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Astronomy & Astrophysics manuscript no. aanda

©ESO 2022

February 14, 2022

The discovery of a radio galaxy of at least 5 Mpc

Martijn S.S.L. Oei1?, Reinout J. van Weeren1, Martin J. Hardcastle2, Andrea Botteon1, Tim W. Shimwell1, Pratik

Dabhade3, Aivin R.D.J.G.I.B. Gast4, Huub J.A. Röttgering1, Marcus Brüggen5, Cyril Tasse6, 7, Wendy L. Williams1, and

Aleksandar Shulevski1

1 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2300 RA Leiden, The Netherlands

e-mail: oei@strw.leidenuniv.nl

2 Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hateld AL10 9AB, United Kingdom

3 Observatoire de Paris, LERMA, Collège de France, CNRS, PSL University, Sorbonne University, 75014 Paris, France

4 Somerville College, University of Oxford, Woodstock Road, Oxford OX2 6HD, United Kingdom

5 Hamburger Sternwarte, University of Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany

6 GEPI & USN, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, 92190 Meudon, France

7 Department of Physics & Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa

February 14, 2022

ABSTRACT

Context. Giant radio galaxies (GRGs, or colloquially ‘giants’) are the Universe’s largest structures generated by individual galaxies.

They comprise synchrotron-radiating AGN ejecta and attain cosmological (Mpc-scale) lengths. However, the main mechanisms that

drive their exceptional growth remain poorly understood.

Aims. To deduce the main mechanisms that drive a phenomenon, it is usually instructive to study extreme examples. If there exist

host galaxy characteristics that are an important cause for GRG growth, then the hosts of the largest GRGs are likely to possess

them. Similarly, if there exist particular large-scale environments that are highly conducive to GRG growth, then the largest GRGs

are likely to reside in them. For these reasons, we aim to perform a case study of the largest GRG available.

Methods. We reprocessed the LOFAR Two-metre Sky Survey (LoTSS) DR2 by subtracting compact sources and performing multi-

scale CLEAN deconvolution at 60′′ and 90′′ resolution. The resulting images constitute the most sensitive survey yet for radio

galaxy lobes, whose diuse nature and steep synchrotron spectra have allowed them to evade previous detection attempts at higher

resolution and shorter wavelengths. We visually searched these images for GRGs.

Results. We discover Alcyoneus, a low-excitation radio galaxy with a projected proper length lp = 4.99 ± 0.04 Mpc. Its jets and

lobes are all four detected at very high signicance, and the SDSS-based identication of the host, at spectroscopic redshift zspec =

0.24674 ± 6 ·10−5, is unambiguous. The total luminosity density at ν = 144MHz is Lν = 8±1 ·1025 W Hz−1, which is below-average,

though near-median (percentile 45±3%), for GRGs. The host is an elliptical galaxy with a stellar mass M? = 2.4±0.4 ·1011 M and

a supermassive black hole mass M• = 4±2 ·108 M, both of which tend towards the lower end of their respective GRG distributions

(percentiles 25±9% and 23±11%). The host resides in a lament of the Cosmic Web. Through a new Bayesian model for radio galaxy

lobes in three dimensions, we estimate the pressures in the Mpc3-scale northern and southern lobe to be Pmin,1 = 4.8±0.3 ·10−16 Pa

and Pmin,2 = 4.9±0.6·10−16 Pa, respectively. The corresponding magnetic eld strengths are Bmin,1 = 46±1 pT and Bmin,2 = 46±3 pT.

Conclusions. We have discovered what is in projection the largest known structure made by a single galaxy — a GRG with a projected

proper length lp = 4.99 ± 0.04 Mpc. The true proper length is at least lmin = 5.04 ± 0.05 Mpc. Beyond geometry, Alcyoneus and

its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass and supermassive black hole mass are

all lower than, though similar to, those of the medial GRG. Thus, very massive galaxies or central black holes are not necessary

to grow large giants, and, if the observed state is representative of the source over its lifetime, neither is high radio power. A low-

density environment remains a possible explanation. The source resides in a lament of the Cosmic Web, with which it might have

signicant thermodynamic interaction. The pressures in the lobes are the lowest hitherto found, and Alcyoneus therefore represents

the most promising radio galaxy yet to probe the warm–hot intergalactic medium.

Key words. galaxies: active – galaxies: individual: Alcyoneus – galaxies: jets – intergalactic medium – radio continuum: galaxies

1. Introduction

Most galactic bulges hold a supermassive (i.e. M• > 106 M)

Kerr black hole (e.g. Soltan 1982) that grows by accreting gas,

dust and stars from its surroundings (Kormendy & Ho 2013).

The black hole ejects a fraction of its accretion disk plasma

from the host galaxy along two collimated, magnetised jets

that are aligned with its rotation axis (e.g. Blandford & Rees

? In dear memory of Pallas. If your name hadn’t been this popular with

asteroid discoverers, you’d now be the giants’ giant — once again looking

down at the sprawling ants below.

1974). The relativistic electrons contained herein experience

Lorentz force and generate, through spiral motion, synchrotron

radiation that is observed by radio telescopes. The two jets

either fade gradually or end in hotspots at the end of diuse

lobes, and ultimately enrich the intergalactic medium (IGM)

with cosmic rays and magnetic elds. The full luminous

structure is referred to as a radio galaxy (RG). Members of a

rare RG subpopulation attain megaparsec-scale proper (and

thus comoving) lengths (e.g. Willis et al. 1974; Andernach

et al. 1992; Ishwara-Chandra & Saikia 1999; Jamrozy et al.

2008; Machalski 2011; Kuźmicz et al. 2018; Dabhade et al.

Article number, page 1 of 18

arXiv:2202.05427v1 [astro-ph.GA] 11 Feb 2022

A&A proofs: manuscript no. aanda

Fig. 1: Joint radio-infrared view of Alcyoneus, a radio galaxy with a projected proper length of 5.0 Mpc. We show a

2048′′ × 2048′′ solid angle centred around right ascension 123.590372° and declination 52.402795°. We superimpose LOFAR

Two-metre Sky Survey (LoTSS) DR2 images at 144 MHz of two dierent resolutions (6′′ for the core and jets, and 60′′ for the

lobes) (orange), with the Wide-eld Infrared Survey Explorer (WISE) image at 3.4 µm (blue). To highlight the radio emission, the

infrared emission has been blurred to 0.5′ resolution.

2020a). The giant radio galaxy (GRG, or colloquially ‘giant’)

denition accommodates our limited ability to infer an RG’s

true proper length from observations: an RG is called a GRG

if and only if its proper length projected onto the plane of the

sky exceeds some threshold lp,GRG, usually chosen to be 0.7 or

1 Mpc. Because the conversion between angular length and

projected proper length depends on cosmological parameters,

which remain uncertain, it is not always clear whether a given

observed RG satises the GRG denition.

Currently, there are about a thousand GRGs known, the major-

ity of which have been found in the Northern Sky. About one

hundred exceed 2 Mpc and ten exceed 3 Mpc; at 4.9 Mpc, the

literature’s projectively longest is J1420-0545 (Machalski et al.

2008). As such, GRGs — and the rest of the megaparsec-scale

RGs — are the largest single-galaxy–induced phenomena in the

Universe. It is a key open question what physical mechanisms

lead some RGs to extend for ∼102 times their host galaxy

diameter. To determine whether there exist particular host

galaxy characteristics or large-scale environments that are

essential for GRG growth, it is instructive to analyse the largest

GRGs, since in these systems it is most likely that all major

favourable growth factors are present. We thus aim to perform

Article number, page 2 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

a case study of the largest GRG available.

As demonstrated by Dabhade et al. (2020b)’s record sample

of 225 discoveries, the Low-frequency Array (LOFAR) (van

Haarlem et al. 2013) is among the most attractive contempo-

rary instruments for nding new GRGs. This Pan-European

radio interferometer features a unique combination of short

baselines to provide sensitivity to large-scale emission, and

long baselines to mitigate source confusion.1 These qualities

are indispensable for observational studies of GRGs, which

require identifying both extended lobes and compact cores and

jets. Additionally, the metre wavelengths at which the LOFAR

operates allow it to detect steep-spectrum lobes far away from

host galaxies. Such lobes reveal the full extent of GRGs, but

are missed by decimetre observatories. Thus, in Section 2, we

describe a reprocessing of the LOFAR Two-metre Sky Survey

(LoTSS) Data Release 2 (DR2) aimed at revealing hitherto

unknown RG lobes — among other goals. An overview of the

reprocessed images, which cover thousands of square degrees,

and statistics of the lengths and environments of the GRGs

they have revealed, are subjects of future publications. For

now, these images allow us to discover Alcyoneus2, a 5 Mpc

GRG, whose properties we determine and discuss in Section 3.

Figure 1 provides a multi-wavelength, multi-resolution view

of this giant. Section 4 contains our concluding remarks.

We assume a concordance inationary ΛCDM model with

parameters M from Planck Collaboration et al. (2020); i.e. M =

(

h = 0.6766,ΩBM,0 = 0.0490,ΩM,0 = 0.3111,ΩΛ,0 = 0.6889

),

where H0 B h · 100 km s−1 Mpc−1. We dene the spectral

index α such that it relates to ux density Fν at frequency ν

as Fν ∝ να. Regarding terminology, we strictly distinguish

between a radio galaxy, a radio-bright structure of relativistic

particles and magnetic elds (consisting of a core, jets, hotspots

and lobes), and the host galaxy that generates it.

2. Data and methods

The LoTSS, conducted by the LOFAR High-band Antennae

(HBA), is a 120–168 MHz interferometric survey (Shimwell et al.

2017, 2019, in prep.) with the ultimate aim to image the full

Northern Sky at resolutions of 6′′, 20′′, 60′′ and 90′′. Its cen-

tral frequency νc = 144 MHz. The latest data release — the

LoTSS DR2 (Shimwell et al. in prep.) — covers 27% of the North-

ern Sky, split over two regions of 4178 deg2 and 1457 deg2; the

largest of these contains the Sloan Digital Sky Survey (SDSS)

DR7 (Abazajian et al. 2009) area. By default, the LoTSS DR2 pro-

vides imagery at the 6′′ and 20′′ resolutions. We show these

standard products in Figure 2 for the same sky region as in

Figure 1. In terms of total source counts, the LoTSS DR2 is

the largest radio survey carried out thus far: its catalogue con-

tains 4.4 · 106 sources, most of which are considered com-

pact. By contrast, the 60′′ and 90′′ imagery, which we dis-

cuss in more detail in Oei et al. (in prep.), is intended to re-

veal extended structures in the low-frequency radio sky, such

1 Source confusion is an instrumental limitation that arises when the

resolution of an image is low compared to the sky density of statisti-

cally signicant sources. It causes angularly adjacent, but physically

unrelated sources to blend together, making it hard or even impossible

to distinguish them (e.g. Condon et al. 2012).

2 Alcyoneus was the son of Ouranos, the Greek primordial god of the

sky. According to Ps.-Apollodorus, he was also one of the greatest of

the Gigantes (Giants), and a challenger to Heracles during the Gigan-

tomachy — the battle between the Giants and the Olympian gods for

supremacy over the Cosmos. The poet Pindar described him as ‘huge

as a mountain’, ghting by hurling rocks at his foes.

123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0

200

400

600

800

1000

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0

100

200

300

400

500

600

specificintensityIν(Jydeg−2)Fig. 2: Alcyoneus’ lobes are easily overlooked in the LoTSS

DR2 at its standard resolutions. We show images at cen-

tral frequency νc = 144 MHz and resolutions θFWHM = 6′′

(top) and θFWHM = 20′′ (bottom), centred around host galaxy

J081421.68+522410.0.

as giant radio galaxies, supernova remnants in the Milky Way,

radio halos and shocks in galaxy clusters, and — potentially

— accretion shocks or volume-lling emission from laments

of the Cosmic Web. To avoid the source confusion limit at

these resolutions, following van Weeren et al. (2021), we used

DDFacet (Tasse et al. 2018) to predict visibilities corresponding

to the 20′′ LoTSS DR2 sky model and subtracted these from the

data, before imaging at 60′′ and 90′′ with WSClean IDG (Of-

fringa et al. 2014; van der Tol et al. 2018). We used -0.5 Briggs

weighting and multiscale CLEAN (Oringa & Smirnov 2017),

with -multiscale-scales 0,4,8,16,32,64. Importantly, we

did not impose an inner (u, v)-cut. We imaged each pointing sep-

arately, then combined the partially overlapping images into a

mosaic by calculating, for each direction, a beam-weighted av-

erage.

Finally, we visually searched the LoTSS DR2 for GRGs, primarily

at 6′′ and 60′′ using the Hierarchical Progressive Survey (HiPS)

system in Aladin Desktop 11.0 (Bonnarel et al. 2000).

Article number, page 3 of 18

A&A proofs: manuscript no. aanda

3. Results and discussion

3.1. Radio morphology and interpretation

During our LoTSS DR2 search, we identied a three-component

radio structure of total angular length φ = 20.8′, visible at all

(6′′, 20′′, 60′′ and 90′′) resolutions. Figure 2 provides a sense of

our data quality; it shows that the outer components are barely

discernible in the LoTSS DR2 at its standard 6′′ and 20′′ reso-

lutions. Meanwhile, Figure 1 shows the outer components at

60′′, and the top panel of Figure 9 shows them at 90′′; at these

resolutions, they lie rmly above the noise. Compared with the

outer structures, the central structure is bright and elongated,

with a 155′′ major axis and a 20′′ minor axis. The outer struc-

tures lie along the major axis at similar distances from the cen-

tral structure, are diuse and amorphous, and feature specic

intensity maxima along this axis.

In the arcminute-scale vicinity of the outer structures, the DESI

Legacy Imaging Surveys (Dey et al. 2019) DR9 does not reveal

galaxy overdensities or low-redshift spiral galaxies, the ROSAT

All-sky Survey (RASS) (Voges et al. 1999) does not show X-ray

brightness above the noise, and there is no Planck Sunyaev–

Zeldovich catalogue 2 (PSZ2) (Planck Collaboration et al. 2016)

source nearby. The outer structures therefore cannot be super-

nova remnants in low-redshift spiral galaxies or radio relics

and radio halos in galaxy clusters. Instead, the outer structures

presumably represent radio galaxy emission. The radio-optical

overlays in Figure 3’s top and bottom panel show that it is im-

probable that each outer structure is a radio galaxy of its own,

given the lack of signicant 6′′ radio emission (solid light green

contours) around host galaxy candidates suggested by the mor-

phology of the 60′′ radio emission (translucent white contours).

For these reasons, we interpret the central (jet-like) structure

and the outer (lobe-like) structures as components of the same

radio galaxy.

Subsequent analysis — presented below — demonstrates that

this radio galaxy is the largest hitherto discovered, with a pro-

jected proper length of 5.0 Mpc. We dub this GRG Alcyoneus.

3.2. Host galaxy identification

Based on the middle panel of Figure 3 and an SDSS DR12 (Alam

et al. 2015) spectrum, we identify a source at a J2000 right as-

cension of 123.590372°, a declination of 52.402795° and a spec-

troscopic redshift of zspec = 0.24674 ± 6 · 10−5 as Alcyoneus’

host. Like most GRG hosts, this source, with SDSS DR12 name

J081421.68+522410.0, is an elliptical galaxy3 without a quasar.

From optical contours, we nd that the galaxy’s minor axis

makes a ∼20° angle with Alcyoneus’ jet axis.

In Figure 4, we further explore the connection between

J081421.68+522410.0 and Alcyoneus’ radio core and jets. From

top to bottom, we show the LoTSS DR2 at 6′′, the Very Large

Array Sky Survey (VLASS) (Lacy et al. 2020) at 2.2′′, and the

Panoramic Survey Telescope and Rapid Response System (Pan-

STARRS) DR1 (Chambers et al. 2016) i-band. Two facts conrm

that the host identication is highly certain. First, for both the

LoTSS DR2 at 6′′ and the VLASS at 2.2′′, the angular separa-

tion between J081421.68+522410.0 and the arc connecting Al-

cyoneus’ two innermost jet features is subarcsecond. Moreover,

the alleged host galaxy is the brightest Pan-STARRS DR1 i-band

3 Based on the SDSS morphology, Kuminski & Shamir (2016) calculate

a probability of 89% that the galaxy is an elliptical.

123.32

123.36

123.4

123.44

123.48

right ascension (°)

52.44

52.46

52.48

52.5

52.52

52.54

declination(°)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)123.64

123.68

123.72

123.76

123.8

right ascension (°)

52.26

52.28

52.3

52.32

52.34

52.36

52.38

declination(°)Fig. 3: Joint radio-optical views show that Figure 1’s outer

structures are best interpreted as a pair of radio galaxy

lobes fed by central jets. On top of DESI Legacy Imaging

Surveys DR9 (g, r, z)-imagery, we show the LoTSS DR2 at var-

ious resolutions through contours at multiples of σ, where σ

is the image noise at the relevant resolution. The top and bot-

tom panel show translucent white 60′′ contours at 3, 5, 7, 9, 11σ

and solid light green 6′′ contours at 4, 7, 10, 20, 40σ. The central

panel shows translucent white 6′′ contours at 5, 10, 20, 40, 80σ.

Article number, page 4 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0

1 · 103

2 · 103

3 · 103

4 · 103

5 · 103

6 · 103

7 · 103

specificintensityIν(Jydeg−2)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0

1 · 103

2 · 103

3 · 103

4 · 103

5 · 103

6 · 103

7 · 103

specificintensityIν(Jydeg−2)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0.0

0.2

0.4

0.6

0.8

1.0

1.2

relativespecificintensityIν(1)Fig. 4: The SDSS DR12 source J081421.68+522410.0 is Al-

cyoneus’ host galaxy. The panels cover a 2.5′ × 2.5′ region

around J081421.68+522410.0, an elliptical galaxy with spectro-

scopic redshift zspec = 0.24674 ± 6 · 10−5. From top to bot-

tom, we show the LoTSS DR2 6′′, the VLASS 2.2′′, and the Pan-

STARRS DR1 i-band — relative to the peak specic intensity of

J081421.68+522410.0 — with LoTSS contours (white) as in Fig-

ure 3 and a VLASS contour (gold) at 5σ.

source within a radius of 45′′ of the central VLASS image com-

ponent.

3.3. Radiative- or jet-mode active galactic nucleus

Current understanding (e.g. Heckman & Best 2014) suggests

that the population of active galactic nuclei (AGN) exhibits a

dichotomy: AGN seem to be either radiative-mode AGN, which

generate high-excitation radio galaxies (HERGs), or jet-mode

AGN, which generate low-excitation radio galaxies (LERGs). Is

Alcyoneus a HERG or a LERG? The SDSS spectrum of the host

features very weak emission lines; indeed, the star formation

rate (SFR) is just 1.6 · 10−2 M yr−1 (Chang et al. 2015). Fol-

lowing the classication rule of Best & Heckman (2012); Best

et al. (2014); Pracy et al. (2016); Williams et al. (2018) based on

the strength and equivalent width of the OIII 5007 Å line, we

conclude that Alcyoneus is a LERG. Moreover, the WISE pho-

tometry (Cutri & et al. 2012) at 11.6 µm and 22.1 µm is below the

instrumental detection limit. Following the classication rule of

Gürkan et al. (2014) based on the 22.1 µm luminosity density,

we arm that Alcyoneus is a LERG. Through automated classi-

cation, Best & Heckman (2012) came to the same conclusion.

Being a jet-mode AGN, the supermassive black hole (SMBH) in

the centre of Alcyoneus’ host galaxy presumably accretes at an

eciency below 1% of the Eddington limit, and is fueled mainly

by slowly cooling hot gas.

3.4. Projected proper length

We calculate Alcyoneus’ projected proper length lp through

its angular length φ and spectroscopic redshift zspec. We for-

mally determine φ = 20.8′ ± 0.15′ from the compact-source–

subtracted 90′′ image (top panel of Figure 9) by selecting the

largest great-circle distance between all possible pairs of pix-

els with a specic intensity higher than three sigma-clipped

standard deviations above the sigma-clipped median. We nd

lp = 4.99 ± 0.04 Mpc; this makes Alcyoneus the projectively

largest radio galaxy known. For methodology details, and for a

probabilistic comparison between the projected proper lengths

of Alcyoneus and J1420-0545, see Appendix A.

3.5. Radio luminosity densities and kinetic jet powers

From the LoTSS DR2 6′′ image (top panel of Figure 4), we mea-

sure that two northern jet local maxima occur at angular dis-

tances of 9.2 ± 0.2′′ and 23.7 ± 0.2′′ from the host, or at pro-

jected proper distances of 36.8 ± 0.8 kpc and 94.8 ± 0.8 kpc.

Two southern jet local maxima occur at angular distances of

8.8± 0.2′′ and 62.5± 0.2′′ from the host, or at projected proper

distances of 35.2 ± 0.8 kpc and 249.9 ± 0.8 kpc.

At the central observing frequency of νc = 144MHz, the north-

ern jet has a ux density Fν = 193±20mJy, the southern jet has

Fν = 110±12mJy, whilst the northern lobe has Fν = 63±7mJy

and the southern lobe has Fν = 44 ± 5 mJy. To minimise con-

tamination from fore- and background galaxies, we determined

the lobe ux densities from the compact-source–subtracted 90′′

image. The ux density uncertainties are dominated by the 10%

ux scale uncertainty inherent to the LoTSS DR2 (Shimwell

et al. in prep.). The host galaxy ux density is relatively weak,

and the corresponding emission has, at νc = 144 MHz and

6′′ resolution, no clear angular separation from the inner jets’

emission; we have therefore not determined it.

