SPECTROSCOPIC DETERMINATION OF THE LUMINOSITY FUNCTION IN THE GALAXY CLUSTERS A2199 AND VIRGO

Cluster galaxies display a well-defined red sequence in color–magnitude diagrams (Visvanathan & Sandage 1977). Cluster members are unlikely to have colors redder than the red sequence unless they are very dusty or have very unusual stellar populations.

Using photometric data from SDSS, the red sequence of cluster galaxies is readily apparent in A2199 (Figure 1). The red sequence can be characterized as g − r = −0.035(r − 12) + 1.0 (solid line in Figure 1). Among bright galaxies (r ≲ 16) with measured redshifts (SDSS, Rines et al. 2002), most photometric red-sequence galaxies are cluster members. As recommended in the SDSS Web pages, we use composite model magnitudes as the best estimates of the galaxy magnitudes. Composite model magnitudes are a linear combination of the best-fit deVaucouleurs and exponential profiles. We correct all magnitudes for Galactic extinction.

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For MMT spectroscopy, we use SDSS photometry to identify candidate cluster members in the magnitude range r = 17–20, or −18.6 < M_r < −15.6 at the distance of A2199. This range samples dwarf galaxies in A2199 and therefore offers an excellent test of the abundance of dwarf galaxies in dense environments.

We obtained optical spectroscopy of A2199 with MMT/Hectospec in 2007 July under marginal observing conditions. Hectospec is a 300-fiber multiobject spectrograph with a circular field of view of 1° diameter (Fabricant et al. 2005). We used the 270 line grating, yielding 6.2 Å FWHM resolution.

We observed A2199 with two configurations and obtained 479 secure redshifts (out of 482 galaxies targeted). We obtained three exposures of 600 s for each configuration to facilitate cosmic ray removal. Targets in the first configuration included all galaxies in the magnitude range 17 < r < 19. Targets in the second configuration included galaxies in the range 17 < r < 20, with rankings assigned to four groups: (1) 17 < r < 19 galaxies on the photometric red sequence or blueward [g − r = −0.035(r − 12) + 1.0], (2) 17 < r < 19 galaxies redward of the red sequence, (3) 19 < r < 20 galaxies on or blueward of the red sequence, and (4) 19 < r < 20 galaxies redward of the red sequence. We consider galaxies to lie on the red sequence or blueward if their colors are no more than 0.1 mag redder than the red sequence (Figure 1). We remove all galaxies with fiber magnitudes r_fib > 21 because they are unlikely to yield reliable redshifts in short exposures with Hectospec. We discuss the impact of this selection in Section 4.1.5. Note that one galaxy with a Hectospec redshift in A2199 (R.A.: 16^h29^m00.39^s, Decl.: +39:36:48.8 J2000) is blended with a star in SDSS so that we do not have reliable photometry for it. As a rough estimate for the magnitude of the galaxy, we subtract the flux for the star (determined from psfMag) from the merged object to find r ≈ 19.6.

The Hectospec field of view covers projected radii R_P ⩽ 1.11h⁻¹₇₀ Mpc, equivalent to R_P = 0.69r₂₀₀ ≈ r₅₀₀ for the parameters given in Rines et al. (2003) or 0.76r₂₀₀ for the parameters given in Rines & Diaferio (2006). There are 32 galaxies in the Hectospec sample that have SDSS spectroscopy. The mean velocity difference is −11.4 ± 8.4 km s⁻¹, and the scatter in the velocity differences is 47.8 km s⁻¹, slightly smaller than the mean uncertainty of 59.6 km s⁻¹ calculated from the formal uncertainties. Table 1 lists the coordinates and redshifts for the Hectospec data. Columns 1 and 2 list the coordinates (J2000), Columns 3 and 4 list the heliocentric velocity cz and the corresponding uncertainty σ_cz, and Column 5 lists the cross-correlation score R (Kurtz & Mink 1998).

