# Mining Mini-Halos with MeerKAT I. Calibration and Imaging

K. S. Trehaeven1, V. Parekh1,2,3, N. Oozeer2,4,1, B. Hugo2,1, O. Smirnov1,2,
G. Bernardi1,2,5, K. Knowles1,2, C. Tasse6,1, K. M. B. Asad1,2,7,8, and S.
Giacintucci9
1Department of Physics and Electronics, Rhodes University, PO Box 94,
Grahamstown, 6140, South Africa
2South African Radio Astronomy Observatory, 2 Fir Street, Black River Park,
Observatory, Cape Town, 7405, South Africa
3National Radio Astronomy Observatory (NRAO), 1003 Lopezville Rd, Socorro, NM
87801, USA
4African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg,
7945, South Africa
5INAF – Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy
6GEPI & USN, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place
Jules Janssen, 92190 Meudon, France
7ARGI, Department of Physical Sciences, Independent University, Bangladesh,
Bashundhara RA, Dhaka, 1229, Bangladesh
8Department of Physics and Astronomy, University of the Western Cape,
Bellville, Cape Town 7535, South Africa
9Naval Research Laboratory, 4555 Overlook Avenue SW, Code 7213, Washington, DC
20375, USA E-mail<EMAIL_ADDRESS>
(Accepted 2023 January 31. Received 2023 January 23; in original form 2022
October 3)

###### Abstract

Radio mini-halos are clouds of diffuse, low surface brightness synchrotron
emission that surround the Brightest Cluster Galaxy (BCG) in massive cool-core
galaxy clusters. In this paper, we use third generation calibration (3GC),
also called direction-dependent (DD) calibration, and point source subtraction
on MeerKAT extragalactic continuum data. We calibrate and image archival
MeerKAT L-band observations of a sample of five galaxy clusters (ACO 1413, ACO
1795, ACO 3444, MACS J1115.8+0129, MACS J2140.2-2339). We use the CARACal
pipeline for direction-independent (DI) calibration, DDFacet and killMS for
3GC, followed by visibility-plane point source subtraction to image the
underlying mini-halo without bias from any embedded sources. Our 3GC process
shows a drastic improvement in artefact removal, to the extent that the local
noise around severely affected sources was halved and ultimately resulted in a
7% improvement in global image noise. Thereafter, using these spectrally
deconvolved Stokes I continuum images, we directly measure for four mini-halos
the flux density, radio power, size and in-band integrated spectra. Further to
that, we show the in-band spectral index maps of the mini-halo (with point
sources). We present a new mini-halo detection hosted by MACS J2140.2-2339,
having flux density $S_{\rm 1.28\,GHz}=2.61\pm 0.31$ mJy, average diameter 296
kpc and $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=1.21\pm 0.36$. We also found a
$\sim$100 kpc southern extension to the ACO 3444 mini-halo which was not
detected in previous VLA L-band observations. Our description of MeerKAT wide-
field, wide-band data reduction will be instructive for conducting further
mini-halo science.

###### keywords:

Galaxies: clusters: general – galaxies: clusters: intracluster medium –
galaxies: haloes – radio continuum: general – techniques: image processing –
techniques: miscellaneous

††pubyear: 2023††pagerange: Mining Mini-Halos with MeerKAT I. Calibration and
Imaging–D

##  1 Introduction

Galaxy clusters are the largest virialised structures in the Universe. They
form at the intersection of the Cosmic Web filaments and often grow by
capturing smaller systems (for example, galaxies or groups) in their
gravitational potential well, in so-called merger events. Clusters host most
of their baryonic mass in the hot, tenuous, intra-cluster medium (ICM, all
abbreviations used in this work are listed in Table 4 of Appendix A), in which
merger events deposit most of their energy. The dynamical state of the ICM can
be probed in several ways, for example through thermal bremsstrahlung
radiation observed in X-rays, as well as via the presence of diffuse radio
emission - which is observed in an ever-increasing number of clusters (Feretti
et al., 2012). This faint, diffuse emission is caused by a pool of non-thermal
ultra-relativistic electrons emitting synchrotron radiation in the presence of
intra-cluster magnetic fields. A combined analysis of the X-ray and radio
cluster environment can inform on the relationship between its thermal and
non-thermal components.

Extended radio emission found within galaxy clusters generally has a low
surface brightness and is classified into three broad phenomena, namely radio
relics, giant radio halos and radio mini-halos; see van Weeren et al. (2019)
for a comprehensive observational review of diffuse radio emission from galaxy
clusters. Specifically, radio mini-halos are generally a few 100 kpc in size,
confined to the cool-core of relaxed (non-merging) massive (> $5\times
10^{14}M_{\odot}$) clusters, typically have irregular morphologies, surround
the Brightest Cluster Galaxy (BCG), and are unpolarized (Giacintucci et al.,
2014a, b). They are difficult to image and just over 40 mini-halos (including
candidates) are known to date (Knowles et al., 2022; Bégin et al., 2022) with
many more predicted for future discoveries (Gitti et al., 2018). The origin of
the synchrotron-emitting electron population is still under debate, however,
it is thought to be closely linked to the cluster thermal component and/or the
BCG. The particle (re)-acceleration mechanism responsible for the production
of mini-halos is argued to either be turbulent (re)-acceleration via cool-core
gas sloshing (Gitti et al., 2012; ZuHone et al., 2013; Richard-Laferrière et
al., 2020; Riseley et al., 2022) and/or hadronic collision models (ZuHone et
al., 2015; Jacob & Pfrommer, 2017a, b).

The MeerKAT telescope (Jonas & MeerKAT Team, 2016) located in the Karoo desert
of South Africa, consists of 64 dishes, each 13.5 m in diameter resulting in a
primary beam of just over one degree in diameter at L-band. Approximately
three-quarters of its collecting area lies within a dense core region of
diameter 1 km; the remaining dishes are spread to a maximum baseline of 7.6
km, giving a minimum angular scale of $\sim$5$\arcsec$ at L-band. This
configuration allows MeerKAT to observe low surface brightness diffuse
emission while simultaneously resolving compact sources. This is particularly
important in the context of radio mini-halos, as the embedded Active Galactic
Nucleus (AGN, which is often only a few arcseconds in size) needs to be
disentangled from the diffuse synchrotron emission in order for a precise
image and study to be made. Additionally, the low system temperature and high
number of correlator frequency channels (in this case 4096) allows for the
precise removal of radio frequency interference (RFI), increasing sensitivity
to low surface brightness emission.

With radio telescopes becoming more powerful, calibration and imaging
techniques have been developed to deal with the added complications of higher
sensitivity data. The standard calibration techniques of cross-calibration
(first generation calibration, or 1GC) and traditional self-calibration
(second generation calibration, or 2GC) only correct for direction-independent
effects (DIEs). However, the wide field-of-view (FoV) and large fractional
bandwidth of modern radio telescopes significantly complicates the calibration
problem by introducing considerable direction-dependent effects (DDEs), such
as the atmospheric conditions in the ionosphere and the telescope’s primary
beam response (Iheanetu et al., 2019; Asad et al., 2021; de Villiers & Cotton,
2022). The additional computational challenges limit the extent to which the
resulting image artefacts can be addressed, thereby limiting overall image
sensitivity. Novel direction-dependent (DD) calibration (hereafter referred to
as third generation calibration, or 3GC) techniques that correct for the
residual errors are being used to rectify this problem - for example, the
faceted approach of killMS and DDFacet (Tasse, 2014a, b; Tasse et al., 2018)
or the source ‘peeling’ of CubiCal (Kenyon et al., 2018). Implementing such
techniques on modern telescopes such as the MeerKAT allows for images of
higher dynamic range (DR - a common measure of the depth or quality of a
continuum image) to be produced, thereby unveiling emission that was otherwise
undetectable in previous observations.

Many science cases require the subtraction of unwanted compact sources, such
as the study of faint transient objects in the image plane (for example
Gondhalekar et al., 2022) and, of course, the study of extended diffuse
sources in which compact emission is blended (as in Savini et al., 2019).
Source subtraction is particularly important for mini-halo studies, as a
significant fraction of the mini-halo area is contaminated by the embedded
AGN, which should be removed to avoid biasing any physical measurements. The
effectiveness of such a technique is difficult to quantify, and interestingly,
3GC adds complexity to the issue which has not been explored in much detail.
It is known that 3GC can worsen the effect of flux suppression/absorption;
flux from faint unmodelled sources can be absorbed by the antenna gains and
often transferred to the brighter modelled sources, thereby decreasing the
observed flux density of the faintest sources and increasing that of the
brighter ones (see for example Grobler et al., 2014; Wijnholds et al., 2016;
Sob et al., 2019). Thus, we note that examining the effect of 3GC on source
subtraction could prove to be a useful exercise, however this is outside the
scope of our current research.

