11institutetext:  INAF-Padova Astronomical Observatory, Vicolo
dell’Osservatorio 5, I-35122 Padova, Italy 22institutetext: Dipartimento di
Fisica e Astronomia, Università di Bologna, via Gobetti 93/2, 40129 Bologna,
Italy 33institutetext: INAF, Istituto di Radioastronomia di Bologna, via
Gobetti 101, 40129 Bologna, Italy 44institutetext: ASTRON, the Netherlands
Institute for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands
55institutetext: Leiden Observatory, Leiden University, PO Box 9513, 2300 RA
Leiden, The Netherlands 66institutetext: Hamburger Sternwarte, Universität
Hamburg, Gojenbergsweg 112, D-21029, Hamburg, Germany 77institutetext: Centre
for Extragalactic Astronomy, Durham University, DH1 3LE, UK 88institutetext:
CSIRO Space & Astronomy, PO Box 1130, Bentley, WA 6102, Australia

#  A LOFAR view into the stormy environment of the galaxy cluster 2A0335+096

A. Ignesti Contact<EMAIL_ADDRESS>G. Brunetti 33 T. Shimwell
4455 M. Gitti 2233 L. Birzan 66 A. Botteon 55 M. Brüggen 66 F. de Gasperin
6633 G. Di Gennaro 66 A. C. Edge 77 C. J. Riseley 223388 H. J. A. Röttgering
55 R. J. van Weeren 55

(Accepted )

###### Abstract

Context. Radio observations represent a powerful probe of the physics
occurring in the intracluster medium (ICM) because they trace the relativistic
cosmic rays in the cluster magnetic fields, or within galaxies themselves. By
probing the low-energy cosmic rays, low-frequency radio observations are
especially interesting because they unveil emission powered by low-efficiency
particle acceleration processes, which are believed to play a crucial role in
the origin of diffuse radio emission.

Aims. We investigate the origin of the radio mini-halo at the center of the
galaxy cluster 2A0335+096 and its connection to the central galaxy and the
sloshing cool core. We also study the properties of the head-tail galaxy GB6
B0335+096 hosted in the cluster to explore the lifecycle of the relativistic
electrons in its radio tails.

Methods. We use new LOw Frequency ARray (LOFAR) observations from the LOFAR
Two-meter Sky Survey at 144 MHz to map the low-frequency emission with a high
level of detail. The new data were combined with archival Giant Metrewave
Radio Telescope and $Chandra$ observations to carry out a multi-wavelength
study.

Results. We have made the first measurement of the spectral index of the mini-
halo ($\alpha=-1.2\pm 0.1$ between 144 MHz and 1.4 GHz) and the lobes of the
central source ($\alpha\simeq-1.5\pm 0.1$ between 144 and 610 MHz). Based on
the low-frequency radio emission morphology with respect to the thermal ICM,
we propose that the origin of the diffuse radio emission is linked to the
sloshing of the cool core. The new data revealed the presence of a Mpc-long
radio tail associated with GB6 B0335+096. The observed projected length is a
factor 3 longer than the expected cooling length, with evidence of flattening
in the spectral index trend along the tail. Therefore, we suggest that the
electrons toward the end of the tail are kept alive by the ICM gentle re-
acceleration.

###### Key Words.:

galaxies: clusters: individual: 2A0335+096; galaxies: jets; radio continuum:
galaxies; radiation mechanisms: non-thermal; methods: observational

##  1 Introduction

Low-frequency ($\sim 100$ MHz) observations probe low-energy particles that,
due to their lower emissivity, have typical life-times of a factor $\sim 3$
longer than the high-energy particles traced by high-frequency ($>$1000 MHz)
radio emission (e.g., Pacholczyk 1970). This difference in time-scales is
crucial when studying the re-acceleration processes of cosmic rays electrons
(CRe). This is because a re-acceleration process, and re-energization in
general, is effective only if the time-scale of the energy gain is smaller
(or, at least, equal) than the time-scale of the energy losses. It follows
that low-energy CRe, due to their longer lifetimes in a given magnetic field
(e.g., 1-10 $\mu$G for galaxy clusters, see Carilli & Taylor 2002), are more
prone to be affected by re-acceleration than the high-energy ones. Therefore,
only low-frequency observations have the potential to explore these phenomena,
hence opening a window on a class of low-efficiency processes related to the
intracluster medium (ICM) microphysics, in form of turbulence or gentle
compression of the radio plasma by the thermal gas (for a review Brunetti &
Jones 2014, and references therein).

In this work we present an analysis of the low-frequency observation of the
galaxy cluster 2A0335+096 carried out with the LOw Frequency ARray (LOFAR, van
Haarlem et al. 2013). 2A0335+096 is a low redshift ($z=0.035$), low-mass
cluster (M${}_{500}=2.27\times 10^{14}$ M⊙; Planck Collaboration et al.
2014)111 M500 and R500 indicate the total mass and radius at a mean over-
density of 500 with respect to the cosmological critical density at redshift
of the cluster, respectively. . We selected this cluster on the basis of its
interesting features. To begin with, 2A0035 hosts a radio mini-halo (MH) at
its center (Sarazin et al. 1995). MHs are diffuse radio sources observed at
the center of relaxed clusters and characterized by steep spectral indices
($-1.5<\alpha<-1.1$)222In this work we define the synchrotron spectrum as
$S\propto\nu^{\alpha}$, where $S$ is the flux density at the frequency $\nu$
and $\alpha$ is the spectral index. and typical radii of 100-150 kpc (e.g.,
Giacintucci et al. 2017; van Weeren et al. 2019, and the references therein).
Usually the radio emission is mostly confined within the cool-core region of
the clusters, thus outlining a connection between the non-thermal ICM
components and the thermal plasma (e.g., Rizza et al. 2000; Gitti et al. 2002;
Mazzotta & Giacintucci 2008; Giacintucci et al. 2014a). This intrinsic
connection is supported by the correlations observed between integrated radio
and X-ray luminosities (Bravi et al. 2016; Gitti et al. 2015, 2018;
Giacintucci et al. 2019; Richard-Laferrière et al. 2020). Recently, the
discovery of very steep-spectrum ($\alpha<-2$), low-frequency emission on
large scales ($\sim 500$ kpc) around a number of MHs (Savini et al. 2018,
2019; Biava et al. 2021) suggested the possibility that low-energy CRe could
actually diffuse over such large scales also in relaxed clusters. The origin
of CRe in the MH volume is still debated, with two proposed scenarios. One is
the leptonic scenario, where the CRe, possibly injected by the active galactic
nucleus (AGN) of the central radio galaxy, are re-accelerated by ICM
turbulence (e.g., Gitti et al. 2002; ZuHone et al. 2013). The other scenario
is the hadronic case, where CRe are produced by collisions between the
relativistic and the thermal protons of the ICM, which can travel to larger
distances due to their lower emissivity (Pfrommer & Enßlin 2004).

