11institutetext: Hamburger Sternwarte, University of Hamburg, Gojenbergsweg
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11email<EMAIL_ADDRESS>22institutetext: DIFA - Università
di Bologna, via Gobetti 93/2, I-40129 Bologna, Italy 33institutetext: INAF -
Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero
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2300 RA Leiden, The Netherlands 77institutetext: INAF, Istituto di Astrofisica
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88institutetext: Center for Astrophysics $|$ Harvard $\&$ Smithsonian, 60
Garden Street, Cambridge, MA 02138, USA 99institutetext: INAF - IASF Milano,
Via A. Corti 12, I-20133, Milano, Italy 1010institutetext: INAF - Padova
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1111institutetext: ASTRON, Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedijk 4, 7991 PD, Dwingeloo, The Netherlands and Leiden Observatory,
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Scienza 5, I-09047 Selargius (CA), Italy

# LOFAR detection of extended emission around a mini-halo in the galaxy
cluster Abell 1413

G. Lusetti 1122 A. Bonafede 22 L. Lovisari 7788 M. Gitti 22 S. Ettori 33 R.
Cassano 44 C. J. Riseley 224455
F. Govoni 1212 M. Brüggen 11 L. Bruno 2244 R. J. van Weeren 66 A. Botteon 44
D. N. Hoang 11
F. Gastaldello 99 A. Ignesti 1010 M. Rossetti 99 T. W. Shimwell 1111

(August 2023)

###### Abstract

Context. The relation between giant radio halos and mini-halos in galaxy
clusters is not understood. The former are usually associated with merging
clusters, while the latter are found in relaxed systems. In the last years,
the advent of low-frequency radio observations has challenged this dichotomy
by finding intermediate objects with a hybrid radio morphology.

Aims. We investigate the presence of diffuse radio emission in the cluster
Abell 1413 and determine its dynamical status to explore the relation between
mini-halos and giant radio halos.

Methods. We use LOFAR HBA observations centered at 144 MHz to study the
diffuse radio emission. To investigate the dynamical state of the system, we
use newly analysed XMM-Newton archival data.

Results.  A1413 shows features that are typically present in both relaxed
(e.g., peaked x-ray surface brightness distribution and little large scale
inhomogeneities) and disturbed (e.g., flatter temperature and metallicity
profiles) clusters. This suggests that A1413 is neither disturbed nor fully
relaxed, and we argue that it is an intermediate-phase cluster. At 144 MHz, we
discover a wider diffuse component surrounding the previously known mini-halo
at the cluster center. By fitting the radio surface brightness profile with a
double-exponential model, we can disentangle the two components. We find an
inner mini-halo with an $e$-folding radius $r_{\rm e,1}=28\pm 5\,\rm kpc$ and
an extended component with $r_{\rm e,2}=290\pm 60\,\rm kpc$. We also evaluated
the point-to-point correlation between the radio and X-ray surface brightness,
finding a sub-linear relation for the outer emission and a super-linear
relation for the mini-halo. The mini-halo and the diffuse emission extend over
different scales and show different features, confirming the double nature of
the radio emission and suggesting that the mechanisms responsible for the re-
acceleration of the radio emitting particle might be different.

###### Key Words.:

Galaxies: clusters: individual: Abell 1413 – Radio continuum: galaxies –
Galaxies: clusters: intracluster medium – X-rays: galaxies: clusters

##  1 Introduction

Galaxy clusters have a size that ranges from a few hundreds kpc to $\sim$ Mpc,
and have typical masses of $10^{14}-10^{15}\,\mathrm{M_{\odot}}$. They can
contain up to $\sim 1000$ galaxies, embedded in the intracluster medium (ICM).
The ICM consists of fully ionized hydrogen and helium plus traces of highly
ionized heavier elements with a temperature $\mathrm{T}\sim
10^{7}-10^{8}\,\mathrm{K}\,(1-10\,\mathrm{keV})$ and particle number densities
steeply declining from $n_{e}\sim 10^{-2}-10^{-3}\,\mathrm{cm^{-3}}$ near the
center to $10^{-4}\mathrm{cm^{-3}}$ in the outskirts. It emits mainly via
thermal bremsstrahlung in the X-ray band (e.g., Forman & Jones 1982; Sarazin
2009).

Co-spatial with the ICM, an increasing number of galaxy clusters is also
observed to host diffuse non-thermal synchrotron emission, with no obvious
connection to the individual cluster members. The synchrotron emission reveals
the presence of magnetic field and relativistic cosmic-ray electrons (CRe) in
the ICM. Extended, central diffuse sources are usually classified as giant
radio halos and mini-halos (see van Weeren et al. 2019, for a review). Both
giant and mini-halos exhibit steep spectrum111The synchrotron radio spectrum
is defined as $S(\nu)\propto\nu^{-\alpha}$. with a spectral index
$\rm\alpha>1$. While giant radio halos are Mpc-sized radio sources which cover
large parts of the cluster volume, mini-halos are typically confined within
the cool-cores ( $\sim 300-500$ kpc) of relaxed galaxy clusters, and are
characterized by higher emissivity (e.g., Cassano et al. 2008; Murgia et al.
2009).

Two emission mechanisms have been proposed to explain these non-thermal radio
sources (see Brunetti & Jones 2014, for a review): (i) the (re-)acceleration
scenario, where CRe are (re-)accelerated by turbulence that develops during
cluster mergers; (ii) the hadronic scenario, where secondary CRe are produced
by proton-proton collisions between relativistic cosmic-ray protons (CRp) and
thermal protons in the ICM. While it has been shown that purely hadronic
models cannot reproduce the emission of giant radio halos (Brunetti et al.
2008, 2017), they remain a viable mechanism for mini-halos (Pfrommer & Enßlin
2004; Jacob & Pfrommer 2017). Statistical studies have shown the connection
between the dynamical status of the cluster and the type of diffuse radio
emission (e.g., Cassano et al. 2010; Cuciti et al. 2021; Cassano et al. 2023).
Galaxy clusters hosting giant halos are mainly characterized by a disturbed
dynamical status, suggesting recent or ongoing mergers (e.g., Cassano et al.
2013; Cuciti et al. 2015). On the other hand, mini-halos are usually found in
relaxed cool-core clusters (Gitti et al. 2004; Cassano et al. 2010; Gitti et
al. 2018) but their origin is unclear since they could be of leptonic or
hadronic origin (e.g., van Weeren et al. 2019). Moreover, in a statistical
study about the occurrence of mini-halos in a sample of massive ($\rm M>6\cdot
10^{14}M_{\odot}$) galaxy clusters, Giacintucci et al. (2017) found that
$\rm\sim 80\%$ of cool-core clusters (classified according to the value of
central entropy $\rm K_{0}<30\,keV\,cm^{2}$) host a mini-halo, suggesting a
clear relation between the origin of mini-halos and the properties of the
cool-core gas.

In the last years, the advent of deep, low-frequency observations has
complicated the picture described above. In fact, a few cool-core clusters
show the presence of a diffuse synchrotron halo or halo-like emission that
extends quite far from the central mini-halo (Bonafede et al. 2014; Venturi et
al. 2017; Sommer et al. 2017; Savini et al. 2018, 2019; Biava et al. 2021;
Riseley et al. 2022b, Bruno et al. submitted), challenging the idea that mini-
halos occur in cool-core clusters and giant halos in non cool-core clusters.
These peculiar galaxy clusters, either, present giant halos (Bonafede et al.
2014; Savini et al. 2018), or show a hybrid morphology in the radio emission
(Biava et al. 2021), even though being classified as relaxed systems based on
their thermal properties. Until now, a somewhat unique case is represented by
the galaxy cluster RX J1720.1+2638 (Savini et al. 2019; Biava et al. 2021),
where the radio emission is composed of a central bright mini-halo plus
fainter and wider emission, extending far from the cluster core. In this
study, the authors showed that the radio diffuse emission presents different
features, suggesting an intrinsic different nature of the two sources. An even
more complex morphology has been found in Abell 2142, a galaxy cluster with an
intermediate dynamical state, already known to host a hybrid radio halo with
two distinct components (Venturi et al. 2017). This object has recently been
studied with low-frequency radio observation by Bruno et al. (submitted), who
detected a third ultra-steep spectrum and wider component. Only a handful of
multi-component diffuse sources only are currently known, but their number may
increase thanks to sensitive low frequency observations.