In this paper, we present the calibration and imaging of mini-halo results
using MeerKAT archival L-band observations for a sample of five galaxy
clusters. We provide a guideline on how to process MeerKAT radio data to
improve image quality via facet-based 3GC and unambiguously study the diffuse
emission after point source subtraction. We describe the sample in Section 2.
Thereafter, in Section 3, we describe our data reduction procedures, including
3GC, source subtraction and spectral imaging. The results of the MeerKAT and
mini-halo data processing are presented in Section 4, with a discussion
thereof provided in Section 5. Finally, a summary and conclusions are given in
Section 6.

Throughout this paper, we adopt the physically motivated mini-halo definition
proposed by Giacintucci et al. (2017), which states that a mini-halo is a
diffuse radio source at the cluster center that (1) does not consist of, or
have any morphological connection to, any emission directly associated with
any embedded galaxies, (2) has a minimum radius of approximately 50 kpc and
(3) a maximum radius of 0.2R500111R500 is the radius that encloses a mean
overdensity of 500 with respect to the critical density at the cluster
redshift. We adopt a flat $\Lambda$CDM cosmology with H0 = 70 km s-1 Mpc-1,
$\Omega_{m}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7. Finally, we describe the
spectral nature of the radio emission as $S_{\nu}\propto\nu^{-\alpha}$, where
$S_{\nu}$ is the measured flux density at frequency $\nu$ and $\alpha$ is the
spectral index.

##  2 Sample selection

Table 1: Physical properties of the cluster sample. Columns: cluster name,
cluster redshift, angular-to-physical scale calculated for our cosmology,
cluster mass taken from Planck Collaboration et al. (2014) unless stated
otherwise, core X-ray morphology, core radio morphology and literature
references.
CC = cool-core, nCC = non-cool-core, MH = mini-halo, cMH = candidate mini-
halo.
* Taken from Piffaretti et al. (2011).
** New detection from this work.
Ref.: (1) Savini et al. (2019); (2) Giacintucci et al. (2017), (3) Kokotanekov et al. (2018); (4) Giacintucci et al. (2019); (5) Giovannini et al. (2020). Name | z | Scale | $M_{\rm 500\,,SZ}$ | X-ray | Radio | Ref
---|---|---|---|---|---|---
|  | (kpc/$\arcsec$) |  $(10^{14}M_{\odot}$) | morphology | morphology |
ACO 1413 | 0.143 | 2.51 | 5.98${}_{-0.40}^{+0.38}$ | nCC | MH | 1, 2
ACO 1795 | 0.062 | 1.20 | 4.54$\pm 0.21$ | CC | cMH | 2, 3
ACO 3444 | 0.254 | 3.96 | 7.6 ${}_{-0.6}^{+0.5}$ | CC | MH | 2, 4
MACS J1115.8+0129 | 0.350 | 4.94 | 6.4 $\pm 0.7$ | CC | MH | 2, 5
MACS J2140.2$-$2339 | 0.313 | 4.59 | 4.7* | CC | MH** | 5
Table 2: Observation details. Columns: cluster name, RA, Dec, observation date, observation time on-target, primary calibrator, secondary calibrator. Name | RA | DEC | Observation date | On target time | Primary | Secondary
---|---|---|---|---|---|---
| (h:m:s, J2000) | (d:m:s, J2000) |  | (minutes) | calibrator | calibrator
ACO 1413 | 11:55:19.40 | 23:24:26.0 | 10 August 2019 | 117 | J0408$-$6545 | J1120+1420
ACO 1795 | 13:48:55.00 | 26:36:01.0 | 16, 17 June 2019 | 240 | 3C286 | 3C286
ACO 3444 | 10:23:54.80 | -27:17:09.0 | 20, 26 July 2019 | 228 | J0408$-$6545 | J1051$-$2023
MACS J1115.8+0129 | 11:15:54.90 | 1:29:56.0 | 2 August 2019 | 93 | J0408$-$6545 | J1058+0133
MACS J2140.2$-$2339 | 21:40:15.20 | -23:39:40.0 | 30 June 2019 | 87 | J1939$-$6342 | J2152$-$2828

A sample of five relaxed galaxy clusters was selected from the literature and
observed by MeerKAT during its first Open Time Call in 2019 to detect and
characterise the central diffuse radio emission in each (proposal ID:
SCI-20190418-KA-01). Table 1 summarizes the physical properties of each
target. The clusters were chosen based on the classification of the core radio
emission detailed in the literature at the time of proposal submission, having
either a confirmed or candidate mini-halo, as well as the declination of the
host cluster. ACO 3444 was a newly confirmed mini-halo when Giacintucci et al.
(2019) studied archival VLA narrow L-band observations. However, because of
the MeerKAT’s greater sensitivity to diffuse emission due to its denser
coverage of short baselines, more diffuse emission may potentially be
detected. Additionally, this cluster lies at an optimal observing declination
for MeerKAT. ACO 1413, ACO 1795 and MACS J1115.8+0129 were the most southerly
candidate mini-halos, having previous radio observations of insufficient
quality to derive physical properties (see Govoni et al., 2009; Giacintucci et
al., 2014a; Pandey-Pommier et al., 2016, respectively); hence, new
observations were proposed utilising the MeerKAT’s angular resolution and
sensitivity to confirm their nature. However, the first and last of which have
since been confirmed by Savini et al. (2019) and Giovannini et al. (2020),
respectively. The last cluster in the sample, MACS J2140.2-2339, had no
previous radio observations and was classified by Cavagnolo et al. (2009) to
be a cool-core, massive cluster. Since Giacintucci et al. (2017) studied a
mass-limited sample of galaxy clusters and concluded that most of the massive,
cool-core clusters possessed a mini-halo, this last cluster with its ideal
southerly declination was identified to potentially be a new MeerKAT-mini-halo
detection. The sample possesses a large declination range; two are near
$\pm$25∘ each - the negative being near the ideal observing range of the
MeerKAT and the positive being near that of its northern limit - and one near
the celestial equator. This large declination range lends to the possibility
of future synergy between MeerKAT/SKA-Mid and LOFAR in mini-halo studies (ACO
1413 had already been observed by LOFAR in Savini et al., 2019, and Riseley et
al. in prep aims to provide an indepth MeerKAT+LOFAR study of the diffuse
radio emission in this very interesting cluster). The sample varies in
redshift from 0.06 to 0.35 and has mass $\gtrsim$ $5\times 10^{14}M_{\odot}$.

##  3 Data Processing

All data were downloaded from the SARAO archive222https://archive.sarao.ac.za
\- all specialized software used in the data reduction is listed in Table 5 of
Appendix B. The observations were conducted with full 4096 channelization, 856
MHz total bandwidth centred at 1283 MHz (L-band), and an 8 second dump rate -
except for MACS J1115.8+0129 which had a dump rate of 2 seconds. Table 2
summarizes the observation details. Primary calibrators were visited once for
10 minutes at the start of each observing block and were used for bandpass
calibration and fluxscaling. Secondary calibrators and the target sources were
then visited alternately for 2 and 15 minutes, respectively, where the
secondary calibrator was used for delay and complex-gain calibration.

The KATDAL (KAT Data Access Library) package was used during download to
convert from MeerKAT Visibility Format (MVF) to the standard CASA (McMullin et
al., 2007) Measurement Set (MS) while applying flags generated by the
telescope’s control and monitoring system. A known issue with early MeerKAT
observations is that of poor calibrator
positions333https://skaafrica.atlassian.net/wiki/spaces/ESDKB/pages/1481015302/Astrometry.
The sources J0408$-$6545 and J1939$-$6342 needed to be corrected by
0.52$\arcsec$ and 2.00$\arcsec$, respectively, which was done using CASA’s
fixvis task. Subsequent flagging, 1GC and 2GC were automated using the CARACal
pipeline (Józsa et al., 2020). First, the pipeline copies all sources into
separate ‘calibrators’ and ‘target’ MS files. The calibrator data was flagged
for the known problematic ranges of the spectral window, autocorrelations and
any shadowed antennas using CASA’s flagdata task. Thereafter, Tricolour (Hugo
et al., 2022) \- an automated flagging software specifically developed for
MeerKAT data - was used to flag persistent RFI with a particularly strict
default flagging strategy from the CARACal MeerKAT
files444https://github.com/caracal-
pipeline/caracal/blob/master/caracal/data/meerkat_files/stalin.yaml to ensure
no erroneous data were carried into calibration. In order to improve image DR
and the detection of faint diffuse emission, we perform 3GC on all data. Below
we briefly describe our calibration process.

