11institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Université Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
22institutetext: Laboratoire Cogitamus, rue Descartes, 75005 Paris, France

# Identification of a Single Plasma Parcel during a Parker Solar Probe-Solar
Orbiter radial alignment

Etienne Berriot 11 Pascal Démoulin, 1122 Olga Alexandrova 11 Arnaud Zaslavsky
11 Milan Maksimovic 11

(January 19, 2024)

Configurations where two spacecraft, such as Parker Solar Probe (PSP) and
Solar Orbiter (SolO), are radially aligned provide opportunities to study the
evolution of a single solar wind parcel during so called plasma line-ups. The
most critical part of such studies is arguably the identification of what can
be considered a same plasma crossing both spacecraft. We present here a method
that allowed us to find what we believe to be the same plasma parcel passing
through PSP ($\sim 0.075$ au) and SolO ($\sim 0.9$ au) after their radial
alignment the 29/04/2021. We started by modeling the plasma propagation in
order to get a first estimation of the plasma line-up intervals. The
identification of the same density structure passing through the two
spacecraft allows to precise and confirm this estimation. Our main finding is
how stable the density structure is, remaining well recognizable from PSP to
SolO despite its $\sim 137$ hours journey in the inner heliosphere. We
moreover found that the studied slow solar wind plasma parcel undergone a
significant acceleration (from $\sim 200$ to $\sim 300$ km/s) during its
propagation.

###### Key Words.:

Solar Wind – Radial Alignment – Parker Solar Probe – Solar Orbiter

##  1 Introduction

The solar corona has a too high temperature to remain in hydrostatic
equilibrium with its surroundings. This leads to an expansion of the Sun’s
atmosphere in the interplanetary medium, creating a supersonic and super-
Alfvénic outflow of plasma we call the solar wind (Parker 1958).

The solar wind is often separated into two categories depending on its speed,
the fast and slow winds. The fast solar wind is thought to arise from coronal
holes (Zirker 1977; McComas et al. 1998) with speeds ranging from $\sim$500
km/s to $\sim$800 km/s. The slow solar wind has speeds ranging from $\sim$150
km/s to $\sim$500 km/s with lower temperatures and higher densities than the
fast wind. The processes giving it birth as well as its exact origins are
however still disputed (see Abbo et al. 2016; Rouillard et al. 2021, and
references therein). There are also still many pending questions regarding the
radial evolution of the solar wind and interplanetary structures it carries
within the heliosphere.

Helios 1 & 2 allowed to study the radial evolution of what can be considered
the same plasma passing through both spacecraft when they were radially
aligned. This method was first employed by Schwenn et al. (1981a, b), giving
it the name ”plasma line-up”. One of the interval they found has been
thoroughly studied in Schwartz & Marsch (1983) when Helios 1 and Helios 2
respectively were situated at $\sim$0.51 au and $\sim 0.72$ au from the Sun.
Their study focuses on the radial evolution of the energy budget and adiabatic
invariants (Chew et al. 1956) in what was identified to be the same plasma
parcel inside a fast solar wind stream. The authors also discuss different
hypotheses and difficulties linked with the mapping and plasma parcel
identification. Their identification has been done considering a constant and
radial propagation speed for the plasma parcel. However, any acceleration, or
non-radial flow can change the time intervals to consider and are therefore
potential sources of uncertainties.

Moreover, the term ”plasma line-up” and ”plasma parcel” are somewhat ill-
defined, as discussed in Schwartz & Marsch (1983). For example, the particles
populations of the solar wind tend to propagate with different velocities. The
protons often exhibit a beam in their velocity distribution function, the
electrons have a thermal speed much higher than their bulk speed, and alpha
particles usually have a different bulk speed’s than the protons (see Marsch
2012, and references therein). Furthermore, halo electrons are probably
governed by non-local scattering mechanisms (Zaslavsky et al. 2024). We
however prefer to keep the general terminology ”plasma parcel/line-up”
throughout this study.

The recently launched Parker Solar Probe (PSP, Fox et al. 2016) and Solar
Orbiter (SolO, Müller, D. et al. 2020) missions are great new opportunities
for such line-up studies. Both of them are orbiting in the inner heliosphere
simultaneously, allowing combined observations and measurements between the
two spacecraft (Velli et al. 2020). Conjoined observations with other
spacecraft such as STEREO-A or BepiColombo are also possible (Hadid et al.
2021).

Figure 1:  On panel (a) are PSP (blue) and SolO (orange) positions the
29/04/2021 (dots) and trajectories (lines) within the ecliptic plane between
the 24/04/2021 and the 04/05/2021 as seen in an inertial reference frame
centered on the Sun. The radial line coming from the Sun and passing through
PSP and SolO for the spacecraft coalignment time $t_{0}$ is indicated by a
black dashed line. Panels (b) and (c) respectively show the longitude $\phi$,
and latitude $\theta$ of PSP (blue) and SolO (orange) as functions of time for
the same interval as in panel (a). We have indicated the spacecraft
coalignment time $t_{0}=$ 00:45 UT on the 29/04/2021 as a vertical black
dashed line on both panels.

Indeed, there have been some recent plasma line-up studies, with PSP (0.1 au)
and SolO (1 au) in Telloni et al. (2021), and with PSP (0.17 au) and
BepiColombo (0.6 au) in Alberti et al. (2022). The purpose of these studies
was the radial evolution of statistical properties of magnetic turbulence.
Telloni et al. (2021) first estimated the intervals for the plasma line-up
assuming a constant speed of 315 km.s-1 for the solar wind (as measured at
PSP). They then assessed the plasma correspondence by calculating the Pearson
correlation coefficient between PSP and SolO magnetic field’s magnitude
measurements at time intervals corresponding to different propagation speeds
([250, 350] km.s-1) to take into account some eventual acceleration. A similar
method for the plasma propagation has been employed in Alberti et al. (2022).
The authors also based their identification on a cross-correlation method
between the two spacecraft magnetic field’s magnitude, with sliding windows
for both PSP and BepiColombo and calculation of the mutual information
coefficient (Shannon 1948a, b; Cover & Thomas 2005). The identifications in
Telloni et al. (2021) and Alberti et al. (2022) suggest nearly zero
acceleration of the solar wind during the propagation, a result not verified
due to the absence of solar wind velocity measurement at the outer spacecraft
in both studies. This is quite surprising as several statistical studies
reported a non-negligible acceleration of the slow solar wind in the inner
heliosphere (Schwenn et al. 1981a; Sanchez-Diaz et al. 2016; Maksimovic et al.
2020; Dakeyo et al. 2022).

The results of plasma line-up studies are highly dependent on the time
intervals taken as being the same parcel of solar wind. It is therefore
crucial to be able to unambiguously identify what can be considered the same
plasma on both spacecraft.

In this paper, we present a new approach that allowed us to identify the same
plasma parcel passing through two radially aligned spacecraft. We study here a
radial alignment between PSP (at $\sim 0.075$ au) and SolO (at $\sim 0.9$ au)
on the 29/04/2021. The identification of the same plasma is done through
several steps. We first present an overview of the spacecraft configuration in
Section 2. Then, in Section 3, we describe our modeling of the solar wind’s
propagation from the inner to outer spacecraft to estimate the time intervals
corresponding to the plasma line-up. Using this estimation, we identify the
same density structure passing through both PSP and SolO in Section 4. We
finally summarize our results and conclude in Section 5.

##  2 Data and Line-up Configuration

The goal of this study is to identify the same plasma parcel at two different
distances from the Sun with PSP and SolO. In order to do this, we had to take
advantage of a configuration where the two spacecraft are quasi radially
aligned as shown in Figure 1(a), where we consider the positions of PSP and
SolO on the 29/04/2021 around 01:00. The plasma parcel is first crossed by
PSP, then propagates outward and is eventually crossed by SolO after some
propagation time $\tau$.

For this configuration, PSP and SolO were respectively situated at
approximately 0.075 au and 0.9 au from the Sun. We show in Figure 1(b) and (c)
the longitude ($\phi$) and latitude ($\theta$) of the two spacecraft around
their radial alignment, defined as the time when they are both at the same
longitude as indicated by the vertical black dashed lines. We will call this
time $t_{0}$. It corresponds to

$t_{0}=\text{29/04/2021 00:45 UTC}$ (1)

and is used as the reference time for the rest of this study. We therefore
define

$t=t_{UTC}-t_{0}$ (2)

where $t_{UTC}$ is the Coordinated Universal Time.

