PASA 2024

# Early Science from POSSUM: Shocks, turbulence, and a massive new reservoir
of ionised gas in the Fornax cluster

C. S. Anderson1,2,3 G. H. Heald2 J. A. Eilek1,4 E. Lenc3 B. M. Gaensler5
Lawrence Rudnick6 C. L. Van Eck5 S. P. O’Sullivan7 J. M. Stil8 A. Chippendale3
C. J. Riseley9,10 E. Carretti10 J. West5 J. Farnes11 L. Harvey-Smith12,13 N.
M. McClure-Griffiths14 Douglas C. J. Bock3 J. D. Bunton3 B. Koribalski3,13 C.
D. Tremblay2 M. A. Voronkov3 K. Warhurst2 1Jansky fellow of the National Radio
Astronomy Observatory, 1003 Lopezville Rd, Socorro, NM 87801 USA  2CSIRO
Astronomy and Space Science, PO Box 1130, Bentley WA 6102, Australia  3ATNF,
CSIRO Astronomy and Space Science, PO Box 76, Epping, New South Wales 1710,
Australia  4Physics Department, New Mexico Tech, Socorro NM 87801 USA  5Dunlap
Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON M5S 3H4, Canada  6Minnesota Institute for Astrophysics,
University of Minnesota, 116 Church St. SE, Minneapolis, MN 55455 USA  7School
of Physical Sciences and center for Astrophysics & Relativity, Dublin City
University, Glasnevin, D09 W6Y4, Ireland  8Department of Physics & Astronomy,
The University of Calgary, 2500 University Drive NW, Calgary AB, T2N 1N4,
Canada  9Dipartimento di Fisica e Astronomia, Università degli Studi di
Bologna, via P. Gobetti 93/2, 40129 Bologna, Italy  10INAF - Istituto di
Radioastronomia, Via Gobetti 101, 40129 Bologna, Italy  11Oxford e-Research
center (OeRC), Department of Engineering Science, University of Oxford,
Oxford, OX1 3QG, UK  12School of Physics, University of New South Wales,
Sydney, NSW 2052, Australia  13Western Sydney University, Locked Bag 1797,
Penrith, NSW, 2751, Australia  14Research School of Astronomy & Astrophysics,
Australian National University, Canberra ACT 2611 Australia

###### Abstract

We present the first Faraday rotation measure (RM) grid study of an individual
low-mass cluster — the Fornax cluster — which is presently undergoing a series
of mergers. Exploiting commissioning data for the POlarisation Sky Survey of
the Universe’s Magnetism (POSSUM) covering a $\sim 34$ square degree sky area
using the Australian Square Kilometre Array Pathfinder (ASKAP), we achieve an
RM grid density of $\sim 25$ RMs per square degree from a 280 MHz band centred
at 887 MHz, which is similar to expectations for forthcoming GHz-frequency
$\sim 3\pi$-steradian sky surveys. These data allow us to probe the extended
magnetoionic structure of the cluster and its surroundings in unprecedented
detail. We find that the scatter in the Faraday RM of confirmed background
sources is increased by $16.8\pm 2.4$ rad m-2 within 1 degree (360 kpc)
projected distance to the cluster centre, which is 2–4 times larger than the
spatial extent of the presently-detectable X-ray-emitting intracluster medium
(ICM). The mass of the Faraday-active plasma is larger than that of the X-ray-
emitting ICM, and exists in a density regime that broadly matches expectations
for moderately-dense components of the Warm-Hot Intergalactic Medium. We argue
that forthcoming RM grids from both targeted and survey observations may be a
singular probe of cosmic plasma in this regime. The morphology of the global
Faraday depth enhancement is not uniform and isotropic, but rather exhibits
the classic morphology of an astrophysical bow shock on the southwest side of
the main Fornax cluster, and an extended, swept-back wake on the northeastern
side. Our favoured explanation for these phenomena is an ongoing merger
between the main cluster and a sub-cluster to the southwest. The shock’s Mach
angle and stand-off distance lead to a self-consistent transonic merger speed
with Mach 1.06. The region hosting the Faraday depth enhancement also appears
to show a decrement in both total and polarised radio emission compared to the
broader field. We evaluate cosmic variance and free-free absorption by a
pervasive cold dense gas surrounding NGC 1399 as possible causes, but find
both explanations unsatisfactory, warranting further observations. Generally,
our study illustrates the scientific returns that can be expected from all-sky
grids of discrete sources generated by forthcoming all-sky radio surveys.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

galaxies: clusters: individual(Fornax) – galaxies: clusters: intracluster
medium – magnetic fields – techniques: polarimetric – radio continuum:
galaxies

##  1 Introduction

The Universe’s baryons are mostly located outside the stellar envelopes of
galaxies, in the vast expanses occupied by clusters of galaxies, and in
filaments of tenuous plasma that connect them. The properties of gas in these
different regimes — including their state of magnetisation — lie at the heart
of theories of cosmic evolution and ecology, but remain difficult to pin down
observationally. For instance, the hot intracluster medium (ICM) contains
about 4% of the baryonic mass of the late-time Universe, which is (for
example) a higher proportion than is contained in stars (de Graaff et al.,
2019). Magnetic fields break the isotropy of viscosity, pressure support, and
thermal conductivity of the ICM, thereby exerting an out-sized influence on
cluster physics and evolution. They can trace ordered and turbulent flows in
plasma (e.g. Anderson et al., 2018), reveal interactions between the ICM and
in-falling gas (e.g. Keshet et al., 2017), embedded galaxies (e.g. Dursi &
Pfrommer, 2008; Pfrommer & Dursi, 2010) and galactic outflows (e.g. Guidetti
et al., 2011, 2012; Anderson et al., 2018), and help reveal how the broader
cosmos became magnetised (e.g. Vazza et al., 2014; Bonafede et al., 2015).

Beyond the ICM, the Warm-Hot Intergalactic Medium (WHIM) must contain around
80% of the Universe’s baryons (de Graaff et al., 2019), though this material
is comparatively unstudied, with only recent claims of detection of its
sparser phases (Nicastro et al., 2017; de Graaff et al., 2019; Tanimura et
al., 2019; Macquart et al., 2020). Simulations suggest that the WHIM will be
found in a diverse set of regimes, occupying relatively dense agglomerations
($\delta\sim 200$, where $\delta$ is the over-density factor of baryons
compared to the cosmic mean, defined by $\delta\equiv\rho/\bar{\rho}-1$, and
in turn, $\rho$ is the baryon number density at a given location, while
$\bar{\rho}$ is the mean baryon number density in the Universe, which is
currently $\bar{\rho}\approx 2\times 10^{-7}$ cm-3; Planck Collaboration et
al., 2016) around galaxy clusters in its densest and hottest manifestations,
and in tenuous filaments between massive galaxies in its sparsest and coolest
regimes ($\delta\sim$ a few) (Davé et al., 2001). The magnetisation of this
material is also consequential, since the predictions of models for cosmic
magneto-genesis differ most strongly here (e.g. Donnert et al., 2018 and
references therein), which is just now beginning to be revealed with
extraordinary new low frequency radio measurements (Govoni et al., 2019;
Botteon et al., 2020), and may also be probed via measurements of radio
polarisation and Faraday rotation (e.g. Akahori et al., 2018; Locatelli et
al., 2018; O’Sullivan et al., 2020).

Faraday rotation is an effective tracer of the distribution and properties of
rarefied, magnetised cosmic plasma, such as the ICM and WHIM (e.g. Cooper &
Price, 1962; Burn, 1966; Conway et al., 1974; Kronberg & Simard-Normandin,
1976; Taylor & Perley, 1993; Farnsworth et al., 2011; O’Sullivan et al., 2013;
Johnston-Hollitt et al., 2015; Gaensler et al., 2015; Anderson, 2016).
Consider that the linear polarisation state of radio emission can be described
by a complex vector $\bm{P}$, related to the Stokes parameters $Q$ and $U$,
the polarisation angle $\psi$, the fractional polarisation $p$ and the total
intensity $I$ as

$\bm{P}=Q+iU=pIe^{2i\psi}$ (1)

After being emitted at a location $L$, linearly polarised radiation will be
Faraday rotated by magnetised thermal plasma along the line of sight (LOS) to
an observer by an amount equal to

$\Delta\psi=\phi(L)\lambda^{2}$ (2)

where $\lambda$ is the observing wavelength, and $\phi$ is the Faraday depth,
given by

$\text{$\phi$}(L)=0.812\int_{L}^{0}n_{e}\bm{B}\cdot\text{d}\bm{s}~{}\text{rad
m}^{-2}$ (3)

and, in turn, $n_{e}$ [cm-3] & $\bm{B}$ [$\mu$G] are the thermal electron
density and magnetic field along the LOS respectively.

The observable polarisation spectrum $\bm{P}(\lambda^{2})$ is obtained by
summing the polarised emission emerging from all possible Faraday depths
within the synthesised beam of the telescope:

$\bm{P}(\lambda^{2})=\int_{-\infty}^{\infty}\bm{F}(\phi)e^{2i\phi\lambda^{2}}d\phi$
(4)

The function $\bm{F}(\phi)$ (the so-called Faraday Dispersion Spectrum,
henceforth FDS) specifies the distribution of polarised emission as a function
of Faraday depth along the LOS. This quasi-Fourier relationship can be
inverted to reconstruct $\bm{F}(\phi)$ given observations of
$\bm{P}(\lambda^{2})$ (Burn, 1966; Brentjens & de Bruyn, 2005). In the common
situation where a point-like source is viewed through an extended reservoir of
magnetised thermal plasma lying in the foreground (which we deal with in this
work), $\bm{F}(\phi)$ shows a single peak at a well-defined Faraday depth
$\phi_{\mathrm{peak}}=\text{argmax}(|\bm{F}(\phi)|)$. This is then equivalent
to the so-called Faraday rotation measure (RM) of the source. Largely for
historic reasons, we continue to use the term “RM” in places in this work,
though the measurements themselves are of $\phi_{\mathrm{peak}}$ — i.e.
extracted through the method of RM synthesis, rather than from the gradient of
the polarisation angle as a function of $\lambda^{2}$. Regardless of
nomenclature, $\phi_{\mathrm{peak}}$ provides a direct measure of the amount
of magnetised thermal plasma that has been traversed along the line of sight.
Ensembles of polarised radio sources can therefore back-illuminate the
magnetoionic structure of extended plasma reservoirs in the foreground.

Applying these ‘RM grid’ techniques to study the ICM or WHIM in individual
galaxy clusters has historically been difficult, because past generations of
radio instrumentation could only recover a relatively low density of polarised
sources over the required multi-square-degree sky areas ($\mathcal{O}(1)$
linearly polarised source per square degree — e.g. Taylor et al., 2009),
combined with the uncertainty of whether any given RM grid source is located
behind a target cluster, inside it, or in the foreground. Fortunately, these
limitations will soon be transcended using data from a new generation of radio
interferometers and optical redshift surveys. In the former domain, the
Australian Square Kilometre Array Pathfinder (ASKAP) can routinely measure
polarised source densities of $\sim 25$ per square degree over tens of square
degrees in just a few hours (this work, West et al. _in prep._), and will soon
survey the entire southern sky at this or greater depth for the Polarisation
Sky Survey of the Universe’s Magnetism (POSSUM; Gaensler et al., 2010). At the
same time, deep pointed observations like the MeerKAT (Jonas, 2009) Fornax
Survey (Serra et al., 2016) may soon recover $\sim$hundreds of polarised
sources per square degree. Both approaches will open up the field of
observational galaxy cluster astrophysics to routine RM grid studies (Heald et
al., 2020).

This paper heralds the dawn of this era by exploiting ASKAP’s unique imaging
capabilities to study the ionised gas in the Fornax cluster. The Fornax
cluster is nearby (20.64 megaparsecs; Mpc; Lavaux & Hudson, 2011), but much
poorer than similarly well-studied clusters like the Virgo and Coma clusters.
It has only $\sim 390$ member galaxies (which are typically low in mass;
Maddox et al., 2019), and has a comparatively low total mass of
$6^{+3}_{-1}\times 10^{13}$ M⊙ (Drinkwater et al., 2001; Nasonova et al.,
2011; Maddox et al., 2019), which is one and two orders-of-magnitude less
massive than the Virgo and Coma clusters respectively. The (presently
detectable) X-ray-emitting ICM is also small, extending asymmetrically outward
from NGC1399 to a mere 15%–30% of the cluster’s 1.96 degree virial radius.
Nevertheless, its halo mass is more representative of that in which the
majority of the galaxies in the Universe reside and evolve (Haan & Braun, 2014
and references therein). The dynamical state of the cluster, and associated
astrophysical processes, are therefore of keen interest.

Our understanding of the dynamics of this system is evolving rapidly. The core
of the Fornax cluster is densely populated with early-type galaxies that
possess relatively uniform properties and low velocity dispersion. This used
to be cited as evidence that the Fornax cluster is virialised and well-evolved
(see Iodice et al., 2019 and references therein), but it is now clear that the
Fornax cluster possesses complex spatial sub-structure both in terms of its
constituent gas and galaxies (e.g. Drinkwater et al., 2001; Paolillo et al.,
2002; Scharf et al., 2005; Su et al., 2017; Venhola et al., 2018; Sheardown et
al., 2018), and is still assembling mass through a series of on-going mergers.
At the largest scales, Drinkwater et al. (2001) argue that there is a genuine
mass partition between northeast and southwest sub-components of the cluster,
which are respectively dominated by the cD-type galaxy NGC 1399 and NGC 1316 —
the host galaxy of the radio source Fornax A — a few degrees ($\sim 1.3$ Mpc)
away. Scharf et al. (2005) cite the swept-back (to the northeast) morphology
of the hot ICM traced by X-rays as possible evidence that these sub-components
are merging at transonic speeds. On smaller scales, NGC 1399 is undergoing a
series of close encounters with the spiral galaxy NGC 1404, as the latter
falls into the cluster potential and interacts with the diffuse gas there
(Machacek et al., 2005). This appears to have induced sloshing in the ICM
which is most apparent in the central $\sim 30$ kpc of the cluster (Su et al.,
2017), and may have generated shocks and turbulence on scales up to more than
a degree (Sheardown et al., 2018).

Thus, the Fornax cluster differs greatly in its properties from other
relatively nearby massive clusters, including Virgo and Coma, and from the
massive clusters from which our canonical understanding of the magnetised ICM
were chiefly derived. While our understanding of the magneto-thermal plasma
structures of even large clusters remain incomplete (e.g. Johnston-Hollitt et
al., 2015; Heald et al., 2020), our ignorance is much more pronounced at the
important low-mass end of the halo distribution. Sensitive new RM grid
experiments can directly reveal the magnetised gas in such environments,
providing sorely needed new data in this sphere. In this work then, our
overarching aim is to search for Faraday RM enhancements to trace the
structure of magnetised ionised gas in an individual low-mass galaxy cluster
for the first time. Our specific goals are to (a) estimate the mass and extent
of any such material, and to compare these estimates to those determined for
ensembles of the more massive clusters previously probed in RM stacking
experiments, (b) to determine whether any material thereby revealed differs in
its properties or extent from that revealed by Bremsstrahlung radiation, thus
establishing the complementarity of the two measurement techniques, and (c)
demonstrate what unique information the RM measurements provide about the
dynamical mass assembly processes that continue to take place in the system.
The spatial density and areal sky coverage of our RM grid, coupled the fact
that it is dominated by confirmed background radio sources (as we will show),
is groundbreaking in this field.

