# A Deep uGMRT view of the ultra steep spectrum radio halo in Abell 521

R. Santra National Centre for Radio Astrophysics, Tata Institute of
Fundamental Research, Pune 411007, India  R. Kale National Centre for Radio
Astrophysics, Tata Institute of Fundamental Research, Pune 411007, India  S.
Giacintucci Naval Research Laboratory, 4555 Overlook Avenue SW, Code 7213,
Washington, DC 20375, USA  M. Markevitch NASA/Goddard Space Flight Center,
Greenbelt, MD 20771, USA  F. De Luca Dipartimento di Fisica, Università di
Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy  INFN,
Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca
Scientifica, 1, Roma, Italy  H. Bourdin  Dipartimento di Fisica, Università di
Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy  INFN,
Sezione di Roma 2, Università di Roma Tor Vergata, Via della Ricerca
Scientifica, 1, Roma, Italy  T. Venturi INAF - IRA, Via Gobetti 101, I-40129
Bologna, Italy; IRA - INAF, via P. Gobetti 101, I-40129 Bologna, Italy  D.
Dallacasa Dipartimento di Fisica e Astronomia, Università di Bologna, via P.
Gobetti 93/2, 40129, Bologna, Italy  R. Cassano INAF - IRA, Via Gobetti 101,
I-40129 Bologna, Italy; IRA - INAF, via P. Gobetti 101, I-40129 Bologna, Italy
G. Brunetti INAF - IRA, Via Gobetti 101, I-40129 Bologna, Italy; IRA - INAF,
via P. Gobetti 101, I-40129 Bologna, Italy  K.D.Buch Digital Backend Group,
Giant Metrewave Radio Telescope, NCRA-TIFR, Pune, 410504, India

###### Abstract

We present the first detailed analysis of the ultra-steep spectrum radio halo
in the merging galaxy cluster Abell 521, based on upgraded Giant Metrewave
Radio telescope (uGMRT) observations. The combination of radio observations
(300-850 MHz) and archival X-ray data provide a new window into the complex
physics occurring in this system. When compared to all previous analyses, our
sensitive radio images detected the centrally located radio halo emission to a
greater extent of $\sim$ 1.3 Mpc. A faint extension of the southeastern radio
relic has been discovered. We detected another relic, recently discovered by
MeerKAT, and coincident with a possible shock front in the X-rays, at the
northwest position of the center. We find that the integrated spectrum of the
radio halo is well-fitted with a spectral index of $-1.86\pm 0.12$. A
spatially resolved spectral index map revealed the spectral index
fluctuations, as well as an outward radial steepening of the average spectral
index. The radio and X-ray surface brightness are well correlated for the
entire and different sub-parts of the halo, with sub-linear correlation slopes
(0.50$-$0.65). We also found a mild anti-correlation between the spectral
index and X-ray surface brightness. Newly detected extensions of the SE relic
and the counter relic are consistent with the merger in the plane of the sky.

Galaxy clusters(584)-Radio continuum emission(1340)-Extragalactic radio
sources(508)-Intracluster medium(858)

††facilities: upgraded Giant Metrewave Radio Telescope (uGMRT), Chandra, XMM-
Newton††software: CASA, numpy, aplpy, matplotlib, ds9,

##  1 Introduction

Galaxy clusters accrete mass via mergers of galaxy groups and other clusters.
These processes result in turbulent motions in the Intra-cluster medium (ICM),
the largest baryonic pool in the universe. Consequently, a notable amount of
energy is released during a merger, accelerates the cosmic ray (CR)
population, and amplifies the ICM magnetic field. The observational footprints
of these events are seen via diffuse radio emission in galaxy clusters (for
reviews; Brunetti & Jones, 2014; van Weeren et al., 2019). These radio sources
have a steep spectra111Sν $\propto$ $\nu^{\alpha}$, where Sν is the flux
density at the frequency at $\nu$ and $\alpha$ stands for spectral index
($\alpha$ <-1). The origin of these synchrotron radio-emitting electrons,
discovered to fill the entire cluster volume is still poorly understood
(Brunetti & Jones, 2014; Botteon et al., 2022a).

Diffuse radio sources in clusters are mainly classified into two classes:
radio halos and radio relics (Feretti et al., 2012). Radio halos are the
centrally located $\sim$ Mpc scale diffuse radio sources and are unpolarized
(Giovannini et al., 2009; Bonafede, 2010). Radio relics are the diffuse
sources located at the cluster peripheries. They are thought to be the tracers
of the outgoing merger shock waves (e.g. van Weeren et al., 2019). Radio
relics show filamentary structures on kpc scales, a high degree of linear
polarization, and aligned magnetic field distribution (e.g. van Weeren et al.,
2010; Kierdorf et al., 2017; Di Gennaro et al., 2021a; Rajpurohit et al.,
2021a). Advancement of both the theoretical models and observations that
resulted in the discovery of many sources (e. g. Knowles et al., 2022; Botteon
et al., 2022b) provide the tools to explore the possible physical mechanism
behind the origin of radio halos and relics.

The currently favored scenario for the generation of the radio halo involves
the re-acceleration of the pre-existing relativistic electrons, due to
turbulence, originating in cluster mergers (re-acceleration models Brunetti et
al., 2001; Petrosian, 2001; Cassano et al., 2006; Brunetti & Lazarian, 2007;
Beresnyak et al., 2013; Brunetti & Lazarian, 2016). Observational signatures
of the radio halos match quite well with the predictions of the turbulent re-
acceleration model (Cuciti et al., 2021; Cassano et al., 2023). Contribution
from secondary electrons from hadronic collisions (e.g Blasi & Colafrancesco,
1999) have been constrained to be subdominant from Fermi-LAT observations and
by arguments based on the energetics of CRs (e.g Bonafede et al., 2012;
Brunetti, 2017; Adam et al., 2021), yet secondary electrons may contribute
significantly if they are re-accelerated by turbulence during mergers (e.g.
Brunetti & Lazarian, 2011a; Nishiwaki & Asano, 2022).

It is widely accepted that the fraction of kinetic energy from the shock fuels
the radio emission of the relic by the Diffusive Shock Acceleration (DSA;
Ensslin et al., 1998; Hoeft & Brüggen, 2007) of the cosmic ray electrons. The
ongoing debate on the acceleration of the electrons is, whether they are
accelerated from a thermal pool (keV) to relativistic energy (GeV) (standard
scenario; Ensslin et al., 1998; Hoeft & Brüggen, 2007) or from a pre-existing
relativistic electron population (re-acceleration; Markevitch et al., 2005;
Kang & Ryu, 2011, 2016). There are a few examples that show some connection
between the relic emission to an Active Galactic Nucleus (AGN) and radio
galaxy (e.g. Bonafede et al., 2014; Shimwell et al., 2015; van Weeren et al.,
2017; Di Gennaro et al., 2018; Stuardi et al., 2019). The standard scenarios
have explained many observable from the relics. But still, some disputes
remain regarding the acceleration of the CRe from the thermal pool via weak
shock (Botteon et al., 2020a).

Radio halos have been observed to have a broad range of spectral indices,
varying from -2.1 to -1.0 (see Feretti et al., 2012; van Weeren et al., 2019,
for reviews). In particular, there are radio halos, detected over the years,
that have shown an integrated spectral index of <-1.5. This population is
known as Ultra-Steep Spectrum Radio Halos (USSRH), which are thought to be
produced during less energetic mergers (involving clusters with masses
<1015M⊙), is the key expectation of the re-acceleration model (e.g. Cassano et
al., 2006, 2010). The prime example is Abell 521 (Brunetti et al., 2008;
Dallacasa et al., 2009; Macario et al., 2013), the cluster with first
discovered a USSRH. The number of steep spectrum radio halos has increased
thanks to low-frequency observations from LOFAR, MWA, GMRT (e. g. Wilber et
al., 2018; Rajpurohit et al., 2021a; Duchesne et al., 2021; Di Gennaro et al.,
2021b; Bruno et al., 2021; Botteon et al., 2022a).

The integrated radio spectrum for the radio halos generally follows a single
power law, although there are a few examples of halos with curved spectrum
(Coma Thierbach et al. 2003, MACSJ017.5+3745 Rajpurohit et al. 2021a). A radio
spectral index is a key tool in the understanding of the shape of the energy
distribution of the relativistic electrons and properties of the turbulence in
ICM. Only for very few clusters, high-resolution resolved spectral maps, and
robust estimations of the spectral index, are available. Some halos show a
uniform spectral index all over the spatial scales, whereas some show small-
scale fluctuations in the resolved spectral maps (Pearce et al., 2017; Botteon
et al., 2020b; Rajpurohit et al., 2022). Resolved spectral maps for the steep
spectrum radio halos are not explored significantly, which can throw light on
the self-similarity behavior (if any) of the global spectral index at local
kpc scales.

Here, we present the deep upgraded Giant Metrewave Radio telescope (uGMRT)
radio observations of the prototype USSRH in Abell 521. For a better
understanding of the diffuse emission, we have also used archival Chandra and
XMM-Newton data sets. For the Chandra analysis, we use the results presented
by Wang 2019. The Chandra data include four archival Chandra datasets (ObsIDs
430, 901, 12880, 13190) with a total exposure of 146 ks on the cluster (after
cleaning for background flares; see Wang 2019 for details
222http://hdl.handle.net/1903/22163). The XMM-Newton observations were
published by Bourdin et al. (2013).

The paper is organized as follows. In Sec. 2 we provide a brief overview of
Abell 521. The uGMRT observations and data analysis procedures are explained
in Sec. 3. In Sec. 4, we show the uGMRT continuum images at different
resolutions and discuss newly discovered features from our study. The results
obtained from spectral analysis and radio vs X-ray correlations are described
in Sec. 5$-$7. Then we summarised our results and findings in Sec. 8.
Throughout this paper, we have adopted a flat $\Lambda$CDM cosmology with H0 =
70 km s-1, $\Omega$ m = 0.3 , $\Omega_{\Lambda}=0.7$. At the redshift of Abell
521, 1′′ corresponds to a linear scale of 4 kpc.

##  2 The galaxy cluster Abell 521

The galaxy cluster Abell 521 (hereafter A521) is a very massive and merging
cluster, located at a redshift of 0.247 (e.g. Arnaud et al., 2000), having a
center at RA = 04:54:6.9, DEC = -10:13:26.2 (Yoon et al., 2020). The mass
(M200) of the cluster is 1.3${}_{-1.3}^{+1.0}$ $\times$ 1015 M⊙ (Yoon et al.,
2020), with a very high X-ray luminosity of LX[0.1-2.4keV] $\sim$ 8 $\times$
1044 erg s-1, and a temperature of 5.9 keV (Bourdin et al., 2013).

Optical and X-ray observations of this cluster have shown high dynamical
activity at the center. Chandra X-ray observations show that this cluster
constitutes two main clumps along the north-west and south-east direction
(Ferrari et al., 2006). The southern gas clump is more diffuse compared to the
northern one. Using the XMM-Newton observations, Bourdin et al. (2013) have
detected two shock fronts (one along the south-east and the other one along
the south-west direction) and two cold fronts at the intersection region of
these two gas clumps. Temperature maps from the XMM-Newton observations
suggest that the cluster has a temperature of $\sim$ 4.5 keV in the northern
part, whereas in the southern it is $\sim$ 4 keV. The interacting region
between these two sub-clumps has a temperature of >7 keV.

Optical spectroscopic observations of 113 member galaxies of A521 have shown a
complex galaxy distribution (Ferrari et al., 2003). There are 7 different
galaxy groups situated in the NW-SE region. Additionally, the velocity
analysis suggests that this cluster is very complex along the line of sight.
Recent HST weak lensing studies by Yoon et al. (2020) have shown that the
cluster is composed of mainly three distinct substructures named North-West
(NW), South-East (SE), and Central (C). Using the available multi-wavelength
observations, they revisited the merging scenario of A521 via numerical
simulations. However, their study did not get a conclusive picture of the
merging scenario in the central region.

