# Properties of an interplanetary shock observed at 0.07 and 0.7 Astronomical
Units by Parker Solar Probe and Solar Orbiter.

Domenico Trotta The Blackett Laboratory, Department of Physics, Imperial
College London, London, SW7 2AZ, UK  Andrea Larosa Department of Physics and
Astronomy, Queen Mary University of London, London, E1 4NS, UK  Georgios
Nicolaou Department of Space and Climate Physics, Mullard Space Science
Laboratory, University College London, Dorking, Surrey, RH5 6NT, UK  Timothy
S. Horbury The Blackett Laboratory, Department of Physics, Imperial College
London, London, SW7 2AZ, UK  Lorenzo Matteini The Blackett Laboratory,
Department of Physics, Imperial College London, London, SW7 2AZ, UK  Heli
Hietala Department of Physics and Astronomy, Queen Mary University of London,
London, E1 4NS, UK  Xochitl Blanco-Cano Departamento de Ciencias Espaciales,
Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad
Universitaria, Ciudad de México, Mexico  Luca Franci The Blackett Laboratory,
Department of Physics, Imperial College London, London, SW7 2AZ, UK  C. H. K
Chen Department of Physics and Astronomy, Queen Mary University of London,
London, E1 4NS, UK  Lingling Zhao Center for Space Plasma and Aeronomic
Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL 35805,
USA  Department of Space Science, University of Alabama in Huntsville,
Huntsville, AL 35899, USA  Gary P. Zank Center for Space Plasma and Aeronomic
Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL 35805,
USA  Department of Space Science, University of Alabama in Huntsville,
Huntsville, AL 35899, USA  Christina M. S. Cohen California Institute of
Technology, Pasadena, CA 91125, USA  Stuart D. Bale Physics Department,
University of California, Berkeley, CA 94720-7300, USA  Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Ronan
Laker The Blackett Laboratory, Department of Physics, Imperial College London,
London, SW7 2AZ, UK  Nais Fargette The Blackett Laboratory, Department of
Physics, Imperial College London, London, SW7 2AZ, UK  Francesco Valentini
Dipartimento di Fisica, University of Calabria, Rende 87036, Italy  Yuri
Khotyaintsev Swedish Institute of Space Physics, Uppsala, Sweden  Rungployphan
Kieokaew Institut de Recherche en Astrophysique et Planétologie, CNRS, UPS,
CNES, 9 Ave. du Colonel Roche 31028 Toulouse, France  Nour Raouafi Johns
Hopkins Applied Physics Laboratory, Laurel, MD 20723, USA  Emma Davies
Austrian Space Weather Office, GeoSphere Austria, 8020 Graz, Austria  Rami
Vainio Department of Physics and Astronomy, University of Turku, FI-20014
Turku, Finland  Nina Dresing Department of Physics and Astronomy, University
of Turku, FI-20014 Turku, Finland  Emilia Kilpua Department of Physics,
University of Helsinki, Helsinki, Finland  Tomas Karlsson KTH Royal Institute
of Technology, Stockholm, Sweden  Christopher J. Owen Department of Space and
Climate Physics, Mullard Space Science Laboratory, University College London,
Dorking, Surrey, RH5 6NT, UK  Robert F. Wimmer-Schweingruber Institute of
Experimental and Applied Physics, Kiel University, 24118 Kiel, Germany

###### Abstract

The Parker Solar Probe (PSP) and Solar Orbiter (SolO) missions opened a new
observational window in the inner heliosphere, which is finally accessible to
direct measurements. On September 05, 2022, a coronal mass ejection
(CME)-driven interplanetary (IP) shock has been observed as close as 0.07 au
by PSP. The CME then reached SolO, which was well radially-aligned at 0.7 au,
thus providing us with the opportunity to study the shock properties at so
different heliocentric distances. We characterize the shock, investigate its
typical parameters and compare its small-scale features at both locations.
Using the PSP observations, we investigate how magnetic switchbacks and ion
cyclotron waves are processed upon shock crossing. We find that switchbacks
preserve their V–B correlation while compressed upon the shock passage, and
that the signature of ion cyclotron waves disappears downstream of the shock.
By contrast, the SolO observations reveal a very structured shock transition,
with a population of shock-accelerated protons of up to about 2 MeV, showing
irregularities in the shock downstream, which we correlate with solar wind
structures propagating across the shock. At SolO, we also report the presence
of low-energy ($\sim$ 100 eV) electrons scattering due to upstream shocklets.
This study elucidates how the local features of IP shocks and their
environments can be very different as they propagate through the heliosphere.

shock waves – waves – turbulence – Sun: heliosphere – Sun: solar wind

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Collisionless shock waves are present in a large number of astrophysical
systems, and are pivotal for efficient energy conversion and particle
acceleration in our universe (e.g., Richardson, 2011; Bykov et al., 2019).
Generally speaking, shocks convert directed flow energy (upstream) into heat
and magnetic energy (downstream), and, in the collisionless case, a fraction
of the available energy is channeled into the production of energetic
particles.

Shocks in the heliosphere are unique, being directly accessible by spacecraft
exploration, thus providing the missing link to the remote observations of
astrophysical systems (Richardson, 2011). Most of our knowledge about the _in-
situ_ properties of collisionless shocks comes from the Earth’s bow shock
(Eastwood et al., 2015), due to its convenient location (Dungey, 1979;
Wilkinson, 2003). Shocks besides the Earth’s bow shock are not as well
observed and understood. Interplanetary (IP) shocks are generated as a
consequence of solar activity phenomena, such as coronal mass ejections (CMEs,
e.g., Gosling et al., 1974; Webb & Howard, 2012; Kilpua et al., 2017) and
stream interaction regions (SIR, e.g., Dessler & Fejer, 1963; Richardson,
2018; Pérez-Alanis et al., 2023). As elucidated in decades of observations, IP
shocks are generally weaker and show larger radii of curvature with respect to
the Earth’s bow shock, thus allowing the study of collisionless shocks in
highly different regimes (see Kilpua et al., 2015).

