BRIGHTEST CLUSTER GALAXIES AT THE PRESENT EPOCH

The dispersion in the luminosities of the BCGs about the mean Hubble relation, measured by the first studies to use BCGs as distance indicators, was typically 0.3–0.4 mag (Sandage 1972a, 1972b; Gunn & Oke 1975). An important refinement of the use of BCG as distance indicators was developed by Hoessel (1980), who showed that BCG metric luminosity, L_m, was correlated with the logarithmic slope, α, of the photometric curve of growth. The L_m–α relation is a form of a luminosity–radius relation that side-steps the difficulties of characterizing the extended envelopes of BCG at large radii and faint isophotal levels. Postman & Lauer (1995, hereafter PL95) reinvestigated the use of BCG as distance indicators, using the L_m–α relation for a full-sky characterization of the linearity of the local Hubble flow (Lauer & Postman 1992) and providing a distant reference frame to measure the relative peculiar velocity of the Local Group (Lauer & Postman 1994). Residuals about the PL95 L_m–α relation were only 0.25 mag (rms).

This paper presents a large full-sky sample of BCGs in Abell clusters over the redshift range 0 < z ⩽ 0.08. The original goal for obtaining this sample was to extend the bulk-flow analysis of Lauer & Postman (1994) to greater distances. That work implied that the Abell clusters within z ⩽ 0.05 participated in a coherent motion in excess of 689 ± 178 km s⁻¹ superimposed on the background cosmological expansion or "Hubble flow" within the volume containing the sample. This analysis will be presented in a separate work. Requirements for measuring accurate bulk flows, however, specify much of the sample definition, observational methodologies, and analysis of the BCG properties undertaken in this work. A full-sky sample allows for the optimal determination of any large-scale bulk mass flow. The relatively low redshift limit of the sample and its overall size is dictated by the scale out to which the BCGs can be used as accurate distance indicators. The observational methodology is driven by the need to obtain highly uniform photometry over the angular and spatial extent of the sample. Much of the analysis is a reinvestigation of the use of BCGs as distance indicators, with a substantially larger sample and new observations that go well beyond the material available to PL95.

Regardless of the bulk-flow analysis, the present sample offers an excellent opportunity to assess the structural properties of BCG to understand their origin, evolution over time, and their particular uniqueness as the luminous endpoint of galaxy formation, problems that were not addressed by the smaller sample and less-complete cluster information available to Postman & Lauer (1995). Oegerle & Hoessel (1991) and Lauer et al. (2007), for example, found that the central stellar velocity dispersions, σ, of BCGs increase very slowly if at all with the total BCG luminosity (also see Bernardi et al. 2007; von der Linden et al. 2007; Liu et al. 2008). Typical BCG σ values are modest for their large luminosities, which may reflect the origin of BCGs in "dry" mergers (Boylan-Kolchin et al. 2006). In contrast, BCGs are unusually extended as compared to giant ellipticals, as is seen in the relation between effective radius, R_e, and total luminosity of the BCGs (Lauer et al. 2007; Bernardi et al. 2007). We will use the structure of BCGs as a probe of the effects of cluster environment on their evolution. In a companion paper (N. E. Chisari et al. 2014, in preparation), we will compare the structure of BCGs to those of other highly luminous elliptical galaxies. The mutual relations between L, σ, and R_e for elliptical galaxies overall are understood as various projections of the "fundamental plane" (Dressler et al. 1987; Djorgovski & Davis 1987). Understanding the relationship of BCGs to the fundamental plane will be explored in N. E. Chisari et al. (2014, in preparation).

The "textbook" picture of a galaxy cluster is that it is a swarm of galaxies anchored by a massive cD residing at rest in the exact center of the potential as marked by hot, X-ray-emitting gas. Early work on the X-ray morphology of galaxy clusters (Jones & Forman 1984) and their velocity structure (Quintana & Lawrie 1982) indeed show that the BCG is likely to be centrally located. There are certainly examples of such clusters in our sample. At the same time, there are also massive galaxy clusters, like Coma (A1656), in which neither the BCG nor second-ranked galaxy, M2, are at the center of the potential. Coma may be the recent merger of two clusters, and this is the point—the position of the BCG with respect to the center of the potential, X-ray emission, may testify to the evolutionary state of both the BCG and the cluster. More recent work (Patel et al. 2006; Hashimoto et al. 2014) shows that the BCG is often displaced from the center of the cluster potential as defined by the X-ray emission. For the present sample of clusters, we have quantified the distribution of projected spatial offsets of the BCGs, finding that it is a steep power law over three decades in radius.

For the BCG to reside at the spatial center of the cluster, it must also be at rest there. It has long been known that there are BCGs with "significant" peculiar velocities within the cluster (Zabludoff et al. 1990, 1993; Malumuth 1992; Oegerle & Hill 2001). PL95 described the overall distribution of BCG peculiar velocities within their sample as a Gaussian with dispersion, σ₁ = 264 km s⁻¹, comparing this to the substantially larger mean cluster velocity dispersion σ_c = 666 km s⁻¹. We attempted to verify this result with our much larger present sample, finding now that the distribution of peculiar velocities is exponential, extending out to galaxies with ΔV₁ > σ_c.

Both the distributions of the spatial and velocity offsets of the BCGs are particularly interesting when compared to the Martel et al. (2014) simulations of galaxy cluster formation and evolution. These simulations emphasize that the location of the BCG within a cluster bears witness to its history of formation from smaller accreted groups and clusters. The dark matter, galaxy, and X-ray-emitting gas distributions in any cluster all have different timescales and physical mechanisms for responding to the accretion or interaction with another cluster. The locations of the BCGs reflect this. Skibba et al. (2011) studied the peculiar velocities and spatial positions of a large sample of clusters and groups, showing that the location of the brightest galaxy in the systems provides a sharp test of the mechanisms that formed them.

The relationship of the structure and luminosity of the BCG to the properties of the cluster has proven to be a multi-faceted problem. Initial work showed the BCG luminosity to be only weakly related to the richness of the clusters (Sandage 1972b, 1975, 1976; Sandage & Hardy 1973). We re-investigated this relation in PL95 and saw no relation between the metric luminosity and cluster richness.

BCG luminosity and structure, however, do appear to be related to the X-ray properties of the clusters. Schombert (1988) found that the envelope luminosity of cD galaxies, a subset of the BCGs, increases with total cluster X-ray luminosity. Edge (1991) and Edge & Stewart (1991) found a strong relationship between BCG luminosity and cluster X-ray temperature, which itself is closely related to the cluster velocity dispersion (Solinger & Tucker 1972). Hudson & Ebeling (1997) and Collins & Mann (1998) also found that BCG luminosity increases with cluster X-ray luminosity. Lastly, Brough et al. (2005) found the structure of the BCG to also correlate with cluster X-ray luminosity, with the BCG envelope becoming more extended in more luminous clusters.

In this paper, we show that both the BCG luminosity, and the radial extent (as characterized by α) of their envelopes (where by "envelope", we mean simply the outer portions of the galaxies) correlate with cluster velocity dispersion. We take this a step further; however, finding that the extent of the envelope is related to both the spatial and velocity positions of the BCG within its hosting cluster. The luminosity and structural difference between the BCG and the second-ranked galaxy, M2, also appears to depend on which galaxy has the smaller peculiar velocity within the cluster or the smaller offset from the center of the cluster potential as marked by X-ray emission. Beers & Geller (1983) found that early type galaxies with extended halos (e.g., D or cD galaxies) lie on significant peaks in the cluster galaxy distribution regardless of whether they are the BCG. We now see how the structure of the BCG itself changes smoothly as a function of how close to the center of the cluster it resides.

We begin in Section 2 with the geometric and redshift selection of the Abell clusters defining the present sample, detailing the imaging observations used both to select the BCG for any given cluster and to provide accurate surface photometry. Spectroscopic observations are presented, which provide BCG redshifts and central stellar velocity dispersions. A crucial part of the sample definition is the derivation of accurate mean redshifts and velocity dispersions for the galaxy clusters. The projected spatial and velocity locations of the BCGs within their clusters is presented in Section 3. The photometric and kinematic properties of the BCGs are presented in Section 4, with particular attention to parametric relations between the metric luminosity and BCG structure. This section also explores the relationship between BCG properties and cluster environment. Additional information about the BCGs is provided by the properties of the second-ranked galaxies, M2, which are presented in Section 5. We summarize what we have learned about the origin and evolution of BCGs in the final section of the paper.

The present sample of BCGs comprises 433 Abell (1958) and Abell–Corwin–Olowin (ACO; Abell et al. 1989) galaxy clusters with mean heliocentric velocities, V < 24,000 km s⁻¹ and galactic latitude, |b| ⩾ 15°. There is no limit on the minimum richness class of the clusters. Table 1 lists the BCG coordinates, and heliocentric velocities, V₁, as well as the cluster velocities, V_c, cluster velocity dispersions, σ_c, the number of galaxy velocities used to compute these, N_g, and the Schlegel et al. (1998) A_B values. The distribution on the sky in galactic coordinates is shown in Figure 1. We use a cosmological model with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3, and Λ = 0.7 throughout this paper.

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Table 1. The BCG and Abell Cluster Sample

Abell	α	δ	V1	Vc	σc	Ng	AB	Notes
	(2000)	(2000)	(km s−1)	(km s−1)	(km s−1)
14	3.7957	−23.8823	19280 ± 26	19397 ± 111	$\phantom{0}474\pm 118$	18	0.094	E473-G005
27	6.1965	−20.7324	15519 ± 0	16037 ± 90	$\phantom{0}339\pm 79$	14	0.094	E539-G013
71	9.4947	+29.6035	22044 ± 33	22044 ± 33	...	2	0.210
74	9.5903	−22.3449	19013 ± 46	19326 ± 56	$\phantom{0}212\pm 59$	14	0.091
75	9.8692	+21.2295	17610 ± 42	17355 ± 326	$\phantom{0}731\pm 511$	5	0.146

Notes. Columns: (1) Abell cluster number, (2) J2000 R.A. of the BCG, (3) J2000 decl. of the BCG, (4) heliocentric velocity of the BCG, (5) mean heliocentric cluster velocity, (6) cluster velocity dispersion (when N > 3), (7) number of cluster galaxies used to derive the mean velocity and dispersion, (8) Schlegel et al. (1998) A_B extinction for the cluster, and (9) notes, which gives the NGC, UGC, IC, and ESO catalog designations of the BCGs where available, and 15 K denotes a BCG originally selected in the LP94 and PL95 sample.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The sample was originally designed to serve as a reference frame to measure the peculiar velocity of the Milky Way. The inferred luminosities of the BCGs serve as distance indicators, following the methodology presented in Lauer & Postman (1994, hereafter LP94). In that paper, a volume-limited frame was constructed from 119 clusters with V < 15,000 km s⁻¹, again using a ±15° galactic "zone of avoidance." The present sample largely includes the LP94 set (as will be qualified further below). For convenience, we will refer to the LP94 set of clusters as the 15 K sample, while its present augmentation is the 24 K sample. The 24 K outer limit of the survey was selected to provide a significantly deeper reference frame than that constructed in LP94, but one that would not be too strongly affected by the limited depth of the Abell and ACO catalogs, which are heavily incomplete beyond z ∼ 0.1 (Postman et al. 1992).

The present sample is drawn from a considerably larger provisional sample defined by us in the early 1990s based on a literature survey of Abell clusters with measured or estimated redshifts. Because we wanted to construct the best sampling of the local volume possible within the limitations of the Abell and ACO catalogs, we were liberal with accepting plausible candidates for the 24 K sample. As we describe below, the final cluster selection is made with a combination of the latest published redshift surveys and radial velocities that we measured ourselves.

