11institutetext:  INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi,
I-50122 Firenze, Italy 11email<EMAIL_ADDRESS>22institutetext: INAF -
Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3,
40129 Bologna, Italy 33institutetext: Max Planck Institute for
Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany
44institutetext: Astronomy Unit, Department of Physics, University of Trieste,
via Tiepolo 11, I-34131 Trieste, Italy 55institutetext: INAF-Osservatorio
Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy
66institutetext: IFPU - Institute for Fundamental Physics of the Universe, Via
Beirut 2, 34014 Trieste, Italy 77institutetext: INFN–Sezione di Trieste,
Trieste, Italy 88institutetext: INAF - Osservatorio Astronomico di Roma, Via
Frascati 33, I-00040 Monteporzio (RM), Italy 99institutetext: National Radio
Astronomy Observatory, P. O. Box 0, Socorro, NM 87801, USA 1010institutetext:
Leiden Observatory, PO Box 9513, 2300 RA Leiden, The Netherlands
1111institutetext: European Southern Observatory (ESO), Karl-Schwarzschild-
Str. 2, D-85748 Garching, Germany 1212institutetext: Dipartimento di Fisica e
Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, I-44122
Ferrara, Italy 1313institutetext: Jansky Fellow of the National Radio
Astronomy Observatory, P. O. Box 0, Socorro, NM 87801, USA 1414institutetext:
Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain
1515institutetext: Universidad de La Laguna, Dpto. Astrofísica, 38206 La
Laguna, Tenerife, Spain 1616institutetext: Instituto de Física, Pontificia
Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile
1717institutetext: Cosmic Dawn Center, Rådmandsgade 62, 2200 København N,
Danimarca 1818institutetext: DTU-Space, Technical University of Denmark,
Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark 1919institutetext: Space
Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21210, USA
2020institutetext: Johns Hopkins University, 3400 N. Charles Street,
Baltimore, MD 21218, USA

# The 700 ks Chandra Spiderweb Field II: Evidence for inverse-Compton and
thermal diffuse emission in the Spiderweb galaxy

P. Tozzi 11 R. Gilli 22 A. Liu 33 S. Borgani 44556677 M. Lepore 11 L. Di
Mascolo 44 A. Saro 44556677 L. Pentericci 88 C. Carilli 99 G. Miley 1010 T.
Mroczkowski 1111 M. Pannella 44 E. Rasia 5566 P. Rosati 1212 C. S. Anderson
1313 A. Calabrò 88 E. Churazov 33 H. Dannerbauer 14141515 C. Feruglio 55 F.
Fiore 55 R. Gobat 1616 S. Jin 17171818 M. Nonino 55 C. Norman 19192020 H.J.A.
Röttgering 99

###### Abstract

Aims. We present the X-ray imaging and spectral analysis of the diffuse
emission around the radio galaxy J1140-2629 (the Spiderweb galaxy) at $z=2.16$
and of its nuclear emission, based on a deep (700 ks) Chandra observation.

Methods. We obtained a robust characterization of the unresolved nuclear
emission, and carefully computed the contamination in the surrounding regions
due to the wings of the instrument point spread function. Then, we quantified
the extended emission within a radius of 12 arcsec. We used the Jansky Very
Large Array radio image to identify the regions overlapping the jets, and
performed X-ray spectral analysis separately in the jet regions and in the
complementary area.

Results. We find that the Spiderweb galaxy hosts a mildly absorbed quasar,
showing a modest yet significant spectral and flux variability on a timescale
of $\sim 1$ year. We find that the emission in the jet regions is well
described by a power law with a spectral index of $\Gamma\sim 2-2.5$, and it
is consistent with inverse-Compton upscattering of the cosmic microwave
background photons by the relativistic electrons. We also find a roughly
symmetric, diffuse emission within a radius of $\sim 100$ kpc centered on the
Spiderweb galaxy. This emission, which is not associated with the jets, is
significantly softer and consistent with thermal bremsstrahlung from a hot
intracluster medium (ICM) with a temperature of $kT=2.0_{-0.4}^{+0.7}$ keV,
and a metallicity of $Z<1.6\,Z_{\odot}$ at 1 $\sigma$ c.l. The average
electron density within 100 kpc is $n_{e}=(1.51\pm 0.24\pm 0.14)\times
10^{-2}$ cm-3, corresponding to an upper limit for the total ICM mass of
$\leq(1.76\pm 0.30\pm 0.17)\times 10^{12}M_{\odot}$ (where error bars are
$1\sigma$ statistical and systematic, respectively). The rest-frame luminosity
$L_{0.5-10{\rm keV}}=(2.0\pm 0.5)\times 10^{44}$ erg s-1 is about a factor of
2 higher than the extrapolated $L-T$ relation for massive clusters, but still
consistent within the scatter. If we apply hydrostatic equilibrium to the ICM,
we measure a total gravitational mass $M(<100\leavevmode\nobreak\ {\rm
kpc})=(1.5^{+0.5}_{-0.3})\times 10^{13}\,M_{\odot}$ and, extrapolating at
larger radii, we estimate a total mass $M_{500}=(3.2^{+1.1}_{-0.6})\times
10^{13}\,M_{\odot}$ within a radius of $r_{500}=(220\pm 30)$ kpc.

Conclusions. We conclude that the Spiderweb protocluster shows significant
diffuse emission within a radius of 12 arcsec, whose major contribution is
provided by inverse-Compton scattering associated with the radio jets. Outside
the jet regions, we also identified thermal emission within a radius of $\sim
100$ kpc, revealing the presence of hot, diffuse baryons that may represent
the embryonic virialized halo of the forming cluster.

###### Key Words.:

galaxies: clusters: general – galaxies: clusters: intracluster medium –
X-rays: AGN – X-rays: galaxies: clusters

##  1 Introduction

Protoclusters are defined as overdense regions in the high-$z$ Universe which
are predicted to evolve into massive, virialized clusters of galaxies by $z=0$
(see Overzier, 2016, for a review). Identifying and studying their properties
is key to studying the formation and evolution of the large-scale structure of
the Universe. In particular, in recent years the scientific community has
mostly focused on the identification of the first virialized, dark-matter-
dominated halos, on the origin and evolution of the hot, diffuse baryons
permeating the potential wells – the intracluster medium (ICM) – and on the
transformational processes that affect star formation and nuclear activity in
the member galaxies. From the observational point of view, a protocluster is
usually identified as a high-z region that is overdense in galaxy counts
compared to the field. However, the dynamical state of the structure and the
virialized mass that its halo ultimately achieves by $z=0$ is usually highly
uncertain (see, e.g., Muldrew et al., 2015). At present, there are no standard
methods to search for protoclusters, and this is the reason why biased-tracer
techniques are often used to identify promising candidates. For example,
several protoclusters have been found around high redshift, powerful radio
galaxies. Spectroscopic follow-up has provided evidence that a significant
fraction of powerful, high-z radio galaxies reside in a protocluster or
overdense regions (see Venemans et al., 2007; Galametz et al., 2012; Wylezalek
et al., 2013). In addition to massive spectroscopic surveys, intensive
multiwavelength campaigns in the radio and X-ray bands are necessary to trace
the many processes occurring in the protocluster member galaxies during their
rapid evolution, such as nuclear activity, feedback into the surrounding
medium, star formation, chemical enrichment, among others.

In this paper we focus on the role of deep X-ray observations. X-ray data are
unique to use when studying the unresolved emission from active galactic
nuclei (AGN), due to the accretion onto nuclear supermassive black holes
(SMBHs), the thermal diffuse emission from the ICM, and the inverse-Compton
(IC) emission from relativistic plasma. To a lesser extent, X-ray observations
can also be used to trace the less intense emission from strongly star-forming
galaxies, but this signal is too weak to be detected beyond $z\sim 1.5$ (see,
e.g., Lehmer et al., 2016). Until now, only a few deep ($t_{\rm exp}\geq 200$
ks) observations of protoclusters have been carried out with Chandra or XMM-
Newton. While the unresolved emission from AGN is relatively easy to detect,
the faint, extended emission associated with the expected proto-ICM has been
very hard to detect, with a few promising candidates (Gobat et al., 2011;
Valentino et al., 2016; Wang et al., 2016), and some other cases where it is
not possible to determine the thermal or relativistic nature of the diffuse
emission unambiguously (Gilli et al., 2019; Champagne et al., 2021). This is
not surprising since the expected thermal emission from $z\gtrsim 2$
protoclusters is predicted to be very faint from semianalytic models and
numerical simulations (see Saro et al., 2009). In addition, the presence of
nonthermal diffuse emission associated with the relativistic electrons in the
radio jets, coupled to the bright, unresolved nuclear emission from
protocluster members, which are both often present, may easily overwhelm the
thermal emission from the proto-ICM. This implies that only high-sensitivity
and angular-resolution ($\sim 1$ arcsec) observations can be used to search
and characterize the diffuse, thermal emission in high-z protocluster and, at
the same time, the unresolved emission from AGN members and the diffuse,
nonthermal emission due to IC from radio jets.

A key piece of information in protoclusters is the identification and
characterization of thermal emission from surrounding diffuse baryons, which
directly traces the thermodynamics of the largest baryonic component and, at
the same time, the virialization status of the halo. As of today, there have
been no observations that can track the origin of the ICM from the birth of
the protocluster, to the formation of an evolved, virialized clusters. In this
short yet intense phase, which is expected to occur in the redshift range
$2<z<3$, the heating of the proto-ICM can be due to several mechanisms, namely
the following: accretion shocks; heating from stellar winds associated with
strong star formation episodes in the protocluster galaxies; and mechanical
heating and turbulence caused by radio jets from the central massive galaxy.
Therefore, a characterization of the proto-ICM, including its morphology,
thermodynamical, and chemical properties, can provide an independent and
direct measure of the heating efficiency of the feedback and of the gas
accretion and circulation into the protocluster potential well. In addition,
feedback processes not only affect the ICM, but can also couple directly with
the member galaxies, as suggested by the recent X-ray observation of a $z\sim
1.7$ protocluster (Gilli et al., 2019).

In this paper we investigate the diffuse emission (both thermal and
nonthermal) and the central nuclear emission of the Spiderweb galaxy
(J1140-2629) protocluster at $z=2.16$, using a deep Chandra ACIS-S observation
($\sim 700$ ks). The central, powerful radio galaxy is embedded in a giant
Ly$\alpha$ halo (Pentericci et al., 1997; Miley et al., 2006), and surrounded
by a $\gtrsim 2$ Mpc–sized overdensity of star-forming galaxies (Ly$\alpha$
and H$\alpha$ emitters), dusty starbursts, and galaxies that are rapidly
migrating toward a nascent red sequence (Zirm et al., 2008). The archetypal
Spiderweb protocluster is expected to evolve into a massive cluster in less
than 1.5 Gyr, with the radio galaxy itself showing the properties of a cD
progenitor (Miley et al., 2006). In a companion paper (Tozzi et al., 2022,
hereafter Paper I), we investigated the nuclear activity in the protocluster
members, finding a high fraction of bright AGN with $L_{0.5-10keV}>3\times
10^{43}$ erg s-1 in spectroscopically confirmed members with ${\rm
log}(M_{*}/M_{\odot})>10.5$, equal to $25.5\pm 4.5$%, and estimating an
enhancing factor of $6.0^{+9.0}_{-3.0}$ for the nuclear activity with respect
to the COSMOS field at comparable redshifts and stellar mass range. Here we
focus on the very central regions of the Spiderweb protocluster, which is a
rapidly evolving, high-density region where complex and intricate phenomena
concur to shape the properties of the nascent brightest cluster galaxy (BCG)
and possibly an embryonic ICM halo. Such diffuse X-ray emission is also
investigated in connection with the radio (JVLA) and submillimeter (ALMA)
spatially resolved signal in other papers of the collaboration (Carilli et
al., 2022; Anderson et al., 2022, Di Mascolo et al. in preparation). Finally,
in a forthcoming paper, the structure and thermodynamics of the ICM will be
investigated by combining X-ray and submillimeter data (Lepore et al. in
preparation).

