
other $\beta$-streams could intersect PSP’s orbit. To do so, using a model we
release $\beta$-meteoroids throughout the orbits of known cometary and
asteroidal bodies over a range of $\beta$ values. We then determine how near
each of these $\beta$-meteoroid trajectories transits to PSP’s orbit, for this
case specifically we focus on orbit 4. Table 4 shows the results of this
investigation, which lists the $\beta$ value and ejection location of
$\beta$-meteoroids that transit nearest to PSP for the top 30 best candidates.
The weight listed is the sum of squares of the difference between
$\beta$-meteoroid and PSP orbits near their closest points. Note, many of
these objects are asteroidal in nature, and with the exception of Phaethon,
not expected to have substantial debris trails.

[ht!] Parent Body Description | MPC ID | Weight | a (au) | e | i (deg) | Lon. Asc. Node (deg) | Arg. of periapsis (deg) | Ejection true anomaly (deg) | $\beta$
---|---|---|---|---|---|---|---|---|---
2017 TC1 | K17T01C | 0.08765 | 2.49370 | 0.96948 | 9.28577 | 274.33215 | 253.11950 | -139.04569 | 0.34
2008 MG1 | K08M01G | 0.18731 | 0.78321 | 0.82277 | 5.71888 | 352.31482 | 108.99051 | 78.03032 | 0.96
2019 MQ2 | K19M02Q | 0.22405 | 1.24427 | 0.79155 | 0.86036 | 140.51356 | 2.58481 | 91.72988 | 0.62
P/2010 H3 (SOHO) | PK10H030 | 0.22689 | 3.06328 | 0.98549 | 23.19750 | 77.30420 | 25.94300 | 128.15754 | 0.95
2017 MM7 | K17M07M | 0.30649 | 2.06194 | 0.96157 | 23.45918 | 250.78365 | 230.54887 | 111.11971 | 0.81
(3200) Phaethon | 03200 | 0.32130 | 1.27137 | 0.88983 | 22.25951 | 265.21769 | 322.18660 | -45.19997 | 0.98
2001 VB | K01V00B | 0.32164 | 2.34292 | 0.89387 | 8.55212 | 299.62327 | 230.74255 | 73.21723 | 0.74
322P/SOHO | 0322P | 0.35598 | 2.50843 | 0.97976 | 11.45970 | 351.46930 | 56.93660 | 151.86599 | 0.99
2014 OX299 | K14OT9X | 0.38085 | 2.20797 | 0.85483 | 6.45584 | 154.49171 | 283.03433 | 39.52130 | 0.99
342P/SOHO | 0342P | 0.38731 | 3.04374 | 0.98338 | 11.97350 | 33.80130 | 67.41560 | 136.49451 | 0.99
2008 EA8 | K08E08A | 0.38746 | 0.95730 | 0.67088 | 1.78545 | 225.49417 | 145.72531 | 25.51899 | 0.29
2011 XA3 | K11X03A | 0.41532 | 1.46668 | 0.92597 | 28.02557 | 273.44955 | 323.92086 | -72.15717 | 0.98
2001 QJ96 | K01Q96J | 0.41914 | 1.59194 | 0.79758 | 5.86449 | 338.63127 | 121.77849 | 33.34591 | 0.54
2019 BE5 | K19B05E | 0.43269 | 0.61011 | 0.65913 | 1.43626 | 309.02706 | 9.75573 | 111.86369 | 0.52
(114158) 2002 VE70 | b4158 | 0.43386 | 1.26635 | 0.92662 | 23.77742 | 39.58359 | 149.56361 | -160.37104 | 0.36
2015 DC200 | K15DK0C | 0.43583 | 1.77711 | 0.86590 | 3.05844 | 224.17269 | 85.72899 | -159.15922 | 0.31
2012 US68 | K12U68S | 0.45501 | 2.50317 | 0.95790 | 25.80834 | 40.49586 | 190.74041 | -174.71960 | 0.29
2019 UJ12 | K19U12J | 0.46491 | 2.42465 | 0.94348 | 27.48587 | 211.04715 | 40.56684 | -61.02572 | 0.99
1996 BT | J96B00T | 0.46584 | 1.20677 | 0.83362 | 12.10530 | 296.85776 | 328.23513 | -84.65248 | 0.95
