11institutetext:  INAF, IASF-Milano, via A. Corti 12, I-20133 Milano, Italy
22institutetext: INAF – Osservatorio Astronomico di Trieste, via Tiepolo 11,
I-34131 Trieste, Italy 33institutetext: IFPU – Institute for Fundamental
Physics of the Universe, via Beirut 2, 34151, Trieste, Italy 44institutetext:
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France 55institutetext: Department of Astronomy, University of
Geneva, ch. d’Écogia 16, CH-1290 Versoix Switzerland 66institutetext: DiSAT,
Università degli Studi dell’Insubria, via Valleggio 11, I-22100 Como, Italy
77institutetext: Astronomy Unit, Department of Physics, University of Trieste,
via Tiepolo 11, 34131 Trieste, Italy 88institutetext: Università degli studi
di Roma ‘Tor Vergata’; Via della ricerca scientifica, 1; 00133 Roma, Italy
99institutetext: INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via
Piero Gobetti 93/3, 40129 Bologna, Italy 1010institutetext: Dipartimento di
Fisica e Astronomia, Universitá di Bologna, via Gobetti 92/3, 40121 Bologna,
Italy 1111institutetext: INAF, Osservatorio Astronomico di Brera, via E.
Bianchi 46, 23807 Merate, Italy 1212institutetext: Dipartimento di Fisica,
Sapienza Universitá di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
1313institutetext: INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127
Bologna, Italy 1414institutetext: Department of Astrophysical Sciences,
Princeton University, Princeton, NJ 08544, USA 1515institutetext: Laboratoire
d’Astrophysique de Marseille, Aix-Marseille Université, CNRS, CNES, Marseille,
France 1616institutetext: Institut d’Astrophysique de Paris, CNRS, Sorbonne
Université, Paris, France 1717institutetext: Jodrell Bank Centre for
Astrophysics, Department of Physics and Astronomy, School of Natural Sciences,
The University of Manchester, Manchester M13 9PL, UK 1818institutetext: Center
for Astrophysics $|$ Harvard $\&$ Smithsonian, 60 Garden Street, Cambridge, MA
02138, USA 1919institutetext: HH Wills Physics Laboratory, University of
Bristol, Tyndall Ave, Bristol, BS8 1TL, UK 2020institutetext: IRAP, Université
de Toulouse, CNRS, CNES, UPS, 9 av du colonel Roche, BP44346, 31028 Toulouse
cedex 4, France 2121institutetext: Dipartimento di Fisica, Universitá degli
Studi di Milano, Via G. Celoria 16, 20133 Milano, Italy

# CHEX-MATE: Constraining the origin of the scatter in galaxy cluster radial
X-ray surface brightness profiles

I. Bartalucci CHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles S. Molendi CHEX-MATE: Constraining the origin of the scatter in
galaxy cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles E. Rasia CHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles G.W.
Pratt CHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles M.
Arnaud CHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles M.
Rossetti CHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles F.
Gastaldello CHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles D. Eckert CHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles M. Balboni CHEX-MATE: Constraining the origin of the scatter in
galaxy cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles S.
Borgani CHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profilesCHEX-MATE: Constraining the origin of the scatter
in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profilesCHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profiles H. Bourdin CHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profilesCHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profiles M.G. Campitiello CHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profilesCHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profiles S. De Grandi CHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profiles M. De Petris CHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profiles R.T. Duffy CHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profiles S. Ettori CHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profiles A. Ferragamo CHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profiles M. Gaspari CHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profiles R. Gavazzi CHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profilesCHEX-MATE: Constraining the origin of the scatter
in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profiles S. Ghizzardi CHEX-MATE: Constraining the origin of the
scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profiles A. Iqbal CHEX-MATE: Constraining the origin of the scatter
in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profiles S.T. Kay CHEX-MATE: Constraining the origin of the scatter
in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profiles L. Lovisari CHEX-MATE: Constraining the origin of the
scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-MATE:
Constraining the origin of the scatter in galaxy cluster radial X-ray surface
brightness profilesCHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles P. Mazzotta CHEX-MATE: Constraining the origin of the scatter in
galaxy cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles B.J. Maughan CHEX-MATE: Constraining the origin of the scatter in
galaxy cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles E. Pointecouteau CHEX-MATE: Constraining the origin of the scatter in
galaxy cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining
the origin of the scatter in galaxy cluster radial X-ray surface brightness
profiles G. Riva CHEX-MATE: Constraining the origin of the scatter in galaxy
cluster radial X-ray surface brightness profilesCHEX-MATE: Constraining the
origin of the scatter in galaxy cluster radial X-ray surface brightness
profilesCHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profiles M.
Sereno CHEX-MATE: Constraining the origin of the scatter in galaxy cluster
radial X-ray surface brightness profilesCHEX-MATE: Constraining the origin of
the scatter in galaxy cluster radial X-ray surface brightness profilesCHEX-
MATE: Constraining the origin of the scatter in galaxy cluster radial X-ray
surface brightness profilesCHEX-MATE: Constraining the origin of the scatter
in galaxy cluster radial X-ray surface brightness profiles

(received – accepted)

We investigate the statistical properties and the origin of the scatter within
the spatially resolved surface brightness profiles of the CHEX–MATE sample,
formed by 118 galaxy clusters selected via the SZ effect. These objects have
been drawn from the Planck SZ catalogue and cover a wide range of masses,
M${}_{500}=[2-15]\times 10^{14}$M⊙, and redshift, z=[0.05,0.6]. We derived the
surface brightness and emission measure profiles and determined the
statistical properties of the full sample and sub-samples according to their
morphology, mass, and redshift. We found that there is a critical scale,
R$\sim 0.4R_{500}$, within which morphologically relaxed and disturbed object
profiles diverge. The median of each sub-sample differs by a factor of $\sim
10$ at $0.05\,R_{500}$. There are no significant differences between mass- and
redshift-selected sub-samples once proper scaling is applied.

We compare CHEX–MATE with a sample of 115 clusters drawn from the The Three
Hundred suite of cosmological simulations. We found that simulated emission
measure profiles are systematically steeper than those of observations. For
the first time, the simulations were used to break down the components causing
the scatter between the profiles. We investigated the behaviour of the scatter
due to object-by-object variation. We found that the high scatter,
approximately 110%, at $R<0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ is due to a
genuine difference between the distribution of the gas in the core of the
clusters. The intermediate scale,
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}=[0.4-0.8]$, is characterised by the
minimum value of the scatter on the order of $0.56,$ indicating a region where
cluster profiles are the closest to the self-similar regime. Larger scales are
characterised by increasing scatter due to the complex spatial distribution of
the gas. Also for the first time, we verify that the scatter due to projection
effects is smaller than the scatter due to genuine object-by-object variation
in all the considered scales.

###### Key Words.:

intracluster medium – X-rays: galaxies: clusters

##  1 Introduction

Galaxy clusters represent the ultimate manifestation of large-scale structure
formation. Dark matter comprises $80\%$ of the total mass in a cluster and is
the main actor of the gravitation assembly prcoess (Voit 2005; Allen et al.
2011; Borgani & Kravtsov 2011). This influences the prevalent baryonic
component represented by a hot and rarefied plasma that fills the cluster
volume, that is, the intracluster medium (ICM). This plasma’s properties are
affected by the individual assembly history and ongoing merging activities.
The study of its observational properties is thus fundamental to study how
galaxy clusters form and evolve. The ideal tool for investigating this
component is X-ray observations, as the ICM emits in this band via thermal
Bremsstrahlung.

The radial profiles of the X-ray surface brightness (SX) of a galaxy cluster
and the derived emission measure (EM) are direct probes of the plasma
properties. These two quantities can be easily measured in the X-ray band and
have played a crucial role in the characterisation of the ICM distribution
since the advent of high spatial resolution X-ray observations (e.g. Vikhlinin
et al. 1999). Neumann & Arnaud 1999 and Neumann & Arnaud 2001 compared SX
profiles with expectations from theory to test the self-similar evolution
scenario and investigate the relation between the cluster luminosity and its
mass and temperature. Arnaud et al. (2001) tested the self-similarity of the
EM profiles of 25 clusters in the [0.3-0.8] redshift range, finding that
clusters evolve in a self-similar scenario, which deviates from the simplest
models because of the individual formation history. The SX and EM profiles
have been used to investigate the properties of the outer regions of galaxy
clusters, both in observations (e.g. Vikhlinin et al. 1999; Neumann 2005;
Ettori & Balestra 2009) and in a suite of cosmological simulations (see e.g.
Roncarelli et al. 2006). These regions are of particular interest because of
the plethora of signatures from the accretion phenomena, but they are hard to
observe because of their faint signal. More recent works based on large
catalogues (see e.g. Rossetti et al. 2017 and Andrade-Santos et al. 2017) have
determined the effects of the X-ray versus the Sunyaev Zel’Dovich (SZ; Sunyaev
& Zeldovich 1980) selection by studying the concentration of the surface
brightness profiles in the central regions of galaxy clusters. Finally, the SX
radial profile represents the baseline for any study envisaging to derive the
thermodynamical properties of the ICM, such as the 3D spatial distribution of
the gas (Sereno et al. 2012, 2017, 2018). This information can be combined
with the radial profile of the temperature, and together, they can be used to
derive quantities such as the entropy (see, e.g. Voit et al. 2005), pressure,
and mass of the galaxy cluster under the assumption of hydrostatic equilibrium
(Ettori et al. 2013; Pratt et al. 2022).

In this paper, we used the exceptional data quality of the 118 galaxy clusters
from the Cluster HEritage project with XMM-Newton - Mass Assembly and
Thermodynamics at the Endpoint of structure formation (CHEX-MATE111xmm-
heritage.oas.inaf.it, PI; S. Ettori and G.W. Pratt). Specifically, we
investigate for the first time the statistical properties of the X-ray surface
brightness and emission measure radial profiles of a sample of galaxy clusters
observed with unprecedented and homogeneous deep XMM-Newton observations. The
sample, being based on the Planck catalogue, is SZ selected and thus predicted
to be tightly linked to the mass of the cluster (e.g. Planelles et al. 2017
and Le Brun et al. 2018), and hence it should yield a minimally biased sample
of the underlying cluster population.

Our analysis is strengthened by the implementation of the results from a mass-
redshift equivalent sample from cosmological and hydrodynamical simulations of
the The Three Hundred collaboration (Cui et al. 2016). We used a new approach
to understand the different components of the scatter, considering the
population (i.e. cluster-to-cluster) scatter and the single object scatter
inherent to projection effects.

In Sect. 2, we present the CHEX–MATE sample. In Sects. 3 and 4, we describe
the methodology used to prepare the data and the derivation of the radial
profiles of the CHEX–MATE and numerical datasets, respectively. In Sect. 5, we
discuss the shape of the profiles. In Sect. 6, we present the scatter within
the CHEX–MATE sample. In Sect. 7, we investigate the origin of the scatter of
the EM profiles, and finally in Sect. 8, we discuss our results and present
our conclusions.

We adopted a flat $\Lambda$-cold dark matter cosmology with
$\Omega_{\mathrm{M}}(0)=0.3$, $\Omega_{\Lambda}=0.7$, H${}_{0}=70$ km Mpc s-1,
E(z)$=(\Omega_{\mathrm{M}}(1+z)^{3}+\Omega_{\Lambda})^{1/2}$, and
$\Omega_{\mathrm{M}}(z)=\Omega_{\mathrm{M}}(0)(1+z)^{3}/$E(z)2 throughout. The
same cosmology was used for the numerical simulations, except for $h=0.6777$.
Uncertainties are given at the 68% confidence level (i.e. $1\sigma$). All the
fits were performed via $\chi^{2}$ minimisation. We characterised the
statistical properties of a sample by computing the median and the $68\%$
dispersion around it. This dispersion was computed by ordering the profiles
according to their $\chi^{2}$ with respect to the median and by considering
the profile at $\pm 34\%$ around it. We use natural logarithm throughout the
work except for where we state otherwise.

##  2 The CHEX–MATE sample

###  2.1 Definition

Figure 1: Distribution of the clusters published in the PSZ2 Planck catalogue
(Planck Collaboration XXVII 2016) in the mass-redshift plane. The masses in
the Planck catalogue were derived iteratively from the
$M_{500}$–$Y_{\textrm{S}Z}$ relation calibrated using hydrostatic masses from
XMM-Newton; they were not corrected for the hydrostatic equilibrium bias. The
magenta and green points represent the Tier 1 and Tier 2 clusters of the
CHEX–MATE sample, respectively (CHEX-MATE Collaboration 2021). The triangles
and squares identify the morphologically relaxed and disturbed clusters,
respectively, which were identified according to the classification scheme in
Campitiello et al. (2022). The two red crosses identify the clusters excluded
from the analysis of this work.

This work builds on the sample defined for the XMM-Newton heritage programme
accepted in AO-17. We briefly report the sample definition and selection
criteria here that are detailed in CHEX-MATE Collaboration (2021). The
scientific objective of this programme is to investigate the ultimate
manifestation of structure formation in mass and time by observing and
characterising the radial thermodynamical and dynamic properties of a large,
minimally biased and S/N-limited sample of galaxy clusters. This objective is
achieved by selecting 118 objects from the Planck PSZ2 catalogue (Planck
Collaboration XXVII 2016), applying an SNR threshold of 6.5 in the SZ
identification, and folding the XMM-Newton visibility criteria.

