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At least 55 records · Page 3

ZTF SN Ia DR2: An environmental study of Type Ia supernovae using host galaxy image decomposition

The second data release of Type Ia supernovae (SNe Ia) observed by the Zwicky Transient Facility has provided a homogeneous sample of 3628 SNe Ia with photometric and spectral information. This unprecedented sample size enables us to better explore our currently tentative understanding of the dependence of the host environment on SN Ia properties. In this paper, we make use of two-dimensional image decomposition to model the host galaxies of SNe Ia. We model elliptical galaxies as well as disc and spiral galaxies with or without central bulges and bars. This allows for the categorisation of SN Ia based on their morphological host environment, as well as the extraction of intrinsic galaxy properties corrected for both cosmological and atmospheric effects, through point-spread-function (PSF) convolution. We find that although this image decomposition technique leads to a significant bias towards elliptical galaxies in our final sample of processed galaxies, the overall results are still robust. By successfully modelling 728 host galaxies, we find that the photometric properties of SNe Ia found in discs and in elliptical galaxies correlate fundamentally differently with their host environment. We identified strong linear relations between light-curve stretch and our model-derived galaxy colour for both the elliptical (16.8σ) and disc (5.1σ) subpopulations of SNe Ia. Lower-stretch SNe Ia are found in redder environments, which we identify as an age and/or metallicity effect. Within the subpopulation of SNe Ia found in disc-containing galaxies, we find a significant linear trend (6.1σ) between light-curve stretch and model-derived local r-band surface brightness, which we link to the age and metallicity gradients found in disc galaxies. SN Ia colour shows little correlation with the host environment, as is seen in the literature. We do identify a possible dust effect in our model-derived surface brightness (3.3σ) for SNe Ia in disc galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

A machine learning approach to galaxy properties: joint redshift–stellar mass probability distributions with Random Forest

We demonstrate that highly accurate joint redshift–stellar mass probability distribution functions (PDFs) can be obtained using the Random Forest (RF) machine learning (ML) algorithm, even with few photometric bands available. As an example, we use the Dark Energy Survey (DES), combined with the COSMOS2015 catalogue for redshifts and stellar masses. We build two ML models: one containing deep photometry in the griz bands, and the second reflecting the photometric scatter present in the main DES survey, with carefully constructed representative training data in each case. We validate our joint PDFs for 10 699 test galaxies by utilizing the copula probability integral transform and the Kendall distribution function, and their univariate counterparts to validate the marginals. Benchmarked against a basic set-up of the template-fitting code bagpipes, our ML-based method outperforms template fitting on all of our predefined performance metrics. In addition to accuracy, the RF is extremely fast, able to compute joint PDFs for a million galaxies in just under 6 min with consumer computer hardware. Such speed enables PDFs to be derived in real time within analysis codes, solving potential storage issues. As part of this work we have developed galpro 1, a highly intuitive and efficient python package to rapidly generate multivariate PDFs on-the-fly. galpro is documented and available for researchers to use in their cosmology and galaxy evolution studies.

79 ASTRONOMY AND ASTROPHYSICS↗

The ALMaQUEST survey IX: the nature of the resolved star forming main sequence

ABSTRACT We investigate the nature of the scaling relations between the surface density of star formation rate (ΣSFR), stellar mass (Σ*), and molecular gas mass ($\Sigma _{\rm H_2}$), aiming at distinguishing between the relations that are primary, i.e. more fundamental, and those which are instead an indirect by-product of the other relations. We use the ALMA-MaNGA QUEnching and STar formation survey and analyse the data by using both partial correlations and random forest regression techniques. We unambiguously find that the strongest intrinsic correlation is between ΣSFR and $\Sigma _{\rm H_2}$ (i.e. the resolved Schmidt–Kennicutt relation), followed by the correlation between $\Sigma _{\rm H_2}$ and Σ* (resolved molecular gas main sequence, rMGMS). Once these two correlations are taken into account, we find that there is no evidence for any intrinsic correlation between ΣSFR and Σ*, implying that star formation rate (SFR) is entirely driven by the amount of molecular gas, while its dependence on stellar mass (i.e. the resolved star forming main sequence, rSFMS) simply emerges as a consequence of the relationship between molecular gas and stellar mass.

