A NEW TEST OF THE STATISTICAL NATURE OF THE BRIGHTEST CLUSTER GALAXIES

To check for the statistical nature of BCGs, we proceed as follows. Based on the observed LD, we generate many realizations of the L_bcg–L_tot correlation, from which we derive the mean L_bcg–L_tot relation that is expected if the BCGs are drawn from the same LD as other cluster galaxies. We can then compare the observed mean L_bcg–L_tot relation and determine if it is consistent with the mean from the mock data.

We show in the middle panel of Figure 4 one such realization. Basically, for each cluster in our sample, we create a corresponding mock cluster by randomly drawing galaxies from the galaxy pool, until the total luminosity of the mock cluster matches that of the observed one (see also Figure 1). Specifically, suppose the first N galaxies give a total luminosity of L_N < L_tot, and the next mock galaxy brings the total luminosity to L_{N + 1} > L_tot. We keep the (N + 1)th galaxy if L_{N + 1} − L_tot < |L_N − L_tot|, otherwise we only use the first N galaxies. Because of this procedure, although the total luminosity for massive (luminous) clusters can be matched fairly well, for low luminosity clusters the difference in L_tot between the mock and real clusters can be large.

The black squares in the middle panels of Figure 4 show the mean in BCG luminosity, $\overline{L_{{\rm bcg,sim}}}$ in this particular ensemble of simulated clusters. The cyan crosses in both the top and middle panels are the mean of $\overline{L_{{\rm bcg,sim}}}$ from 200 Monte Carlo realizations of the mock cluster ensemble, which we denote as $\left< \overline{L_{{\rm bcg,sim}}}\right>$. It is found that for luminous clusters (e.g., L_tot > 3.7 × 10¹¹h⁻²₇₀ L_☉), the observed $\overline{L_{{\rm bcg,obs}}}$ is higher than $\left< \overline{L_{{\rm bcg,sim}}}\right>$ from mock clusters (e.g., compare the magenta squares with the cyan crosses in the top panel). For low-luminosity clusters, the observed and mock values are comparable.

We list in Table 1 the definition of various terms employed in this and the following subsections.

Table 1. Definition of Terms

Notation	Meaning
$\overline{L_{{\rm bcg,obs}}}$	mean BCG luminosity of the real clusters (in 15 Ltot bins)
$\overline{L_{{\rm bcg,sim}}}$	mean BCG luminosity of one Monte Carlo ensemble (in 15 Ltot bins)
$\left< \overline{L_{{\rm bcg,sim}}}\right>$	average of $\overline{L_{{\rm bcg,sim}}}$ over 200 Monte Carlo ensembles; this is the statistical expectation
dobs	$\equiv \log \overline{L_{{\rm bcg,obs}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$
dsim	$\equiv \log \overline{L_{{\rm bcg,sim}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$

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By comparing the observed and mock L_bcg–L_tot correlations we infer that BCGs in real clusters have systematically higher luminosities than the mock BCGs. One might easily imagine a systematic measurement error that systematically gives too large a luminosity for bright galaxies. But that would not produce the effect we find. One would need a systematic error that increased the brightness of BCGs but had a less dramatic effect on second brightest galaxies in other (more luminous) clusters of the same intrinsic luminosity. Here we quantify the significance of the difference between the two populations.

Using 200 realizations of the mock cluster ensemble (each containing 494 clusters), we calculate $\left< \overline{L_{{\rm bcg,sim}}}\right>$, as well as the difference between logarithms of the mean BCG luminosity $\overline{L_{{\rm bcg,sim}}}$ from individual realizations and $\left< \overline{L_{{\rm bcg,sim}}}\right>$, $d_{{\rm sim}}= \log \overline{L_{{\rm bcg,sim}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$. By dividing the cluster sample into 15 bins in L_tot, each containing roughly equal number of clusters, for each mock cluster ensemble we have 15 evaluations of the statistic d_sim. One such example is shown in the lower panel of Figure 4 (open red symbols). We see that d_sim roughly scatters about zero. We expect that for mock clusters, the distribution of d_sim should follow a Gaussian, which is shown as the blue histogram in the top panel of Figure 5. The green curve is a Gaussian fit to the histogram.

