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

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 di ff use 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 observations centered at 144 MHz to study the di ff use 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 di ff use 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 e , 1 = 28 ± 5kpc and an extended component with r e , 2 = 290 ± 60kpc. 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 di ff use emission extend over di ff erent scales and show di ff erent 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 di ff erent.


Introduction
Galaxy clusters have a size that ranges from a few hundreds of kpc to, approximately, Mpc, and have typical masses of 10 14 −10 15 M .They can contain up to 1000 galaxies, embedded in the intracluster medium (ICM).The ICM consists of fully ionized hydrogen and helium and traces of highly ionized heavier elements with a temperature T ∼ 10 7 −10 8 K (1−10 keV) and particle number densities steeply declining from n e ∼ 10 −2 −10 −3 cm −3 near the center to 10 −4 cm −3 in the outskirts.It emits mainly via thermal bremsstrahlung in the X-ray band (e.g., Forman & Jones 1982;Sarazin 2009).
Co-spatially with the ICM, an increasing number of galaxy clusters is also observed to host diffuse nonthermal 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 (CRes) 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 spectrum 1 with a spectral index α > 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 (∼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 nonthermal radio sources (see Brunetti & Jones 2014 for a review): (i) the (re-)acceleration scenario, where CRes are (re-) accelerated by turbulence that develops during cluster mergers; (ii) the hadronic scenario, where secondary CRes are produced by proton-proton collisions between relativistic cosmicray protons (CRps) 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(Brunetti et al. , 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. 2010Cassano et al. , 2023;;Cuciti et al. 2021).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(Gitti et al. , 2018;;Cassano et al. 2010), 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 (M > 6 × 10 14 M ) galaxy clusters, Giacintucci et al. (2017) found that ∼80% of cool-core clusters (classified according to the value of central entropy 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 recent 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. 2018Savini et al. , 2019;;Biava et al. 2021;Riseley et al. 2022b;Bruno et al. 2023a), challenging the idea that mini halos occur in cool-core clusters and giant halos in noncool-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 they are classified as relaxed systems based on their thermal properties.Until now, a somewhat unique case has been 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 intrinsically 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. (2023a), 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.
In this work, we present the galaxy cluster Abell 1413 (hereafter A1413; Planck name: PSZ2 G226.18+76.79),which is a nearby (z = 0.143) massive (M 500 = 5.95 +0.24  −0.25 × 10 14 M , Planck Collaboration XXVII 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 wavelengths showed that the properties of A1413 are intermediate between classical relaxed and merging clusters.From the morphological point of view, it shows a high ellip-ticity 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 ∼ 0.35 at large radii, increasing up to ∼0.8 at the cluster center r < 1 .While fitting the XMM-Newton image, they found values quickly converging toward ∼ 0.27, which is fully compatible with the global value of ∼ 0.29 derived by Pratt & Arnaud (2002) over the radial range [3 −13 ].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 a spherically symmetric distribution more rapidly.However, Castagné et al. (2012) also stressed that A1413 seems a rather extreme example of this effect.Moreover, by analyzing the velocity distribution of the cluster, they found a velocity offset of 450 ± 210 km s −1 (i.e., 2σ significance) between the velocity of the central cD galaxy (Paturel et al. 1989), which is also among the brightest cluster galaxies (BCGs), 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 also been 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 (∼50 kpc) of t cool = 4.2 ± 0.3 Gyr (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, such as elongated X-ray isophotes and a relatively flat X-ray profile near the center.
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 halo 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 ± 8 keV cm 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 ∼210 kpc and a total flux density of 40 ± 7 mJy.In a recent study of Riseley et al. (2023), 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 A132, page 2 of 14 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 analyzed XMM-Newton data, which include a more accurate background treatment and improved calibration.This paper is organised as follows.In Sect. 2 we give an overview of the observations and data reduction.The X-ray and radio results are presented in Sect.3. We discuss the results from X-ray and radio analysis jointly in Sect. 4. In Sect.5, we summarise our conclusions.
Throughout the paper we adopt a cosmology with H 0 = 70 km s −1 Mpc −1 , Ω Λ = 0.7 and Ω m = 0.3.Thus, 1 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.

