# A Spectropolarimetric Study of Pictor A Radio Galaxy with MeerKAT

Lexy A. L. Andati1, Lerato M. Baidoo1,2, Athanaseus J. T. Ramaila1,3, Oleg M.
Smirnov1,3, Sphesihle Makhathini4,1, and Richard A. Perley5,1
1Centre for Radio Astronomy Techniques and Technologies, Department of Physics
and Electronics, Rhodes University, PO Box 94, Makhanda 6140, South Africa
2Dunlap Institute for Astronomy and Astrophysics, University of Toronto,
Toronto, ON M5S 3H4, Canada
3South African Radio Astronomy Observatory, 2 Fir Street, Black River Park,
Observatory, 7925, South Africa
4School of Physics, University of the Witwatersrand, 1 Jan Smuts Avenue,
Johannesburg 2000, South Africa
5National Radio Astronomy Observatory, Socorro, NM 87801, USA E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the results of a polarimetric study from our new high-sensitivity
L-band (0.8–1.7 GHz) observation of Pictor A with the MeerKAT radio telescope.
We confirm the presence of the radio jet extending from the nucleus to the
western hotspot of this source. Additionally, we show the radio emission
expected to be coincident with previously observed X-ray emission in the radio
lobes, confirming that the emission mechanism is of inverse Compton origin, as
suggested by a previous study. Our spectropolarimetric analysis using the RM-
Synthesis technique reveals a relatively uniform mean RM distribution across
the lobes of Pictor A, with most lines-of-sight exhibiting single-peaked
Faraday spectra. However, a number of the lines-of-sight exhibit single peaked
spectra with a wide base or multiple peaks, suggesting the presence of
multiple Faraday components or a Faraday thick structure along Pictor A’s
lines-of-sight. We also confirm the asymmetry in RM variability and
depolarization between the two lobes of this source which were reported in a
previous study.

###### keywords:

Polarization – galaxies:magnetic fields – galaxies: interactions – galaxies:
individual Pictor A

††pubyear: 2023††pagerange: A Spectropolarimetric Study of Pictor A Radio
Galaxy with MeerKAT–A

##  1 Introduction

Pictor A is the fifth-brightest discrete radio source in the Southern sky. It
has a radio power of $3.79\times 10^{26}$ W Hz-1 (Robertson, 1973) and is a
FR-II galaxy (Fanaroff & Riley, 1974), with structure consisting of two radio
lobes, hotspots at the edge of each lobe, a core at its centre, and radio
jets. Its host galaxy is centred at right ascension (RA, J2000)
5${}^{\textrm{h}}$19${}^{\textrm{m}}$50${}^{\textrm{s}}$ and declination (Dec)
-45∘46′$44$″, and is identified as a broad-line galaxy (Halpern & Eracleous,
1994; Lewis et al., 2010), with disc-like morphology (Inskip et al., 2010) of
either Sa (Lauberts, 1982) or S0 (Loveday, 1996) type. Pictor A is at a
redshift of $0.035$ (Schmidt, 1965), such that 1″angular scale corresponds to
0.697 kpc in linear scale.111Assuming a $\Lambda$CDM cosmological model, with
$H_{\textrm{0}}=70$ km s-1 $\Omega_{\textrm{m}}=0.3$ and $\Omega_{\Lambda}$ =
0.7. This radio source has an angular size of $\sim$8 arcmin in the East-West
(EW) direction, and $\sim$4 arcmin in the North-South (NS) direction, which
corresponds to $\sim$340 kpc EW and $\sim$170 kpc NS in linear size,
respectively (e.g. see fig. 3 of Perley et al., 1997).

The radio lobes of Pictor A are known for their remarkably round shape – with
ellipticity > 0.9 (Perley et al., 1997), which is unusual for FR-II galaxies.
The lobes of FR-II galaxies are usually found to be elongated along a specific
direction (e.g. the direction of jet propagation). Thus, Perley et al. (1997)
supposes that the lobes of Pictor A may be relaxed in a uniform environment –
hence its uniform expansion in all directions. Moreover, these lobes showed no
edge brightening, except for filamentary structures particularly in the
western lobe consisting of enhanced surface brightness of 0.5 mJy arcsec-2.
The presence of such filamentary structure is common in radio galaxies; for
example, they are seen in the lobes of Cygnus A (Perley et al., 1984), Fornax
A (Fomalont et al., 1989; Anderson et al., 2018), 4C 12.03, CGCG 044-046 and
CGCG 021-063 (Fanaroff et al., 2021).

The X-ray emission spectra of Pictor A’s lobes are best modelled using a
power-law model with a photon index of 1.57$\pm$0.04 (Hardcastle et al.,
2016), indicating that the dominant source of emission is the inverse Compton
scattering of CMB photons off the relativistic electron population. Fitting a
two-component model (power-law and thermal) to the region absent of radio
emission, the authors found a ‘soft thermal emission’ component of temperature
0.33$\pm$0.07 keV (also noted by Zirbel 1997) and a rather steep photon index
of 2.07. The results were inconclusive, with no compelling evidence for
including a thermal component.

The western hotspot (WHS hereon) is one of the brightest radio and X-ray
hotspots known (Wilson et al., 2001; Hardcastle et al., 2004; Tingay et al.,
2008). The radio hotspot is identified to be coincident with a compact, very
bright (magnitude of 19.5) and highly polarized ($>$30 per cent) source of
optical emission (Roeser & Meisenheimer, 1987), and with an extremely bright
X-ray hotspot (Wilson et al., 2001). In the X-ray, Hardcastle et al. (2016)
determined this hotspot to be variable on scales of months to a year. The
eastern lobe has two hotspots (Prestage, 1985; Perley et al., 1997).

The western radio jet was first observed by Perley et al. (1997), with no
evidence of a counter-jet (the eastern jet). In the X-ray, the western jet was
observed first by Wilson et al. (2001), and the counter-jet by Hardcastle &
Croston (2005); Hardcastle et al. (2016), while in the optical, jet knots
coincident with the known radio and X-ray jet were imaged by (Gentry et al.,
2015). The western jet (in the X-ray) extends from the core out to a distance
of 250″(174 kpc, Hardcastle et al., 2016). The jet emission responsible for
the observed X-rays is purely synchrotron radiation (Hardcastle et al., 2016).
At 34 and 49 kpc from the core, the X-ray jet is observed to be variable
(Marshall et al., 2010; Hardcastle et al., 2016).

The radio emission across the lobes of Pictor A is linearly polarized and is
undergoing Faraday rotation. Perley et al. (1997) presents this source’s most
detailed polarimetric study using the Very Large Array (VLA) telescope at 90,
20, 6, 3.6 and 2 cm and at a few arcseconds resolution. Perley et al. (1997)
found significant polarization across the source, ranging between 30 to 60 per
cent along the lobe edges and between 10 to 20 per cent within the central
regions of the lobes. The authors derived the mean rotation measures (RM) of
both lobes to be $43.5\pm 1.4$ rad m-2, which is consistent with previous
studies. The challenge faced by the VLA observations was that the source was
too far South. Therefore, Pictor A never exceeded an elevation of 10∘ due to
the telescope’s latitude. This resulted in a significantly limited observation
time ($\sim$4 hrs), hence reduced sensitivity and low fidelity of the output
images. Furthermore, the foreshortening of the N-S baselines due to the low
source elevation resulted in an elongated synthesised beam.

In this work, we present a study of Pictor A using high-sensitivity L-band
(0.8–1.7 GHz) data from the MeerKAT telescope. The location of this telescope
made it a suitable instrument for the observation of this source. In Sec. 3.1,
we present the total intensity features of Pictor A from our observations,
including the radio jet and the ‘missing’ accompanying radio emission expected
to be co-spatial with previously observed IC/CMB emission (Hardcastle et al.,
2016). Sec. 3.2 presents a new spectropolarimetric analysis of Pictor A using
newer diagnostic methods: RM-synthesis and $\mathit{QU}$-fitting. This is
followed by a summary and conclusion.

##  2 Observations and Data Reduction

Pictor A was observed in December 2019 using the MeerKAT L-band system in full
polarization 4K mode, whereby each channel width was 208.984 kHz. The specific
details of the observation are summarised in Table 1.

Table 1: Observational setup Observation Details for Pictor A
---
Dates (UTC) | from: 2019 Dec 18 16:46
| to: 2019 Dec 19 03:48
Antennas | 62
Band | L
Frequencies | 856 MHz – 1.7 GHz
Bandwidth | 856 MHz
Channels | 4096
Time on target |  $\sim$ 9 Hrs
Scan duration |  $\sim$ 15 min
Integration time | 8 sec
Bandpass calibrator | J0408-6545
Gain calibrator | J0538-4405
Polarization Angle calibrator | J0521+1638

###  2.1 Calibration

The standard calibration procedures were undertaken using the CARACal pipeline
(Józsa et al., 2020) to obtain our polarimetry-ready
data.222https://caracal.readthedocs.io/en/latest/ The procedures consisted of
first-generation calibration (1GC) followed by the second-generation
calibration (2GC), (see Smirnov, 2011a, for a description of 1GC and 2GC). The
1GC comprises mostly of cross-calibration, whereby a well-known calibrator is
used to correct the data of the target source, while the 2GC consists mostly
of self-calibration (selfcal). The following calibration strategy was used:

  1. 1.