The MH in 2A0335+096 was first imaged at 1.4 GHz and 5.5 GHz by Sarazin et al.
(1995). It surrounds the central radio galaxy and extends for only $\sim
100^{\prime\prime}$ ($\sim$ 70 kpc), thus making it one of the smallest MH
known (Giacintucci et al. 2017, 2019). The detection of extended radio
emission at 144 MHz surrounding the central galaxy has already been presented
in Kokotanekov et al. (2017). The central radio galaxy presents two
components: there is an inner source (Donahue et al. 2007; Giacintucci et al.
2019) and a pair of large-scale radio lobes filling the X-ray cavities (Bîrzan
et al. 2020). There is also a third patch of extended emission, which is
interpreted as a fossil lobe from an older AGN outburst, detected at $\sim
25^{\prime\prime}$($\sim$18 kpc) from the cluster center (Giacintucci et al.
2019). Sarazin et al. (1995) discussed the possibility that part of the radio
emission comes also from a companion galaxy close to the cluster center.

From the thermal point of view, 2A0335+096 has a central temperature of 3.6
keV and a central entropy of 7.1 keV cm-2 (Cavagnolo et al. 2009; Giacintucci
et al. 2017). In the X-ray band it presents two cavities, which coincide with
the position of the radio lobes. In addition, there is a cold front located at
$\sim$40 kpc from the cluster center, which suggests that the cool core is
sloshing in the gravitaional well (Mazzotta et al. 2003; Ghizzardi et al.
2006; Sanders et al. 2009).

Evidences of the stormy nature of 2A0335+096 come also from optical and ALMA
observations. Vantyghem et al. (2016) and Olivares et al. (2019) present
detailed studies of cold and warm filaments at the cluster center. The gas
appears to be distributed over several clumpy and dusty filaments trailing the
X-ray cavities, which support the idea that the gas has cooled from low-
entropy ICM that could have been lifted out of the cluster core, becoming
thermally unstable. Both the molecular and the warm gas present disturbed
kinematics, mainly due to the projection of several inflowing or outflowing
filaments, or due to the presence of turbulence induced by bubbles, jets, or
sloshing.

The stormy environment in which we observe the MH, with the cavities inflated
by the AGN and the sloshing, would entail a perturbed, turbulent environment,
at least in the central region of the ICM and possibly beyond. Therefore,
motivated by the recent findings of low-frequency diffuse emission extending
beyond cool cores (Savini et al. 2018, 2019; Biava et al. 2021), we obtained
8.3 hrs of LOFAR HBA data in co-observing with the LOFAR Two Metre Sky Survey
(LoTSS; Shimwell et al. 2017, 2019). The new radio images were combined with
archival Giant Metrewave Radio Telescope (GMRT) and Chandra observations to
investigate the origin of the MH.

Interestingly, 2A0335+096 also hosts the head-tail radio galaxy GB6 B0335+096
($z=0.038$), an FR I galaxy (Fanaroff & Riley 1974) with an extended radio
tail (Patnaik & Singh 1988; Sebastian et al. 2017). Tailed FR I radio galaxies
are characterized by large-scales, low-brightness emission bent by the
environmental pressure (thermal or ram-pressure) in the same direction,
forming structures similar to tails. These radio galaxies are classified in
two classes: narrow-angle tailed sources (NAT), with small angles between the
tails (O’Dea & Owen 1985), and wide-angle tailed sources (WAT), with a larger
angle between the tails (e.g., Miley 1980; Feretti & Venturi 2002). Based on
ROSAT observations of head tail radio sources (including GB6 B0335+096), Edge
& Rottgering (1995) studied the connection between the enhanced X-ray emission
and radio core emissions, concluding that the X-ray enhancement likely results
from nuclear emission rather than substructure in the extended cluster gas.
The study of radio tails can provide useful insights into the properties of
the ICM, specifically on the interplay between the relativistic jets and the
thermal ICM (e.g., Terni de Gregory et al. 2017; Müller et al. 2021), as well
as allowing us to probe the CRe life-cycle in the ICM. For the case of
2A0335+096, the new LOFAR data permit us to ascertain if turbulence can extend
outside the cool core and interact with the CRe released by the galaxy.

In this work we used the standard concordance cosmology parameters
$H_{0}=70\,\rm km\,s^{-1}\,Mpc^{-1}$, ${\Omega}_{M}=0.3$ and
${\Omega}_{\Lambda}=0.7$. At the redshift of 2A0335+096, $z$=0.035, this
yields333http://www.astro.ucla.edu/$\\#$7Ewright/CosmoCalc.html 1′′= 0.694 kpc
and a luminosity distance of 145 Mpc.

