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Perseus Cluster

The Perseus Cluster, also known as Abell 426, is a massive galaxy cluster located approximately 250 million light-years from Earth in the constellation Perseus, containing thousands of galaxies bound by gravity within a vast cloud of hot intracluster gas. At its center lies the active galaxy NGC 1275, which harbors a supermassive black hole that drives powerful outflows and jets, contributing to the cluster's dynamic structure. With a total mass exceeding 660 trillion times that of the Sun, it ranks among the most massive nearby galaxy clusters and is a key laboratory for studying dark matter, galaxy evolution, and intracluster medium processes. This cluster, part of the larger Pisces-Perseus supercluster, spans about 11.6 million light-years across and exhibits one of the most luminous emissions in the sky due to its multimillion-degree gas heated by and from the central . Notable features include enormous cavities or "bubbles" in the gas, carved by relativistic jets from , and pressure waves—effectively sound waves at frequencies far below human hearing—that propagate through the medium, offering insights into energy transport in clusters. Observations across wavelengths, from radio to , reveal its role in cosmic , with the cluster's of around 5,366 km/s indicating its via the Hubble flow.

Overview

Discovery and historical observations

The Perseus Cluster, cataloged optically as Abell 426, was first identified in 1958 by George O. Abell through a systematic survey of rich galaxy clusters on the Society-Palomar Observatory Sky Survey plates. Abell's catalog highlighted Abell 426 as a prominent concentration of galaxies in the constellation Perseus, though its full significance as a massive cluster became apparent only with later multiwavelength observations. The cluster's X-ray emission was detected for the first time on March 1, 1970, during an rocket flight equipped with proportional counters launched from . This serendipitous discovery, reported by Fritz et al., identified a strong source (designated Per XR-1) centered near the galaxy , marking the initial evidence of hot intracluster gas in the Perseus Cluster. Subsequent confirmation came from the satellite, launched in 1970, which in 1972 observations revealed the source as extended and the brightest -emitting cluster in the sky, with emission spanning several arcminutes and associated with the cluster's core. These data, analyzed by Forman et al., demonstrated that the luminosity originated from diffuse hot rather than solely the central . Advancing into the late 1970s, the Einstein Observatory (launched 1978) provided the first high-resolution X-ray images of the Perseus Cluster in 1979, resolving the emission peak around —the central galaxy hosting the radio source 3C 84—and hinting at cooling gas in the core through spectral analysis with its Solid State Spectrometer. The 1990s brought detailed mapping with the ROSAT satellite (1990–1999), whose Position Sensitive Proportional Counter and High-Resolution Imager observations in the early 1990s uncovered an extensive X-ray halo extending over 1.3 degrees, revealing substructure in the and confirming the cluster's role as a prototype for cooling-flow systems. Böhringer et al.'s 1993 ROSAT study emphasized the halo's symmetry and brightness, attributing it to thermal emission from a massive hot gas envelope. A pivotal milestone occurred in 2003 with a deep exposure of nearly 200 kiloseconds on the cluster core, led by Fabian et al., which imaged unprecedented details including outward-propagating fronts approximately 30 kpc from and concentric "ripples" in the extending to 50 kpc. These features, collected using Chandra's Advanced CCD Imaging Spectrometer, suggested dynamic interactions such as sound waves generated by recurrent outbursts from the central , providing the first direct evidence of active feedback mechanisms regulating the . This observation built on the timeline of major telescopes: Uhuru for initial detection (1970–1973), Einstein for imaging (1978–1981), ROSAT for halo mapping (1990s), and for high-resolution dynamics (2000s onward).

Location and basic parameters

The Perseus Cluster lies in the constellation Perseus, with equatorial coordinates of right ascension 03h 18m 50.3s and declination +41° 30′ 08″ (J2000 epoch). These coordinates mark the central position as defined in the Abell catalog for this rich cluster of galaxies. The cluster spans an angular diameter of 863 arcminutes, reflecting its extensive apparent size on the sky due to its irregular morphology and proximity. It is characterized by a richness class of 2, indicating a substantial population of galaxies within the standard magnitude limits for cluster classification, and a Bautz–Morgan classification of II-III, which denotes an intermediate level of dominance by the brightest central galaxies. Over 1,000 have been identified as members, contributing to its status as one of the richer nearby clusters. Overall, the Perseus Cluster exhibits a rich, irregular structure with a prominent bright core centered on the dominant galaxy NGC 1275. At an estimated distance of 73.6 Mpc, it provides a key nearby laboratory for studying cluster dynamics.