We remark in Figure 1(b) that PSP’s longitude is varying much more than SolO’s
one. This is because PSP is orbiting way faster than SolO as it is about 12
times closer to the Sun. Indeed, at $t_{0}$, $\omega_{PSP}\simeq 1.25\times
10^{-5}$ rad/s and $\omega_{SolO}\simeq 1.95\times 10^{-7}$ rad/s, so
$\omega_{PSP}/\omega_{SolO}\sim 64$, with $\omega_{PSP}$ and $\omega_{SolO}$
the angular speeds of PSP and SolO respectively.

We also point that there is a latitude difference of $\Delta\theta\sim 3$° at
$t_{0}$ when the two spacecraft are coaligned in longitude. At SolO’s distance
($\sim 0.9$ au), this difference corresponds to a length
$l_{\Delta\theta}\simeq 7\times 10^{6}$ km or $0.05$ au. This
$l_{\Delta\theta}$ imposes a lower bound on the scale of the plasma parcel we
can expect to observe on both spacecraft.

For this study, we analyzed the protons densities $N_{p}$, and bulk velocities
$\mathbf{V}_{p}$ as well as the magnetic field $B$ measured by PSP and SolO.
For the protons, on PSP those are obtained with measurements of the SPAN-ion
instrument (Livi et al. 2021), part of the SWEAP (Kasper et al. 2016) suite,
while on SolO, we are using measurements of the Proton and Alpha particle
Sensor (PAS, Owen et al. 2020). Magnetic field measurements are from the
FIELDS instrument (Bale et al. 2016) on PSP and the MAG instrument on SolO
(Horbury et al. 2020).

Figure 2: Simplified and not to scale schematic of the propagation method for
a purely radial plasma speed. PSP’s position at a considered time $t_{in}$ is
indicated by a blue dot. We represented with green dots, and at subsequent
times ($t_{1},t_{2},t_{3}$), the position of the plasma parcel passing by PSP
at $t_{in}$ and propagating outward with $\mathbf{V}$ (green vector). We also
represented the position of SolO (orange dots) and the distance between the
parcel and SolO (purple double arrows) at the same times. The simplified
trajectories of PSP and SolO are shown by blue and orange dashed lines
respectively. The plasma parcel trajectory is shown by the black dashed line.

##  3 Ballistic Propagation model

In order to identify the time intervals corresponding to a plasma line-up, one
first needs to model the propagation of the plasma parcel from the inner
spacecraft to the outer one. There are several ways of doing this modeling, we
propose the following method.

We begin with the configuration shown in Figure 1, for which the two
spacecraft are quasi-aligned. The positions of PSP and SolO as functions of
time $t$ will be noted $\mathbf{R}_{PSP}(t)$ and $\mathbf{R}_{SolO}(t)$
respectively. We then define the position of the plasma parcel,
$\mathbf{R}(t,t_{in})$, at every moment $t$ following its crossing of the
inner spacecraft at a time $t_{in}$:

$\mathbf{R}(t,t_{in})=\mathbf{R}_{in}(t_{in})+\int_{t_{in}}^{t}\mathbf{V}(t^{\prime},t_{in})dt^{\prime}$
(3)

where $\mathbf{V}(t^{\prime},t_{in})$ is the plasma propagation velocity,
which can have any profile, and by definition
$\mathbf{R}_{in}(t_{in})=\mathbf{R}_{PSP}(t=t_{in})$.

Figure 3:  Outcome of the propagation method for a radial and constant
velocity $\mathbf{V}=(V_{R}=V_{in},0,0)$ in RTN coordinates. Panel (a) shows
the radial proton speed recorded by PSP ($V_{p,PSP}(t)$ in blue) and
$V_{in}(t_{in})$ (in black) calculated as averages of $V_{p,PSP}$ over $\Delta
t=1$ h centered on each considered $t_{in}$. Panels (b) and (c) respectively
show the minimum distance $d_{min}$ between the plasma parcel and SolO, and
the corresponding propagation times $\tau$ and $t_{out}$, all functions of
$t_{in}$. The vertical lines in panels (b) and (c) correspond to the $t_{in}$
for which $d_{min}$ is minimum (called $d_{MIN}$). Panel (d) is the solar wind
radial speed recorded by SolO ($V_{p,SolO}(t)$), for a time interval
corresponding to $0$ h $\leq t\leq 200$ h. All the time origins are set at
$t_{0}$, corresponding to the radial alignment time of the spacecraft
(Equation (1)).

We calculate the distance between the plasma parcel and the outer spacecraft
as

$d(t,t_{in})=|\mathbf{R}_{SolO}(t)-\mathbf{R}(t,t_{in})|$ (4)

The process is schematized in Figure 2. After some travel time $\tau(t_{in})$,
this distance passes by a minimum $d_{min}(t_{in})$ at a time
$t=t_{out}(t_{in})$. We define the position of the plasma parcel after the
travel as $\mathbf{R}_{out}(t_{in})\equiv\mathbf{R}(t=t_{out},t_{in})$. We
next iterate the above computation for different $t_{in}$. Each considered
time $t_{in}$ is therefore linked to a time $t_{out}(t_{in})$ through the
relation:

$t_{out}(t_{in})=t_{in}+\tau(t_{in})$ (5)

This allows to describe, for every considered solar wind parcel, when the
plasma crossed by the inner spacecraft (at $t_{in}$) gets the closest to the
outer one (at $t_{out}(t_{in})$), as well as the corresponding distance
$d_{min}(t_{in})$ and propagation time $\tau(t_{in})$.

###  3.1 Propagation Method with Constant Velocity

As a first step, we consider the plasma’s propagation velocity to be constant
and radial over its journey from the inner to the outer spacecraft. We define
the parcel’s propagation velocity as
$\mathbf{V}_{in}\equiv\langle\mathbf{V}_{p,PSP}\rangle$, where
$\langle\mathbf{V}_{p,PSP}\rangle$ is the average protons velocity measured by
PSP over a time interval $\Delta t$ centered on $t_{in}$. We chose ${\Delta
t}=1$ h as to get a more relevant estimation of the plasma propagation’s speed
by averaging velocity fluctuations related to the turbulent cascade. The
computations were realized with a temporal resolution of 5 min for $t_{in}$
and of 1 min for $t$ (defining the position of the plasma parcel).

We show in Figure 3 the results of this propagation method for the line-up
configuration of Figure 1. Figure 3(a) shows the radial protons bulk speed
measured by PSP $V_{p,PSP}(t)$ with a blue curve, and $V_{in}(t_{in})$ with a
black curve. Figure 3(b) gives the estimated distance between the plasma
parcel and the outer spacecraft after the propagation, $d_{min}(t_{in})$,
while on Figure 3(c) are the corresponding propagation time $\tau(t_{in})$ and
time at the outer spacecraft $t_{out}(t_{in})$.

The closest approach is well defined: $d_{min}$ has a clear minimum
$d_{MIN}=\text{min}(d_{min})\simeq 7\times 10^{6}$ km, indicated by the
vertical dashed line. Therefore the time intervals to look at the inner and
outer spacecraft should respectively be around the associated $t_{in}\simeq
2.9$ h and $t_{out}\simeq 180$ h. We obtained $d_{MIN}\simeq 7\times 10^{6}$
km, which is close to $l_{\Delta\theta}$ (due to the spacecraft latitude
difference around $t_{0}$) estimated in the end of Section 2. This is expected
since the corresponding $t_{in}(\simeq 2.9$ h) and $\tau(\simeq 180$ h) are
respectively much lower than the orbiting period of PSP and SolO, therefore
the spacecraft latitude difference $\Delta\theta$ does not change
significantly from its value at $t=t_{0}$. Due to the large radial difference
between the spacecraft, the speed variations observed at PSP (Figure 3(a)),
and therefore on $V_{in}$ make the propagation time $\tau(t_{in})$ vary a lot
(145 h $\lesssim\tau\lesssim$ 185 h) in the range of $t_{in}$ considered, see
Figure 3(c).