###  3.1 Cross-calibration (1GC)

The calibrator sources were used to derive corrections for various
instrumental and propagation effects that corrupt the astrophysical signals
(such as antenna gain fluctuations and atmospheric conditions). These
corrections are applied to the target to mitigate the observational errors,
which is known as cross-calibration. Firstly, CARACal modelled the primary
calibrators using the MeerKAT local sky models (lsm) of each. In the case of
3C2865553C286 does not have a MeerKAT lsm yet, we used CASA’s setjy,
externally from CARACal, to set the model using the polarisation information
from Perley & Butler (2017) and the updated polarimetry properties from Table
7.2.7 from the NRAO website
666https://science.nrao.edu/facilities/vla/docs/manuals/obsguide/modes/pol.
The pipeline then derived delay, complex-gain and bandpass amplitude- and
phase-corrections using CASA’s gaincal and bandpass tasks. The bandpass
solutions from the primary calibrator were applied to the secondary calibrator
on-the-fly while its delay and complex-gain corrections were calculated. The
derivation was repeated after a mild automated flagging. The final secondary
gain solutions were scaled to that of the primary using CASA’s fluxscale task.
The complex-bandpass solutions of the primary calibrator and the delay and
flux-scaled complex-gain solutions of the secondary were applied to the target
data which was then frequency-averaged over five channels to save on storage
space and future processing time. Finally, this cross-calibrated target data
was flagged similarly to the calibrator data but with a slightly more relaxed
flagging strategy777https://github.com/caracal-
pipeline/caracal/blob/master/caracal/data/meerkat_files/gorbachev.yaml.

###  3.2 Direction-independent self-calibration (2GC)

2GC is the process of using the target data itself to improve the instrumental
complex-gain corrections derived earlier and hence improve image DR. We used
the CARACal defaults of WSClean (Offringa et al., 2014) for imaging and
CubiCal (Kenyon et al., 2018) to calculate self-calibration solutions. Three
rounds of phase-only self-calibration were performed. In each iteration, we
imaged a 2.5∘ squared area spanning 6000 pixels (scale 1.5$\arcsec$/pixel)
using multi-frequency synthesis (MFS, Conway et al., 1990; Sault & Conway,
1999), multi-scale cleaning (Offringa & Smirnov, 2017), robust 0 weighting
(Briggs, 1995), a 2nd-order spectral polynomial fit, applying the built-in
WSClean automasking routine (halving the masking threshold in each iteration,
finishing on 3$\sigma$) and terminating the deconvolution after one million
minor iterations. After each imaging round, we fed the model data into CubiCal
to calculate phase-only self-calibrated corrected data. We solved for the gain
solutions in 30 equal time chunks per calibration round over the entire
bandwidth while flagging visibilities per each baseline and correlation using
a median absolute residual (MAD) filter. Whilst the global image noise (root
mean square of the residual image) steadily decreased, the reduction of severe
artefacts from the 2nd to 3rd rounds of self-calibration was minimal; thus,
2GC was stopped at this point.

###  3.3 Direction-dependent self-calibration (3GC)

3GC is the process of correcting the DD terms in the Radio Interferometer
Measurement Equation (RIME) caused by DDEs (see for example Noordam & Smirnov,
2010; Smirnov, 2011a, b, c, d; Smirnov & Tasse, 2015, for comprehensive
discussions on generic DDEs). In the case of MeerKAT, the major effects are
that of primary beam rotation and pointing errors. These manifest as radial
artefacts around bright sources that lie towards the half-power point (flanks
or sidelobes) of the telescope’s primary
beam888https://skaafrica.atlassian.net/wiki/spaces/ESDKB/pages/1484128294/Dynamic+range+considerations.
Instead of considering these sources on a case-by-case basis through the
differential gain calibration or ‘source peeling’ of CubiCal, we chose to
calibrate over the entire field via the facet-based approach of killMS (kMS,
Tasse, 2014a, b) and DDFacet (DDF, Tasse et al., 2018). Riseley et al. (2022)
applied a similar approach to MeerKAT L-band observations of the MS
1455.0+2232 mini-halo. Additionally, we used the holographic MeerKAT L-band
primary beam models generated via the eidos package to correct for the primary
beam attenuation and rotation (Asad et al., 2021; de Villiers & Cotton, 2022).
The kMS software performs 3GC on a tessellated DI image by exploiting
Wirtinger complex differentiation (Tasse, 2014b) to directly estimate the
physical DD RIME terms. We used its non-linear Kalman Filter algorithm (KAFCA,
Tasse, 2014a) to solve for stable and well-conditioned estimates within a 5
minute time and 10 channel frequency solution interval in as many directions
as there were tessellation tiles. DDF is a facet-based imager that can take
into account kMS solutions to perform wide-band, wide-field DD spectral
deconvolution in the apparent and intrinsic (primary beam corrected) flux
scale. Note that we describe all images in which primary beam corrections were
performed as being intrinsic.

The image tessellation mentioned above was determined using the MakeModel.py
script999https://github.com/saopicc/DDFacet/tree/master/SkyModel of the DDF
software. Since 3GC is very sensitive to the signal-to-noise (SNR) of the
data, we experimented with the number of tessellation tiles in each cluster
case. We found that 4-6 tiles each centred on one of the brightest field
sources responsible for radial artefacts gave a good improvement in image
noise after 3GC. Once the tessellation pattern was defined, the 3GC procedure
was analogous to that of 2GC (i.e. imaging a rigorous model then deriving the
calibration solutions and applying them in a further imaging round).

DDF offers a deconvolution algorithm called subspace deconvolution (SSD),
which was shown in Tasse et al. (2018) to work well in conjuction with kMS to
produce 3GC intrinsic images. SSD jointly and independently deconvolves
subsets/islands of pixels (defined using a mask) very efficiently, making it
excellent for studying extended emission in large images. We used the SSD2
deconvolver on default settings (unless otherwise stated) throughout the 3GC
imaging process - which simply adds to the original functionality to choose
the order of the spectral polynomial fitted and a few other deconvolution
parameters.

To produce the deepest DR images possible, imaging was split into an
automasked initial deconvolution from which a rigorous mask was externally
generated and applied in a direct continuation. Each initial deconvolution ran
for two major cycles with a 5$\sigma$ automasking threshold, while each
subsequent externally masked deconvolution ran for a single major cycle. After
each deconvolution, an updated mask was generated with the tightest possible
threshold (visually inspected to ensure only real sources were masked) and
applied in the next step. The breizorro tool was used to generate all external
masks, as it estimates a local noise rms so that a deeper mask can be achieved
on smaller scales. The external masks had a threshold of 6-8$\sigma$ depending
on the field, which could then be made even tighter after the 3GC solutions
were applied. All images were produced using MFS with 10 sub-bands, robust 0
weighting, fitting a 4th-order spectral polynomial, superimposing a 20x20
square facet grid onto the tessellation pattern and re-evaluating the primary
beam correction every 5 minutes in each facet to produce both apparent and
intrinsic images throughout the process.

The basic procedure is summarized as follows:

  1. 1.

Tessellate the field;

  2. 2.

Initial deconvolution;

  3. 3.

Deeper deconvolution;

  4. 4.

Derive kMS DD gain solutions;

  5. 5.

Apply kMS solutions in deconvolution;

  6. 6.

Final deconvolution.

###  3.4 Source subtraction

In most cases, the central BCG - which may have a complex morphology with
jets/lobes - blends with the diffuse mini-halo emission. These sources need to
be removed in order to obtain an uncontaminated and unbiased detection of the
underlying mini-halo. Most modern mini-halo studies choose to perform
visibility-plane subtraction (for example Biava et al., 2021; Riseley et al.,
2022). This involves subtracting the modelled compact sources - which can be
generated from a high-resolution (HR) image - directly from the visibilities
and then re-imaging the residual data at the original resolution, thereby
analysing the mini-halo without bias from the embedded sources. This approach
introduces an additional uncertainty component into the measurement-taking
process (Equation 1), which scales with the size and strength of the
subtracted sources. We performed visibility-plane subtraction on the final 2GC
and 3GC images produced in Section 3.3.

All imaging, deconvolution and masking parameters outlined in Section 3.3 were
followed for the subtraction, except where otherwise stated. The kMS solutions
derived in Step 4 of Section 3.3 were applied throughout the subtraction
procedure. HR deconvolutions were performed at robust -2 while applying a uv-
cut to model only the compact emission present in the field, giving images at
a $\sim$4$\arcsec$ resolution. The uv-cut was chosen to be the minimum size of
a mini-halo (100 kpc at the cluster redshift, as per the definition adopted
from Giacintucci et al., 2017) and specified as a hard cut in the visibility
space using the \--Selection-UVRangeKm parameter in DDF. All sources expected
to be subtracted were included in the mask derived from the initial HR
deconvolution. We used a generic 4$\sigma$ threshold for all clusters to mask
the field sources and manually inserted masked pixels that corresponded to
confirmed point-like sources in the immediate vicinity of the mini-halo that
were not picked up by the generic mask; a generic threshold any tighter than
that stated above caused significant artefacts to be included in the mask. It
is expected that any emission too faint to be captured by the HR mask is left
unsubtracted. The deep HR model derived from the deeper deconvolution needs to
be (re)-predicted in DDF with \--Output-Mode Predict while specifying the same
uv-cut that will be used in the imaging of the diffuse emission. This ensures
that the shorter baselines that were previously cut are included into the
calculation of the subtraction model, i.e. so that all spatial scales are
considered during every step of modelling, subtraction and (re)-imaging. Once
this final model was set, we used the msutils method of Stimela (Makhathini,
2018) to perform the subtraction operation. The resulting column was then
imaged at robust 0, giving an $\sim$8$\arcsec$ resolution. External masks were
thereafter generated using the same thresholds from Step 2 of Section 3.3,
ultimately giving the final source-subtracted (SRC-SUB) 2GC/3GC images.