###  1.1 Abell 1413

The galaxy cluster Abell 1413 (hereafter A1413; Planck name: PSZ2
G226.18+76.79) is a nearby (z=0.143) massive
($M_{500}=5.95^{+0.24}_{-0.25}\times 10^{14}\,M_{\odot}$, Planck Collaboration
et al. 2016) system. It is well-studied in the optical (Castagné et al. 2012),
X-ray (Pratt & Arnaud 2002) and radio (Govoni et al. 2001; Giacintucci et al.
2017; Savini et al. 2019) bands. These previous studies at different
wavelenghts showed that the properties of A1413 are intermediate between
classical relaxed and merging clusters. From the morphological point of view,
it shows a high ellipticity confirmed by both optical (Castagné et al. 2012)
and X-ray (Pratt & Arnaud 2002) studies. The cluster is elongated
approximately in the North-South direction (Fig. 7 of Castagné et al. 2012).
Studying the galaxy number density distribution, Castagné et al. (2012)
estimated the value of the cluster ellipticity $\epsilon\sim 0.35$ at large
radii, increasing up to $\sim 0.8$ at the cluster center $r<1\arcmin$. While
fitting the XMM-Newton image, they found values quickly converging toward
$\epsilon\sim 0.27$, which is fully compatible with the global value of
$\epsilon\sim 0.29$ derived by Pratt & Arnaud (2002) over the radial range
$[3\arcmin-13\arcmin]$. They argue that this discrepancy at large radii of the
optical and X-ray ellipticity values is not surprising as the collisional gas
relaxes more quickly within the cluster potential than the non-collisional
galaxies and reaches more rapidly a spherically symmetric distribution.
However, Castagné et al. (2012) also stressed that A1413 seems a rather
extreme example of this effect. Moreover, by analysing the velocity
distribution of the cluster, they found a velocity offset of $450\pm 210\,\rm
km\,s^{-1}$(i.e., $2\sigma$ significance) between the velocity of the central
cD galaxy (Paturel et al. 1989), also being the brightest cluster galaxies
(BCG), and the mean cluster velocity. These authors argue that BCGs velocity
offsets are frequently found in clusters that are distant from equilibrium,
for example during significant merging events. If this BCG velocity offset is
confirmed, it would indicate that A1413 may potentially be in a non-fully
relaxed dynamical state.

On the other hand, early X-ray studies (Pratt & Arnaud 2002; Vikhlinin et al.
2005; Baldi et al. 2007) classified this cluster as a relaxed system, with no
obvious signatures of a recent merger. The lack of shocks or cold fronts has
been also confirmed by Botteon et al. (2018), who explored a combination of
different analysis approaches of X-ray observations to firmly detect and
characterize edges in a sample of non-cool-core, massive galaxy clusters. The
XMM-Newton observation (Pratt & Arnaud 2002) does not provide strong evidence
of a cool-core region, in contrast with the temperature profiles obtained with
Chandra (Vikhlinin et al. 2005), with a cooling time in the innermost bin
($\rm\sim 50\,kpc$) of $t_{\rm cool}=4.2\pm 0.3$ Gyrs (Baldi et al. 2007).
These authors suggested that this could be related to the better resolution of
Chandra compared to XMM-Newton. Finally, using common X-ray morphological
parameters, such as the concentration parameter ($c$, Santos et al. (2008))
and the centroid shift ($w$, Poole et al. (2006); Mohr et al. (1995)) A1413
has been classified as a relaxed system (Lovisari et al. 2017; Campitiello et
al. 2022; Yuan et al. 2022). However, it also has a mixed morphology, i.e.
elongated X-ray isophotes and a relatively flat X-ray profile near the centre.

The first investigation of the radio emission in A1413 was performed by Govoni
et al. (2009), who found a candidate mini-halo at a frequency of 1.4 GHz.
Later, A1413 was included in a mini-halos statistical study of Giacintucci et
al. (2017), who classified clusters according to the value of the specific
central entropy $K_{0}$. Interestingly, A1413 was the only non-cool-core
cluster ($K_{0}=64\pm 8\,\rm keVcm^{2}$) of the whole sample hosting a
(candidate) mini-halo. The diffuse radio emission was then studied at a
frequency of 144 MHz, in a study of non-merging galaxy clusters (Savini et al.
2019), confirming the presence of a mini-halo with a linear size of $\sim$ 210
kpc and a total flux density of $40\pm 7$ mJy. In a recent study of Riseley et
al. (accepted), the mini-halo was studied with a combination of LOFAR at 144
MHz and MeerKAT at 1283 MHz observations. They measured a linear size of at
least 449 kpc (and up to 584 kpc) at 1283 MHz, twice as large as previously
measured by Govoni et al. (2009). From these past studies, a controversial and
intriguing picture of A1413 emerges, making it a very interesting object. In
this paper we present new radio observations performed with the LOFAR High
Band Antennae (HBA) at 144 MHz. This is combined with archival and newly
analysed XMM-Newton data, which includes a more accurate background treatment
and improved calibration.

This paper is organized as follows: in Section 2 we give an overview of the
observations and data reduction. The X-ray and radio results are presented in
Section 3. We discuss the results from X-ray and radio analysis jointly in
Section 4. In Section 5 we summarise our conclusions.

Throughout the paper we adopt a $\Lambda$CDM cosmology with
$\mathrm{H_{0}=70\,km\,s^{-1}Mpc^{-1}}$, $\Omega_{\Lambda}=0.7$ and
$\Omega_{\rm m}=0.3$. Thus, $1^{\prime\prime}$ corresponds to a physical scale
of $2.51$ kpc at the redshift of A1413. A summary of the properties of A1413
collected from literature is reported in Table 1.

Table 1: Properties of A1413 derived from literature: redshift (z: Böhringer et al. 2000), equatorial coordinates (NASA ExtragalCatalog, Bade et al. 1998), mass within the radius that defines the sphere within which the cluster’s density is 500 times the critical density at the cluster’s redshift ($M_{500}$; Planck Collaboration et al. 2016), central entropy ($K_{0}$; Giacintucci et al. 2017), X-ray morphological parameter ($c,w$; Campitiello et al. 2022). z | $0.1427$
---|---
R.A. (h, m, s) | $11\,55\,18.9$
$\text{Dec }(^{\circ},^{\prime},^{\prime\prime})$ | $+23\,24\,31$
$M_{500}\,(10^{14}\rm M_{\odot})$ | $5.95^{+0.24}_{-0.25}$
$K_{0}\,(\rm keV\,cm^{2})$ | $64\pm 8$
$c$ | $0.44^{+0.04}_{-0.04}$
$w$ | $0.04^{0.02}_{0.01}\,\cdot 10^{-1}$

Figure 1: Left panel: High resolution 144 MHz image with beam
$\mathrm{4^{\prime\prime}\times 6^{\prime\prime}}$ (shown in the bottom left
corner) and noise $\mathrm{\sigma_{\rm rms}=0.15\,mJy/beam}$. Sources that
have been subtracted are labeled with letters: two central AGN A, B; one
point-like north source C and one extended south source D. Right panel: Low
resolution source-subtracted 144MHz image with beam
$\mathrm{35^{\prime\prime}\times 35^{\prime\prime}}$ (shown in the bottom left
corner) and noise $\mathrm{\sigma_{\rm rms}=0.27\,mJy/beam}$. For both images,
the contour levels start at $2\sigma_{\rm rms}$ and are spaced with a factor
of $\mathrm{\sqrt{2}}$. The $-3\sigma_{\rm rms}$ contours are red coloured,
while $2\sigma_{\rm rms}$ contours are dotted green coloured.

##  2 Observations and data reduction

In order to investigate the radio emission and dynamical state of A1413 we
used LOFAR data at 144 MHz together with newly analysed archival data from
XMM-Newton telescope, the latter being also part of the CHEX-MATE program
(CHEX-MATE Collaboration et al. 2021). Moreover, we compare the LOFAR
observation of the source with VLA image of this source presented in Govoni et
al. (2009).

In the following sections, we describe these observations and the data
reduction procedures. A summary of the observational details is reported in
Table 2.

Table 2: Observational overview: LOFAR HBA and VLA observations (top); X-ray observation (bottom). Telescope | Central frequency | Configuration | On source time | Observation ID | Observation Date | Reference
---|---|---|---|---|---|---
LOFAR | 144 MHz | HBA_DUAL | 8+8 hrs | 591537,602890 | 04.05.2017,09.08.2017 | (1)
VLA | 1.4 GHz | C array | 8 hrs |  | 23.10.2006 | (2)
Telescope | Energy range | Filter | On source time | Observation ID | Observation Date | Reference
XMM-Newton | [0.5-14] keV | THIN1 | 82.5 ks | 0502690201 | 11.12.2007, rev. # 1466 | (1),(3)
References: (1) this work; (2) Govoni et al. (2009); (3) Pratt & Arnaud
(2002).

###  2.1 XMM-Newton

A1413 was observed with XMM-Newton four times between 2007 and 2008: all four
observations are longer than 60 ks each. For this work, we retrieved the
longest and less flared one, OBS ID 0502690201, from the XMM-Newton data
archive (see Table 2). Data consists of 82.5 ks archival observation time
during XMM-Newton revolution 1466 (2007 December 11), obtained with the THIN1
filters and are processed with XMM-SAS222https://www.cosmos.esa.int/web/xmm-
newton/sas v19.0.0 and the XMM-
ESAS333https://heasarc.gsfc.nasa.gov/docs/xmm/xmmhp_xmmesas.html software
package (Snowden et al. 2008), through a dedicated pipeline that follows the
procedure described in Lovisari & Reiprich (2019) and which is briefly
described in this section. The pipeline processes the X-ray data from the raw
data, the Observation Data File (ODF), passing through data calibration and
filtering, imaging, and spectral fitting.