Our paper is organised as follows. We describe our observations, calibration,
and imaging procedures in Section 2, and our polarimetric analysis in Section
3. Analysis of ancillary redshift data is presented in Section 4. We present
our results, discussion, and conclusions in Sections 5, 6 and 7
(respectively). In this paper, we adopt a distance to NGC 1399 of D = 20.64
Mpc (Lavaux & Hudson, 2011), which yields an image scale of 100
parsecs/arcsec. Cardinal directions are referred to by their usual
abbreviations — e.g. NE, SW, S for northeast, southwest, and south,
respectively. We use the spectral index convention $S\propto\nu^{\alpha}$.

##  2 Observations, calibration and imaging

Table 1: Summary of observations

Target | Fornax cluster region
---|---
Scheduling block ID | 8279
Date of observations | 23rd March 2019
Field centre (J2000) |  $03^{h}29^{m}30^{s}$, $-34^{d}58^{m}30^{s}$
Field centre (J2000, Gal. $l,b$) | 235.988∘, $-$55.484∘
No. telescope pointings | 1
Total integration time | 6 hours
Full-band sensitivity † | 30 $\muup$Jy beam-1
Recorded polarisations |  $XX$, $XY$, $YX$, $YY$
Beam footprint | square_6x6
Beam pitch | 0.9 deg
Number of valid beams | 30 paired X and Y pol. (of 36)
FWHM of formed beams at 887 MHz‡ | 1.59 deg
Total sky coverage |  $\sim 34$ deg2
Number of antennas | 32
Antenna diameter | 12 m
Shortest baseline | 22.4 m
Longest baseline | 6.4 km
Angular resolution (robust $=0$) | 11$\times$14 arcsec
Largest recoverable angular scale‡‡ | 30 arcmin
Frequency range (central frequency) | 747–1027 MHz (887 MHz)
Frequency resolution | 1 MHz
$\lambda^{2}$ range | 0.085–0.161 m2
Resolution in $\phi$ space a | 46 rad m-2
Largest recoverable $\phi$-scale‡‡ a | 37 rad m-2
Largest recoverable |$\phi$| a | 510 rad m-2

  * •

† Measured per Stokes parameter in multi-frequency synthesis images generated
with a Briggs’ robust weighting value of 0.0. ‡ At centre frequency of band.
‡‡ At greater than 50% sensitivity. a Calculated from equations in Section 6
of Brentjens & de Bruyn (2005).

We observed the Fornax cluster region during commissioning tests of the ASKAP
radio telescope (DeBoer et al., 2009; Johnston et al., 2007; Schinckel & Bock,
2016). ASKAP consists of $36\times 12$-m antennas, each equipped with a Phased
Array Feed (PAF), yielding a $\sim 30$ square degree instantaneous field of
view (depending on frequency) and high survey speed. We observed a single such
pointing for six hours with a square_6x6 beam footprint (McConnell et al.,
2016), using a beam pitch (horizontal and vertical angular separation) of 0.9
degrees, covering the frequency range 747–1027 MHz (averaging to 1 MHz
frequency channels on-the-fly from ASKAP’s native 18 kHz spectral resolution),
achieving a band-averaged sensitivity of 30 $\mu$Jy beam-1 per Stokes
parameter. The ASKAP array configuration is shown in Figure 2. Of the 36 beams
formed for our observations, six in the south-west corner of the mosaic
suffered from beamforming errors leading to low sensitivity, and were
discarded from further analysis. It is coincidental that the bright radio
galaxy Fornax A appears in the vicinity of these beams for our observations.
The sky position of the remaining valid beams are indicated in Figure 2. We
note that the field centre was chosen to satisfy the competing demands of
several science teams working in this sky area, and so while the field
incorporates the Fornax Cluster, Fornax A and several other radio galaxies, it
is not centred on any of them. Further details are summarised in Table 1.

We flagged and calibrated our data in the Common Astronomy Software
Applications (CASA; McMullin et al., 2007) package. We flagged radio frequency
interference manually, which is feasible only because of the exceptionally
RFI-quiet conditions at the Murchison Radio-astronomy Observatory (e.g.
Indermuehle et al., 2018). We calibrated the flux scale, instrumental
bandpass, and (on-axis) polarisation leakage ‘D-terms’ using standard methods
applied to observations of the (unpolarised) standard calibrator source PKS
B1934–638. The frequency-dependent instrumental XY-phase was nulled out at the
beamforming stage using the ASKAP on-dish calibration system (ODC; Chippendale
& Anderson, 2019). The off-axis polarimetric instrumental response was not
corrected for this work. However, after the on-beam-axis D-term corrections
are applied, and for the frequency and beam pitch employed, we estimate that
the leakage from Stokes $I$ to $Q$ and $U$ is usually less than 1%, though it
can be worse in isolated areas. We discuss this more in Section 3. The
absolute polarisation angle was also left uncalibrated, but the inter-beam and
inter-channel relative polarisation angle is guaranteed to be consistent by
the beamforming procedure, which we subsequently verified by comparing
polarisation spectra of sources observed in images of adjacent beams. The
absolute flux scale is uncertain by up to 10%, and this can vary as a function
of position in the field (McConnell et al., 2019).

Figure 1: Offset ASKAP antenna positions in meters from antenna ak01
(Longitude: 116.631424∘ E, Latitude: -26.697000∘; McConnell et al.,
2019222https://confluence.csiro.au/download/attachments/733676544/ASKAP_sci_obs_guide.pdf)
for the full ASKAP array. The inset panel zooms in on E-W offsets of -100m to
+60m, and N-S offsets of -70m to +90m. Figure 2: The position (and FWHM at our
maximum frequency of 1027 MHz) of formed ASKAP beams used in this work (white
circles), overlaid on a map of the local root-mean-squared (RMS) noise in
peak-$P$ (see Figure 3 and Section 3 for an enlarged version containing
further detail). The centre of the Fornax cluster is indicated with a red
cross-hair, and the lobes of Fornax A are partially visible to the south west.

We imaged the data with WSClean (Offringa et al., 2014). For all Stokes
parameters, we generated image cubes with $5000\times 5000$ pixels, a pixel
scale of 2.5 arcseconds, a Briggs (1995) robust weighting value of 0, local
noise estimation with automatic CLEAN thresholding and masking (at 1 $\sigma$
and 3$\sigma$ respectively), and joined-channel CLEANing with 8 MHz
channelisation. We performed two rounds of phase-only self-calibration using
CASA with a solution interval of 300 seconds, and then one round of phase and
amplitude self-calibration with a solution interval of 60 seconds. We
experimented with shorter solution intervals, but this produced little effect.
We then re-imaged and cleaned Stokes $I$ MFS maps independently, an then the
Stokes $Q$ and $U$ datacubes using WSClean’s ‘join polarisations’ and ‘squared
channel joining’ modes. The individual beam images were then linearly
mosaicked for all channels and Stokes parameters using the SWarp package
(Bertin et al., 2002), employing a scaled-width circular Gaussian beam, whose
full-width-half-maximum scales as $(1.09/12)\lambda$ (McConnell et al.,
2019)333https://confluence.csiro.au/download/attachments/733676544/ASKAP_sci_obs_guide.pdf,
truncated at the 10% power point. We then smoothed to the spatial resolution
of our lowest frequency channel — $18\times 14$ arcseconds — then re-gridded
to a common pixel grid, and concatenated together to form Stokes I, Q and U
datacubes with dimensions RA, Decl, $\lambda^{2}$.

We note that in the final linear mosaics, nine or more beams contribute to the
data values at any point located inside the outer ring of beam centres, and
that to our knowledge, the cluster centre is not in a ‘special’ or otherwise
noteworthy location with respect to the positions of the beam centres (see
Figure 2).

##  3 Polarimetric analysis

We calculated the Faraday Dispersion Spectrum (FDS) over the range -200 to
+200 rad m-2 using RM synthesis444https://github.com/brentjens/rm-synthesis,
version 1.0-rc4 (Burn, 1966; Brentjens & de Bruyn, 2005) applied to the Stokes
Q and U data cubes with equal weighting per image channel. The result is a
complex-valued FDS datacube with dimensions RA, Decl, and $\phi$.

Figure 3: The local root-mean-squared (RMS) noise in the peak-$P$ map. This is
supplied in lieu of the peak-$P$ map itself, which renders point sources
effectively invisible for our high resolution, large area map. This RMS map
was generated by running a square sliding window of width and height both
equal to five synthesised beamwidths over the peak-$P$ map, and calculating
the RMS values of the pixels inside the window. The image shown here has a
square root stretch applied. Linearly polarised radio sources are visible as a
marked increase in the local RMS value. In source-free regions, the RMS is
typically $\sim 30$ $\mu$Jy beam-1, except at the mosaic edges, and in the
vicinity of bright sources, where the faint imprint of the synthesised beam
manifests as narrow, diagonal fan–like structures. The centre of the Fornax
cluster is indicated with a red cross-hair. Fornax A is partially visible in
the bottom-right corner of the map, where six beams are missing due to
beamforming errors. The white dashed box approximately indicates the region
shown in Figure 8. The white dashed line indicates an angular radius of one
degree, while the white dotted line indicates the 705 kpc (1.96 degree) virial
radius of the cluster.

Figure 4: The calculated dirty (i.e. no rmclean (Heald et al., 2009)
performed; see Section 3) FDS (first column), corresponding Stokes $Q$ (red)
and $U$ (blue) spectra (second column), peak-$P$ image (third column), and
total intensity image (fourth column)

, for selected sources showing a range of polarised signal-to-noise. For
columns 1 & 2, the horizontal axes range from -200 to +200 rad m-2 for the FDS
plots (first column), and 0.08 to 0.16 m2 for the Stokes ($Q$,$U$) plots
(second column); note that tick labels are included on the bottom-most
horizontal axes only. The vertical axes limits are all scaled to the maximum
amplitude of the data points in individual plots. The J2000 name, right
ascension, declination, and band-averaged polarised signal-to-noise ratio (SN)
are all written in the respective FDS plots. The error bars on the ($Q$,$U$)
data points indicate the standard deviation measured per image channel from
the Stokes ($Q$,$U$) datacubes in a small region adjacent to each source. The
peak polarised intensity of the sources are (from left to right, top to
bottom) 212, 81, 3.5, 1.9, 0.7, and 0.4 mJy/beam/RMSF. Note that because the
FDS have not been deconvolved with rmclean, the emission-free regions of the
FDS cannot be used as a guide to the underlying noise level. The RMSF (which
is common to all of our sources, given our method) is plotted as a magenta
dot-dashed line in the top-most FDS plot, scaled to the magnitude of the
accompanying FDS. Note that the bottom-most source is a possible example of a
source with multiple FDS emission peaks, but the $6\sigma$ S/N of the
secondary peak is barely significant due to polarisation bias (Macquart et
al., 2012; Hales et al., 2012). The peak-$P$ and total intensity images
presented in columns 3 & 4 each span $6.9\times 11.3$ arcminutes, and are
presented with a logarithmic image scaling. The red and white cross-hairs
indicate the position at which the polarised data were extracted for the
source in question. The first two rows illustrate how we have sampled
independent lines of sight towards a resolved radio source; see discussion in
Section 3.

We generated a map of the peak polarised intensity (peak-$P$) across the field
from the FDS cube using Miriad’s (Sault et al., 1995) moment function (see
Figure 3, which shows the associated RMS noise map, since the resolution and
sky coverage of our observations would render most sources invisible in the
peak-$P$ map itself). We then identified polarised radio sources in the field
by applying the Aegean Software Tools source-finding package
555https://github.com/PaulHancock/Aegean, version 2.0.2 (Hancock et al., 2012,
2018) to our multi-frequency synthesis (MFS) Stokes $I$ map and the peak-$P$
map, using seedclip [floodclip] values666The Aegean algorithm incorporates the
generic Floodfill algorithm. In an image, source detections are ‘seeded’ at
all locations where the pixel values exceeds the seedclip value. The algorithm
then walks outwards from these locations, incorporating all adjacent pixels
that exceed the floodclip value into the source model. Details are supplied in
Hancock et al. (2012). of 5 [4] and 7 [5] respectively. With these settings,
the nominal false detection rate (FDR) expected from Aegean is $\sim 2\%$ and
$\sim 0.5\%$ for the total intensity and peak-$P$ images respectively, though
we note that the true FDR will be somewhat higher because (a) the peak-$P$
image contains complicated non-Gaussian noise structure (e.g. see Hales et
al., 2012), which has the effect of pushing noise peaks to higher apparent
significance under the assumption of Gaussian noise statistics, and (b) the
peak-$P$ and total intensity images each contain residual deconvolution
artifacts around bright sources. We ameliorated these relatively high
effective FDRs by cross-matching the source-finding results from the peak-$P$
and total intensity images. Assuming that the peak-$P$ and total intensity
images and associated source finding results are statistically independent,
the resulting nominal baseline FDR is a negligible $\sim 0.01$%. Most
importantly though, the cross-matching provided almost total suppression of
false detections from deconvolution artifacts near bright sources, as
determined by careful visual examination of the results. Our source-finding
parameters yield a $7\sigma$ signal-to-noise cut in band-averaged linear
polarisation, which is required for reliable RM measurements (e.g. Macquart et
al., 2012).

For this background RM grid experiment, we extracted the peak polarised
intensity (peak-$P$) and associated sensitivity (see Figure 3), and the peak
Faraday depth of the source ($\phi_{\mathrm{peak}}$), from the dominant peak
in the Faraday spectrum from each source. In practice, almost all of sources
only had a single peak in the FDS. The quality of the FDS (and the associated
Stokes $Q$ and $U$ spectra) are good throughout the mosaic; representative
examples of FDS and their associated (Q,U) vs. $\lambda^{2}$ spectra are shown
in Figure 4, which were selected at random to span the range of polarised
signal-to-noise of the sources included in our RM grid sample. The full
catalogue is provided online777¡insert web address here¿.

Since the vast majority of sources detected were spatially unresolved or
nearly so, we extracted the Stokes $I$, peak-$P$, and $\phi_{\mathrm{peak}}$
values of each source at the location of the brightest pixel in the peak-$P$
map. For the few heavily-resolved sources in the map (i.e. PKS B0336–35, which
is actually comprised of the radio source inside NGC 1399, and a physically
un-associated source several arcminutes to the NE — see Killeen et al., 1988),
we extracted the aforementioned quantities at the central coordinate location
of the Gaussian emission components comprising the islands outputted by
Aegean. This results in samples of a suitable number of independent lines of
sight towards these resolved sources. The polarisation state of some of these
sources will be dominated by off-axis polarisation leakage. Since a robust,
frequency-dependent, off-axis polarisation calibration procedure has not been
finalised for ASKAP, we proceeded by identifying and eliminating such sources
from our sample. We estimated the position-dependent, frequency-_independent_
, Stokes $I\rightarrow Q$ and $I\rightarrow U$ leakages using field sources,
as described in Appendix A. The results are that the leakages are lower than
1% in most of the mosaic, but are greater in some areas, and in particular,
the mosaic edges and corners (see Appendix A). We eliminated sources from our
sample that:

  1. 1.

were located more than $3.5^{\circ}$ from the mosaic centre

  2. 2.

were not located inside the half power point (at 1027 MHz) of at least one
formed beam

  3. 3.

had measured fractional Stokes $q,u$ (we define $q=Q/I$, $u=U/I$, and use this
nomenclature henceforth) values within $3\sigma$ uncertainty of our leakage
map predictions at that location

  4. 4.

had measured Stokes $q$ and $u$ values that were simultaneously within a
factor of 2 of our leakage map predictions at that location

In combination, the first two criteria ensure that sources are observed close
to at least one beam centre, that multiple beams contribute to the final
mosaic at the source locations, and that regions of high $I\rightarrow U$
leakage found outside the centres of the corner beams in the square_6x6 beam
footprint are excluded from our analysis (see Appendix A, Figure 13). In turn,
this ensures that the off-axis response is averaged down, and that the
polarisation of the sources are measured in multiple beams that can be
evaluated for consistency. The latter two criteria (respectively) ensure that
the polarisation state of a source is not either (a) consistent with pure
instrumental leakage, or (b) dominated by instrumental leakage. We note that
we also tested other methods to exclude spurious leakage-dominated sources,
including local cuts on fractional polarisation based on predictions from our
leakage maps, and uniform cuts on sources with fractional polarisations as
high as 1.5%. Our results were not significantly affected by the choice of
method. After the cuts listed above, our sample consists of 870 linearly
polarised sources, with a median fractional polarisation of 4.8% (uncorrected
for Ricean polarisation bias; see e.g. Hales et al., 2012, and noting the
additional upward bias on the sample median fractional polarisation imposed by
our polarised intensity cutoff; see Figure 5). Figure 5 plots the distribution
of polarised vs. total intensity for this sample, which appear broadly
consistent with distributions derived from several similar $\sim$GHz-frequency
studies (Feain et al., 2009; Banfield et al., 2014; Hales et al., 2014).
Moreover, the sample yields an average polarised source density of 27 per
square degree. This is consistent with several predictions for the number
density of linearly polarised sources for GHz-frequency surveys with similar
depth and resolution (Stil et al., 2014; Rudnick & Owen, 2014a, b), and is
somewhat _lower_ than others (e.g. Hales et al., 2014). We are therefore
confident that spurious leakage-dominated sources do not significantly
contaminate our sample, and do not affect our analysis or conclusions in any
important way.