We next look at the solar wind recorded by SolO for a very large range of
times to see how well the hypothesis of constant speed is verified. We thus
show on Figure 3(d) the proton radial speed recorded by SolO for $0\leq t\leq
200$ h. Unfortunately, there is no SWA data available from the
$\sim$05/02/2021 23:50 to the $\sim$05/04/2021 05:00, which translates to
$90\leq t\leq 125$ h. Even though, no matter the considered $t$, the observed
proton’s speed at SolO is higher than at PSP. This is consistent with results
of precedent studies that reported an acceleration of the slow wind in the
inner heliosphere (Sanchez-Diaz et al. 2016; Maksimovic et al. 2020; Dakeyo et
al. 2022). We therefore have to take this acceleration into account in the
propagation method.

###  3.2 Propagation Method with Constant Acceleration

Using only the data at both spacecraft, the in situ measured speeds can only
define an average acceleration during the travel time. In the absence of other
data, the simplest solution is to consider a constant acceleration constrained
by the protons velocities measured by PSP and SolO. However, this acceleration
$\mathbf{a}$ is not directly accessible from the two set of measurements as it
first requires to link $t_{in}$ and $t_{out}(t_{in})$ through the minimization
of $d(t,t_{in})$ (Equation (4)) for each $t_{in}$. This is done below by
scanning a range of $\mathbf{a}$, constrained by the observations. In order to
simplify the notations, we will mostly omit explicit mentions of the
dependencies to $t_{in}$ in the rest of this paper.

We first consider the plasma propagation with an arbitrary constant
acceleration $\mathbf{a}$. For each $t_{in}$, the positions and speeds of the
plasma parcel at every time $t>t_{in}$ following the crossing of the inner
spacecraft are given as

$\displaystyle\mathbf{R}(t)=$
$\displaystyle\mathbf{R}_{in}+(t-t_{in})\mathbf{V}_{in}+\frac{(t-t_{in})^{2}}{2}\,\mathbf{a}$
$\displaystyle\mathbf{V}(t)=$
$\displaystyle\mathbf{V}_{in}+(t-t_{in})\,\mathbf{a}$ (6)

After the propagation this model provides

$\displaystyle\mathbf{R}_{out}=$
$\displaystyle\mathbf{R}_{in}+\tau\mathbf{V}_{in}+\frac{\tau^{2}}{2}\mathbf{a}$
(7) $\displaystyle\mathbf{V}_{out}=$
$\displaystyle\mathbf{V}_{in}+\tau\,\mathbf{a}$ (8)

with $\tau$ the propagation time, $\mathbf{R}_{out}=\mathbf{R}(t=t_{out})$,
and $\mathbf{V}_{out}=\mathbf{V}(t=t_{out})$. We use Equation (8) to write the
acceleration as

$\mathbf{a}=\frac{\mathbf{V}_{out}-\mathbf{V}_{in}}{\tau}$ (9)

Replacing this in Equation (7), we get

$\tau=\frac{2\,||\mathbf{R}_{out}-\mathbf{R}_{in}||}{||\mathbf{V}_{out}+\mathbf{V}_{in}||}\,.$

Combining the two last equations, we finally can express the acceleration as

$\mathbf{a}=\frac{||\mathbf{V}_{out}+\mathbf{V}_{in}||}{2\,||\mathbf{R}_{out}-\mathbf{R}_{in}||}\left(\mathbf{V}_{out}-\mathbf{V}_{in}\right)$
(10)

If we indeed cross the same plasma on both spacecraft, then after the
propagation we should have
$\mathbf{R}_{out}\approx\mathbf{R}_{SolO}(t=t_{out})$ and
$\mathbf{V}_{out}\approx\mathbf{V}_{p,SolO}(t=t_{out})$. $\mathbf{V}_{in}$ can
be estimated using $\mathbf{V}_{p,PSP}(t)$ around $t_{in}$ and
$\mathbf{R}_{in}=\mathbf{R}_{PSP}(t=t_{in})$. The acceleration written in
Equation (10) is therefore a function of the observed quantities only,
provided that $t_{in}$ and $t_{out}$ are defined. However, it does not
incorporate the association between $t_{in}$ and $t_{out}$ established by
minimizing $d(t,t_{in})$ of Equation (4).

Figure 4:  Propagation method outcomes using a constant acceleration $a$
constrained by the observed plasma speeds at the inner and outer spacecraft.
All quantities are shown as functions of $t_{in}$. The colors on panel (a) are
values of $\Delta V=\langle V_{p,SolO}\rangle-V_{out}$ and the black curve is
the radial acceleration for the minimum of $|\Delta V|$. The method’s outcomes
shown in panels (b), (c) and (d) are all corresponding to propagations
considering this acceleration. On panel (b) are, in blue $V_{in}$ calculated
as in Figure 3, in orange $\langle V_{p,SolO}\rangle$ defined as the proton
bulk speed measured by SolO and averaged for $\Delta t=$1 h centered on
$t_{out}$, in green $V_{out}$, which is the theoretical speed given by
Equation (8). The corresponding $d_{min}$, $\tau$ and $t_{out}$ are shown in
panels (c) and (d) respectively.

The results are expected to be close to the spacecraft alignment, so $R_{i}n$
and $R_{o}ut$ are weakly dependent of the precise $t_{i}n$ and $t_{o}ut$
values. We therefore calculate $||\mathbf{R}_{out}-\mathbf{R}_{in}||$ using
the position of PSP around $t_{in}$ and the position of SolO around $t_{out}$
(both associated to $d_{MIN}$) estimated using a constant velocity (Figure 3).
This approximation is a posteriori justified by the fact that the
$||\mathbf{R}_{out}-\mathbf{R}_{in}||$ estimated using a constant velocity and
the one estimated with a constant acceleration only differ by $\sim 1\%$. The
acceleration given by Equation (10) is indeed mostly changing with $t_{in}$
due to variations of $\mathbf{V}_{in}$ and $\mathbf{V}_{out}$.

As previously, we consider a purely radial plasma propagation and
$V_{in}=\langle V_{p,PSP}\rangle$ averaged over $\Delta t=1$ h around
$t_{in}$. The acceleration and speeds are noted as scalars representing the
radial component of the vectors. Since the radial acceleration cannot be
derived directly from observations, we define a maximum acceleration $a_{max}$
as

$a_{max}=\frac{V_{out,max}^{2}-V_{in,min}^{2}}{2\,||\mathbf{R}_{out}-\mathbf{R}_{in}||}$
(11)

where we fix $V_{out,max}=480$ km/s and $V_{in,min}=180$ km/s close to the
maximum and minimum of the radial plasma speeds respectively measured by SolO
and PSP during the considered time periods. Then, for each considered
$t_{in}$, we compute the plasma’s propagation for 75 accelerations uniformly
spaced between 0 and $a_{max}$. By minimizing $d(t,t_{in})$ for each $a$ value
we obtain the parameters ($t_{out}$, $\tau$, $d_{MIN}$) as functions of
$t_{in}$ and $a$.

We next need to define the $a$ value which is the most compatible with the
observed velocity at SolO for each $t_{in}$ value. The above model provides
$V_{out}=V(t=t_{out})$ using Equation (6) projected on the radial axis. Using
SolO measurements, we also define (as for PSP) a proton bulk speed $\langle
V_{p,SolO}\rangle$ averaged over $\Delta t=1$ h around each obtained
$t=t_{out}$. We note that because of the data gap at SolO (straight line in
Figure 3(d)), we considered the missing data as interpolated values between
the nearest available measurements. We therefore define

$\Delta V=\langle V_{p,SolO}\rangle-V_{out}$ (12)

Finally, for each $t_{in}$ we select the acceleration $a$ that minimizes
$|\Delta V|$ so that the propagation model with a constant acceleration is
most consistent with the observations at both spacecraft.