IP shocks play an important role for the overall heliosphere energetics, due
to their ability to accelerate particles to high energies and influence the
plasma environment in their surroundings (see Reames, 1999, for a review).
Interestingly, the nature of such an interaction between IP shocks and the
complex, turbulent solar wind is still largely unknown (Guo et al., 2021).
Several studies addressed the interaction of IP shocks with various kinds of
pre-existing structures (e.g., Nakanotani et al., 2021; Zank et al., 2021;
Pitna et al., 2023) as well as fully-developed turbulence (Zhao et al., 2021;
Pitňa et al., 2017; Zank et al., 2021). From this point of view, important
insights have also been provided by means of numerical modelling, looking at
large-scale, MagnetoHydroDynamics (MHD) behaviour (Giacalone, 2005; Wijsen et
al., 2023) as well as the small-scale, kinetic behaviour of the shock system
(Trotta et al., 2021; Nakanotani et al., 2022). A complex picture is emerging,
where shocks strongly influence the plasma environment in which they
propagate, while they are strongly influenced by the self-induced and pre-
existing irregularities they encounter (Kajdič et al., 2021; Turc et al.,
2023). Such a picture, corroborated by theoretical studies of the shock-
turbulence interaction (Zank et al., 2015, 2021), and has been tested in the
framework of Earth’s bow shock (Sundberg et al., 2016; Schwartz et al., 2022).

IP shocks have also been shown to evolve and change their behaviour at
different heliocentric distances, as elucidated by early _Helios_ observations
in the inner heliosphere (Lai et al., 2012), and _Ulysses_ between 1 and 5 AU
(e.g., Burton et al., 1992; Zhao et al., 2018), making the study of IP shocks
in different heliospheric environments particularly interesting. From this
point of view, the Parker Solar Probe (PSP, Fox et al., 2016) and Solar
Orbiter (SolO, Müller et al., 2020) missions are opening a novel observational
window for IP shocks at previously unexplored, small heliocentric distances.
Therefore, the early stages of shock evolution can be probed, and it is
possible to investigate how collisionless shocks influence the plasma
environment close to the Sun. Examples of studies where such new observational
capabilities have been exploited are the new multi-spaceraft observations of
Solar Energetic Particle events (Dresing et al., 2023), the CMEs exhibiting
ion dropouts as observed by PSP (Giacalone et al., 2021), and the recent
crossing of a CME leg by PSP at a distance of 14 Solar Radii (McComas et al.,
2023). A thorough investigation of the event reported in this paper,
addressing the interaction between the CME and the heliospheric current sheet
with a combination of remote and in-situ PSP observations is reported in Romeo
et al. (2023). Long et al. (2023) investigated the presence and eruption of a
magnetic flux rope for this event, also combining remote-sensing and direct
observations. SolO _in-situ_ measurements were also recently used together
with _Wind_ (at 1 au) to understand how shocks interact with and influence
their environments at different heliocentric distances (e.g., Zhao et al.,
2021). Recently, joint PSP-SolO observations elucidated the evolution of a
slow plasma parcel from the solar corona to the inner heliosphere (Adhikari et
al., 2022), SolO also provided important insights into the production and
dynamics of energetic particles at IP shocks (Yang et al., 2023), exploiting
the unprecedented quality of energetic particle measurements of the Energetic
Particle Detector (EPD, Rodríguez-Pacheco et al., 2020) suite.

In this work, we study a CME-driven IP shock seen by PSP at very small
heliocentric distances (0.07 AU). We characterize the shock parameters and
environment, and study how the shock interacts with upstream switchbacks
(SBs), i.e., Alfvénic fluctuations typical of the inner heliosphere. SB
observations have been pivotal for the PSP mission, and are emerging as a
fundamental building block of the solar wind at low heliocentric distances
(e.g. Bale et al., 2019; Kasper et al., 2019; Dudok de Wit et al., 2020;
Raouafi et al., 2023; Bale et al., 2023), making the study of their
interaction with collisionless shock waves novel and of particular interest.
The CME was later observed by SolO, at a heliocentric distance of 0.7 au. SolO
was well radially-aligned with PSP, thus providing us with the opportunity to
study the associated shock at larger heliocentric distances and therefore
later evolution time (see Figure 1). The shock at SolO is characterized by a
much more structured transition, and is stronger with respect to the PSP
crossing.

The paper is organized as follows: in Section 2 we describe the datasets used
and the basic parameter estimation techniques; in Section 3 we present and
discuss the results, and the conclusions are reported in Section 4.

##  2 Data

This study focuses on the analysis of _in-situ_ magnetic field and plasma data
for PSP and SolO. At PSP, we use magnetic field data from the FIELDS
instrument (Bale et al., 2016), while proton density, bulk flow velocity and
temperature are obtained using the Solar Probe ANalyzer-Ions (SPAN-I, Livi et
al., 2022), which is part of the Solar Wind Electrons Alphas and Protons
(SWEAP) investigation suite (Kasper et al., 2016). The electron density was
also estimated using the quasi-thermal noise technique (Moncuquet et al.,
2020).

This work uses the entire _in-situ_ SolO instrument suite. The magnetic field
is measured by the flux-gate magnetometer (MAG, Horbury et al., 2020). Ion
bulk flow speed, plasma density and temperature and electron distribution
functions are obtained from the Solar Wind Analyser suite (SWA, Owen et al.,
2020). The plasma density was also estimated using the SolO Radio and Plasma
Waves instrument (RPW, Maksimovic, M. et al., 2020). The SolO Energetic
Particle Detector (EPD, Rodríguez-Pacheco et al., 2020) is used to investigate
the properties of energetic particles at SolO. Throughout the shock event
studied here, we find a discrepancy between the density estimated from the
moments of the distribution function measured by Proton and Alpha Sensor (PAS)
of the SolO SWA suite and the density estimated from the spacecraft potential
using RPW (Khotyaintsev et al., 2021) (see Figure 2). After analysing the
density computed as a moment of the electron distribution function (see
Nicolaou et al., 2021), we decided to use the density estimated from RPW for
our analysis.

##  3 Results

Figure 1: Spacecraft configuration at 17:30 UT of Sep. 05, 2022 (immediately
after the IP shock crossed PSP). In this plot (generated using the Solar
MAgnetic Connection HAUS tool (Solar-MACH, Gieseler et al., 2023)), the Sun is
at the center, the Earth is represented by the green circle, and the dashed
(solid) lines indicate the spacecraft radial (along a Parker Spiral with a
nominal speed of 400 km/s) connection.

###  3.1 Event Overview

On September 5, 2022, a CME erupted from the Sun and into interplanetary
space. The CME was not Earth directed as the eruption happened on the far side
of the Sun. An overview of the orbital configuration is shown in Figure 1.