A static mask was applied to the already known RFI-affected regions in the
MeerKAT L-band within the calibrator fields. Additional flagging was then
performed to remove any remaining RFI (1200 $\sim$ 1230 MHz, 1280 $\sim$ 1300
MHz, 1560 $\sim$ 1610 MHz) and the band edges (856 $\sim$ 880 MHz and 1700
$\sim$ 1800 MHz). Furthermore, antennas and scans whose signals seemed
corrupted from visual inspection were also excised at this step.

  2. 2.

Delay, complex gain, bandpass calibration, and flux scaling solutions were
derived using J0538-4405 and J0408-6545.

  3. 3.

For polarization calibration, J0521+1638 was used to calibrate the
polarization position angle, while the bandpass calibrator (J0408-6545) was
used for leakage corrections.

  4. 4.

The obtained solutions were first applied to all the calibrator sources, where
they were inspected for validity. Once determined to be suitable, they were
transferred to the target source, marking the end of 1GC.

  5. 5.

Our first image of the target source generated from these calibrated data was
used as our model image for our initial selfcal.

Self-calibration proceeded in two stages:

  * •

During the first stage, we performed two rounds and phase-and-delay selfcal,
using Stokes $\mathit{I}$ models only. In particular, we used the CubiCal
calibration software (Kenyon et al., 2018), selecting a ‘unislope’ solver with
a solution interval of 4096 channels in frequency (the full band) and a single
integration in time. In effect, this solves for independent X and Y phase
offsets and a common phase slope (i.e. delay term) per each solution interval.

  * •

For the second stage, we performed phase-and-amplitude self-calibration using
Stokes $\mathit{I,Q}$ and $\mathit{U}$ models. This used CubiCal’s G complex
2-by-2 solver with a solution interval of 64 channels in frequency and 8
integrations in time. No Stokes $\mathit{V}$ models were used.

However, Pictor A’s extremely bright western hotspot introduced major
artefacts into our images and, thus, posed a great challenge for our 2GC. As
similarly noted by Hardcastle et al. (2016), it required different calibration
solutions from the rest of the source. We applied the differential gains
(Smirnov, 2011b) iteratively to improve our models for this hotspot to obtain
better images. The specific steps undertaken for calibration and high dynamic
range imaging will be highlighted in a different paper: Ramaila et al. (in
prep).

Figure 1: An MFS image of Pictor A from our new MeerKAT L-band observations
centred at 1.28 GHz with a bandwidth of 856 MHz and $\sim$7.5″resolution. The
contour levels shown start from 4 mJy beam-1 and increase by a factor of 2.
The off-source noise is 22 $\mu$Jy beam-1. The source emission shows a single
hotspot at the extreme end of the western lobe and two hotspots at the
extremity of the eastern lobe. The jet radio knots on the west side of the
source are seen with a clear path from the Active Galactic Nucleus (AGN) to
the hotspot.

During imaging, we averaged our data in frequency, ensuring that our channel
widths were small enough to avoid bandwidth depolarization and large enough to
maintain a good signal-to-noise ratio, thus improving sensitivity. We used a
maximum channel width, $\delta\nu$, of $\sim$ 10.7 MHz, resulting in 80 sub-
band images. We did not apply primary beam corrections to the whole field
overall as it seemed unnecessary for these data since our interest was the
central source (Pictor A) whose angular size is $\sim$8 arcmin – well within
the MeerKAT main beam of $1.2$ degrees. However, differential gains were
applied to the WHS.

All Imaging was done with the wsclean imaging and deconvolution software
(Offringa et al., 2014) in multi-scale mode because of extended emission of
this source. Upon inspection, multiple channels exhibiting abnormal spectral
variations were identified and discarded. These channels coincided with the
parts of our bandwidth that were heavily flagged because of GSM-induced radio
frequency interference (RFI), perhaps leading to a reduced signal-to-noise.
The remaining images were combined to form Stokes $\mathit{I}$, $\mathit{Q}$
and $\mathit{U}$ cubes which were all convolved to the same beam (11″by 10″)
using the spimple Python package.333https://github.com/landmanbester/spimple
The resulting images were then used for polarimetric analysis of Pictor A.

We estimated an off-source noise of $21.85\mu$Jy beam-1 for the $\mathit{I}$
multi-frequency synthesis (MFS) image, while the per-channel noise ranged
between 95–278 $\mu$Jy beam-1 in $\mathit{Q}$, and 41–233 $\mu$Jy beam-1 in
$\mathit{U}$. The noise close to the WHS increases by almost an order of
magnitude for each image.

###  2.2 Extracting Spectropolarimetric Information

To study the polarimetric composition of Pictor A, we applied the Rotation
Measure synthesis (RM-synthesis) technique first introduced by Burn (1966) and
later refined by Brentjens & de Bruyn (2005). RM-synthesis mitigates the
$n\pi$ ambiguity problem that plagues the traditional linear fitting of the
polarization angle against $\lambda^{2}$. In this context, erroneous values of
RM and zeroth-wavelength position angle may be obtained due to polarization
angle wrapping (or jumps) into multiple turns, hence the ambiguity. An in-
depth discussion of RM-synthesis and its concepts can be found at Brentjens &
de Bruyn (2005); Heald (2009); Ferrière et al. (2021). Here, we briefly define
some concepts related to our application of the technique. Following Burn
(1966), we define complex polarization as $P=\mathit{Q}+i\mathit{U}$.

Polarization amplitude and angle can be derived from Stokes $\mathit{Q}$ and
$\mathit{U}$ images respectively as follows:

$\displaystyle|P|$ $\displaystyle=\sqrt{\mathit{Q}^{2}+\mathit{U}^{2}}\text{
and },$ (1) $\displaystyle\chi$
$\displaystyle=0.5\arctan\left(\mathit{\frac{U}{Q}}\right)\text{[rad]}.$ (2)

As polarized emission is radiated from a source towards the observer, it
encounters a variety of media that can perturb the intrinsic properties of the
signal. In particular, when the emission passes through regions of magneto-
ionized plasma, its polarization angle gets rotated due to the birefringent
effect. This phenomenon is known as Faraday rotation. In the presence of
Faraday rotation, the observed polarization angle, $\chi$, is given by:

$\chi=\chi_{0}+\mathrm{RM}\lambda^{2},$ (3)

where $\lambda$ is the observation wavelength, $\chi_{0}$ is the intrinsic
polarization angle, and RM is known as the rotation measure. However, since
multiple intervening magnetized media (both Faraday rotating and synchrotron
emitting) could exist along a line-of-sight, Burn (1966) introduced the notion
of Faraday depth, $\phi$, given as:

$\phi(L)=0.81\int_{\text{L}}^{\text{obs}}n_{e}\text{ {B} }\cdot
d\mathbf{l}\text{ [rad m ${}^{-2}$] },$ (4)

where $n_{e}$ is the electron density in cm -3, B is the magnetic field in
$\mu$G, and $d\mathbf{l}$ is the infinitesimal path length in parsecs, and L
is a specific point along the line-of-sight. This gives the RM contribution
within a specific distance along the line-of-sight to an observer.

For a more general representation, contributions from the magnetised plasma
and the multiple sources of Faraday rotation along the line-of-sight are taken
into account by summing them up. Practically, we can only observe at
$\lambda^{2}>0$ at discretely sampled wavelengths. This means that the
observed $P(\lambda^{2})$ is weighted by a sampling function $W(\lambda^{2})$
– whose values are only valid at the sampled wavelengths. Furthermore,
Brentjens & de Bruyn show that RM-synthesis operates within two Fourier-
related spaces, namely the Faraday depth ($\phi$) space and lambda
($\lambda^{2}$). Taking the above limitations into consideration, the observed
polarized intensity in $\lambda^{2}$ and $\phi$ is represented as follows:

$\displaystyle\tilde{P}(\lambda^{2})$
$\displaystyle=W(\lambda^{2})P(\lambda^{2})=W(\lambda^{2})\int_{-\infty}^{+\infty}F(\phi)e^{2i\phi(\lambda^{2}-\lambda^{2}_{0})}d\phi,$
(5) $\displaystyle\tilde{F}(\phi)$ $\displaystyle=F(\phi)\star
R(\phi)=K\int_{-\infty}^{+\infty}\tilde{P}(\lambda^{2})e^{-2i\phi(\lambda^{2}-\lambda^{2}_{0})}d\lambda^{2},$
(6)

where $\tilde{P}$ and $\tilde{F}$ represent the observed quantities of $P$ and
$F$, and $F(\phi)$ is the Faraday dispersion function (FDF henceforth): the
polarized emission at each Faraday depth, and

$K=\left(\int_{\infty}^{\infty}W(\lambda^{2})d\lambda^{2}\right)^{-1}.$

From Eqn. 6, the observed (dirty) Faraday spectra, $\tilde{F}$, results from
the convolution of the true (clean/deconvolved) spectra $F(\phi)$ and
$R(\phi)$ is the rotation measure transfer function (RMTF), a complex-valued
function given by:

$R(\phi)=K\int_{-\infty}^{+\infty}W(\lambda^{2})e^{-2i\phi(\lambda^{2}-\lambda^{2}_{0})}d\lambda^{2}.$
(7)

$\lambda^{2}$ is derotated to a reference $\lambda^{2}_{0}$ in Eqn. 7 to
mitigate difficulties in estimating the correct polarization angle at the peak
of $|F(\phi)|$ due to the rapid rotation of the real and imaginary components
of the RMTF.