Due to cosmological redshifting, the conversion between ux

Article number, page 5 of 18

A&A proofs: manuscript no. aanda

123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

spectralindexα(1)Fig. 5: The LoTSS–VLASS spectral indexmap reveals Alcy-

oneus’ at-spectrum core and steeper-spectrum jets. We

show all directions where both the LoTSS and VLASS image

have at least 5σ signicance. In black, we overlay the same

LoTSS contours as in Figures 3 and 4. The core spectral index

is α = −0.25 ± 0.1 and the combined inner jet spectral index is

α = −0.65 ± 0.1.

density and luminosity density depends on the spectral indices

α of Alcyoneus’ luminous components. We estimate the spec-

tral indices of the core and jets from the LoTSS DR2 6′′ and

VLASS 2.2′′ images. After convolving the VLASS image with

a Gaussian to the common resolution of 6′′, we calculate the

mean spectral index between LoTSS’ νc = 144MHz and VLASS’

νc = 2.99 GHz. Using only directions for which both images

have a signicance of at least 5σ, we deduce a core spectral

index α = −0.25 ± 0.1 and a combined inner jet spectral index

α = −0.65±0.1. The spectral index uncertainties are dominated

by the LoTSS DR2 and VLASS ux scale uncertainties. We show

the full spectral index map in Figure 5. We have not determined

the spectral index of the lobes, as they are only detected in the

LoTSS imagery.

The luminosity densities of the northern and southern jet at

rest-frame frequency ν = 144 MHz are Lν = (3.6 ± 0.4) ·

1025 W Hz−1 and Lν = (2.0 ± 0.2) · 1025 W Hz−1, respectively.

Following Dabhade et al. (2020a), we estimate the kinetic power

of the jets from their luminosity densities and the results of the

simulation-based analytical model of Hardcastle (2018). We nd

Qjet,1 = 1.2 ± 0.1 · 1036 W and Qjet,2 = 6.6 ± 0.7 · 1035 W,

so that the total kinetic jet power is Qjets B Qjet,1 + Qjet,2 =

1.9 ± 0.2 · 1036 W. Interestingly, this total kinetic jet power is

lower than the average Qjets = 3.7 ·1036 W, and close to the me-

dian Qjets = 2.2 · 1036 W, for low-excitation giant radio galaxies

(LEGRGs) in the redshift range 0.18 < z < 0.43 (Dabhade et al.

2020a).

Because the lobe spectral indices are unknown, we present lu-

minosity densities for several possible values of α in Table 1.4

(Because of electron ageing, α will decrease further away from

the core.)

4 The inferred luminosity densities have a cosmology-dependence;

our results are ∼6% higher than for modern high-H0 cosmologies.

1024

1025

1026

1027

1028

luminosity density Lν(ν = 144 MHz) (W Hz

−1)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

239 literature GRGs

Alcyoneus

Fig. 6: Alcyoneus has a low-frequency luminosity density

typical for GRGs. We explore the relation between GRG pro-

jected proper length lp and total luminosity density Lν at rest-

frame frequency ν = 144 MHz. Total luminosity densities in-

clude contributions from all available radio galaxy components

(i.e. the core, jets, hotspots and lobes). Literature GRGs are from

Dabhade et al. (2020b), and are marked with grey disks, while

Alcyoneus is marked with a green star. Translucent ellipses in-

dicate -1 to +1 standard deviation uncertainties. Alcyoneus has

a typical luminosity density (percentile 45 ± 3%).

Table 1: Luminosity densities Lν (in 1024 W Hz−1) of Alcyoneus’

lobes for three potential spectral indices α at rest-frame fre-

quency ν = 144 MHz, assuming a Planck Collaboration et al.

(2020) cosmology.

α = −0.8 α = −1.2 α = −1.6

Northern lobe

12 ± 1

13 ± 1

14 ± 1

Southern lobe

8.3 ± 0.8

9.0 ± 0.9

9.9 ± 1

Assuming α = −1.2, Alcyoneus total luminosity density at

ν = 144 MHz is Lν = 7.8 ± 0.8 · 1025 W Hz−1. In Figure 6, we

compare this estimate to other GRGs’ total luminosity density at

the same frequency, as found by Dabhade et al. (2020b) through

the LoTSS DR1 (Shimwell et al. 2019). Interestingly, Alcyoneus

is not particularly luminous: it has a low-frequency luminosity

density typical for the currently known GRG population (per-

centile 45 ± 3%).

3.6. True proper length: relativistic beaming

Following Hardcastle et al. (1998a), we simultaneously con-

strain Alcyoneus’ jet speed u and inclination angle θ from the

jets’ ux density asymmetry: the northern-to-southern jet ux

density ratio J = 1.78 ± 0.3.5 We assume that the jets prop-

agate with identical speeds u in exactly opposing directions

(making angles with the line-of-sight θ and θ + 180°), and have

statistically identical relativistic electron populations, so that

they have a common synchrotron spectral index α. Using α =

−0.65 ± 0.1 as before, and

β B

u

c

; β cos θ =

J

1

2−α − 1

J

1

2−α + 1

,

(1)

5 Because J is obtained through division of two independent normal

random variables (RVs) with non-zero mean, J is an RV with an un-

correlated noncentral normal ratio distribution.

Article number, page 6 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

we nd β cos θ = 0.106 ± 0.03. Because cos θ ≤ 1, β is bounded

from below by βmin = 0.106 ± 0.03.

From detailed modelling of ten Fanaro–Riley (FR) I radio

galaxies (which have jet luminosities comparable to Alcy-

oneus’), Laing & Bridle (2014) deduced that initial jet speeds are

roughly β = 0.8, which decrease until roughly 0.6 r0, with r0 be-

ing the recollimation distance. Most of Laing & Bridle (2014)’s

ten recollimation distances are between 5 and 15 kpc, with the

largest being that of NGC 315: r0 = 35 kpc. Because the lo-

cal specic intensity maxima in Alcyoneus’ jets closest to the

host occur at projected proper distances of 36.8 ± 0.8 kpc and

35.2±0.8 kpc, the true proper distances must be even larger. We

conclude that the observed jet emission presumably comes from

a region further from the host than r0, so that the initial stage

of jet deceleration — in which the jet speed is typically reduced

by several tens of percents of c — must already be completed.

Thus, βmax = 0.8 is a safe upper bound.

Taking βmax = 0.8, θ is bounded from above by θmax = 82.4± 2°

(θ ∈ [0, 90°]), or bounded from below by 180° − θmax = 97.6 ±

2° (θ ∈ [90°, 180°]).6 If we model Alcyoneus’ geometry as a

line segment, and assume no jet reorientation, Alcyoneus’ true

proper length l and projected proper length lp relate as

l =

lp

sin θ

;

l ≥ lmin =

lp

sin θmax

.

(2)

We bound l from below: lmin = 5.04 ± 0.05 Mpc. A triangu-

lar prior on β between βmin and βmax with the mode at βmax

induces a skewed prior on l; the 90% credible interval is l ∈

[5.0 Mpc, 5.5 Mpc], with the mean and median being 5.2 Mpc

and 5.1 Mpc, respectively. A at prior on β between βmin and

βmax also induces a skewed prior on l; the 90% credible inter-

val is l ∈ [5.0 Mpc, 7.1 Mpc], with the mean and median being

5.6 Mpc and 5.1 Mpc, respectively. The median of l seems par-

ticularly well determined, as it is insensitive to variations of the

prior on β.

In Appendix B, we explore the inclination angle conditions un-

der which Alcyoneus has the largest true proper length of all

known (> 4 Mpc) GRGs.

3.7. Stellar and supermassive black hole mass

Does a galaxy or its central black hole need to be massive in

order to generate a GRG?

Alcyoneus’ host has a stellar mass M? = 2.4 ± 0.4 · 1011 M

(Chang et al. 2015). We test whether or not this is a typical stel-

lar mass among the total known GRG population. We assem-

ble a literature catalogue of 1013 GRGs by merging the com-

pendium of Dabhade et al. (2020a), which is complete up to

April 2020, with the GRGs discovered in Galvin et al. (2020),

Ishwara-Chandra et al. (2020), Tang et al. (2020), Bassani et al.

(2021), Brüggen et al. (2021), Delhaize et al. (2021), Masini et al.

(2021), Kuźmicz & Jamrozy (2021), Andernach et al. (2021) and

Mahato et al. (2021). We collect stellar masses with uncertainties

from Chang et al. (2015), which are based on SDSS and WISE

photometry, and from Salim et al. (2018), which are based on

GALEX, SDSS and WISE photometry. We give precedence to

the stellar masses by Salim et al. (2018) when both are avail-

able. We obtain stellar masses for 151 previously known GRGs.

The typical stellar mass range is 1011 – 1012 M, the median

6 Taking βmax = 1 instead, θ is bounded from above by θmax = 83.9±2°

(θ ∈ [0, 90°]), or bounded from below by 180° − θmax = 96.1 ± 2° (θ ∈

[90°, 180°]).

1011

1012

stellar mass M? (M)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

151 literature GRGs

Alcyoneus

106

107

108

109

1010

1011

supermassive black hole mass M• (M)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

189 literature GRGs

Alcyoneus

Fig. 7: Alcyoneus’ host has a lower stellar and supermas-

sive black hole mass than most GRG hosts. We explore

the relations between GRG projected proper length lp and host

galaxy stellar mass M? (top panel) or host galaxy supermas-

sive black hole mass M• (bottom panel). Our methods allow de-

termining these properties for a small proportion of all litera-

ture GRGs only. Literature GRGs are marked with grey disks,

while Alcyoneus is marked with a green star. Translucent el-

lipses indicate -1 to +1 standard deviation uncertainties. Alcy-

oneus’ host has a fairly typical — though below-average — stel-

lar mass (percentile 25±9%) and supermassive black hole mass

(percentile 23 ± 11%).

M? = 3.5 · 1011 M and the mean M? = 3.8 · 1011 M. Strik-

ingly, the top panel of Figure 7 illustrates that Alcyoneus’ host

has a fairly low (percentile 25±9%) stellar mass compared with

the currently known population of GRG hosts.

For the GRGs in our literature catalogue, we also estimate

SMBH masses via the M-sigma relation. We collect SDSS DR12

stellar velocity dispersions with uncertainties (Alam et al. 2015),

and apply the M-sigma relation of Equation 7 in Kormendy

& Ho (2013). Alcyoneus’ host has a SMBH mass M• = 3.9 ±

1.7 ·108 M. We obtain SMBH masses for 189 previously known

GRGs. The typical SMBH mass range is 108 – 1010 M, the me-

dian M• = 7.9 · 108 M and the mean M• = 1.5 · 109 M.

Strikingly, the bottom panel of Figure 7 illustrates that Alcy-

oneus’ host has a fairly low (percentile 23 ± 11%) SMBH mass

compared with the currently known population of GRG hosts.

We note that Alcyoneus is the only GRG with lp > 3Mpc whose

host’s stellar mass is known through Chang et al. (2015) or Salim

et al. (2018), and whose host’s SMBH mass can be estimated

Article number, page 7 of 18

A&A proofs: manuscript no. aanda

through its SDSS DR12 velocity dispersion. These data allow

us to state condently that exceptionally high stellar or SMBH

masses are not necessary to generate 5-Mpc–scale GRGs.

3.8. Surrounding large-scale structure

Several approaches to large-scale structure (LSS) classication,

such as the T-web scheme (Hahn et al. 2007), partition the mod-

ern Universe into galaxy clusters, laments, sheets and voids. In

this section, we determine Alcyoneus’ most likely environment

type.

We conduct a tentative quantitative analysis using the SDSS

DR7 spectroscopic galaxy sample (Abazajian et al. 2009). Does

Alcyoneus’ host have fewer, about equal or more galactic neigh-

bours in SDSS DR7 than a randomly drawn galaxy of similar

r-band luminosity density and redshift? Let r (z) be the comov-

ing radial distance corresponding to cosmological redshift z. We

consider a spherical shell with the observer at the centre, inner

radius max {r(z = zspec) − r0, 0} and outer radius r(z = zspec)+r0.

We approximate Alcyoneus’ cosmological redshift with zspec

and choose r0 = 25 Mpc. As all galaxies in the spherical shell

have a similar distance to the observer (i.e. distances are at most

2r0 dierent), the SDSS DR7 galaxy number density complete-

ness must also be similar throughout the spherical shell.7 For

each enclosed galaxy with an r-band luminosity density be-

tween 1 − δ and 1 + δ times that of Alcyoneus’ host, we count

the number of SDSS DR7 galaxies Ncomoving radius R around it — regardless of luminosity den-

sity, and excluding itself. Alcyoneus’ host has an SDSS r-band

apparent magnitude mr = 18.20; the corresponding luminosity

density is Lν (λc = 623.1 nm) = 3.75 · 1022 W Hz−1. We choose

δ = 0.25; this yields 9,358 such enclosed galaxies.

In Figure 8, we show the distribution of N5 Mpc and R = 10 Mpc. We verify that the distributions

are insensitive to reasonable changes in r0 and δ. Note that

there is no SDSS DR7 galaxy within a comoving distance

of 5 Mpc from Alcyoneus’ host. The nearest such galaxy,

J081323.49+524856.1, occurs at a comoving distance of 7.9 Mpc:

the nearest ∼2, 000 Mpc3 of comoving space are free of galactic

neighbours with Lν (λc) > 5.57 ·1022 W Hz−1.8 In the same way

as in Section 3.1, we verify that the DESI Legacy Imaging Sur-

veys DR9, RASS and PSZ2 do not contain evidence for a galaxy

cluster in the direction of Alcyoneus’ host. The nearest galaxy

cluster, according to the SDSS-III cluster catalogue of Wen et al.

(2012), instead lies 24′ away at right ascension 123.19926°, dec-

lination 52.72468° and photometric redshift zph = 0.2488. It has

an R200 = 1.1 Mpc and, according to the DESI cluster catalogue

of Zou et al. (2021), a total mass M = 2.2·1014 M. The comoving

distance between the cluster and Alcyoneus’ host is 11Mpc. All

in all, we conclude that Alcyoneus does not reside in a galaxy

cluster. Meanwhile, there are ve SDSS DR7 galaxies within a

comoving distance of 10 Mpc from Alcyoneus’ host: this makes

it implausible that Alcyoneus lies in a void. Finally, one could

interpret Naround a galaxy. For R = 10 Mpc, just 17% of galaxies in the

shell with a similar luminosity density as Alcyoneus’ host have

a higher LSS total matter density. Being on the high end of the

7 For r0 = 25 Mpc, this is a good approximation, because the shell is

cosmologically thin: 2r0 = 50 Mpc roughly amounts to the length of a

single Cosmic Web lament.

8 This is the luminosity density that corresponds to the SDSS r-band

apparent magnitude completeness limit mr = 17.77 (Strauss et al.

2002).

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

№ SDSS DR7 galaxies within some comoving distance N0.0

0.1

0.2

0.3

0.4

0.5

probability(1)Alcyoneus’ host

similarly luminous SDSS DR7 galaxies in shell

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

№ SDSS DR7 galaxies within some comoving distance N0.00

0.05

0.10

0.15

0.20

probability(1)Alcyoneus’ host

similarly luminous SDSS DR7 galaxies in shell

Fig. 8: Like most galaxies of similar r-band luminosity

density and redshift, Alcyoneus’ host has no galactic

neighbours in SDSS DR7 within 5 Mpc. However, within

10 Mpc, Alcyoneus’ host has more neighbours than most

similar galaxies. For all 9,358 SDSS DR7 galaxies with an r-

band luminosity density between 75% and 125% that of Al-

cyoneus’ host and a comoving radial distance that diers at

most r0 = 25 Mpc from Alcyoneus’, we count the number of

SDSS DR7 galaxies NR = 5 Mpc (top panel) and R = 10 Mpc (bottom panel). The top

panel indicates that Alcyoneus does not inhabit a galaxy clus-

ter; the bottom panel indicates that Alcyoneus does not inhabit

a void.

density distribution, but lying outside a cluster, Alcyoneus most

probably inhabits a lament of the Cosmic Web.

3.9. Proper lobe volumes

We determine the proper volumes of Alcyoneus’ lobes with a

new Bayesian model. The model describes the lobes through a

pair of doubly truncated, optically thin cones, each of which

has a spatially constant and isotropic monochromatic emis-

sion coecient (MEC) (Rybicki & Lightman 1986). We allow

the 3D orientations and opening angles of the cones to dier,

as the lobes may traverse their way through dierently pres-

sured parts of the warm–hot intergalactic medium (WHIM):

e.g. the medium near the lament axis, and the medium near

the surrounding voids. By adopting a spatially constant MEC,

we neglect electron density and magnetic eld inhomogeneities

as well as spectral-ageing gradients; by adopting an isotropic

MEC, we assume non-relativistic velocities within the lobe so

that beaming eects are negligible. Numerically, we rst gener-

ate the GRG’s 3D MEC eld over a cubical voxel grid, and then

calculate the corresponding model image through projection,

including expansion-related cosmological eects. Before com-

parison with the observed image, we convolve the model image

with a Gaussian kernel to the appropriate resolution. We exploit

the approximately Gaussian LoTSS DR2 image noise to formu-

Article number, page 8 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

late the likelihood, and assume a at prior distribution over the

parameters. Using a Metropolis–Hastings (MH) Markov chain

Monte Carlo (MCMC), we sample from the posterior distribu-

tion.9

In the top panel of Figure 9, we show the LoTSS DR2 compact-

source-subtracted 90′′ image of Alcyoneus. The central region

has been excluded from source subtraction, and hence Alcy-

oneus’ core and jets remain. (However, when we run our MH

MCMC on this image, we do mask this central region.) In the

middle panel, we show the highest-likelihood (and thus max-

imum a posteriori (MAP)) model image before convolution. In

the bottom panel, we show the same model image convolved

to 90′′ resolution, with 2σ and 3σ contours of the observed im-

age overlaid. We provide the full parameter set that corresponds

with this model in Table C.1.

The posterior mean, calculated through the MH MCMC sam-

ples after burn-in, suggests the following geometry. The north-

ern lobe has an opening angle γ1 = 10 ± 1°, and the cone

truncates at an inner distance di,1 = 2.6 ± 0.2 Mpc and at an

outer distance do,1 = 4.0 ± 0.2 Mpc from the host galaxy. The

southern lobe has a larger opening angle γ2 = 26 ± 2°, but its

cone truncates at smaller distances of di,2 = 1.5 ± 0.1 Mpc and

do,2 = 2.0± 0.1 Mpc from the host galaxy. These parameters x

the proper volumes of Alcyoneus’ northern and southern lobes.

We nd V1 = 1.5 ± 0.2 Mpc3 and V2 = 1.0 ± 0.2 Mpc3, respec-

tively (see Equation C.15).10

How are the lobes oriented? Figure 1 provides a visual hint

that the lobes are subtly non-coaxial. The posterior indicates

that the position angles of the northern and southern lobes are

ϕ1 = 307±1° and ϕ2 = 139±2°, respectively. The position angle

dierence is thus ∆ϕ = 168±2°: although close to ∆ϕ = 180°, we

can reject coaxiality with high signicance. Interestingly, the

posterior also constrains the angles that the lobe axes make with

the plane of the sky: |θ1 − 90°| = 51± 2° and |θ2 − 90°| = 18± 7°.

Again, the uncertainties imply that the lobes are probably not

coaxial. We stress that these inclination angle results are tenta-

tive only. Future model extensions should explore how sensitive

they are to the assumed lobe geometry (by testing other shapes

than just truncated cones, such as ellipsoids).

One way to validate the model is to compare the observed

lobe ux densities of Section 3.5 to the predicted lobe ux

densities. According to the posterior, the MECs of the north-

ern and southern lobes are jν,1 = 17 ± 2 Jy deg−2 Mpc−1 and

jν,2 = 18 ± 3 Jy deg−2 Mpc−1. Combining MECs and volumes,

we predict northern and southern lobe ux densities Fν,1(νc) =

63 ± 4 mJy and Fν,2(νc) = 45 ± 5 mJy (see Equation C.16).

We nd excellent agreement: the relative dierences with the

observed results are 0% and 2%, respectively.

9 For a detailed description of the model parameters, the MH MCMC

and formulae for derived quantities, see Appendix C.

10 As a sanity check, we compare our results to those from a less rig-

orous, though simpler ellipsoid-based method of estimating volumes.

By tting ellipses to Figure 9’s top panel image, one obtains a semi-

minor and semi-major axis; the half-diameter along the ellipsoid’s third

dimension is assumed to be their mean. This method suggests a north-

ern lobe volume V1 = 1.4 ± 0.3 Mpc3 and a southern lobe volume

V2 = 1.1 ± 0.3 Mpc3. These results agree well with our Bayesian

model results. (If the half-diameter along the third dimension is instead

treated as an RV with a uniform distribution between the semi-minor

axis and the semi-major axis, the estimates remain the same.)