The final set of 433 clusters are those of the larger candidate set for which we observed the BCG, and had sufficient redshift information to determine that the cluster was within the 24 K redshift limit. There are 38 additional clusters that had well-determined redshifts that placed them within the 24 K volume, but for which we were unable to image the BCG or sufficient candidates to be confident of the BCG selection. There are 15 more clusters with non-elliptical BCGs, which were also excluded from the sample. These clusters are listed in Table 2. These two sets include five clusters observed as part of the 15 K sample, but that are now deleted from the present sample for a variety of reasons. With the present richer data set, we now find that for two of the 15 K clusters we in fact observed a small foreground group in front of a rich cluster, selected the M2 rather than the correct BCG in one cluster, and that the BCG is non-elliptical in the remaining two clusters.

The cluster space density as a function of redshift is shown in Figure 2. The cluster counts in each bin have been weighted by the established Abell cluster galactic latitude selection function: P(|b|) = dex(0.3[1 − csc|b|]) (Bahcall & Soneira 1983). The volume computation accounts for the ±15° galactic-plane zone of avoidance. The average cluster comoving space density over the range z ⩽ 0.08 is (5.97 ± 0.28) × 10⁻⁶ Mpc⁻³, including the 53 missed and non-elliptical BCG clusters. The space density of Abell clusters used in this study is relatively constant, with the exception of the last bin where a significant decline is seen. The positive deviation in the first bin, centered at z = 0.015, is not statistically significant (∼2.5σ).

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The completeness of the Abell catalog as a function of richness, redshift, and galactic latitude has been extensively studied. The trends are such that the completeness is lower at lower richness, higher redshift, and lower galactic latitude. More specifically, Postman et al. (2002) show that the detection efficiency as a function of richness class in the Abell catalog is ∼55% for RC = 0, ∼75% for RC = 1 and is essentially 100% for RC ⩾ 2. The richness class distribution in the current 24 K sample is 55% RC = 0, 35% RC = 1, and 10% RC ⩾ 2. Our study here is immune to the known completeness trends so long as the properties of the BCG in the detected Abell clusters are representative of those in the clusters that were missed during the construction of the northern and southern Abell catalogs.

2.1.1. A BCG Hubble Diagram

As a further illustration of the sample geometry and its utility as a probe of the Hubble flow within the local volume, we show a Hubble diagram derived from our BCG sample in Figure 3. The details of the sample selection, reduction, and analysis of the photometry needed to generate this figure are the subject of much of the rest of this paper. For now, the relevant details are that the velocities are mean cluster velocities in the cosmic microwave background (CMB) frame, and the photometry is the apparent metric luminosity, R_m, of the BCGs, but with extinction and k-corrections applied. The photometry has also been corrected to α = 0.5 using the relationship (Equation (8)) between metric luminosity and log α, a parameter measuring the slope of the photometric curve of growth at the metric radius, r_m.

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The Hubble diagram shows that the number of galaxies per velocity interval, rises with distance as ∼D², as expected for a survey with a roughly constant cluster density with redshift. The sharp cutoff at the 24 K velocity limit is also evident. The rms scatter about the nominal theoretical relation specified by Ω_m = 0.3 and Λ = 0.7 is 0.271 mag. We can constrain any departures from the expected Hubble flow as a function of redshift by binning the residuals about the Hubble relation. Since our sample is full sky, this effectively tests for monopole variations in the Hubble flow with distance. Figure 3 shows the mean residuals in shells of 4000 km s⁻¹ starting at 4000 km s⁻¹, expressed as ΔH/H₀. All but the innermost shell are consistent with ΔH/H₀ = 0 at the <2.0% level; the innermost shell is also consistent with ΔH/H₀ = 0, but with a poorer 5.7% error due to the small number of BCGs interior to 8000 kms⁻¹. The present result is completely consistent with the BCG Hubble diagram derived from the earlier 15 K sample (Lauer & Postman 1992), but with errors nearly a factor of two smaller.

We define the BCG to be the brightest member (in the R_C band) of the cluster within a 14.3 kpc radius⁴ "metric aperture" centered on the galaxy (and with close or embedded companions photometrically subtracted), with the proviso that the galaxy must also be an elliptical. The use of metric BCG luminosity as a distance indicator was initially advanced by Gunn & Oke (1975), and developed further by Hoessel (1980) and PL95. As demonstrated in PL95, our particular choice of the metric aperture minimizes the scatter in the average BCG luminosity. The aperture is large enough to include a large fraction of the total luminosity of the BCG but avoids the difficulty of measuring a total magnitude for the BCG, which requires surface photometry at very faint levels and large angular radii in a rich-cluster environment. This problem has limited the accuracy of a number of recent studies of BCGs. The photometry provided by both the SDSS and Two Micron All Sky Surveys (2MASS), for example, strongly underestimates the total luminosity of low-z BCGs (Lauer et al. 2007). The SDSS photometry suffers from over-subtraction of the sky background, while the 2MASS total magnitudes are based on a profile model that fails to include the extensive envelopes of the galaxies.

As with the definition of the cluster sample, we were liberal with observing all plausible BCG candidates for any given cluster, making the final choice only when all the observations were in hand. The initial selection of BCG candidates was done visually from digitized sky-survey plates, augmented with velocity information when available. Unless one galaxy was strongly dominant and known to be in the cluster from its redshift, we would typically select several bright elliptical galaxies for imaging and spectroscopic observations, with the final selection based on CCD aperture photometry and knowledge of the cluster redshift.

As noted in PL95, the BCGs in the 15 K sample were often displaced in angle and/or velocity from the nominal cluster center, thus we attempted to select all bright elliptical galaxies within the nominal Abell radius of the cluster, rather than the brightest "central" galaxy. The Abell radius is 1.5 h⁻¹ Mpc or 2.1 Mpc for the present cosmological parameters.⁵ One of the questions that will be considered in the later sections is the extent to which the BCG is in fact displaced from the cluster spatial and velocity centroids, allowing for the possibility that the BCG may be significantly offset from either is critical to the BCG selection.

The selection of the BCG can be complex and different surveys may disagree on which galaxy is the BCG in any given cluster. As one example, we compared our selection to those from von der Linden et al. (2007), who extracted their sample from the SDSS-based C4 cluster catalog (Miller et al. 2005) using isophotal magnitudes for BCG luminosity. Of the 429 C4 clusters selected by von der Linden et al. (2007) that should be within our redshift limit, only 44 clusters are in our sample.⁶ Of the 44 cluster matches, we agreed on the BCG in 33% or 75% of the clusters. In 8 of the 11 clusters remaining, von der Linden et al. (2007) selected a galaxy that we classified as M2, the second-ranked galaxy, based on our photometry. As noted in the next section, this choice may depend on the size of the metric aperture, but we concluded that our M2 would be the BCG based on total flux (see below) in only three of the eight cases. Lastly, for one cluster, A1142, the C4 catalog identified two clusters, with the BCG for one corresponding to our M2 for A1142.

2.2.1. A Subset of Bright M2 Members

2.3.1. Observations

2.3.2. Image Reduction and Surface Photometry

2.3.3. Cross-validation of the Photometry

To provide additional validation of the galaxy photometry, observations of several galaxies were repeated in multiple runs, as was also done in PL95. We also re-imaged most of the 15 K BCGs in the course of obtaining the 24 K sample. This not only allows the accuracy of any given aperture measurement to be confirmed, but also provides a test for any systematic differences in the photometric zeropoints between the various runs. Because we were concerned with obtaining consistent photometry over the full sky, care was taken to ensure that extensive cross-validation observations between the north/south hemispheres and spring/fall observing seasons were obtained. Figure 4 shows the measured differences in the metric magnitude for galaxies observed across different runs. The data allowed 48 separate inter-run comparisons to be performed, assembled from 260 overlap galaxy observations obtained over the course of the imaging part of the survey. This provided 62% of the 78 potentially unique comparisons among the 13 runs, thus densely populating the run/run cross-correlation matrix.

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The total rms for all differences between pairs of duplicate galaxy metric magnitudes is 0.0195 mag, which implies that the average error in any single measure is 0.0138 mag, a factor of $\sqrt{2}$ smaller. By fitting for an average photometric offset correction for each run (done as a simultaneous least-squares fit to the entire ensemble of overlap observations), we can reduce the total difference rms to 0.0159 mag, or 0.0112 mag for any single observation. Because the average photometric correction for any single run is small, 0.0104 mag, we have chosen not to apply the corrections in the present work; however, systematic differences between runs are more important for the measurement of large-scale deviations from the Hubble flow and will be considered in our use of the present sample as a velocity reference frame.

We obtained long-slit spectra of all BCG candidates in the sample over the course of 14 observing runs, spanning a 5 yr timeframe, at NOAO's CTIO and KPNO. The CTIO observations were done primarily using the Blanco 4 m Telescope, except for the first two runs, which used the 1.5 m telescope. All KPNO runs were done using the Goldcam spectrograph on the 2.1 m telescope. Table 5 summarizes the instrumental parameters. The slit width was set to 2''. For most observations, two or three independent exposures were obtained (and co-added for further analysis), although in some cases only a single exposure was acquired. The exposure times for each individual exposure varied depending on telescope aperture and the estimated target redshift. As the overall objective was to use the spectra to obtain both a measurement of the redshift and the internal stellar velocity dispersion, we set integrations to achieve a minimum signal-to-noise ratio (S/N) of 20 per pixel in the final co-added one-dimensional (1D) spectrum. A total of 842 co-added spectra were obtained for 689 unique galaxies.

Over the course of the survey, we repeatedly observed 13 bright nearby galaxies as radial velocity reference standards. These observations were designed to provide a cross-check on our redshift measurement accuracy over the duration of the program. The mean absolute value of the velocity difference between the reference galaxy redshifts from different runs was 32 km s⁻¹ (〈Δv/v〉 = 0.005) with an rms scatter of 38 km s⁻¹. We also observed a subset of BCGs multiple times, both from this survey and from the earlier LP94 survey to serve as cross-checks between different observing runs and telescopes. For the velocity dispersion estimates, a series of spectra were repeatedly obtained of 27 K-giant stars.

The two-dimensional spectra were corrected for basic instrumental signatures. Bad columns were identified and interpolated over. Bias subtraction was done by first using the overscan region to determine the mean DC level, which was subtracted from the full frame. Bias structure removal was then performed using a series of zero-duration exposures acquired before the start of each night. Quartz lamp exposures were co-added and normalized to provide a flat-field correction frame. Any cosmic ray hits that extended for more than 2 pixels were manually identified and interpolated over where possible. Smaller cosmic ray hits were dealt with during co-addition of the extracted 1D spectra.

The 1D spectra were extracted and wavelength-calibrated using IRAF's NOAO onedspec package. The extraction was done using a third-order Legendre polynomial function to allow the aperture center to track any significant spectral curvature along the dispersion axis. The average spectrum extraction aperture width was 9''(rms 25), which is significantly larger than the typical FWHM seeing (∼15) for any given observation. The 9''width corresponds, on average, to a projected physical width of 10 kpc (rms 3.6 kpc). The background level was estimated in two 15 pixel wide regions on either side of the source spectrum with a 15 pixel gap between the center of the source spectrum and the start of the background sampling regions. Two iterations of 3σ rejection were done during both spectrum tracing and background level determination to reduce susceptibility to cosmic rays. Occasionally, spectra for other galaxies (in addition to the BCG) fell along the slit. We extracted these spectra as well in hopes of providing additional redshift information for the clusters.