The preceding discussion assumes that $Q$ and $U$ are independent of frequency
except for the effects of Faraday rotation. In practice, there is an intrinsic
spectral dependence (Brentjens & de Bruyn, 2005, Sec.3). In order to correct
for this, we applied RM synthesis to the fractional quantities $q=Q/I$ and
$u=U/I$ rather than to Q and U. We used the individual sub-band Stokes
$\mathit{I}$ obtained from multiscale deconvolution to obtain these maps. We
attempted spectral fitting on the Stokes $\mathit{I}$ image cube to obtain a
Stokes $\mathit{I}$ model. However, using the model Stokes $\mathit{I}$ showed
no significant improvements in the behaviour of the fractional polarization.
Therefore, we used the originally obtained Stokes $\mathit{I}$ cubes.

The maximum Faraday depth, $|\phi_{\mathrm{max}}|$, we can probe is dependent
on the channel width $\delta\lambda^{2}$ of our observations as follows:

$|\phi_{\mathrm{max}}|=\frac{\sqrt{3}}{\delta\lambda^{2}}.$ (8)

We mentioned in the previous section that our data were frequency averaged,
yielding 80 sub-band images for the spectropolarimetric analysis. These
frequency averaging bins were determined by deriving the maximum allowable
channel width, $\delta\nu$, using the following equation (a modification of
Eqn. 4 of Sebokolodi et al., 2020):

$\delta\nu=\frac{\Delta\chi\nu}{2\lambda^{2}\mathrm{RM}},$

where $\nu$ is the centre of the observational bandwidth, $\lambda$ is its
corresponding wavelength, $\Delta\chi$ is the per-channel maximum allowable
rotation of the plane of polarization in radians, and RM is the maximum
rotation measure of the source. Pictor A’s maximum rotation measure was shown
by Perley et al. (1997) as $\sim$ 100 rad m-2. Therefore, using a condition
that the maximum allowable rotation is 5 degrees and a centre frequency of
1.28 GHz, we derived $\delta\nu$ to be $\sim$ 10.2 MHz, which we adjusted to
$\sim$ 10.7 MHz. By converting the chosen $\delta\nu$ to $\delta\lambda^{2}$,
we then obtained a $\phi_{\text{max}}$ of $\sim$1981 rad m-2.

On the other hand, the expected Faraday depth resolution, $\delta\phi$ (the
full width half maximum, FWHM, of the RMTF), is dependent on the bandwidth in
$\lambda^{2}$ space,
$\Delta(\lambda^{2})=\lambda_{\text{max}}^{2}-\lambda_{\text{min}}^{2}$, as
follows (Schnitzeler et al., 2009):

$\delta\phi=\frac{3.8}{\Delta(\lambda^{2})}.$ (9)

The RMTF resolution for our data is $\sim$ 41 rad m-2, and its structure is
illustrated in Fig. 2. Due to a large gap in the data (resulting from flagging
frequencies that were significantly affected by RFI), our RMTF exhibits large
sidelobes.

Figure 2: The RMTF of our data with a resolution (FWHM) of 41 rad m-2. The
solid, dashed, and dotted lines represent the amplitude, real and imaginary
components, respectively. The high sidelobes are a result of missing
frequencies that were flagged out due to RFI.

We performed RM-CLEAN (Heald, 2009) on the derived Faraday spectra to extract
more accurate values of RM. The RM-CLEAN algorithm is similar to the standard
imaging CLEAN algorithm (Högbom, 1974). Peaks in the Faraday spectrum are
iteratively identified, scaled, and subtracted until a minimum threshold or a
maximum number of iterations is achieved. The clean components are then
convolved by an idealised RMTF, e.g. a Gaussian function of width equivalent
to that of the RMTF, resulting in a ‘cleaned’ Faraday spectrum. Besides
deconvolution, RM-CLEAN attempts to mitigate problems arising from a large
first sidelobe that could mimic a peak and cause confusion in cases where
polarized emission is detected at multiple depths along a line-of-sight (Heald
et al., 2009).

##  3 Results

###  3.1 Total Intensity

Figure 3: The Pictor A Chandra X-ray image from Hardcastle et al. (2016) made
with 0.5–5.0 keV passband data (in greyscale) overlaid with radio emission
contours from our data (in blue) at $\sim$7.5″resolution. Contours start from
0.6 mJy beam-1 similar to Fig 1 of Hardcastle et al. (2016), and separated by
a factor of $\sqrt{2}$. The X-ray image was convolved to a resolution similar
to our radio data and is shown in a log scale to emphasize the low-level
emission. The orange polygons show a region previously identified by
Hardcastle et al. (2016) as a region with missing radio flux emission. This
radio emission is visible in the image. The X-ray jet is also excellently
aligned with the radio jet.

####  3.1.1 Hotspots

In Fig. 1, we show the total intensity map of Pictor A obtained from our
L-band data at 7.5″resolution – the highest possible resolution for our
observations. Our data show a single hotspot at the extremity of the western
lobe, herein the western hotspot, and two hotspots at the extremity of the
eastern lobe, herein the eastern hotspot (EHS). The presence of these hotspots
is consistent with previous observations of Pictor A. The WHS is exceptionally
bright, with a peak brightness of 7.2 Jy beam-1 ($\sim$118 mJy arcsec-2),
while the inner EHS shows a peak brightness of 1.62 Jy beam-1 ($\sim$26.5 mJy
arcsec-2), and the outer shows 1.01 Jy beam-1 ($\sim$ 16.5 mJy arcsec-2). At
our resolution, these hotspots and radio core remain unresolved.

According to the dentist drill model (Scheuer, 1982), the inner hotspot is the
primary hotspot (most recent), and the outer hotspot is the secondary hotspot
(older). This model implies that the eastern jet was first directed towards
the secondary hotspot and then later changed direction towards the primary
hotspot. The change in the jet direction occurred on time-scales shorter than
the time taken for the secondary hotspot to dissipate due to expansion losses.
Perhaps this may explain the differences in peak brightness of the hotspots –
with the primary hotspot being brighter than the secondary – but this
inference remains inconclusive and subjective.

From the spectral index analysis presented in section 3.1.4, the spectral
index of the primary hotspot (inner) is $-0.76$ and of the secondary (outer)
hotspot is $-0.79$ with standard deviations of 0.01 and 0.021 respectively (we
define the relationship between spectral index, $\alpha$, and integrated flux
density, $S$, as $S_{\nu}\propto\nu^{-\alpha}$). Assuming that the spectral
index provides a good representation of the emission’s age, the spectral index
of the hotspots suggests that the primary hotspot is younger than the
secondary – consistent with the prediction by the Dentist drill model.
However, the difference in the spectral indices is minimal – comparable to the
spread (standard deviation) in the spectral index distribution within the
hotspots; thus, it cannot provide strong evidence for the age difference
between them.

####  3.1.2 Lobes

The notably round radio lobes are also visible from Fig. 1 are consistent with
those seen in previous studies, except that our data reveals more diffuse
emission due to higher sensitivity. The eastern lobe has an integrated flux
density of 20.9 Jy, and the western lobe of 20.5 Jy, leading to a flux density
lobe ratio of the eastern to western lobe of $\sim 1.02$. Radio filaments
within the lobes are also observed similar to those seen by Perley et al.
(1997). These filamentary structures are known to form as a result of shearing
and strengthening of weak magnetic fields due to the dynamo amplification
effect in filamentary regions (see Clarke, 1996; Donnert et al., 2018, for MHD
simulation and details of the dynamo effect). The ‘un-jetted’ lobe has also
been shown to be associated with a slightly steeper spectral index (see also
our spectral indices in Sec 3.1.4) and is thought to be spatially smaller
owing to higher pressure from its denser environment (Perley et al., 1997).
This high pressure was speculated to enhance its magnetic field, leading to a
higher observed spectral age (Liu & Pooley, 1991a; Liu & Pooley, 1991b).
Similar to Perley et al. (1997), we find the length ratio of the western to
the eastern lobe to be 1.19 and a comparable ratio between the widths,
indicating that the western lobe is larger. However, this could also be a
projection effect.

