Spectroscopic Determination of the Faint End of the Luminosity Function in the Nearby Galaxy Clusters A2199 and Virgo
ABSTRACT We report a new determination of the faint end of the galaxy luminosity function in the nearby clusters Virgo and Abell 2199 using data from SDSS and the Hectospec multifiber spectrograph on the MMT. The luminosity function of A2199 is consistent with a single Schechter function to M_r=-15.6 + 5 log h_70 with a faint-end slope of alpha=-1.13+/-0.07. The LF in Virgo extends to M_r=-13.5= M^*+8 and has a slope of alpha=-1.28+/-0.06. The red sequence of cluster members is prominent in both clusters, and almost no cluster galaxies are redder than this sequence. We show that selecting objects on the red sequence and blueward produces a steeply rising faint-end. A large fraction of photometric red-sequence galaxies lie behind the cluster. We compare our results to previous estimates and find poor agreement with estimates based on statistical background subtraction but good agreement with estimates based on photometric membership classifications (e.g., colors, morphology, surface brightness). We conclude that spectroscopic data are critical for estimating the faint end of the luminosity function in clusters. The faint-end slope we find is consistent with values found for field galaxies, weakening any argument for environmental evolution in the relative abundance of dwarf galaxies. However, dwarf galaxies in clusters are significantly redder than field galaxies of similar luminosity or mass, indicating that star formation processes in dwarfs do depend on environment. Comment: to appear in AJ, includes tables of probable and spectroscopically confirmed Virgo cluster members
arXiv:0710.1082v3 [astro-ph] 3 Mar 2008
Draft version March 3, 2008
Preprint typeset using LATEX style emulateapj v. 10/09/06
SPECTROSCOPIC DETERMINATION OF THE LUMINOSITY FUNCTION IN THE GALAXY CLUSTERS
A2199 AND VIRGO
Kenneth Rines and Margaret J. Geller
Smithsonian Astrophysical Observatory, 60 Garden St, MS 20, Cambridge, MA 02138; email@example.com
Draft version March 3, 2008
We report a new determination of the faint end of the galaxy luminosity function in the nearby
clusters Virgo and Abell 2199 using data from SDSS and the Hectospec multifiber spectrograph on the
MMT. The luminosity function of A2199 is consistent with a single Schechter function to Mr= −15.6
+ 5 log h70 with a faint-end slope of α = −1.13 ± 0.07 (statistical). The LF in Virgo extends to
Mr ≈ −13.5 ≈ M∗+ 8 and has a slope of α = −1.28 ± 0.06 (statistical). The red sequence of
cluster members is prominent in both clusters, and almost no cluster galaxies are redder than this
sequence. A large fraction of photometric red-sequence galaxies lie behind the cluster. We compare our
results to previous estimates and find poor agreement with estimates based on statistical background
subtraction but good agreement with estimates based on photometric membership classifications (e.g.,
colors, morphology, surface brightness). We conclude that spectroscopic data are critical for estimating
the faint end of the luminosity function in clusters. The faint-end slope we find is consistent with
values found for field galaxies, weakening any argument for environmental evolution in the relative
abundance of dwarf galaxies. However, dwarf galaxies in clusters are significantly redder than field
galaxies of similar luminosity or mass, indicating that star formation processes in dwarfs do depend
Subject headings: galaxies: clusters — galaxies: elliptical and lenticular, cD — galaxies: kinematics
and dynamics — cosmology: observations
The luminosity function of galaxies is fundamental to
understanding galaxy formation and evolution. The lu-
minosity function differs dramatically from the expected
mass function of dark matter halos, indicating that bary-
onic physics is very important for understanding galaxies.
In particular, a well-determined luminosity function en-
ables accurate modeling linking the masses of dark mat-
ter haloes to galaxy luminosities (e.g., Vale & Ostriker
2006; Yang et al. 2007, and references therein). These
empirical models provide a powerful test of any model of
galaxy formation and evolution.
Early studies of the luminosity function used the large
galaxy density in clusters as a tool for measuring the
shape of the luminosity function (e.g., Sandage et al.