Figure 5: Linearly polarised (un-debiased) vs. total flux density for the 870
sources in our sample. The red points represent sources inside a projected
cluster-centric distance of 1 degree, while the blue points represent the
converse. This distinction becomes relevant in Section 5.1.1. The dashed
diagonal lines are lines of constant fractional polarisation (from top left to
bottom right: 100%, 10%, 1%, 0.1%).

The Galactic contribution to Faraday rotation measure is of order 10 rad m-2
in this region (refer to Table 1 for Galactic coordinates), but varies over
the field (Anderson et al., 2015, and see below). We attempted to remove this
foreground contribution by fitting and subtracting a 2nd degree polynomial
surface to the position-dependent $\phi_{\mathrm{peak}}$ values of our sources
to yield $\phi_{\mathrm{peak,res}}$ — the residual peak Faraday depth. Sources
located within 1.5 degrees projected distance of the Fornax A radio core were
excluded from the fit. We tested alternative fitting approaches, such as using
a planar surface instead of the 2nd degree polynomial surface, and both the
planar and 2nd degree polynomial surfaces after excluding data points located
within 1.5 degrees of the cluster centre (the reason for which will become
apparent in Section 5.1). In all cases, the residual RMs did not differ
substantially in the vicinity of the cluster or throughout the larger field.
Denoting the right ascension and declination a given source in decimal degrees
in the J2000 epoch as $x$ and $y$, we define a 2nd degree polynomial surface
as $p(x,y)=\sum_{i,j}c_{i,j}*x^{i}*y^{j}$ with $i,j\leq 2$, then fitting over
the field as described above, we derive best-fit model coefficients of
$c_{0,0}=-4.88001154\times 10^{5}$, $c_{0,1}=-2.76553274\times 10^{4}$,
$c_{0,2}=-3.91350846\times 10^{2}$, $c_{1,0}=1.84508441\times 10^{4}$,
$c_{1,1}=1.04492833\times 10^{3}$, $c_{1,2}=1.47768421\times 10^{1}$,
$c_{2,0}=-1.74583394\times 10^{2}$, $c_{2,1}=-9.88225943$,
$c_{2,2}=-0.139675724$. Over the field, this model is essentially consistent
with the all-sky Galactic RM model derived by Hutschenreuter & Enßlin (2020),
both qualitatively and quantitatively. In both cases, the model has a mean
value of $\sim+10$ rad m-2, showing a slight gradient running almost directly
N-S through the field, from $\sim+20$ rad m-2 for the northernmost sources
down to $\sim+3$ rad m-2 for the southernmost sources. Our model is
systematically $\sim 5$ rad m-2 lower than the Hutschenreuter & Enßlin (2020)
model in the very northern-most part of our field, but the discrepancy is
generally less than or equal to the Hutschenreuter & Enßlin (2020) model
uncertainty in this region, and we have verified that the difference does not
affect our results or conclusions in any case.

The uncertainty in $\phi_{\mathrm{peak,res}}$ was calculated as per Brentjens
& de Bruyn (2005), based on the peak-$P$ value of each source, the RMS noise
measured from band-averaged maps of Stokes $Q$ and $U$ in an adjacent source-
free region (typically 30 $\mu$Jy beam-1 per Stokes parameter), and the
rotation measure spread function (RMSF, which is the point-spread function in
Faraday depth space; see Brentjens & de Bruyn, 2005) width measured directly
from our data accounting for our channelisation scheme. We have multiplied the
uncertainties in $\phi_{\mathrm{peak,res}}$ by an additional factor of 1.2, to
capture the aggregate effect of uncorrected widefield polarisation leakage
(see Section 5.2 of Ma et al., 2019).

##  4 Ancillary analysis

###  4.1 Cluster-relative line-of-sight source positions from redshift
catalogue cross-matching

The vast majority of radio sources brighter than $\sim 1$mJy at $\sim$GHz-
frequencies are powerful AGN that lie at far greater distance than the Fornax
cluster (e.g. Magliocchetti et al., 2000; Gendre & Wall, 2008; de Zotti et
al., 2010). Nevertheless, it is desirable to confirm this for the sources used
in our particular RM grid experiment, for reasons described in Section 1.

Maddox et al. (2019) have compiled a catalogue of reliable spectroscopic
redshifts towards the Fornax cluster, drawn from deep optical imaging surveys
in the literature as well as their own data. This catalogue is complete within
a degree of NGC 1399, down to brightness levels typical of faint ultra-compact
dwarf galaxies and globular clusters. The objects in our sample are all
brighter than 0.6 mJy/beam in total radio intensity (see Figure 5) and will,
if located in the Fornax cluster redshift range ($600<cz<3000$ km s-1), have
optical counterparts at least as bright as a star-forming galaxy (e.g.
Padovani, 2016). It follows that any of our sample sources inside the Fornax
cluster will have a spectroscopic redshift in the Maddox _et al._ catalogue.

We cross-matched our sample against the Maddox _et al._ catalogue for objects
which (1) lie within 1 degree of the centre of NGC 1399 (this radius becomes
relevant in Section 5.1.1), (2) had redshifts consistent with lying inside the
Fornax cluster volume, and (3) were not associated with ‘Galactic stars’ or
‘globular clusters’ in Maddox _et al._. We used an initial matching radius of
90 arcseconds to account for possible offsets between steep spectrum double
radio sources and their optical counterparts (e.g. Hammond et al., 2012). This
produced three candidate matches, apart from the radio source associated with
NGC 1399 itself. However, each candidate was then found to be a single,
isolated, spatially-unresolved radio source. For such sources, a matching
radius equal to the synthesised primary beam width (10 arcseconds) is more
appropriate, but even a 30 arcsecond matching radius eliminates all of the
initial candidate matches. This complete lack of optical redshift counterparts
confirms that our entire sample (with the exception of the radio source hosted
by NGC 1399) lies beyond the Fornax cluster, and that their RMs are
accumulated along lines-of-sight that traverse the entire distance through the
cluster.

Relaxing the cluster volume redshift range constraint, but otherwise cross-
matching using the same methods against the full Maddox _et al._. catalogue,
we obtain 21 matches satisfying $0.0048<z<2.41$, with $<z>=0.65$. Six of these
sources are located at a projected distance of less than one degree, and have
$0.197<z<1.606$ and $<z>=0.902$. Outside one degree, there is one source
inside the cluster redshift range with $z=0.0048$. The remainder of this sub-
sample satisfies $0.079<z<2.41$ and $<z>=0.55$. Therefore, to the extent
possible, we confirm that our sample sources typically reside well beyond the
Fornax cluster, though somewhat closer than is typical for moderately bright
(>10 mJy/beam) radio sources in the NRAO VLA Sky Survey (NVSS, for which
$<z>\approx 1.2$; see Brookes et al., 2008 and Figure 11 of de Zotti et al.,
2010). This is not unexpected, given the complex differences in strategy used
by the redshift surveys involved. Finally, a 2-sample Kolmogorov-Smirnov test
shows that the redshift distributions of cross-matched sources located inside
versus outside 1 degree do not differ significantly from each other (D-value =
0.25, p-value = 0.89). Again, this scale becomes relevant in Section 5.1.1. We
also looked for any significant dependence of RM on redshift, but given the
small size of the sample, we were unable to provide any reasonable
constraints.

##  5 Results

###  5.1 Faraday depths are enhanced near the cluster

####  5.1.1 Global trend in Faraday depth versus projected distance

Figure 6 plots $\phi_{\mathrm{peak,res}}$ as a function of distance from the
centre of the cD-type galaxy NGC 1399, which we take to be the position of the
centre of the cluster (Drinkwater et al., 2001; i.e. located at
$03^{h}38^{m}28.68^{s}$, $-35^{d}27^{m}07.14^{s}$ (J2000), and indicated on
Figure 3 with a red cross-hair). Note that our sample selection criteria (with
respect to the mosaic beam centres; see Section 3) mean that sources are
truncated eastward of RA = 03h45m, 1.33 degrees east of the cluster centre at
the same declination.

The left-hand panel of Figure 6 shows an increase in the dispersion of
$\phi_{\mathrm{peak,res}}$ within $\sim 1$ degree (360 kpc) of the cluster
centre ($\sigma=20.5$ rad m-2), compared to sources outside this radius
($\sigma=11.8$ rad m-2). Figure 7 shows a similar result, but with finer
granularity. Here, we have calculated the median of
|$\phi_{\mathrm{peak,res}}$| in a sliding window of maximum width 0.5 degrees
as a function of the cluster-centric radius of the outer bound of this window.
A sharp decrease in the median of |$\phi_{\mathrm{peak,res}}$| is clearly
evident as the outer bound of the sliding window crosses 1 degree, and has
dropped to a more-or-less constant lower value at 1.5 degrees, indicating that
a sharp transition in the degree of observed Faraday rotation occurs at the
former distance.

The difference in Faraday rotation inside vs. outside 1 degree cannot be
accounted for by the measurement uncertainties. A 2-sample K-S test applied to
the cumulative distributions for these sub-samples (also shown in Figure 6)
confirms that they differ both substantially and significantly, yielding D-
and p-values of 0.32 and $3\times 10^{-7}$ respectively. The sharp break in
$\phi_{\mathrm{peak,res}}$ dispersion, and the relative enhancement in this
dispersion towards smaller cluster-centric radii, persist regardless of how
the data are binned or the dispersion is parameterised — see for example the
half-interdecile range plot calculated in 0.56 degree (200 kpc) -wide bins in
Figure 6. Thus, we take the mean Faraday depth contribution of the cluster
plasma to be
$\sigma_{\phi_{\mathrm{peak,res,cluster}}}=\sqrt{20.5^{2}-11.8^{2}}=16.8\pm
2.4$ rad m-2, where the quoted uncertainty range corresponds to the 95%
confidence interval calculated via bootstrap re-sampling.

Apart from their difference in $\phi_{\mathrm{peak,res}}$ dispersion, the
distributions are otherwise similar — the skew and excess kurtosis of both do
not significantly differ from the values expected for the Normal distribution
(i.e. zero), for example.

Figure 6: Foreground-corrected Faraday depth ($\phi_{\mathrm{peak,res}}$)
versus projected distance from NGC 1399. The foreground was removed as
described in Section 5.1. Data points within one degree (indicated by the red
shaded region) show an excess dispersion, as described in Section 5.1. Sources
that are located inside the Fornax cluster volume, instead of behind it, are
indicated with magenta crosses (see Section 4). Note that all such sources are
in fact sub-components of the central radio source in NGC 1399 (following from
our approach for dealing with heavily resolved sources, discussed in Section
3).

The blue step plot shows the half-interdecile range (i.e. the interdecile
range divided by two) for the data points located within each 0.56 degree (200
kpc) -wide step. The vertical blue bars indicate the associated 90% confidence
interval for the underlying population distribution in each bin, calculated
using bootstrap re-sampling. The right-most axes show normalised cumulative
histograms of $\phi_{\mathrm{peak}}$ for sources located within (red) and
outside (black) a projected distance of one degree. The red shading highlights
the difference between these distributions.

Figure 7: The median of |$\phi_{\mathrm{peak,res}}$| in a sliding window of
width 0.5 degrees as a function of the cluster-centric radius of the outer
bound of this window (blue line). The blue-shaded region indicates the 95%
confidence interval on this value, calculated as $\pm
1.58\times\text{IQR}/\sqrt{n}$ (McGill et al., 1978), where IQR and $n$ are
the interquartile range and number of measurements (respectively) of
|$\phi_{\mathrm{peak,res}}$| in the sliding window. A sharp and significant
decrease in the plotted values is evident when the outer bound of the window
passes a cluster-centric radius of 1 degree, which is marked with a vertical
red dashed line. The width of the sliding window is indicated by the gray
shaded region. Figure 8: A map of $\phi_{\mathrm{peak,res}}^{2}$ across the
field, employing nearest neighbour interpolation, as described in Section 5.1.
The region shown is indicated in its broader context in Figure 3 with a white
dashed box. Each cell contains a single polarised source, and is colorised by
the sources’ value of $\phi_{\mathrm{peak,res}}^{2}$. The extent of X-ray
emission from the Fornax cluster ICM as seen by _Chandra_ (0.3–1.5 keV
bandpass; light blue contours; smoothed to 2.5 arcminute resolution; Scharf et
al., 2005) and the _ROSAT_ Position-Sensitive Proportional Counter (PSPC;
0.1—2.4 keV; smoothed to 3 arcminute resolution; pink contours; Jones et al.,
1997) is indicated. White contours show Fornax A. The white dashed circle
indicates 1∘ projected distance — the projected distance inside which the
variance of $\phi_{\mathrm{peak,res}}$ was found to be enhanced in Figure 6.
The white dashed ellipse roughly indicates where
$\phi_{\mathrm{peak,res}}^{2}$ values appear to be elevated in a contiguous
region near Fornax A. The white dotted line indicates the 1.96 degree (705
kpc) virial radius of the cluster (Iodice et al., 2017). The blacked-out
polygons indicate sources which fall more than $3.5^{\circ}$ from the mosaic
centre, and which are therefore excluded from our polarimetric analysis
(Section 3).

####  5.1.2 Spatial morphology of the Faraday depth enhancements

The morphology of the $\phi_{\mathrm{peak,res}}$ enhancement is revealed in
Figure 8, where we interpolate (using the nearest neighbour method) and plot
$\phi_{\mathrm{peak,res}}^{2}$ as a function of position. Contours showing the
observable extent of the X-ray-emitting ICM as seen by _Chandra_ (Scharf et
al., 2005) and _ROSAT_ (Jones et al., 1997) are overlaid, as are contours for
the radio galaxy Fornax A (see Norris _et al. submitted_). We note that the
_Chandra_ and _ROSAT_ X-ray maps are smoothed to 2.5 and 3 arcminutes
respectively to reveal the faint diffuse emission. Particularly for the
_ROSAT_ contours, the apparent degree of extension towards the southwest is
affected by the smoothing required combined with the presence of bright point
sources unrelated to the cluster medium. The ICM is therefore more asymmetric
around NGC 1399 — more ‘swept back’ towards northeast — than the _ROSAT_
contours initially seem to imply (echoing the morphology of the _Chandra_
contours). The dispersion in $\phi_{\mathrm{peak,res}}$ is clearly enhanced in
the vicinity of the main cluster, extending to a radius of one degree in most
directions, and to 1.5–2 degrees towards the N and NW. The radial extent of
the enhanced region exceeds that of the currently observable X-ray emitting
ICM by a factor of 2–4 depending on azimuthal bearing, but lies within the
1.96 degree (705 kpc) virial radius of the cluster (Iodice et al., 2017;
indicated on Fig. 8).