123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)Fig. 9: Alcyoneus’ lobe volumes can be estimated by com-

paring the observed radio image to modelled radio im-

ages. Top: LoTSS DR2 compact-source-subtracted 90′′ image of

Alcyoneus. For scale, we show the stellar Milky Way disk (di-

ameter: 50 kpc) and a 10 times inated version; the spiral galaxy

shape follows Ringermacher & Mead (2009). Middle: Highest-

likelihood model image. Bottom: The same model image con-

volved to 90′′ resolution, with 2σ and 3σ contours of the ob-

served image overlaid.

Article number, page 9 of 18

A&A proofs: manuscript no. aanda

3.10. Lobe pressures and the local WHIM

From Alcyoneus’ lobe ux densities and volumes, we can in-

fer lobe pressures and magnetic eld strengths. We calculate

these through pysynch11 (Hardcastle et al. 1998b), which uses

the formulae rst proposed by Myers & Spangler (1985) and re-

examined by Beck & Krause (2005). Following the notation of

Hardcastle et al. (1998b), we assume that the electron energy

distribution is a power law in Lorentz factor γ with γmin = 10,

γmax = 104 and exponent p = −2; we also assume that the

kinetic energy density of protons is vanishingly small com-

pared with that of electrons (κ = 0), and that the plasma ll-

ing factor is unity (φ = 1). Assuming the minimum-energy

condition (Burbidge 1956), we nd minimum-energy pressures

Pmin,1 = 4.8±0.3·10−16 Pa and Pmin,2 = 4.9±0.6·10−16 Pa for the

northern and southern lobes, respectively. The corresponding

minimum-energy magnetic eld strengths are Bmin,1 = 46±1 pT

and Bmin,2 = 46 ± 3 pT. Assuming the equipartition condi-

tion (Pacholczyk 1970), we nd equipartition pressures Peq,1 =

4.9 ± 0.3 · 10−16 Pa and Peq,2 = 4.9 ± 0.6 · 10−16 Pa for the

northern and southern lobes, respectively. The corresponding

equipartition magnetic eld strengths are Beq,1 = 43± 2 pT and

Beq,2 = 43 ± 2 pT. The minimum-energy and equipartition re-

sults do not dier signicantly.

From pressures and volumes, we estimate the internal energy

of the lobes E = 3PV . We nd Emin,1 = 6.2 ± 0.5 · 1052 J,

Emin,2 = 4.3 ± 0.6 · 1052 J, Eeq,1 = 6.3 ± 0.5 · 1052 J and Eeq,2 =

4.4± 0.6 · 1052 J. Next, we can bound the ages of the lobes from

below by neglecting synchrotron losses, and assuming that the

jets have been injecting energy in the lobes continuously at the

currently observed kinetic jet powers. Using ∆t = EQ−1

jet , we

nd ∆tmin,1 = 1.7 ± 0.2 Gyr, ∆tmin,2 = 2.1 ± 0.4 Gyr, and identi-

cal results when assuming the equipartition condition. Finally,

we can obtain a rough estimate of the average expansion speed

of the radio galaxy during its lifetime u = lp(∆t)−1. We nd

u = 2.6 ± 0.3 · 103 km s−1, or about 1% of the speed of light.

Several other authors (Andernach et al. 1992; Lacy et al. 1993;

Subrahmanyan et al. 1996; Parma et al. 1996; Mack et al. 1998;

Schoenmakers et al. 1998, 2000; Ishwara-Chandra & Saikia 1999;

Lara et al. 2000; Machalski & Jamrozy 2000; Machalski et al.

2001; Saripalli et al. 2002; Jamrozy et al. 2005; Subrahmanyan

et al. 2006, 2008; Saikia et al. 2006; Machalski et al. 2006, 2007,

2008; Safouris et al. 2009; Malarecki et al. 2013; Tamhane et al.

2015; Sebastian et al. 2018; Heesen et al. 2018; Cantwell et al.

2020) have estimated the minimum-energy or equipartition

pressure of the lobes of GRGs embedded in non-cluster envi-

ronments (i.e. in voids, sheets or laments of the Cosmic Web).

We compare Alcyoneus to the other 151 GRGs with known lobe

pressures in the top panel of Figure 10.12 Alcyoneus rearms

the negative correlation between length and lobe pressure (Jam-

rozy & Machalski 2002; Machalski & Jamrozy 2006), and has the

lowest lobe pressures found thus far. Alcyoneus’ lobe pressure

is in fact so low, that it is comparable to the pressure in dense

and hot parts of the WHIM: for a baryonic matter (BM) density

11 The pysynch code is publicly available online: https://github.

com/mhardcastle/pysynch.

12 We have included all publications that provide pressures, energy

densities or magnetic eld strengths. Note that some authors assume

γmin = 1, we assume γmin = 10 and Malarecki et al. (2013) assume

γmin = 103. If possible, angular lengths were updated using the LoTSS

DR2 at 6′′ and redshift estimates were updated using the SDSS DR12.

All projected proper lengths have been recalculated using our Planck

Collaboration et al. (2020) cosmology. When authors provided pres-

sures for both lobes, we have taken the average.

0.7

1.0

2.0

3.0

4.0

5.0

projected proper length lp (Mpc)

10−16

10−15

10−14

10−13

10−12

lobepressureP(Pa)Alcyoneus

151 literature GRGs

Alcyoneus

10−1

100

101

102

baryonic matter density ρBM (ρc,0 ΩBM,0)

10−19

10−18

10−17

10−16

10−15

10−14

pressureP(Pa)GRG B2147+816

GRG 3C 236

GRG J1420-0545, GRG J0331-7710

GRG Alcyoneus

ideal gas at T = 1 · 107 K

ideal gas at T = 5 · 106 K

ideal gas at T = 1 · 106 K

ideal gas at T = 5 · 105 K

Fig. 10: Of all GRGswith known lobe pressures, Alcyoneus

is the most plausible candidate for pressure equilibrium

with the WHIM. In the top panel, we explore the relation be-

tween length and lobe pressure for Alcyoneus and 151 literature

GRGs. In the bottom panel, we compare the lobe pressure of Al-

cyoneus (green line) with the lobe pressures of the largest four

similarly analysed GRGs (grey lines) and with WHIM pressures

(red lines).

ρWHIM = 10 ρc,0ΩBM,0 and TWHIM = 107 K, PWHIM = 4·10−16 Pa.

Here, ρc,0 is today’s critical density, so that ρc,0ΩBM,0 is today’s

mean baryon density. See the bottom panel of Figure 10 for

a more extensive comparison between Pmin (green line) and

PWHIM (red lines). For comparison, we also show the lobe pres-

sures of the four other thus-analysed GRGs with lp > 3 Mpc

(grey lines). These are J1420-0545 of lp = 4.9 Mpc (Machal-

ski et al. 2008), 3C 236 of lp = 4.7 Mpc (Schoenmakers et al.

2000), J0331-7710 of lp = 3.4 Mpc (Malarecki et al. 2013) and

B2147+816 of lp = 3.1 Mpc (Schoenmakers et al. 2000).

Although proposed as probes of WHIM thermodynamics for

decades, the bottom panel of Figure 10 demonstrates that even

the largest non-cluster literature GRGs are unlikely to be in

pressure equilibrium with their environment. Relying on results

from the Overwhelmingly Large Simulations (OWLS) (Schaye

et al. 2010), Malarecki et al. (2013) point out that baryon den-

sities ρBM > 50 ρc,0 ΩBM,0, which are necessary for pressure

equilibrium in these GRGs (see the intersection of grey and red

lines in the bottom panel of Figure 10), occur in only 1% of

the WHIM’s volume. By contrast, Alcyoneus can be in pres-

sure equilibrium with the WHIM at baryon densities ρBM ∼

20 ρc,0 ΩBM,0, and thus represents the most promising inter-

galactic barometer of its kind yet.13

13 At Alcyoneus’ redshift, this density amounts to a baryon overden-

sity of ∼10.

Article number, page 10 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

Why do most, if not all, observed non-cluster GRGs have over-

pressured lobes? The top panel of Figure 10 suggests that GRGs

must grow to several Mpc to approach WHIM pressures in their

lobes, and such GRGs are rare. However, the primary reason is

the limited surface brightness sensitivity of all past and current

surveys. Alcyoneus’ lobes are visible in the LoTSS, but not in

the NRAO VLA Sky Survey (NVSS) (Condon et al. 1998) or in

the Westerbork Northern Sky Survey (WENSS) (Rengelink et al.

1997). Their pressure approaches that of the bulk of the WHIM

within an order of magnitude. Lobes with even lower pressure

must be less luminous or more voluminous, and thus will have

even lower surface brightness. It is therefore probable that most

GRG lobes that are in true pressure equilibrium with the WHIM

still lie hidden in the radio sky.

4. Conclusion

1. We reprocess the LoTSS DR2, the latest version of the LO-

FAR’s Northern Sky survey at 144 MHz, by subtracting an-

gularly compact sources and imaging at 60′′ and 90′′ reso-

lution. The resulting images (Oei et al. in prep.) allow us to

explore a new sensitivity regime for radio galaxy lobes, and

thus represent promising data to search for unknown GRGs

of large angular length. We present a sample in forthcoming

work.

2. We discover the rst 5 Mpc GRG, which we dub Alcyoneus.

The projected proper length is lp = 4.99 ± 0.04 Mpc, while

the true proper length is at least lmin = 5.04 ± 0.05 Mpc.

We condently associate the 20.8′ ± 0.15′ radio structure

to an elliptical galaxy with a jet-mode AGN detected in the

DESI Legacy Imaging Surveys DR9: the SDSS DR12 source

J081421.68+522410.0 at J2000 right ascension 123.590372°,

declination 52.402795° and spectroscopic redshift 0.24674±

6 · 10−5.

3. Alcyoneus has a total luminosity density at ν = 144 MHz

of Lν = 8± 1 · 1025 W Hz−1, which is typical for GRGs (per-

centile 45 ± 3%). Alcyoneus’ host has a fairly low stellar

mass and SMBH mass compared with other GRG hosts (per-

centiles 25±9% and 23±11%). This implies that — within the

GRG population — no strong positive correlation between

radio galaxy length and (instantaneous) low-frequency ra-

dio power, stellar mass or SMBH mass can exist.

4. The surrounding sky as imaged by the LoTSS, DESI Legacy

Imaging Surveys, RASS and PSZ suggests that Alcyoneus

does not inhabit a galaxy cluster. According to an SDSS-III

cluster catalogue, the nearest cluster occurs at a comoving

distance of 11 Mpc. A local galaxy number density count

suggests that Alcyoneus instead inhabits a lament of the

Cosmic Web. A low-density environment therefore remains

a possible explanation for Alcyoneus’ formidable size.

5. We develop a new Bayesian model that parametrises in

three dimensions a pair of arbitrarily oriented, optically

thin, doubly truncated conical radio galaxy lobes with con-

stant monochromatic emission coecient. We then gen-

erate the corresponding specic intensity function, taking

into account cosmic expansion, and compare it to data as-

suming Gaussian image noise. We use Metropolis–Hastings

Markov chain Monte Carlo to optimise the parameters,

and thus determine northern and southern lobe volumes of

1.5±0.2 Mpc3 and 1.0±0.2 Mpc3, respectively. In total, the

lobes have an internal energy of ∼1053 J, expelled from the

host galaxy over a Gyr-scale period. The lobe pressures are

4.8 ± 0.3 · 10−16 Pa and 4.9 ± 0.6 · 10−16 Pa, respectively;

these are the lowest measured in radio galaxies yet. Nev-

ertheless, the lobe pressures still exceed a large range of

plausible WHIM pressures. Most likely, the lobes are still

expanding — and Alcyoneus’ struggle for supremacy of the

Cosmos continues.

Acknowledgements. M.S.S.L. Oei warmly thanks Frits Sweijen for coding the

very useful https://github.com/tikk3r/legacystamps.

M.S.S.L. Oei, R.J. van Weeren and A. Botteon acknowledge support from the

VIDI research programme with project number 639.042.729, which is nanced

by The Netherlands Organisation for Scientic Research (NWO). M. Brüggen

acknowledges support from the Deutsche Forschungsgemeinschaft under Ger-

many’s Excellence Strategy — EXC 2121 ‘Quantum Universe’ — 390833306. W.L.

Williams acknowledges support from the CAS–NWO programme for radio as-

tronomy with project number 629.001.024, which is nanced by The Nether-

lands Organisation for Scientic Research (NWO).

The LOFAR is the Low-frequency Array designed and constructed by ASTRON.

It has observing, data processing, and data storage facilities in several coun-

tries, which are owned by various parties (each with their own funding sources),

and which are collectively operated by the ILT Foundation under a joint scien-

tic policy. The ILT resources have beneted 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 Sci-

ence and Higher Education, Poland; the Istituto Nazionale di Astrosica (INAF),

Italy.

The National Radio Astronomy Observatory is a facility of the National Science

Foundation operated under cooperative agreement by Associated Universities,

Inc. CIRADA is funded by a grant from the Canada Foundation for Innovation

2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario,

British Columbia, Alberta, Manitoba and Quebec.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the

Participating Institutions, the National Science Foundation, and the U.S. De-

partment of Energy Oce of Science. The SDSS-III web site is http://www.

sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium

for the Participating Institutions of the SDSS-III Collaboration including the

University of Arizona, the Brazilian Participation Group, Brookhaven National

Laboratory, Carnegie Mellon University, University of Florida, the French Par-

ticipation Group, the German Participation Group, Harvard University, the In-

stituto de Astrosica de Canarias, the Michigan State/Notre Dame/JINA Partic-

ipation Group, Johns Hopkins University, Lawrence Berkeley National Labora-

tory, Max Planck Institute for Astrophysics, Max Planck Institute for Extrater-

restrial Physics, New Mexico State University, New York University, Ohio State

University, Pennsylvania State University, University of Portsmouth, Princeton

University, the Spanish Participation Group, University of Tokyo, University of

Utah, Vanderbilt University, University of Virginia, University of Washington,

and Yale University.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been

made possible through contributions by the Institute for Astronomy, the Uni-

versity of Hawaii, the Pan-STARRS Project Oce, the Max-Planck Society and

its participating institutes, the Max Planck Institute for Astronomy, Heidel-

berg and the Max Planck Institute for Extraterrestrial Physics, Garching, The

Johns Hopkins University, Durham University, the University of Edinburgh, the

Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics,

the Las Cumbres Observatory Global Telescope Network Incorporated, the Na-

tional Central University of Taiwan, the Space Telescope Science Institute, the

National Aeronautics and Space Administration under Grant No. NNX08AR22G

issued through the Planetary Science Division of the NASA Science Mission Di-

rectorate, the National Science Foundation Grant No. AST-1238877, the Univer-

sity of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National

Laboratory, and the Gordon and Betty Moore Foundation.

This publication makes use of data products from the Wide-eld Infrared Sur-

vey Explorer, which is a joint project of the University of California, Los An-

geles, and the Jet Propulsion Laboratory/California Institute of Technology,

funded by the National Aeronautics and Space Administration.

The Legacy Surveys consist of three individual and complementary projects:

the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404;

PIs: David Schlegel and Arjun Dey), the Beijing–Arizona Sky Survey (BASS;

NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall

z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS,

BASS and MzLS together include data obtained, respectively, at the Blanco tele-

scope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok tele-

scope, Steward Observatory, University of Arizona; and the Mayall telescope,

Kitt Peak National Observatory, NOIRLab. The Legacy Surveys project is hon-

ored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt

Peak), a mountain with particular signicance to the Tohono O’odham Na-

tion. NOIRLab is operated by the Association of Universities for Research in

Article number, page 11 of 18

A&A proofs: manuscript no. aanda

Astronomy (AURA) under a cooperative agreement with the National Science

Foundation. This project used data obtained with the Dark Energy Camera

(DECam), which was constructed by the Dark Energy Survey (DES) collabo-

ration. Funding for the DES Projects has been provided by the U.S. Depart-

ment of Energy, the U.S. National Science Foundation, the Ministry of Science

and Education of Spain, the Science and Technology Facilities Council of the

United Kingdom, the Higher Education Funding Council for England, the Na-

tional Center for Supercomputing Applications at the University of Illinois at

Urbana-Champaign, the Kavli Institute of Cosmological Physics at the Univer-

sity of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio

State University, the Mitchell Institute for Fundamental Physics and Astron-

omy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao

Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao

Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Con-

selho Nacional de Desenvolvimento Cientico e Tecnologico and the Ministe-

rio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft

and the Collaborating Institutions in the Dark Energy Survey. The Collaborat-

ing Institutions are Argonne National Laboratory, the University of California

at Santa Cruz, the University of Cambridge, Centro de Investigaciones Ener-

geticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago,

University College London, the DES-Brazil Consortium, the University of Ed-

inburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi Na-

tional Accelerator Laboratory, the University of Illinois at Urbana-Champaign,

the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Fisica d’Altes En-

ergies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Uni-

versität München and the associated Excellence Cluster Universe, the Univer-

sity of Michigan, NSF’s NOIRLab, the University of Nottingham, the Ohio State

University, the University of Pennsylvania, the University of Portsmouth, SLAC

National Accelerator Laboratory, Stanford University, the University of Sussex,

and Texas A&M University. BASS is a key project of the Telescope Access Pro-

gram (TAP), which has been funded by the National Astronomical Observato-

ries of China, the Chinese Academy of Sciences (the Strategic Priority Research

Program “The Emergence of Cosmological Structures” Grant # XDB09000000),

and the Special Fund for Astronomy from the Ministry of Finance. The BASS

is also supported by the External Cooperation Program of Chinese Academy

of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Sci-

ence Foundation (Grant # 11433005). The Legacy Survey team makes use of data

products from the Near-Earth Object Wide-eld Infrared Survey Explorer (NE-

OWISE), which is a project of the Jet Propulsion Laboratory/California Institute

of Technology. NEOWISE is funded by the National Aeronautics and Space Ad-

ministration. The Legacy Surveys imaging of the DESI footprint is supported

by the Director, Oce of Science, Oce of High Energy Physics of the U.S. De-

partment of Energy under Contract No. DE-AC02-05CH1123, by the National

Energy Research Scientic Computing Center, a DOE Oce of Science User

Facility under the same contract; and by the U.S. National Science Foundation,

Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.

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Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

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Article number, page 13 of 18

A&A proofs: manuscript no. aanda

4.7

4.8

4.9

5.0

5.1

5.2

projected proper length lp (Mpc)

0

5

10

15

20

25

probabilitydensity(Mpc−1)h = 0.677 | ΩM,0 = 0.311 | ΩΛ,0 = 0.689

J1420-0545

Alcyoneus

Fig. A.1: Alcyoneus’ projected proper length just exceeds

that of J1420-0545. The probability that Alcyoneus (green) has

a larger projected proper length than J1420-0545 (grey) (Machal-

ski et al. 2008) is 99.9%. For both GRGs, we take into account

uncertainty in angular length and spectroscopic redshift, as well

as the possibility of peculiar motion along the line of sight.

Appendix A: J1420-0545 comparison

We verify that Alcyoneus is the largest known radio galaxy (RG)

in projection by comparing it with J1420-0545 (Machalski et al.

2008), the literature’s record holder.

The angular lengths of Alcyoneus and J1420-0545 are φ =

20.8′ ± 0.15′ and φ = 17.4′ ± 0.05′, respectively. For J1420-

0545, we adopt the angular length reported by Machalski et al.

(2008) because it lies outside the LoTSS DR2 coverage. The spec-

troscopic redshifts of Alcyoneus and J1420-0545 are zspec =

0.24674 ± 6 · 10−5 and zspec = 0.3067 ± 5 · 10−4, respectively.

For both giants, we assume the peculiar velocity along the line

of sight up to be a Gaussian random variable (RV) with mean

0 and standard deviation 100 km s−1, similar to conditions in

low-mass galaxy clusters.

Equations A.1 describe how to calculate the cosmological red-

shift RV z via the peculiar velocity redshift RV zp:

βp B

up

c

; zp =

√

1 + βp

1 − βp

− 1; z =

1 + zspec

1 + zp

− 1.

(A.1)

Here, c is the speed of light in vacuo. Finally, we calculate the

projected proper length RV lp = rφ (z,M) · φ. Here, rφ is the

angular diameter distance RV, which depends on cosmological

model parameters M. Propagating the uncertainties in angular

length φ, spectroscopic redshift zspec and peculiar velocity

along the line of sight up through Monte Carlo simulation,

the projected proper lengths of Alcyoneus and J1420-0545 are

lp = 4.99 ± 0.04 Mpc and lp = 4.87 ± 0.02 Mpc, respectively.

We show the two projected proper length distributions in

Figure A.1. The probability that Alcyoneus has the largest

projected proper length is 99.9%. This result is insensitive to

plausible changes in cosmological parameters; for example,

the high-H0 (i.e. H0 > 70 km s−1 Mpc−1) cosmology with M =

(

h = 0.7020,ΩBM,0 = 0.0455,ΩM,0 = 0.2720,ΩΛ,0 = 0.7280

)

yields a probability of 99.8%.