The key quantity $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, defined as the mass
enclosed within the radius $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ of the cluster
where its average total matter density is 500 times the critical density of
the Universe, is measured by the Planck collaboration using the MMF3 SZ
detection algorithm detailed in Planck Collaboration XXVII (2015). This
algorithm measures the $Y_{\textrm{S}Z}$ flux associated to each detected
cluster, and it is used to derive the $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$
using the $M_{500}$–$Y_{\textrm{S}Z}$ relation calibrated in Arnaud et al.
(2010), assuming self-similar evolution. We note that while the clusters’
precise mass determination is one of the milestones of the multi-wavelength
coverage of the CHEX-MATE programme, in this paper we consider the radii and
mass values directly from the Planck catalogue. The impact of this choice will
be discussed in Sect. 5.5.

The CHEX–MATE sample is split in two sub-samples according to the cluster
redshift.
Tier 1 provides a local sample of 61 objects in the [0.05-0.2] redshift range
in the northern sky (i.e. DEC ¿ 0), and their
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ span the $[2-9]\times 10^{14}M_{\odot}$
mass range. These objects represent a local anchor for any evolution study.
Tier 2 offers a sample of the massive clusters,
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}>7.25\times 10^{14}M_{\odot}$ in the
[0.2-0.6] redshift range. These objects represent the culmination of cluster
evolution in the Universe.

The distribution in the mass and redshift plane of the CHEX–MATE sample and
its sub-samples are shown in Fig. 1. The exposure times of these observations
were optimised to allow the determination of spatially resolved temperature
profiles at least up to $R_{500}$ with a precision of $15\%$.

The clusters PSZ2 G028.63+50.15 and PSZ2 G283.91+73.87 were excluded from the
analysis presented in this work since their radial analysis could introduce
large systematic errors without increasing the statistical quality of the
sample. Indeed, the former system presents a complex morphology (see
Schellenberger et al. 2022 for a detailed analysis), and it has a background
cluster at $z=0.38$ within its extended emission. The latter is only $\sim 30$
arcmin from M87, and thus its emission is heavily affected by the extended
emission of Virgo. The basic properties of the final sample of 116 objects are
listed in Table LABEL:tab:500_prop.

###  2.2 Sub-samples

We defined CHEX–MATE sub-samples based on key quantities: mass, redshift, and
morphological status. The analysis of the morphology of the CHEX–MATE clusters
sample is described in detail in Campitiello et al. (2022). The authors use a
combination of morphological parameters (see Rasia et al. 2013 for the
definition of these parameters) to classify the clusters as morphologically
relaxed, disturbed, or mixed. Following the criteria described in Sect. 8.2 of
Campitiello et al. (2022), the authors identified the 15 most relaxed and 25
most disturbed clusters. We adopted their classification in this paper and
refer to the former group as morphologically relaxed clusters and the latter
group as disturbed clusters.

We defined the sub-samples of nearby and distant clusters considering the 85
and 31 clusters at $z\leq 0.33$ and $z>0.33$, respectively, the value 0.33
being the mean redshift of the sample. Similarly, we built the sub-samples of
high- and low-massive clusters considering the 40 and 76 clusters with
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}\leq 5\times 10^{14}M_{\odot}$ and
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}>5\times 10^{14}M_{\odot}$, respectively.

##  3 Data analysis

###  3.1 Data preparation

####  3.1.1 XMM-Newton data

The clusters used in this work were observed using the European Photon Imaging
Camera (EPIC; Turner et al. 2001 and Strüder et al. 2001). The instrument
comprises three CCD arrays, namely, MOS1, MOS2, and pn, that simultaneously
observe the target. Datasets were reprocessed using the Extended-Science
Analysis System (ESAS222cosmos.esa.int/web/xmm-newton; Snowden et al. 2008)
embedded in SAS version 16.1. The emchain and epchain tools were used to apply
the latest calibration files made available January $2021$ and produce pn out-
of-time datasets. Events in which the keyword PATTERN is greater than four for
the MOS$1$ and MOS$2$ cameras and greater than 12 for the pn camera were
filtered out from the analysis. The CCDs showing an anomalous count rate in
the MOS1 and MOS2 cameras were also removed from the analysis. Time intervals
affected by flares were removed using the tools mos-filter and pn-filter by
extracting the light curves in the [2.5-8.5] keV band and removing the time
intervals where the count rate exceeded $3\sigma$ times the mean count rate
from the analysis. Point sources were filtered from the analysis following the
scheme detailed in Section 2.2.3 of Ghirardini et al. (2019), which we
summarise as follows. Point sources were identified by running the SAS wavelet
detection tool ewavdetect on $[0.3-2]$ keV and $[2-7]$ keV images obtained
from the combination of the three EPIC cameras and using wavelet scales in the
range of 1–32 pixels and an S/N threshold of five, with each bin width being
$\sim 2$ arcsec. The PSF and sensitivity of XMM-Newton depends on the off-axis
angle. For this reason, the fraction of unresolved point sources forming the
Cosmic X-ray Background (CXB; Giacconi et al. 2001) is spatially dependent. We
used a threshold in the LogN-LogS distribution of detected sources, below
which we deliberately left the point source in the images to ensure a constant
CXB flux across the detector. Catalogues produced from the two energy band
images were then merged. At the end of the procedure, we inspected the
identified point sources by eye to check for false detections in CCD gaps. We
also identified extended bright sources other than the cluster itself by eye
and removed them from the analysis. We identified 13 clusters affected by at
least one sub-structure within $R_{200}^{\mathrm{Y_{\textrm{S}Z}}}$ that were
masked by applying circular masks of $\sim 3$ arcmin radius on average.

####  3.1.2 Image preparation

We undertook the following procedures to generate the images from which we
derived the profiles. Firstly, we extracted the photon count images in the
$[0.7-1.2]$ keV band for each camera, this energy band maximises the source to
background ratio (Ettori et al. 2010). An exposure map for each camera folding
the vignetting effect was produced using the ESAS tool eexpmap.

The background affecting the X-ray observations was due to a sky and
instrumental component. The former was from the local Galactic emission and
the CXB (Kuntz & Snowden 2000), and its extraction is described in detail in
Sect. 3.3. The latter was due to the interaction of high energy particles with
the detector. We followed the strategy described in Ghirardini et al. (2019)
to remove this component by producing background images that accounted for the
particle background and the residual soft protons.

The images, exposure, and background maps of the three cameras were merged to
maximise the statistic. The pn exposure map was multiplied by a factor to
account for the ratio of the effective area MOS to pn in the $[0.7-1.2]$ keV
band when merging the exposure maps. This factor was computed using XSPEC by
assuming a mean temperature and using the hydrogen column absorption value,
${\mathrm{N_{H}}}$, reported in Table LABEL:tab:500_prop. Henceforth, we refer
to the combined images of the three cameras and the background maps simply as
the observation images and the particle background datasets, respectively.

Figure 2: Comparison of the statistical properties of the CHEX–MATE EM
profiles divided in redshift and mass selected sub-samples. Top-left panel:
Median of the ${\mathrm{EM_{S}}}$ profiles centred on the X-ray peak scaled
for self-similar evolution using Eq. 4. The dispersion is shown with the black
solid line and the grey envelope. The magenta and green solid lines represent
the median of the low-mass, $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}\leq 5\times
10^{14}$M⊙, and high-mass, $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}>5\times
10^{14}$M⊙, sub-samples, respectively. The red and blue dotted lines represent
the medians of the low-, $z\leq 0.33$, and high-redshift, $z>0.33$, sub-
samples, respectively. Bottom-left panel: Ratio of the sub-samples medians
with respect to the full CHEX–MATE sample median. The dotted horizontal lines
represent the 25% plus and minus levels. Middle panels: Comparison of the
statistical properties of the $\widetilde{\mathrm{EM}}$ profiles scaled to
account also for mass and redshift evolution divided in mass selected sub-
samples. On the top panel we show the same as in the top left panel except the
medians and the dispersion were computed using profiles scaled with Eq. 5. On
the bottom we show the ratio between the medians of the sub-samples as respect
to the median of the full sample. The dotted horizontal lines in the lower-
middle panel represent the plus and minus 5% levels. Right panels: Comparison
of the statistical properties of the scaled $\widetilde{\mathrm{EM}}$ profiles
divided in redshift selected sub-samples. On the top panel we show the same as
in the top left panel except that the medians and the dispersion were computed
using the profiles scaled with Eq. 5. On the bottom we show the ratio between
the medians of the sub-samples as respect to the median of the full CHEX–MATE
sample.

###  3.2 Global quantities

####  3.2.1 Average temperature

We estimated the average temperature, Tavg, of each cluster by applying the
definition of the temperature of a singular isothermal sphere with mass
$M_{500}$ as described in Appendix A of Arnaud et al. (2010):

$\mathrm{T}_{\mathrm{avg}}=0.8\times\mathrm{T_{500}}=0.8\times\frac{\mu
m_{p}GM_{500}^{\mathrm{Y_{\textrm{S}Z}}}}{2R_{500}^{\mathrm{Y_{\textrm{S}Z}}}},$
(1)

where $\mu=0.59$ is the mean molecular weight, $m_{p}$ is the proton mass, $G$
is the gravitational constant, and the 0.8 factor represents the average value
of the universal temperature profile derived by Ghirardini et al. (2019) with
respect to T500. These temperatures are reported in Table LABEL:tab:500_prop.

####  3.2.2 Cluster coordinates

We produced point source free emission images by filling the holes from the
masking procedure with the local mean emission estimated in a ring around each
excluded region by using the tool dmfilth. We then performed the vignetting
correction by dividing them for the exposure map. We used these images to
determine the peak by identifying the maximum of the emission after the
convolution of the map with a Gaussian filter with $\sim 10$ arcsec width. The
centroid of the cluster was determined by performing a weighted-mean of the
pixel positions using the counts as weight within a circular region centred on
the peak and with its radius as $R_{500}^{Y_{SZ}}$. This has been done to
avoid artefact contamination near the detector edges. The coordinates obtained
are reported in Table LABEL:tab:500_prop.

###  3.3 Radial profiles

####  3.3.1 Surface brightness profiles

Azimuthal mean profiles. The surface brightness radial profiles, S$(\Theta)$,
were extracted using the following technique. We defined concentric annuli
centred on the X-ray peak and the centroid. The minimum width was set to
$4^{\prime\prime}$ and was increased using a logarithmic factor. In each
annulus, we computed the sum of the photons from the observation image as well
as from the particle background datasets. The particle background-subtracted
profile was divided by the exposure folding the vignetting in the same annulus
region. We estimated the sky background component as the average count rate
between $R_{200}=1.49R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and $13.5$ arcmin and
subtracted it from the profile. If $R_{200}$ was outside the field of view, we
estimated the sky background component using the XMM-Newton-ROSAT background
relation described in Appendix B. The sky background-subtracted profiles were
re-binned to have at least nine counts per bin after background subtraction.
We corrected the profiles for the PSF using the model developed by Ghizzardi
(2001). We refer hereafter to these profiles as the mean SX profiles.

Azimuthal median profiles. We also computed the surface brightness radial
profiles considering the median in each annulus following the procedure
detailed in Section 3 of Eckert et al. (2015). This procedure has been
introduced to limit the bias caused by the emission of sub-clumps and sub-
structures too faint to be identified and masked (e.g. Roncarelli et al. 2013;
Zhuravleva et al. 2013). Briefly, we applied the same binning scheme and point
source mask to the particle background dataset to perform the background
subtraction. Employing the procedure of Cappellari & Copin (2003) and Diehl &
Statler (2006), we first produced Voronoi-binned maps to ensure 20 counts per
bin on average to apply the Gaussian approximation. We then extracted the
surface brightness median profile with the same annular binning of the mean
profile, considering in each radial bin the median count rate of the Voronoi
cells, whose centre lies within the annulus. The sky background was estimated
with the same approach used for the mean profile except that we estimated the
median count rate. Finally, the sky background-subtracted profiles were re-
binned using the same 3$\sigma$ binning of the mean profiles. The four
resulting types of surface brightness profiles are shown in 18. We were able
to measure the profiles beyond $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ for 107 of
the 116 (i.e. $\sim 92\%$) CHEX–MATE objects.

We report in Table 1 the median relative errors at fixed radii to illustrate
the excellent data quality. From now on, we refer to these profiles as the
median SX profiles, and throughout the paper, we use these profiles centred on
the X-ray peak unless stated otherwise.

Table 1:  Average of the relative errors of the CHEX–MATE EM profiles.

Radius $[R_{500}^{\mathrm{Y_{\textrm{S}Z}}}]$ | Average relative error [%] | Number of profiles used
---|---|---
0.2 | 1.7 | 116
0.5 | 2.1 | 116
0.7 | 3.0 | 116
1.0 | 6.0 | 107

Notes: We used the EM median profiles centred on the X-ray peak. We also
report the number of profiles that have been used to compute the relative
error in the third column.