1985).The obvious drawback of this method is that
the luminosity function in dense environments may
differ from that in more typical galaxy environments
(Binggeli et al. 1988; Driver et al. 1994; de Propris et al.
1995). Environmental trends in the luminosity function
may reflect differences in galaxyformation in different en-
vironments (Tully et al. 2002; Benson et al. 2003). For
instance, tidal stripping or “threshing” of larger galax-
ies may produce dwarf galaxies (Bekki et al. 2001), or
dwarf galaxies may be formed in tidal tails of intrac-
tions among giant galaxies (Barnes & Hernquist 1992).
Alternatively, the denser environments of protoclusters
may have shielded low-mass galaxies from the ultraviolet
radiation responsible for reionization (Tully et al. 2002;
Benson et al. 2003).
Many studies suggest an environmental influence on
the LF; others provide no such evidence. The main diffi-
Electronic address: firstname.lastname@example.org
culty in resolving this important issue is the challenge of
determining cluter membership for faint galaxies where
background galaxy counts are large.
Because few deep spectroscopic surveys of clusters ex-
tend into the dwarf galaxy regime (Mr?-18; for excep-
tions, see Mobasher et al. 2003; Christlein & Zabludoff
2003; Mahdavi et al. 2005, and references therein), clus-
ter membership is usually determined via statistical
subtraction of background galaxies (e.g., Popesso et al.
2006; Jenkins et al. 2007; Milne et al. 2007; Adami et al.
2007; Yamanoi et al. 2007; Barkhouse et al. 2007, and
references therein). Because galaxy number counts in-
crease much more steeply than cluster member counts
(even for very steep faint-end slopes), small system-
atic uncertainties in background subtraction can pro-
duce large uncertainties in the abundance of faint cluster
Here,we use MMT/Hectospec spectroscopy and
data from the Sloan Digital Sky Survey (SDSS,
Stoughton et al. 2002) to estimate the luminosity func-
tion (LF) in the clusters Abell 2199 and Virgo. These
data enable very deep sampling of the luminosity func-
tion. In particular, we report an estimate of the faint-end
slope of the luminosity function with much smaller sys-
tematic uncertainties than most previous investigations.
We demonstrate that photometric properties of galaxies
such as color and surface brightness correlate well with
cluster membership (in agreement with many previous
studies). Very few galaxies redder than the red sequence
are cluster members.
We discuss the photometric and spectroscopic data in
§2. We present the luminosity functions in §3. We com-
pare our results to previous studies and discuss possible
systematic effects and uncertainties in §4. We conclude
2Rines & Geller
in §5. An Appendix details the construction of our cat-
alog of confirmed and probable Virgo cluster members.
We assume cosmological parameters of Ωm=0.3,
ΩΛ=0.7, H0=70 h70km s−1Mpc−1.
at the distance of A2199 is 1′′=0.61 h−1
70kpc at the distance of Virgo.
The spatial scale
2.1. Abell 2199
The nearby X-ray cluster Abell 2199 (e.g., Rines et al.
2002, and references therein) offers an excellent opportu-
nity for probing the LF in a rich, nearby cluster. A2199 is
significantly more massive than Virgo (Rines & Diaferio
2006) and X-ray data suggest that it is a relaxed cluster
(Markevitch et al. 1999). The center of A2199 is domi-
nated by NGC 6166, a massive cD galaxy (Kelson et al.
2.1.1. Optical Imaging
Cluster galaxies display a well-defined red sequence
in color-magnitude diagrams (Visvanathan & Sandage
1977). Cluster mmebers are unlikely to have colors red-
der 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 galax-
ies are cluster members. As recommended in the SDSS
web pages, we use composite model magnitudes as the
best estimates of the galaxy magnitudes.
model magnitudes are a linear combination of the best-
fit deVaucouleurs and exponential profiles. We correct
all magnitudes for Galactic extinction.
2.1.2. Optical Spectroscopy
For MMT spectroscopy, we use SDSS photometry to
identify candidate cluster members in the magnitude
range r=17-20, or −18.6 < Mr< −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 spec-
trograph with a circular field of view of 1◦diameter
(Fabricant et al. 2005). We used the 270-line grating,
yielding 6.2˚ A FWHM resolution.
