Coma Cluster Research Articles

A near-infrared study of the obscured 3C129 galaxy cluster

We present a catalogue of 261 new infrared selected members of the 3C 129 galaxy cluster. The cluster, located atz ≈ 0.02, forms part of the Perseus-Pisces filament and is obscured at optical wavelengths due to its location in the zone of avoidance. We identified these galaxies using theJ- andK-band imaging data provided by the UKIDSS Galactic Plane Survey within an area with a radius of 1.1° centred on the X-ray emission of the cluster atℓ, b ≈ 160.52° ,0.27°. A total of 26 of the identified galaxy members have known redshifts 24 of which are from our 2016 Westerbork H I survey and two are from optical spectroscopy. An analysis of the galaxy density at the core of the 3C 129 cluster shows it to be less dense than the Coma and Norma clusters, but comparable to the galaxy density in the core of the Perseus cluster. From an assessment of the spatial and velocity distributions of the 3C 129 cluster galaxies that have redshifts, we derived a velocity ofcz = 5227 ± 171 km s−1andσ = 1097 ± 252 km s−1for the main cluster, with a substructure in the cluster outskirts atcz = 6923 ± 71 km s−1withσ = 422 ± 100 km s−1. The presence of this substructure is consistent with previous claims based on the X-ray analysis that the cluster is not yet virialised and may have undergone a recent merger.

Pairs of giant shock waves (N-waves) in merging galaxy clusters

ABSTRACT When a subcluster merges with a larger galaxy cluster, a bow shock is driven ahead of the subcluster. At a later merger stage, this bow shock separates from the subcluster, becoming a ‘runaway’ shock that propagates down the steep density gradient through the cluster outskirts and approximately maintains its strength and the Mach number. Such shocks are plausible candidates for producing radio relics in the periphery of clusters. We argue that, during the same merger stage, a secondary shock is formed much closer to the main cluster centre. A close analogue of this structure is known in the usual hydrodynamics as N-waves, where the trailing part of the ‘N’ is the result of the non-linear evolution of a shock. In merging clusters, spherical geometry and stratification could further promote its development. Both the primary and the secondary shocks are the natural outcome of a single merger event and often both components of the pair should be present. However, in the radio band, the leading shock could be more prominent, while the trailing shock might conversely be more easily seen in X-rays. The latter argument implies that for some of the (trailing) shocks found in X-ray data, it might be difficult to identify their ‘partner’ leading shocks or the merging subclusters, which are farther away from the cluster centre. We argue that the Coma cluster and A2744 could be two examples in a post-merger state with such well-separated shock pairs.

DISCRETE X-RAY SOURCES IN GALAXY CLUSTERS

Current studies of extragalactic X-ray sources using data from orbital observatory XMM- Newton are at the front line of X-ray astronomy. Resolution capacity of the observatory instruments enable to detect separate X-ray sources in the close galaxies. In this study we used data from recently published 4XMM-DR9 catalog. This catalog contains 550124 unique sources covering 2,85% of the sky. The main type of extended extragalactic X-ray source is hot halo of galaxy cluster. Such objects are important for astrophysics in the tasks of revealing dark matter distribution, galaxy formation and different cosmology studies. At the same time, some images of the galaxy clusters contain small scale pointlike X-ray sources scattered around the clusters or their halos. In many cases the nature of such objects is not clear yet and require more detailed studies. Furthermore, energy flux detection threshold of such objects is limited to a great extend not only by the instruments on board of the XMM- Newton observatory but the natural sky background radiation, which is at the level of 10 −13 Wm −2 sr −1 both for XMM cameras and the sky background. Using Hyperleda database we revealed 16 galaxy clusters among the bright XMM sources. Images of the two galaxy clusters demonstrate bright galaxy cores. These clusters are galaxy group NGC 507 and Coma cluster where nucleus of galaxy NGC 4889 was clearly detected. We also analyzed X-ray images of 14 other X-ray clusters but had not found AGNs there. Out of the above mentioned 16 clusters in the 7 of them we have found 30 pointlike X-ray sources. These sources are presumably mainly AGNs within these clusters. From the other side, they could be also more distant X-ray objects.