The paper is organized as follows. In Section 2 we summarize the properties of
the Spiderweb galaxy and its immediate surroundings on the basis of previous
multiwavelength observations. In Section 3 we briefly recall the X-ray data
acquisition and reduction, which has already been detailed in Paper I. In
Section 4 we present the ray-tracing simulations we used to accurately model
the unresolved nuclear emission, a step needed to estimate the contamination
of the AGN to the extended emission and, therefore, exploit the imaging
capability of Chandra when studying the morphology of such emission. In
Section 5 we present a detailed analysis of the nuclear emission, including
its variability. In Section 6 we finally investigate the nature of the diffuse
X-ray emission surrounding the Spiderweb galaxy, spatially separating the
thermal and nonthermal contribution. In Section 7 we focus on the nature of
the thermal diffuse component, and discuss the implication of the presence of
a virialized halo. Finally, our conclusions are summarized in Section 8.
Throughout this paper, we adopt the seven-year Wilkinson Microwave Anisotropy
Probe cosmology with $\Omega_{\Lambda}=0.73$, $\Omega_{m}=0.27$, and
$H_{0}=70.4$ km s-1 Mpc-1 (Komatsu et al., 2011). In this cosmology, at
$z=2.156$, 1 arcsec corresponds to 8.473 kpc, the Universe is 3.13 Gyr old,
and the lookback time is 77% of the age of the Universe. Therefore, the size
of a Chandra ACIS pixel (0.492 arcsec) corresponds to $4.237$ kpc. These
values are especially relevant in the following section where we
interchangeably report distances in pixels, arcsecs, or kiloparsecs. Quoted
error bars correspond to a 1 $\sigma$ confidence level, unless noted
otherwise.

##  2 The Spiderweb galaxy: Previous observational campaign and main results

The Spiderweb Galaxy (J1140-2629) and its environment have been extensively
targeted in the last 25 years, with 54 papers published about the Spiderweb
field including observations in radio, sub-mm, infra red (IR), optical and
X–ray wavelengths. In this section we summarize the main results that have
been obtained on the Spiderweb Galaxy focusing on its immediate environment,
while a review on the properties of the protocluster galaxy population has
been presented in Paper I. In the last subsection we point out the open
science cases that are addressed by this work.

###  2.1 Discovery and peculiarities

The Spiderweb Galaxy was identified as a high-z radio galaxy (HzRG) while
targeting ultra-steep spectrum radio sources (Roettgering et al., 1994) in an
ESO Key Program. The peculiarity of this object was immediately noticed thanks
to the unusually clumpy and bent radio morphology, and the exceptionally high
rotation measure ($\sim 6200$ rad/m), showing that the radio synchrotron jet
is severely affected by a surrounding, dense ($10^{-1}-10^{-2}$cm-3) external
medium (Carilli et al., 1997; Pentericci et al., 1997; Athreya et al., 1998).
From the optical point of view, the Spiderweb Galaxy has been classified as a
narrow emission line galaxy at $z=2.16$ (see Roettgering et al., 1997). A
clumpy morphology has been observed also in the optical band, and HST ACS
observations revealed that this is due to $\sim 10$ star-forming satellite
galaxies moving with peculiar velocities of several hundred km s-1, with a
distribution resembling that of flies trapped in a spiderweb (Miley et al.,
2006), and probably in the process of merging with the central radio galaxy
within a few hundreds of Myr. Nevertheless, the most striking feature is
constituted by a spectacular 200 kpc Ly$\alpha$ halo with a luminosity of
$42.5\times 10^{44}$ erg s-1 elongated along the direction of the jets. The
total stellar mass inferred from the K-band luminosity is $10^{12}M_{\odot}$
(Pentericci et al., 1997), and it has been confirmed by Spitzer data (Seymour
et al., 2007). The large majority of the stellar mass appears to be already in
place well before $z=2.16$, while the satellite galaxies have measured stellar
masses in the range $10^{8}<M_{*}<10^{10}M_{\odot}$, contributing only $\sim
1/10$ of the total mass of the central galaxy.

The optical and NIR spectrum of the nucleus of J1140-2629 shows a broad,
spatially unresolved H$\alpha$ line (blended with [NII]) with a FWHM of $\sim
14900$ km s-1 (Nesvadba et al., 2011). The presence of broad nuclear lines is
rare for optical counterparts of high redshift radio sources, which typically
show narrow emission lines and are classified as TypeII AGN. Other emission
lines which have been detected include [OII], [NeIII], Hβ narrow, [OIII],
[OI], [NII], [SII], and a strong CIV and HeII (see Humphrey et al., 2008).
Such an optical spectrum is consistent with photoionization from the AGN,
powered by a SMBH with estimated mass $M_{BH}=2\times 10^{10}M_{\odot}$,
putting J1140-2629 a factor of $\sim 2$ above the local $M-\sigma$ relation
(Nesvadba et al., 2011).

Taken together, the observed characteristics suggest that the Spiderweb is
indeed a massive radio galaxy forming at the center of a dynamically evolving
protocluster region, that is likely to evolve into a BCG (see Pentericci et
al., 1997). These findings sparkled a wide interest in this object and its
surroundings, as one of the most promising targets where the evolutionary
transition from a collapsing large scale structure to a gravitationally bound
halo heavily affects the entire galaxy population.

###  2.2 Star formation activity in and around J1140-2629

The Spiderweb Galaxy shows a star formation rate of $15.6\pm 0.7M_{\odot}$/yr
from dust-uncorrected optical/UV emission, as opposed to a total of $\sim
50M_{\odot}$/yr contributed from all the ”flies” (Hatch et al., 2009). IR
observations with Spitzer (De Breuck et al., 2010), Herschel and LABOCA
(Seymour et al., 2012) provide, however, a different picture, finding that the
contribution to the $8-1000\mu\,$m luminosity is $1.17\pm 0.27\times
10^{13}L_{\odot}$ and $0.79\pm 0.09\times 10^{13}L_{\odot}$ for an AGN and a
starburst component, respectively. This implies a 20% Eddington accretion rate
onto the SMBH and, most important, a star formation rate (SFR) equal to
$1390\pm 150M_{\odot}$/yr, dramatically higher than the value derived from the
UV/optical rest-frame111Despite the source appears as unresolved in the
8$\,\mu$m to 250$\,\mu$m Herschel bands, the IR flux includes also the
surrounding star-forming galaxies (the so called flies), and the possible
diffuse star formation, and, in the 350, 500, and 870$\,\mu$m bands, also
additional member galaxies out to 30 arcsec from the protocluster center. This
value, therefore, should be considered an upper limit (Seymour et al., 2012)..
Such a large value is confirmed by other works based on spectroscopic analysis
of IR data (Ogle et al., 2012; Rawlings et al., 2013), and it cannot be
reconciled with the dust-corrected estimate based on the rest-frame UV and the
simplistic assumption of a uniform dust screen by Hatch et al. (2008),
implying that most of the star formation in the Spiderweb must be highly
obscured at rest-UV wavelengths.

The diffuse UV-continuum light surrounding the Spiderweb Galaxy shows a
distribution similar to the $Ly\alpha$ halo, and it is likely due to a young
stellar population (Hatch et al., 2008) originated by a diffuse, in situ star
formation of $57\pm 8\,M_{\odot}$/yr, or $\sim 140\,M_{\odot}$/yr if corrected
by dust extinction. This interpretation is not free from complexities, though.
The presence of CIV and HeII emission lines suggest an ionization source
harder than stellar continuum, implying a lower limit of only $\sim 7$% of the
$Ly\alpha$ emission associated with star formation (Hatch et al., 2008).

A key point here is the presence of a large gas reservoir that feeds both the
central and the diffuse star formation. The molecular gas mass inferred from
observation of the redshifted CO(1-0) line with ATCA (Emonts et al., 2013) is
$M_{H_{2}}=6\times 10^{10}M_{\odot}$ (for $M_{H2}/L^{{}^{\prime}}_{CO}=0.8$).
This reservoir is able to sustain a SFR of $\sim 1400M_{\odot}/$yr for only
$\sim 40$ Myr, while the doubling-mass time would be as long as 0.8 Gyr. A
deeper ATCA observation (Emonts et al., 2016), coupled with JVLA, was able to
identify an even larger amount of molecular gas $M_{H_{2}}=1.5\pm 0.4\times
10^{11}M_{\odot}$ (assuming $\alpha_{CO}=4\,M_{\odot}$ (K km s-1 pc2)-1),
associated with the $Ly\alpha$ halo, showing that there is enough molecular
gas to fuel the star formation within the IGM up to $z\sim 1.6$.

Summing the $3\times 10^{9}M_{\odot}$ in ionized gas found by Nesvadba et al.
(2006), the gas masses of various components of the ISM are about an order of
magnitude lower than the stellar mass, suggesting that this is already quite
an evolved galaxy with a gas fraction in line for a main sequence galaxy at
$z\sim 2$. Again, this suggests that we might be witnessing a transition from
a very active starburst phase triggered by a merging event to a radio-feedback
dominated phase, leading to the more quiescent existence of a BCG.

How this transition will be happening is still very much debated and unclear.
On the one hand we have the expectation that this transition should be short
lived, and that strong nuclear activity is indeed able to rapidly quench star
formation, to be consistent with the lack of bright IR sources at X-ray
luminosities comparable to the Spiderweb ($L_{X}>10^{45}$ erg s-1) in the
GOODS-North deep field (Page et al., 2012). On the other hand, results
obtained with improved statistics (Harrison et al., 2012), and actually
extending a trend well established at lower X-ray luminosities, seem to
suggest that high-accretion, X-ray luminous, nuclear activity phases do not
show any obvious impact on the star formation rate and stellar mass growth of
the host galaxies and that super massive black hole and galaxy growth rather
coexist over cosmic times. In this perspective, it is key to investigate a
large sample of massive, high-z transitional galaxies similar to the Spiderweb
to assess the relative contribution of radiative and mechanical feedback.

###  2.3 Diffuse gas and feedback effects in the halo of J1140-2629

The X-ray band is a promising spectral window to explore the Spiderweb,
particularly when considering the expected X-ray brightening of high-z radio
galaxies driven by the cosmic microwave background (CMB) photons (see Hodges-
Kluck et al., 2021). The discovery of extended X-ray emission, that was not
possible to constrain in an early ROSAT HRI image (Carilli et al., 1998), was
measured to reach a luminosity of $\sim 3\times 10^{44}$ erg s-1 in the 2-10
keV band aligned along the direction of the jet on a scale of $\sim 20$
arcsec, thanks to the first shallow Chandra exposure of about $\sim 30$ ks
(Carilli et al., 2002). From these preliminary data it was not possible to
spectrally verify whether the dominant emission mechanism is thermal
bremsstrahlung from hot gas or Inverse Compton due to a relativistic electron
population.

The nature of the diffuse X-ray emission is clearly a key ingredient in any
picture aimed at constraining the presence of a virialized halo and the
feedback processes in the diffuse medium around the Spiderweb Galaxy.
Nevertheless, signatures of preheating have been searched for at other
wavelengths. Integral field unit spectroscopy with SPIFFI (SINFONI) at VLT,
found evidences for strong outflows with velocities on the order of $\sim
2000$ km s-1, and total kinetic power in the range $2\times 10^{46}$-$3\times
10^{47}$erg s-1 (Nesvadba et al., 2006, 2011). The high ratio of kinetic over
radiative power may indicate that J1140-2629 is about to complete the
transition to the radio-mode feedback regime (see Fabian, 2012). This picture
is reinforced by the gas kinematics derived from the [OIII]$\lambda 5007$
line, that highlights a structure reminiscent of ”bubbles” intended as
expanding spheres or conical outflows (Humphrey et al., 2008), implying a
relatively efficient interaction between the AGN and the interstellar medium.
With a sufficient coupling, the jet has enough energy to totally remove the
ISM of the host within $\sim 5\times 10^{8}$ yr.

###  2.4 Dynamical state of the Spiderweb Complex

A first dynamical estimate of the total mass from the member galaxy velocity
dispersion provided a value of $\sim 2.3\times 10^{14}M_{\odot}$, contributed
by two subgroups of $1.7$ and $0.6\times 10^{14}M_{\odot}$ and virial radii of
1.1 and 0.8 Mpc, respectively, assuming they are both virialized (Kurk et al.,
2004). However, the entire system, as well as the two subgroups separately,
seem far from being virialized, not only for the lack of evident extended,
thermal X-ray emission, but also for the lack of a well-formed red-sequence in
the J-K vs K color magnitude diagram (despite this does not necessarily follow
or precede virialization). Thanks to near-infrared spectroscopy with SINFONI
at VLT, Kuiper et al. (2011) found that the velocity distribution within 60
kpc of the Spiderweb Galaxy is consistent with bimodality, with no global peak
resembling that of a virialized halo. They described the system as the merger
of two halos that would predict, in case of virialization of both subhalos, a
rest-frame 2-10 keV luminosity of $2\times 10^{43}$ erg s-1. Finally Shimakawa
et al. (2014) identified a region possibly virialized with a $R_{200}\sim
0.53$ Mpc and $\sigma_{cl}\sim 680$ km s-1, while the inclusion of all the
spectroscopic members would give $\sigma_{cl}\sim 880$ km s-1, reflecting some
velocity structures particularly in the outer regions. The inferred mass is
consistent with that of a progenitor of present-day most massive class of
galaxy clusters.