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We can similarly calculate $d_{{\rm obs}} = \log \overline{L_{{\rm bcg,obs}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$ for the real clusters. We show d_obs as a function of L_tot in the lower panel of Figure 4 as the solid blue points; it is apparent that d_obs correlates positively with cluster luminosity. The errorbars of d_obs are derived from 200 bootstrap resampling of the observed L_bcg–L_tot correlation. The distribution of d_obs is further shown as the red dashed histogram in the top panel of Figure 5 and is clearly different from a Gaussian. A Kolmogorov–Smirnov (K-S) test shows that the probability of d_obs to be drawn from the same distribution of d_sim is $\mathcal {P}=0.79\%$.

We examine in more detail the distribution of d in high and low luminosity clusters in the other two panels of Figure 5. The distribution of d for the high (low)-luminosity subsample is shown in the middle (bottom) panel. As we note above, in high luminosity clusters the real BCGs are on average more luminous than the mock ones, resulting positive d_obs, and there is only a 0.028% probability that they are drawn from the same LD, based on a K-S test. On the other hand, $\overline{L_{{\rm bcg,obs}}}$ in low luminosity clusters scatter about $\left< \overline{L_{{\rm bcg,sim}}}\right>$, and d_obs and d_sim have a much higher probability ($\mathcal {P}=54.5\%$) to be drawn from the same distribution. These results are recorded in Table 2 (the column under "galaxy pool;" for the meaning of the other columns, see Section 4.2).

Next we compare our results with those from the TR tests. Based on the whole cluster sample, we find that σ₁ = 0.58, $\overline{\Delta }=0.62$, and σ_Δ = 0.49, resulting in $T_1=\sigma _1/\overline{\Delta } = 0.93$ and $T_2=\sigma _\Delta /\overline{\Delta }=0.78$. Recall that if the BCGs are "statistical," we would have obtained T₁ ⩾ T_1,lim = 1 and T₂ ≳ T_2,lim = 0.82 (Section 1; but see below). Note that these lower limits are derived under rather general assumptions about the form of the LD (TR). For example, assuming the underlying LD follows the Schechter (1976) form, the limits are T_1,lim,sch ≈ 1.16 and T_2,lim,sch ≈ 0.88.

Dividing the clusters into high and low luminosity subsamples (using the same division luminosity $L_{{}\rm div}$ as in Section 3.2), we find $(\sigma _1, \overline{\Delta }, \sigma _{\Delta }, T_1, T_2)=(0.39,0.55,0.43,0.70,0.79)$ and (0.54, 0.65, 0.50, 0.84, 0.77), respectively. That T₁ for the whole sample is larger than T₁ for both high and low luminosity subsamples is due to the weak, positive correlation between L_bcg and L_tot.

Both our test and the TR tests indicate that, using the whole cluster sample, the BCGs are not drawn from the same LD as other cluster galaxies. Separating the low luminosity clusters from the luminous ones, our test suggests that this conclusion is mainly driven by the BCGs in the luminous clusters. The T₁ statistic implies a larger deviation from the statistical expectation for BCGs in high luminosity clusters (T_1,lim − T₁ = 0.30), compared to their counterparts in less luminous clusters (T_1,lim − T₁ = 0.16).

It is important to quantify how significant are the deviations of the T_i (i = 1, 2) values we observe from the statistical limits. Without assuming any particular form of the LD, we evaluate the significance empirically, using the "galaxy pool" from all the member galaxies. More specifically, we construct ensembles of mock clusters following the method described in Section 3.1; we can then derive the mean and dispersion of the two statistics from the distributions of T₁ and T₂ based on the mock clusters (see Figure 6). For all clusters in our sample, when the BCGs are generated statistically, the mean and standard deviation of the two statistics are $\overline{T_1}=1.01$, $\sigma _{T_1} = 0.05$, and $\overline{T_2}=0.90$, $\sigma _{T_2} = 0.03$. We find that about 3.5% of the Monte Carlo realizations have T₁ as low as the observed value (0.93), but only 0.007% have T₂ as low as 0.78, the observed value. We denote these fractions as p₁ ≡ p₁(⩽T_1,obs) and p₂ ≡ p₂(⩽T_2,obs). For high luminosity clusters, ($\overline{T_1}, \sigma _{T_1}, p_1, \overline{T_2}, \sigma _{T_2}, p_2)=(0.96,0.08,0.014\%,0.84,0.06,14\%)$. For low luminosity ones, the values are (0.90, 0.05, 8.8%, 0.88, 0.04, 0.07%).