###  2.1 Previous radio studies

A521 is widely observed in radio wavelengths. Brunetti et al. (2008) reported
the discovery of a radio halo with the legacy GMRT (240, 325, 610 MHz). They
estimated an integrated spectral index of the halo of $\sim$ -1.9, which is
very steep compared to the typical observed spectral index of a radio halo.
Dallacasa et al. (2009) have reported the detection of the radio halo at 1.4
GHz, using the VLA (D-array and BnC array) and obtained a very steep spectral
index of $\sim$ -1.86 $\pm$ 0.08. Macario et al. (2013) had also reported the
spectral index of the halo to be very steep $\sim$ -1.81 $\pm$ 0.02.

Arnaud et al. (2000) had first presented the image of the radio relic using
the archival National Radio Astronomy Observatory Very Large Array Sky Survey
(NVSS) (Condon et al., 1998). Since then, the relic has been observed at radio
frequencies ranging from 150 to 4800 MHz with legacy GMRT and VLA. The radio
relic is detected up to 8.4 GHz, with a good significance (Giacintucci et al.,
2008). The measured integrated spectral index of the relic over a frequency
range of 325 to 4800 MHz is -1.48 $\pm$ 0.01. Chandra observations of A521 at
0.5$-$4 keV also suggested the presence of an X-ray surface brightness edge, a
shock front at that position of the relic (Giacintucci et al., 2008). The
shock front was later confirmed by Bourdin et al. (2013).

##  3 Observations and Data analysis

Table 1: Summary of uGMRT observations. | Band 3 | Band 4
---|---|---
Frequency range (MHz) | 300-500 | 550-750
No. of channels | 2048 | 2048
Channel width (kHz) | 97.65 | 97.65
Bandwidth (MHz) | 200 | 200
On source time (Hr) | 9 + 9 | 9 + 9
Polarisation | RR, LL | RR, LL
Flux calibrator | 3C147 | 3C147
Largest angular scale | 1920′′ | 1020′′
Table 2: Summary of the radio images. | Name | Beam Size (′′) | Robust |  uv range |  uvtaper (′′) | Map RMS ($\mu$Jybeam-1)
---|---|---|---|---|---|---
uGMRT band3 | IMG1 | 8′′ $\times$ 7′′ | 0.5 | None | None | 24.9
| IMG2 | 6′′ $\times$ 6′′ | -0.5 | >0.2k$\lambda$ | 5′′ | 28.7
| IMG3 | 10′′ $\times$ 10′′ | -0.5 | >0.2k$\lambda$ | 10′′ | 31.7
| IMG4 | 15′′ $\times$ 15′′ | 0 | >0.2k$\lambda$ | 15′′ | 34.0
| IMG5 | 20′′ $\times$ 20′′ | -0.5 | >0.2 k$\lambda$ | 20′′ | 38.6
uGMRT band 4 | IMG6 | 5′′ $\times$ 5′′ | 0.5 | None | None | 8.5
| IMG7 | 6′′ $\times$ 6′′ | 0.0 | >0.2k$\lambda$ | 5′′ | 11.0
| IMG8 | 10′′ $\times$ 10′′ | -0.5 | >0.2k$\lambda$ | 10′′ | 16.7
| IMG9 | 15′′ $\times$ 15′′ | 0 | >0.2 k$\lambda$ | 15′′ | 21.6
| IMG10 | 20′′ $\times$ 20′′ | 0 | >0.2 k$\lambda$ | 20′′ | 25.6

###  3.1 uGMRT

A521 was observed with the uGMRT at bands 3 and 4 using the GMRT Wideband
Backend (GWB) (Reddy et al., 2017). The observations in each band were taken
in two sessions, placed on consecutive days. We used real-time RFI filtering
(e.g. Buch et al., 2019, 2022, 2023) in GWB, targeted for mitigating broadband
RFI. We used it in the mode where the RFI samples (those that exceeded the
3$\sigma$ threshold in voltages) were replaced with digital noise. Band 3
observations were done on July 6 (with filter) and July 7 (without a filter)
and band 4 observations were done on July 8 (with filter) and July 9 (without
a filter) in 2018. In Table 1 we have summarized the observational parameters.
Further details on the effect of the RFI filtering are provided in Appendix A.

We processed the uGMRT data with CASA using the
CAPTURE333https://github.com/ruta-k/CAPTURE-CASA6 pipeline (Kale & Ishwara-
Chandra, 2021). After this initial flagging, the flux density of the primary
calibrator was set according to the flux scale of Perley - Butler 2017 (Perley
& Butler, 2017). After following the standard calibration routines, the
calibrated target source data were split and were further flagged using the
automated flagging modes. To reduce the volume of the data but still prevent
bandwidth smearing, 10 frequency channels ($\sim$ 1 MHz) were averaged. The
procedure was identical for bands 3 and 4. The target visibilities were imaged
using the CASA task tclean with wide-field and wide-band imaging algorithms.
At the stage of imaging and self-calibration, the target source measurement
set was divided into 8 frequency sub-bands, each sub-band containing 20
channels. It divides the whole 200 MHz bandwidth into 8 spectral windows and
does imaging and self-calibration on the sub-banded data file.

We assume the flux density uncertainty is $10\%$ for both the uGMRT
frequencies (Chandra & Kanekar, 2017). The flux density uncertainty ($\Delta
f$) is defined by:

$\Delta f=\sqrt{(0.1f)^{2}+(N_{\rm beam}(\sigma_{\rm rms})^{2}}),$ (1)

where $f$ is the flux density, $N_{\rm beam}$ is the number of beams, and
$\sigma_{\rm rms}$ is the rms noise. We have used the task
ugmrtpb444https://github.com/ruta-k/uGMRTprimarybeam-CASA6 for the primary
beam correction of all the images.

Figure 1: uGMRT band 3 full resolution image IMG1 obtained with robust$=0.5$
is shown here. The rms is $\sigma_{\rm rms}$ $=$ 24.9 $\mu$Jybeam-1 and the
beam is 8′′ $\times$ 7′′ . We have labeled the extended sources; R1 the relic
situated at the South-east position of the cluster, R2 the newly discovered
relic, “R1-TL” the faint extension of the R1 relic, and the Wide Angle-Tailed
galaxy (WAT). At the central region, the low surface brightness radio halo
emission is seen. The boxed region indicates the radio galaxy J0454-1016a.
Figure 2: Multi-wavelength view for the central region of the A521 is shown.
The yellow intensity shows the image of the A521 field in the DSS2 red. The
intensity in blue shows the Chandra image at 0.5$-$4 keV from Wang (2019),
which is smoothed up to 5′′ using a Gaussian kernel. The radio emission from
the uGMRT band 3 image is shown in red.

###  3.2 Removal of the discrete sources

The diffuse radio emission in A521 is contaminated by weak/faint discrete
sources like unresolved point sources and radio galaxies in the images which
bring complications in the measurement of the flux densities of the diffuse
sources. For the removal of the discrete sources at band 3 and band 4, their
models were created by applying a uv-cut of >5 k$\lambda$ (angular scale of
40′′). In both frequencies, we have kept the uv cut-off range the same to
match the spatial scales. In Appendix B, we have presented the high-resolution
point source images for the A521 cluster field in both uGMRT bands 3 and 4
frequencies.

After creating the model image for the point sources, the clean components
corresponding to the “point” sources were subtracted from the observed data.
After subtraction, the data file was then imaged again using tclean with a uv
baseline of <10 k$\lambda$, to highlight the extended emission. We used the
multi-scale deconvolver to properly map the large-scale diffuse emission. We
have checked the consistency of the point source subtraction process, by
calculating the flux density of the radio halo with that obtained by
algebraically subtracting the flux density of the embedded sources from the
total (halo $+$ sources), finding a good agreement (5% for uGMRT band 3 and
band 4). During the measurement of the errors regarding the flux density
estimation of the extended sources, we used the equation 1, with one extra
term for the point source subtraction error ($\sigma_{\rm sub}$), 5% for the
uGMRT, added in quadrature.

###  3.3 X-ray observations

Using the XMM-Newton pointing ObsID 0603890101, we generated a surface
brightness and projected temperature map. XMM-Newton data have been pre-
processed in the framework of the CHEX-MATE project (CHEX-MATE Collaboration
et al., 2021), as detailed in De Luca et al., (in prep). This pre-processing
includes, in particular, spatial and spectral modeling of the sky and
instrumental backgrounds, together with the detection and masking of point
sources. The XMM-Newton surface brightness map is derived from wavelet
analysis of the photon image that we adapted for spatial variations in the
effective area and a background noise model. The photon image has been
denoised via the 4-$\sigma$ soft-thresholding of a variance-stabilized wavelet
transform (Starck et al., 2009; Zhang et al., 2008), that is especially suited
to process low photon counts. The XMM-Newton temperature map has been computed
using a spectral-imaging algorithm that combines spatially weighted likelihood
estimates of the projected intra-cluster temperature with a curvelet analysis,
that is especially suited to recover curvilinear features expected near shocks
or cold fronts. This algorithm can be seen as an adaptation of the spectral-
imaging algorithm presented in Bourdin et al. (2015) and recently applied also
in Oppizzi et al. (2022). Briefly, temperature log-likelihoods are first
computed from spectral analyses in each pixel of the maps, then spatially
weighted with kernels that correspond to the negative and positive parts of
B3-spline wavelets functions. This allows us to derive wavelet coefficients of
the temperature features and their expected fluctuation, which we derive from
spatially weighted Fisher information. We use these wavelet coefficients to
compute a curvelet transform that typically analyses features of apparent size
in the range of [3.5, 60] arcsec. We finally reconstruct the temperature map
from a de-noised curvelet transform that we inferred from a 4-sigma soft
thresholding of the curvelet coefficients. For each pixel of the temperature
map, we associate a temperature uncertainty that results from a parametric
bootstrap re-sampling. Specifically, combining the best-fit temperature and
surface brightness maps with the effective area and background noise models,
we generate mock XMM-Newton event lists whose statistical properties mimic the
real pointing and derive bootstrap temperature maps from curvelet analyses of
these mock event lists. The temperature uncertainty map derives from the
standard deviations of temperature values registered in each pixel of 100
bootstrap maps. It can be noticed that temperature uncertainties increase at
the same time as the best-fit temperature values increase and the X-ray
surface brightness decreases.

Figure 3:  Top left: Total power uGMRT band 3 image (IMG4), with a beam size
of 15′′, is shown. The previously discovered diffuse emissions are also well
recovered in our uGMRT images. The tail of the relic R1 (extended faint
emission) is labelled in yellow with a naming convention of “R1-TL”.
Similarly, the newly discovered relic, named R2 (and better observable with
this resolution), labelled in yellow. Contours level are drawn at
[1,2,4,8…]$\times$ 3$\sigma_{\rm rms}$, where $\sigma_{\rm rms}$ = 34.0
$\mu$Jybeam-1. Top right:Similar to the precedent panel but at 650MHz. The
spacing of the contour levels is similar with $\sigma_{\rm rms}$ = 21.6
$\mu$Jybeam-1. Bottom left: The cluster central diffuse sources are shown in a
low resolution (20′′) image (IMG5) at 400 MHz. The interested sources are
mentioned above. The RMS noise of the image is 38.6 $\mu$Jybeam-1 and the
contour spacing is similar to the above panel. Bottom right: Here we have
shown the 20′′ image (IMG10) of the cluster field at 650 MHz. The RMS noise of
the image is 25.6 $\mu$Jybeam-1.

##  4 Results: continuum images

In Figure 1 we have shown the uGMRT band 3 continuum image of the A521 cluster
field. At the central region, we have detected the arc-shaped relic (R1),
southeast of the cluster center. The central regions are dominated by the
point sources and the low surface brightness radio halo emission. Figure 2
shows a multi-wavelength view of the cluster. The central radio halo emission
is morphologically similar to the Chandra X-ray thermal emission (blue). Many
point sources are seen in the central regions and all over the field at the
optical wavelength (yellow). In the following sections, we will discuss the
previously known and newly discovered diffuse radio emissions at the cluster
central regions.