To measure the luminosity function of the brighter galaxies in A2199, we use redshifts for 207 galaxies measured either in the SDSS or the literature sources compiled in the NASA/IPAC Extragalactic Database (NED).¹ Many of these redshifts are from the Cluster And Infall Region Nearby Survey (CAIRNS), which is complete to $M_{K_s}\approx -22.55 + 5$ log h₇₀ ≈ M* + 2 (Rines et al. 2002, 2004), corresponding to r ≈ 16. SDSS is nominally complete to r = 17.77 (Strauss et al. 2002), although fiber collisions are more problematic for completeness in dense regions such as nearby clusters. There are 12 galaxies in the range 16 < r < 17 that do not have redshifts in either the SDSS or NED. Artificially including all these galaxies as members does not change the LF significantly.

Figure 1 shows that very few galaxies redward of the red sequence are members of A2199. Seven of the nine galaxies above our nominal cutoff for red-sequence galaxies are ⩽0.05 mag redder than the cutoff (five are ⩽0.02 mag redder), suggesting that our cutoff might be too restrictive. Inspection of the two remaining galaxies reveals that their g − r colors are likely overestimated due to deblending problems in the SDSS: the g − r colors based on the SDSS fiber magnitudes place both objects onto the red sequence. Figure 2 shows a similar trend for the Virgo cluster; nearly all galaxies redder than the red sequence are background galaxies. The exceptions are either deblending problems or low surface brightness galaxies for which accurate colors are difficult to obtain (Section 2.2). This result suggests that the red sequence is a remarkably well-defined limit for the intrinsic colors of cluster galaxies. There are no large populations of edge-on spiral galaxies or dusty starbursts that have unusually red colors.

One striking aspect of Figures 1 and 2 is that many galaxies that lie on the photometric red sequence (and blueward) lie well behind the cluster.

We quantify these trends in Figure 5. We divide the galaxies into three populations: red sequence galaxies within ±0.1 mag of our assumed red sequence, and "very red" and "blue" galaxies for galaxies outside this color range. The upper left panel of Figure 5 shows the membership fraction as a function of apparent magnitude for these three classes as well as for the total population. Using these membership fractions, we define the cluster population by assuming that these membership fractions are constants for each population.

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The lower left panel of Figure 5 shows the fraction of the total cluster population in each of the three classes. Galaxies on the red sequence dominate the cluster population at all magnitudes. Because most of the member galaxies in the "very red" population lie close to the cutoff, the fraction of "very red" cluster members is almost certainly overestimated. Similarly, the membership fraction of the very red population should be regarded as an upper limit.

The right-hand panels of Figure 5 show these fractions as functions of projected clustrocentric radius. As expected, the membership fractions generally decline with radius, and the fraction of blue galaxies increases with radius (and hence decreasing density, e.g., Abraham et al. 1996; Balogh et al. 2004; Tanaka et al. 2004; Rines et al. 2005). One surprising feature of the upper right panel of Figure 5 is that the membership fraction of blue galaxies does not decline monotonically with radius but instead reaches a minimum and then increases until it crosses the trend for red galaxies.

Based on Figures 1 and 5, we estimate the total LF in 0.5 mag bins by applying the membership fraction of spectroscopically observed galaxies to the three color populations. Figure 6 shows the LF of A2199. The upper solid line shows the counts of galaxies in 0.5 mag bins when restricted to the "red sequence" and "blue" cuts defined above. The raw counts of galaxies on or bluer than the red sequence show a significant upturn at M_r = −17, similar to the upturn seen by Popesso et al. (2006).

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The dashed line at 17 < r < 20 shows the counts of spectroscopic members, where members have 7000 < cz < 11, 000 km s⁻¹ (Rines et al. 2002). Note that the contrast in redshift space between cluster members and background galaxies is large; the exact velocity limits are not a significant source of uncertainty (Rines et al. 2002; Rines & Diaferio 2006). The thick solid line in Figure 6 shows the resulting LF.

The LF of A2199 can be well fit by a function with the form proposed by Schechter (1976). This function has the form

$$\begin{equation} \left. \frac{dn}{dM} \right|_M \propto 10^{0.4(1+\alpha)(M_*-M)} \,{\rm exp} [- 10^{0.4(M_*-M)}], \end{equation} \tag{ 1 }$$

where M* is the characteristic magnitude of the LF and α is the faint-end slope. We fit the A2199 LF by minimizing χ² and excluding the brightest cluster galaxy (BCG). We find M* = −21.11^+0.21_−0.25 and α = −1.13^+0.07_−0.06 (thin solid line in Figure 6). Both parameters are similar to those found for field galaxies in the SDSS (Blanton et al. 2003) and the Century Survey (Brown et al. 2001). Blanton et al. (2005) investigate the field LF to fainter absolute magnitudes with the SDSS and find that it is consistent with steeper faint-end slopes of α ≈ −1.4. The LF of A2199 is therefore similar to or perhaps shallower than the field LF. We discuss systematic uncertainties in the LF of A2199 below. An independent, shallower (M_V < −17) study of the A2199 LF found α = −1.12 ± 0.06 (Andreon 2007), in excellent agreement with our result.