##  2 Data processing and analysis

###  2.1 Radio data

Figure 1: Multi-frequency view of 2A 0335+096. Top left: GMRT image at 610
MHz, $\sigma$=84 $\mu$Jy beam-1, resolution $6^{\prime\prime}\times
4^{\prime\prime}$; Top right: LOFAR low-resolution image, $\sigma$=517 $\mu$Jy
beam-1, resolution $52^{\prime\prime}\times 16^{\prime\prime}$; Bottom left:
LOFAR mid-resolution image, $\sigma$=231 $\mu$Jy beam-1, resolution
$14^{\prime\prime}\times 5^{\prime\prime}$; Bottom right: LOFAR high-
resolution image, $\sigma$=400 $\mu$Jy beam-1, resolution
$6^{\prime\prime}\times 4^{\prime\prime}$; We report the -3, 3, 6, 24,
96$\sigma$ levels in the LOFAR images and the -3, 3, 6, 12, 24$\sigma$ in the
GMRT image. The maps are reported in units of Jy.

We present the analysis of the LOFAR observation LC11$\\_$014 (PI Ignesti) in
the 120-168 MHz band. We reduced the dataset using the direction-dependent
calibration and imaging pipeline ddf-
pipeline444https://github.com/mhardcastle/ddf-pipeline v. 2.2 . This pipeline
was developed by the LOFAR Surveys Key Science Project and corrects for
ionospheric and beam model errors in the data. The latest version of the
pipeline is described in Tasse et al. (2021). The entire data processing
procedure makes use of prefactor (van Weeren et al. 2016; Williams et al.
2016; de Gasperin et al. 2019), killMS (Tasse 2014; Smirnov & Tasse 2015) and
DDFacet (Tasse et al. 2018) and results in a thermal noise-limited image of
the entire LOFAR field-of-view. To further refine the processing, we then
performed additional phase and amplitude self-calibration cycles in a small
region centered on the source, where the direction-dependent errors are
assumed to be negligible (see van Weeren et al. 2021, for further details on
this procedure).

We produced images at a central frequency of 144 MHz with different
resolutions by using WSClean v2.6 (Offringa et al. 2014) and using several
different Briggs weightings (Briggs 1995), tapering configurations, and multi-
scale cleaning (Offringa & Smirnov 2017). An inner uvcut of 80$\lambda$,
corresponding to an angular scale of 43′, was applied to the data to remove
the shortest spacings where calibration is more challenging. We corrected for
the systematic offset of the LOFAR flux density scale produced by inaccuracies
in the LOFAR HBA beam model by multiplying it with a factor of 0.8284 . This
aligns it with the flux-scale of the point-sources mapped in the first
alternative data release from the TIFR GMRT Sky Survey (TGSS-ADR1, Intema et
al. 2017). Following LoTSS, we adopt a conservative calibration error of
20$\%$, which dominates the uncertainties on the LOFAR flux densities. The
resulting images, with resolutions between 6 and 50 arcseconds, are shown in
Figure 1. 2A0335+096 has a low declination of +10∘ that resulted in a noise
above 200 $\mu$Jy beam-1, which is consistent with that archived at higher
declination (e.g., Shimwell et al. 2019) when accounting for the projection of
the LOFAR stations. We note the presence of artifacts surrounding the bright
sources, such as the AGN of GB6 B0335+096 (Figure 1, panel b), which are due
to imperfections in our calibration.

For a spectral analysis we make use of a GMRT image of 2A0335+096 that we made
from an archival observation at 610 MHz (cycle 17, project 17$\\_$016, PI
Raychaudhury). The time on target is 75 mins and the data were processed using
the SPAM pipeline (see Intema et al. 2009, 2017, for details). We produced new
images at 610 MHz by using WSClean (Figure 1, panel a). For the GMRT images we
adopted a calibration error of 5$\%$ (Chandra et al. 2004).

###  2.2 X-ray data

Figure 2:  Top: LOFAR contours at 144 MHz of the mid-resolution image (panel
b, Figure 1, $\sigma$=231 $\mu$Jy beam-1, resolution $14^{\prime\prime}\times
5^{\prime\prime}$ ) at the 3, 24, 96$\sigma$ levels on top of the background-
subtracted, exposure-corrected Chandra image in the $0.5-2.0$ keV band;
Bottom-left: ggm-processed X-ray image produced with $\sigma=4$ and the 144
MHz emission contours on top; Bottom-right: Residual unsharp-masked image
obtained by combining X-ray maps smoothed with 3 and 30 pixel-wide Gaussians
with the 144 MHz emission contours on top. The white arrows in the bottom
panels point to the spiral pattern and the cavities described in Section 3.1 .

In order to study the interplay between the relativistic plasma and the
thermal ICM, we produced new images in the 0.5-2.0 keV band to search for
substructures in the ICM associated with the radio emission (in form of
cavities or edge in the surface brightness) and to study the spatial
correlation between radio and X-ray emissions (see Section 2.4). For this
purpose, we selected the Chandra observation 7939 (PI Sanders), which is
currently the deepest archival observation (total exposure time 49.5 ks). We
reprocessed the Chandra datasets with CIAO 4.10 and CALDB 4.8.1 to correct for
known time-dependent gain and for charge transfer inefficiency. In order to
filter out strong background flares, we also applied screening of the event
files555http://cxc.harvard.edu/ciao/guides/acis_data.html. For the background
subtraction, we used the CALDB $``$Blank-Sky” files normalized to the count
rate of the source image in the 10-12 keV band. The background-subtracted,
exposure corrected image in the $0.5-2.0$ keV band is presented in the top
panel of Figure 2. We further processed the image to investigate the presence
of sub-structures in the ICM. We produced a new image with the ggm filter
(Sanders et al. 2016), which emphasizes the gradients in the surface
brightness, by setting the parameter $\sigma=4$ (Figure 2, bottom-left panel).
Finally, to highlight the presence of cavities in the ICM, we produced also an
unsharp-masked image by combining X-ray maps smoothed with a 3 and 30 pixel-
wide Gaussians (Figure 2, bottom-right panel).