The extracted 1D spectra were wavelength-calibrated by extracting identical regions of the companion arc lamp spectra obtained either just before or just after each galaxy spectrum. Helium–neon–argon arc lamps were used for these observations. The IRAF dispcor routine was used to perform the wavelength calibration. We typically used a third-order polynomial wavelength solution. The wavelength calibration was checked both by looking at the fit residuals provided in the IRAF identify and reidentify routines, and by confirming that the prominent night sky emission lines appeared at their proper central wavelengths. A final co-added 1D spectrum for each object observed on a given night was then produced from the individual wavelength-calibrated 1D spectra using the IRAF scombine routine. Any cosmic ray artifacts that may have survived the co-addition were manually removed via interpolation using IRAF's splot routine.

2.4.1. Redshift Measurements

2.4.2. Velocity Dispersion Measurements

We measured central stellar velocity dispersions from the extracted 1D spectra using a "direct" penalized pixel-fitting method, as implemented the IDL code pPXF⁹ (Cappellari & Emsellem 2004). We use the pPXF code in combination with single-burst stellar population synthesis models (Vazdekis et al. 2010) based on the empirical MILES stellar library (Sánchez-Blázquez et al. 2006), with a Chabrier-style initial mass function (model "Mbi1.30").¹⁰ The templates span a range of metallicities (−1.71 < [Z/H] < +0.22) and single-burst ages (1 Gyr < age < 17.8 Gyr). As part of the fitting process, pPXF finds the linear combination of templates that best reproduces the galaxy spectrum. These models are convolved with the instrumental resolution for each observing run, which is modeled using a low-order polynomial fit to the width of the arclines, and typically varies as a function of wavelength. We allow pPXF to fit for four velocity moments (V, σ, h3, and h4) and use a fourth-order multiplicative polynomial to account for continuum mismatch due to imperfect spectral flux-calibration. We mask regions covering possible strong emission lines (the Balmer lines and [O iii] λ4959,5007), and run the fit iteratively; on the first run, we identify ±4σ outliers in the fit residuals as noise spikes, mask them out, then rerun the velocity fits.

We further experimented with a number of model parameter choices that might introduce systematic effects into the σ measurements. The massive BCGs in our sample have non-solar abundance ratios (in particular, they are enhanced in Mg, CN, and C₂, e.g., Graves et al. 2007; Greene et al. 2013), while the stellar population templates have solar-scale abundance patterns. We experimented with masking the absorption line regions strongly affected by these non-solar abundances, but this had a negligible impact on the resulting σ measurements. We also investigated the effects of using the pPXF "BIAS" keyword to push the velocity solution toward low values of h3 and h4. This also had a negligible effect on our derived σ values. Fitting for only two velocity moments (i.e., setting h3 = h4 = 0) in some cases produced a modest increase in the derived σ values, as did using a higher-degree multiplicative polynomial for the continuum adjustment. There was little effect for galaxies with σ ∼ 150 km s⁻¹, but increases of ∼20 km s⁻¹ for galaxies with σ ∼ 350 km s⁻¹ were seen. We elected to use the fourth-order multiplicative polynomial and unbiased four moment velocity fits for our final measurements.

By far the largest systematic effect was the choice of rest-frame wavelength interval used in the velocity fits, with σ values biased by up to ∼40 km s⁻¹ when comparing different wavelength intervals. Our observations were not acquired with uniform wavelength coverage (see Table 5). Accordingly, we define three rest-frame wavelength ranges to use in our analysis. The "full" range of 4000–6200 Å was available for 9 of our 14 observing runs. We also defined a more limited "blue" range of 4350–5200 Å for runs CT92F, CT93S, and KP92S, and a "red" range of 4750–5700 Å for runs CT94F and CT94S. For the nine runs with full wavelength coverage, we measured σ from the "blue," "red," and "full" wavelength separately, in order to calibrate the effect of differing wavelength coverage as described below.

Due to imperfect data archiving, we were only able to retrieve pixel-by-pixel error arrays for a subset of the observing runs. The pPXF code uses error spectra both to penalize low S/N or bad pixels in the fitting process and to estimate errors in the derived parameters, such as σ. We were able to use the iterative outlier rejection described above to mask bad pixels, with all other pixels being assigned equal weight. In order to estimate the uncertainties in measured values of σ, we again resorted to iterative use of the pPXF code. In the first run, all pixels were simply assigned equal (and arbitrary) errors. pPXF outputs the residuals between the best-fitting template combination and the observed spectrum, which have Gaussian scatter about zero. We used the width of the scatter as an estimate of the typical flux error per pixel, then reran pPXF using this value as the input error for all pixels to propagate through the resulting uncertainties in σ. Where the true error spectra were available, we could compare these estimated errors in σ with the true errors; the difference in error estimates was Gaussian with a mean offset of 0.63 km s⁻¹ and width of 0.75 km s⁻¹. This means that where we could compare them, the bootstrapped error estimates agreed with the true statistical error estimates to within 1–2 km s⁻¹. This made us confident that bootstrapped errors could be used reliably for observations whose error spectra had been lost. Overall, the typical uncertainty in our σ measurements is ∼14 km s⁻¹, but varies substantially between observations, depending on the spectral S/N and wavelength coverage.

For runs with full wavelength coverage, we compared the σ measurements from the full 4000–6200 Å range to those derived from the more limited blue (4350–5200 Å) or red (4750–5700 Å) wavelength ranges in the same spectra, as shown in Figure 5. This calibration demonstrates that when only the blue coverage is available the resulting σ measurements are unbiased. The mean offset between the blue and full coverage σ values is −3.2 km s⁻¹, with rms scatter of 19.6 km s⁻¹ and no clear trend with σ. In contrast, measurements made with only the red coverage show substantial bias; the mean offset is +9.3 km s⁻¹ but increases to ∼30 km s⁻¹ for the highest σ galaxies, with similar scatter of 20.7 km s⁻¹. To put all of our targets onto the same effective system, we fit a line that defines the "correction" from the red coverage onto the full coverage values. This correction is applied to the σ measurements from the CT94F and CT94S runs, which only have the red wavelength coverage. No correction is applied to the runs with blue coverage.

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The spectroscopic observations include many repeat measurements of individual targets, usually in different runs. These can be used to test the internal consistency of our σ measurements. Using only σ measurements made from the full spectral coverage, we find that differences between repeat measurements of galaxies are Gaussian distributed with a standard deviation of 16.5 km s⁻¹. This is comparable to the expected typical statistical error of 14 km s⁻¹, suggesting that the σ measurements are stable across the various runs. This is not a trivial statement, given that the observations use two different telescopes, different instruments and instrumental configurations, and span multiple years including instrument upgrades. Comparing repeat observations on a run-by-run basis, the runs showing the largest mean offsets from the rest are KP92S (−18.6 km s⁻¹ for 16 galaxies), CT93S (17.3 km s⁻¹ for 4 galaxies), and KP94F (−12.3 km s⁻¹ for 12 galaxies). Notice that none of these deviant runs are the "corrected" runs with only red wavelength coverage. All other runs show offsets that are <10 km s⁻¹ from the aggregate.

Where multiple observations are available, rather than averaging the individual measurements, we assign a "best" measurement for each galaxy. For the vast majority of the sources, the various observations agree within the estimated 3σ errors; for these sources, the best measurement is the one with the smallest formal error in σ (i.e., that measured from the highest-S/N spectrum). For the six galaxies where repeat measurements show catastrophic disagreement (>3σ), we choose the best observation based on the following criteria: full wavelength coverage is preferred over limited red or blue coverage, spectra with noticeable flux calibration or sky subtraction issues are disfavored, higher S/N is preferred over low S/N, and better wavelength coverage is preferred over higher S/N. These measurements and calibrations result in a sample of 689 σ "best" measurements among our galaxies.

Finally, 78 of our galaxies are in the SDSS spectroscopic survey, making it possible to compare our σ measurements to those from the SDSS spectroscopic pipeline (Bolton et al. 2012; see Adelman-McCarthy et al. 2008 for a comparison of different velocity dispersion algorithms in SDSS). This comparison is shown in Figure 6. The overlapping galaxy sample is mostly from the KP96S run (black points), with a few from KP93S (green points) and one from KP92F (orange point). The solid line shows the one-to-one relation. The inset panel shows a histogram of the differences between our σ and the SDSS σ measurements. These have a Gaussian distribution with a mean offset of 19.9 km s⁻¹ and width of 19.3 km s⁻¹. The scatter compares favorably with the estimated statistical errors in the measurement (∼17 km s⁻¹ for the typical errors from our observations and those from the SDSS combined in quadrature) and the internal consistency of our repeat measurements (16.5 km s⁻¹). However, there is a significant systematic offset of ∼20 km s⁻¹. The larger angular apertures (∼9'') of the spectral extractions used in this work relative to the 3'' fiber apertures used in SDSS may explain some of the systematic shift if the trend of increasing stellar velocity dispersion with increasing radius, like that seen in the BCG in A383 (Newman et al. 2011), is typical. We do not attempt to "correct" our values onto the SDSS system, but note that work combining σ measurements from different sources and spectral reduction pipelines must take such systematic variations into account. Dispersion values for the BCGs and M2 galaxies are listed in Tables 7 and 8, respectively.

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Table 7. BCG Parameters

Abell	α	Mm	Mm(2rm)	Mm(4rm)	σ	rx	Ref
					(km s−1)	(kpc)
0014	0.983	−22.872	−23.564	−24.069	219 ± 15	73	R
0027	0.289	−22.298	...	...	...	...	...
0071	0.397	−22.517	−22.740	...	262 ± 14	...	...
0074	0.447	−22.927	−23.223	−23.461	315 ± 6	...	...
0075	0.455	−22.728	...	...	295 ± 12	...	...

Notes. Columns: (1) Abell cluster number, (2) α at the metric radius, (3) absolute metric luminosity, (4) absolute metric luminosity within 2r_m, (5) absolute metric luminosity within 4r_m, (6) galaxy stellar velocity dispersion, (7) radial offset of the BCG from the X-ray cluster center, (8) X-ray reference, R = ROSAT, C = Chandra. The CMB frame has been adopted for the calculation of metric luminosities.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 8. M2 Parameters

Abell	α	Mm	σ	rx
			(kms−1)	(kpc)
0027	0.542	−22.229	224 ± 20	...
0071	0.323	−22.340	223 ± 21	...
0074	0.408	−22.669	...	...
0075	0.528	−22.316	...	...
0076	0.397	−22.601	...	313

Notes. Columns: (1) Abell cluster number, (2) α at the metric radius, (3) absolute metric luminosity, (4) galaxy stellar velocity dispersion, (5) radial offset of the BCG from the X-ray cluster center, The CMB frame has been adopted for the calculation of metric luminosities.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Cluster redshifts were derived based on galaxy velocities drawn from the database maintained by the NASA Extragalactic Database, augmented by velocities measured for the BCG candidates by us, and SDSS DR7 spectroscopy (Abazajian et al. 2009) where available. The "biweight" estimator of Beers et al. (1990) was used to calculate the mean cluster redshifts, V_c, and velocity dispersions, σ_c. The initial calculations used galaxies within ±3000 km s⁻¹ of the nominal BCG or estimated cluster redshifts, and the cluster Abell radius. While the biweight statistic is designed to be robust in the presence of background or foreground contamination, we still considered it prudent to remove obvious background contamination or other complexities, such as overlap with nearby clusters or groups. This was done by ad hoc inspection of the velocity maps and histograms for each cluster. A second robust statistic introduced by Beers et al. (1990) was used to estimate the cluster velocity dispersion for clusters with four or more velocities.

Table 1 lists the final number, N_g, of galaxy velocities used to compute the mean velocity and dispersion. This parameter is used as a general marker for the quality of both parameters. For some evaluations of the peculiar velocities of the BCGs within their clusters, we will restrict the analysis to clusters with N_g ⩾ 50, to minimize the effects of the error in the mean velocity. For analyses requiring accurate σ_c, we require the clusters to have N_g ⩾ 25.