Figure 6 shows an extrapolation of the A2199 LF to fainter magnitudes. The thick solid line at faint magnitudes shows the LF inferred by assuming that the membership fraction f_mem for the three color bins remains constant at fainter apparent magnitudes, using the faintest bin with spectroscopy to estimate these values of f_mem. Because f_mem decreases steeply with apparent magnitude, this extrapolation is an approximate upper limit on the LF. We make two additional estimates of the extrapolated LF in A2199. The first assumes that all galaxies with r_fib > 21 in the "blue" and "red-sequence" bins are cluster members. This extrapolated LF yields a more conservative upper limit to the A2199 LF; this extrapolation contains a factor of two fewer galaxies than the Popesso LF at M_r ≈ −14. The final estimate of the extrapolated LF assumes that f_mem at a given absolute magnitude is the same as in the Virgo cluster (Figure 7; Section 3.2.1). Interestingly, the values of f_mem for Virgo and A2199 agree at the absolute magnitudes where they overlap, perhaps because the data extend to similar fractions of r₂₀₀ for the two clusters. This Virgo-based extrapolation of the LF is consistent with extending the best-fit Schechter function. Thus, the range of lines at faint magnitudes in Figure 6 encompasses the full range of reasonable extrapolations of the LF in A2199.

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Figure 7 shows the membership fraction of the three galaxy populations (red sequence, very red, and blue) in the Virgo cluster. The large deficit of galaxies in the immediate background of Virgo (Ftaclas et al. 1984) means that the membership cuts in redshift space are robust. The scatter in g − r color increases with increasing apparent magnitude (Figure 2), although part of this increased scatter reflects less reliable photometry for low-surface-brightness galaxies (Figure 3). Conselice et al. (2003) find a similar increase in scatter of colors of fainter galaxies in the Perseus cluster; they interpret this result as evidence that faint galaxies have more complex star-formation histories than bright galaxies. To avoid the larger photometric uncertainties in these faint galaxies, we show the radial trends in Figure 7 only for galaxies with r < 16 so that these can be compared directly with A2199.

Many of the trends apparent in Figure 5 for A2199 are also apparent for Virgo. Membership fractions decline with radius for all populations, and f_mem ≲ 0.1 for very red galaxies. Virgo members are dominated by red-sequence galaxies, and the blue fraction increases (weakly) with radius. Again, the membership fractions of blue galaxies and red-sequence galaxies are quite similar at large projected radii.

Figure 8 shows the LF of the Virgo cluster within 1 Mpc of M87. The bright end of the LF is poorly constrained due to the difficulty of bright galaxy photometry in SDSS (Adelman-McCarthy et al. 2007). Open squares show the "minimal" LF including only spectroscopically confirmed cluster members. Figure 7 shows the membership fraction f_mem as a function of apparent magnitude. Note that f_mem decreases from unity at bright magnitudes to ∼0.03 at M_r ≈ −14. Therefore, any attempt to measure the cluster LF at such faint magnitudes using statistical background subtraction requires a determination of the background population to percent level accuracy and uniformity.

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The number of background galaxies exceeds the number of Virgo galaxies at r ≈ 15 (first suggested by Shapley & Ames 1929, and confirmed here), or M_r ≈ −16.5, close to the absolute magnitude where the cluster LF from the stacked SDSS analysis indicates an upturn in the cluster LF (Popesso et al. 2005, 2006). There is a pronounced dip in the Virgo LF at r ≈ 14. Because galaxies at these magnitudes may have inaccurate photometry from the SDSS pipeline (Adelman-McCarthy et al. 2007), we caution that this dip may be an artifact of the pipeline photometry.