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 the XMM-Newton telescope, the latter also being part of the CHEX-MATE program (CHEX-MATE Collaboration 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.

XMM-Newton
A1413 was observed with XMM-Newton four times between 2007 and 2008.All four observations were longer than 60 ks each.For this work, we retrieved the longest and least flared one, OBS ID 0502690201, from the XMM-Newton data archive (see Table 2).Data consist of 82.5 ks of archival observation time during XMM-Newton revolution 1466 (2007 December 11), obtained with the THIN1 filters and processed with XMM-SAS2 v19.0.0 and the XMM-ESAS3 software package (Snowden et al. 2008) and through a dedicated pipeline that follows the procedure described in Lovisari & Reiprich (2019) and 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 tasks.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 ≤ 12) and single and double events for pn (i.e., PATTERN ≤ 4).We also performed bright pixels and hot column 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 Sect.2.1.1)were removed using a two-stage filtering process (see Lovisari et al. 2011 for details); these are summarised in the following.First, the light curve with 100 s bins is inspected in the [10][11][12] keV band for  keV for pn, since particle background dominates at high energy.The histogram of the light curve is then fit with a Poisson distribution, and time intervals with a higher count rate, i.e., a count rate that deviates by more than 2σ 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.This ensures a safety check for possible flares with soft spectra (e.g., Nevalainen et al. 2005).
Filtering the light curve does not ensure complete deletion of 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 suitable code of De Luca & Molendi (2004).By comparing area-corrected count rates in the in-and out-of-field-of-view (FoV) regions of the detector, we find that the SP contamination is low4 , with a ratio of 1.12 ± 0.02 for all the cameras.Nevertheless, we included SP contamination as an extra model component when dealing with spectral fitting (Sect.2.1.2).Anomalous behaviour of CCD 7 for MOS2 was detected via the mos-filter task (Kuntz & Snowden 2008) and excluded from this analysis, together with CCD 6 of MOS1, which was damaged by a meteorite in the early phase of the emission and thus is not available.
Point-like sources were detected with the edetect-chain task and visually inspected to distinguish between real point sources and extended cluster substructures before removing them from the event files.Even if it was identified as a 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.

Background treatment
The 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 Survey5 (RASS) diffuse background maps in a region beyond the cluster virial radius (between 1 and 2 deg) and fit it simultaneously with XMM-Newton data, as described in Lovisari & Reiprich (2019).We described the unresolved emission from AGNs via an absorbed power-law model with its slope set to 1.41 (De Luca & Molendi 2004); an absorbed ∼0.2 keV thermal component representing the Galactic halo (GH) emission, and an unabsorbed ∼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 AGN's 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 E few MeV) and soft protons (SP, with 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 closed6 (FWC) observations to estimate the intensity of QPB components.The model used for FWC observations consists of a broken power law with additional fluorescence lines (e.g., Fe, Al, Si, Au).Since FWC observations are taken from a different time period to our observation and have a different exposure, they need to be normalised, comparing them with the out-FoV 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 Sect.2.1, SP components are usually highly variable in time (ranging from ∼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 lightcurve screening procedures, which are summarized in Sect.2.1.Nonetheless, a more constant SP component 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).

Spectral analysis
The spectral analysis is performed with the XSPEC (Arnaud et al. 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 centered on the peak of the X-ray emission of coordinates RA(J2000) 11 h 55 m 17 s .85 and Dec (J2000) +23 • 24 m 18 s .85,and they extend until R 500 ∼ 1224 kpc (Planck Collaboration XXIX 2014).The sizes of the annuli were determined by requiring a minimum width of 30 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 1σ uncertainty in the spectrally resolved temperature (and consequently in the fit temperature profiles) is <10% (see Appendix B of Lovisari & Reiprich 2019 for more details on how uncertainties vary as a 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) N H ∼ 1.97 × 10 20 cm −2 value estimated using the Swift online tool7 .For the apec model (Smith et al. 2001), the parameters are plasma temperature [keV], metallicity, redshift (kept fixed at z = 0.1427), and the normalization n e n H dV, with D A being the angular distance of the source and n e ,n H the electron and proton densities in units of cm −3 .In each annulus, the MOS and pn spectra were fit simultaneously and their normalizations left free to vary; this was done 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.