Saro et al. (2009) analyzed at $z=2.2$ cosmological hydrodynamical simulations
of two protocluster regions, which form by $z=0$ two clusters with virial
masses of $\sim 10^{14}$ and $\sim 10^{15}M_{\odot}$. The line-of-sight
velocity dispersion of galaxies in the largest halo is roughly in agreement
with the total observed dispersion in the Spiderweb Complex, thus suggesting
that it is consistent with being the progenitor of a massive cluster by $z=0$
and that a diffuse atmosphere of hot gas already in equilibrium should be
already present. Their prediction for the 0.5-2 keV flux from the ICM is $\sim
10^{-15}$ and $\sim 10^{-14}$ erg s-1 cm-2 for subhalo temperatures of 2 and 5
keV, respectively. These findings underline the importance of confirming the
thermal nature of at least part of the diffuse X-ray emission around the
Spiderweb Galaxy.

###  2.5 Open issues and role of the new X-ray observations

Previous works on the Spiderweb Complex found a clear overdensity in discrete
source counts in all the observed bands, from sub-mm to the X-ray, that ranges
from a factor of $\sim 2-6$ to $\sim 100$, depending on the source selection
and on the specific region considered within the complex. In Paper I we
explored the galaxy population of the Spiderweb Protocluster, finding an X-ray
AGN fraction enhanced by a factor of $6.0^{+9.0}_{-3.0}$ with respect to the
COSMOS field at comparable redshifts and stellar mass range.

In this work we focus on other key aspects concerning the Spiderweb Galaxy and
its immediate surroundings: i) the nature of the diffuse emission; ii) the
presence of diffuse thermal emission possibly associated with the
virialization of a central halo; iii) the nuclear activity in the Spiderweb
Galaxy. Exploring these aspects is mandatory to explore more general aspects
concerning galaxy evolution, such as: the main process (gravitational or
feedback) responsible for the heating of the proto-ICM; the compresence of hot
and cold gas within the halo as a typical phase of the formation of BCG
progenitors; the eventual evolution of the Spiderweb protocluster into a
virialized structure by $z=0$. The new set of data obtained in the last couple
of years in the Spiderweb field, and in particular the deep Chandra
observation presented in this work, allows us to better address all the
science goals listed here. The presence of hot, diffuse baryons around the
Spiderweb Galaxy and its relation to its nuclear activity are discussed in
other papers based on the combination of Chandra X-ray and JVLA Radio data
(Carilli et al., 2022; Anderson et al., 2022). Moreover, the ICM of the
Spiderweb Galaxy is also investigated with ALMA SZ data (Di Mascolo et al. in
preparation). The main results and the data products can be found on the
project webpage.222http://www.arcetri.inaf.it/spiderweb/.

##  3 X-ray data reduction

The Spiderweb Galaxy was observed with a Chandra Large Program for 700 ks with
ACIS-S granted in Cycle 20 (PI P. Tozzi). The observations were completed in
the period November 2019 - August 2020, splitted in 21 separate Obsid. To this
data set we add the first X-ray observation with ACIS-S, dating back to June
2000 for a total of 39.5 ks. All the Obsid are listed in Table 1 of Tozzi et
al. (2021), where we also describe the followed standard data reduction
procedure. We used the latest release of the Chandra Calibration Database at
the time of writing (CALDB 4.9.3) We briefly recall that we run the task
acis_process_events with the parameter apply_cti=yes to flag background
events, most likely associated with cosmic rays, and, by rejecting them,
obtain a significant reduction of the background, thanks to the VFAINT mode of
data acquisition.

This step is particularly relevant in this work since the background may
significantly affect the detection of the low surface-brightness, diffuse
emission surrounding the Spiderweb Galaxy. The small price to pay is the
removal of some of photons from bright sources. We find that slightly more
than 100 net counts in the total band are removed at the position of the
nucleus of J1140-2629 across the 22 exposures. This indicates that the source,
which is by far the brightest in the field, suffers a small amount of pile-up.
However, we have verified a posteriori that this effect does not impact the
spectral analysis of the nucleus and therefore it is not worth to give up the
VFAINT cleaning to recover less than 1% of the flux in the nucleus.

The final total exposure time after data reduction and excluding the dead-time
correction amounts to 715 ks (corresponding to the LIVETIME keyword in the
header of Chandra fits files), including the first observation. The 22 level-2
event files are merged together with the tool reproject_obs, using the
reference coordinates of Obsid 21483. The soft (0.5-2 keV) and hard (2-7 keV)
band images at full resolution in the immediate surroundings ($90\times 70$
arcsec2) of the Spiderweb Galaxy are shown in Figure 1. It is possible to
notice the larger background in the hard band, and to appreciate the
enhancement of the diffuse emission along the radio emission associated with
the jets, shown with red contours (after Carilli et al., 2022). Fairly
isotropic extended mission is also present far from the jets in the soft band.
Clearly, a meaningful analysis of the extended emission must take into account
all the complexity arising from the overlapping of four components: emission
from the jets, isotropic extended emission, unresolved nucleus, and
background. The choice of the extraction regions corresponding to different
components is based on a detailed imaging analysis as described in the next
Section. Finally, to perform spectral analysis, we extracted the spectra and
computed the ancillary response file (ARF) and redistribution matrix file
(RMF) for each observation separately with the command mkarf and mkacisrmf.
Our default spectral analysis uses a local background that, given the small
extent of the sources, is directly extracted from a source-free region on the
same CCD-7 chip.

Figure 1: Close-up of the Spiderweb Galaxy ($90\times 70$ arcsec by side) at
full angular resolution, in the soft (0.5-2 keV, left) and hard (2-7 keV,
right) band. Extended emission is clearly visible in both bands. Red contours
show radio emission observed in the 10 GHz band with the JVLA (see Carilli et
al., 2022) at levels of $0.03$, $0.2$, $2$ and $20$ mJy/beam. It is possible
to appreciate the larger background in the hard band, and the enhancement of
the diffuse emission along the radio jets. Also, the extended emission appear
to be stronger and more isotropic in the soft band. Blue circles correspond to
X-ray emitting, spectroscopically confirmed members, while red circles to
spectroscopically confirmed members with no X-ray emission (see Tozzi et al.,
2022).

##  4 X-ray properties of the Spiderweb Galaxy: Imaging analysis

The X-ray images of the Spiderweb Galaxy show a prominent, dominant unresolved
component due to the central AGN, and a clear diffuse emission limited to a
radius of $\sim 12$ arcsec. Due to the brightness of the unresolved emission,
the small extension and the composite nature of the diffuse emission, a
detailed imaging and morphological analysis of the image is required to
account for multiple components. The major contribution is due to the
unresolved nucleus, that, due to its intensity, contaminates the surrounding
regions up to radii of several arcsec. Therefore, as a first step, we perform
ray-tracing simulations to obtain an accurate modalization of the unresolved
AGN, assuming the same instrumental setup of the real data.

Figure 2: Simulated images of an unresolved source with the same observational
characteristics (position, roll angle, exposure time and observation date) of
the merged image of the Spiderweb Galaxy, in the 0.5-2 keV (left) and 2-7 keV
(right) band. The inner red circle (2 arcsec radius) is used to extract the
AGN spectrum, while the outer green circle (12 arcsec radius) encompasses the
diffuse emission. The color scale is the same on both bands.

###  4.1 Ray-tracing simulations

To perform accurate ray-tracing simulations, we used the Chandra Ray Tracer
(Chart) tool333https://cxc.cfa.harvard.edu/ciao/PSFs/chart2/.. First, we
created spectral files with Sherpa to reproduce the spectrum of the Spiderweb
nucleus. The spectral fits were performed with Xspec 12.11.1 (Arnaud, 1996)
for each Obsid in order to track possible fluctuations in the flux and
spectral shape as well. To model the nuclear emission, we considered the
emission in the extraction region within a radius of 2 arcsec, which is
clearly dominated by the AGN, but it still may include some minor, unknown
contribution from diffuse emission. Despite this, the background subtraction
in this first step was computed simply rescaling the emission in an annulus
with inner and outer radii of 3′′ and 5′′, respectively, centered on the
nucleus. This background subtraction can be considered as an amenable
approximation to the non-AGN component within 2 arcsec, and it is sufficient
to obtain spectra accurate enough to compute the expected image of the point
spread function (PSF), which has a mild dependence on the photon energy444See,
for example, https://cxc.cfa.harvard.edu/ciao/PSFs/psf_central.html and
references therein.. The spectral fits of the AGN obtained in this first step
are not discussed, and the detailed description of the spectral analysis, with
all the components properly included, are presented in Section 5. Spectral
analysis of the diffuse emission, both thermal and nonthermal, is presented in
Section 6.

The best-fit spectral parameters of a power law emission with intrinsic
absorption obtained for each Obsid are given as input for the image
simulations. We perform 10 ray tracing simulations for each one of the 22
Obsid. We pay attention to normalize the exposure time of the simulations to
the actual effective exposure time of each Obsid after data reduction.
Eventually, we create the evt2 file corresponding to a given simulation with
Marx555https://space.mit.edu/cxc/marx/ .. We merge the ten files of each Obsid
to obtain an event file corresponding to an exposure $10\times$ larger than in
the real data. Finally, we merge all the simulated Obsid by reprojecting the
evt2 files onto the reference frame of Obsid 21483 as in the reduction of real
data. We obtain the soft and hard band images of the simulated source directly
from the reprojected and merged file, dividing the outcome by a factor of 10
to obtain an image with the same nominal exposure time, but a Poisson noise a
factor of $\sim 3$ lower. The images of the merged PSF in the soft and hard
bands are shown in Figure 2, where the extraction region is shown as a red
circle (with a radius of 2 arcsec), while the approximate extent of the
diffuse emission is shown as a green circle (with a radius of 12 arcsec). The
color scale is the same in both bands. The large majority of the signal is
within a radius of 2 arcsec, however the wings of the PSF contribute a
significant fraction of the total intensity in the region between 2 and 12
arcsec, with the PSF having slightly more extended wings in the hard band.
From the simulated image we can robustly estimate that in the soft band 94.8%
of the source emission is within 2 arcsec, and 4.0% between 2 and 12 arcsec,
while only 1.2% is beyond 12 arcsec. In the hard band these values are 91.7%,
5.2% and 3.1%, respectively. From a preliminary estimate, the counts
contributed in our data (full exposure) by the AGN in the 2-12 arcsec region,
where the diffuse emission dominates, amounts to $\sim 300$ net photons in the
0.5-7 keV band. Since we measure a total of $\sim 900$ net counts in the 0.5-7
keV band in the 2-12 arcsec region, this implies that a non-negligible
fraction ($\sim 0.33$) of the diffuse signal is due to the wings of the AGN
emission.

###  4.2 Reconstruction of the diffuse emission

The next step is to accurately normalize the simulated image to the actual
nuclear emission. We note, in fact, that the normalization of the spectra used
for the ray-tracing simulations, were based on an approximate background
subtraction. Therefore, while we trust the distribution of the emission
according to the combined PSF, the overall normalization can be offset by a
few percent. While this translates in an almost negligible correction to the
normalization of the AGN emission, it is very important to estimate the
relation between the AGN normalization and the diffuse component expected in
the inner circle of 2 arcsec (corresponding to $\sim 17$ kpc), for example, to
infer the degree of ”cool coreness”. In both cases, we note that in this step
we consider both thermal and nonthermal components in the extended emission,
since it is impossible, without spectral analysis, to disentangle the two
components.

Therefore, we proceed as follows. We measure the emission in an annulus with
inner and outer radius of 3 and 5 arcsec, respectively, that would correspond
to the standard choice in case of an isolated, unresolved sources (see Tozzi
et al., 2001). Within this annulus we find 1.5% and 1.9% of the total emission
from the nucleus in the soft and hard bands, respectively. In addition, we
also estimate the instrumental (plus unresolved extragalactic X-ray
background) from an annulus with inner and outer radius of 16 and 29.5 arcsec,
respectively. This last region (from where two unresolved sources have been
previously identified and removed) has negligible contribution from the
extended emission and the AGN emission, if any. Therefore, we estimate the
instrumental background per pixel as $b_{i}=B_{i}/A_{B}$, where $B_{i}$ is the
photometry in the 16-29.5 arcsec annulus and $A_{B}$ is the area of the
annulus. The surface brightness of the diffuse emission in the range 3-5
arcsec can be written as $d=[D-(A_{d}\times b_{i})-p_{d}\times
S_{AGN}]/A_{d}$, where $D$ is the total photometry in the 3′′-5′′ annulus,
$A_{d}$ the area of the annulus, and $p_{d}$ is the fraction of the actual
total signal from the AGN falling in the annulus, (computed from the simulated
PSF images). Finally, the total photometry of the nuclear emission is
$S_{AGN}=[C_{AGN}-A_{AGN}\times b_{i}-n_{d}\times A_{AGN}\times d]/p_{AGN}$,
where $C_{AGN}$ is the photometry in the 2 arcsec extraction region, $A_{AGN}$
is the area of the extraction region, and $p_{AGN}$ is the fraction of the AGN
emission falling into the extraction region (computed again from the simulated
PSF images). This system can be easily solved, apart from the unknown, free
parameter $n_{d}$ that is defined as the ratio of the average surface
brightness within a radius of 2 arcsec with respect to the average surface
brightness in the range 3-5 arcsec. We use the parameter $n_{d}$ to quantify
our ignorance on the surface brightness of the diffuse emission in the 2
arcsec extraction region in terms of multiple of the average surface
brightness observed in the 3.5′′-5′′ annulus.