A phylogenetic analysis of galaxies in the Coma Cluster and the field: a new approach to galaxy evolution

ABSTRACT We propose a phylogenetic approach (PA) as a novel and robust tool to detect galaxy populations (GPs) based on their chemical composition. The branches of the tree are interpreted as different GPs and the length between nodes as the internal chemical variation along a branch. We apply the PA using 30 abundance indices from the Sloan Digital Sky Survey to 475 galaxies in the Coma Cluster and 438 galaxies in the field. We find that a dense environment, such as Coma, shows several GPs, which indicates that the environment is promoting galaxy evolution. Each population shares common properties that can be identified in colour–magnitude space, in addition to minor structures inside the red sequence. The field is more homogeneous, presenting one main GP. We also apply a principal component analysis (PCA) to both samples, and find that the PCA does not have the same power in identifying GPs.

The number of globular clusters around the iconic UDG DF44 is as expected for dwarf galaxies

ABSTRACT There is a growing consensus that the vast majority of ultradiffuse galaxies (UDGs) are dwarf galaxies. However, there remain a few UDGs that seem to be special in terms of their globular cluster (GC) systems. In particular, according to some authors, certain UDGs exhibit large GC populations when compared to expectations from their stellar (or total) mass. Among these special UDGs, DF44 in the Coma cluster is one of the better-known examples. DF44 has been claimed to have a relatively high number of GCs, $N_{\mathrm{ GC}}=74^{+18}_{-18}$, for a stellar mass of only $3\times 10^8\, \mathrm{ M}_{ \odot}$ which would indicate a much larger dark halo mass than dwarfs of similar stellar mass. In this paper, we revisit this number and, contrary to previous results, find $N_{\mathrm{ GC}}=21^{+7}_{-9}$ assuming that the distribution of the GCs follows the same geometry as the galaxy. If we assume that the GCs around DF44 are distributed in a (projected) circularly symmetric way and, if we use a less strict criterion for the selection of the GCs, we find $N_{\mathrm{ GC}}=18^{+23}_{-12}$. Making use of the MGC–Mhalo relation, this number of GCs suggests a dark matter halo mass of $M_{\mathrm{ halo}}=1.1^{+0.4}_{-0.5} \times 10^{11} \mathrm{ M}_{\odot}$, a value which is consistent with the expected total mass for DF44 based on its velocity dispersion, $\sigma =33^{+3}_{-3}$ km s−1. We conclude that the number of GCs around DF44 is as expected for regular dwarf galaxies of similar stellar mass and DF44 is not extraordinary in this respect.

Probing Magnetic Field Morphology in Galaxy Clusters with the Gradient Technique

Abstract Magnetic fields in the intracluster medium affect the structure and the evolution of galaxy clusters. However, their properties are largely unknown, and measuring magnetic fields in galaxy clusters is challenging, especially on large scales outside of individual radio sources. In this work, we probe the plane-of-the-sky orientation of magnetic fields in clusters using the intensity gradients. The technique is a branch of the gradient technique (GT) that employs emission intensity maps from turbulent gas. We utilize Chandra X-ray images of the Perseus, M87, Coma, and A2597 galaxy clusters, and the VLA radio observations of the synchrotron emission from Perseus. We find that the fields predominantly follow the sloshing arms in Perseus, which is in agreement with numerical simulations. The GT-predicted magnetic field shows signatures of magnetic draping around rising bubbles driven by supermassive black hole feedback in the centers of cool-core clusters, as well as draping around substructures merging with the Coma cluster. We calculate the mean-field orientation with respect to the radial direction in these clusters. In the central regions of cool-core clusters, the mean orientation of the magnetic fields is preferentially azimuthal. There is broad agreement between the magnetic field of Perseus predicted using the X-ray and radio data. Further numerical studies and better future observations with higher resolution and larger effective area will help reduce the uncertainties of this method.