LOFAR
The data used in this work are part of the LOFAR Two-metre Sky Survey (LoTSS; Shimwell et al. 2017Shimwell et al. , 2019Shimwell et al. , 2022)), a sensitive (rms ∼ 100 µJy beam), high-resolution (full width at half maximum (FWHM) of the synthesised beam of ∼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 an 8 h observation book-ended by 10 min scans of the calibrator (3c395 for the observations used in this work), at the central frequency of 144 MHz and 48 MHz in 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, which cover the 120-168 MHz band) and returns fully direction-dependent calibrated data.After calibration, the so-called extraction and selfcalibration 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 two data sets centered on the target that do not need any further beam correction.These two data sets are selfcalibrated again to obtain 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 re-imaged the self-calibrated data sets 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 us to downweight the secondary lobes, and inner uv-cut fixed at b min > 80λ  to avoid interference on short baseline, i.e., to filter out emission on angular scales larger than 43 .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 (∼0.5R 500 ) from the cluster center.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 multistep procedure.First, we produced images at high resolution (5 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 was created as a subtraction of the selected model components' uvdata from the original data.Given the slight extension of sources A and D, we performed an additional subtraction step, imaging the subtracted data sets at an intermediate ∼10 resolution.This allowed us to definitively remove all the sources, as they do not leave any residual during inspection of the final uv-subtracted data set.Eventually, the final data set is imaged with a negative Briggs weighting with R = −0.25 and tapering down the baselines longer than 10 kλ (i.e., using a Gaussian taper with a FWHM of 20 ) to gain sensitivity on 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 h pointed observation, while we combined two pointings for a total of 16 h of observation.Moreover, we used improved calibration techniques that allowed us to produce images with a higher level of sensitivity.For example, our 15 resolution image has a σ rms improved by more than a factor of two with respect to the one of Savini et al. (2019), with σ rms = 450 µJy beam at ∼20 -resolution.On the other hand, Riseley et al. (2023) used the same set of observations presented here and achieved comparable sensitivity.However, these authors undertook independent post-processing.
Throughout the paper, the error ∆ S associated with a flux density measurement, S , is estimated as where N beam = N pixel /A beam is the number of independent beams in the source area, and σ c indicates the systematic calibration uncertainty of the flux density, with a typical value of 10% for LOFAR HBA observations (Shimwell et al. 2022).Moreover, we point out that different statistical significance is used, depending on the kind of analysis.In the assessment of the radio diffuse emission's proprieties, such as size, flux density, and power, we employ the standard 3σ rms detection limit (see Sects.3.5 and 3.7).This also allows fair comparison with other works on the same cluster (e.g., Govoni et al. 2001;Riseley et al. 2023 cited in the paper).However, as explained in Sect.4, when performing surface brightness measurements, a 2σ rms is applied to avoid biased results.

Results
As presented in Sect. 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 the 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 analyzed the radio surface brightness in 1D and 2D to classify the morphology of the radio source and define its properties.

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 ∼ 75 kpc width) shows a drop from 7.43 ± 0.12 to 7.11 ± 0.01 keV.On the other hand, at larger radii the temperature profile decreases as expected for a 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, which is 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.Figure 2 shows that both properties do not clearly follow the mean profile of either relaxed or 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.A132, page 6 of 14

X-ray − optical offset
For dynamically relaxed systems, we expect the BCG to be at the center of the gravitational potential well (van den Bosch et al. 2005).Several X-ray studies have proven the 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 RA (J2000) 11 h 55 m 17 s .87,Dec (J2000) +23 • 24 m 16 s .02(Rawle et al. 2012) and correspond to the brightest pixel of the optical image, which was retrieved from the 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 7.2 kpc = 0.006 R 500 ) is still consistent with a relaxed system.Moreover, we point out that this offset corresponds to an angular size of ∼3 , which is slightly bigger than the X-ray image's pixel size (∼2 ), as used to determine the X-ray peak.