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Additionally, we can estimate the distribution of the observed T_i via bootstrap resampling. The overlapping area under the "observed" and the "statistical" distributions (the latter from the mock cluster ensembles, as mentioned above) then gives an estimate of the likelihood that the observed T_i is consistent with the statistical expectation. We denote the likelihood as q_i ≡ q_i(T_i,obs∩T_i,stat), i = 1, 2. For the whole cluster sample, we find (q₁, q₂) = (8.8%, 1.4%), For high and low luminosity clusters, the results are (0.35%, 23.4%) and (14.9%, 0.64%), respectively.

For clarity, these results are summarized in Table 3. We present the results for the whole sample, as well as the high and low luminosity subsamples. For each (sub)sample, the numbers under "obs." are the observed values, and the triplet of numbers under "stat." denotes the mean and standard deviation of the TR statistics based on 10⁵ ensembles of mock data sets, and p_i (i = 1, 2). Looking at p₁ for the high and low luminosity clusters, one would conclude that BCGs in the faint clusters are much more likely to be statistical. One would draw the opposite conclusion if considering the T₂ statistic, however. It is therefore not clear if the TR tests give a consistent picture (e.g., dependence on L_tot) for the degree of deviation of BCGs from the galaxy population in clusters.

Using a large group sample from SDSS, Yang et al. (2008) examined the gap statistics. Although they did not explicitly calculate the values of T_i, using the results presented in their Figure 7, we can roughly estimate that for clusters with virial mass log M ≈ 14.8, (T₁, T₂) ∼ (1.4, 0.7), and for clusters with log M ≈ 14.0, (T₁, T₂) ∼ (0.6, 0.8). We see that the T_i values have opposite trends with cluster mass: based on T₁, the BCGs in low mass systems are special; using T₂ one would be lead to the opposite conclusion.

In their study of LRG-selected groups and clusters, Loh & Strauss (2006) found that $(\sigma _1, \overline{\Delta }, \sigma _{\Delta }, T_1, T_2)=(0.30,0.87,0.52,0.35,0.59)$ at z ≈ 0.12. They also examined the richness dependence of the TR statistics. For the richest systems (75%–100% quartile in the richness distribution) they found (T₁, T₂) = (0.75, 0.71); for those with richness in the 25%–50% quartile (with about 2–3 galaxies on the red sequence), (T₁, T₂) = (0.27, 0.34). The results of Loh & Strauss (2006) suggest a dependence of the TR statistics on richness that is opposite to our findings (if their richness estimator correlates well with the total luminosity).

The main cause for the differences seems to be in the magnitude gap, Δ. Although for the richest systems, Loh & Strauss (2006) obtained $\overline{\Delta }\sim 0.55$, a value comparable to ours, their finding of $\overline{\Delta }\sim 1$ for the much poorer systems (25%–50% quartile in richness) is much higher than our value. Because Loh & Strauss (2006) looked for galaxy concentrations around LRGs using (primarily) photometric data, we suspect many of their low-richness systems may not be real, but simply chance projections of galaxies with similarly red colors (see purity of the maxBCG algorithm; see Koester et al. 2007). On the other hand, we note that Yang et al. (2008) also found that $\overline{\Delta }\gtrsim 1$ for their low mass systems (log M ∼ 13).

These results seem to suggest that, when considered together, the two TR statistics may not unambiguously determine the statistical nature of BCGs with respect to cluster mass (or its proxy, such as total galaxy number or luminosity).

The disagreements among the different studies may be attributed to several factors (such as the way BCGs are selected, the way background contamination is treated, or even the group/cluster selection). A possible resolution would be to compare the selection of BCG and G2 on a cluster-by-cluster basis, for the clusters that are present in multiple catalogs.

We have applied a new method to confirm that the BCGs are special. This approach can be used to check if G2s are statistical, that is, are their LD consistent with that of the overall cluster galaxy population. The results can then be used to answer the second question we set out to address: are BCGs different from other luminous, red, cluster galaxies?