The raw data are initially processed using the cifbuild and odfingest task.
The former provides an up-to-date list of the relevant calibration files,
while the latter produces extended ODF summary file needed for subsequent data
processing. Then we run emchain and epchain tasks to convert the raw data into
calibrated photon event files for the MOS and pn camera, respectively. We
applied standard filtering procedures to discriminate events based on their
pattern: only single, double, triple and quadruple events for MOS (i.e.
PATTERN $\leq 12$) and single and double for pn (i.e. PATTERN $\leq 4$). We
also performed bright pixels and hot columns removal (i.e. FLAG==0) and
removed the contamination by pn Out-of-Time events. Observation periods
affected by high background due to soft protons (SP; see Section 2.1.1) were
removed using a two-stage filtering process (see Lovisari et al. 2011 for
details), summarised in the following. First, the light curve with 100 s bins
is inspected in [10-12] keV band for MOS and [12-14] keV for pn, since
particle background dominates at high energy. The histogram of the light curve
is then fitted with a Poisson distribution and time intervals with a higher
count rate, i.e. a count rate that deviates by more than 2$\sigma$ from the
mean, are excluded (see Pratt & Arnaud 2002, Appendix A for a precise
description). To obtain the final Good Time Intervals (GTI), a second-stage
filtering is performed, this time using the full [0.3-10] keV band in 10 s
bins, as a safety check for possible flares with soft spectra (e.g. Nevalainen
et al. 2005).

Filtering the light curve does not ensure to delete completely the SP
contributions: continuous and constant flares (i.e. flares that last for long
time intervals), without evident spikes, can still remain. We therefore
estimate the amount of residual SP flare contamination by using the suited
code of De Luca & Molendi (2004). By comparing area-corrected count rates in
the in-FOV and out-of-FOV regions of the detector, we find that the SP
contamination is low444https://www.cosmos.esa.int/web/xmm-newton/epic-scripts
with a ratio of $1.12\pm 0.02$ for all the cameras. Nevertheless, we included
SP contamination as an extra model component, when dealing with spectral
fitting (Section 2.1.2). Anomalous behaviour of CCD 7 for MOS2 was detected
via mos-filter task (Kuntz & Snowden 2008) and excluded from this analysis,
together with CCD 6 of MOS1, that was damaged by a meteorite in the early
phase of the emission and thus, not available.

Point-like sources were detected with the edetect-chain task and visually
inspected to discriminate between real point sources and extended cluster
substructures before removing them from the event files. Even if identified as
point-source, we did not remove the BCG region because the cluster emission is
so bright that the contribution of the AGN is negligible and can be modeled as
a power law during the spectral fitting.

####  2.1.1 Background treatment

XMM-Newton EPIC background is made of several different components, each one
with temporal, spectral, and spatial variations. It can be generally divided
between particle and photon background.

The main component of the photon background is the cosmic X-ray background
(CXB), primarily consisting of unresolved cosmological sources, the Local
Bubble and the Galactic Halo. To model the CXB we used ROSAT All-Sky
Survey555Since in ROSAT observations, particle background is negligible, these
can be considered as pure sky data. RASS diffuse background maps can be
downloaded at the HEASARC webpage https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/xraybg/xraybg.pl. (RASS) diffuse background maps in a region beyond
the cluster virial radius (between 1 and 2 degrees) and fitted it
simultaneously with XMM-Newton data, as described in Lovisari & Reiprich
(2019). We described the unresolved emission from AGN via an absorbed power-
law model with its slope set to 1.41 (De Luca & Molendi 2004); an absorbed
$\sim 0.2$ keV thermal component representing the Galactic Halo (GH) emission,
and an unabsorbed $\sim 0.1$ keV component representing the Local Hot Bubble
(LHB). During the fitting procedure, the temperature values of the LHB and GH
thermal components are free to vary, while the slope of the AGNs’ power law is
fixed.

The particle background includes signal generated by interactions of particles
with the detectors and depends on the energy of the hitting particles. It can
be divided between the quiescent particle background (QPB, with $\rm E\gtrsim
few\,MeV$) and soft protons (SP, with $\rm E<100\,keV$). QPB consists of a
continuum component and fluorescent lines produced by the interaction of
penetrating high energy particle with the detectors. We used Filter Wheel
Closed666FWC exposures are taken with closed filters that avoid X-ray photons
to hit the detector and thus, are dominated by the instrumental background.
(FWC) observations to estimate the intensity of QPB components. The model used
for FWC observations consists in a broken power-law with additional
fluorescence lines (e.g. Fe, Al, Si, Au). Since FWC observations are taken in
different time period from our observation and have a different exposure, they
need to be normalized, comparing them with the out-of-field-of-view data
(unexposed corners of the CCD chips are a measure of the particle background
level in each observation.). SP background is produced by relatively low
energy protons accelerated in the Earth magnetosphere, passing through the
telescope optics and depositing their energy directly on the CCDs. As
described in Section 2.1, SP components are usually highly variable in time
(ranging from $\rm\sim 100\,s$ to hours), and the enhancement in the count
rate (referred to as ”flares”) can be more than three orders of magnitude.
This flare component is usually filtered via light-curve screening procedures,
summarised in Section 2.1. Nonetheless, a more constant SP components can
remain and contaminate the observations. To take into account this residual SP
contamination, we added a power-law component, folded only with the
Redistribution Matrix File, to the background modeling. We left both the slope
and normalization free to vary for each detector and in the region of
interest. This approximately accounts for the proton vignetting (which differs
from the photon one, and for which there is no model available).

####  2.1.2 Spectral analysis

The spectral analysis is performed with the XSPEC (Arnaud 1996) package
v12.11.1, in the [0.5-12] keV and [0.5-14] keV energy range for MOS and pn,
respectively. All the regions used for the spectral profiles were centred on
the peak of the X-ray emission of coordinates R.A.(J2000)
$11^{h}55^{m}17^{s}.85$, DEC. (J2000) $+23^{\circ}24^{m}18^{s}.85$, and extend
until $\rm R_{500}\sim 1224\,kpc$ (Planck Collaboration et al. 2014). The size
of the annuli have been determined by requiring a minimum width of
$30^{\prime\prime}$ and a fixed S/N=50. The first requirement ensures that
most of the flux (i.e. $>80$%, Zhang et al. 2009) comes from the selected
region (due to the XMM–Newton PSF, some photons scatter from one annulus to
another); the second that the $\mathrm{1\sigma}$ uncertainty in the spectrally
resolved temperature (and consequently in the fitted temperature profiles) is
¡10% (see Appendix B of Lovisari & Reiprich 2019 for more details of how
uncertainties vary as function of S/N and cluster temperature). The cluster
emission is modeled with an apec single-temperature thermal plasma model with
an absorption fixed at the total (neutral and molecular; see Willingale et al.
2013) $\rm N_{H}\sim 1.97\cdot 10^{20}\,cm^{-2}$ value estimated using the
SWIFT online tool777https://www.swift.ac.uk/analysis/nhtot/index.php. For the
apec model (Smith et al. 2001) the parameters are: plasma temperature [keV],
metallicity, redshift (that is kept fixed at z=0.1427), and the normalization
$\frac{10^{-14}}{4\pi[D_{A}(1+z)]^{2}}\int n_{e}n_{H}dV$, with $D_{A}$ the
angular distance of the source, $n_{e}$,$n_{H}$ the electron and proton
densities in units of $\rm cm^{-3}$. In each annulus, the MOS and pn spectra
were fitted simultaneously and their normalizations left free to vary, in
order to account for the calibration offsets between the different detectors
(e.g., Madsen et al. 2017). The background is modeled together with the
cluster emission by performing a joint fit with the above explained RASS
spectrum and FWC observations components, and cluster emission extracted from
each region.

###  2.2 LOFAR

The data used in this work are part of the LOFAR Two-metre Sky Survey (LoTSS;
Shimwell et al. 2017, 2019, 2022), a sensitive ($\rm rms\sim 100\,\mu
Jy/beam$), high-resolution (FWHM of the synthesised beam of $\mathrm{\sim
6"}$), low-frequency (120-168 MHz) radio survey of the northern sky. LoTSS
observations are conducted with LOFAR (LOw Frequency Array: van Haarlem et al.
(2013)) High Band Antennas (HBA), in the DUAL_INNER mode configuration, each
pointing consists of 8 hrs observation book-ended by 10-min scans of the
calibrator ($\rm 3c395$ for the observations used in this work), at the
central frequency of 144 MHz and 48 MHz total bandwidth. A1413 is covered by
two pointings, P177$+$22, and P180$+$22.