The global $\phi_{\mathrm{peak,res}}$ enhancement appears to be comprised of
two smaller sub-regions: the first, a circular sector of angle $\sim
90^{\circ}$, with its vertex located near NGC 1399, and its two enclosing
radii oriented slightly clockwise of N-S and E-W (respectively), and the
second region, a $\sim 0.5^{\circ}$-wide strip centred $\sim 0.75^{\circ}$ SW
of NGC 1399, slightly concave towards it. The sharp decline in
$\phi_{\mathrm{peak,res}}$ dispersion at 1 degree cluster-centric radius is
particularly evident at the outer edge of this feature. This ‘two region’
interpretation is bolstered when the sign of the $\phi_{\mathrm{peak,res}}$
values are considered (see Figure 9). The SW enhancement shows predominantly
negative $\phi_{\mathrm{peak,res}}$ values, indicating that the magnetic field
in this structure is oriented predominantly away from the observer.
Conversely, the NE enhancement does not reveal an obvious bias towards either
positive or negative values, implying that the magnetic field in this region
is more isotropic.

Figure 9: _Left:_ As for Figure 8, but zoomed on the main Fornax cluster, and
with $|\phi_{\mathrm{peak,res}}^{2}|<200$ rad2m-4 masked (appearing black).
The two distinct regions of enhancement described in the main text are
delineated by white dashed lines. _Right:_ As for the left panel, but showing
$\text{sign}(\phi_{\mathrm{peak,res}})~{}\times~{}\phi_{\mathrm{peak,res}}^{2}$.

We note the existence of another possible region of enhanced
$\phi_{\mathrm{peak,res}}$ dispersion, which runs adjacent to the Fornax A
lobes to their NE, and which is not obviously attributable to calibration or
image artifacts. While comprised of only six sources, the
$\phi_{\mathrm{peak,res}}^{2}$ values are clearly elevated above the typical
surrounding values for field sources — values for four of the sources lie at
$>3\sigma$, with two more at $>2\sigma$. The probability of this happening in
a contiguous region is correspondingly small.

###  5.2 Flux densities of discrete sources are reduced near the cluster

Figure 10: Binned polarised source counts, polarised and total radio flux, and
fractional polarisation statistics, calculated in equal $\pi$-square-degree
annular bins centred on NGC 1399, and plotted against the bounding radius of
each annulus. Details of the experiment are described in Appendix B. _Panel 1
(top):_ The observed polarised source counts in each annular bin, expressed as
the average number per square-degree. The grey bands indicate confidence
intervals of 67% (dark grey), 95% (mid grey), and 99.7% (light grey) around
the expected count of 27 polarised sources per square degree (black horizontal
line). _Panel 2:_ Integrated polarised (red) and total (blue) flux for ASKAP
(joined dots), NVSS (‘+’ symbols) and GLEAM (‘x’ symbols; total intensity
only) in each annular bin, with the NVSS and GLEAM fluxes scaled to those
expected in the ASKAP frequency band by assuming a spectral index of $-0.7$.
The fractional uncertainty on the plotted quantities is generally less than
1%, and so are not indicated. The shaded areas indicate confidence intervals
of 67% (dark shading), 95% (medium shading), and 99.7% (light shading) for the
ASKAP-derived polarised (red-shaded) and total intensity (blue-shaded)
quantities, which have each been truncated horizontally in the plot for
clarity. _Panel 3:_ As for panel 2, but here for the median flux in each
annulus rather than its sum. _Panel 4 (bottom):_ The median fractional
polarisation of sources in each annulus, calculated on a per-source basis.
Confidence intervals are represented as above. Figure 11: Normalised
integrated source counts versus the logarithm of polarised (dashed lines) and
total (solid lines) flux density inside (red lines) and outside (black lines)
1 degree projected cluster-centric radius.

An apparent paucity of bright polarised sources in the vicinity of NGC 1399
(see the RMS map shown in Figure 3) motivated us to consider whether small-
scale structure in the cluster medium could be depolarising background sources
(e.g. Burn, 1966; Murgia et al., 2004; Bonafede et al., 2011). A 2-sample
Kolmogorov–Smirnov test comparing the polarised intensity distribution of our
sample inside versus outside 1 degree results in a D-value of 0.22 with a
p-value of 0.001, suggesting the lower apparent polarised flux near the
cluster is statistically significant at the $\sim 3.3\sigma$ level. The
question is then whether sources near the cluster are fewer, fainter, or both,
and whether depolarisation or some other mechanism is responsible for this.

We investigated this by calculating the polarised source counts, and the
integrated and median polarised and total flux, in equal-area ($\pi$-square
degrees) annular bins centred on NGC 1399. The experiment is described in more
detail in Appendix B, while the results are shown in Figure 10. The polarised
source counts (top panel of Figure 10) show a $2\sigma$ decrement inside one
degree radius (averaging 17 polarised sources per square degree), a $2\sigma$
_enhancement_ between 1 and 2 degrees radius (averaging 34 polarised sources
per square degree), but are consistent with our average 27 polarised sources
per square degree thereafter. Taken on their own, these deviations are not
statistically significant in a sky area the size of our mosaic. However, the
plots of the integrated and median flux densities (middle and lower panels of
Figure 10, respectively) mirror the behaviour of the source count plot, and
here the deviations are significant. The integrated polarised and total fluxes
(respectively) show $\sim 3\sigma$ and $\sim 2\sigma$ decrements relative to
the expectation from the broader mosaic inside 1 degree radius, corresponding
to a 50% decrement in polarised flux in the former case. The data then show a
$\sim 2.5\sigma$ and $\sim 4\sigma$ _enhancement_ in the integrated polarised
and total flux (respectively) between 1 and 2 degrees radius. In total
intensity only, the enhancement continues to fall outside the 99.7% confidence
interval until a radius of 2.3 degrees, beyond which it returns to low-
significance deviations from the expectation value, along with the polarised
flux. The median polarised and total fluxes also show $\sim 2.5\sigma$ and
$\sim 3\sigma$ decrements (respectively) inside 1 degree radius, but less
significant deviations in other bins. This may indicate that the flux
decrement inside 1 degree is driven by the bulk of the sources therein, while
the flux enhancement from 1–2.3 degrees may be driven by a smaller proportion
of brighter sources. Finally, the median fractional polarisation shows no
evidence for a decrement at small cluster-centric radii, as was found by
Bonafede et al. (2011) for a sample of clusters, who attributed the effect to
depolarisation by turbulent cells in the ICM as per our motivating hypothesis
above. Instead, we find that the median fractional polarisation is $\sim 5$%,
which is consistent with the values that Bonafede et al. (2011) derive at
cluster-centric distances larger than 1.5 effective core radii (see figure 2
of that work). However, this comparison comes with the caveats that (a) our
polarised sources at small cluster-centric radii is insufficient to probe the
core ICM regions where Bonafede et al. (2011) observed the depolarisation
effect, and (b) our RM grid sources are confirmed to lie exclusively behind
the Fornax cluster, whereas Bonafede et al. (2011)’s sample was a
heterogeneous mixture of sources embedded in the clusters being measured, as
well as in the background (though see their Appendix A.1 for arguments that
this cannot explain their results). In the future, it might be possible to
probe closer to the core of the Fornax cluster using the central radio source
hosted in NGC 1399 (e.g. Killeen et al., 1988), at which point a more detailed
comparison would be appropriate.

We note that the decrement and enhancement structures described above are
visually apparent in Figure 3 (in polarisation). The results are unusual for
the Stokes $I$ emission in particular, for it is not obvious how emission or
transmission processes linked to the cluster could produce them (see
discussion in Section 6.7). A mundane possibility is that, given the fact that
the data were collected during ASKAP’s Early Science phase, the effect could
be instrumental. We rule this out conclusively by including data from the
GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM; 72–231
MHz; Wayth et al., 2015; Hurley-Walker et al., 2017) and NVSS (1.4 GHz; Condon
et al., 1998) survey catalogues in the middle panel of Figure 10. We applied
the same spatial truncations to these data as to our sample, and then scaled
their fluxes to those expected in the ASKAP band assuming a spectral index of
$-0.7$. Evidently, the modified total intensity data from GLEAM and NVSS track
the ASKAP data closely, as does the polarised intensity data from NVSS, ruling
out instrumental effects. A final question is whether these effects are
observed over a range of flux densities. In Figure 11, we plot the normalised
integrated source counts versus flux density, which we calculate both inside
and outside 1 degree cluster-centric radius, for both polarised and total
intensity. The data show that the decrement inside 1 degree radius persists
over a wide range in flux density.

Thus, we draw the following conclusions: (1) the polarised source counts, the
polarised flux densities, and the total flux densities _all_ show decrements
inside 1 degree projected radius from the cluster; (2) the average magnitude
of the decrement is $\sim 50\%$ for the polarised flux, and $\sim 30\%$ for
the total intensity; (3) between 1 and 2.3 degrees projected radius, there is
a surfeit of flux in one or both quantities; (4) none of these results can be
attributed to a normalisation problem, instrumental effect, or the modest
$\sim 50$% increase in image noise that occurs in a small $\sim 20\times 10$
arcminute region around PKS B0336–35 (see Figure 3 and Section 3); (5) the
coexistent decrements in Stokes $I$ and $P$ are inconsistent with our initial
hypothesis that depolarisation by turbulent cells in the Fornax ICM could
cause the decrement in $P$ (cf. Bonafede et al., 2011). We discuss these
results further in Section 6.7.

###  5.3 Summary of observational results

We summarise our observational results as follows:

  * •

The distribution of peak Faraday depths for discrete radio sources shows an
excess scatter of $\sigma_{\phi_{\mathrm{peak,res,cluster}}}=16.8\pm 2.4$ rad
m-2 within 1 degree (360 kpc) of the Fornax cluster centre. In more spatially-
limited areas, the excess can be traced out to the $\sim 1.96$ degree (705
kpc) virial radius of the cluster.

  * •

The projected area of the Faraday depth enhancement extends 2–4 times the
projected distance of the X-ray emitting ICM, though is mostly contained
within the cluster’s virial radius.

  * •

The global Faraday depth enhancement naturally divides into two distinct
(projected) morphological sub-regions: (1) A triangular region extending from
its vertex near NGC 1399 towards its apparent base about $\sim 1.5^{\circ}$ to
the NE, with a mix of both positive and negative $\phi_{\mathrm{peak,res}}$
values, and (2) a banana-shaped strip of width $\sim 0.5^{\circ}$ and length
$\sim 2.5^{\circ}$, curving slightly around NGC 1399, but centred $\sim
0.75^{\circ}$ to its SW, and having predominantly negative
$\phi_{\mathrm{peak,res}}$ values.

  * •

On average, the areal total and polarised radio emission density is $\sim
30\%$ and $\sim 50\%$ lower within 1 degree (360 kpc) of the Fornax cluster
(respectively) compared to outside this radius. Cumulative source counts
versus flux density show that the emission decrement persists over the full
range of flux densities that are effectively probed by our observations and
sample.

##  6 Discussion

###  6.1 Location of the Faraday screen along the line of sight

There are two possible locations for the excess Faraday rotating material
identified in Section 5.1: In our Galaxy, or in the Fornax cluster. The field
contains intervening Galactic emission in H i, H$\alpha$ and soft X-rays. The
interstellar medium thereby traced is Faraday-active, appearing capable of
inducing mild depolarisation via _differential_ Faraday rotation in spatially-
limited regions due to its arcsecond-scale ionisation structure (Anderson et
al., 2015). The question is whether degree-scale Galactic foregrounds can
enhance variance in _net_ Faraday rotation over arcminute scales. We consider
this unlikely, because:

  * •

Anderson et al. (2015) showed that in this field (i.e. centred on Galactic
$l,b$ of 235.988∘, $-$55.484∘), over spatial scales of several arcminutes, the
expected contribution of the Galactic foreground to RM variance is only $\sim
50$ rad2m-4 (see Figure 36 and Section 7.2.1 of that work). The
$282^{+86}_{-75}$ rad2m-4 variance in the vicinity of the cluster is $\sim
6\pm 2$ times higher than this.

  * •

We cannot identify any foreground structures in maps of Parkes Galactic All
Sky Survey (GASS; McClure-Griffiths et al., 2009) H i column density, _Planck_
spectral brightness, _ROSAT_ soft X-rays, or H$\alpha$ photon flux that map to
precisely the same region in which the Faraday depth enhancements occur
(though structure in such emission is present in the field — see Figure 37 of
Anderson et al., 2015).

  * •

The prior probability that a degree-scale Galactic Faraday depth enhancement
would happen to centre precisely on a pre-existing and well-defined position
of interest is not well known, but must surely be small.

In what follows then, we assume that the Faraday active material is physically
associated with the Fornax cluster.

###  6.2 Mass and phase of the ionised cluster gas

Little is concretely known about the structure of magnetised plasma in the
periphery of low-mass galaxy clusters, including the strength and
characteristic scale-length of the turbulent and regular magnetic fields in
the rarefied plasma. In the case of the Fornax cluster, our difficulties are
compounded by the fact that the Fornax cluster is not relaxed, but is being
disturbed by ongoing mass-assembly processes. This can distribute $\mu$G-level
magnetic fields throughout cluster volumes (e.g. Xu et al., 2009), inducing
significant variability in the Faraday rotation signal of clusters with
otherwise similar properties (Xu et al., 2011), and degeneracies in the
interpretation of such data (Johnson et al., 2020). Nevertheless, following
Anderson et al. (2018), we can crudely estimate the baryonic gas mass
generating the Faraday depth enhancement as follows. The peak Faraday depth of
a column of thermal electrons threaded by a uniform magnetic field is (e.g.
Heiles & Haverkorn 2012):

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle 0.81n_{e}B_{u,||}L$ (5)
$\displaystyle=$ $\displaystyle 26N_{e,20}B_{u,||}~{}\text{rad m}^{-2}$

where $B_{u,||}$ is the strength of the magnetic field projected along the
line of sight [$\mu$G], and $N_{e,20}$ is the electron column density in units
of $10^{20}$ cm-2. From Figure 8, we estimate the solid angle occupied by the
Faraday depth enhancements to be $4.1\times 10^{7}$ square arcseconds,
corresponding to a projected area $A=3.8\times 10^{48}$ cm2 at the distance of
the cluster. Assuming the Faraday-active plasma is dominated by ionised
hydrogen and helium nuclei with a typical ICM abundance ratio of 9:1
(respectively), and thus a mass density of $1.17n_{e}m_{p}$ (where $m_{p}$ is
the mass of a proton), the total mass of baryonic material is roughly:

$\displaystyle M_{\text{Thermal}}$ $\displaystyle\approx$ $\displaystyle
1.17\times 10^{20}N_{e,20}Am_{\text{p}}$ (6)

Defining the characteristic Faraday depth of the material as
$\phi_{\mathrm{peak,res,char}}$, then combining Eqns. 5 and 6 yields:

$\displaystyle M_{\text{Thermal}}$ $\displaystyle\approx$ $\displaystyle
1.39\times
10^{10}\bigg{(}\frac{\phi_{\mathrm{peak,res,char}}}{B_{u,||}}\bigg{)}~{}M_{\odot}$
(7)

Using
$\phi_{\mathrm{peak,res,char}}\sim\sigma_{\phi_{\mathrm{peak,res,cluster}}}\approx
17$ rad m-2, we get $M_{\text{Thermal}}\approx 2.4\times
10^{11}\big{(}\frac{1}{B\mu\text{G}}\big{)}~{}M_{\odot}$, which for
comparison, is approximately three times the mass of the X-ray-emitting hot
ICM (but see Johnson et al., 2020 for discussion of inherent uncertainties in
such measurements). If there are typically $N$ magnetic field reversals along
the line of sight to sources in our sample, $M_{\text{Thermal}}$ will be
larger by a factor of $\sim N^{1/2}$.