The stereotypical image of a galaxy cluster has the BCG centrally located, both in projected angular coordinates and radial velocity relative to other cluster members. Studies of clusters with rich enough velocity sampling such that an accurate mean cluster redshift can be estimated, however, show that the BCG may often have a significant "peculiar velocity" with respect to their hosting cluster (Zabludoff et al. 1990, 1993; Malumuth 1992; Oegerle & Hill 2001). PL95 obtained the distribution of BCG ΔV₁ ≡ (V₁ − V_c)/(1 + z) for 42 clusters with 20 or more member velocities, finding that the ΔV₁ followed a Gaussian distribution with $\sigma _{\Delta V_1}=264 \,{\rm km \,s^{-1}},$ once the error in cluster mean redshift was accounted for. This value is ∼0.4 of the typical 1D cluster velocity dispersion, σ_c = 666 km s⁻¹, of the same subset of clusters. To test the hypothesis that the BCG peculiar velocities may be related to their masses, merger histories, and ages, we have derived the BCG peculiar velocity distribution function and investigated the relationship of this parameter to other BCG properties. The distribution function is considered in this section, while the relationship of BCG peculiar velocities to other BCG properties will be considered later in the paper.

Figure 7 shows the absolute values of ΔV₁ as function of cluster velocity dispersion for the 178 clusters in the present sample that have 50 or more member velocities. With this level of velocity information, the error in the mean cluster velocity is ∼100 km s⁻¹ or less, allowing relatively small ΔV₁ to be detected. As can be seen, most BCGs have ΔV₁ well in excess of this error threshold.

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The median peculiar velocity increases with σ_c. A power-law fit shows

$$\begin{equation} |\Delta V_1|=152\pm 15\left({\sigma _c\over 600 \,{\rm km \,s^{-1}}}\right)^{0.66\pm 0.26} \,{\rm km \,s^{-1}}. \end{equation} \tag{ 1 }$$

Coziol et al. (2009) studied the distribution of BCG peculiar velocities in a large sample of Abell clusters and also found |ΔV₁| to increase with σ_c, although they did not quantify the trend.

Figure 8 shows the binned distribution of ΔV₁ normalized by the cluster velocity dispersion. Normalizing by σ_c largely removes any dependence of the amplitude of the peculiar velocity on the properties of the cluster itself. Since we are restricting this analysis to clusters with 50 redshifts or more, the errors on ΔV₁/σ_c will be ${<}1/\sqrt{50}\approx 0.14.$ In normalized units, the mean ΔV₁/σ_c = 0.04 ± 0.04, with an rms dispersion of 0.49—note that this number measures a different statistical property of the distribution than does the median peculiar velocity plotted in Figure 7. The two BCGs with the largest ΔV₁/σ_c values are those in A2399 and A3764, which have |ΔV₁| of 1191 and 1180 km s⁻¹, respectively, or normalized values of 1.67 and 1.51.

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The distribution of |ΔV₁|/σ_c is exponential in form. Since the line-of-sight velocity distributions of galaxies in clusters are well known to be Gaussian (Yahil & Vidal 1977), a random draw of any non-BCG cluster member would, of course, echo this expectation. An exponential velocity distribution specific to the BCGs is thus surprising. The best-fitting exponential distribution is

$$\begin{equation} \ln \left(N\right)=-2.54\pm 0.18 |\Delta V_1|/\sigma _c + 3.76\pm 0.20, \end{equation} \tag{ 2 }$$

where N is the number of clusters per bin of width 0.1 in |ΔV₁|/σ_c. The implied exponential scale length (the reciprocal of the slope given above) is thus 0.39 ± 0.03 in |ΔV₁|/σ_c; this form and scale implies that the median |ΔV₁|/σ_c is 0.26. Note that for small |ΔV₁|/σ_c, the observed distribution represents the convolution of an unknown intrinsic peculiar-velocity distribution with the error distributions of the BCG and cluster redshifts, plus the errors in the cluster dispersions. However, the observed distribution appears to be smooth and simple in form, thus the intrinsic distribution is likely to transition smoothly from BCGs with |ΔV₁|/σ_c ≈ 0 to those with large |ΔV₁|/σ_c, where the peculiar velocity of any given BCG is clearly significant.

The present distribution appears to be similar to that measured by Coziol et al. (2009), although they did not characterize it with any functional form. We tested the likelihood that the distribution was non-Gaussian using an Anderson–Darling (AD) test (Stephens 1974), finding that a Gaussian distribution is strongly rejected. For the observed rms distribution of ΔV₁/σ_c of 0.46, the Gaussian is rejected at the 8 × 10⁻⁶ significance level. Deletion of the two galaxies that have the largest relative peculiar velocities decreases the rms value of the distribution to 0.44, and the AD test allows a Gaussian at 6 × 10⁻⁵ significance.

It has long been known that the difference in peculiar velocities of pairs of galaxies is exponentially distributed on small scales, both in observations of redshift-space distortions of the two-point correlation function and in simulations (Fisher et al. 1994; Marzke et al. 1995). This effect has been explained in terms of the number, rather than mass, weighting of galaxies in pair statistics (Diaferio & Geller 1996; Juszkiewicz et al. 1998). We are unaware of an equivalent study of BCG peculiar velocities in clusters. Reid et al. (2014), in their analysis of redshift-space distortions of SDSS/BOSS galaxies, identified massive halos at z ≈ 0.55 in a ΛCDM N-body simulation containing 2048³ particles in a box 677.7 h⁻¹ Mpc on a side. They measured the difference in peculiar velocity between the most dense spherical region of radius 0.2 times the virial radius, and the cluster overall. B. A. Reid (2014, private communication) finds that the distribution of this difference is accurately exponential in a variety of halo mass bins corresponding to rich clusters. While it is unclear whether the Reid et al. identification of the highest-density region in each halo is a good proxy for the BCG, this result is intriguing, and it would be interesting to explore more detailed cluster simulations in which individual subhalos can be identified.

The exponential distribution may reflect a dispersion in the ages of the clusters, the timing of when the BCG was captured by the cluster, or may simply be due to the superposition of Gaussian distributions of different velocity dispersions, weighted by the BCG number distribution (Diaferio & Geller 1996). BCGs in the tail of the exponential distribution may be those in which the BCG arrived to the cluster in the merger of a group or subcluster relatively recently, and not yet completely relaxed. These ideas could also be explored in simulations, or by looking for correlations between BCG peculiar velocity and signatures of merging in their host clusters.

The X-ray emission from the intracluster medium (ICM) provides insight into processes that govern the formation and evolution of the BCGs. Numerous investigations (Edge 1991; Edge & Stewart 1991; Hudson & Ebeling 1997; Collins & Mann 1998; Stott et al. 2012) find significant positive correlations between the total luminosity of the BCG and the X-ray luminosity and X-ray temperature. Schombert (1988) finds that the envelope luminosity of cD galaxies, a subset of the BCGs, increases with total cluster X-ray luminosity. Stott et al. (2012) find that the steepness of the L_X–T_X relation in galaxy clusters correlates with the stellar masses and X-ray offsets of their BCGs. Clusters in which the offset between the BCG position and the peak of the X-ray surface brightness distribution is small tend to be the most regular, most massive systems (Allen 1998; Smith et al. 2005; Hudson et al. 2010). Haarsma et al. (2010) find that ∼90% of local (z < 0.2) clusters host a BCG within ∼30 kpc of the X-ray peak, although their sample is small, and unlikes us, they include a criterion of proximity to the X-ray peak for selecting the BCG from among candidates of "similar" brightness. Brough et al. (2005) find that the structure of the BCG correlates with cluster X-ray luminosity, with the BCG envelope becoming more extended in more luminous clusters.

Our sample is well suited to characterize the precise form of the distribution of the spatial offset, r_x, of the BCG from the peak of the ICM X-ray emission, and to assess whether the spatial offset correlates with the velocity offset of the BCG relative to the mean cluster velocity.¹¹ In addition, the availability of robust BCG profile shape measurements allows us to determine if the BCG stellar light profile is influenced by the spatial offset.

We cross-correlated our BCG catalog with the ROSAT-based X-ray Brightest Abell-type Cluster Survey (XBACS; Ebeling et al. 1996). XBACS is a flux-limited catalog derived from the ROSAT All-Sky Survey (RASS; Voges et al. 1999). Of the 283 Abell cluster sources listed in the VizieR version of the XBACS sample, 111 are in common with our current survey. An additional 70 clusters in our current sample have X-ray peak positions from the analysis of the RASS done by Ledlow et al. (2003), which extended to an X-ray flux limit that is ∼seven times lower than that used to derive the XBACS. Chandra X-Ray Observatory data were obtained as well for 48 of the ROSAT clusters from the Archive of Chandra Cluster Entropy Profile Tables (Cavagnolo et al. 2009), which we use in preference to the ROSAT peak positions given the superb Chandra angular resolution. We also searched the literature for XMM data but only 13 of the clusters in our sample have XMM data, too few to provide independent cross-checks on the ROSAT and Chandra samples. We thus focus our X-ray analyses on the above subsample of 174 clusters in our survey, using ROSAT data for 127 clusters and Chandra data for 47 clusters. We note that while Chandra is not a survey mission, the Abell clusters in our survey that have Chandra X-ray temperatures and luminosities that span the same range as the ROSAT temperatures and luminosities. No significant biases are introduced by including the Chandra data in our study of BCG dependence on the X-ray properties of their host clusters.

ROSAT and Chandra have very different on-axis point-spread functions (PSFs): FWHM of ∼05 for Chandra versus ∼1' for ROSAT. We used a sample of 101 Abell clusters (not limited by the redshift limits of our current survey) with observations from both observatories to measure the typical difference, θ_x, between the ROSAT peak position and the Chandra peak position. The distribution of θ_x is shown in Figure 9. The median and mean differences between the ROSAT X-ray position and the Chandra X-ray position are 43''and 69'', respectively. The median and mean difference between the ROSAT and the Chandra peak X-ray positions in projected physical distance units derived from cluster redshift information are 68 kpc and 121 kpc, respectively.

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We characterize the distribution of differences between the location of the Chandra and ROSAT X-ray peaks on the assumption that the distribution can be modeled as a circularly symmetric Gaussian. This will be almost entirely due to the large ROSAT PSF, but it may be more compact than that given that the appropriate source of variance is the error in the ROSAT centers, rather than the width of the ROSAT PSF itself. The distribution of the total angular differences between the X-ray peaks will be a Rayleigh distribution:

$$\begin{equation} p(\theta _x)=\sigma _x^{-2} \exp \left[{-\theta _x^2\over 2\sigma _x^2}\right] \theta _x d\theta _x, \end{equation} \tag{ 3 }$$

where σ_x is the dispersion of the Gaussian, as well as the peak location of the above distribution. We fitted this form to θ_x in 10''bins, limited to the six bins around the peak of the distribution shown in Figure 9 to avoid the effects of extreme outliers. We measure σ_x = 25'' ± 3'', or an implied FWHM of 60'' for the underlying position-error Gaussian. The adopted distribution given by Equation (3) for this σ_x is also plotted in Figure 9.

Figure 10 shows the offset between the BCG position and ROSAT position as a function of the velocity offset between the BCG and the cluster for the 174 clusters in our sample with ROSAT data. The somewhat triangular shape of the distribution of points shows that there is an overall correlation between the BCG spatial and velocity offsets. In particular, |ΔV₁|/σ_c > 1 occurs only for clusters with r_x > 200 kpc, while all r_x < 40 kpc clusters have |ΔV₁|/σ_c < 0.5. Stated qualitatively, BCG that are close to the X-ray center in projection always have relatively small velocities. Of course, with a large enough sample, there should be BCGs with high velocities seen in projection against the center; however, r_x < 40 kpc corresponds to a very small portion of the projected area of the galaxy clusters.