323P/SOHO | 0323P | 0.50569 | 2.58246 | 0.98484 | 5.33740 | 324.37460 | 353.04870 | 168.73639 | 0.95
2015 CG13 | K15C13G | 0.50891 | 2.50631 | 0.91341 | 6.27430 | 124.67063 | 238.05924 | -124.39949 | 0.32
2019 YV2 | K19Y02V | 0.53408 | 1.22723 | 0.89717 | 6.45997 | 129.14947 | 107.71557 | -110.13753 | 0.88
2015 HG9 | K15H09G | 0.53418 | 2.13351 | 0.84329 | 5.23836 | 225.43773 | 218.68841 | 38.03775 | 0.38
2015 KO120 | K15KC0O | 0.55629 | 1.88183 | 0.93466 | 2.19757 | 240.67466 | 199.03772 | 87.47979 | 0.98
(195426) 2002 GU51 | j5426 | 0.61320 | 2.53303 | 0.92217 | 6.83110 | 233.34100 | 200.73095 | 53.08246 | 0.74
2016 EL56 | K16E56L | 0.62822 | 2.02813 | 0.85579 | 11.35514 | 334.61334 | 70.21392 | 72.42659 | 0.32
2014 JS54 | K14J54S | 0.63082 | 2.07434 | 0.89409 | 3.36280 | 235.05906 | 206.53006 | 46.05289 | 0.94
2007 PR10 | K07P10R | 0.64163 | 1.23182 | 0.89242 | 20.92875 | 335.15196 | 190.69424 | 61.85728 | 0.40
2013 AT72 | K13A72T | 0.66963 | 0.67829 | 0.53565 | 15.01660 | 118.18749 | 167.76605 | -45.29174 | 0.86
2P/Encke | 0002P | 0.68005 | 2.21528 | 0.84800 | 11.76460 | 334.55170 | 186.56250 | -38.62496 | 0.41
Table 4: Top 30 possible $\beta$-streams for Orbit 4 & 5 post-perihelion
enhancement.

## References

  * Altobelli et al. (2006) Altobelli, N., Grün, E., & Landgraf, M. 2006, A&A, 448, 243, doi: 10.1051/0004-6361:20053909
  * Andersson et al. (2015) Andersson, L., Weber, T. D., Malaspina, D. M., et al. 2015, Science, 350, 0398
  * Auer (2001) Auer, S. 2001, in Interplanetary Dust, ed. E. Grün, B. Gustafson, S. Dermott, & H. Fechtig. (Springer), 387–438
  * Bale et al. (2016) Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, Space Science Reviews, 204, 49
  * Bale et al. (2020) Bale, S. D., Goetz, K., Bonnell, J. W., et al. 2020, Dust impact voltage signatures on Parker Solar Probe: influence of spacecraft floating potential.  https://arxiv.org/abs/2006.00776
  * Battams & Knight (2017) Battams, K., & Knight, M. M. 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375, 20160257
  * Battams et al. (2020) Battams, K., Knight, M. M., Kelley, M. S. P., et al. 2020, The Astrophysical Journal Supplement Series, 246, 0
  * Berg & Grün (1973) Berg, O. E., & Grün, E. 1973, Space Research XIII, 1047
  * Berg et al. (1973) Berg, O. E., Richardson, F. F., & Burton, H. 1973, in NASA Special Publication, Vol. 330, Apollo 17: Preliminary Science Report, 16
  * Blaauw (2017) Blaauw, R. C. 2017, Planetary and Space Science, 143, 83
  * Burns et al. (1979) Burns, J. A., Lamy, P. L., & Soter, S. 1979, Icarus, 40, 1, doi: 10.1016/0019-1035(79)90050-2
  * Christou et al. (2015) Christou, A. A., Killen, R. M., & Burger, M. H. 2015, Geophysical Research Letters, 42, 7311
  * Collette et al. (2014) Collette, A., Grün, E., Malaspina, D. M., & Sternovsky, Z. 2014, Journal of Geophysical Research-Space Physics, 119, 6019
  * Collette et al. (2016) Collette, A., Malaspina, D. M., & Sternovsky, Z. 2016, Journal of Geophysical Research-Space Physics, 121, 8182