###  2.3 Spectral index map

Figure 3:  Spectral index map between 144 and 610 MHz (top) and the
corresponding error map (bottom). We report the 3, 12, 48, 192$\sigma$
contours of the 144 MHz images ($\sigma$=750 $\mu$Jy beam-1) produced to map
the spectral index.

We combine the 144 and 610 MHz images to map the spectral index of the central
source and the head tail galaxy. To do this we first produced images at both
frequencies with the same $uv$-range (200-54000 $\lambda$) to compare fluxes
from data with the same range of spatial scales. Then we smoothed both images
to a resolution of 30′′, which was close to the resolution of the LOFAR image
produced with the aforementioned $uv$-cut, and we regridded the GMRT image to
match the LOFAR one to be sure that each pixel covers the same angular region.
The final noise levels of the low-resolution images are 750 $\mu$Jy beam-1 for
the 144 MHz image and 410 $\mu$Jy beam-1 for the 610 MHz one. The two images
were compared on a pixel-by-pixel basis and the spectral index $\alpha$ was
computed as:

$\alpha=\frac{\text{log}\left(\frac{S_{610}}{S_{144}}\right)}{\text{log}\left(\frac{610}{144}\right)}\pm\frac{1}{{\text{log}\left(\frac{610}{144}\right)}}\sqrt{\left(\frac{\sigma_{S,610}}{S_{610}}\right)^{2}+\left(\frac{\sigma_{S,144}}{S_{144}}\right)^{2}}\text{
,}$ (1)

where $S$ and $\sigma_{S}=\sqrt{(f\cdot S)^{2}+\sigma^{2}}$ (where $f$ is the
flux density scale uncertainty and $\sigma$ is the noise level) are the flux
densities and the corresponding errors at the two frequencies, 144 and 610
MHz.

We computed the spectral index (and the corresponding error) for each pixel
with a surface brightness above the 3$\sigma$ noise level. The results are
shown in Figure 3, where we present the spectral index map and the
corresponding error map.

###  2.4 Point-to-point analysis

Figure 4:  Top: result of the single-mesh point-to-point analysis. The slope
of the best-fitting power-law $I_{\text{R}}\propto I_{\text{X}}^{k}$ is
reported in the legend, and the insert presents the sampling grid overlaid on
the low-resolution 144 MHz map (Figure 1). The blue-shaded region indicate the
90$\%$ confidence interval of the fit. Bottom: Slope index distributions
produced by the MCptp analysis at 0.144 MHz, 1.4 and 5.5 GHz (from Ignesti et
al. 2020). The mean and standard deviation of each distribution is shown in
the labels.

The resolution of the LOFAR data permitted us to search for a spatial
correlation between the MH radio, $I_{\text{R}}$, and ICM X-ray,
$I_{\text{X}}$, surface brightness. To asses this we performed a point-to-
point (ptp) analysis with the Point-to-point TRend EXtractor (PT-REX, Ignesti
2022). At higher frequency, the MH in 2A0335+096 exhibits a strong correlation
between $I_{\text{R}}$ and $I_{\text{X}}$, in the form of $I_{\text{R}}\propto
I_{\text{X}}^{k}$ where $k=1.33\pm 0.23$ and $1.01\pm 0.15$ at 1.4 and 5.5 GHz
respectively. This is in agreement with the others MHs (Ignesti et al. 2020).
This previous analysis was done at a resolution of the order of 18-20 ′′. To
sample the 144 MHz radio surface brightness with a comparable resolution we
used the low-resolution LOFAR image ($52^{\prime\prime}\times
16^{\prime\prime}$, Figure 1). In order to focus on the MH, we masked the
central AGN, the lobes, and the putative bridge between the MH and the radio
tail.

At lower frequency, the correlation is less strong. A a single-mesh (SM)
analysis resulted in a super-linear scaling ($k_{\text{SM}}=1.12\pm 0.19$,
Figure 4, top panel), whilst the Monte Carlo ptp analysis (we refer to Ignesti
2022, for further details), revealed that, when testing 100 different grid
configurations, the correlation can assume a broad distribution of slopes
(Figure 4, bottom panel), although the preference was for $k=0.72\pm 0.42$. A
sub-linear relation would be typical of giant radio halos (e.g., Ignesti et
al. 2020; Bruno et al. 2019; Rajpurohit et al. 2021; Biava et al. 2021). A
similar slope has already been reported for the MH in the Phoenix cluster
($k=0.84\pm 0.23$, Timmerman et al. 2021). However, in the case of 2A0335+096
our fit is poorly-constrained, likely due to the presence of the AGN lobes and
the radio tail that, even though they were masked, are bright enough to
contaminate the surrounding diffuse emission resulting in a off-center
increase of $I_{\text{R}}$ (i.e., a flattening of the spatial correlation).
Hence, at present, we are unable to reliably infer further information on the
CRe dynamics from the point-to-point correlation. On the other hand, it shows
how the SM ptp analysis could be biased when performed on sources with complex
morphology or multiple components.

##  3 Results and discussion

The new LOFAR observation allows us to explore the impact of ICM turbulent re-
acceleration on the lifecycle of the low-energy CRe. Here we analyze the
central source, which consists of several distinct components (Figure 5), and
the radio galaxy GB6 B0335+096.

Figure 5: LOFAR contours of the mid-resolution image (panel b, Figure 1,
$\sigma$=231 $\mu$Jy beam-1, resolution $14.43^{\prime\prime}\times
5.05^{\prime\prime}$) where we label the components of the central source as
given in the text.