A fast-forward shock, driven by the CME, was detected by PSP as close as 0.07
AU ($\sim$ 15 $\mathrm{R}_{\sun}$) at 17:27:19 UT, making this observation the
closest to the Sun to date. An overview of the _in-situ_ magnetic field and
plasma quantities is shown on the left-hand side of Figure 2. A sharp
transition in magnetic field, with a jump from about 600 to about 1600 nT
occurs at the shock. The velocity profile across the shock transition shows
two interesting features, namely a large flow speed deflection in the
transverse direction $\mathrm{V_{T}}$, and a steady rise in $\mathrm{V_{R}}$
further downstream, up to about 17:35. We have marked this time as the end of
the downstream (sheath) region of the event, i.e., the point where the leading
edge of the CME flux rope likely starts (see also Romeo et al., 2023).
However, we note that the boundary between the sheath and CME flux rope is not
clear for this event. The properties of the CME sheath at such low
heliocentric distances (and therefore early evolutionary stages) are very
interesting and will be object of future work, comparing them with
substructures and properties of CMEs at larger heliocentric distances (Kilpua
et al., 2017).

Figure 2: Magnetic field magnitude, its components in the RTN system, ion bulk flow velocity, density and temperature as measured by PSP (left) and later on by SolO (right). The plots show an overview of the event (a,c) with a zoom around the shock transition at both spacecraft (b,d). The dashed magenta line marks the shock transition, and the blue shaded area in panel (b) indicates the approximate end of the CME sheath as observed by PSP. Table 1: In-situ shock parameters at PSP (left) and SolO (right). In the table are displayed, from top to bottom, the shock arrival time, Carrington longitude and latitude at the time of shock arrival, spacecraft heliocentric distance at the time of shock arrival, shock normal vector in $\mathrm{RTN}$ coordinates, shock normal angle $\langle\theta_{Bn}\rangle$, magnetic compression ratio $\langle\rm{r}_{B}\rangle$, gas compression ratio $\langle\rm{r}\rangle$, shock speed $v_{\mathrm{sh}}$, upstream plasma beta $\beta_{\mathrm{up}}$, fast magnetosonic and Alfvénic Mach numbers ($\rm{M_{\rm{fms}}}$ and $\rm{M_{\rm{A}}}$, respectively). The shock speed $v_{\mathrm{sh}}$ is along the shock normal and computed in the spacecraft rest frame using the mass flux conservation law. | Parker Solar Probe | Solar Orbiter
---|---|---
Shock Time [UT] | 05 Sep 2022, 17:27:19 | 06 Sep 2022, 10:00:51
Carrington longitude [∘] | 232 | 251.9
Carrington latitude [∘] | -1.8 | -3.6
Heliocentric dist. [AU] | 0.07 | 0.7
$\langle\hat{\mathrm{n}}_{\mathrm{RTN}}\rangle$ | [0.5 -0.8 0.2] | [0.6 -0.2 0.7]
$\langle\theta_{Bn}\rangle$ [∘] | 53 | 51
$\langle\rm{r}_{B}\rangle$ | 2.3 | 1.9
$\langle\rm{r}\rangle$ | 1.6 | 2
$\langle v_{\rm{sh}}\rangle$ [km/s] | 1520 | 942
$\beta_{\mathrm{up}}$ | 0.1 | 0.6
$\rm{M_{\rm{fms}}}$ | 3.8 | 3.2
$\rm{M_{\rm{A}}}$ | 3.9 | 3.8

We carried out a comprehensive characterization of the shock parameters
locally observed by PSP. Given the density drops visible in the upstream
region in Figure 2(b), due to the core of the ion distribution function not
being well-inside the SPAN-i field of view (Livi et al., 2022), we estimated
the density using the spacecraft quasi-thermal noise (QTN) for our shock
analysis. Using a systematic collection of averaging windows spanning a few
seconds to three minutes (with the technique described in Trotta et al.
(2022a)) upstream/downstream of the shock, we used the Mixed Mode 3 method
(MX3 Paschmann & Schwartz, 2000) to estimate the average shock normal $\langle
n\rangle=[0.5,-0.8,0.2]$, though similar values are obtained with the other MX
methods as well as magnetic coplanarity. Propagation direction deviating
clearly from the radial direction is comptatible with the picture in which PSP
is crossing a flank of the CME event. Our estimation of $\theta_{Bn}$ , i.e.,
the angle between the shock normal and the upstream magnetic field, a crucial
parameter influencing shock behaviour, reveals that we are in the presence of
an oblique ($\theta_{Bn}$ $\sim 53^{\circ}$), supercritical shock with
moderate Alfvénic and fast magnetosonic Mach numbers ($\mathrm{M_{A}\simeq
3.9\,,\,M_{fms}\simeq 3.8}$, respectively) with respect to other IP shocks
observed at or near 1 au (Kilpua et al., 2015). Furthermore, we find that the
shock propagates at the very high speed of about 1500 km/s in the spacecraft
frame and along the estimated shock normal. At PSP, upstream of the shock the
magnetic field is mostly radial and interspersed with one-sided Alfvénic
fluctuations (Gosling et al., 2009) of the magnetic field and ion bulk
velocity, known as magnetic switchbacks, with angular deflections in the
magnetic field of moderate amplitude, as expected for very low heliocentric
distances (e.g. Jagarlamudi et al., 2023). SBs are a crucial feature of the
solar wind in the inner heliosphere, as extensively shown in previous
literature (Dudok de Wit et al., 2020; Krasnoselskikh et al., 2020; Larosa et
al., 2021; Liang et al., 2021; Pecora et al., 2022; Liu et al., 2023;
Jagarlamudi et al., 2023; Raouafi et al., 2023). Here, we have the opportunity
to study how such structures are processed by shock waves, a crucial point of
this work summarized in Section 3.2.

Figure 3: Magnetic fluctuations in the shock environment at PSP (left) and
SolO (right). The Figure shows, from top to bottom: magnetic field magnitude
and its components, trace wavelet spectrogram of the magnetic field
(colormap), the normalised magnetic helicity $\sigma_{m}$ spectrogram, with
the proton cyclotron frequency in the interval (black line), magnetic field
power spectral densities (PSDs) computed in the 10 minutes upstream (green)
and downstream (orange) of the shock.