We proceed in a similar fashion as in the previous sections. The main difference is that we are now comparing the observed L_g2–L_tot correlation with the corresponding mean relation from mock clusters. Requiring that mock clusters must have at least two members causes a small modification to the way mock clusters are constructed, which mainly affects the low luminosity clusters. Therefore, we will set a lower luminosity limit at L_lim = 1.6 × 10¹¹h⁻²₇₀ L_☉ and define the low luminosity subsample (234 systems) with L_lim ⩽ L_tot ⩽ L_div, where L_div = 3.7 × 10¹¹h⁻²₇₀ L_☉. The 124 cluster high-luminosity subsample remains the same as before.

In Figure 7, we show the distribution of the quantity d for G2s, which is analogous to Figure 5 for the BCGs. The bottom, middle, and top panels are the results for the low-luminosity subsample, high-luminosity subsample, and the combined sample (358 systems), respectively. In all samples considered, we find that G2s are consistent with the statistical expectation. K-S tests suggest that $\mathcal {P}=26\%$, 55%, and 24% for all, high-, and low-luminosity clusters, respectively. Our results are not sensitive to the exact value of L_lim.

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This result immediately suggests that BCGs are distinct from G2s, and very likely, other luminous member galaxies.

This simple exercise also demonstrates one advantage of our method over the TR tests: while the latter by definition could not determine the statistical nature of G2s, our method can in principle be applied to even the third- or lower-ranked member galaxies.

We first check the sensitivity of our results on the galaxy selection criteria, by repeating our test using a much more stringent set of conditions to assign cluster memberships for red galaxies. In addition to the basic requirements (u − r ⩾ 2.2, r_p ⩽ 17.7, projected within 0.8h⁻¹₇₀ Mpc), for the spectroscopic members, we only include galaxies whose (1) redshifts satisfy c|z − z_cl| ⩽ v, where v ≡ min(2σ, 1500 km s^-1); (2) g − r and r − i colors fall in the range $\bar{c}-1.5\sigma _{c}$ to $\bar{c}+2.5\sigma _{c}$, where $\bar{c}$ and σ_c are the mean and dispersion of the g − r and r − i colors from the galaxies at redshifts broadly consistent with the cluster; and (3) "concentration parameter" c_m ≡ r₉₀/r₅₀ ⩾ 2.6, where r₉₀ and r₅₀ are radii that enclose 90% and 50% of the Petrosian flux, respectively; this last condition aims to select galaxies with early-type morphology (Strateva et al. 2001). As for the photometric members, along with the filters in the r-band flux, u − r color, spatial distribution, and morphology, we require that their g − r and r − i colors to lie between $\bar{c}-\sigma _{c}$ and $\bar{c}+1.5\sigma _{c}$, where $\bar{c}$ and σ_c are now derived from the spectroscopic members.

These criteria reduce our cluster and galaxy sample; only 344 clusters containing 2819 galaxies are included. We find that the probability $\mathcal {P}$ that d_obs and d_sim are drawn from the same distribution is 1.5%, 0.15%, and 67% for the whole, high-luminosity, and low-luminosity (sub)samples. These values confirm the results found in Section 3.2.

We note in Section 2 that 20% (≈97/494) of the BCGs are photometrically identified. Some of these may be foreground/background galaxies. To evaluate the effect of possible contamination due to these photometric BCGs, we repeat our analysis using the 397 clusters whose BCGs are spectroscopically confirmed members (with the membership assignment criteria of Section 2). Based on the galaxy pool constructed from the 3661 galaxies in these clusters, we find that $\mathcal {P}=1.7\%$, 0.28%, and 27%, for the whole, high-luminosity, and low-luminosity (sub)samples.

Although the absolute values of $\mathcal {P}$ changes somewhat with respect to our nominal values recorded in Table 2, the trend remains clear that the BCGs in high luminosity clusters are much less likely to be drawn from the same LD as other cluster members when compared to their counterparts in lower luminosity clusters.

