# Prediction of the in situ coronal mass ejection rate for solar cycle 25:
Implications for Parker Solar Probe in situ observations

Christian Möstl Space Research Institute, Austrian Academy of Sciences,
Schmiedlstraße 6, 8042 Graz, Austria  Institute of Geodesy, Graz University of
Technology, Steyrergasse 30, 8010 Graz, Austria  Andreas J. Weiss Space
Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz,
Austria  Institute of Geodesy, Graz University of Technology, Steyrergasse 30,
8010 Graz, Austria  Institute of Physics, University of Graz,
Universitätsplatz 5, 8010 Graz, Austria  Rachel L. Bailey Space Research
Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria
Zentralanstalt für Meteorologie und Geodynamik, Hohe Warte 38, 1190 Vienna,
Austria  Martin A. Reiss Space Research Institute, Austrian Academy of
Sciences, Schmiedlstraße 6, 8042 Graz, Austria  Institute of Geodesy, Graz
University of Technology, Steyrergasse 30, 8010 Graz, Austria  Tanja
Amerstorfer Space Research Institute, Austrian Academy of Sciences,
Schmiedlstraße 6, 8042 Graz, Austria  Jürgen Hinterreiter Space Research
Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria
Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz,
Austria  Maike Bauer Space Research Institute, Austrian Academy of Sciences,
Schmiedlstraße 6, 8042 Graz, Austria  Institute of Physics, University of
Graz, Universitätsplatz 5, 8010 Graz, Austria  Scott W. McIntosh National
Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA  Noé
Lugaz Space Science Center, Institute for the Study of Earth, Oceans, and
Space, University of New Hampshire, Durham, NH, USA  David Stansby Mullard
Space Science Laboratory, University College London, Holmbury St. Mary, Surrey
RH5 6NT, UK

(Received 2020 July 28; Revised 2020 September 15; Accepted 2020 September 17)

###### Abstract

The Parker Solar Probe (PSP) and Solar Orbiter missions are designed to make
groundbreaking observations of the Sun and interplanetary space within this
decade. We show that a particularly interesting in situ observation of an
interplanetary coronal mass ejection (ICME) by PSP may arise during close
solar flybys ($<0.1$ AU). During these times, the same magnetic flux rope
inside an ICME could be observed in situ by PSP twice, by impacting its
frontal part as well as its leg. Investigating the odds of this situation, we
forecast the ICME rate in solar cycle 25 based on 2 models for the sunspot
number (SSN): (1) the forecast of an expert panel in 2019 (maximum SSN = 115),
and (2) a prediction by McIntosh et al. (2020, maximum SSN = 232). We link the
SSN to the observed ICME rates in solar cycles 23 and 24 with the Richardson
and Cane list and our own ICME catalog, and calculate that between 1 and 7
ICMEs will be observed by PSP at heliocentric distances $<0.1$ AU until 2025,
including 1$\sigma$ uncertainties. We then model the potential flux rope
signatures of such a double-crossing event with the semi-empirical 3DCORE flux
rope model, showing a telltale elevation of the radial magnetic field
component $B_{R}$, and a sign reversal in the component $B_{N}$ normal to the
solar equator compared to field rotation in the first encounter. This holds
considerable promise to determine the structure of CMEs close to their origin
in the solar corona.

editorials, notices — miscellaneous — catalogs — surveys

††journal: ApJ

##  1 Introduction

Coronal mass ejections (CMEs) form a major link between a star and planetary
bodies within the astrosphere. In the solar system, the knowledge of how many
CMEs erupt during the solar cycle and how often we see their manifestations at
spacecraft observing the in situ solar wind are of great interest in order to
assess the potential of CMEs as drivers of major geomagnetic storms
(Richardson et al., 2000; Zhang et al., 2007). If a spacecraft observes a CME
in situ, we call the full interval of disturbed solar wind an interplanetary
coronal mass ejection (ICMEs, for an exact definition see Rouillard, 2011).