X-ray surface brightness profile
We computed the radial surface brightness (SB) profiles 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 S /N = 10 and minimum width of 2 .
The profile is fit with a double β-model (LaRoque et al. 2006) in the following form: where r c1 ,r c2 are the core radii, Σ 01 , Σ 02 are the central surface brightnesses of the two components, and β represents the ratio of specific kinetic energies of galaxies and gas (e.g., Gitti et al. 2012).This is the extension of the isothermal β model (Cavaliere & Fusco-Femiano 1976), commonly used to describe the surface brightness profile of the ICM.In fact, when fitting the profile over the radial range with a single β 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 r c ), producing non-random strong residuals.Thus, a second component was needed and 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).

Cooling time
From the temperature and density profiles, the cooling time of each region can be estimated as where γ = 5/3 is the adiabatic index, H is the enthalpy, µ ∼ 0.6 is the molecular weight for a fully ionised plasma, and Λ(T ) is the cooling function (Sutherland & Dopita 1993).Given the cooling profile, we can define the cooling radius, i.e., the radius at which the cooling time is less 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 age 7.7 Gyr (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 cool , we qualitatively fit the cooling-time profile with a power-law relation and selected r cool as the intersection between the bestfitting power law and t age .Thus, the cooling region for A1413 is within r cool = 0.6 arcmin = 90 kpc (Fig. 4, green vertical line).Hudson et al. (2010) argued that the central cooling time t 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 cool < 1 Gyr for strong cool-core clusters; t cool ∼ 1−7.7 Gyr for weak cool-core clusters; t 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 cool = 6.4 ± 1.3 Gyr within r < 0.5 ∼ 75 kpc.

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 is hosting a central mini halo.Thanks to the improved calibration strategy and our deeper LOFAR HBA observations, we detect low-surface-brightness emission that extends further out from the cooling region, only visible in the LoTSS images.It is elongated in the north-south direction, as happens for the X-ray, and has a maximum linear size of D ∼ 800 kpc (3σ rms -contour), on scales typical of giant radio halos (see source-subtracted A132, page 7 of 14 35 -resolution image of Fig. 1, right).West of the cluster, a head-tail radio galaxy with optical counterpart at z = 0.144 (i.e., consistent with the cluster's redshift) is present, which appears to be connected with the outer radio emission, though we cannot 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 the superposition of two different sources (mini halo plus halo emission), as was recently found in a few other clusters (RX J1720.1+2638 of Biava et al. 2021, Abell 2142of Bruno et al. 2023a).