The basic procedure is summarized below:

  1. 1.

Initial deconvolution at HR;

  2. 2.

HR masking;

  3. 3.

Deeper deconvolution;

  4. 4.

Re-predict HR model;

  5. 5.

Subtraction;

  6. 6.

Initial deconvolution at standard resolution;

  7. 7.

Final deconvolution.

The Aladin Sky Atlas (Bonnarel et al., 2000; Boch & Fernique, 2014) was used
to overlay varies catalogues on the images to identify individual sources when
needed. The only differences between the 2GC and 3GC subtraction were the
implementation of the kMS solutions and the masking thresholds.

###  3.5 Spectral imaging

There are no spectral index maps available for the mini-halos in our sample,
and so we generate them for the first time with this MeerKAT data. We used the
sub-band images to analyse both the unsubtracted and SRC-SUB in-band spectral
properties of each mini-halo. To do this, we repeated the 3GC and source
subtraction procedures from Step 5 of Section 3.3 with three equal sub-bands,
centred at 998, 1283 and 1569 MHz, and fitting a 3rd-order spectral
polynomial. For the robust 0 images, to ensure the same uv-range was imaged
across all sub-bands, we applied an inner and outer taper of $\sim$5$\arcmin$
and $\sim$14$\arcsec$, respectively. We then convolved each sub-band image to
a common 15$\arcsec$ circular beam, except for the SRC-SUB ACO 1413 images. In
this case, significant oversubtraction of the embedded sources occurred in the
high sub-band image and so, to salvage a detection, we convolved to
30$\arcsec$. Similarly, for the HR images, to determine in-band spectral
properties of the embedded sources, we applied an inner taper corresponding to
the uv-cut and an outer taper $\sim$15$\arcsec$. The resulting sub-band images
were convolved to a 7$\arcsec$ resolution; except for ACO 1413, which were
convolved to a 12$\arcsec$ because of its high northern declination and poorer
spatial resolution, particularly in the low-band image. We used brats (Harwood
et al., 2013; Harwood et al., 2015) to make the in-band spectral index maps,
with a $5\sigma$ cutoff and a 10% calibration uncertainty.

###  3.6 ACO 1795

Strong artefacts centred around the cluster BCG limited image quality and
prevented us from detecting any nearby diffuse emission (see Figure 6 in
Appendix C). This BCG is known to be particularly strong; $917\pm 46$ mJy from
its FIRST image (Giacintucci et al., 2014a) and $975\pm 97.5$ mJy from our
best 2GC and 3GC images (left and right panels of Figure 6). Its strong
sidelobes could not be sufficiently corrected by the calibration methods used
for the rest of the clusters.

To improve the accuracy of the 2GC calibration solutions described in Section
, we used QuartiCal (CubiCal’s successor, currently in public beta release) to
chain together Jones terms during individual rounds of DI self-calibration. In
each round, for three rounds, we simultaneously derived and applied a delay
solution and (per channel) complex-gain solutions, where the latter was
refined down to 60 equal time intervals by the third round. For more
controlled cleaning, since we noticed some artefacts appearing in the CARACal
model images, we performed manual masking with breizorro similarly to that
described in Section 3.3, again finishing on 3$\sigma$. We found significant
improvements over the CARACal images but still no diffuse emission could be
detected. Furthermore, no improvements were ever found after 3GC. We suspect
polarisation leakage effects compounded by the poor uv-sampling distribution
and asymmetric PSF due to the target’s high northern declination
($+26^{\circ}$) to be the cause of the residual BCG sidelobe modelling errors.
The calibration may have improved if standard primary and secondary
calibrators were observed together with 3C286. Further investigation is
required to determine if synergy between MeerKAT/SKA-Mid and LOFAR is possible
for this cluster. We exclude ACO 1795 from all further discussions. Below we
describe the results for the remaining clusters.

##  4 Results

This section presents the mini-halo cluster images and associated statistical
and physical properties. We give qualitative statistics and depictions (Table
6 and Figures 7-10 in Appendix D) of the improvements in image quality after
using the 3GC procedure outlined in Section 3.3. In Figures 1-4, for each
mini-halo, we show: (i) a central cutout of our standard resolution (robust 0,
$\sim$8$\arcsec$) images before and after point source subtraction in panels
a) and b), (ii) a lower resolution (LR, $15\arcsec$) image of b) in panel c)
from which we derive flux density and size measurements, (iii) an in-band
spectral index and associated uncertainty map of the unsubtracted mini-halo in
panels d) and e). Lastly, we give an in-band integrated spectrum of each SRC-
SUB mini-halo, shown in Figure 5.

Table 3: Source measurements. Columns: cluster name, source, flux density at
1.28 GHz, average mini-halo diameter (to the nearest kpc), largest linear size
(to the nearest kpc), spectral index, k-corrected radio power at 1.4 GHz.
∗ Sub-band maps convolved to 30$\arcsec$.
∗∗ Spectral index of the unresolved source S1+S2.
∗∗∗ A large subtraction uncertainty in the flux density measurements due to the strong AGN causes a large uncertainty in the integrated spectrum. Cluster | Source |  $S_{\rm 1.28\,GHz}$ (mJy) |  $D_{\rm MH}$ (kpc) |  LLS (kpc) | $\alpha$ |  $P_{\rm 1.4\,GHz}$ ($\times 10^{24}$ W/Hz)
---|---|---|---|---|---|---
ACO 1413 | Mini-halo | $2.05\pm 0.27$ | 185 | 211 | $1.52\pm 0.46^{*}$ | $0.104\pm 0.014$
BCG (S1) | $0.37\pm 0.05$ |  |  | $0.89\pm 0.26^{**}$ | $0.018\pm 0.003$
S2 | $2.86\pm 0.29$ |  |  | $0.89\pm 0.26^{**}$ | $0.143\pm 0.015$
ACO 3444 | Mini-halo | $12.10\pm 1.71$ | 372 | 412 | $1.53\pm 0.44$ | $2.40\pm 0.40$
BCG | $2.25\pm 0.24$ |  |  | $0.61\pm 0.27$ | $0.38\pm 0.04$
MACS J1115.8+0129 | Mini-halo | $7.91\pm 2.59$ | 375 | 499 | $1.00\pm 1.10^{***}$ | $3.00\pm 1.20$
BCG | $8.73\pm 0.88$ |  |  | $0.78\pm 0.26$ | $3.15\pm 0.40$
MACS J2140.2-2339 | Mini-halo | $2.61\pm 0.31$ | 296 | 390 | $1.21\pm 0.36$ | $0.79\pm 0.11$
BCG | $1.43\pm 0.15$ |  |  | $0.72\pm 0.31$ | $0.39\pm 0.05$

Table 6 shows that all images improved after 3GC, evident by the reduction in
global noise levels and greater DR values. We define $DR^{(1)}$ and $DR^{(2)}$
as the maximum pixel value of the unsubtracted image divided by, respectively,
its global image rms and the modulus of its minimum pixel value. For the
sample, the overall improvement is shown by the global noise and $DR^{(1)}$
values improving by an average of 7% each, and the factor of three improvement
in the $DR^{(2)}$ values showing specifically that the kMS solutions well
characterised the deep negative DI point spread function (PSF) sidelobes. The
3GC HR images experienced a similar improvement in $DR^{(2)}$, but also a
slight noise amplification in some facets covering the first negative sidelobe
of the primary beam causing the global noise to remain near that of the 2GC
images. Further investigation into what causes this effect is needed;
nevertheless, our science case was not affected. A handful of weak PyBDSF-
detected (Mohan & Rafferty, 2015) sources (< 5 mJy) in the 2GC fields
experienced flux suppression, decreasing by up to an order of magnitude and
resulting in an average flux density difference between 2GC and 3GC of 8% -
further investigation into these outliers is needed. Nevertheless, this
difference did not affect our science case as the measured 2GC and 3GC mini-
halo flux densities were well consistent within the measured uncertainties. We
also note that the 2GC image of MACS J2140.2-2339 was largely absent of any
artefacts; therefore, 3GC was not expected to significantly improve this
field.

