11institutetext: INAF - Istituto di Radioastronomia, Via Gobetti 101, 40129,
Bologna, Italy 22institutetext: INFN-Sezione di Roma, Piazzale Aldo Moro, 2 -
c/o Dipartimento di Fisica, Edificio G. Marconi, I-00185 Roma, Italy
33institutetext: INAF - Osservatorio Astronomico di Roma, Via Frascati 33,
I-00078 Monteporzio Catone, Italy 44institutetext: INAF - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, 40129,
Bologna, Italy 55institutetext: INAF - Osservatorio Astrofisico di Catania,
Via Santa Sofia 78, 95123, Catania, Italy) 66institutetext: Leiden
Observatory,Leiden University, POBox 9513, 2300RA Leiden, The Netherlands
77institutetext: Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK 88institutetext:
Dipartimento di Fisica e Astronomia, Università di Bologna, via P. Gobetti
93/2, I-40129, Bologna, Italy 99institutetext: Centre for Astrophysics
Research, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
1010institutetext: INAF - IAPS, Via Fosso del Cavaliere 100, 00133, Rome,
Italy 1111institutetext: Dipartimento di Fisica, Universitá degli Studi di
Torino, via Pietro Giuria 1, I-10125 Torino, Italy 1212institutetext: Centre
for Extragalactic Astronomy, Department of Physics, Durham University, Durham
DH1 3LE, UK 1313institutetext: Institute for Computational Cosmology,
Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK
1414institutetext: ASTRON, Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedijk 4, 7991 PD, Dwingeloo, The Netherlands 1515institutetext: GEPI
& ORN, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen,
92190 Meudon, France 1616institutetext: Department of Physics & Electronics,
Rhodes University, PO Box 94, Grahamstown, 6140, South Africa
1717institutetext: School of Physical Sciences, The Open University, Walton
Hall, Milton Keynes, MK7 6AA, UK 1818institutetext: RAL Space, STFC Rutherford
Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK

# LOFAR HBA Observations of the Euclid Deep Field North (EDFN)

M. Bondi 11<EMAIL_ADDRESS>R. Scaramella 2233 G. Zamorani 44 P. Ciliegi
44 F. Vitello 1155 M. Arias 66 P.N. Best 77 M. Bonato 11 A. Botteon 11 M.
Brienza 4488 G. Brunetti 11 M.J. Hardcastle 99 M. Magliocchetti 1010 F.
Massaro 1111 L.K Morabito 12121313 L. Pentericci 33 I. Prandoni 11 H.J.A.
Röttgering 66 T.W. Shimwell 141466 C. Tasse 15151616 R.J. van Weeren 66 G.J.
White 17171818

We present the first deep (72 hours of observations) radio image of the Euclid
Deep Field North (EDFN) obtained with the LOw-Frequency ARray (LOFAR) High
Band Antenna (HBA) at 144 MHz. The EDFN is the latest addition to the LOFAR
Two-Metre Sky Survey (LoTSS) Deep Fields and these observations represent the
first data release for this field. The observations produced a
$6^{\prime\prime}$ resolution image with a central r.m.s. noise of $32\,\mu$Jy
beam-1. A catalogue of $\sim 23,000$ radio sources above a signal-to-noise
ratio (SNR) threshold of 5 is extracted from the inner circular 10 deg2
region. We discuss the data analysis and we provide a detailed description of
how we derived the catalogue of radio sources and on the issues related to
direction-dependent calibration and their effects on the final products.
Finally, we derive the radio source counts at 144 MHz in the EDFN using
catalogues of mock radio sources to derive the completeness correction
factors. The source counts in the EDFN are consistent with those obtained from
the first data release of the other LoTSS Deep Fields (ELAIS-N1, Lockman Hole
and Bootes), despite the different method adopted to construct the final
catalogue and to assess its completeness.

###### Key Words.:

surveys - catalogs - radio continuum: general - radio continuum: galaxies

##  1 Introduction

In the last twenty years it has been demonstrated that panchromatic surveys
(from X-rays to radio wavelengths) of selected regions of the sky (e.g. Le
Fèvre et al., 2005; Scoville et al., 2007; Driver et al., 2011; Grogin et al.,
2011; Chen et al., 2016; Franco et al., 2018; Pentericci et al., 2018; Weaver
et al., 2022; Treu et al., 2022; Finkelstein et al., 2022) are fundamental to
provide the database on which to build a consensus on galaxy formation and
evolution (e.g. Conselice, 2014; Somerville & Davé, 2015; Förster Schreiber &
Wuyts, 2020). Observations at different wavelengths can reveal different
aspects of the structure and properties of the galaxies, such as the
distribution of stars, gas, and dust, as well as the presence of active
galactic nuclei (AGN). For example, optical and infrared observations can
provide information on the stars within a galaxy, while radio and X-ray
observations can reveal the presence of AGN and the energetic processes
associated with them. By combining data at multiple wavelengths, astronomers
can hence obtain a comprehensive picture of how galaxies form and evolve over
time.

Observations in the radio waveband are extremely important to study the
physical processes connected to star formation, the properties of supermassive
black holes and the interplay between star formation and AGN across cosmic
time. The radio continuum emission is not affected by dust extinction and
therefore provides a dust-unbiased star formation tracer (e.g. Condon, 1992;
Haarsma et al., 2000; Seymour et al., 2008; Smolčić et al., 2009; Novak et
al., 2017; van der Vlugt et al., 2022; Cochrane et al., 2023). Moreover, only
at radio frequencies one can reliably identify and probe low-luminosity jet-
mode AGN hosted by the most massive galaxies, and therefore investigate the
effects of feedback on their growth (e.g. Croton et al., 2006; Best et al.,
2006; Hardcastle et al., 2007; Smolčić, 2009; Best & Heckman, 2012; Heckman &
Best, 2014; Smolčić et al., 2017b; Kondapally et al., 2022; Mingo et al.,
2022). Present and future radio surveys can reach the depth needed to detect
star-forming galaxies as well as quasar-mode AGN which are typically faint in
the radio band (Bonzini et al., 2012; Padovani, 2016; Best et al., 2023).
However, most of the existing deep radio observations required to study these
source populations at high redshifts are usually limited to small regions of
the sky, ranging from tens of square arcminutes to a few square degrees (e.g.
Hopkins et al., 1998, 2003; Ciliegi et al., 1999; Prandoni et al., 2000, 2018;
Bondi et al., 2003, 2007; Huynh et al., 2005; Schinnerer et al., 2007, 2010;
Tasse et al., 2007; Owen & Morrison, 2008; Miller et al., 2008, 2013; Morrison
et al., 2010; Smolčić et al., 2017a; Murphy et al., 2017; Owen, 2018; Muxlow
et al., 2020; van der Vlugt et al., 2022; Algera et al., 2022; Heywood et al.,
2022; D’Amato et al., 2022; Hale et al., 2023). In this context, large area
(e.g. tens of degrees) surveys down to unprecedented depths are planned with
new and upgraded facilities (e.g. Jarvis et al., 2016). The LOw-Frequency
ARray (LOFAR, van Haarlem et al., 2013) plays a key role in this framework by
combining a wide field of view with high sensitivity and angular resolution.
The recently published second data release of the LOFAR Two-Metre Sky Survey
(LoTSS, Shimwell et al., 2017, 2019, 2022) has publicly released images
covering 5,700 square degrees in the northern sky centred at approximately
12h45m +44∘30′ and 1h00m +28∘00′ and spanning 4178 and 1457 square degrees,
respectively. The observations are carried out at the central frequency of 144
MHz and each pointing is observed for $\sim 8$ hours. The images have a median
$1\sigma$ r.m.s. sensitivity of $83\,\mu$Jy beam-1 at $6^{\prime\prime}$
resolution. When completed the LoTSS observations will cover the whole
Northern sky. To complement the LoTSS observations, deeper pointings (the
LoTSS Deep Fields, Best et al., 2023) in regions already covered by extensive
and deep multi-wavelengths ancillary observations, are being carried out with
LOFAR with the aim to reach an rms noise of $\simeq 10\,\mu$Jy beam-1 over a
sky area of $\simeq 50$ deg2. The LoTSS Deep Fields data release 1 (DR1)
accounts for about 1/3 of the integration time for 3 deep fields (Lockman
Hole, Bootes and ELAIS-N1) and provides radio images and catalogues (Tasse et
al., 2021; Sabater et al., 2021), near-infrared optical identifications
(Kondapally et al., 2021), photometric redshifts (Duncan et al., 2021), and
host galaxies classification and properties (Best et al., 2023).

The North Ecliptic Pole (NEP) region is the fourth field of the LoTSS Deep
Fields project (Best et al., 2023), but was not included in the first LoTSS-
Deep data release as the data were obtained later. This field was chosen
because the NEP is the location of the Euclid Deep Field North (EDFN), one of
the deep fields observed by the Euclid mission (Euclid Collaboration et al.,
2022), and the only one in the Northern sky. Euclid observations will provide
sub-arcsecond near-IR imaging down to $H=26$ mag over a 20 deg2 field centred
at RA$=269.73$ deg and DEC$=+66.02$ deg. This paper complements the LoTSS Deep
Fields DR1 presenting the LOFAR $6^{\prime\prime}$ resolution image at 144 MHz
and the radio source catalogue obtained from the first 72 hours in the EDFN.
Other publications in preparation will present the near-IR/optical
identifications of the radio sources and will focus on the results obtained
with the inclusion of the LOFAR International Stations that allow improvement
of the angular resolution. The LOFAR observations of the EDFN have been
completed in summer 2023, totalling around 400 hours of observations (expected
final noise $\sim 12\,\mu$Jy beam-1) and this complete dataset is now being
processed and analysed.