  * Collette et al. (2015) Collette, A., Meyer, G., Malaspina, D. M., & Sternovsky, Z. 2015, Journal of Geophysical Research-Space Physics, 120, 5298
  * Czechowski & Mann (2010) Czechowski, A., & Mann, I. 2010, ApJ, 714, 89, doi: 10.1088/0004-637X/714/1/89
  * Fahr et al. (1981) Fahr, H. J., Ripken, H. W., & Lay, G. 1981, Astronomy and Astrophysics, 102, 359
  * Fox et al. (2016) Fox, N., Velli, M. C., Bale, S. D., et al. 2016, Space Science Reviews, 204, 7
  * Grün et al. (1980) Grün, E., Pailer, N., FECHTIG, H., & Kissel, J. 1980, Planetary and Space Science, 28, 333
  * Grun & Zook (1980) Grun, E., & Zook, H. 1980, Solid Particles in the Solar System, 293
  * Grün et al. (1985) Grün, E., Zook, H. A., Fechtig, H., & Giese, R. H. 1985, Icarus, 62, 244, doi: 10.1016/0019-1035(85)90121-6
  * Grün et al. (1993) Grün, E., Zook, H. A., Baguhl, M., et al. 1993, Nature, 362, 428, doi: 10.1038/362428a0
  * Grün et al. (1997) Grün, E., Staubach, P., Baguhl, M., et al. 1997, Icarus, 129, 270
  * Gruntman (1996) Gruntman, M. 1996, Journal of Geophysical Research, 101, 15555
  * Guo et al. (2021) Guo, Y., Thompson, P., Wirzburger, J., et al. 2021, Acta Astronautica, 179, 425
  * Gurnett et al. (1983) Gurnett, D. A., Grun, E., Gallagher, D., Kurth, W. S., & Scarf, F. L. 1983, Icarus, 53, 236, doi: 10.1016/0019-1035(83)90145-8
  * Hajduková Jr et al. (2017) Hajduková Jr, M., Koten, P., Kornoš, L., & Tóth, J. 2017, Planetary and Space Science, 143, 89
  * Hill et al. (2020) Hill, M. E., Mitchell, D. G., Allen, R. C., et al. 2020, The Astrophysical Journal Supplement Series, 246, 65
  * Hoffman et al. (1975) Hoffman, H.-J., Fechtig, H., Grün, E., & Kissel, J. 1975, Planetary and Space Science, 23, 985 , doi: https://doi.org/10.1016/0032-0633(75)90186-5
  * Hoffmann et al. (1975) Hoffmann, H.-J., Fechtig, H., Grün, E., & Kissel, J. 1975, Planetary and Space Science, 23, 215 , doi: https://doi.org/10.1016/0032-0633(75)90080-X
  * Holzer (1977) Holzer, T. E. 1977, Reviews of Geophysics and Space Physics, 15, 467
  * Horányi et al. (2015) Horányi, M., Szalay, J. R., Kempf, S., et al. 2015, Nature, 522, doi: 10.1038/nature14479
  * Ishimoto & Mann (1998) Ishimoto, H., & Mann, I. 1998, Planetary and Space Science, 47, 225
  * Jenniskens et al. (2018) Jenniskens, P., Baggaley, J., Crumpton, I., et al. 2018, Planetary and Space Science, 154, 21 , doi: https://doi.org/10.1016/j.pss.2018.02.013
  * Jewitt & Li (2010) Jewitt, D. C., & Li, J. 2010, The Astronomical Journal, 140, 1519, doi: 10.1088/0004-6256/140/5/1519
  * Jewitt et al. (2013) Jewitt, D. C., Li, J., & Agarwal, J. 2013, The Astrophysical Journal Letters, 771, L36
  * Kasper et al. (2016) Kasper, J. C., Abiad, R., Austin, G., et al. 2016, Space Science Reviews, 204, 131
  * Kellogg et al. (2016) Kellogg, P. J., Goetz, K., & Monson, S. J. 2016, Journal of Geophysical Research-Space Physics, 121, 966
  * Knight et al. (2016) Knight, M. M., Fitzsimmons, A., Kelley, M. S. P., & Snodgrass, C. 2016, The Astrophysical Journal Letters, 823, 1