###  3.1 The central radio source

LOFAR’s sensitivity to large-scale emission provides us with a detailed view
of the structure of the central radio source that we can explore at various
resolutions. The mid-resolution LOFAR image (Figure 1, panel b, resolution
$14^{\prime\prime}\times 5^{\prime\prime}$) shows the bright central emission
of the radio galaxy and the northern, southern and relic lobes embedded within
the MH. The Chandra X-ray image in 0.5-2.0 keV shows the cavities produced by
the radio-mode AGN feedback at the center of the cluster. In particular, the
unsharp-masked image (Figure 2, bottom-right panel) highlights that the cavity
reported by Mazzotta et al. (2003) is coincident with the northern radio lobe,
while the opposite radio lobe is located close to the southern cavity. In the
high-resolution LOFAR image (Figure 1, panel c, resolution
$6.13^{\prime\prime}\times 3.82^{\prime\prime}$), due to lower surface
brightness sensitivity, we detect only the AGN lobes, plus the relic lobe in
the north west (Giacintucci et al. 2019). Finally, at the lowest resolution
(Figure 1, panel a, resolution $52^{\prime\prime}\times 16^{\prime\prime}$),
where the surface brightness sensitivity is highest, we observe the previously
discussed aspects but also see a putative bridge of radio emission between the
central source and the radio tail. The apparent discrepancy between the
spectral indices at the two ends of the putative radio bridge (Figure 3)
challenges the idea that the same relativistic plasma may extend from the
central source to the radio tail and, hence, suggests that this feature is a
blend of their emission produced by the resolution of the image.

Here we focus on the study of the MH. A detailed analysis of a LOFAR
observation of the central AGN and its lobes can be found in Bîrzan et al.
(2020). In the low-resolution LOFAR image (Figure 1, panel a), we estimated
the MH radius following Cassano et al. (2015), i.e.
$R_{\text{MH}}=\sqrt{R_{\text{min}}\times
R_{\text{max}}}=\sqrt{75^{\prime\prime}\times
127^{\prime\prime}}=97.6^{\prime\prime}\simeq 67$ kpc, where $R_{\text{min}}$
and $R_{\text{max}}$ are the radii of the circular regions centered on the AGN
that inscribes and circumscribes, respectively, the MH. This is consistent
with the observed radius at 1.4 GHz (70 kpc Giacintucci et al. 2014b), and
generally smaller than other MHs ($100-150$ kpc). However, due to the high
noise of the images and the presence of a bright, extended radio galaxy at the
center of the MH, we cannot be certain that we are able to measure the full
extent of the system.

In the mid- and low-resolution images we measured an integrated flux density
of the central region (mini-halo plus central radio galaxy and lobes) at 144
MHz of 1.15$\pm$0.23 Jy within the 3$\sigma$ contours and bounded by the
6$\sigma$ contour in the direction of the radio tail. In order to estimate the
flux density of the diffuse emission only, we first subtracted the
contributions of the AGN lobes (0.75$\pm$0.15 Jy) and of the north-west relic
lobe (0.17$\pm$0.03 Jy), both measured within the 3$\sigma$ contour of the
high-resolution image (Figure 1, panel c). The resulting net integrated MH
flux density is 0.23$\pm$0.05 Jy, where the error is computed by accounting
for the $20\%$ flux density scale uncertainty as well as the error from the
noise and the net area ( $\sigma_{S}=[(f\cdot S)^{2}+\sigma^{2}]^{1/2}$). This
entails an average MH surface brightness of 1.23$\pm$0.27 mJy beam-1 with a
beam size of $14^{\prime\prime}\times 5^{\prime\prime}$. Then we estimated the
’missing’ MH flux density corresponding to the AGN and relic regions in the
mid-resolution image by multiplying the MH average surface brightness by their
areas (estimated as the area encompassed by the 12$\sigma$ contours). We
computed an additional contribution of 77.7$\pm 16.9$ mJy associated with the
MH emission, that results in a total flux density of 387.7$\pm$53.9 mJy.

By comparing this measure with the flux density of the MH at 1.4 GHz reported
in Giacintucci et al. (2014b), which is 21.1$\pm$2.1 mJy, we estimated a
spectral index $\alpha=-1.2\pm 0.1$ between 144 MHz and 1.4 GHz. This entails
a radio power at 144 MHz of $P=4\pi D_{L}^{2}S(1+z)^{1+\alpha}=9.9\times
10^{23}$ Watt Hz-1, where $D_{L}=145$ Mpc is the cluster luminosity distance.
We did not compute the flux density and radio power at 610 MHz due to the low
quality of the current observation.

We note that the MH in 2A0335+096 is a low mass, local example of this class
in terms of radio power at 1.4 GHz, as well as the size (Giacintucci et al.
2014b, 2017). Unfortunately, the the resolved MH spectral index could not be
reliably mapped (Figure 3) because of the insufficient sensitivity of the GMRT
observation. We were however able to measure the spectral index of the AGN
lobes and the relic lobe, which show steep spectra with $\alpha\simeq-1.5\pm
0.1$. Interestingly, the Chandra GGM image presented in the bottom-left panel
of Figure 2 shows that the southern part of the MH is encompassed by a bright,
spiral-like edge in the X-ray surface brightness. This structure corresponds
with the front already reported by previous studies (Mazzotta et al. 2003;
Ghizzardi et al. 2006; Sanders & Fabian 2006; Sanders et al. 2009) and it can
be interpreted as evidence of sloshing of the cool core. Interestinlgy,
Mazzotta & Giacintucci (2008); ZuHone et al. (2013) discussed the possibility
that, by injecting turbulence in the ICM, the sloshing could be able to
originate MHs.