This paper is organized as follows. In Sections 2 and 3 we describe the 72
hours duration LOFAR observations and summarise the data calibration and
imaging procedures, respectively. Section 4 contains a detailed description of
the methods used to test the reliability of the LOFAR data products, including
the refinement of the amplitude calibration, the compilation of the final
radio source catalogue and an analysis of the properties of the radio sources
at different distances from the field centre. The generation of mock samples
of realistic radio sources used to derive the completeness factors to be
applied to the source counts is described in Section 5. Finally, Section 6
presents the radio source counts obtained in the EDFN and a brief comparison
with those obtained from the other LoTSS Deep Fields.

Throughout this paper we adopt the spectral index convention
$S_{\nu}\propto\nu^{-\alpha}$.

##  2 LOFAR observations

LOFAR observed the EDFN for 72 hours at 144 MHz during cycle 12 (proposal
LC12_027, P.I. van Weeren). The proposal combined EDFN observations with those
of the galaxy cluster Abell 2255 (Botteon et al., 2022). This was possible
because the targets, EDFN and Abell 2255, are about 5 degrees apart on the sky
(resulting in minimal sensitivity losses due to the LOFAR HBA tile beam), and
with the adopted setup we could split the LOFAR beam to observe both targets.
The observations presented in this paper were obtained using a single pointing
centre, shifted by $\simeq 30^{\prime\prime}$ with respect to the EDFN
positions.

The 72 hours were split over 9 nights in the period June to November 2019 (see
Table 1). The observations were carried out with the high-frequency band
antennas in configuration HBA_DUAL_INNER which provides a uniform shape of the
primary beam over the whole of the LOFAR Dutch stations (i.e only using the
inner 24 tiles of the 48 tiles on the remote stations). All data sets were
recorded with an integration time of 1s, a 48 MHz bandwidth centred at 144 MHz
and a channel width of 3.05 kHz. The data were then passed through the
standard LOFAR pre-processing pipeline (Heald et al., 2010) which performed
the Dysco compression to reduce the data size (Offringa, 2016), the RFI
flagging using the AOflagger tool (Offringa et al., 2012) and averaged down
the data to a channel width of 12.2 kHz. Time resolution remains unchanged.

All the observations were preceded and followed by a $\sim 15$ minute run on
the calibrators 3C295 and 3C48, respectively. The latter source was selected
as primary amplitude calibrator for the data sets. During the nine epochs of
observations, the number of observing stations, including the international
stations, varied from 47 to 51: two nights had one international station
missing and one night had 4 stations (two international and two core stations)
missing. In the standard LOFAR data analysis aimed to produce a $\sim
6^{\prime\prime}$ angular resolution image only the data from the 38 Dutch
stations (with baseline lengths in the range 0.15–100 km) are processed. The
international stations are flagged out at an early stage of processing to
reduce the sizes of intermediate data products since they are not used.

Table 1: EDFN Observations

SAS ID | Date | Duration | DI rms
---|---|---|---
|  | (s) | (mJy beam-1)
L720376 | 2019-06-07 20:03:59 | 29180 | 0.26
L725452 | 2019-06-22 19:00:01 | 29170 | 0.25
L726706 | 2019-06-28 18:00:01 | 30060 | 0.25
L727108 | 2019-07-03 18:00:01 | 29170 | 0.28
L728072 | 2019-07-08 17:51:01 | 29180 | 0.32
L733075 | 2019-08-09 16:30:00 | 29180 | 0.26
L746862 | 2019-09-28 12:00:01 | 29180 | 0.45
L747611 | 2019-10-04 12:41:01 | 29170 | 0.25
L751364 | 2019-11-15 09:11:00 | 29140 | 0.23
111Col.1: unique LOFAR SAS id; Col.2: starting date and hour of the
observation (format yy-mm-dd hh:mm:ss); Col.3: duration of observation in
seconds; Col.4: r.m.s. of the direction independent image obtained using a
smaller bandwidth.

##  3 Data calibration and imaging

The data calibration and imaging were performed using the OCCAM infrastructure
for High Performance Computing (HPC) run by the Competence Centre for
Scientific Computing (C3S), a joint interdepartmental advanced research centre
of the Turin University and the Italian National Institute for Nuclear Physics
(INFN) (Aldinucci et al., 2017).

The data were downloaded from the LOFAR Long Term Archive and copied over to
the OCCAM system. The data are stored using the Dysco data compression format
(Offringa, 2016) that allows to reduce the data volume by roughly a factor of
4. Each night of observation amounts to $\sim 4.2$ TB, including the
calibrator scans. For the data reduction we used one OCCAM Fat Node, running
48 cores with 768 GB of RAM available.

For the calibration and imaging we followed the same steps used for the Bootes
and Lockman Hole deep fields and explained in details in Tasse et al. (2021).
Here we briefly summarise these steps. Direction independent calibration was
performed using the PREFACTOR version 3 pipeline (de Gasperin et al., 2019;
van Weeren et al., 2016; Williams et al., 2016)222https://github.com/lofar-
astron/prefactor using 3C48 as amplitude calibrator. The PREFACTOR calibrator
pipeline was run individually for each of the nine nights and the resulting
bandpasses were checked for possible antenna malfunctions. Then, the PREFACTOR
target pipeline was run for each night producing the calibration tables to be
applied in the next data reduction step. Direction dependent calibration was
performed using the DDF-pipeline (Shimwell et al., 2019; Tasse et al.,
2021),333https://github.com/mhardcastle/ddf-pipeline which combines the solver
KillMS444https://github.com/saopicc/killMS (Tasse, 2014b, a; Smirnov & Tasse,
2015) and the imager DDFacet555https://github.com/saopicc/DDFacet (Tasse et
al., 2018). First, each night was processed to an early stage of reduction,
that stopped after producing the direction independent corrected image. These
images were checked for evident issues, such as poor ionospheric conditions
and processing failures. Seven nights produced images with very similar r.m.s.
noise values with the remaining two with slightly higher but still acceptable
values (the noise values are reported in Tab. 1). The night with the best
noise (day 2019-11-15) was fully processed with DDF-pipeline to produce the
sky model and, finally, the DDF-pipeline was run combining all the nine nights
together using this sky model as a starting model for the self-calibration
direction-dependent stage.

Figure 1: LOFAR image of the Euclid Deep Field North: the image shows only a
sub-region $\sim 0.6\times 0.3$ deg2 centred at RA=270.3086 deg and DEC=65.882
deg

###  3.1 Pipeline products

After the completion of the DDF-pipeline we obtained the deep (72 hours) full
resolution (6 arcsec) Stokes I radio image centred on the EDFN. The size of
the image is an input parameter for the pipeline and is usually set to
$20,000\times 20,000$ pixels (1 pixel $=$ 1.5 arcsec). In processing the EDFN
data sets we set this size to a slightly larger value ($21,500\times 21,500$
pixels, $\sim 9\times 9$ deg2) to mitigate artefacts associated with a few
bright sources at the edges of the field. A lower resolution Stokes I image
(20 arcsec) was also produced by the pipeline, but this image was not used as
part of the analysis presented here. The final image (before the refinement of
the amplitude scale, see Sec. 4.2) has a central r.m.s noise of 36 $\mu$Jy
beam-1. A $\sim 0.6\times 0.3$ deg2 inset of the final image is shown in Fig.
1.

A preliminary catalogue of the whole field was produced from the full
resolution image using PyBDSF (Mohan & Rafferty, 2015). PyBDSF fits individual
Gaussian components in regions selected on the basis of the local noise,
extracting sources that can either be composed of a single or multiple
components. Very extended radio sources or sources whose brightness spatial
distribution can not be properly modelled by Gaussian components are
effectively recovered using wavelets. The parameters used to run PyBDSF on the
EDFN are the same used for the other Deep Fields and listed in Table C.1 in
Sabater et al. (2021). PyBDSF found almost 50,000 sources above a signal-to-
noise ratio (SNR) threshold of 5 over the whole imaged area.

##  4 Data analysis

In this section we describe the methods adopted to test the reliability of the
data products, refining the amplitude calibration, producing the final
catalogue of radio sources and classifying all the sources as resolved or
unresolved.

###  4.1 Direction-dependent effects

We decided to investigate the possible effects on the catalogued source
parameters introduced by the choice of a different field faceting geometry and
of a different sky model during the direction-dependent calibration. For this
reason the EDFN datasets were independently reduced in Leiden using a
different computer infrastructure. During this procedure the image obtained
from day 2019-06-07 was adopted as sky model along with a different faceting
geometry. The two final images, the one obtained by us and the one obtained in
Leiden, were then compared. For each image we measured the median noise in
annular regions at increasing distances from the centre and found that the
median values were consistent to within 1% (corresponding to $\sim 0.3$
$\mu$Jy beam-1) up to a radius of 2.5 deg from the field centre. Beyond this
distance the difference in the median noise increases, reaching 4% at 4 deg
from the field centre. Then, we used the same version of PyBDSF with the same
set of parameters on the two images obtaining two catalogues. The two
catalogues (hereafter dubbed as Leiden catalogue and Turin catalogue) were
cross-matched using a matching radius of 3 arcsec, and only the 2918 matched
sources with SNR$>50$ in both catalogues were selected. These sources were
then split according to their distance $r$ from the field centre in four
annular regions: $r<1$ deg, $1<r<2$ deg, $2<r<3$ deg and $3<r<4$ deg. For each
group of matched sources we calculated the ratio between the total (peak)
fluxes obtained in the Turin and Leiden catalogues, and we derived the median
and the scaled median absolute deviation (MAD) in each of the four distance
intervals.