The images at the native $\sim$8$\arcsec$ resolution show the detailed
structure of the mini-halos; however, to enhance the surface brightness
sensitivity we convolved these to a low-resolution (LR) $15\arcsec$ circular
beam, as shown in panel c) of Figures 1-4. From these LR images, we measure
the flux density, radio power and size within the $3\sigma$ contours (see
Table 3). We followed Giacintucci et al. (2014a) in calculating the
uncertainty on the flux density measurements as the quadrature sum of the
systematic, statistical and subtraction uncertainties. The latter is given by:

$\sigma_{\rm sub}^{2}=\sum_{\mathclap{s=1}}^{N}\left(I_{\rm MH,\,s}\times
N_{\rm beam,\,s}\right)^{2}$ (1)

where $I_{\rm MH,\,s}$ is the mean residual surface brightness of the mini-
halo within the $s^{\rm th}$ source region, and $N_{\rm beam,\,s}$ is the
number of beams within that region. Combining these uncertainties gives:

$\sigma_{\rm MH}=\sqrt{\left(\sigma_{\rm cal}S_{\rm
MH}\right)^{2}+\left(rms\sqrt{N_{\rm beam}}\right)^{2}+\sigma_{\rm sub}^{2}}$
(2)

where $\sigma_{\rm cal}$ is the percentage calibration uncertainty, $S_{\rm
MH}$ is the measured mini-halo flux density, rms is the local noise given in
each the caption of each respective image and $N_{\rm beam}$ is the number of
beams contained within the mini-halo region. We found an average calibration
uncertainty of 10% across our sample after cross-matching and comparing the
3GC flux densities of the PyBDSF-detected sources > 1 mJy within the primary
beam of the MeerKAT images, scaled using a spectral index of 0.7, to those
from the NVSS catalogue (Condon et al., 1998). Since mini-halos usually have
non-spherical morphologies, we followed Cassano et al. (2007) in measuring the
average diameter as

$D_{\rm MH}=\sqrt{D_{\rm min}\times D_{\rm max}}$ (3)

where $D_{\rm min}$ and $D_{\rm max}$ are the minimum and maximum diameters of
the $3\sigma$ contours. Note that $D_{\rm max}$ is the same as the largest
linear size (LLS). We measure the BCG flux and associated uncertainty in a
similar manner but derived from the HR images, and are listed in Table 3 as
well.

During the spectral analysis, we measured the flux densities within the
$5\sigma$ contours of each sub-band image. We fit these to a power-law least
squares regression across the central frequencies to produce an in-band
integrated spectrum for each SRC-SUB mini-halo and BCG - Table 3 shows the
fitted slopes. Note that we choose to display the unsubtracted spectral index
maps while we use the SRC-SUB integrated spectrum to describe the average
mini-halo spectral index. This is because characterizing the spectral effects
of our subtraction method on the residual diffuse emission is beyond the scope
of this work. The resulting large spectral uncertainties (which are absolute
in nature) limits our discussion on the science extracted from the spectral
analyses.

##  5 Discussion

The 3GC procedure discussed in Section 3.3 improved image quality in each case
by reducing artefacts and thereby lowering the background rms noise level,
allowing for more significant emission to be detected. We then described the
2GC/3GC visibility-plane point source subtraction and applied it to our
cluster sample in Section 3.4 to image the central mini-halos without bias
from the surface brightness of the BCG. Section 4 gave the resulting
statistics that evaluated the reduction and the physical quantities describing
the mini-halo systems. This section discusses these result in the context of
previous literature findings. We used
SIMBAD101010http://simbad.cds.unistra.fr/simbad/ to identify individual
sources of interest unless otherwise stated.

It is difficult to quantify the effectiveness of a subtraction procedure. In
our case, some strong sources were oversubtracted while many more fainter
sources were undersubtracted, seen in Figures 1-4 by the residual positive and
negative surface brightness regions. Overall, the subtraction induced minimal
error, increasing the noise of the SRC-SUB images by an average of 4%. We
notice that for most of our sample, before and after point source subtraction,
the mini-halo orientation roughly follows that of the BCG. The role of the AGN
in mini-halo formation is currently still uncertain; however, strong
correlations between mini-halo radio power and BCG/X-ray cavity power suggest
a clear physical connection (Richard-Laferrière et al., 2020). Further high
resolution and sensitive observations, similar to those presented in Ignesti
et al. (2022); Perrott et al. (2023); Bégin et al. (2023), are required to
gain a deeper understanding of the connection between the embedded radio
galaxy and mini-halo and their orientation as we found in our study.

Figure 1: ACO 1413 3GC mini-halo. a) beam size
$(12.2\arcsec,5.9\arcsec,-4.5^{\circ})$, local rms $1\sigma=$ 11.2
$\mu$Jy/beam, S1 marks the BCG, S2 and a small blue cross indicate other
projected galaxies. b) SRC-SUB image, beam size and color-scale same as a),
local rms $1\sigma=$ 12.2 $\mu$Jy/beam. c) SRC-SUB 15$\arcsec$ LR image, local
rms $1\sigma=25.1$ $\mu$Jy/beam. d) unsubtracted spectral index map and e)
uncertainty map. The beam is shown in cyan in the bottom left corner of the
images. Dashed blue contours show the $2\sigma$ level and the solid blue
contours start at $3\sigma$ and increase by a factor of 2. Dashed black
contours show the $-3\sigma$ level. White contours show the emission from the
HR image, which has beam size $(7.6\arcsec,3.1\arcsec,-2.6^{\circ})$ and local
rms $1\sigma=$ 23.0 $\mu$Jy/beam. Green contours in d) show the surface
brightness of the SRC-SUB mid-band image and start at $5\sigma$ and increase
by a factor of 2, where $1\sigma=25.7$ $\mu$Jy/beam. The white plus indicates
the BCG position.

###  5.1 ACO 1413

Giacintucci et al. (2017) classified ACO 1413 as a (modest) non-cool-core
cluster using Chandra data (Obs. ID 5002). Govoni et al. (2009) first showed
the presence of a candidate mini-halo in ACO 1413 using VLA 1.4 GHz
C-configuration data (maximum baseline 3.4 km with L-band resolution
14$\arcsec$), making it the first and only non-cool-core cluster to host a
candidate mini-halo. Savini et al. (2019) later used a 144 MHz LOFAR
observation to confirm its detection.

Our MeerKAT wide-FoV images of ACO 1413, shown in Figure 7 of Appendix D, give
a depiction of the improvement from 2GC to 3GC imaging. The DI image displays
a significant snowflake-like artefact around a bright double radio source
located at (RA, DEC) = (11:52:26.5, 23:13:47.3) and (11:52:26.2, 23:12:54.6),
designated as NVSS J115226+231347 and NVSS J115226+231255, with flux densities
measured from the final 3GC intrinsic images $\sim$260 mJy and $\sim$175 mJy,
respectively. The sources are located at a distance of $\sim$40$\arcmin$ from
the phase centre (just outside the main lobe of the primary beam) and create
significant artefacts affecting the image quality. These artefacts are
dramatically reduced after 3GC, with much of the snowflake spikes going down
to noise level, drastically improving image quality. The global rms noise
improved by 11% from 8.3 $\mu$Jy/beam to 7.4 $\mu$Jy/beam. To depict how
valuable 3GC can be if a source of interest is affected by such artefacts, we
show a zoom of an FR-II galaxy located $\sim$33$\arcmin$ from the phase centre
in the insets of Figure 7. In this comparison, the 2GC image displays a
snowflake-spike piercing the radio lobes and nucleus; however, the 3GC image
shows these spikes disappeared, giving a clear depiction of the source
morphology. The local rms near this source improves by just under a factor of
two, from 22 $\mu$Jy/beam to 12 $\mu$Jy/beam after 3GC.

Figure 2: ACO 3444 3GC mini-halo. a) beam size
$(6.8\arcsec,5.9\arcsec,-16.0^{\circ})$, local rms $1\sigma=8.5$ $\mu$Jy/beam,
S1 marks the BCG, blue crosses indicate individual sources that were manually
masked at HR. b) SRC-SUB image, beam size and color-scale same as a), local
rms $1\sigma=9.3$ $\mu$Jy/beam. c) SRC-SUB 15$\arcsec$ LR image, local rms
$1\sigma=27.8$ $\mu$Jy/beam. d) unsubtracted spectral index map and e)
uncertainty map. The beam is shown in cyan in the bottom left corner of the
images. Blue contours start at $3\sigma$ and increase by a factor of 2. Dashed
black contours show the $3\sigma$ level. White contours show the emission from
the HR image, which has beam size $(3.9\arcsec,3.2\arcsec,-9.6^{\circ})$ and
local rms $1\sigma=$ 20.9 $\mu$Jy/beam. Green contours in d) show the surface
brightness of the SRC-SUB mid-band image and start at $5\sigma$ and increase
by a factor of 2, where $1\sigma=34.6$ $\mu$Jy/beam. The white plus indicates
the BCG position.