The Virgo cluster LF is fit by a faint-end slope of α = −1.28 ± 0.06 (statistical). We hold M*_r = −21.32 fixed at the value in the field (Blanton et al. 2003) due to the problem with bright galaxy photometry in the SDSS (Adelman-McCarthy et al. 2007). The faint-end slope increases by less than 1σ (α = −1.24) if we fit to the "minimal" LF.

For comparison, we show the composite cluster LF from Popesso et al. (2006), which has a steep upturn at faint magnitudes. We find no such upturn in the Virgo cluster. Figure 8 also shows the field LF determined from SDSS data (Blanton et al. 2005). This field LF is slightly steeper than the Virgo LF, although the Virgo LF is consistent with the faint-end slope of α = −1.32 found for the raw counts in Blanton et al. (2005, i.e., with no correction for missing LSB galaxies). The faint-end slope we measure for Virgo is significantly steeper than that in A2199. Because of the different ranges of absolute magnitude sampled, it would be premature to conclude that there is true variation in the faint-end slope among clusters.

One remarkable aspect of our new determination of the Virgo LF is that it lies very close to the one determined by Sandage et al. (1985), who found a faint-end slope of α ≈ −1.25 from the raw counts and α = −1.30 after correcting for incompleteness at the surface-brightness limit of the survey. Another interesting feature of our determination of the Virgo LF is that it requires assumptions about the membership of low-surface-brightness galaxies. Sandage et al. (1985) assumed a relation between absolute magnitude and surface brightness (based on galaxies in the RSA catalog) to conclude that only low-surface-brightness galaxies are Virgo members. The extensive SDSS spectroscopy demonstrates that the vast majority of high surface brightness galaxies are indeed in the background, but the low surface brightnesses of the presumed Virgo members prevents them from having well-measured redshifts even with SDSS spectroscopy.

Figure 1 shows that there are essentially no cluster galaxies significantly redward of the red sequence. At least some of the apparent discrepancy between the LF in A2199 and claims of steeper slopes can be explained by the treatment of red galaxies, in particular those redward of the red sequence. Some studies contain no color information (Sandage et al. 1985; Phillipps et al. 1998). They are thus unable to identify background galaxies by using the red sequence.

Jenkins et al. (2007) use SDSS colors to select candidate members of the Coma Cluster from an Infrared Array Camera (IRAC) object catalog using a generous color cut of g − r < 2, despite the fact that there are no spectroscopically confirmed members redward of g − r = 1.0 (their Figure 11). They then use the optical spectroscopy of Mobasher et al. (2003) to determine the fraction of galaxies that are Coma members in each IRAC Channel 1 (CH1) magnitude bin. If the spectroscopic target selection were independent of r − Ch1 color, this procedure would yield an accurate estimate of the membership fraction in each IRAC bin. However, the spectroscopic targets are strongly biased to lie blueward of the large population of faint red galaxies (their Figure 14). Thus, this membership fraction is biased high and the resulting LF is probably artificially steep.

A caveat to the conclusion that nearly all "very red" galaxies are background galaxies is the larger scatter in the color-magnitude diagram at faint magnitudes (Section 3.2.1; Conselice et al. 2003). This increased scatter may cause very faint cluster members to be scattered redward of the red sequence.

The deep gravitational potentials of clusters produce gravitational lensing of background galaxies. Lensing distortions are usually measured from systematic distortions in the shapes of faint galaxies which lie behind the cluster. A fascinating application of gravitational lensing to the LF problem was implemented by Medezinski et al. (2007). Using deep HST photometry of A1689, they showed that the weak lensing signal depends on the location of faint galaxies relative to the cluster's red sequence. The shear signal is stronger for galaxies redward of the red sequence than it is for galaxies on the red sequence. The diminution of the shear signal for galaxies on the red sequence is directly correlated with the fraction of galaxies that are cluster members and therefore unlensed. The amplitude of the diminution in a given magnitude bin is then a measurement of the membership fraction in that bin. The LF estimated with this procedure has a significantly shallower slope (α = −1.05 ± 0.1) than many estimates based on background subtraction.

Figures 1 and 2 support this approach. Few galaxies redward of the red sequence are cluster members. Thus, the lensing signal measured from these galaxies suffers little dilution from cluster members.