###  4.1 Radio halo emission

We have detected the radio halo emission both at 400 and 650 MHz, shown at
different resolutions in Figure 3. The radio halo is not circular in shape
rather it is very elongated. The elongation is along the southeast to the
northwest, which was supposed to be the merger axis (Ferrari et al., 2003). In
our sensitive high-resolution images, we have detected 33 discrete sources
(>3.5 $\sigma_{\rm rms}$) in the halo region. The total flux density of the 32
sources (excluding J0454-1016a, squared boxed region in Figure 1) is 22.5
$\pm$ 1.7, which is $\sim$ 33 % of the total flux density (radio halo $+$
individual sources)at 400 MHz. In Figure 4 we have shown the point source
subtracted image at 400 MHz (left) and 650 MHz (right). Radio emissions from
both the radio halo and relic were recovered well. Similar to Macario et al.
(2013) we have also detected the “spur” region connected to the radio relic at
the southeast position in our low-resolution images. This is still unexplained
whether the “spur” region is a part of the radio relic or the centrally
located radio halo. At 400 MHz, the radio halo has been detected up to the
Largest Linear Size (LLS) of 1.3 $\times$ 1.0 Mpc2, whereas at 650 MHz it is
1.03 $\times$ 0.85 Mpc2. Macario et al. (2013) had reported the size of the
radio halo to be $\sim$ 1.35 Mpc at 153 MHz. The radio halo is more extended
at lower frequencies. The radio halo brightness shows a rapid decline in
surface brightness at the outermost regions. The overall radio surface
brightness is very smooth over the spatial scale of the radio halo at this
resolution.

The radio relic emission penetrates the southern part of the radio halo
emission, which we define as the ”bridge” region. This component was also
detected significantly in previous studies from 150 to 1400 MHz. It is
extremely difficult to distinguish the contribution of radio halo and radio
relic emissions at a given observed frequency. This bridge region’s properties
will be discussed in Section 6. Many examples of overlapping emissions from
relics, bridges, and halos are discussed in the literature, including the
radio halo in toothbrush clusters (e.g. van Weeren et al., 2016), halo in CIZA
J2242.8+5301 (Hoang et al., 2017) and radio halo in Abell 3667 (de Gasperin et
al., 2022).

###  4.2 New discoveries

Our deep and sensitive uGMRT observations have revealed some new insights into
the A521 cluster central field. In Figure 3 we have presented the central
region of this cluster, where we have labeled the different extended sources
in yellow color. The R1 is the brightest relic which was discovered to the
southeast of the cluster center and studied in the literature (Arnaud et al.,
2000; Ferrari et al., 2006; Giacintucci et al., 2006, 2008). “R1-TL” is the
low surface brightness extension of the relic, which is uncovered at 400 MHz
at a significant level. Similarly, our new deep radio images revealed another
relic “R2” northwest of the cluster center, first reported by Knowles et al.
(2022) using the MeerKAT.

Table 3: Flux density estimates for the radio halo. Frequency (MHz) | Flux density (mJy) | Ref.
---|---|---
74 | 1500 | B.08
153 | 328$\pm$ 66 | M.13
240 | 152$\pm$15 | B.08
330 | 90$\pm$ 7 | B.08
402 | 45$\pm$5 | This work
610 | 15$\pm$ 3.5 | B.08
650 | 15.31 $\pm$ 3 | This work
1400 | 6.4$\pm$ 0.6 | D. 09

Figure 4: Left: We have shown the point source subtracted image of the cluster
central field at 400 MHz. The resolution of the image is 22′′ $\times$ 22′′.
The yellow circle, having a size of $\sim$ 1 Mpc represents the regions of the
radio halo emission, adopted from Brunetti et al. (2008). The spur emission, a
bridge region between the radio halo and relic is labeled in yellow. Contours
level are drawn at [1,2,4,8…]$\times$ 3$\sigma_{\rm rms}$, where $\sigma_{\rm
rms}$ = 45 $\mu$Jybeam-1 is the noise level. Right: The point source
subtracted image is shown at 650 MHz. The spur emission is also visible here.
The RMS noise of the image is 30 $\mu$Jybeam-1 and the spacing of the contour
level is similar to the left image.

Brunetti et al. (2008) had also detected a very small portion of this “R1-TL”
in the 240 MHz legacy GMRT image of the A521 cluster. This low surface
brightness extended region is visible in low-resolution images (15′′, 25′′) at
400 MHz (in Figure 3). At 650 MHz we have also detected the extension. In this
region, some of the point sources are seen both in the optical and radio
images. Therefore we have tried to see if this emission is seen after the
point source subtraction. At 400 MHz, we have detected this “R1-TL” at a
significant level of 12$\sigma_{\rm rms}$, shown in Figure 4. In Figure 5 we
have shown the publicly available MeerKAT image (at 1.28 GHz) from MGCLS
survey (Knowles et al., 2022). The faint extension is also seen in the MeerKAT
image. In Figure 6, the uGMRT 400 MHz image (beam size: 10′′ $\times$ 10′′) is
overlaid on the DSS2 optical image (grey scale). Several point sources are
present in the “R1-TL” region. At 400 MHz the linear size of the extension is
$\sim$ 1 Mpc, with a width of $\sim$ 300 kpcs. The continuous decrement of the
surface brightness along the arc of the relic is still unexplained, especially
after a certain length. Considering this “R1-TL” extension, the total length
of the relic arc is $\sim$ 2.2 Mpc. Till now only very few relics (sausage
Hoang et al. 2017, Toothbrush van Weeren et al. 2016, ZwCl 0008.8+5215 van
Weeren et al. 2011b) have been detected with such a large extent.

Figure 5: Left: MeerKAT 1.28 GHz image of the A521 cluster is shown in colour
and contours. The MeerKAT image has a beam size of 8.5′′ $\times$ 7.5′′.
Contours levels are drawn at [1,2,4,8]$\times$ 3$\sigma_{\rm rms}$, where
$\sigma_{\rm rms}=3$ $\mu$Jybeam-1 is the noise level, quoted from Knowles et
al. (2022). Right: uGMRT 400 MHz emission (full resolution, IMG1) is overlaid
(white contour) on the MeerKAT image (color). Contour levels are at a similar
level, with $\sigma_{\rm rms}=24.9$ $\mu$Jybeam-1.

Our uGMRT band 3 and band 4 images revealed the presence of another relic at
the northwest position of the cluster. In Figure 3 (top panel), we have
labeled the position of the newly discovered relic, R2. In Figure 5 (right
panel) we have overlaid the 400 MHz (in white contours) on the MeerKAT image
(in colors), where the relic R2 is seen clearly up a larger extent compared to
our analyzed images. Thanks to the superb sensitivity of the image, MeerKAT
has been able to detect the R2 relic up to such a low surface brightness
extent. In both our 400 and 650 MHz this relic is detected up to the Largest
linear size of $\sim$ 650 kpc, and 450 kpc respectively, due to the
sensitivity limit of our data quality. In Figure 4 the R2 relic is seen at a
significant level for both bands 3 and 4 in the low-resolution image (beam
size: 22′′ $\times$ 22′′)

Figure 6: DSS2 image cutout of the A521 SE relic field is shown (in grey), overlaid with the radio emission at 400 MHz, with a beam size of 10′′ (IMG3). Contours level are drawn at [1,2,4,8…]$\times$ 3$\sigma_{\rm rms}$, where $\sigma_{\rm rms}=31.7$ $\mu$Jybeam-1 is the noise level. In the relic region, the point sources embedded in the radio emission are also seen here, similar to the (Giacintucci et al., 2006). Table 4: Flux density estimates for the two radio relics. Relic | Freq. (MHz) | Flux density (mJy) | Ref.
---|---|---|---
R1 | 74 | 660 | G.08
| 153 | 299.9 $\pm$ 59 | M.13
| 235 | 181.8$\pm$ 10 | G.08
| 327 | 115.14$\pm$6 | G.08
| 402 | 87.8$\pm$10 | This work
| 610 | 42$\pm$ 2 | G.08
| 650 | 41.6 $\pm$ 3 | This work
| 1410 | 14.1$\pm$ 1 | G. 08
| 4890 | 2.02 $\pm$ 0.2 | G.08
R2 | 400 | 3.45 $\pm$ 0.34 | This work
| 650 | 2.01 $\pm$ 0.2 | This work
| 1280 | 0.800$\pm$ 0.09 | K.22

Figure 7: Left: Integrated radio spectral index is shown for the radio halo
between 150-1400 MHz. The black dashed line shows the overall spectrum of the
halo follows a single power law. The blue points denote the results from our
work, the other values are taken from the literature. The annotated figure
shows the 400 MHz radio emission (in grey). Right: The integrated spectrum of
both the radio relics, R1 and R2 are shown. The black and green dashed lines
represent the overall spectrum of the relics R1 and R2. The annotated figure
(color) shows the emission of the two relics (green box) at 400 MHz.

##  5 Spectral analysis

To provide new insights into the origin of the diffuse radio sources, in this
section, we present the integrated spectrum, radial profile, and resolved
spectral index map, using our uGMRT observations at bands 3 and 4.

###  5.1 Integrated spectrum

Using the point source subtracted image, we have calculated the flux density
of the halo and estimated the integrated spectral index, combined with the
available literature values. We have tabulated the flux density values in
Table 3. The integrated flux density values of the radio halo are calculated
over the same circular region, used in Brunetti et al. (2008).

In Figure 7 the resulting integrated spectrum is shown for the radio halo in
A521. The overall spectrum of the radio halo is well fitted with the single
power law between the frequency range of 150 - 1400 MHz, having an integrated
radio spectral index of -1.86 $\pm$ 0.12. Therefore, our results are in line
with the previous results, which showed that the radio halo in the A521 is an
ultra-steep spectrum radio halo. The value of the spectral index quoted here
has not used the 610 MHz flux density value during the estimation of the
spectral index due to the uncertainty and poor recovery of the extended radio
halo at the cluster central position. The inclusion of the 610 MHz flux
density value will lead to a spectral index of -1.92 $\pm$ 0.13, which is also
very steep for a radio halo in A521. The flux density value at 74 MHz is
reported from Brunetti et al. (2008), but due to the uncertainty in the
measurement of the flux density, we have not accounted for this value during
the fitting. We have combined the flux densities from the previous studies
(Flux density values are taken from; Brunetti et al., 2008; Dallacasa et al.,
2009; Macario et al., 2013) with ours. So, before doing the estimation, all
the flux density values were converted into a common flux density scale,
Perley - Butler 2017 (Perley & Butler, 2017). There are only a handful of
clusters, where sensitive wide-band radio observations have been done and the
radio halos are found to be a USSRH. Some of them (MACS J0717.5+3745;
Rajpurohit et al. 2021a) also show high-frequency steepening in the integrated
spectrum, while the rest show a single power law spectrum.

We have also estimated the integrated spectral index of the R1 and R2 relics
by combining the previously available flux density values and our analysis.
The integrated spectrum of the R1 relic is well fitted with a single power-law
of spectral index -1.46 $\pm$ 0.03. For the stationary shock, the integrated
spectral index, $\alpha_{\rm int}$, is steeper by 0.5 from the injection
spectral index, $\alpha_{\rm inj}$,

$\alpha_{\rm int}=\alpha_{\rm inj}-0.5$ (2)

Assuming the Diffusive Shock Acceleration to be the origin of the particle
acceleration in the relic region, the Mach number (MS) is related to the
injection spectral index $\alpha_{\rm inj}$ as

$\rm M_{\rm S}=\sqrt{\frac{2\alpha_{\rm inj}-3}{2\alpha_{\rm inj}+1}}=2.30\pm
0.02$ (3)

Giacintucci et al. (2008) had reported the integrated spectral index of the
relic to be -1.48 $\pm$ 0.01, with a Mach number of shock is 2.27 $\pm$ 0.02.
Using the equation 3, we have estimated the Mach number to be 2.30 $\pm$ 0.02,
consistent with the previous work. We have calculated the integrated spectral
index value of the R2 relic using the uGMRT (point source subtracted image)
and MeerKAT images. The spectrum is well represented by a single power law
with a spectral index of -1.34 $\pm$ 0.03 over a frequency range of 400-1280
MHz. Assuming DSA to be the origin of the R2 relic also, the Mach number
estimated from the integrated radio spectral index is 2.67$\pm$ 0.09. Although
constraining the acceleration efficiency at these shocks is beyond the aim of
our paper, we stress that weak shocks with Mach number $\sim$ 2.2$-$2.7 sit
exactly in the transition region where direct DSA from thermal pool becomes
energetically difficult (e.g. Botteon et al., 2020a)

Figure 8: The radial profile of the radio surface brightness is shown. The
annotated image shows the 400 MHz radio emission (in grey) and the contours
(in red), starting with 3$\sigma_{\rm rms}$ $\times$ [1,2,4,8….]. Each of the
red squares denotes the azimuthally averaged radio surface brightness values,
within the annular regions. The error bars represent the error on the mean.
The blue line shows the two-dimensional elliptical model fitted to the radio
surface brightness profile.