A major result of previous studies is that ICMEs drive all major geomagnetic
storms with Dst $<-250$ nT, with the Disturbance storm time (Dst) index as a
measure of the severity of geomagnetic storms at Earth. Love et al. (2015)
found that a geomagnetic disturbance with a Dst of roughly $-1000$ nT,
generally accepted to be the order of magnitude of the Carrington event
(Carrington, 1859) and the May 1921 storm (Love et al., 2019), happens once in
100 years. Given the average ICME rate at Earth (e.g. Richardson et al., 2000)
with about 20 events per year, we can easily see that in $10^{2}$ years about
$2\times 10^{3}$ ICMEs impact the Earth, out of which a single one is expected
to elicit a Carrington magnitude geomagnetic storm. CMEs are now widely
considered to be a natural hazard, as the probability of a Carrington-type
event is $10\%$ per solar cycle and knowing the ICME rate at Earth and other
solar system locations is a basic need in order to study, forecast and
mitigate their effects.

The idea for this paper originated from simulations that we have done with our
semi-empirical 3DCORE model which can describe the magnetic flux ropes inside
ICMEs (Möstl et al., 2018). When the Parker Solar Probe (PSP, Fox et al.,
2016) spacecraft passes close to the Sun in the upcoming years, an ICME flux
rope could be crossed twice by PSP as it moves extremely fast during those
solar flybys. We have thus wondered how often this situation might happen
during the PSP primary mission until 2025, and thus considered to forecast the
ICME rate for the next solar cycle to calculate how many such events we could
expect. This paper is thus essentially divided into 2 parts: the first where
we predict the ICME rate for solar cycle 25, and the second where we model a
CME encounter during a PSP solar flyby in an idealized way.

Richardson et al. (2001), Jian et al. (2006) and other works by the same
authors have clarified many questions on the occurrence rate of ICMEs (for a
summary see Webb & Howard, 2012). However, during a solar cycle the rate of
ICMEs impacting different points in the heliosphere can vary considerably from
location to location. Using accurate statistics to forecast the occurrence
rate of ICMEs as potentially measured by PSP in the innermost heliosphere and
upper corona is critical. In a previous study, we compiled the ICMECAT catalog
(Möstl et al., 2017). It is an aggregate of different individual catalogs
along with our own identified ICMEs from data provided by the missions STEREO,
Wind, Venus Express, Ulysses, and MESSENGER. The catalog contains data from
2007 onwards and was originally constructed to validate CME prediction models
at various targets within the inner heliosphere. We present here an updated
ICMECAT to include ICMEs at the aforementioned missions until the end of 2019,
have added events observed by the MAVEN spacecraft at times when it sampled
the solar wind.

In section 2, we show that our combined ICMECAT catalog allows an assessment
of the ICME rate better than previous studies, as multi-point locations give a
better ICME sample size. We can thus determine the random spread in ICME
numbers per year at different in situ locations directly from data. The ICME
rate is a stochastic phenomenon, as the Sun does not produce CMEs in a steady
way but CMEs erupt from the Sun at a high rate during the presence of active
regions, followed by times with no or few CMEs, or erupt with slow speeds from
regions of the otherwise quiet Sun. The ICME rate does not depend on
heliocentric distance, as it is well established that CMEs expand self-
similarly (Yashiro et al., 2004), which means they retain the same angular
width as they propagate away from the Sun.

Section 3 estimates the number of ICMEs observed at Earth and other
heliospheric locations including the Parker Solar Probe, Solar Orbiter (Müller
et al., 2013) and Bepi Colombo up until 2030. To this end, we combine the
relationship that we derived between the ICME rate and the Sunspot number with
two different predictions for the magnitude of solar cycle 25, assuming one of
these solar cycle predictions holds roughly true. Additionally, the PSP
mission, launched in August 2018, will perform many close approaches to the
Sun as close as 0.05 AU, which will give us an unprecedented opportunity to
observe an ICME flux rope with high relative tangential speed between the
spacecraft and ICME. PSP will attain speeds of up to 180 km s-1, which is
close to the lower end of the speed distribution of CMEs. We show that this
observing situation may lead to a double-crossing of the same ICME flux rope
by PSP.

We further investigate the expected magnetic signatures of such an
observation, aided by modeling with the 3DCORE semi-empirical flux rope
technique. The 3DCORE method allows both forward modeling and fitting of 3D
magnetic field flux rope signatures in retrospect. It is based on a global
torus shape with elliptical cross section, a Gold Hoyle field (Gold & Hoyle,
1960) and drag-based kinematics (Vršnak et al., 2013). Here, we use the new
version of Weiss et al. (2020) with a highly streamlined numerical calculation
pipeline, considerably improving the Möstl et al. (2018) 3DCORE prototype.