Physical properties

Mass and size

The Perseus Cluster possesses a total mass of (8.6 \pm 0.9) \times 10^{14} solar masses within r_{200}, derived from analyses of X-ray gas dynamics under the assumption of hydrostatic equilibrium and corroborated by weak gravitational lensing measurements of the cluster's mass profile. These methods reveal a gravitationally bound system dominated by dark matter, with the intracluster medium and galaxies contributing lesser fractions to the overall mass budget. The cluster's dark matter component accounts for about 85% of the total mass, as inferred from comparisons between dynamical, X-ray, and luminous matter distributions. Recent Euclid observations (as of May 2025) confirm this mass estimate within r_{200} = 1.96 Mpc. The spatial structure of the Perseus Cluster is defined by a core radius of about 200 kpc, marking the scale over which the intracluster gas density profile flattens in the central regions, and an r_{200} of 1.96 Mpc, encompassing the volume where the mean overdensity relative to the is 200. These dimensions highlight the cluster's extended halo, with the core exhibiting enhanced density due to cooling processes near the central galaxy, while the r_{200} boundary delineates the transition to infalling material. Dynamical estimates of the cluster's mass further support these findings through observations of member motions. The line-of-sight velocity dispersion of galaxies is approximately 1170 km/s, reflecting the depth of the well. Applying the provides an independent mass calculation, assuming the cluster is in approximate dynamical : M = \frac{3 \sigma^2 R}{G} where \sigma is the velocity dispersion, R is the virial radius, and G is the . Substituting \sigma \approx 1170 km/s and R \approx 1.96 Mpc into this relation yields a estimate on the order of $10^{15} solar masses, aligning closely with and lensing results and underscoring the reliability of these complementary approaches. Among nearby clusters at redshifts z < 0.1, the Perseus Cluster ranks as one of the most massive, its scale and binding energy influencing large-scale structure formation in the local cosmic web. This prominence makes it a key laboratory for studying gravitational dynamics in massive systems.

Redshift and distance

The Perseus Cluster exhibits a spectroscopic redshift of z = 0.01790 \pm 0.00016, measured from optical spectra of numerous member galaxies, providing a precise indicator of its recession due to the expansion of the universe. This redshift corresponds to a heliocentric recession velocity of 5258 km/s, as determined from recent spectroscopic surveys. The distance to the cluster is derived from this redshift using in the form v = H_0 d, with the Hubble constant H_0 \approx 70 km/s/Mpc, yielding a luminosity distance of approximately 75 Mpc (about 245 million light-years). More recent place it at 72 Mpc (as of May 2025). More precisely, the distance accounts for the standard \LambdaCDM cosmological model with parameters \Omega_m = 0.3 and \Omega_\Lambda = 0.7. The comoving distance D_c is given by the line-of-sight integral D_c = \int_0^z \frac{c \, dz'}{H(z')}, where H(z') = H_0 \sqrt{\Omega_m (1 + z')^3 + \Omega_\Lambda}, resulting in a value consistent with 72 Mpc for the adopted parameters. Distance estimates carry uncertainties of about 5–10%, primarily arising from the ongoing tension between measurements of H_0 (e.g., 67 km/s/Mpc from cosmic microwave background data versus 73 km/s/Mpc from local distance ladder methods). Independent validations can employ Cepheid variables or the Tully-Fisher relation on individual member galaxies to cross-check the redshift-based distance. This relatively nearby position renders the Perseus Cluster an ideal target for high-resolution multi-wavelength studies.