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Figure 11 shows the radial distribution of r_x for the subset of BCGs with cluster X-ray centers as a plot of BCG surface density as a function of radius. The normalization is set so that the cumulative integral over the surface density for each cluster is unity (each cluster has a single BCG). Generating this figure requires understanding the effects of the angular resolution on the X-ray peak locations obtained with ROSAT as well as how to incorporate the high-resolution Chandra data into the sample. As it happens, however, this is only a minor issue as the ROSAT data are used mainly at larger radii and the Chandra data are used exclusively at small radii.

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The solid points in Figure 11 represent the entire X-ray sample, using both Chandra and ROSAT together, as radial bins of r_x. The innermost bin extends from the origin to 50 kpc, thus enclosing most of the ROSAT PSF. Subsequent bins are rings starting at 50 kpc, with inner and outer limits increasing geometrically by a factor of two. The data follow a simple power law, which as we discuss below, appears to be the best form for the overall BCG spatial distribution. A log–log line fitted to the points gives the surface density as

$$\begin{eqnarray} \log _{10}\rho (r_x) &=& (-2.33\pm 0.08)\log _{10}\left [{r_x\over 1\, {\rm kpc}}\right ] \nonumber\\ && +5.30\pm 0.19 \log _{10}[{\rm N Mpc^{-2}}], \end{eqnarray} \tag{ 4 }$$

which is shown in the figure as the red line.

We used Monte Carlo simulations to understand the effects of the limiting ROSAT resolution on the apparent density profile, and to verify that the profile incorporating ROSAT data was consistent with the center of the distribution inferred from Chandra alone. For an assumed surface-density profile, we drew 10⁴ "BCGs" for each cluster in the X-ray subset. For clusters observed by only by ROSAT, we scaled the angular position-error Gaussian to the appropriate physical resolution, given the redshift of the cluster, and drew a point at random from the circularly symmetric Gaussian centered at a radius drawn from the density distribution. For Chandra, we simply drew galaxies directly from the assumed profile. The form we tested included an inner quadratic core to suppress the singular number integral as r_x → 0, implied by the γ < −2 power law:

$$\begin{equation} \rho (r_x)=\rho _0\left(1+\left({r_x\over a}\right)^2\right)^{\gamma /2}, \end{equation} \tag{ 5 }$$

where a specifies the core scale, and ρ₀ the central surface density.

We did not attempt to derive a formally, but simply compared the quality of the fits obtained by varying a geometrically in the sequence of a = 5, 10, 20, and 40 kpc. We fixed γ = −2.33, given its excellent description of the outer profile, where the ROSAT PSF would have little effect. As it happens, even with the large 50 kpc outer limit of the innermost bin using the full X-ray sample, we required a ⩽ 10 kpc to obtain a satisfactory fit to the central point. The "observed" profile for a = 10 kpc incorporating the blurring of the ROSAT offsets is shown in Figure 11 as the solid blue line, while the intrinsic unblurred profile is shown as the dotted line. The effects of the ROSAT resolution is evident on the simulated profile, but with a = 10 kpc, the form given by Equation (5) just matches the central point. It is also noteworthy that the profile incorporating ROSAT data is indeed compatible the profile inferred from Chandra data alone.

The subset of Chandra observations underscores the conclusion that any core in the BCG r_x distribution must be extremely small. Figure 11 shows the implied surface density from the Chandra clusters alone. Again the Chandra points were incorporated into the bins representing the full sample, but here we can use considerably finer radial bins, given the superb Chandra angular resolution. The inner Chandra bin extends from the origin to 4 kpc, with the next two bins covering 4–16 and 16–64 kpc. The innermost Chandra bin is fully an order of magnitude above the density implied by the a = 10 kpc profile. The pure power-law fit given in Equation (4) did not incorporate the central Chandra-only points, but its inward extrapolation clearly falls only slightly above them. In short, the BCG r_x distribution shows no sign of any core or decrease in slope as r_x → 0.

3.2.1. Some BCGs Have Large X-Ray Offsets

A radial integral over Equation (4) gives the cumulative distribution of the BCGs away from the cluster X-ray center, which is shown as an inset in Figure 11. The integral starts at 1 kpc to avoid the central divergence, and continues out to include a few clusters with r_x > 1 Mpc. The steep power-law form in Equation (4) unifies two superficially different pictures of where BCGs are located in their hosting clusters. The median r_x implied by this distribution is only ∼10 kpc, and is consistent with the common impression that most BCGs reside close to the center of the X-ray gas, and presumably to the center of the cluster potentials. At the same time, the distribution also includes BCGs with large displacements from the X-ray defined center; 15% of the BCGs in the present sample have r_x > 100 kpc, with the largest offsets reaching ∼1 Mpc. Because the finding that some BCGs may be greatly displaced from the center of the cluster potential is strongly at odds with the paradigm that BCGs should be centrally located (at least in relaxed clusters), we review the evidence for BCGs with large r_x and their import for understanding the formation of BCGs and clusters.

As noted in the Introduction, the first large survey of the X-ray morphology of galaxy clusters (Jones & Forman 1984) showed that the majority of systems have well-defined X-ray cores largely coincident with the position of a bright galaxy, typically the BCG. However, the same study also showed that if the cluster sample was sorted by X-ray core radius, the ensemble showed a smooth progression to clusters with large X-ray cores not coincident with any particular galaxy. To underscore this point, we note two well-studied rich clusters that have long been known to have BCGs markedly displaced from the peak of the X-ray emission. A1367, part of the present sample, is among the first examples found of a cluster with regular X-ray morphology, but with a large offset between the BCG (NGC 3842 in this case) and X-ray center (Bechtold et al. 1983); Table 7 gives r_x = 354 kpc for this cluster. The Coma Cluster (A1656) is the classic example of a rich galaxy cluster, yet it also is a system with a large BCG/X-ray offset, having r_x = 256 kpc. White et al. (1993) analyzed a deep ROSAT image of Coma, and in conjunction with the positions and X-ray morphology of its two bright central elliptical galaxies, NGC 4889 (the BCG) and NGC 4874 (M2), concluded that Coma was produced in a still ongoing merger of two massive clusters.

Martel et al. (2014) emphasize that the BCG offset from the cluster center of mass (which may be different from the location of the peak X-ray emission), as well as the velocity offset discussed in the previous section, is a signature of the assembly of galaxy clusters by hierarchical merging. The BCG itself may be introduced into the cluster as part of an infalling group. A key point is that the galaxies, X-ray gas, and the dark matter halo of the cluster all have strongly different mechanisms and timescales for relaxing after cluster mergers. While we might expect the X-ray morphology of the cluster to be disturbed by strong or recent mergers with smaller clusters or groups, it is likely that the regularity of the X-ray gas distribution is re-established before any new BCG introduced by the merger is dynamically "captured" by the central potential. The regularity of the X-ray morphology plus the amplitude of |ΔV₁/σ_c| and r_x in fact may provide means to constrain the recent merger history of clusters.

Previous studies of the location of BCGs with respect to the peak of the X-ray emission have produced diverse results. Patel et al. (2006) measured r_x for a sample of 49 clusters and found r_x > 100 kpc in 16 systems, or 33% of the sample, a fraction considerably larger than the 15% that we found. The positional accuracy of their centers is low, so a large fraction of the measured offsets with r_x ∼ 100 kpc may really be significantly smaller; however their r_x distribution has a long tail extending to three clusters with r_x > 500 kpc. In contrast, Haarsma et al. (2010) find r_x > 100 kpc for just one cluster out of their small sample of 33, although, as we noted above, they also included proximity to the X-ray peak as a criterion for selecting their BCGs in the first place. It does appear that there is an important distinction between searching broadly within the cluster for the BCG versus selecting the brightest galaxy within the core of the X-ray emission. Hashimoto et al. (2014) explicitly limited their search for the BCG to within 500 kpc of the X-ray peak, but found offsets out to this limit.

While we have used extensive imaging and velocity observations to cast a wide net for the BCG in any cluster, we have relied on the literature to provide the matching X-ray centers. In order to understand the reliability of the largest BCG/X-ray offsets seen in our sample, we obtained archival Chandra or ROSAT images for the subset of clusters with r_x > 500 kpc. Of the clusters with X-ray centers available, we initially identified 29 clusters with r_x > 500 kpc. Of these, we accepted 22 clusters as credible systems with r_x of this amplitude. Our criterion was that the X-ray center had to fall within the extended X-ray source closest to the BCG that was associated with galaxies consistent with the cluster redshift.

Of the seven clusters rejected, three were cases in which the X-ray emission was from either a foreground or background system seen in projection close to the nominal cluster, which itself had no detectable X-ray emission. Since we thus had no valid X-ray center, these clusters were dropped from the set with X-ray data. In four clusters, the clusters were either binary, with the BCG clearly associated with a different X-ray component than we had assumed, or the X-ray emission from a projected cluster at different redshift had been selected over the X-ray emission from the nominal cluster. In these cases we remeasured the r_x with respect to the revised centers. In one cluster, A0548, r_x decreased, but still remained >500 kpc. In the end we conclude that 22/174 or 12% of the sample has r_x > 500 kpc. We show X-ray maps for four examples of clusters with r_x ∼ 1 Mpc in Figure 12. The clusters have well-defined central X-ray emission, but their BCGs are well outside of it.

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In addition to these observational tests, we are encouraged by the cluster formation simulations of Martel et al. (2014), which produce ensembles of clusters that exhibit both the large |ΔV₁/σ_c| and r_x seen in the present sample. Martel et al. argue that their simulations support cluster formation by the "merging group scenario." As various groups merge with the cluster over the age of the universe, the identity of the BCG may change several times. Newly arrived BCGs can be marked by high |ΔV₁/σ_c| and large offsets from the center of their clusters, which we characterize with r_x.

The history of many of the more massive clusters simulated by Martel et al. (2014) show significantly long periods during which the BCG lies at a projected distance of more than 500 kpc or even 1 Mpc from the cluster center. The typical value of |ΔV₁/σ_c| is found to range over 0.15–0.31 for Abell-like clusters, in excellent agreement with the median value of 0.26 found for our sample in the previous section. The maximum value of |ΔV₁/σ_c| seen in the simulations may briefly exceed 1.5 in the early stages of a merger, again in good agreement with the observational limits on the BCG peculiar velocities. We argue later in this paper that additional lines of evidence support the merging-group scenario.

Figure 13 shows the distribution of M_m for the present sample, where we have applied the extinction and k-corrections outlined in Section 2.3.2 to the observed R_C surface photometry. The distribution is well fitted by a Gaussian with mean −22.844 ± 0.016, and standard deviation of σ_L = 0.337 mag. This is good agreement with σ_L = 0.327 mag measured from the 15 K sample in PL95. We have excluded three extremely faint BCGs of the total sample of 433 BCGs from the Gaussian fit and most of the analysis that follows. The lowest-luminosity bin plotted in Figure 13 is −21.7 > M_m > −21.8, which contains a single BCG.¹² The three BCGs in question, those in A3188, A3599, and A3685, all have M_m > −21.46, which is yet fainter by ∼σ_L; all are fainter than the mean M_m by >4σ_L. A3599 and A3685 have very few cluster members with velocities and may not be real systems.