####  3.3.2 Emission measure profiles

We computed the $\mathrm{EM}$ radial profiles using Equation 1 of Arnaud et
al. (2002):

$\mathrm{EM(r)}=\mathrm{S}(\Theta)\frac{4\pi(1+z)^{4}}{\epsilon{(T,z)}},$ (2)

where $\Theta=r/\mathrm{d_{A}}(z),$ with $\mathrm{d_{A}}(z)$ as the angular
diameter distance, and $\epsilon$ is the emissivity integrated in the
E${}_{1}=0.7$ keV and E${}_{2}=1.2$ keV band and is defined as

$\epsilon(T,z)=\int_{E_{1}}^{E_{2}}\Sigma(E)e^{-\sigma(E)N_{H}}f_{T}((1+z)E)(1+z)^{2}dE,$
(3)

where $\Sigma$(E) is the detector effective area at energy $E$, $\sigma(E)$ is
the absorption cross section, $N_{\mathrm{H}}$ is the hydrogen column density
along the line of sight, and $f_{T}((1+z)E)$ is the emissivity at energy
$(1+z)E$ for a plasma at temperature $T$. The $\epsilon$ factor was computed
using an absorbed Astrophysical Plasma Emission Code (APEC) within the XSPEC
environment. The absorption was calculated using the phabs model folding the
Hydrogen absorption column reported in Table LABEL:tab:500_prop. The
dependency of $\epsilon$ on temperature and abundance in the band we used to
extract the profile is weak (e.g. Lovisari & Ettori 2021). Therefore, for APEC
we used the average temperature, $k\mathrm{T_{avg}}$, of the cluster within
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and the abundance fixed to 0.25
(Ghizzardi et al. 2021) with respect to the solar abundance table of Anders &
Grevesse (1989). Finally, we used the redshift values reported in Table
LABEL:tab:500_prop. We obtained EM azimuthal mean and azimuthal median
profiles centred on the X-ray peak and centroid, converting the respective
surface brightness profiles. The EM profiles were first scaled considering
only the self-similar evolution scenario, $\mathrm{EM_{S}}$, as in Arnaud et
al. (2002):

$\mathrm{EM_{S}}(r,T,z)=E(z)^{-3}\times\left(\frac{kT_{\mathrm{avg}}}{10}\right)^{-1/2}\times\mathrm{EM}(r),$
(4)

where $x$=r/$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and Tavg is the average
temperature of the cluster, as in Equation 1. The left panel of Fig. 2 shows
the median of the $\mathrm{EM_{S}}$ profiles centred on the X-ray peak as well
as its $68\%$ dispersion. In the same plot, we also show the medians of the
sub-samples introduced in Sect 2.2.

Their ratio with respect to the CHEX-MATE median shown in the bottom panel
demonstrates that the employed re-scaling is not optimal since for all sub-
samples there are variations with respect to the median that range between 10%
and 50% at all scales. We therefore tested another re-scaling following Pratt
et al. (2022) and Ettori et al. (2022), who point out how the mass dependency
is not properly represented by the self-similar scenario and had a small
correction also with respect to the redshift evolution. The final scaling that
we considered is given by the following:

$\mathrm{\widetilde{EM}}(r,T,z)=E(z)^{-3.17}\left(\frac{kT_{\mathrm{\mathrm{avg}}}}{10\;\mathrm{keV}}\right)^{-1.38}\times\mathrm{EM}(r).$
(5)

The effect of this scaling on the mass and redshift residual dependency is
shown in the middle and right panel of Fig. 2, respectively. The medians of
the sub-samples show little variations in relation to the whole sample within
the order of a few percentage points on average. We show the individual scaled
median radial profiles centred on the X-ray peak in Fig. 3 together with the
$68\%$ dispersion. The discussion of the difference between the relaxed and
disturbed sub-samples is detailed in Section 5.3.

Figure 3: Scaled emission measure median profiles centred on the X-ray peak.
The blue and red solid lines indicate morphologically relaxed and disturbed
clusters, respectively. The profiles extracted from clusters with mixed
morphology are shown with black solid lines. The selection criteria was based
on the classification made by Campitiello et al. (2022) in their Section 8.2.
The grey-shaded envelope represents the dispersion at the $68\%$ level.

##  4 Cosmological simulations data

The main scientific goal of this paper is to investigate the origin of the
diversity of the EM profiles. The main source of the scatter between the
profiles is expected to be due to a genuine different spatial distribution of
the ICM related to the individual formation history of the cluster. The other
sources that impact the observed scatter are related to how we observe
clusters. There are systematic errors associated with X-ray analysis and
observing clusters in projection. This latter point is of crucial importance
when computing the scatter within a cluster sample. For instance, a system
formed by two merging halos of similar mass will appear as a merging system if
the projection is perpendicular to the merging axis but will otherwise appear
regular if the projection is parallel. In this work, we employed cosmological
simulations from the The Three Hundred collaboration (Cui et al. 2018) to
evaluate this effect.

Figure 4: Mass distribution of the CHEX–MATE and The Three Hundred samples.
These are shown with black empty and green polygons, respectively. The gap
between $6-7\times 10^{14}M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ is an artefact
from the CHEX–MATE sample being divided into two tiers, shown in Fig. 1. The
shift between the two distributions is due to the fact that the CHEX–MATE
masses are assumed to be 0.8 lower than the true mass due to the hydrostatic
bias. For more details, refer to Section 4.

Specifically, we study the GADGET-X version of The Three Hundred suite. This
is composed of re-simulations of the 324 most massive clusters identified at
$z=0$ within the dark matter-only MULTIDARK simulation Klypin et al. (2016),
and thus it constitutes an ideal sample of massive clusters from which to
extract a CHEX-MATE simulated counterpart. The cosmology assumed in the
MULTIDARK simulation is that of the Planck collaboration XIII (2016) and is
similar to what is assumed in this paper. The adopted baryon physics include
metal-dependent radiative gas cooling, star formation, stellar feedback,
supermassive black hole growth, and active galactic nuclei (AGN) feedback
(Rasia et al. 2015). To cover the observational redshift range, the simulated
sample was extracted from six different snapshots corresponding to
$z=0.067,0.141,0.222,0.333,0.456$, and $0.592$. For each observed object, in
addition to the redshift, we matched the cluster mass $M_{500}$ imposed to be
close to $M_{500}^{\mathrm{Y}_{SZ}}/0.8$. With this condition, we followed the
indication of the Planck collaboration (see Planck Collaboration XX 2014) that
assumed a baseline mass bias of 20% ($1-b=0.8$). We also checked whether the
selected simulated clusters have a strikingly inconsistent morphological
appearance, such as a double cluster associated to a relaxed system. In such
cases, we considered the second closest mass object. In the final sample, the
standard deviation of the
$M_{500,\mathrm{sim}}/(M_{500}^{\mathrm{Y_{\textrm{S}Z}}}/0.8)$ is equal to
$0.037$. Due to the distribution of the CHEX-MATE sample in the mass-redshift
place, we allowed a few Tier 2 clusters to be matched to the same simulated
clusters taken from different cosmic times. Even with this stratagem, which
will not impact the results of this investigation, we observed that a very
massive cluster at $z=0.4$ remained unmatched. The final simulated sample thus
includes 115 objects.

The simulation sample mass distribution is shown in Fig. 4. For each simulated
cluster, we generated 40 EM maps centred on the cluster total density peak and
integrating the emission along different lines of sight for a distance equal
to $6R_{500}$ using the Smac code (Dolag et al. 2005; Ansarifard et al. 2020).
Henceforth, we refer to these maps as ”sim EM” and they are in units of [Mpc
cm-6].

Figure 5: Example of the creation of the X-ray mock images. Left panel: EM map
of a simulated cluster of our sample. The white circle encompasses R500. Right
panel: Mock X-ray background-subtracted image in the [0.5-2] keV band of the
same object shown in the left panel after we applied the procedures simulating
typical X-ray observation effects. These are described in detail in Section
4.1.

###  4.1 X-ray mock images of simulated clusters

We produced mock X-ray observations by applying observational effects to the
The Three Hundred maps. Firstly, we transformed the EM in surface brightness
maps by inverting Equation 2. The emissivity factor $\epsilon$(T,z) was
computed using the same procedure as in Section 3.3.2. The absorption was
fixed to the average value of the CHEX–MATE sample, N${}_{\mathrm{H}}=2\times
10^{20}$ cm-2, and the average temperature was computed by using the $M_{500}$
of the cluster and applying Equation 1. The instrumental effects were
accounted for by folding in the pn instrumental response files computed at the
aimpoint. We produced the count rate maps by multiplying the surface
brightness maps by the median exposure time of the CHEX–MATE programme,
$4\times 10^{4}$ s, and by the size of the pixel in arcminutes2. We added to
these maps a spatially non-uniform sky background whose count rate is $\langle
cr_{sky}\rangle=5.165\times 10^{-3}$ [ct/s/arcmin2], as measured by pn in the
$[0.5-2]$ keV band. We then included the XMM-Newton vignetting as derived from
the calibration files, and we simulated the PSF effect by convolving the map
with a Gaussian function with a width of ten arcsec. Finally, we drew a
Poisson realisation of the expected counts in each pixel and produced a mock
X-ray observation. We divided the field of view into square tiles with sides
of 2.6 arcmin within which we introduced $3\%$ variations to the mean sky
background count rate to mimic the mean variations of the sky on the field of
view of XMM-Newton. We multiplied these maps by 1.07 and 0.93 to create over-
and underestimated background maps, respectively, which account for the
systematic error related to the background estimation. We randomly chose the
over- or underestimated map and subtracted it from the mock X-ray observation.
After the subtraction, we corrected for the vignetting by using a function
obtained through the fit of the calibration values to those we randomly added
a $1\pm 0.05$ factor to mimic our imprecision in the calibration of the
response as a function of the off-axis angle.

The typical effects introduced by the procedures described above are shown in
Fig. 5. The EM map produced using the simulation data is shown in the left
panel where there is a large sub-structure in the west sector and a small one
in the south-west sector within R500. The right panel shows the mock X-ray
image where the degradation effects are evident. The spatial features within
the central regions were lost due to the PSF. Despite the resolution loss, the
ellipsoidal spatial distribution of the ICM is clearly visible, and the
presence of features such as the small sub-structure in the south-west are
still visible. The emission outside R500 is dominated by the background, and
the small filament emission in the south-west was too faint to remain visible.
The large sub-structure is still evident, but the bridge connecting it to the
main halo has become muddled into the background.

###  4.2 Simulation emission measure profiles

Figure 6: Comparison between the Sim and Simx profiles. Top panel: Comparison
between the medians of the Sim and Simx $\widetilde{\mathrm{EM}}$ profiles
extracted from random projections. These are shown with black solid and dotted
lines, respectively. The grey solid lines represent the Sim profiles. Bottom
panel: Ratio between the Sim over the Simx median. The black solid and dotted
lines correspond to the identity and the $\pm$ 2% levels, respectively.

We extracted the EM profiles from the The Three Hundred maps by computing the
median EM of all the pixels within concentric annuli, the bin width being $2$
arcsec. These annuli are centred on the map centre (i.e. the peak of the halo
total density). We obtained the $\widetilde{\mathrm{EM}}$ profiles by applying
this process to our sample of 115 simulated clusters and for each of the 40
projections, and we scaled them according to Equation 5. From hereon, we refer
to these profiles as the ”Sim” profiles. Similarly, we extracted the X-ray
mock profiles, henceforth the ”Simx” profiles, from the synthetic X-ray maps.
These are shown along one randomly selected projection with a grey solid line
in Fig. 6. We show the emission measure profile projected along only one line
of sight because the results along the other projections are similar.

The comparison of the sample medians of the Sim and Simx profiles is shown in
Fig. 6. The two sample medians are in excellent agreement, up to $\sim
0.7R_{500}$. Beyond that radius, the Simx median is flatter. This is an effect
of the PSF, which redistributes on larger scales the contribution of sub-halos
and local inhomogeneities. However, the fact that the medians are similar
after the application of the X-ray effects is likely due to the combination of
the good statistics of the CHEX–MATE programme and the procedure used to
derive each cluster EM profile, which considers the medians of all pixels. The
former ensures that the extraction of profiles is not affected by large
statistical scatter, at least up to $R_{500}$, and the latter tends to hamper
effects related to the presence of sub-structures.

There are key differences between the analysis of the Simx and CHEX–MATE
profiles despite our underlying strategy of applying the same procedures. For
example, the centre used in the simulations introduces a third option with
respect to the X-ray centroid and peak. Furthermore, in simulated clusters, we
computed the azimuthal median on pixels instead of on the Voronoi cells. We
expected that the centre offsets would affect the profiles at small scales,
$R<0.1$R500, as shown in the left panel of Fig. 7. Finally, the X-ray analysis
masks the emission associated to sub-halos, while this is not possible in
simulations, as the development of an automated procedure to detect extended
sources in the large number of images of our simulations, $4600=115\times 40$,
was beyond the scope of this paper. The impact of this difference on the
scatter is discussed in Appendix A.

##  5 The profile shape

In this section, we study the shape of the emission measure profiles by
checking the impact of the centre definition (as in Sect. 3.3.1) and of the
radial profile procedure (as described in Sect. 3.3). Subsequently, we compare
the sample median profiles of the relaxed and disturbed sub-samples and
compare the CHEX-MATE median profile with the literature and the The Three
Hundred simulations.

###  5.1 The impact of the profile centre

Figure 7:  Ratio between the medians of the EM profiles obtained using the
X-ray peak or the centroid as centre and using the azimuthal average or
median. Left panels: Ratio between the medians of the profiles centred on the
peak and centroid. The top and bottom panels show the ratio computed using the
azimuthal median and azimuthal mean $\widetilde{\mathrm{EM}}$ profiles,
respectively. The black solid lines represent the median of the ratio
considering the whole sample. The blue and red solid lines show the ratio
considering only the morphologically relaxed and disturbed clusters,
respectively. The grey solid lines indicate the identity line, and the dotted
lines represent the plus and minus 5% levels. Right panels: Same as the left
panels except we show the ratio between the medians of the azimuthal median
and mean $\widetilde{\mathrm{EM}}$ profiles. The top and bottom panels show
the ratio computed using the profiles extracted with the X-ray peak and the
centroid as centre, respectively. The legend of the solid and dotted lines is
the same as in the left panels except for the fact that we show only the minus
5% level.