The material extends well beyond the presently-detectable hot ICM, typically
to a distance of 360 kpc. This yields an average thermal electron density
inside this volume of $n_{e}\approx 4.3\times
10^{-5}\big{(}\frac{1}{B\mu\text{G}}\big{)}N^{1/2}f^{-1}$ cm-3, where $f$ is
the volume-filling factor. If $B$, $N$, and $f$ are equal to 1, it represents
a baryonic over-density of $\delta\sim 215$ — a regime which is obviously very
different to the hot ICM, for which $\delta\gg 1000$ for the inner regions of
the cluster, but which drops off precipitously to $\delta\sim 50$ by 180 kpc
from the cluster centre, and which is projected to drop by a further two
orders of magnitude to negligible levels by 360 kpc (based on _ROSAT_
measurements of soft X-ray emission; see Sections 2.6 and 2.6 of Paolillo et
al. 2002). Instead, it is comparable to the radially-averaged characteristic
density of the intergalactic medium expected to inhabit $\sim
10^{13}~{}M_{\odot}$ galaxy groups and poor clusters (Davé et al., 2010; Haan
& Braun, 2014), and the moderately-dense phases of the WHIM thought to
surround galaxy clusters (Davé et al., 2001; Yoon et al., 2012).
Alternatively, it is possible (or perhaps likely) that the magnetic field
strength $B$ is lower than 1 $\mu$G, or that the number of magnetic field
reversals along typical lines of sight $N$ is large, or that the volume
filling factor $f$ is small, or some combination of the above. But it each
case, this would tend to raise our total gas mass estimate. While this might
bring the baryonic over-density more into line with expectations for a hot ICM
phase in a cluster environment, its radial distribution would be rendered even
more inconsistent with direct measurements of radial gas densities in this hot
phase reported by Paolillo et al. (2002). Thus, we claim to have directly
detected a moderately-dense phase of the diffuse WHIM via Faraday rotation,
which is either too cool, too diffuse, or both, to have been detected by X-ray
observatories to date (though this may be addressed with future observations
by eROSITA for example; Merloni et al., 2012). This appears to be made
possible by cluster dynamic processes that act to organize and amplify the
embedded magnetic field. We discuss this further in Section 6.3.

###  6.3 Structure and origin of the Faraday depth enhancements

The morphology of the Faraday depth enhancement (Section 5.1.2; Figures 9,12)
is reminiscent of the bow shocks seen around merging cluster cores (e.g.
Markevitch & Vikhlinin, 2007), and astrophysical bow shocks more generally. If
a shock system does exist, it could be either a stationary bow-shock caused by
interaction between the NE and SW sub-clusters (Drinkwater et al., 2001;
Scharf et al., 2005), or a propagating shock, set up by merger activity in the
main (i.e. NE) sub-cluster itself (Sheardown et al., 2018). We consider these
possibilities in turn.

####  6.3.1 NE-SW sub-cluster merger

Drinkwater et al. (2001) proposed that the Fornax cluster can be partitioned
into NE and SW clumps, which are likely merging at speeds between 100 and 500
km s-1. Scharf et al. (2005) interpreted the swept-back (to the NE) morphology
of the X-ray-emitting main cluster ICM as evidence to support this.
Extrapolating X-ray-derived temperature and density measurements from Jones et
al. (1997) to the approximate radius of the shock front (1 degree) yields an
estimated sound speed of $c_{s}=1480\sqrt{T/10^{8}~{}\text{K}}\approx 420$ km
s-1 (Sarazin, 2002), and an Alfvén speed of $c_{A}=\sqrt{B^{2}/\rho}\approx
320$ km s-1. The merger speed $V$ may therefore be transonic, in which case a
bow shock could form in diffuse gas between the merging components.

We propose that the features apparent in our maps of
$\phi_{\mathrm{peak,res}}^{2}$ (Figure 9) may correspond to the canonical
features of a stationary shock leading a blunt object as illustrated in Figure
12 (left panel). That is, in this instance (a) a concave shock front centred
around the X-ray-emitting ICM, whose point of closest approach is located 360
kpc to its SW, (b) a contact discontinuity between the main cluster gas and
that in a more diffuse extended envelope, located perhaps one third of the way
from NGC 1399 to the leading shock, and (c) a triangular extension of the
X-ray-emitting ICM to the NE, representing a less-dense and less-hot phase of
the magneto-ionised ICM stripped from the NE cluster. The boundary of this
region may be associated with magnetic fields draped over the X-ray emitting
ICM and amplified there — expected to appear for any super-Alfvénic merger in
magnetised media (Lyutikov, 2006).

Figure 12: Physical features for the NE-SW sub-cluster merger (left panel) and
NGC 1404 merger (right panel) scenarios, described in Section 6.3. _Left
panel:_ Light blue filled circle: X-ray emitting ICM; Light blue arrows:
Turbulent eddies in stripped ICM; Black lines: Canonical features of
astrophysical shocks, as discussed in the main text; X-ray emitting ICM; Red
dotted line: Shock stand-off distance $d_{s}$; Purple dashed line: Projected
angle of shock front. Note that the angle between the red-dotted and purple-
dashed lines is the Mach angle referred to in Section 6.3.1. _Right panel:_
Large light blue filled circle: NGC 1399; Small light blue filled circle: NGC
1404; Light blue blobs: Wake features generated by NGC 1404 in-fall, as
described in the main text; Red line and arrows: Path of in-falling NGC 1404;
Black line: Detached moving bow shock generated by NGC 1404 in-fall.

If the features described in the previous paragraph are correctly identified,
then since the radius of the X-ray-emitting medium is small compared with the
length of the shock front, we can estimate the Mach number of the shock via
its Mach angle and stand-off distance (see Figure 12). The Mach angle is given
by $\mu=\text{sin}^{-1}({1/M_{s}})$, where $M_{s}$ is the sonic Mach number.
From visual inspection of Figure 9, $\mu\approx 70\pm 5^{\circ}$ yielding
$M_{s}=1.06^{+0.04}_{-0.03}$, placing the merger in the transonic regime. The
shock stand-off distance $d_{s}$ is a sensitive and independent measure and
test of this claim (Sarazin, 2002) — it depends only on the value of $M_{s}$
and the effective shape of the supersonic object. A prediction for the stand-
off distance versus Mach number is given by Sarazin (2002, Figure 4). For our
best estimate of $M_{s}=1.06$, we expect $d_{s}/R\approx 7$, where $R$ is the
radius of curvature of the dense core. For the dense X-ray-emitting ICM, we
estimate $R\approx 7$ arcminutes in the SW direction using X-ray contours
provided by Paolillo et al. (2002) and Scharf et al. (2005) (also see Figure
5), and so expect the bow shock front to be located $\sim 49$ arcminutes SW of
the cluster. This is essentially consistent with the location of the proposed
shock front in our maps (1 degree to the SW of the cluster core).

This sub-cluster merger model predicts several features that could be used to
confirm or refute it in future, and which might then be used to measure the
properties of the gas. First, a generic feature of super-Alfvénic mergers is
that magnetic fields should be ‘collected’ and amplified along a thin conic
layer by the dense cluster ICM upon its passage through the more extended
diffuse envelope. This is known as magnetic draping (Lyutikov, 2006), and is
expected to occur for a wide range of merger speeds and magnetic field
configurations. The expected minimum thickness for the magnetic field
amplification layer is $\sim 2Rc_{A}/V\approx 10$ arcminutes (Lyutikov, 2006),
presumably draped over the leading edge of the dense core $\sim 7$ arcminutes
to the SW of NGC 1399. At such a pressure-balance surface, we expect that (a)
a spatial discontinuity in X-ray emission may be observed (e.g. Machacek et
al., 2005), and (b) the magnetic pressure will roughly equal the ram pressure
in the zone of plasma compression, such that $B^{2}\sim 8\pi\rho V^{2}$. For
the NE-SW merger speed and ICM densities cited above, this implies a magnetic
field strength of $\sim 8~{}\mu$G. Both expectations are interesting, for they
have been raised by Su et al. (2017) in the context of a sloshing cold front
identified in this system at approximately the expected location (identified
as the "F2" front in that work), and the hypothesis that such a cold front can
be stabilised against Kelvin-Helmholtz instabilities via magnetic tension, if
only strong enough fields can be generated. The stabilisation requires a
magnetic field running locally parallel to the front with strength $\sim
10\mu$G (Vikhlinin et al., 2001), which is evidently similar to the value we
obtain for ram-pressure balance in the system. Thus, in our merger model, we
suggest that the cold front identified by Su et al. (2017) may act as the
effective surface of ram-pressure balance with a diffuse ionised medium
through which the dense core of the cluster moves with transonic speed. The
cold front might stabilize itself via the very magnetic draping and
compression that the NE-SW sub-cluster merger induces.

A second feature that might be sought to furnish evidence for or against our
model is as follows. In the wake of blunt objects embedded in a hydrodynamic
flow where the Reynolds number exceeds $\sim 50$ (most likely applicable
throughout galaxy clusters — e.g. Zhuravleva et al., 2019), vortex shedding
occurs (e.g. Lienhard, 1966). This terms refers to a phenomenon where vortices
are created behind a blunt object embedded in a fluid flow, and are
periodically jettisoned into the flow. The vortex shedding frequency ($f$) is
related to $L$ and the flow speed of the external fluid ($U$) by the Strouhal
number ($St$) as $St=(fL)/U$, where $St\approx 0.2$ over a wide range of
Reynolds numbers (e.g. Sakamoto & Haniu, 1990; Gruszecki et al., 2010). This
defines a characteristic longitudinal scale for the vortex separation, while
the transverse scale of the vortices is comparable to that of the blunt object
($L$). Thus, in the wake of the dense Fornax cluster ICM, we expect to see
coherent vortical structures ranging in size from $\sim 14$–70 arcminutes
(84–420 kpc). This may also correspond to an injection scale for a subsequent
turbulent cascade, whose detailed character and volume filling factor will
depend on the effective Reynolds number of the medium (among other factors)
(e.g. Subramanian et al., 2006; Zhuravleva et al., 2019). These observational
signatures are not detectable in our present observations, but may be in
pending deep observations of the region (see Section 7).

Finally, we note that in Section 5.1.2, we proposed that an enhancement in
Faraday depth can be seen running adjacent to the NE of the long axis of the
Fornax A lobes. Ekers et al. (1983) previously argued that the Fornax A system
was moving to the N or NE due to a $\sim$ southerly offset of a diffuse radio
bridge between the lobes and its radio core. Both this feature, and the
purported Faraday depth enhancement to the NE, might be naturally explained as
a consequence of gas and magnetic field compression in the sub-cluster merger
model.

####  6.3.2 NGC 1404 in-fall

Recently, Sheardown et al. (2018) simulated the in-fall and merger of the
galaxy NGC 1404 to the main Fornax cluster, and highlighted several features
that bear an intriguing correspondence with features in our data. Their
preferred simulated in-fall scenario produces: (a) a detached bow shock from
NGC 1404’s previous orbit in the cluster potential, having a Mach number in
the range 1.3–1.5, currently propagating outwards between 450 and 750 kpc S
and SW of NGC 1399, and (b) regions of relatively cold turbulent material
distributed through much of the cluster, but particularly towards the NE,
again stirred up by NGC 1404’s previous orbits through the system (see Figure
12, right panel).

Qualitatively, these predictions are similar to the sub-cluster merger
scenario above, and can explain our results almost equally well. A minor point
of disagreement is that our putative shock is located 20% closer to NGC 1399
than their lower limit on this distance. Nevertheless, we are reluctant to
claim that our results strongly favour either one of the models with the
present data. In the near future, the MeerKAT Fornax Survey (Serra et al.,
2016) will obtain much deeper observations, and may be able to trace (via
Faraday rotation or depolarisation) the trails of cold gas and turbulence that
Sheardown et al. (2018) predict will follow in the wake behind NGC 1404.

###  6.4 Comparison with cluster stacking experiments

The excess scatter ($\sigma_{\phi_{\mathrm{peak,res,cluster}}}=16.8$ rad m-2)
and spatial extent (360 kpc radius) of the RM enhancement are both smaller
than reported for ensembles of galaxy clusters in RM stacking experiments.
Clarke et al. (2001) report a standard deviation in RM of 114 rad m-2 for
sources located within the projected radius of galaxy clusters, compared to 15
rad m-2 for a control sample, which we note is similar to our value of 11.8
rad m-2 for our field sources. More recently, Böhringer et al. (2016) report a
value of 120 rad m-2 for sources located within the projected radius of
clusters. However, Böhringer et al. (2016) also show that this scatter is
driven mainly by massive clusters: When their sample is split by the median
X-ray luminosity of their associated cluster (corresponding to a total mass of
$1.4\times 10^{14}$ M⊙), the standard deviation in RM increases to $158\pm 34$
rad m-2 for sources associated with clusters in the upper half of the mass
range, and decreases to $62\pm 11$ rad m-2 for the converse sources. Our lower
derived value of of $16.8\pm 2.4$ rad m-2 for the RM scatter seems broadly
consistent with this coarse mass dependence, given that the total mass of the
Fornax cluster is only $6^{+3}_{-1}\times 10^{13}$ M⊙ (Drinkwater et al.,
2001; Nasonova et al., 2011; Maddox et al., 2019) inside a few Mpc, which is
half the mass cut value used by Böhringer et al. (2016). It is also comparable
to results from simulations for unrelaxed clusters of approximately the same
mass (see Figure 14 of Xu et al., 2011, for example). However, in light of our
conclusions about the nature of the ionised gas from Section 6.2, Böhringer
_et. al_ ’s statement that the relationship between RM scatter and cluster
mass confirms "that the observed excess scatter in the RM in the lines of
sight of galaxy clusters is due to the cluster ICM" may require a minor
qualification, in the sense that some portion of the RM scatter may be driven
by plasma regimes that differ from the canonical hot ICM.

The Clarke et al. (2001) and Böhringer et al. (2016) studies, and results from
other recent works of simulation and observation (Marinacci et al., 2018 and
references therein), show that cluster-based RM enhancements are typically a
strong function of radius. The central 200 kpc of the cluster ensemble shows
|RM| values often exceeding 100 rad m-2, but these values decrease roughly
exponentially until they become indistinguishable from field source RMs beyond
$\sim 800$ kpc. The behaviour of the Fornax cluster RMs appear to be
qualitatively different: The degree of RM scatter is quite uniform out to a
projected radius of 360 kpc, at which point it drops precipitously to roughly
half its value (Figures 6 and 7), and thereafter remains constant with
projected radius. This qualitative difference could be due to several factors,
possibly including that (a) the plasmas traced by our RMs compared to the
stacking experiments differ in their phase or properties at various fiducial
cluster-centric radii, most plausibly because the Fornax cluster is
exceptionally poor in terms of the number of member galaxies, gas mass, and
total mass (b) the Fornax ICM is being disturbed by ongoing mass assembly, or
(c) clusters that contribute to stacking experiments may individually show
similar behaviour, but this signal is averaged out in the stacking. Further
observations of a range of individual clusters — now possible with ASKAP and
other instruments — are obviously the key to exploring these possibilities.