Data have been calibrated following the LoTSS-DR2 pipeline (see Shimwell et
al. 2022; Tasse et al. 2021 for details), which works on the direction-
independent calibration products (i.e. 24 measurement sets for each pointing,
that cover the 120-168 MHz band) and returns fully direction-dependent
calibrated data. After calibration, the so called ”extraction and self-
calibration” procedure (van Weeren et al. 2021) is performed: a region around
the target of interest is selected and all the sources outside that region are
subtracted using the direction-dependent gains obtained by the pipeline. The
station beam is corrected for directly in the visibility space, and the phase
center of the observation is shifted toward the direction of the target. This
way, we are left with 2 data sets centered on the target that do not need any
further beam correction. These 2 data sets are self-calibrated again to
obtained more accurate calibration solutions in the region of interest. Self-
calibration and imaging are done with DPPP and WSClean 2.10.0 (Offringa et al.
2014).

We have re-imaged the self-calibrated datasets at different resolutions to
classify the kind of radio emission and its properties. We always applied a
Briggs (Briggs 1995) weighting scheme with a negative robust parameter, as it
allows to down-weigh the secondary lobes, and inner $uv$-cut fixed at
$\mathrm{b_{min}>80\lambda}$ to avoid interference on short baseline, i.e. to
filter out emission on angular scale larger than $43\arcmin$. This is a
standard procedure used in LoTSS analysis (e.g., Botteon et al. 2022). We
performed images at different resolutions: high-resolution (HR) images to
subtract point-sources which overlaps to the diffuse emission and low-
resolution (LR) images to gain sensitivity to the diffuse emission (see Table
3 for different imaging parameters used in this work).

On the HR image (Fig. 1, left) we identified different sources, named with
letters (A, B, C, D), within a radius of 500 kpc ($\rm\sim 0.5R_{500}$) from
the cluster centre. The emission from these sources contaminates the diffuse
emission that we want to study, and thus it has been subtracted. The tailed
galaxy on the right, with optical counter part at $z=0.144$, is instead quite
extended and difficult to remove and disentangle from the diffuse radio
emission. Therefore, we decided to not subtract it. The source subtraction is
performed directly from the visibilities through a multi-step procedure.
First, we produced images at high resolution ($5\arcsec$ FWHM) to identify
these compact sources (A, B, C, D in Fig. 1, left). We selected the
corresponding clean components in the model image and, through an anti-Fourier
transform, converted them into visibilities. Finally, a new data set is
created as subtraction of the selected model components $uv-$data from the
original data. Given the slight extension of source A and D, we performed an
additional subtraction step, imaging the subtracted data sets at an
intermediate $\sim 10\arcsec$ resolution. This allowed to definitively remove
all the sources, as they do not leave residual when inspecting the final
$uv-$subtracted data set. Eventually, the final data set is imaged maintaining
a negative Briggs weighting with $\mathrm{R=-0.25}$ and tapering down the
baselines longer than $\mathrm{10\,k\lambdaup}$ (i.e. using a Gaussian taper
with a FWHM of 20″) to gain sensitivity to the diffuse emission (see Fig. 1,
right).

To ensure a fair comparison with other LOFAR works on the same cluster, we
point out that we used different observations respect to the previous LOFAR
study of Savini et al. (2019). In particular, these authors used a single
8-hours pointed observation, while we combined two pointings for a total of
16-hours observation. Moreover, we used improved calibration techniques that
allowed to produce images with higher sensitivity. For example, our
$15\arcsec$-resolution image has $\rm\sigma_{rms}$ improved more than a factor
of two respect to the one of Savini et al. (2019), with
$\rm\sigma_{rms}=450\leavevmode\nobreak\ \mu Jy/beam$ at $\sim
20\arcsec$-resolution. On the other hand, Riseley et al. (accepted) used the
same set of observations presented here and achieved comparable sensitivity.
However, these authors undertaken an independent postprocessing.

Throughout the paper, the error $\Delta_{S}$ associated with a flux density
measurement, $S$, is estimated as

$\Delta_{S}=\sqrt{(\sigma_{c}\cdot S)^{2}+N_{\rm beam}\cdot\sigma_{rms}^{2}},$
(1)

where $\rm N_{beam}=N_{pixel}/A_{beam}$ is the number of independent beams in
the source area, and $\rm\sigma_{c}$ indicates the systematic calibration
uncertainty of the flux density, with a typical value of 10% for LOFAR HBA
observations (Shimwell et al. 2022).

Table 3: Radio images used in this work. Telescope | Freq | Briggs weighting | Taper FWHM | Resolution | $\rm\sigma_{rms}$ | Reference
---|---|---|---|---|---|---
| [MHz] |  | $[\arcsec]$ | [″$\times$″] | [mJy/beam] |
HBA | 144 | $-1$ | - | $4\times 6$ | 0.15 |  Fig. 1 (left)
HBA | 144 | $-0.25$ | $20$ | $35\times 35$ | 0.27 |  Fig. 1 (right)
HBA | 144 | $-0.25$ | $10$ | $15\times 15$ | 0.18 |  Fig. 5 (left)

##  3 Results

As presented in Section 1.1, A1413 is a peculiar galaxy cluster, showing
intermediate characteristics between relaxed and merging clusters. Past radio
observations have revealed a mini-halo at its center (Govoni et al. 2009;
Savini et al. 2019). In this section, we present the main results of our X-ray
and radio analysis. We used shape of the surface brightness (SB), temperature,
and metallicity profiles, as well as the central cooling time to determine the
cluster’s dynamical state. Moreover, we analysed the radio surface brightness
in 1D and 2D to classify the morphology of the radio source and define its
properties.

Figure 2: Left panel: Projected temperature profile and Right panel: metals
abundance profile obtained from spectral fitting. Both profiles are compared
with different averaged values of relaxed systems sample (pink) and disturbed
ones (yellow) from Lovisari & Reiprich (2019).

###  3.1 X-ray azimuthal spectral profiles

The projected temperature profile is shown in Fig. 2 (left). It is well known
that for relaxed cool-core galaxy clusters, the temperature profile shows a
clear drop in the central region (e.g., Vikhlinin et al. 2005; Pratt et al.
2007). However, for A1413, the temperature does not show the drop in the
central region clearly: just the innermost bin ($0.5\arcmin\sim 75\,\rm kpc$
width) shows a drop from $7.43\pm 0.12$ to $7.11\pm 0.01$ keV. On the other
hand, at larger radii the temperature profile decreases as expected for
typical, relaxed cluster. This is consistent with the analysis of Pratt &
Arnaud (2002), even if we point out that it could be biased by the low
resolution of XMM (as pointed out in Vikhlinin et al. 2005).
We also obtained the metal abundance of the apec model, which is a mean
abundance gathering the possible emission from different elements (e.g., C, N,
O, Ne, Mg, Al, Si, S, Ar, Ca, Fe, Ni), where Fe generally dominates. The
profile is displayed in Fig. 2 (right) and does not present clear peak in the
cluster core, typical of relaxed objects. We compared the observed profiles to
average temperature and metallicity profiles for relaxed (pink) and disturbed
(yellow) galaxy groups and clusters studied by Lovisari & Reiprich (2019), as
a reference. Fig. 2 shows that both properties do not clearly follow either
the mean profile of relaxed nor of disturbed clusters. In the core, where
actually relaxed and disturbed cluster profiles differ, A1413 sits in between
the two distributions. This supports the scenario that A1413 is an
intermediate phase system.

###  3.2 X-ray $-$ optical offset

For dynamically relaxed systems, we expect the BCG to be at the centre of the
gravitational potential well (van den Bosch et al. 2005). Several X-ray
studies have proven the connection/correlation between the X-ray peak$-$BCG
offset and the disturbed dynamical state (e.g., Hudson et al. 2010; Mann &
Ebeling 2012; Rossetti et al. 2016; Pasini et al. 2021), making this indicator
a robust and reliable diagnostic for assessing the dynamical condition of
clusters. We find a small offset of about $7.2$ kpc between the X-ray peak
with the optical BCG. The coordinates of the central galaxy were available
R.A. (J2000) $11^{h}55^{m}17^{s}.87$, DEC. (J2000)
$+23^{\circ}24^{m}16^{s}.02$ (Rawle et al. 2012) and correspond to the
brightest pixel of the optical image, retrieved from HST (F606W filter)
archive. Despite this offset, recent statistical studies by Rossetti et al.
(2016), show that an offset less than $0.02\,R_{500}$ (as in this case $\rm
7.2\,kpc=0.006\,R_{500}$) is still consistent with a relaxed system. Moreover
we point out, this offset corresponds to angular size of $\sim 3\arcsec$,
which is slightly bigger than the X-ray image’s pixel size ($\sim 2\arcsec$),
used to determine the X-ray peak.