The impact of the choice of the centre for the profile extraction is crucial
for any study that builds on the shape of profiles, such as the determination
of the hydrostatic mass profile (see Pratt et al. 2019 for a recent review).
The heterogeneity of morphology and the exquisite data quality of the
CHEX–MATE sample offer a unique opportunity to assess how the choice of the
centre affects the overall shape of the profile.

We show in the top part of the left panel in Fig. 7 the ratio between the
medians of the $\widetilde{\mathrm{EM}}$ azimuthal median profiles centred on
the peak and those centred on the centroid. The colours of the lines
respectively refer to the entire sample and the morphologically relaxed and
disturbed sub-samples. The bottom panel shows a similar ratio where the
azimuthal mean profiles are considered. From the figure, we noticed that the
results obtained using mean or median profiles are similar, with the exception
of the outskirts of the disturbed systems, which will be discussed below. On
average, the relaxed sub-sample shows little deviation from one at all radial
scales, as would be expected since for these systems the X-ray peak likely
coincides with the centroid. The variations of the disturbed objects are up to
$10\%$ in the centre, where the profiles centred on the X-ray peak are
steeper, and about $5\%$ in the [0.15-0.5]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$
region, where the centroid profiles have greater emission. These variations
are not reflected in the entire CHEX–MATE sample, despite the fact that it
includes approximately $87\%$ of the disturbed and morphologically mixed
systems. Indeed, in this case, all deviations are within 2%, implying that the
choice of referring to the X-ray peak does not influence the shape of the
sample median profile.

###  5.2 Mean versus median

We proceeded by testing the radial profile procedure (Sect. 3.3.1) next,
comparing the azimuthal median and the azimuthal mean profiles (Fig. 7) and
centring both on either the X-ray peak (top panel) or, for completeness, on
the centroid (bottom panel). As expected from the previous results, there are
little differences between the two panels. Overall, we noticed that the
azimuthal mean profiles are greater than the azimuthal medians, implying that
greater density fluctuations are present at all scales and that they play a
larger role in the outskirts where a larger number of undetected clumps might
be present. The differences between the two profiles are always within 5% for
the relaxed systems. The deviations are more important for the disturbed
objects, especially when centred on the X-ray peak. This last remark implies
that the regions outside $\sim 0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ of the
CHEX–MATE disturbed objects not only have greater density fluctuations but are
also spherically asymmetric in their gas distribution; otherwise, the same
mean-median deviations would be detected when considering the centroid as
centre. The global effect on the CHEX–MATE sample is that the median profiles
are about 7% lower than the mean profiles at
$R>0.3-0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. We noticed that similar results
were obtained by Eckert et al. (2015) (cfr. Fig.6). This test confirmed that
our choice of using the azimuthal medians for each cluster profile is more
robust for our goal of describing the overall CHEX–MATE radial profiles.

###  5.3 The median CHEX–MATE profiles and the comparison between relaxed and
disturbed systems

Figure 8:  Comparison of the statistical properties of the CHEX–MATE
$\widetilde{\mathrm{EM}}$ profiles with morphologically selected sub-samples
and the X-COP and REXCESS samples. Top-left panel: Median of the
$\widetilde{\mathrm{EM}}$ peak median profiles of the CHEX–MATE sample. Its
dispersion is shown with the black solid line and the grey envelope. The blue
and red solid lines represent the median of the profiles derived from the
morphologically relaxed and disturbed objects, respectively. Bottom-left
panel: Ratio of the median of the morphologically relaxed and disturbed
$\widetilde{\mathrm{EM}}$ profiles over the median of the full CHEX–MATE
sample. The same colour coding as above is used. The dotted lines represent
0.8 and 1.2 values and are shown for reference. Top-right panel: Same as the
left panel except the median of the azimuthal mean profiles is shown. The
median is indicated with the dotted black line. The median of the X-COP
(Ghirardini et al. 2019) and REXCESS (Croston et al. 2008)
$\widetilde{\mathrm{EM}}$ profiles are shown with green and orange solid
lines, respectively. Bottom-left panel: Ratio between the median of X-COP and
REXCESS samples to the CHEX–MATE median. The REXCESS profiles were extracted
performing an azimuthal average. For this reason, we show the ratio between
the median of the REXCESS profiles and the median of the CHEX–MATE azimuthal
mean profiles.

In Fig. 2, we show the behaviour of the CHEX–MATE $\widetilde{\mathrm{EM}}$
median as well as the medians of the mass and redshift sub-samples. In the
left panel of Fig. 8, we compare the medians of the relaxed and disturbed sub-
samples whose individual profiles are shown in Fig. 3. The former is
approximately two times greater than the median of the whole sample at $R\sim
0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and is not within the dispersion. The
morphologically disturbed clusters are on average within the dispersion, being
$70\%$ smaller than the whole sample median at
$R<0.2R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The morphologically relaxed
profiles become steeper than the disturbed profiles at
$R>0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. A similar behaviour has been
observed in several works, such as Arnaud et al. (2010), Pratt et al. (2010),
Maughan et al. (2012), and Eckert et al. (2012) (cfr. Figure 4), when
comparing cool core systems with non-cool core systems.

Combining these results with those of Fig. 2, we concluded that CHEX–MATE Eq.
5 provides reasonable mass normalisation and captures the evolution of the
cluster population well. The large sample dispersion seen in the cluster cores
is linked to the variety of morphologies present in the sample. The medians of
the relaxed and disturbed sub-samples indeed differ by more than a factor of
ten at R$<0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. At around
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, we also noticed some different
behaviours in our sub-sample: The most massive objects are approximately 25%
larger than the least massive ones, and the morphologically disturbed clusters
are 50% larger than the relaxed ones (see also Sayers et al. 2022).

###  5.4 Comparison with other samples

In this section, the statistical properties of the CHEX–MATE profiles are
compared to SZ and X-ray selected samples at $z\lesssim 0.3$ with similar mass
ranges in order to investigate the impact of different selection effects. The
SZ-selected sample is the XMM-Newton Cluster Outskirts Project (X-COP; Eckert
et al. 2017) sample that contains 12 SZ-selected clusters in the [0.05-0.1]
redshift range and has a total mass range similar to CHEX–MATE but with a
greater median mass ($\sim 6\times 10^{14}$M⊙). The individual
$\widetilde{\mathrm{EM}}$ profiles for X-COP were computed using the same
procedure as described in this work. The profiles were scaled by applying
Equation 5, with $T_{\mathrm{avg}}$ given by Equation 1, using the masses
presented in Table 1 of Ettori et al. (2019).

We also compare the CHEX–MATE profile properties to the X-ray selected
Representative XMM-Newton Cluster Structure Survey (REXCESS; Böhringer et al.
2007) sample, which is composed of 31 X-ray selected clusters in the
[0.05-0.3] redshift range, with a mass range spanning the [1-8]$\times
10^{14}$M⊙ and the median mass being $2.7\times 10^{14}\,$M⊙. The REXCESS
$\widetilde{\mathrm{EM}}$ profiles were obtained from the surface brightness
presented in Appendix A of (Croston et al. 2008). These profiles were computed
using the azimuthal average in each annulus. For this reason, we compare the
REXCESS profiles with the mean CHEX–MATE profiles. The REXCESS profiles were
scaled using Equation 5 with $T_{\mathrm{avg}}$ from Pratt et al. (2009).

The median and its dispersion for each of these samples were computed using
the procedure described above, and their comparison with CHEX–MATE is shown in
the right panel of Fig. 8. Both sample medians present an overall good
agreement that is within $10\%$ at $R>0.2R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$.
The X-COP median is $25\%$ more peaked in the central regions at
$R<0.2R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ with respect to both CHEX–MATE and
REXCESS. Nevertheless, the X-COP median is well within the dispersion of the
CHEX–MATE sample and variations of such order are expected in the core where
the $\widetilde{\mathrm{EM}}$ values are comprised in the wide range
$\sim[6,30]\times 10^{-5}$ cm -6 Mpc.

Figure 9: Top panel: Simx $\widetilde{\mathrm{EM}}$ profiles and their median.
The profiles are shown with grey solid lines and their median with a black
dashed line. The Simx profiles were extracted from a randomly chosen line of
sight for each cluster. The red solid line identifies the median of the
CHEX–MATE sample. The black solid line is the median of the CHEX–MATE sample
assuming a 20% bias on the mass (i.e. each profile has been scaled for
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}/0.8^{1/3}$; see Sect. 4 for details).
Bottom panel: Ratio of the median of the CHEX–MATE with and without correction
for hydrostatic bias and the median of the Simx simulations. The CHEX-MATE
median with the correction is shown with the red line and the median without
it is shown with the black line.

Figure 10: Results of the fit of the CHEX–MATE $\widetilde{\mathrm{EM}}$
profiles using broken power laws. Left panel: Fit of the median CHEX–MATE
$\widetilde{\mathrm{EM}}$ profiles shown with grey lines with a power law in
four radial bins. The bins are $[0.2-0.4],[0.4-0.6],[0.6-0.8]$, and $[0.8-1]$
and are in units of $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. For each radial bin,
the black solid line represents the best fit of the power law shown in Eq. 6.
The magenta envelope was obtained considering the dispersion of the fitted
parameters (A and $\alpha$ in Eq. 6). The blue and red solid lines represent
the best fit of the profiles of the morphologically relaxed and disturbed
clusters, respectively. Right panel: Median values of the power law indexes,
$\alpha$, obtained from the fit of the CHEX–MATE, X-COP, REXCESS, and Simx
samples in the four radial bins shown in the left panel. For each value we
report its dispersion.

###  5.5 Comparison with simulations

The 115 Simx $\widetilde{\mathrm{EM}}$ profiles extracted from random
projections for each cluster are shown together with their median value in
Fig. 9. The median of the CHEX–MATE sample is also shown. Overall, the CHEX-
MATE median is flatter than the medians of Simx in the
[0.06-1]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range, and specifically it
is $\sim 50\%$ smaller in the centre and $\sim 50\%$ larger in the outskirts.
Part of the difference in the external regions might be caused by the re-
scaling of the observational sample. Indeed, each CHEX–MATE profile has been
scaled using the $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ derived from
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, which is expected to be biased low by
20%. Factoring in this aspect, a more proper re-scaling should be done with
respect to $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}/(0.8^{1/3})$. The agreement
between the CHEX–MATE data and the simulations increases at
$R>0.5R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, with relative variations of about
40% in the [0.2-1] $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. These
considerations do not have any repercussion on the central regions, which
remain larger in the simulated profiles, confirming the results found in
Campitiello et al. (2022) and Darragh-Ford et al. (2023).

Providing precise measures of the mass of the observed clusters is one of the
goals of the CHEX–MATE collaboration. For this paper, it is sufficient to
prove that The Three Hundred clusters have a profile in reasonable agreement
with the observed sample in order to employ them for the study of the
$\widetilde{\mathrm{EM}}$ scatter.

###  5.6 Measuring the slopes

We measured the slopes of the CHEX–MATE $\widetilde{\mathrm{EM}}$ profiles
adopting the technique described in Section 3.1 of Ghirardini et al. (2019).
Briefly, we considered four radial bins in the [0.2-1]$\times
R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range and with widths equal to
$0.2R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. We excluded the innermost bin
[0.-0.2]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ because of the very high
dispersion of the profiles within this region. We measured the slope,
$\alpha$, and normalisation, $A$, of each profile by performing the fit within
each radial bin using the following expression:

$Q(x)=Ax^{\alpha}e^{\pm\sigma_{int}},$ (6)

where $x=R/R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and $\sigma_{int}$ is the
intrinsic scatter. The error on each parameter was estimated via a Monte Carlo
procedure, producing 100 realisations of each profile. The left panel of Fig.
10 shows the power law computed using the median of the $\alpha$ and A within
each radial bin.

The fit of the [0.2-0.4]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ bin revealed that
there is a striking difference between the morphologically relaxed and
disturbed objects. This result is notable because the considered region is far
from the cooling region at $\sim 0.1-0.15R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$.
The median power law index of the morphologically relaxed object profiles is
$\alpha_{rel}=2.57\pm 0.15$ and is not consistent with the morphologically
disturbed one, which is $\alpha_{dis}=1.37\pm 0.2$ at more than the $3\sigma$
level. That is, the shape of the most disturbed and relaxed objects differ at
least up to $0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. However, the fitted power
law is within the dispersion of the full sample, whose median index is
$\alpha=2.02\pm 0.36$. The median values in the
$[0.4-0.6]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ region are $\alpha_{rel}=3\pm
0.22$ and $\alpha_{dis}=2.3\pm 0.2$ for the morphologically relaxed and
disturbed objects, respectively. The indexes are consistent at the $2\sigma$
level, implying that the profiles are still affected by the morphology in the
centre. The overall scenario changes at
$R>0.6R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The power law index of the
morphologically relaxed and disturbed objects are consistent with the median
obtained from fitting the whole sample. Ettori & Balestra (2009) found that
the average slope of a sample of 11 clusters at $0.4R_{200}(\sim
0.6R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$) and $0.7R_{200}(\sim
R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$) is $3.15\pm 0.46$ and $3.86\pm 0.7$,
respectively. These values are consistent within $1\sigma$ with our
measurements.