Let us comment next on one effect our cluster selection may have on the results. The requirement that the clusters need to host at least two galaxies with M_r ⩽ −20 potentially excludes systems dominated by a single ∼M_* galaxy, such as the "fossil groups" (which are defined to have Δ ⩾ 2 and extended X-ray emissions). These systems are believed to evolve in isolation, with the last major merger with other galactic systems being long enough in the past that a dominant central galaxy can result from the dynamic friction (Ponman et al. 1994). The BCGs in these groups would then deviate significantly from the statistical extreme, which makes them distinct from the other BCGs in low luminosity groups included in our sample. One might be concerned that the exclusion of fossil groups from our cluster sample may have contributed to our conclusion that, for the lower luminosity systems, there is no statistically significant deviation of BCGs from the statistical distribution of all cluster galaxies. However, given the rarity of the fossils (number density ∼2 × 10⁻⁶h³₇₀ Mpc⁻³; e.g., La Barbera et al. 2009; Voevodkin et al. 2010), we expect there to be at most ∼20 such systems in the volume we sample (z = 0.03–0.077, ≈5700 deg²), and therefore our conclusion should be robust.

We first investigate whether the way galaxy pool is constructed affects our results. Instead of pouring galaxies from all clusters into one pool, we can create two pools separately for galaxies in high and low luminosity clusters. High and low luminosity mock clusters are then created from the respective pools. We find that $\mathcal {P}=0.98\%$ and 92% for the high and low luminosity clusters, respectively. The trend is clear that BCGs in low luminosity clusters are far more likely to be the statistical extreme than their cousins in high luminosity systems.

Second, we note that instead of utilizing the galaxy pool, we can construct mock clusters using fits to the observed LD of cluster galaxies. The LDs for the whole cluster sample, and for the high- and low-luminosity subsamples, are measured by the method developed in Lin & Mohr (2004), to which we refer the reader for more details. The results are shown in Figure 8. From top to bottom are the LDs for the high luminosity clusters, the whole sample, and the low luminosity clusters.

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When BCGs are included, the Schechter function is not a good description of the LDs. Instead, the bright end may be more appropriately described by a log-normal distribution. Therefore, in fitting the LDs we consider both a bright end-truncated Schechter function plus a Gaussian,

$$\begin{equation} \phi (M) dM = \left\lbrace \begin{array}{@{}ll} \frac{\ln 10}{2.5} \phi _* \mathcal {M}^{\alpha +1} \exp \left(- \mathcal {M} \right) dM & \mbox{if $M\ge M_{{\rm ch}}$}\\ \tilde{\phi }_{*} \exp \left(-\frac{(M-\tilde{M}_{*})^2}{2\sigma _{M}^{2}} \right) dM & \mbox{otherwise,} \end{array} \right. \end{equation} \tag{ 1 }$$

where $\mathcal {M}=10^{-0.4 (M-M_*)}$ and a single Schechter function. The best-fit parameters are given in Table 4.

Table 4. LD Parameters in r-band

Sample	Single Schechter			Gaussian+Schechtera
	α	ϕ*	M*	α	ϕ*	M*	Mch	$\tilde{\phi }_{*}$	$\tilde{M}_{*}$	σ2M
All	−1.01	8.53	−21.92	−0.93	10.25	−21.70	−22.40	1.38	−22.25	0.32
High	−1.17	12.62	−22.13	−0.99	18.30	−21.73	−22.60	1.90	−22.57	0.25
Low	−0.93	6.34	−21.78	−0.73	8.92	−21.28	−22.05	1.35	−21.98	0.33

Note. ^aSee Equation (1) for the definition of the parameters.

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Using the Gaussian+Schechter function fit to the whole sample as the LD, we find that for the whole, high-, and low-luminosity samples, $\mathcal {P}=0.47\%$, 0.013%, and 39%, respectively. If using the Schechter function fit to the LD of the whole sample, $\mathcal {P}=0.57\%$, 0.015%, and 44% for the three (sub)samples, respectively. These results are recorded in the third and fourth columns of Table 2.

We thus conclude that our results are robust irrespective of the way the mock clusters are constructed. Another implication from this exercise is that our method does not rely on largely complete spectroscopic data set like SDSS; as long as (1) one can determine the LD accurately (e.g., via a statistical background subtraction method; Lin et al. 2004), and (2) the cluster membership of BCGs can be reliably assigned, our test can be applied.

We remark in Section 1 concerning the difficulty in the measurement of the total light of the BCGs. Here we examine the effect of underestimation of BCG luminosity (and more generally, luminosity of galaxies with large angular extents) on both our test and the TR tests. We address this issue by using two methods to obtain the best estimates of the true galaxy magnitudes for our galaxy sample. The first one is a re-measurement of the galaxy photometry from the reduced SDSS images for every galaxy in our sample. The second one employs a statistical correction to the official SDSS photometry.