Since it is likely to have an increase in the central surface brightness with
respect to the outer annulus, we expect $n_{d}\geq 1$. In the case of a
strong, small-scale cooling flow the spectral shape may change on the scale of
$\sim 1$ arcsec (corresponding to $\sim 8.5$ kpc), and, therefore, the
parameter $n_{d}$ may be different in the soft and in the hard band. If we
consider, as a reference, the strong cool-core cluster CL1415 at $z\sim 1$
(Santos et al., 2012), and the same physical regions we are using here, we
find that the surface brightness in the inner region is 3.3 and 2.7 times
larger than that in the annulus in the soft and hard bands, respectively.
Despite the fact that CL1415 is a mature, massive cluster observed at much
lower redshift, we adopt values in the range 1-6 for the parameter $n_{d}$,
where $n_{d}=1$ correspond to a constant surface brightness, $n_{d}\sim 3.5$
correspond to a fully developed cool core, and $n_{d}\sim 6$ to an
exceptionally peaked cool core.

We compute the total photometry of the AGN in the soft and hard bands as a
function of the enhancement factor $n_{d}$, and use these values to normalize
the simulated images and compare them with the real data. The elative
normalization of the diffuse emission within a radius of 2 arcsec (the $n_{d}$
parameter), nevertheless, has a marginal impact on the AGN emission. In the
soft band the uncertainty on the AGN normalization is at most 100 net counts
for a strong cool core with $n_{d}=3$ and 200 net counts for an extreme cool
core with $n_{d}=6$ (corresponding to a relative uncertainty of 1.8% and 3.5%,
respectively). In the hard band the uncertainty is much less, due to the lower
surface brightness measured on the 3′′-5′′ annulus. The hard-band net counts
expected from the diffuse emission within 2 arcsec are in the range 8-15,
corresponding to an uncertainty of 0.2%-0.4% in the normalization of the
nuclear hard emission.

If we focus on the estimated flux from diffuse emission within 2 arcsec, we
find that the energy flux associated with diffuse emission within a radius of
2 arcsec is $(1.17\pm 0.09)\times 10^{-15}\times(n_{d}/3)$ erg s-1 cm-2 and
$(2.9\pm 0.4)\times 10^{-16}\times(n_{d}/3)$ erg s-1 cm-2 in the soft and hard
(2-10 keV) bands, respectively. These values have been obtained using the
conversion factors at the aimpoint listed in Table 3 of Paper I. The lower
value in the hard band is due to the low hardness ratio of the emission
measured in the 3′′-5′′ annulus, which is measured to be $HR=-0.85$, and it
provides a first, preliminary hint that the diffuse emission may be dominated
by thermal bremsstrahlung at least in this region.

Figure 3: Background-subtracted soft band images of the Spiderweb Galaxy after
removing the simulated image of the AGN. The image has been smoothed with a
Gaussian kernel with a sigma of 1 pixel, slightly degrading the effective
resolution. The normalization of the surface brightness of the diffuse
emission within a radius of 2 arcsec is free and it is parametrized with
$n_{d}$ ranging from 1 (upper left image) to 6 (lower-right image). There are
a few slightly negative pixels in the image, so that the average pixel value
far from the Spiderweb Galaxy is zero. The unresolved source close to the
center is an AGN that has not been subtracted from the image.

Assuming that the diffuse emission in the hard band is negligible, we focus on
the soft band and attempt to reconstruct an image of the Spiderweb Galaxy
removing the dominant AGN emission. This can be obtained simply by subtracting
the normalized PSF image from the real data. Clearly, the slope of the PSF
image on the scale of 1 arcsec is so steep that an uncertainty on the order of
1 pixel on the relative astrometry of the real data and of the ray-tracing
simulation does not allow an accurate subtraction, and may result in a
distorted distribution of net counts. After the subtraction, in fact, the
resulting image shows a small number of pixels with negative values (about
ten), all very close to the center. Such negative values can be eliminated
with a simple redistribution, consisting in averaging iteratively the maximum
and minimum pixel values within the inner 2 arcsec, untill no negative pixels
are left (about 10 iterations). This procedure preserves the photometry, but
the final distribution of the pixel values is not accurate on the scale of
$\sim 1$ arcsec. Finally, we smooth the images with a 2D Gaussian kernel with
a size of 1 pixel, to obtain a less noisy image without loss of information,
and to subtract the average instrumental plus sky background as estimated from
the 16′′-29.5′′ annulus.

The background-subtracted, soft-band images of the Spiderweb Galaxy after the
removal of the AGN emission are shown in Figure 3 for $n_{d}$ ranging from 1
to 6. As previously mentioned, the presence of a hole in the center for
$n_{d}<3$ is an artifact of the method used to reconstruct the image, and
reflects our ignorance on the distribution of surface brightness on scales on
the order of 1 arcsec in the regions dominated by the AGN emission. As a
consequence, we focus only on the global photometry within a radius of 2
arcsec, ignoring anomalies in the surface brightness distribution. The images
shown in Figure 3, however, still allow us to appreciate that the choice
$n_{d}\geq 4$ corresponds to an almost uniform, prominent core in the
innermost 17 kpc, nicely matching the surface brightness level robustly
measured outside 2 arcsec (where the effects of a sub-pixel mismatch are much
smaller due to the rapidly flattening shape of the Chandra PSF). As previously
discussed, the choice $n_{d}=4$ corresponds to a prominent, but still
plausible, cool core. Thus, we adopt $n_{d}=4$ as a reference value to
estimate the central ICM density, stressing that the exact value has a mild
impact on all the conclusions derived in this work.

Figure 4: Left: Azymuthally averaged profile of the AGN simulated image
(normalized to $n_{d}=4$, red line) compared to the soft-band Spiderweb
profile (blue line). The horizontal line shows the instrumental plus
unresolved X-ray background level. Right: same in the hard (2-7 keV) band.
Figure 5: Azymuthally averaged profile of the diffuse emission (after
subtraction of the AGN emission and of the background) at different $n_{d}$
values. As expected, only the innermost bin is affected by the choice of
$n_{d}$, while the indirect effects on the normalization of the AGN
contamination, thus on the overall normalization of the diffuse emission, is
negligible. The black line is the simulated AGN emission for comparison.

To further support this choice, we look directly at the surface brightness
profile of the simulated AGN and the real data, in both bands. We note that
the effect of the normalization of the AGN image corresponding to different
values of $n_{d}$ is affecting only the first bin of the profile ($r<1$
arcsec), while all the other bins (at $r>1$ arcsec) are practically
unaffected. In the left panel of Figure 4 we show the azymuthally averaged
profile of the AGN simulated image (for the specific choice $n_{d}=4$)
compared to the Spiderweb profile in the soft band. We notice that the data
show signal in excess of the unresolved nuclear source at $r\geq 1.5$ arcsec,
while in the hard band (shown in the right panel), the excess starts to appear
only at $r\geq 2.5$ arcsec. Since the measurement of the diffuse emission at
$r\geq 1.5$ arcsec is practically independent of the $n_{d}$ value, we note
that an approximately constant increase of the surface brightness toward the
center is obtained for $3<n_{d}<4$. This is shown in Figure 5, where we plot
the azymuthally averaged surface brightness profile of the extended emission
in the soft band (that is, the difference of the data and the PSF model shown
in the left panel of Figure 4). Unfortunately, at the moment we do not have
compelling argument to strongly constrain the value of $n_{d}$. The
implications of values different from $n_{d}=4$, as adopted in this work, will
be investigated in Lepore et al. (in preparation).

From the right panel of Figure 4, we also notice that the hard band emission
is less extended than the soft emission, that is detected well above the
background up to 12 arcsec ($\sim 100$ kpc) from the nucleus of the Spiderweb
Galaxy. Instead, in the hard band there is no excess at $r\leq
2^{\prime\prime}$, and the diffuse emission is less significant and limited to
the range $3^{\prime\prime}<r<7^{\prime\prime}$. This suggests that the
diffuse emission is contributed by two distinct components. The one with
larger hardness ratio is less extended and is most likely associated with the
IC from the relativistic population of electrons in the jet. The soft
component is more extended up, to a distance of $\sim 100$ kpc from the
center, and is possibly associated with hot gas. In the following section we
proceed with the spectral analysis of the AGN-dominated emission included
within a radius of 17 kpc, while in Section 6 we investigate the diffuse
emission at radii $>2^{\prime\prime}$, after accounting for the background and
AGN contamination.

##  5 X-ray properties of the Spiderweb Galaxy: spectral analysis of the
central AGN

###  5.1 Global nuclear emission

Compared to the shallower X-ray observation presented in Carilli et al.
(2002), our new, deep exposure allows us to obtain a more detailed spectral
analysis of the nuclear emission, and, in particular, to investigate the
contribution of the diffuse component, on the basis of the accurate imaging
analysis presented in the previous Section. In the following, we perform our
reference analysis of spectra extracted separately from each Obsid, and for
which we consider the respective Obsid ARF and RMF calibration files.
Ultimately, this info are considered all together during the fitting
procedure. This approach allows us to keep track of the different effective
area corresponding to each Obsid. Eventually, we also analyze the spectrum
obtained by merging the single Obsid spectra, using cumulative ARF and RMF
files obtained by weighting single Obsid files by the corresponding exposure
times. We find that this approach gives results that are in agreement with our
reference spectral analysis. Given the brightness of the source, each Obsid
has at least 300 net counts within the extraction region (a 2′′ radius
circle). Therefore, we compute the ARF and RMF files for each one of the 22
Obsid.

We fit the nuclear emission with Xspec 12.11.1 (Arnaud, 1996) over the 0.5-9.0
keV energy range, adopting the simplest possible model, as done for the
analysis of the protocluster members in Paper I. The AGN emission is,
therefore, described by an intrinsically absorbed power law, using the model
components zwabs and powerlaw. The galactic absorption is described with the
model tbabs666See
https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node268.html ., and its
value is fixed to $3.18\times$ 1020 cm-2 according to the HI map of the Milky
Way (HI4PI Collaboration et al., 2016).

As previously discussed, the contribution from diffuse emission within 2
arcsec cannot be constrained accurately but it must be parametrized by the
enhancement factor $n_{d}$, that represents the ratio of the surface
brightness within 2 arcsec over that measured in the 3-5 arcsec annulus. As a
preliminary test, we perform a straightforward spectral analysis assuming two
background files: the first including only the instrumental plus unresolved
background, sampled in the annular region between 16 and 24 arcsec; the second
including instrumental background and extended emission contribution directly
sampled in the 3-5 arcsec annulus. The last choice corresponds to the
assumption that the surface brightness of the diffuse emission is constant
within 5 arcsec with no central enhancement ($n_{d}=1$). We find that the fit
obtained in the second case significantly improves, with $\Delta C_{stat}\sim
23$. This simply confirms that we must account for some contribution from
diffuse emission within 2 arcsec, possibly with $n_{d}>1$, as we already shown
from the imaging analysis. At the same time, the best-fit spectral parameters
of the AGN emission model are in agreement within 1 $\sigma$ irrespective of
the background treatment, confirming that the contribution of the diffuse
emission is very low compared to the nuclear emission. This is expected also
from photometry, since we measure 9710 net counts in the 0.5-7 keV band within
2 arcsec, while, from the imaging analysis, we expect only $95\times(n_{d}/3)$
and $8\times(n_{d}/3)$ net counts in the soft and hard bands, respectively,
from diffuse emission. Overall, within a radius of 2 arcsec, we expect a
fraction of $0.015\times(n_{d}/3)$ of the total emission to be not associated
with the AGN in the soft band, and only $0.003\times(n_{d}/3)$ in the hard
band. Such contribution is so small that $n_{d}$ cannot be constrained by
spectral analysis.