We show in the right panel of Fig. 10 the comparison between the median
$\alpha$ computed from the CHEX–MATE sample in each radial bin with the same
quantity obtained using REXCESS and X-COP. There is an excellent agreement in
all the considered radial bins. Interestingly, there is also a good agreement
in the shape between a sample selected in X-ray (REXCESS) and SZ. One could
expect to see more differences in the central parts, as X-ray selection should
favour peaked clusters.

The comparison with the median $\alpha$ obtained using the Simx profiles is
also shown in the right panel of Fig. 10. The Simx median $\alpha$ is
systematically greater than the median of the observed sample. As discussed in
Sect. 5.5, the bias introduced by using $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$
might play a role when comparing CHEX–MATE to Simx and partly contributes to
this systematic difference. However, we stress the fact that the slopes are
consistent within $1\sigma$ in the four radial bins.

Campitiello et al. (2022) find similar results when comparing the
concentration of surface brightness profiles within fixed apertures of
simulated and CHEX–MATE clusters. This quantity measures how concentrated the
cluster core is with respect to the outer regions (i.e. more concentrated
clusters show a steeper profile). The concentration of the simulations is
systematically higher by approximately $20-30\%$ (cfr. Table 1 of Campitiello
et al. 2022).

##  6 The EM radial profile scatter

###  6.1 Computation of the scatter

Departures from self-similarity are linked to individual formation history as
well as non-gravitational processes such as AGN feedback (outflows, jets,
cavities, shocks) and feeding (cooling, multi-phase condensation; e.g. Gaspari
et al. 2020). The additional terms used to obtain the
$\widetilde{\mathrm{EM}}$ profiles can partly account for these effects. The
scatter of these profiles offers the opportunity to quantify such departures,
and the CHEX–MATE sample is ideal to achieve this goal since the selection
function is simple and well understood.

We computed the intrinsic scatter of the CHEX–MATE radial profiles by applying
the following procedure. First, we interpolated each scaled profile on a
common grid formed by ten logarithmically spaced radial bins in the
$[0.05-1.1]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. We used the model
for which the observed distribution of the points, $S_{obs}$, in each radial
bin is the realisation of an underlying normal distribution, $S_{true}$, with
log-normal intrinsic scatter, $\sigma_{int}$:

$\displaystyle\ln\mathrm{S_{true}}\sim\mathrm{Normal}(\ln\mu,\sigma_{\mathrm{int}}),$
(7)

with $\mu$ as the mean value of the distribution. We set broad priors on the
parameters we are interested in:

$\displaystyle\ln\mu\sim\mathrm{Normal}(\ln\langle\mathrm{\widetilde{EM}(r)}\rangle,\sigma=10),$
(8) $\displaystyle\sigma_{int}\sim\mathrm{Half-Cauchy}(\beta=1.0),$ (9)

where $\langle\mathrm{\widetilde{EM}(r)}\rangle$ is the mean value of the
interpolated EM profiles at the radius r. We assumed a Half-Cauchy
distribution for the scatter, as this quantity is defined as positive. Since
$\sigma(\ln X)=\sigma(X)/X$, the intrinsic scatter in linear scale becomes:

$\sigma_{lin}=\sigma_{int}*\mu,$ (10)

and the total scatter, $\sigma_{tot}$, is the quadratic sum of $\sigma_{lin}$
and the statistical scatter $\sigma_{stat}$:

$\sigma_{tot}=\sqrt{\sigma_{lin}^{2}+\sigma_{stat}^{2}}.$ (11)

The observed data were then assumed to be drawn from a normal realisation of
the mean value and total scatter:

$\mathrm{S_{obs}}\sim\mathrm{Normal}(\mu,\sigma_{\mathrm{tot}}).$ (12)

We determined the intrinsic scatter $\sigma_{int}$ and its $1\sigma$ error by
applying the No U-Turn Sampler (NUTS) as implemented in the Python package
PyMC3 (Salvatier et al. 2016) and using 1,000 output samples.

Figure 11:  Scatter of the $\widetilde{\mathrm{EM}}$ CHEX–MATE, X-COP, and
REXCESS profiles. Top-left panel: Comparison between the scatter of the
CHEX–MATE sample and the morphologically selected sub-samples. Black dotted
lines identify the $\pm 1\sigma$ scatter between the $\widetilde{\mathrm{EM}}$
profiles of the CHEX–MATE sample. The scatter between the profiles of
morphologically relaxed and disturbed clusters are shown with blue and red
envelopes, respectively. The width of the envelopes corresponds to the
$1\sigma$ uncertainty. Top-right panel: Comparison between the scatter of the
CHEX–MATE sample and the mass selected sub-samples. Green and magenta
envelopes represent the scatter between the $\widetilde{\mathrm{EM}}$ profiles
of the low- and high-mass sub-samples, respectively. Bottom-left panel:
Investigation of evolution of the scatter. Blue and red envelopes represent
the scatter between the profiles of the hi-mass clusters,
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}>5\times 10^{14}M_{\odot}$, of the low- and
high- redshift samples, respectively. Bottom-right panel: Comparison between
the scatter of the CHEX–MATE with X-COP (Ghirardini et al. 2019) and REXCESS
(Croston et al. 2008) samples. The X-COP and REXCESS scatters are shown with
green and orange envelopes, respectively. We recall that the scaling of the
X-COP and REXCESS profiles was performed using temperatures obtained
differently than those for CHEX–MATE, see Section 5.4 for details.

Our sample contains nine objects for which we were not able to measure the
profile above $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. Six of these objects are
less massive than $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}\lesssim 4\times
10^{14}$M⊙ and are classified as ”mixed morphology” objects. We investigated
the impact of excluding these profiles from the computation of the scatter by
comparing the scatter computed within $0.9R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$
using the full sample with the scatter computed excluding the nine objects. We
noticed that this exclusion reduces the scatter by a factor of
approximately$15\%$ at $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ starting from
$\sim 0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. We argue that the reduction of
the scatter is linked to the fact that the nine clusters contribute positively
to the total scatter being morphologically mixed. For this reason, we
corrected for this effect, defining a correction factor, $cf$, that quantifies
the difference in the scatter due to the exclusion of these profiles. We
computed the ratio between the scatter including and excluding the nine
profiles in the [$0.06-0.9R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$] radial range,
where we extracted the profiles for the whole CHEX–MATE sample. We fitted this
ratio via the mpcurvefit routine using a two degree polynomial function of the
form $cf(r)=ar^{2}+br+c$ and obtained the coefficients
$[a,b,c]=[0.410,-0.117,0.993]$. We multiplied the scatter of the whole sample
by $cf$ in the [$0.06-1]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. From
hereon, we refer to this scatter as the ”corrected intrinsic scatter”.

###  6.2 The CHEX–MATE scatter

The corrected intrinsic scatter of the $\widetilde{\mathrm{EM}}$ profiles is
reported in the top-left panel of Fig. 11. The scatter computed using the
profiles centred either on the centroid or on the peak gave consistent
results.

The intrinsic scatter of the scaled $\widetilde{\mathrm{EM}}$ profiles
substantially depends on the scale considered. In the central regions, the
large observed scatter of $\sim 0.8$, at $R\sim
0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ reflects the complexity of the cluster
cores in the presence of non-gravitational phenomena, such as cooling and AGN
feedback. On top of that, merging events are known to redistribute gas
properties between the core and the outskirts, which flattens the gas density
profiles in cluster cores. The scatter reaches a minimum value of $\sim 0.2$
in the [0.3-0.7]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range, where the
scatter remains almost constant. This result confirms the behaviour observed
in the left panel of Fig. 8, where the dispersion of the profiles shown is
minimal in this radial range and the scaled profiles converge to very similar
values. The scatter increases at $R>0.7R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$
from 0.2 to 0.35.

The scatter of the morphologically disturbed and relaxed clusters considered
separately are shown in the top-left panel of Fig. 11. The scatter of the
morphologically disturbed clusters is higher but consistent at
$R<0.3R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ with the relaxed one. This is
expected, as the scatter originates from the combination of non-gravitational
processes in the core and merging phenomena. This reinforces the scenario in
which the differences between the $\widetilde{\mathrm{EM}}$ profiles of
relaxed and disturbed objects disappear in cluster outskirts, as already shown
with the study of the shapes in Sect. 5. The dependency of the scatter on
cluster mass was identified by comparing the scatter between high- and low-
mass objects, as shown in the top-right panel of Fig. 11. No significant
differences could be seen. We investigated the evolution of the scatter by
comparing the most massive clusters,
$M_{500}^{\mathrm{Y_{\textrm{S}Z}}}>5\times 10^{14}M_{\odot}$, in the low- and
high-redshift samples. This is shown in the bottom-left panel of the figure,
and as for the mass sub-samples, we found no significant differences except in
the very inner core at $R<0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, where the
local objects indicate larger variation.

Figure 12: EM maps of two of the simulated clusters used in this work
projected along the lines of sight X, Y, and Z. The top row shows a cluster
whose morphology appears roundish in the three projections considered. On the
bottom we show, on the contrary, a cluster whose morphology is particularly
complex and appears different in each of the three projections. We refer to
the cluster in the top row as regular and the latter as irregular. The white
circle indicates $R_{500}$.

###  6.3 Comparison with other samples

We computed the scatters of the profiles of the REXCESS and X-COP sample
following the same procedure we used for the CHEX–MATE sample. These are shown
in the bottom-right panel of Fig. 11. The width of the envelope corresponds to
1$\sigma$ uncertainty. Overall, the CHEX–MATE, REXCESS, and X-COP scatters are
consistent at the [0.07-0.6] $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial
range. This excellent agreement is due to the fact that the samples are
representative of the wide plethora of $\widetilde{\mathrm{EM}}$ profile
shapes in the core of clusters. There is slight disagreement at a larger scale
between the samples, with the CHEX–MATE scatter being lower at more than
$1\sigma$, which could be do to the re-scaling. This is one important issue
that will be investigated in forthcoming papers recurring also to multi-
wavelength data.

##  7 Investigating the origin of the scatter

###  7.1 Simulation scatters

In this section, we turn our attention to the The Three Hundred dataset. The
cosmological simulations allowed us to break down the sample scatter, or total
scatter, into two components: the genuine cluster-to-cluster scatter, which
would be the sample scatter measured between the true 3D profiles of the
objects, and the projection scatter. The latter measures the differences that
various observers across the Universe would detect when looking at the same
object from distinct points of view.

In this work, we scaled the CHEX–MATE EM profiles using the results of Pratt
et al. (2022) and Ettori et al. (2022), which were derived using empirical ad-
hoc adaptation of the self-similar scaling predictions. However, the same
scaling is less suitable for the simulations, which agree better with the
self-similar evolution of Eq. 4 since this expression minimises their scatter.
For this reason, all the scatters presented from this point on were derived
from EM profiles scaled assuming only self-similar evolution both for The
Three Hundred and CHEX–MATE samples.

###  7.2 The projection scatter term

The evaluation of this term requires the knowledge of the 3D spatial
distribution of the ICM. A perfectly spherical symmetric object would appear
identical from all perspectives, and the projection scatter would be equal to
zero. On the other hand, an object whose ICM spatial distribution presents a
complex morphology will produce a large projection scatter. This can be
visualised by looking at the three EM maps obtained for three orthogonal lines
of sight for two objects of the The Three Hundred collaboration in Fig. 12. In
detail, the cluster shown in the top rows is roundish and does not show
evident traces of merging activity within a radius of R=R500 (white circle).
The cluster in the bottom row, however, exhibits a complex morphology due to
ongoing merging activities and the presence of sub-structures, which cause it
to appear different in the three projections.

This complexity is reflected in the projection scatters shown in Fig. 13,
which was computed considering the 40 lines of projection for the two objects
and not only the three shown with the images. The scatters are similar within
$R<0.2R_{500}$. At R¿0.4R500, the irregular cluster scatter diverges, while
the one of the regular object remains almost constant. In particular, in the
case of the irregular object, $\sim 0.4R_{500}$ corresponds to the position of
the big sub-structure visible in the bottom row of the left panel of Fig. 12.
Interestingly, the Simx projection scatter increases rapidly at R$\sim
0.8R_{500}$ also for the regular cluster, while the Sim one remains mostly
constant. This difference in the behaviour is due to the deliberate 7% over-
and underestimated background correction explained in Section 4.2. The over-
and underestimation of the background yields profiles that are steeper or
flatter than the correct profiles, respectively, and hence they increase the
scatter between the profiles. This effect is particularly important at R$\sim
R_{500}$ because the cluster signal reaches the background level.

We calculated the projection scatter between the 40 projections for each of
the 115 objects from The Three Hundred sample, and these profiles are shown in
Fig. 14. On average, the scatter starts from a value of $0.15$ at $R\sim
0.1R_{500}$ and then reaches the value of $0.3$ at $R_{500}$, with a rapid
increase from 0.2 to 0.3 at $R\sim 0.9R_{500}$. This rapid increase is due to
the complex spatial ICM distribution at large radii. There are approximately
five outliers that exhibit a larger scatter from the envelope and a complex
behaviour. These clusters are characterised by the presence of sub-structures
that happen to be behind or in front of the main halo of the cluster along
some lines of projection. For this reason, the sub-structure emission is not
visible as it blends with the emission from the core of the cluster. On the
contrary, if the sub-halo is on a random position as respect to the main halo
it will appear as a sub-structure in different position depending on the
projection. In this case, the cluster morphology is complex. For these
reasons, the resulting profile for these clusters can show remarkable
differences depending on the line of sight.