Member galaxies

Central galaxy NGC 1275

NGC 1275, also known as , serves as the brightest and most massive member of the , acting as its central dominant galaxy. Classified as an early-type cD elliptical galaxy, it exhibits a stellar mass of approximately M_\star = (1.3 \pm 0.2) \times 10^{12} \, M_\odot and an apparent visual magnitude of V \approx 12.7. This morphological type is characteristic of galaxies at the cores of rich clusters, featuring an extended envelope that envelops surrounding smaller galaxies. At its core, NGC 1275 harbors a supermassive black hole with a mass estimated at approximately $3 \times 10^7 \, M_\odot, powering an active galactic nucleus classified as a . The black hole drives significant accretion activity, manifesting as broad emission lines in its optical spectrum and contributing to the galaxy's role in regulating the cluster's intracluster medium. This active nucleus aligns with the peak brightness in X-ray observations of the cluster core. NGC 1275 is the host of the prominent radio source , which includes relativistic jets emanating from the nucleus and forming extended radio lobes spanning roughly 100 kpc. These structures arise from the black hole's energetic outflows, with the jets exhibiting complex morphologies on parsec to kiloparsec scales. Star formation continues within NGC 1275 at a mild rate of about 1–2 M_\odot yr^{-1}, primarily occurring in filamentary structures traced by Hα emission that extend several kiloparsecs from the core. These filaments, suspended in a magnetic field, are likely triggered by interactions between the galaxy's interstellar medium and the surrounding hot cluster gas, fostering young star clusters along their lengths. Spectroscopic and imaging evidence indicates that NGC 1275 is in a post-merger phase, having recently accreted a smaller companion galaxy, which has disrupted its morphology and supplied cold gas to fuel a cooling flow. This merger event, involving a high-velocity system approaching at ~3000 km s^{-1}, enhances the supply of molecular gas and supports ongoing dynamical activity in the cluster core.

Galaxy population and notable members

The Perseus Cluster hosts over 1,000 member galaxies within its virial radius. The galaxy population is dominated by early-type galaxies, including ellipticals and lenticulars, which constitute about 73% of classified members brighter than B ≈ 20 mag, while late-type spirals and irregulars make up the remaining 27% and are more abundant in the outer regions. This morphological segregation reflects the cluster's dense environment, where early-types prevail in the core due to environmental processes that quench star formation in late-types. The spatial distribution of galaxies is highly concentrated toward the core, following a projected radial density profile modeled by a Hubble law with core radius R_c ≈ 216 kpc and slope β ≈ 0.9. This structure, combined with a line-of-sight velocity dispersion of σ_c ≈ 1040 km s⁻¹, indicates a virialized system in dynamical equilibrium. Among the notable non-central members, NGC 1272 is a prominent head-tail radio galaxy whose jets are bent by ram pressure from the intracluster medium as it moves through the cluster. IC 310 stands out as a blazar candidate exhibiting variable radio emission and a one-sided jet structure, suggesting relativistic beaming. Similarly, NGC 1265 is recognized for its wide-angle tail radio morphology, with extended lobes tracing interactions with the surrounding gas. Dwarf galaxies form a significant component of the population, with recent imaging surveys utilizing Subaru Hyper Suprime-Cam data identifying around 500 early-type dwarf candidates in the cluster core, including ultra-diffuse galaxies characterized by low surface brightness and extended envelopes. Recent observations with the Euclid telescope have identified approximately 1,100 dwarf galaxies, including many ultra-faint members previously undetected. These dwarfs are predominantly ellipticals, often nucleated, and contribute to understanding the faint-end luminosity function in dense environments.

Intracluster medium

X-ray emissions and luminosity

The Perseus Cluster exhibits the highest X-ray luminosity among known galaxy clusters, with a total luminosity of approximately $4 \times 10^{45} erg s^{-1} in the 0.1-10 keV band, primarily arising from the hot intracluster medium (ICM). This makes it the brightest X-ray-emitting cluster in the sky, as confirmed by deep Chandra observations that resolve the extended emission structure. The observed flux in the 2-10 keV band is about $6.3 \times 10^{-11} erg s^{-1} cm^{-2} within a 3 arcmin radius of the core, based on detailed spectral mapping. The X-ray spectrum of the cluster is dominated by thermal bremsstrahlung emission from the ICM, a process where free electrons collide with ions, producing a continuum spectrum that peaks in the soft X-ray regime. Superimposed on this continuum are emission lines from metals such as iron (Fe) and silicon (Si), which arise from collisional excitation and recombination in the hot plasma; these lines provide diagnostics for the ICM's chemical enrichment history. The total luminosity can be approximated using the bremsstrahlung formula for optically thin plasma: L_X \propto n_e^2 T^{1/2} V where n_e is the electron density, T is the gas temperature, and V is the emitting volume. Applying this to Perseus, with typical core densities of n_e \approx 0.1 cm^{-3} and temperatures around 7 keV over a volume corresponding to the core radius of ~100 kpc, yields luminosities consistent with observations, highlighting the dense, hot gas's role in sustaining the emission. The extended X-ray halo dominates the total output, with surface brightness peaking sharply at the cluster core due to the high gas density there, while discrete point sources—primarily active galactic nuclei (AGN) in member galaxies—contribute only about 10% to the overall flux. These point sources, including the central AGN in , are resolved in high-resolution Chandra images and subtracted to isolate the diffuse ICM emission. The X-ray cavities associated with radio bubbles slightly depress the extended emission in specific regions, but the overall luminosity remains dominated by the thermal halo.