At SolO (Fig. 2 right), the shock crossing appears much more structured
compared to the PSP observation. Strong transverse flow deflections are still
present, with a strong increase in $\mathrm{V_{N}}$ at the shock transition.
The plasma density, as measured from the SWA-PAS and RPW instruments, yields
two different values. For the shock parameter estimation, we used the density
as measured by the RPW, since it is in agreement with the value obtained from
EAS electron moments (Figure 8). However, locally computed shock parameters do
not change dramatically when using the density derived from PAS. Steep
upstream enhancements of magnetic field magnitude are found ahead of the shock
(09:52, 09:56 UT), compatible with very rarely observed shocklets at strong IP
shocks (Wilson et al., 2009; Trotta et al., 2023a). A high degree of magnetic
field structuring is also found downstream of the shock, indicating a high
level of complexity for this shock crossing. At SolO, the shock parameters
estimation shows values very close to the ones observed at PSP (see Table 1).
However, such values are obtained using very local averaging windows for the
upstream/downstream quantities ($\sim$ 10 seconds to 1 minute). The overall
behaviour of the shock at SolO indicates a high level of variability. Indeed,
using larger averaging windows (about 6 to 10 minutes up/downstream), we find
parameters compatible with a quasi-parallel shock transition ($\theta_{Bn}$
$\sim 30^{\circ}$) with high Mach numbers ($\mathrm{M_{A}\sim M_{fms}\sim
7}$). It is worth mentioning that, in addition to the local variability
examined here, a different behaviour of the two shocks at PSP and SolO is
expected, due to the crossing of the event at two different points in space
and time. The longitudinal separation of PSP and SolO is of about 20∘, and
depending on the CME width, the shock will be crossed in different locations.
However, such a comparison of shock parameters, summarized in Table 1, is
useful when addressing the evolution of the whole event, and it may be
important to support remote-sensing observations of the event as well as
modelling of the CME evolution.

It is interesting, at this point, to characterize the environment in which the
shock at PSP and SolO are propagating and how it is influenced by the shock
passage. To this end, we studied the magnetic fluctuations in the $\sim$ 15
minutes before/after the shock arrival, with the relevant analyses shown in
Figure 3.

Figure 4: PSP observations of magnetic switchbacks transmission across the
shock. (a) to (c): magnetic field (blue) and ion bulk flow velocity (plum)
components. (d) and (e): pitch angle distributions for electrons with 2072 and
355 eV energy, respectively. (f) to (h): zoom around the shock crossing
(shaded area in top panels). Scatterplots of radial and tangential (i),
tangential and normal (j) and radial to normal (k) components of the magnetic
field for the 4 minutes upstream of the shock crossing (green) and 4 minutes
downstream of it (orange).

The shock crossing at PSP is characterized by a sharp transition, with no
signs of upstream wave foreshock, as elucidated by the trace wavelet
spectrogram in Figure 3(b). This is compatible with other observations of IP
shocks, particularly at such oblique geometries (e.g. Blanco-Cano et al.,
2016). It is worth noting that micro-instabilities of the shock front
happening at ion kinetic scales, such as rippling (Trotta et al., 2023b) may
still be present, albeit not well resolved by PSP. The level of magnetic
fluctuations is enhanced in the downstream sheath region, lasting about 6
minutes. Such an enhancement in magnetic fluctuations is further shown in
Figure 3(d), where the power spectral density in the 5 minutes
upstream/downstream of the shock has been computed (green and orange lines,
respectively). Here, it can be seen that the level of fluctuations increases
by a factor of 4, and the downstream spectra show a flattening around the ion
cyclotron frequency, a behaviour compatible with the spectral behaviour of IP
shocks (Pitňa et al., 2021), with some observations of turbulence in the
Earth’s bow shock environment (e.g. Sahraoui et al., 2020) and also compatible
with modelling of turbulence transmission across shocks (see Fig. 18 in Zank
et al., 2021). However, because the downstream flow speed is much larger than
the upstream, and the shock changes the plasma parameters abruptly, the level
of fluctuations may be overestimated. While the increase in the level of the
frequency spectrum has been documented in previous literature (Zhao et al.,
2021; Park et al., 2023), it remains to be understood if IP shocks inject new
turbulence or they simply modify the background plasma properties. However,
preliminary analysis shows that the spectral break happens around the ion skin
depth $d_{i}$, compatible with statistical work carried out by Park et al.
(2023). Further studies on the matter, relevant also for understanding the
behaviour of turbulence at very low heliocentric distances (Zank et al., 2020)
are further complicated by the geometrical constraints imposed by the single-
spacecraft nature of the observations. These will be the objects of future
work.

Figure 5: PSP observations of: (a)-(d): Magnetic field magnitude (black),
magnetic field components (blue) and ion bulk flow speed components (plum) as
a function of distance from the shock, where the downstream quantities have
been decompressed using the Rankine-Hugoniot relations with the average shock
parameters reported in Table 1. (e)-(g): Magnetic field excursion scatter
plots as in Figure 4, performed on the decompressed quantities. Note that the
scales in panels (e)-(g) are different from the ones used in Figure 4.

To further investigate the wave properties across the shock, we computed the
normalised magnetic helicity, defined as

$\sigma_{m}=\frac{2\Im\left(\tilde{B_{T}^{\star}}\tilde{B_{N}}\right)}{|\tilde{B_{R}}|^{2}+|\tilde{B_{T}}|^{2}+|\tilde{B_{N}}|^{2}},$
(1)

where $\mathrm{B}$ indicates the magnetic field components, the $\tilde{}$
represents the wavelet- transformed quantities and $\star$ represents the
complex conjugation operation (Matthaeus et al., 1982). Upstream of the shock
at PSP, we observe a clear signature of consistently high $\sigma_{m}$ at ion
scales, compatible with ion cyclotron wave activity (see the red patches by
the ion plasma frequency line in Figure 3(c)). Ion cyclotron waves, crucial
components of the solar wind as elucidated by PSP observations (Telloni et
al., 2019; Verniero et al., 2020), have been shown to be very important for
energy dissipation and solar wind heating (e.g. Woodham et al., 2018; Bowen et
al., 2022). In our observations, we note that the magnetic helicity signature
of ion cyclotron waves found upstream of the shock is lost in the shock
downstream, as discussed in Section 3.2, together with an explanation of why
such a behaviour is observed.

The magnetic fluctuations environment at SolO is different than that found at
PSP (right-hand side of Figure 3). Here, the shock transition appears much
more complex, especially due to the presence of larger scale structures at the
SolO shock with respect to the PSP one. In the wavelet spectrum of the
magnetic field, enhanced power extending to small ($\sim$ 1 Hz) scales is
found corresponding to the upstream shocklet activity. The downstream appears
populated with strong compressive and non compressive magnetic field
fluctuations, indicative of the fact that the shock propagated through a very
structured portion of the solar wind, as it can be noted from the magnetic
field behaviour in the 10 minutes downstream of the shock (Blanco-Cano et al.,
2019; Kropotina et al., 2021). The spectral behaviour of turbulence in being
transmitted from the shock upstream to downstream is similar to the one
observed for the shock at PSP, and compatible with previous multi- spacecraft
studies of turbulence processed by IP shocks (Zhao et al., 2021). It is
interesting to note that, in the IP case, an inertial range is always
recovered downstream of IP shocks, in contrast with some observations of
Earth’s magnetosheath where an inertial range is not observed downstream of
Earth’s bow shock (Sahraoui et al., 2020), often interpreted as the Earth’s
bow shock “resetting” the turbulent cascade. Finally, we note that some
features of ion-cyclotron wave activity are seen upstream of the SolO event,
though they are much less clear than the PSP observation.