Appendix B: Inclination angle comparison

Under what conditions is Alcyoneus not only the largest GRG

in the plane of the sky, but also in three dimensions? To an-

swer this question, we compare Alcyoneus to the ve previ-

ously known GRGs with projected proper lengths above 4 Mpc,

0

15

30

45

60

75

90

inclination angle Alcyoneus θ (°)

0

15

30

45

60

75

90

inclinationanglechallengerθc(°)θmax,c (θ) for five GRGs

lp,c = 4.87 Mpc

lp,c = 4.72 Mpc

lp,c = 4.60 Mpc

lp,c = 4.35 Mpc

lp,c = 4.11 Mpc

Fig. A.2: When is Alcyoneus not only the largest GRG in

the plane of the sky, but also in three dimensions? Alcy-

oneus’ inclination angle θ is not well determined, and there-

fore the full range of possibilities is shown on the horizontal

axis. To surpass Alcyoneus in true proper length, a challenger

must have an inclination angle (vertical axis) of at most Alcy-

oneus’ (grey dotted line). More specically, as a function of θ,

we show the maximum inclination angle for which challengers

with a projected proper length lp,c > 4 Mpc trump Alcyoneus

(coloured curves). The shaded areas of parameter space repre-

sent regimes with a particularly straightforward interpretation.

One can imagine populating the graph with ve points (located

along the same vertical line), representing the ground-truth in-

clination angles of Alcyoneus and its ve challengers. If any of

these points fall in the red-shaded area, Alcyoneus is not the

largest GRG in 3D. If all points fall in the green-shaded area,

Alcyoneus is the largest GRG in 3D.

which we dub challengers. A challenger surpasses Alcyoneus in

true proper length when

lc > l, or

lp,c

sin θc

>

lp

sin θ

, or sin θc <

lp,c

lp

sin θ,

(B.1)

where lc, lp,c and θc are the challenger’s true proper length,

projected proper length and inclination angle, respectively. Be-

cause the arcsine is a monotonically increasing function, a chal-

lenger surpasses Alcyoneus if its inclination angle obeys

θc < θmax,c (θ) , where θmax,c (θ) B arcsin

(

lp,c

lp

sin θ

)

.

(B.2)

In Figure A.2, we show θmax,c (θ) for the ve challengers with

lp,c ∈ {4.11 Mpc, 4.35 Mpc, 4.60 Mpc, 4.72 Mpc, 4.87 Mpc}

(coloured curves). Alcyoneus is least likely to be the longest

GRG in 3D when its true proper length equals its projected

proper length; i.e. when θ = 90°. The challengers then surpass

Alcyoneus in true proper length when their inclination angles

are less than 55°, 61°, 67°, 71° and 77°, respectively. For θ < 90°,

the conditions are more stringent.

Article number, page 14 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

The third and fourth largest challengers, whose respec-

tive SDSS DR12 host names are J100601.73+345410.5 and

J093139.03+320400.1, harbour quasars in their host galaxies. If

small inclination angles distinguish quasars from non-quasar

AGN, as proposed by the unication model (e.g. Hardcastle &

Croston 2020), these two challengers may well be the longest

radio galaxies in three dimensions.

Appendix C: Lobe volumes with truncated double

cone model

Appendix C.1: Synopsis

We build a Metropolis–Hastings Markov chain Monte Carlo

(MH MCMC) model, similar in spirit to the model of Boxelaar

et al. (2021) for galaxy cluster halos, in order to formalise the

determination of RG lobe volumes from a radio image. To this

end, we introduce a parametrisation of a pair of 3D radio galaxy

lobes, and explore the corresponding parameter space via the

Metropolis algorithm.14 For each parameter tuple encountered

during exploration, we rst calculate the monochromatic emis-

sion coecient (MEC) function of the lobes on a uniform 3D

grid representing a proper (rather than comoving) cubical vol-

ume. The RG is assumed to be far enough from the observer

that the conversion to a 2D image through ray tracing simpli-

es to summing up the cube’s voxels along one dimension, and

applying a cosmological attenuation factor. This factor depends

on the galaxy’s cosmological redshift, which is a hyperparame-

ter. We blur the model image to the resolution of the observed

image, which is also a hyperparameter. Next, we calculate the

likelihood that the observed image is a noisy version of the

proposed model image. The imaged sky region is divided into

patches with a solid angle equal to the PSF solid angle; the noise

per patch is then assumed to be an independent Gaussian RV.

These RVs have zero mean and share the same variance, which

is another hyperparameter — typically obtained from the ob-

served image. We choose a uniform prior over the full phys-

ically realisable part of parameter space. The resulting poste-

rior, which contains both geometric and radiative parameters,

allows one to calculate probability distributions for many inter-

esting quantities, such as the RG’s lobe volumes and inclination

angle. The inferences depend weakly on cosmological parame-

ters M. Furthermore, their reliability depends signicantly on

the validity of the model assumptions.

Appendix C.2: Model

Appendix C.2.1: Geometry

We model each lobe in 3D with a truncated right circular cone

with apex O ∈ R3, central axis unit vector â ∈ S2 and opening

angle γ ∈ [0, π2 ], as in Figure 9. The lobes share the same O,

which is the RG host location. Each central axis unit vector can

be parametrised through a position angle ϕ ∈ [0, 2π) and an in-

clination angle θ ∈ [0, π]. Each cone is truncated twice, through

planes that intersect the cone perpendicularly to its central axis.

Thus, each truncation is parametrised by the distance from the

apex to the point where the plane intersects the central axis.

The two inner (di,1, di,2 ∈ R≥0) and two outer (do,1, do,2 ∈ R≥0)

truncation distances are parameters that we allow to vary inde-

pendently, with the only constraint that each inner truncation

14 The more general Metropolis–Hastings variant need not be consid-

ered, as we work with a symmetric proposal distribution.

distance cannot exceed the corresponding outer truncation dis-

tance.

Appendix C.2.2: Radiative processes

The radiative formulation of our model is among the simplest

possible. The radio emission from the lobes is synchrotron ra-

diation. We consider the lobes to be perfectly optically thin:

we neglect synchrotron self-absorption. The proper MEC is as-

sumed spatially constant throughout a lobe, though possibly

dierent among lobes; this leads to parameters jν,1, jν,2 ∈ R≥0.

The relationship between the specic intensity Iν (in direction

r̂ at central frequency νc) and the MEC jν (in direction r̂ at cos-

mological redshift z and rest-frame frequency ν = νc (1 + z)) is

Iν (r̂, νc) =

∫ ∞

0

jν (r̂, z (l) , νc (1 + z (l)))

(1 + z (l))3

dl ≈ jν (ν)∆l (r̂)

(1 + z)3

,

(C.1)

where l represents proper length. The approximation is valid

for a lobe with a spatially constant MEC that is small enough

to assume a constant redshift for it. ∆l(r̂) is the proper length

of the line of sight through the lobe in direction r̂. The inferred

MECs jν,1 (ν) , jν,2 (ν) thus correspond to rest-frame frequency ν.

Appendix C.3: Proposal distribution

In order to explore the posterior distribution on the parameter

space, we follow the Metropolis algorithm. The Metropolis al-

gorithm assumes a symmetric proposal distribution.

Appendix C.3.1: Radio galaxy axis direction

To propose a new RG axis direction given the current one whilst

satisfying the symmetry assumption, we perform a trick. We

populate the unit sphere with N ∈ N≥1 points (interpreted

as directions) drawn from a uniform distribution. Of these N

directions, the proposed axis direction is taken to be the one

closest to the current axis direction (in the great-circle distance

sense). Note that this approach evidently satises the criterion

that proposing the new direction given the old one is equally

likely as proposing the old direction given the new one. Also

note that the distribution of the angular distance between cur-

rent and proposed axis directions is determined solely by N.

In the following paragraphs, we rst review how to perform

uniform sampling of the unit two-sphere. More explicitly than

in Scott & Tout (1989), we then derive the distribution of the

angular distance between a reference point and the nearest of

N uniformly drawn other points. The result is a continuous uni-

variate distribution with a single parameter N and nite support

(0, π). Finally, we present the mode, median and maximum like-

lihood estimator of N. As far as we know, these properties are

new to the literature.

Uniform sampling of S2 Let us place a number of points

uniformly on the celestial sphere S2. The spherical coordinates

of such points are given by the RVs (Φ,Θ), where Φ denotes

position angle and Θ denotes inclination angle. As all posi-

tion angles are equally likely, the distribution of Φ is uniform:

Φ ∼ U[0, 2π). In order to eect a uniform number density, the

probability that a point lies within a rectangle of width dϕ and

height dθ in the (ϕ, θ)-plane equals the ratio of the solid angle of

Article number, page 15 of 18

A&A proofs: manuscript no. aanda

the corresponding sky patch and the sphere’s total solid angle:

P(ϕ ≤ Φ < ϕ + dϕ, θ ≤ Θ < θ + dθ) = sin θ dϕ dθ

4π

.

(C.2)

The probability that the inclination angle is found somewhere

in the interval [θ, θ + dθ), regardless of the position angle, is

therefore

P(θ ≤ Θ < θ + dθ) = dFΘ(θ) = fΘ(θ)dθ

=

∫ 2π

0

sin θ dθ

4π

dϕ =

1

2

sin θ dθ,

(C.3)

where FΘ is the cumulative distribution function (CDF) of Θ,

and fΘ the associated probability density function (PDF). So,

fΘ(θ) =

1

2

sin θ; FΘ(θ) B

∫ θ

0

fΘ(θ′) dθ′ =

1 − cos θ

2

.

(C.4)

Nearest-neighbour angular distance distribution Pick a

reference point and stochastically introduce N other points in

above fashion, which we dub its neighbours. We now derive the

PDF of the angular distance to the nearest neighbour (NNAD).

Let (ϕref , θref) be the coordinates of the reference point and let

(ϕ, θ) be the coordinates of one of the neighbours. Without

loss of generality, due to spherical symmetry, we can choose to

place the reference point in the direction towards the observer:

θref = 0. (Note that ϕref is meaningless in this case.) The angular

distance between two points on S2 is given by the great-circle

distance ξ. For our choice of reference point, we immediately

see that ξ(ϕref , θref , ϕ, θ) = θ. Because θ is a realisation of Θ, ξ

too can be regarded as a realisation of an RV, which we call Ξ.

Evidently, the PDF fΞ(ξ) = fΘ(ξ) and the CDF FΞ(ξ) = FΘ(ξ).

Now consider the generation of N points, whose angular dis-

tances to the reference point are the RVs {Ξi} B {Ξ1, ...,ΞN}.

The NNAD RV M is the minimum of this set: M B min{Ξi}.

What are the CDF FM and PDF fM of M?

FM(µ) B P(M ≤ µ) = P(minimum of {Ξi} ≤ µ)

= P(at least one of the set {Ξi} ≤ µ)

= 1 − P(none of the set {Ξi} ≤ µ)

= 1 − P(all of the set {Ξi} > µ).

(C.5)

Because the {Ξi} are independent and identically distributed,

FM(µ) = 1 −

N∏

i=1

P (Ξi > µ)

= 1 − PN(Ξ > µ) = 1 − (1 − FΞ(µ))N .

(C.6)

By substitution, the application of a trigonometric identity and

dierentiation to µ, we obtain the CDF and PDF of M:

FM (µ) = 1 − cos2N

(

µ

2

)

; fM(µ) = N sin

(

µ

2

)

cos2N−1

(

µ

2

)

.

(C.7)

In Figure C.1, we show this PDF for various values of N.

The mode of M (i.e. the most probable NNAD), µmode, is the solu-

tion to d fM

dµ (µmode) = 0. The median of M, µmedian, is the solution

to FM(µmedian) = 12 . Hence,

µmode = arccos

(

1 − 1

N

)

; µmedian = arccos

(

21−

1

N − 1

)

.

(C.8)

As common sense dictates, both equal π2 for N = 1 and tend to

0 as N → ∞. We nd the mean of M through integration by

parts:

E [M] B

∫ π

0

µ fM(µ) dµ =

∫ π

0

µ dFM(µ)

=

[

µFM(µ)

]π

0

−

∫ π

0

FM(µ) dµ

=

∫ π

0

cos2N

(

µ

2

)

dµ = 2

∫ π

2

0

cos2N (µ) dµ.

(C.9)

Again via integration by parts,

E [M] = π

N∏

k=1

2k − 1

2k

=

π

22N

(

2N

N

)

.

(C.10)

Maximum likelihood estimation A typical application is

the estimation of N in the PDF fM(µ | N) (Equation C.7) us-

ing data. Let us assume we have measured k NNADs, denoted

by {µ1, ..., µk}. Let the joint PDF or likelihood be

L(N) B

k∏

i=1

fM (µi | N)

=

( N

2N

)k

k∏

i=1

sin µi (cos µi + 1)N−1.

(C.11)

To nd NMLE, we look for the value of N that maximises L(N).

To simplify the algebra, we could however equally well max-

imise a k-th of the natural logarithm of the likelihood, or the

average log-likelihood l̂ B k−1 lnL(N), because the logarithm is

a monotonically increasing function:

l̂(N) B

1

k

lnL(N) = lnN − N ln 2

+

1

k

k∑

i=1

ln sin µi + (N − 1) ln(cos µi + 1).

(C.12)

We nd NMLE by solving dl̂

dN (NMLE) = 0. This leads to

NMLE =

ln 2 − 1k

k∑

i=1

ln(cos µi + 1)

−1

.

(C.13)

An easy limit to evaluate is the case when µ1, ..., µk → 0. In

such case, cos µi → 1, and so 1k

∑k

i=1 ln(cos µi + 1)→ ln 2. Then,

NMLE → (0+)−1 → ∞. This is expected behaviour: when all

measured NNADs approach 0, the number of points distributed

on the sphere must be approaching innity.

Appendix C.3.2: Other parameters

The other proposal parameters are each drawn from indepen-

dent normal distributions centred around the current parameter

values. These proposal distributions are evidently symmetric,

but have support over the full real line, so that forbidden pa-

rameter values can in principle be proposed. As a remedy, we set

the prior probability density of the proposed parameter set to 0

when the proposed opening angle is negative or exceeds π2 rad,

at least one of the proposed MECs is negative, or when at least

one of the proposed inner truncation distances is negative or

Article number, page 16 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

0

1

2

3

4

5

6

7

8

nearest-neighbour angular distance µ (°)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

probabilitydensityfM(µ)(deg−1)N = 1 · 103 | n = 0.024 deg−2

N = 2 · 103 | n = 0.048 deg−2

N = 3 · 103 | n = 0.073 deg−2

N = 4 · 103 | n = 0.097 deg−2

N = 5 · 103 | n = 0.121 deg−2

N = 6 · 103 | n = 0.145 deg−2

Fig. C.1: Probability density functions (PDFs) of the nearest-neighbour angular distance (NNAD) RV M between some xed point

and N other points distributed randomly over the celestial sphere. As the sphere gets more densely packed, the probability of

nding a small M increases. For each N, we provide the mean point number density n.

exceeds the corresponding proposed outer truncation distance.

In such cases, the posterior probability density is 0 too, as it is

proportional to the prior probability density. Consequently, the

Metropolis acceptance probability vanishes and the proposal is

rejected. We do not enter forbidden regions of parameter space.

The condition of detailed balance is still respected: probability

densities for transitioning towards the forbidden region are 0,

just as probability densities for being in the forbidden region.

Appendix C.4: Likelihood

We assume the likelihood to be Gaussian. To avoid dimension-

ality errors, we multiply the likelihood by a constant before we

take the logarithm:

ln

(

L ·

(

σ

√

2π

)Nr)

= − Nr

2σ2Np

Np∑

i=1

(

Iν,o [i] − Iν,m [i]

)2 .

(C.14)

Here, σ is the image noise, Nr ∈ R≥0 is the number of reso-

lution elements in the image, Np ∈ N is the number of pixels

in the image, and Iν,o [i] and Iν,m [i] are the i-th pixel values of

the observed and modelled image, respectively. For simplicity,

one may multiply the likelihood by a constant factor (or, equiva-

lently, add a constant term to the log-likelihood): the acceptance

ratio will remain the same, and the MH MCMC runs correctly.

Appendix C.5: Results for Alcyoneus

We apply the Bayesian model to the 90′′ LoTSS DR2 image of Al-

cyoneus, shown in the top panel of Figure 9. Thus, the hyperpa-

rameters are z = 0.24674, νc = 144MHz (so that ν = 180MHz),

θFWHM = 90′′, N = 750 and σ =

√

2 · 1.16 Jy deg−2. We set

the image noise to

√

2 times the true image noise to account

for model incompleteness. This factor follows by assuming that

the inability of the model to produce the true lobe morphol-

ogy yields (Gaussian) errors comparable to the image noise. To

speed up inference, we downsample the image of 2,048 by 2,048

pixels by a factor 16 along each dimension. We run our MH

MCMC for 10,000 steps, and discard the rst 1,500 steps due to

burn-in.

Table C.1 lists the obtained maximum a posteriori probabil-

ity (MAP) estimates and posterior mean and standard deviation

(SD) of the parameters.

Table C.1: Maximum a posteriori probability (MAP) estimates

and posterior mean and standard deviation (SD) of the param-

eters from the Bayesian, doubly truncated, conical radio galaxy

lobe model of Section 3.9.

parameter

MAP estimate

posterior mean and SD

ϕ1

307°

307 ± 1°

ϕ2

140°

139 ± 2°

|θ1 − 90°|

54°

51 ± 2°

|θ2 − 90°|

25°

18 ± 7°

γ1

9°

10 ± 1°

γ2

24°

26 ± 2°

di,1

2.7 Mpc

2.6 ± 0.2 Mpc

do,1

4.3 Mpc

4.0 ± 0.2 Mpc

di,2

1.6 Mpc

1.5 ± 0.1 Mpc

do,2

2.0 Mpc

2.0 ± 0.1 Mpc

jν,1 (ν)

17 Jy deg−2 Mpc−1

17 ± 2 Jy deg−2 Mpc−1

jν,2 (ν)

22 Jy deg−2 Mpc−1

18 ± 3 Jy deg−2 Mpc−1

The proper volumes V1 and V2 are derived quantities:

V =

π

3

tan2 γ

(

d3o − d3i

)

,

(C.15)

just like the ux densities Fν,1 (νc) and Fν,2 (νc) at central fre-

quency νc:

Fν (νc) =

jν (ν)V

(1 + z)3 r2φ (z)

.

(C.16)

Together, V and Fν(νc) imply a lobe pressure P and a magnetic

eld strength B, which are additional derived quantities that we

calculate through pysynch.

Table C.2 lists the obtained MAP estimates and posterior mean

and SD of the derived quantities.

Article number, page 17 of 18

A&A proofs: manuscript no. aanda

5

10

15

20

25

30

monochromatic emission coefficient jν (ν)

(

Jy deg−2 Mpc−1

)

0.5

1.0

1.5

2.0

2.5

3.0

properlobevolumeV(Mpc3)Fν (νc) = 63 mJy

Fν (νc) = 44 mJy

northern lobe

southern lobe

Fig. C.2: OurBayesianmodel yields strongly correlated es-

timates for jν (ν) and V that reproduce the observed lobe

ux densities. We show MECs jν (ν) at ν = 180 MHz and

proper volumes V of Metropolis–Hastings Markov chain Monte

Carlo samples for the northern lobe (purple dots) and south-

ern lobe (orange dots). The curves represent all combinations

( jν (ν) ,V) that correspond to a particular ux density at the

LoTSS central wavelength νc = 144 MHz. We show the ob-

served northern lobe ux density (purple curve) and the ob-

served southern lobe ux density (orange curve).

Table C.2: Maximum a posteriori probability (MAP) estimates

and posterior mean and standard deviation (SD) of derived

quantities from the Bayesian, doubly truncated, conical radio

galaxy lobe model of Section 3.9.

derived quantity MAP estimate

posterior mean and SD

∆ϕ

167°

168 ± 2°

V1

1.5 Mpc3

1.5 ± 0.2 Mpc3

V2

0.8 Mpc3

1.0 ± 0.2 Mpc3

Fν,1 (νc)

63 mJy

63 ± 4 mJy

Fν,2 (νc)

44 mJy

45 ± 5 mJy

Pmin,1

4.7 · 10−16 Pa

4.8 ± 0.3 · 10−16 Pa

Pmin,2

5.4 · 10−16 Pa

5.0 ± 0.6 · 10−16 Pa

Peq,1

4.8 · 10−16 Pa

4.9 ± 0.3 · 10−16 Pa

Peq,2

5.4 · 10−16 Pa

5.0 ± 0.6 · 10−16 Pa

Bmin,1

45 pT

45 ± 1 pT

Bmin,2

48 pT

46 ± 3 pT

Beq,1

42 pT

43 ± 1 pT

Beq,2

45 pT

43 ± 3 pT

Emin,1

6.3 · 1052 J

6.2 ± 0.4 · 1052 J

Emin,2

3.7 · 1052 J

4.4 ± 0.6 · 1052 J

Eeq,1

6.4 · 1052 J

6.3 ± 0.4 · 1052 J

Eeq,2

3.8 · 1052 J

4.4 ± 0.6 · 1052 J

The uncertainties of the parameters and derived quantities

reported in Tables C.1 and C.2 are not necessarily independent.

To demonstrate this, we present MECs and volumes from the

MH MCMC samples in Figure C.2. MECs and volumes do not

vary independently, because their product is proportional to

ux density (see Equation C.16); only realistic ux densities

correspond to high-likelihood model images.

Finally, we explore a simpler variation of the model, in

which we force the lobes to be coaxial. In such a case, the true

proper length l and projected proper length lp are additional

derived quantities:

l =

do,1 + do,2

cos γ

;

lp = l sin θ.

(C.17)

For Alcyoneus, this simpler model does not provide a good t

to the data.