Figure 13: Projection scatter of the regular and irregular clusters in Fig.
12. The scatters are shown with black and red solid lines, respectively. The
black and red envelops represent the dispersion. The black and red dotted
lines refer to the projection scatter computed using the Simx profiles.

###  7.3 The total scatter term

Figure 14: Comparison between the total and projection scatters of the Simx
profiles. The scatters are shown with solid green and magenta lines,
respectively.

The total scatter term measures the differences between the cluster EM
profiles within a sample. We recall that each of our simulated objects is seen
along 40 lines of sight. With this possibility in hand, we created 40
realisations of the same sample of 115 objects and computed the scatter for
each realisation. The 40 total scatters of the Simx profiles are shown in Fig.
14 .

The average high value of $0.9$ at $R<0.3R_{500}$ of the total scatter
captures the wide range of the profile shapes within the inner core. The
scatter reaches its minimum value of $\sim 0.4$ at $R\sim
0.5R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and then rapidly increases afterwards,
due to the presence of sub-structures in the outskirts and the phenomena
related to merging activities as well as the background subtraction discussed
in Section 7.2.

###  7.4 Comparison between total and projection

Direct comparison of the projection and total scatter terms in the simulated
sample allowed us to investigate the origin of the scatter as predicted by
numerical models. The two scatters are shown in Fig. 14. The total scatter is
almost eight times greater than the projection at $R\sim 0.1R_{500}$ and
rapidly decreases to be only two times greater at $R\sim 0.3R_{500},$ as shown
in Fig. 14. This indicates that differences between clusters dominate with
respect to the variations from the projection along different lines of sight
at such scale.

The total scatter is only $20\%$ greater than the projection term in the
$[0.4-0.8]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. This scale is
where the cluster differences are smaller. At
$R>0.8R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, both scatters increase, implying
that merging phenomena and sub-structures are impacting the distribution of
the gas. Furthermore, we argue that the deliberate background over- and under-
subtraction discussed in Section 7.2 contributes to increasing both scatter
terms by enlarging the distribution of the profiles where the signal of the
cluster reaches the background level. The total scatters obtained using the
Sim are similar to the ones obtained using Simx up to $0.9R_{500}$ but remain
below $\sim 0.45$ at $R_{500}$.

###  7.5 Simulation versus observations

We can break down the contributions to the scatter in the CHEX–MATE sample by
using the numerical simulation scatter terms as a test bed. The Simx total and
projection scatter medians and their dispersion are shown in Fig. 15. The
CHEX–MATE scatter dispersion is also shown in the same figure. We recall that
it is computed using the EM profiles scaled according to the self-similar
model using Equation 2 and is greater than the one shown in Fig. 11 due to the
residual dependency on mass and redshift discussed in Section 3.3.2 in the
$[0.2-0.8]$ radial range. However, the scatters reach the value of
approximately $0.4$ at $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, indicating that
the differences between the profiles are dominated by clumpy patches in the
ICM distribution due to sub-halos and filamentary structures.

The CHEX–MATE and Simx total scatters are in excellent agreement at
R$<0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and marginally consistent within
$2\sigma$ in the [0.1-0.3] radial range. Generally speaking, they exhibit the
same behaviour, rapidly declining from the maximum value of the scatter of 1.2
to 0.4. The projection scatter on the other hand is at a minimum value of 0.1
and is almost constant up to $0.3R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. This
result implies that the observed scatter between the ${\mathrm{EM_{S}}}$
profiles within $0.5R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ is dominated by
genuine differences between objects and not by the projection along one line
of sight, as explained in Sect. 7.4. In other words, we are not limited by the
projection on the plane of the sky when studying galaxy cluster population
properties at such scales.

The total and CHEX–MATE scatters reach the minimum value, approximately $0.4$,
in the [$0.3-0.5]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range and remain
almost constant within these radii. This minimum value quantifies the narrow
distribution of the profiles shown in Fig. 3 and Fig. 6. Furthermore, the
slopes between morphologically relaxed and disturbed objects become consistent
at such radii, as shown in the right panel of Fig. 10. This suggests that the
differences between EM profiles are minimum at such intermediate scales
despite their morphological statuses, mass or redshift. As for the inner
regions, the projection term increases mildly in these regions and provides a
small contribution.

The The Three Hundred scatters rapidly increase at
$R>0.5R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, with the total scatter reaching the
value of approximately $0.7$ at $R_{500}$. The projection scatter reaches the
CHEX–MATE scatter at $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. We argue that this
effect is due to a combination of not masking the sub-structures when
extracting the Simx profiles and the deliberate wrong background subtraction
discussed in Sect. 7.2. Indeed, the use of median profiles reduces this
effect, and we discuss this effect in detail in Appendix A, where we show that
the use of azimuthal median profiles is efficient for removing part of these
spatial features. The fact that the scatter terms increase in a similar manner
despite the use of median profiles reinforces that this behaviour is likely
related to analysis techniques rather than genuine differences within the
profiles and projection effects.

Figure 15: Comparison between the scatter of the CHEX–MATE sample and the
total and projection terms of the simulations. The CHEX–MATE scatter is shown
using dashed black lines. The medians of the total and projection terms are
represented with green and magenta solid lines, respectively. Their 68%
dispersions are represented using envelopes coloured accordingly.

##  8 Discussion and conclusion

We have studied the properties of the SX and EM radial profiles of the
CHEX–MATE sample, which comprises 116 SZ selected clusters observed for the
first time with deep and homogeneous XMM-Newton observations. Our main
findings are as follows:

  * •

The choice of making the centre between the peak and the centroid for
extraction of the SX profiles yields consistent results in the
[0.05-1]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. Significant
differences can be seen within $\sim 0.05R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$.

  * •

The use of azimuthal average and median techniques to extract the profiles
impacts the overall profile normalisation by a factor of $5\%$ on average. The
shape is mostly affected at $R>0.8R_{500}^{\mathrm{Y_{\textrm{S}Z}}},$ with
azimuthal averaged profiles being flatter at this scale.

  * •

The EM profiles exhibit a dependency on the mass and a mild dependency on
redshift, which is not accounted for by the computed scaling according to the
self-similar scenario, as found also by Pratt et al. (2022) and Ettori et al.
(2022).

  * •

Morphologically disturbed and relaxed cluster $\widetilde{\mathrm{EM}}$
profiles have different normalisations and shapes within $\sim
0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The differences at larger radii are on
average within $10\%$ and are consistent within the dispersion of the full
sample.

  * •

The shape and normalisation of the $\widetilde{\mathrm{EM}}$ profiles present
a continuum distribution within the
[0.2-0.4]$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range. The extreme cases
of morphologically relaxed and disturbed objects are characterised by power
law indexes, $\alpha=2.51\pm 0.13$ and $\alpha=1.38\pm 0.2$, respectively,
that are not consistent at the $3\sigma$ level. The picture changes at
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}>0.4$, where the slopes of these extremes
becomes marginally consistent at $1\sigma$ in the
$[0.4-0.6]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial bin. The slopes in the
last bin are in excellent agreement.

  * •

The scatter of the CHEX–MATE sample depends on the scale. The scatter maximum
is $\sim 1.1$ within $0.3R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, reflecting the
wide range of profile shapes within the cluster cores that range from the flat
emission of disturbed objects to the peaked emission of the relaxed clusters.
The scatter decreases towards its minimum value, 0.2, at
$0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and increases rapidly to 0.4 at
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. This result is coherent with the overall
picture of a characteristic scale, R$\sim
0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, at which the differences between
profiles in terms of shape and normalisation are minimum. The increase of the
scatter at $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ is expected, as this is the
scale at which merging related phenomena and patchy distribution of the ICM
become important.

  * •

The scatters of the morphologically relaxed cluster and the disturbed cluster
are different within $0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, the former being
smaller. Above this radius, they are in excellent agreement between themselves
and with the entire sample as well, implying that the properties of EM
profiles in the outer parts are not affected by the properties in the core.
There are no differences in the scatter of the sub-samples formed by high- and
low-mass objects, and we found no evolution of the scatter for high-mass
objects.

The overall emerging picture is that there is a characteristic scale, R$\sim
0.4R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, where the differences between profiles
in terms of shape and normalisation are minimum. The exceptional data quality
has allowed us to provide to the scientific community the scatter of SX and
$\widetilde{\mathrm{EM}}$ radial profiles of a representative cluster sample
with an unprecedented precision of approximately $5\%$.

The results from observations were compared to a sample drawn from the
numerical simulation suite The Three Hundred formed by 115 galaxy clusters
selected to reflect the CHEX–MATE mass and redshift distribution. For each
cluster, we computed the EM along 40 randomly distributed lines of sight,
which allowed for the investigation of projection effects for the first time.
Our main findings can be summarised as follows:

  * •

The properties derived using the Sim or the Simx profiles are similar within
$R_{500}$, confirming the statistical quality of the mock X-ray images, which
were calibrated to match the CHEX–MATE average statistical quality.

  * •

The simulation $\widetilde{\mathrm{EM}}$ profiles appear systematically
steeper than those from observations. The hydrostatic bias might play a key
role in explaining this difference. The scaling of the CHEX–MATE profiles by
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}/(0.8^{1/3})$, assuming a $20\%$ bias,
alleviates these differences, and the ratio between the CHEX–MATE and
simulation medians becomes closer to one, with the exception of the centre
where simulations typically have a greater gas density.

  * •

The total scatter of the simulation sample follows the same behaviour as that
of the observations up to $0.6R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and then
increases more rapidly to an average value of approximately $0.7,$ whereas the
observation reaches the value of $0.4$ at
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The comparison with the projection
scatter at such scales hints at a contribution from projection effects on the
order of $0.3$.

  * •

The projection scatter allowed us to study the spherical symmetry of clusters.
This term slightly increases from approximately $0.1$ at
$0.1R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ up to approximately $0.3$ at $\sim
R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$, exhibiting a rapid gradient at
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. This term is smaller than the total in
the entire $[0.1-0.9]R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial range
considered, and its dispersion is on the order of $10\%$. This implies that
the difference we observe between objects is due to a genuine difference in
the gas spatial distribution.

  * •

The background subtraction process becomes crucial at $R_{500}$ for
determining of the profile shape at $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The
deliberate over- or underestimation significantly contributes to increasing
both the total and projection scatter at such large scales. Furthermore, the
rapid increase of both scatters can be also explained by the fact that sub-
structures are not masked in simulated images.

The large statistics offered by the simulation dataset allowed us for the
first time to investigate the origin of the scatter and break down the
components, namely the projection and total terms, and study them as a
function of $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. The overall picture emerging
is that there are three regimes amongst the scatter:

  * •

$[0.1-0.4]$: The differences between profiles are genuinely due to a different
distribution of the gas and also influenced by feedback processes and their
implementation (see, e.g., Gaspari et al. 2014), which translates into a
plethora of profile shapes and normalisations.

  * •

$[0.4-0.6]$: In this range, the scatter is sensitive to the scaling applied,
suggesting that this is the scale where clusters are closer to being within
the self-similar scenario.

  * •

$[0.6-1]$: The CHEX–MATE scatter and the total scatter increase at such scales
and are greater by a factor of approximately two than the projection, showing
that profile differences are genuine and not due to projection effects. The
emission of sub-structures and filamentary structures and the correct
determination of the background play a crucial role in determining the shape
of the profiles at such scales.

We were able to investigate the origin of the scatter by combining the
statistical power of the CHEX–MATE sample not only because of the great number
of objects observed with sufficient exposure time to measure surface
brightness profiles above $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ but also
because of the sample’s homogeneity and the uniqueness of the simulation
sample. The latter allowed us to discriminate the scatter due to genuine
differences between profiles and those related to projection. The CHEX–MATE
sample allowed us to measure the scatter up to
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ with the sufficient precision to clearly
discriminate the contribution from the projection term at all scales.

###### Acknowledgements.

The authors thank the referee for his/her comments. We acknowledge financial
contribution from the contracts ASI-INAF Athena 2019-27-HH.0, “Attività di
Studio per la comunità scientifica di Astrofisica delle Alte Energie e Fisica
Astroparticellare” (Accordo Attuativo ASI-INAF n. 2017-14-H.0), and from the
European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant
agreement n. 871158). This research was supported by the International Space
Science Institute (ISSI) in Bern, through ISSI International Team project #565
(Multi-Wavelength Studies of the Culmination of Structure Formation in the
Universe). The results reported in this article are based on data obtained
with XMM-Newton, an ESA science mission with instruments and contributions
directly funded by ESA Member States and NASA. GWP acknowledges financial
support from CNES, the French space agency.

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##  Appendix A The impact of sub-structures in simulations

The presence of sub-structures within the region of extraction of the radial
profiles modifies the shape of the surface brightness and emission measure
profiles. This translates into an increase of the scatter between them. In
this work, we are interested in the distribution of the gas within the cluster
halo filtering the contribution of sub-structures whose emission is detectable
within or near $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. This filtering is
achieved by masking the sub-structures in observations. The same procedure is
difficult to apply to simulations.

Generally speaking, automatic detection algorithms in X-ray analyses are
calibrated to detect point source emission only, as the detection of extended
sources would cause the algorithm to also detect the cluster emission itself.
For this reason, the identification of extended emission associated to sub-
structures is done via eye inspection, but this approach cannot be taken with
large datasets comprised of thousands of maps, such as the one we used in this
work. The fact that we do not mask sub-structures in the simulated maps
constitutes one of the main differences between the X-ray analysis and The
Three Hundred analysis. However, we could qualitatively investigate the impact
of sub-structures on the scatter by comparing the results obtained following
the procedures of Section 7 that used the azimuthal average and median
profiles shown in Fig. 16.