###  3.2 The shock at Parker Solar Probe

Figure 6: The IP shock interaction with ion cyclotron waves. (a): magnetic
field magnitude around the shock crossing by PSP. (b) Angle between local
magnetic field and radial direction ($\theta_{\mathrm{BR}}$ , blue) and ion
velocity ($\theta_{\mathrm{BV}}$ , yellow). (c): Reduced magnetic helicity for
the interval (as in Figure 3).

In this section, we focus on the PSP observations of the shock interacting
with pre-existing fluctuations in the inner-heliospheric solar wind and its
features. Particular interest is focused on how switchbacks (SBs) are
processed by the IP shock wave.

In Figure 4 we observe the high degree of correlation between the magnetic and
velocity fields. The presence of moderate amplitude SBs and their one-sided
nature (Gosling et al., 2009), is evident especially in the $\mathrm{B_{R}}$
and $\mathrm{V_{R}}$ components (Figure 4(a)) upstream of the shock. In the
downstream (sheath) region, the high degree of correlation is preserved, as
elucidated by the Figure zoom (panels f, g, h). These observations reveal that
the Alfvénic nature of the SBs is preserved downstream of the shock. However,
the downstream SBs present larger, shock-driven variations in the field
magnitude. Small deviations from a constant B state have been already observed
at SBs boundaries (Krasnoselskikh et al., 2020; Farrell et al., 2020; Larosa
et al., 2021). Here we show a different compressive effect, resulting from the
propagation across the IP shock. As discussed in Section 3.1, the downstream
region is characterized by stronger fluctuations driven by the shock,
including small-scale, compressive fluctuations that are not Alfvénic.

From this point of view, it is interesting to note that the SBs signature is
not completely lost upon shock passage, but rather the SBs are processed by
the shock. Considering the Rankine-Hugoniot relations describing the change in
plasma parameters across MHD shocks (e.g. Krehl, 2015), we expect, for the
SBs, a compression along the shock normal direction, and a stretch along the
two shock transverse directions. The extremely short duration of the sheath
poses a strong limitation on a more quantitative study of how SB are affected
by the shock. For instance, the Z parameter would be helpful in addressing
such an issue (Dudok de Wit et al., 2020). Such an investigation will be
carried out in future works focusing on several shocks at low heliocentric
distances.

To further investigate how the shock affects the upstream SBs, we study the
magnetic field excursion in the regions upstream and downstream of the shock
(Figure 4(i-m)), as done in other studies elucidating the nature of
fluctuations in the solar wind (e.g., Matteini et al., 2015). Two effects
arise in the magnetic field fluctuations after the shock crossing. First of
all, due the change in background field magnitude, the arc of polarization
upon which the magnetic field fluctuates become wider, as particularly evident
in the $\mathrm{B_{R}}$-$\mathrm{B_{T}}$ scatter plot (Figure 4(i)). It is
also possible to observe that downstream magnetic field fluctuations cover a
larger area in the magnetic field excursion space, due to the presence of
compressive fluctuations in the shock downstream. From Figure 4(i-j-k), it is
possible to note that the arc of polarization is displaced from the R-T plane
into both the R-T and R-N plane when comparing the shock upstream to the
downstream, consistent with compression along the normal and the stretching
along the perpendicular directions. The shock-SBs interaction may have
important consequences for the SBs’ ability to propagate without dispersive
effects, and therefore it may affect the SBs lifetime, since the constant B
magnitude is a necessary condition for the unperturbed propagation of large
amplitude Alfvén waves (Barnes & Hollweg, 1974).

Figure 7: (a): Energetic ion fluxes in the $\sim$0.05 - 2 MeV energy range as
observed by the Sun telescope of EPD-EPT instrument of Solar Orbiter. (b):
Local $\theta_{\mathrm{BR}}$ (blue) and $\theta_{Bn}$ (red) angles. (c)
Magnetic field magnitude (black) and its components. The dashed magenta line
indicates the shock crossing time.

The above assessment of the SBs features upstream/downstream of the shock
suffers from limitations related to the single-spacecraft nature of the
observations. First, it is known that the features of SBs (deflections,
duration) tend to have broad distributions (e.g., Larosa et al., 2021), making
it difficult to estimate the extent to which the processed, downstream SBs
were comparable to their upstream, unshocked counterparts before they crossed
the shock. Furthermore, how plasma is sampled by PSP throughout the
observation introduces a geometrical constraint on the observations, where
fluctuations are sampled differently from upstream to downstream. This latter
point is further expanded in Figure 6(b) and related discussion. Another
crucial caveat stems from the downstream flow speed in the spacecraft frame
being much larger than that upstream, allowing more plasma to flow over the
spacecraft per unit time, causing the downstream SBs to appear shorter than
their upstream counterparts, an effect that appears in addition to the
compression induced by the shock. Furthermore, the field increase due to the
presence of the shock influences the discussion of the downstream SBs
deflection amplitude (Figure 4(i)).

For this reason, we decided to “unshock” the magnetic field and plasma time
series across the PSP shock crossing. This analysis is carried out in two
steps. First, using the mean upstream and downstream flow speeds, we plot the
PSP measurements in units of distance from shock using the Taylor hypothesis
(Taylor, 1938). Then, we used the Rankine-Hugoniot jump conditions to remove
the shock compression. Here, knowing the downstream quantities and the shock
parameters as estimated from the spacecraft crossing (see Table 1), we derived
the upstream quantities, i.e., we estimated the magnetic field and plasma
conditions prior to the shock passage. Further details about this method of
decompression can be found in the Appendix A. The resulting, decompressed
timeseries is shown in Figure 5. As can be seen in panels (a)-(d), the jump in
the magnetic field and speed is greatly reduced with this technique. It is
possible to see that, some compression in magnetic field persists even after
the plasma has been “unshocked”. This is due to the decompression method
having several limitations. First, the shock parameters are assumed to be
fixed throughout the decompression, and are estimated using a single
spacecraft technique, inherently affected by uncertainties (Trotta et al.,
2022a). The shock variability, particularly important for supercritical
shocks, on a variety of scales (e.g. Marghitu et al., 2017; Kajdič et al.,
2019, 2021) may therefore introduce fluctuations in the decompression.
Finally, as discussed in the appendix, the decompression method makes use of
the Rankine-Hugoniot jump conditions without including waves and/or turbulence
(e.g., Zank et al., 2002; Gedalin, 2023). This technique can be further
improved, in future studies, to mitigate the above limitations.