Temperature, composition, and metals

The intracluster medium (ICM) in the Perseus Cluster exhibits a high average temperature of approximately 6–7 keV, equivalent to 60–80 million Kelvin, as derived from X-ray spectral analyses of the hot plasma. This thermal state reflects the gravitational potential of the cluster, where the gas is heated to extreme temperatures through shock heating during cluster formation and mergers. The temperature profile reveals a characteristic cool core structure, with central regions near the dominant galaxy cooling to about 2–3 keV over scales of tens of kiloparsecs, while the outskirts extend to hotter phases around 7–8 keV beyond 100 kpc. This gradient arises from radiative cooling dominating in the dense core, contrasted by adiabatic compression and shocks in the outer envelope, as mapped by deep and observations. The ICM's composition is dominated by fully ionized hydrogen and helium, comprising roughly 90% of the gas mass, with the remaining fraction consisting of heavier elements (metals) such as oxygen, silicon, sulfur, and iron, primarily synthesized in core-collapse and Type Ia supernovae within member galaxies over cosmic time. The overall metallicity is sub-solar, with an iron abundance of Z ≈ 0.3–0.5 Z_⊙, reflecting enrichment from multiple generations of stellar nucleosynthesis distributed across the cluster volume. Measurements indicate a relatively uniform metal distribution in the outskirts at Z ∼ 0.3 Z_⊙, but with enhancements toward the core up to ∼0.6 Z_⊙, likely driven by outflows and galactic winds from that transport enriched material into the surrounding ICM. Radial abundance gradients, decreasing outward from the core, have been precisely quantified using high-resolution X-ray spectroscopy from and , highlighting the role of central feedback in local metal redistribution. Radiative cooling in the ICM proceeds efficiently in the cool core, where the classical mass deposition rate is estimated at 200–300 M_⊙ yr⁻¹ based on the gas density, temperature, and cooling function within the central ∼100 kpc. However, this flow is not fully realized, as active galactic nucleus (AGN) feedback from NGC 1275 injects mechanical energy via radio jets and outflows, partially offsetting cooling and preventing runaway accumulation of cold gas. This balance maintains the observed multiphase structure, with suppressed cooling rates inferred from spectral modeling of X-ray emission lines. ===== END CLEANED SECTION =====

Dynamical features

Radio bubbles and cavities

The radio bubbles in the originate from powerful jets emanating from the active galactic nucleus in the central galaxy , which inflate large cavities in the surrounding intracluster medium filled with relativistic plasma. These structures were first clearly imaged as X-ray cavities using observations, revealing depressions in the thermal X-ray emission where the hot intracluster gas has been displaced. The cavities are characterized by sharp, bright rims of compressed gas, indicating the dynamic interaction between the expanding bubbles and the ambient medium. The inner pair of bubbles has diameters of approximately 20–30 kpc, while outer "ghost" cavities extend to larger scales, with the northwestern one measuring about 25 kpc across. The energy injected into these bubbles to inflate them against the intracluster pressure is estimated at around $10^{60} erg for the pair, corresponding to the PdV work done during expansion. For relativistic plasma filling the bubbles, the total enthalpy is calculated as $4PV, where P is the external pressure and V is the bubble volume, providing an estimate of the available energy for heating the surrounding gas. The bubbles are associated with the radio source 3C 84, whose prominent lobes emit synchrotron radiation from relativistic electrons in magnetic fields, prominently detected at 1.4 GHz with the Very Large Array. These radio lobes align closely with the X-ray cavities, confirming that the relativistic plasma traced by radio emission occupies the underdense regions carved out by the jets. The synchrotron emission spectra indicate ages of order $10^7 years for the inner lobes, consistent with episodic AGN activity. As buoyant structures, these bubbles rise through the intracluster medium, releasing their stored energy via PdV work and potentially through weak shocks and viscous dissipation, thereby heating the gas and suppressing radiative cooling flows in the cluster core. This AGN feedback mechanism balances the cooling luminosity of approximately $10^{44}–$10^{45} erg s^{-1} in the Perseus core, preventing runaway star formation. Oscillations during bubble inflation and rise generate propagating sound waves in the intracluster medium.