###  6.5 Comparison with studies of individual low-mass galaxy clusters

What studies do exist of the magnetised ICM in individual low-mass clusters
generally involve mapping the RMs across the lobes of the central dominant
radio galaxy (e.g. Perley et al., 1984; Laing et al., 2008; Guidetti et al.,
2010; Govoni et al., 2017). This provides an exquisite view of magnetic
structure in the inner-most regions of the ICM, but little information beyond
the range where the the radio lobes are bright enough to calculate reliable
RMs (10s–100 kpc). In addition, the fields are likely modified by interactions
with the radio galaxies themselves (Guidetti et al., 2012). RM grid studies
avoid both of these problems, since the sources are not in physical proximity
to the medium being probed, and generally provide abundant information beyond
100 kpc, but limited information inside this distance. While there are not yet
sufficient data to connect the results of these experiments across the gap in
scales, some brief comments are nevertheless warranted.

Laing et al. (2008); Guidetti et al. (2010) and Govoni et al. (2017) mapped
the RMs of centrally-located radio lobes to study the magnetised ICM of poor
clusters with $\sim$ hundreds of member galaxies, masses in the range
4–$9\times 10^{13}~{}M_{\odot}$ (Komossa & Böhringer, 1999; Nikogossyan et
al., 1999), and central magnetic fields strengths of 3–10 $\muup$G, making
them broadly comparable to the Fornax cluster. These studies measure RM
dispersions of 0–40 rad m-2, which our measured enhancement of 17 rad m-2
falls squarely inside. For those studies however, the RM dispersion measured
against the lobes generally drops to a few rad m-2 by cluster-centric
distances of 100 kpc (see Figures 9 and 12 of Laing et al., 2008 and Guidetti
et al., 2010 respectively, and Figure 14 of Govoni et al., 2010), whereas the
Faraday depth enhancement in the Fornax cluster is maintained out to at least
360 kpc. The studies cited above also attributed the enhanced Faraday
dispersion to the cluster ICM, and indeed, the scale-size of X-ray emission
from the hot ICM and region of enhanced Faraday depth are well matched, unlike
the case for the Fornax cluster. It is not clear how to reconcile these
results with the present data. It is possible that the Fornax cluster is being
fed with more warm-hot gas from filaments of large-scale structure, or that
similar reservoirs of gas exist in the vicinity of the other clusters but have
not been detected yet, or that this gas exists but can only be detected in
certain situations — in the case of the Fornax cluster we speculate, because
its ongoing merger status organises the magnetic field structure to as to
induce a detectable Faraday rotation signature over a sufficiently large area
of sky. Further observations of a larger sample of similar clusters are
clearly needed.

Laing et al. (2008); Guidetti et al. (2010) speculate that the outer scale of
magnetic field fluctuations — of order 70 kpc in both cases — could be set by
the characteristic separation of interacting cluster members. If this picture
is correct, then the $\sim 3$ degree separation of the merging NE and SW
Fornax sub-clusters might have been used to predict the $\sim$degree-scale
Faraday depth enhancements that we have detected in the cluster _a priori_. In
turn, this may suggest that clusters undergoing similar mergers of massive
sub-components, such as the nearby Antlia (e.g. Caso & Richtler, 2015, and
references therein) and Centaurus (Churazov et al., 1999) clusters, may
represent particularly valuable targets for future RM grid studies, since the
detectable x-ray-emitting ICM may not provide the whole picture. Finally, in
Section 6.3.1, we predicted that if the Faraday depth enhancement to the NE of
the cluster is associated with vortex shedding of the cluster gas, we might
expect to see coherent associated RM structures ranging in size from $\sim
14$–70 arcminutes (84–420 kpc). This scale-size range is apparently
characteristically larger than the RM structures induced by quiescent
evolution in individual poor, relaxed, low-mass clusters, and so if RM
structure is observed on these scales in the Fornax and other clusters, it
might act as a signpost of ongoing merger and gas-stripping activity.

###  6.6 The RM grid as a sensitive tracer of ‘missing’ baryons?

Given our conclusions that the Faraday-active ionised gas is associated with
the WHIM, it is interesting to consider whether Faraday rotation measure grids
(e.g. Gaensler et al., 2004) and associated advanced analysis techniques (e.g.
Vernstrom et al., 2019; O’Sullivan et al., 2020; Stuardi et al., 2020) might
effectively probe such material. The thermal electron density drops rapidly
with cluster-centric radius in typical ICM models, and the efficiency of
Bremsstrahlung as a tracer of this gas drops more precipitously, since its
emissivity is proportional to $n_{e}^{2}$. In contrast, the Faraday effect is
proportional to $n_{e}BL$, assuming no significant separation of magnetic
fields and thermal electrons along the line of sight. It is not clear what
structure magnetic fields generally have at the periphery of clusters, but it
is clear that certain processes can amplify magnetic fields, increase their
degree of order, push their auto-correlation lengths to larger characteristic
scales, or simply inject large-scale magnetic fields to begin with (e.g. sub-
cluster mergers; e.g. Govoni et al., 2005; Bonafede et al., 2009; Girardi et
al., 2016, magnetic draping; e.g. Lyutikov, 2006, shocks; e.g. van Weeren et
al., 2010; Brüggen et al., 2011, subsonic gas motions and shear flows; e.g.
Keshet et al., 2010; Zuhone & Roediger, 2016; Donnert et al., 2018, also this
work). Finally, the periphery of clusters provides Mpc-scale path lengths
through which polarised emission can propagate through such plasmas.
Therefore, it seems reasonable to conclude that Faraday rotation measures
should provide a sensitive means of detecting and studying rarefied warm
plasma phases, assuming a sufficient density of background sources over a wide
enough area. The current generation of radio telescopes will soon provide this
routinely for many galaxy clusters.

###  6.7 Evaluation of the decrements and enhancements in source counts and
flux observed near the Fornax cluster

Following the results and arguments presented in Section 5.2, there are two
two possible causes for the polarised and total intensity emission decrements
observed inside one degree cluster-centric radius: (1) scattering or
absorption by foreground material, or (2) a genuine paucity of radio emission
due to cosmic variance.

The former possibility requires that cool, dense gas lies along the line of
sight. While it can be shown that the required gas could plausibly exist in
the cluster, perhaps in the form of cloudlets of $\sim 10^{4}$ K photoionised
gas that often envelop elliptical galaxies (e.g. Gauthier et al., 2010; Thom
et al., 2012; Prochaska et al., 2013; Farnes et al., 2017; Lan & Fukugita,
2017; Berg et al., 2019; Pradeep et al., 2019; Werner et al., 2019; Liang &
Remming, 2020), we can test this scenario using the frequency-dependence of
the effect. Consider that the free-free optical depth is given by

$\tau_{\text{ff}}\approx 3.27\times
10^{-7}\bigg{(}\frac{T_{e}}{10^{4}~{}\text{K}}\bigg{)}^{-1.35}\bigg{(}\frac{\nu}{\text{GHz}}\bigg{)}^{-2.1}\bigg{(}\frac{\text{EM}}{\text{pc
cm}^{-6}}\bigg{)}$ (8)

where $T_{e}$ is the electron temperature, $\nu$ is the observing frequency,
and EM is the emission measure.

We therefore cross-matched the discrete sources from our sample found within 1
degree of the cluster centre with radio surveys over a wide range of
frequencies (including the GLEAM survey (72–231 MHz; Wayth et al., 2015;
Hurley-Walker et al., 2017) survey, the VLA Low-frequency Sky Survey redux
(VLSSr; 74 MHz; Lane et al., 2014), the TIFR GMRT Sky Survey (TGSS; 150 MHz;
Intema et al., 2017), the Westerbork in the Southern Hemisphere (WISH; 352
MHz; De Breuck et al., 2002) survey, the Molonglo Reference Catalogue (MRC;
408 MHz; Large et al., 1981), the Sydney University Molonglo Sky Survey
(SUMSS; 843 MHz; Bock et al., 1999; Mauch et al., 2003), the NVSS (1.4 GHz;
Condon et al., 1998), the Parkes-MIT-NRAO (PMN; 4.85 GHz; Gregory et al.,
1994; Wright et al., 1994) survey, the Australia Telescope Parkes-MIT-NRAO
(ATPMN; 5 and 8 GHz; McConnell et al., 2012) follow-up survey, and the
Australia Telescope 20 GHz (AT20G; Murphy et al., 2010)) to reconstruct their
total intensity spectra. The result is that no sources brighter than 10
mJy/beam at 887 MHz in this sub-sample possess a spectral peak at frequencies
above $\sim 100$ MHz, and we conclude that the available evidence does not
support the absorption hypothesis.

Considering the cosmic variance scenario in turn, randomly sampling $\pi$
square degrees of sky down to $\sim$mJy beam-1 sensitivity (at which we
continue to see the emission decrement of up to $\sim 50\%$; see Figure 11)
yields a $1\sigma$ uncertainty in source counts of only $\sim 15\%$ from
cosmic variance (Heywood et al., 2013, Figure 2), corresponding to only a
$\sim 0.1\%$ probability that our results can be ascribed to cosmic variance
alone. Nevertheless, decrements of similar depth and angular size have been
observed in NVSS which, it has been claimed, may be associated with supervoids
in the large-scale structure of the Universe (e.g. Rudnick et al., 2007).

The enhancement in polarised and total radio flux, which appears to arc around
the cluster at projected radii between 1 and $\sim 2$ degrees, appears even
more difficult to explain. The handful of redshift cross-matches in this
region indicate distances much greater than the cluster, making it appear to
be a chance effect. We are unaware of known clusters in these locations, but
redshift coverage here is currently relatively sparse.

Nevertheless, remarkable large-scale cosmic structure does appear to exist in
both the foreground and background of the Fornax cluster: Purportedly, a dense
filament of dark matter extends along the entire line of sight between the
Fornax cluster and the Milky Way Galaxy (see Figure 4 of Hong et al., 2020),
while the $\sim 190\times 90\times 140$ Mpc diameter Sculptor void lurks
immediately behind (Tully et al., 2019). At a near-side distance of only $\sim
30$ Mpc, the latter possesses an apparent angular diameter which is far too
large to explain our source decrement/enhancement structures, which cover less
than 10 square degrees. Nevertheless, such voids appear to contain significant
substructure, and are linked to other voids in a sponge-like lattice which
cannot yet be mapped out in detail to the typical distance of powerful radio
sources. We speculate that a lattice of connected under-dense regions could
result in smaller-scale ‘tunnels’, through which the average density and
brightness of radio sources is lower on average, and perhaps chance alignments
of higher-density filaments or walls where the opposite is true. With the
available evidence failing to support scenarios involving instrumental
effects, depolarisation, or absorption, we tentatively conclude that cosmic
variance due to such large scale structure may be the most viable explanation
for our results. Over the next several years, the Evolutionary Map of the
Universe (EMU; Norris et al., 2011) and POSSUM surveys will map the radio sky
with an unprecedented combination of sky coverage, depth, and detail. This
will shed much new light on the 3-dimensional distribution of radio sources
and large-scale cosmic structure, as well as on Faraday rotation associated
with the gas in this cosmic web. Our result can then be revisited and
reinterpreted if necessary.

##  7 Conclusion

We have conducted the first Faraday rotation measure grid study of an
individual low mass galaxy cluster, achieving a polarised source density of 27
per square degree using the revolutionary new survey capabilities of the ASKAP
telescope. Our key results are that:

  * •

The distribution of peak Faraday depths for confirmed background radio sources
shows an excess dispersion of $\sim 17$ rad m-2 within 1 degree (360 kpc) of
the Fornax cluster centre, and in more spatially-limited regions out to the
705 kpc virial radius of the cluster. This is between 2 and 4 times farther
than the projected distance of the currently observable X-ray emitting ICM.

  * •

We estimate that the mass of the Faraday-active gas is $2.4\times
10^{11}~{}M_{\odot}$, which is approximately triple the mass of the hot ICM so
far detected in X-rays, but with a low average density of $n_{e}\approx
2.7\times 10^{-5}$ cm-3. This represents a baryonic over-density (compared to
the cosmic average) of $\delta\sim 215$ found at cluster-centric radii out to
360 kpc. This appears to be markedly different from the hot ICM phase in
Fornax, for which $\delta\gg 1000$ in the core of the cluster, but which is
observed to drop to $\delta\sim 50$ by 180 kpc, and projected to drop to
negligible densities much beyond this.

  * •

Morphologically, the Faraday depth enhancement divides into two regions. We
interpret one as an astrophysical shock front, and the other as the turbulent
main-cluster wake, in a scenario in which NE and SW sub-clusters are merging
at transonic speeds. Alternatively, it may be that the protracted merger of
the dominant cluster galaxy NGC 1399 and the in-falling galaxy NGC 1404 has
produced a detached shock moving to the SW, and has redistributed cold
turbulent gas from the cluster centre and the ISM of NGC 1404 throughout the
cluster.

  * •

On average, the total and polarised radio emission areal density is $\sim
30\%$ and $\sim 50\%$ lower (respectively) within 1 degree of the Fornax
cluster compared to outside this radius. Consistent with this, cumulative
source counts versus flux density show the $\sim 50\%$ emission decrement
persists over a large range of flux densities effectively probed by our
observations and sample. Depolarisation, instrumental effects and image
artifacts, and free-free absorption by a cold and dense gas, are all ruled out
as possible causes or found to be unlikely. Cosmic variance also appears to be
an unlikely explanation, but one which we tentatively favour in lieu of more
compelling evidence. This result should be examined using future, deeper
observations.

We argued that the Faraday-active gas is associated with a moderately dense
phase of the WHIM, in which ongoing merger processes in the cluster continue
to amplify and organize magnetic fields, thereby providing ideal conditions to
trace this material with the Faraday effect. In particular, a shock system
traced by Faraday RMs may confirm that the NE and SW Fornax appear to be
undergoing a transonic merger, as previously described.

These results demonstrate that deep, wide-field RM grid studies have the
capacity to reveal the gas in and around galaxy clusters in a diverse set of
regimes. This being the case, impending deep all-sky linear polarisation
surveys like POSSUM, the VLA Sky Survey (VLASS; Lacy et al., 2019), the
POlarised GLEAM survey (POGS; Riseley et al., 2018, 2020), and the LOFAR Two-
Metre Sky Survey (LoTSS; Shimwell et al., 2017) will all help to revolutionize
our understanding of large samples of such objects, while much deeper targeted
observations like the MeerKAT Fornax Survey (Serra et al., 2016) will reveal a
wealth of structure in individual clusters that has gone heretofore unseen.

Finally, in addition to the RM grid analysis presented here, Anderson et al.
(2015) demonstrated the value of _depolarisation_ grids to study small-scale
magnetoionic structure along the line of sight. In this work, we initially
postulated that ICM depolarisation may be responsible for the reduced counts
and polarised flux we observed in the vicinity of the Fornax cluster, but then
recognised that this signal was largely reflected in total intensity emission,
too. It remains unclear where this underdensity comes from, but it is
reasonable to assume that such fluctuations will be found on these angular
scales elsewhere on the sky, for similarly uncertain reasons (e.g. see Rudnick
et al., 2007). Now that deep polarised source grids are becoming available,
which provide RM and depolarisation information at areal densities of $\sim$
several tens per square-degree, we must be aware of these fluctuations, and
pursue a better understanding of the clustering scales for extragalactic radio
sources.