The preservation of the SBs Alfvénicity is evident from Figure 5 (a)-(d),
where the downstream behaviour of the magnetic field and velocity fluctuations
is remarkably similar to the upstream one. The analysis of magnetic field
excursions further strengthens this point, where compatible behaviour between
downstream and upstream fluctuations is found. The magnetic field excursion
space covered by the downstream fluctuations is still larger, due to some
compressive features downstream of the shock not fully removed by the Rankine-
Hugoniot decompression. However, we observe that downstream of the shock we
find larger deflections of the magnetic field. It is also worth noting that,
after decompression, the difference in the $\mathrm{B_{R}}$-$\mathrm{B_{N}}$
plane becomes less evident.

A totally different behaviour from that found for SBs is found for the
upstream ion cyclotron waves discussed in Section 3.1. The signature of these
waves, a clearly visible feature in the reduced magnetic helicity spectrogram
(red patches of $\sigma_{m}=1$, near ion plasma frequency, Figure 3), is
completely lost downstream of the shock. We address this interesting feature
in Figure 6, where we study how PSP crosses the wave environment surrounding
the shock. To this end, we compute the angle between the local magnetic field
and the radial direction, $\theta_{\mathrm{BR}}$, and the angle between the
local magnetic field and the local velocity $\theta_{\mathrm{BV}}$, where
$\mathbf{V}=\mathbf{V}-\mathbf{V_{PSP}}$, where $\mathbf{V}_{\mathbf{PSP}}$ is
the velocity of PSP in the inertial RTN frame. In Figure 6(b) ion cyclotron
waves are clearly observed when $\theta_{\mathrm{BV}}$ is close to 180 degrees
and lost when $\theta_{\mathrm{BV}}$ is of the order of 90 degrees. This
behaviour could be simply due to an observational bias. As shown by Bowen et
al. (2020), this phenomenon is the result of measuring a quasi-parallel wave
vector at oblique angles combined with the higher amplitude in the
perpendicular direction of the anisotropic turbulent fluctuations.
Furthermore, the conditions downstream of the shock inhibit the possibility of
resonant beam-field interaction, due to the strongly turbulent environment
diffusing the beam in velocity space (e.g., Valentini et al., 2010). We
analysed the ion velocity distribution functions as measured by SPANi, albeit
their being complicated by the limited field of view of the instrument, we did
not find any beam signature downstream of the shock, suggesting that the
production mechanisms of ion cyclotron waves are indeed suppressed. A detailed
investigation of such features is out of scope for this work and will be the
object of further studies.

###  3.3 The shock at Solar Orbiter

In this Section, we discuss the event as observed by SolO. The shock crossed
the spacecraft at 10:00:51 on Sep 6, 2022. The shock parameters, reported in
Table 1, are compatible with the PSP observations. However, as discussed in
Section 3.1, shock variability plays a major role in the SolO observations.
Here, the shock transition appears far more structured, with an overall
behaviour compatible with strong shocks.

Such interesting aspects of the SolO shock observations are investigated in
Figure 7. The structuring observed in the shock surroundings is particularly
evident in the 20 minutes downstream of the shock, populated with many sharp
changes of magnetic field (Figure 7). First of all, using the EPD-EPT
instrument, we address how the production of energetic particles is affected
by such disturbed conditions. As can be seen in Figure 7(a), energetic ions of
up to $\sim$ 2 MeV are accelerated by the shock, with fluxes rising as the
shock crossing is approached, a behaviour typical of ion acceleration at IP
shocks (Giacalone, 2012; Lario et al., 2022). Interestingly, some of the
fluxes (e.g. the ones for $\sim$0.33 and $\sim$ 0.7 MeV, green and yellow
lines, respectively) have their peak at a time which is not coincident with
the shock crossing time at SolO, but instead is downstream of the shock. This
behaviour, consistent with the statistical study presented in Lario et al.
(2003), has implications on the production and propagation of shock
accelerated particles (Perri et al., 2022), and may well be due to additional
acceleration mechanisms downstream of the shock (Zank et al., 2015; Zhao et
al., 2018; Kilpua et al., 2023).

A crucial feature found for the energetic particle population is the
fluctuations found in their fluxes immediately downstream of the shock with
typical timescales of 1 to 2 minutes, particularly evident for the lower
energy channels (0.05 to 0.35 MeV, dark blue to green). Such fluctuations
correlate well with the magnetic field structuring (Figure 7(a)-(c)), thus
suggesting that particle acceleration is indeed happening in an irregular
fashion for this IP shock, where additional acceleration may be provided by
the magnetic structures (Zhao et al., 2018; Trotta et al., 2022b; Nakanotani
et al., 2021). To quantify the variability associated with the shock crossing,
the $\theta_{Bn}$ and local $\theta_{\mathrm{BR}}$ angles have been estimated
using the local magnetic field, the radial direction and the average shock
normal as reported in Table 1. It is evident from this analysis, shown in in
Figure 7(b), that the shock geometry changes significantly on the timescales
examined here, moving from quasi-perpendicular to quasi-parallel geometries
and thus supporting the picture of irregular particle acceleration.
Complementing the theme of shock variability, it is worth discussing the role
of plasma structures in particle acceleration. The typical gyroradii of the
particles showing such irregular behaviour have been estimated using the local
magnetic field, $\mathrm{r}_{L}\sim 10^{4}$ km. Using the Taylor’s hypothesis,
such lengths compare to 1/10th of the typical size of the downstream
structures, thus suggesting that trapping of accelerated particles may play a
role into the variability observed in the particle fluxes, an important source
of extra particle acceleration beyond the shock (see Zank et al., 2015; Zhao
et al., 2018; Trotta et al., 2020; Roux et al., 2019; Pezzi et al., 2022).

Figure 8: Overview of electron behaviour at SolO. Top: EAS electron pitch
angle distributions for particles with energy larger than 70 eV. Middle:
Magnetic field measurements in RTN.Bottom: electron density computed from EAS
measurements.

