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 N<R (R) within a sphere of comoving 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 N<R (R) for R = 5 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 N<R (R) as a proxy for the LSS total matter density around 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 N<R (R = 5 Mpc) (1) 0.0 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 N<R (R = 10 Mpc) (1) 0.00 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 N<R (R) within a sphere of comoving radius R = 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. 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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 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