Figure 2: Median peak brightness ratio (left panel), median total flux density
ratio (middle panel) and median absolute value separation (right panel) in the
four annular regions at increasing distance $r$ from the pointing centre for
matched sources in the Turin and Leiden catalogues with SNR$>50$. The error
bars are the scaled median absolute deviation (MAD) values.

The main results obtained from this comparison are shown in Fig.2 and can be
summarised as follows:

  1. 1.

the median values of the peak brightness ratios and total flux density ratios
do not significantly change with increasing distance from the field centre and
are consistent with a value of 1, meaning there is no systematic scale offset
in the amplitudes of the two images.

  2. 2.

The MAD associated to each median value increases with $r$. Within $r<2$ deg
the MAD is $<0.05$ for both the peak brightness ratios and total flux ratios.
Beyond this distance the peak brightness ratios show a larger dispersion up to
a MAD$=0.16$ in the range $3<r<4$ deg compared to a MAD$=0.09$ for the total
flux density ratios in the same distance range.

  3. 3.

We also checked whether the position of the radio sources in the Leiden and
Turin catalogues could depend on $r$. The right panel in Fig.2 shows the
median separation in arcsec as a function of distance $r$. We remind the
reader that the two catalogues were matched using a maximum distance $r=3$
arcsec, half the size of the restoring beam. There is a trend of larger
differences in the source positions between the two catalogues with increasing
distance from the field centre, but the median shifts are small,
$\\!\\!\\!\phantom{\leq}\smash{\buildrel\over{\lower
2.67502pt\hbox{$\buildrel\lower
2.14001pt\hbox{$\displaystyle<$}\over{\sim}$}}}\,\,0.2$ arcsec for $r<2$ deg
and even at the largest distance probed by this test the median shift is $\sim
0.6$ arcsec, less than half of the pixel size.

Summarising, the peak brightness and flux density ratios are consistent to
$\leq 5\%$ within a radius $r=2$ deg from the field centre. However, it should
be noted that at larger distances from the field centre these uncertainties
increase significantly, reaching values of about $16\%$ and $9\%$ at distance
$r\simeq 4$ deg for the peak brightness and total flux density, respectively.
The two images have been obtained using the same pipeline but on different
computer infrastructures, slightly different singularity images, different
starting sky models and faceting geometry. We note that we cannot exclude that
the differences between the two images might be caused, at least to some
extent, by different software versions or different computer hardware used to
run the pipeline by us and in Leiden, but this is rather unlikely. The larger
dispersion observed in the measured peak brightness ratios with respect to the
total flux density ratios, and the general trend for the dispersion to be
larger at larger distances suggest that the differences between the two images
are likely caused by residual smearing and uncertainties in the direction-
dependent calibration deriving from using different faceting patterns and
starting sky models in the two images (see also the discussion in Sec. 4.4).

###  4.2 Absolute flux density calibration

We follow the method described in Sabater et al. (2021) to adjust the absolute
flux density scale using the external radio catalogues available from the
literature. We compared the flux density of sources in common between our
catalogue and the external catalogues using only sources with distance $r<3$
deg from the field centre and adopting the constraints listed in Section 3.5
from Sabater et al. (2021) to avoid the introduction of biases due to the
different depths and angular resolution of the external catalogues. The
external catalogues we used are the VLASSr at 74 MHz (Lane et al., 2014), the
TGSS at 150 MHz (Intema et al., 2017), the 6C at 151 MHz (Hales et al., 1990),
the WENSS at 350 MHz (Rengelink et al., 1997), the NVSS at 1.4 GHz (Condon et
al., 1998) and a WSRT pointed observation at 1.4 GHz (White et al., 2010).

The result is shown in Fig.3: each point is the median of the flux density
ratios obtained from the LOFAR sources matched with each of the other external
catalogues and the red dashed line is the linear best fit to these points.
Assuming perfect a-priori calibration of the LOFAR array after the direction-
dependent corrections, and of all other surveys plotted in Fig. 3, and that a
simple power-law spectral index is appropriate, the line should pass through
the ratio value of 1 at 144 MHz. As can be seen from the inset in Fig.3, the
best fit line has a value of $0.88\pm 0.04$. This is the scale factor that
needs to be applied to the final image to correct the absolute amplitude
scale. The value we obtained for the EDFN is similar to those previously
derived for other deep fields that are in the range 0.80-0.92 (Sabater et al.,
2021). We scaled the radio image by a factor of 0.88 to set the final LOFAR
image of the EDFN to the correct amplitude scale.

Figure 3: Calibration of the EDFN flux density scale. The flux density scale
after DDF-pipeline is set to unity and shown as a blue cross. The flux density
ratios with respect to the external catalogues from the literature and their
errors are shown in different colours. The red dashed line is the linear fit
to the data. The inset shows a zoomed view close to the LOFAR HBA central
frequency.

###  4.3 The final catalogue of the Euclid Deep Field North

Due to the intrinsic nature of the cleaning process, radio images can be
contaminated by spurious sources in the proximity of real bright radio
objects. Furthermore, radio sources can have morphologies too complex and/or
extended to be properly recovered by source finding algorithms which only
provide a catalogue of radio components. Moving from a catalogue of radio
components to a catalogue of radio sources requires some additional steps that
will be discussed below.

Having the final radio image set to the correct flux scale we run again PyBDSF
to obtain a new catalogue of radio sources, using the same parameters as
before. Then we selected only the sources within a 10 deg2 circular area
($r=1.784$ deg) centred on the Euclid Deep Field North position (RA=17:58:55.9
DEC=66:01:03.7), and throughout the rest of the paper we will consider only
this area. This is the region that was originally selected for the EDFN when
the observations presented in this paper were planned, and only more recently
the area was extended to 20 deg2. It is worth saying that producing a
catalogue from these observations covering the whole 20 deg2 of the EDFN is
not convenient, considering that the noise increases with the distance from
the pointing centre and the effects discussed in Sec. 4.1. The more recent
observations of the EDFN adopted a different pointing strategy that will
increase sensitivity and accuracy over a larger area allowing to properly
investigate the full 20 deg2 field. In the final rescaled image the central
r.m.s. noise is $32\,\mu$Jy beam-1, increasing to $\sim 45\,\mu$Jy beam-1 at
$r\simeq 1.8$ deg due to the primary beam correction.

PyBDSF is a tool designed to decompose a radio image in islands and extract
the components as a set of Gaussians, shapelets or wavelets, above a given
threshold in SNR (Mohan & Rafferty, 2015). Two or more radio components found
inside the same island can be grouped together to form a single radio source
on the basis of the distance between the components and the brightness
distribution along the line joining the centre of the components. The outcome
of this procedure is recorded in the output catalogue by the S_Code parameter
that is used to define the source structure: “S_Code=S” for a single-Gaussian
source that is the only component in the island , “S_Code=C” for a single-
Gaussian source in an island with other sources, “S_Code=M” for a multi-
Gaussian source. The raw catalogue produced by PyBDSF needs to be checked
against known issues that can affect it and we briefly discuss them below,
together with the solutions we adopted. We supported our analysis by using the
unWISE catalogue in the overlapping 10 deg2 region (Lang, 2014; Meisner et
al., 2017b, a). The unWISE Catalogue is derived from the unWISE coadds of the
Wide-field Infrared Survey Explorer (WISE, Wright et al., 2010) images, and
includes two billion sources over the entire sky at 3.4 and 4.6 microns. We
used the catalogue produced after 5 years of WISE imaging (Schlafly et al.,
2019). The unWISE catalogue has two advantages over the existing WISE
catalogue (AllWISE): first, it is based on significantly deeper imaging, and
second, it features improved modelling of crowded regions using the
crowdsource666 https://github.com/schlafly/crowdsource analysis pipeline to
simultaneously determine the positions and fluxes of all sources in the unWISE
coadds.

Figure 4: LOFAR radio images at 144 MHz of three multi-component sources each
originally classified as a single radio source by PyBDSF. Cyan large circles
identify the individual Gaussian components, the black cross is the derived
radio position of the source, and the green circles mark the position of
unWISE sources inside the region. The left panel shows a case where deblending
is necessary as all the three components are very likely distinct radio
sources. The middle panel is an example of correct association of different
components to a single radio source, while the right panel is an uncertain
case.

Firstly, we check for spurious sources. Bright sources, in particular those
where the direction dependent calibration did not perform well, can produce
artefacts associated with calibration uncertainties that can exceed the local
SNR threshold. PyBDSF can generally deal with this issue adapting the window
size over which to calculate the local r.m.s. around bright sources, but in
some cases this is not effective. These calibration artefacts produce a
typical pattern with spikes irradiating from the bright source. We visually
checked the region within a radius of $90$ arcsec around the sources with
total flux $S_{T}>15$ mJy looking for sources with $5<{\rm SNR}<10$ and
without a counterpart in the unWISE catalogue (within a matching radius of $3$
arcsec). When such a source was found to be located along one of the spikes,
it was considered a spurious source and it was removed from the catalogue.
According to this criterion we removed 110 sources ($\sim 0.5$ percent).