Another extremely interesting feature of the SolO observations is the steep
magnetic field enhancements observed in the 10 minutes upstream of the shock
(Figure 8). Such steep enhancements are compatible with upstream shocklets,
typically arising from the interplay between shock-reflected, energetic ions
and upstream waves which, in the case of Earth’s bow shock, are the Ultra Low
Frequency (ULF) waves routinely found in the foreshock (Stasiewicz et al.,
2003). Shocklets are important as they influence the plasma environment
upstream of the shock transition, thus pre-conditioning the shock inflow
(Lucek et al., 2008). Most of our knowledge of shocklets is built on
observations upstream of Earth’s bow shock (Plaschke et al., 2018), being
rarely observed at IP shocks with only three cases previously reported (Lucek
& Balogh, 1997; Wilson et al., 2009; Trotta et al., 2023a), making this
observation particularly interesting.

Two shocklets were observed upstream of the shock at SolO, compatible with the
fact that, when estimated at $\sim$ 20 minute timescales, the shock Mach
number is high and the geometry quasi-parallel (see discussion in Section
3.1). In Figure 8, we show how the shocklets influence the low-energy electron
population using SolO-EAS measurements. First of all, shocklets are associated
with rises in the electron density, identified by the magenta arrows in Figure
8(c), a behaviour recently found also at Venus (Collinson et al., 2023).
Furthermore, by analysing the pitch-angle distributions of electrons with
energy greater than 70 eV, we found that electrons are more isotropic in pitch
angle space in the vicinity of the shocklets, probably due to pitch-angle
scattering induced by the compressive structures, as observed in the Earth’s
bow shock environment (Wilson III et al., 2013). This behaviour, namely the
pre-conditioning of the upstream particle population, may have important
consequences for electron injection and acceleration to higher energies (Katou
& Amano, 2019; Dresing et al., 2022; Wijsen et al., 2023).

##  4 Conclusions

The September 05, 2022 CME event provided an opportunity to study a CME-driven
IP shock at unprecedentedly low heliocentric distance, with PSP crossing the
event at 0.07 au. Furthermore, well-radially aligned SolO at 0.7 au provided
important insights about the evolution of the event.

We focused on the small-scale behaviour of CME-driven shocks associated with
this event. The PSP spacecraft observed the shock very early in its evolution.
The shock had moderate Mach numbers ($\mathrm{M_{A}}\sim\mathrm{M_{fms}}\sim
4)$), as inferred using local shock parameter estimation. The average shock
normal was found to be $\mathrm{\hat{n}}=(0.5,-0.8,0.2)$, significantly
departing from the radial direction, indicating that the crossing of the shock
likely happened on one flank of the CME event. This has also important
implications when addressing the joint PSP-SolO observations, as due the shock
will be crossed in different loactions, with features depending on the CME
width and spacecraft longitudinal separation. The shock at PSP has a notably
short sheath region due to its early evolutionary stage, an interesting
property that may have fundamental implications for the possibility of
accelerating particles to high energies, as preliminary shown in Cohen et al.
(2023). We however emphasize that as this event was crossed at the flanks of
the sheath - CME flux rope boundary cannot be determined accurately.

We studied how switchbacks, fundamental constituents of the solar wind as
elucidated by several, previous PSP observations (e.g. Bale et al., 2019;
Kasper et al., 2019; Dudok de Wit et al., 2020), are processed in the shock
crossing. We found that the SBs are compressed along the shock normal
direction and stretched along the other two directions. Interestingly, many
SBs properties are preserved in the shock downstream. Furthermore, SBs with
larger magnetic field deflections were found in the shock downstream, an
important ingredient to consider when addressing their statistical properties.
Such statistical properties of how SBs are processed by IP shocks will be
object of future investigation, now increasingly possible thanks to the novel
PSP ShOck Detection Algorithm (SODA) IP shock list
https://parker.gsfc.nasa.gov/shocks.html. This behaviour has been investigated
in detail by looking at the decompressed magnetic field and plasma quantities,
obtained using the Rankine-Hugoniot shock jump conditions. This technique of
decompression is useful to estimate how quantities were before their
interaction with the shock. It can also be used to yield an estimate of how
the ambient plasma conditions could change due to the passage of a shock with
a given set of parameters, as will be shown in future work.

How ion cyclotron waves are transmitted across the shock was also addressed.
This study is particularly interesting due to the role that ion cyclotron
waves play for energy dissipation (e.g., Bowen et al., 2022). We observe that
the signature of pre-existing ion cyclotron waves, identified in the shock
upstream at PSP, disappears downstream of the shock (Figure 6). This may be
due to the sudden change in plasma conditions at the shock, injecting strong
fluctuations in the downstream, making the conditions for ion cyclotron waves
propagation less favourable. Through the analysis of the
$\theta_{\mathrm{BV}}$ and $\theta_{\mathrm{BR}}$ angles, we also found
unfavourable conditions to detecting ion cyclotron activity downstream (Bowen
et al., 2020), due to the change of the mean magnetic field direction upon the
shock passage.

SolO observations of the same event are extremely interesting to address the
role of evolution for the CME shock region in its propagation to larger
heliocentric distances. On the large scales, we note that the large transverse
flow deflections are still present, with a $\mathrm{V_{N}}$ increase
comparable to the one observed in $\mathrm{V_{R}}$.

The shock environment at SolO is much more disturbed than the one observed at
PSP. A shock parameter estimation using very short averaging windows ($\sim$ 1
minute, thus addressing the very local shock properties) yields similar values
with respect to the PSP observations (see Table 1). However, analysing the 30
minutes across the shock transition, we find an environment compatible with a
quasi-parallel shock and relatively high Mach number, propagating in a very
structured portion of the solar wind. Two shocklets, structures that grow
favourably upstream of high Mach number, quasi-parallel shocks, were found
upstream of the shock, a rare observation for the interplanetary case.
Signature of $\sim 100$ eV electrons scattering was found corresponding to the
shocklets, an important ingredient to be considered for pre-conditioning of
the upstream particle population at the shock. It is worth noting that these
shocklets, with duration much larger than those observed at Earth’s bow shock,
probably arise from pre-existing upstream waves and not from shock-generated
upstream waves, a behaviour observed also for the case in Trotta et al.
(2023a). Therefore, this observation points to the idea that the origin of the
IP shocklets is different than those observed in the Earth’s foreshock.

Energetic ions up to 2 MeV were found in association with the shock at SolO. A
detailed analysis of the high time resolution EPD-EPT energetic particle
fluxes reveal structuring corresponding to the magnetic field structures
processed by the shock, indicating a potential role of trapping as an extra
source of energy for particle acceleration.