As a next step, we search for the best spectral modelization of the diffuse
emission, fitting the emission within the 3-5 arcsec annulus after
instrumental background subtraction and after accounting for 1.5% of the
nuclear emission expected in that region. We find that statistically
equivalent fits are obtained with a thermal mekal model or power law, with
$kT\sim 3.0$ keV and $\Gamma\sim 2.7$, respectively. On the other hand, the
fit considerably gets worse if $\Gamma=2$ is assumed (with $\Delta
C_{stat}\sim 16$). This shows us that the diffuse component in this region is
better described by a soft, thermal emission than by a nonthermal, $\Gamma\sim
2$ power-law emission. Therefore, we describe the weak diffuse component
within 2 arcsec by using a mekal model with $kT=3$ keV.

Incidentally, we note that we do not consider here a nuclear plus diffuse
contribution possibly associated with star formation. The total SFR associated
with the core of the Spiderweb Galaxy plus the star-forming satellite galaxies
(the so-called flies) has been estimated to range from $70M_{\odot}$/yr (Hatch
et al., 2009) to $1400M_{\odot}$/yr (Seymour et al., 2012), and is expected to
be significantly obscured. If we maximize the star-formation (SF) contribution
with an unabsorbed spectrum with $\Gamma\sim 2.0$, we find that in our data we
expect $\sim 2.9\times({\rm SFR}/100M_{\odot}/yr)$ net counts in the 0.5-7.0
keV range. Considering the maximum value of $1400M_{\odot}$/yr, we estimate an
upper limit of 40 net counts to the SF contribution within 2 arcsec, which is
about 0.3% of the AGN component and, therefore, can be safely ignored. The
possible contribution from SF associated with the Ly$\alpha$ halo, has been
estimated to be in the range $60-140M_{\odot}$/yr, and is also expected to be
completely negligible in the annular region between 2′′ and 12′′ that we
consider in the Section 6.

Figure 6: Left: Best-fit spectral slope $\Gamma$ of the nuclear emission of
the Spiderweb as a function of the enhancement factor $n_{d}$ (red points).
The best-fit values obtained with direct background subtraction are shown as
blue and green points, where blue is for the background sampled in the 2.6-4.6
arcsec annulus, and green in the 16-24 arcsec annulus. Right: Same for the
best-fit value of the intrinsic absorption $N_{H}$.

Now we focus on the spectral analysis of the nuclear emission. In Figure 6 we
show the best fit values as a function of the enhancement factor $n_{d}$ that
paramterizes the normalization of the diffuse component within 2 arcsec. We
see that the spectral slope $\Gamma$ is hardly affected, since it is robustly
constrained by the hard-band signal at energies larger than 2 keV, where the
contribution of diffuse emission is estimated to be very small. However, when
compared to the values obtained with direct background subtraction, with and
without the inclusion of the foreground as estimated in the 3-5 arcsec
annulus, we find that the 1 $\sigma$ uncertainties almost double, even though
it is still below 2%. This marginal but noticeable effect is simply due to the
fact that the inclusion of a small thermal contribution makes the model more
similar to the data, resulting in a milder dependence of $C_{stat}$ on the
parameters of the fit, hence larger error bars. We also note that the
inclusion of a thermal component with a fixed $n_{d}$ does not imply
additional free parameters (in other words, not only the normalization but
also the shape, namely temperature and metallicity, are kept frozen). When we
turn to the intrinsic absorption $N_{H}$ (right panel of Figure 6), we notice
that the best fit value increases slightly with $n_{d}$. We do expect this
trend since, when the thermal component increases, the nuclear emission in the
soft band is correspondingly reduced, and this is compensated with a slightly
larger intrinsic absorption. Here the typical 1 $\sigma$ error bar is constant
at the level of 5%. As expected, these results do not change if we describe
the diffuse emission with a $\Gamma\sim 2.7$ power law instead of a $\sim 3$
keV mekal model. To summarize, the investigation of the effect of the diffuse
emission within a radius of 2 arcsec on the spectral analysis of the nuclear
emission, shows that the properties of the nuclear emission are marginally
affected by the parameter $n_{d}$. According to the discussion in Section 4.2
based on our imaging analysis, in the following we assume a value $n_{d}=4$
for reference, bearing in mind that different $n_{d}$ values would not impact
the results presented in this work.

Figure 7: Merged spectrum (folded with the instrument spectral response) of
the nucleus of the Spiderweb Galaxy with the best fit model (first column of
Table 1, with $n_{d}=4$). The residuals are shown in the bottom panel. There
is no significant excess in the soft band, showing that diffuse emission is
properly accounted for within the limits of the spectrum quality.  Table 1:
Best-fit values for the reference spectral analysis of the nuclear emission of
the Spiderweb Galaxy assuming $n_{d}=4$. Fluxes are corrected for Galactic
absorption, while luminosities are corrected both for Galactic and intrinsic
absorption

.

parameter | aperture (2′′) | PSF correction
---|---|---
$\Gamma$ | $1.883_{-0.012}^{+0.016}$ | $1.85\pm 0.02$
$N_{H}$ | $4.23_{-0.17}^{+0.28}\times 10^{22}$ | $4.2_{-0.2}^{+0.3}\times 10^{22}$
$F_{S}$ | $5.64\pm 0.17\times 10^{-14}$ | $5.95\pm 0.18\times 10^{-14}$
$F_{H}$ | $1.38\pm 0.02\times 10^{-13}$ | $1.51\pm 0.02\times 10^{-13}$
$L_{0.5-2keV}$ | $3.10\pm 0.09\times 10^{45}$ | $3.27\pm 0.10\times 10^{45}$
$L_{2-10keV}$ | $4.29\pm 0.05\times 10^{45}$ | $4.68\pm 0.05\times 10^{45}$

We also consider aperture correction that introduces a mild spectral
distortion due to the different PSF in the soft and hard bands. From the AGN
ray-tracing simulations, we estimate that total soft and hard fluxes and
luminosities are larger by a factor $1.05$ and $1.09$, respectively, compared
to that measured within 2 arcsec. Fluxes are corrected only for the Galactic
absorption, while luminosities are corrected for both Galactic and intrinsic
absorption. Error on fluxes and luminosities account for the uncertainty
associated with the full range $n_{d}=1-6$ and are 3% in the soft and 1% in
the hard band. The effect of the PSF correction on the spectral shape is
instead very mild, with the intrinsic slope $\Gamma$ being harder by less than
2%. We verified a posteriori that the fit is not sensitive to the assumed
value for the parameter $N_{HGal}$ that describes the Galactic absorption. If
left free, $N_{HGal}$ has a best-fit value of $3.2_{-0.9}^{+1.6}\times
10^{20}$ cm-2, right on top of the value based on the HI map of the Milky Way
(HI4PI Collaboration et al., 2016). Finally, we search for the presence of a
neutral iron line at 6.4 keV rest-frame, corresponding to 2.03 keV in the
observing frame, but we found no signal. The mild obscuration corresponds to a
source dominated by the transmitted emission, while significant iron line are
expected when the reflected component is dominant. If we perform a blind
search for emission lines in the range 1-7 keV, we do not find any significant
candidate. The merged spectrum and the best-fit model, along with the
residuals, are shown in Figure 7. The residuals are very regular with no
significant departure from pure noise. In particular, we do not find
significant residuals in the soft band, showing that the diffuse emission is
properly accounted for within the limits of the spectrum quality. Our best-fit
model for the nuclear spectrum of the Spiderweb Galaxy is shown in Table 1,
with and without aperture correction in terms of normalization and spectral
shape.

In conclusion, an X-ray obscuration at the level of $4.2\times 10^{22}$ cm-2
can be provided by the surrounding galactic medium and circumgalactic medium
as observed in high-z galaxies in the CDFS (Circosta et al., 2019; D’Amato et
al., 2020a), or in the BCG of a $z\sim 1.7$ protocluster (D’Amato et al.,
2020b). Therefore it is plausible that the obscuration is not dominated by a
torus, and that the AGN in the Spiderweb Galaxy is an unabsorbed, TypeI
quasar, as also suggested by the detection of broad lines. This would be
unusual since radio galaxies are typically associated with obscured TypeII
AGN. However, the peculiar phase of the Spiderweb Galaxy may justify the
copresence of a strong quasar and a strong radio activity.

###  5.2 Spectral variability

We can investigate the variability of the AGN in the Spiderweb Galaxy on a
scale of $\sim 9$ months, which is the time span over which the observations
of the Large Program have been taken (November 2019 - August 2020). In
addition, we have one measurement at a distance of $\sim 20$ years (in 2000),
and the timescale provided by the single Obsid duration, from a minimum of 5
to a maximum of 14 hours.

Figure 8: Scatter plot of the best fit values of the intrinsic absorption
$N_{H}$ and the spectral slope $\Gamma$ for each Obsid. Error bars refer to 1
$\sigma$ c.l. on a single parameter. Shaded area show the 1 $\sigma$ interval
for the best fit value of $N_{H}$ from the global fit.

First we investigate the spectral variability. We fit separately each Obsid
and we plot the distribution of best fit values for $N_{H}$ and $\Gamma$. We
note that for the fits of each Obsid we do not apply the small PSF correction
to the spectral shape. As we show in Figure 8, the distribution of the best-
fit values in the $N_{H}$-$\Gamma$ plane appears to be roughly consistent with
the global best-fit, with the apparent correlation due to the well-known
degeneracy of $N_{H}$ and $\Gamma$. If we compute a simple reduced $\chi^{2}$
with respect to the average value (after removing the unique outlier with
$N_{H}>10^{23}$ cm-2 obtained for Obsid=22922), we find $\chi^{2}\sim 0.98$
and $\chi^{2}\sim 1.21$ for 21 degrees of freedom, for $\Gamma$ and $N_{H}$,
respectively. In both cases we are not able to reject the null hypothesis of a
constant value also in the case of $N_{H}$, for which we have a probability
slightly less than $0.25$ to obtain a larger $\chi^{2}$. If we do not remove
the unique strong outlier (corresponding to Obsid=22922), we are able to
reject the null hypothesis of a constant $N_{H}$ on the entire observing
period with a probability less than $0.05$ to obtain a larger $\chi^{2}$. To
summarize, from a simple $\chi^{2}$ test on the entire data set, we get only a
weak hint, if any, of spectral variability.

Figure 9: Best-fit values of $\Gamma$ for each Obsid as a function of the
Mission Timeline $MET$ (years). The first point has been moved to $MET=21.5$
to be shown in the same linear plot. The horizontal shaded line shows the
best-fit value of the cumulative spectrum with PSF correction, as listed in
Table 1.

Then, in Figure 9 and 10 we plot the best-fit values of $\Gamma$ and $N_{H}$
as a function of the Mission Timeline (MET). In Figure 9 we can see that
$\Gamma$ is consistent with the best-fit value of the spectral analysis of the
cumulative spectrum for each Obsid within 1 $\sigma$, as confirmed by a simple
$\chi^{2}$ test. On the other hand, in Figure 10 we find that $N_{H}$ shows a
higher level of fluctuations, and that the lowest values of $N_{H}$ seems to
occur in contiguous periods within a narrow time frame. We repeat the plot
after freezing the spectral slope to the average best-fit value $\Gamma=1.88$.
This reduces the noise around the average value, removing the degeneracy
between $N_{H}$ and $\Gamma$, but the period with low $N_{H}$ is still visible
(see Figure 10, lower panel). Actually, when $\Gamma$ is frozen, the reduced
$\chi^{2}$ with respect to a constant $N_{H}$ value increases to
$\chi^{2}=1.65$ despite the smaller scatter, due to the smaller error bars,
after the removal of the single outlier (Obsid 22922). To quantify the
spectral variability of $N_{H}$ we discard the outlier and focus on three
contiguous periods. One is when a low $N_{H}$ is measured, corresponding to
$22.179<{\rm MET}<22.323$, while the rest consists in all the observations
before ${\rm MET}=22.179$ and after ${\rm MET}=22.323$. We obtain
$N_{Hhigh}=(5.02\pm 0.32)\times 10^{22}$ cm-2 and $N_{Hlow}=(2.95\pm
0.24)\times 10^{22}$ cm-2, with a difference of a factor $\sim 1.7$ of $\sim
5$ $\sigma$777We note that the statistical errors of the two measurements
refer to the average, central value in the corresponding period, and do not
reflect the dispersion observed across the measurement in each Obsid shown in
Figure 10.. To summarize, we conclude that we see a modest but significant
spectral variation corresponding to a change in the intrinsic absorption by a
factor $\sim 1.7$ on a timescale of $\sim 1$ year. This also implies that part
of the intrinsic absorption happens close to the SMBH and it is associated
with a clumpy obscuring torus, while part of the absorption may still be
associated with the intergalactic medium, as suggested in Section 5.1.