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Hoessel (1980) observed a sample of BCGs to refine their use as "standard candles" in cosmological probes, finding that L_m correlated with the physical concentration of the galaxies. Elliptical galaxies have long been known to have a relationship between total luminosity and effective radius, which is reflected in the relationship between the metric luminosity and radial scale as well. Hoessel (1980) expressed the physical concentration of the BCGs in terms of α, the logarithmic slope of the variation of L_m with the physical radius of the aperture, r, evaluated at the metric radius:

$$\begin{equation} \alpha \equiv d\log L_m / d\log r \left |_{r_m}.\right. \end{equation} \tag{ 6 }$$

In addition to serving as a measure for the concentration of the BCGs, α also relates the error in L_m to a corresponding distance error, σ_D = σ_L/(2 − α) for using BCGs as standard candles. When α = 0, all the galaxy light is contained within the aperture, and L_m is the total luminosity, while when α = 2, the surface brightness distribution is constant with radius, and the metric luminosity provides no information on distance.

The L_m–α relationship observed by Hoessel (1980) showed that α initially increases steeply with L_m, then plateaus for the more luminous BCGs. LP94 and PL95 confirmed this behavior, and quantified it by fitting a quadratic relationship between L_m and α, which they used as a distance indicator. For this application, α is used to predict L_m, which in turn is used as a "standard candle" to infer the distance to the BCGs—in this case, L_m must be the dependent, rather than independent variable. The L_m–α relationship for the present sample is plotted in Figure 14. A quadratic form fitted to all BCG but the three with M_m > −21.5 gives

$$\begin{equation} M_m=-21.35\pm 0.13-(4.12\pm 0.43)\alpha +(2.46\pm 0.36)\alpha ^2. \end{equation} \tag{ 7 }$$

We note that we weight all points equally in this fit and those that follow, unless we indicate otherwise. This is because the observed scatter around the relation of Equation (7) is over an order of magnitude larger than the formal observational errors. The residuals about the relation are 0.267 mag rms in M_m in the CMB frame, essentially identical to the CMB-frame residuals of 0.261 mag for the 15 K sample of LP94. In passing, we note that the quadratic form gives significantly smaller residuals in L_m than does a simple linear relation in α, as we first showed in PL95. The bulk flow of the Abell cluster sample with respect to the CMB was derived by LP94 by finding the average BCG peculiar velocity field that minimized residuals in the L_m–α relationship.

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The quadratic form of the L_m–α relationship derived by PL95 is also shown in Figure 14. The earlier relation falls ∼0.1 mag below the present relationship. The majority of this offset (0.06 mag) is likely to be due to our present use of Schlegel et al. (1998, hereafter SFD) extinctions rather than the Burstein & Heiles (1982) values used by PL95; the average A_B values from SFD are 0.10 mag greater than those of Burstein & Heiles for the 15 K sample.

The quadratic form has the unattractive feature, however, of reaching maximum L_m at α ∼ 0.8, and then predicting fainter L_m as α increases beyond this point where there are few BCGs to elucidate its behavior. This motivated us to introduce a new form, fitting L_m to a linear function of log α. This new form closely parallels the quadratic form over most of the domain, but is monotonic and may better represent the handful of BCGs with α > 0.9. This form has the additional advantage of requiring only two, rather than three parameters. For the present sample with M_m < −21.5, we measure

$$\begin{equation} M_m=-23.26\pm 0.03\ -\ (1.597\pm 0.104)\log _{10}\alpha. \end{equation} \tag{ 8 }$$

The residuals in M_m are 0.271 mag rms in the CMB frame, only 0.004 mag larger than those for the form given in Equation (7). This form is also plotted in Figure 14 and only strongly differs from the quadratic form for α > 1, where we have very few galaxies.

Figure 15 plots the relationship between M_m and σ. While the measurement errors in σ are subdominant to the intrinsic scatter in this relationship, they are not completely negligible, and we have incorporated them into our fitting procedure, following the methodology described in Hogg et al. (2010) and Kelly (2011).

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If M_m is treated as the independent variable, the relationship derived from a simple least-squares fit for the 369 galaxies with M_m < −21.5 and measured σ is

$$\begin{equation} \log _{10}\left({\sigma \over {\rm 300\, km \,s^{-1}}}\right)= -(0.275\pm 0.023)\left({M_m \over 2.5}\right)-2.55\pm 0.21, \end{equation} \tag{ 9 }$$

with 0.052 ± 0.02 scatter in log σ, corresponding to ±12% in σ. Conversely, if σ is the independent variable, then

$$\begin{eqnarray} M_m &=& {-}22.956\pm 0.015-2.5(1.09\pm 0.08) \nonumber\\ && \times \log _{10}\left({\sigma \over {\rm 300 \,km \,s^{-1}}}\right), \end{eqnarray} \tag{ 10 }$$

with an intrinsic scatter of 0.278 ± 0.011 in M_m, implying that σ is just as good as α is for predicting M_m.

The slope in Equation (9) is essentially the same as the classic Faber & Jackson (1976) result of σ∝L^1/4 for normal elliptical galaxies. This suggests that the central portions of the BCGs enclosed within r_m may have a "normal" relationship between σ and L, in contrast to that between total BCG L and σ. As noted in the Introduction, Oegerle & Hoessel (1991) and Lauer et al. (2007) found that σ is only weakly correlated with BCG total luminosity. This behavior may reflect the putative formation of BCGs by dry mergers of less luminous elliptical galaxies. Simulations of this process shows that σ remains essentially constant over dry mergers, with the effective radius, R_e, growing rapidly with L (Boylan-Kolchin et al. 2006). This associated steepening of the R_e–L relation has also been seen in BCGs (Lauer et al. 2007).

We also fitted the Faber–Jackson relationship with M_m measured at 2 times and 4 times the nominal metric radius to test the hypothesis that the relation between BCG L and σ becomes shallower as r_m increases to include a larger fraction of total galaxy luminosity, When the metric radius is doubled, we find

$$\begin{eqnarray} \log _{10}\left({\sigma \over {\rm 300\, km \,s^{-1}}}\right) &=& {-}(0.205\pm 0.022)M_m(2r_m)/2.5 \nonumber\\ && -1.94\pm 0.20 \end{eqnarray} \tag{ 11 }$$

for the 352 BCGs that have both valid σ and photometry at 2r_m, with an intrinsic scatter of 0.054 ± 0.02 in log σ. For M_m measured at 4r_m the sample decreases to 200 BCGs, and we measure

$$\begin{eqnarray} \log _{10}\left({\sigma \over {\rm 300\, km \,s^{-1}}}\right) &=& {-}(0.147\pm 0.027)M_m(4r_m)/2.5 \nonumber\\ && -1.44\pm 0.26, \end{eqnarray} \tag{ 12 }$$

with an intrinsic scatter of 0.053 ± 0.03 in log σ. In short, the slope decreases from ∼1/4 to ∼1/5 and then ∼1/6 as r_m is doubled twice. As even at 4r_m the integrated luminosity of the BCGs is still increasing, the relationship for σ and total luminosity would be yet shallower.

With the large 24 K sample and improved knowledge of galaxy and cluster parameters over what was available to PL95, we can now better investigate sources of residual scatter in the L_m–α or L_m–σ relationships. In fact, given the fundamental plane relationships between total L, σ, and R_e (Dressler et al. 1987; Djorgovski & Davis 1987) for ordinary elliptical galaxies, it is not surprising to find that a multi-parameter "metric plane" relationship between L_m, σ, and α, has smaller scatter than those between any two of these parameters.

The points in the L_m–α relationship shown in Figure 14 are color-coded by σ, showing a strong gradient such that at any α, higher L_m is correlated with higher σ. Likewise, the color-coding of the points by α in the L_m–σ plot in Figure 15 show that at any σ, higher L_m is correlated with higher α. This behavior is shown more explicitly in Figure 16, which plots the M_m residuals from the mean L_m–α relationship given by Equation (8), as a function of σ. A strong correlation is clearly evident.

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Use of α and σ together to predict M_m for the 368 galaxies with M_m < −21.5 and measured σ gives the relationship

$$\begin{eqnarray} M_m &=& -23.31\pm 0.03 -(1.43\pm 0.09)\log _{10}\alpha \nonumber\\ && {-} (2.20\pm 0.17)\log _{10}\left({\sigma \over {\rm 300 \,km \,s^{-1}}}\right). \end{eqnarray} \tag{ 13 }$$

The intrinsic scatter is 0.214 ± 0.010 in M_m, a marked improvement over the L_m–α and L_m–σ relationships. Figure 17 plots observed M_m as a function of M_m(α, σ) estimated from this relationship. This metric plane is not identical to a fundamental plane relation, but with L_m and α serving as proxies for L and R_e, it encodes similar structural information. A detailed comparison between the metric plane and a true fundamental plane relation for the present sample is the subject of our second paper (N. E. Chisari et al. 2014, in preparation).

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4.5.1. BCGs and the Bulk Properties of Clusters

As noted in the Introduction, the structure and luminosity of BCGs may be tied to the properties of the clusters as traced by the temperature and luminosity of the associated X-ray-emitting gas. The present 24 K sample suggests that BCG luminosity is also correlated with cluster velocity dispersion, which can serve as a proxy for cluster X-ray luminosity, L_X, given the relationship between the two cluster parameters (Solinger & Tucker 1972; Quintana & Melnick 1982; Wu et al. 1999; Mahdavi & Geller 2001).

Figure 18 plots cluster velocity dispersion as a function of both M_m and α for the 259 clusters having 25 or more galaxy redshifts, the minimum number needed for accurate measurement of the velocity dispersion. While there is considerable scatter in the velocity dispersion at any M_m or α, there is a clear correlation such that the median dispersion increases by nearly a factor of two over the range of both parameters. The relation between α and σ_c in particular echoes the relation between the shallowness of the BCG surface brightness profile and cluster X-ray luminosity seen by Schombert (1988) and Brough et al. (2005).

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Both correlations seen in Figure 18 raise the question of whether or not σ_c offers any independent information that can reduce some of the scatter in the L_m–α relationship. Hudson & Ebeling (1997) argued that there is an L_m–α–L_X relationship, which offers better prediction of L_m than using α alone and in doing so reduces the significance of the LP94 bulk-flow amplitude. If we take σ_c as a proxy for L_X, we should thus expect to see a L_m–α–σ_c relationship if the Hudson & Ebeling (1997) is highly significant.

Figure 19 plots the residuals of the L_m–α relationship, given by Equation (8), as a function of cluster velocity dispersion. The residuals of the L_m–α relationship do show a barely significant correlation with cluster velocity dispersion in the sense that brighter residuals in L_m are still associated with clusters with higher velocity dispersion. The measured slope is −0.23 ± 0.10 mag dex⁻¹ in dispersion, such that the mean M_m residuals decrease by ∼0.15 mag over the sample range of cluster velocity dispersion.

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The L_m–α–σ relation (Equation (13)), however, "soaks up" this residual dependence on cluster velocity dispersion. When α and σ are used to predict L_m, the remaining dependence on cluster velocity dispersion decreases to ∼0.07 mag dex⁻¹ with no significance over the sample range in dispersion. Notably, the L_m residuals from this relation are already slightly better than those of the Hudson & Ebeling (1997) L_m–α–L_X relationship. The present L_m–α–σ relation thus offers a BCG-based distance indicator with the effects of the cluster environment removed.

Both the metric luminosity and structure of the BCGs may also be related to the peculiar velocities of the BCGs within their hosting clusters. Figure 20 plots the peculiar velocities of the BCGs normalized by cluster velocity dispersion as a function of BCG L_m and α for the subset of clusters with 25 or more members. The median peculiar velocity steadily decreases with increasing α. The trend of median |ΔV₁/σ| with M_m is less clear; however, the most luminous BCGs have relatively smaller peculiar velocities.