The new LOFAR observation allows us to study the nature of the MH and its
interplay with the central AGN. The spectral index map (Figure 3) show that
the northern and southern AGN lobes have steep spectra ($\alpha\simeq-1.5\pm
0.1$). Meanwhile, the surrounding diffuse radio emission appears distinct from
this morphologically but it also has a flatter666These spectra are estimated
using different bands, namely 144-610 MHz for the lobes and 144-1400 MHz for
the MH. The fact that the MH results to be flatter despite the larger band
suggests that its spectrum is overall flatter than the lobes one. spectrum
($\alpha=-1.2\pm 0.1$). This might suggest that the CRe which are powering the
MH were not simply injected from the lobes and later propagated to larger
scales without any re-acceleration because, in this case, we would expect the
MH spectrum to be steeper (i.e. the CRe to be older) than the lobes. Instead,
the flatter spectrum could either indicate that there is no relation between
the AGN and the MH, or the action of turbulent re-acceleration affecting the
CRe lifecyle. Obvious sources of such turbulence would be the AGN mechanical
feeding and sloshing, the second one supported also by the close spatial
correspondance between the radio emission and the sloshing region (Figure 2,
bottom-left panel).

Alternatively, one might assume that the CRp generated by the AGN escape the
lobes at late times (e.g., Ehlert et al. 2018) and propagate around the core
generating CRe via the proton-proton cascade that, in turn, will generate
diffuse radio emission. The analyzed data do not allow us to check for
spectral steepening in the MH, which might test this possibility (e.g.,
Brunetti & Jones 2014), hence this remains a viable scenario.

###  3.2 GB6 B0335+096

The most striking feature observed by LOFAR is the giant radio tail of GB6
B0335+096, which extends for more than 900 kpc ($\sim 22^{\prime}$). This is
more than double than what was observed in previous works ($\sim 400$ kpc
Patnaik & Singh 1988; Sarazin et al. 1995; Sebastian et al. 2017).
Interestingly, the surface brightness of the new low-frequency continuation is
almost uniform up to the end of the tail. The fact that only LOFAR could
observe this low-frequency continuation suggests that these structures may
actually be common in many systems, thus they are going to be found more and
more efficiently with the advances of the LOFAR surveys (Shimwell et al. 2019;
de Gasperin et al. 2021). The high- and mid-resolution images (Figure 1,
panels a, b) show that the tail already bends within 100 kpc from the AGN. We
detect hard X-ray 2-7 keV emission coming from the flat-spectrum core in GB6
B0335+096 (Figure 6), this implies that the AGN is currently active.

Figure 6: LOFAR contours of the high-resolution image at 144 MHz (panel c,
Figure 1, $\sigma$=400 $\mu$Jy beam-1, resolution $6.13^{\prime\prime}\times
3.82^{\prime\prime}$) at the 3, 24, 48, 96$\sigma$ levels on top of the
Chandra 0.5-2.0 keV (left), 2.0-7.0 (middle) and PanSTARR r-band (right)
images.

The extent of this tail is intriguing. The maximum life-time of low-energy CRe
emitting at 144 MHz, which can be estimated by assuming the minimal energy
loss magnetic field $B=B_{\text{CMB}}/\sqrt{3}=2.02$ $\mu$G (where
$B_{\text{CMB}}=3.25(1+z)^{2}$ $\mu$G is the Cosmic Microwave Background
equivalent field) is $\tau\simeq 3\times 10^{8}$ yr. Under these conditions,
given an estimated velocity of GB6 B0335+096 ($v_{g}=\sqrt{2}\times\text{line-
of-sight (los) velocity}=960$ km s-1, Sebastian et al. 2017) the expected
length of a radio-emitting trail left behind would be $l\simeq
v\cdot\tau\simeq 300$ kpc, which is a factor $\sim 3$ shorter than what we
observe. Moreover, the real difference is likely larger because we can measure
only the projected length of the tail, which is the lower limit of this actual
length. Furthermore, the actual magnetic field differs from the minimal energy
loss condition, which entails that the CR lifetime $\tau$, and hence the radio
tail, would be shorter. The long radio tail of GB6 B0335+096 could therefore
be explained by either a galactic velocity significantly higher than inferred
from the los velocity, or an extended CRe life-time due to the action of some
physical process taking place in the tail.

We investigated the radio tail with simple numerical calculations of the
expected emission of an aging radio plasma. Our simple model allowed us to
estimate the flux density profile and the corresponding decline of the
spectral index along the tail. In order to compare the flux densities at 144
and 610 MHz, we used the images produced for the spectral analysis (Section
2.3) with matching uv-ranges (200-54000 $\lambda$) and resolutions
(30$\times$30 arcsec). Based on the 3$\sigma$ noise level of the 144 MHz
emission, we used the PT-REX code to sample the tail with 19 boxes,
$60^{\prime\prime}\times 60^{\prime\prime}$ in size. Then we measured the flux
densities at 144 and 610 MHz in each of them, producing a profile of the
emission along the tail (Figure 7). In this profile the 610 MHz emission is
present up to box number 8. Where both LOFAR and GMRT detected the emission
above the 3$\sigma$ level we computed a simple spectral index (Equation 1).
Where we do not detect emission at 610 MHz, we used the 2$\sigma$ level of the
GMRT image to derive an upper limit for the spectral index. By assuming that
the tail is parallel to the plane of the sky, our $60^{\prime\prime}\times
60^{\prime\prime}$ sampling can be roughly converted to a spatial scale, which
sets a lower limit for its actual length. Under these assumptions, each box is
separated by 45 kpc. Assuming that the plasma left behind by the galaxy does
not move with respect to the ICM, the time elapsed since the injection can be
estimated as $\tau\simeq d/v$, where $d$ is the distance from the AGN and $v$
is the velocity of the plasma with respect to the AGN. For simplicity, we
neglected the bulk motion of the jet and we assumed $v=v_{\text{gal}}$, where
$v_{\text{gal}}$ is the galaxy velocity. It follows that the space grid
defined by our sampling can be converted in a ’time grid’ with the dynamic age
of each chunk of the radio tail.