Figure 10: Upper panel: Best-fit values of $N_{H}$ for each Obsid as a
function of the Mission Timeline $MET$ (years). The first point has been moved
to 21.5 to be shown in the same linear plot. The horizontal shaded line shows
the best-fit value of the cumulative spectrum with PSF correction, as listed
in Table 1. Lower panel: Same, but with best-fit values of $N_{H}$ obtained
with $\Gamma$ frozen to 1.88. The vertical dashed lines bracket the period
with low intrinsic absorption as discussed in the text.

###  5.3 Flux variability

We compute the soft and hard band flux values in each Obsid keeping
$\Gamma=1.88$ frozen, and correcting for the Galactic absorption. The result
is shown in Figure 11. We note that for simplicity here we report the fluxes
within 2 arcsec, without correcting for the PSF effects in each band, since
our discussion would not be affected by this correction. We notice a
significant variability on a scale of a few months in the observing frame. If
we fit the flux as a function of the time with a constant, the reduced
$\chi^{2}$ is $\sim 3.0$ in both bands for 21 degrees of freedom (for
consistency, we conservatively removed Obsid 22922 that appears to be an
outlier both for shape and normalization of the spectrum). Also if we remove
the lowest and highest points, we still obtain a reduced $\chi^{2}\sim 2$ with
19 degrees of freedom. The rms fluctuation is $12$% in both bands, therefore
very mild, however it is statistically significant. In addition, we note that
the flux variability seems to have a temporal behavior different from the
spectral variation (or change in $N_{H}$). This can be seen by the lack of
correlation between the best-fit $N_{H}$ values and the percentage variation
of the flux, shown in Figure 12. Therefore, we conclude that we are able to
detect variability both in the normalization of the spectrum and in the
spectral shape, and that there are no signs of correlation between the two
phenomena.

Figure 11: Soft (red points) and hard (blue points) flux values measured in
each Obsid within a radius of 2 arcsec as a function of the Mission Timeline.
Fluxes are corrected for the Galactic absorption. Horizontal lines show the
average values.  Figure 12: Best-fit values of $N_{H}$ plotted against the
ratio of the flux in each Obsid over the average flux (red and blue dots
correspond to the soft and hard bands, respectively).

##  6 X-ray properties of the Spiderweb Galaxy: Diffuse emission

###  6.1 Imaging analysis

Figure 13: Left: Background-subtracted soft band image of the Spiderweb Galaxy
after AGN subtraction (with $n_{d}=4$). The image has been smoothed with a
Gaussian kernel with a sigma of 1 pixel, slightly degrading the effective
resolution. The green and magenta boxes correspond to the regions used for the
east and west jet spectral analysis, respectively. Two AGN circular extraction
regions (in blue) are removed (the AGN, with a radius of 2 arcsec, and a
nearby AGN with a radius of 1.5 arcsec). The large green circle is the region
used for the spectral analysis of the isotropic diffuse emission, excluding
both jet regions and the AGNs. Red contours show radio emission observed in
the 10 GHz band with the JVLA (Carilli et al., 2022) at levels of $0.03$,
$0.2$, $2$ and $20$ mJy/beam.

In Figure 13 we show the extraction regions used for our analysis of the
diffuse emission, overplotted on the background-subtracted images after
removing the central AGN (we used $n_{d}=4$ for display). Consistently with
the radio and X-ray combined analysis of the jets presented in Carilli et al.
(2022) and Anderson et al. (2022), we select two box regions for the east and
west jet regions. These regions cover all the radio emission observed at 10
GHz (see Carilli et al., 2022), include most of the diffuse emission in the
soft band (see Figure 13, left) and almost the entire diffuse emission in the
hard band (see Figure 13, right). The extraction region for the analysis of
the diffuse emission outside the jet region is obtained by selecting a circle
centered on the Spiderweb Galaxy with a radius of 12 arcsec, corresponding to
$\sim 100$ physical kpc at $z=2.156$. From this circular region, we remove two
boxes corresponding to the east and west jets, and two circular regions
corresponding to the central AGN (a circle with a radius of 2 arcsec) and to
the source XID 7 (a circle with a radius of 1.5 arcsec) that is found to be a
companion quasar (see Paper I). The boxy shape is chosen to follow at best the
jet shape and to keep, at the same time, a simple geometry. The instrumental
plus unresolved extragalactic X-ray background is sampled from an annulus with
inner and outer radius of 16 and 29.5 arcsec, respectively, which is the same
used for the analysis of the nuclear emission.

In Figure 13, it is possible to appreciate that the diffuse emission in the
soft band is clearly visible outside both jet regions, out to a radius of 12′′
from the nucleus, while the shape of the diffuse emission in the hard band is
almost entirely elongated along the radio jets, with no emission elsewhere. On
the basis of this preliminary imaging analysis, we argue that the emission
along the jets is dominated by IC emission from the relativistic jet
population, while the more isotropic soft emission may be dominated by thermal
emission from shocked gas. The diffuse emission along the jets is clearly
harder, as expected in case of a power law, nonthermal emission, while the
diffuse emission outside the jet regions has an hardness ratio $HR\sim-1$,
being entirely soft, as expected in the case of thermal emission with a
relatively low temperature, or a very steep power law. Incidentally, this
result may provide another piece of evidence that inverse Compton losses are
important in high-z radio sources, and that this effect is likely to be a
strong contributor to the well-known correlation between ultra-steep spectrum
and redshift for radio sources (Tielens et al., 1979; Blumenthal & Miley,
1979; Morabito & Harwood, 2018). On the other hand, another plausible
mechanism for a steep radio spectrum may be associated with an increased
ambient density (Klamer et al., 2006). Clearly, both mechanisms may be at play
in the Spiderweb Galaxy. A systematic exploration of high-z radio galaxies in
a variety of environments would be key to understand the physics behind the
correlation between redshift and their spectral properties. This relevant
aspect is not explored in this work, but it constitutes a clear science case
for future Chandra programs.

While the nature of the diffuse emission is further investigated in our
spectral analysis, as a preliminary step, we obtain the photometry in the
three different regions. We find $207\pm 19$ and $228\pm 21$ net counts in the
0.5-7 keV band in the west and east jet regions, respectively, after
subtracting the background. In the circular region with a radius of 12 arcsec,
after removing the west jets, east jet, and AGN regions, we measure $447\pm
53$ net counts in the 0.5-7 keV band. We note that the isotropic, diffuse
component, where we expect to find thermal emission, overlaps with the west
and east jet regions, while the AGN wings overlaps with all the diffuse
components. Therefore, we have to take into account two and three components
to perform spectral analysis of the isotropic diffuse emission and the jet
emission, respectively. While we already have established the shape and
normalization of the AGN emission, we need to measure first the diffuse,
isotropic emission (accounting for AGN contamination), and finally the
emission in the two jet regions (accounting for AGN contamination and
isotropic emission).

Table 2: Best-fit values for the reference spectral analysis of the diffuse emission. We note that in this table, flux and luminosity values of the isotropic (thermal) emission have not been corrected by the geometrical factor, but directly correspond to the best-fit values obtained from the extraction region. Error bars refer to 1 $\sigma$ c.l. on a single parameter. region |  $kT$ (mekal) | $F_{S}$ | $F_{H}$ | $L_{0.5-2keV}$ | $L_{2-10keV}$
---|---|---|---|---|---
| keV | erg s-1 cm-2 | erg s-1 cm-2 | erg s-1 | erg s-1
isotropic | $1.98_{-0.43}^{+0.70}$ | $(2.5\pm 0.6)\times 10^{-15}$ | $(2.4\pm 0.9)\times 10^{-16}$ | $(7.7\pm 1.8)\times 10^{43}$ | $(7.3\pm 1.7)\times 10^{43}$
region | $\Gamma$ | $F_{S}$ | $F_{H}$ | $L_{0.5-2keV}$ | $L_{2-10keV}$
| - | erg s-1 cm-2 | erg s-1 cm-2 | erg s-1 | erg s-1
west jet | $1.90\pm 0.20$ | $(1.32\pm 0.21)\times 10^{-15}$ | $(1.78\pm 0.28)\times 10^{-15}$ | $(4.1\pm 0.7)\times 10^{43}$ | $(5.5\pm 0.9)\times 10^{43}$
east jet | $2.51\pm 0.21$ | $(1.89\pm 0.25)\times 10^{-15}$ | $(1.02\pm 0.13)\times 10^{-15}$ | $(1.18\pm 0.15)\times 10^{44}$ | $(6.4\pm 0.8)\times 10^{43}$

###  6.2 Spectral analysis of the diffuse emission: Isotropic component

Figure 14: Left: Spectrum (folded with the instrument spectral response) of
the isotropic diffuse emission (after removing the central, AGN-dominated
region, and the jet regions) fitted with a thermal mekal model plus the AGN
contamination. Right: Unfolded spectrum with the two model components shown
with dotted lines. The AGN contamination (absorbed power law) is dominant at
energies $>1.5$ keV, while the thermal component is dominant below $1$ keV.

The spectral fits are performed with Xspec 12.11.1 (Arnaud, 1996) over the
energy range 0.5–9.0 keV of the merged spectrum. In fact, we do not use single
Obsid for the fits, since several observations would have less than 10 net
counts, so that the spectral shape is completely undetermined in each separate
Obsid. In this case, we use cumulative ARF and RMF files obtained by weighting
single Obsid files by the corresponding exposure times. Galactic absorption is
described with the model tbabs and its value is fixed to $3.18\times 10^{20}$
cm-2. We used Cash statistics (Cash, 1979) applied to the source plus
background, which is preferable for low S/N spectra (Nousek & Shue, 1989). We
assume that the emission observed within 12 arcsec is approximately symmetric
and centered on the Spiderweb Galaxy. We fit the diffuse emission outside the
jet regions, after removing the AGN as previously described, with a mekal
model and a power law model.

From the ray-tracing simulation, we find that $3.3$% of the soft nuclear
emission inside 2 arcsec is found in the extraction region. Therefore, we add
this component modeling it with an absorbed power law with the parameters
frozen to the best-fit values found in the AGN extraction region, including
the correction for the PSF effect that makes the AGN contamination slightly
harder than the spectrum measured within 2 arcsec. This last correction
implies that $\Gamma=1.85$ is a suitable expectation. We also do not include
the neutral iron line component, since it has been found to be negligible (see
Section 5.1). The Galactic absorption is clearly the same for both components.
Therefore, after accounting for a fixed component associated with the AGN
contamination, we perform a fit of the isotropic diffuse emission, that, as
thoroughly discussed in the imaging analysis (Section 4.2) is almost entirely
included in the soft band.

The unabsorbed power law model has a best-fit slope of $\Gamma=3.27\pm 0.43$
for a flux of $(2.7\pm 0.7)\times 10^{-15}$ erg s-1 cm-2 and $(4.9\pm
2.6)\times 10^{-16}$ erg s-1 cm-2 in the 0.5–2 keV and 2–10 keV bands. If,
instead, we assume a thermal model, we find that the fit improves with a
$\Delta C\sim 10$. We obtain a best-fit temperature of $1.98_{-0.43}^{+0.70}$
keV and metallicity of $Z=0.26_{-0.26}^{+1.35}Z_{\odot}$, or better, a
$1\sigma$ upper limit $Z<1.6Z_{\odot}$. 888Based on our experience with lower
redshift ICM, and considering the strong K-correction, we estimate that the
signal should be higher by a factor $>3$ before obtaining a statistically
significant measurement of the iron emission line complex. This would be of
paramount relevance to investigate the chemical volution of the ICM well
beyond its current limit at $z\sim 1$ (see Mernier et al., 2018; Liu et al.,
2020). This is clearly a relevant scientific goals of future, high-angular
resolution X-ray missions, such as Lynx (The Lynx Team, 2018) and AXIS
(Mushotzky & AXIS Team, 2019; Marchesi et al., 2020).. We remark that the 1
$\sigma$ error bars (or upper limits) are obtained by varying a single
parameter, keeping the other parameters frozen to their best-fit values. The
measured fluxes are $(2.5\pm 0.6)\times 10^{-15}$ erg s-1 cm-2 and $(2.4\pm
0.9)\times 10^{-16}$ erg s-1 cm-2 in the 0.5–2 keV and 2–10 keV bands,
consistent with those found for the power law model. The 0.5-10 keV rest-frame
luminosity is $L_{0.5-10{\tt keV}}=(1.50\pm 0.36)\times 10^{44}$ erg s-1. To
correct for the excluded area (the jet regions and the AGN extraction region),
we can assume an approximately constant surface brightness and apply a
geometric correction factor of 1.22, obtaining $L_{0.5-10{\tt keV}}=(1.83\pm
0.44)\times(1+0.03\,n_{d})\times 10^{44}$ erg s-1, that, for our reference
choice $n_{d}=4$, is equal to $L_{0.5-10{\tt keV}}=(2.0\pm 0.5)\times 10^{44}$
erg s-1. A more accurate study including a detailed analysis of the surface
brightness distribution will be presented in a forthcoming paper (Lepore et
al., in preparation).