Systematically Measuring Ultradiffuse Galaxies in H i: Results from the Pilot Survey

Abstract We present neutral hydrogen (H i) observations using the Robert C. Byrd Green Bank Telescope (GBT) of 70 optically detected UDG candidates in the Coma region from the Systematically Measuring Ultra-Diffuse Galaxies survey (SMUDGes). We detect H i in 18 targets, confirming nine to be gas-rich UDGs and the remainder to be foreground dwarfs. None of our H i-detected UDGs are Coma Cluster members and all but one are in low-density environments. The H i-detected UDGs are bluer and have more irregular morphologies than the redder, smoother candidates not detected in H i, with the combination of optical color and morphology being a better predictor of gas richness than either parameter alone. There is little visual difference between the gas-rich UDGs and the foreground dwarfs in the SMUDGes imaging, and distances are needed to distinguish between them. We find that the gas richnesses of our H i-confirmed UDGs and those from other samples scale with their effective radii in two stellar mass bins, possibly providing clues to their formation. We attempt to place our UDGs on the baryonic Tully–Fisher relation (BTFR) using optical ellipticities and turbulence-corrected H i line widths to estimate rotation velocities, but the potential systematics associated with fitting smooth Sérsic profiles to clumpy, low-inclination disks of low surface brightness precludes a meaningful analysis of potential BTFR offsets. These observations are a pilot for a large campaign now under way at the GBT to use the H i properties of gas-rich UDGs to quantitatively constrain how these galaxies form and evolve.

NGC 4869 in the Coma Cluster: Twist, Wrap, Overlap, and Bend

Abstract The upgraded Giant Metrewave Radio Telescope (GMRT) has been used to image the head–tail radio galaxy NGC 4869 in the Coma cluster with an angular resolution of 6.″26 at 250–500 MHz and 2.″18 at the 1050–1450 MHz bands. The archival legacy GMRT data have also been used to image the source with angular resolutions from 4.″9 to 21.″8 at 610 MHz, 325 MHz, 240 MHz, and 150 MHz. We find that the ∼200 kpc scale radio morphology consists of five distinct regions with the clear presence of a pinch at ≈1.′4 (=38.8 kpc) and a ridge at ≈3.′4 (=94.2 kpc) from the head. The sharp bend by ∼70° at ∼3.′5 (=97 kpc) from the head is possibly due to projection effects. The radio spectra show progressive spectral steepening as a function of distance from the head and there is possibly re-acceleration of the synchrotron electrons and perhaps also magnetic field regeneration in the 6″–208″ (=2.8–96.1 kpc) region of the jet. We report a steep spectrum sheath layer enveloping a flat spectrum spine, hinting at a transverse velocity structure with a fast-moving spine surrounded by a slow-moving sheath layer. We also derive the lifetimes of the radiating electrons and equipartition parameters. A plausible explanation for the characteristic feature, a ridge of emission perpendicular to the direction of tail is the flaring of a straight, collimated radio jet as it crosses a surface brightness edge due to Kelvin–Helmholtz instabilities.

Upgraded GMRT Observations of the Coma Cluster of Galaxies: The Observations

Abstract We have used the upgraded Giant Metrewave Radio Telescope to map the Coma cluster of galaxies at 250–500 MHz and 1050–1450 MHz bands. These 6.″26 and 2.″18 resolutions observations allow detailed radio structures to be determined of all detected radio sources that show both discrete pointlike and extended morphologies. We present images of a subset of 32 brightest (S ν ≳ 30 mJy) and dominant sources, and several sources show discrete pointlike radio morphologies. We find the steepening of the spectra consistent with synchrotron cooling in the majority of sources and the median for spectral indices is −0.78, suggesting that ∼59% sources have steep spectra. The nature and the statistical properties of the radio sources in the Coma cluster will be discussed in subsequent papers.

Ultraviolet and X-ray properties of Coma’s ultra-diffuse galaxies

ABSTRACT Many ultra-diffuse galaxies (UDGs) have been discovered in the Coma cluster, and there is evidence that some, notably Dragonfly 44, have Milky Way-like dynamical masses despite dwarf-like stellar masses. We used X-ray, ultraviolet (UV), and optical data to investigate the star formation and nuclear activity in the Coma UDGs, and we obtained deep UV and X-ray data (Swift and XMM–Newton) for Dragonfly 44 to search for low-level star formation, hot circumgalactic gas, and the integrated emission from X-ray binaries. Among the Coma UDGs, we find UV luminosities consistent with quiescence but NUV − r colours indicating star formation in the past Gyr. This indicates that the UDGs were recently quenched. The r-band luminosity declines with projected distance from the Coma core. The Dragonfly 44 UV luminosity is also consistent with quiescence, with SFR$\lt 6\times 10^{-4} \, \mathrm{M}_{\odot }$ yr−1, and no X-rays are detected down to a sensitivity of 1038 erg s−1. This rules out a hot corona with a $M \gt 10^8 \, \mathrm{M}_{\odot }$ within the virial radius, which would normally be expected for a dynamically massive galaxy. The absence of bright, low-mass X-ray binaries is consistent with the expectation from the Galaxy total stellar mass, but it is unlikely if most low-mass X-ray binaries form in globular clusters, as Dragonfly 44 has a very large population. Based on the UV and X-ray analysis, the Coma UDGs are consistent with quenched dwarf galaxies, although we cannot rule out a dynamically massive population.