##  2 Methods

We start by describing our ICME catalog, and proceed to establish a
correlation with the sunspot number as a proxy for solar activity, allowing us
to derive a forecast of the ICME rate for the next solar cycle.

###  2.1 ICME catalog

Figure 1: ICMECAT data coverage and impact frequency. (a) ICME event
detections as a function of time and heliocentric distance. The spacecraft is
given by the color code in the legend. (b) Number of ICMECAT events per year
for each spacecraft, corrected for in situ observation gaps. White dots
indicate the mean for each year with the error bars as the standard deviation.
The sunspot number from SIDC is plotted for orientation with respect to the
solar cycle progress.

First, we present the most comprehensive catalog of interplanetary coronal
mass ejections to date, including a total of 739 events at MESSENGER in the
cruise phase and in orbit around Mercury, at VEX in orbit around Venus, at
Wind at the Sun-Earth L1 point, at STEREO-A/B in the solar wind near 1 AU, at
Ulysses during its last ecliptic pass in 2007, and at MAVEN in orbit around
Mars since September 2014. Our update here extends the ICMECAT catalog first
shown by Möstl et al. (2017), which originally contained 668 events. The time
range now spans 1 January 2007 - 31 December 2019.

The ICME times were taken from various authors (see Möstl et al., 2017) or
newly included in the lists by ourselves if we had data intervals for which no
ICME categorization had been performed by other authors to date (valid for
MESSENGER, VEX, Ulysses, and MAVEN). All ICME parameters were then
consistently recalculated from the data. The ICMECAT contains parameters for
each event based on magnetic field and bulk plasma observations, when
available. From the up to 30 parameters determined for each event, in this
study we use only the existence of the event and its ICME start time. Further
details on how the ICMECAT was established are described in Möstl et al.
(2017). The most recent catalog version is available on our
webpage111https://helioforecast.space/icmecat, and an archived release of the
catalog used in this paper, in many different formats and including a citable
DOI, can be found on figshare (see data sources in the acknowledgements).

Figure 1a shows the ICME event coverage with each dot being an ICME observed
at the indicated time and heliocentric distance, with a color code
distinguishing each spacecraft. We use Wind data for the continuously observed
near Earth space at the Sun–Earth L1 point. The plot further shows a decent
general coverage of the inner heliosphere $<1$ AU during solar cycle 24, with
more sparse event detections towards the end of the cycle as the VEX and
MESSENGER missions ended in 2014 and 2015, respectively.

Figure 1b demonstrates the number of ICMECAT events per year for each
spacecraft. For each data point, the number of ICMEs per year was normalized
according to the data availability for the given year, thus longer data gaps
were identified in the magnetic field data and excluded in the calculation.
For years with little data availability, the histogram entry was set to NaN.
The plot shows a clear rise of the yearly ICME rate (ICR) from solar minimum
to maximum, as well as a decrease in the declining phase to the current very
low rates during solar minimum. The number of ICMEs has risen from an average
of 5 events during solar minimum to around 30 events per year in solar
maximum, or about 2.5 events per month. The spread in the ICME rate at
different observatories is given as an error bar on Figure 1b. The standard
variation in the ICME rate from including all observatories varied from about
$\pm 2$ ICMEs in solar minimum to $\pm 10$ events in solar maximum.

###  2.2 Correlation of ICME rate and sunspot number

Figure 2: Linear fit of yearly averaged sunspot number, from a 13 month
running mean (solar cycle 23: black dots, solar cycle 24: white dots) versus
the yearly average ICME rate. A $1\sigma$ confidence interval is plotted,
corresponding to the error bars in Equation (1). The two dashed horizontal
lines indicate ICME rates of 26 and 52 per year, corresponding to one ICME
every two weeks or one ICME every week at a heliospheric location,
respectively.

In order to predict the ICR for solar cycle 25 at a position in the inner
heliosphere such as the Parker Solar Probe, we need to establish a
relationship between a solar cycle progress indicator and the ICR (Kilpua et
al., 2011; Richardson, 2013). We choose the sunspot number (SSN) as it is the
standard parameter for such analyses.