Figure 9: Left: Spectral index map for the radio halo in A521 at 22′′ is shown
between 400 MHz and 650 MHz. The black contours are drawn at 3$\sigma_{\rm
rms}$ $\times$ [1,2,4,8..] from the 400 MHz image. The relic is showing a
spectral steepening towards the cluster center. While the radio halo region
shows fluctuations in the spectral indices. Right: The corresponding error map
is shown here. The procedures for the estimation of the error map are
mentioned in Section 5.3.

###  5.2 Radial profile of the radio surface brightness

To characterize the radio halo properties, we have fitted the surface
brightness model and checked how the averaged radio surface brightness is
varying with distance. As suggested by Murgia et al. (2009) and very recently
by Boxelaar et al. (2021), we modeled the surface brightness variation by
taking into account the asymmetric and elongated shape of the radio halo. The
fitting procedures mentioned in Boxelaar et al. (2021) allow one to fit the
elliptical and skewed (asymmetrical) models. The surface brightness profile is
given by:

$I(r)=I_{0}\exp^{-G(r)}$ (4)

where I0 is the surface brightness in the central region, and the G(r) is the
radial function given by: G(r) =
$\left(\frac{|r|^{2}}{r_{e}^{2}}\right)^{0.5}$ for the circular shape, G(r) =
$\left(\frac{x^{2}}{r_{1}^{2}}+\frac{y^{2}}{r_{2}^{2}}\right)^{0.5}$ for
elliptical models. Here re denotes the e-folding radius and r2 = x2 \+ y2.

Table 5: Radio halo surface brightness profile fitting parameters Model | $\chi_{r}^{2}$ | I0 ($\mu$Jyarcsec-2) | r1 (kpc) | r2 (kpc)
---|---|---|---|---
Circular | 1.76 | 1.28 $\pm$ 0.04 | 380 $\pm$ 40 |
Elliptical | 1.15 | 1.42$\pm$ 0.03 | 420$\pm$ 22 | 360 $\pm$ 20

In Figure 8, we have shown the radial profile of the radio surface brightness
at 400 MHz and the model fitted to that. The total radio-emitting regions are
divided into concentric elliptical annular regions to estimate the azimuthally
averaged surface brightness of the radio halo. The central annular region is
drawn at 22′′, which is the beam size of the image. The cluster center is
chosen to be the center of the annuli. The red data points in Figure 8 are the
estimations of the averaged surface brightness from each of these annular
regions. Then we fitted the elliptical and circular surface brightness model
on the estimated averaged radio surface brightness and estimated the
parameters, using the Bayesian analysis. In Table 5 we have reported the
parameters obtained from the fitting. The central radio brightness is $\sim$
1.35 $\mu$Jyarcsec-2. The extension of the radio halo is detected up to $\sim$
3r1 and 2.5r2. In following this, we refer to the halo “core” region as the
emission contained within the ellipse, having major and minor axis as 0.5r1
and 0.5r2 respectively, while the outer halo region is the emissions contained
within the region beyond the core.

Figure 10: Histogram of the spectral index distribution for the halo in A521
between 400 MHz and 650 MHz is shown. The values for the spectral indices were
obtained by sampling the halo into square boxes of size 22′′. The red solid
line is the median value of the spectral index is $\alpha_{\rm med}$ = -1.75.
The two dashed line shows the standard deviation around the median value, with
$\sigma=0.45$. The blue dashed-dotted line shows the integrated spectral index
value.

###  5.3 Spectral index map of the halo

Spatially-resolved spectral index maps are a very useful tool in understanding
the different physical processes related to mergers happening at the local
scale and the origin behind the radio halo emission. Our uGMRT band 3 and band
4 images have allowed us to study the resolved spectral map for the radio halo
and radio relic region. Despite some problems with the band 4 point source
subtracted image, the resolved spectral index map for the central region is
obtained with fair accuracy.

The spectral index maps are created using the point source subtracted
visibilities in band 3 and band 4. The images were made with a similar inner
uv cut-off (>0.2 k$\lambda$) and uniform weighting to match the spatial scale
of the radio emission at both 400 and 650 MHz. Then both the images are
convolved to a common resolution of 22′′, such that the radio halo emission
can be seen properly in both frequencies. The resolution of the image is
chosen such that there is an optimization between probing the spectral
variation in the radio-emitting region and having a good Signal to Noise
Ratio. For each image, the pixel value of less than 3$\sigma_{\rm rms}$ is
blanked. Flux density values are obtained 1000 times from a Gaussian
distribution that has the mean as the estimated flux density value and the
standard deviation as the background rms of the images. The errors regarding
the spectral indices are obtained from the standard deviation of the extracted
1000 values.

Figure 9 shows the resolved spectral index map (left panel) and corresponding
error map (right panel) for the radio halo. A local variation of the spectral
index is visible. In the northern part, some flat spectral regions are seen.
The central part of the radio halo shows some uniform spectral index regions.
The southern part of the radio halo is seen to have a very patchy distribution
of the spectral index. Fluctuations in the spectral index over the radio halo
can occur due to the non-uniformity of the magnetic field distribution and
different acceleration efficiencies (e.g. Brunetti et al., 2001; Petrosian,
2001; Brunetti & Lazarian, 2007). However, artificial patches in a resolved
spectral index map can also be originated due to unmatched uv-coverage in the
different frequencies.

In Figure 10 we have shown the distribution for the spectral index over the
radio halo extent. The median spectral index is $-1.75$ and the standard
deviation is $\sigma=0.45$. According to Vacca et al. (2014); Pearce et al.
(2017), if the variations in the spectral index are the result of measurement
errors, the median error value is expected to be comparable or similar to the
standard deviation. The estimated median error is 0.35, which is lower than
the dispersion of our distribution, which indicates small-scale fluctuations
over the extent of the radio halo. Also, this is a projected spectral index.
It is averaged along the Line Of Sight (LOS) and the “intrinsic” scatter in
the volume is larger ($\sim$ $\sqrt{N}$), where N is the number of beams/cells
along the LOS. Fluctuations in the resolved spectral map are seen in some
recent studies for the halos in Abell 2255 (Botteon et al., 2020b), Abell 520
(Hoang et al., 2021), MACSJ017.5+3745 (Rajpurohit et al., 2021a), and A2256
(Rajpurohit et al., 2022). The median spectral index of -1.75 is slightly
flatter compared to the integrated spectral index. However, the integrated
spectrum is calculated based on the flux density estimation, which is
brightness weighted, and the resolved spectral maps are created by giving each
pixel the same relevance.

We show the spectral index as a function of distance from the cluster center
in Figure 11. By sampling the total radio halo into different annular regions,
the average spectral index values were estimated. The cluster center is chosen
to be the center of the annulus, and we have checked the radial profile of
average spectral index by shifting the center 20 $-$ 30 ′′ from its original
position to check the robustness of the profile. In all cases, the steepening
is present along the radially outward direction. Only the measurement errors
are shown in the error bars. The radio halo in A521 has a distinct outward
radial spectral steepening. It begins at -1.4 and rises to -2.2 as we move
towards the outskirts. The steepening of the spectral index along outwards is
expected either due to the decline of the magnetic field (for a constant
acceleration efficiency) or due to the decline of the acceleration efficiency
(i.e. less acceleration at the larger distance from the center) (Brunetti et
al., 2001). Insufficient uv-coverage at the upper frequency can also lead to a
radial steepening. However, we have a sufficient uv-range at both frequencies
to map the extended emission of the A521.

Observations have shown a variety regarding the radial variation of the
spectral index. An outward radial spectral steepening is seen for Abell 2744,
Pearce et al. (2017) and MACS J0717.5+3745 Rajpurohit et al. (2021a);
uniformity in the spectral index is seen for the Toothbrush cluster van Weeren
et al. (2016); Rajpurohit et al. (2018); de Gasperin et al. (2020), the
Sausage cluster Hoang et al. (2017). No firm detection of the radial spectral
steepening has been made in A2163 (Feretti et al., 2004), A2219 Orrù et al.
(2006). However, the studies of A2163 and A2219 are very old and done at
different (very poor) resolutions.

Figure 11: Here the spectral index profile across the radio halo region is
shown. The annotated figure shows the annular regions, which sampled the radio
halo. The average spectral indices estimated from each annulus region are
fitted in red data points. There is an indeed clear radial steepening in the
spectral index value of the radio halo with the increasing radius.

Figure 12: Upper left: Chandra image of A521 at 0.5-4 keV band is shown,
smoothed with a Gaussian Full-width half maximum of 5′′, also overlaid (green
contours) with radio emission at 400 MHz. The contours level starts from
3$\sigma_{\rm rms}$ $\times$ [1,2,4..], where $\sigma_{\rm rms}$ = 45
$\mu$Jy/bm. The X-ray image is not point-source subtracted. The surface
brightness gradient is shown as a color scale. The yellow triangle indicates
the position, where a shock font has been detected by Bourdin et al. (2013).
Upper right: Here the XMM-Newton surface brightness map (0.5$-$2.5 keV) is
shown overlaid with the same radio map. The radio emission is very much
correlated with the thermal emission from the ICM. Lower left: temperature map
extracted from the XMM-Newton surface brightness map is shown, overlaid with
the radio emission at 400 MHz. Most of the radio emission occurs at the
comparable hotter part of the cluster. The yellow diamond regions show the
interaction region. Lower right: Here the error map for the temperature is
shown. At the edge region, the errors are high compared to the central
regions.

##  6 X-ray and Radio correlation

All models for the origin of the radio halos predict similarity between the
X-ray and radio component of the ICM (e.g. Brunetti & Jones, 2014). Emission
from the radio halos typically follows the X-ray distribution of the ICM
(Govoni et al., 2001). However, there are a few clusters whose radio emission
does not trace the thermal emission with the same morphology (Giacintucci et
al., 2005). Morphological similarity between the X-ray and radio can probe the
interactions and different physical processes between the thermal and non-
thermal plasma of the ICM.

In Figure 12 we show the thermal X-ray emission (both from the Chandra and
XMM-Newton) of the A521, overlaid with radio emission at 400 MHz. Overall, the
radio emission from the halo follows the X-ray emission, despite the
elongation of the radio halo along the merger axis. The position of the second
relic (R2) is also seen in the peripheral region of the cluster and it is
associated with low surface brightness ICM thermal emission, similar to relic
R1. A significant part of the radio halo seems to be strongly correlated to
thermal gas. However, after a certain distance ($\sim$ 600 kpc) from the
cluster center (marked yellow triangle), the radio emission rapidly drops,
even if the X-ray emissions are still detected in that region. Bourdin et al.
(2013) have detected a shock front at that location. We have shown the XMM-
Newton temperature map and overlay our 400 MHz radio emission on that in
Figure 12. The northern part of the cluster is cooler compared to the southern
one. The former shows morphological similarity with the radio halo emission.
At the interacting region between the northern and southern clumps, patches of
the high-temperature region are seen (yellow diamond), as also reported in
Bourdin et al. (2013). This hot gas may be a result of the compression between
the two cold fonts. The bridge region between the R1 and halo shows a high
temperature ($\sim$ 10 keV) region. A temperature drop across the shock front
(situated in SE to NW) is also seen in Figure 12.

We also study the radial variation of radio and X-ray surface brightness
profiles. To investigate the surface brightness variation, we divided the
radio halo into concentric annular regions (centered at the cluster center).
In each annular region, the mean surface brightness and standard deviations
are calculated. The relic R1, R2, and discrete unresolved sources (from X-ray
images) were masked and excluded from the calculations. The resulting radial
profiles are shown in Figure 13a. The figure shows that the surface brightness
of radio (red squares) and X-ray (blue squares) correlates well, close to the
central region. Non-thermal components decay at a slower rate than thermal
gas. Similarly, we investigated the radial profiles of surface brightness in
the halo’s northern (middle panel) and southern (right panel). Both sub-
clusters exhibit a similar pattern, with the non-thermal components
experiencing a less steep decline than the thermal components.