In Figs. 1 and 3, we also note a region akin to what Carilli & Barthel (1996)
call ‘radio plumes’. In Pictor A’s case, plumes appear midway between the
lobes on both the far south and north of the lobe emission. These plumes
appear as emission ‘spilling’ from the central lobe emission. Carilli &
Barthel (1996) attribute plumes to the displacement of radio-emitting plasma
away from the galactic core due to pressure gradients from the intracluster
medium. Hardcastle et al. (2016) also pointed out that diffuse X-ray emission
appeared to extend beyond the visible lowest VLA data radio contour at 0.6 mJy
(see fig. 1 of Hardcastle et al., 2016) at the ‘waist’ of the source. This
contradicts expectations since the X-ray lobe emission is suggested to be
inverse Compton (IC) in origin; thus, we expect to see the spatial
superposition of the X-ray and radio. The missing radio emission was
attributed to either 1) the limited sensitivity of the VLA at the time, 2) the
X-ray emission being thermal in origin within these regions, or 3) the radio
emission being too faint or of steep-spectrum.

Fig. 3 shows an X-ray image of Pictor A in greyscale from Hardcastle et al.
(2016) overlaid with contours from our radio data. The lowest radio contour
here is 0.6 mJy beam-1, with consecutive contours separated by a factor of
$\sqrt{2}$. The previously unseen radio emission is visible (marked by the
orange polygons), consistent with the suggestion that X-ray lobe emission is
predominantly of IC/CMB origin. However, Hardcastle et al. (2016) pointed out
that the soft-thermal emission cannot be ruled out.

####  3.1.3 Radio Jets

Pictor A’s jets are remarkably visible in the X-rays (e.g. see Fig. 3 and
Wilson et al., 2001; Hardcastle et al., 2016), the western jet clearly shows
to extend from the core and terminates at the hotspot ($\sim$ 174 kpc). The
jets emission was determined to be of synchrotron origin (Hardcastle et al.,
2016). While the western radio jet was slightly noticeable in the study of
Perley et al. (1997) using the VLA data, the authors remarked that it was very
faint and was best visible on a TV monitor. We show for the first time in Fig.
1 a more pronounced western radio jet from the radio nucleus to the WHS.
Although this radio jet remains unresolved, it is collimated and consists of a
series of elongated knots, similar to those observed by Carilli & Barthel
(1996) on the jets of Cygnus A. These knots were also partially observed in
optical frequencies using the Hubble Space Telescope (HST) by Gentry et al.
(2015). This radio jet has a position angle (as per the IAU convention) of
$\sim 281\pm 1^{\circ}$ and runs from the centre of the radio core to about
1.93′, after which its emission diffuses into the western lobe. Though the jet
becomes indistinguishable at this point, tracing it along this position angle
leads directly to the WHS.

By contrast, the counter-jet is still not detected in radio images. Although
discernible in the X-ray, it seems to diffuse into the lobe emission near the
nucleus and hotspots (Hardcastle et al., 2016). Our findings are consistent
with those of the previous studies; there is no discernible counter-jet. This
apparent jet single-sidedness in a radio galaxy like Pictor A’s (in Fig. 1)
could result from the Doppler boosting effect that makes the jet closer to the
observer appear brighter and more visible.

Figure 4: A novel collimated feature reminiscent of a radio jet observed from
our data. The top panel highlights this feature marked by the orange polygon.
The green contours show the radio emission, while the straight lines show the
possible extension of the paths of this feature. The bottom panel shows the
X-ray image superposed with the same radio contours as the top panel. The
straight lines are also drawn at the same position.

Fig. 4 shows a feature near and North-West of the AGN reminiscent of a radio
jet due to its semblance in total intensity and collimation to the jet’s
emission rather than the neighbouring lobe emission. Since this feature is not
directly aligned with the WHS and the core, it is difficult to tell
conclusively. Nevertheless, we drew two possible lines based on the knot
extension to trace its extension towards the lobe’s extremity. To our
knowledge, this feature was not seen in previous radio images and may be of
interest in the future.

####  3.1.4 Spectral Index

As a result of the excision of data affected by RFI, we generated our spectral
index map using a frequency range smaller than that of the MeerKAT telescope’s
L-band, 0.89–1.64 GHz, using the spimple tool. We consider $0>\alpha\geq-0.7$
as flat and $\alpha<-0.7$ as steep.

Figure 5: The spectral index map generated using channelised images between
0.88 and 1.65 GHz at a resolution of 11″by 10″(top panel). The spectral index
colour bar ranges from $-0.5$ to $-1.2$. Spectral steepening is seen towards
the waist of Pictor A, while the core shows a flat spectral index. The bottom
panel shows errors associated with spectral index fitting, whereby higher
errors are associated with the lobe edges.

The top panel in Fig. 5 reveals that both lobes have fairly similar spectral
indices, with a mean of $\left\langle\alpha\right\rangle=-0.86$. Furthermore,
the EHS pair displays a steeper spectral index of
$\left\langle\alpha\right\rangle=-0.76$ and $-0.79$ for the inner and outer
hotspots, respectively, compared to that of the WHS
$\left\langle\alpha\right\rangle=-0.64$, while the radio core has
$\left\langle\alpha\right\rangle=-0.48$. These values are summarised in Table
2. In that table, we also indicate the number of independent beams that fit
within the specified region of Pictor A rounded off to the nearest integer.
They were obtained by dividing the region area by the beam’s area.

Table 2: Mean spectral indices for selected regions within Pictor A. The mean frequency is 1.21 GHz. The last column of this table shows the number of independent beams that fit in the specified regions within the source. Region | Mean $\alpha$ | Std Deviation $\alpha$ | Beams
---|---|---|---
E-lobe | $-0.88\pm 2.0\times 10^{-4}$ | $0.09$ | 291
W-lobe | $-0.85\pm 3.7\times 10^{-4}$ | $0.08$ | 382
E-Hotspot (inner) | $-0.76\pm 6.4\times 10^{-6}$ | $0.010$ | 2
E-Hotspot (outer) | $-0.79\pm 3.0\times 10^{-5}$ | $0.021$ | 3
W-Hotspot | $-0.64\pm 2.0\times 10^{-4}$ | $0.085$ | 11
Core | $-0.48\pm 2.0\times 10^{-5}$ | $0.27$ | 5
Plume | $-1.02\pm 7.0\times 10^{-4}$ | $0.074$ | 38
Waist | $-0.94\pm 3.1\times 10^{-4}$ | $0.12$ | 223

In contrast, the source’s waist region displays significant spectral
steepening away from the core and towards its periphery along the north and
south edges. These spectral indices align well with the observations of Perley
et al. (1997), who stated a value of $\left\langle\alpha\right\rangle=-0.8$
for the lobes and noted that the eastern lobe has a steeper spectrum than the
western lobe. However, we found a maximum of $\alpha\sim-1.3$ along the waist
compared to Perley et al.’s $-1.7$. Errors associated with our spectral index
fitting are shown in the bottom panel of Fig. 5. These are the random errors
propagated from deconvolution and are calculated as:

$\alpha_{\text{err}}=\sqrt{\left(\frac{\Delta
I_{0}}{I_{0}}\right)^{2}+\left(\frac{\Delta I_{1}}{I_{1}}\right)^{2}},$

where $\Delta I_{0}$ and $\Delta I_{1}$ are the RMS errors measured from the
first and second Taylor-coefficient maps. Higher error areas appear to be
associated with the lobe’s edges. We attribute this to low signal-to-noise
along those regions coupled with edge effects. The ‘stripy’ structure on both
maps results from residual calibration artefacts.

This type of spectral variation is typical of FR-II sources. Usually, the
hotspots and core exhibit a flatter spectrum, while the lobe exhibits a
generally steeper spectrum. For Pictor A, this is true except for the eastern
hotspots, with spectral indices comparable to the lobes. Flat spectral indices
indicate the presence of younger electrons, and conversely, a steeper spectral
index indicates older electrons. Our map shows that the age of electrons
increases away from the core in the N-S direction.

Spectral ageing is thought to occur due to more energetic electrons
radiatively losing their energy faster in the absence of acceleration,
resulting in spectral cooling and a steepened spectrum. Models such as the
Kardashev–Pacholczyk (KP) and Jaffe–Perola (JP) have been proposed to predict
the energy spectral shapes resulting from spectral ageing (see Harwood et al.,
2013, for a summary and references therein for a complete treatment). These
models assume a fixed magnetic field strength across the source and a single
injection electron distribution at the jet termination points. However, to
factor in a more realistic magnetic field structure, Tribble (1993) introduced
more advanced spectral ageing models that assume a Gaussian randomly varying
magnetic field that allowed electrons to diffuse with the varying field
strengths. Identifying the more accurate model without deeply understanding
the underlying physical processes is difficult. Unfortunately, the relatively
limited bandwidth of our current data confines the extent to which we can
probe the spectral behaviour of this source. Thus, studies with a broader
bandwidth coverage will be invaluable in better characterising the spectral
indices of Pictor A.

###  3.2 Polarimetry

####  3.2.1 Linear Polarized Intensity and Fractional Polarization

Fig. 6 shows the linear polarization (top) and degree of polarization across
the lobes of Pictor A at 1.65 GHz and 11″by 10″resolution. We compute the
polarized brightness using Eqn. 1. High linear polarization in Pictor A is
found at the hotspots, within the western lobe’s filamentary structure and
close to the EHS. This is similar to the findings of Perley et al. (1997).
There seems to be no correlation between filaments in the total intensity
image and those in the polarized brightness distribution.