79 ASTRONOMY AND ASTROPHYSICS↗

Semi-analytic spectral fitting: simultaneously modelling the mass accumulation and chemical evolution in MaNGA spiral galaxies

We develop a novel semi-analytic spectral fitting approach to quantify the star formation histories (SFHs) and chemical enrichment histories (ChEHs) of individual galaxies. We construct simple yet general chemical evolution models that account for gas inflow and outflow processes as well as star formation, to investigate the evolution of merger-free star-forming systems. These models are fitted directly to galaxies’ absorption-line spectra, while their emission lines are used to constrain current gas phase metallicity and star formation rate. We apply this method to spiral galaxies selected from the SDSS-IV Mapping Nearby Galaxies at Apache Point Observatory survey. By fitting the co-added absorption-line spectra for each galaxy, and using the emission-line constraints on present-day metallicity and star formation, we reconstruct both the SFHs and the ChEHs for all objects in the sample. We can use these reconstructions to obtain archaeological measures of derived correlations such as the mass–metallicity relation at any redshift, which compare favourably with direct observations. We find that both the SFHs and ChEHs have strong mass dependence: massive galaxies accumulate their stellar masses and become enriched earlier. This mass dependence causes the observed flattening of the mass–metallicity relation at lower redshifts. The model also reproduces the observed gas-to-stellar mass ratio and its mass dependence. Moreover, we are able to determine that more massive galaxies have earlier gas infall times and shorter infall time-scales, and that the early chemical enrichment of low-mass galaxies is suppressed by strong outflows, while outflows are not very significant in massive galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

The role of radio loudness in Eigenvector 1 and the Baldwin Effect of [O iii ] λ5007

ABSTRACT The radio emission is an important observable of quasars, but its relationship to Eigenvector 1 and the [O iii] λ5007 Baldwin Effect is not fully understood. We explore these issues based on a sample of 1800 quasars taken from the Sloan Digital Sky Survey Data Release 7. We employ a new approach of selecting subsamples in the plane of fundamental physical parameters of the black hole mass and Eddington ratio, so as to reduce variables and complexity in analyses. Based on these subsamples, we investigate the relationship between radio loudness R and Eigenvector 1 and find that radio loudness is correlated with [O iii] λ5007 emission, but has no clear relationship with optical Fe ii emission, which indicates that the radio power is probably not a driver of Eigenvector 1 but merely a secondary process. In addition, we also investigate the impact of radio loudness on the Baldwin Effect of [O iii] λ5007. We find that when the radio loudness is not strong (log R < 2), the Baldwin Effect of [O iii] λ5007 is clear, while in the samples of extreme radio loudness (log R ≥ 2), the Baldwin Effect of [O iii] λ5007 becomes weaker or even disappears. We suggest that both radio loudness and luminosity influence the relative strength of [O iii].

79 ASTRONOMY AND ASTROPHYSICS↗

The effect of spiral arms on the Sérsic photometry of galaxies

The Sérsic profile is a widely used model for describing the surface brightness distribution of galaxies. Spiral galaxies, however, are qualitatively different from a Sérsic model. Aims. The goal of this study is to assess how accurately the total flux and half-light radius of a galaxy with spiral arms can be recovered when fitted with a Sérsic profile. Methods. I selected a sample of bulge-dominated galaxies with spiral arms. Using photometric data from the Hyper Suprime-Cam survey, I estimated the contribution of the spiral arms to their total flux. Then I generated simulated images of galaxies with similar characteristics, fitted them with a Sérsic model, and quantified the error on the determination of the total flux and half-light radius. Results. Spiral arms can introduce biases on the photometry of galaxies in a way that depends on the underlying smooth surface brightness profile, the location of the arms, and the depth of the photometric data. A set of spiral arms accounting for 10% of the flux of a bulge-dominated galaxy typically causes the total flux and the half-light radius to be overestimated by 15% and 30%, respectively. This bias, however, is much smaller if the galaxy is disk-dominated. Conclusions. Galaxies with a prominent bulge and a non-zero contribution from spiral arms are the most susceptible to biases in the total flux and half-light radius when fitted with a Sérsic profile. If photometric measurements with high accuracy are required, then measurements over finite apertures are to be preferred over global estimates of the flux.