###  3.3 X-ray surface brightness profile

We computed the radial surface brightness profiles (SB) in the [0.7–2] keV
energy band (Fig. 3). This is the energy range where, for temperatures as
those observed in A1413, the SB has a very small dependence on the $kT$ and it
also provides an optimal ratio of the source and background flux in XMM-Newton
data. The annuli for the SB profile are again centered on the X-ray peak and
have been determined by requiring a fixed $\rm S/N=10$ and minimum width of
$2^{\prime\prime}$.

The profile is fitted with a double $\beta$-model (LaRoque et al. 2006) in the
form

$\Sigma=\Sigma_{01}\Bigg{[}1+\Big{(}\frac{r}{r_{c1}}\Big{)}^{2}\Bigg{]}^{-3\beta_{1}+1/2}+\Sigma_{02}\Bigg{[}1+\Big{(}\frac{r}{r_{c2}}\Big{)}^{2}\Bigg{]}^{-3\beta_{2}+1/2},$
(2)

where $r_{c1}$,$r_{c2}$ are the core radii, $\Sigma_{01}$, $\Sigma_{02}$ are
the central surface brightnesses of the two components and $\beta$ represents
the ratio of specific kinetic energies of galaxies and gas (e.g., Gitti et al.
2012). This is the extension of the isothermal $\beta$-model (Cavaliere &
Fusco-Femiano 1976), commonly used to describe the surface brightness profile
of ICM. In fact, when fitting the profile over the radial range with a single
$\beta$-model, we found the presence of a central excess. This biases the best
fit results toward a small core radius and overestimates the true profiles in
the region outside the excess ($r\gg r_{c}$), producing non-random strong
residuals. Thus, a second component was needed and it is typically required to
reproduce the sharp X-ray peak in the core of relaxed clusters (e.g., Mohr et
al. 1999; Pratt & Arnaud 2002; Pasini et al. 2019).

Figure 3: X-ray surface brightness profile, together with the best fit
double-$\beta$ model of Eq. 2. The best fit parameters are
$\Sigma_{01}=0.56\pm 0.05\rm\,cst/s/arcmin^{2}$, $r_{c1}=189.62\pm
0.41\rm\,kpc$, $\beta_{1}=2.17\pm 0.91$ and $\Sigma_{02}=0.35\pm
0.04\rm\,cst/s/arcmin^{2}$, $r_{c2}=257.04\pm 0.25\rm\,kpc$,
$\beta_{2}=0.80\pm 0.04$ with $\chi^{2}/d.o.f\simeq 1.64$. The vertical line
indicates $R_{500}$. The surface brightness values have errors of the order of
1$-$2% , thus error bars on y-axis are not visible in this plot.

###  3.4 Cooling time

From the temperature and density profiles, the cooling time of each region can
be estimated as

$t_{\rm
cool}=\frac{H}{\Lambda(T)n_{e}n_{p}}=\frac{\gamma}{\gamma-1}\frac{kT(r)}{\mu
Xn_{e}(r)\Lambda(T)},$ (3)

where $\gamma=5/3$ is the adiabatic index, $H$ is the enthalpy, $\mu\sim 0.6$
is the molecular weight for a fully ionized plasma and $\Lambda(T)$ is the
cooling function (Sutherland & Dopita 1993). Given the cooling profile, we can
to define the cooling radius, i.e. the radius at which the cooling time is
shorter than the age of the system. The latter can be estimated assuming the
cluster’s age to be equal to the look-back time at $z=1$, that is, $t_{\rm
age}\simeq 7.7$ Gyrs (e.g., Bîrzan et al. 2004; Gitti et al. 2012), as at this
time many clusters appear to be relaxed (Fig. 4, red line). To obtain $r_{\rm
cool}$ we qualitatively fitted the cooling time profile with a power-law
relation and selected $r_{\rm cool}$ as the intersection between the best-
fitting power law and $t_{\rm age}$. Thus, the cooling region for A1413 is
within $r_{\rm cool}=\rm 0.6\,arcmin=90\,kpc$ (Fig. 4, green vertical line).

Hudson et al. (2010) argued that the central cooling time $t_{\rm cool}$ is
the best parameter, for low-redshift clusters, to identify cool-core and non-
cool-core clusters. They divided clusters into three types: $t_{\rm cool}<1$
Gyr for strong cool-core clusters; $t_{\rm cool}\sim\,1-7.7$ Gyr for weak
cool-core clusters; $t_{\rm cool}>7.7$ Gyr for non-cool-core clusters.
Following the definition of a cool-core based on the cooling time given by
Hudson et al. (2010), A1413 appears to be a weak cool-core cluster with a
central cooling time of $t_{\rm cool}=6.4\pm 1.3$ Gyr within $r<0.5\arcmin\sim
75\,\rm kpc$.

Figure 4: Cooling time profile for A1413. The red solid line represents the
look-back time at z=1, $\rm t_{age}=7.7\,Gyr$. The power-law fit with $\rm
t_{cool}=9.27R^{1.84}$ allows to define the cooling radius (green vertical
line), namely the region within which $\rm t_{cool}<t_{age}$.

###  3.5 Radio morphology

Previous studies of A1413 at 1.4 GHz (Govoni et al. 2009) and 144 MHz (Savini
et al. 2019) frequencies confirmed the cluster hosting a central mini-halo.
Thanks to the improved calibration strategy and our deeper LOFAR HBA
observations, we detect low surface brightness emission which extends further
out from the cooling region, only visible in the LoTSS images. It is elongated
in the North-South direction, as it happens for the X-ray, and has a maximum
linear size of $\rm D\sim 800\,kpc$ ($3\sigma_{\rm rms}$ contour), on scales
typical of giant radio halos (see source-subtracted $35\arcsec$-resolution
image of Fig. 1, right). West of the cluster, a head-tail radio galaxy with
optical counter part at $z=0.144$ (i.e., consistent with the cluster’s
redshift) is present, which appear to be connected with the outer radio
emission, though we can not exclude projection effects. Given the new
discovery, we analysed the radio emission of the cluster to understand whether
we detected a giant radio halo or as the superposition of two different
sources (mini-halo plus halo emission), as recently found in few other
clusters (RX J1720.1+2638 of Biava et al. 2021, Abell 2142 of Bruno et al.
(submitted)).

###  3.6 Radio surface brightness profiles

The exponential profile has been customarily used to fit the SB profile of
giant and mini-halos (e.g. Murgia et al. 2009; Botteon et al. 2022; Bonafede
et al. 2022; Cuciti et al. 2022). This is a simple model with only two free
parameters, and it provides a reasonable description of many radio halos. We
studied the radio surface brightness making use of two approaches: we first
performed a 1D analysis following the fitting procedure of Murgia et al.
(2009), and then the more recent 2D method developed by Boxelaar et al.
(2021).

However, we point out that this is not a physical model, and Botteon et al.
(2022, 2023) showed that the exponential is not always a good model for
representing halos. In particular, when halos present substructures in the
radio emission (such as edges and filaments), the exponential smooth profile
leaves strong residuals.

Following Murgia et al. (2009), we calculated the radio brightness profile,
averaged in concentric annuli of $\mathrm{15\arcsec}$ width, i.e. the FWHM of
image beam, centred on the peak of the radio halo (Fig. 5). In fact, to
increase the number of points for the fit, we used the source-subtracted
$\mathrm{15\arcsec}$ resolution image, which is the most sensitive to the
central mini-halo emission. To test the double morphology of the radio source,
we extended the procedure of Murgia et al. (2009) using a double exponential
law of the form

$I(r)=I_{0,1}e^{-r/r_{\rm e,1}}+I_{0,2}e^{-r/r_{\rm e,2}},$ (4)

where the parameters are the central radio surface brightnesses
$I_{0,1},I_{0,2}$ and the corresponding $e$-folding radii $r_{\rm e,1},r_{\rm
e,2}$, the length-scales at which the surface brightness drops to $I_{0}/e$.
We masked all the discrete sources in the cluster outskirt which would
possibly overlap with the extended emission (red regions of Fig. 5, left) and
stopped the profile where average surface brightness is 2 times the detection
limit in each annulus $\sigma_{\rm annulus}=\sigma_{\rm rms}/\sqrt{N_{\rm
beam}}$ (dashed black line in Fig. 5, left), that corresponds to $0.4R_{500}$
from the cluster centre. The resulting profile is shown in Fig. 5 (right) and
the best fit parameters are listed in Table 4. We see a clear evidence of an
inner brighter component and a second broader emission, with a surface
brightness one order of magnitude lower than that of the inner one. This
double-exponential fit allows us to set a boundary on the extension of the
inner component. Motivated by the empirical evidence that halos do not extend
indefinitely, it is customary to adopt a source radius of $3r_{\rm e}$ (Murgia
et al. 2009; Botteon et al. 2022; Bruno et al. 2023), which contain 80% of the
flux that would be obtain when integrating the model up to infinity
$S_{\nu}(<3r_{\rm e})=0.8S_{\nu}^{\infty}$. Thus, we fix the size of the mini-
halo at three times the $e$-folding radius of the first model component,
$R_{\rm MH}=3r_{\rm e,1}=84$ kpc. The second component results to have a
extension of $R_{\rm ext}=3r_{\rm e,2}=870$ kpc, which corresponds to a total
size $D_{\rm ext}\sim 1.7$ Mpc. This means, double the size based on the
$3\sigma_{\rm rms}$-contours of Section 3.5. We notice that the fitted
$e$-folding radius for the inner component is smaller that the value of
Riseley et al. (accepted), who measured a total extent of 343 kpc (i.e. a
radius of $\sim$ 170 kpc) at 144 MHz, using the same LOFAR observations.
However, we point out that those authors focus their study on the inner
component only, using a higher $3\sigma_{\rm rms}$ threshold and a more
robust-weighted imaging parameters for a better spatially-resolved spectral
analysis. When using common imaging settings, the results are consistent.