The spectrum fitted with a thermal component is shown in Figure 14. In the
right panel we show that the AGN contamination is dominant at energies $>1.5$
keV, while the thermal component is higher below 1 keV. We find that the AGN
contamination contributes 25% and 92% of the signal in the soft and hard
bands, respectively. From a statistical point of view, the thermal model is
marginally better than a power law model ($\Delta C\sim 10$ with an additional
degree of freedom). However, the slope of the power law model is softer than
the AGN emission at 3$\sigma$ c.l. This implies that the diffuse emission
cannot be simply ascribed to contamination from the AGN, and that it is
unusually steep for nonthermal emission mechanisms. Therefore, adding these
aspects to the statistical improvement obtained with the thermal model, we
conclude that the diffuse almost isotropic emission detected within 12 arcsec
around the Spiderweb Galaxy is very likely due to thermal emission from
diffuse hot baryons. Unfortunately, as already mentioned, we cannot detect the
emission line complex of the H-like and He-like iron ions because of the low
S/N and for the unfortunate rest-frame position of the iron lines at $\sim
2.1-2.2$ keV. Moreover, the large K-correction combined with the loss of
sensitivity below 1 keV, makes it unfeasible to use the iron L-shell complex
to efficiently constrain metallicity and temperature. However, the formal
best-fit value for the metallicity $Z=0.26_{-0.26}^{1.35}\,Z_{\odot}$ leaves
room for enrichment above solar abundance. Finally, we would like to point out
that the presence of ICM is strongly corroborated by the successful detection
of SZ signal in the same position and shape of the X-ray diffuse emission,
providing a completely independent and remarkably consistent measurement of
the ICM (Di Mascolo et al., in preparation).

A final note on the ICM temperature: we notice that the best fit value of
$1.98_{-0.43}^{+0.70}$ keV is somewhat lower than the $\sim 3$ keV component
we assumed to account for the diffuse emission within a radius of 2 arcsec
when fitting the AGN spectrum. In fact, one may expect the opposite, if any,
with a lower temperature associated with denser, central ICM as in normal cool
core. However, we must bear in mind that, apart from the unavoidable large
uncertainties on its spectral shape, the diffuse component overlapping the
outshining AGN emission is necessarily a mix of thermal emission associated to
the central ICM and nonthermal emission associated with the bases of the jets,
and therefore it is actually expected to be harder. Needless to say, it is
impossible to speculate about the share of thermal and nonthermal emission
within a distance of 2 arcsec ($\sim 17$ physical kpc).

If we consider the best estimate for an evolved L-T relation, for a
temperature of 2 keV, we obtain $L_{X}=1.0\times 10^{44}$ erg s-1 and
$L_{X}=0.7\times 10^{44}$ from Reichert et al. (2011) and Pratt et al. (2009),
respectively, where $L_{X}$ is the bolometric X-ray luminosity. In both cases
we applied the phenomenologically estimated evolution correction
$E(z)^{-0.23}$ at $z=2.156$. Therefore, our estimate of $L_{0.5-10{\tt
keV}}=(2.0\pm 0.5)\times 10^{44}$ erg s-1 (after including the geometrical
correction and the core emission with $n_{d}=4$) is about a factor of 2 above
the expected average relation, but still in agreement with it given the large
intrinsic scatter observed particularly at low temperatures. This result, if
coupled to the forthcoming results from the SZ data, is consistent with the
observation of a virialized halo, as we further discuss in Section 7.

Figure 15: Left: Spectrum (folded with the instrument spectral response) of
the emission in the west jet region (including the AGN contamination and the
residual thermal emission from the ICM) fitted with a power law. Right:
Unfolded spectrum with the three components in the model shown with dotted
lines. The lowest dotted line corresponds to the thermal emission from the
overlapping ICM, while the line in the middle accounts for the AGN wings
contamination. The IC from the relativistic population in the west jet is
shown by the unabsorbed power law that is dominating at any energy.

Figure 16: Left: Spectrum (folded with the instrument spectral response) of
the emission in the east jet region (including the AGN contamination and the
residual thermal emission from the ICM) fitted with a power law. Right:
Unfolded spectrum with the three components in the model shown with dotted
lines. The lowest dotted line corresponds to the thermal emission from the
overlapping ICM, while the line in the middle accounts for the AGN wings
contamination. The IC from the relativistic population in the east jet is
shown by the unabsorbed power law that is dominating at any energy.

###  6.3 Spectral analysis of the diffuse emission: Jet regions

To fit the diffuse component in the west and east jet regions, we use a power
law emission as a baseline model, being more appropriated for IC emission, as
expected along radio jets. On the other hand, we do not include any intrinsic
absorption, but only the Galactic absorption. As for the AGN contamination, we
apply the same procedure described in Section 6.2. In this case, however, we
also need to include the presence of the thermal diffuse emission that is
expected to be present also in the jet regions. The uncertainty associated
with the thermal component within the two box regions is, however, quite
large, since the surface brightness distribution is not known and it may be
affected by the jet itself. Therefore, we vary the contribution from the
thermal component when constraining the nonthermal emission in the jet regions
to explore its effect on the best-fit parameters.

First we focus on the west jet region, which is a rectangle of $4.9\times 5.8$
arcsec2 centered on RA=11:40:48.03,DEC=-26:29:10.89. The AGN contamination is
obtained by rescaling the emission measured within 2 arcsec by the factor
$4.38\times 10^{-3}$. The diffuse thermal component in this region is obtained
rescaling the emission measured in the diffuse extraction region by a factor
of $7.7\times 10^{-2}$, which corresponds to assuming a constant surface
brightness. The best-fit value for the slope of the IC emission is
$\Gamma_{west}=1.90\pm 0.20$. We investigate how the slope of the emission
depends on our subtraction of the thermal component that we expect in that
region. We find that the best-fit $\Gamma$ becomes slightly softer when
reducing the normalization of the thermal component, up to a value of
$\Gamma=2.00\pm 0.20$ when the thermal component is ignored. The stability of
the fit is not surprising, since in this region the fit is driven by the
prominent emission in the hard band, and therefore is modestly affected by the
uncertainty in the soft band. If we attempt to fit the emission with a thermal
model, we obtain an unrealistic temperature of $9.3_{-2.5}^{+3.7}$ keV, and
the quality of the fit decreases with $\Delta C_{stat}\sim 9$.

We find fluxes of $(1.32\pm 0.21)\times 10^{-15}$ erg s-1 cm-2 and $(1.78\pm
0.28)\times 10^{-15}$ erg s-1 cm-2 in the 0.5-2 and 2-10 keV bands,
respectively, after correcting for Galactic absorption. These values
correspond to luminosities of $(4.1\pm 0.7)\times 10^{43}$ erg s-1 and
$(5.5\pm 0.9)\times 10^{43}$ erg s-1 in the 0.5-2 and 2-10 keV rest-frame
bands, respectively. The spectrum in the west jet region folded with the
spectral response, and unfolded, including the three components used for the
fit, is shown in Figure 15. We note that the IC emission is more than 3 times
larger than the AGN contamination across the entire energy range, while the
thermal component is almost negligible.

We then focus on the east jet region, which is a rectangle of $7.0\times 5.1$
arcsec2 centered on RA=11:40:48.76, DEC=-26:29:09.29. The AGN contamination is
obtained by rescaling the emission measured within 2 arcsec by the factor
$5.79\times 10^{-3}$. The diffuse thermal component in this region is obtained
rescaling the emission measured in the diffuse extraction region by a factor
of $9.2\times 10^{-2}$, which corresponds again to a constant surface
brightness. The best-fit value for the slope of the IC emission is
$\Gamma_{east}=2.51\pm 0.21$. As in the previous case, we investigate how the
slope of the emission depends on the thermal component that we expect in that
region. We find that the best-fit $\Gamma$ becomes slowly softer when reducing
the normalization of the thermal component, up to a value of $\Gamma=2.57\pm
0.22$ when the thermal component is ignored. If we attempt to fit the emission
with a thermal model, instead, we obtain a temperature of $4.2_{-0.7}^{+1.0}$
keV, and the quality of the fit significantly decreases with $\Delta
C_{stat}\sim 20$. Therefore, we conclude that also in this regions the
emission associated with the east jet is not consistent with being thermal,
despite it appears to be significantly softer than the emission found in the
west jet region. We thus conclude that also in the east jet region the diffuse
emission is dominated by nonthermal processes associated with IC from the
relativistic electrons. In this regions we find fluxes of $(1.89\pm
0.25)\times 10^{-15}$ erg s-1 cm-2 and $(1.02\pm 0.13)\times 10^{-15}$ erg s-1
cm-2 in the 0.5-2 and 2-10 keV bands, respectively. We find luminosity of
$(1.18\pm 0.15)\times 10^{44}$ erg s-1 and $(6.4\pm 0.8)\times 10^{43}$ ergs
s-1 cm-2 in the 0.5–2 and 2–10 keV rest-frame bands, respectively. The folded
and unfolded spectra in the east jet region, including the three components
used for the fit, are shown in Fig. 16.

On a more quantitative ground, the difference in spectral slope with respect
to the west jet region is $\Delta\Gamma=0.61\pm 0.29$, and therefore is
significant at 2$\sigma$ c.l. This difference can be interpreted in two ways.
The first possible explanation is the presence of an additional thermal
component associated with some gas shocked by the east jet itself up to
temperatures of few keV. However, this would imply an amount of hot gas at
least comparable if not larger than that expected from the ICM within the east
jet box, which seems unlikely. A second explanation can be provided by the
presence of an older population of relativistic electrons. In fact, this is
the scenario that is discussed in much greater details in Carilli et al.
(2022) and Anderson et al. (2022) where the synergy between the Chandra X-ray
data and the JVLA radio data is fully exploited.

##  7 Nature of the isotropic diffuse emission and protocluster dynamical
state

As a first-cut analysis of the diffuse isotropic emission, we can consider a
flat ICM distribution at least in the core region. This drastically simplified
picture is consistent with a beta profile (Cavaliere & Fusco-Femiano, 1978)
with a large $\sim 100$ kpc core, and reflects our ignorance of the actual ICM
distribution. On the other hand, since the X-ray emission depends on the
square of the electron density, a homogeneous gas distribution would rather
provide an upper limit to the true ICM mass. Nevertheless, since we focus only
on the innermost 100 kpc, the impact of the radial dependence of the ICM
density is expected to be modest, and we expect an upper limit close to the
actual mass value within the inner 12 arcsec. A more accurate treatment of the
distribution of the ICM, including the mild decrease of the surface brightness
with radius, and a possible asymmetry, will be presented in a forthcoming
paper (Lepore et al. in preparation).

Therefore, by assuming a constant gas density within to 12 arcsec, from the
spectrum of the diffuse thermal emission we measure an average electron
density of $n_{e}=(1.44\pm 0.23)\times 10^{-2}$ cm-3, for a total ICM mass of
$M_{ICM}\leq(1.67\pm 0.25)\times 10^{12}M_{\odot}$ within a radius of physical
100 kpc. However, this value does not include the emission from the excised
jet regions, plus the central 2 arcsec circle. As we already discussed, we
treated this aspect assuming a simple geometrical correction applied to the
surface brightness, corresponding to a factor of 1.22. Since the electron
density is proportional to the square root of the emission, this corresponds
to a factor of $\sim 1.10$ for the density. Therefore, the geometrically
corrected values are $n_{e}=(1.58\pm 0.24)\times 10^{-2}$ cm-3 and
$M_{ICM}\leq(1.84\pm 0.30)\times 10^{12}M_{\odot}$. A more conservative value
would be a simple mean with the addition of a systematic uncertainty
bracketing the cases of no correction (corresponding to ICM-empty jet regions)
and geometrical correction (ICM-filled jet regions): $n_{e}=(1.51\pm 0.24\pm
0.14)\times 10^{-2}$ cm-3 and $M_{ICM}\leq(1.76\pm 0.30\pm 0.17)\times
10^{12}M_{\odot}$. The error bars here corresponds to the statistical errors
on the normalization, and to the systematics associated with the geometrical
correction.