As said above, when more than one Gaussian component is found within an island
and a set of conditions are fulfilled, PyBDSF attempts to group together the
components into a single radio source. This process cannot be perfect. For
instance, two or more distinct radio sources can be grouped into a single
multi-component source due to the limited resolution of the LOFAR
observations. For this reason we visually inspected all the catalogued radio
sources classified as “M” or “C” by PyBDSF, again using the unWISE catalogue
to support our analysis. Fig. 4 shows the three typical cases we encountered.
In each panel, the radio image is shown in colours, the cyan circles are
centred at the positions of the different radio Gaussian components, the black
cross is the final position of the radio source derived from a brightness
weighted average of the position of the individual components, and the green
small circles indicate the positions of the unWISE sources in the area. In the
left panel, the three Gaussian components are grouped to form a single radio
source by PyBDSF, but the comparison of the radio and unWISE source positions
strongly suggests that we have three different radio sources. There were
around 550 such cases where sources originally classified as a single source
had to be split in two or more individual sources, in these cases we replace
the single source in the radio catalogue with the individual Gaussian
components that have SNR$>5$. Deblended sources with SNR$<5$ were not included
in the radio catalogue. In the example shown in Fig. 4 all the three Gaussian
components have SNR $>5$ and therefore the single entry is replaced by three
entries with appropriate parameters in the final radio catalogue. The middle
panel shows an example where there is an unWISE source close to the radio
source position and no unWISE sources near the peak of the individual
components: in all these cases we considered the PyBDSF outcome as reliable.
In the right panel, unWISE sources are found both at the radio source position
and close to the peak of the individual components: such cases are uncertain
and we decided not to change the PyBDSF output.

Another typical case of PyBDSF failure, in combining the appropriate
components into a single radio source, is when components that belong to the
same radio source are catalogued as distinct sources. This can happen when the
lobes of a very extended radio galaxy are separated by tens of arcseconds or
even arcminutes. Each of the two lobes (and a radio core, if present) can be
classified as a distinct source by PyBDSF, with each lobe usually classified
as a multi-component source. We visually inspected all the sources classified
with “S_Code=M” or “S_Code=C”, and we identified the lobe components of
extended radio galaxies. Whenever necessary we grouped together the lobe
components into a single source in the catalogue. We also searched for a
possible radio core previously identified as a separate source and we assigned
it to the extended radio source. The total flux of the source was assumed to
be the sum of the fluxes of all grouped components, and its position the one
of the radio core. When no radio core is detected, the radio position is set
equal to the flux density weighted average of the position of the lobes. For
these 131 sources ($\sim 0.6\%$ of the total) the size parameters in the
catalogue are set to -1 and the error associated with the total flux density
is derived combining the errors of the individual components. As a result of
this process we obtain a final catalogue 777The data associated with this
article are released at:
http:/lofar-surveys.org over the 10 deg2 field centred on the EDFN containing
23,333 sources ($93\%$ classified as “S_Code=S”) with SNR$>5$.

It is worth mentioning that the errors given in the catalogue are only those
derived by PyBDSF from the fitting procedure. In particular, as far as
concerns the peak brightness and total flux density values, these errors do
not include the effects investigated in Sec. 4.1 and Sec. 4.2. Combining the
error associated with the linear fit in Sec. 4.2 with the error on peak
brightness or total flux density for sources within 2 degrees from the field
centre we have a likely uncertainty of around 6 percent in the flux density
scale, a value consistent with that quoted for other deep fields (Sabater et
al., 2021).

Figure 5: The upper panel shows the total-to-peak flux ratio vs signal-to-
noise ratio plot for all the sources in the catalogue. The line
$S_{T}/S_{p}=1$ is drawn. The lower panel shows the same plot using $S^{\rm
cor}_{p}$ instead of $S_{p}$. $S^{\rm cor}_{p}$ is the corrected peak flux
obtained by multiplying $S_{p}$ by a radially dependent correction factor (see
Sec. 4.4 for details). The upper black line separates resolved from unresolved
sources according to the relation $S_{T}/S^{\rm cor}_{p}>(1+3/{\rm SNR})$. The
lower black line shows the mirrored relation.

Figure 6: Distribution of $S_{T}/S_{p}$ values for sources in different bins
of distance from the field centre. Only sources with SNR$>40$ are considered.
The plots are truncated at $S_{T}/S_{p}=1.5$ for a better visualization. For
each distance bin we fit a Gaussian function to the region of low
$S_{T}/S_{p}$ values.

###  4.4 Resolved and unresolved sources

The ratio between measured total and peak fluxes ($S_{T}/S_{p}$) is a proxy
for the extension of a radio source since it is equivalent to the ratio
between the fitted source full width half maximum (FWHM) axes, $\theta_{\rm
maj}$ and $\theta_{min}$, and the restoring beam FWHM axes, $b_{\rm maj}$ and
$b_{\rm min}$:

$S_{T}/S_{p}=\theta_{\rm maj}\theta_{min}/b_{\rm maj}b_{\rm min}$ (1)

In Fig. 5 (upper panel) we plot $S_{T}/S_{p}$ versus the SNR for all the
sources in the final catalogue. Values of $S_{T}/S_{p}<1$ are due to
statistical errors affecting the flux density measurements and these errors
should equally affect the $S_{T}/S_{p}>1$ region. In the following we use
$S_{T}/S_{p}$ to classify the sources as resolved or unresolved. It is also
clear that high SNR sources systematically lay above the $S_{T}/S_{p}=1$ line
(shown in black in Fig. 5). This is a clear indication that the distribution
of $S_{T}/S_{p}$ is affected by an offset that artificially increases the
derived ratios. Such an offset has been previously found, for instance, in
measurements derived from images affected by bandwidth smearing (e.g. Bondi et
al., 2008) and in the LOFAR observations of the Lockman Hole and ELAIS-N1,
where it has been interpreted as due to a combination of residual facet-
dependent calibration errors and PSF modelling (Mandal et al., 2021).

Mandal et al. (2021) accounted for the statistical errors in $S_{T}/S_{p}$ by
defining a lower envelope of the $S_{T}/S_{p}$ distribution shown in Fig. 5,
as:

$S_{T}/S_{p}=A/(1+B/{\rm SNR})$ (2)

where the parameter A represents the aforementioned offset. The lower envelope
can then be mirrored around the $S_{T}/S_{p}=A$ axis to get the upper
envelope:

$S_{T}/S_{p}=A\times(1+B/{\rm SNR})$ (3)

All the sources above the upper envelope are classified as resolved, while
those below are considered unresolved. Mandal et al. (2021) derived single A
and B values for each of the LoTSS Deep Fields.

We decided to use a slightly different approach. To investigate and quantify
the offset A we split all the sources with SNR$>40$ in the final catalogue
according to their distance from the field centre ($r$) into 5 different
subsamples: $r<0.7$ deg, $0.7<r<1.0$ deg, $1.0<r<1.25$ deg, $1.25<r<1.6$ deg,
and $r>1.6$ deg. For each subsample we plot the distribution of the ratio
$S_{T}/S_{p}$ in Fig. 6. The plots are truncated at $S_{T}/S_{p}=1.5$, but the
tail of extended sources continues up to value of $\sim 50$. Each distribution
shows a Gaussian-like region at low $S_{T}/S_{p}$ values that we fit with a
Gaussian (the black curves in Fig. 6).

We found that the Gaussian region of the $S_{T}/S_{p}$ distribution has an
offset with respect to the $S_{T}/S_{p}=1$ value and that this offset changes
for sources at different distances from the field centre. More distant sources
show a higher value (A $\sim 1.12$) than the sources closer to the field
centre (A $\sim 1.03$). In Tab. 8 we list for each subsample the range in $r$,
the number of sources in that distance range, and the A values derived from
the Gaussian region of the $S_{T}/S_{p}$ distributions.

The radio catalogue lists sources in a 10 deg2 circular region ($r<1.784$ deg)
and, given the spectral (97.6 kHz) and time (8s) averaging of the data, we can
expect bandwidth and time smearing up to $20\%$ at the edges of the field and
around $8\%$ at 1 deg from the field centre (Bridle & Schwab, 1999). DDFacet,
the imaging code within DDF-pipeline is specifically designed to minimize the
decorrelation and smearing effects using facet-dependent corrections during
the deconvolution (Tasse et al., 2018). Such a radially dependent residual
smearing was already noted by Shimwell et al. (2019, see their Fig.10) in the
LoTSS images. DDFacet derives, for each facet, its own PSF to be used during
deconvolution, but large facets (spanning a significant range in $r$) can
still produce smearing at the levels we observe. Moreover, it is worth noting
that facets at larger distances tend to be larger. As we have shown in Sec.
4.1, the accuracy of the flux measurements decreases with increasing distance
from the field centre, and this effect is stronger for peak fluxes compared to
total fluxes as expected in case of smearing. Mandal et al. (2021) arrived at
the same conclusion, with the only difference that they used a single A value,
averaged over distance. We conclude that the observed total-to-peak flux ratio
offset is caused by a residual smearing effect, which is not totally corrected
by DDFacet.