Figure 5: Left panel: Medium resolution ($15\arcsec\times 15\arcsec$) image at
144 MHz. Masked regions are shown in red, together with the green annuli where
the surface brightness has been extracted. The contour levels start at
$2\sigma_{\rm rms}$, where $\mathrm{\sigma_{\rm rms}=0.18\,mJy/beam}$, and are
spaced with a factor of $\mathrm{\sqrt{2}}$. The $-3\sigma_{\rm rms}$ contours
are red coloured. Right panel: Radio surface brightness profile. The solid
line represents the resulting double exponential model, while the single
components are plotted as reference. The central mini-halo component has
$r_{\rm e,1}=28\pm 5$ kpc, while the second broader component $r_{\rm
e,2}=290\pm 60$ kpc. The vertical line indicates $R_{500}$ and the dashed
black line the $1\sigma_{\rm annulus}$ detection limit for each annulus.

We then analysed the low resolution ($\mathrm{35^{\prime\prime}\times
35^{\prime\prime}}$) image, where the broader extended emission can be
detected at higher signal to noise (Fig. 1 (right)), using Halo-Flux Density
CAlculator888 https://github.com/JortBox/Halo-FDCA (Halo-FDCA, Boxelaar et al.
2021) fitting procedure. This method fits the surface brightness profile of
halos to a 2D-model using Bayesian Inference and allows to extend the
exponential 1D-model, taking into account different shapes of halos (circular,
elliptical or skewed). Moreover, it is able to perform flux density and size
estimation less affected by the image noise. The general exponential model
used by Boxelaar et al. (2021) is

$I(\textbf{r})=I_{0}e^{-G(\textbf{r})},$ (5)

where $I_{0}$ is the central surface brightness value and $G(\textbf{r})$ the
function that takes different forms depending on the complexity of the model.
In the circular model case, $G(\textbf{r})=|{\textbf{r}|/r_{\rm e}}$ recalls
the exact 2D version of the exponential profile used in Murgia et al. (2009).
Since Halo-FDCA is not built for a double component emission, we additionally
masked the central MH region to derive flux coming from the extended emission
only (Fig. 9). In masked regions, Halo-FDCA performs an extrapolation of the
model. The masked circular MH region has a size $R_{\rm MH}=\rm 84\,kpc$ based
on the 1D-fitting results. Despite the clear elliptical shape of the extended
radio emission, elongated in the North-South direction (Fig. 1, right), Halo-
FDCA does not find the elliptical model to be a better representation of the
source (see Fig. 9 and Table 5). From the circular 2D model, we get the
$e$-folding radius $r_{\rm e}^{\rm FDCA}=224\pm 16\,\rm kpc$ (i.e. $R_{\rm
ext}^{\rm FDCA}=3r_{\rm e}^{\rm FDCA}\sim 675$ kpc, which corresponds to
$D_{\rm ext}^{\rm FDCA}\sim 1.3$ Mpc) and a flux of $S^{\rm FDCA}=\rm 106\pm
8\,mJy$ integrated until $3r_{\rm e}^{\rm FDCA}$. The best-fit results are
listed in Tab. 5.

The two fitting methods give compatible results for the extended component:
$r_{\rm e,2}=290\pm 60$ kpc and $r_{\rm e}^{\rm FDCA}=224\pm 16\,\rm kpc$, for
the 1D and Halo-FDCA $e$-folding radius, respectively. Moreover, both methods
find the second component to be wider compared to the $800\rm\,kpc$ size based
on the $\rm 3\sigma_{\rm rms}$ contour of Fig. 1, with a total linear size
$D_{\rm ext}>1.2\,\rm Mpc$. This suggests the presence of a second very low
surface brightness radio emission filling the cluster volume, overlapped with
the previously known central mini-halo.

Table 4:  Summary of the main properties of the radio emission in A1413. | $\nu$ | $D$ | $S_{3\sigma}$ | $P_{3\sigma}$ | $I_{0,12}$ | $r_{e,12}$ | $\rm\chi$
---|---|---|---|---|---|---|---
MHz | $\mathrm{kpc}$ | mJy | W/Hz | $\rm\mu Jy/arcsec^{2}$ | $\mathrm{kpc}$ |
Halo | $144$ | $(800,500)$ | $37\pm 6$ | $1.9\pm 0.3\,10^{24}$ | $1.6\pm 0.4$ | $290\pm 60$ | 1.8
Mini-Halo | $144$ | $220^{\star}$ | $23\pm 2$ | $(1.2\pm 0.1)\times 10^{24}$ | $23\pm 5$ | $28\pm 5$ |
| $1440$ | $220$ | $1.9\pm 0.7$ | $(1.0\pm 0.1)\times 10^{23}$ |  |  |
Col. 1: Name of the radio source;  Col. 2: size measured above $\rm 3\sigma_{\rm rms}$ contour. For the halo, the two measures refer to $(D_{max},D_{min})$ in the north-south and west-east direction, respectively. Listed result at 1440 MHz from Govoni et al. (2009). ⋆: Since in LOFAR observations it is impossible to disentangle the mini-halo, we use 200 kpc as a reference (see Section 3.7).  Col. 3: Flux density measured within the $\rm 3\sigma_{\rm rms}$ contour;  Col. 4: Radio power with the k-correction applied, assuming $\rm\alpha=-1.2$;  Col. 5: Central surface brightness of the double-exponential profile of Eq. 4;  Col. 6: $e$-folding radius of the double-exponential profile of Eq. 4;  Col. 7: reduced $\chi^{2}$ of the fit. ———————————————————————————————————————————- Table 5:  Summary of the Halo-FDCA results for the extended emission in A1413.  | $\nu$ | $I_{0}$ | $r_{e,1}$ | $r_{e,2}$ | $r_{e,3}$ | $r_{e,4}$ | $S_{<3r_{e}}$ | $P_{<3r_{e}}$ | $\rm\chi$ | Model | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
MHz | $\rm\mu Jy/arcsec^{2}$ | kpc | kpc | kpc | kpc | mJy | $\rm W/Hz$ |  |  |
Halo | $144$ | $2.6\pm 0.2$ | $224\pm 16$ | - | - | - | $106\pm 8$ | $(6.0\pm 0.5)\times 10^{24}$ | 0.823 | circle |  Fig. 9 (top panel)
|  | $2.6\pm 0.2$ | $228\pm 17$ | $224\pm 16$ | - | - | $107\pm 8$ | $(6.0\pm 0.5)\times 10^{24}$ | 0.828 | ellipse |  Fig. 9 (middle panel)
|  | $2.6\pm 0.2$ | $295\pm 40$ | $227\pm 35$ | $148\pm 25$ | $255\pm 34$ | $109\pm 8$ | $(6.0\pm 0.5)\times 10^{24}$ | 0.769 | skewed |  Fig. 9 (bottom panel)
Col. 1: Name of the radio source;  Col. 2: Frequency;  Col. 3: Central surface
brightness obtained by Halo-FDCA, Eq. 5;  Col. 4: $e$-folding radius obtained
by Halo-FDCA (circular model),  Col. 5: (elliptical model),  Col. 6: (skewed
model)  Col. 7: Flux density measured by Halo-FDCA the $\rm 3r_{e}$;  Col. 8:
Radio power with the k-correction applied, assuming $\rm\alpha=-1.2$;  Col. 9:
Reduced $\chi^{2}$;  Col. 10: Reference image.
———————————————————————————————————————————- Figure 6: Image at 1.5 GHz with
beam of $\rm 15\arcsec\times 15\arcsec$. The contour levels start at
$2\sigma_{\rm rms}$, where $\mathrm{\sigma_{\rm rms}=0.035\,mJy/beam}$, and
are spaced with a factor of $\mathrm{\sqrt{2}}$. The $-3\sigma_{\rm rms}$
contours are red coloured, while $2\sigma_{\rm rms}$ contours are dotted green
coloured. This image, shown here to provide context, is derived and presented
in Govoni et al. (2009).  Figure 7: Background subtracted and exposure
corrected XMM-Newton image of A1413 in [0.7-2] keV energy band. Radio contours
from $35\arcsec$ LOFAR HBA image at 144 MHz Fig. 1 (right) are overplotted.