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 we used 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. (2022Botteon et al. ( , 2023) ) showed that the exponential one 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 15 in width, i.e., the FWHM of an image beam, centered 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 15 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 where the parameters are the central radio surface brightnesses I 0,1 , I 0,2 and the corresponding e-folding radii r e,1 , r e,2 , the length scales at which the surface brightness drops to I 0 /e.We masked all the discrete sources in the cluster outskirts that would possibly overlap with the extended emission (red regions of Fig. 5, left) and stopped the profile where average surface brightness is two times the detection limit in each annulus σ annulus = σ rms / √ N beam (dashed black line in Fig. 5, left), which corresponds to 0.4R 500 from the cluster center.The resulting profile is shown in Fig. 5 (right) and the best-fit parameters are listed in Table 4.We see clear evidence of an inner brighter component and a second broader emission, with a surface brightness being 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 e (Murgia et al. 2009;Botteon et al. 2022;Bruno et al. 2023b), which contains 80% of the flux that would be obtained when integrating the model up to infinity S ν (< 3r e ) = 0.8S ∞ ν .Thus, we fixed the size of the mini halo to three times the e-folding radius of the first model component, R = 3r e,1 = 84 kpc.The second component results to have a extension of R ext = 3r e,2 = 870 kpc, which corresponds to a total size of D ext ∼ 1.7 Mpc.This means double the size based on the 3σ rms contours of Sect.3.5.We notice that the fit e-folding radius for the inner component is smaller that the value of Riseley et al. (2023), who measured a total extent of 343 kpc (i.e., a radius of ∼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 3σ rms threshold and more robustly weighted imaging parameters for a better spatially resolved spectral analysis.When using common imaging settings, the results are consistent.We also notice the presence of several 2σ rms contours with filamentary morphology around the central mini halo of Fig. 5 (left), which we attribute to the extended halo component.To exclude the possibility of chance occurrences, we analysed the surface brightness profiles of several other radio sources within the same field of view.This examination revealed the absence of plateau-like features similar to that depicted in Fig. 5 (right), confirming its association with the extended radio component.
We then analyzed the low-resolution (35 × 35 ) image, where the broader extended emission can be detected at a higher signal-to-noise ratio (Fig. 1, right) using a Halo-Flux Density CAlculator8 (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 us 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 estimations that are less affected by the image noise.The general exponential model used by Boxelaar et al. (2021) is where I 0 is the central surface brightness value and G(r) the function that takes different forms depending on the complexity of the model.In the circular model case, G(r) = |r|/r 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 mini halo region to derive flux coming from the extended emission only (Fig. A.1).
In masked regions, Halo-FDCA performs an extrapolation of  Notes.Column 1: Name of the radio source; Col. 2: Size measured above 3σ rms contour.For the halo, the two measures refer to (D max , D min ) in the north-south and west-east directions, respectively.Listed result at 1440 MHz from Govoni et al. (2009). ( )Since in LOFAR observations it is impossible to disentangle the mini halo, we used the 220 kpc region as a reference (see Sect. 3.7).Column 3: Flux density measured within the 3σ rms contour; Col. 4: Radio power with the k correction applied, assuming α = −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 χ 2 of the fit.  5.
The two fitting methods give compatible results for the extended component: r e,2 = 290 ± 60 kpc and r FDCA e = 224 ± 16 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 kpc size based on the 3σ rms contour of Fig. 1, with a total linear size of D ext > 1.2 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.

Mini halo integrated spectral index
The radio mini halo emission detected by Govoni et al. (2009;Fig. 6) is about 1.5 ( 220 kpc) in size and has a flux density of S 1.4 GHz = 1.9 ± 0.7 mJy, which corresponds to a power of P 1.4 GHz = 1.0 × 10 23 W Hz −1 (see Table 4).We caution that this flux-density value should be considered as an upper limit to the actual flux density of the mini halo 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 mini halo, as shown by our LOFAR highresolution image (Sect.1, left, source B).To estimate the integrated spectral index of the mini halo emission, we re-imaged the LOFAR data at the same 15 × 15 resolution and with the common inner uv-cut of 180λ, minimum baseline of VLA image.To be consistent with Govoni et al. (2009), we used a circular region with r ∼ 0.7 ∼ 110 kpc and find a total flux density of S 144 MHz = 23 ± 2 mJy (Table 4).The integrated spectral A132, page 9 of 14  Fig. 6.Radio image at 1.5 GHz with beam of 15 ×15 .The contour levels start at 2σ rms , where σ rms = 0.035 mJy beam −1 , and they are spaced with a factor of √ 2. The −3σ rms contours are red, while 2σ rms contours are dotted green.This image, shown here to provide context, is derived from and presented in Govoni et al. (2009).
index between 144 MHz and 1.5 GHz is α = 1.1 ± 0.2.We 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 tried to put a lower limit on 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 considered a rectangular area that follows the 3σ rms contours of the 800 kpc scale emission (Fig. 1, right) and excluded the central r < 0.7 .Unfortunately, this does not allow us to firmly constrain the spectral index of the external emission, as we find α > 0.75 at 1σ rms .Riseley et al. (2023) produced resolved spectral map combining MeerKAT at 1283 MHz with LOFAR at 145 MHz data and found that the innermost region (R 50 kpc, which corresponds to ∼2r 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 Sect.4, where X-ray properties are also considered.