In order to separate resolved from unresolved sources, we first correct the
source peak fluxes in the final catalogue using the A values listed in Tab. 8
according to the distance of the source from the field centre. The values in
Tab. 8 are reasonably well fitted by a parabola given by
$A(r)=1.025+(r^{2}/35)$, where $r$ is the distance from the field centre in
degree, and we use this expression to interpolate the corrected peak fluxes
($S^{\rm cor}_{p}$) at each distance. The lower panel in Fig. 5 shows the new
$S_{T}/S^{\rm cor}_{p}$ versus SNR plot. The $S_{T}/S^{\rm cor}_{p}$
distribution, for the sources at high SNRs, has a Gaussian region centred at
$S_{T}/S^{\rm cor}_{p}=1$. Then we assume for the lower envelope:

$S_{T}/S^{\rm cor}_{p}=1/(1+3.0/{\rm SNR})$ (4)

where $B=3.0$ was chosen to be comparable with the equivalent functions in
Mandal et al. (2021) and satisfying the criterion that only $\simeq 1$ percent
of the points are outliers (i.e. sources below the lower envelope). Mirroring
Eq. 4 we obtain the selection function for the resolved sources:

$S_{T}/S^{\rm cor}_{p}=1+3.0/{\rm SNR}$ (5)

Adopting this relation we obtain that $\simeq 24\%$ of the sources are
classified as resolved. This value is consistent with the values of resolved
sources derived in other LOFAR deep fields as Lockman Hole and Elais-N1
(Mandal et al., 2021), and with the fraction of $24\%$ of resolved sources
found in the VLA-VIRMOS VLT Deep Survey (Bondi et al., 2003), a survey made
with the VLA at 1.4 GHz, but with the same angular resolution (6 arcsec) of
LOFAR HBA images. For the purposes of deriving the radio number counts (see
Sec. 6) we will use the total flux density ($S_{T}$) for the $\sim 5,500$
resolved sources and the corrected peak flux ($S^{\rm cor}_{p}$) for the $\sim
17,800$ unresolved sources. Needless to say that the method adopted to
classify a source as resolved or unresolved is an approximation. One
limitation is that we assume that all the sources below the upper envelope of
Fig. 5 but with $S_{\rm T}/S^{\rm cor}_{p}>1$ are unresolved and this is
likely not be entirely true. Another one is how to fit the lower envelope. For
all these reasons the classification as resolved or unresolved is used only on
a statistical basis and the total flux density derived from the classification
is used only to calculate the source counts and not listed in the catalogue.
The fluxes in the catalogue are those derived by fitting and assembling
Gaussian components by PyBDSF or manually for the very extended sources.
Having said that, we note that using samples of mock sources to derive the
source counts corrections allows to correct, if not entirely, at least for
part of these limitations as we explain in the next section.

Table 2: $S_{T}/S_{p}$ offset values

Distance interval | Number | A($r$)
---|---|---
(deg) |  |
$r<0.70$ | 232 | 1.030
$0.70<r<1.00$ | 270 | 1.050
$1.00<r<1.25$ | 266 | 1.055
$1.25<r<1.60$ | 394 | 1.070
$r>1.60$ | 230 | 1.120
888Col.1: Distance from the field centre bin; Col.2: number of SNR$>40$
sources in the distance bin; Col.3: Offset value of $S_{T}/S_{p}$ derived from
the peak of the Gaussian fit.

##  5 Radio source counts corrections

The observed number counts of radio sources must be corrected for several
effects such as:

  * •

varying noise distribution in the radio image due to the presence of bright
sources and the effect of the primary beam correction.

  * •

systematic effects in the source extraction procedure that could affect the
measured peak brightness and total flux density.

  * •

the resolution bias: number counts are derived in bins of total flux density
from samples of radio sources that are selected on the basis of their peak
brightness. For this reason the completeness of the radio sample depends also
on the intrinsic angular size distribution.

  * •

the Eddington bias (Eddington, 1913), also referred to as noise bias.

  * •

the contribution of spurious sources.

In order to take into account the combined effects of the first four effects
we used mock samples of radio sources as described below (see Sec. 5.2). The
contamination by spurious sources is not modelled by the simulations, since
the mock samples are inserted in the same residual image obtained after
subtraction of all the components. The fraction of spurious sources is derived
in the next subsection.

Table 3: False detection rates

SNR | Fraction range
---|---
| (%)
5.0–5.5 | 2.6–3.6
5.5–6.0 | 1.2–1.8
6.0–6.5 | 0.7–1.1
6.5–7.0 | 0.4–1.1
7.0–8.0 | 0.6–1.1
8.0–10.0 | 0.5–0.9
10.0–20.0 | 0.1–0.4
999Col.1: SNR bin; Col.2: range of false detection rates, the upper value is
obtained using all sources with SNR$>5$ in the inverted image, while for the
lower value only those without a bright source within $90^{\prime\prime}$ are
used.

###  5.1 False detection rate

The false detection rate is associated to spurious sources that are detected
above the SNR threshold value. To derive an estimate of the fraction of
spurious sources still present in the catalogue we ran PyBDSF on the inverted
(i.e. multiplied by $-1$) LOFAR image with the same settings used to produce
the catalogue. Since there is no negative emission in the radio continuum
emission of the sky, every detection above $5\sigma$ in the inverted image is
due to a noise peak. Therefore, assuming the noise distribution is symmetric
around zero we expect a similar number of noise peaks above $5\sigma$ (false
detection rate) in the positive image as well. This assumption may not be
valid near the bright sources due to the combination of residual calibration
errors and the shape of the sidelobes of the beam (i.e. the peak of the
positive sidelobes is higher than the absolute value of the peak of the
negative sidelobes). Therefore, the number of negative peaks has to be
considered as a lower limit for the number of spurious sources. For this
reason in Sec. 4.3 we previously removed spurious radio sources with SNR$>5$
produced or contaminated by the secondary lobes of nearby bright objects
($S_{T}>$ 15 mJy). PyBDSF returned 236 negative peaks above $5\sigma$ within
the 10 deg2 region. Among these 236 negative peaks, 101 are found within a
distance of $90^{\prime\prime}$ from a bright source in the positive image.
This number is close to the number of sources (110) that were excluded as
spurious in Sec. 4.3. We derive the fraction of false detections using both
the 135 sources far from bright sources (as a lower limit) and the whole
sample of 236 sources. The sources were binned in SNR groupings alongside true
detections and the false detection rates, defined as the number of false
detections divided by the number of recovered sources in each SNR bin, are
reported in Table 9. In general, the false detection rate values derived are
rather small. We expect $\sim 3\%$ of false detection rate for sources in the
range $5.0\leq{\rm SNR}<5.5$ and about $\sim 1.5\%$ in the range $5.5\leq{\rm
SNR}<6.0$. For sources with SNR $>10$ the false detection rate is $<0.5\%$, so
practically negligible. We applied a correction due to the false detection
rate to the differential source counts listed in Table 10 and Fig. 7 using all
the 236 false detections split into the appropriate total flux density bins.
We note that the differences in the source counts deriving from using 135 or
236 false detections are much smaller than the error bars.

###  5.2 Completeness and noise bias corrections

To derive the completeness and noise bias corrections affecting the EDFN radio
catalogue we follow the same approach used, for instance, for the COSMOS-3GHz
catalogue (Smolčić et al., 2017a). This procedure requires generating a
realistic sample of mock radio sources that will be added to a radio image
with the same noise properties of the real one. Then, the new image containing
the mock sources is processed with the same algorithm that yielded the radio
source catalogue. Sources in the input and recovered catalogues are then split
into bins of total flux density, and the numbers of injected versus recovered
sources in each total flux density bin are compared. The image on which the
mock sources are injected is the residual image produced by PyBDSF after the
real sources have been extracted. To limit the effects of confusion and
blending of different radio sources (the effect we, partially, correct for
using the UnWISE images as described in Sec. 4.3 for the real sources) we set
a minimum distance of 12 arcsec between two mock sources. With the expression
realistic sample we mean a sample of mock sources that follows a flux density
and angular size distribution as close as possible to those of real radio
sources as described in the next subsections.

####  5.2.1 Mock radio sources catalogue: flux density distribution

To simulate the flux density distribution we used a 7-th order polynomial
function fitting the differential source counts derived from the other LoTSS
Deep Fields and the TGSS-ADR1 (Intema et al., 2017) to better constrain the
bright end of the counts (eq. 13 and Table 4 in Mandal et al., 2021). The mock
catalogue is generated down to a flux density of $90\,\mu$Jy (roughly
$3\sigma$) well below the $5\sigma$ detection threshold used to produce our
radio catalogue. We generated three different realisations of the mock source
catalogue that were individually processed and compared. Each mock catalogue
contains $\simeq 55,000$ sources in the 10 deg2 area.