Panels a) and b) of Figure 1 show a central cutout comparing the 3GC vs 3GC
SRC-SUB images for this cluster. Savini et al. (2019), in their Figure 6,
first noted that there are in fact two sources embedded within this mini-halo,
and these are clearly visible in this image; S1 being the BCG with optical
counterpart MCG+04-28-097 and S2 being the radio source FIRST
J115518.6+232422. A localised peak south-west of these sources is also
visible, marked with a small blue cross in Figure 1a, identified as galaxy
2MASS J11551712+2323527 and is blended with the mini-halo. The morphology is
similar to that shown in the unsubtracted image of Savini et al. (2019). The
result of the subtraction, Figure 1b, shows only the contribution of diffuse
emission in the field. The mini-halo is visible in this image and is shown to
have an irregular morphology. It is extended from north to south, similar to
the BCG orientation, with the north having fainter filamentary-like emission
that is only detected at the $2\sigma$ level.

The 3GC SRC-SUB LR image with circular beam size of 15$\arcsec$ is shown in
pancel c) of Figure 1. Once smoothed, the morphology is comparable to the
corresponding SRC-SUB image in Savini et al. (2019), even though our image is
at a slightly higher resolution. From this image, we determine a flux density
of $S_{\rm 1.28\,GHz}=2.05\pm 0.27$ mJy for this mini-halo, and an average
diameter of 185 kpc with LLS of 211 kpc. These values are consistent with
those presented in Govoni et al. (2009) and Savini et al. (2019) when scaled
with the measured in-band spectral index of $1.52\pm 0.46$ (see below).
Additionally, the k-corrected radio power scaled to 1.4 GHz is $P_{\rm
1.4\,GHz}=(0.104\pm 0.014)\times 10^{24}$ W/Hz. This is the faintest mini-halo
in our sample.

Figure 3: MACS J1115.8+0129 3GC mini-halo. a) beam size
$(8.5\arcsec,6.5\arcsec,-16.2^{\circ})$, local rms $1\sigma=10.2$
$\mu$Jy/beam, S1 marks the BCG. b) SRC-SUB image, beam size same and color-
scale as a), local rms $1\sigma=10.9$ $\mu$Jy/beam. c) SRC-SUB 15$\arcsec$ LR
image, local rms $1\sigma=24.6$ $\mu$Jy/beam. d) unsubtracted spectral index
map and e) uncertainty map. The beam is shown in cyan in the bottom left
corner of the images. Blue contours start at $+3\sigma$ and increase by a
factor of 2. Dashed black contours show the $3\sigma$ level. White contours
show the sources from the HR image, which has beam size
$(4.5\arcsec,4.0\arcsec,-39.3^{\circ})$ and local rms $1\sigma=$ 28.6
$\mu$Jy/beam. Green contours in d) show the surface brightness of the SRC-SUB
mid-band image and start at $5\sigma$ and increase by a factor of 2, where
$1\sigma=31.8$ $\mu$Jy/beam. The white plus indicates the BCG position.

The spectral analysis of this mini-halo is limited in both sensitivity and
resolution due to the cluster’s high declination and inconsistent subtraction
across the sub-bands. The high-band SRC-SUB image at native resolution showed
at the location of source S2 a $\sim$50 kpc region of negative surface
brightness where the low- and mid-band images had non-negative brightness. To
salvage a measurement of the in-band spectrum, we convolve the subtracted
images to 30$\arcsec$ and compute the flux density inside the $3\sigma$
contours. We ultimately obtain an average spectral index of $\alpha^{\rm
1.5\,GHz}_{\rm 1\,GHz}=1.52\pm 0.46$, which is consistent with the value of
1.3 given in Savini et al. (2019). When imaging the HR sub-band images, the
low-band image had a very poor resolution and so could not reliably measure
the index of each embedded source, S1 and S2. These sources blended to give
one unresolved source at a 12$\arcsec$ resolution and thus the spectral index
quoted in Table 3 of $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=0.89\pm 0.26$ is of
this combined source. The spectral index and uncertainty maps of the
unsubtracted mini-halo, which could remain at a 15$\arcsec$ resolution, are
shown in panels d) and e) of Figure 1. Sources S1 and S2 obscure the view of
the underlying mini-halo spectral distribution, which is evident for all our
spectral index maps. Nevertheless, the spectral index of the entire radio core
region is reliable within a central $\sim$100 kpc diameter and, outside the
embedded source regions, a possible gradient from north to south is visible,
with a flatter spectrum of $\sim$0.4 in the north and a steeper spectrum of
$\sim$1.3 in the south. This southern region may be a glimpse of the
underlying mini-halo as it is not contaminated by any embedded sources and is
roughly consistent with the integrated spectrum. The blended infrared source
mentioned above causes the flatter spectrum visible in the south-west.

###  5.2 ACO 3444

Venturi et al. (2007) reported a candidate mini-halo in ACO 3444 based on a
610 MHz GMRT observation. Giacintucci et al. (2017, 2019) provided
confirmation through the analysis of archival VLA 1.4 GHz DnC+BnA
configuration data, with images restored to 5$\arcsec$ and 11$\arcsec$.

Figure 4: MACS J2140.2-2339 3GC mini-halo. a) beam size
$(7.8\arcsec,6.2\arcsec,-8.6^{\circ})$, local rms $1\sigma=8.0$ $\mu$Jy/beam,
S1 marks the BCG. b) SRC-SUB image, beam size same and color-scale as a),
local rms $1\sigma=8.7$ $\mu$Jy/beam. c) SRC-SUB 15$\arcsec$ LR image, local
rms $1\sigma=20.4$ $\mu$Jy/beam. d) unsubtracted spectral index map and e)
uncertainty map. The beam is shown in cyan in the bottom left corner of the
images. Blue contours start at $+3\sigma$ and increase by a factor of 2.
Dashed black contours show the $3\sigma$ level. White contours show the
sources from the HR image, which has beam size
$(4.3\arcsec,3.2\arcsec,-7.1^{\circ})$ and local rms $1\sigma=$ 20.0
$\mu$Jy/beam. Green contours in d) show the surface brightness of the SRC-SUB
mid-band image and start at $5\sigma$ and increase by a factor of 2, where
$1\sigma=22.8$ $\mu$Jy/beam. The white plus indicates the BCG position.

Figure 8 shows a comparison between the 2GC and 3GC images for this cluster.
Significant concentric artefacts that affect the entire north-central region
of the image originate from a set of three bright point-like sources located
$\sim$30$\arcmin$ from the phase centre. These sources are positioned from
left to right as seen in Figure 8 at (RA, DEC) = (10:22:29.4, -26:47:34.5),
(10:23:27.1, -26:50:09.7), (10:24:07.4, -26:44:17.5), designated as WISEA
J102407.38-264417.8 (NED), NVSS J102327-265009 and NVSS J102229-264734,
respectively, and have a maximum flux density measured from the 3GC intrinsic
image to be $\sim$270 mJy. The PSF sidelobes and further artefacts have been
reduced by the kMS gain solutions and are now almost at the level of the
background noise. The global noise rms improved by 8% from 6.4 $\mu$Jy/beam to
5.9 $\mu$Jy/beam. The inset of Figure 8 shows a zoom of the third
contaminating source mentioned above; its local rms improved by just over a
factor of two, from $\sim$52 $\mu$Jy/beam to $\sim$24 $\mu$Jy/beam.

Figures 2a, b show the comparison between the 3GC vs 3GC SRC-SUB images. The
embedded BCG (marked S1, optical counterpart 2MASX J10235019-2715232) and
mini-halo are clearly visible. Also visible are five compact sources on the
eastern edge of the mini-halo that obscure its boundary, identified from NED
to be individual infrared sources and indicated in Figure 2a by small blue
crosses. These sources had to be manually added to the HR mask. To directly
compare our MeerKAT SRC-SUB image to the VLA image shown in Figure 5b of
Giacintucci et al. (2019), which had twice as much on-target exposure time, we
restored it to the same 11$\arcsec$ beam size and found that our image has a
local rms 2.6 times deeper and shows the mini-halo extended further south by
$\sim$100 kpc. This detection shows the ability of the MeerKAT to detect, with
shorter tracks, more low surface brightness L-band emission than what was
achieved with the VLA. The extension is partially seen in Figure A4 of Venturi
et al. (2007), suggesting that it may have a particularly steep spectrum.

Figure 2c shows the LR 3GC SRC-SUB image. Residual diffuse emission,
coincident with the northernmost infrared source, blends with the mini-halo
but its contribution to the determined size and flux density is negligible. We
found a flux density of $S_{\rm 1.28\,GHz}=12.10\pm 1.71$, and an average
diameter of 372 kpc with LLS of 412 kpc. The flux density is consistent with
that presented in Giacintucci et al. (2019); however, the extension adds
another $\sim$130 kpc onto the reported diameter and $\sim$60 kpc onto the
Venturi et al. (2007) LLS. Even though a larger size is observed, a similar
flux density is expected because of the extension’s low surface brightness.
Using the measured in-band spectral index (see below), the k-corrected radio
power is $P_{\rm 1.4\,GHz}=(2.40\pm 0.40)\times 10^{24}$ W/Hz, consistent with
Giacintucci et al. (2019). However, the BCG flux density is significantly
higher than that reported in Giacintucci et al. (2019).