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4.5.2. The Structure of BCGs and Their Positions Within the Clusters

Figure 21 shows that α is also correlated with the offset of the BCG within the cluster relative to the X-ray center. The right panel shows that α is clearly larger for BCGs closer to the center of their clusters, while left panel of the figure shows that M_m is largely unrelated to the position of the BCG within the cluster. The former result appears to be consistent with the weak correlation discovered by Ascaso et al. (2011) between the effective radii of the BCGs and their spatial offsets, with larger BCGs being positioned closer to the centers of the clusters.

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One possibility is that the increase of α may be dominated by high-speed or non-merging interactions with other galaxies in the cluster. The interactions would preferentially take place more often at the center of the cluster and would add energy to the stellar envelope of the BCGs, causing them to become more extended. Without actual mergers, however, little stellar mass is added to the BCGs, thus no overall luminosity growth occurs as the structure of the BCGs becomes more extended. A second hypothesis is that merging does occur as the BCG dwells within the cluster center, but the density of stars within the metric aperture does not increase in the process. Hausman & Ostriker (1978) argued that dry mergers in fact would cause little central growth of the BCGs, a phenomenon that is also seen in merger simulations (Boylan-Kolchin et al. 2006). Recent theoretical and observational work (Hopkins et al. 2009; van Dokkum et al. 2010) indeed argues that growth of massive galaxies since redshifts ∼two is mainly in their outer envelopes.

Since both L_m and α are related to the position of the BCGs with respect to both the spatial and velocity centroids of the clusters, we also examined the combined effect of the last two parameters. Figure 22 revisits the plot of the BCG spatial location within the cluster, r_x, versus the normalized absolute BCG peculiar velocity, |ΔV₁|/σ_c, which was first shown for the full sample of BCGs with X-ray centers and accurate mean velocities in Figure 10. We now split the BCGs with M_m < −22.5 into two luminosity bins,¹³ each of which is split further into two halves by whether or not the galaxies are above or below the mean relation for α, given M_m,

$$\begin{equation} \alpha (M_m)=(-0.256\pm 0.019)(M_m+22.5)+0.484\pm 0.009. \end{equation} \tag{ 14 }$$

These four subsets are shown as individual panels in Figure 22, with the columns separating BCGs with higher (left) or lower (right) than average α, given M_m, and the rows corresponding to the two luminosity bins, with the brightest BCGs plotted in the top row. In any panel, we additionally note (red symbols) BCGs with α residuals in excess of the rms residual, σ_α about the mean relation; σ_α = 0.13 for the full BCG sample.

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Examination of the individual panels suggests that r_x and |ΔV₁|/σ_c indeed both work together to moderate the structure of the BCGs. The upper left panel in Figure 22 contains the most luminous (M_m < −23.0) and extended BCGs. There are no BCGs with |ΔV₁|/σ_c > 0.7, and the BCGs with Δα > σ_α, have |ΔV₁|/σ_c < 0.3, or peculiar velocities less than half of those of the BCGs in this luminosity range with 0 < Δα < σ_α. Moreover, these large Δα galaxies have r_x < 100 kpc, while the BCGs with 0 < Δα < σ_α can have r_x an order of magnitude larger.

The galaxies in the upper right panel of Figure 22 are just as luminous as the BCGs in the panel to their left; however, this subset now has three galaxies with |ΔV₁|/σ_c > 0.7, and a paucity of galaxies with r_x < 10 kpc. The three galaxies with Δα < −σ_α have r_x an order of magnitude larger than those with σ_α < Δα < 0.

BCGs in the −23.0 < M_m < −22.5 luminosity bin largely echo the behavior exhibited by the more luminous BCGs, although the BCGs with Δα > σ_α have |ΔV₁|/σ_c about twice as large. The BCGs in this luminosity range with Δα < σ_α (right-middle panel) avoid the centers of their clusters even more, however, with no galaxies having r_x < 50 kpc. The BCGs with Δα < −σ_α have r_x an order of magnitude yet larger.

The overall picture, again, is that α is larger for the BCGs that lie closer to the spatial and velocity center of their hosting clusters. BCGs with r_x < 10 kpc and |ΔV₁|/σ_c < 0.5 are highly likely to have markedly higher α, with Δα > σ_α as compared to BCGs of the same luminosity. Conversely, galaxies with |ΔV₁|/σ_c > 1 always have large r_x and Δα < 0. It is striking that the BCGs with r_x ∼ 50 kpc have larger α than do BCGs at larger distances from the center—these galaxies are still well displaced from the X-ray center, but yet are deep enough within the potential such that α has already been affected. Conversely, position within the cluster seems to have little effect on L_m, as we already saw in Figures 20 and 21.

While α is dependent on the spatial and velocity locations of the BCGs within the cluster, the metric plane scatter seems to be independent of both. Figure 23 shows the residuals of the metric plane (Equation (13)) as a function of |ΔV₁|/σ_c (upper panel), and for those objects with X-ray data, as a function of r_x (lower panel). There is no evidence for a bias or increased scatter for objects with large offsets. The metric plane thus again implicitly accounts for the environmental effects of the clusters, regardless of whether the BCGs reside in the center of the cluster or in its outskirts.

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We imaged the second-ranked galaxies, M2, in ∼41% of the clusters in the total sample. As we described in Section 2.2.1, we did this mainly when the identity of the BCG during the initial definition of the sample was ambiguous, thus we are less likely to have data on M2 in cases where the BCG is dominant, i.e., considerably brighter than M2. We assess how strongly our observational procedures bias our derived M_m(M2) − M_m(BCG) distribution by randomly selecting 30 clusters in our 15 K sample for which we did not observe the second-rank galaxy. Here, M_m(M2) and M_m(BCG) are the metric luminosities of the M2 galaxy and the BCG, respectively. For these 30 clusters, we used the SDSS r-band images to derive M_m(M2) − M_m(BCG) using the same photometric techniques as described in Section 2.3.2. Under the assumption that this randomly selected sample of 30 clusters is representative of the clusters for which we did not initially observe the second-ranked galaxy, we derive a corrected M_m(M2) − M_m(BCG) distribution by drawing M_m(M2) − M_m(BCG) values from the SDSS sample for the remaining ∼59% of the clusters in our sample. The results are shown in Figure 24. The clusters for which we did observe M2 in our survey have a M_m(M2) − M_m(BCG) distribution that is peaked at lower values than the corrected M_m(M2) − M_m(BCG) distribution. We show, for comparison, the M₂ − M₁ distribution from Smith et al. (2010) and from galaxies drawn from 2000 realizations of clusters with a Schechter (1976) luminosity function. A Kolmogorov–Smirnov test rejects consistency between our corrected M_m(M2) − M_m(BCG) distribution and that from the Smith et al. (2010) study at the 99% confidence level. Our selection procedure clearly introduces a significant bias that cannot be completely compensated for. However, the Smith et al. (2010) clusters are an X-ray selected sample, and thus may favor more luminous BCGs; their corresponding M2s may thus follow a different luminosity-offset distribution from ours as well. While this rules out performing analyses that require a complete distribution of M_m(BCG) versus M_m(M2), we can, however, compare the structure of the BCG and M2 on a per-cluster basis. It also appears that our M2 sample is close to complete for M_m(M2) − M_m(BCG) ⩽ 0.3 mag.

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Figure 25 shows the distribution of the metric absolute magnitude, M_m, of the present M2 sample compared to the Gaussian representation of the BCG luminosity function shown in Figure 13.¹⁴ Interestingly, the largest differences between the M2 and BCG distributions are seen at the bright end, rather than the fainter end, where one might expect the selection effects to be the most severe. There are few highly luminous M2s, despite a deliberate effort to include such galaxies in the sample. For example, while there are 139 BCGs that have M_m < −23, there are only nine M2s that exceed this threshold.

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Figure 26 plots the BCG versus M2 luminosities for each cluster, as well as a histogram of the difference in the metric magnitude between the BCG and M2 in finer bins than shown in Figure 24. In the majority of cases, the M2 systems imaged are within a few tenths of a magnitude of the BCGs. Indeed, the M2 in any given cluster may be more luminous than many of the BCGs in other clusters. We observed M2s in 54 of the 139 clusters with BCGs with M_m ⩽ −23, or 39% of the systems, a fraction essentially identical to that for all clusters. At the same time, only 17/54 or ∼30% of this subset have M2s within 0.3 mag of the BCG M_m, a level at which we consider the M2 to be a close "rival" of the BCG, while 60/98 or ∼60% of the BCGs with M2s observed and −23 < M_m ⩽ −22.5 have an M2 that is a close rival. The difference between the two subsets is readily evident in Figure 26. This result is consistent with the classic result from Sandage & Hardy (1973) that the most luminous BCGs are associated with relatively faint M2s. The large luminosity differences between the BCG and M2 for the brightest BCGs are also consistent with the arguments of Tremaine & Richstone (1977), Loh & Strauss (2006), and Lin et al. (2010) that these galaxies are "special" and are not drawn from a standard luminosity function.

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Figure 27 shows that the projected offsets of the BCGs and their corresponding M2s from the X-ray center of their clusters follow different distributions. There are essentially no M2s closer to the X-ray center than the BCG once r_x(BCG) < 100 kpc. Even when r_x(BCG) > 100 kpc, there are only a handful of M2s with r_x(M2) < 50 kpc; of these there are only three M2s with luminosities within 0.3 mag of their BCGs. In short, even though we have searched for and selected BCGs at large distances from the X-ray centers of their clusters, we are not overlooking a large population of bright "central" galaxies that might plausibly be better choices for the dominant galaxy in the cluster.

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There are other ways in which the population of M2 galaxies differs from that of the BCGs besides just being less luminous. Figure 28 compares the L_m–α properties of the two sets. In this case, we plot M_m as the independent variable, instead of assigning this role to α as we did in Section 4.2. Intriguingly, this figure echoes the conclusion suggested by Figures 25 and 26 that the differences between M2 and the BCGs are most important for the brightest BCGs. For M_m > −22.75, the M2 and BCG galaxies have essentially indistinguishable distributions of α as a function of M_m. It is also worth noting that given the large scatter in the L_m–α relation, many of the M2s may actually have larger α values than do their corresponding BCGs. For M_m < −23, however, while the mean α continues to increase for BCGs, it does not for the M2s. The few M2s with M_m < −23 have significantly lower α on average than the corresponding set of BCGs. Considering the results from both the previous and following sections, this implies that the bright M2s are galaxies that are not likely to be close to the X-ray center of their clusters. Bright M2s with more "normal" α would be more centrally located, and thus vulnerable to merging with the BCG. The median r_x for M2 with M_m ⩽ −22.5 is 327 kpc, indeed showing that the brightest M2s are most likely to be found markedly displaced from the center of the cluster.

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Additional insight into the differences between M2 and the BCG comes from considering their relative velocity and spatial displacements within their clusters. Figure 29 plots luminosity difference between M2 and the BCG, M_m(M2) − M_m(BCG) as a function of either the peculiar velocity of the BCG with the cluster (left panel) or that of M2 (right panel) for clusters with 25 or more members (so as to minimize the contribution of the error in the mean redshift to the peculiar velocities). It is striking that M_m(M2) − M_m(BCG) < 0.5 mag for nearly all clusters in which the BCG ΔV₁ > 400 km s⁻¹. This suggests a picture in which the BCGs with high peculiar velocities are relatively recent additions to the cluster that have not had enough time to undergo the relaxation interactions that would move them closer to the cluster velocity centroid. In this case, the M2 is likely to be the "former BCG," reflecting its rank prior to the infall of a more luminous galaxy. However, a fraction of these M2s will ultimately merge with the "new" BCG once the latter's peculiar velocity has been reduced enough to make merging interactions possible. The "new" M2 will be less luminous than the old M2 that merged with the BCG, thus increasing the BCG–M2 luminosity difference for that cluster.