The relationship between CMEs, ICMEs and solar activity indices such as the
SSN has been studied extensively in the past (Webb & Howard, 1994; Gopalswamy
et al., 2010; Li et al., 2018; Lamy et al., 2019). In general, the CME rate as
measured in coronagraphs follows solar activity indices such as the number and
complexity of active regions, reflected by the sunspot number, but Lamy et al.
(2019) find the association of CMEs with streamers is the clearest link.
However, these associations remain a nuanced problem, and many factors such as
progress in the solar cycle, the hemisphere where an eruption originates,
coronal magnetic field changes (e.g. Gopalswamy et al., 2014) and CMEs from
stealth events or filament eruptions affect the relationship. In this study we
concentrate on a linear fit between the ICME and SSN without considering other
influences for a first approximation.

Figure 2 shows the yearly mean SSN from SIDC222http://sidc.be/silso/home for
solar cycles 23 (SC23) and 24 (SC24), with SC23 defined as the years 1996 to
2008 and SC24 covering 2009 to 2019, for simplicity. This yearly mean was
generated from a 13-month running mean of the daily SSN. We plot this against
the ICME rate in the Richardson & Cane (2010)
list333http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm.

A linear relationship is fitted as:

$ICR=(0.20\pm 0.03)*SSN+8.4\pm 7.0,$ (1)

with ICR in a unit of events per year, and SSN given as a yearly average. The
error in the slope was taken from a standard procedure (scipy.stats.linregress
in python) and the error in the intercept term is the average standard
deviation between the linear fit and the data points, yielding a $1\sigma$
confidence interval.

With the Richardson & Cane (2010) list, we thus find a considerable spread in
the yearly ICME rate compared to the fit that arises from the stochastic
nature of CME eruptions and ICME impacts. Nevertheless, the Pearson
correlation coefficient is 0.84, and it is also clear from physical reasoning
that in a simplified picture a higher sunspot number results in more active
regions, more CMEs, and more ICMEs. Thus, we consider this relationship robust
in representing an approximate connection between the SSN and the ICR.

##  3 Results

We now proceed to predict the ICR for solar cycle 25 and calculate the number
of ICMEs the Parker Solar Probe spacecraft is expected to encounter for
different heliocentric distances until the end of the nominal mission in 2025.
We then model the magnetic field signatures of possible double flux rope
encounters close to the Sun.

###  3.1 ICME rate prediction for solar cycle 25

Figure 3: Observations and forecasts for the sunspot number and ICME rates.
(a) The Richardson and Cane (black and white dots) ICME rate is shown for
solar cycle 23/24, as well as the ICME rate in ICMECAT (white squares)
including the spread of the ICME rate per year obtained from ICMECAT as error
bars. For solar cycle 25, a Hathaway function for a mean solar cycle is given
(green solid line) as well as the predicted SSN by PP19 (blue solid line)
including their $1\sigma$ uncertainty. The derived ICME rate is shown as grey
dots, including uncertainties as error bars (see text). (b) The MC20 sunspot
number prediction, which is clearly above a mean solar cycle model, and the
derived ICME rate, is shown in a similar style. Note that the scaling of the
axes is similar on both panels.

Given the linear relationship between the ICR and SSN, we can now proceed to
predict the ICR for solar cycle 25. Overviews of solar cycle prediction
methods are given by Hathaway (2015) and Petrovay (2020). In 2019, an expert
panel consisting of NOAA, NASA and ISES issued a consensus prediction that is
published on the NOAA website444https://www.swpc.noaa.gov/products/solar-
cycle-progression and available as a JSON file (downloaded in May 2020). The
predicted SSN values are available in monthly resolution, and we fit those and
another two time series that are given for the lower and upper uncertainty
ranges with a Hathaway et al. (1994) function of the form:

$SSN(t)=A\left(\frac{t-t_{0}}{b}\right)^{3}\left[exp\left(\frac{t-t_{0}}{b}\right)^{2}-c\right]^{-1}$
(2)

The fit parameters emerge as $A=256~{}[228.2,~{}273.3]$,
$b=60.3~{}[54.4,~{}67.4]$, $c=0.35~{}[0.48,~{}0.34]$, with the values for the
functions with the uncertainty ranges given in brackets (first the lower, then
the upper error boundary). Variable $t_{0}$ is a start time that is returned
here by the fit as 2019 September 6. This SSN model peaks at a value of 115 in
July 2025. For comparison, the mean maximum SSN for all solar cycles since
1755 is 179, when smoothing the SSN with a 13-month running mean. Thus, the
panel forecasts a cycle with a maximum that is a factor 0.64 or about 2/3 of
the historic average. This peak would almost exactly match solar cycle 24,
which had the fourth lowest maximum SSN since 1755. We call this the panel
prediction 2019 (PP19).