Simply put, the reason that the SDSS photometric pipeline has difficulties dealing with the photometry of large galaxies is because it is defaulted to use the "local" sky flux measured within regions of 256 × 256 pixels (17 × 17) in the CCD frames. In the case that a large fraction of a given region is occupied by galaxies, the sky background level will be biased high, and in turn the galaxy magnitude will be lower than the true value. A remedy is to use the "global" sky level measured in frames of 2048 × 1498 pixels (135 × 98). Following this line of reasoning, von der Linden et al. (2007) devised a way to recover the true magnitude, given the crowdedness of a field, and the relative values of the local and global sky background.

As our first approach, we adopt the methodology of von der Linden et al. (2007) to apply corrections to the magnitudes of the galaxies and change the cluster luminosity accordingly, and then use the modified galaxy catalogs (and the galaxy pool) to conduct our test and the TR tests. It is found that $\mathcal {P}=1.2\%$, 0.012%, and 49%, for the whole, high luminosity, and low luminosity cluster (sub)samples, respectively. As for the TR tests, for the whole sample, (T₁, p₁, T₂, p₂) = (0.99, 11.1%, 0.77, 2 × 10⁻⁴); for the high and low luminosity subsamples, (T₁, p₁, T₂, p₂) = (0.68, 5 × 10⁻⁵, 0.77, 2.4%) and (0.84, 7.1%, 0.77, 3 × 10⁻⁴), respectively.

Using an independent photometric reduction software, Hyde & Bernardi (2009) quantified the degree of underestimation of the magnitudes of large galaxies by the official SDSS pipeline. Their tests suggest that, for early-type galaxies with effective radius ≈10'', the mean magnitude deficit is about 0.25 mag, with a 68% scatter about 0.04 mag. They provided a fitting function that gives the mean deficit $\overline{\Delta m}$ as a function of the angular size θ, which we use as the basis of our second method for recovering the true galaxy magnitudes. For every galaxy in our sample, we assume its magnitude was underestimated by the SDSS pipeline by $\overline{\Delta m}(\theta)+\delta m$, where δm is a Gaussian random variate with dispersion of 0.04 mag, and θ is determined from the de Vaucouleur fits to the galaxy surface brightness profile (see Hyde & Bernardi 2009 for details).

For the galaxy and cluster catalogs thus modified, we find that (T₁, T₂) = (0.99, 0.77) for the whole sample, and the probability to obtain T₁ and T₂ from the galaxy pool as low as the "observed" values is 11.3% and 8 × 10⁻⁵, respectively. On the other hand, the probability that d_obs and d_sim are drawn from the same distribution is $\mathcal {P}=1.2\%$. Breaking the sample into high- and low-luminosity subsamples, the results are (T₁, p₁, T₂, p₂) = (0.68, 5 × 10⁻⁵, 0.78, 2.2%) for the luminous clusters and (0.84, 7.5%, 0.77, 2 × 10⁻⁴) for the faint clusters. From our test we find that BCGs in high and low luminosity clusters have probabilities of $\mathcal {P}=0.012\%$ and $\mathcal {P}=49\%$ to be statistical.

These results suggest that both our test and the TR tests are insensitive to the uncertainties in galaxy photometry due to sky subtraction.

It is also important to check if the low luminosity clusters are systematically at lower redshifts than the luminous ones. If this were true, the issue with sky subtraction of the SDSS pipeline may affect the photometry of BCGs in low luminosity clusters (hereafter LLCBCG) more strongly than that of BCGs in high luminosity clusters (hereafter HLCBCG), causing the photometry of LLCBCG to be underestimated more than the case for HLCBCG. This would in turn affect our finding that LLCBCG are consistent with being drawn from the same LD as other cluster galaxies (Section 3.2). We have shown in Section 2 that the high and low luminosity clusters have very similar redshift distribution, and therefore we do not think the photometry of LLCBCG and HLCBCG is treated differently.

