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.
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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).
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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
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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.
References
Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS,
182, 543
Alam, S., Albareti, F. D., Prieto, C. A., et al. 2015, ApJSS, 219, 12
Andernach, H., Feretti, L., Giovannini, G., et al. 1992, A&AS, 93, 331
Andernach, H., Jiménez-Andrade, E. F., & Willis, A. G. 2021, Galaxies, 9
Bassani, L., Ursini, F., Malizia, A., et al. 2021, MNRAS, 500, 3111
Beck, R. & Krause, M. 2005, Astronomische Nachrichten, 326, 414
Best, P. N. & Heckman, T. M. 2012, MNRAS, 421, 1569
Best, P. N., Ker, L. M., Simpson, C., Rigby, E. E., & Sabater, J. 2014, MNRAS, 445,
955
Blandford, R. D. & Rees, M. J. 1974, MNRAS, 169, 395
Bonnarel, F., Fernique, P., Bienaymé, O., et al. 2000, A&AS, 143, 33
Boxelaar, J. M., van Weeren, R. J., & Botteon, A. 2021, Astronomy and Comput-
ing, 35, 100464
Brüggen, M., Reiprich, T. H., Bulbul, E., et al. 2021, A&A, 647, A3
Burbidge, G. R. 1956, ApJ, 124, 416
Cantwell, T. M., Bray, J. D., Croston, J. H., et al. 2020, MNRAS, 495, 143
Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints,
arXiv:1612.05560
Chang, Y.-Y., van der Wel, A., da Cunha, E., & Rix, H.-W. 2015, ApJS, 219, 8
Condon, J. J., Cotton, W. D., Fomalont, E. B., et al. 2012, ApJ, 758, 23
Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
Cutri, R. M. & et al. 2012, VizieR Online Data Catalog, II/311
Dabhade, P., Mahato, M., Bagchi, J., et al. 2020a, A&A, 642, A153
Dabhade, P., Röttgering, H. J. A., Bagchi, J., et al. 2020b, A&A, 635, A5
Delhaize, J., Heywood, I., Prescott, M., et al. 2021, MNRAS, 501, 3833
Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168
Galvin, T. J., Huynh, M. T., Norris, R. P., et al. 2020, MNRAS, 497, 2730
Gürkan, G., Hardcastle, M. J., & Jarvis, M. J. 2014, MNRAS, 438, 1149
Hahn, O., Porciani, C., Carollo, C. M., & Dekel, A. 2007, MNRAS, 375, 489
Hardcastle, M. J. 2018, MNRAS, 475, 2768
Hardcastle, M. J., Alexander, P., Pooley, G. G., & Riley, J. M. 1998a, MNRAS, 296,
445
Hardcastle, M. J. & Croston, J. H. 2020, New A Rev., 88, 101539
Hardcastle, M. J., Worrall, D. M., & Birkinshaw, M. 1998b, MNRAS, 296, 1098
Heckman, T. M. & Best, P. N. 2014, ARA&A, 52, 589
Heesen, V., Croston, J. H., Morganti, R., et al. 2018, MNRAS, 474, 5049
Ishwara-Chandra, C. H. & Saikia, D. J. 1999, MNRAS, 309, 100
Ishwara-Chandra, C. H., Taylor, A. R., Green, D. A., et al. 2020, MNRAS, 497,
5383
Jamrozy, M., Konar, C., Machalski, J., & Saikia, D. J. 2008, MNRAS, 385, 1286
Jamrozy, M. & Machalski, J. 2002, in Astronomical Society of the Pacic Con-
ference Series, Vol. 284, IAU Colloq. 184: AGN Surveys, ed. R. F. Green, E. Y.
Khachikian, & D. B. Sanders, 295
Jamrozy, M., Machalski, J., Mack, K. H., & Klein, U. 2005, A&A, 433, 467
Kormendy, J. & Ho, L. C. 2013, ARA&A, 51, 511
Kuminski, E. & Shamir, L. 2016, ApJS, 223, 20
Kuźmicz, A. & Jamrozy, M. 2021, ApJS, 253, 25
Kuźmicz, A., Jamrozy, M., Bronarska, K., Janda-Boczar, K., & Saikia, D. J. 2018,
ApJS, 238, 9
Lacy, M., Baum, S. A., Chandler, C. J., et al. 2020, PASP, 132, 035001
Lacy, M., Rawlings, S., Saunders, R., & Warner, P. J. 1993, MNRAS, 264, 721
Laing, R. A. & Bridle, A. H. 2014, MNRAS, 437, 3405
Lara, L., Mack, K. H., Lacy, M., et al. 2000, A&A, 356, 63
Machalski, J. 2011, MNRAS, 413, 2429
Machalski, J. & Jamrozy, M. 2000, A&A, 363, L17
Machalski, J. & Jamrozy, M. 2006, A&A, 454, 95
Machalski, J., Jamrozy, M., & Zola, S. 2001, A&A, 371, 445
Machalski, J., Jamrozy, M., Zola, S., & Koziel, D. 2006, A&A, 454, 85
Machalski, J., Koziel-Wierzbowska, D., & Jamrozy, M. 2007, AcA, 57, 227
Machalski, J., Kozieł-Wierzbowska, D., Jamrozy, M., & Saikia, D. J. 2008, ApJ,
679, 149
Mack, K. H., Klein, U., O’Dea, C. P., Willis, A. G., & Saripalli, L. 1998, A&A, 329,
431
Mahato, M., Dabhade, P., Saikia, D.