We show the results of the above procedure in Figure 4. On panel (a), the
colors indicate values of $\Delta V$ with the black curve being the
acceleration associated to the minimum of $|\Delta V|$ at each $t_{in}$. Panel
(b) shows the corresponding $V_{in}$ in blue, $\langle V_{p,SolO}\rangle$ in
orange and $V_{out}$ in green. The minimum of $|\Delta V|$ is therefore the
difference between the green and orange lines. Its standard deviation is
$\simeq 2$ km/s. This small value, an order of magnitude lower than the
fluctuations of $V_{in}$ and $\langle V_{p,SolO}\rangle$, shows that the
acceleration bins are numerous enough.

The minimum distance $d_{min}$, obtained for the minimum of $|\Delta V|$, is
shown in Figure 4(c). As in Figure 3, the vertical dashed line indicates the
value $t_{in}$ obtained for the minimum distance $d_{MIN}$. Here,
$t_{in}=2.25$ h which is smaller by only $\sim 40$ min as compared to the
result obtained above with a constant velocity. Considering a purely radial
propagation speed and neglecting 3D effects, the plasma line-up implies that
the longitudes $\phi$ of the inner and outer spacecraft have to be the same at
$t_{in}$ and $t_{out}$, respectively. Using the longitude of the spacecraft
line-up as the longitude origin, this condition approximately writes

$\omega_{out}\,t_{out}=\omega_{in}\,t_{in}$ (13)

where $\omega_{out}$ and $\omega_{in}$ are the longitudinal angular velocities
of the spacecraft around the Sun (in an inertial reference frame). This
implies that a variation of $t_{out}$, say by $\delta t_{out}$ due to a
different transit time $\tau$, has an effect $\delta
t_{in}=(\omega_{out}/\omega_{in})\delta t_{out}$ on $t_{in}$. So, the more
different are the spacecraft distances from the Sun, the more different are
their angular velocities, and therefore, the smaller $\delta t_{in}$ is. With
the present PSP and SolO configuration the ratio
$\omega_{out}/\omega_{in}\approx 1/64$. This explains why in the studied case,
the $t_{in}$ associated with the plasma line-up is weakly dependent on the
plasma velocity profile.

Figure 4(d) shows $\tau$ and $t_{out}$. These $t_{out}$ values are fortunately
not in the data gap shown in Figure 3(d), so the resulting outcomes are
directly linked to SolO observations. $\tau$ and $t_{out}$ fluctuate much less
than with a constant speed hypothesis (Figure 3(c)). More precisely, for a 10
hours time interval of $t_{in}$, the $t_{out}$ variations are in a time
interval of 15 h for a constant acceleration, as compared to 50 h in the case
of a constant velocity. Hence, the constant acceleration model (constrained by
both spacecraft data) provides a more reliable estimation of the time at SolO
for an approximate plasma line-up. The constant acceleration model implies
also a significantly lower propagation time $\tau$, by almost 50 h as compared
to the constant velocity model (as expected due to the observed increase of
plasma velocity).

Figure 4(c) shows that $d_{MIN}=7\times 10^{6}$ km (the minimum value of
$d_{min}$), is well defined. This $d_{MIN}$ is close to the one obtained with
a constant velocity (Figure 3) and the distance estimated in Section 2 due to
the spacecraft latitude difference around the spacecraft alignment
($l_{\Delta\theta}\sim 7\times 10^{6}$ km). The fact that $d_{MIN}\simeq
l_{\Delta\theta}$ confirms that, when considering a purely radial propagation
velocity, this minimum is mainly constrained by the spacecraft orbits and not
by the plasma dynamics. Moreover, the $d_{min}$ curve with constant
accelerations, Figure 4(c), is smoother than with constant velocities (Figure
3(b)). This is a consequence of a narrower range of $t_{out}$ as shown in
Figure 4(d).

Figure 5:  Proton density $N_{p}$ (panels (a) and (c)) and magnetic field’s
magnitude $B$ (panels (b) and (d)) measured by PSP (blue) around $t_{in}$ and
Solar Orbiter (orange) around $t_{out}$ when the minimum distance $d_{MIN}$ is
achieved.

##  4 Same Plasma Identification

###  4.1 Data selection

The propagation model provides a first estimation of the time intervals
corresponding to the plasma line-up. This estimation however has to be
precised and confirmed. To do so, we propose to search for a same structure
passing through both spacecraft. Once found, this structure can then be used
as a marker to unambiguously define the same plasma.

On Figure 5 are the proton density $N_{p}$ and magnetic field’s magnitude $B$
measurements taken by PSP (panels (a, b)) and SolO (panels (c, d)) around the
times corresponding to $d_{MIN}$, so $t_{in}=2.25$ h and $t_{out}\sim 135$ h
as determined above, see Figure 4(d). We therefore chose to consider $t\in
2.25\pm 4$ h for PSP, and $t\in 135\pm 10\leavevmode\nobreak\ $h for SolO to
take into account uncertainties linked to the propagation model. We remark on
PSP measurements the presence of a density enhancement and simultaneous anti-
correlated magnetic field’s depletion for $t\in[-0.5,1.5]\leavevmode\nobreak\
$h (indicated by red vertical dashed lines). We note that a visual comparison
with SolO’s data already allows the identification of a very similar structure
around $t\in[137,139]\leavevmode\nobreak\ $h, also indicated with two red
vertical dashed lines.

The proton density is indeed typically a useful parameter to look as it often
exhibits well identifiable spatial structures. Density enhancements have been
reported in the solar wind (especially the slow one), either using remote-
sensing instruments (e.g. Sheeley et al. 1997; Rouillard et al. 2010a; Viall &
Vourlidas 2015) or in situ measurements (e.g. Viall et al. 2008; Rouillard et
al. 2010b; Stansby & Horbury 2018). These structures, that are thought to
emerge near the tip of coronal helmet streamers, are also generally believed
to be well conserved during their propagation in the heliosphere and simply
carried by the surrounding solar wind, ”like leaves in the wind” as put by
Sheeley et al. (1997).

Moreover mesoscale structures, such as density enhancements, typical radial
size at 1 au are $\sim 5\times 10^{3}-10^{7}$ km (Viall et al. 2021, and
references therein). Then, the larger density structures have sizes comparable
to the minimum distance $d_{min}$ ($\in[1,1.5]\times 10^{7}$ km for
$t_{in}\in[-0.5,1.5]$ h) evaluated for a propagation with constant
acceleration. This implies that, since SolO is situated at $\sim 0.9$ au in
the studied case, one of those density structures could be large enough to
eventually be crossed by both spacecraft. We note that $d_{min}$ is most
probably lower due to non-radial propagation effects (Appendix A). Moreover, a
direct comparison between the radial sizes of such structure (determined as
$V_{p,R}\times\delta t$, with $\delta t$ the duration of the structure) and
$d_{min}$ is only pertinent for structures with similar radial and ortho-
radial extensions.

###  4.2 Association with Cross-Correlations

Figure 6:  Cross-correlation for measurements of proton density $N_{p}$ (black
curves) and magnetic field magnitude $B$ (red curves) for PSP and SolO using
different measures. Panel (a) shows the Pearson correlation coefficient
defined by Equation (14), panel (b) the covariance, and panel (c)
$1/\chi_{X,Y}$ defined by Equation (17). The abscissa $\tau$ is the travel
time between PSP and SolO. The time interval considered at PSP is the one
shown between dashed lines on Figure 5(a,c). The same time duration, 2 h, is
used for the moving time window at SolO.

The goal is here to see if the structure identified at PSP (between two red
vertical dashed lines in Figure 5 (a,b)) has also passed through SolO. A
common way to quantitatively assess the correspondence between the
measurements at two spacecraft is through the use of a cross-correlation
method. Therefore, using three different coefficients, we computed the cross-
correlation between PSP measurements for $t\in[-0.5,1.5]\leavevmode\nobreak\
$h, and intervals of same temporal lengths ($T=2$ h) at SolO for time shifts
$125$ h $\leq\tau\leq 145$ h. We chose a time step of 0.1 h between each
$\tau$, defining the intervals at SolO used for the cross-correlation. We show
the results in Figure 6, where we computed these coefficients for both the
proton density $N_{p}$ (black curves) and the magnetic field amplitude $B$
(red curves).