Ultradiffuse Galaxies in the Coma Cluster: Probing Their Origin and AGN Occupation Fraction

Abstract Ultradiffuse galaxies (UDGs) exhibit low surface brightness, but their optical extent is comparable to that of Milky Way-type galaxies. In this work, we utilize Chandra X-ray observations of 404 UDGs in the Coma cluster and address two crucial goals. First, we constrain the formation scenario of UDGs by probing the X-ray emission originating from diffuse gas and from the population of unresolved low-mass X-ray binaries (LMXBs) residing in globular clusters (GCs). It is expected that both the luminosity of the hot gas and the number of GCs, and hence the luminosity from GC-LMXBs, are proportional to the total mass of the dark matter halo. We do not detect statistically significant emission from the hot gas or from GC-LMXBs. The upper limits on the X-ray luminosities suggest that the bulk of the UDGs reside in low-mass dark matter halos, implying that they are genuine dwarf galaxies. This conclusion agrees with our previous results obtained for isolated UDGs, arguing that UDGs are a homogeneous population of galaxies. Second, we constrain the AGN occupation fraction of UDGs, i.e., the fraction of UDGs that are occupied by an active galactic nucleus, for the first time. To this end, we cross-correlate the position of detected X-ray sources in the Coma cluster with the position of UDGs. We identify two UDGs that have a luminous X-ray source at 3.″0 and 3.″2 from their center, which could be off-center AGNs. However, Monte Carlo simulations suggest that one of these sources could be the result of spatial coincidence with a background AGN. Therefore, we place an upper limit of ≲0.5% on the AGN occupation fraction of UDGs.

The Importance of Having an Extended Point-spread Function in Low Surface-brightness Science

Abstract Obtaining and characterizing the extended point-spread function (PSF) is essential to extract accurate photometric information from deep imaging surveys. In this contribution, we present the extended PSFs of the Sloan Digital Sky Survey (SDSS). By stacking images of individual stars with different brightness, two-dimensional SDSS PSFs extending over 8 arcmin in radius for all the SDSS filters have been obtained. We demonstrate how to use the PSF models for correcting the scattered light field produced by the brightest stars in the Coma cluster central region field. This example shows the enormous importance of extended PSFs in the study of the low-surface brightness Universe, especially with the upcoming of ultra-deep surveys such as the Large Synoptic Survey Telescope (LSST). All PSF models and the scripts used to obtain them are publicly available and further details can be found in Infante-Sainz et al. (2020).

Search for the sub-stellar lithium depletion boundary in the open star cluster Coma Berenices

Aims. We mainly aim to search for the lithium depletion boundary (LDB) among the sub-stellar population of the open star cluster Coma Berenices. Methods. Since the number of brown dwarf candidates in Coma Ber available in the literature is scarce, we carried out a search for additional candidates photometrically using colour–magnitude diagrams combining optical and infrared photometry from the latest public releases of the following large-scale surveys: the United Kingdom InfraRed Telescope Infrared Deep Sky Survey (UKIRT/UKIDSS), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), the Sloan Digital Sky Survey (SDSS), and AllWISE. We checked astrometric consistency with cluster membership using Gaia DR2. A search for Li in three new and five previously known brown dwarf candidate cluster members was performed via spectroscopic observations using the OSIRIS instrument at the 10.4 m Gran Telescopio de Canarias (GTC). Results. A couple dozen new photometric candidate brown dwarfs located inside the tidal radius of Coma Ber are reported, but none of these are significantly fainter and cooler than previously known members. No LiI resonance doublet at 6707.8 Å was detected in any of eight Coma Ber targets in the magnitude range J = 15–19 and G = 20–23 observed with the GTC. Spectral types and radial velocities were derived from the GTC spectra. These values confirm the cluster membership of four L2–L2.5 dwarfs, two of which are new in the literature. Conclusions. The large Li depletion factors found among the four bona fide sub-stellar members in Coma Ber implies that the LDB must be located at spectral type later than L2.5 in this cluster. Using the latest evolutionary models for brown dwarfs, a lower limit of 550 Myr on the cluster age is set. This constraint has been combined with other dating methods to obtain an updated age estimate of 780 ± 230 Myr for the Coma Ber open cluster. Identification of significantly cooler sub-stellar cluster members in Coma Ber awaits the advent of the Euclid wide survey, which should reach a depth of about J = 23; this superb sensitivity will make it possible to determine the precise location of the sub-stellar LDB in this cluster and to carry out a complete census of its sub-stellar population.