Finally, we underline that this event is a very good example of the novel
observational window provided by missions exploring the inner heliosphere such
as PSP and SolO, exploited in this study to highlight the fact that the
features of IP shock environments can be very different as they propagate
through the heliosphere, with important consequences on the modelling effort
accuracy and possibility of prediction associated with such energetic events.

##  5 acknowledgments

This work has received funding from the European Unions Horizon 2020 research
and innovation programme under grant agreement No. 101004159 (SERPENTINE,
www.serpentine-h2020.eu). AL is supported by STFC Consolidated Grant
ST/T00018X/1. CHKC is supported by UKRI Future Leaders Fellowship MR/W007657/1
and STFC Consolidated Grants ST/T00018X/1 and ST/X000974/1. H.H. is supported
by the Royal Society University Research Fellowship URF/R1/180671. N.D. is
grateful for support by the Academy of Finland (SHOCKSEE, grant No. 346902).
The PSP/FIELDS experiment was developed and is operated under NASA contract
NNN06AA01C. Work at IRAP is supported by CNRS, UPS, and CNES.E.E.D
acknowledges funding by the European Union (ERC, HELIO4CAST, 101042188). Views
and opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. XBC is grateful to PAPIIT DGAPA
grant IN110921. This work was supported by the UK Science and Technology
Facilities Council (STFC) grant ST/W001071/1.L.F. .is supported by the Royal
Society University Research Fellowship No. URF\R1\231710. RFWS thanks the
German Space Agency (DLR) for its support of SOLO/EPD under grant 50OT2002.

##  Appendix A Rankine-Hugoniot decompression method

Heliospheric shocks crossing spacecraft can be directly measured. An
hypothesis that is often made when interpreting spacecraft measurements is the
Taylor hypothesis (Taylor, 1938), linking time-variations in the spacecraft
measured quantities to spatial variations. Thus, upon shock crossing, it is
possible to address the properties of the upstream/downstream shock
environments (see Figure 2). However, it is often the case that it would be
interesting to address the shock downstream plasma before it was shocked, as
discussed in Section 3.2. Here we present a technique providing a proxy to
address the plasma condition before the shock propagation.

Figure 9: Overview of the procedure of magnetic field decompression.

The Rankine-Hugoniot jump conditions (e.g., Burgess & Scholer, 2015) have been
widely used to model the plasma properties across a shock. Here, the shock is
treated as a planar, one-dimensional, time stationary structure. Following an
integration of the MHD equations for mass, momentum and energy conservation,
and the divergence-free condition for the magnetic field, the Rankine-Hugoniot
relations linking the properties of upstream and downstream plasmas can be
written as follows:

$\displaystyle\frac{\rho_{m2}}{\rho_{m1}}$ $\displaystyle=\mathrm{r_{gas}}$
(A1) $\displaystyle\frac{V_{n2}}{V_{n1}}$
$\displaystyle=\frac{1}{\mathrm{r_{gas}}}$ (A2)
$\displaystyle\frac{V_{t2}}{V_{t1}}$
$\displaystyle=\frac{M^{2}_{A1}-1}{M^{2}_{A1}-\mathrm{r_{gas}}}$ (A3)
$\displaystyle\frac{B_{n2}}{B_{n1}}$ $\displaystyle=1$ (A4)
$\displaystyle\frac{B_{t2}}{B_{t1}}$
$\displaystyle=\mathrm{r_{gas}}\frac{M^{2}_{A1}-1}{M^{2}_{A1}-\mathrm{r_{gas}}}$
(A5) $\displaystyle\frac{P_{2}}{P_{1}}$
$\displaystyle=\mathrm{r_{gas}}+\frac{(\gamma-1)\mathrm{r_{gas}}V_{1}^{2}}{2V_{S}1}\left(1-\frac{V_{2}^{2}}{V_{1}^{2}}\right).$
(A6)

The above equations for density, bulk flow velocity, magnetic field and
pressure ($\rho_{m},V,B$ and $P$, respectively) are expressed in the
deHoffmann-Teller frame, i.e., a frame aligned to the shock normal that moves
at a speed such that the upstream convective electric field
($\mathbf{E}_{1}=-\mathbf{V}_{1}\times\mathbf{B}_{1}$) vanishes (de Hoffmann &
Teller, 1950). Here, the subscripts 1 and 2 are referred to the upstream and
downstream states, respectively. The subscripts $n$ and $t$, instead, indicate
the shock normal and tangential directions. $\mathrm{r_{gas}}$ is the shock
gas compression ratio, and $M_{A1}\equiv V_{n1}\sqrt{\mu_{0}\rho_{m1}}/B_{n1}$
is the Alfvénic Mach number. $V_{S1}$ is the upstream sound speed.

Often, the Rankine-Hugoniot equations are used to address shock parameters.
What we do in our decompression technique is instead to use the downstream
measurements and the shock parameters to compute the upstream conditions, thus
“unshocking” the plasma according to Equations A1-A6. The procedure, given a
time-series of spacecraft measurements, is performed as follows, and displayed
in Figure 9. First, the data is rotated in a shock normal frame. Here, we
choose the $nlm$ frame, where the $n$ direction is aligned with the shock
normal (computed using the MX3 method (Paschmann & Schwartz, 2000)), $m$ is
perpendicular both to the shock normal and to the upstream magnetic field, and
$l$ completes the triad. A boost is performed to then move to the deHoffman-
Teller frame. Then, Equations A1-A6 are used to derive the upstream quantities
given the downstream measurements and the shock parameters, i.e., the
decompression is performed. The data is then returned in the spacecraft frame.
Finally, using the mean upstream and downstream flow speeds, it is possible to
shock the measurements in units of distance from shock (see Figure 5).

This technique, while giving a proxy for the plasma conditions before shock
processing, has several limtations. The decompression is performed assuming
that the shock parameters do not change throughout the event, thus neglecting
the inherent variability of shock system. Shock parameters are estimated from
single spacecraft measurements, and therefore are associated to uncertainties.
The Rankine-Hugoniot relations used to decompress the plasma assume a one-
dimensional, planar, time-stationary MHD shock with laminar
upstream/downstream regions, which is notoriously a stringent assumption for
heliosphere shocks, that are characterised by several space/time variabilities
and propagate in the turbulent solar wind.

For the above reasons, it is readily understood that the decompression is more
reliable for closer measurements with respect to the shock crossing. In Figure
9, it is possible to see how unphysical results are introduced by the large
structure embedded in the plasma happening around 17:33 UT. Future
improvements of the diagnostic for example including wave transmission will be
object of future work.

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