The main result of this study is that BCGs as a whole have an LD that is distinct from that of the majority of red cluster galaxies (those with M_r ⩽ −20). This conclusion is primarily due to the high luminosities of HLCBCG, which has only 0.03% of probability to be drawn from the LD of all galaxies. On the contrary, LLCBCG are more likely to be simply the statistical extreme of the LD of all galaxies.

With the help of mock clusters, and the elements that come into the calculation of TR tests, one gains some insight into the BCG formation process. For high luminosity clusters, we find that $(\sigma _1,\sigma _\Delta,\overline{\Delta })=(0.39,0.43,0.55)$ from the real data, while the corresponding values from the mock data set are (0.42, 0.37, 0.44). Therefore, although σ₁ in real and mock clusters are comparable, in real clusters $\overline{\Delta }$ is higher that that in mock ones (suggesting that real BCGs are on average 0.11 mag brighter than the statistical extreme). Physics of BCG formation drives the magnitude gap to be larger than the statistical expectation.

To explain the origin of HLCBCG, one therefore needs to invoke galactic mergers (e.g., Ostriker & Tremaine 1975; Hausman & Ostriker 1978; Lin & Mohr 2004; Cooray & Milosavljević 2005; Vale & Ostriker 2008), which is naturally expected within the hierarchical structure formation paradigm (e.g., Dubinski 1998; De Lucia & Blaizot 2007). However, the details of the mergers (e.g., major mergers between BCG and very luminous galaxies, minor mergers between BCG and M_*-type galaxies, or the "galactic cannibalism"¹²) remain to be understood. For example, while Lin & Mohr (2004) suggested that major mergers are a viable route for forming HLCBCG, Vale & Ostriker (2008) were more in favor of minor mergers.

In principle, the extent of late mergers can be constrained by the L_bcg–L_tot correlation, or statistics related to the magnitude gap between BCG and second-ranked galaxy (e.g., Milosavljević et al. 2006; Yang et al. 2008). We have developed a simple merger model for cluster galaxies and will present constraints on the importance of mergers based on the observations obtained in this paper in a future publication (Y.-T. Lin & J. P. Ostriker 2010, in preparation).

After confirming that BCGs are indeed different from the bulk of cluster galaxy population, it is natural to ask if BCGs are different from other luminous/massive cluster galaxies (e.g., M ≲ M_* − 1). It has long been known that BCGs follow different scaling relations from other early-type galaxies (e.g., Oegerle & Hoessel 1991; Lauer et al. 2007; Desroches et al. 2007; Bernardi et al. 2007), in the sense that the BCGs are larger, less dense, and have lower velocity dispersion, compared to other early type galaxies of the same luminosity. However, at present it is not clear if the structure of the BCGs (in terms of Sersic fits) is indeed different from non-BCG early-type galaxies of comparable color and stellar mass (see Guo et al. 2009).

We address this question somewhat indirectly; in Section 3.4, it is shown that the LD of G2s in high luminosity clusters are consistent with that of the whole cluster galaxy population. (Our method could not be robustly applied to examine the statistical nature of G2s in low luminosity clusters, unfortunately.) Although we do not check the third-ranked brightest members, we believe a similar result will emerge for them. Effectively, we suggest that among the most luminous, red, cluster members, only BCGs in high luminosity clusters show significant deviations from the statistical extreme, and therefore these galaxies are a distinct population.

Our results suggest both BCGs in low luminosity/mass clusters and G2s in luminous/massive clusters are the statistical extreme of the galaxy LD. As the latter are very likely BCGs (in clusters or groups that merge with the current host clusters) themselves at earlier stages of their lives, this finding seems to give a self-consistent picture of the luminous galaxy evolution within the hierarchical cluster evolution scenario.

The results and reasoning presented in Sections 3.4 and 5.1 suggest that LRGs that are BCGs in high luminosity (massive) clusters are different from other LRGs (e.g., those that are LLCBCG and G2s or lower-ranked galaxies in high luminosity clusters), and call for very careful modeling and selection of the LRGs.

Suppose the finding that HLCBCG and LLCBCG are two distinct populations continues to hold toward higher redshifts. In a flux-limited survey, the LRGs at higher redshifts would be intrinsically more luminous than those at lower-z. Therefore, a larger fraction of LRGs at higher-z would be composed of HLCBCG, when compared to LRGs at lower-z. When accounting for the Malmquist bias present in the LRG sample, one cannot simply assume that BCGs follow the same LD as LRGs as a whole, otherwise the inferred luminosity would be lower than the true value. This potential systematic effect would be larger toward higher-z.