Article number, page 18 of 18

©ESO 2022

February 14, 2022

The discovery of a radio galaxy of at least 5 Mpc

Martijn S.S.L. Oei1?, Reinout J. van Weeren1, Martin J. Hardcastle2, Andrea Botteon1, Tim W. Shimwell1, Pratik

Dabhade3, Aivin R.D.J.G.I.B. Gast4, Huub J.A. Röttgering1, Marcus Brüggen5, Cyril Tasse6, 7, Wendy L. Williams1, and

Aleksandar Shulevski1

1 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2300 RA Leiden, The Netherlands

e-mail: oei@strw.leidenuniv.nl

2 Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hateld AL10 9AB, United Kingdom

3 Observatoire de Paris, LERMA, Collège de France, CNRS, PSL University, Sorbonne University, 75014 Paris, France

4 Somerville College, University of Oxford, Woodstock Road, Oxford OX2 6HD, United Kingdom

5 Hamburger Sternwarte, University of Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany

6 GEPI & USN, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, 92190 Meudon, France

7 Department of Physics & Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa

February 14, 2022

ABSTRACT

Context. Giant radio galaxies (GRGs, or colloquially ‘giants’) are the Universe’s largest structures generated by individual galaxies.

They comprise synchrotron-radiating AGN ejecta and attain cosmological (Mpc-scale) lengths. However, the main mechanisms that

drive their exceptional growth remain poorly understood.

Aims. To deduce the main mechanisms that drive a phenomenon, it is usually instructive to study extreme examples. If there exist

host galaxy characteristics that are an important cause for GRG growth, then the hosts of the largest GRGs are likely to possess

them. Similarly, if there exist particular large-scale environments that are highly conducive to GRG growth, then the largest GRGs

are likely to reside in them. For these reasons, we aim to perform a case study of the largest GRG available.

Methods. We reprocessed the LOFAR Two-metre Sky Survey (LoTSS) DR2 by subtracting compact sources and performing multi-

scale CLEAN deconvolution at 60′′ and 90′′ resolution. The resulting images constitute the most sensitive survey yet for radio

galaxy lobes, whose diuse nature and steep synchrotron spectra have allowed them to evade previous detection attempts at higher

resolution and shorter wavelengths. We visually searched these images for GRGs.

Results. We discover Alcyoneus, a low-excitation radio galaxy with a projected proper length lp = 4.99 ± 0.04 Mpc. Its jets and

lobes are all four detected at very high signicance, and the SDSS-based identication of the host, at spectroscopic redshift zspec =

0.24674 ± 6 ·10−5, is unambiguous. The total luminosity density at ν = 144MHz is Lν = 8±1 ·1025 W Hz−1, which is below-average,

though near-median (percentile 45±3%), for GRGs. The host is an elliptical galaxy with a stellar mass M? = 2.4±0.4 ·1011 M and

a supermassive black hole mass M• = 4±2 ·108 M, both of which tend towards the lower end of their respective GRG distributions

(percentiles 25±9% and 23±11%). The host resides in a lament of the Cosmic Web. Through a new Bayesian model for radio galaxy

lobes in three dimensions, we estimate the pressures in the Mpc3-scale northern and southern lobe to be Pmin,1 = 4.8±0.3 ·10−16 Pa

and Pmin,2 = 4.9±0.6·10−16 Pa, respectively. The corresponding magnetic eld strengths are Bmin,1 = 46±1 pT and Bmin,2 = 46±3 pT.

Conclusions. We have discovered what is in projection the largest known structure made by a single galaxy — a GRG with a projected

proper length lp = 4.99 ± 0.04 Mpc. The true proper length is at least lmin = 5.04 ± 0.05 Mpc. Beyond geometry, Alcyoneus and

its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass and supermassive black hole mass are

all lower than, though similar to, those of the medial GRG. Thus, very massive galaxies or central black holes are not necessary

to grow large giants, and, if the observed state is representative of the source over its lifetime, neither is high radio power. A low-

density environment remains a possible explanation. The source resides in a lament of the Cosmic Web, with which it might have

signicant thermodynamic interaction. The pressures in the lobes are the lowest hitherto found, and Alcyoneus therefore represents

the most promising radio galaxy yet to probe the warm–hot intergalactic medium.

Key words. galaxies: active – galaxies: individual: Alcyoneus – galaxies: jets – intergalactic medium – radio continuum: galaxies

1. Introduction

Most galactic bulges hold a supermassive (i.e. M• > 106 M)

Kerr black hole (e.g. Soltan 1982) that grows by accreting gas,

dust and stars from its surroundings (Kormendy & Ho 2013).

The black hole ejects a fraction of its accretion disk plasma

from the host galaxy along two collimated, magnetised jets

that are aligned with its rotation axis (e.g. Blandford & Rees

? In dear memory of Pallas. If your name hadn’t been this popular with

asteroid discoverers, you’d now be the giants’ giant — once again looking

down at the sprawling ants below.

1974). The relativistic electrons contained herein experience

Lorentz force and generate, through spiral motion, synchrotron

radiation that is observed by radio telescopes. The two jets

either fade gradually or end in hotspots at the end of diuse

lobes, and ultimately enrich the intergalactic medium (IGM)

with cosmic rays and magnetic elds. The full luminous

structure is referred to as a radio galaxy (RG). Members of a

rare RG subpopulation attain megaparsec-scale proper (and

thus comoving) lengths (e.g. Willis et al. 1974; Andernach

et al. 1992; Ishwara-Chandra & Saikia 1999; Jamrozy et al.

2008; Machalski 2011; Kuźmicz et al. 2018; Dabhade et al.

Article number, page 1 of 18

arXiv:2202.05427v1 [astro-ph.GA] 11 Feb 2022

A&A proofs: manuscript no. aanda

Fig. 1: Joint radio-infrared view of Alcyoneus, a radio galaxy with a projected proper length of 5.0 Mpc. We show a

2048′′ × 2048′′ solid angle centred around right ascension 123.590372° and declination 52.402795°. We superimpose LOFAR

Two-metre Sky Survey (LoTSS) DR2 images at 144 MHz of two dierent resolutions (6′′ for the core and jets, and 60′′ for the

lobes) (orange), with the Wide-eld Infrared Survey Explorer (WISE) image at 3.4 µm (blue). To highlight the radio emission, the

infrared emission has been blurred to 0.5′ resolution.

2020a). The giant radio galaxy (GRG, or colloquially ‘giant’)

denition accommodates our limited ability to infer an RG’s

true proper length from observations: an RG is called a GRG

if and only if its proper length projected onto the plane of the

sky exceeds some threshold lp,GRG, usually chosen to be 0.7 or

1 Mpc. Because the conversion between angular length and

projected proper length depends on cosmological parameters,

which remain uncertain, it is not always clear whether a given

observed RG satises the GRG denition.

Currently, there are about a thousand GRGs known, the major-

ity of which have been found in the Northern Sky. About one

hundred exceed 2 Mpc and ten exceed 3 Mpc; at 4.9 Mpc, the

literature’s projectively longest is J1420-0545 (Machalski et al.

2008). As such, GRGs — and the rest of the megaparsec-scale

RGs — are the largest single-galaxy–induced phenomena in the

Universe. It is a key open question what physical mechanisms

lead some RGs to extend for ∼102 times their host galaxy

diameter. To determine whether there exist particular host

galaxy characteristics or large-scale environments that are

essential for GRG growth, it is instructive to analyse the largest

GRGs, since in these systems it is most likely that all major

favourable growth factors are present. We thus aim to perform

Article number, page 2 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

a case study of the largest GRG available.

As demonstrated by Dabhade et al. (2020b)’s record sample

of 225 discoveries, the Low-frequency Array (LOFAR) (van

Haarlem et al. 2013) is among the most attractive contempo-

rary instruments for nding new GRGs. This Pan-European

radio interferometer features a unique combination of short

baselines to provide sensitivity to large-scale emission, and

long baselines to mitigate source confusion.1 These qualities

are indispensable for observational studies of GRGs, which

require identifying both extended lobes and compact cores and

jets. Additionally, the metre wavelengths at which the LOFAR

operates allow it to detect steep-spectrum lobes far away from

host galaxies. Such lobes reveal the full extent of GRGs, but

are missed by decimetre observatories. Thus, in Section 2, we

describe a reprocessing of the LOFAR Two-metre Sky Survey

(LoTSS) Data Release 2 (DR2) aimed at revealing hitherto

unknown RG lobes — among other goals. An overview of the

reprocessed images, which cover thousands of square degrees,

and statistics of the lengths and environments of the GRGs

they have revealed, are subjects of future publications. For

now, these images allow us to discover Alcyoneus2, a 5 Mpc

GRG, whose properties we determine and discuss in Section 3.

Figure 1 provides a multi-wavelength, multi-resolution view

of this giant. Section 4 contains our concluding remarks.

We assume a concordance inationary ΛCDM model with

parameters M from Planck Collaboration et al. (2020); i.e. M =

(

h = 0.6766,ΩBM,0 = 0.0490,ΩM,0 = 0.3111,ΩΛ,0 = 0.6889

),

where H0 B h · 100 km s−1 Mpc−1. We dene the spectral

index α such that it relates to ux density Fν at frequency ν

as Fν ∝ να. Regarding terminology, we strictly distinguish

between a radio galaxy, a radio-bright structure of relativistic

particles and magnetic elds (consisting of a core, jets, hotspots

and lobes), and the host galaxy that generates it.

2. Data and methods

The LoTSS, conducted by the LOFAR High-band Antennae

(HBA), is a 120–168 MHz interferometric survey (Shimwell et al.

2017, 2019, in prep.) with the ultimate aim to image the full

Northern Sky at resolutions of 6′′, 20′′, 60′′ and 90′′. Its cen-

tral frequency νc = 144 MHz. The latest data release — the

LoTSS DR2 (Shimwell et al. in prep.) — covers 27% of the North-

ern Sky, split over two regions of 4178 deg2 and 1457 deg2; the

largest of these contains the Sloan Digital Sky Survey (SDSS)

DR7 (Abazajian et al. 2009) area. By default, the LoTSS DR2 pro-

vides imagery at the 6′′ and 20′′ resolutions. We show these

standard products in Figure 2 for the same sky region as in

Figure 1. In terms of total source counts, the LoTSS DR2 is

the largest radio survey carried out thus far: its catalogue con-

tains 4.4 · 106 sources, most of which are considered com-

pact. By contrast, the 60′′ and 90′′ imagery, which we dis-

cuss in more detail in Oei et al. (in prep.), is intended to re-

veal extended structures in the low-frequency radio sky, such

1 Source confusion is an instrumental limitation that arises when the

resolution of an image is low compared to the sky density of statisti-

cally signicant sources. It causes angularly adjacent, but physically

unrelated sources to blend together, making it hard or even impossible

to distinguish them (e.g. Condon et al. 2012).

2 Alcyoneus was the son of Ouranos, the Greek primordial god of the

sky. According to Ps.-Apollodorus, he was also one of the greatest of

the Gigantes (Giants), and a challenger to Heracles during the Gigan-

tomachy — the battle between the Giants and the Olympian gods for

supremacy over the Cosmos. The poet Pindar described him as ‘huge

as a mountain’, ghting by hurling rocks at his foes.

123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0

200

400

600

800

1000

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0

100

200

300

400

500

600

specificintensityIν(Jydeg−2)Fig. 2: Alcyoneus’ lobes are easily overlooked in the LoTSS

DR2 at its standard resolutions. We show images at cen-

tral frequency νc = 144 MHz and resolutions θFWHM = 6′′

(top) and θFWHM = 20′′ (bottom), centred around host galaxy

J081421.68+522410.0.

as giant radio galaxies, supernova remnants in the Milky Way,

radio halos and shocks in galaxy clusters, and — potentially

— accretion shocks or volume-lling emission from laments

of the Cosmic Web. To avoid the source confusion limit at

these resolutions, following van Weeren et al. (2021), we used

DDFacet (Tasse et al. 2018) to predict visibilities corresponding

to the 20′′ LoTSS DR2 sky model and subtracted these from the

data, before imaging at 60′′ and 90′′ with WSClean IDG (Of-

fringa et al. 2014; van der Tol et al. 2018). We used -0.5 Briggs

weighting and multiscale CLEAN (Oringa & Smirnov 2017),

with -multiscale-scales 0,4,8,16,32,64. Importantly, we

did not impose an inner (u, v)-cut. We imaged each pointing sep-

arately, then combined the partially overlapping images into a

mosaic by calculating, for each direction, a beam-weighted av-

erage.

Finally, we visually searched the LoTSS DR2 for GRGs, primarily

at 6′′ and 60′′ using the Hierarchical Progressive Survey (HiPS)

system in Aladin Desktop 11.0 (Bonnarel et al. 2000).

Article number, page 3 of 18

A&A proofs: manuscript no. aanda

3. Results and discussion

3.1. Radio morphology and interpretation

During our LoTSS DR2 search, we identied a three-component

radio structure of total angular length φ = 20.8′, visible at all

(6′′, 20′′, 60′′ and 90′′) resolutions. Figure 2 provides a sense of

our data quality; it shows that the outer components are barely

discernible in the LoTSS DR2 at its standard 6′′ and 20′′ reso-

lutions. Meanwhile, Figure 1 shows the outer components at

60′′, and the top panel of Figure 9 shows them at 90′′; at these

resolutions, they lie rmly above the noise. Compared with the

outer structures, the central structure is bright and elongated,

with a 155′′ major axis and a 20′′ minor axis. The outer struc-

tures lie along the major axis at similar distances from the cen-

tral structure, are diuse and amorphous, and feature specic

intensity maxima along this axis.

In the arcminute-scale vicinity of the outer structures, the DESI

Legacy Imaging Surveys (Dey et al. 2019) DR9 does not reveal

galaxy overdensities or low-redshift spiral galaxies, the ROSAT

All-sky Survey (RASS) (Voges et al. 1999) does not show X-ray

brightness above the noise, and there is no Planck Sunyaev–

Zeldovich catalogue 2 (PSZ2) (Planck Collaboration et al. 2016)

source nearby. The outer structures therefore cannot be super-

nova remnants in low-redshift spiral galaxies or radio relics

and radio halos in galaxy clusters. Instead, the outer structures

presumably represent radio galaxy emission. The radio-optical

overlays in Figure 3’s top and bottom panel show that it is im-

probable that each outer structure is a radio galaxy of its own,

given the lack of signicant 6′′ radio emission (solid light green

contours) around host galaxy candidates suggested by the mor-

phology of the 60′′ radio emission (translucent white contours).

For these reasons, we interpret the central (jet-like) structure

and the outer (lobe-like) structures as components of the same

radio galaxy.

Subsequent analysis — presented below — demonstrates that

this radio galaxy is the largest hitherto discovered, with a pro-

jected proper length of 5.0 Mpc. We dub this GRG Alcyoneus.

3.2. Host galaxy identification

Based on the middle panel of Figure 3 and an SDSS DR12 (Alam

et al. 2015) spectrum, we identify a source at a J2000 right as-

cension of 123.590372°, a declination of 52.402795° and a spec-

troscopic redshift of zspec = 0.24674 ± 6 · 10−5 as Alcyoneus’

host. Like most GRG hosts, this source, with SDSS DR12 name

J081421.68+522410.0, is an elliptical galaxy3 without a quasar.

From optical contours, we nd that the galaxy’s minor axis

makes a ∼20° angle with Alcyoneus’ jet axis.

In Figure 4, we further explore the connection between

J081421.68+522410.0 and Alcyoneus’ radio core and jets. From

top to bottom, we show the LoTSS DR2 at 6′′, the Very Large

Array Sky Survey (VLASS) (Lacy et al. 2020) at 2.2′′, and the

Panoramic Survey Telescope and Rapid Response System (Pan-

STARRS) DR1 (Chambers et al. 2016) i-band. Two facts conrm

that the host identication is highly certain. First, for both the

LoTSS DR2 at 6′′ and the VLASS at 2.2′′, the angular separa-

tion between J081421.68+522410.0 and the arc connecting Al-

cyoneus’ two innermost jet features is subarcsecond. Moreover,

the alleged host galaxy is the brightest Pan-STARRS DR1 i-band

3 Based on the SDSS morphology, Kuminski & Shamir (2016) calculate

a probability of 89% that the galaxy is an elliptical.

123.32

123.36

123.4

123.44

123.48

right ascension (°)

52.44

52.46

52.48

52.5

52.52

52.54

declination(°)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)123.64

123.68

123.72

123.76

123.8

right ascension (°)

52.26

52.28

52.3

52.32

52.34

52.36

52.38

declination(°)Fig. 3: Joint radio-optical views show that Figure 1’s outer

structures are best interpreted as a pair of radio galaxy

lobes fed by central jets. On top of DESI Legacy Imaging

Surveys DR9 (g, r, z)-imagery, we show the LoTSS DR2 at var-

ious resolutions through contours at multiples of σ, where σ

is the image noise at the relevant resolution. The top and bot-

tom panel show translucent white 60′′ contours at 3, 5, 7, 9, 11σ

and solid light green 6′′ contours at 4, 7, 10, 20, 40σ. The central

panel shows translucent white 6′′ contours at 5, 10, 20, 40, 80σ.

Article number, page 4 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0

1 · 103

2 · 103

3 · 103

4 · 103

5 · 103

6 · 103

7 · 103

specificintensityIν(Jydeg−2)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0

1 · 103

2 · 103

3 · 103

4 · 103

5 · 103

6 · 103

7 · 103

specificintensityIν(Jydeg−2)123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)0.0

0.2

0.4

0.6

0.8

1.0

1.2

relativespecificintensityIν(1)Fig. 4: The SDSS DR12 source J081421.68+522410.0 is Al-

cyoneus’ host galaxy. The panels cover a 2.5′ × 2.5′ region

around J081421.68+522410.0, an elliptical galaxy with spectro-

scopic redshift zspec = 0.24674 ± 6 · 10−5. From top to bot-

tom, we show the LoTSS DR2 6′′, the VLASS 2.2′′, and the Pan-

STARRS DR1 i-band — relative to the peak specic intensity of

J081421.68+522410.0 — with LoTSS contours (white) as in Fig-

ure 3 and a VLASS contour (gold) at 5σ.

source within a radius of 45′′ of the central VLASS image com-

ponent.

3.3. Radiative- or jet-mode active galactic nucleus

Current understanding (e.g. Heckman & Best 2014) suggests

that the population of active galactic nuclei (AGN) exhibits a

dichotomy: AGN seem to be either radiative-mode AGN, which

generate high-excitation radio galaxies (HERGs), or jet-mode

AGN, which generate low-excitation radio galaxies (LERGs). Is

Alcyoneus a HERG or a LERG? The SDSS spectrum of the host

features very weak emission lines; indeed, the star formation

rate (SFR) is just 1.6 · 10−2 M yr−1 (Chang et al. 2015). Fol-

lowing the classication rule of Best & Heckman (2012); Best

et al. (2014); Pracy et al. (2016); Williams et al. (2018) based on

the strength and equivalent width of the OIII 5007 Å line, we

conclude that Alcyoneus is a LERG. Moreover, the WISE pho-

tometry (Cutri & et al. 2012) at 11.6 µm and 22.1 µm is below the

instrumental detection limit. Following the classication rule of

Gürkan et al. (2014) based on the 22.1 µm luminosity density,

we arm that Alcyoneus is a LERG. Through automated classi-

cation, Best & Heckman (2012) came to the same conclusion.

Being a jet-mode AGN, the supermassive black hole (SMBH) in

the centre of Alcyoneus’ host galaxy presumably accretes at an

eciency below 1% of the Eddington limit, and is fueled mainly

by slowly cooling hot gas.

3.4. Projected proper length

We calculate Alcyoneus’ projected proper length lp through

its angular length φ and spectroscopic redshift zspec. We for-

mally determine φ = 20.8′ ± 0.15′ from the compact-source–

subtracted 90′′ image (top panel of Figure 9) by selecting the

largest great-circle distance between all possible pairs of pix-

els with a specic intensity higher than three sigma-clipped

standard deviations above the sigma-clipped median. We nd

lp = 4.99 ± 0.04 Mpc; this makes Alcyoneus the projectively

largest radio galaxy known. For methodology details, and for a

probabilistic comparison between the projected proper lengths

of Alcyoneus and J1420-0545, see Appendix A.

3.5. Radio luminosity densities and kinetic jet powers

From the LoTSS DR2 6′′ image (top panel of Figure 4), we mea-

sure that two northern jet local maxima occur at angular dis-

tances of 9.2 ± 0.2′′ and 23.7 ± 0.2′′ from the host, or at pro-

jected proper distances of 36.8 ± 0.8 kpc and 94.8 ± 0.8 kpc.

Two southern jet local maxima occur at angular distances of

8.8± 0.2′′ and 62.5± 0.2′′ from the host, or at projected proper

distances of 35.2 ± 0.8 kpc and 249.9 ± 0.8 kpc.

At the central observing frequency of νc = 144MHz, the north-

ern jet has a ux density Fν = 193±20mJy, the southern jet has

Fν = 110±12mJy, whilst the northern lobe has Fν = 63±7mJy

and the southern lobe has Fν = 44 ± 5 mJy. To minimise con-

tamination from fore- and background galaxies, we determined

the lobe ux densities from the compact-source–subtracted 90′′

image. The ux density uncertainties are dominated by the 10%

ux scale uncertainty inherent to the LoTSS DR2 (Shimwell

et al. in prep.). The host galaxy ux density is relatively weak,

and the corresponding emission has, at νc = 144 MHz and

6′′ resolution, no clear angular separation from the inner jets’

emission; we have therefore not determined it.