Discussion: 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 nonthermal 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 also extend the sample 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 show a linear or sublinear scaling, while mini halos tend to have a superlinear one.
The correlation is generally investigated via a power-law relationship of the following kind: 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 nonthermal plasma could reflect the particle reacceleration mechanism, and it could be used to discriminate between models of halo formation (Govoni et al. 2001).
To compute these correlations, we used the LinMix9 package (Kelly 2007), which uses a Bayesian approach to linear regression and allows us to consider measurement uncertainties on both quantities.The threshold for upper limits is fixed at 2σ rms in the radio image following Botteon et al. (2020).These authors noted that the introduction of a high threshold combined with a large intrinsic scatter can introduce a bias in the correlation and demonstrated that a threshold of 2σ rms is the right choice.The strength of the correlation is measured by the Pearson r p and Spearmann r s correlation coefficients.The former assesses the linear relation between the two variables, while the latter determines their monotonic relation (whether linear or not).
Based on the two-component emission found with the surface brightness analysis of Sect.3.6, we performed point-topoint correlation separately for both the extended and mini-halo emission, using the radio images at 144 MHz presented in Figs. 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 rms, which can be assumed as an estimate of the statistical error.The size of the boxes was chosen to be equal to 1.5 times the area of the radio beam (see Table 6).This grid resolution also allows it to be minimally affected by the PSF of XMM-Newton, which for the combined MOS+pn images is ∼10 (FWHM).First, we performed the correlation on the 35 -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, which includes the radio emission and excludes other external radio sources, in order to select a 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 were 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 selections (blue boxes) and upper limits (cyan boxes) are shown in the top left corner of each plot of Fig. 8.
The two surface brightnesses are positively correlated r p = 0.87, r s = 0.85 and have a sublinear scaling A = 0.81 +0.07 −0.08 .This is in agreement with other literature works for giant halos (e.g., Botteon et al. 2020;Rajpurohit et al. 2021;Riseley et al. 2022a).We notice four points, spatially corresponding to the central brighter component, that appear distinctly separated from the extended component ones on the correlation.To exclude that those points are biasing the correlation toward steeper values, we performed it again, removing these four central boxes.This also allowed us to study the correlation properties of the outer component only.We still find a sublinear (A = 0.78 +0.08 −0.07 )) strong (r p = 0.83, r s = 0.82) correlation consistent with the previous one.The absence of statistical difference between the two results confirms that the correlation is determined by the extended emission and reinforces the robustness of our initial findings.
The correlation indicates that the decline of the nonthermal 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 studied the central mini-halo component, making use of the higher 15 -resolution radio image (Fig. 5), which is the most sensitive to the central emission and allows us to better sample the inner region.Differently from the sublinear or linear scalings that are reported in the literature for giant radio halos, but consistently with the sample of mini halos studied by Ignesti et al. (2020), we find instead a superlinear A = 1.27 +0.16 −0.14 correlation for the central emission (Fig. 8, right).Also in this case, there is clear evidence of a spatial correlation between radio and X-ray emission, with r p = 0.93, r s = 0.91 (Table 6).This superlinear correlation is also consistent with analysis of Riseley et al. (2023), 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 superlinear 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 nonthermal component is more concentrated around the central AGN.This is consistent with the hadronic scenario that predicts a superlinear 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 superlinear scaling of the X-ray and radio surface brightnesses.
The different trends of the two radio sources further strengthen the idea that the radio emission of A1413 is composed of two different components.They might also mean that the mechanisms responsible for the re-acceleration of the radioemitting particle are different.