Ripples, sound waves, and mergers

The X-ray ripples in the Perseus Cluster were first identified in a deep Chandra observation conducted in 2003, revealing low-amplitude, quasi-periodic concentric arcs in the intracluster medium (ICM) surface brightness beyond approximately 20 kpc from the center. These arcs, visible in unsharp-masked images, exhibit a characteristic wavelength of about 11 kpc and represent pressure variations of ±5 to ±10% extending out to 50 kpc or more. These ripples are interpreted as propagating sound waves in the ICM, generated by the periodic inflation of radio bubbles from the central active galactic nucleus in NGC 1275. The waves arise from pressure perturbations during bubble expansion, with a corresponding period of approximately 9.6 million years derived from the ripple wavelength divided by the sound speed. The propagation speed of these sound waves is around 1,200 km/s, consistent with the ICM sound speed given by c_s = \sqrt{\frac{\gamma k T}{\mu m_H}}, where k is Boltzmann's constant, T is temperature, \mu is the mean molecular weight, and m_H is the hydrogen mass. Evidence for cluster-scale mergers in the Perseus Cluster includes dynamical substructures such as the sloshing cold front, a spiral-like feature in the ICM indicating past interactions with an infalling subgroup that displaced the core gas. A 2024 weak lensing study revealed evidence of a ~3:1 mass ratio off-axis major merger, with a subcluster centered on (mass ~1.7 × 10^{14} M_⊙) located ~430 kpc west of the core, explaining a prominent cold front ~700 kpc east and a mass bridge traced by galaxies. This sloshing motion has generated velocity substructures, including turbulent velocities of about 400 km/s associated with Kelvin-Helmholtz instabilities at the cold front edges. Such substructures, including evidence of a major off-axis merger from recent weak lensing studies, highlight the cluster's complex dynamical history. In 2022, NASA produced an audio representation of these sound waves by scaling the extremely low-frequency ripples—corresponding to a B-flat note 57 octaves below middle C—to the human hearing range, allowing auditory exploration of the ICM's acoustic phenomena. This sonification draws directly from the pressure wave patterns identified in Chandra data, emphasizing their role in energy transport within the cluster.

Recent studies

Euclid observations and galaxy discs

The Euclid mission's Early Release Observations (ERO), conducted in September 2023 as part of performance verification, targeted the core of the , imaging a 0.7 deg² region within 0.25 r_{200} and achieving depths of I_E = 27.3 mag for point sources and surface brightness limits of μ_{I_E} = 30.1 mag arcsec^{-2}, enabling detection of low-surface-brightness features in the cluster environment. This dataset, analyzed using photometric redshifts from the validated against 130 spectroscopic redshifts, refined the cluster member catalog to 1220 galaxies, including 1083 dwarfs that effectively doubled the previously known dwarf population and incorporated approximately 500 new faint members through enhanced completeness at M(I_E) ≤ -11.33 mag. Analysis of late-type galaxies in the ERO data revealed truncated disc profiles characteristic of ram pressure stripping, with four Type II (down-bending) profiles identified among 102 massive disc galaxies classified via bulge-disc decomposition, where three core-proximate examples—such as —exhibit abrupt truncations at surface brightness levels around 24.5 mag arcsec^{-2}, accompanied by redder colors beyond the break indicative of gas removal in the intracluster medium. These disc-dominated systems, comprising about 25% of the sample with bulge-to-total ratios B/T ≤ 0.1 and exponential profiles (Sérsic index n ≈ 1), display inward color gradients (redder cores) spanning 0.3 mag in rest-frame g - i, reflecting environmental quenching of star formation. Euclid's high-resolution imaging via tools like AutoProf and AstroPhot extended surface brightness profiles to beyond 28 mag arcsec^{-2}, uncovering faint dwarf galaxies and tidal disturbances in ~20% of members, including minor merger signatures like satellite streams in 50% of cases, which contribute to Type III (up-bending) profiles in interacting systems such as . Integration of Euclid ERO data with Hubble Space Telescope (HST) imaging, covering ~30% of the field for resolved stellar populations, and Chandra X-ray observations of the intracluster medium provided a multiwavelength perspective on core dynamics, linking disc truncations to hot gas pressures exceeding 10^{-10} dyn cm^{-2} and revealing how tidal features align with X-ray cavities from active galactic nucleus feedback in NGC 1275. This combined view highlights the role of cluster environment in reshaping galaxy morphologies, with Euclid's near-infrared sensitivity (Y_E, J_E, H_E bands) enhancing photometric accuracy for stellar mass estimates and confirming the transformation of spirals into lenticulars.