The −12 mag Dip in the Galaxy Luminosity Function of Hickson Compact Groups*

Abstract We present the galaxy luminosity functions (LFs) of four Hickson Compact Groups using image data from the Subaru Hyper Suprime-Cam. A distinct dip appeared in the faint ends of all the LFs at M g ∼ −12. A similar dip was observed in the LFs of the galaxy clusters Coma and Centaurus. However, LFs in the Virgo, Hydra, and the field had flatter slopes and no dips. As the relative velocities among galaxies are lower in compact groups than in clusters, the effect of galaxy–galaxy interactions would be more significant in compact groups. The M g ∼ −12 dip of compact groups may imply that frequent galaxy–galaxy interactions would affect the evolution of galaxies, and the dip in LF could become a boundary between different galaxy populations.

A complete view of the outskirts of the Coma cluster

ABSTRACT We present a new extended XMM–Newton mosaic of the nearby Coma cluster, which covers the cluster out to the virial radius with nearly complete azimuthal coverage. This large mosaic is combined with the Planck Sunyaev Zel’dovich effect observations to recover the thermodynamic properties of the intracluster medium in an azimuthally averaged profile and 36 angular sectors, producing the highest spatial resolution view of the thermodynamics of the outskirts of a galaxy cluster in its entirety. Beyond r500, our clumping-corrected entropy measurements along the less disturbed directions are statistically consistent with the power-law entropy profile predicted by non-radiative simulations, and the gas mass fraction agrees with the mean cosmic baryon fraction. However, there is a clear entropy deficit in the outskirts to the south-west, coinciding with where Coma connects to a cosmic web filament that joins it to Abell 1367. The low entropy to the south-west extends from 0.5 to 1.0r200, and is consistent with what is expected from simulations of a filamentary gas stream penetrating into the cluster as it continues to accrete matter from the cosmic web. We also find that the radial profiles of the recovered quantities become increasingly asymmetric in the outskirts, particularly along the more disturbed directions, consistent with the predictions of cosmological simulations.

The past history of galaxy clusters told by their present neighbours

ABSTRACT Galaxy clusters can play a key role in modern cosmology, provided their evolution is properly understood. However, observed clusters give us only a single timeframe of their dynamical state. Therefore, finding present observable data of clusters that are well correlated to their assembly history constitutes an inestimable tool for cosmology. Former studies correlating environmental descriptors of clusters to their formation history are dominated by halo mass–environment relations. This paper presents a mass-free correlation between the present neighbour distribution of cluster-size haloes and the latter mass assembly history. From the Big Multidark simulation, we extract two large samples of random haloes with masses ranging from Virgo to Coma cluster sizes. Additionally, to find the main environmental culprit for the formation history of the Virgo cluster, we compare the Virgo-size haloes to 200 Virgo-like haloes extracted from simulations that resemble the local Universe. The number of neighbours at different cluster-centric distances permits discriminating between clusters with different mass accretion histories. Similarly to Virgo-like haloes, clusters with numerous neighbours within a distance of about two times their virial radius experience a transition at z ≈ 1 between an active period of mass accretion, relative to the mean, and a quiet history. In contrary, clusters with few neighbours share an opposite trend: from passive to active assembly histories. Additionally, clusters with massive companions within about four times their virial radius tend to have recent active merging histories. Therefore, the radial distribution of cluster neighbours provides invaluable insights into the past history of these objects.