Figure 16: Comparison between the total scatters computed using the azimuthal
mean and median profiles. Top: Medians of the total from The Three Hundred
scatters computed using the azimuthal mean and median EM profiles. These are
shown with green and grey lines, respectively. The $68\%$ dispersion is shown
with the coloured envelopes. Bottom: Ratio between the median of the total
scatters computed using the azimuthal averaged profiles over the median
computed using the azimuthal median profiles. The dashed-dotted lines indicate
the identity line and the $\pm 20\%$ levels.

The bottom panel shows that the scatters are nearly identical, within
$0.2R_{500}$, and are on average around the $20\%$ level in the
[0.2-0.6]$R_{500}$ radial range. The scatter of the mean profiles increases
rapidly above that radius. The same behaviour is observed for the scatter of
the median even if the increase is less rapid, as shown by the ratio in the
bottom panel at R$>0.6R_{500}$.

The azimuthal median in a given annulus does not completely remove the
emission from the extended sub-structure, which can only be achieved by
masking it. However, we argue that the scales at which the sub-structures
become important, R$>0.6R_{500}$, correspond to annuli whose size is typically
larger than the size of a sub-halo. For this reason, the azimuthal median is
marginally affected.

Indeed, sub-structure masking is a key difference between observations and
simulations, and it does affect the computation of the total scatter. However,
we suggest that using the median profiles is an effective way to reduce the
impact of sub-structures at the scales at which they are important. For this
reason, the rapid increase of the total scatter at $\sim R_{500}$ is more
likely to be due to a genuine difference between the profiles and to the
background subtraction effect discussed in Section 7.2.

##  Appendix B ROSAT-XMM-Newton background relation

Figure 17: Calibration of the sky background count rate between XMM-Newton and
ROSAT-PSPC. Left panel: Relation between the sky background count rate as
measured using the XMM-Newton mean SX profiles and ROSAT-PSPC in the R5,
[0.56-1.21] keV, energy band. The black points represent the clusters for
which $R_{200}^{\mathrm{Y_{\textrm{S}Z}}}$ is less than 12 arcmin and that
have been used to fit the relations. The grey points are the clusters filling
the field of view, their $R_{200}^{\mathrm{Y_{\textrm{S}Z}}}$ being greater
than 12 arcmin. The solid line represents the cross correlation obtained via
the linear regression. The dashed lines represent the intrinsic scatter of the
relation. Right panel: Same as the left panel except for the fact that the
XMM-Newton count rate is measured using the median SX profiles.

The determination of the sky background level was performed in a region free
from cluster emission. In this work, we used the annular region between
$R_{200}^{\mathrm{Y_{X}}}$ and 13.5 arcminutes to measure the photon count
rate associated with the sky background. We considered that we had sufficient
statistics for the background estimation if the width of this region is at
least 1.5 arcmin (i.e. $R_{200}^{\mathrm{Y_{\textrm{S}Z}}}<12$ arcmin). The
$R_{200}^{\mathrm{Y_{\textrm{S}Z}}}$ of nearby clusters at z $\lesssim 0.2$
are generally larger than $12$ arcmin, and it was not possible to define a sky
background region unless offset observations were available. For this reason,
we predicted the sky background for these objects using the ROSAT All-Sky
Survey diffuse background maps obtained with the Position Sensitive
Proportional Counters (PSPC).

We determined the ROSAT photon count rate, ROSAT${}_{\textrm{cr}}$, for each
CHEX–MATE object in the R5 band, [$0.73-1.56$] keV, within an annular region
centred on the X-ray peak and with the minimum and maximum radius being
$R_{200}^{\mathrm{Y_{\textrm{S}Z}}}$ and 1.5 degrees, respectively, using the
sxrbg tool (Sabol & Snowden 2019). We then calibrated the relation between the
ROSAT${}_{\textrm{cr}}$ and the XMM-Newton background sky count rate,
XMM${}_{\textrm{cr}}$, for clusters whose $R_{200}^{\mathrm{Y_{X}}}$ was less
than 12 by performing a linear regression using the linmix package (Kelly
2007):

$\mathrm{XMM_{cr}}=\alpha+\beta\times\mathrm{ROSAT_{cr}}.$ (13)

The results of the linear regression for the mean and median profiles are
shown in the left and right panels of Fig. 17, respectively. The values of the
linear minimisation $y=\alpha+\beta*x$ and the intrinsic scatter $\epsilon$
are reported in Table 2. We used these relations to estimate the XMM-Newton
sky background for the objects where $R_{200}^{\mathrm{Y_{X}}}$ is greater
than 12 arcmin in the CHEX–MATE sample.

Table 2:  Results of the linear minimisation of the ROSAT-R5 vs XMM-Newton sky
background count rate shown in Equation 13.

Parameter | Val |  | Val
---|---|---|---
| Mean |  | Median
$\alpha$ [$10^{-5}$ ct/s] | 3.974 |  | 2.346
$\beta$ | 2.730 |  | 2.630
$\epsilon$ [$10^{-5}$ ct/s] | 2.375 |  | 2.902

Notes: The term $\epsilon$ represents the intrinsic scatter.

##  Appendix C Power law fit

We report in Table 3 the results of the fit of the median
$\widetilde{\mathrm{EM}}$ profiles centred on the X-ray peak profiles using
the power law shown in Eq. 6 and described in Sect. 5. The fit was performed
using the mean value of each bin as the pivot for the radius, that is, 0.3,
0.5, 0.7, and 0.9 for the [0.2-0.4], [0.4-0.6], [0.6-0.8], and [0.8-1]
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ radial bins, respectively.

Table 3:  Results of the power law fit shown in Equation 6.

Radial bin | $\alpha_{\mathrm{CHX}}$ | $\alpha_{\mathrm{Simx}}$ |  $\alpha_{\mathrm{CHX}}$ MR |  $\alpha_{\mathrm{CHX}}$ MD | ACHX | ASimx | ACHX MR | ACHX MD
---|---|---|---|---|---|---|---|---
[$R_{500}$] |  |  |  |  | [ $10^{-6}\mathrm{cm}^{-6}$ Mpc]
0.2-0.4 | $2.01\pm 0.36$ | $2.37\pm 0.36$ | $2.51\pm 0.13$ | $1.38\pm 0.20$ | $1.88\pm 0.41$ | $2.04\pm 0.27$ | $2.13\pm 0.38$ | $1.75\pm 0.31$
0.4-0.6 | $2.58\pm 0.33$ | $2.98\pm 0.38$ | $2.93\pm 0.22$ | $2.25\pm 0.18$ | $0.59\pm 0.11$ | $0.50\pm 0.11$ | $0.55\pm 0.05$ | $0.66\pm 0.14$
0.6-0.8 | $3.03\pm 0.27$ | $3.54\pm 0.56$ | $3.17\pm 0.32$ | $3.00\pm 0.20$ | $0.22\pm 0.04$ | $0.18\pm 0.05$ | $0.19\pm 0.04$ | $0.27\pm 0.04$
0.8-1.0 | $3.27\pm 0.36$ | $3.55\pm 0.99$ | $3.44\pm 0.42$ | $3.49\pm 0.27$ | $0.10\pm 0.02$ | $0.07\pm 0.02$ | $0.07\pm 0.02$ | $0.12\pm 0.02$

Notes: The letters MR and MD in columns 4, 5, 8, and 9 stand for
morphologically relaxed and disturbed, respectively.

##  Appendix D Surface brightness profiles

We show in 18 the surface brightness profiles of the CHEX–MATE sample that we
extracted as described in Section 3.3.1. Thedotted line shown in the top-left
panel indicates $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ and highlights the data
quality of the sample as most of the profiles extend beyond that radius.

Figure 18: Surface brightness radial profiles of the CHEX–MATE sample. Left
column: Azimuthal averaged surface brightness profiles of the CHEX–MATE sample
centred on the X-peak and the centroid in the top and bottom panels,
respectively. Blue and red solid lines represent morphologically relaxed and
disturbed clusters, respectively. The black solid vertical line identifies
$R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$. Right column: Same as left column but
for profiles extracted computing the azimuthal median.