In the spectral analysis of ACO 3444, the southern extension progressively
recedes in the higher parts of the band - also suggesting a steep spectrum.
The remaining SRC-SUB mini-halo gives a spectral index $\alpha^{\rm
1.5\,GHz}_{\rm 1\,GHz}=1.53\pm 0.44$. The BCG has $\alpha^{\rm 1.5\,GHz}_{\rm
1\,GHz}=0.61\pm 0.27$. The spectral index maps of the unsubtracted mini-halo
are shown in panels d) and e) of Figure 2, where much of the southern
extension is undetected. The eastern edge is flatter due to the contaminating
infrared sources. No preferential gradient is evident as much of the region
outside the BCG has an index consistent with the integrated spectrum.

###  5.3 MACS J1115.8+0129

Giovannini et al. (2020) reported the presence of a mini-halo in MACS
J1115.8+0129 through a short JVLA 1.5 GHz D-configuration observation (maximum
baseline 1.03 km with L-band resolution 46$\arcsec$). In this lower-resolution
data, the central source blended with the diffuse emission. Hence, the
properties of any diffuse structures could not be determined and the authors
simply give the total flux density of the entire radio core region
(compact+diffuse emission). They compare this total flux density to that of
the same region’s FIRST counterpart and cite the $\sim$7 mJy difference as
evidence for a radio mini-halo.

Figure 9 shows a comparison between the 2GC and 3GC images for MACS
J1115.8+0129. No major artefacts are present in the 2GC image but several
sources inside the primary beam cause some minor artefacts. One such source
identified as GB6 B1114+0136, located $\sim$22$\arcmin$ south-east of the
phase centre, is highlighted by the inset and the comparison shows the
reduction in artefacts after 3GC. Its local rms value improves by just under a
factor of two, from 22 $\mu$Jy/beam to 12 $\mu$Jy/beam. The artefacts of a few
other sources similarly reduce (except for a source $\sim$27$\arcmin$ south of
the phase centre), such that the global noise level improves by 7% from 8.2
$\mu$Jy/beam to 7.6 $\mu$Jy/beam.

Figure 5: 3GC SRC-SUB mini-halo integrated spectra. All sub-band images were
convolved to 15$\arcsec$ and flux densities measured within a $5\sigma$
contour, except for ACO 1413 which were convolved to 30$\arcsec$ and measured
within $3\sigma$.

Figures 3a, b show the 3GC vs 3GC SRC-SUB images for this cluster. The BCG
(optical counterpart SDSS J111551.90+012955.0) is shown to have radio lobes
orientated in the north-west/south-east direction, which are removed in the
SRC-SUB image showing the residual diffuse emission. After such subtraction,
the mini-halo is shown to have a similar orientation. We are limited by
sensitivity to confirm the full extent of its morphology; however, we note
that there appears to be a hint of a radio arm extending from the south,
seemingly misaligned with the rest of the mini-halo.

Figure 3c shows the 3GC SRC-SUB image at a 15$\arcsec$ resolution. The
convolution causes some nearby faint emission to blend with the mini-halo
along the direction of its LLS, thus the sizes for this mini-halo quoted in
Table 3 may be slightly overestimated. The strong embedded AGN results in
large measurement uncertainties for the mini-halo flux density, integrated
spectrum and radio power, which are dominated by the subtraction uncertainty.
We determine a flux density of $S_{\rm 1.28\,GHz}=7.91\pm 2.59$, and an
average diameter of 375 kpc with LLS of 499 kpc. The flux density of the mini-
halo is consistent with the missing flux cited by Giovannini et al. (2020),
and the BCG flux density is consistent with their FIRST measurement. Using the
measured in-band spectral index (below), the k-corrected radio power of the
mini-halo is $P_{\rm 1.4\,GHz}=(3.00\pm 1.20)\times 10^{24}$ W/Hz, making it
the most powerful in our sample.

The integrated spectra gives $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=1.0\pm 1.1$
for the mini-halo and $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=0.78\pm 0.26$ for
the BCG. The spectral index and its uncertainty map are shown in panels d) and
e) of Figure 3 and shows a very interesting spectral distribution; a
significant spectral flattening is visible horizontally across the mini-halo
and reaches its minimum directly east and west of the AGN core, at a spectral
index of $\sim$0.5. The steepest regions are directly north and south of the
AGN core, with spectral index $\sim$1.5.

###  5.4 MACS J2140.2-2339

We present a new mini-halo detection in the galaxy cluster MACS J2140.2-2339.
This is an X-ray luminous and massive cool-core cluster at redshift z = 0.313
(Ebeling et al., 2010), with $L_{X,500}=11.1\times 10^{44}$ erg s-1 and
$M_{500}=4.7\times 10^{14}M_{\odot}$, making it one of the lowest mass
clusters to host a confirmed mini-halo. Yu et al. (2018) produced a high-
resolution image of the cluster BCG using a 1.5 GHz JVLA observation in
A-configuration (maximum baseline 36.4 km with L-band resolution
1.3$\arcsec$), measured a flux density of $1.39\pm 0.03$ mJy and quoted a
$3.8\pm 0.5$ mJy flux density from the NVSS counterpart. Similar to MACS
J1115.8+0129, Giovannini et al. (2020) observed this cluster with a short JVLA
1.5 GHz observation in D-configuration; however, a useful image could not be
produced due to strong RFI. They cite the difference in flux density between
the NVSS and JVLA A-configuration measurements of Yu et al. (2018) as evidence
for possible extended radio emission around the BCG. The MeerKAT’s southern
location, high sensitivity and array layout make it the ideal telescope to
observe this source and disentangle the compact and possible diffuse radio
emission.

Figure 10 shows a comparison between the 2GC and 3GC images. Similar to MACS
J1115.8+0129, no strong sources causing any significant artefacts are present
in the map. Further, there are no major DD errors present, except for minor
errors around a source identified as PMN J2142-2329 and highlighted by the
inset. The 3GC procedure reduced these artefacts and improved the local rms
near this source by just under a factor of 1.75, from 33 $\mu$Jy/beam to 19
$\mu$Jy/beam. Otherwise, no significant improvements could be made over the
2GC image.

Figures 4a, b compare the 3GC vs 3GC SRC-SUB images. Diffuse emission is
clearly evident around the BCG (optical counterpart 2MASX J21401517-2339398),
and satisfies all the criteria defining a radio mini-halo set out in
Giacintucci et al. (2017). Thus, we classify this source as a new mini-halo
detection. The BCG and high-redshift galaxy on the western edge (CLASH
MS2137-2353 830, z = 5.9, Bradley et al., 2014) are removed after subtraction,
providing an uncontaminated view of the mini-halo, which seems to be smoothly
orientated in the east-west direction with no AGN jets influencing its
morphology. There seems to be a hint of a pair of radio arms that are
symmetrically opposite the core.

Figure 4c shows the $15\arcsec$ LR 3GC SRC-SUB image. We determine a flux
density of $S_{\rm 1.28\,GHz}=2.61\pm 0.31$, consistent with the difference in
measurements from Yu et al. (2018). We determine an average diameter of 296
kpc with LLS of 390 kpc. The k-corrected mini-halo radio power is $P_{\rm
1.4\,GHz}=(0.79\pm 0.11)\times 10^{24}$ W/Hz. The measured BCG flux density is
consistent with that given in Yu et al. (2018).

The integrated spectra gives $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=1.21\pm 0.36$
for the mini-halo, and $\alpha^{\rm 1.5\,GHz}_{\rm 1\,GHz}=0.72\pm 0.31$ for
the BCG. The spectral index maps of the unsubtracted data are shown in panels
d) and e) of Figure 4. The BCG and western infrared source obscure much of the
underlying mini-halo spectral distribution. However, we can see that the
spectral index monotonically increases radially outwards from $\sim$0.75 to
$\sim$1.50.

##  6 Summary and Conclusions

In this paper, we have described the application of 3GC to MeerKAT L-band
continuum data for a sample of five relaxed galaxy clusters. We used CARACal
for DI calibration and DDF+kMS for DD calibration, and presented Stokes I
continuum and spectral index maps of the central radio mini-halos. Our 3GC
procedure drastically improved image quality, reducing the global image noise
levels and improving the DR(1) values by 7%, DR(2) by almost a factor of three
and the local noise around severe artefact-affected sources by a factor of
two. We presented the new detection of a mini-halo in MACS J2140.2-2339, and a
$\sim$100 kpc southern extension to the ACO 3444 mini-halo which was not
present in previous VLA L-band observations. We were unable to image the
candidate mini-halo in ACO 1795 due to significant image artefacts caused by
its strong BCG. The point source subtraction procedure allowed for the
underlying mini-halos to be viewed but simultaneously increased the global
image noise by 4%. We presented the flux density, size and in-band spectral
analysis of these mini-halos, noting that the spectral index maps are
generated from the unsubtracted mini-halo images. The described calibration
procedures of our MeerKAT wide-band data will be useful for future studies of
extended diffuse radio sources.