If we blindly apply the hydrostatic equilibrium relation of Vikhlinin et al.
(2006, see their Equation 7) for a temperature of $kT=1.98^{+0.7}_{-0.4}$ keV
out to 100 kpc (physical), we find a mass of $M(<100\,{\tt
kpc})=(1.5^{+0.5}_{-0.3})\times 10^{13}\,M_{\odot}$ by assuming a roughly
isothermal distribution (consistent with a total density decreasing
approximately as $r^{-2}$ beyond 100 kpc and, therefore, a linearly increasing
total mass). The implied upper limit to the ICM mass fraction over the total
mass is, therefore, $f_{ICM}\leq(0.12\pm 0.04)$. Incidentally, this value is
somewhat higher, and only marginal compatible, with the baryonic fraction
measured at the group scale (Eckert et al., 2021), possibly indicating that
mechanical feedback has not yet depleted the ICM reservoir in the core.
However, this is only a preliminary hint that must be verified after a more
careful modelization of the actual ICM distribution.

Applying instead the self-similar mass-temperature scaling relation at a
redshift of $z=2.156$ (therefore, assuming that $M_{500}\propto
T^{\alpha}E(z)^{-1}$ with $\alpha\sim 3/2$) calibrated on data, we find
$M_{500}=(3.0\pm 1.1)\times 10^{13}\,M_{\odot}$ within a radius
$r_{500}=(220\pm 30)$ kpc (see also Ettori et al., 2004, that provides
consistent results). The hydrostatic-equilibrium mass value linearly
extrapolated to $r_{500}$ provides $M(<220\,{\tt
kpc})=(3.2^{+1.1}_{-0.6})\times 10^{13}\,M_{\odot}$, in agreement with the
scaling relation. We find consistent results also when scaling self-similarly
with redshift the $M-T$ relation observed in local groups by Lovisari et al.
(2015), obtaining $M_{500}=(2.7\pm 1.3)\times 10^{13}M_{\odot}$. We note that
the agreement toward a value $M_{500}=3.0\times 10^{13}M_{\odot}$ essentially
relies on the assumption of self similar scaling with redshift, which is
approximately tested only up to $z\sim 1$. If the dependence of $M_{500}$ on
redshift at a given temperature is significantly milder than the self-similar
behavior $M_{500}\propto(1+z)^{-3/2}$ for $z>1$, the extrapolated mass of the
halo can be more than a factor of two larger (see, e.g., Bulbul et al., 2019).

To interpret our findings we rely on the cosmological hydrodynamic simulations
used to predict the properties of proto-cluster regions at $z\sim 2$ (Saro et
al., 2009). These simulations show evidences of ongoing assembly of a
dominating central galaxy, with a pattern remarkably similar to that observed
for the Spiderweb complex. Saro et al. (2009) also find that the projected
galaxy velocity dispersion, the observed level of star formation and the
stellar mass of the dominant galaxy suggest that this region should trace the
progenitor of a rich cluster, whose mass by $z=0$ would be $\simeq
10^{15}\,M_{\odot}$. It is found that such a structure at $z\sim 2$ should
already contain a diffuse X-ray emitting atmosphere of hot gas in hydrostatic
equilibrium, which is chemically enriched at a level comparable to that of
nearby galaxy clusters. To compare the predictions of Saro et al. (2009) with
our analysis of the isotropic, thermal diffuse emission in the Spiderweb
galaxy, we rescale the values shown in their Table from 1.7 to 2 keV for the
simulation C1, corresponding to a final mass by $z=0$ of
$h^{-1}10^{14}\,M_{\odot}$. We find that they predict a total luminosity
$L_{X}$ which is lower by a factor of 2 with respect to the measured
$L_{X}=(2.0\pm 0.5)\times 10^{44}$ erg s-1, similarly to what we noticed in
the comparison with the average $L_{X}-T$ relation. On the other hand, both
the temperature and the luminosity found for the simulation C2, corresponding
to a final mass by $z=0$ of $2\times 10^{15}\,h^{-1}M_{\odot}$, are in excess
with respect to what we observe. From this simple comparison, we argue that
the final mass of the Spiderweb complex may be few$\times 10^{14}M_{\odot}$
rather than $\geq 10^{15}M_{\odot}$, but clearly this is very hard to confirm,
given the large dispersion expected in mass accretion histories from
simulations (see Chiang et al., 2013). Remarkably, recent simulations from the
same group show that the SZ signal is more consistent with that of a
progenitor of a few$\times 10^{14}\,M_{\odot}$ cluster at $z=0$, in line with
our X-ray analysis (Di Mascolo et al. in preparation).

When considering the dynamical mass estimates based on the velocity dispersion
of the member galaxies, we note that the observed bimodality suggests the
presence of two possibly virialized halos, whose virial masses are estimated
to be $9\times 10^{13}\,M_{\odot}$ and $3\times 10^{13}\,M_{\odot}$ in
Pentericci et al. (2000), and $17\times 10^{13}\,M_{\odot}$ and $6\times
10^{13}\,M_{\odot}$ in Kurk et al. (2004). In Shimakawa et al. (2014) the
dynamical mass of the core is estimated to be $1.71\times 10^{14}\,M_{\odot}$.
In all cases, the mass of the largest halo is in excess if compared to our
measurement. We argue that the Spiderweb core is caught in a phase of
previrialization or very close to the first phase of virialization, making it
unfeasible to perform a meaningful comparison of the dynamical mass from
velocity dispersion to the mass inferred from the hydrostatic equilibrium of
the ICM. We might instead observe an overlap of two recently virialized halos,
each with its X-ray emitting ICM. The possible presence of multiple halos is
further discussed in Di Mascolo et al. (in preparation) based on SZ data
analysis.

Yet another process that might affect the morphology in SZ and X-ray bands is
the cooling of the ICM. The average cooling time in the inner 100 kpc of the
Spiderweb hot halo is $t_{c}=(1.5\pm 0.4)$ Gyr. This is twice shorter than the
age of the Universe $t_{age}\sim 3$ Gyr at $z=2.156$, but still longer than
the dynamic time roughly estimated as $0.2\times t_{age}\sim 0.6$ Gyr. This
implies that, even for a relatively well shaped dark matter halo, the ICM
distribution might be very sensitive to the history of the assembly
(especially for lower mass subhalos). This may help in explaining the marginal
features of a somewhat irregular and off-centered distribution of the thermal
diffuse emission, that are not addressed in the present work, but will be
thoroughly discussed in a forthcoming paper (Lepore et al., in preparation).

At present, we conclude that the diffuse, isotropic emission observed around
the Spiderweb Galaxy is due to an embryonic halo corresponding to a total mass
$\sim 3.0\times 10^{13}\,M_{\odot}$. The proto-ICM appears to have an
approximately isotropic distribution, showing no correlation with the
direction of the jets, making it unlikely that direct feedback can provide a
dominant source of heating. Nevertheless, the coexistence of radio-mode
feedback and a complex dynamical status, makes it very hard to provide a
unique and self-consistent modelization of the Spiderweb protocluster. We are
planning to provide a more accurate spatially resolved analysis of the X-ray
data and combine them with SZ data, and to increase the numerical effort for a
better comparison with simulations. We also plan to add ALMA SZ data deeper
than those currently available. Furthermore, the next necessary step is to
increase the sample of $z>2$ protoclusters with comparable properties,
hopefully with an observational coverage similar in quality and depth to that
of the archetypal Spiderweb.

##  8 Conclusions

A deep Chandra X–ray observation of the field of the Spiderweb galaxy
(J1140-2629) at $z=2.156$ enabled us to investigate the properties of the
central AGN and of the diffuse emission in detail. Our main conclusions are
summarized as follows:

  * •

we find that the Spiderweb galaxy hosts a mildly absorbed quasar, showing a
modest yet significant spectral and flux variability on a timescale of $\sim
1$ year.

  * •

We firmly identify significant extended emission out to a radius of 12 arcsec
(100 kpc), which appears to be made of a thermal and a nonthermal component,
and it is significantly contaminated by the wings of the strong AGN emission.

  * •

After accurately accounting for all the three components plus the instrumental
background, we were able to construct an azimuthally averaged profile of the
diffuse emission.

  * •

We find that the emission in the jet regions (overlapping the radio extended
emission) is significantly harder than the rest of the extended emission, with
$\Gamma\sim 2-2.5$, and consistent with IC upscattering of the CMB photons by
the relativistic electrons.

  * •

We find a roughly symmetric, diffuse emission in a circular region with a
radius of $\sim 100$ kpc centered on the Spiderweb galaxy. This emission is
significantly softer than that in the jet regions, and it is consistent with
thermal emission from hot ICM with a temperature of $kT=2.0_{-0.4}^{+0.7}$
keV, with an upper limit on the level of ICM enrichment of $Z<1.6Z_{\odot}$ at
1$\sigma$ c.l.

  * •

The diffuse emission, if assuming a constant density within 100 kpc,
corresponds to an average electron density of $n_{e}=(1.51\pm 0.24\pm
0.14)\times 10^{-2}$ cm-3. If we make the simple assumption of an
approximately flat ICM distribution, we obtain an upper limit to the ICM mass
of $\leq(1.76\pm 0.30\pm 0.17)\times 10^{12}M_{\odot}$ within the same radius
(error bars are $1\sigma$ statistical and systematic, respectively).

  * •

The total rest-frame luminosity of the ICM, $L_{0.5-10{\rm keV}}=(2.0\pm
0.5)\times 10^{44}$ erg s-1, is about a factor of 2 higher than the central
value of the extrapolated $L-T$ relation for massive clusters, but still
consistent within the scatter.

  * •

If we apply hydrostatic equilibrium to the ICM, we estimate a total mass of
$M(<100\,{\rm kpc})=(1.5^{+0.5}_{-0.3})\times 10^{13}\,M_{\odot}$. Applying
the average scaling relation at a redshift of $z=2.156$, we estimate a total
mass of $M_{500}=(3.2^{+1.1}_{-0.6})\times 10^{13}\,M_{\odot}$ within a radius
of $r_{500}=(220\pm 30)$ kpc.

We conclude that the Spiderweb galaxy is hosting a bright, mildly absorbed AGN
showing limited but significant variability. Significant diffuse emission is
detected within a radius of 12 arcsec, and it is shown to be dominated by IC
scattering associated with the radio jets. Outside the jet regions, we also
identify thermal emission showing that the protocluster is hosting a $\sim
100$ kpc core of hot, diffuse baryons that may represent the embryonic
virialized halo of the forming cluster, and, based on numerical simulations,
it is expected to evolve into a massive halo with a mass a few times
$10^{14}M_{\odot}$ by $z=0$. The origin and the detailed properties of the ICM
identified in the Spiderweb galaxy will be discussed in a paper presenting
complementary SZ results (Di Mascolo et al. in preparation) and in another
paper further investigating the spatial properties of the X-ray diffuse
emission (Lepore et al. in preparation). The main results of the
multiwavelength survey of the Spiderweb galaxy field and the corresponding
data products can be found on the project
webpage999http://www.arcetri.inaf.it/spiderweb/..

Finally, our results demonstrate the central role of a sharp Chandra-like PSF
for detailed analysis of the ICM properties of high-z protoclusters. They also
pave the way for future X–ray studies of distant proto-clusters to be carried
out with the advent of the next generation of X–ray satellites, such as Lynx
(The Lynx Team, 2018), AXIS (Mushotzky & AXIS Team, 2019; Marchesi et al.,
2020), and, to some extent, the survey and time domain mission STAR-X
101010The Survey and Time-domain Astrophysical Research Explorer, or STAR-X,
is targeted for launch in 2028. The mission has been recently selected in the
Phase A of the Medium Explorer category. (Zhang et al., 2022).

###### Acknowledgements.

This work was carried out during the ongoing COVID-19 pandemic. The authors
would like to acknowledge the health workers all over the world for their role
in fighting in the frontline of this crisis. We thank the anonymous referee
for detailed comments and positive criticism that helped improving the quality
of the paper. S.B., P.T., E.R. and R.G. acknowledge financial contribution
from the agreement ASI-INAF n.2017-14-H.0. M.N. acknowledges
INAF-1.05.01.86.20. L.D.M., M.P., and A.S. are supported by the ERC-StG
’ClustersXCosmo”’ grant agreement 716762. A.S. is also supported by the FARE-
MIUR grant ’ClustersXEuclid’ R165SBKTMA, and by INFN InDark Grant. S.B.
acknowledges partial financial support from the Indark INFN Grant. HD
acknowledges financial support from AEI-MCINN under the grant with reference
PID2019-105776GB-I00/DOI:10.13039/501100011033, and from the ACIISI,
Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the
European Regional Development Fund (ERDF) under grant with reference
PROID2020010107. We thank Hans Moritz Günther for help with the use of the
MARX software. We thank Malgorzata Sobolewska for assistance during the
Chandra observations.

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