  * Krüger et al. (2020) Krüger, H., Strub, P., Sommer, M., et al. 2020, Astronomy and Astrophysics, 643, A96
  * Kuchner & Stark (2010) Kuchner, M. J., & Stark, C. C. 2010, AJ, 140, 1007, doi: 10.1088/0004-6256/140/4/1007
  * Laakso et al. (1989) Laakso, H., Grard, R., Pedersen, A., & Schwehm, G. 1989, Advances in Space Research, 9, 269, doi: 10.1016/0273-1177(89)90273-1
  * Landgraf et al. (2003) Landgraf, M., Krüger, H., Altobelli, N., & Grün, E. 2003, Journal of Geophysical Research (Space Physics), 108, 8030, doi: 10.1029/2003JA009872
  * Leinert et al. (1978a) Leinert, C., Hanner, M., Link, H., & Pitz, E. 1978a, Astronomy and Astrophysics, 64, 119
  * Leinert et al. (1978b) Leinert, C., Hanner, M., & Pitz, E. 1978b, Astronomy and Astrophysics, 63, 183
  * Leinert et al. (1981) Leinert, C., Richter, I., Pitz, E., & Planck, B. 1981, A&A, 103, 177
  * Malaspina et al. (2014) Malaspina, D. M., Horanyi, M., Zaslavsky, A., et al. 2014, Geophysical Research Letters, 41, 266
  * Malaspina et al. (2015) Malaspina, D. M., O’Brien, L. E., Thayer, F., Sternovsky, Z., & Collette, A. 2015, Journal of Geophysical Research-Space Physics, 120, 6085
  * Malaspina et al. (2016) Malaspina, D. M., Ergun, R. E., Bolton, M., et al. 2016, Journal of Geophysical Research-Space Physics, 121, 5088
  * Malaspina et al. (2020) Malaspina, D. M., Szalay, J. R., Pokorný, P., et al. 2020, Observations of Inner Heliospheric Dust Variability
  * Mann & Czechowski (2020) Mann, I., & Czechowski, A. 2020, Astronomy and Astrophysics, 1
  * Mann et al. (2004) Mann, I., Kimura, H., Biesecker, D. A., et al. 2004, Space Science Reviews, 110, 269
  * Mann et al. (2019) Mann, I., Nouzák, L., Vaverka, J., et al. 2019, Annales Geophysicae, 37, 1121
  * Markwardt (2009) Markwardt, C. B. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 411, Astronomical Data Analysis Software and Systems XVIII, ed. D. A. Bohlender, D. Durand, & P. Dowler, 251
  * McComas et al. (2007) McComas, D. J., Velli, M., Lewis, W. S., et al. 2007, Reviews of Geophysics, 45, 131
  * McComas et al. (2016) McComas, D. J., Angold, N., Bale, S. D., et al. 2016, Space Science Reviews, 204, 187
  * McComas et al. (2019) McComas, D. J., Christian, E. R., Cohen, C. M. S., et al. 2019, Nature, 576, 223
  * Mozer et al. (2020) Mozer, F. S., Agapitov, O. V., Bale, S. D., et al. 2020, The Astrophysical Journal Supplement Series, 246, 50
  * Nesvorný et al. (2010) Nesvorný, D., Youdin, A. N., & Richardson, D. C. 2010, AJ, 140, 785, doi: 10.1088/0004-6256/140/3/785
  * Page et al. (2020) Page, B., Bale, S. D., Bonnell, J. W., et al. 2020, Examining Dust Directionality with the Parker Solar Probe FIELDS Instrument
  * Pokorný & Brown (2016) Pokorný, P., & Brown, P. G. 2016, Astronomy & Astrophysics, 592, A150, doi: 10.1051/0004-6361/201628134
  * Pokorný et al. (2019) Pokorný, P., Janches, D., Sarantos, M., et al. 2019, Journal of Geophysical Research: Planets, 124, 752