As part of a weak lensing analysis of systems in the SDSS, Mandelbaum et al. (2005) found that SDSS catalogs contain a deficit of objects near bright galaxies. They conclude that this deficit is produced by overestimation of the sky background because of the stellar halos of the bright galaxies. Because bright galaxies are significantly more common in fields containing galaxy clusters than in offset fields, there should be fewer objects in cluster fields. This deficit could lead to an apparent deficit of faint galaxies in cluster fields and hence potentially mask a steep LF.

We noticed a related effect which may counteract this deficit during the visual inspection of Virgo cluster galaxies. Many large galaxies, especially those with low surface brightnesses, have small pieces detected as separate objects. These detections have little effect on the photometry of the "parent" galaxies because they are usually >3 mag fainter than the parent galaxy. However, these pieces are included in the SDSS Galaxy tables. If these pieces of galaxies are not removed, they could produce an artificial excess of faint galaxies in cluster fields.

The VCC assigned membership via the photometric and morphological properties of galaxies. It is quite striking that our Virgo LF reproduces their results even after the extensive spectroscopy of the SDSS. We saw in Section 2.3 (Figure 3) that Virgo galaxies are cleanly separated from background galaxies in the distribution of apparent magnitude versus central surface brightness. The relatively tight relation between absolute magnitude and surface brightness assumed by Sandage et al. (1985) enabled them to efficiently select Virgo members and determine the Virgo LF accurately. The SDSS spectroscopy confirms that most of the galaxies omitted from the VCC due to photometric properties are indeed background galaxies. At faint magnitudes, the SDSS spectroscopy is not sufficient to confirm the Virgo membership of many of the "possible" members from the VCC (due to their low surface brightness).

The study of the NGC 5846 group by Mahdavi et al. (2005) uses a similar approach to the one used here: using spectroscopy of a subsample to estimate the fraction of galaxies identified as probable members by their photometric properties. They find that all galaxies they classify as "probable" members are in fact members, while about half of the "possible" members are true members.

One longstanding problem in determining the LF is the low surface brightness of dwarf galaxies (e.g., Impey et al. 1988). Indeed, some investigators have used the correlation between magnitude and surface brightness as a way of identifying likely cluster members (Conselice et al. 2002; Mieske et al. 2007). However, low-surface-brightness galaxies can be difficult to detect, especially in shallow photometric surveys such as SDSS (see the discussion in Blanton et al. 2005) and they are challenging objects for accurate spectroscopy.

We show the distribution of surface brightness as a function of apparent magnitude for A2199 in Figure 9. At the faintest magnitudes probed (r = 19–20), there are several galaxies excluded from spectroscopic targeting due to their low surface brightnesses, fiber magnitudes r_fib ⩾ 21. These low-surface-brightness galaxies lie on both the red sequence and on the locus of surface brightness and apparent magnitude traced by cluster members (Figure 9). It therefore seems likely that many of these low-surface-brightness galaxies are members of A2199. Including these galaxies steepens the LF slightly, and this systematic uncertainty is large enough that the LF could be consistent with the LF of Popesso et al. (2006). Our analysis in Section 3.1.2 suggests that the faint end of the A2199 LF is likely intermediate between the slope of α = −1.13 found for our spectroscopic sample and α = −1.4 for field galaxies (Blanton et al. 2005). It is quite possible that the difference in the measured values of α for A2199 and Virgo is created by missing dwarf galaxies in A2199, either from the SDSS photometric pipeline or from our spectroscopic cutoff of r_fib < 21.

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It is curious to note that the distinction between A2199 members and background galaxies (Figure 9) is not nearly as clean as in Virgo (Figure 3). This difference probably indicates that there is lower contrast between A2199 and "near-background" galaxies (those within Δz ≈ 0.01) than in the Virgo cluster.

In the Virgo cluster, there have been many searches for low-surface-brightness galaxies missing from the VCC (e.g., Impey et al. 1988; Trentham & Tully 2002; Sabatini et al. 2003), including studies with deep CCD imaging using the Subaru 8 m telescope. Because these studies have failed to reveal a much larger population of low-surface-brightness galaxies, Trentham & Hodgkin (2002) conclude: "the major concern is now that the sample may be missing a sample of high surface brightness galaxies that we have culled from the sample because we think that they are background galaxies." With the extensive spectroscopy available in SDSS DR6, the lingering concern over high-surface-brightness galaxies can be laid to rest.