The spectral analysis presented here would greatly be furthered by follow-up
MeerKAT UHF-band observations. Additionally, a multi-wavelength study of this
sample in which the radio and X-ray core characteristics are compared would
further the understanding of various correlations between the thermal and non-
thermal properties of the host galaxy clusters in the context of the broader
population of mini-halos and possible production mechanisms.

## Acknowledgements

The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory, which is a facility of the National Research Foundation, an
agency of the Department of Science and Innovation. The financial assistance
of the South African Radio Astronomy Observatory (SARAO) towards this research
is hereby acknowledged (www.sarao.ac.za). V. P. acknowledges the South African
Research Chairs Initiative of the Department of Science and Technology and
National Research Foundation who supported this research. V.P. also
acknowledges the financial assistance of the South African Radio Astronomy
Observatory (SARAO) towards this research. Oleg Smirnov’s research is
supported by the South African Research Chairs Initiative of the Department of
Science and Technology and National Research Foundation. S. G. acknowledges
that the basic research in radio astronomy at the Naval Research Laboratory is
supported by 6.1 Base funding.

## Data Availability

The raw data underlying this article is available in SARAO archive, at
https://archive.sarao.ac.za.

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##  Appendix A Abbreviations

List of abbreviations used in this work.

Table 4: List of specialised abbreviations. Columns: contraction, full phrase. Contraction | Phrase | Contraction | Phrase
---|---|---|---
1GC | First generation calibration | lsm | local sky model
2GC | Second generation calibration | kMS | killMS
3GC | Third generation calibration | DDF | DDFacet
DIE | Direction independent effect | SSD | Sub-Space Deconvolution
DDE | Direction dependent effect | HR | High resolution
DR | Dynamic range | LR | Low resolution
KATDAL | Karoo Array Telescope Data Access Library | SRC-SUB | Source subtracted
CARACal | Containerized Automated Radio Astronomy Calibration | LLS | Largest linear size

##  Appendix B Software

List of software used in this work.

Table 5: Software links. Columns: software, web link. Software | Link | Software | Link
---|---|---|---
KATDAL | https://github.com/ska-sa/katdal | DDFacet | https://github.com/saopicc/DDFacet
CARACal | https://github.com/caracal-pipeline/caracal | eidos | https://github.com/ratt-ru/eidos
Tricolour | https://github.com/ska-sa/tricolour | breizorro | https://github.com/ratt-ru/breizorro
WSClean | https://gitlab.com/aroffringa/wsclean | Stimela | https://github.com/ratt-ru/Stimela
CubiCal | https://github.com/ratt-ru/CubiCal | brats | http://www.askanastronomer.co.uk/brats
killMS | https://github.com/saopicc/killMS | QuartiCal | https://github.com/ratt-ru/QuartiCal

##  Appendix C ACO 1795 images

Figure 6 shows our highest fidelity images of ACO 1795, each of which
highlights the BCG and its strong artefacts obscuring the cluster core. We
suspect these artefacts are not direction-dependent in nature and are in fact
due to polarisation leakage effects, compounded by a very asymmetric PSF, as
described in Section 3.6. The highest fidelity image (with the least artefacts
and lowest noise) was produced with QuartiCal and WSClean. The 3GC solutions
reduced the artefacts in the corresponding DDF 2GC image similar to that seen
in ACO 1413, but could not improve upon the WSClean image. A detailed
comparison between the two imagers is outside the scope of this work.

Figure 6: ACO 1795 images. Left: best 2GC image we could produce, where
QuartiCal self-calibrated visibilities were imaged with WSClean for three
rounds of chained delay and complex-gain solutions. Middle: visibilities from
the Left imaged with DDFacet as per described in Section 3.3 until Step 3.
Right: visibilities from the Middle after 3GC kMS solutions have been applied
similar to the other fields. The black dashed circle shows the MeerKAT primary
beam and the black dashed square indicates the central 500 kpc box around the
BCG as shown by the insets. All images are on the same colorscale and
convolved to a (15$\arcsec$, 7$\arcsec$, 0∘). The global rms noise from left
to right is: 6.7, 11.2, 8.7 $\mu$Jy/beam, respectively. The local rms noise in
each inset from left to right is: 34.9, 91.0, 58.5 $\mu$Jy/beam, respectively.

##  Appendix D 2GC and 3GC Field of Views

Table 6 gives the 2GC and 3GC global noise and DR values for each cluster
except ACO 1795. Figures 7-10 show comparisons between the 2GC and 3GC field
artefacts for each observation except ACO 1795, centred at a position that
highlights the artefacts with respect to the phase centre. If no artefacts
require focus, the image is centred on the phase centre. The colorscale is the
same for each pair of images.

Table 6: Image properties. Columns: Image corresponding to each cluster, global root mean square, and global dynamic ranges (DRs) calculated with respect to the root mean square and minimum pixel values of the unsubtracted images. Image |  RMS ($\mu$Jy/beam) |  DR(1) (max/rms) |  DR(2) $\|$max/min$\|$
---|---|---|---
| 2GC | 3GC | 2GC | 3GC | 2GC | 3GC
ACO 1413 | 8.3 | 7.4 | 10024 | 11284 | 87 | 354
SRC-SUB | 8.5 | 7.9 |  |  |  |
ACO 3444 | 6.4 | 5.9 | 26828 | 29017 | 110 | 319
SRC-SUB | 6.8 | 6.1 |  |  |  |
MACS J1115.8+0129 | 8.2 | 7.6 | 4902 | 5224 | 92 | 265
SRC-SUB | 9.1 | 7.9 |  |  |  |
MACS J2140.2-2339 | 7.2 | 7.1 | 12208 | 12394 | 198 | 386
SRC-SUB | 7.3 | 7.1 |  |  |  |
Figure 7: ACO 1413 2GC vs 3GC artefacts. Left: 2GC image, beam size
(12.3$\arcsec$, 5.9$\arcsec$, -4.6∘), global rms noise 8.3 $\mu$Jy/beam.
Right: 3GC image, beam (12.2$\arcsec$, 5.9$\arcsec$, -4.5∘), global rms noise
7.4 $\mu$Jy/beam. A blue cross marks the phase centre of the images. The thin
dashed black arc indicates the primary beam of the telescope. The bold dashed
black rectangle marks a FR-II galaxy that is contaminated by an artefact with
an arrow pointing to a zoomed inset of this source displayed in the inverse
colorbar. The local rms of this source decreased by just under a factor of two
in the 2GC to 3GC images. Figure 8: ACO 3444 2GC vs 3GC artefacts. Left: 2GC
image, beam size (7.4$\arcsec$, 6.0$\arcsec$, -10.3∘), global rms noise 6.4
$\mu$Jy/beam. Right: 3GC image, beam (6.8$\arcsec$, 5.9$\arcsec$, -16.0∘),
global rms noise 5.9 $\mu$Jy/beam. A blue cross marks the phase centre of the
images. The thin dashed black arc indicates the primary beam of the telescope.
The bold dashed black rectangle marks a bright point source with an arrow
pointing to a zoomed inset of this source displayed in the inverse colorbar.
The local rms decreased by just over a factor of two. Figure 9: MACS
J1115.8+0129 2GC vs 3GC artefacts. Left: 2GC image, beam size (8.6$\arcsec$,
6.6$\arcsec$, -16.7∘), global apparent rms noise 8.2 $\mu$Jy/beam. Right: 3GC
image, beam (8.5$\arcsec$, 6.5$\arcsec$, -16.2∘), global rms noise 7.6
$\mu$Jy/beam. A blue cross marks the phase centre of the images. The thin
dashed black arc indicates the primary beam of the telescope. The bold dashed
black rectangle marks a bright point source with an arrow pointing to a zoomed
inset of this source displayed in the inverse colorbar. The local rms
decreased by just under a factor of two. Figure 10: MACS J2140.2-2339 2GC vs
3GC artefacts. Left: 2GC image, beam size (7.9$\arcsec$, 6.3$\arcsec$, -9.7∘),
global apparent rms 7.2 $\mu$Jy/beam. Right: 3GC image, beam (7.8$\arcsec$,
6.2$\arcsec$, -8.6∘), global rms noise 7.1 $\mu$Jy/beam. A blue cross marks
the phase centre of the images. The thin dashed black arc indicates the
primary beam of the telescope. The bold dashed black rectangle marks a bright
point source with an arrow pointing to a zoomed inset of this source displayed
in the inverse colorbar. The local rms decreased by just under a factor of
two.