Figure 6(a) shows the cross-correlation function based on the standard Pearson
correlation coefficient $\rho_{X,Y}$, defined as

$\rho_{X,Y}(\tau)=\frac{\langle\delta X(t)\,\delta
Y(t+\tau)\rangle}{\sqrt{\langle\delta X(t)^{2}\rangle}\sqrt{\langle\delta
Y(t+\tau)^{2}\rangle}}$ (14)

with

$\displaystyle\delta X(t)$ $\displaystyle=$ $\displaystyle X(t)-\langle
X(t)\rangle$ $\displaystyle\delta Y(t+\tau)$ $\displaystyle=$ $\displaystyle
Y(t+\tau)-\langle Y(t+\tau)\rangle$ (15)

for $X(t)$ and $Y(t+\tau)$ the same physical parameter recorded by PSP and
SolO respectively. The brackets $\langle...\rangle$ denote averages over $T$
(=2 h here). Due to its normalization, this coefficient is not affected by the
amplitudes of $X$ and $Y$, and therefore tends to give high correlation
between structures of different amplitudes. This explains why we observe
numerous high values peaks in Figure 6(a).

On Figure 6(b), is the cross-correlation function based on the covariance
$\sigma_{X,Y}$ defined as

$\sigma_{X,Y}(\tau)=\langle\delta X(t)\,\delta Y(t+\tau)\rangle$ (16)

and which we normalized by its maximum over all the tested intervals
(max($\sigma_{X,Y}$)). That way, due to their large variance, the biggest
structures will therefore tend to naturally give higher values for this
coefficient.

On Figure 6(c) is the cross-correlation function based on the inverse of the
chi-square coefficient $1/\chi_{X,Y}$, also normalized by its maximum over all
the tested intervals, with $\chi_{X,Y}$ defined as :

$\chi_{X,Y}(\tau)=\sqrt{\langle\left(\delta X_{c}(t)-\delta
Y_{c}(t+\tau)\right)^{2}\rangle}\,.$ (17)

$\chi_{X,Y}$ is named from its similarity with the statistical chi-square and
should be minimum when the signals are the most similar. We therefore chose to
show $1/\chi_{X,Y}$ to more easily compare it with cross-correlation functions
based on other coefficients. We use the typical radial variation summarized
with a power law of solar radial distance to define the normalised quantities
$\delta X_{c}(t)=\delta X(t)\,(R_{X}/R_{0})^{\varepsilon}$ and $\delta
Y_{c}(t+\tau)=\delta Y(t+\tau)\,(R_{Y}/R_{0})^{\varepsilon}$, where $R_{X}$ &
$R_{Y}$ are the distance between the spacecraft and the Sun, and $R_{0}=1$ au
is the distance taken for normalisation. We fixed $\varepsilon=2$ for the
density and $\varepsilon=1.6$ for the magnetic field’s magnitude in order to
take into account the plasma’s nearly spherical expansion. Due to its spiral
shape, the interplanetary magnetic field magnitude falls of less rapidly than
$R^{-2}$. We therefore choose to correct $B$ by a factor $(R/R_{0})^{1.6}$, in
accordance with previous statistical studies using Helios 1 data (Musmann et
al. 1977; Schwenn & Marsch 1990). This correction is needed as the solar wind
expansion changes strongly the magnitude of the studied plasma parameters with
solar distance. Note that, because of the subtraction $\delta X_{c}(t)-\delta
Y_{c}(t+\tau)$, Equation (17) results are sensitive to the choice of
$\varepsilon$ (or any other normalization chosen for the signals). However,
$1/\chi_{XY}$ is an important quantity since it shows much less numerous
peaks, with the main one being more outstanding and narrower than with the two
other correlation coefficients Finally, the three above coefficients are all
linear and any non-linear evolution of the plasma will therefore not correctly
be taken into account.

Table 1: Correlation coefficients between PSP and SolO measurements for the proton density $N_{p}$ and the magnetic field’s magnitude $B$ at $\tau=137.6$ h (see dashed line in Figure 6 and Section 4.2). Due to the normalization, a value of 1 for the coefficients using $\sigma_{X,Y}$ and $1/\chi_{X,Y}$ indicates an absolute maximum over the tested intervals.  | Correlation coefficients
---|---
Physical Quantity | $\rho_{X,Y}$ | $\frac{\sigma_{X,Y}}{\text{max}(\sigma_{X,Y})}$ | $\frac{1/\chi_{X,Y}}{\text{max}(1/\chi_{X,Y})}$
$Np$ | 0.90 | 1 | 1
$B$ | 0.81 | 0.97 | 1

All these cross-correlation functions show either absolute or local maxima for
Np and B at $\tau=137.6$ h (Figure 6). We show those values in Table 1. The
fact that all three coefficients have such high values, for both $N_{p}$ and
$B$, at $\tau=137.6$ h supports the correspondence between the two structures
shown within the red vertical dashed lines in Figures 5 (a,b) and (c,d).

As we can see on Figure 6, the correlation coefficients also exhibit local
maxima for some other time shifts. The most remarkable one being for $\tau\sim
130$ h. Going back to Figure 5(c, d), we see that there are another density
structure and associated magnetic field’s depletion around $t=130$ h. The
corresponding correlation values however are lower and a following visual
inspection does not allow to find a clear correspondence between the
structures. We therefore disregard this density structure as being the same
than the one observed at PSP.

###  4.3 Justifications and limitations of the correlation method

In the previous subsection, several parameters involved in the correlation
estimation were fixed. We justify below this choice and present a more general
way of computing the different correlation coefficients. We point that, in
general $X$ and $Y$ can be described as functions of several free parameters:

$\displaystyle X=$ $\displaystyle X(t^{*},T_{X},\delta t_{X})$ $\displaystyle
Y=$ $\displaystyle Y(t^{*}+\tau,T_{Y},\delta t_{Y})$

with $t^{*}$ denoting the center of the time interval at the inner spacecraft,
$\tau$ the time shift between $t^{*}$ and the center of the time interval at
the outer spacecraft (propagation time). $T_{X}$ and $T_{Y}$ are the lengths
of the time intervals, and we also introduce $\delta t_{X}$ and $\delta
t_{Y}$, the time resolution over which to resample the $X$ and $Y$ data sets
respectively. The evaluation of the correlation coefficient however requires
that of $X$ and $Y$ have the same number of values $n$. This adds a
constraint:

$\frac{T_{X}}{\delta t_{X}}=\frac{T_{Y}}{\delta t_{Y}}=n$

Therefore, any correlation coefficient $C_{X,Y}$ between two measurements $X$
and $Y$ should in general be computed as a function of all the free
parameters:

$C_{X,Y}=C_{X,Y}(t^{*},\tau,T_{X},T_{Y},n).$

In Section 4.2, we fixed and constrained these parameters, using hypotheses
discussed below, as to get physically relevant results.

As previously specified, we have set $t\in[-0.5,1.5]\leavevmode\nobreak\ $h,
fixing $t^{*}=0.5$ h and $T_{X}=2$ h. The goal was to select a prominent
structure passing through one of the spacecraft, to then search and find it at
the other spacecraft. We selected the structure on PSP first because, for the
plasma line-up, $t_{in}$ is better defined than the associated $t_{out}$,
which is more dependent on the model selected to describe the plasma velocity
(Section 3.2).

We also supposed that the selected structure had a global uniform evolution
with a homogeneous acceleration during the propagation. As we will see later
(Figure 8, Section 4.4), this implies that the structure’s temporal duration
is similar at both spacecraft. This is why the time window $T$ (= 2 h here)
used for the cross-correlation is the same on both spacecraft $T_{X}=T_{Y}=T$.

Regarding the temporal resolution, we have $T_{X}=T_{Y}$, so $\delta
t_{X}=\delta t_{Y}=\delta t$, and we chose $\delta t=20$ s. We note that in
our case the cross-correlation results were weakly dependent on $\delta t$ as
long as $\delta t\ll T$. Finally, the above constraints leave only $\tau$ as a
free parameter.

Written in this form, the set of Equations (14, 16, 17) are valid for the
special case of $T_{X}=T_{Y}$. Those Equations can however simply be written
in their generalized form, for whatever signals $X$ and $Y$ respectively
constituted of $n$ samples $X_{i}$ and $Y_{i}$, by replacing the bracket
$\langle..\rangle$ by $\sum_{i}^{n}(...)$ and $X(t)\,\&\,Y(t+\tau)$ by
$X_{i}\,\&\,Y_{i}$.