Conclusions
In this work, we present a composite X-ray and radio analysis of the galaxy cluster A1413.For this purpose, XMM-Newton and LOFAR 144 MHz data were first analyzed separately.They were then combined to quantify the connection between the thermal and nonthermal components of the ICM.The main results can be summarized as follows.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 is expected in the inner regions of a strong cool-core A132, page 11 of 14 cluster.Moreover, the metallicity profile is flatter than expected for a very relaxed system.However, the surface brightness profile is well fit with a double β model, which typically accounts for the central brightness excess of coolcore clusters.Finally, from the calculation of the cooling time, we obtain t cool = 6.4±1.3Gyr within r < 0.5 ∼ 75 kpc, and a cooling region of r cool ∼ 90 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. A1413 was known to host a mini-halo (Govoni et al. 2009;Giacintucci et al. 2017;Savini et al. 2019;Riseley et al. 2023).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 3σ 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. We performed 1D surface brightness fitting using the double exponential model of Eq. ( 4).We find an inner brighter component extended up to R = 84 kpc, which is consistent with other mini halos, and a broader component with R 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-fit values of R fdca ext = 3r fdca e ∼ 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 kpc size based on the 3σ rms contour of Fig. 1, with a total linear size of D ext > 1.2 Mpc. 4. We performed a point-to-point correlation analysis between the X-ray and radio surface brightness, which confirms the spatial connection between the nonthermal and thermal emission in galaxy clusters.We investigated a possible correlation for both the halo and the mini-halo regions separately.We found a sublinear correlation I R ∝ I On the other hand, superlinear correlations are usually found for minihalos, indicating that the ICM nonthermal component is more concentrated around the central AGN.This is consistent with the hadronic scenario.Nevertheless, the reacceleration model cannot be excluded, as depending on the CRe distribution, it could also produce a superlinear 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. 2023a).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 less energetic merger events.In this context, we argue that minor merger 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 with mini halos.

Fig. 1 .
Fig. 1.High-and low-resolution radio images of A1413 at 144 MHz.Left panel: high resolution 144 MHz image with 4 × 6 beam (shown in the bottom left corner) and noise σ rms = 0.15 mJy beam −1 .Sources that have been subtracted are labelled with letters: two central AGNs A, B; one point-like north source C and one extended south source D. Right panel: low-resolution source-subtracted 144 MHz image with 35 × 35 beam (shown in the bottom left corner) and noise σ rms = 0.27 mJy beam −1 .For both images, the contour levels start at 2σ rms and are spaced with a factor of √ 2. The −3σ rms contours are red, while 2σ rms contours are dotted green.

Fig. 4 .
Fig. 4. Cooling time profile for A1413.The red solid line represents the look-back time at z = 1, t age = 7.7 Gyr.The power-law fit with t cool = 9.27R 1.84 allows us to define the cooling radius (green vertical line), namely the region within which t cool < t age .

Fig. 5 .
Fig. 5. 1D radio surface-brightness profile.Left panel: medium resolution (15 × 15 ) source-subtracted 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 2σ rms , where σ rms = 0.18 mJy beam −1 , and are spaced with a factor of √ 2. The −3σ rms contours are red.Right panel: radio surface-brightness profile.The solid line represents the resulting double exponential model, while the single components are plotted as a reference.The central mini-halo component has r e,1 = 28 ± 5 kpc, while the second broader component has r e,2 = 290 ± 60 kpc.The vertical line indicates R 500 , and the dashed black line indicates the 1σ annulus detection limit for each annulus.

Fig. 8 .
Fig. 8. Spatial correlation between X-ray and radio surface brightness.Left panel: I R −I X relation of the halo in A1413, extracted in square boxes with width of 35 .The horizontal gray dash-dotted lines indicate the σ rms = 298 µJy beam −1 in the radio maps.Right panel: I R − I X relation of the mini halo in A1413, extracted in square boxes with a width of 23 .The horizontal gray dash-dotted lines indicate the σ rms = 203 µJy beam −1 in the radio maps.For both relations, upper limits (cyan points) refer to cells where the radio surface brightness is below the 2σ rms level.The best-fit (green dashed line) is reported with the corresponding 1σ confidence interval.Red dashed best-fit line (left panel) refers to the correlation without the four central boxes.
inner mini-halo component.Sublinear or linear scalings are reported in the literature for giant radio halos and can be explained by the re-acceleration scenario.

Table 1 .
Properties of A1413 derived from literature.

Table 3 .
Radio images used in this work.
Fig. 2. 1D thermodynamical profiles obtained from spectral fitting, up to R 500 .Left panel: projected temperature profile.Right panel: metal abundance profile.Both profiles are compared with different averaged values of relaxed system samples (pink) and disturbed ones (yellow) from Lovisari & Reiprich (2019).

Table 4 .
Summary of main properties of radio emission in A1413.

Table 5 .
Summary of Halo-FDCA results for extended emission in A1413.