####  5.2.2 Mock radio sources catalogue: angular size distribution

The most challenging aspect in generating mock samples of realistic radio
sources is to assign to each source its own angular size. This is because the
intrinsic source angular size distribution of sub-mJy radio sources is still
not well known. The modelling of the intrinsic angular size distribution of
the radio sources is necessary to correct for the resolution bias which can
severely affect the radio source counts, since these are a function of the
total flux density, while the completeness of a radio catalogue is typically
based on the signal-to-noise ratio and, therefore, on the peak brightness.
Clearly, such an effect is more severe for catalogues derived from
observations with an angular resolution
$\theta\\!\\!\\!\phantom{\leq}\smash{\buildrel\over{\lower
2.67502pt\hbox{$\buildrel\lower
2.14001pt\hbox{$\displaystyle<$}\over{\sim}$}}}\,\,1^{\prime\prime}$, but it
needs to be taken into account in the lower resolution ($\theta\simeq
6^{\prime\prime}$) LOFAR observations as well. For a more detailed description
of the methods used to overcome this issue we refer to Bondi et al. (2008) and
Smolčić et al. (2017a).

To allow for an easier comparison with the results obtained in the other LoTSS
Deep Fields we adopted the radio source angular size distribution modelled by
eqs (7) and (9) in Mandal et al. (2021) with $m=0.3$. Using this distribution
we assigned to each source its own angular size and we modelled the mock
sources as circular Gaussian. It is worth noting that real radio sources
usually are not circular Gaussians. The effects on the radio source counts
produced by a parent source population with more realistic source morphologies
have been recently investigated by Hale et al. (2023). Using rather low
resolution ($\simeq 8^{\prime\prime}$) images, Hale et al. (2023) find that
the source counts are not strongly affected by different source models. This
is not surprising given the low resolution, that is comparable to that of our
LOFAR image. At these resolutions the bulk of the radio sources detected in a
deep field are unresolved or slightly resolved and the circular Gaussian
approximation is appropriate. Such an approximation is not valid anymore for
catalogues of radio sources derived from images at higher resolution and in
such a case a more detailed source modelling is necessary (e.g. Smolčić et
al., 2017a).

Figure 7: Euclidean normalized radio source counts at 144 MHz, obtained for
the Euclid Deep Field North plotted together with those obtained from other
LOFAR Deep Fields (see symbols) from Mandal et al. (2021). The black line is
the 7-th order polynomial fit from Mandal et al. (2021). The lower panel is a
zoom of the region between 0.2 and 2 mJy showing also with empty circles the
uncorrected normalized source counts for the EDFN. Table 4: Differential
source counts in the EDFN

Smin | Smax |  $\Delta$S | S | NS | $n$ | $n_{\rm cor}$ | erry | $C_{\rm sim}$
---|---|---|---|---|---|---|---|---
0.26 | 0.36 | 0.10 | 0.31 | 6075 | 32.65 | 39.36 | 2.15 | 1.21
0.36 | 0.51 | 0.15 | 0.43 | 5079 | 42.25 | 45.42 | 2.51 | 1.08
0.51 | 0.73 | 0.22 | 0.61 | 3479 | 47.74 | 46.36 | 2.66 | 0.97
0.73 | 1.03 | 0.30 | 0.87 | 1983 | 48.05 | 45.28 | 2.83 | 0.95
1.03 | 1.45 | 0.42 | 1.22 | 1119 | 45.67 | 46.73 | 3.35 | 1.04
1.45 | 2.06 | 0.61 | 1.73 | 658 | 43.98 | 43.71 | 3.61 | 1.00
2.06 | 2.91 | 0.85 | 2.45 | 401 | 45.94 | 44.80 | 4.44 | 0.99
2.91 | 4.11 | 1.20 | 3.46 | 273 | 52.53 | 51.57 | 6.02 | 1.00
4.11 | 5.82 | 1.71 | 4.89 | 176 | 56.53 | 56.02 | 7.90 | 1.02
5.82 | 8.23 | 2.41 | 6.92 | 152 | 82.51 | 78.73 | 11.57 | 0.98
8.23 | 11.60 | 3.37 | 9.77 | 122 | 112.16 | 112.62 | 18.17 | 0.98
11.60 | 23.30 | 11.70 | 16.44 | 175 | 170.17 | 171.87 | 23.93 | 1.01
101010Columns are as follows: $S_{\rm min}$ and $S_{\rm max}$ are the minimum
and maximum values of the flux density interval (in mJy), respectively;
$\Delta$S is the flux density interval (in mJy); x is the geometric mean of
$S_{\rm min}$ and $S_{\rm max}$; NS is the number of observed radio sources in
each bin; $n$ and $n_{\rm corr}$ denote the normalized source counts (in Jy3/2
sr-1) obtained from $N_{S}$ before and after correcting for the false
detections and completeness factors, respectively; erry is an estimate of the
errors associated to the corrected differential source counts (in Jy3/2 sr-1)
that takes into account both the Poissonian contribution and the deviations
produced by different simulations; $C_{\rm sim}$ is the average correction
factor derived from the simulations.

####  5.2.3 Mock radio sources catalogue: injection and recovering

We generated three different catalogues of mock radio sources using the flux
density and angular size distributions as described above. The mock sources
were inserted, as circular Gaussian components, in the residual image and
PyBDSF was run with the same parameters used for the real image to recover the
mock sources. The catalogues of retrieved mock sources with SNR$\geq 5$ were
cross matched with their respective input catalogues producing the recovered
mock sources matched catalogues (simply referred as matched catalogues in the
following). We then used the method discussed in Sec. 4.4 to classify the mock
sources in the matched catalogues as resolved or unresolved. The mock sources
are not affected by residual calibration errors or bandwidth smearing and,
therefore, the $S_{T}/S_{p}$ versus SNR diagram shows no systematic offset.
This means A=1 in eqs. (2) and (3). For the B parameter in eqs. (2) and (3) we
use B=3.0, the same value adopted for the real sources. Summarising, for the
matched catalogues we use the same selection function used for the real
sources (after the correction for the radially dependent offset). The
percentage of resolved mock sources in the matched catalogue is 23%, very well
consistent with the 24% we have found for the real sources.

Finally, the mock sources are split into different bins of flux density. For
the input catalogues, the flux density is that generated using the input flux
density distribution, whilst for the matched catalogue the flux density is the
measured flux density for the mock sources classified as resolved or the peak
flux density for the unresolved ones. The ratio, for each flux density bin,
between the number of sources in input and the number of retrieved sources in
the same flux density bin is the correction factor that needs to be applied to
the source number counts. We find good consistency, within $1\sigma$, in the
values of the correction factors among the three different catalogues of mock
sources and therefore we adopted an average correction factor for each flux
density bin. The final values of the correction factors are reported in Tab.
10.

As discussed in Sec. 4.4 the method adopted to classify a source as resolved
or unresolved is an approximation and somewhat arbitrary. Therefore, we used
also slightly different B values and/or shape of the selection function (Eq.
2) to derive alternative versions of the matched catalogues for the mock radio
sources. Then, we compared the corrections factors we obtained for these
catalogues with those originally derived. The differences in the correction
factors obtained from different selection functions are taken into account in
an additional fractional error quantified as $\sim 5\%$.

##  6 Differential source counts

The Euclidean normalized source counts at 144 MHz obtained for the EDFN are
presented in Fig. 7 and listed in Tab. 10. The errors associated to the number
counts contain the term due to the Poisson statistics for the number of
sources in each flux density bin, plus the fixed term of 5% as discussed
above. For the purposes of this paper it is useful to compare the source
counts derived from the EDFN with those derived from the other LoTSS Deep
Fields (Mandal et al., 2021), also shown in Fig. 7, since we used a different
method to assemble the final source catalogue and to derive the counts’
corrections. A more comprehensive analysis on the source counts is postponed
until the full set of observations of the EDFN is completed and processed. The
lower panel of Fig. 7 is a zoom of the region between 0.2 mJy and 2 mJy at 144
MHz (corresponding to about 0.04 mJy and 0.4 mJy at 1.4 GHz assuming a
spectral index $\alpha=0.7$), where it is quite common to find significant
differences in the source counts from different surveys at the depths we are
achieving. Such differences are usually interpreted as a mixing of cosmic
variance and uncertainty in the derived correction factors. In this panel, for
comparison, we also show the uncorrected normalized source counts (empty
circles with no errors).

The source counts obtained for the EDFN are in good agreement with those
derived from the other LoTSS Deep Fields and this is reassuring considering
the differences in obtaining them. One major difference between the source
counts derived from the EDFN and the other LoTSS Deep Fields is the method
applied to derive the correction factors for completeness. For the EDFN we
produced mock catalogues of radio sources that were added to the residual
radio image and then retrieved with the same procedure adopted for the real
ones, while for the other LoTSS Deep Fields Mandal et al. (2021) derived the
correction factors using a more theoretical approach.

In addition, the counts for Elais N-1, Lockman Hole and Bootes have been
obtained using a sample of optically identified and fully deblended radio
sources, while for the EDFN only a “first order” deblending was performed
using the unWISE catalogue. Most of the sources that went through deblending
had fluxes around $\sim 1$-2 mJy and the higher value of the source counts in
this flux density range suggests that it is possible that not all the sources,
that needed to be deblended, were actually deblended. It is worth noting that
the number of these potentially not deblended sources is rather small: about
50 sources that were not deblended can explain the somewhat higher value of
the counts in the EDFN around $\sim 1$-2 mJy. Clearly, the same effect would
lead to an underestimate of the source counts at fluxes
$\\!\\!\\!\phantom{\leq}\smash{\buildrel\over{\lower
2.67502pt\hbox{$\buildrel\lower
2.14001pt\hbox{$\displaystyle<$}\over{\sim}$}}}\,\,1$ mJy producing the sort
of discontinuity in the shape of the EDFN source counts around 1 mJy. This
effect was indeed noted in Mandal et al. (2021) when comparing source counts
obtained from raw catalogues (containing radio sources which were not
optically identified and not checked and corrected for deblending) and final
catalogues (containing only optically identified radio sources with source
deblending when needed). Summarising, the EDFN source counts are generally
between those derived from the ELAIS-N1 field, that are systematically
slightly lower than the other two deep fields, and those from the Lockman Hole
and Bootes fields. The observed field-to-field differences in the source
counts at sub-mJy levels are typically around a few percent and these can be
justified both by the different methods adopted to derive the source counts
(e.g. EDFN with respect to other LoTSS deep fields) and the expectations from
sample variance for surveys covering areas
$\\!\\!\\!\phantom{\leq}\smash{\buildrel\over{\lower
2.67502pt\hbox{$\buildrel\lower
2.14001pt\hbox{$\displaystyle<$}\over{\sim}$}}}\,\,10$ deg2 (Heywood et al.,
2013; Prandoni et al., 2018).