Table 4:  Observational, morphological, and global properties of the CHEX–MATE sample. Planck name | z | X-peak | Centroid | NH | $R_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ | $M_{500}^{\mathrm{Y_{\textrm{S}Z}}}$ | Morphology
---|---|---|---|---|---|---|---
|  | RA pippopippo DEC | RA pippopippo DEC |  |  |  |
|  | [J2000] | [J2000] | [$10^{20}$cm-2] | [arcmin] | [$10^{14}$ M⊙] |
PSZ2 G075.71+13.51 | $0.056$ |  $19\mathrel{\mathop{\mathchar 58\relax}}21\mathrel{\mathop{\mathchar 58\relax}}12.415$ $43\mathrel{\mathop{\mathchar 58\relax}}56\mathrel{\mathop{\mathchar 58\relax}}50.623$ |  $19\mathrel{\mathop{\mathchar 58\relax}}21\mathrel{\mathop{\mathchar 58\relax}}9.426$ $43\mathrel{\mathop{\mathchar 58\relax}}57\mathrel{\mathop{\mathchar 58\relax}}57.118$ | $1.91$ | $22.020$ | $8.740$ | M
PSZ2 G068.22+15.18 | $0.057$ |  $18\mathrel{\mathop{\mathchar 58\relax}}57\mathrel{\mathop{\mathchar 58\relax}}37.625$ $38\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}31.119$ |  $18\mathrel{\mathop{\mathchar 58\relax}}57\mathrel{\mathop{\mathchar 58\relax}}42.458$ $38\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}12.868$ | $3.31$ | $13.548$ | $2.142$ | M
PSZ2 G040.03+74.95 | $0.061$ |  $13\mathrel{\mathop{\mathchar 58\relax}}59\mathrel{\mathop{\mathchar 58\relax}}15.106$ $27\mathrel{\mathop{\mathchar 58\relax}}58\mathrel{\mathop{\mathchar 58\relax}}34.080$ |  $13\mathrel{\mathop{\mathchar 58\relax}}59\mathrel{\mathop{\mathchar 58\relax}}14.381$ $27\mathrel{\mathop{\mathchar 58\relax}}59\mathrel{\mathop{\mathchar 58\relax}}6.593$ | $1.48$ | $12.980$ | $2.342$ | M
PSZ2 G033.81+77.18 | $0.062$ |  $13\mathrel{\mathop{\mathchar 58\relax}}48\mathrel{\mathop{\mathchar 58\relax}}52.939$ $26\mathrel{\mathop{\mathchar 58\relax}}35\mathrel{\mathop{\mathchar 58\relax}}26.919$ |  $13\mathrel{\mathop{\mathchar 58\relax}}48\mathrel{\mathop{\mathchar 58\relax}}52.646$ $26\mathrel{\mathop{\mathchar 58\relax}}35\mathrel{\mathop{\mathchar 58\relax}}44.943$ | $1.69$ | $15.848$ | $4.463$ | R
PSZ2 G057.78+52.32 | $0.065$ |  $15\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}59.000$ $36\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6\mathrel{\mathop{\mathchar 58\relax}}40.699$ |  $15\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}57.836$ $36\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}7\mathrel{\mathop{\mathchar 58\relax}}34.614$ | $5.67$ | $12.147$ | $2.316$ | M
PSZ2 G105.55+77.21 | $0.072$ |  $13\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}4.980$ $39\mathrel{\mathop{\mathchar 58\relax}}13\mathrel{\mathop{\mathchar 58\relax}}26.314$ |  $13\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}8.588$ $39\mathrel{\mathop{\mathchar 58\relax}}13\mathrel{\mathop{\mathchar 58\relax}}55.277$ | $3.43$ | $10.901$ | $2.196$ | M
PSZ2 G042.81+56.61 | $0.072$ |  $15\mathrel{\mathop{\mathchar 58\relax}}22\mathrel{\mathop{\mathchar 58\relax}}29.473$ $27\mathrel{\mathop{\mathchar 58\relax}}42\mathrel{\mathop{\mathchar 58\relax}}27.922$ |  $15\mathrel{\mathop{\mathchar 58\relax}}22\mathrel{\mathop{\mathchar 58\relax}}25.737$ $27\mathrel{\mathop{\mathchar 58\relax}}43\mathrel{\mathop{\mathchar 58\relax}}22.702$ | $5.36$ | $13.499$ | $4.219$ | M
PSZ2 G031.93+78.71 | $0.072$ |  $13\mathrel{\mathop{\mathchar 58\relax}}41\mathrel{\mathop{\mathchar 58\relax}}48.706$ $26\mathrel{\mathop{\mathchar 58\relax}}22\mathrel{\mathop{\mathchar 58\relax}}21.527$ |  $13\mathrel{\mathop{\mathchar 58\relax}}41\mathrel{\mathop{\mathchar 58\relax}}50.464$ $26\mathrel{\mathop{\mathchar 58\relax}}22\mathrel{\mathop{\mathchar 58\relax}}46.747$ | $5.14$ | $11.643$ | $2.718$ | M
PSZ2 G287.46+81.12 | $0.073$ |  $12\mathrel{\mathop{\mathchar 58\relax}}41\mathrel{\mathop{\mathchar 58\relax}}17.552$ $18\mathrel{\mathop{\mathchar 58\relax}}34\mathrel{\mathop{\mathchar 58\relax}}26.322$ |  $12\mathrel{\mathop{\mathchar 58\relax}}41\mathrel{\mathop{\mathchar 58\relax}}17.618$ $18\mathrel{\mathop{\mathchar 58\relax}}33\mathrel{\mathop{\mathchar 58\relax}}48.907$ | $2.87$ | $11.329$ | $2.563$ | M
PSZ2 G040.58+77.12 | $0.075$ |  $13\mathrel{\mathop{\mathchar 58\relax}}49\mathrel{\mathop{\mathchar 58\relax}}23.507$ $28\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6\mathrel{\mathop{\mathchar 58\relax}}23.519$ |  $13\mathrel{\mathop{\mathchar 58\relax}}49\mathrel{\mathop{\mathchar 58\relax}}23.701$ $28\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}5\mathrel{\mathop{\mathchar 58\relax}}58.161$ | $1.06$ | $11.082$ | $2.569$ | M
PSZ2 G057.92+27.64 | $0.076$ |  $17\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}14.333$ $32\mathrel{\mathop{\mathchar 58\relax}}59\mathrel{\mathop{\mathchar 58\relax}}29.115$ |  $17\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}14.092$ $32\mathrel{\mathop{\mathchar 58\relax}}59\mathrel{\mathop{\mathchar 58\relax}}27.068$ | $5.60$ | $11.084$ | $2.659$ | M
PSZ2 G006.49+50.56 | $0.077$ |  $15\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}56.232$ $\phantom{0}5\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}40.654$ |  $15\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}56.030$ $\phantom{0}5\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}40.937$ | $1.55$ | $15.168$ | $7.045$ | R
PSZ2 G048.10+57.16 | $0.078$ |  $15\mathrel{\mathop{\mathchar 58\relax}}21\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}8.350$ $30\mathrel{\mathop{\mathchar 58\relax}}38\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}7.594$ |  $15\mathrel{\mathop{\mathchar 58\relax}}21\mathrel{\mathop{\mathchar 58\relax}}15.377$ $30\mathrel{\mathop{\mathchar 58\relax}}38\mathrel{\mathop{\mathchar 58\relax}}16.582$ | $4.34$ | $11.898$ | $3.540$ | D
PSZ2 G172.74+65.30 | $0.079$ |  $11\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}39.825$ $40\mathrel{\mathop{\mathchar 58\relax}}50\mathrel{\mathop{\mathchar 58\relax}}23.218$ |  $11\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}39.716$ $40\mathrel{\mathop{\mathchar 58\relax}}50\mathrel{\mathop{\mathchar 58\relax}}21.227$ | $2.10$ | $10.226$ | $2.388$ | M
PSZ2 G057.61+34.93 | $0.080$ |  $17\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}9\mathrel{\mathop{\mathchar 58\relax}}47.051$ $34\mathrel{\mathop{\mathchar 58\relax}}27\mathrel{\mathop{\mathchar 58\relax}}14.010$ |  $17\mathrel{\mathop{\mathchar 58\relax}}9\mathrel{\mathop{\mathchar 58\relax}}47.446$ $34\mathrel{\mathop{\mathchar 58\relax}}27\mathrel{\mathop{\mathchar 58\relax}}23.362$ | $1.43$ | $11.727$ | $3.705$ | M
PSZ2 G243.64+67.74 | $0.083$ |  $11\mathrel{\mathop{\mathchar 58\relax}}32\mathrel{\mathop{\mathchar 58\relax}}51.251$ $14\mathrel{\mathop{\mathchar 58\relax}}27\mathrel{\mathop{\mathchar 58\relax}}15.521$ |  $11\mathrel{\mathop{\mathchar 58\relax}}32\mathrel{\mathop{\mathchar 58\relax}}50.980$ $14\mathrel{\mathop{\mathchar 58\relax}}28\mathrel{\mathop{\mathchar 58\relax}}24.900$ | $1.69$ | $11.226$ | $3.624$ | M
PSZ2 G080.16+57.65 | $0.088$ |  $15\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}1\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}7.943$ $47\mathrel{\mathop{\mathchar 58\relax}}16\mathrel{\mathop{\mathchar 58\relax}}38.232$ |  $15\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}59.165$ $47\mathrel{\mathop{\mathchar 58\relax}}16\mathrel{\mathop{\mathchar 58\relax}}51.937$ | $2.91$ | $9.474$ | $2.513$ | D
PSZ2 G044.20+48.66 | $0.089$ |  $15\mathrel{\mathop{\mathchar 58\relax}}58\mathrel{\mathop{\mathchar 58\relax}}20.468$ $27\mathrel{\mathop{\mathchar 58\relax}}13\mathrel{\mathop{\mathchar 58\relax}}48.074$ |  $15\mathrel{\mathop{\mathchar 58\relax}}58\mathrel{\mathop{\mathchar 58\relax}}21.094$ $27\mathrel{\mathop{\mathchar 58\relax}}13\mathrel{\mathop{\mathchar 58\relax}}43.075$ | $3.79$ | $14.133$ | $8.773$ | M
PSZ2 G114.79-33.71 | $0.094$ |  $\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}36.995$ $28\mathrel{\mathop{\mathchar 58\relax}}39\mathrel{\mathop{\mathchar 58\relax}}36.116$ |  $\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}36.917$ $28\mathrel{\mathop{\mathchar 58\relax}}39\mathrel{\mathop{\mathchar 58\relax}}42.461$ | $2.47$ | $10.198$ | $3.788$ | M
PSZ2 G056.77+36.32 | $0.095$ |  $17\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}2\mathrel{\mathop{\mathchar 58\relax}}42.568$ $34\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}3\mathrel{\mathop{\mathchar 58\relax}}36.225$ |  $17\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}2\mathrel{\mathop{\mathchar 58\relax}}41.971$ $34\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}3\mathrel{\mathop{\mathchar 58\relax}}32.091$ | $2.04$ | $10.535$ | $4.338$ | M
PSZ2 G049.32+44.37 | $0.097$ |  $16\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}30.894$ $29\mathrel{\mathop{\mathchar 58\relax}}53\mathrel{\mathop{\mathchar 58\relax}}30.425$ |  $16\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}32.043$ $29\mathrel{\mathop{\mathchar 58\relax}}53\mathrel{\mathop{\mathchar 58\relax}}35.492$ | $1.38$ | $9.866$ | $3.763$ | M
PSZ2 G080.37+14.64 | $0.098$ |  $19\mathrel{\mathop{\mathchar 58\relax}}26\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}9.543$ $48\mathrel{\mathop{\mathchar 58\relax}}33\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0.121$ |  $19\mathrel{\mathop{\mathchar 58\relax}}26\mathrel{\mathop{\mathchar 58\relax}}10.342$ $48\mathrel{\mathop{\mathchar 58\relax}}32\mathrel{\mathop{\mathchar 58\relax}}54.284$ | $1.29$ | $9.205$ | $3.127$ | M
PSZ2 G099.48+55.60 | $0.105$ |  $14\mathrel{\mathop{\mathchar 58\relax}}28\mathrel{\mathop{\mathchar 58\relax}}38.408$ $56\mathrel{\mathop{\mathchar 58\relax}}51\mathrel{\mathop{\mathchar 58\relax}}36.620$ |  $14\mathrel{\mathop{\mathchar 58\relax}}28\mathrel{\mathop{\mathchar 58\relax}}34.413$ $56\mathrel{\mathop{\mathchar 58\relax}}53\mathrel{\mathop{\mathchar 58\relax}}9.043$ | $1.76$ | $8.271$ | $2.749$ | D
PSZ2 G080.41-33.24 | $0.107$ |  $22\mathrel{\mathop{\mathchar 58\relax}}26\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}5.889$ $17\mathrel{\mathop{\mathchar 58\relax}}21\mathrel{\mathop{\mathchar 58\relax}}51.623$ |  $22\mathrel{\mathop{\mathchar 58\relax}}26\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}4.592$ $17\mathrel{\mathop{\mathchar 58\relax}}22\mathrel{\mathop{\mathchar 58\relax}}24.699$ | $5.57$ | $9.028$ | $3.774$ | M
PSZ2 G113.29-29.69 | $0.107$ |  $\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}45.583$ $32\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}53.523$ |  $\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}11\mathrel{\mathop{\mathchar 58\relax}}45.554$ $32\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}52.973$ | $1.59$ | $8.857$ | $3.573$ | M
PSZ2 G053.53+59.52 | $0.113$ |  $15\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}12.594$ $33\mathrel{\mathop{\mathchar 58\relax}}30\mathrel{\mathop{\mathchar 58\relax}}29.924$ |  $15\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}11.521$ $33\mathrel{\mathop{\mathchar 58\relax}}30\mathrel{\mathop{\mathchar 58\relax}}12.360$ | $2.06$ | $9.580$ | $5.209$ | M
PSZ2 G046.88+56.48 | $0.115$ |  $15\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}8.181$ $29\mathrel{\mathop{\mathchar 58\relax}}53\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6.523$ |  $15\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}9.401$ $29\mathrel{\mathop{\mathchar 58\relax}}53\mathrel{\mathop{\mathchar 58\relax}}31.977$ | $4.02$ | $9.402$ | $5.104$ | D
PSZ2 G204.10+16.51 | $0.122$ |  $\phantom{0}7\mathrel{\mathop{\mathchar 58\relax}}35\mathrel{\mathop{\mathchar 58\relax}}47.144$ $15\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6\mathrel{\mathop{\mathchar 58\relax}}48.825$ |  $7\mathrel{\mathop{\mathchar 58\relax}}35\mathrel{\mathop{\mathchar 58\relax}}47.232$ $15\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6\mathrel{\mathop{\mathchar 58\relax}}58.589$ | $1.16$ | $7.978$ | $3.706$ | M
PSZ2 G192.18+56.12 | $0.124$ |  $10\mathrel{\mathop{\mathchar 58\relax}}16\mathrel{\mathop{\mathchar 58\relax}}22.500$ $33\mathrel{\mathop{\mathchar 58\relax}}38\mathrel{\mathop{\mathchar 58\relax}}19.329$ |  $10\mathrel{\mathop{\mathchar 58\relax}}16\mathrel{\mathop{\mathchar 58\relax}}24.056$ $33\mathrel{\mathop{\mathchar 58\relax}}38\mathrel{\mathop{\mathchar 58\relax}}18.491$ | $3.28$ | $7.801$ | $3.620$ | M
PSZ2 G098.44+56.59 | $0.132$ |  $14\mathrel{\mathop{\mathchar 58\relax}}27\mathrel{\mathop{\mathchar 58\relax}}24.855$ $55\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}55.015$ |  $14\mathrel{\mathop{\mathchar 58\relax}}27\mathrel{\mathop{\mathchar 58\relax}}22.753$ $55\mathrel{\mathop{\mathchar 58\relax}}44\mathrel{\mathop{\mathchar 58\relax}}59.931$ | $3.92$ | $6.800$ | $2.826$ | M
PSZ2 G273.59+63.27 | $0.134$ |  $12\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}24.093$ $\phantom{0}3\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}40.725$ |  $12\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}0\mathrel{\mathop{\mathchar 58\relax}}23.716$ $\phantom{0}3\mathrel{\mathop{\mathchar 58\relax}}20\mathrel{\mathop{\mathchar 58\relax}}31.335$ | $0.82$ | $8.353$ | $5.465$ | M
PSZ2 G217.09+40.15 | $0.136$ |  $\phantom{0}9\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}5.764$ $14\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}25.122$ |  $\phantom{0}9\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}\phantom{0}6.079$ $14\mathrel{\mathop{\mathchar 58\relax}}10\mathrel{\mathop{\mathchar 58\relax}}19.625$ | $1.34$ | $7.369$ | $3.890$ | M
PSZ2 G218.59+71.31 | $0.137$ |  $11\mathrel{\mathop{\mathchar 58\relax}}29\mathrel{\mathop{\mathchar 58\relax}}52.386$ $23\mathrel{\mathop{\mathchar 58\relax}}48\mathrel{\mathop{\mathchar 58\relax}}39.122$ |  $11\mathrel{\mathop{\mathchar 58\relax}}29\mathrel{\mathop{\mathchar 58\relax}}51.064$ $23\mathrel{\mathop{\mathchar 58\relax}}48\mathrel{\mathop{\mathchar 58\relax}}49.086$ | $7.13$ | $7.219$ | $3.759$ | D
PSZ2 G179.09+60.12 | $0.137$ |  $10\mathrel{\mathop{\mathchar 58\relax}}40\mathrel{\mathop{\mathchar 58\relax}}44.656$ $39\mathrel{\mathop{\mathchar 58\relax}}57\mathrel{\mathop{\mathchar 58\relax}}10.577$ |  $10\mathrel{\mathop{\mathchar 58\relax}}40\mathrel{\mathop{\mathchar 58\relax}}44.381$ $39\mathrel{\mathop{\mathchar 58\relax}}57\mathrel{\mathop{\mathchar 58\relax}}12.733$ | $1.03$ | $7.265$ | $3.839$ | M
PSZ2 G226.18+76.79 | $0.143$ |  $11\mathrel{\mathop{\mathchar 58\relax}}55\mathrel{\mathop{\mathchar 58\relax}}17.827$ $23\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}16.924$ |  $11\mathrel{\mathop{\mathchar 58\relax}}55\mathrel{\mathop{\mathchar 58\relax}}18.153$ $23\mathrel{\mathop{\mathchar 58\relax}}24\mathrel{\mathop{\mathchar 58\relax}}15.915$ | $7.24$ | $8.129$ | $5.974$ | M