It is thus important to take into account the difference in the LD of HLCBCG and LLCBCG when creating mock LRG catalogs for BAO surveys. Unfortunately, the BCGs in this study are at lower redshifts (z < 0.1) even compared to SDSS and SDSS-II LRG studies (at z ∼ 0.3; e.g., Eisenstein et al. 2005), and therefore the LDs we measure (see the Gaussian fits in Table 4) are not readily applicable. Accordingly our study strongly motivates for a systematic investigation on the L_bcg–L_tot correlation at z > 0.3, either through direct observations (similar to the methodology used in this paper, utilizing spectroscopic data from, e.g., BOSS or GAMA surveys), through the halo occupation distribution analysis (e.g., Zheng et al. 2009), or the non-parametric method of Vale & Ostriker (2006).

The question of whether the LD of BCGs is drawn from the same LD as other cluster galaxies has been extensively discussed over the last four decades. Here we propose a simple new test to examine the statistical nature of the BCGs and to supplement the classical tests proposed by Tremaine & Richstone (1977).

Our basic idea is to shuffle the cluster galaxies and see how likely the observed LD of BCGs can be reproduced. The procedure is shown in Figure 1. Using a sample of 494 clusters from the C4 catalog at z = 0.03–0.077, we combine all member galaxies with M_r ⩽ −20 to form a big pool of galaxies (5980 in total). We then randomly pick up galaxies from the pool to create 494 mock clusters, whose total luminosities L_tot are matched to that of the real clusters as close as possible. This way we can compare the correlation between the BCG luminosity L_bcg and L_tot of real and mock clusters (see Figure 4). Of course, there are more than one way to create an ensemble of mock clusters, so we repeat this Monte Carlo process to generate 200 mock ensembles.

The averaged BCG luminosity from all 200 realizations of mock data sets, $\left< \overline{L_{{\rm bcg,sim}}}\right>$, gives the expected value when BCGs are statistical, as a function of cluster luminosity. The differences between the logarithms of mean mock BCG luminosity ($\overline{L_{{\rm bcg,sim}}}$) from each ensemble and $\left< \overline{L_{{\rm bcg,sim}}}\right>$ represent the degree of scatter expected if BCGs are stochastically selected from a global LD. In Figure 5, we compare the distribution of $d_{{\rm sim}}= \log \overline{L_{{\rm bcg,sim}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$ with that of $d_{{\rm obs}}= \log \overline{L_{{\rm bcg,obs}}} - \log \left< \overline{L_{{\rm bcg,sim}}}\right>$ and find that the real BCGs are more luminous than the statistical expectation; there is 0.8% of probability for d_obs to be drawn from the same distribution of d_sim (see Table 2). Separating the clusters into high- and low-luminosity subsamples (with a division luminosity of L_div = 3.7 × 10¹¹h⁻²₇₀ L_☉), we conclude that the difference is mainly coming from BCGs in high luminosity clusters (only 0.03% chance to be statistical). The luminosities of BCGs in low luminosity clusters are roughly consistent with the global LD.

We extend the analysis to discuss the statistical nature of the second brightest galaxies (G2s) and find strong evidence that (in high luminosity clusters), G2s have a LD similar to that of the whole cluster galaxy population.

We also apply the tests proposed by TR to our cluster sample and record the results in Table 3. The two statistics (T₁ and T₂) both suggest a small probability for BCGs to be statistical, consistent with our findings. However, examining the TR statistics for the high and low luminosity clusters reveals confusing trends. According to the T₁ statistic, BCGs in the high luminosity clusters are much less likely to be statistical than their counterparts in the low luminosity clusters. However, the T₂ statistic suggests the opposite conclusion.

Our results confirm previous findings that BCGs in high luminosity clusters are a distinct population from other cluster galaxies and tentatively supports the physical mechanism of cannibalism (Ostriker & Tremaine 1975), which is essentially dynamical friction. We also suggest that BCGs are distinct from other non-BCG LRGs. As the majority of LRGs are the brightest member in groups and clusters, such an effect should be taken into account in selecting a homogeneous sample of LRGs for ongoing and future BAO surveys.