Filaments, dark matter, and merger evidence

Recent deep Chandra X-ray observations of the Perseus Cluster core have enabled the direct detection of X-ray emission counterparts to the prominent Hα optical filaments surrounding the central galaxy NGC 1275. These filaments, previously observed only in optical wavelengths as cool gas structures, now reveal faint X-ray signatures at approximately 1 keV, extracted using advanced component separation techniques like the proximal gradient method for multichannel analysis (pGMCA) to isolate them from the dominant hot intracluster medium. The X-ray maps highlight thread-like structures of relatively cool gas weaving through the cluster center, consistent with multiphase gas dynamics driven by active galactic nucleus feedback. Complementary Chandra imaging from 2025 displays these cool gas threads in blue hues against the hotter X-ray background, spanning scales of hundreds of kiloparsecs and illustrating the intricate filamentary network in the Perseus core. This detection confirms that the Hα filaments are embedded in a broader multiphase environment, where cooler phases (around 10^4 K) coexist with hotter X-ray emitting gas (around 10^7-10^8 K), providing evidence for ongoing cooling and uplift processes. In 2025, observations validated Candidate Dark Galaxy-2 (CDG-2) as an almost entirely dark system within the , characterized by four globular clusters and lacking a detectable stellar body. Initially identified through Subaru Hyper Suprime-Cam imaging, follow-up with and data revealed only faint diffuse emission around the clusters, yielding a total galaxy luminosity of L_{V, \mathrm{gal}} = 6.2 \pm 3.0 \times 10^6 L_{\odot}, with globular clusters contributing about 16-33% of the light. The system's dark matter halo dominates with a mass fraction exceeding 99.94%, marking CDG-2 as the faintest known galaxy identified primarily by its globular cluster population and highlighting the role of such ultra-diffuse objects in cluster environments. Weak-lensing analysis from Subaru Hyper Suprime-Cam data in 2025 uncovered evidence for a missing merger companion in the , a subcluster remnant centered near NGC 1264 with a mass of M_{200} = 1.70_{-0.59}^{+0.73} \times 10^{14} M_{\odot}, located about 430 kpc west of the main core. This infaller, undetected in prior optical surveys due to its low baryonic content, explains observed substructures and gas swirls through a simulated off-axis 3:1 mass ratio merger occurring roughly 5 billion years ago, involving multiple core passages that stirred the intracluster medium. A statistically significant (>3σ) bridge connects this subcluster to the Perseus core, tracing the gravitational remnants of the interaction and confirming an irregular distribution shaped by the merger. Supporting these findings, 2024 Gemini North imaging of the Perseus Cluster revealed its galaxies aligned along filaments, underscoring the irregular mass distribution inferred from the merger dynamics. These recent observations tie into broader dynamical features like sloshing cold fronts, providing confirmatory evidence for the cluster's violent assembly history without altering foundational merger models. In October 2025, XRISM/Resolve observations, combining 2025 data (745 ks total exposure), produced extended gas kinematic maps of the Perseus core, revealing high velocity dispersions of 300 km/s in the eastern region and a dipole-like bulk velocity pattern (±200-300 km/s) along the east-west direction, indicative of rotational motions from a recent merger. The nonthermal pressure fraction was estimated at 7-13%, with turbulent dissipation energy matching the gravitational potential from mergers, suggesting a turbulent cascade driven by at least two merger events since z1, the most recent linked to the IC 310. These findings constrain the to the merger plane at ~30°-50° and highlight the role of turbulence in the dynamics.

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