The degree of polarization (fractional polarization), $p$, was given as
$|P|/\mathit{I}$. We obtained images of $p$ from the sub-band $\mathit{I}$
maps as described in Sec. 2.2. No Ricean bias correction was made for any of
the images. High degrees of polarization are seen around the edges of Pictor
A, and a single filamentary structure located at around RA
$5^{\mathrm{h}}19^{\mathrm{m}}41^{\mathrm{s}}$ and Dec
$-45^{\circ}46^{\prime}00\arcsec$. In this figure, the colour bar is
representative of the degree of polarization. Levels of the fractional
polarization over most of the lobes’ regions are below 10 per cent, whereby
the western lobe demonstrates more regions with higher fractional polarization
values than the eastern lobes. As noted by Perley et al., there seems to be no
correlation between the regions of higher total intensity and fractional
polarization. Their observations at 7.5″resolution showed that some regions
displayed up to 70 per cent fractional linear polarization. From our data in
Fig. 6, a few regions within the western lobe also appear to show a degree of
polarization approaching 60 per cent.

Figure 6: The linear polarized intensity across Pictor A (top panel) and the
degree of polarization (bottom panel) at 1.65 GHz, whereby the hotspots and
radio filaments show the highest amounts of linear polarization. Both images
are at 11″by 10″in resolution, and the top panel’s units are in Jy beam-1.

####  3.2.2 Line-of-Sight Fractional Polarization

Fig. 7 shows polarization structures for the individual lines-of-sight (LoS)
across the lobe of Pictor A. We have 2389 lines-of-sight across Pictor A,
spatially separated by approximately 6″. These lines-of-sight were chosen
using the criteria of having a signal-to-noise ratio (SNR) above 50 in total
intensity image at 1 GHz. For illustration, we can only show eight of the
lines-of-sight. The left column is the degree of polarization as a function of
wavelength-squared, and the middle column consists of the unwrapped
polarization angle as a function of wavelength-squared in blue points and the
fitted linear model in green dashed lines. The third column is the cleaned
Faraday spectrum (cleaned FDF) in black superimposed by the dirty FDF in red
dashed line. In this plot, the top three rows highlight examples of lines-of-
sight with a single peak in the FDF, the fourth and fifth rows show examples
with double peaks, and the last three rows show examples with multiple peaks.
The lobes of Pictor A are dominated by single-peaked FDF.

Figure 7: The polarization behaviour of example lines-of-sight across the
lobes of Pictor A. Left column: fractional polarization vs $\lambda^{2}$.
Middle column: polarization angle vs $\lambda^{2}$ superimposed by a linear
fit (green dashed line). Right column: the RM-cleaned Faraday spectra are
represented. RM values shown on the legend of this column indicate the
position in Faraday depth of the Faraday spectrum’s highest peak. The RMTF is
shown in Fig. 2.

Quantitatively, $\sim$43.5 per cent of the LoS exhibited Faraday spectra with
a single dominant peak such as those illustrated in row one of Fig. 7.
Additionally, we found that for these single-peaked LoS, $\sim$78 per cent
showed a maximum fractional polarization greater than 10 per cent, while
$\sim$33 per cent showed fractional polarization of more than 20 per cent. On
the other hand, $\sim$32.6 per cent of the single-peaked LoS spectra showed a
wide base and/or smaller peaks on either side (such as those of rows two and
three of Fig. 7). We also found that $\sim$23.5 per cent of the spectra showed
double peaks (e.g. those in rows 4, 5 and 6 of Fig. 7), while spectra of some
a few of the boundary regions associated with more than two peaks (rows 7-8 in
the same figure). The remaining $\sim$0.3 per cent of Los spectra were deemed
to be noisy. Shown in Fig. 8 are the locations of these various categories of
spectra, whereby LoS spectra with single, single wide base, and multiple peaks
are represented by green, blue and orange respectively. Noisy regions are
represented by purple. Fig. 8 also highlights the positions of the lines-of-
sight exemplified in Fig. 7 using yellow outlines. These LoS have also been
labelled accordingly. Pictor A’s lines-of-sight are available at
https://pica.ratt.center/ displayed using the PolarVis tool (Andati et al.,
2021), which allows the visualization of various diagnostic plots associated
with the specific lines-of-sight.444https://github.com/Mulan-94/polarvis The
lines-of-sight and their corresponding interactive plots were generated using
Scrappy, another Python-based tool that semi-automates generation of the LoS
depending on a user’s specifications.555https://github.com/Mulan-94/scrappy We
probed further into regions showing RM gradients and double peaks to determine
if these effects were real. This is discussed in Sec. 3.2.6.

Figure 8: Locations of the various categories of LoS Faraday spectra exhibited
by Pictor A. Most of the lines-of-sight show Faraday spectra with a single
dominant peak (shown in green). In other cases, this single peak has a wide
base or an additional smaller peak on either side (in blue). This source also
exhibits multiple peaks in some regions (represented by orange). Noisy lines-
of-sight are represented by purple. Lines-of-sight outlined in yellow are
those exemplified by Fig. 7, and are labelled accordingly.

The behaviour of fractional polarization vs $\lambda^{2}$ is quite complex,
even for single-peaked spectra. In some cases, repolarization appears to occur
at longer wavelengths with highly oscillatory structure. The position angle vs
$\lambda^{2}$ mostly exhibits linearity except for the double and multi-peaked
spectra. On the other hand, the double and multi-peaked Faraday spectra show a
wide range of behaviour: in some lines-of-sight the components are well
resolved, and in some cases, the components are blended. The spectra of the
last row resemble that of a Burn-slab (Burn, 1966) – a region in which
polarized emission and Faraday rotation occur simultaneously. It is evident
that the polarization behaviour across the lobes of Pictor A is complex.

####  3.2.3 Depolarization

Depolarization refers to decreased observed fractional polarization as a
function of resolution or frequency. It can result from effects internal to
the source (due to mixed thermal and magneto-ionized gas) or external to the
source (such as beam depolarization) as highlighted by Burn (1966); Sokoloff
et al. (1998) and manifests more prominently at the longer wavelengths (lower
frequencies). Depolarization can be quantified using a ratio of fractional
polarization measurements at two different frequencies or resolutions – i.e.
the depolarization ratio.

Figure 9: The depolarization ratio obtained by taking a ratio of the
fractional polarization map at 0.88 GHz and 1.65 GHz, both at 11″by
10″resolution. The eastern lobe shows more depolarization than the western
lobe. Here, values > 1 (towards bright red) show re-polarization, while those
< 1 (towards deep blue) show depolarization. Only data where the
depolarization error to depolarization ratio was > 0.6 are shown. Higher
errors are associated with the eastern lobe.

In Fig. 9, we show the depolarization ratio map obtained by dividing the
fractional polarization map at 0.88 GHz with a map at 1.65 GHz, both at 11″by
10″resolution. The depolarization is expected to be stronger at longer
wavelengths; thus, a depolarization ratio of $<$ 1 (the bluer) suggests a
depolarized region, while $>$1 (redder) suggests a re-polarizing region. Here,
we show only regions where the fractional depolarization error666I.e. the
standard propagated depolarization error ratio divided by the depolarization
ratio. was less than 0.6. We find that the eastern lobe shows more
depolarization than the western lobe, particularly in the extreme lobe region
– consistent with the results of Perley et al. (1997), with some patches of
re-polarization. This map bears a striking boundary separating the two lobes.
The depolarization or re-polarization may be caused by unresolved Faraday
structures (beam depolarization) or internal mixing of thermal and ionized gas
within the lobes. Our data are insufficient to determine the actual cause for
the depolarization and re-polarization within the lobes of Pictor A. However,
these depolarization ratios must be interpreted cautiously, particularly
because of our narrow bandwidth and significant variations in fractional
polarization vs $\lambda^{2}$ (left column of Fig. 7). The re-polarization may
not necessarily be real or true but a result of limited bandwidth.

Figure 10: The binned depolarization ratio with distance from the radio core.
Data points were obtained by subdividing Pictor A into rectangular regions of
30″in width along the jet axis and calculating the mean, median and standard
deviation of data within those regions. The standard deviation is shown in the
filled coloured patches, while the solid and dashed lines indicate the mean
and median, respectively. For clarity, we also show a dotted line at a
depolarization ratio of 1. Negative distances indicate a leftward direction
from the radio core, while positive ones indicate a rightward direction. The
vertical dashed line indicates the core’s position.

Supporting depolarization profiles of both lobes are also shown in Fig. 10 as
a function of distance from the radio core, obtained by sub-dividing the
source into box regions of 30″in width777Except for the bin encompassing the
radio core, whereby the width is $\sim$ 20″. The radio core region was
excluded from these bins. along the jet axis and calculating a simple mean,
median, and standard deviation from our depolarization map within the bins.
The western lobe of this source shows consistent behaviour across the lobe,
with values centred around a depolarization ratio of 1, indicating little to
no depolarization. Contrastingly, the eastern lobe shows a more erratic spread
with no particular trend with increasing core distance.