In summary, we have shown that the cross-correlation method can be misleading
as high values for one (or several) coefficient do not necessarily imply that
the same structure has been crossed by both spacecraft. Therefore, such method
has to be employed carefully, ideally on several (relevant) physical
quantities, and using different coefficients. Finally, although it can be
helpful, the use of such method alone is not enough for a rigorous
identification of the same plasma at two spacecraft. The identification can
only be confirmed through a physical analysis of the measurements.

Figure 7: Proton density structure (a) and magnetic field magnitude (b),
corrected by a factor $(R/R_{0})^{2}$ and $(R/R_{0})^{1.6}$ respectively, to
take into account the expansion from PSP (blue) to SolO (orange). Quantities
are plotted as functions of $t$ for PSP and as functions of $t-\tau$ for SolO
over a 2 h time interval (with $\tau=137.6$ h as precised by several cross
correlation methods, see Section 4.2). An average over 20 s is applied to the
data to better highlight the global behaviour of the density structures.

###  4.4 Local comparison of the structures

We show in Figure 7 the proton density and the magnetic field magnitude
measurements of PSP and SolO taking into account the propagation time
$\tau=137.6$ h, adjusted using a cross-correlation methods, as described in
Section 4.2. The measurements on PSP are plotted as functions of $t$ over a 2
h time interval, (between the two red vertical dashed lines in Figure 5). SolO
measurements have been plotted as functions of $t-\tau$ as to get comparable
signals, also over a 2 hour time interval. We correct $N_{p}$ by a factor
$(R/R_{0})^{2}$ and $B$ by a factor $(R/R_{0})^{1.6}$, with $R$ the spacecraft
distance to the Sun and $R_{0}=1$ au (as done in the definition of
$\chi_{X,Y}$, see Section 4.2).

Figure 8:  Schematic illustration of a 1D structure propagation. The structure
is delimited by its rear and front boundaries (with dashed and dotted lines,
respectively). These boundaries are supposed to have the same acceleration
profile (so the same forces are acting on them). We represent here a general
case with arbitrary acceleration and deceleration. Their normalized position
(r) is shown as a function of normalized time (t). The red horizontal arrows
represent the time interval of the structure at a given radial position (as
typical measurements provided by spacecraft). Green vertical arrows represent
the spatial length of the structure at a given time.

We note that there is not only a global correspondence, as found in Figure 6,
but a finer scale correspondence is also present within the structures
themselves (Figure 7). This is especially noticeable for the plasma density.
Of particular interest are four sub-structures, of duration between 0.1 and
0.3 h, detected at both spacecraft and numbered in Figure 7(a). We also note
that the density enhancement ends in both observations (at $t\,\&\,t-\tau\sim
1.1$ h) with a sharp decrease of similar short duration and magnitude (taking
into account the spherical expansion).

With radial velocities between 200 and 300 km/s, the four density sub-
structure durations translate to spatial sizes $L$ between $0.07$ and
$0.5\times 10^{6}$ km, so significantly below the estimated minimum distance
$d_{MIN}\sim 7\times 10^{6}$ km ($\sim 2\times 10^{6}$ km when including non-
radial propagation, see Appendix A). Since $L$ and $d_{MIN}$ are sizes
estimations in two orthogonal directions (R and N), we conclude that, in order
to be observed at both spacecraft, the finer observed structures could be
elongated by at least a factor 4 in the N direction (about orthogonal to the
ecliptic) as compared to the radial direction. However, the difficulty is to
estimate precisely enough the total amount of solar wind deflection in
latitude with only velocity data measured at two locations of the trajectory
(see Appendix A). Thus, the estimation of the extension ratio in the $R$ and
$N$ directions of those finer density structures remains uncertain.

The physical parameters in Figure 7 are shown as functions of time. This way
of comparing measurements may appear arbitrary, but, in the case of a
structure varying along the radial direction, it appears as the most relevant
one, as follows. We show on Figure 8 a schematic of the radial distance $r$ as
a function of time $t$ between two points representing the front (dashed line)
and rear (dotted line) boundaries of the structure. These two points have the
same profile $r(t)$ up to a shift in time as represented in Figure 8. The
spatial extent of the structure at a given time (green arrows) is increasing
(or decreasing) with time because of the plasma acceleration (or
deceleration). However, for each given position, the time extent of the
structure (red arrows) remains constant regardless of the acceleration
profile. Therefore, if the acceleration profile is the same along the
structure, and neglecting the spacecraft radial speeds as compared to the
solar wind’s, the time difference between two parts of the structure should
remain the same at the inner and outer spacecraft. Considering the observed
structure to vary along the radial direction, this justifies the comparison in
Figure 7 of PSP and SolO data within the same time interval (only shifted by
the transit time).

The studied plasma structure, shown in Figure 7, has indeed a similar duration
of $\sim 1.5$ hours on the two spacecraft. Taking into account the mean
relative radial plasma velocity (with respect to the spacecraft) this
corresponds to a radial spatial scale of $\sim 1.5\times 10^{6}$ km and $\sim
2.4\times 10^{6}$ km at PSP and SolO respectively, so about a factor $1.6$ of
radial expansion between the two spacecraft. This radial expansion is however
much smaller than the expected azimuthal and latitudinal expansions ($\sim
R_{out}/R_{in}=12$). Moreover, a radial extension due to the solar wind
acceleration would in general induce an additional decrease of $N_{p}$, as for
a spherical expansion, the continuity equation implies

$\partial_{R}(N_{p}V_{p}R^{2})=0\Rightarrow N_{p}V_{p}\propto R^{-2}$

with $\partial_{R}$ the partial derivative in the radial direction. This in
general would need to be taken into account when renormalizing the
measurements in order to compare them. We however found that in our case, this
decrease was, coincidentally, compensated by a compression due to the
formation of a stream interaction region (SIR), as we will present in the next
article.

##  5 Conclusions and Perspectives

We presented in this paper how we identified what we believe to be the same
plasma (plasma line-up) passing through PSP ($\sim 0.075$ au) and SolO ($\sim
0.9$ au) after their radial alignment the 29/04/2021. We began by modeling the
plasma propagation considering a purely radial velocity, first with a constant
speed using only PSP measurements (Section 3.1), and then with a constant
acceleration (Section 3.2) constrained by the measurements of both spacecraft.
This led to a first estimation of the time intervals corresponding to the
plasma line-up. A visual inspection paired with the use of a cross-correlation
method finally allowed us to identify (Section 4) the same density structure
passing through both PSP and SolO.

Our main finding here is how well conserved the identified density structure
is despite its $\sim 137$ h journey from PSP ($0.075$ au) to SolO ($0.9$ au).
We even are able to associate substructures with temporal scales of about
$10-20$ min. There is also a (somewhat weaker) correspondence between the
magnetic field magnitude at both spacecraft. This may be simply a consequence
of a total pressure equilibrium with the plasma pressure mainly modulated by
its plasma density.

In general, identifying the same structure might not always be possible, even
if the spacecraft are passing through the ”same plasma”. Indeed, in order to
recognize the same plasma structure, several hypotheses have to be fulfilled,
as follows.

  * •

The structure has to exist before reaching the inner spacecraft.

  * •

Not only the structure should not be destroyed during its propagation, but it
should also keep its identity enough as to be unambiguously recognizable
between the two spacecraft. For example, Borovsky (2021) presents some solar
wind structures that are believed to not be destroyed under the effects of
turbulence.

  * •

The structure has to be large enough to pass through both spacecraft.
Typically, mesoscale structures as described in Viall et al. (2021) (with
radial sizes ranging from $\sim 5\times 10^{3}$ km to $\sim 10^{7}$ km)
usually have the right lengths for this, depending on the latitude difference
between the two spacecraft.

Density enhancements generally fulfill these three conditions as they are
believed to be created by reconnection near the tip of helmet streamers. At
least part of them are conserved during their propagation within the
heliosphere, and they can be large enough to pass through the two spacecraft
as described in Section 4.