The ram pressure stripped radio tails of galaxies in the Coma cluster

ABSTRACT Previous studies have revealed a population of galaxies in galaxy clusters with ram pressure stripped (RPS) tails of gas and embedded young stars. We observed 1.4 GHz continuum and H i emission with the Very Large Array in its B-configuration in two fields of the Coma cluster to study the radio properties of RPS galaxies. The best continuum sensitivities in the two fields are 6 and 8 µJy per 4 arcsec beam, respectively, which are 4 and 3 times deeper than those previously published. Radio continuum tails are found in 10 (8 are new) out of 20 RPS galaxies, unambiguously revealing the presence of relativistic electrons and magnetic fields in the stripped tails. Our results also hint that the tail has a steeper spectrum than the galaxy. The 1.4 GHz continuum in the tails is enhanced relative to their H α emission by a factor of ∼7 compared to the main bodies of the RPS galaxies. The 1.4 GHz continuum of the RPS galaxies is also enhanced relative to their infrared emission by a factor of ∼2 compared to star-forming galaxies. The enhancement is likely related to ram pressure and turbulence in the tail. We furthermore present H i detections in three RPS galaxies and upper limits for the other RPS galaxies. The cold gas in D100’s stripped tail is dominated by molecular gas, which is likely a consequence of the high ambient pressure. No evidence of radio emission associated with ultra-diffuse galaxies is found in our data.

The Far-ultraviolet Variations of 17 Com

Abstract All the archival International Ultraviolet Explorer spectra of 17 Com, the most massive member of the open cluster Coma Berenices, are used to show that the far-ultraviolet spectral energy distribution shows large variations at wavelengths shorter than 1800 Å. The large far-ultraviolet variations are most likely due to spots or belts of overabundant elements (chromium in particular) which come and go into the line of sight as the star rotates.

An expanded catalogue of low surface brightness galaxies in the Coma cluster using Subaru/Suprime-Cam

ABSTRACT We present a catalogue of low surface brightness (LSB) galaxies in the Coma cluster obtained from deep Subaru/Suprime-Cam V- and R-band imaging data within a region of $\mathord {\sim }4$ deg2. We increase the number of LSB galaxies presented in Yagi et al. (2016) by a factor of $\mathord {\sim }3$ and report the discovery of 29 new ultradiffuse galaxies (UDGs). We compile the largest sample of UDGs with colours and structural parameters in the Coma cluster. While most UDGs lie along the red-sequence relation of the colour–magnitude diagram, $\mathord {\sim }16$ per cent are outside (bluer or redder) the red-sequence region of Coma cluster galaxies. Our analyses show that there is no special distinction in the basic photometric parameters between UDGs and other LSB galaxies. We investigate the clustercentric colour distribution and find a remarkable transition at a projected radius of $\mathord {\sim }0.6$ Mpc. Within this cluster core region and relative to the red-sequence of galaxies, LSB galaxies are on average redder than co-spatial higher surface brightness galaxies at the 2σ level, highlighting how vulnerable LSB galaxies are to the physical processes at play in the dense central region of the cluster. The position of the transition radius agrees with expectations from recent cosmological simulation of massive galaxy clusters within which ancient infalls are predicted to dominate the LSB galaxy population.

Multi-resolution Filtering: An Empirical Method for Isolating Faint, Extended Emission in Dragonfly Data and Other Low Resolution Images

We describe an empirical, self-contained method to isolate faint, large-scale emission in imaging data of low spatial resolution. Multi-resolution filtering (MRF) uses independent data of superior spatial resolution and point source depth to create a model for all compact and high surface brightness objects in the field. This model is convolved with an appropriate kernel and subtracted from the low resolution image. The halos of bright stars are removed in a separate step and artifacts are masked. The resulting image only contains extended emission fainter than a pre-defined surface brightness limit. The method was developed for the Dragonfly Telephoto Array, which produces images that have excellent low surface brightness sensitivity but poor spatial resolution. We demonstrate the MRF technique using Dragonfly images of a satellite of the spiral galaxy M101, the tidal debris surrounding M51, two ultra-diffuse galaxies in the Coma cluster, and the galaxy NGC 5907. As part of the analysis we present a newly-identified very faint galaxy in the filtered Dragonfly image of the M101 field. We also discuss variations of the technique for cases when no low resolution data are available (self-MRF and cross-MRF). The method is implemented in mrf, an open-source MIT licensed Python package (https://github.com/AstroJacobLi/mrf).