79 ASTRONOMY AND ASTROPHYSICS↗

Cosmological implications of ROSAT observations of groups and clusters of galaxies

We have combined ROSAT Position Sensitive Proportional Counter (PSPC) and optical observations of a sample of groups and clusters of galaxies to determine the fundamental parameters of these systems (e.g., the dark matter distribution, gas mass fraction, baryon mass fraction, mass-to-light ratio, and the ratio of total-to-luminous mass). Imaging X-ray spectroscopy of groups and clusters show that the gas is essentially isothermal beyond the central region, indicating that the total mass density (mostly dark matter) scales as rho(sub dark) varies as 1/r squared. The density profile of the hot X-ray emitting gas is fairly flat in groups with rho(sub gas) varies as 1/r and becomes progressively steeper in hotter richer systems, with rho(sub gas) varies as 1/r squared in the richest clusters. These results show, that in general, the hot X-ray-emitting gas is the most extended mass component in groups and clusters, the galaxies are the most centrally concentrated component, and the dark matter is intermediate between the two. The flatter density rofile of the hot gas compared to the dark matter produces a gas mass fraction that increases with radius within each object. There is also a clear trend of increasing gas mass fraction (from 2% to 30%) between elliptical galaxies and rich clusters due to the greater detectable extent of the X-ray emission in richer systems. For the few systems in which the X-ray emission can be traced to the virial radius (where the overdensity delta is approximately equal 200), the gas mass fraction (essentially the baryon mass fraction) approaches a roughly constant value of 30%, suggesting that this is the true primordial value. Based on standard big bang nucleosynthesis, the large baryon mass fraction implies that Omega = 0.1 - 0.2. The antibiased gas distribution suggests that feedback from galaxy formation and hydrodynamics play important roles in the formation of structure on the scale of galaxies to rich clusters. All the groups and clusters in our sample have mass-to-light ratios of M/L(sub V) approximately 100 - 150 solar mass/solar luminosity, which strongly contrasts with the traditional view that the mass-to-light ratio of rich clusters is significantly greater than individual galaxies or groups with M/L(sub V) approximately 250 - 300 solar mass/solar luminosity. We also show that M/L(sub V is essentially constant within the virial radius of clusters (where delta is greater than or approximately 200), which is consistent with the peaks formalism of biased galaxy formation. While the mass-to-light ratios of groups and clusters are comparable (indicating a constant mass fraction of optically luminous material), the ratio of the total mass-to-luminous mass (gas plus stars) monotonically decreases between galaxies and clusters. The decrease in M(sub total)/M(sub lum) arises from two factors: (1) the composition of baryonic matter varies from a predominance of optically luminous material (stars) on the scale of galaxies (approximately 10 kpc) to a predominance of X-ray luminous material (hot gas) on the scale of rich clusters (approximately 1 Mpc), and (2) the hot gas has a more extended spatial distribution than the gravitating matter. The observed decrease M(sub total)/M(sub lum) between galaxies and clusters indicates that the universe actually becomes `brighter' on mass scales between 10(exp 12) and 10(exp 15) solar mass, in the sense that a greater fraction of the gravitating mass is observable.

David, Laurence P.↗

The impact of environment on size: Galaxies are 50% smaller in the Fornax Cluster compared to the field

Size is a fundamental parameter for measuring the growth of galaxies and the role of the environment on their evolution. However, the conventional size definitions used for this purpose are often biased and miss the diffuse, outermost signatures of galaxy growth, including star formation and gas accretion. We address this issue by examining low surface brightness truncations or galaxy ‘edges’ as a physically motivated tracer of size based on star formation thresholds. Our total sample consists of ∼900 galaxies with stellar masses ranging from 10 5 M ⊙ < M ⋆ < 10 11 M ⊙ . This sample of nearby cluster, group satellite, and nearly isolated field galaxies was compiled using multi-band imaging from the Fornax Deep Survey, deep IAC Stripe 82, and Dark Energy Camera Legacy Surveys. We find that the edge radii scale as R edge ∝ M ⋆ 0.42 , with a very small intrinsic scatter (∼0.07 dex). The scatter is driven by the morphology and environment of galaxies. In both the cluster and field, early-type dwarfs are systematically smaller by approximately 20% compared to late-type dwarfs. However, galaxies in the Fornax cluster are the most impacted. At a fixed stellar mass, edges in the cluster can be found at about 50% smaller radii, and the average stellar surface density at the edges is a factor of two higher, ∼1 M ⊙ /pc 2 . Our findings support the rapid removal of loosely bound neutral hydrogen (H I ) in hot, crowded environments, which truncates galaxies outside-in earlier, preventing the formation of more extended sizes and lower density edges. Our results highlight the importance of deep imaging surveys to the study of low surface brightness imprints of the large-scale structure and environment on galaxy evolution.