We then compare our observations with the expected synchrotron emissivity by
taking into account only the radiative cooling of CRe which is dominated by
synchrotron and Inverse Compton energy losses (e.g., Pacholczyk 1970). Our
estimate is conservative because we do not include the energy losses due to
adiabatic expansion of the plasma, which would cause the tail emissivity to
steepen faster, and we assume a uniform magnetic field. We computed
numerically the synchrotron emissivity spectrum $j(\xi)$, where
$\xi=\nu/\nu_{\text{br}}$ and $\nu_{\text{br}}$ is the break frequency, for
$\xi=0.1-10$ to properly model the spectral steepening. We assumed an
injection index $\alpha=-0.6$, based on the spectral index observed at the
beginning of the tail, and the minimal energy loss magnetic field
$B=B_{\text{CMB}}/\sqrt{3}\simeq 2$ $\mu$G.

Figure 7:  Left: 3, 48, 96, 192$\sigma$ levels of the 144 MHz radio emission
($\sigma$=750 $\mu$Jy beam-1, resolution 30$\times$30 arcsec) with the
sampling cells on top. Right: Observed and model profiles of flux density
(top), spectral index (middle) and break-frequency (bottom). The upper-limits
are computed based on the 2$\sigma$ level of the 610 MHz map. In the top and
middle panels are reported the predictions based on the two different galactic
velocities.

To compare this model emissivity spectrum with our observation, we first
derive the corresponding values of $\xi$ at 144 and 610 MHz by evaluating the
$\nu_{\text{br}}$ for each box. This can be done from the radiative time of a
relativistic particle population, $t_{\text{rad}}$, which can be expressed as:

$t_{r}\simeq 3.2\times
10^{10}\frac{B^{1/2}}{B^{2}+B_{\text{CMB}}^{2}}\frac{1}{\sqrt{\nu(1+z)}}\text{
yr,}$ (2)

where the magnetic fields are expressed in $\mu$G and the observed frequency
$\nu$ is in units of MHz (e.g., Miley 1980). Under the assumption that the
radiative age of the plasma coincides with the time elapsed since the
injection, i.e. $t_{\text{rad}}\simeq\tau$, we can derive a putative
$\nu_{\text{br}}$ for each chunk, and thus the corresponding values of $\xi$
at the two frequencies, $\xi_{144}$ and $\xi_{610}$.

By combining the predicted synchrotron spectrum with the time-scale described
by the break frequency, we can compare the observed spectrum with the
prediction in the case of pure radiative ageing. Under the assumption that
each cell defines a cylindrical section of the tail with a radius of 22.5 kpc
and height of 45 kpc (i.e., corresponding with the size of the sampling
boxes)777Such a simple assumption does not well describe the geometry at the
end of the tail where it curves toward east (cells number 10-15). However,
more advanced geometrical models including the rotation of the cells would not
affect affect our conclusions. and by using the flux density at 144 MHz
measured in the first cell to normalize the model spectrum $j(\xi)$, we can
produce the expected flux density and spectral index profiles along the tail.

We compared the observed spectral index profile with extreme two predictions
produced by assuming $v_{\text{gal}}=$ 960 and 1600 km s-1. The first one is
computed by assuming that the los velocity is $1/\sqrt{2}$ of the total
velocity (Sebastian et al. 2017), whereas the second one is an upper limit
estimated by forcing the model to reproduce the spectral index decline within
the first half of the tail. We note that both velocities are higher than the
cluster velocity dispersion $\sigma\simeq 600$ km s-1 (calculated on the basis
of the M${}_{500}=2.27\times 10^{14}$ M⊙ and R${}_{500}=0.92$ Mpc reported in
Giacintucci et al. 2017), which is not unusual for head-tail galaxies (e.g.,
Venkatesan et al. 1994). Under our assumptions, the higher the velocity of the
galaxy, the younger the age of the plasma in the tail, hence the slower the
emissivity and the spectral index decline. Finally, to estimate the state of
the plasma aging, i.e. the change of $\nu_{\text{br}}$, along the tail, we
used the model synchrotron spectrum to sample the expected spectral index for
values of $\nu_{\text{br}}$ within 1-5000 MHz. Then, by comparing via
interpolation the observed spectral index with our predictions, we evaluated
the break frequency profile along the tail. We report the observed and
predicted profiles in Figure 7

Neither of the two models can reproduce the observations. They both predict
that the flux density and the spectral index trends would be steeper than
observed, with the emission that would fall below the detection limit within
$\sim$500 kpc. In the first 500 kpc (i.e. where we still observe the 610 MHz
emission), the tail shows several bright knots of emission, which can be
observed at both frequencies and they appear as a change of slope of the
otherwise declining trend in flux density. Because we normalize the spectrum
at the flux density observed at the beginning of the tail, we expect these
knots, which are not included in our simple model, to be the main cause of the
difference between the observations and the predictions in the cells number
$4-8$. The question is whether these features were produced by phases of more
powerful AGN activity or by external processes affecting the tail. However,
beyond 500 kpc (cell number 8) the observed flux density differs from the
model by more than 2 orders of magnitude, with the 144 MHz emission declining
dramatically more slowly than expected. The break frequency profile (Figure 7,
bottom panel) shows a decline in the first half of the tail. Then a sudden
flattening occurs in the cells 7, 8 and 9 at values of 250 MHz, followed by an
apparent, inverted trend up to the end of the tail. Because the break
frequency of the spectrum is an indicator of the plasma age, observing its
values to be constant for $\sim 100$ kpc (and likely way further, as suggested
by the flatter decline of the 144 MHz emission beyond cell number 9) indicates
that in that region the radiative energy losses are balanced by some external
process that is providing energy to the radiative electrons.