Due to cosmological redshifting, the conversion between ux

Article number, page 5 of 18

A&A proofs: manuscript no. aanda

123.56

123.58

123.6

123.62

right ascension (°)

52.39

52.4

52.41

52.42

declination(°)−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

spectralindexα(1)Fig. 5: The LoTSS–VLASS spectral indexmap reveals Alcy-

oneus’ at-spectrum core and steeper-spectrum jets. We

show all directions where both the LoTSS and VLASS image

have at least 5σ signicance. In black, we overlay the same

LoTSS contours as in Figures 3 and 4. The core spectral index

is α = −0.25 ± 0.1 and the combined inner jet spectral index is

α = −0.65 ± 0.1.

density and luminosity density depends on the spectral indices

α of Alcyoneus’ luminous components. We estimate the spec-

tral indices of the core and jets from the LoTSS DR2 6′′ and

VLASS 2.2′′ images. After convolving the VLASS image with

a Gaussian to the common resolution of 6′′, we calculate the

mean spectral index between LoTSS’ νc = 144MHz and VLASS’

νc = 2.99 GHz. Using only directions for which both images

have a signicance of at least 5σ, we deduce a core spectral

index α = −0.25 ± 0.1 and a combined inner jet spectral index

α = −0.65±0.1. The spectral index uncertainties are dominated

by the LoTSS DR2 and VLASS ux scale uncertainties. We show

the full spectral index map in Figure 5. We have not determined

the spectral index of the lobes, as they are only detected in the

LoTSS imagery.

The luminosity densities of the northern and southern jet at

rest-frame frequency ν = 144 MHz are Lν = (3.6 ± 0.4) ·

1025 W Hz−1 and Lν = (2.0 ± 0.2) · 1025 W Hz−1, respectively.

Following Dabhade et al. (2020a), we estimate the kinetic power

of the jets from their luminosity densities and the results of the

simulation-based analytical model of Hardcastle (2018). We nd

Qjet,1 = 1.2 ± 0.1 · 1036 W and Qjet,2 = 6.6 ± 0.7 · 1035 W,

so that the total kinetic jet power is Qjets B Qjet,1 + Qjet,2 =

1.9 ± 0.2 · 1036 W. Interestingly, this total kinetic jet power is

lower than the average Qjets = 3.7 ·1036 W, and close to the me-

dian Qjets = 2.2 · 1036 W, for low-excitation giant radio galaxies

(LEGRGs) in the redshift range 0.18 < z < 0.43 (Dabhade et al.

2020a).

Because the lobe spectral indices are unknown, we present lu-

minosity densities for several possible values of α in Table 1.4

(Because of electron ageing, α will decrease further away from

the core.)

4 The inferred luminosity densities have a cosmology-dependence;

our results are ∼6% higher than for modern high-H0 cosmologies.

1024

1025

1026

1027

1028

luminosity density Lν(ν = 144 MHz) (W Hz

−1)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

239 literature GRGs

Alcyoneus

Fig. 6: Alcyoneus has a low-frequency luminosity density

typical for GRGs. We explore the relation between GRG pro-

jected proper length lp and total luminosity density Lν at rest-

frame frequency ν = 144 MHz. Total luminosity densities in-

clude contributions from all available radio galaxy components

(i.e. the core, jets, hotspots and lobes). Literature GRGs are from

Dabhade et al. (2020b), and are marked with grey disks, while

Alcyoneus is marked with a green star. Translucent ellipses in-

dicate -1 to +1 standard deviation uncertainties. Alcyoneus has

a typical luminosity density (percentile 45 ± 3%).

Table 1: Luminosity densities Lν (in 1024 W Hz−1) of Alcyoneus’

lobes for three potential spectral indices α at rest-frame fre-

quency ν = 144 MHz, assuming a Planck Collaboration et al.

(2020) cosmology.

α = −0.8 α = −1.2 α = −1.6

Northern lobe

12 ± 1

13 ± 1

14 ± 1

Southern lobe

8.3 ± 0.8

9.0 ± 0.9

9.9 ± 1

Assuming α = −1.2, Alcyoneus total luminosity density at

ν = 144 MHz is Lν = 7.8 ± 0.8 · 1025 W Hz−1. In Figure 6, we

compare this estimate to other GRGs’ total luminosity density at

the same frequency, as found by Dabhade et al. (2020b) through

the LoTSS DR1 (Shimwell et al. 2019). Interestingly, Alcyoneus

is not particularly luminous: it has a low-frequency luminosity

density typical for the currently known GRG population (per-

centile 45 ± 3%).

3.6. True proper length: relativistic beaming

Following Hardcastle et al. (1998a), we simultaneously con-

strain Alcyoneus’ jet speed u and inclination angle θ from the

jets’ ux density asymmetry: the northern-to-southern jet ux

density ratio J = 1.78 ± 0.3.5 We assume that the jets prop-

agate with identical speeds u in exactly opposing directions

(making angles with the line-of-sight θ and θ + 180°), and have

statistically identical relativistic electron populations, so that

they have a common synchrotron spectral index α. Using α =

−0.65 ± 0.1 as before, and

β B

u

c

; β cos θ =

J

1

2−α − 1

J

1

2−α + 1

,

(1)

5 Because J is obtained through division of two independent normal

random variables (RVs) with non-zero mean, J is an RV with an un-

correlated noncentral normal ratio distribution.

Article number, page 6 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

we nd β cos θ = 0.106 ± 0.03. Because cos θ ≤ 1, β is bounded

from below by βmin = 0.106 ± 0.03.

From detailed modelling of ten Fanaro–Riley (FR) I radio

galaxies (which have jet luminosities comparable to Alcy-

oneus’), Laing & Bridle (2014) deduced that initial jet speeds are

roughly β = 0.8, which decrease until roughly 0.6 r0, with r0 be-

ing the recollimation distance. Most of Laing & Bridle (2014)’s

ten recollimation distances are between 5 and 15 kpc, with the

largest being that of NGC 315: r0 = 35 kpc. Because the lo-

cal specic intensity maxima in Alcyoneus’ jets closest to the

host occur at projected proper distances of 36.8 ± 0.8 kpc and

35.2±0.8 kpc, the true proper distances must be even larger. We

conclude that the observed jet emission presumably comes from

a region further from the host than r0, so that the initial stage

of jet deceleration — in which the jet speed is typically reduced

by several tens of percents of c — must already be completed.

Thus, βmax = 0.8 is a safe upper bound.

Taking βmax = 0.8, θ is bounded from above by θmax = 82.4± 2°

(θ ∈ [0, 90°]), or bounded from below by 180° − θmax = 97.6 ±

2° (θ ∈ [90°, 180°]).6 If we model Alcyoneus’ geometry as a

line segment, and assume no jet reorientation, Alcyoneus’ true

proper length l and projected proper length lp relate as

l =

lp

sin θ

;

l ≥ lmin =

lp

sin θmax

.

(2)

We bound l from below: lmin = 5.04 ± 0.05 Mpc. A triangu-

lar prior on β between βmin and βmax with the mode at βmax

induces a skewed prior on l; the 90% credible interval is l ∈

[5.0 Mpc, 5.5 Mpc], with the mean and median being 5.2 Mpc

and 5.1 Mpc, respectively. A at prior on β between βmin and

βmax also induces a skewed prior on l; the 90% credible inter-

val is l ∈ [5.0 Mpc, 7.1 Mpc], with the mean and median being

5.6 Mpc and 5.1 Mpc, respectively. The median of l seems par-

ticularly well determined, as it is insensitive to variations of the

prior on β.

In Appendix B, we explore the inclination angle conditions un-

der which Alcyoneus has the largest true proper length of all

known (> 4 Mpc) GRGs.

3.7. Stellar and supermassive black hole mass

Does a galaxy or its central black hole need to be massive in

order to generate a GRG?

Alcyoneus’ host has a stellar mass M? = 2.4 ± 0.4 · 1011 M

(Chang et al. 2015). We test whether or not this is a typical stel-

lar mass among the total known GRG population. We assem-

ble a literature catalogue of 1013 GRGs by merging the com-

pendium of Dabhade et al. (2020a), which is complete up to

April 2020, with the GRGs discovered in Galvin et al. (2020),

Ishwara-Chandra et al. (2020), Tang et al. (2020), Bassani et al.

(2021), Brüggen et al. (2021), Delhaize et al. (2021), Masini et al.

(2021), Kuźmicz & Jamrozy (2021), Andernach et al. (2021) and

Mahato et al. (2021). We collect stellar masses with uncertainties

from Chang et al. (2015), which are based on SDSS and WISE

photometry, and from Salim et al. (2018), which are based on

GALEX, SDSS and WISE photometry. We give precedence to

the stellar masses by Salim et al. (2018) when both are avail-

able. We obtain stellar masses for 151 previously known GRGs.

The typical stellar mass range is 1011 – 1012 M, the median

6 Taking βmax = 1 instead, θ is bounded from above by θmax = 83.9±2°

(θ ∈ [0, 90°]), or bounded from below by 180° − θmax = 96.1 ± 2° (θ ∈

[90°, 180°]).

1011

1012

stellar mass M? (M)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

151 literature GRGs

Alcyoneus

106

107

108

109

1010

1011

supermassive black hole mass M• (M)

0.7

1.0

2.0

3.0

4.0

5.0

projectedproperlengthlp(Mpc)Alcyoneus

189 literature GRGs

Alcyoneus

Fig. 7: Alcyoneus’ host has a lower stellar and supermas-

sive black hole mass than most GRG hosts. We explore

the relations between GRG projected proper length lp and host

galaxy stellar mass M? (top panel) or host galaxy supermas-

sive black hole mass M• (bottom panel). Our methods allow de-

termining these properties for a small proportion of all litera-

ture GRGs only. Literature GRGs are marked with grey disks,

while Alcyoneus is marked with a green star. Translucent el-

lipses indicate -1 to +1 standard deviation uncertainties. Alcy-

oneus’ host has a fairly typical — though below-average — stel-

lar mass (percentile 25±9%) and supermassive black hole mass

(percentile 23 ± 11%).

M? = 3.5 · 1011 M and the mean M? = 3.8 · 1011 M. Strik-

ingly, the top panel of Figure 7 illustrates that Alcyoneus’ host

has a fairly low (percentile 25±9%) stellar mass compared with

the currently known population of GRG hosts.

For the GRGs in our literature catalogue, we also estimate

SMBH masses via the M-sigma relation. We collect SDSS DR12

stellar velocity dispersions with uncertainties (Alam et al. 2015),

and apply the M-sigma relation of Equation 7 in Kormendy

& Ho (2013). Alcyoneus’ host has a SMBH mass M• = 3.9 ±

1.7 ·108 M. We obtain SMBH masses for 189 previously known

GRGs. The typical SMBH mass range is 108 – 1010 M, the me-

dian M• = 7.9 · 108 M and the mean M• = 1.5 · 109 M.

Strikingly, the bottom panel of Figure 7 illustrates that Alcy-

oneus’ host has a fairly low (percentile 23 ± 11%) SMBH mass

compared with the currently known population of GRG hosts.

We note that Alcyoneus is the only GRG with lp > 3Mpc whose

host’s stellar mass is known through Chang et al. (2015) or Salim

et al. (2018), and whose host’s SMBH mass can be estimated

Article number, page 7 of 18

A&A proofs: manuscript no. aanda

through its SDSS DR12 velocity dispersion. These data allow

us to state condently that exceptionally high stellar or SMBH

masses are not necessary to generate 5-Mpc–scale GRGs.

3.8. Surrounding large-scale structure

Several approaches to large-scale structure (LSS) classication,

such as the T-web scheme (Hahn et al. 2007), partition the mod-

ern Universe into galaxy clusters, laments, sheets and voids. In

this section, we determine Alcyoneus’ most likely environment

type.

We conduct a tentative quantitative analysis using the SDSS

DR7 spectroscopic galaxy sample (Abazajian et al. 2009). Does

Alcyoneus’ host have fewer, about equal or more galactic neigh-

bours in SDSS DR7 than a randomly drawn galaxy of similar

r-band luminosity density and redshift? Let r (z) be the comov-

ing radial distance corresponding to cosmological redshift z. We

consider a spherical shell with the observer at the centre, inner

radius max {r(z = zspec) − r0, 0} and outer radius r(z = zspec)+r0.

We approximate Alcyoneus’ cosmological redshift with zspec

and choose r0 = 25 Mpc. As all galaxies in the spherical shell

have a similar distance to the observer (i.e. distances are at most

2r0 dierent), the SDSS DR7 galaxy number density complete-

ness must also be similar throughout the spherical shell.7 For

each enclosed galaxy with an r-band luminosity density be-

tween 1 − δ and 1 + δ times that of Alcyoneus’ host, we count

the number of SDSS DR7 galaxies N

sity, and excluding itself. Alcyoneus’ host has an SDSS r-band

apparent magnitude mr = 18.20; the corresponding luminosity

density is Lν (λc = 623.1 nm) = 3.75 · 1022 W Hz−1. We choose

δ = 0.25; this yields 9,358 such enclosed galaxies.

In Figure 8, we show the distribution of N

are insensitive to reasonable changes in r0 and δ. Note that

there is no SDSS DR7 galaxy within a comoving distance

of 5 Mpc from Alcyoneus’ host. The nearest such galaxy,

J081323.49+524856.1, occurs at a comoving distance of 7.9 Mpc:

the nearest ∼2, 000 Mpc3 of comoving space are free of galactic

neighbours with Lν (λc) > 5.57 ·1022 W Hz−1.8 In the same way

as in Section 3.1, we verify that the DESI Legacy Imaging Sur-

veys DR9, RASS and PSZ2 do not contain evidence for a galaxy

cluster in the direction of Alcyoneus’ host. The nearest galaxy

cluster, according to the SDSS-III cluster catalogue of Wen et al.

(2012), instead lies 24′ away at right ascension 123.19926°, dec-

lination 52.72468° and photometric redshift zph = 0.2488. It has

an R200 = 1.1 Mpc and, according to the DESI cluster catalogue

of Zou et al. (2021), a total mass M = 2.2·1014 M. The comoving

distance between the cluster and Alcyoneus’ host is 11Mpc. All

in all, we conclude that Alcyoneus does not reside in a galaxy

cluster. Meanwhile, there are ve SDSS DR7 galaxies within a

comoving distance of 10 Mpc from Alcyoneus’ host: this makes

it implausible that Alcyoneus lies in a void. Finally, one could

interpret N

shell with a similar luminosity density as Alcyoneus’ host have

a higher LSS total matter density. Being on the high end of the

7 For r0 = 25 Mpc, this is a good approximation, because the shell is

cosmologically thin: 2r0 = 50 Mpc roughly amounts to the length of a

single Cosmic Web lament.

8 This is the luminosity density that corresponds to the SDSS r-band

apparent magnitude completeness limit mr = 17.77 (Strauss et al.

2002).

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

№ SDSS DR7 galaxies within some comoving distance N

0.1

0.2

0.3

0.4

0.5

probability(1)Alcyoneus’ host

similarly luminous SDSS DR7 galaxies in shell

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

№ SDSS DR7 galaxies within some comoving distance N

0.05

0.10

0.15

0.20

probability(1)Alcyoneus’ host

similarly luminous SDSS DR7 galaxies in shell

Fig. 8: Like most galaxies of similar r-band luminosity

density and redshift, Alcyoneus’ host has no galactic

neighbours in SDSS DR7 within 5 Mpc. However, within

10 Mpc, Alcyoneus’ host has more neighbours than most

similar galaxies. For all 9,358 SDSS DR7 galaxies with an r-

band luminosity density between 75% and 125% that of Al-

cyoneus’ host and a comoving radial distance that diers at

most r0 = 25 Mpc from Alcyoneus’, we count the number of

SDSS DR7 galaxies N

panel indicates that Alcyoneus does not inhabit a galaxy clus-

ter; the bottom panel indicates that Alcyoneus does not inhabit

a void.

density distribution, but lying outside a cluster, Alcyoneus most

probably inhabits a lament of the Cosmic Web.

3.9. Proper lobe volumes

We determine the proper volumes of Alcyoneus’ lobes with a

new Bayesian model. The model describes the lobes through a

pair of doubly truncated, optically thin cones, each of which

has a spatially constant and isotropic monochromatic emis-

sion coecient (MEC) (Rybicki & Lightman 1986). We allow

the 3D orientations and opening angles of the cones to dier,

as the lobes may traverse their way through dierently pres-

sured parts of the warm–hot intergalactic medium (WHIM):

e.g. the medium near the lament axis, and the medium near

the surrounding voids. By adopting a spatially constant MEC,

we neglect electron density and magnetic eld inhomogeneities

as well as spectral-ageing gradients; by adopting an isotropic

MEC, we assume non-relativistic velocities within the lobe so

that beaming eects are negligible. Numerically, we rst gener-

ate the GRG’s 3D MEC eld over a cubical voxel grid, and then

calculate the corresponding model image through projection,

including expansion-related cosmological eects. Before com-

parison with the observed image, we convolve the model image

with a Gaussian kernel to the appropriate resolution. We exploit

the approximately Gaussian LoTSS DR2 image noise to formu-

Article number, page 8 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

late the likelihood, and assume a at prior distribution over the

parameters. Using a Metropolis–Hastings (MH) Markov chain

Monte Carlo (MCMC), we sample from the posterior distribu-

tion.9

In the top panel of Figure 9, we show the LoTSS DR2 compact-

source-subtracted 90′′ image of Alcyoneus. The central region

has been excluded from source subtraction, and hence Alcy-

oneus’ core and jets remain. (However, when we run our MH

MCMC on this image, we do mask this central region.) In the

middle panel, we show the highest-likelihood (and thus max-

imum a posteriori (MAP)) model image before convolution. In

the bottom panel, we show the same model image convolved

to 90′′ resolution, with 2σ and 3σ contours of the observed im-

age overlaid. We provide the full parameter set that corresponds

with this model in Table C.1.

The posterior mean, calculated through the MH MCMC sam-

ples after burn-in, suggests the following geometry. The north-

ern lobe has an opening angle γ1 = 10 ± 1°, and the cone

truncates at an inner distance di,1 = 2.6 ± 0.2 Mpc and at an

outer distance do,1 = 4.0 ± 0.2 Mpc from the host galaxy. The

southern lobe has a larger opening angle γ2 = 26 ± 2°, but its

cone truncates at smaller distances of di,2 = 1.5 ± 0.1 Mpc and

do,2 = 2.0± 0.1 Mpc from the host galaxy. These parameters x

the proper volumes of Alcyoneus’ northern and southern lobes.

We nd V1 = 1.5 ± 0.2 Mpc3 and V2 = 1.0 ± 0.2 Mpc3, respec-

tively (see Equation C.15).10

How are the lobes oriented? Figure 1 provides a visual hint

that the lobes are subtly non-coaxial. The posterior indicates

that the position angles of the northern and southern lobes are

ϕ1 = 307±1° and ϕ2 = 139±2°, respectively. The position angle

dierence is thus ∆ϕ = 168±2°: although close to ∆ϕ = 180°, we

can reject coaxiality with high signicance. Interestingly, the

posterior also constrains the angles that the lobe axes make with

the plane of the sky: |θ1 − 90°| = 51± 2° and |θ2 − 90°| = 18± 7°.

Again, the uncertainties imply that the lobes are probably not

coaxial. We stress that these inclination angle results are tenta-

tive only. Future model extensions should explore how sensitive

they are to the assumed lobe geometry (by testing other shapes

than just truncated cones, such as ellipsoids).

One way to validate the model is to compare the observed

lobe ux densities of Section 3.5 to the predicted lobe ux

densities. According to the posterior, the MECs of the north-

ern and southern lobes are jν,1 = 17 ± 2 Jy deg−2 Mpc−1 and

jν,2 = 18 ± 3 Jy deg−2 Mpc−1. Combining MECs and volumes,

we predict northern and southern lobe ux densities Fν,1(νc) =

63 ± 4 mJy and Fν,2(νc) = 45 ± 5 mJy (see Equation C.16).

We nd excellent agreement: the relative dierences with the

observed results are 0% and 2%, respectively.

9 For a detailed description of the model parameters, the MH MCMC

and formulae for derived quantities, see Appendix C.

10 As a sanity check, we compare our results to those from a less rig-

orous, though simpler ellipsoid-based method of estimating volumes.

By tting ellipses to Figure 9’s top panel image, one obtains a semi-

minor and semi-major axis; the half-diameter along the ellipsoid’s third

dimension is assumed to be their mean. This method suggests a north-

ern lobe volume V1 = 1.4 ± 0.3 Mpc3 and a southern lobe volume

V2 = 1.1 ± 0.3 Mpc3. These results agree well with our Bayesian

model results. (If the half-diameter along the third dimension is instead

treated as an RV with a uniform distribution between the semi-minor

axis and the semi-major axis, the estimates remain the same.)

123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)123.2

123.4

123.6

123.8

124.0

right ascension (°)

52.2

52.3

52.4

52.5

52.6

declination(°)Milky Way

× 1 × 10

0.0

5.0

10.0

15.0

20.0

25.0

specificintensityIν(Jydeg−2)Fig. 9: Alcyoneus’ lobe volumes can be estimated by com-

paring the observed radio image to modelled radio im-

ages. Top: LoTSS DR2 compact-source-subtracted 90′′ image of

Alcyoneus. For scale, we show the stellar Milky Way disk (di-

ameter: 50 kpc) and a 10 times inated version; the spiral galaxy

shape follows Ringermacher & Mead (2009). Middle: Highest-

likelihood model image. Bottom: The same model image con-

volved to 90′′ resolution, with 2σ and 3σ contours of the ob-

served image overlaid.