Astronomy & Astrophysics↗

Photometric Redshifts and Galaxy Clusters for DES DR2, DESI DR9, and HSC-SSP PDR3 Data

Photometric redshift (photoz) is a fundamental parameter for multi-wavelength photometric surveys, while galaxy clusters are important cosmological probes and ideal objects for exploring the dense environmental impact on galaxy evolution. We extend our previous work on estimating photoz and detecting galaxy clusters to the latest data releases of the Dark Energy Spectroscopic Instrument (DESI) imaging surveys, Dark Energy Survey (DES) and Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) imaging surveys and make corresponding catalogs publicly available for more extensive scientific applications. The photoz catalogs include accurate measurements of photoz and stellar mass for about 320, 293 and 134 million galaxies with r < 23, i < 24 and i < 25 in DESI DR9, DES DR2 and HSC-SSP PDR3 data, respectively. The photoz accuracy is about 0.017, 0.024 and 0.029 and the general redshift coverage is z < 1, z < 1.2 and z < 1.6, respectively for those three surveys. Furthermore, the uncertainty of the logarithmic stellar mass that is inferred from stellar population synthesis fitting is about 0.2 dex. With the above photoz catalogs, galaxy clusters are detected using a fast cluster-finding algorithm. A total of 532,810, 86,963 and 36,566 galaxy clusters with the number of members larger than 10 is discovered for DESI, DES and HSC-SSP, respectively. Their photoz accuracy is at the level of 0.01. The total mass of our clusters is also estimated by using the calibration relations between the optical richness and the mass measurement from X-ray and radio observations. The photoz and cluster catalogs are available at ScienceDB (https://www.doi.org/10.11922/sciencedb.o00069.00003) and PaperData Repository (https://doi.org/10.12149/101089).

79 ASTRONOMY AND ASTROPHYSICS↗

RRab variables with identical light-curve shapes at different pulsation periods

In this paper, we report on the detection of RRab stars with quasi-identical-shape light curves but period differences as large as 0.05–0.21 d using the Galactic bulge data of the OGLE-IV survey. We have examined stars with shorter periods than the Oosterhoff I ridge of the bulge. These stars generally have smaller amplitudes and larger Fourier phase-differences than the typical bulge RRab stars have at the same period. Many of these ‘anomalous’ stars have good-quality light curves without any sign of the Blazhko modulation. Examining their Fourier parameters revealed that several of these stars show very similar light curve to the typical bulge RR Lyrae. We found hundreds of quasi-identical-shape light-curve pairs with different periods between the ‘anomalous’- and the ‘normal’-position RRab stars based on the OGLE I-band data. The OGLE V-band, and the archive VVV and MACHO surveys K s -, b- and r -band data of these stars were also checked for light-curve-shape similarity. Finally, 149 pairs with identical-shape light curves in each available photometric band were identified. Calculating the physical properties of the variables using empirical formulae, on average, −0.5 dex, −0.13 mag, 0.67, and 165 K differences between the [Fe/H], M V , R/R ⊙ , and T eff values of the members of the pairs are derived, being the short-period stars less metal-poor, fainter, smaller, and hotter than the long-period variables. To explain the existence of variables with different physical properties and pulsation periods but with identical-shape light curves is a challenging task for modelling.