## Acknowledgements

The authors would like to thank the anonymous reviewer for their time, and for
their very helpful comments. Our paper has benefited significantly from this
input. C. S. A. is a Jansky Fellow of the National Radio Astronomy
Observatory. C. S. A. thanks Phil Edwards for helpful feedback which has
improved this paper, and Joe Callingham for curating the multi-survey spectral
measurements discussed in Section 4.2. The Australian SKA Pathfinder is part
of the Australia Telescope National Facility which is managed by CSIRO.
Operation of ASKAP is funded by the Australian Government with support from
the National Collaborative Research Infrastructure Strategy. ASKAP uses the
resources of the Pawsey Supercomputing centre. Establishment of ASKAP, the
Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are
initiatives of the Australian Government, with support from the Government of
Western Australia and the Science and Industry Endowment Fund. We acknowledge
the Wajarri Yamatji people as the traditional owners of the Observatory site.
The POSSUM project has been made possible through funding from the Australian
Research Council, the Natural Sciences and Engineering Research Council of
Canada, the Canada Research Chairs Program, and the Canada Foundation for
Innovation. The Dunlap Institute is funded through an endowment established by
the David Dunlap family and the University of Toronto. B.M.G. acknowledges the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC) through grant RGPIN-2015-05948, and of the Canada Research Chairs
program. CIRADA is funded by a grant from the Canada Foundation for Innovation
2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario,
British Columbia, Alberta, Manitoba and Quebec. C. J. R. acknowledges
financial support from the ERC Starting Grant “DRANOEL”, number 714245. NMc-G
acknowledges funding from the Australian Research Council via grant
FT150100024. S.P.O. acknowledges financial support from the Deutsche
Forschungsgemeinschaft (DFG) under grant BR2026/23. Partial support for LR is
provided by U.S. National Science Foundation grant AST-1714205 to the
University of Minnesota. J. M. S. acknowledges the support of the Natural
Sciences and Engineering Research Council of Canada (NSERC) Discovery Grants
program.

##  Appendix A Estimating the uncorrected off-axis polarisation leakage

The magnitude of uncorrected off-axis Stokes $I\rightarrow Q$ and
$I\rightarrow U$ polarisation leakage can be estimated for ASKAP data using
field sources themselves as probes, by adopting a modified version of the
approach described by Lenc et al. (2017), as follows.

Within the $\sim 34$ square degrees covered by the ASKAP footprint, we detect
18640 sources in Stokes $I$ — approximately 550 sources per square degree on
average. We extracted the Stokes $I$, $Q$, and $U$ values at the location of
each of each of these sources in multi-frequency synthesis mosaics of the
field, and calculated the fractional Stokes quantities $q=Q/I$ and $u=U/I$.
These data points provide a local point-probe of the effective frequency-
independent leakage between the relevant Stokes parameters in our linear
mosaics. We call them ‘local leakage estimators’ here.

We create maps of the Stokes $I\rightarrow Q$ and $I\rightarrow U$
polarisation leakage over the ASKAP footprint by first forming a dense,
regular grid of coordinate locations covering the field. At each coordinate
location, we then:

  1. 1.

select all Stokes I sources with a full-band (i.e. 288 MHz) $\text{S/N}>10$ in
an $0.4^{\circ}$ radius aperture around the point, corresponding to one
quarter of the FWHM of the formed beams, and typically netting $\sim 300$
source components

  2. 2.

calculate the inverse-variance-weighted mean of Stokes [Q,U]/I for the sample,
providing an initial estimate of the local polarisation leakage, which will be
affected true polarised sources

  3. 3.

calculate the standardized residuals (SRs) of Stokes [Q,U]/I from the initial
leakage estimate for each source component — that is, the number of standard
deviations that each source’s Stokes [Q,U]/I value lies away from the initial
leakage estimate in both the positive and negative directions

  4. 4.

identify sources whose SR falls in the upper and lower 5% of the SR
distribution, and remove these from subsequent calculations. These $\sim 30$
source components are assumed to be genuinely polarised sources (i.e. their
polarisation is inconsistent with the leakage model). The remaining ensemble
of sources are assumed to be polarised only insomuch as they are affected by
the local polarisation leakage, which will be roughly similar in manner and
degree, and can therefore provide a reasonable estimate of such.

  5. 5.

perform step (2) again with the trimmed sample, which results in our final
estimate of the local polarisation leakage at the coordinate location in
question

The resulting maps for Stokes $I\rightarrow Q$ and $I\rightarrow U$
polarisation leakage over the full ASKAP beam footprint are shown in Figure
13. Before highlighting the salient features of these maps, we first describe
a set of simulations that we undertook to assess the accuracy of our estimates
and methods. The simulator was set up to answer the following question: For a
given grid location in our mosaic, and given (a) specified Stokes
$I\rightarrow[Q,U]$ leakage value, (b) a randomly generated population of
sources with realistic distributions of Stokes $I$, $Q$, and $U$ values and
fractional polarisations, (c) the sensitivity of our observations, and (d) the
leakage estimation protocol described above, what is the resulting
distribution in the derived leakage values for 10,000 independent realisations
of the simulation? The results were that our median estimated leakage value
was almost identical to the specified input value ($0.98\times$ the specified
input value, to be precise), while the standard deviation was 0.5% — that is,
the uncertainty on the fractional leakage maps in Figure 13 is $\sim 0.005$.

We now comment on the salient features of Figure 13. As expected, the
strongest leakage occurs at the mosaic edges. This is primarily due to the
lack of adjacent beams contributing to mosaic beyond the outer ring, meaning
that (a) the measurements reflect leakage values at much larger beam-centric
radii than generally contribute to the mosaic within the outer ring of beams,
and (b) the leakage experiences no averaging down in the linear combination of
beams either. Quadrupolar symmetry is evident in both maps, which follows from
the classic quadrupolar ‘clover leaf’ leakage pattern displayed by the
individual beam responses (offset by $45^{\circ}$ for Stokes q and u), which
is characteristic of linear feeds. In the body of the mosaic, the leakage is
typically less than 1%, though values of up to 2% are commonly observed in
isolated regions, and can become as severe as 5% for a handful of sources. We
eliminated leakage-dominated sources from our sample as described in Section
2.

Figure 13: Position-dependent Stokes $I\rightarrow Q$ (top) and $I\rightarrow
U$ (bottom) leakage maps, derived as described in the main text. The centre of
the Fornax cluster is indicated with a red cross-hair. The white dashed circle
indicates an angular radius of one degree, the smaller of the two white dotted
circles indicates the 705 kpc ($1.96^{\circ}$) virial radius of the cluster,
and the larger of the two white dotted circles indicates $3.5^{\circ}$
distance from the centre of the mosaic. We do not consider sources in the main
analysis of this paper, in order to reject regions of large $I\rightarrow U$
leakage from the corner beams. The white dots are the formed beam centres. For
reference, the half-width-at-half-maximum of the formed beams is approximately
equal to the separation of the beam centres (see Figure 2).

##  Appendix B Details of the binning experiment from Section 5.2

For the equal-area annular binning experiment described in Section 5.2, the
bin radii cannot be calculated analytically, because some of the annuli
overlap truncated regions near the edge of our mosaic (see Sections 2 and 3).
Therefore, to simplify the experiment, and to ensure that the annular bounding
radii were as uniformly spaced as possible, we started by selecting an
appropriate sub-set of our sample. Consider the following logical statements
about the possible locations of sources in our mosaic. A source may be: (1)
located outside a projected cluster-centric radius of 10 arcminutes, (2)
located outside a projected cluster-centric radius of 10 arcminutes, but
inside a cluster-centric radius of 1.014 degrees (corresponding to the first
$\pi$ square degree annulus), (3) located outside a cluster-centric radius of
1.014 degrees, (4) located eastward of the western-most source satisfying
condition 2 above, (5) located northward of the southern-most source
satisfying condition 2 above, (6) located in the southeast quadrant relative
to the cluster centre. We selected sources for this experiment satisfying the
following logical combination of these conditions: (1 and 2) or ( (3 and 4 and
5) and not 6). The positions of the selected sources, and the NVSS and GLEAM
sources also used in the experiment, are shown in Figure 14. Note that
condition 1 ensures that we exclude known Fornax cluster sources from the
analysis, and that these selection criteria were applied after, and in
addition to, the selection criteria described in Sections 2 and 3.

Next, we calculated the bin radii corresponding to an effective (i.e. post-
spatial-truncation) enclosed area of $\pi$ square degrees in each annulus. We
did so by generating a dense grid of 107 points over the entire mosaic area,
being random in location but uniform in their average spatial density. We then
applied the same spatial truncations to these points as to the sample. By
tallying up these random points, it is trivial to define the bin radii
corresponding to the desired $\pi$ square degree effective bin areas. The
resulting bounding radii for the annular bins are located 0.167, 1.014, 1.737,
2.324, 2.826, 3.313, 3.764 and 4.309 degrees from the cluster centre, and are
shown in Figure 14.

Figure 14: Locations of the sources and bins used in the experiment described
in Section 5.2 and Appendix B. Dots indicate the locations of polarised ASKAP
(large red), polarised NVSS (medium blue), and GLEAM (small green) sources
after the spatial truncations described in this appendix. The black dashed
lines indicate the locations of the annular bins, each incorporating an
effective area of $\pi$ square degrees (with the exception of the central and
outer bins, which are not included in the analysis). The red ‘x’ indicates the
position of the Fornax cluster centre.

The outcome of the experiments depend on the confidence intervals plotted in
Figure 10. Note that our aim is to establish whether the _observed_ properties
of sources behind the cluster differ significantly from those of the broader
radio source population, based on our estimates of the mean and variance of
the latter. We are _not_ trying to estimate the characteristics of the broader
radio source population from the statistics in our individual $\pi$ square
degree annuli, nor the probability that the measurements in each bin could
fluctuate towards the expectation value of the broader radio population.
Therefore, the appropriate confidence intervals are attached to the sky model,
not to the individual data points (for which the observed counts, median and
sum have zero or negligible uncertainties for the purpose of this analysis),
and it is the degree to which the latter fall outside the former that provide
the measure of statistical significance which is appropriate for our aims.

We estimated the expectation values and confidence intervals for the broader
population statistics differently depending on the statistic in question. For
the polarised source counts (plotted in the uppermost panel in Figure 10), we
assume the probability distribution corresponds to a spatial Poisson point
process with an expected source count of 27 polarised sources per square-
degree — this value being the average polarised source density outside 2
degrees radius from the cluster centre. The confidence intervals then follow
from the standard properties of Poisson processes. For the sums and medians of
the polarised and total flux in each annular bin (middle and lower panels of
Figure 10), we derived the confidence intervals from a bootstrap analysis of
our full sample. That is, we randomly select (with replacement) $10^{5}$
independent sub-samples of 76 radio sources — the number of polarised sources
closer than 1 degree to the cluster centre — situated further than 2 degrees
from the cluster centre in the mosaic. The relevant confidence intervals
follow from the percentiles of these data. Ideally, we should also impose the
constraint that each sample be taken from a contiguous $\pi$ square degree
area, but our mosaic is too small to provide a sufficient number of
independent samples for this purpose. Moreover, we caution that our mosaic
region may not be representative of the broader extragalactic sky, which would
affect our calculated confidence intervals. Future ASKAP surveys are needed to
provide crucial new information about the global clustering properties of
radio sources, particularly in polarisation, at this combination of survey
depth, resolution, and frequency (see Sections 6.7 and 7).