###  6.1 Spatial correlation between the X-ray and radio surface brightness

To date, point-to-point analysis between the radio surface brightness and the
X-ray surface brightness has been done for many radio halos, mostly at 1.4
GHz. For giant radio halos the correlation slope predominantly shows sub-
linear to linear values (e.g. Govoni et al., 2001; Giacintucci et al., 2005;
Rajpurohit et al., 2018; Hoang et al., 2019; Cova et al., 2019; Xie et al.,
2020; Rajpurohit et al., 2021b; Bruno et al., 2021; Hoang et al., 2021;
Bonafede et al., 2022; Rajpurohit et al., 2022). This relationship between the
X-ray (IX) and radio surface brightness (IR) is generally described in the
form of a power law:

$\log(\rm I_{\rm R})=a+b\log(\rm I_{\rm X})$ (5)

where b is the slope of the correlation. The correlation slope b=1 is for a
perfect linear correlation. The value of the b determines whether the thermal
components (gas density, temperature) of the ICM decay faster (b <1) than the
non-thermal components (relativistic electron populations, magnetic field) or
vice-versa (b >1).

Figure 13: Left: Radio (red squares) and X-ray (blue squares) surface
brightness radial profiles within the halo region, normalized at their maximum
are shown. Inset at the top right corner is the X-ray Chandra image, overlaid
with the radio contours (black contours) at 400 MHz. The annular regions in
cyan are used to estimate the averaged radio and X-ray surface brightness
profiles. The blue square indicates the position of the BCG, which is masked
during the calculation. The error bars represented here are purely
statistical. Centre: The similar radial profile is shown here for the northern
part of the halo. The semi-circle (cyan) annular regions are used to estimate
the averaged surface brightness values. Right: The radial profiles for the
southern part of the radio halo is shown with the corresponding regions in the
inset. Figure 14: The grey-scale image shows the uGMRT 400 MHz image. Each of
the square boxes has a size of 22′′. The red boxes sample the northern part of
the radio halo, whereas the blue ones are for the southern part. The magenta
square boxes are used to estimate the surface brightness values at the bridge
region between the relic R1 and the radio halo.

High-resolution, sensitive uGMRT, Chandra X-ray observations allow us to do a
detailed investigation of the interplay between the thermal and non-thermal
components of the ICM. For the analysis, we have used the 400 MHz, and 650 MHz
uGMRT point-source subtracted images. Chandra X-ray image is smoothed to a
resolution of 5′′. We have made use of the publicly available software PT-
REX555https://github.com/AIgnesti/PT-REX (Point-to-point TRend EXtractor)
(Ignesti et al., 2020) to sample the radio and X-ray emissions in radio halo
regions and exclude the relics and the discrete unresolved sources. The
properties of the bridge region between the relic R1 and the halo are unclear,
therefore we have done a correlation study separately for this region. For
estimation of the radio and X-ray surface brightness, we have chosen a grid
size of 22′′ (88 kpc). To retain a good signal-to-noise ratio, we have
included only regions with above 3$\sigma_{\rm rms}$, below that, are included
as the upper limit. The radio brightness is expressed in the units of Jy
arcsec-2 and X-ray in units of Counts s-1arcsec-2.

Due to the threshold and large intrinsic scatter of the data, a sophisticated
linear regression method should be taken into account the Malmquist bias for
both variables. Here we have used the LinMix666For more information on LinMix
check https://linmix.readthedocs.io/en/latest/src/linmix.html (Kelly, 2007) to
determine the best-fitting parameters from the observed data set. LinMix is an
MCMC hierarchical approach, that considers the uncertainties of both variables
and incorporates the upper limits (non-detection) of the y-axis, allowing the
estimation of intrinsic scatter ($\sigma_{\rm int}$). The strength of the
correlation is measured using the Pearson (rp) and Spearman (rs) correlation
coefficients.

In Table 6 we have summarised all the best-fit parameters and correlation
coefficients for each frequency of our analysis. The radio (IR) and X-ray (IX)
surface brightness show a significant positive correlation both at 400 and 650
MHz. A sub-linear slope of $b_{\rm 400MHz}$ = 0.61 $\pm$ 0.06, $b_{\rm
650MHz}$ = 0.55 $\pm$ 0.08, is obtained at 400 and 650 MHz respectively.
Point-to-point correlation studies for the ultra-steep spectrum radio halo
(Abell 2256; Rajpurohit et al. 2022 and MACSJ1149+222.3; Bruno et al. 2021)
shows a similar range of values.

Figure 15:  Radio Vs X-ray surface brightness (on a logarithmic scale) plots
for the halo in A521 at 400 (left panel) and 650 MHz (right panel) are shown.
The LinMix best-fit line is shown in the solid black (entire halo), blue
dashed (southern), and red dashed-dotted (northern) lines. The red and blue
square boxes show the detection of the radio halo above 3$\sigma_{\rm rms}$.
The 2$\sigma_{\rm rms}$ are considered as the upper limits and indicated by
arrows. The green dashed line shows the 1$\sigma_{\rm rms}$ line at both
frequencies.

As discussed in Section 6, the cluster is composed of two main gas clumps
(northern and southern). In Figure 15 we show our test to search for a
correlation between the northern and southern clumps. Our best-fit parameters
are consistent with a sub-linear slope of the radio-X-ray surface brightness,
regardless of the radio frequency we consider. The correlation is stronger in
the northern clump, and is also steeper than the southern clump, being in all
cases sub-linear. In the bridge region (shown in the magenta boxes in Figure
14) the linear regression shows that the radio and X-ray surface brightness is
very weakly correlated, with a linear correlation coefficient of 0.30. Due to
the uncertain boundary of the bridge region, it is still very hard to draw any
strong conclusion on the properties of the bridge region.

In past, the frequency evolution of the correlation slope has been studied for
many radio halos, exceptional case: Abell 520 (Hoang et al., 2019),
MACSJ017.5+3745 (Rajpurohit et al., 2021a), where the investigation for the IR
\- IX correlation have been done for multiple frequencies. They report a
significant variation of correlation slope over frequency, indicating the
presence of a high-frequency cut-off. However, there are cases where the slope
remains constant as a function of frequency: Abell 2744 (Rajpurohit et al.,
2021b), Abell 2256 (Rajpurohit et al., 2022). Similarly, in A521 we discovered
that the correlation slope is nearly constant across the frequency range of
our study. However, the distance between the frequencies in our study is not
very big.

It is worth checking the effect of the box size on the correlation slope by
using the lower-resolution images. We thus repeated the fitting procedures
using the LinMix for 28′′ and 35′′ resolution images. We have created two new
grids, sampling the total radio halo emission. We have obtained a slope of
$b_{28^{\prime\prime}}$ = 0.58 $\pm$ 0.09, $b_{35^{\prime\prime}}$ = 0.56
$\pm$ 0.1. The results indicate that the correlation slope does not change
very much with changing the box size. Rajpurohit et al. (2021b) have shown the
effect of the grid size on the correlation slope in Abell 2744, where the
slope becomes flatter at higher resolutions and steeper at lower resolutions.

Table 6: Best-fit slopes and Spearman (rs) and Pearson (rp) correlation coefficients of the radio and X-ray surface brightness data are summarized. | Frequency(MHz) | 3$\sigma$ | 2$\sigma$
---|---|---|---
|  | slope (b) | $\sigma_{\rm int}$ | rs | rp | slope (b) | $\sigma_{\rm int}$ | rs | rp
Entire halo | 400 | 0.55 $\pm$ 0.07 | 0.02 | 0.72 | 0.76 | 0.61 $\pm$ 0.06 | 0.04 | 0.68 | 0.71
| 650 | 0.50 $\pm$ 0.08 | 0.02 | 0.65 | 0.72 | 0.55 $\pm$ 0.08 | 0.02 | 0.65 | 0.70
Northern | 400 | 0.55 $\pm$ 0.06 | 0.02 | 0.80 | 0.83 | 0.65 $\pm$ 0.06 | 0.03 | 0.72 | 0.74
| 650 | 0.53$\pm$0.08 | 0.01 | 0.77 | 0.73 | 0.66 $\pm$ 0.09 | 0.02 | 0.76 | 0.81
Southern | 400 | 0.50 $\pm$ 0.07 | 0.02 | 0.80 | 0.81 | 0.54 $\pm$ 0.1 | 0.07 | 0.65 | 0.70
| 650 | 0.50 $\pm$ 0.09 | 0.2 | 0.73 | 0.75 | 0.50 $\pm$ 0.08 | 0.02 | 0.69 | 0.68

####  6.1.1 Radial variation of the correlation strength

We investigated the relationship between radio and X-ray surface brightness at
various sub-regions (core and the outer region) shown in Figure 16. The
boundaries of the core and outer regions were defined in Section 5.2. Because
the radio halo is much more extended at 400 MHz, we used that image as well as
the same Chandra X-ray image. The slope is 0.80$\pm$ 0.09 in the core regions
and 0.62$\pm$0.06 in the outer regions. Although we saw a sub-linear
correlation strength in both regions, a gradient in the slope can still be
seen. Bonafede et al. (2022) discovered an increase in correlation slope for
the Coma cluster. While Rajpurohit et al. (2022) has also reported variation
in the correlation slope across sub-regions. In our case, we discovered
different slopes in the two sub-regions at increasing radii. For some cool
core clusters that host a hybrid halo, that is, a mini halo and halo-type
component, such as RXC J1720.1+2638 (Biava et al., 2021), MS 1455.0+2232
(Riseley et al., 2022), a steeper slope is found in the core region followed
by flattening in the outermost region. The above-mentioned behavior can be
possible due to the complex interplay between the thermal and non-thermal
components of the ICM in a different way for the central and outer regions.

Figure 16: We have shown the X-ray vs radio surface brightness at 400 MHz for
the core and outer region of the halo. The annotated figure shows the uGMRT
400 MHz emission (in grey). Different regions are shown in square boxes with
different color-coded. The blue solid line is the best-fit line for the outer
regions and the red dotted line is the best-fit line for the core region.

Figure 17: Left: X-ray surface brightness versus spectral index relation for
the A521 radio halo is shown at 400 MHz. The cyan lines represent the
posterior of the MCMC chain and the blue line indicates the best fit for the
regression. Right: $\alpha$$-$T relation for the A521 radio halo. The
temperature extracted from the XMM-Newton temperature map is shown in Figure
12, (bottom left).

###  6.2 Spatial correlation in the spectral index vs X-ray surface brightness

We also checked the point-to-point relationship between the spectral index and
the X-ray surface brightness in A521. We used the same sampling method
described in the previous section to obtain the spectral index and X-ray
surface brightness values. We extracted the surface brightness using the same
Chandra image as in Section 6. We fitted the data with LinMix to do linear
regression between the spectral index and X-ray surface brightness and to test
the significance of the correlations. We have assumed the relation between the
spectral index and X-ray surface brightness in the form of:

$\alpha=\rm a+\rm b\log(\rm I_{\rm X})$ (6)

The spectral index and the X-ray surface brightness appear to have a mild
anti-correlation (Figure 17). To test the significance of the correlation
between these two quantities, we estimated the Spearman and Pearson
correlation coefficients for the given data set. Spearman coefficient is
$\approx$ -0.37, with a slope of 0.46$\pm$0.10. The weak anti-correlation
indicates that the brighter X-ray-emitting regions correspond to flatter
spectral indices. This is consistent with the radial spectral steepening in
the radio halo region.

To our best knowledge, the studies for the correlation between the spectral
index and the X-ray surface brightness have been done for a very small number
of halos like A2744 (Rajpurohit et al., 2021b), A2255 (Botteon et al., 2020b)
and CIG0217+70 (Hoang et al., 2019), A2256 (Rajpurohit et al., 2022). A2255
has a mildly positive correlation between the two quantities. However, in
A2744, both positive and negative correlations were discovered, indicating the
presence of multi-component radio halo and complex merger systems. While there
is no significant correlation between the spectral index and radio surface
brightness for CIG0217+70. A recent study by Rajpurohit et al. (2022) found a
moderate anti-correlation between these two quantities for the A2256, which
contains an ultra-steep spectrum radio halo. Based on the correlation
coefficients, a weak negative correlation between the spectral index and X-ray
surface brightness can be seen. However, further investigations are needed to
establish any strong trend.