McIntosh et al. (2020) introduced a much higher SSN prediction based on
timings of terminator events, which signal the end of a solar magnetic cycle
(Leamon et al., 2020). We call this the MC20 forecast. We created a Hathaway
function with parameters $A=444~{}[396,~{}492]$, $b=60$, $c=0.8$, including
the $1\sigma$ error range given by these authors, which fit the MC20
prediction of a maximum SSN of 232 in April 2025. This maximum SSN is very
nearly twice as high as for PP19, and it is a factor 1.3 higher than the
historic average. This forecast would clock in as the 7th strongest solar
cycle since 1755. The start time $t_{0}$ here is set as 2020 February 1. Note
that a difference in $t_{0}$ of a few months will not affect our final results
for the expected ICME rates at the Parker Solar Probe spacecraft. Given the
large discrepancy between the two forecasts, we further work with both and
check the results for each predicted ICME rate individually.

Figure 3a shows the ICR prediction for solar cycle 25 based on the Hathaway
model curve for the SSN given by PP19 and then further derived from a linear
relationship between SSN and ICR. For comparison, we plot the average solar
cycle peaking with an SSN of 179 as a green solid line. It is seen that the
PP19 prediction is clearly below an average solar cycle. Additionally, we show
the yearly Richardson and Cane ICR as well as the ICMECAT ICR with a standard
deviation of the rate at different in situ locations. Figure 3b shows the
prediction for solar cycle 25 using the Hathaway function consistent with the
MC20 SSN prediction.

We have also plotted an uncertainty range for the ICR predictions in Figure 3
based on 3 different sources: (1) the given uncertainties in the SSN
prediction, (2) the error in the slope of the linear fit from the SSN and ICR
correlation, (3) and the standard deviation from the ICMECAT yearly rate.
These errors are added together as squares first, and then the root of their
sum is taken as the final error range, as the sources of errors are all
independent variables. We see that with all these uncertainties taken into
account, the MC20 ICR peak is between 40 and 70 ICMEs per year, which would
even exceed the observed rates in SC23, while the PP19 ICR would remain around
the low values observed in SC24, peaking between 20 and 40 ICMEs per year. We
now figure out how these results affect the observations by the Parker Solar
Probe close to the Sun.

###  3.2 Implications for Parker Solar Probe close encounters

Figure 4: Predicted monthly ICME rate for solar cycle 25 (right y-axis) and
the heliocentric distance (left y-axis) for (a) Parker Solar Probe, and (b)
for Solar Orbiter (black) and Bepi Colombo (orange). ICME rates based on the
PP19 (violet) and MC20 (coral) sunspot number predictions are shown, including
an uncertainty range from several causes (see text). The times terminate at
the end times of the available spice kernels from each spacecraft, and the
position for Bepi Colombo is continued at Mercury after orbit insertion in
December 2025.

Figure 4 shows the heliocentric distances for PSP, Solar Orbiter and Bepi
Colombo and the monthly ICME rate determined from the previous section
including error bars. For PSP in in Figure 4a, this covers the nominal part of
the mission, while for Solar Orbiter and Bepi Colombo in Figure 4b, with the
latter to be inserted into orbit around Mercury in December 2025, we extended
the plot to the end of 2029. To convert the results of the ICR from a yearly
to daily time resolution we used a 2nd order spline fit to interpolate on the
yearly values. The figure clearly shows that there is little overlap between
the PP19 and MC20 based forecasts for the ICR.

With a predicted daily ICR for solar cycle 25 in our hands given in Hathaway
function format, we can now easily calculate for each day until the end of
2025 how much time PSP spends below a given heliocentric distance and how many
ICMEs it is expected to observe there. Table 1 gives an overview of the total
number of ICMEs expected at PSP below several heliocentric distances for both
SSN models. These numbers are valid from 2020 July 1 to 2025 August 31, when
PSP will have completed 24 solar flybys. For the total mission time since July
2020 and for all heliocentric distances, the PP19 prediction gives $109\pm 57$
ICME events, whereas the model based on MC20 expects $186\pm 67$ total ICMEs,
which is almost a factor of 2 higher. Below $0.3$ AU, which was never visited
by spacecraft before the launch of PSP, the numbers range from 7 to 34, when
we include error bars on both the PP19 and MC20 forecasts. The most important
result for our purposes is that PSP is expected to observe between 1 and 7
ICMEs below 0.1 AU, which is sufficiently high to consider the special
observing geometry of these encounters further in the next section.