79 ASTRONOMY AND ASTROPHYSICS↗

Interactions in Massive Colliding Wind Binaries

The most massive stars (M> 60 Solar Mass) play crucial roles in altering the chemical and thermodynamic properties of their host galaxies. Stellar mass is the fundamental stellar parameter that determines their ancillary properties and which ultimately determines the fate of these stars and their influence on their galactic environs. Unfortunately, stellar mass becomes observationally and theoretically less well constrained as it increases. Theory becomes uncertain mostly because very massive stars are prone to strong, variable mass loss which is difficult to model. Observational constraints are uncertain too. Massive stars are rare, and massive binary stars (needed for dynamical determination of mass) are rarer still: and of these systems only a fraction have suitably high orbital inclinations for direct photometric and spectroscopic radial-velocity analysis. Even in the small number of cases in which a high-inclination binary near the upper mass limit can be identified, rotational broadening and contamination of spectral line features from thick circumstellar material (either natal clouds or produced by strong stellar wind driven mass loss from one or both of he stellar components) biases the analysis. In the wilds of the upper HR diagram, we're often left with indirect and circumstantial means of determining mass, a rather unsatisfactory state of affairs.

Corcoran, M.↗

A measurement of the distance to the Galactic centre using the kinematics of bar stars

The distance to the Galactic centre R 0 is a fundamental parameter for understanding the Milky Way, because all observations of our Galaxy are made from our heliocentric reference point. The uncertainty in R 0 limits our knowledge of many aspects of the Milky Way, including its total mass and the relative mass of its major components, and any orbital parameters of stars employed in chemo-dynamical analyses. While measurements of R 0 have been improving over a century, measurements in the past few years from a variety of methods still find a wide range of R 0 being somewhere within 8.0 to $8.5\, \mathrm{kpc}$. The most precise measurements to date have to assume that Sgr A* is at rest at the Galactic centre, which may not be the case. In this paper, we use maps of the kinematics of stars in the Galactic bar derived from APOGEE DR17 and Gaia EDR3 data augmented with spectrophotometric distances from the astroNN neural-network method. These maps clearly display the minimum in the rotational velocity v T and the quadrupolar signature in radial velocity v R expected for stars orbiting in a bar. From the minimum in v T , we measure $R_0 = 8.23\pm 0.12\, \mathrm{kpc}$. We validate our measurement using realistic N-body simulations of the Milky Way. We further measure the pattern speed of the bar to be $\Omega _\mathrm{bar} = 40.08\pm 1.78\, \mathrm{km\, s}^{-1}\,\mathrm{kpc}^{-1}$. Because the bar forms out of the disc, its centre is manifestly the barycentre of the bar+disc system and our measurement is therefore one of the most robust and accurate measurements of R 0 to date.

79 ASTRONOMY AND ASTROPHYSICS↗

A unified exploration of the chronology of the Galaxy

The Milky Way has distinct structural stellar components linked to its formation and subsequent evolution, but disentangling them is non-trivial. With the recent availability of high-quality data for a large numbers of stars in the Milky Way, it is a natural next step for research in the evolution of the Galaxy to perform automated explorations with unsupervised methods of the structures hidden in the combination of large-scale spectroscopic, astrometric, and asteroseismic data sets. We determine precise stellar properties for 21 076 red giants, mainly spanning 2–15 kpc in Galactocentric radii, making it the largest sample of red giants with measured asteroseismic ages available to date. We explore the nature of different stellar structures in the Galactic disc by using Gaussian mixture models as an unsupervised clustering method to find substructure in the combined chemical, kinematic, and age subspace. The best-fitting mixture model yields four distinct physical Galactic components in the stellar disc: the thin disc, the kinematically heated thin disc, the thick disc, and the stellar halo. We find hints of an age asymmetry between the Northern and Southern hemisphere, and we measure the vertical and radial age gradient of the Galactic disc using the asteroseismic ages extended to further distances than previous studies.

79 ASTRONOMY AND ASTROPHYSICS↗

What the UV SED Tells us About Stellar Populations and Galaxies

The UV SED parameter b as in f(sub 1) ~ 1(sup b), is commonly used to estimate fundamental properties of high-redshift galaxies including age and metallicity. However, sources and processes other than age and metallicity can influence the value of b. We use the local starforming dwarf galaxy, I Zw 18, in a case study to investigate uncertainties in age and metallicity inferred from b due errors or uncertainties in: mode of star formation (instantaneous starburst vs. continuous SF), dust extinction, nebular continuous emission (2-photon emission, Balmer continuum flux), and presence of older stars.