Table 7: Linmix best fitting slopes and Spearman and Pearson coefficient for the data plotted in Figure 17 |  $\alpha$-IX |  $\alpha$-T
---|---|---
Correlation slope | 0.46 $\pm$ 0.10 | 0.37$\pm$ 0.09
Spearman Coff. | -0.37 | 0.43
Pearson Coff. | -0.35 | 0.39

###  6.3 Correlation between the spectral index and ICM temperature

The XMM-Newton map was used for the analysis because it covers a larger area
than Chandra. The resulting plot is in Figure 17 (right panel). We used LinMix
to look for any possible correlations. The Spearman correlation coefficient is
0.43. This indicates a very weak positive correlation between the spectral
index and the ICM temperature. The positive correlation between these two
quantities strengthens the fact that a fraction of the gravitational energy
dissipated during cluster mergers heats up the thermal plasma as well as
accelerates relativistic particles in the ICM.

A comparison between the resolved spectral index and ICM temperature values
has been studied in Orrù et al. (2006) for A2744, where they reported that
flatter spectrum emission would trace higher temperature regions. However, the
results were questioned by the Pearce et al. (2017) using deeper radio and
X-ray observation and found no strong correlation between these two. A mild
anti-correlation has been found for radio halo in A2255 Botteon et al.
(2020b). While we have found a weak positive correlation for the radio halo in
A521. High-temperature regions are expected to trace the heated regions,
excited by turbulence, hence, the gas dynamics and the turbulent energy flux,
that is dumped into the particles will correlate well with flat spectral index
regions. However, Kale & Dwarakanath (2010) argued about the validity of any
possible correlation and anti-correlation between the spectral index and ICM
temperature due to the different cooling time scales for the thermal gas and
non-thermal plasma. The other caveat about these correlations is the effect of
plasma instability and micro-physics, which is not well understood in
theoretical models. The plasma collisionality may play a crucial role in the
acceleration efficiency (e.g. Brunetti & Lazarian, 2011b). An increase in
temperature may affect collisionality and mean free path of (thermal)
particles which potentially could drive effects on the effective acceleration
rate. It is also possible to imagine situations where decreasing
collisionality (hot regions) might decrease the acceleration rate.

###  6.4 X-ray shock at R2?

Using the Chandra image, we searched for an X-ray brightness edge at the
position of the newly-detected radio relic R2, which would indicate a possible
shock front. The Chandra image binned to $8^{\prime\prime}$ pixels to
emphasize low surface brightness features in the cluster outskirts is shown in
the left panel of Fig. 18. It indeed shows a subtle edge (green arrow)
coincident with the brightest and sharpest segment of the relic (Figure 3,
upper-left panel). To quantify this edge and see if it is consistent with a
shock front, we extracted an X-ray radial brightness profile in the white
sector shown in Figure 18 (left). The sector is centered on the curvature
center of the relic and covers the most prominent northern half of the relic.
We used an image binned to $1^{\prime\prime}$ pixels and masked out the X-ray
point sources in the sector. The X-ray profile is shown in the right panel of
Figure 18. A brightness edge is clearly seen at $r=226^{\prime\prime}$,
corresponding to the outer edge of the relic, which is shown by a dashed green
vertical line. The brightness edge has the characteristic shape of a spherical
density discontinuity projected on the sky plane (e.g. Markevitch & Vikhlinin,
2007). To quantify this discontinuity, we modeled the brightness profile close
to the edge ($\pm$ factor 2 from the radius of the edge) using a spherical gas
density distribution with the center of symmetry at the center of the sector
and the gas density described by two power laws on the two sides of the edge
and a sharp jump at the edge, with the power laws and the position and
amplitude of the jump being free parameters (Markevitch et al., 2000). The
best-fit model is shown in red in the right panel of Figure 18. A continuous
density profile is inconsistent with the data at a $\sim 3\sigma$ confidence;
the density jump amplitude is $2.4^{+2.3}_{-0.8}$ at 90% (where we did not
apply the precise conversion factor between the emission measure and X-ray
brightness, which is uncertain without the temperature measurements but
sufficiently close to 1 for the Chandra instrument response in this energy
band and our large error bars). If this density jump is a shock front — which
is plausible but cannot be determined with certainty without the temperature
measurements on both sides of the edge — it would correspond to $M=2.1$ or
$M>1.4$ at 90%. The X-ray Mach number (MX is lower as compared to the radio
Mach number (M = 2.67$\pm$0.09) at R2. The discrepancy between the radio and
X-ray Mach numbers may indicate a non-uniform distribution of the Mach number
over the shock surface (Ryu et al., 2019; Wittor et al., 2021). The radio
observations trace the higher Mach number regions, where the particles get
more accelerated. Also, the radio Mach number is estimated assuming the
standard DSA to be the possible origin of the R2. The estimated Mach numbers
of the shock are low ($\sim$ 2.5) to generate the observed emission of the
relic in a standard DSA scenario (Botteon et al., 2020a). Therefore a shock
re-acceleration of supra-thermal electrons might be needed to understand the
relic emission.

Figure 18: Left: Chandra image of A521 in the 0.5$-$4.0 keV energy band binned
to 8′′ pixels. The image is background subtracted and divided by the exposure
map. A subtle edge (marked by a green arrow) is coincident with the brightest
and sharpest segment of the relic R2 (Figure 3, upper-left panel). The white
sector is used for the radial profile shown in the right panel. Right: X-ray
0.5$-$4.0 keV surface brightness profile across the edge in the region of the
relic R2, extracted in the white sector indicated in the right panel. Error
bars are 1$\sigma$. The red line is a fit with a shock discontinuity. The
dashed green vertical line marks the position of the outer edge of the relic.

##  7 Discussion

###  7.1 Spectral properties and re-acceleration models

A strong indication of spectral fluctuations and radial spectral steepening
may be reconciled with the absence of a clear spectral curvature in the
integrated spectrum assuming in-homogeneous turbulent re-acceleration.
Homogeneous turbulent re-acceleration models predict the spectral steepening
to occur at frequencies ($\nu_{s}$) larger than a few times the critical
frequency ($\nu_{c}$) of the high energy electrons (Cassano et al., 2006),
$\nu_{s}$ $\sim$ $\xi$$\nu_{c}$, where $\xi$ was 6-8 estimated by Cassano et
al. (2012). However, if the magnetic field intensity, acceleration time scale
vary in the emitting volume (in-homogeneous case) and along the line of sight,
the spectrum may get stretched in frequency and the curvature becomes less
evident (Rajpurohit et al., 2021a; Donnert et al., 2013). The similar in-
homogeneous behavior of the magnetic field and acceleration time may lead to
the observed spectral fluctuations in the radio halo regions. Radial spectral
steepening demonstrates the presence of a high-frequency break in the
spectrum, however, the in-homogeneous conditions might smooth the effect of
the break in the integrated spectrum. This combined with other results (e.g.
Rajpurohit et al., 2021a), demands further theoretical steps to understand the
spectral shape of radio halos.

The turbulent re-acceleration model has predicted that a large number of radio
halos (at GHz frequencies) should host the ultra-steep spectrum radio halos
(Cassano et al., 2006; Brunetti et al., 2008). Macario et al. (2013) have
taken a homogeneous (homogeneous in the emitting volume) re-acceleration model
to explain the integrated spectrum of A521. The curved spectra had shown a
gradual steepening of the spectrum at high frequencies, although the
steepening can not be very drastic since the observations for A521 are limited
up to 1.4 GHz (Dallacasa et al., 2009). However, they have used the 74 MHz
upper limit, which we decided to avoid. The steepening frequency ($\nu_{s}$)
of the synchrotron emission is given by:

$\nu_{s}\propto\tau_{\rm acc}^{-2}\frac{\rm B}{(\rm B^{2}+\rm B_{\rm\rm
IC}^{2})^{2}},$ (7)

where B is the magnetic field and $\tau_{\rm acc}$ is the re-acceleration time
(Brunetti & Lazarian, 2007). If $\tau_{\rm acc}$ is constant, the steepening
frequency depends on the magnetic field. The highest frequency will be emitted
at a field $\rm B_{\rm cr}=\frac{B_{\rm IC}}{\sqrt{3}}$, where the CRe has a
maximum lifetime at a given frequency. With B <Bcr, a radial steepening of the
spectral index can be seen at distances larger from the cluster center, at a
lower observing frequency. Assuming a spectral steepening above 1.4 GHz (due
to the stretch of curvature in frequency domain by in-homogeneous ICM
properties) implies the minimum acceleration time (Eq.8 in Rajpurohit et al.
2021a) in the emitting volume is $\sim$ 157 Myr, providing that the magnetic
field fluctuations in that region are B $\sim$ Bcr.

The limits obtained for the turbulent acceleration timescale can be used to
constrain the energy density and scale of the turbulence. This will depend on
the adopted re-acceleration model, as different models consider the nonlinear
interactions between the turbulence and the particles. Stochastic re-
acceleration model (Brunetti & Lazarian, 2016), based on the scattering of
particles in super-Alfvenic in-compressible turbulence, connects the
steepening frequency and the turbulent Mach number via:

$M_{\rm t}\sim 0.2\frac{2000\rm km\rm s^{-1}}{\rm
c_{s}}\left(\frac{\nu_{s}}{1\rm GHz}\right)^{1/6}\left(\frac{L}{10^{2}\rm
kpc}\right)^{1/2}(1+z)^{\frac{11}{6}},$ (8)

where cs is the sound speed in that medium and $L$ is the injection length
scale. Using eq 8, the value of $M_{\rm t}^{2}$ to be 0.3 and the ratio of the
turbulent energy to the thermal energy $\epsilon_{\rm t}/\epsilon_{\rm ICM}$
$\sim$ 1/3$\Gamma M_{\rm t}^{2}$ $\sim$ 0.19, considering the adiabatic index
$\Gamma=5/3$ and injection scale $\sim$ 100 kpc. The above calculation assumes
that turbulence and particle acceleration occur everywhere (filling factor of
1). However, there can be situations with a filling factor <1, leading to a
smaller energy budget for the turbulence. The obtained value is consistent
with results found in the high-resolution cosmological simulations of the ICM
(e.g. Miniati, 2015; Vazza et al., 2017).

Figure 19: Distribution of the IR vs IX correlation slope for the different
radio halos. The cyan color shows the statistics for the giant radio halos,
whereas the value line is for the mini halos. We have collected the data for
22 giant radio halos and 9 mini halos. The USSRH populations are shown in the
white hatch line. The giant radio halos mostly fall in <1 region, having a
sub-linear slope.

###  7.2 Thermal and non-thermal correlation

Despite the complex morphology of the radio halo in A521, the radio and X-ray
surface brightness show a positive correlation at both frequencies. In Figure
19, we have shown the distribution of the radio vs X-ray surface brightness
correlation slope. We emphasize that the sample space is very low, but a
significant number of giant radio halos ($\sim$ 22) have shown a sub-linear
correlation slope. Whereas the USSRH spans over a very narrow range of
correlation slopes.