The asymmetric depolarization observed, where the jet-sided lobe depolarizes
slower than its counterpart, has previously been defined as the Laing-
Garrington effect (Laing, 1988; Garrington et al., 1988). This is an
orientation effect that affects the lobe further from the observer because its
emission undergoes more depolarization effects resulting from passing through
a longer path length along the magneto-ionic halo of its host galaxy.
Asymmetric depolarization is a common feature in FR-II sources (e.g. Goodlet
et al., 2004; Ishwara-Chandra et al., 1998).

Assuming the eastern lobe is further, its higher observed depolarization could
result from external Faraday dispersion. This means that its emission may go
through various turbulent magneto-ionized cells along a line-of-sight towards
the observer, which causes depolarization and a resulting low polarized
signal. Another possible cause of depolarization is the intermixing of thermal
and synchrotron-emitting gas within the lobes. The significant depolarization
of this lobe between 20 and 6 cm noted by Perley et al. means that low-
frequency depolarization possibly further contributes to the observed lower
average degrees of polarization in the eastern lobe visible in Fig. 6. As
such, higher frequency observations are required to properly characterise the
depolarization behaviour of Pictor A’s lobes.

####  3.2.4 Faraday Rotation Measures

We performed pixel-by-pixel RM-synthesis across the entirety of Pictor A,
resulting in the RM map shown in the top panel of Fig. 11. Its colour bar
ranges between $0$ and $60$ rad m-2. Each pixel value in this map represents
the Faraday depth corresponding to the highest peak of the cleaned Faraday
spectrum. We only show rotation measures in regions of the source whereby
$p_{\text{err}}:p>0.6$, in addition to masking out unphysical values (i.e.
where $p\neq[0,1]$). Our lobe edges are the regions where the total intensity
falls below 4 mJy beam-1.

Figure 11: Top panel: The rotation measure map of Pictor A overlaid by Stokes
$\mathrm{I}$ radio contours. The RMs across the western lobe are generally
smooth, while those across the eastern lobe exhibit more variability. Bottom
panel: The polarized intensity map obtained from integrating the cleaned FDF
(Eqn. 11 of Burn (1966)), superimposed by the projected magnetic field vectors
(polarization vectors rotated by 90∘) whose length is set to 1. The field
orientation is well aligned along the lobe edges and the filamentary
structures. Both maps are at 11″by 10″resolution.

Generally, the $\mathrm{RM}$ distribution seems relatively similar across the
entire source, whereby our derived mean RM for the source was 48 rad m-2 with
a standard deviation of 10.2 rad m-2. The per lobe RM distribution showed that
RMs are centred around $49.7$ for the eastern lobe and $46.7$ rad m-2 for the
western lobe, with widths of $14.4$ and $4.5$ rad m-2 respectively. However,
patches of significant and sudden RM gradients are observed, especially within
the eastern lobe, forming boundary-like structures. A few regions also exhibit
negative RMs, such as those near the waist region towards the eastern lobe.

Perley et al. found the average RM to be approximately 43.5$\pm$1.4 rad m-2,
with the RM for each lobe falling within the margin of error in the L-band.
For a more direct comparison, we used our defined lobe boundaries and the VLA
L-band RM maps made available to us to extract the per lobe RM distribution
from the VLA data. This test yielded RMs for the VLA data centred around 55
and 52 rad m-2 and widths of $10.8$ and $6.4$ rad m-2 for the eastern and
western lobes, respectively. Furthermore, the eastern lobe displays a broader
variation than the western lobe in good agreement with the observations of
Perley et al.. The difference in our means may be attributed to our high
spatial sensitivity. The spread in the RM distribution is not significantly
different for each distribution. Simple statistics were also performed based
on this RM map to characterise the rotation measure behaviour with distance
from the core as illustrated in Fig. 12. The variability of the eastern lobe’s
RMs is more erratic than the western lobe, with regions closer to the core
showing more variation.

Haverkorn (2015) showed that Galactic contributions within spatially co-
located RM largely correlate, making the estimation of contribution on
individual lines-of-sight possible. Therefore, we estimated the Galactic
contribution towards Pictor A (galactic longitude of $251.6^{\circ}$ and
latitude $-34.63^{\circ}$) by taking the mean and standard deviation of
Galactic RM measurements derived by Hutschenreuter et al. (2022) (these
authors used Bayesian inferencing to cater for noisy data). We found a mean of
23.57 rad m-2 and mean standard deviation of $10.87$ rad m-2 estimated over a
5∘ region. Our Galaxy may contribute to the mean RM, but it cannot explain the
small-scale fluctuations, suggesting that some fraction of the observed RMs
may result from the intergalactic medium, X-ray gas or other unknown
intervening material.

####  3.2.5 Magnetic Fields

The bottom panel of Fig. 11 shows the orientation of the projected magnetic
field (electric field rotated by 90∘) overlaid on top of the cleaned and
integrated Faraday spectra map (obtained by integrating Eqn. 11 of Burn
(1966)). We set the length of the field vectors to 1 for demonstration
purposes. The magnetic field vectors align along the edges of the lobes. Such
alignment is common in radio galaxies (see Taylor et al., 1990; Eilek & Owen,
2002; Heald et al., 2009; Sebokolodi et al., 2020). Work by Laing (1980)
ascribes this effect to the shearing and compression of originally random
magnetic fields, making them tangential to the source’s boundaries. They argue
that the coexistence of aligned magnetic fields and regions of a high
fractional polarization only suggests the constriction of the magnetic field
to planes along a line-of-sight and not necessarily an originally uniform
magnetic field. Moreover, the projected magnetic field vectors also align
along the highly linearly polarized filamentary structures within the lobes,
especially in the western lobe.

Figure 12: Binned rotation measure profiles with increasing distance from the
radio core of Pictor A. Separation of these bins and the distances indicated
are similar to those stated for Fig. 10. RMs in the western lobe show a
decreasing variation with increasing distance from the radio core.

####  3.2.6 RM Gradients and Multiple-Peaked Faraday Spectra

In this section, we investigate the cause of the observed RM gradients across
Pictor A radio lobes and the presence of Faraday spectra with a single-peak
and a wide base, or multiple-peaks. We find that many LoS with sudden RM
changes show multiple high FDF peaks, making it difficult to differentiate
from sidelobes. Furthermore, they had much lower polarized emission (the
lowest polarized SNR per LoS ranged between 0.4 and 2). The largely flagged
out $\lambda^{2}$ data samples could also contribute to this issue because a
reduction in the number of $\lambda^{2}$ samples has been shown to increase
the RMTF sidelobes (see Fig.7 of Brentjens & de Bruyn, 2005). A combination of
low SNR and large gaps in $\lambda^{2}$ resulted in higher and erratic side
lobes on the deconvolved Faraday spectra (due to sidelobes on the RMTF as
illustrated in Fig. 2), thus, unstable solutions.

However, there are regions of high polarized SNR (up to two or three orders of
magnitude) showing double/multiple peaks and high RM gradients. These often
consist of one dominant peak and a secondary one of lesser intensity, while
others show more than two peaks. Spectra showing more than a single peak
appear to show "more" complex behaviour in the fractional polarization as a
function of wavelength. In most cases, it is difficult to determine whether
the LoS depolarizes at longer wavelengths, particularly for those showing
repolarization. We will need low-frequency data to properly determine if these
lines-of-sight depolarize indefinitely with increasing $\lambda^{2}$. The
ideal type of data would be, for example, similar to that of the wideband
polarimetric study of Cygnus A by Sebokolodi et al. (2020), which consisted of
a very broad frequency coverage (2–18 GHz) finely sampled along $\lambda^{2}$
space. Thus, the authors could characterize the decaying behaviour of
fractional polarization for various lines-of-sight (see Figs. 4 and 2 of that
work for demonstrations of the various categories of $p(\lambda^{2})$ and the
RMTF from their data). Their well-sampled $\lambda^{2}$ space resulted in
minimum sidelobes. Using our data, it is impossible to properly characterize
the behaviour of $p(\lambda^{2})$.

We further investigated whether the multiple peaks were real or whether our
high RMTF sidelobe resulted in a false positive peak. To do this, we used the
$\mathit{QU}$-fitting method (see for example Feain et al., 2009; Farnsworth
et al., 2011; O’Sullivan et al., 2012; Sun et al., 2015; Miyashita, 2019), to
fit specific models to the data in $\lambda^{2}$, and determine whether the
modelling favoured a model with double peaks or not. Since most of the sight
lines across Pictor A exhibited single peaked clean FDF, at least a single
Faraday rotating screen was anticipated. Therefore, we explored a single
polarized emitting component model coupled with a depolarizing component. For
our models, we assumed the external Faraday depolarizing screen (FDS) as the
main cause of the observed depolarization. The above Faraday effect is
modelled using the following:

$p=p_{0}e^{-2\sigma^{2}_{RM}\lambda^{4}}e^{2i(\chi_{0}+RM\lambda^{2})},$ (10)

where the first exponential term represents depolarization and
$\sigma_{\mathrm{RM}}$ is the Faraday dispersion (O’Sullivan et al., 2012). We
also explored two emitting components, each with an associated depolarization
effect. This is done by summing up the contributions of each component, i.e.
summing up Eqn. 10 as $p_{\mathrm{t}}=p_{\mathrm{1}}+p_{\mathrm{2}}$.