J.,
et al. 2021, arXiv e-prints,
arXiv:2111.11905
Malarecki, J. M., Staveley-Smith, L., Saripalli, L., et al. 2013, MNRAS, 432, 200
Masini, A., Celotti, A., Grandi, P., Moravec, E., & Williams, W. L. 2021, A&A,
650, A51
Myers, S. T. & Spangler, S. R. 1985, ApJ, 291, 52
Oei, M. S. S. L., van Weeren, R. J., de Gasperin, F., et al. in prep.
Oringa, A. R., McKinley, B., Hurley-Walker, N., et al. 2014, MNRAS, 444, 606
Oringa, A. R. & Smirnov, O. 2017, MNRAS, 471, 301
Pacholczyk, A. G. 1970, Radio astrophysics. Nonthermal processes in galactic
and extragalactic sources
Parma, P., de Ruiter, H. R., Mack, K. H., et al. 1996, A&A, 311, 49
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A27
Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
Pracy, M. B., Ching, J. H. Y., Sadler, E. M., et al. 2016, MNRAS, 460, 2
Rengelink, R. B., Tang, Y., de Bruyn, A. G., et al. 1997, A&AS, 124, 259
Ringermacher, H. I. & Mead, L. R. 2009, MNRAS, 397, 164
Rybicki, G. B. & Lightman, A. P. 1986, Radiative Processes in Astrophysics
Safouris, V., Subrahmanyan, R., Bicknell, G. V., & Saripalli, L. 2009, MNRAS, 393,
2
Saikia, D. J., Konar, C., & Kulkarni, V. K. 2006, MNRAS, 366, 1391
Salim, S., Boquien, M., & Lee, J. C. 2018, ApJ, 859, 11
Saripalli, L., Subrahmanyan, R., & Udaya Shankar, N. 2002, ApJ, 565, 256
Schaye, J., Dalla Vecchia, C., Booth, C. M., et al. 2010, MNRAS, 402, 1536
Schoenmakers, A. P., Mack, K. H., de Bruyn, A. G., et al. 2000, A&AS, 146, 293
Schoenmakers, A. P., Mack, K. H., Lara, L., et al. 1998, A&A, 336, 455
Scott, D. & Tout, C. A. 1989, MNRAS, 241, 109
Sebastian, B., Ishwara-Chandra, C. H., Joshi, R., & Wadadekar, Y. 2018, MNRAS,
473, 4926
Shimwell, T. W., Hardcastle, M. J., & Tasse, C. in prep.
Shimwell, T. W., Röttgering, H. J. A., Best, P. N., et al. 2017, A&A, 598, A104
Shimwell, T. W., Tasse, C., Hardcastle, M. J., et al. 2019, A&A, 622, A1
Soltan, A. 1982, MNRAS, 200, 115
Strauss, M. A., Weinberg, D. H., Lupton, R. H., et al. 2002, AJ, 124, 1810
Subrahmanyan, R., Hunstead, R. W., Cox, N. L. J., & McIntyre, V. 2006, ApJ, 636,
172
Subrahmanyan, R., Saripalli, L., & Hunstead, R. W. 1996, MNRAS, 279, 257
Subrahmanyan, R., Saripalli, L., Safouris, V., & Hunstead, R. W. 2008, ApJ, 677,
63
Tamhane, P., Wadadekar, Y., Basu, A., et al. 2015, MNRAS, 453, 2438
Tang, H., Scaife, A. M. M., Wong, O. I., et al. 2020, MNRAS, 499, 68
Tasse, C., Hugo, B., Mirmont, M., et al. 2018, A&A, 611, A87
van der Tol, S., Veenboer, B., & Oringa, A. R. 2018, A&A, 616, A27
Article number, page 12 of 18
Martijn S.S.L. Oei et al.: The discovery of a radio galaxy of at least 5 Mpc
van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2
van Weeren, R. J., Shimwell, T. W., Botteon, A., et al. 2021, A&A, 651, A115
Voges, W., Aschenbach, B., Boller, T., et al. 1999, A&A, 349, 389
Wen, Z. L., Han, J. L., & Liu, F. S. 2012, ApJS, 199, 34
Williams, W. L., Calistro Rivera, G., Best, P. N., et al. 2018, MNRAS, 475, 3429
Willis, A. G., Strom, R. G., & Wilson, A. S. 1974, Nature, 250, 625
Zou, H., Gao, J., Xu, X., et al. 2021, ApJS, 253, 56
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
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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