  * Pokorný & Kuchner (2019) Pokorný, P., & Kuchner, M. 2019, ApJ, 873, L16, doi: 10.3847/2041-8213/ab0827
  * Pusack et al. (2021) Pusack, A., Malaspina, D. M., Szalay, J. R., et al. 2021, PSJ
  * Ryabova & Rendtel (2018) Ryabova, G. O., & Rendtel, J. 2018, Monthly Notices of the Royal Astronomical Society: Letters, 475, L77
  * Schwadron et al. (2000) Schwadron, N. A., Geiss, J., Fisk, L. A., et al. 2000, Journal of Geophysical Research, 105, 7465
  * Soja et al. (2019) Soja, R. H., Grün, E., Strub, P., et al. 2019, Astronomy and Astrophysics, 628, A109
  * Steel & Elford (1986) Steel, D. I., & Elford, W. G. 1986, Monthly Notices of the Royal Astronomical Society (ISSN 0035-8711), 218, 185
  * Stenborg et al. (2020) Stenborg, G., Howard, R., Hess, P., & Gallagher, B. 2020, Astronomy and Astrophysics, doi: 10.1051/0004-6361/202039284
  * Sterken et al. (2012) Sterken, V. J., Altobelli, N., Kempf, S., et al. 2012, A&A, 538, A102, doi: 10.1051/0004-6361/201117119
  * Sterken et al. (2015) Sterken, V. J., Strub, P., von Steiger, R., & Frisch, P. C. 2015, The Astrophysical Journal, 812, 1
  * Strub et al. (2019) Strub, P., Sterken, V. J., Soja, R., et al. 2019, Astronomy and Astrophysics, 621, A54
  * Szalay et al. (2020a) Szalay, J. R., Pokorný, P., & Horányi, M. 2020a, The Astrophysical Journal, 890, L11, doi: 10.3847/2041-8213/ab7195
  * Szalay et al. (2019) Szalay, J. R., Pokorný, P., Horanyi, M., et al. 2019, Planetary and Space Science, 165, 194, doi: 10.1016/j.pss.2018.11.001
  * Szalay et al. (2020b) Szalay, J. R., Pokorný, P., Bale, S. D., et al. 2020b, The Astrophysical Journal Supplement Series, 246, 27, doi: 10.3847/1538-4365/ab50c1
  * Tsurutani et al. (2003) Tsurutani, B. T., Clay, D. R., Zhang, L. D., et al. 2003, Geophysical Research Letters, 30, 2134
  * Tsurutani et al. (2004) Tsurutani, B. T., Clay, D. R., Zhang, L. D., et al. 2004, Icarus, 167, 89, doi: 10.1016/j.icarus.2003.08.021
  * Vaverka et al. (2019) Vaverka, J., Pavlů, J., Nouzák, L., et al. 2019, Journal of Geophysical Research-Space Physics, 124, 8179
  * Vaverka et al. (2018) Vaverka, J., Nakamura, T., Kero, J., et al. 2018, Journal of Geophysical Research-Space Physics, 123, 6119
  * Vourlidas et al. (2016) Vourlidas, A., Howard, R. A., Plunkett, S. P., et al. 2016, Space Science Reviews, 204, 83
  * Wehry et al. (2004) Wehry, A., Krüger, H., & Grün, E. 2004, A&A, 419, 1169, doi: 10.1051/0004-6361:20035613
  * Wehry & Mann (1999) Wehry, A., & Mann, I. 1999, A&A, 341, 296
  * Wilck & Mann (1996) Wilck, M., & Mann, I. 1996, Planetary and Space Science, 44, 493
  * Wimmer-Schweingruber & Bochsler (2003) Wimmer-Schweingruber, R. F., & Bochsler, P. 2003, Geophysical Research Letters, 30, 1077
  * Zaslavsky et al. (2012) Zaslavsky, A., Meyer-Vernet, N., Mann, I., et al. 2012, Journal of Geophysical Research (Space Physics), 117, 5102, doi: 10.1029/2011JA017480
  * Zook & Berg (1975) Zook, H. A., & Berg, O. E. 1975, Planetary and Space Science, 23, 183 , doi: https://doi.org/10.1016/0032-0633(75)90078-1