Table 1: Total number of ICMEs expected to be observed by PSP in situ. Each row gives the predicted total number of ICME encounters by PSP below the given heliocentric distance, from 2020 July 1 until 2025 August 31.  R [AU] | PP19 | MC20
---|---|---
$all$ | $109\pm 57$ | $186\pm 67$
$<0.3$ | $15\pm 8$ | $25\pm 9$
$<0.2$ | $8\pm 4$ | $14\pm 5$
$<0.1$ | $3\pm 2$ | $5\pm 2$

###  3.3 Modeling the PSP double observation of ICME flux ropes

Figure 5: Modeling a potential ICME flux rope double crossing event by a
virtual Parker Solar Probe with 3DCORE. (a) The 3DCORE flux rope is shown in
red at a simulation time of $t_{launch}+4$ hours, when PSP enters the flux
rope for the first time, with the PSP trajectory indicated as a dashed line.
The blue flux rope represents the simulation time of $t_{launch}+26$ hours,
approximately when the virtual PSP spacecraft starts to cross the leg of the
flux rope. Panel (b) shows the same model configuration in a view from solar
north, and in panel (c) only the blue flux rope in an edge-on view at a 90
degree angle to the CME direction is given, with the position of PSP at
simulation time of 4 (26) hours shown as a red (blue) dot. This demonstrates a
considerable movement of PSP for an imaging observer during the solar close
encounters. This figure is available as an animation in the online version of
the article and at https://www.youtube.com/watch?v=VNC2lsw-UtU. Figure 6: (a)
Magnetic field components in RTN coordinates as observed by the virtual PSP
spacecraft as it moves along its trajectory (solid lines), compared to the
stationary case (dashed lines). The time is given in hours since the launch
time $t_{launch}$. (b) Similar measurements during the leg encounter, where
the time is shifted compared to panel (a) by + 22 hours. Note that the virtual
stationary spacecraft only impacts the flux rope apex, and not the leg. The
axis ranges are similar in both panels and cover 5 hours of in situ
observation.

We now ask ourselves what happens when PSP observes a CME with high relative
speed during close solar flybys at $<0.1$ AU and show a simulated ICME
encounter of such a situation. As first presented by Möstl et al. (2018) and
recently updated by Weiss et al. (2020, submitted to ApJS), 3DCORE is a semi-
empirical 3D flux rope model for the magnetic field inside a CME. It can be
used for both fitting available in situ observations and for forward modeling
of synthetic magnetic field measurements. It uses the force-free constant
twist Gold-Hoyle flux rope model (Gold & Hoyle, 1960) in an approximated 3D
configuration to calculate the magnetic field components, which are shown here
in radial-tangential-normal RTN coordinates, similar to how PSP magnetic field
observations are available. The geometrical shape is described as a tapered
torus attached to the Sun, and here we use a circular cross-section, although
elliptically flattened cross-sections are available within the model framework
too. The flux rope expands according to empirical relationships for the decay
of the flux rope magnetic fields, and moves with a drag based modeling
approach (Vršnak et al., 2013).

For the simulated PSP in situ ICME encounter, we assume it happens during its
12th solar flyby on 2022 June 1, reaching at minimum distance to the Sun of
0.061 AU or 13.1 solar radii. Between late 2021 and 2023 there is a series of
7 PSP orbits with the same perihelion distance. We place the flux rope apex at
5 solar radii, from where it launches at the time $t_{launch}=$ 2022 June 1
20:00 UT. The CME direction, defined by the flux rope apex direction, is
chosen as such that it points towards the flyby perihelion at 145 degree
longitude and 2.5 degree latitude in HAE coordinates. The initial CME speed is
400 km s-1. The flux rope axis inclination is 0 degree to the solar equatorial
plane, and it has a left-handed magnetic field chirality, with its axis
pointing to solar east. This makes it a south-east-north (SEN) type of
magnetic flux rope (Bothmer & Schwenn, 1998). It propagates into a uniform 400
km s-1 background solar wind, and all other parameters are chosen to represent
an average ICME, as it would reach a radial size of 0.24 AU and a maximum
total magnetic field strength of 12 nT at 1 AU.