###  3.7 Mini-halo integrated spectral index

The radio mini-halo emission detected by Govoni et al. (2009) (Fig. 6) is
about $1.5^{\prime}$ ($\simeq$ 220 kpc) in size, and has a flux density of
$\mathrm{S_{MH,1.4GHz}=1.9\pm 0.7\,mJy}$, which corresponds to a power of
$\mathrm{P_{MH,1.4GHz}=1.0\cdot 10^{23}\,W/Hz}$ (see Table 4). We caution that
this flux density value should be considered as an upper limit to the actual
flux density of the MH at 1.5 GHz. Indeed, just the contribution from a single
unresolved source (source A in our analysis) detected in the FIRST survey
(Becker et al. 1995) was considered, whereas an additional point source is
embedded within the MH, as shown by our LOFAR high-resolution image (Fig. 1
left, source B). To estimate the integrated spectral index of the MH emission,
we have re-imaged the LOFAR data at the same $\mathrm{15^{\prime\prime}\times
15^{\prime\prime}}$ resolution and with the common inner $uv$-cut of
$180\lambda$, minimum baseline of VLA image. To be consistent with Govoni et
al. (2009), we use a circular region with $\mathrm{r\sim 0.7\arcmin\sim
105\,kpc}$ and find total flux density of $\mathrm{S_{MH,144MHz}=23\pm
2\,mJy}$ (Tab. 4). The integrated spectral index between 144 MHz and 1.5 GHz
is $\alpha_{\rm MH}=1.1\pm 0.2$. Note that the spectral index might be
steeper, considering that the flux reported by Govoni et al. (2009) could be
slightly overestimated, as explained above.

As the extended halo emission is not detected in the VLA image, we try to put
a lower limit to its average spectral index, comparing the mean surface
brightness of the extended emission at 144 MHz with the rms noise of VLA
image. To do this, we consider a rectangular area that follows the
$3\sigma_{\rm rms}$ contours of the 800 kpc scale emission (Fig. 1 right) and
exclude the central $\mathrm{r<0.7^{\prime}}$. Unfortunately, this does not
allow us to firmly constrain the spectral index of the external emission, as
we find $\alpha>0.75$ at $1\sigma_{\rm rms}$. Riseley et al. (accepted),
produced resolved spectral map combining MeerKAT at 1283 MHz with LOFAR at 145
MHz data and found that the innermost region ($R\lesssim 50$ kpc, which
corresponds to $\sim 2r_{\rm e,1}$ of our inner component) shows a flatter
spectral index around $-0.7$ to $-1$, while there is a hint of larger-scale
diffuse emission with a typical spectral index of around $-1.1$ or steeper.
The different spectral index of the two regions may suggest the presence of
two different types of radio emission co-existing in the same cluster. Further
investigation about the hybrid morphology of the radio emission is performed
in Section 4.1, where also X-ray properties are considered.

##  4 Discussion

###  4.1 Spatial correlation between X-ray and radio surface brightness

Quantitative studies to investigate the point-to-point correlation between the
radio and X-ray brightness distribution of giant halos can provide useful
information about the connection between the thermal and non-thermal emission
(e.g. Govoni et al. 2001; Feretti et al. 2001; Giacintucci et al. 2005).
However, most of the literature studies were limited by the resolution and
sensitivity of the radio and X-ray observations. Now, thanks to deep and
sensitive observations, these studies can be carried out in more detail and
extending the sample also to mini-halos (Ignesti et al. 2020; Botteon et al.
2020; Rajpurohit et al. 2021; Biava et al. 2021; Duchesne et al. 2021;
Bonafede et al. 2022; Riseley et al. 2022a, b). These recent studies have
shown different trends for giant and mini-halos: halos turned out to show a
linear or sub-linear scaling, while mini-halos tend to have a super-linear
one.

The correlation is generally investigated via a power-law relationship of the
kind:

$\log{I_{R}}=A\,\log{I_{X}}+B,$ (6)

where the slope of the scaling $A$ determines whether the radio brightness
(i.e. the magnetic field strength and CRe density) declines faster ($A>1$)
than the X-ray brightness (i.e. the thermal gas density), or vice versa
($A<1$). The connection between thermal and non-thermal plasma could reflect
the particle re-acceleration mechanism, and it could be used to discriminate
between models of halo formation (Govoni et al. 2001).

Table 6: LinMix best fit parameter for $I_{R}-I_{X}$ correlation of reference figure Region | Grid | $A$ | $r_{p}$ | $r_{S}$ | Ref.
---|---|---|---|---|---
Halo | $35^{\prime\prime}\times 35^{\prime\prime}$ | $0.81^{+0.07}_{-0.08}$ | $0.87$ | $0.85$ |  Fig. 8 (left)
| $35^{\prime\prime}\times 35^{\prime\prime}$ | $0.78^{+0.08}_{-0.07}$ | $0.83$ | $0.82$ |  Fig. 8 (left)
Mini-halo | $23^{\prime\prime}\times 23^{\prime\prime}$ | $1.27^{+0.16}_{-0.14}$ | $0.93$ | $0.91$ |  Fig. 8 (right)
Col. 1: Considered radio source;  Col. 2: Grid boxes size;  Col. 3: fitting
slopes ($A$);  Col. 4: Pearson ($r_{p}$), and  Col. 5: Spearmann ($r_{s}$)
correlation coefficients;  Col. 6: Reference plot. ————–

To compute these correlations we used the
LinMix999https://linmix.readthedocs.io/en/latest/index.html package (Kelly
2007), which uses a Bayesian approach to linear regression and allows to
consider measurement uncertainties on both quantities. The threshold for
upper-limits is fixed at $2\sigma_{\rm rms}$ in the radio image following
Botteon et al. (2020). The strength of the correlation is measured by the
Pearson $r_{p}$ and Spearmann $r_{s}$ correlation coefficients. The former
assessing the linear relation between the two variables, while the latter
determining their monotonic relation (whether linear or not).

Based on the two-component emission found with the surface brightness analysis
of Section 3.6, we performed point-to-point correlation separately for both
the extended and mini-halo emission, using the radio image at 144 MHz
presented in Fig. 1 (right) and Fig. 5 (left) together with the XMM-Newton
images presented in Fig. 7. Following Govoni et al. (2001), we constructed a
grid covering the cluster region and calculated for each box the mean radio
and X-ray quantities, as well as the root-mean-square (rms), which can be
assumed as an estimate of the statistical error. The size of the boxes has
been chosen to be equal to the 1.5 times the area of the radio beam (see Table
6). This grid resolution also allows to be minimally affected by the PSF of
XMM-Newton, which for the combined MOS+pn images is $\sim 10^{\prime\prime}$
(FWHM).

First, we performed the correlation on the 35$\arcsec$-resolution image (Fig.
1, right) to study the properties of the extended emission. The total
considered area was chosen based on the lowest X-ray brightness level, that
includes the radio emission and excludes other external radio sources, in
order to select an homogeneous and not radio-biased area. Both point-like and
extended (the tailed galaxy) contaminating sources in the radio image were
identified and cells containing them, removed. Presumably, the tailed galaxy
also influences some contiguous regions south of the tail, thus, a safe choice
is to exclude these regions too. Final selection (blue boxes) and upper limits
(cyan boxes) are shown in the top left corner of each plot of Fig. 8.

Figure 8:  Left:  $I_{R}-I_{X}$ relation of the halo in A1413, extracted in
square boxes with width of $35^{\prime\prime}$. The horizontal grey dash-
dotted lines indicate the $\sigma_{\rm rms}=298\,\mu Jy/$beam in the radio
maps. Right: $I_{R}-I_{X}$ relation of the mini-halo in A1413, extracted in
square boxes with width of $23^{\prime\prime}$. The horizontal grey dash-
dotted lines indicate the $\sigma_{\rm rms}=203\,\mu Jy/$beam in the radio
maps. For both relations, upper limits (cyan points) refer to cells where the
radio surface brightness is below the $2\sigma_{\rm rms}$ level. The best-fit
(green dashed line) is reported with the corresponding $1\sigma$ confidence
interval. Red dashed line best-fit line (left panel) refers to the correlation
without the 4 central boxes.

The two surface brightnesses are positively correlated $r_{p}=0.87,r_{s}=0.85$
and have a sub-linear scaling $A=0.81^{+0.07}_{-0.08}$ We notice that the four
central boxes, which spatially correspond to the central brighter component,
are distinctly separated from the ones of the extended component. Thus, we
also calculate the correlation removing these four central points. We still
find a sub-linear ($A=0.78^{+0.08}_{-0.07}$) strong ($r_{p}=0.83,r_{s}=0.82$)
correlation, consistent with the previous one. The correlation indicates that
the decline of the non-thermal radio component is shallower than the thermal
one: this slope can be explained by re-acceleration models, while it is a
challenge for hadronic models (as explained by Govoni et al. 2001).