Furthermore, the correspondence of the plasma structures at both spacecraft is
best when the density measurements are compared as functions of time. This
implies that the structure mostly has radial gradients and was accelerated
with a somewhat homogeneous velocity profile along it during the propagation
(Figure 8). The discrepancies between the two recorded signals could moreover
not only be due to the structure’s evolution, but also to non-radial
gradients. We remind that the two spacecraft orbit with different azimuthal
velocities and are at different latitudes around the studied intervals.
Moreover, in the present case, the solar wind also developed non-radial
velocity components during its propagation (Figure 9).

The association of plasmas passing through two spacecraft solely using their
in situ measurements can be difficult since the provided data represent 1D
temporal cuts through temporally evolving 3-dimensional plasmas. All the
interpretations are therefore very dependent of the underlying assumptions
made in order to compare these cuts. Here, 3D MHD numerical simulations
constrained with all possible data (in situ and remote sensing) would be of
great help to further confirm the association and to help understanding the
physics involved. It could also give a more physically relevant estimation of
the solar wind propagation than a ballistic model.

The constant acceleration obtained in Section 3.2 is in agreement with the
average velocity profiles derived in Maksimovic et al. (2020) using Helios 1 &
2 measurements as close as 0.3 au. The more recent study of Dakeyo et al.
(2022) using PSP and Helios 1 & 2 data however shows that, in average, the
slow solar wind tends to have a steeper acceleration closer to the Sun (within
$[0.1,0.3]$ au). The modeling considering a constant acceleration might
therefore no hold in a general case, and the fact that it is well relevant for
our observations is probably due to the peculiarity of the identified
intervals.

Lastly, the difference in latitude between the two spacecraft ($\sim 3$)
imposes a minimum distance $d_{MIN}\simeq l_{\Delta\theta}\simeq 7\times
10^{6}$ km (Section 3) limiting how close the plasma measured by the inner
spacecraft can get to the outer one. A more realistic estimation considering
the latitudinal plasma deflection (Appendix A) gives $d_{MIN}\simeq 2\times
10^{6}$ km. In fact, this deflection might have played a major role by
bringing the plasma closer to the outer spacecraft. It is very likely that the
structure would have missed SolO otherwise, making the plasma line-up
identification not possible. The density enhancement, as well as its
substructures, should indeed be more extended and coherent in the north-south
direction than this minimum distance $d_{MIN}$.

Finally, we only considered here the proton density $N_{p}$ and magnetic
field’s magnitude $B$. However, other physical parameters also show
interesting behaviours, requiring a deeper analysis which will be presented in
the following study. We will see that the identified intervals correspond to
crossings of the heliospheric current and plasma sheets (HCS and HPS). A SIR
also developed during the plasma propagation, engulfing the HCS $\&$ HPS and
sweeping up the density structure.

## References

  * Abbo et al. (2016) Abbo, L., Ofman, L., Antiochos, S. K., Hansteen, V. H., et al. 2016, Space Science Reviews, 201, 55
  * Alberti et al. (2022) Alberti, T., Milillo, A., Heyner, D., Hadid, L. Z., et al. 2022, The Astrophysical Journal, 926, 174
  * Bale et al. (2016) Bale, S., Goetz, K., Harvey, P., et al. 2016, Space Science Reviews, 204
  * Borovsky (2021) Borovsky, J. E. 2021, Frontiers in Astronomy and Space Sciences, 8, 131
  * Chew et al. (1956) Chew, G. F., Goldberger, M. L., & Low, F. E. 1956, Proceedings of the Royal Society of London Series A, 236, 112
  * Cover & Thomas (2005) Cover, T. M. & Thomas, J. A. 2005, Entropy, Relative Entropy, and Mutual Information (John Wiley & Sons, Ltd), 13–55
  * Dakeyo et al. (2022) Dakeyo, J.-B., Maksimovic, M., Démoulin, P., Halekas, J., & Stevens, M. L. 2022, The Astrophysical Journal, 940, 130
  * Fox et al. (2016) Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Science Reviews, 204, 7
  * Hadid et al. (2021) Hadid, L. Z., Génot, V., Aizawa, S., et al. 2021, Frontiers in Astronomy and Space Sciences, 8
  * Horbury et al. (2020) Horbury, T. S., O’Brien, H., Carrasco Blazquez, I., et al. 2020, A&A, 642, A9
  * Kasper et al. (2016) Kasper, J., Abiad, R., Austin, G., et al. 2016, Space Science Reviews, 204, 131
  * Livi et al. (2021) Livi, R., Larson, D. E., Kasper, J. C., et al. 2021, ESS Open Archive, 105, essoar.10508651
  * Maksimovic et al. (2020) Maksimovic, M., Bale, S. D., Berčič, L., et al. 2020, ApJS, 246, 62
  * Marsch (2012) Marsch, E. 2012, Space Sci. Rev., 172, 23
  * McComas et al. (1998) McComas, D. J., Bame, S. J., Barraclough, B. L., et al. 1998, Geophysical Research Letters, 25, 1
  * Müller, D. et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642
  * Musmann et al. (1977) Musmann, G., Neubauer, F. M., & Lammers, E. 1977, Journal of Geophysics Zeitschrift Geophysik, 42, 591
  * Owen et al. (2020) Owen, C. J., Bruno, R., Livi, S., et al. 2020, A&A, 642, A16
  * Parker (1958) Parker, E. N. 1958, APJ, 128, 664
  * Rouillard et al. (2010a) Rouillard, A., Davies, J., Lavraud, B., et al. 2010a, Journal of Geophysical Research, 115
  * Rouillard et al. (2010b) Rouillard, A., Lavraud, B., Davies, J., et al. 2010b, Journal of Geophysical Research, 115
  * Rouillard et al. (2021) Rouillard, A. P., Viall, N., Pierrard, V., et al. 2021, in Solar Physics and Solar Wind, ed. N. E. Raouafi & A. Vourlidas, Vol. 1, 1
  * Sanchez-Diaz et al. (2016) Sanchez-Diaz, E., Rouillard, A. P., Lavraud, B., et al. 2016, Journal of Geophysical Research: Space Physics, 121, 2830
  * Schwartz & Marsch (1983) Schwartz, S. J. & Marsch, E. 1983, Journal of Geophysical Research: Space Physics, 88, 9919
  * Schwenn & Marsch (1990) Schwenn, R. & Marsch, E. 1990, Physics of the Inner Heliosphere I. Large-Scale Phenomena. (Springer-Verlag Berlin, Heidelberg)
  * Schwenn et al. (1981a) Schwenn, R., Mohlhauser, K. H., Marsch, E., & Rosenbauer, H. 1981a, in Solar Wind 4, 126
  * Schwenn et al. (1981b) Schwenn, R., Mohlhauser, K. H., & Rosenbauer, H. 1981b, in Solar Wind 4, 118
  * Shannon (1948a) Shannon, C. E. 1948a, The Bell System Technical Journal, 27, 379
  * Shannon (1948b) Shannon, C. E. 1948b, The Bell System Technical Journal, 27, 623
  * Sheeley et al. (1997) Sheeley, N. R., Wang, Y.-M., Hawley, S. H., et al. 1997, The Astrophysical Journal, 484, 472
  * Stansby & Horbury (2018) Stansby, D. & Horbury, T. S. 2018, A&A, 613, A62
  * Telloni et al. (2021) Telloni, D., Sorriso-Valvo, L., Woodham, L. D., et al. 2021, The Astrophysical Journal Letters, 912, L21
  * Velli et al. (2020) Velli, M., Harra, L., Vourlidas, A., et al. 2020, Astronomy & Astrophysics, 642
  * Viall et al. (2021) Viall, N. M., DeForest, C. E., & Kepko, L. 2021, Frontiers in Astronomy and Space Sciences, 8
  * Viall et al. (2008) Viall, N. M., Kepko, L., & Spence, H. E. 2008, Journal of Geophysical Research: Space Physics, 113
  * Viall & Vourlidas (2015) Viall, N. M. & Vourlidas, A. 2015, The Astrophysical Journal, 807, 176
  * Zaslavsky et al. (2024) Zaslavsky, A., Kasper, J. C., Kontar, E. P., et al. 2024, arXiv e-prints, arXiv:2401.04222
  * Zirker (1977) Zirker, J. B. 1977, Reviews of Geophysics, 15, 257

##  Appendix A Consistency with a Non-radial Propagation

In Section 3, the plasma line-up is calculated considering the solar wind
propagation to be purely radial. This is not necessarily always justified as
some phenomena can cause the solar wind to travel with non radial components.
We indeed show on Figure 9(a,b) that the protons bulk velocity has non-zero
non-radial components around the identified plasma structures (especially for
SolO). We will see in the next study that this is due to the formation of a
SIR. We therefore investigate below the effect of these non-radial components
on the modeled propagation (Section 3.2). The goal here is to see if, when
considering these effects, the results are still consistent with the structure
association done in Section 4.2.