Once the LoTSS Deep Field project will be completed and fully analysed, it
will deliver images and catalogues of radio sources down to $\sim 60\,\mu$Jy
beam-1 ($5\sigma$) at 144 MHz (corresponding to $\sim 10\,\mu$Jy beam-1 at 1.4
GHz) over a region of $\sim 50$ deg2. Considering the range of angular
resolutions that LOFAR with the international stations can probe, as well as
the multi-wavelengths ancillary observations already available or that will be
soon available for these fields, in the next few years it is reasonable the
LoTSS Deep Fields will allow us to make a major leap in our comprehension of
the processes related to galaxy and black-hole formation and co-evolution over
cosmic time from a radio perspective.

##  7 Summary

In this paper we present the image and catalogue derived from the first 72
hours of LOFAR observations at 144 MHz covering the central 10 deg2 region of
the Euclid Deep Field North. The image has an angular resolution of
$6^{\prime\prime}$ and a central r.m.s. sensitivity of $32\,\mu$Jy beam-1.

The main results presented in this paper can be summarised as follows:

  * •

We compared images obtained from the same data sets but produced with
different faceting geometry and starting sky models in the direction-dependent
calibration pipeline (DDF-pipeline). We found that for sources within $r<2$
deg from the field centre, the peak or total flux density are usually
consistent within $\\!\\!\\!\phantom{\leq}\smash{\buildrel\over{\lower
2.67502pt\hbox{$\buildrel\lower
2.14001pt\hbox{$\displaystyle<$}\over{\sim}$}}}\,\,5\%$, while sources at
$r>2$ deg can have larger differences up to $\sim$ 10%-15% for sources at
$r>3$ deg. Whilst the two images were obtained running DDF-pipeline on
different computers with different individual software package versions, the
radial dependency of the dispersion of the measured flux ratios in the two
images suggests that a different faceting geometry could be responsible for
the observed trend.

  * •

From the inner 10 deg2 circular region ($r<1.784$ deg) we derived a $5\sigma$
catalogue listing about 23,000 radio sources. We used the unWISE 5 year
catalogue to check for possible blended sources in all the objects that were
classified as multiple or complex components by PyBDSF. We also visually
inspected the extended sources to properly assign radio lobes and diffuse
components to the correct radio source.

  * •

We performed a detailed analysis of the properties of the derived catalogue of
radio sources finding a radial dependent effect on the $S_{T}/S_{p}$ ratio. We
interpreted this as an underestimate of the peak brightness with increasing
distance from the field centre as due to residual direction-dependent
calibration errors. For the purpose of deriving the radio source counts we
corrected the peak brightness of the sources classified as unresolved.

  * •

Samples of mock sources following realistic flux density and angular size
distributions were used to derive the completeness correction factors to be
applied to the observed source counts. An estimate of the source false
detection rate was obtained as well. We found $\sim 3\%$ of false detection
rate for sources in the range $5.0\leq{\rm SNR}<5.5$ and about $\sim 1.5\%$ in
the range $5.5\leq{\rm SNR}<6.0$. For sources with SNR $>10$ the false
detection rate is $<0.5\%$ and practically negligible.

  * •

The final source counts obtained in the EDFN are consistent with those
obtained for the other LoTSS Deep Fields (ELAIS-N1, Lockman Hole and Bootes)
and, in particular, they lay between those derived from ELAIS-N1 and those
obtained for the other two fields. In our analysis we also explored the
contribution to the errors associated to the source counts produced by
slightly different selection functions to separate resolved from unresolved
sources. Such differences can be quantified in a 5% additional noise factor
that we factored into the final errors shown in Fig. 7 and listed in Tab. 10.

###### Acknowledgements.

MBond, IP, MBona, MBr, MM, LP, RS acknowledge support from INAF under the
Large Grant 2022 funding scheme (project ”MeerKAT and LOFAR Team up: a Unique
Radio Window on Galaxy/AGN co-Evolution”. PNB is grateful for support from the
UK STFC via grant ST/V000594/1. MBri acknowledges support from the ERC-Stg
”DRANOEL”, no. 714245, from the agreement ASI-INAF n. 2017-14-H.O and from the
PRIN MIUR 2017PH3WAT ”Blackout”. MA acknowledges support from the VENI
research programme with project number 202.143, which is financed by the
Netherlands Organisation for Scientific Research (NWO). LKM is grateful for
support from the Medical Research Council [MR/T042842/1]. RJvW acknowledges
support from the ERC Starting Grant ClusterWeb 804208. GJW gratefully
acknowledges an Emeritus Fellowship from The Leverhulme Trust. This paper is
based on data obtained with the International LOFAR Telescope (ILT) under
project codes LC12_027. AB acknowledges financial support from the European
Union - Next Generation EU. LOFAR 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 which 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 OCCAM
supercomputing facility run by the Competence Centre for Scientific Computing
an interdepartmental advanced research centre, that specializes in High
Performance Computing (HPC). The unWISE coadded images and catalogue are based
on data products from the Wide-field Infrared Survey Explorer, which is a
joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, and NEOWISE, which
is a project of the Jet Propulsion Laboratory/California Institute of
Technology. WISE and NEOWISE are funded by the National Aeronautics and Space
Administration. For the purpose of open access, the author has applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