Double component models showed reasonably better fits; this was determined
using the Bayes’ factor method to avoid overfitting. This is not surprising,
especially given the structure we see in fractional polarization, which is not
expected for a simple uniform foreground screen. A physical interpretation of
these results is that multiple Faraday depths or a Faraday thick structure
exists along most sight lines across the source. Variations across the
observing beam could also lead to the observed behaviour. We note that models
with more than two Faraday rotation components were not tested.

####  3.2.7 Do the Missing Frequencies Affect RM Measurements?

One lingering question was whether missing frequency samples affected our RM
measurements. As an additional check, we simulated data at 80 evenly spaced
channels and set known fiduciary values of fractional polarization, RM, and
Stokes $\mathit{I}$ spectral variation. This formed our control dataset, to
which Gaussian random noise was added to make it more realistic. We then set
the test dataset up to match our actual observed data by excluding the same
frequencies that were excised from our channelised images. Specific details
about the setup of this experiment are highlighted in Appendix A.

Figure 13: Example plots showing the result of experimental
$\mathit{QU}$-fitting on simulated $\mathit{Q}$ and $\mathit{U}$ to ensure
that missing frequencies within our data do not affect results. A pair of rows
represents a single line-of-sight, with test data on the top panel and the
control data on the bottom. Both $\mathit{QU}$-fitting and RM-synthesis could
correctly locate the peak amplitude of the emitting components, even with some
missing frequency chunks.

Inspection of the results of $\mathit{QU}$-fitting and RM synthesis showed
that in both the control and test data cases, it was still possible to recover
the correct peak and its corresponding location despite missing $\lambda$ data
(some examples are shown in Fig. 13). However, $\mathit{QU}$-fitting within
test cases where the maximum degree of polarization was less than 20 per cent,
and two Faraday components had difficulty locating the secondary (lesser) FDF
peak position. On the other hand, the strongest FDF peak was correctly
identified in almost all cases. Furthermore, the sidelobes were considerably
higher, which caused confusion in the validity of a peak.

Two possible interpretations of the multiple peaks observed arise. The first
is that there is a single complex continuous Faraday structure (i.e. with a
wide Faraday depth) along the line-of-sight of Pictor A. Rudnick & Cotton
(2023) (in sec. 4 of their work) have shown that the resolution of the RMTF
approximated by Brentjens & de Bruyn (2005) incorrectly identifies a
continuous tophat Faraday distributions as two narrow-peaked components
located at the opposite edges of the original distributions. The second
possibility is that two or more Faraday components exist along this LoS. At
this point, it is impossible to definitively determine the correct
interpretation of the two; wideband data with higher Faraday depth resolution
could illuminate this issue better.

##  4 Summary and Conclusions

Using MeerKAT L-band data, we have generated a high-sensitivity map of Pictor
A shown in Fig. 1. We observe the bright but unresolved WHS, the double EHS
and the radio core. Furthermore, we have confirmed the presence of the kpc
scale radio jet from the radio core to the WHS. This radio jet is in good
alignment with the X-ray jet from Hardcastle et al. (2016). However, the
counter-jet remains undetected.

Hardcastle et al. (2016) speculated that if the X-ray emission within the
lobes of Pictor A is inverse Compton in nature, it must be co-spatial with
synchrotron emission. From the low sensitivity VLA radio observations by
Perley et al. (1997), X-ray emission appeared to extend beyond the radio
emission, which was strange. However, from our observations, the diffuse
emission extending away from the core is now clearly visible (see Fig. 3),
with some emission extending beyond the X-ray emission. Thus, we conclude that
X-ray emission within the lobes of Pictor A could result from inverse Compton
scattering.

Pictor A has also shown spectral variation typical of any FR-II source; it
exhibits a flat spectral index in the radio core and hotspots, but spectral
steepening occurs within the lobes and especially away from the radio core
towards the source edges (see Fig. 5 and Table 2 for a summary of the spectral
indices). More accurate model fitting and characterisation using wideband data
is necessary to distinguish between different variants of spectral ageing
models and to determine their parameters.

We have additionally performed a spectropolarimetric study of the lobes of
Pictor A (Sec. 3.2.4), where we show that this source exhibits mostly smooth
and positive rotation measures within both its lobes, with an average RM of
48.06 rad m-2 and a standard deviation of 10.19 rad m-2. However, the eastern
lobe is associated with a wider RM variation, more sudden RM changes and
higher depolarization than the western lobe. We have estimated the average RM
in the eastern lobe to be 50 rad m-2, while the western lobe averages 47 rad
m-2. The Galactic contribution was estimated to be $\mathrm{23.57\pm 16.12}$
rad m-2 in the direction of Pictor A. Although Galactic contributions could
dominate our observed rotation measures, narrowing down the various
contributions with our current data is difficult. This characteristically
smooth RM behaviour could indicate a large-scale Faraday rotating screen in
the foreground of Pictor A. However, this could also be caused by our data’s
low Faraday depth resolution directly linked to our limited bandwidth (see
Fig. 2), which limits our view of smaller Faraday depth scales.

Furthermore, we have demonstrated that lines-of-sight across Pictor A show
varied Faraday spectra, with some showing single, double or multiple peaks.
Lines-of-sight with more than one peak indicate the presence of multiple
polarized components at different Faraday depths. It is difficult, however, to
establish their depolarization characteristics, as the behaviour of fractional
polarization is highly variable with wavelength. This directly stems from the
limited frequency coverage and missing frequency chunks of our data. Moreover,
limited frequency coverage further lowers our depth resolution. Despite this,
we have also demonstrated through $\mathit{QU}$-fitting that the missing
frequency chunks do not affect our observed rotation measures (Sec. 3.2.7) and
that the multiple peaks observed in our data are possibly real, caused by
either the existence of multiple Faraday components or the inability of the
RMTF to correctly distinguish a single continuous Faraday complex structure
(Sec. 3.2.6).

Therefore, wideband observations will prove helpful in resolving the rotation
measure variations and determining whether the observed smoothness across
Pictor A is intrinsic to the source or its environs. For example, Sebokolodi
et al. (2020) showed that at higher spatial and depth resolution, Cygnus A
exhibited higher degrees of polarization and, hence, more polarization
structure. Therefore, with better depth and spatial resolutions, the origin of
the depolarization and repolarization patches and the RM gradients seen in
Figs. 9 and 11 could be explained better.

## Acknowledgements

We would like to thank M. J. Hardcastle for kindly making the Chandra X-ray
images of Pictor A available to us. LALA also thanks U. A. Sob and C.
Russeeawon for many helpful consultations and support.

The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory, which is a facility of the National Research Foundation, an
agency of the Department of Science and Innovation. OS and LA’s research is
supported by the National Research Foundation of South Africa (grant number
81737). The financial assistance of the South African Radio Astronomy
Observatory (SARAO) towards this research is hereby acknowledged
(http://www.sarao.ac.za).

## Data Availability

The data are available from the SARAO archive under proposal number
SCI-20190418-LS-01. The sub-band Stokes I, Q and U spectral images, and their
corresponding MFS images are available at https://doi.org/10.48479/kg33-es24.