Heap, Sara R.↗

The Size Evolution of Passive Galaxies: Observations From the Wide-Field Camera 3 Early Release Science Program

We present the size evolution of passively evolving galaxies at z approximately 2 identified in Wide-Field Camera 3 imaging from the Early Release Science program. Our sample was constructed using an analog to the passive BzK galaxy selection criterion, which isolates galaxies with little or no ongoing star formation at z greater than approximately 1.5. We identify 30 galaxies in approximately 40 arcmin(sup 2) to H less than 25 mag. By fitting the 10-band Hubble Space Telescope photometry from 0.22 micrometers less than approximately lambda (sub obs) 1.6 micrometers with stellar population synthesis models, we simultaneously determine photometric redshift, stellar mass, and a bevy of other population parameters. Based on the six galaxies with published spectroscopic redshifts, we estimate a typical redshift uncertainty of approximately 0.033(1+z).We determine effective radii from Sersic profile fits to the H-band image using an empirical point-spread function. By supplementing our data with published samples, we propose a mass-dependent size evolution model for passively evolving galaxies, where the most massive galaxies (M(sub ∗) approximately 10(sup 11) solar mass) undergo the strongest evolution from z approximately 2 to the present. Parameterizing the size evolution as (1 + z)(sup - alpha), we find a tentative scaling of alpha approximately equals (−0.6 plus or minus 0.7) + (0.9 plus or minus 0.4) log(M(sub ∗)/10(sup 9 solar mass), where the relatively large uncertainties reflect the poor sampling in stellar mass due to the low numbers of highredshift systems. We discuss the implications of this result for the redshift evolution of the M(sub ∗)-R(sub e) relation for red galaxies.

fundamental parameters↗

The fundamental plane correlations for globular clusters

In the parameter space whose axes include a radius (core, or half-light), a surface brightness (central, or average within the half-light radius), and the central projected velocity dispersion, globular clusters lie on a two-dimensional surface (a plane, if the logarithmic quantities are used). This is analogous to the 'fundamental plane' of elliptical galaxies. The implied bivariate correlations are the best now known for globular clusters. The derived scaling laws for the core properties imply that cluster cores are fully virialized, homologous systems, with a constant (M/L) ratio. The corresponding scaling laws on the half-light scale are differrent, but are nearly identical to those derived from the 'fundamental plane' of ellipticals. This may be due to the range of cluster concentrations, which are correlated with other parameters. A similar explanation for elliptical galaxies may be viable. These correlations provide new empirical constraints for models of globular cluster formation and evolution, and may also be usable as rough distance-indicator relations for globular clusters.

Djorgovski, S.↗

Fundamental Parameters of ∼30,000 M dwarfs in LAMOST DR1 Using Data-driven Spectral Modeling

M dwarfs are the most common type of star in the Galaxy, and because of their small size are favored targets for searches of Earth-sized transiting exoplanets. Current and upcoming all-sky spectroscopic surveys, such as the Large Sky Area Multi Fiber Spectroscopic Telescope (LAMOST), offer an opportunity to systematically determine physical properties of many more M dwarfs than has been previously possible. Here, we present new effective temperatures, radii, masses, and luminosities for 29,678 M dwarfs with spectral types M0–M6 in the first data release (DR1) of LAMOST. We derived these parameters from the supervised machine-learning code, The Cannon, trained with 1388 M dwarfs in the Transiting Exoplanet Survey Satellite Cool Dwarf Catalog that were also present in LAMOST with high signal-to-noise ratio (>250) spectra. Our validation tests show that the output parameter uncertainties are strongly correlated with the signal-to-noise of the LAMOST spectra, and we achieve typical uncertainties of 110 K in T{sub eff} (∼3%), 0.065 R{sub ⊙} (∼14%) in radius, 0.054 M{sub ⊙} (∼12%) in mass, and 0.012 L{sub ⊙} (∼20%) in luminosity. The model presented here can be rapidly applied to future LAMOST data releases, significantly extending the samples of well-characterized M dwarfs across the sky using new and exclusively data-based modeling methods.

79 ASTRONOMY AND ASTROPHYSICS↗

What determines the morphological fractions in clusters of galaxies?

A reexamination of Dressler's (1980) sample of nearly 6000 galaxies in 55 clusters shows that the morphology-clustercentric radius relation is more fundamental than the morphology-local density relation. This conclusion is supported by improved correlations when the projected clustercentric radius is used as the independent parameter, and by a comparison of galaxies with the same normalized clustercentric radii by different values of the projected local density.

Whitmore, Bradley C.↗