Article number, page 9 of 18

A&A proofs: manuscript no. aanda

3.10. Lobe pressures and the local WHIM

From Alcyoneus’ lobe ux densities and volumes, we can in-

fer lobe pressures and magnetic eld strengths. We calculate

these through pysynch11 (Hardcastle et al. 1998b), which uses

the formulae rst proposed by Myers & Spangler (1985) and re-

examined by Beck & Krause (2005). Following the notation of

Hardcastle et al. (1998b), we assume that the electron energy

distribution is a power law in Lorentz factor γ with γmin = 10,

γmax = 104 and exponent p = −2; we also assume that the

kinetic energy density of protons is vanishingly small com-

pared with that of electrons (κ = 0), and that the plasma ll-

ing factor is unity (φ = 1). Assuming the minimum-energy

condition (Burbidge 1956), we nd minimum-energy pressures

Pmin,1 = 4.8±0.3·10−16 Pa and Pmin,2 = 4.9±0.6·10−16 Pa for the

northern and southern lobes, respectively. The corresponding

minimum-energy magnetic eld strengths are Bmin,1 = 46±1 pT

and Bmin,2 = 46 ± 3 pT. Assuming the equipartition condi-

tion (Pacholczyk 1970), we nd equipartition pressures Peq,1 =

4.9 ± 0.3 · 10−16 Pa and Peq,2 = 4.9 ± 0.6 · 10−16 Pa for the

northern and southern lobes, respectively. The corresponding

equipartition magnetic eld strengths are Beq,1 = 43± 2 pT and

Beq,2 = 43 ± 2 pT. The minimum-energy and equipartition re-

sults do not dier signicantly.

From pressures and volumes, we estimate the internal energy

of the lobes E = 3PV . We nd Emin,1 = 6.2 ± 0.5 · 1052 J,

Emin,2 = 4.3 ± 0.6 · 1052 J, Eeq,1 = 6.3 ± 0.5 · 1052 J and Eeq,2 =

4.4± 0.6 · 1052 J. Next, we can bound the ages of the lobes from

below by neglecting synchrotron losses, and assuming that the

jets have been injecting energy in the lobes continuously at the

currently observed kinetic jet powers. Using ∆t = EQ−1

jet , we

nd ∆tmin,1 = 1.7 ± 0.2 Gyr, ∆tmin,2 = 2.1 ± 0.4 Gyr, and identi-

cal results when assuming the equipartition condition. Finally,

we can obtain a rough estimate of the average expansion speed

of the radio galaxy during its lifetime u = lp(∆t)−1. We nd

u = 2.6 ± 0.3 · 103 km s−1, or about 1% of the speed of light.

Several other authors (Andernach et al. 1992; Lacy et al. 1993;

Subrahmanyan et al. 1996; Parma et al. 1996; Mack et al. 1998;

Schoenmakers et al. 1998, 2000; Ishwara-Chandra & Saikia 1999;

Lara et al. 2000; Machalski & Jamrozy 2000; Machalski et al.

2001; Saripalli et al. 2002; Jamrozy et al. 2005; Subrahmanyan

et al. 2006, 2008; Saikia et al. 2006; Machalski et al. 2006, 2007,

2008; Safouris et al. 2009; Malarecki et al. 2013; Tamhane et al.

2015; Sebastian et al. 2018; Heesen et al. 2018; Cantwell et al.

2020) have estimated the minimum-energy or equipartition

pressure of the lobes of GRGs embedded in non-cluster envi-

ronments (i.e. in voids, sheets or laments of the Cosmic Web).

We compare Alcyoneus to the other 151 GRGs with known lobe

pressures in the top panel of Figure 10.12 Alcyoneus rearms

the negative correlation between length and lobe pressure (Jam-

rozy & Machalski 2002; Machalski & Jamrozy 2006), and has the

lowest lobe pressures found thus far. Alcyoneus’ lobe pressure

is in fact so low, that it is comparable to the pressure in dense

and hot parts of the WHIM: for a baryonic matter (BM) density

11 The pysynch code is publicly available online: https://github.

com/mhardcastle/pysynch.

12 We have included all publications that provide pressures, energy

densities or magnetic eld strengths. Note that some authors assume

γmin = 1, we assume γmin = 10 and Malarecki et al. (2013) assume

γmin = 103. If possible, angular lengths were updated using the LoTSS

DR2 at 6′′ and redshift estimates were updated using the SDSS DR12.

All projected proper lengths have been recalculated using our Planck

Collaboration et al. (2020) cosmology. When authors provided pres-

sures for both lobes, we have taken the average.

0.7

1.0

2.0

3.0

4.0

5.0

projected proper length lp (Mpc)

10−16

10−15

10−14

10−13

10−12

lobepressureP(Pa)Alcyoneus

151 literature GRGs

Alcyoneus

10−1

100

101

102

baryonic matter density ρBM (ρc,0 ΩBM,0)

10−19

10−18

10−17

10−16

10−15

10−14

pressureP(Pa)GRG B2147+816

GRG 3C 236

GRG J1420-0545, GRG J0331-7710

GRG Alcyoneus

ideal gas at T = 1 · 107 K

ideal gas at T = 5 · 106 K

ideal gas at T = 1 · 106 K

ideal gas at T = 5 · 105 K

Fig. 10: Of all GRGswith known lobe pressures, Alcyoneus

is the most plausible candidate for pressure equilibrium

with the WHIM. In the top panel, we explore the relation be-

tween length and lobe pressure for Alcyoneus and 151 literature

GRGs. In the bottom panel, we compare the lobe pressure of Al-

cyoneus (green line) with the lobe pressures of the largest four

similarly analysed GRGs (grey lines) and with WHIM pressures

(red lines).

ρWHIM = 10 ρc,0ΩBM,0 and TWHIM = 107 K, PWHIM = 4·10−16 Pa.

Here, ρc,0 is today’s critical density, so that ρc,0ΩBM,0 is today’s

mean baryon density. See the bottom panel of Figure 10 for

a more extensive comparison between Pmin (green line) and

PWHIM (red lines). For comparison, we also show the lobe pres-

sures of the four other thus-analysed GRGs with lp > 3 Mpc

(grey lines). These are J1420-0545 of lp = 4.9 Mpc (Machal-

ski et al. 2008), 3C 236 of lp = 4.7 Mpc (Schoenmakers et al.

2000), J0331-7710 of lp = 3.4 Mpc (Malarecki et al. 2013) and

B2147+816 of lp = 3.1 Mpc (Schoenmakers et al. 2000).

Although proposed as probes of WHIM thermodynamics for

decades, the bottom panel of Figure 10 demonstrates that even

the largest non-cluster literature GRGs are unlikely to be in

pressure equilibrium with their environment. Relying on results

from the Overwhelmingly Large Simulations (OWLS) (Schaye

et al. 2010), Malarecki et al. (2013) point out that baryon den-

sities ρBM > 50 ρc,0 ΩBM,0, which are necessary for pressure

equilibrium in these GRGs (see the intersection of grey and red

lines in the bottom panel of Figure 10), occur in only 1% of

the WHIM’s volume. By contrast, Alcyoneus can be in pres-

sure equilibrium with the WHIM at baryon densities ρBM ∼

20 ρc,0 ΩBM,0, and thus represents the most promising inter-

galactic barometer of its kind yet.13

13 At Alcyoneus’ redshift, this density amounts to a baryon overden-

sity of ∼10.

Article number, page 10 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

Why do most, if not all, observed non-cluster GRGs have over-

pressured lobes? The top panel of Figure 10 suggests that GRGs

must grow to several Mpc to approach WHIM pressures in their

lobes, and such GRGs are rare. However, the primary reason is

the limited surface brightness sensitivity of all past and current

surveys. Alcyoneus’ lobes are visible in the LoTSS, but not in

the NRAO VLA Sky Survey (NVSS) (Condon et al. 1998) or in

the Westerbork Northern Sky Survey (WENSS) (Rengelink et al.

1997). Their pressure approaches that of the bulk of the WHIM

within an order of magnitude. Lobes with even lower pressure

must be less luminous or more voluminous, and thus will have

even lower surface brightness. It is therefore probable that most

GRG lobes that are in true pressure equilibrium with the WHIM

still lie hidden in the radio sky.

4. Conclusion

1. We reprocess the LoTSS DR2, the latest version of the LO-

FAR’s Northern Sky survey at 144 MHz, by subtracting an-

gularly compact sources and imaging at 60′′ and 90′′ reso-

lution. The resulting images (Oei et al. in prep.) allow us to

explore a new sensitivity regime for radio galaxy lobes, and

thus represent promising data to search for unknown GRGs

of large angular length. We present a sample in forthcoming

work.

2. We discover the rst 5 Mpc GRG, which we dub Alcyoneus.

The projected proper length is lp = 4.99 ± 0.04 Mpc, while

the true proper length is at least lmin = 5.04 ± 0.05 Mpc.

We condently associate the 20.8′ ± 0.15′ radio structure

to an elliptical galaxy with a jet-mode AGN detected in the

DESI Legacy Imaging Surveys DR9: the SDSS DR12 source

J081421.68+522410.0 at J2000 right ascension 123.590372°,

declination 52.402795° and spectroscopic redshift 0.24674±

6 · 10−5.

3. Alcyoneus has a total luminosity density at ν = 144 MHz

of Lν = 8± 1 · 1025 W Hz−1, which is typical for GRGs (per-

centile 45 ± 3%). Alcyoneus’ host has a fairly low stellar

mass and SMBH mass compared with other GRG hosts (per-

centiles 25±9% and 23±11%). This implies that — within the

GRG population — no strong positive correlation between

radio galaxy length and (instantaneous) low-frequency ra-

dio power, stellar mass or SMBH mass can exist.

4. The surrounding sky as imaged by the LoTSS, DESI Legacy

Imaging Surveys, RASS and PSZ suggests that Alcyoneus

does not inhabit a galaxy cluster. According to an SDSS-III

cluster catalogue, the nearest cluster occurs at a comoving

distance of 11 Mpc. A local galaxy number density count

suggests that Alcyoneus instead inhabits a lament of the

Cosmic Web. A low-density environment therefore remains

a possible explanation for Alcyoneus’ formidable size.

5. We develop a new Bayesian model that parametrises in

three dimensions a pair of arbitrarily oriented, optically

thin, doubly truncated conical radio galaxy lobes with con-

stant monochromatic emission coecient. We then gen-

erate the corresponding specic intensity function, taking

into account cosmic expansion, and compare it to data as-

suming Gaussian image noise. We use Metropolis–Hastings

Markov chain Monte Carlo to optimise the parameters,

and thus determine northern and southern lobe volumes of

1.5±0.2 Mpc3 and 1.0±0.2 Mpc3, respectively. In total, the

lobes have an internal energy of ∼1053 J, expelled from the

host galaxy over a Gyr-scale period. The lobe pressures are

4.8 ± 0.3 · 10−16 Pa and 4.9 ± 0.6 · 10−16 Pa, respectively;

these are the lowest measured in radio galaxies yet. Nev-

ertheless, the lobe pressures still exceed a large range of

plausible WHIM pressures. Most likely, the lobes are still

expanding — and Alcyoneus’ struggle for supremacy of the

Cosmos continues.

Acknowledgements. M.S.S.L. Oei warmly thanks Frits Sweijen for coding the

very useful https://github.com/tikk3r/legacystamps.

M.S.S.L. Oei, R.J. van Weeren and A. Botteon acknowledge support from the

VIDI research programme with project number 639.042.729, which is nanced

by The Netherlands Organisation for Scientic Research (NWO). M. Brüggen

acknowledges support from the Deutsche Forschungsgemeinschaft under Ger-

many’s Excellence Strategy — EXC 2121 ‘Quantum Universe’ — 390833306. W.L.

Williams acknowledges support from the CAS–NWO programme for radio as-

tronomy with project number 629.001.024, which is nanced by The Nether-

lands Organisation for Scientic Research (NWO).

The LOFAR is the Low-frequency Array designed and constructed by ASTRON.

It has observing, data processing, and data storage facilities in several coun-

tries, which are owned by various parties (each with their own funding sources),

and which are collectively operated by the ILT Foundation under a joint scien-

tic policy. The ILT resources have beneted 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 Sci-

ence and Higher Education, Poland; the Istituto Nazionale di Astrosica (INAF),

Italy.

The National Radio Astronomy Observatory is a facility of the National Science

Foundation operated under cooperative agreement by Associated Universities,

Inc. CIRADA is funded by a grant from the Canada Foundation for Innovation

2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario,

British Columbia, Alberta, Manitoba and Quebec.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the

Participating Institutions, the National Science Foundation, and the U.S. De-

partment of Energy Oce of Science. The SDSS-III web site is http://www.

sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium

for the Participating Institutions of the SDSS-III Collaboration including the

University of Arizona, the Brazilian Participation Group, Brookhaven National

Laboratory, Carnegie Mellon University, University of Florida, the French Par-

ticipation Group, the German Participation Group, Harvard University, the In-

stituto de Astrosica de Canarias, the Michigan State/Notre Dame/JINA Partic-

ipation Group, Johns Hopkins University, Lawrence Berkeley National Labora-

tory, Max Planck Institute for Astrophysics, Max Planck Institute for Extrater-

restrial Physics, New Mexico State University, New York University, Ohio State

University, Pennsylvania State University, University of Portsmouth, Princeton

University, the Spanish Participation Group, University of Tokyo, University of

Utah, Vanderbilt University, University of Virginia, University of Washington,

and Yale University.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been

made possible through contributions by the Institute for Astronomy, the Uni-

versity of Hawaii, the Pan-STARRS Project Oce, the Max-Planck Society and

its participating institutes, the Max Planck Institute for Astronomy, Heidel-

berg and the Max Planck Institute for Extraterrestrial Physics, Garching, The

Johns Hopkins University, Durham University, the University of Edinburgh, the

Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics,

the Las Cumbres Observatory Global Telescope Network Incorporated, the Na-

tional Central University of Taiwan, the Space Telescope Science Institute, the

National Aeronautics and Space Administration under Grant No. NNX08AR22G

issued through the Planetary Science Division of the NASA Science Mission Di-

rectorate, the National Science Foundation Grant No. AST-1238877, the Univer-

sity of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National

Laboratory, and the Gordon and Betty Moore Foundation.

This publication makes use of data products from the Wide-eld Infrared Sur-

vey Explorer, which is a joint project of the University of California, Los An-

geles, and the Jet Propulsion Laboratory/California Institute of Technology,

funded by the National Aeronautics and Space Administration.

The Legacy Surveys consist of three individual and complementary projects:

the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404;

PIs: David Schlegel and Arjun Dey), the Beijing–Arizona Sky Survey (BASS;

NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall

z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS,

BASS and MzLS together include data obtained, respectively, at the Blanco tele-

scope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok tele-

scope, Steward Observatory, University of Arizona; and the Mayall telescope,

Kitt Peak National Observatory, NOIRLab. The Legacy Surveys project is hon-

ored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt

Peak), a mountain with particular signicance to the Tohono O’odham Na-

tion. NOIRLab is operated by the Association of Universities for Research in

Article number, page 11 of 18

A&A proofs: manuscript no. aanda

Astronomy (AURA) under a cooperative agreement with the National Science

Foundation. This project used data obtained with the Dark Energy Camera

(DECam), which was constructed by the Dark Energy Survey (DES) collabo-

ration. Funding for the DES Projects has been provided by the U.S. Depart-

ment of Energy, the U.S. National Science Foundation, the Ministry of Science

and Education of Spain, the Science and Technology Facilities Council of the

United Kingdom, the Higher Education Funding Council for England, the Na-

tional Center for Supercomputing Applications at the University of Illinois at

Urbana-Champaign, the Kavli Institute of Cosmological Physics at the Univer-

sity of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio

State University, the Mitchell Institute for Fundamental Physics and Astron-

omy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao

Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao

Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Con-

selho Nacional de Desenvolvimento Cientico e Tecnologico and the Ministe-

rio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft

and the Collaborating Institutions in the Dark Energy Survey. The Collaborat-

ing Institutions are Argonne National Laboratory, the University of California

at Santa Cruz, the University of Cambridge, Centro de Investigaciones Ener-

geticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago,

University College London, the DES-Brazil Consortium, the University of Ed-

inburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi Na-

tional Accelerator Laboratory, the University of Illinois at Urbana-Champaign,

the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Fisica d’Altes En-

ergies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Uni-

versität München and the associated Excellence Cluster Universe, the Univer-

sity of Michigan, NSF’s NOIRLab, the University of Nottingham, the Ohio State

University, the University of Pennsylvania, the University of Portsmouth, SLAC

National Accelerator Laboratory, Stanford University, the University of Sussex,

and Texas A&M University. BASS is a key project of the Telescope Access Pro-

gram (TAP), which has been funded by the National Astronomical Observato-

ries of China, the Chinese Academy of Sciences (the Strategic Priority Research

Program “The Emergence of Cosmological Structures” Grant # XDB09000000),

and the Special Fund for Astronomy from the Ministry of Finance. The BASS

is also supported by the External Cooperation Program of Chinese Academy

of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Sci-

ence Foundation (Grant # 11433005). The Legacy Survey team makes use of data

products from the Near-Earth Object Wide-eld Infrared Survey Explorer (NE-

OWISE), which is a project of the Jet Propulsion Laboratory/California Institute

of Technology. NEOWISE is funded by the National Aeronautics and Space Ad-

ministration. The Legacy Surveys imaging of the DESI footprint is supported

by the Director, Oce of Science, Oce of High Energy Physics of the U.S. De-

partment of Energy under Contract No. DE-AC02-05CH1123, by the National

Energy Research Scientic Computing Center, a DOE Oce of Science User

Facility under the same contract; and by the U.S. National Science Foundation,

Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.

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Article number, page 12 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

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Article number, page 13 of 18

A&A proofs: manuscript no. aanda

4.7

4.8

4.9

5.0

5.1

5.2

projected proper length lp (Mpc)

0

5

10

15

20

25

probabilitydensity(Mpc−1)h = 0.677 | ΩM,0 = 0.311 | ΩΛ,0 = 0.689

J1420-0545

Alcyoneus

Fig. A.1: Alcyoneus’ projected proper length just exceeds

that of J1420-0545. The probability that Alcyoneus (green) has

a larger projected proper length than J1420-0545 (grey) (Machal-

ski et al. 2008) is 99.9%. For both GRGs, we take into account

uncertainty in angular length and spectroscopic redshift, as well

as the possibility of peculiar motion along the line of sight.

Appendix A: J1420-0545 comparison

We verify that Alcyoneus is the largest known radio galaxy (RG)

in projection by comparing it with J1420-0545 (Machalski et al.

2008), the literature’s record holder.

The angular lengths of Alcyoneus and J1420-0545 are φ =

20.8′ ± 0.15′ and φ = 17.4′ ± 0.05′, respectively. For J1420-

0545, we adopt the angular length reported by Machalski et al.

(2008) because it lies outside the LoTSS DR2 coverage. The spec-

troscopic redshifts of Alcyoneus and J1420-0545 are zspec =

0.24674 ± 6 · 10−5 and zspec = 0.3067 ± 5 · 10−4, respectively.

For both giants, we assume the peculiar velocity along the line

of sight up to be a Gaussian random variable (RV) with mean

0 and standard deviation 100 km s−1, similar to conditions in

low-mass galaxy clusters.

Equations A.1 describe how to calculate the cosmological red-

shift RV z via the peculiar velocity redshift RV zp:

βp B

up

c

; zp =

√

1 + βp

1 − βp

− 1; z =

1 + zspec

1 + zp

− 1.

(A.1)

Here, c is the speed of light in vacuo. Finally, we calculate the

projected proper length RV lp = rφ (z,M) · φ. Here, rφ is the

angular diameter distance RV, which depends on cosmological

model parameters M. Propagating the uncertainties in angular

length φ, spectroscopic redshift zspec and peculiar velocity

along the line of sight up through Monte Carlo simulation,

the projected proper lengths of Alcyoneus and J1420-0545 are

lp = 4.99 ± 0.04 Mpc and lp = 4.87 ± 0.02 Mpc, respectively.

We show the two projected proper length distributions in

Figure A.1. The probability that Alcyoneus has the largest

projected proper length is 99.9%. This result is insensitive to

plausible changes in cosmological parameters; for example,

the high-H0 (i.e. H0 > 70 km s−1 Mpc−1) cosmology with M =

(

h = 0.7020,ΩBM,0 = 0.0455,ΩM,0 = 0.2720,ΩΛ,0 = 0.7280

)

yields a probability of 99.8%.