We then study the central mini-halo component, making use of the higher
15$\arcsec$-resolution radio image (Fig. 5), that is the most sensitive to the
central emission. Differently from the sub-linear or linear scalings that are
reported in the literature for giant radio halo, but consistently with the
sample of MHs studied by Ignesti et al. (2020), we find instead a super-linear
$A=1.27^{+0.16}_{-0.14}$ correlation for the central emission (Fig. 8, right).
Also in this case, there is a clear evidence of a spatial correlation between
radio and X-ray emission, with $r_{p}=0.93,r_{s}=0.91$ (Table 6). This super-
linear correlation is also consistent with analysis of Riseley et al.
(accepted), who found $A=1.20^{+0.13}_{-0.11}$ using LOFAR data at 144 MHz and
$A=1.63^{+0.10}_{-0.10}$ reproducing the correlation with MeerKAT data at 1283
MHz. The super-linear scaling between $I_{R}$ and $I_{X}$ suggests that the
number density of emitting electrons and magnetic field rapidly declines from
the center to the external regions. In particular, the radio emission is more
peaked than the thermal emission, indicating that the ICM non-thermal
component is more concentrated around the central AGN. This is consistent with
the hadronic scenario that predicts a super-linear scaling between the surface
brightness for a radially decreasing magnetic field (e.g. Govoni et al. 2001).
We note, however, that the re-acceleration model cannot be excluded. Depending
on the CRe distribution, it could also produce a super-linear scaling of the
X-ray and radio surface brightness.

The different trends of the two radio sources further strengthens the idea
that the radio emission of A1413 is composed by two different components. They
might also mean that the mechanisms responsible for the re-acceleration of the
radio emitting particle are different.

##  5 Conclusions

In this work, we have presented a composite X-ray and radio analysis of the
galaxy cluster A1413. For this purpose, XMM-Newton and LOFAR 144 MHz data were
first analysed separately. They were then combined to quantify the connection
between the thermal and non-thermal components of the ICM. The main results
can be summarised as follows:

  1. 1.

The X-ray morphology of the cluster on large scales is not spherically
symmetric, confirming the ellipticity found in previous work in, both, optical
and X-ray bands. This morphology suggests the occurrence of a merger that has
left a clear, though minor, imprint on the gas distribution along the N-S axis
of the cluster. Other dynamical features indicate that A1413 is an
intermediate phase cluster: The temperature profile in the core does not
decline sharply as it is expected in the inner regions of a strong cool-core
cluster. Moreover, the metallicity profile is flatter than expected for a very
relaxed system. However, the surface brightness profile is well-fitted with a
double $\beta$-model, which typically accounts for the central brightness
excess of cool-core clusters. Finally, from the calculation of the cooling
time, we obtain $t_{\rm cool}=6.4\pm 1.3\,\rm Gyr$ within $r<0.5\arcmin\sim
75\,\rm kpc$, and a cooling region $r_{\rm cool}\sim 90\,\rm kpc$. We argue
that such a system with a moderate cooling time, an elevated central entropy
and a slightly decreasing central temperature profile can be classified as a
weak cool-core system (Hudson et al. 2010).

  2. 2.

A1413 was known to host a mini-halo (Govoni et al. (2009); Giacintucci et al.
(2017); Savini et al. (2019) Riseley et al. (accepted)). The LOFAR HBA data at
144 MHz led to the discovery of an extended low surface brightness emission,
with an LLS of about 800 kpc in the N-S direction based on the $3\sigma_{\rm
rms}$-contours, so far detected only at low frequencies. To date, this is one
of the few objects that show a wider emission surrounding a previously known
central mini-halo and with a size typical of giant radio halos.

  3. 3.

We perform 1D surface brightness fitting using the double exponential model of
Eq. 4. We find an inner brighter component extended up to $R_{\rm MH}=84$ kpc,
which is consistent with other mini-halos, and a broader component with
$R_{\rm ext}=870$ kpc. Additionally, we performed a 2D fitting of the extended
component only using the Halo-FDCA software. To do that, we masked the
internal region, and we obtained best fitting values of $R_{\rm ext}^{\tiny
FDCA}=3r_{\rm e}^{\tiny FDCA}\sim 675$ kpc for the circular model (see Table
5). Both methods give consistent results and find the second component to be
wider than the $800\rm\,kpc$ size based on the $\rm 3\sigma_{\rm rms}$ contour
of Fig. 1, with a total linear size $D_{\rm ext}>1.2\,\rm Mpc$.

  4. 4.

We performed a point-to-point correlation analysis between the X-ray and radio
surface brightness, which confirms the spatial connection between the non-
thermal and thermal emission in galaxy clusters. We investigated a possible
correlation for, both, the halo and the mini-halo regions separately. We found
a sub-linear correlation $I_{R}\propto I_{X}^{0.78^{+0.08}_{-0.07}}$ when
analysing the extended part and a super-linear $I_{R}\propto
I_{X}^{1.27^{+0.16}_{-0.14}}$ correlation for the inner mini-halo component.
Sub-linear or linear scalings are reported in the literature for giant radio
halos, and can be explained by re-acceleration scenario. On the other hand,
super-linear correlations are usually found for mini-halos, indicating that
the ICM non-thermal component is more concentrated around the central AGN.
This is consistent with the hadronic scenario. Nevertheless, re-acceleration
model cannot be excluded, as depending on the CRe distribution, it could also
produce a super-linear scaling.

The properties of this cluster show all the signs of an intermediate-state
cluster, dynamically not fully relaxed and with diffuse radio emission outside
the central cooling region, as would be expected for cool-core objects. Both
the SB fitting and the $I_{R}-I_{X}$ correlations confirm the presence of two
components as found in other clusters with the same dynamical characteristics
(e.g., RX J1720.1$+$2638 Biava et al. 2021, A2142, Bruno et al. (submitted)).
The mini-halo and the diffuse emission extend over different scales and show
different features, suggesting that the mechanisms responsible for the re-
acceleration of the radio-emitting particles are different. The outer
component might probe turbulent re-acceleration induced by a less energetic
merger events. In this context, we argue that minor mergers events can induce
particle acceleration on large scales without fully disrupting the cluster
core. Considering its hybrid radio morphology and intermediate X-ray
properties, A1413 is a peculiar object that challenges the long lasting
picture of disturbed galaxy clusters hosting giant radio halos and relaxed
ones having mini-halos.

###### Acknowledgements.

This research project made use of the following Python packages: APLpy
(Robitaille & Bressert 2012), Astropy (Astropy Collaboration et al. 2013) and
NumPy (van der Walt et al. 2011). CJR and AB acknowledge financial support
from the ERC Starting Grant ‘DRANOEL’, number 714245. LL acknowledges
financial contribution from the INAF grant 1.05.12.04.01. MB acknowledges
support from the Deutsche Forschungsgemeinschaft under Germany’s Excellence
Strategy - EXC 2121 ”Quantum Universe” - 390833306 and from the BMBF ErUM-Pro
grant 05A2023. RJvW acknowledges support from the ERC Starting Grant
ClusterWeb 804208. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array
designed and constructed by ASTRON. It has observing, data processing, and
data storage facilities in several countries, which are owned by various
parties (each with their own funding sources), and that are collectively
operated by the ILT foundation under a joint scientific policy. The ILT
resources have benefited from the following recent major funding sources:
CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-
NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business,
Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science
and Technology Facilities Council, UK; Ministry of Science and Higher
Education, Poland; The Istituto Nazionale di Astrofisica (INAF), Italy. This
research made use of the Dutch national e-infrastructure with support of the
SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. The Jülich
LOFAR Long Term Archive and the German LOFAR network are both coordinated and
operated by the Jülich Supercomputing Centre (JSC), and computing resources on
the supercomputer JUWELS at JSC were provided by the Gauss Centre for
Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for
Computing (NIC). This research made use of the University of Hertfordshire
high-performance computing facility and the LOFAR-UK computing facility
located at the University of Hertfordshire and supported by STFC
[ST/P000096/1], and of the Italian LOFAR IT computing infrastructure supported
and operated by INAF, and by the Physics Department of Turin university (under
an agreement with Consorzio Interuniversitario per la Fisica Spaziale) at the
C3S Supercomputing Centre, Italy.

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##  Appendix A Flux Density CAlculator results

Figure 9:  FDCA-Halo results for the radio halo in A1413 at 144 MHz frequency.
Left panel: Original image with the contaminating regions masked out. Middle
panel: Circular model map. Right panel: Residual image. The contour shows the
2 $\sigma_{\rm rms}$ level of the model. The red contours show the masked
regions, the contamination sources are visible.