Figure 5 gives an overview of the modeling results, visualizing the flux rope
shape (red) at $t_{0}+4$ hours, just as the CME apex impacts PSP. About 1 day
later, at about $t_{0}+26$ h, PSP passes through the model magnetic flux rope
(blue) for the 2nd time, this time through the flux rope leg. This means that
PSP may essentially make a multipoint in situ CME observation with a single
spacecraft, sampling the CME at more than one position within the flux rope
and at different times. Clearly, the details of such an encounter would depend
on the CME parameters, in particular regarding speed, shape, direction and
orientation, with respect to the PSP position and timing. We further want to
illustrate what is expected to happen in an idealized situation.

In Figure 6a, the signature (solid lines) of a virtual spacecraft that
observes the CME along the real PSP trajectory is seen, exhibiting a typical
flux rope signature of an expanding low-inclination flux rope of the SEN type,
with $B_{T}$ making a unipolar excursion to the $-T$ direction (solar east)
and $B_{N}$ reversing its sign from south to north, while $B_{R}$ stays near
zero. The asymmetry in the total field arises from the flux rope expansion
during the measurement as well as the aging of the flux rope as it passes over
PSP (Farrugia et al., 1993), and other physical processes such as deflection
and reconnection with the surrounding solar wind are neglected here. Due to
the small heliocentric distance, where the CME flux rope has not yet expanded
much, the observation only takes around 1.5 hours, and peaks at a total field
magnitude of roughly $10^{3}$ nT. PSP moves here at a maximum speed of 161 km
s-1 during perihelion. We compare this to a virtual PSP spacecraft that
observes the CME magnetic field while staying still at the CME apex position
in Figure 6a, shown as dashed lines. There is only very little difference
between the moving and stationary cases, from which no additional information
on the magnetic field structure of the flux rope can be gained.

However, about 26 hours after the CME launch, the virtual spacecraft
intercepts the CME flux rope a second time, and in Figure 6b we see that now
the $B_{N}$ component, though staying near zero, is essentially reversed
compared to the first encounter, as the virtual PSP now observes the flux rope
from the inside to the outside - compared to an outside-to-inside trajectory
during the apex encounter. The radial $B_{R}$ component is strongly elevated
and almost similar to the total field as the axial field of the flux rope
points away from the Sun at the western leg. The sign of the $B_{T}$ component
is similar to the first encounter, but the magnitude is strongly decreased.
This observation lasts for about 2 hours in the simulation, as during one day
the flux rope has already somewhat expanded, even though the flux rope cross-
section diameter narrows from the apex to the leg in the 3DCORE model. Without
modeling such as shown here the connection between two such possible
observations as a double encounter of the same flux rope would likely not be
made.

Simultaneous imaging of a double-crossing event by coronagraphs and
heliospheric imagers onboard SOHO, STEREO-A, PSP or Solar Orbiter could
further substantiate the interpretation that PSP indeed crosses a CME flux
rope twice. When multiple viewpoints are available, forward modelling
techniques (e.g. Thernisien et al., 2006) provide results on CME direction,
speed and longitudinal extent. They should help in discriminating whether the
two in situ signatures proposed in Figure 6 may arise from the same CME flux
rope or if rather two flux ropes from different CMEs are observed. This
becomes more important later in the PSP mission as the spacecraft perihelia
become smaller while the CME rate increases (see Figure 4), likely leading to
multiple CMEs erupting in quick succession during solar maximum (e.g.
Schrijver & Title, 2011).

The very different magnetic signatures that could be observed during both PSP
flux rope encounters would be an extremely valuable hint to constrain the 3D
CME magnetic field structure, and may help in clarifying pressing questions on
the origins of CMEs. On the other hand, a lack of the 2nd encounter of the CME
flux rope, where the CME was clearly observed in coronagraphs to have the
morphology of a flux rope with its symmetry axis close to the equator (e.g.
Thernisien et al., 2006; Vourlidas et al., 2013; Wood et al., 2017), would
raise questions on the general validity of the flux rope picture, and how the
magnetic field lines at the flux rope leg close in on the corona and whether
they are already detached from the Sun. However, from bi-directional electron
streaming observations, it is expected that for such small heliocentric
distances only very few magnetic field lines would have already separated from
the Sun (e.g. Shodhan et al., 2000; Nieves-Chinchilla et al., 2020). A
significant deviation of the measurements with the prediction from 3DCORE
would indicate that additional physical processes not captured by this version
of the model are central to the coronal evolution of CMEs, for example a
deflection in direction, a different expansion law or magnetic reconnection
with the background solar wind.