## References

  * Akahori et al. (2018) Akahori T., Ideguchi S., Aoki T., Takefuji K., Ujihara H., Takahashi K., 2018, PASJ, 70, 115
  * Anderson (2016) Anderson C., 2016, PhD thesis, The University of Sydney
  * Anderson et al. (2015) Anderson C. S., Gaensler B. M., Feain I. J., Franzen T. M. O., 2015, ApJ, 815, 49
  * Anderson et al. (2018) Anderson C. S., Gaensler B. M., Heald G. H., O’Sullivan S. P., Kaczmarek J. F., Feain I. J., 2018, ApJ, 855, 41
  * Banfield et al. (2014) Banfield J. K., Schnitzeler D. H. F. M., George S. J., Norris R. P., Jarrett T. H., Taylor A. R., Stil J. M., 2014, MNRAS, 444, 700
  * Berg et al. (2019) Berg M. A., et al., 2019, ApJ, 883, 5
  * Bertin et al. (2002) Bertin E., Mellier Y., Radovich M., Missonnier G., Didelon P., Morin B., 2002, in Bohlender D. A., Durand D., Handley T. H., eds, Astronomical Society of the Pacific Conference Series Vol. 281, Astronomical Data Analysis Software and Systems XI. p. 228
  * Bock et al. (1999) Bock D. C. J., Large M. I., Sadler E. M., 1999, AJ, 117, 1578
  * Böhringer et al. (2016) Böhringer H., Chon G., Kronberg P. P., 2016, A&A, 596, A22
  * Bonafede et al. (2009) Bonafede A., et al., 2009, A&A, 503, 707
  * Bonafede et al. (2011) Bonafede A., Govoni F., Feretti L., Murgia M., Giovannini G., Brüggen M., 2011, A&A, 530, A24
  * Bonafede et al. (2015) Bonafede A., et al., 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14). p. 95 (arXiv:1501.00321)
  * Botteon et al. (2020) Botteon A., et al., 2020, MNRAS, 499, L11
  * Brentjens & de Bruyn (2005) Brentjens M. A., de Bruyn A. G., 2005, A&A, 441, 1217
  * Briggs (1995) Briggs D. S., 1995, in American Astronomical Society Meeting Abstracts. p. 1444
  * Brookes et al. (2008) Brookes M. H., Best P. N., Peacock J. A., Röttgering H. J. A., Dunlop J. S., 2008, MNRAS, 385, 1297
  * Brüggen et al. (2011) Brüggen M., Bykov A., Ryu D., Röttgering H., 2011, Space Science Reviews, 166, 187–213
  * Burn (1966) Burn B. J., 1966, MNRAS, 133, 67
  * Caso & Richtler (2015) Caso J. P., Richtler T., 2015, A&A, 584, A125
  * Chippendale & Anderson (2019) Chippendale A., Anderson C., 2019, 019, On-Dish Calibration of XY Phase for ASKAP’s Phased Array Feeds, https://www.atnf.csiro.au/projects/askap/ACES-memos. Commonwealth Scientific and Industrial Research Organisation, https://www.atnf.csiro.au/projects/askap/ACES-memos
  * Churazov et al. (1999) Churazov E., Gilfanov M., Forman W., Jones C., 1999, The Astrophysical Journal, 520, 105
  * Clarke et al. (2001) Clarke T. E., Kronberg P. P., Böhringer H., 2001, ApJ, 547, L111
  * Condon et al. (1998) Condon J. J., Cotton W. D., Greisen E. W., Yin Q. F., Perley R. A., Taylor G. B., Broderick J. J., 1998, AJ, 115, 1693
  * Conway et al. (1974) Conway R. G., Haves P., Kronberg P. P., Stannard D., Vallee J. P., Wardle J. F. C., 1974, MNRAS, 168
  * Cooper & Price (1962) Cooper B. F. C., Price R. M., 1962, Nature, 196, 761
  * Davé et al. (2001) Davé R., et al., 2001, ApJ, 552, 473
  * Davé et al. (2010) Davé R., Oppenheimer B. D., Katz N., Kollmeier J. A., Weinberg D. H., 2010, MNRAS, 408, 2051
  * De Breuck et al. (2002) De Breuck C., Tang Y., de Bruyn A. G., Röttgering H., van Breugel W., 2002, A&A, 394, 59
  * DeBoer et al. (2009) DeBoer D. R., et al., 2009, IEEE Proceedings, 97, 1507
  * Donnert et al. (2018) Donnert J., Vazza F., Brüggen M., ZuHone J., 2018, Space Sci. Rev., 214, 122
  * Drinkwater et al. (2001) Drinkwater M. J., Gregg M. D., Colless M., 2001, ApJ, 548, L139
  * Dursi & Pfrommer (2008) Dursi L. J., Pfrommer C., 2008, ApJ, 677, 993
  * Ekers et al. (1983) Ekers R. D., Goss W. M., Wellington K. J., Bosma A., Smith R. M., Schweizer F., 1983, A&A, 127, 361
  * Farnes et al. (2017) Farnes J. S., Gaensler B. M., Purcell C., Sun X. H., Haverkorn M., Lenc E., O’Sullivan S. P., Akahori T., 2017, MNRAS, 467, 4777
  * Farnsworth et al. (2011) Farnsworth D., Rudnick L., Brown S., 2011, AJ, 141, 191
  * Feain et al. (2009) Feain I. J., et al., 2009, ApJ, 707, 114
  * Gaensler et al. (2004) Gaensler B. M., Beck R., Feretti L., 2004, New A Rev., 48, 1003
  * Gaensler et al. (2010) Gaensler B. M., Landecker T. L., Taylor A. R., POSSUM Collaboration 2010, in American Astronomical Society Meeting Abstracts #215. p. 515
  * Gaensler et al. (2015) Gaensler B., et al., 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), p. 103
  * Gauthier et al. (2010) Gauthier J.-R., Chen H.-W., Tinker J. L., 2010, ApJ, 716, 1263
  * Gendre & Wall (2008) Gendre M. A., Wall J. V., 2008, MNRAS, 390, 819
  * Girardi et al. (2016) Girardi M., et al., 2016, MNRAS, 456, 2829
  * Govoni et al. (2005) Govoni F., Murgia M., Feretti L., Giovannini G., Dallacasa D., Taylor G. B., 2005, A&A, 430, L5
  * Govoni et al. (2010) Govoni F., et al., 2010, A&A, 522, A105
  * Govoni et al. (2017) Govoni F., et al., 2017, A&A, 603, A122
  * Govoni et al. (2019) Govoni F., et al., 2019, Science, 364, 981
  * Gregory et al. (1994) Gregory P. C., Vavasour J. D., Scott W. K., Condon J. J., 1994, ApJS, 90, 173
  * Gruszecki et al. (2010) Gruszecki M., Nakariakov V. M., Van Doorsselaere T., Arber T. D., 2010, Phys. Rev. Lett., 105, 055004
  * Guidetti et al. (2010) Guidetti D., Laing R. A., Murgia M., Govoni F., Gregorini L., Parma P., 2010, A&A, 514, A50
  * Guidetti et al. (2011) Guidetti D., Laing R. A., Bridle A. H., Parma P., Gregorini L., 2011, MNRAS, 413, 2525
  * Guidetti et al. (2012) Guidetti D., Laing R. A., Croston J. H., Bridle A. H., Parma P., 2012, MNRAS, 423, 1335
  * Haan & Braun (2014) Haan S., Braun R., 2014, MNRAS, 440, L21
  * Hales et al. (2012) Hales C. A., Gaensler B. M., Norris R. P., Middelberg E., 2012, MNRAS, 424, 2160
  * Hales et al. (2014) Hales C. A., Norris R. P., Gaensler B. M., Middelberg E., 2014, MNRAS, 440, 3113
  * Hammond et al. (2012) Hammond A. M., Robishaw T., Gaensler B. M., 2012, preprint,  (arXiv:1209.1438)
  * Hancock et al. (2012) Hancock P. J., Murphy T., Gaensler B. M., Hopkins A., Curran J. R., 2012, MNRAS, 422, 1812
  * Hancock et al. (2018) Hancock P. J., Trott C. M., Hurley-Walker N., 2018, PASA, 35, e011
  * Heald et al. (2009) Heald G., Braun R., Edmonds R., 2009, A&A, 503, 409
  * Heald et al. (2020) Heald G., et al., 2020, Galaxies, 8, 53
  * Heiles & Haverkorn (2012) Heiles C., Haverkorn M., 2012, Space Sci. Rev., 166, 293
  * Heywood et al. (2013) Heywood I., Jarvis M. J., Condon J. J., 2013, MNRAS, 432, 2625
  * Hong et al. (2020) Hong S. E., Jeong D., Hwang H. S., Kim J., 2020, arXiv e-prints, p. arXiv:2008.01738
  * Hurley-Walker et al. (2017) Hurley-Walker N., et al., 2017, MNRAS, 464, 1146
  * Hutschenreuter & Enßlin (2020) Hutschenreuter S., Enßlin T. A., 2020, A&A, 633, A150
  * Indermuehle et al. (2018) Indermuehle B. T., Harvey-Smith L., Marquarding M., Reynolds J., 2018, in Observatory Operations: Strategies, Processes, and Systems VII. p. 107042S, doi:10.1117/12.2311917
  * Intema et al. (2017) Intema H. T., Jagannathan P., Mooley K. P., Frail D. A., 2017, A&A, 598, A78
  * Iodice et al. (2017) Iodice E., et al., 2017, ApJ, 839, 21
  * Iodice et al. (2019) Iodice E., et al., 2019, A&A, 623, A1
  * Johnson et al. (2020) Johnson A. R., Rudnick L., Jones T. W., Mendygral P. J., Dolag K., 2020, ApJ, 888, 101
  * Johnston-Hollitt et al. (2015) Johnston-Hollitt M., et al., 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), p. 92
  * Johnston et al. (2007) Johnston S., et al., 2007, PASA, 24, 174
  * Jonas (2009) Jonas J. L., 2009, IEEE Proceedings, 97, 1522
  * Jones et al. (1997) Jones C., Stern C., Forman W., Breen J., David L., Tucker W., Franx M., 1997, ApJ, 482, 143
  * Keshet et al. (2010) Keshet U., Markevitch M., Birnboim Y., Loeb A., 2010, ApJ, 719, L74
  * Keshet et al. (2017) Keshet U., Kushnir D., Loeb A., Waxman E., 2017, ApJ, 845, 24
  * Killeen et al. (1988) Killeen N. E. B., Bicknell G. V., Ekers R. D., 1988, ApJ, 325, 180
  * Komossa & Böhringer (1999) Komossa S., Böhringer H., 1999, A&A, 344, 755
  * Kronberg & Simard-Normandin (1976) Kronberg P. P., Simard-Normandin M., 1976, Nature, 263, 653
  * Lacy et al. (2019) Lacy M., Chandler C., Kimball A., Myers S., Nyland K., Witz S., 2019, in Teuben P. J., Pound M. W., Thomas B. A., Warner E. M., eds, Astronomical Society of the Pacific Conference Series Vol. 523, Astronomical Data Analysis Software and Systems XXVII. p. 217
  * Laing et al. (2008) Laing R. A., Bridle A. H., Parma P., Murgia M., 2008, MNRAS, 391, 521
  * Lan & Fukugita (2017) Lan T.-W., Fukugita M., 2017, ApJ, 850, 156
  * Lane et al. (2014) Lane W. M., Cotton W. D., van Velzen S., Clarke T. E., Kassim N. E., Helmboldt J. F., Lazio T. J. W., Cohen A. S., 2014, MNRAS, 440, 327
  * Large et al. (1981) Large M. I., Mills B. Y., Little A. G., Crawford D. F., Sutton J. M., 1981, MNRAS, 194, 693
  * Lavaux & Hudson (2011) Lavaux G., Hudson M. J., 2011, MNRAS, 416, 2840
  * Lenc et al. (2017) Lenc E., et al., 2017, PASA, 34, e040
  * Liang & Remming (2020) Liang C. J., Remming I., 2020, MNRAS, 491, 5056
  * Lienhard (1966) Lienhard J., 1966, Synopsis of Lift, Drag, and Vortex Frequency Data for Rigid Circular Cylinders. Bulletin (Washington State University. College of Engineering. Research Division), Technical Extension Service, Washington State University, https://books.google.com/books?id=qSlQGgAACAAJ
  * Locatelli et al. (2018) Locatelli N., Vazza F., Domínguez-Fernández P., 2018, Galaxies, 6, 128
  * Lyutikov (2006) Lyutikov M., 2006, MNRAS, 373, 73
  * Ma et al. (2019) Ma Y. K., Mao S. A., Stil J., Basu A., West J., Heiles C., Hill A. S., Betti S. K., 2019, MNRAS, 487, 3454
  * Machacek et al. (2005) Machacek M. E., Nulsen P., Stirbat L., Jones C., Forman W. R., 2005, ApJ, 630, 280
  * Macquart et al. (2012) Macquart J.-P., Ekers R. D., Feain I., Johnston-Hollitt M., 2012, ApJ, 750, 139
  * Macquart et al. (2020) Macquart J. P., et al., 2020, Nature, 581, 391
  * Maddox et al. (2019) Maddox N., Serra P., Venhola A., Peletier R., Loubser I., Iodice E., 2019, MNRAS, 490, 1666
  * Magliocchetti et al. (2000) Magliocchetti M., Maddox S. J., Wall J. V., Benn C. R., Cotter G., 2000, MNRAS, 318, 1047
  * Marinacci et al. (2018) Marinacci F., et al., 2018, MNRAS, 480, 5113
  * Markevitch & Vikhlinin (2007) Markevitch M., Vikhlinin A., 2007, Phys. Rep., 443, 1
  * Mauch et al. (2003) Mauch T., Murphy T., Buttery H. J., Curran J., Hunstead R. W., Piestrzynski B., Robertson J. G., Sadler E. M., 2003, MNRAS, 342, 1117
  * McClure-Griffiths et al. (2009) McClure-Griffiths N. M., et al., 2009, ApJS, 181, 398
  * McConnell et al. (2012) McConnell D., Sadler E. M., Murphy T., Ekers R. D., 2012, MNRAS, 422, 1527
  * McConnell et al. (2016) McConnell D., et al., 2016, PASA, 33, e042
  * McConnell et al. (2019) McConnell D., et al., 2019, , ASKAP Science Observation Guide, version 1.0. Commonwealth Scientific and Industrial Research Organisation
  * McGill et al. (1978) McGill R., Tukey J. W., Larsen W. A., 1978, The American Statistician, 32, 12
  * McMullin et al. (2007) McMullin J. P., Waters B., Schiebel D., Young W., Golap K., 2007, CASA Architecture and Applications. p. 127
  * Merloni et al. (2012) Merloni A., et al., 2012, arXiv e-prints, p. arXiv:1209.3114
  * Murgia et al. (2004) Murgia M., Govoni F., Feretti L., Giovannini G., Dallacasa D., Fanti R., Taylor G. B., Dolag K., 2004, A&A, 424, 429
  * Murphy et al. (2010) Murphy T., et al., 2010, MNRAS, 402, 2403
  * Nasonova et al. (2011) Nasonova O. G., de Freitas Pacheco J. A., Karachentsev I. D., 2011, A&A, 532, A104
  * Nicastro et al. (2017) Nicastro F., Krongold Y., Mathur S., Elvis M., 2017, Astronomische Nachrichten, 338, 281
  * Nikogossyan et al. (1999) Nikogossyan E., Durret F., Gerbal D., Magnard F., 1999, A&A, 349, 97
  * Norris et al. (2011) Norris R. P., et al., 2011, PASA, 28, 215
  * O’Sullivan et al. (2013) O’Sullivan S. P., et al., 2013, ApJ, 764, 162
  * O’Sullivan et al. (2020) O’Sullivan S. P., et al., 2020, arXiv e-prints, p. arXiv:2002.06924
  * Offringa et al. (2014) Offringa A. R., McKinley B., Hurley-Walker et al., 2014, MNRAS, 444, 606
  * Padovani (2016) Padovani P., 2016, A&A Rev., 24, 13
  * Paolillo et al. (2002) Paolillo M., Fabbiano G., Peres G., Kim D.-W., 2002, ApJ, 565, 883
  * Perley et al. (1984) Perley R. A., Bridle A. H., Willis A. G., 1984, ApJS, 54, 291
  * Pfrommer & Dursi (2010) Pfrommer C., Dursi L. J., 2010, Nature Physics, 6, 520
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
  * Pradeep et al. (2019) Pradeep J., Narayanan A., Muzahid S., Nagai D., Charlton J. C., Srianand R., 2019, MNRAS, 488, 5327
  * Prochaska et al. (2013) Prochaska J. X., et al., 2013, ApJ, 776, 136
  * Riseley et al. (2018) Riseley C. J., et al., 2018, PASA, 35, 43
  * Riseley et al. (2020) Riseley C. J., et al., 2020, PASA, 37, e029
  * Rudnick & Owen (2014a) Rudnick L., Owen F. N., 2014a, ApJ, 785, 45
  * Rudnick & Owen (2014b) Rudnick L., Owen F. N., 2014b, ApJ, 786, 160
  * Rudnick et al. (2007) Rudnick L., Brown S., Williams L. R., 2007, ApJ, 671, 40
  * Sakamoto & Haniu (1990) Sakamoto H., Haniu H., 1990, Journal of Fluids Engineering, 112, 386
  * Sarazin (2002) Sarazin C. L., 2002, The Physics of Cluster Mergers. pp 1–38, doi:10.1007/0-306-48096-4_1
  * Sault et al. (1995) Sault R. J., Teuben P. J., Wright M. C. H., 1995, in Shaw R. A., Payne H. E., Hayes J. J. E., eds, Astronomical Society of the Pacific Conference Series Vol. 77, Astronomical Data Analysis Software and Systems IV. p. 433 (arXiv:astro-ph/0612759)
  * Scharf et al. (2005) Scharf C. A., Zurek D. R., Bureau M., 2005, ApJ, 633, 154
  * Schinckel & Bock (2016) Schinckel A. E. T., Bock D. C.-J., 2016, in Proc. SPIE Ground-based and Airborne Telescopes VI. p. 99062A, doi:10.1117/12.2233920
  * Serra et al. (2016) Serra P., et al., 2016, in MeerKAT Science: On the Pathway to the SKA. p. 8 (arXiv:1709.01289)
  * Sheardown et al. (2018) Sheardown A., et al., 2018, ApJ, 865, 118
  * Shimwell et al. (2017) Shimwell T. W., et al., 2017, A&A, 598, A104
  * Stil et al. (2014) Stil J. M., Keller B. W., George S. J., Taylor A. R., 2014, ApJ, 787, 99
  * Stuardi et al. (2020) Stuardi C., et al., 2020, A&A, 638, A48
  * Su et al. (2017) Su Y., et al., 2017, ApJ, 851, 69
  * Subramanian et al. (2006) Subramanian K., Shukurov A., Haugen N. E. L., 2006, MNRAS, 366, 1437
  * Tanimura et al. (2019) Tanimura H., et al., 2019, MNRAS, p. 2717
  * Taylor & Perley (1993) Taylor G. B., Perley R. A., 1993, ApJ, 416, 554
  * Taylor et al. (2009) Taylor A. R., Stil J. M., Sunstrum C., 2009, ApJ, 702, 1230
  * Thom et al. (2012) Thom C., et al., 2012, ApJ, 758, L41
  * Tully et al. (2019) Tully R. B., Pomarède D., Graziani R., Courtois H. M., Hoffman Y., Shaya E. J., 2019, ApJ, 880, 24
  * Vazza et al. (2014) Vazza F., Brüggen M., Gheller C., Wang P., 2014, MNRAS, 445, 3706
  * Venhola et al. (2018) Venhola A., et al., 2018, A&A, 620, A165
  * Vernstrom et al. (2019) Vernstrom T., Gaensler B. M., Rudnick L., Andernach H., 2019, ApJ, 878, 92
  * Vikhlinin et al. (2001) Vikhlinin A., Markevitch M., Murray S. S., 2001, ApJ, 551, 160
  * Wayth et al. (2015) Wayth R. B., et al., 2015, PASA, 32, e025
  * Werner et al. (2019) Werner N., McNamara B. R., Churazov E., Scannapieco E., 2019, Space Sci. Rev., 215, 5
  * Wright et al. (1994) Wright A. E., Griffith M. R., Burke B. F., Ekers R. D., 1994, ApJS, 91, 111
  * Xu et al. (2009) Xu H., Li H., Collins D. C., Li S., Norman M. L., 2009, ApJ, 698, L14
  * Xu et al. (2011) Xu H., Li H., Collins D. C., Li S., Norman M. L., 2011, ApJ, 739, 77