As in Section 3.2, we suppose a constant acceleration during the plasma
travel. We consider here the RTN referential of the inner spacecraft (PSP) at
$t_{in}$. Since the equations used to model the propagation are written in
vector form, we can extend the method by also scanning a relevant range of
acceleration in the T and N directions. The maximum acceleration values can be
derived from Equation (10) following the same analysis leading to Equation
(11). This would require scanning a 3D space of parameters
($a_{R},a_{T},a_{N}$) to find the minimum of $d_{min}$. While numerically well
achievable, we provide below an analytical approach to derive approximate
estimates.

As a first approximation, and in order to highlight the different effects of
the non-radial components, we consider independently the $T$ and $N$ velocity
components. We therefore compare 4 different cases

$\displaystyle\mathbf{V}(t)=$ $\displaystyle(V_{R}(t),0,0)$
$\displaystyle\mathbf{V}(t)=$ $\displaystyle(V_{R}(t),V_{T}(t),0)$
$\displaystyle\mathbf{V}(t)=$ $\displaystyle(V_{R}(t),0,V_{N}(t))$
$\displaystyle\mathbf{V}(t)=$ $\displaystyle(V_{R}(t),V_{T}(t),V_{N}(t))$

with

$V_{j}(t)=V_{in,j}+(t-t_{in})\,a_{j}$

for $j=R,T,N$.

Figure 9:  Panels (a) and (b) show the $Vp,_{T}$ and $Vp,_{N}$ proton bulk
speeds (red and purple curves respectively) measured at PSP and SolO for the
same time intervals as in Figure 7. Panels (c) and (d) are the outcomes of the
propagation method, as in Figure 4, including some non-radial components. Grey
curves correspond to purely radial propagation, red and purple curves
correspond respectively to propagation with non-zero T and N velocities and
acceleration, and black ones to propagation with both T and N components. For
each propagation vector, the minimum distance $d_{MIN}$ is marked by vertical
dashed lines of matching color. The radial velocities and acceleration are the
same as the ones in Figure 4.

As a second approximation, the radial acceleration ($a_{R}$) and starting
velocity ($V_{R,in}$) are kept as the ones obtained by minimizing $|\Delta V|$
in Section 3.2, see Figure 4. We also consider
$\frac{||\mathbf{V}_{out}+\mathbf{V}_{in}||}{2||\Delta\mathbf{R}||}\simeq\frac{V_{out,R}+V_{in,R}}{2||\Delta\mathbf{R}||}$,
since the radial velocity is about one order of magnitude larger than the
transverse velocity components. This is equivalent to assuming that the
propagation time of the plasma from the inner spacecraft to the nearest
position of the outer spacecraft is mostly due to the radial component. The
non-radial velocity components are large in the studied case (presence of a
SIR, see next study) as compared to the usual solar wind. Still, as shown
below, the inclusion of these non-radial components does not affect
significantly $t_{in}$, $\tau$ and $t_{out}$ so that the plasma line-up is
achieved with similar sets of data for both spacecraft.

Using Equation (10), the non radial acceleration components are written as

$a_{j}=\frac{V_{out,R}+V_{in,R}}{2||\Delta\mathbf{R}||}\left(V_{out,j}-V_{in,j}\right)$
(18)

with $j=T,N$. Since the association between the measurements at PSP and SolO
has already been done, we consider the non-radial speeds as
$V_{(in,out),j}=\langle V_{p,(PSP,SolO),j}\rangle$ averaged over the time
interval corresponding to the density structures. $\langle
V_{p,(PSP,SolO),j}\rangle$ are shown with horizontal dashed lines in Figure
9(a,b). The primary goal here is to evaluate if a non-radial propagation
challenges the correspondence between the two density structures. This also
makes the non-radial speeds and acceleration less dependent on $t_{in}$ and
$t_{out}$, allowing an easier interpretation of their effects on the
propagation.

We show $d_{min}$ and the corresponding $\tau$ as functions of $t_{in}$ for
the different considered velocity vectors in Figure 9(c,d). Grey curves
correspond to a purely radial propagation, the same as shown in Figure 4. Red
and purple curves correspond to radial propagations, to which we respectively
added non-zero $T$ and $N$ components. Black curves correspond to a
propagation with the three $R$, $T$ and $N$ components. Figure 9(c) shows that
in this case, $\tau$ varies by at most $\sim 1$ h when considering a non-
radial propagation, justifying a posteriori keeping the $a_{R}$ found in
Section 3.2 and the use of Equation (18).

Modifying $V_{T}$ affects mostly the $t_{in}$ associated with $d_{MIN}$. Since
the plasma has a finite propagation speed, a small longitude difference is
needed between the two spacecraft in order for the outer one to intercept the
closest possible plasma parcel. In this case, the plasma also propagates along
the azimuthal direction, so, $\mathbf{V}(t)$ has now a small longitudinal
angle and depending on this angle’s sign, the longitude difference needed
between the two spacecraft decreases (positive angle) or increases (negative
angle). This shifts the relationship between $d_{min}$ and $t_{in}$ and almost
does not change $\tau$ or $d_{MIN}$ since the change in the travel distance
and latitude difference, respectively, are small. The shift is quite small due
to PSP’s proximity to the Sun, making its longitude vary a lot as compared to
SolO’s one (see Figure 1), covering rapidly the change of longitude difference
due to the inclusion of a finite $V_{T}$.

Modifying $V_{N}$ changes $d_{min}$, so $d_{MIN}$, see Figure 9(c). As shown
in Figure 1, PSP and SolO have a small latitudinal difference $\Delta\theta$.
As we already discussed in Section 3, this difference implies a distance
$d_{MIN}\sim l_{\Delta\theta}$ when considering a purely radial propagation.
Adding a small positive $V_{N}$ reduces this distance because the plasma
covers a part of the latitude difference during its propagation, lowering
$d_{MIN}$ to $\sim 2\times 10^{6}$ km. However, if the added $V_{N}$ is too
big (such that at $t_{out}$ the plasma has a higher latitude than SolO), then,
increasing $V_{N}$ would make $d_{min}$ and $d_{MIN}$ increase again (this is
not the case with present data).

In summary, each velocity and acceleration component modify mostly one of the
outcome’s parameter: $\tau$ (and $t_{out}$) for $V_{R}$ & $a_{R}$,
$t_{in}(d_{min}=d_{MIN})$ for $V_{T}$ & $a_{T}$ and $d_{MIN}$ for $V_{N}$ &
$a_{N}$. The estimations for non-radial propagations are consistent with the
structure’s identification (Section 4) and do not change much the predicted
time intervals for the plasma line-up. In fact the inclusion a positive
$V_{N}$ lowers $d_{MIN}$, reducing the estimated structures latitudinal
extension. The inclusion of a positive $V_{T}$ reduces the $t_{in}$ associated
to $d_{MIN}$, bringing it closer to the observations, also reducing the needed
structures longitudinal extension. The inclusion of both $T$ and $N$ component
implies a $d_{min}$ curve (black in Figure 9(c)) with, a close $d_{MIN}$ as
when including $V_{N}\,\&\,a_{N}$ (purple Figure 9 (c)), and at a nearby
$t_{in}$ as when including $V_{T}\,\&\,a_{T}$ (red Figure 9 (c)). It is
however difficult to check the exact extent of the deflection as the real
propagation speed profile is certainly more complex than what we considered.
Moreover $V_{T}$ and $V_{N}$ are also varying along the structure, probably
causing some distortion which would need to be taken into account.