Unlike A2199, the difference between the Virgo LF and that of Popesso et al. (2006) cannot be explained by missing LSB galaxies. The difference between the two LFs becomes significant at r ≈ 15 and is greater than a factor of 2 at r ≈ 16. At these bright magnitudes, the incompleteness due to surface brightness is not significant (Figure 4). Also, galaxies in Virgo missed by the SDSS due to surface brightness would be missed by the SDSS in the number counts used by Popesso et al. (2006).

Our efforts to identify additional LSB galaxies in Virgo (see the Appendix) turned up a number of new galaxies, although these additional galaxies have little effect on the LF. The LF from the Virgo catalog prior to these additions (essentially the spectroscopic catalog with the addition of bright galaxies and LSB spectroscopic targets with failed or no spectra) has a best-fit faint-end slope of α = −1.24, or less than 1σ different from our final estimate. The issue of low-surface-brightness galaxies remains a serious systematic uncertainty fainter than M_r ≈ −14 (beyond the limit of the measurements presented here). A systematic search for LSB galaxies in the SDSS imaging but below the detection threshold of the photometric pipeline would be very instructive, but such a search lies beyond the scope of this work.

One aspect of recent claims about the faint-end slope of the LF in clusters is difficult to explain with the effects described in Section 4.1.3. Several studies have claimed that the faint-end slope is steeper at large clustrocentric radii and shallower in the cluster core (R_p ∼ 300h⁻¹₇₀ kpc; e.g., Popesso et al. 2006; Barkhouse et al. 2007). The potential excess from pieces of galaxies described in Section 4.1.3 would naturally produce a faint-end slope which increased continuously toward cluster centers: the density of bright galaxies generally increases toward cluster centers. The number of galaxy pieces should do the same. The shallower faint-end slope in cluster centers suggested by Popesso et al. (2006) and Barkhouse et al. (2007) is therefore difficult to explain with this potential systematic bias.

Some recent studies claim that LF fits also yield brighter M* at larger clustrocentric radii. The direction of the changes in both parameters is along the well-known degeneracy between α and M* in Schechter function fits. There is, however, an important systematic effect in these studies that is often ignored. The BCGs are usually explicitly excluded from the LF fits. Because BCGs tend to lie near cluster centers, this procedure acts to produce an apparent brightening of M* with radius.

Figure 10 shows the LF as a function of radius in Virgo and A2199. We again show the LFs of Popesso et al. (2006); Blanton et al. (2005); Trentham & Tully (2002) for comparison (with arbitrary normalization). The LF does not have a strong dependence on clustrocentric radius, counter to claims by Popesso et al. (2006) and Barkhouse et al. (2007).

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There is some evidence that M* is brighter at smaller radii in Virgo, but A2199 shows the opposite trend. The trend of brighter M* at smaller radii (and hence denser environments) is expected due to the tendency of very luminous galaxies to inhabit the densest environments (e.g., Hogg et al. 2003; Zehavi et al. 2005). Note that the dip in the Virgo LF at r ≈ 14 appears to be confined to the annulus 0.5 < R_P < 1.0 Mpc. A more detailed treatment of the photometry of all Virgo members would be required to test the significance of this dip.

One motivation for determining the LF in clusters is to assess the total amount of starlight in a given cluster. Recent studies of intracluster light in nearby clusters have shown that it contributes a significant fraction of the total cluster light (e.g., Gerhard et al. 2007).

If we adopt the faint-end slope from Virgo, the contribution to total galaxy light from galaxies fainter than M*_r + 2 (M*_r + 3) is 27.4 ± 3.0% (14.6 ± 2.5%). Thus, the uncertainty in the total cluster starlight due to faint galaxies is significantly smaller than the uncertainty due to intracluster light, which may contribute 5–50% of the total stars in a cluster (e.g., Mihos et al. 2005; Krick & Bernstein 2007; Gonzalez et al. 2007).

Future studies of intracluster light which extend to very low surface brightness will be very useful for quantifying the number and luminosity of LSB galaxies missed in existing surveys. Similarly, future surveys of intracluster planetary nebulae and other tracers should provide upper limits on the fraction of galaxian light in clusters contributed by LSB galaxies below the detection limits of galaxy surveys.