##  4 Conclusions

We have shown that double crossings of ICME flux ropes during close solar
flybys of the Parker Solar Probe (PSP) spacecraft are possible. To this end,
we have made an update of the largest available database of interplanetary
coronal mass ejections (ICMEs), with 739 events cataloged since 2007. Together
with the ICME list by Richardson & Cane (2010), which includes all ICMEs
measured in the solar wind near Earth since 1996, we have made a linear fit as
an approximate relationship of the yearly ICME rate with the yearly, smoothed
sunspot number as a proxy for solar activity, including an uncertainty arising
from the stochastic nature of ICME observations that results in the variation
of the yearly ICME rate for different places in the inner heliosphere.

We have then forecasted the ICME rate for the next solar cycle 25 based on two
different predictions for the sunspot number, and determined that between 1
and 7 CMEs are expected to be observed by PSP at distances $<0.1$ AU to the
Sun. We then modeled the consequences for possible in situ observations by the
PSP during a close solar flyby by combining our semi-empirical flux rope model
3DCORE with the predicted PSP flight trajectory in an idealized setting. The
simulated in situ measurements show that although PSP crosses the flux rope at
high relative speed, the observation of the magnetic field components near the
flux rope apex stays almost identical as compared to the measurements made by
a stationary spacecraft. However, for the modeled flux rope with a low
inclination to the solar equatorial plane, PSP crosses the leg about a day
later, which in our simulation gives a telltale reversal in the sign of the
$B_{N}$ component and a strongly increased $B_{R}$ field compared to the apex
encounter. In reality, we expect that a higher inclination of the flux rope
will decrease in the probability for PSP to observe a double crossing, while a
flatter cross section would increase its chances, but a parameter study to
look in detail at the influence of different parameter settings goes beyond
the scope of the current paper. Additionally, imaging data from PSP itself,
SOHO, STEREO-A or Solar Orbiter may provide context to further prove that
indeed the same CME flux rope was crossed by PSP twice.

In summary, we think these are highly interesting results which could strongly
constrain models for predicting the magnetic field components of CMEs, an
unsolved problem in space weather prediction, and could even shed light on the
early stages of CME eruption and formation (see also Al-Haddad et al., 2019).
Future studies may look into more complex physics surrounding the eruption
process of the CME such as channeling (Möstl et al., 2015) or deflection (e.g.
Gopalswamy et al., 2009), and the interaction of the flux rope with the
background solar wind (e.g. Ruffenach et al., 2015), to see whether there are
unexpected features observed concerning magnetic reconnection or ICME
deformations during the putative double-crossing events. The results presented
in this work therefore give an indicator of the type of measurements combined
with simulations that we hope to make in the upcoming years.

It has not escaped our notice that the McIntosh et al. (2020) based prediction
for the ICME rate would lead to the the highest rates of ICME impacts at Earth
in 40 years, which, if true, will make mitigation of destructive effects of
space weather on technological assets increasingly urgent for humankind.

C.M., A.J.W, R.L.B, M.A., T.A., M.B and J.H. thank the Austrian Science Fund
(FWF): P31265-N27, P31521-N27, P31659-N27. D.S. is supported by STFC
consolidated grant number ST/5000240/1. N.L. acknowledges NASA grant
80NSSC20K0700. Sunspot numbers were thankfully provided by SILSO Royal
Observatory of Belgium, Brussels (http://sidc.be/silso/home). This research
used version 1.1.0 (doi: 10.5281/zenodo.3604353) of the SunPy open source
software package (SunPy Community et al., 2020). This research made use of
Astropy (http://www.astropy.org) a community-developed core Python package for
Astronomy (Astropy Collaboration et al., 2013; Price-Whelan et al., 2018).
This research made use of HelioPy, a community-developed Python package for
space physics [doi:10.5281/zenodo.3739113]. The version of the ICMECAT catalog
used in this paper is permanently available at
https://doi.org/10.6084/m9.figshare.6356420.v3. The code to produce the
results in this paper is accessible in two jupyter notebooks at
https://github.com/helioforecast/Papers/tree/master/Moestl2020_PSP_rate and
the code, figures and animations for this paper are additionally archived at
https://doi.org/10.6084/m9.figshare.12563765.

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