The correlation slope provides the information for the different acceleration
processes and the magnetic field distribution (e.g. Govoni et al., 2001;
Brunetti & Jones, 2014; Storm et al., 2015). Most of the thermal energy
content of the ICM results from the dissipation of the kinetic energy of the
gas, falling to the cluster, and a part of the thermal energy channeled into
the non-thermal components. In the turbulent re-acceleration model, the
synchrotron emissivity can be expressed as

$\epsilon_{\rm R}\propto\eta_{\rm acc}F\frac{\rm B^{2}}{\rm B^{2}+\rm B_{\rm
IC}^{2}}$ (9)

Where $\eta_{\rm acc}$ is the acceleration efficiency (fraction of turbulent
energy flux converted into relativistic electrons) and $F$ is the turbulent
energy flux in a unit volume:

$F=\frac{1}{2}\rho\frac{dv^{3}}{L}$ (10)

If we assume a constant temperature and constant Mach number, from equation 9,
the synchrotron emissivity will be:

$\epsilon_{\rm R}=\epsilon_{\rm R}(0)\frac{\rm X(r)}{\rm
X(0)}\left(\frac{\epsilon_{\rm X}(r)}{\epsilon_{\rm
X}(0)}\right)^{\frac{1}{2}}\frac{D_{\rm pp}(r)}{D_{\rm
pp}(0)}\frac{1+\left(\frac{\rm B_{\rm IC}}{\rm
B(0)}\right)^{2}}{1+\left(\frac{\rm B_{\rm IC}}{\rm
B(0)}\right)^{2}\left(\frac{\epsilon_{\rm X}(0)}{\epsilon_{\rm
X}(r)}\right)^{\frac{1}{2}}}$ (11)

where $\epsilon_{\rm R},\epsilon_{\rm X}$ are the radio and X-ray emissivities
($\epsilon_{\rm X}\propto\rm n_{\rm ICM}^{2}$), BIC is the equivalent CMB
magnetic field, and X is the ratio of the energy density of the cosmic rays to
the thermal gas. In terms of turbulent Mach number, the above equations can be
written as:

$\epsilon_{\rm R}(r)\propto\rm X(r)\frac{\rm M_{\rm
t}^{\alpha}(r)}{L(r)}\epsilon_{\rm
X}(r)^{\frac{1}{2}}\frac{1}{1+\left(\frac{\rm B_{\rm IC}}{\rm
B(0)}\right)^{2}\left(\frac{\epsilon_{\rm X}(0)}{\epsilon_{\rm
X}(r)}\right)^{\frac{1}{2}}}$ (12)

where Mt is the turbulent Mach number and $\alpha=4$ for the Transit-time-
damping (Brunetti & Lazarian, 2007) and $\alpha=3$ for the acceleration by in-
compressible turbulence (Brunetti & Lazarian, 2016). Therefore, different re-
acceleration models would lead to sublinear slopes depending on the ICM
conditions. Assuming X(r) and Mt constant with radius, equation 12 would lead
to a sub-linear ($\epsilon_{\rm R}\propto\epsilon_{\rm X}^{0.5}$) and linear
correlations for B(0)2/B${}_{\rm IC}^{2}$ >>1 and B(0)2/B${}_{\rm IC}^{2}$
<<1, respectively.

A change in the correlation slope (flattens outside and steeper in the core
region) has been obtained in the radio halo in A521. According to equation 12,
a possible scenario to explain the observed behavior is to assume the increase
in the ratio of the CR energy to the thermal energy (X(r)) with the increasing
radial distance from the cluster center. This is expected due to the lowering
in the electron energy losses. Also, an increase of the Mach number with
radius would make the radio-X ray scaling flatter in the external regions
(equation 12). However, a Mach number increasing significantly with radius
would make the synchrotron spectrum flatter in the outskirts and this
situation is ruled out by the observed radial spectral steepening.

###  7.3 Merging scenario

A521 is dynamically active at the central region (Yoon et al., 2020). Ferrari
et al. (2006) had interpreted the Chandra X-ray observation, by suggesting
two-step merging events; pre-merger scenario and a post-merger scenario. HST
Weak Lensing studies of A521 have shown that the cluster consists of three
main clumps C, NW, SE (Yoon et al., 2020). The C clump consists of two more
substructures, CN and CS. In Figure 20 we have shown the schematic diagram of
the Merging process.

The double relic in A521 confirmed a binary merger of two clumps at the
cluster central region. In a binary merger, two “equatorial” shocks will move
outward in the equatorial plane, perpendicular to the merger axis. Following
the passage of the dark matter core, two “merger” shocks will travel in
opposite directions along the merger axis. van Weeren et al. (2011b)
demonstrated that the collision of two similar mass clumps results in
diametrically opposite relics with similar size and transverse width and
luminosity. However, for a different mass ratio, the size and luminosity of
the relics changes; the brighter relic will be situated ahead of the lower
mass clump and the fainter will be in front of the higher mass clump. Yoon et
al. (2020) has quoted of the masses of two sub clumps; M${}_{\rm CN}=3.5\times
10^{14}$M⊙, M${}_{\rm CS}=1.5\times 10^{14}$M⊙. The ratio of the LLS for the
R1 and R2 relic is $\sim$ 3, whereas the mass ratio between the mass clumps
(CN and CS) is $\sim$ 2\. A similar value of mass ratio explains the observed
emissions for the Sausage relic (van Weeren et al., 2011a). Because of the
lack of X-ray observations of the NW and SE clumps, the merging scenario
favors an off-axis collision between the CS and CN subclumps. Later in the
phase, the ICM gains angular momentum and, as a result, lags behind. In the
X-ray, it appears as a binary distribution, while the two DM halos begin their
second infall. The observed features, such as double relics and hot
intermediate regions, support a binary merging picture between two mass
clumps.

Figure 20: The schematic diagram for the merging scenario of cluster A521 is
shown. The position of the R1 and R2 and the Mach number of the shock are
labeled. The two clumps CN and CS are colliding with each other along the
southeast and northwest direction. The red dashed lines are the equatorial
shock wave outwards in the equatorial plane.

Some clusters have a radio halo and double relics, while others do not.
Bonafede et al. (2017) argued that clusters with double relics and a radio
halo had a different time-since-merger ( tmerger) than clusters without a
radio halo. It should be noted that the tmerger derived here is not the time
after the core passage. Taking the distance between the two relics as a proxy
of merger time and estimating the shock velocity of $\sim$ 1500 km.s-1, the
time-since-merger is $\sim$ 0.45 Gyr. Brunetti et al. (2008) had shown that
for the observed radio halo emissions, the electrons should be accelerated on
a time scale of $\sim$ 0.1 Gyr. Therefore the turbulence generated during the
merger will get the time to develop. Steep spectrum radio halos are predicted
as an outcome of the less energetic merger or high energy losses (e.g. Cassano
et al., 2010). A binary merger between the two small clumps (CN and CS), in a
bigger potential driven by the rest of the cluster, is well suited for the
origin of the less energetic turbulence.

##  8 Summary and Conclusions

We present here the first deep uGMRT observations (300 - 850) MHz of the
galaxy cluster A521. These sensitive uGMRT observations enable us to obtain
detailed spectral characteristics of the radio halo. Previous studies of this
radio halo at frequencies below 1 GHz were limited by the telescopes’ poor
resolution and insufficient uv-coverage. Our deep observations, combined with
available X-ray observations (Chandra, XMM-Newton), provide profound physical
insights into the thermal and non-thermal connections in the ICM, as well as
the origin of the radio halo. The overall findings are summarised as follows:

  1. 1.

We present the first deep and high resolution (7′′ \- 22′′) uGMRT radio images
of the galaxy cluster A521. The extended radio halo emission at the central
region and radio relic emission at the southeast outskirt regions recovered
well, with an LLS of the radio halo is 1.3 Mpc at 400 MHz and 1 Mpc at 650 MHz
respectively.

  2. 2.

Our deep and sensitive uGMRT images have revealed the presence of another
relic, R2, at the northwest position of the cluster. The LLS of the R2 is 650
and 450 kpc at 400 and 650 MHz respectively. This new relic appears to be
associated with a candidate shock front seen in the Chandra X-ray image. We
have also detected a low surface brightness extension ($\sim$ 1 Mpc) of the R1
relic, R1-TL, which makes the total length of the relic to be $\sim$ 2.2 Mpc.

  3. 3.

The radio halo emission follows a single power law between 153 to 1400 MHz
with an integrated spectral index of -1.86 $\pm$ 0.12, consistent with the
previously obtained values. Using our uGMRT analysis and previously published
literature values, we have obtained the integrated spectral index of the R1
relic to be -1.46 $\pm$ 0.03. We have also reported the integrated spectral
index of the R2 relic to be -1.34 $\pm$ 0.03.

  4. 4.

The spatially-resolved spectral index map may suggest fluctuation over the
extent of the radio halo. The median spectral index estimated from small
regions is within the error bars of the integrated spectral index, indicating
the self-similarity of the global spectral index to the local kpc scales.
Resolved spectral maps also reveal a spectral steepening of the radio halo
along the outward directions. This behavior incorporates the in-homogeneous
conditions (variation in B, acceleration time) in the emitting volume.

  5. 5.

The non-thermal emission from the radio halo has been found to have a strong
morphological correlation with the thermal emission from the ICM, indicating
that the hot gas and non-thermal plasma have a close relationship. The radio
emission is mostly concentrated in the hotter regions of the cluster, as seen
in the XMM-Newton temperature map. Furthermore, a comparison of the radial
profiles of radio (IR) and X-ray (IX) surface brightness has revealed a slower
decline of non-thermal components than the thermal ones, a possible indication
of the radial declination of the magnetic field.

  6. 6.

A tight sub-linear correlation between the radio and X-ray surface brightness
has been discovered through a point-to-point analysis across the entire extent
of the radio halo. This correlation was investigated using different fitting
methods and thresholds, and the slope was found to be relatively constant
across the frequencies studied. There are slight changes in the correlation
slope at the core (IR $\propto$ I${}_{\rm X}^{0.80\pm 0.09}$) and outer (IR
$\propto$ I${}_{\rm X}^{0.62\pm 0.06}$) regions, which may be attributed a
balance between the increase in energy density of the CRe and decrease of the
magnetic field strength.

  7. 7.

A weak anti-correlation (rs = -0.37) has been observed between the spectral
index and X-ray surface brightness over the spatial extent of the radio halo.
This finding is in agreement with the radial spectral steepening and the
dissipation of gravitational energy into the non-thermal component of the ICM.

  8. 8.

A weak positive correlation (rs = 0.43) was found between the average
temperature of the ICM and the spectral index. This observation is consistent
with the idea that regions with higher temperatures will have flatter spectral
indices: the energy dissipated by gravitational forces in the ICM will
accelerate the seed electrons. However, accounting for the plasma
collisionality and the microphysics (poorly understood) may affect the
correlation.

Several radio halo observations support the turbulent re-acceleration model,
including spectral index fluctuations across its spatial extent, a radial
steepening in the spectral index, and a strong sub-linear correlation between
radio and X-ray surface brightness. Because of the high energy budget of
thermal protons in the ICM, as demonstrated by Brunetti et al. (2008), the
pure hadronic model faces challenges. The R1 radio relic shows significant
substructures in the uGMRT images, making it a candidate for future low and
high-frequency polarization studies to understand magnetic field orientation
in the shock plane better. Also characterizing the properties at the shock
downstream region is one of the frontier goals for the future study of the
relics. The detection of another relic at the northwest position has placed
this cluster in a poorly understood class of objects featuring radio halo and
double relic systems. Investigating the complexity of radio emission through
high-resolution MHD simulations is necessary for the future.

We thank the anonymous referee for their comments that have improved the
clarity of the paper. RS acknowledges Alessandro Ignesti for his help
regarding the technical issues of PT-REX. RS also thanks Luca Bruno and
Francesco de Gasperin for their quick replies to many useful queries. RS and
RK acknowledge the support of the Department of Atomic Energy, Government of
India, under project no. 12-R&D-TFR-5.02-0700. SG acknowledges that the basic
research in radio astronomy at the Naval Research Laboratory is supported by
6.1 Base funding. F.D. and H.B. acknowledge the financial contribution from
the European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant
agreement n.871158). We thank the staff of the GMRT that made these
observations possible. The GMRT is run by the National Centre for Radio
Astrophysics (NCRA) of the Tata Institute of Fundamental Research (TIFR). The
scientific results reported in this article are based in part on data obtained
from the Chandra Data Archive. This work made use of observations obtained
with XMM-Newton, an ESA science mission funded by ESA Member States and the
USA (NASA). This research had made use of the NASA/IPAC Extragalactic Database
(NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration.