To explain the long tail, mechanisms of re-energization can be invoked. One
possibility is that turbulence is generated in the tail by instabilities
driven by the interplay of the tail with the external medium (i.e. gentle re-
acceleration, see de Gasperin et al. 2017; Wilber et al. 2019; Botteon et al.
2021; Müller et al. 2021, for similar studies). In this scenario, the re-
energization would allow CRe to emit for longer than predicted in our simple
model, thus explaining the discrepancy. Alternatively, the observed profiles
could be explained by compression due to the interaction of the tail with a
weak shock. This would locally flatten in the spectral index, which may be in
agreement with the uniform spectral index observed in cells number $7-9$. This
latter scenario would also be characterized by the presence of a discontinuity
in the X-ray surface brightness in the proximity of the tail. However,
searching by extracting surface brightness profiles across the tail is
complicated as the tail is placed along the edge of the CCD of the Chandra
observation (Figure 2, top panel). On the other hand, a shock-like
acceleration would extend the radiative life of both the high- and low- energy
CRe, i.e. would result in a strong flattening of the spectral index, thus
resulting in extended ($>$500 kpc) emission at both frequencies. However, the
144 MHz emission is far more extended than the 610 MHz emission. This
indicates that the efficiency of the re-energizing process is a factor $\sim
2$ too low (Equation 2) to compensate the energy losses observed at 610 MHz,
but, still, its time-scale is consistent with the time-scale of the energy
losses of the low-energy CRe emitting at 144 MHz, i.e. $\sim 10^{8}$ yrs.
Therefore we conclude that we are dealing with a low-efficient phenomenon and
the ICM gentle re-acceleration appears as most likely process to power the
low-energy CRe population, counter the radiative cooling and explain the
presence of 144 MHz emission at such a large distance from the AGN.

##  4 Conclusions

Due to its capability to probe the low-energy CRe, LOFAR unveiled a series of
remarkable features in 2A0335+096. We focused our analysis on two aspects: the
nature of the central extended radio emission and the properties of the radio
tail of GB6 B0335+096. For the first aspect, we could estimate the spectral
index of the diffuse radio emission, $\alpha=-1.2\pm 0.1$, which is flatter
than the spectral index we observe in the lobes of the central radio galaxy
($\alpha\simeq-1.5\pm 0.1$). We concluded that the diffuse radio emission may
be connected to the AGN (as primary source of CR), but the spectral index
suggests that additional processes, either due to turbulent re-acceleration or
a hadronic cascade, played a role in the origin of the MH. On the basis of the
new 144 MHz images and the sloshing features highlighted by the GGM analysis,
we suggest that turbulent re-acceleration may have played a role. However,
with the analyzed data it is difficult to discriminate the two models.

For GB6 B0335+096, the new observations reveal that the radio tail is
significantly extended at low frequencies ($\sim$900 kpc). This raises the
questions of how the relativistic particles could still be emitting at the
level observed this far from the AGN. In order to address this question, we
studied the profiles of flux density and spectral index along the tail trying
to reproduce them with a simple ageing model. This led to the conclusion that
a re-acceleration mechanism has to be invoked to explain the observations. The
fact that this continuation is revealed only at low frequency arises the
possibility that such phenomena may be actually common in many systems, and
hence that they will be found more and more frequently with the progress of
the LOFAR surveys. Future multi-frequency and polarimetric studies will be
necessary to obtain complementary constraints on the CRe properties and the
magnetic field geometry and further investigate the structures of these radio
tails and their interplay with the ICM.

## Acknowledgments

We thank the anonymous Referee for helping improving the presentation of this
work and H. T. Intema for providing the reduced GMRT observation of
2A0335+096. AI acknowledges the Italian PRIN-Miur 2017 (PI A. Cimatti). We
acknowledge financial contribution from the agreement ASI-INAF n.2017-14-H.0
(PI Moretti). MB acknowledges support from the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306.
RJvW and AB acknowledges support from the VIDI research programme with project
number 639.042.729, which is financed by the Netherlands Organisation for
Scientific Research (NWO). ACE acknowledges support from STFC grant
ST/P00541/1. FdG acknowledges support from the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306.
GDG acknowledges support from the Alexander von Humboldt Foundation. CJR
acknowledges financial support from the ERC Starting Grant ‘DRANOEL’, number
714245. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed
and constructed by ASTRON. It has observing, data processing, and data storage
facilities in several countries, which are owned by various parties (each with
their own funding sources), and that are collectively operated by the ILT
foundation under a joint scientific policy. The ILT resources have benefited
from the following recent major funding sources: CNRS-INSU, Observatoire de
Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science
Foundation Ireland (SFI), Department of Business, Enterprise and Innovation
(DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities
Council, UK; Ministry of Science and Higher Education, Poland; The Istituto
Nazionale di Astrofisica (INAF), Italy. This research made use of the Dutch
national e-infrastructure with support of the SURF Cooperative (e-infra
180169) and the LOFAR e-infra group. The Jülich LOFAR Long Term Archive and
the German LOFAR network are both coordinated and operated by the Jülich
Supercomputing Centre (JSC), and computing resources on the supercomputer
JUWELS at JSC were provided by the Gauss Centre for Supercomputing e.V. (grant
CHTB00) through the John von Neumann Institute for Computing (NIC). This
research made use of the University of Hertfordshire high-performance
computing facility and the LOFAR-UK computing facility located at the
University of Hertfordshire and supported by STFC [ST/P000096/1], and of the
Italian LOFAR IT computing infrastructure supported and operated by INAF, and
by the Physics Department of Turin university (under an agreement with
Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing
Centre, Italy. We thank the staff of the GMRT that made these observations
possible. GMRT is run by the National Centre for Radio Astrophysics of the
Tata Institute of Fundamental Research. This research made use of Astropy, a
community-developed core Python package for Astronomy (Astropy Collaboration
et al. 2013, 2018), and APLpy, an open-source plotting package for Python
(Robitaille & Bressert 2012).

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