## References

  * Aldinucci et al. (2017) Aldinucci, M., Bagnasco, S., Lusso, S., et al. 2017, Journal of Physics: Conference Series, 898, 082039
  * Algera et al. (2022) Algera, H. S. B., Hodge, J. A., Riechers, D. A., et al. 2022, ApJ, 924, 76
  * Best & Heckman (2012) Best, P. N. & Heckman, T. M. 2012, MNRAS, 421, 1569
  * Best et al. (2006) Best, P. N., Kaiser, C. R., Heckman, T. M., & Kauffmann, G. 2006, MNRAS, 368, L67
  * Best et al. (2023) Best, P. N., Kondapally, R., Williams, W. L., et al. 2023, MNRAS, 523, 1729
  * Bondi et al. (2008) Bondi, M., Ciliegi, P., Schinnerer, E., et al. 2008, ApJ, 681, 1129
  * Bondi et al. (2007) Bondi, M., Ciliegi, P., Venturi, T., et al. 2007, A&A, 463, 519
  * Bondi et al. (2003) Bondi, M., Ciliegi, P., Zamorani, G., et al. 2003, A&A, 403, 857
  * Bonzini et al. (2012) Bonzini, M., Mainieri, V., Padovani, P., et al. 2012, ApJS, 203, 15
  * Botteon et al. (2022) Botteon, A., van Weeren, R. J., Brunetti, G., et al. 2022, Science Advances, 8, eabq7623
  * Bridle & Schwab (1999) Bridle, A. H. & Schwab, F. R. 1999, in Astronomical Society of the Pacific Conference Series, Vol. 180, Synthesis Imaging in Radio Astronomy II, ed. G. B. Taylor, C. L. Carilli, & R. A. Perley, 371
  * Chen et al. (2016) Chen, C.-C., Smail, I., Ivison, R. J., et al. 2016, ApJ, 820, 82
  * Ciliegi et al. (1999) Ciliegi, P., McMahon, R. G., Miley, G., et al. 1999, MNRAS, 302, 222
  * Cochrane et al. (2023) Cochrane, R. K., Kondapally, R., Best, P. N., et al. 2023, MNRAS, 523, 6082
  * Condon (1992) Condon, J. J. 1992, ARA&A, 30, 575
  * Condon et al. (1998) Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
  * Conselice (2014) Conselice, C. J. 2014, ARA&A, 52, 291
  * Croton et al. (2006) Croton, D. J., Springel, V., White, S. D. M., et al. 2006, MNRAS, 365, 11
  * D’Amato et al. (2022) D’Amato, Q., Prandoni, I., Gilli, R., et al. 2022, A&A, 668, A133
  * de Gasperin et al. (2019) de Gasperin, F., Dijkema, T. J., Drabent, A., et al. 2019, A&A, 622, A5
  * Driver et al. (2011) Driver, S. P., Hill, D. T., Kelvin, L. S., et al. 2011, MNRAS, 413, 971
  * Duncan et al. (2021) Duncan, K. J., Kondapally, R., Brown, M. J. I., et al. 2021, A&A, 648, A4
  * Eddington (1913) Eddington, A. S. 1913, MNRAS, 73, 359
  * Euclid Collaboration et al. (2022) Euclid Collaboration, Scaramella, R., Amiaux, J., et al. 2022, A&A, 662, A112
  * Finkelstein et al. (2022) Finkelstein, S. L., Bagley, M. B., Ferguson, H. C., et al. 2022, arXiv e-prints, arXiv:2211.05792
  * Förster Schreiber & Wuyts (2020) Förster Schreiber, N. M. & Wuyts, S. 2020, ARA&A, 58, 661
  * Franco et al. (2018) Franco, M., Elbaz, D., Béthermin, M., et al. 2018, A&A, 620, A152
  * Grogin et al. (2011) Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35
  * Haarsma et al. (2000) Haarsma, D. B., Partridge, R. B., Windhorst, R. A., & Richards, E. A. 2000, ApJ, 544, 641
  * Hale et al. (2023) Hale, C. L., Whittam, I. H., Jarvis, M. J., et al. 2023, MNRAS, 520, 2668
  * Hales et al. (1990) Hales, S. E. G., Masson, C. R., Warner, P. J., & Baldwin, J. E. 1990, MNRAS, 246, 256
  * Hardcastle et al. (2007) Hardcastle, M. J., Evans, D. A., & Croston, J. H. 2007, MNRAS, 376, 1849
  * Heald et al. (2010) Heald, G., McKean, J., Pizzo, R., et al. 2010, arXiv e-prints, arXiv:1008.4693
  * Heckman & Best (2014) Heckman, T. M. & Best, P. N. 2014, ARA&A, 52, 589
  * Heywood et al. (2013) Heywood, I., Jarvis, M. J., & Condon, J. J. 2013, MNRAS, 432, 2625
  * Heywood et al. (2022) Heywood, I., Jarvis, M. J., Hale, C. L., et al. 2022, MNRAS, 509, 2150
  * Hopkins et al. (2003) Hopkins, A. M., Afonso, J., Chan, B., et al. 2003, AJ, 125, 465
  * Hopkins et al. (1998) Hopkins, A. M., Mobasher, B., Cram, L., & Rowan-Robinson, M. 1998, MNRAS, 296, 839
  * Huynh et al. (2005) Huynh, M. T., Jackson, C. A., Norris, R. P., & Prandoni, I. 2005, AJ, 130, 1373
  * Intema et al. (2017) Intema, H. T., Jagannathan, P., Mooley, K. P., & Frail, D. A. 2017, A&A, 598, A78
  * Jarvis et al. (2016) Jarvis, M., Taylor, R., Agudo, I., et al. 2016, in MeerKAT Science: On the Pathway to the SKA, 6
  * Kondapally et al. (2022) Kondapally, R., Best, P. N., Cochrane, R. K., et al. 2022, MNRAS, 513, 3742
  * Kondapally et al. (2021) Kondapally, R., Best, P. N., Hardcastle, M. J., et al. 2021, A&A, 648, A3
  * Lane et al. (2014) Lane, W. M., Cotton, W. D., van Velzen, S., et al. 2014, MNRAS, 440, 327
  * Lang (2014) Lang, D. 2014, AJ, 147, 108
  * Le Fèvre et al. (2005) Le Fèvre, O., Vettolani, G., Bottini, D., et al. 2005, The Messenger, 119, 30
  * Mandal et al. (2021) Mandal, S., Prandoni, I., Hardcastle, M. J., et al. 2021, A&A, 648, A5
  * Meisner et al. (2017a) Meisner, A. M., Lang, D., & Schlegel, D. J. 2017a, AJ, 154, 161
  * Meisner et al. (2017b) Meisner, A. M., Lang, D., & Schlegel, D. J. 2017b, AJ, 153, 38
  * Miller et al. (2013) Miller, N. A., Bonzini, M., Fomalont, E. B., et al. 2013, ApJS, 205, 13
  * Miller et al. (2008) Miller, N. A., Fomalont, E. B., Kellermann, K. I., et al. 2008, ApJS, 179, 114
  * Mingo et al. (2022) Mingo, B., Croston, J. H., Best, P. N., et al. 2022, MNRAS, 511, 3250
  * Mohan & Rafferty (2015) Mohan, N. & Rafferty, D. 2015, PyBDSF: Python Blob Detection and Source Finder, Astrophysics Source Code Library, record ascl:1502.007
  * Morrison et al. (2010) Morrison, G. E., Owen, F. N., Dickinson, M., Ivison, R. J., & Ibar, E. 2010, ApJS, 188, 178
  * Murphy et al. (2017) Murphy, E. J., Momjian, E., Condon, J. J., et al. 2017, ApJ, 839, 35
  * Muxlow et al. (2020) Muxlow, T. W. B., Thomson, A. P., Radcliffe, J. F., et al. 2020, MNRAS, 495, 1188
  * Novak et al. (2017) Novak, M., Smolčić, V., Delhaize, J., et al. 2017, A&A, 602, A5
  * Offringa (2016) Offringa, A. R. 2016, A&A, 595, A99
  * Offringa et al. (2012) Offringa, A. R., van de Gronde, J. J., & Roerdink, J. B. T. M. 2012, A&A, 539, A95
  * Owen (2018) Owen, F. N. 2018, ApJS, 235, 34
  * Owen & Morrison (2008) Owen, F. N. & Morrison, G. E. 2008, AJ, 136, 1889
  * Padovani (2016) Padovani, P. 2016, A&A Rev., 24, 13
  * Pentericci et al. (2018) Pentericci, L., McLure, R. J., Garilli, B., et al. 2018, A&A, 616, A174
  * Prandoni et al. (2000) Prandoni, I., Gregorini, L., Parma, P., et al. 2000, A&AS, 146, 41
  * Prandoni et al. (2018) Prandoni, I., Guglielmino, G., Morganti, R., et al. 2018, MNRAS, 481, 4548
  * Rengelink et al. (1997) Rengelink, R. B., Tang, Y., de Bruyn, A. G., et al. 1997, A&AS, 124, 259
  * Sabater et al. (2021) Sabater, J., Best, P. N., Tasse, C., et al. 2021, A&A, 648, A2
  * Schinnerer et al. (2010) Schinnerer, E., Sargent, M. T., Bondi, M., et al. 2010, ApJS, 188, 384
  * Schinnerer et al. (2007) Schinnerer, E., Smolčić, V., Carilli, C. L., et al. 2007, ApJS, 172, 46
  * Schlafly et al. (2019) Schlafly, E. F., Meisner, A. M., & Green, G. M. 2019, ApJS, 240, 30
  * Scoville et al. (2007) Scoville, N., Aussel, H., Brusa, M., et al. 2007, ApJS, 172, 1
  * Seymour et al. (2008) Seymour, N., Dwelly, T., Moss, D., et al. 2008, MNRAS, 386, 1695
  * Shimwell et al. (2022) Shimwell, T. W., Hardcastle, M. J., Tasse, C., et al. 2022, A&A, 659, A1
  * Shimwell et al. (2017) Shimwell, T. W., Röttgering, H. J. A., Best, P. N., et al. 2017, A&A, 598, A104
  * Shimwell et al. (2019) Shimwell, T. W., Tasse, C., Hardcastle, M. J., et al. 2019, A&A, 622, A1
  * Smirnov & Tasse (2015) Smirnov, O. M. & Tasse, C. 2015, MNRAS, 449, 2668
  * Smolčić (2009) Smolčić, V. 2009, ApJ, 699, L43
  * Smolčić et al. (2017a) Smolčić, V., Novak, M., Bondi, M., et al. 2017a, A&A, 602, A1
  * Smolčić et al. (2017b) Smolčić, V., Novak, M., Delvecchio, I., et al. 2017b, A&A, 602, A6
  * Smolčić et al. (2009) Smolčić, V., Schinnerer, E., Zamorani, G., et al. 2009, ApJ, 690, 610
  * Somerville & Davé (2015) Somerville, R. S. & Davé, R. 2015, ARA&A, 53, 51
  * Tasse (2014a) Tasse, C. 2014a, arXiv e-prints, arXiv:1410.8706
  * Tasse (2014b) Tasse, C. 2014b, A&A, 566, A127
  * Tasse et al. (2018) Tasse, C., Hugo, B., Mirmont, M., et al. 2018, A&A, 611, A87
  * Tasse et al. (2007) Tasse, C., Röttgering, H. J. A., Best, P. N., et al. 2007, A&A, 471, 1105
  * Tasse et al. (2021) Tasse, C., Shimwell, T., Hardcastle, M. J., et al. 2021, A&A, 648, A1
  * Treu et al. (2022) Treu, T., Roberts-Borsani, G., Bradac, M., et al. 2022, ApJ, 935, 110
  * van der Vlugt et al. (2022) van der Vlugt, D., Hodge, J. A., Algera, H. S. B., et al. 2022, ApJ, 941, 10
  * van Haarlem et al. (2013) van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2
  * van Weeren et al. (2016) van Weeren, R. J., Williams, W. L., Hardcastle, M. J., et al. 2016, ApJS, 223, 2
  * Weaver et al. (2022) Weaver, J. R., Kauffmann, O. B., Ilbert, O., et al. 2022, ApJS, 258, 11
  * White et al. (2010) White, G. J., Pearson, C., Braun, R., et al. 2010, A&A, 517, A54
  * Williams et al. (2016) Williams, W. L., van Weeren, R. J., Röttgering, H. J. A., et al. 2016, MNRAS, 460, 2385
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868