Figure 21: upper panel: Here we have shown the full resolution uGMRT band3
images, for the unfiltered, filtered, and combined data sets. The beam size of
each image (left: 8.19 ” $\times$ 7.78′′, middle: 8.54′′ $\times$ 7.51′′,
right: 8.51′′ $\times$ 7.5′′) is given in the left bottom corner. The contours
start from 3$\sigma_{\rm rms}$ $\times$ [1,2,3…], where $\sigma_{\rm rms}$ =
28, 24.6, 25 $\mu$Jybeam-1 for the unfiltered, filtered and combined image
respectively. Lower panel: Here we have shown the same but at 650 MHz. The
beam size of each image (left: 5.4 ” $\times$ 4.7′′, middle: 6.0′′ $\times$
4.6′′, right: 5.8′′ $\times$ 4.7′′) is given in the left bottom corner. The
contours start at a similar significance as the upper panel, where
$\sigma_{\rm rms}$ = 10, 8, 8.5 $\mu$ Jybeam-1. The RFI-filtered image shows a
better recovery of extended emission.

Figure 22: Left: Ratio of the flux densities between the filtered and unfiltered data sets for the different sources are shown as a distance from the phase center. The dotted line at 1, indicates the similarity of flux densities for two cases. The error bars are purely statistical and contain the 10% flux density error and measurement error. Middle: The similar image is shown here, but for the ratio of flux density between the combined and filtered data set. The values are not very much scattered around the 1 at both frequencies. Right: Here we have shown the same plot for the ratio between the combined and unfiltered one. Regions distant from the phase center show the variations Table 8:  List of the discrete unresolved sources in the radio halo and relic region. The positions are reported in the units of degree and flux densities are in units of mJy. The first block reports the sources in the radio halo, the second block is in relic R2, and the third block is for the sources in relic R1. Most of the sources are compact and unresolved. Name | RA | Dec | Fν,400MHz | Fν,650MHz
---|---|---|---|---
J0454-1012a | 73.585 | -10.207 | 0.91$\pm$0.03 | 0.54$\pm$0.02
J0454-1012b | 73.598 | -10.211 | 0.48$\pm$0.02 | 0.22$\pm$0.02
J0454-1012c | 73.597 | -10.210 | 0.38$\pm$0.02 | 0.12$\pm$0.01
J0454-1013a | 73.598 | -10.217 | 0.57$\pm$0.04 | 0.30$\pm$0.02
J0454-1014a | 73.597 | -10.235 | 0.49$\pm$0.03 | 0.27$\pm$0.02
J0454-1012d | 73.566 | -10.215 | 0.4$\pm$0.02 | 0.28$\pm$0.01
J0454-1013b | 73.557 | -10.224 | 0.27$\pm$0.01 | 0.20$\pm$0.02
J0454-1014b | 73.562 | -10.234 | 1.36$\pm$0.08 | 1.143$\pm$0.01
J0454-1015a | 73.557 | -10.261 | 0.47$\pm$0.02 | 0.32$\pm$0.02
J0454-1015b | 73.556 | -10.253 | 0.38$\pm$0.08 | 0.2$\pm$0.02
J0454-1015c | 73.566 | -10.251 | 0.59$\pm$0.04 | 0.25$\pm$0.02
J0454-1015d | 73.563 | -10.234 | 0.5$\pm$0.06 | 0.32$\pm$0.01
J0454-1013c | 73.557 | -10.224 | 0.45$\pm$0.03 | 0.16$\pm$0.01
J0454-1013d | 73.541 | -10.223 | 0.8$\pm$0.04 | 0.15$\pm$0.01
J0454-1013e | 73.527 | -10.221 | 2.58$\pm$0.3 | 1.13$\pm$0.18
J0454-1014c | 73.508 | -10.235 | 0.45$\pm$0.01 | 0.26$\pm$0.01
J0454-1016a | 73.510 | -10.269 | 0.34$\pm$0.02 | 0.22$\pm$ 0.02
J0454-1016b | 73.520 | -10.267 | 0.33$\pm$0.02 | 0.14$\pm$0.01
J0454-1015e | 73.526 | -10.266 | 0.5$\pm$ 0.01 | 0.3$\pm$0.01
J0454-1014d | 73.548 | -10.242 | 0.47$\pm$ 0.02 | 0.38$\pm$ 0.01
J0454-1015f | 73.538 | -10.252 | 2.24$\pm$0.07 | 1.94$\pm$0.02
J0454-1011a | 73.541 | -10.186 | 1.22$\pm$ 0.04 | 0.68$\pm$0.02
J0454-1011b | 73.536 | -10.188 | 0.78$\pm$0.05 | 0.5$\pm$0.01
J0454-1011c | 73.504 | -10.187 | 0.54$\pm$0.03 | 0.21$\pm$0.02
J0454-1012e | 73.514 | -10.216 | 1.02$\pm$0.08 | 0.94$\pm$0.05
J0454-1013f | 73.502 | -10.218 | 0.45$\pm$0.05 | 0.21$\pm$ 0.02
J0454-1016c | 73.520 | -10.267 | 0.26$\pm$ 0.02 | 0.15$\pm$0.01
J0453-1014e | 73.491 | -10.240 | 0.48$\pm$ 0.02 | 0.17$\pm$0.01
J0453-1014f | 73.486 | -10.242 | 0.72$\pm$ 0.02 | 0.33$\pm$0.01
J0453-1015g | 73.489 | -10.262 | 0.46$\pm$ 0.08 | 0.25$\pm$0.02
J0453-1016d | 73.494 | -10.273 | 0.41$\pm$ 0.03 | 0.2$\pm$0.01
J0454-1016e | 73.510 | -10.269 | 0.41$\pm$ 0.03 | 0.12$\pm$0.02
J0454-1016f | 73.516 | -10.282 | 0.41$\pm$ 0.02 | 0.2$\pm$0.01
J0454-1016g | 73.494 | -10.274 | 0.37$\pm$ 0.01 | 0.2$\pm$0.01
J0453-1012 | 73.487 | -10.211 | 1.3 $\pm$ 0.04 | 0.87$\pm$0.02
J0453-1014 | 73.491 | -10.201 | 1.1 $\pm$ 0.06 | 0.55 $\pm$ 0.06
J0454-1015a | 73.594 | -10.265 | 1.19$\pm$ 0.05 | 0.92$\pm$0.01
J0454-1015b | 73.585 | -10.255 | 0.51$\pm$ 0.03 | 0.31$\pm$0.02
J0454-1017a | 73.593 | -10.286 | 1.54$\pm$ 0.09 | 1.08$\pm$0.03
J0454-1016a | 73.587 | -10.279 | 0.9$\pm$ 0.09 | 0.53$\pm$0.02
J0454-1018 | 73.572 | -10.301 | 0.53 $\pm$ 0.01 | 0.13$\pm$0.01
J0454-1017b | 73.590 | -10.284 | 0.43 $\pm$ 0.05 | 0.15$\pm$0.01
J0454-1017c | 73.585 | -10.286 | 0.40 $\pm$ 0.03 | 0.13$\pm$0.01
Table 9: Comparison of the flux densities for the point sources RA | DEC | Fν,400MHz(mJy) | Fν,650MHz(mJy)
---|---|---|---
|  | unfil. | Fil. | Combined | Unfil. | Fil. | Combined
04:54:09 | -10:15:08 | 1.99$\pm$ 0.05 | 2.1$\pm$0.04 | 2.08$\pm$0.05 | 1.52$\pm$0.01 | 1.54$\pm$0.03 | 1.52$\pm$0.02
04:54:03 | -10:12:59 | 1.06$\pm$0.06 | 1.08$\pm$0.07 | 1.09$\pm$0.07 | 0.68$\pm$0.04 | 0.71$\pm$0.03 | 0.7$\pm$0.04
04:53:54 | -10:24:07 | 5.97$\pm$ 0.08 | 6.14$\pm$0.08 | 6.40 $\pm$ 0.08 | 3.90$\pm$0.08 | 4.05$\pm$0.06 | 4.01$\pm$ 0.08
04:52:26 | -10:38:20 | 4.98$\pm$0.13 | 5.25$\pm$0.13 | 5.12$\pm$ 0.12 | 0.99$\pm$0.04 | 1.05$\pm$0.04 | 1.06$\pm$ 0.04
04:53:37 | -09:48:51 | 1.57$\pm$0.02 | 1.65$\pm$0.02 | 1.66$\pm$0.02 | 0.54$\pm$ 0.007 | 0.54$\pm$0.01 | 0.54$\pm$ 0.007
04:54:21 | -9:50:41 | 4.66$\pm$ 0.03 | 4.84$\pm$ 0.05 | 4.82$\pm$ 0.05 | 1.50$\pm$ 0.017 | 1.62 $\pm$ 0.02 | 1.61$\pm$ 0.017
04:54:28 | -10:17:23 | 0.85$\pm$ 0.04 | 1.02$\pm$ 0.04 | 1.04$\pm$ 0.04 | 0.87$\pm$0.03 | 0.88$\pm$ 0.02 | 0.87$\pm$ 0.03
04:55:38 | -10:00:01 | 0.83$\pm$ 0.02 | 0.89$\pm$ 0.02 | 0.91 $\pm$ 0.03 | 0.23 $\pm$0.01 | 0.23 $\pm$ 0.01 | 0.23 $\pm$ 0.01
04:53:46 | -10:11:09 | 1.7 $\pm$ 0.03 | 1.7 $\pm$ 0.03 | 1.72 $\pm$ 0.03 | 0.92 $\pm$ 0.02 | 0.95$\pm$ 0.01 | 0.91 $\pm$ 0.01
04:52:38 | -9:56:12 | 3.39 $\pm$ 0.03 | 3.55 $\pm$ 0.08 | 3.51 $\pm$ 0.08 | 1.3 $\pm$ 0.04 | 1.4 $\pm$ 0.03 | 1.39 $\pm$ 0.04

##  Appendix A Effect of the online RFI filtering

Broadband (BB) RFI (narrow high spikes in the time domain) occurs due to the
power-line discharge and these are very strong at low frequencies. uGMRT has
developed and implemented a real-time RFI filter (see for further details Buch
et al., 2022, 2023) to mitigate the BB RFI in the pre-correlation domain (for
each antenna and each polarization). We have presented a comparison between
the RFI-filtered (hereafter, filtered) and unfiltered data sets at bands 3 and
4. On average, $2-3\%$ of the raw voltage samples were flagged and replaced by
the filter at both bands. Both the filtered and unfiltered data sets had
similar calibration strategies. The flagging in the filtered data was lower
(at the central antennas) than in the unfiltered data. The parameters for the
imaging steps were similar for the two data sets. We have shown the unfiltered
(left), filtered (middle), and combined (right) images (grey scales) for both
band 3 and band 4, overlaid with the contours of each image in Figure 21.
Compared to unfiltered data, the centrally located radio halo is well
recovered in the filtered data.

The ratio of flux densities for the various point sources as a function of
their distances from the cluster center is shown in Figure 22. The ratio of
flux densities of point sources between the filtered and unfiltered data sets
is shown in the left image, where the variation in the flux densities is not
high. The variation is seen clearly in the regions away from the phase center.
A similar pattern can be seen in the other cases. In the filtered and
unfiltered data sets, the flux density of the point sources has not changed
drastically however, the extended emission is better recovered in the filtered
data set. Therefore, the less flagging at central square antennas (Figure 23)
due to the application of the RFI filter is significant.

Figure 23: The flagging percentage as a function of the baseline is shown. The
central baselines are less flagged in the filtered data, in both frequencies,
leading to a better recovery of extended emissions.

##  Appendix B Point source Image of the A521

We have shown the point source image of the A521 cluster field in Figure 24.
Table LABEL:table5 we have presented the flux densities (both band 3 and 4) of
the point sources in the central region (specifically in the central radio
halo and peripheral relic region). In table. LABEL:table9, the reported values
of the flux densities are used in Fig 22. Giacintucci et al. (2006) had listed
the radio sources in the 30′ $\times$ 30′ region. We have cross-checked their
table with ours. At the same frequency, our image is 5 times more sensitive
than theirs.

Figure 24: Left: We have shown the point source image (color) at 400 MHz and
the corresponding point source subtraction low-resolution image (green
contours). The magenta plus signs show the unresolved sources that have been
removed in the central region. The contours are drawn at 3$\sigma_{\rm rms}$
$\times$ [1,2,3], with $\sigma_{\rm rms}=45\mu$ Jybeam-1. Right: Here the
similar is shown at 650 MHz. The green contours start with similar values,
with a $\sigma_{\rm rms}=30\mu$ Jybeam-1. The point sources are listed in the
Table LABEL:table5.

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