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The effect of the BCG on M2 is also visible when M_m(M2) − M_m(BCG) is plotted as function of the BCG X-ray offset (Figure 30). The main signature is that there are no M2s with M_m(M2) − M_m(BCG) < 0.2 mag once the BCG gets within 20 kpc of the X-ray center. Conversely, there appears to be no relationship between M_m(M2) − M_m(BCG) and r_x for BCGs falling outside this radius. The figure also ratifies the conclusion discussed in the context of Figure 27 that there are essentially no M2s closer to the center than the BCG once r_x(BCG) < 100 kpc.

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The structure of M2 also appears to be affected by its proximity to the mean velocity of its hosting cluster, as is the case for the BCGs. Figure 31 compares α between the BCG and M2 in clusters (with 25 or more galaxy velocities) in which M2 is a close rival of the BCG (M_m(M2) − M_m(BCG) < 0.3 mag). On average, the M2s should have smaller α, given their smaller L_m; however, the figure shows that this is clearly over-ridden by the effect of the proximity of the galaxies to the velocity centroid of the cluster. The M2s have slightly larger α when |ΔV₂| < |ΔV₁|, that is when M2 has the smaller peculiar velocity, while the BCGs nearly always have large α when the situation is reversed. The α differences are not symmetrical with the difference in absolute peculiar velocities, given the initial bias for the BCGs to have higher α, but the effect is clear. The processes that increase α as the BCG approaches the center of the cluster are also in play for M2.

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Likewise, when M_m(M2) − M_m(BCG) < 0.3 mag, the galaxy of the two that is closest to the X-ray center of the cluster is more likely to have the higher α. Figure 32 plots the difference in α between the BCG and M2 as function of the difference between each galaxy's offset from the X-ray center. For the 24 clusters with M_m(M2) − M_m(BCG) < 0.3 (and X-ray positions available) and the BCG closer to the X-ray center than M2 (right half of the figure), α(M2) − α(BCG) = −0.14 ± 0.4. When M2 is closer to the center (left half of the figure), the situation is reversed, with α(M2) − α(BCG) = +0.15 ± 0.5 for the 17 clusters in this subset.

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The finding that rival M2s have larger α than their corresponding BCGs despite their smaller metric luminosities when closer to the X-ray or velocity centers can the BCGs implies that they will strongly deviate from the L_m–α relation. This is shown in Figure 33, which plots the BCG versus M2 luminosity residuals from the L_m–α relation (Equation (8)) for clusters in which the BCG has M_m(M2) − M_m(BCG) < 0.3 mag. The set of clusters is divided into four subsets depending on which of the BCG or M2 is closer to the velocity or X-ray center. When the M2 is further from either center, both the M2 and BCG luminosity residuals average to zero, showing that the M2s act as essentially lower-luminosity BCGs. Interestingly, the residuals of both galaxies appear to be correlated for this subset. When M2 is closer to either the velocity of X-ray center, however, M_m(M2) is significantly dimmer than predicted from their large α values. M2s closer to the cluster center than their matching BCGs cannot be simply considered to be less luminous examples of a BCG.

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The initial survey of BCG photometric parameters shows essentially identical results to those presented in PL95. The distribution of M_m is Gaussian with a dispersion of 0.337 mag, and the L_m–α relation has scatter of 0.267 mag in the CMB frame. We now also offer a form that predicts M_m from a linear relationship in log α, rather than the quadratic form of PL95, due to its monotonic behavior with α; it also provides an acceptable fit to the data. It is notable that the random scatter in M_m has not been reduced from that presented in PL95, despite new photometry obtained for most of the 15 K BCGs, and improved cluster velocities.

The present work goes beyond PL95 by including central stellar velocity-dispersion observations of the BCGs. Use of metric luminosities obtained in different aperture sizes allows us to track the flattening of the Faber & Jackson (1976) relationship between L and σ as the aperture grows to include a larger fraction of the total galaxy luminosity. The relationship between L_m and σ using our standard 14.3 kpc metric radius resembles the classic σ∝L^1/4 relation, while increasing r_m 4 times larger to 57.1 kpc yields a much shallower σ∝L^1/6 relation. The flatter Faber–Jackson relation for BCGs has been discussed extensively in the literature, but this shows that it is likely a consequence of the extremely large envelopes of the BCGs.

The residuals of the L_m–α relationship correlate strongly with σ, thus motivating the development of a three-parameter "metric plane" relationship between L_m, α, and σ analogous to the "fundamental plane" relations. Use of α and σ together predicts M_m to a precision of 0.206 mag, a substantial improvement over the L_m–α relationship. The metric plane relation also implicitly removes correlations between the residuals of the L_m–α relation and the cluster velocity dispersion.

A first step in understanding the relationship of BCGs to their galaxy clusters is to ask where they are located in the clusters. In answering this question, we have produced quantitative distribution functions for both the projected spatial and peculiar velocities of the BCGs with the cluster, where the spatial offset is defined with respect to the cluster X-ray center, and the velocity with respect to the mean cluster velocity. The spatial offset, r_x follows a steep power law with γ = −2.33 over three decades in radius; there is no evidence for any core at scales >10 kpc.

The absolute normalized peculiar velocities |ΔV₁|/σ_c of BCGs within their clusters follows an exponential distribution, with scale length 0.39. The spatial and velocity offsets are correlated. Large |ΔV₁|/σ_c always corresponds to large r_x and small r_x always corresponds to small |ΔV₁|/σ_c.

These results raise an important caveat in understanding the relation between the BCGs and their hosting clusters. While BCGs do prefer to reside near the central regions of galaxy clusters, BCGs with r_x > 100 kpc or |ΔV₁|/σ_c > 0.5 are common. These outlying BCGs further follow the same metric plane as do those closer to the center of their parent clusters. This has important consequences for understanding the relationship between "intercluster light" (ICL) and the extended envelopes of BCGs. BCGs are often simplistically assumed to always reside at both the spatial and velocity center of ICL, such that it becomes ambiguous as to where the BCG ends and the ICL picks up (see the discussion in Lauer et al. 2007). While this may be true in some cases, it will not be true in general.

Our BCGs are defined by metric luminosity, while some authors choose the brightest galaxy close to the X-ray center, even if it turns out to be the M2. The latter definition would risk the possibility of missing BCG that are still being accreted by rich galaxy clusters at the present epoch. There is no question that there are first-ranked galaxies in many clusters that are offset by large velocities and projected separations for the X-ray defined centers. These are most easily understood as recently accreted additions to the cluster. The identity of the galaxy that occupies the first-ranked position will change as new galaxies are brought in, and may have changed a number of times in a given cluster over cosmological history, as is found in the simulations of De Lucia & Blaizot (2007) and Martel et al. (2014).

The relationships that we have observed between the properties of the BCGs and their clusters support the picture that the bulk of any given BCG is largely assembled outside of the cluster before the galaxy is accreted by the cluster. The strongest effect is that α is clearly moderated by both r_x and |ΔV₁|/σ_c, such that α increases monotonically as the BCG lies closer and closer to the center of the cluster. BCGs with the largest α for their L_m always reside close to the X-ray and velocity center of the cluster. Conversely, BCGs with the smallest α given L_m are strongly displaced from the center, often with r_x ≫ 100 kpc and |ΔV₁|/σ_c > 0.5. However, L_m is only weakly, if at all, related to the position of the BCG within the cluster.

We conclude that the envelopes of the BCGs are expanded by and perhaps even grown by interactions that become increasingly important as the BCG spends the majority of the time in ever denser regions of the clusters and dynamical relaxation reduces its peculiar velocity. The fact that L_m does not vary with spatial or velocity offset from the center of the cluster argues that the denser central body of the BCG, however, is less affected the same processes. Some BCGs, even at low redshifts, have been recently accreted into the outskirts of the clusters. Even though they may not have the classic extended envelopes associated with, say, massive cD galaxies, their L_m is already high enough that they can claim first rank over all other galaxies within the cluster.

It is here that the properties of the M2 galaxies are important. When M2 is close in luminosity to the BCG, we see that which of the BCG or M2 has higher α depends on which is closer to the mean cluster velocity or the center of the cluster potential as marked by X-ray emission. Clearly, the processes that act on the BCG as a function of its location in the cluster also act on M2. In fact, the luminosity differences between the two sets of galaxies reveal the nature of the competition for first rank. While our sample of M2 is incomplete, we only see large luminosity differences between them when |ΔV₁| is relatively small. We hypothesize that when the BCG initially enters the cluster with a high |ΔV₁|, the previous BCG, now demoted to M2, can still survive as a close rival until the BCG undergoes enough interactions to be captured into the central potential of the cluster. Conversely, when it is the M2 that has a high peculiar velocity or has an orbit that keeps it mainly in the outskirts of the cluster, it can persist as close rival in luminosity to the BCG.

Throughout the narrative our results raise the question of the extent to which BCGs grow their luminosity inside versus outside of the their present hosting clusters. We see that there are some BCGs with large L_m, meaning that the assembly of their stellar mass is essentially complete, but that also have large r_x and large |ΔV₁|/σ_c, which means that they are relatively recent arrivals to their hosting clusters. In the clusters for which this is so, there may also be a centrally located M2 that many would pick as the "true" BCG; certainly, that galaxy is more likely to have the greatly extended envelope that is often associated with the classical pictures of BCGs. The inference is that most of the stellar mass of any given BCG, or at least its portion within the metric radius, grew from the merger of progenitor galaxies outside the rich-cluster environment. Such a scenario was suggested by Merritt (1985), who noted that the lower velocity dispersions of galaxy groups, rather than rich clusters, made them attractive as the birthplace of future BCGs. The BCG may represent the merging terminus of many of the galaxies within the group. The merged system may later be accreted by a rich cluster, but then may be subjected to only minor interactions or mergers once it arrives there. These would add energy to its envelope but little stellar mass.

On the other hand, in this study, we see that BCG L_m increases markedly with σ_c, and as we noted earlier in the Introduction, BCG luminosity also tracks cluster X-ray luminosity and temperature. One might presume that this only reflects the likelihood that today's rich clusters once had the richest retinue of surrounding groups, but there is evidence that major mergers can take place within clusters (Lauer 1988). The most luminous BCGs are unlikely to have strong M2 rivals for the position of first rank. Faint M2 galaxies are further uniquely associated with BCGs with modest |ΔV₁|/σ_c. Whatever level of minor merging that may have taken place, this result suggests that many of the current BCGs have cannibalized their closest rivals, an event that would have been a major merger. It is also a feature of dry merging that this may cause little luminosity growth within the metric aperture (Hausman & Ostriker 1978; Boylan-Kolchin et al. 2006), while greatly extending the envelope. The growth in α as the BCGs become more centrally located argue that this is happening.

The question of whether most BCG luminosity growth occurs inside or outside of rich clusters intersects with the issue of whether or not BCGs are "special." Lin et al. (2010) have argued that only the BCGs in the most X-ray luminous clusters exhibit luminosities high enough such that they cannot be explained as simply being drawn from a standard cluster luminosity function. This is thus a partial contradiction of the analysis of Tremaine & Richstone (1977) that suggested that all BCGs were created by special processes. In this work we see that the M2s are most structurally similar to their corresponding BCGs when the BCGs have more modest metric luminosities. It is when M_m starts to grow beyond ∼ − 22.5 that we begin to see their properties diverge. We conclude that if BCGs were born in relatively small groups, it is their accretion into rich clusters that later shapes and grows them to their final form and special luminosities. We will explore these themes further in follow-up papers that will elucidate the relation of the present sample of BCGs to the properties of giant elliptical galaxies and measure the ongoing rate of merger interactions in BCGs.