## References

  * Andati et al. (2021) Andati L. A. L., Smirnov O. M., Makhathini S., Sebokolodi L. M., 2021, in ASP Conf. Ser. ADASS XXXI. (arXiv:2312.07645)
  * 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
  * Brentjens & de Bruyn (2005) Brentjens M. A., de Bruyn A. G., 2005, A & A, 441, 1217
  * Burn (1966) Burn B. J., 1966, MNRAS, 133, 67
  * Carilli & Barthel (1996) Carilli C. L., Barthel P. D., 1996, A & A Rev., 7, 1
  * Clarke (1996) Clarke D. A., 1996, in Energy Transp. Radio Galaxies Quasars. p. 311
  * Donnert et al. (2018) Donnert J., Vazza F., Brüggen M., ZuHone J., 2018, Space Sci. Rev., 214, 122
  * Eilek & Owen (2002) Eilek J. A., Owen F. N., 2002, ApJ, 567, 202
  * Fanaroff & Riley (1974) Fanaroff B. L., Riley J. M., 1974, MNRAS, 167, 31P
  * Fanaroff et al. (2021) Fanaroff B., et al., 2021, MNRAS, 505, 6003
  * Farnsworth et al. (2011) Farnsworth D., Rudnick L., Brown S., 2011, AJ, 141, 191
  * Feain et al. (2009) Feain I., et al., 2009, ApJ, 707, 114
  * Ferrière et al. (2021) Ferrière K., West J. L., Jaffe T. R., 2021, MNRAS, 507, 4968
  * Fomalont et al. (1989) Fomalont E. B., Ebneter K. A., van Breugel W. J. M., Ekers R. D., 1989, ApJ, 346, L17
  * Garrington et al. (1988) Garrington S. T., Leahy J. P., Conway R. G., Laingt R. A., 1988, Nature, 331, 147
  * Gentry et al. (2015) Gentry E. S., et al., 2015, ApJ, 808, 92
  * Goodlet et al. (2004) Goodlet J. A., Kaiser C. R., Best P. N., Dennett-Thorpe J., 2004, MNRAS, 347, 508
  * Halpern & Eracleous (1994) Halpern J. P., Eracleous M., 1994, ApJ, 433, L17
  * Hardcastle & Croston (2005) Hardcastle M. J., Croston J. H., 2005, MNRAS, 363, 649
  * Hardcastle et al. (2004) Hardcastle M. J., Harris D. E., Worrall D. M., Birkinshaw M., 2004, ApJ, 612, 729
  * Hardcastle et al. (2016) Hardcastle M. J., et al., 2016, MNRAS, 455, 3526
  * Harwood et al. (2013) Harwood J. J., Hardcastle M. J., Croston J. H., Goodger J. L., 2013, MNRAS, 435, 3353
  * Haverkorn (2015) Haverkorn M., 2015, in Lazarian A., de Gouveia Dal Pino E. M., Melioli C., eds, , Magnetic Fields in Diffuse Media. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 483–506, doi:10.1007/978-3-662-44625-6_17
  * Heald (2009) Heald G., 2009, Proc. IAU, 4, 591
  * Heald et al. (2009) Heald G., Braun R., Edmonds R., 2009, A & A, 503, 409
  * Högbom (1974) Högbom JA., 1974, A & A Supplement, 15, 417
  * Hutschenreuter et al. (2022) Hutschenreuter S., et al., 2022, A & A, 657, A43
  * Inskip et al. (2010) Inskip K. J., Tadhunter C. N., Morganti R., Holt J., Ramos Almeida C., Dicken D., 2010, MNRAS, 407, 1739
  * Ishwara-Chandra et al. (1998) Ishwara-Chandra C. H., Saikia D. J., Kapahi V. K., McCarthy P. J., 1998, MNRAS, 300, 269
  * Józsa et al. (2020) Józsa G. I. G., et al., 2020, Astrophys. Source Code Libr., p. ascl:2006.014
  * Kenyon et al. (2018) Kenyon J., Smirnov O., Grobler T., Perkins S., 2018, MNRAS, 478, 2399
  * Laing (1980) Laing R. A., 1980, MNRAS, 193, 439
  * Laing (1988) Laing R. A., 1988, Nature, 331, 149
  * Lauberts (1982) Lauberts A., 1982, ESO/Uppsala Survey of the ESO(B) Atlas. European Southern Observatory Garching
  * Lewis et al. (2010) Lewis K. T., Eracleous M., Storchi-Bergmann T., 2010, ApJ, 187, 416
  * Liu & Pooley (1991a) Liu R., Pooley G., 1991a, MNRAS, 249, 343
  * Liu & Pooley (1991b) Liu R., Pooley G., 1991b, MNRAS, 253, 669
  * Loveday (1996) Loveday J., 1996, MNRAS, 278, 1025
  * Marshall et al. (2010) Marshall H. L., et al., 2010, ApJ, 714, L213
  * Miyashita (2019) Miyashita Y., 2019, Galaxies, 7, 69
  * O’Sullivan et al. (2012) O’Sullivan S. P., et al., 2012, MNRAS, 421, 3300
  * Offringa et al. (2014) Offringa A. R., et al., 2014, MNRAS, 444, 606
  * Perley et al. (1984) Perley R. A., Dreher J. W., Cowan J. J., 1984, ApJ, 285, L35
  * Perley et al. (1997) Perley R. A., Roser H.-J., Meisenheimer K., 1997, A & A, 328, 12
  * Prestage (1985) Prestage R. M., 1985, PhD thesis, University of Edinburgh, UK, doi:1842/27209
  * Robertson (1973) Robertson J. G., 1973, Aust. J. Phys., 26, 403
  * Roeser & Meisenheimer (1987) Roeser H.-J., Meisenheimer K., 1987, ApJ, 314, 70
  * Rudnick & Cotton (2023) Rudnick L., Cotton W. D., 2023, MNRAS, 522, 1464
  * Scheuer (1982) Scheuer PAG., 1982, IAU Symp, 97, 163
  * Schmidt (1965) Schmidt M., 1965, ApJ, 141, 1
  * Schnitzeler et al. (2009) Schnitzeler D. H. F. M., Katgert P., de Bruyn A. G., 2009, A & A, 494, 611
  * Sebokolodi et al. (2020) Sebokolodi M. L. L., Perley R., Eilek J., Carilli C., Smirnov O., Laing R., Greisen E. W., Wise M., 2020, ApJ, 903, 36
  * Smirnov (2011a) Smirnov O. M., 2011a, A & A, 527, A107
  * Smirnov (2011b) Smirnov O. M., 2011b, A & A, 527, A106
  * Sokoloff et al. (1998) Sokoloff D. D., Bykov A. A., Shukurov A., Berkhuijsen E. M., Beck R., Poezd A. D., 1998, MNRAS, 299, 189
  * Sun et al. (2015) Sun X. H., et al., 2015, AJ, 149, 60
  * Taylor et al. (1990) Taylor GB., Perley RA., Inoue M., Kato T., Tabara H., Aizu K., 1990, ApJ, 360, 41
  * Tingay et al. (2008) Tingay S. J., Lenc E., Brunetti G., Bondi M., 2008, AJ, 136, 2473
  * Tribble (1993) Tribble P. C., 1993, MNRAS, 261, 57
  * Wilson et al. (2001) Wilson A. S., Young A. J., Shopbell P. L., 2001, ApJ, 547, 740
  * Zirbel (1997) Zirbel E. L., 1997, ApJ, 476, 489

##  Appendix A Data Simulations and QU-fitting specifics

As highlighted in Sec. 3.2.7, data was simulated for 100 lines-of-sight to
investigate the effect of missing frequency chunks on our data. We replicated
the presence of two Faraday peaks through data simulated using some fiduciary
RM and fractional polarization values. The first step was using the 80
frequency channels corresponding to our observation frequency to generate the
channelised Stokes $\mathit{I}$ data, with an arbitrary peak of 4 Jy using the
following function:

$\mathit{I}_{\text{obs}}=\mathit{I}_{\text{true}}(\frac{\nu}{\nu_{0}})^{\alpha},$
(11)

where $\alpha$ is the spectral index. A Gaussian random noise was added to
make the data more realistic. Using set values of initial fractional
polarization, RM and polarization angle, we generated our ‘observed’
fractional polarization data using the model in Eqn. 10. We also ensured that
there is a single dominating peak by setting $p_{0}=0.5p_{1}$ (see the bottom
panel of Fig. 14). We used the fractional Stokes relationship: $p=q+iu$ to
derive the values of $\mathit{Q}$ and $\mathit{U}$ from their fractional
analogues ($\mathit{q,u}$) and our simulated Stokes $\mathit{I}$ data.
Furthermore, we included noise values to make the data more realistic. We then
stored the fractional polarization, RM and polarization angle for both
components and the $\mathit{Q}$ and $\mathit{U}$ data for each frequency
channel.

The next step was determining if the peaks and their positions could be
recovered accurately, even with missing samples, using $\mathit{QU}$-fitting
and RM-synthesis. Therefore, our control experiment involved the data when all
available samples were contained. Our test data were similar to the control
one, except for the exclusion of channels flagged out in our original data.
This was done to replicate our scenario better. This process was repeated 100
times, representing one hundred different lines-of-sight.

We chose RM values ranging between $-150$ and $150$ rad m-2, as this is the
maximum Faraday depth scale our data is sensitive to. These data are shown in
Fig. 14. The polarization angles are between $[-\frac{\pi}{2},\frac{\pi}{2}]$.
The bottom panel of Fig. 15 shows the distribution of the artificial noise
added to make the $\mathit{I,Q}$, and $\mathit{U}$ data more realistic. The
exact values are shown at the top panel of the same figure.

Figure 14: The top panel shows the distribution of RM difference between our
two simulated emitting components. The range of our RMs was between $-150$ and
150 rad m-2. The bottom panel shows the actual input Faraday depth values;
$\phi_{0}$ is illustrated by the blue dashed line with a cross marker, and
$\phi_{1}$ is illustrated by the red solid line with plus sign markers. The
black solid with point markers shows their difference. Figure 15: The input
fractional polarization values are shown in the top panel. The dominant
fractional polarization component is shown in the blue solid line with cross
markers, while the secondary component is in the red dashed lines with plus
sign markers. This component is always half of the dominant component. _Bottom
panel_ : Distribution of the input noise added to the simulated I, Q and U to
make it more realistic. All lines-of-sight had the same noise across the band.