Appendix B: Inclination angle comparison

Under what conditions is Alcyoneus not only the largest GRG

in the plane of the sky, but also in three dimensions? To an-

swer this question, we compare Alcyoneus to the ve previ-

ously known GRGs with projected proper lengths above 4 Mpc,

0

15

30

45

60

75

90

inclination angle Alcyoneus θ (°)

0

15

30

45

60

75

90

inclinationanglechallengerθc(°)θmax,c (θ) for five GRGs

lp,c = 4.87 Mpc

lp,c = 4.72 Mpc

lp,c = 4.60 Mpc

lp,c = 4.35 Mpc

lp,c = 4.11 Mpc

Fig. A.2: When is Alcyoneus not only the largest GRG in

the plane of the sky, but also in three dimensions? Alcy-

oneus’ inclination angle θ is not well determined, and there-

fore the full range of possibilities is shown on the horizontal

axis. To surpass Alcyoneus in true proper length, a challenger

must have an inclination angle (vertical axis) of at most Alcy-

oneus’ (grey dotted line). More specically, as a function of θ,

we show the maximum inclination angle for which challengers

with a projected proper length lp,c > 4 Mpc trump Alcyoneus

(coloured curves). The shaded areas of parameter space repre-

sent regimes with a particularly straightforward interpretation.

One can imagine populating the graph with ve points (located

along the same vertical line), representing the ground-truth in-

clination angles of Alcyoneus and its ve challengers. If any of

these points fall in the red-shaded area, Alcyoneus is not the

largest GRG in 3D. If all points fall in the green-shaded area,

Alcyoneus is the largest GRG in 3D.

which we dub challengers. A challenger surpasses Alcyoneus in

true proper length when

lc > l, or

lp,c

sin θc

>

lp

sin θ

, or sin θc <

lp,c

lp

sin θ,

(B.1)

where lc, lp,c and θc are the challenger’s true proper length,

projected proper length and inclination angle, respectively. Be-

cause the arcsine is a monotonically increasing function, a chal-

lenger surpasses Alcyoneus if its inclination angle obeys

θc < θmax,c (θ) , where θmax,c (θ) B arcsin

(

lp,c

lp

sin θ

)

.

(B.2)

In Figure A.2, we show θmax,c (θ) for the ve challengers with

lp,c ∈ {4.11 Mpc, 4.35 Mpc, 4.60 Mpc, 4.72 Mpc, 4.87 Mpc}

(coloured curves). Alcyoneus is least likely to be the longest

GRG in 3D when its true proper length equals its projected

proper length; i.e. when θ = 90°. The challengers then surpass

Alcyoneus in true proper length when their inclination angles

are less than 55°, 61°, 67°, 71° and 77°, respectively. For θ < 90°,

the conditions are more stringent.

Article number, page 14 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

The third and fourth largest challengers, whose respec-

tive SDSS DR12 host names are J100601.73+345410.5 and

J093139.03+320400.1, harbour quasars in their host galaxies. If

small inclination angles distinguish quasars from non-quasar

AGN, as proposed by the unication model (e.g. Hardcastle &

Croston 2020), these two challengers may well be the longest

radio galaxies in three dimensions.

Appendix C: Lobe volumes with truncated double

cone model

Appendix C.1: Synopsis

We build a Metropolis–Hastings Markov chain Monte Carlo

(MH MCMC) model, similar in spirit to the model of Boxelaar

et al. (2021) for galaxy cluster halos, in order to formalise the

determination of RG lobe volumes from a radio image. To this

end, we introduce a parametrisation of a pair of 3D radio galaxy

lobes, and explore the corresponding parameter space via the

Metropolis algorithm.14 For each parameter tuple encountered

during exploration, we rst calculate the monochromatic emis-

sion coecient (MEC) function of the lobes on a uniform 3D

grid representing a proper (rather than comoving) cubical vol-

ume. The RG is assumed to be far enough from the observer

that the conversion to a 2D image through ray tracing simpli-

es to summing up the cube’s voxels along one dimension, and

applying a cosmological attenuation factor. This factor depends

on the galaxy’s cosmological redshift, which is a hyperparame-

ter. We blur the model image to the resolution of the observed

image, which is also a hyperparameter. Next, we calculate the

likelihood that the observed image is a noisy version of the

proposed model image. The imaged sky region is divided into

patches with a solid angle equal to the PSF solid angle; the noise

per patch is then assumed to be an independent Gaussian RV.

These RVs have zero mean and share the same variance, which

is another hyperparameter — typically obtained from the ob-

served image. We choose a uniform prior over the full phys-

ically realisable part of parameter space. The resulting poste-

rior, which contains both geometric and radiative parameters,

allows one to calculate probability distributions for many inter-

esting quantities, such as the RG’s lobe volumes and inclination

angle. The inferences depend weakly on cosmological parame-

ters M. Furthermore, their reliability depends signicantly on

the validity of the model assumptions.

Appendix C.2: Model

Appendix C.2.1: Geometry

We model each lobe in 3D with a truncated right circular cone

with apex O ∈ R3, central axis unit vector â ∈ S2 and opening

angle γ ∈ [0, π2 ], as in Figure 9. The lobes share the same O,

which is the RG host location. Each central axis unit vector can

be parametrised through a position angle ϕ ∈ [0, 2π) and an in-

clination angle θ ∈ [0, π]. Each cone is truncated twice, through

planes that intersect the cone perpendicularly to its central axis.

Thus, each truncation is parametrised by the distance from the

apex to the point where the plane intersects the central axis.

The two inner (di,1, di,2 ∈ R≥0) and two outer (do,1, do,2 ∈ R≥0)

truncation distances are parameters that we allow to vary inde-

pendently, with the only constraint that each inner truncation

14 The more general Metropolis–Hastings variant need not be consid-

ered, as we work with a symmetric proposal distribution.

distance cannot exceed the corresponding outer truncation dis-

tance.

Appendix C.2.2: Radiative processes

The radiative formulation of our model is among the simplest

possible. The radio emission from the lobes is synchrotron ra-

diation. We consider the lobes to be perfectly optically thin:

we neglect synchrotron self-absorption. The proper MEC is as-

sumed spatially constant throughout a lobe, though possibly

dierent among lobes; this leads to parameters jν,1, jν,2 ∈ R≥0.

The relationship between the specic intensity Iν (in direction

r̂ at central frequency νc) and the MEC jν (in direction r̂ at cos-

mological redshift z and rest-frame frequency ν = νc (1 + z)) is

Iν (r̂, νc) =

∫ ∞

0

jν (r̂, z (l) , νc (1 + z (l)))

(1 + z (l))3

dl ≈ jν (ν)∆l (r̂)

(1 + z)3

,

(C.1)

where l represents proper length. The approximation is valid

for a lobe with a spatially constant MEC that is small enough

to assume a constant redshift for it. ∆l(r̂) is the proper length

of the line of sight through the lobe in direction r̂. The inferred

MECs jν,1 (ν) , jν,2 (ν) thus correspond to rest-frame frequency ν.

Appendix C.3: Proposal distribution

In order to explore the posterior distribution on the parameter

space, we follow the Metropolis algorithm. The Metropolis al-

gorithm assumes a symmetric proposal distribution.

Appendix C.3.1: Radio galaxy axis direction

To propose a new RG axis direction given the current one whilst

satisfying the symmetry assumption, we perform a trick. We

populate the unit sphere with N ∈ N≥1 points (interpreted

as directions) drawn from a uniform distribution. Of these N

directions, the proposed axis direction is taken to be the one

closest to the current axis direction (in the great-circle distance

sense). Note that this approach evidently satises the criterion

that proposing the new direction given the old one is equally

likely as proposing the old direction given the new one. Also

note that the distribution of the angular distance between cur-

rent and proposed axis directions is determined solely by N.

In the following paragraphs, we rst review how to perform

uniform sampling of the unit two-sphere. More explicitly than

in Scott & Tout (1989), we then derive the distribution of the

angular distance between a reference point and the nearest of

N uniformly drawn other points. The result is a continuous uni-

variate distribution with a single parameter N and nite support

(0, π). Finally, we present the mode, median and maximum like-

lihood estimator of N. As far as we know, these properties are

new to the literature.

Uniform sampling of S2 Let us place a number of points

uniformly on the celestial sphere S2. The spherical coordinates

of such points are given by the RVs (Φ,Θ), where Φ denotes

position angle and Θ denotes inclination angle. As all posi-

tion angles are equally likely, the distribution of Φ is uniform:

Φ ∼ U[0, 2π). In order to eect a uniform number density, the

probability that a point lies within a rectangle of width dϕ and

height dθ in the (ϕ, θ)-plane equals the ratio of the solid angle of

Article number, page 15 of 18

A&A proofs: manuscript no. aanda

the corresponding sky patch and the sphere’s total solid angle:

P(ϕ ≤ Φ < ϕ + dϕ, θ ≤ Θ < θ + dθ) = sin θ dϕ dθ

4π

.

(C.2)

The probability that the inclination angle is found somewhere

in the interval [θ, θ + dθ), regardless of the position angle, is

therefore

P(θ ≤ Θ < θ + dθ) = dFΘ(θ) = fΘ(θ)dθ

=

∫ 2π

0

sin θ dθ

4π

dϕ =

1

2

sin θ dθ,

(C.3)

where FΘ is the cumulative distribution function (CDF) of Θ,

and fΘ the associated probability density function (PDF). So,

fΘ(θ) =

1

2

sin θ; FΘ(θ) B

∫ θ

0

fΘ(θ′) dθ′ =

1 − cos θ

2

.

(C.4)

Nearest-neighbour angular distance distribution Pick a

reference point and stochastically introduce N other points in

above fashion, which we dub its neighbours. We now derive the

PDF of the angular distance to the nearest neighbour (NNAD).

Let (ϕref , θref) be the coordinates of the reference point and let

(ϕ, θ) be the coordinates of one of the neighbours. Without

loss of generality, due to spherical symmetry, we can choose to

place the reference point in the direction towards the observer:

θref = 0. (Note that ϕref is meaningless in this case.) The angular

distance between two points on S2 is given by the great-circle

distance ξ. For our choice of reference point, we immediately

see that ξ(ϕref , θref , ϕ, θ) = θ. Because θ is a realisation of Θ, ξ

too can be regarded as a realisation of an RV, which we call Ξ.

Evidently, the PDF fΞ(ξ) = fΘ(ξ) and the CDF FΞ(ξ) = FΘ(ξ).

Now consider the generation of N points, whose angular dis-

tances to the reference point are the RVs {Ξi} B {Ξ1, ...,ΞN}.

The NNAD RV M is the minimum of this set: M B min{Ξi}.

What are the CDF FM and PDF fM of M?

FM(µ) B P(M ≤ µ) = P(minimum of {Ξi} ≤ µ)

= P(at least one of the set {Ξi} ≤ µ)

= 1 − P(none of the set {Ξi} ≤ µ)

= 1 − P(all of the set {Ξi} > µ).

(C.5)

Because the {Ξi} are independent and identically distributed,

FM(µ) = 1 −

N∏

i=1

P (Ξi > µ)

= 1 − PN(Ξ > µ) = 1 − (1 − FΞ(µ))N .

(C.6)

By substitution, the application of a trigonometric identity and

dierentiation to µ, we obtain the CDF and PDF of M:

FM (µ) = 1 − cos2N

(

µ

2

)

; fM(µ) = N sin

(

µ

2

)

cos2N−1

(

µ

2

)

.

(C.7)

In Figure C.1, we show this PDF for various values of N.

The mode of M (i.e. the most probable NNAD), µmode, is the solu-

tion to d fM

dµ (µmode) = 0. The median of M, µmedian, is the solution

to FM(µmedian) = 12 . Hence,

µmode = arccos

(

1 − 1

N

)

; µmedian = arccos

(

21−

1

N − 1

)

.

(C.8)

As common sense dictates, both equal π2 for N = 1 and tend to

0 as N → ∞. We nd the mean of M through integration by

parts:

E [M] B

∫ π

0

µ fM(µ) dµ =

∫ π

0

µ dFM(µ)

=

[

µFM(µ)

]π

0

−

∫ π

0

FM(µ) dµ

=

∫ π

0

cos2N

(

µ

2

)

dµ = 2

∫ π

2

0

cos2N (µ) dµ.

(C.9)

Again via integration by parts,

E [M] = π

N∏

k=1

2k − 1

2k

=

π

22N

(

2N

N

)

.

(C.10)

Maximum likelihood estimation A typical application is

the estimation of N in the PDF fM(µ | N) (Equation C.7) us-

ing data. Let us assume we have measured k NNADs, denoted

by {µ1, ..., µk}. Let the joint PDF or likelihood be

L(N) B

k∏

i=1

fM (µi | N)

=

( N

2N

)k

k∏

i=1

sin µi (cos µi + 1)N−1.

(C.11)

To nd NMLE, we look for the value of N that maximises L(N).

To simplify the algebra, we could however equally well max-

imise a k-th of the natural logarithm of the likelihood, or the

average log-likelihood l̂ B k−1 lnL(N), because the logarithm is

a monotonically increasing function:

l̂(N) B

1

k

lnL(N) = lnN − N ln 2

+

1

k

k∑

i=1

ln sin µi + (N − 1) ln(cos µi + 1).

(C.12)

We nd NMLE by solving dl̂

dN (NMLE) = 0. This leads to

NMLE =

ln 2 − 1k

k∑

i=1

ln(cos µi + 1)

−1

.

(C.13)

An easy limit to evaluate is the case when µ1, ..., µk → 0. In

such case, cos µi → 1, and so 1k

∑k

i=1 ln(cos µi + 1)→ ln 2. Then,

NMLE → (0+)−1 → ∞. This is expected behaviour: when all

measured NNADs approach 0, the number of points distributed

on the sphere must be approaching innity.

Appendix C.3.2: Other parameters

The other proposal parameters are each drawn from indepen-

dent normal distributions centred around the current parameter

values. These proposal distributions are evidently symmetric,

but have support over the full real line, so that forbidden pa-

rameter values can in principle be proposed. As a remedy, we set

the prior probability density of the proposed parameter set to 0

when the proposed opening angle is negative or exceeds π2 rad,

at least one of the proposed MECs is negative, or when at least

one of the proposed inner truncation distances is negative or

Article number, page 16 of 18

Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc

0

1

2

3

4

5

6

7

8

nearest-neighbour angular distance µ (°)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

probabilitydensityfM(µ)(deg−1)N = 1 · 103 | n = 0.024 deg−2

N = 2 · 103 | n = 0.048 deg−2

N = 3 · 103 | n = 0.073 deg−2

N = 4 · 103 | n = 0.097 deg−2

N = 5 · 103 | n = 0.121 deg−2

N = 6 · 103 | n = 0.145 deg−2

Fig. C.1: Probability density functions (PDFs) of the nearest-neighbour angular distance (NNAD) RV M between some xed point

and N other points distributed randomly over the celestial sphere. As the sphere gets more densely packed, the probability of

nding a small M increases. For each N, we provide the mean point number density n.

exceeds the corresponding proposed outer truncation distance.

In such cases, the posterior probability density is 0 too, as it is

proportional to the prior probability density. Consequently, the

Metropolis acceptance probability vanishes and the proposal is

rejected. We do not enter forbidden regions of parameter space.

The condition of detailed balance is still respected: probability

densities for transitioning towards the forbidden region are 0,

just as probability densities for being in the forbidden region.

Appendix C.4: Likelihood

We assume the likelihood to be Gaussian. To avoid dimension-

ality errors, we multiply the likelihood by a constant before we

take the logarithm:

ln

(

L ·

(

σ

√

2π

)Nr)

= − Nr

2σ2Np

Np∑

i=1

(

Iν,o [i] − Iν,m [i]

)2 .

(C.14)

Here, σ is the image noise, Nr ∈ R≥0 is the number of reso-

lution elements in the image, Np ∈ N is the number of pixels

in the image, and Iν,o [i] and Iν,m [i] are the i-th pixel values of

the observed and modelled image, respectively. For simplicity,

one may multiply the likelihood by a constant factor (or, equiva-

lently, add a constant term to the log-likelihood): the acceptance

ratio will remain the same, and the MH MCMC runs correctly.

Appendix C.5: Results for Alcyoneus

We apply the Bayesian model to the 90′′ LoTSS DR2 image of Al-

cyoneus, shown in the top panel of Figure 9. Thus, the hyperpa-

rameters are z = 0.24674, νc = 144MHz (so that ν = 180MHz),

θFWHM = 90′′, N = 750 and σ =

√

2 · 1.16 Jy deg−2. We set

the image noise to

√

2 times the true image noise to account

for model incompleteness. This factor follows by assuming that

the inability of the model to produce the true lobe morphol-

ogy yields (Gaussian) errors comparable to the image noise. To

speed up inference, we downsample the image of 2,048 by 2,048

pixels by a factor 16 along each dimension. We run our MH

MCMC for 10,000 steps, and discard the rst 1,500 steps due to

burn-in.

Table C.1 lists the obtained maximum a posteriori probabil-

ity (MAP) estimates and posterior mean and standard deviation

(SD) of the parameters.

Table C.1: Maximum a posteriori probability (MAP) estimates

and posterior mean and standard deviation (SD) of the param-

eters from the Bayesian, doubly truncated, conical radio galaxy

lobe model of Section 3.9.

parameter

MAP estimate

posterior mean and SD

ϕ1

307°

307 ± 1°

ϕ2

140°

139 ± 2°

|θ1 − 90°|

54°

51 ± 2°

|θ2 − 90°|

25°

18 ± 7°

γ1

9°

10 ± 1°

γ2

24°

26 ± 2°

di,1

2.7 Mpc

2.6 ± 0.2 Mpc

do,1

4.3 Mpc

4.0 ± 0.2 Mpc

di,2

1.6 Mpc

1.5 ± 0.1 Mpc

do,2

2.0 Mpc

2.0 ± 0.1 Mpc

jν,1 (ν)

17 Jy deg−2 Mpc−1

17 ± 2 Jy deg−2 Mpc−1

jν,2 (ν)

22 Jy deg−2 Mpc−1

18 ± 3 Jy deg−2 Mpc−1

The proper volumes V1 and V2 are derived quantities:

V =

π

3

tan2 γ

(

d3o − d3i

)

,

(C.15)

just like the ux densities Fν,1 (νc) and Fν,2 (νc) at central fre-

quency νc:

Fν (νc) =

jν (ν)V

(1 + z)3 r2φ (z)

.

(C.16)

Together, V and Fν(νc) imply a lobe pressure P and a magnetic

eld strength B, which are additional derived quantities that we

calculate through pysynch.

Table C.2 lists the obtained MAP estimates and posterior mean

and SD of the derived quantities.

Article number, page 17 of 18

A&A proofs: manuscript no. aanda

5

10

15

20

25

30

monochromatic emission coefficient jν (ν)

(

Jy deg−2 Mpc−1

)

0.5

1.0

1.5

2.0

2.5

3.0

properlobevolumeV(Mpc3)Fν (νc) = 63 mJy

Fν (νc) = 44 mJy

northern lobe

southern lobe

Fig. C.2: OurBayesianmodel yields strongly correlated es-

timates for jν (ν) and V that reproduce the observed lobe

ux densities. We show MECs jν (ν) at ν = 180 MHz and

proper volumes V of Metropolis–Hastings Markov chain Monte

Carlo samples for the northern lobe (purple dots) and south-

ern lobe (orange dots). The curves represent all combinations

( jν (ν) ,V) that correspond to a particular ux density at the

LoTSS central wavelength νc = 144 MHz. We show the ob-

served northern lobe ux density (purple curve) and the ob-

served southern lobe ux density (orange curve).

Table C.2: Maximum a posteriori probability (MAP) estimates

and posterior mean and standard deviation (SD) of derived

quantities from the Bayesian, doubly truncated, conical radio

galaxy lobe model of Section 3.9.

derived quantity MAP estimate

posterior mean and SD

∆ϕ

167°

168 ± 2°

V1

1.5 Mpc3

1.5 ± 0.2 Mpc3

V2

0.8 Mpc3

1.0 ± 0.2 Mpc3

Fν,1 (νc)

63 mJy

63 ± 4 mJy

Fν,2 (νc)

44 mJy

45 ± 5 mJy

Pmin,1

4.7 · 10−16 Pa

4.8 ± 0.3 · 10−16 Pa

Pmin,2

5.4 · 10−16 Pa

5.0 ± 0.6 · 10−16 Pa

Peq,1

4.8 · 10−16 Pa

4.9 ± 0.3 · 10−16 Pa

Peq,2

5.4 · 10−16 Pa

5.0 ± 0.6 · 10−16 Pa

Bmin,1

45 pT

45 ± 1 pT

Bmin,2

48 pT

46 ± 3 pT

Beq,1

42 pT

43 ± 1 pT

Beq,2

45 pT

43 ± 3 pT

Emin,1

6.3 · 1052 J

6.2 ± 0.4 · 1052 J

Emin,2

3.7 · 1052 J

4.4 ± 0.6 · 1052 J

Eeq,1

6.4 · 1052 J

6.3 ± 0.4 · 1052 J

Eeq,2

3.8 · 1052 J

4.4 ± 0.6 · 1052 J

The uncertainties of the parameters and derived quantities

reported in Tables C.1 and C.2 are not necessarily independent.

To demonstrate this, we present MECs and volumes from the

MH MCMC samples in Figure C.2. MECs and volumes do not

vary independently, because their product is proportional to

ux density (see Equation C.16); only realistic ux densities

correspond to high-likelihood model images.

Finally, we explore a simpler variation of the model, in

which we force the lobes to be coaxial. In such a case, the true

proper length l and projected proper length lp are additional

derived quantities:

l =

do,1 + do,2

cos γ

;

lp = l sin θ.

(C.17)

For Alcyoneus, this simpler model does not provide a good t

to the data.

Article number, page 18 of 18