We observed A2199 with two configurations and ob-
tained 479 secure redshifts (out of 482 galaxies targeted).
We obtained 3 exposures of 600s for each configuration
to facilitate cosmic ray removal. Targets in the first con-
figuration 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.
Fig.1.— Color-magnitude diagram for A2199.
quence is clearly visible (solid line). Small dots denote galaxies
with SDSS photometry and no spectroscopy. Squares indicate spec-
troscopically confirmed A2199 members from Hectospec (filled)
and SDSS (open), and crosses indicate spectroscopically confirmed
background galaxies. Dashed lines indicate the color-magnitude
cuts we adopt for the red sequence of A2199. Note that essen-
tially all galaxies redder than the red-sequence with spectra are
The red se-
We consider galaxies to lie on the red sequence or blue-
ward if their colors are no more than 0.1 mag redder
than the red sequence (Figure 1). We remove all galaxies
with fiber magnitudes rfib>21 because they are unlikely
to yield reliable redshifts in short exposures with Hec-
tospec. We discuss the impact of this selection in §4.1.5.
Note that one galaxy with a Hectospec redshift in A2199
(RA: 16h29m00.39s, DEC: +39:36:48.8 J2000) is blended
with a star in SDSS so that we do not have reliable pho-
tometry 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
RP ≤ 1.11h−1
70Mpc, equivalent to RP = 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 differ-
ence 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−1calculated from the formal
uncertainties. Table 1 lists the coordinates and redshifts
for the Hectospec data. Columns 1 and 2 list the coordi-
nates (J2000), Columns 3 and 4 list the heliocentric ve-
locity cz and the corresponding uncertainty σcz, and Col-
umn 5 lists the cross-correlation score R (Kurtz & Mink
To measure the luminosity function of the brighter
galaxies in A2199, we use redshifts for 207 galaxies mea-
sured either in 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 MKs≈-
22.55 + 5 log h70 ≈ M∗+ 2 (Rines et al. 2002, 2004),
Faint End of the Luminosity Function3
Spectroscopic Data for A2199a
(J2000)(km s−1)(km s−1)
aThe complete version of this table is in the electronic edi-
tion of the Journal. The printed edition contains only a
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
SDSS or NED. Artificially including all these galaxies as
members does not change the LF significantly.
We use recently released SDSS spectroscopy from Data
Release 6 (DR6; Adelman-McCarthy et al. 2007) to de-
termine the luminosity function in the Virgo cluster,
the closest large galaxy cluster. Virgo’s proximity (d ≈
17 Mpc, Tonry et al. 2001; Mei et al. 2007) allows the
deepest possible probes of the luminosity function. How-
ever, Virgo is clearly unrelaxed dynamically, as shown in
the lumpiness of the galaxy distribution (Binggeli et al.
1985) and of the X-ray gas (Bohringer et al. 1994).
The SDSS DR6 data cover virtually the entire sky
within a projected radius of 1 Mpc from the central
galaxy M87.We focus our efforts on this region, al-
though data are available in a strip extending to much
larger radius. Galaxies within 1 Mpc of M87 are almost
all contained within the main “A” cluster (Binggeli et al.
1985). The radial range covered is similar to that cov-
ered in A2199 both in physical units and in overdensity:
1 Mpc in Virgo is ≈ 0.65r200(McLaughlin 1999).
Many previous studies have estimated the luminosity
function in Virgo (e.g., Binggeli et al. 1985; Impey et al.
1988; Phillipps et al. 1998; Trentham & Hodgkin 2002;
Sabatini et al. 2003), but none have complete spec-
troscopy to the depth of SDSS. Without complete spec-
troscopy, previous investigations have relied on either
statistical methods of background subtraction or on al-
ternative membership indicators including morphology
or surface brightness.
The proximity of the Virgo cluster presents challenges
for constructing a robust photometric catalog containing
galaxies of varying size, morphology, and surface bright-
ness. We detail the construction of our new Virgo cluster
catalog in the Appendix.
Figure 2 shows the color-magnitude diagram of galax-
ies in the Virgo cluster and in the background. Colors
are from the fiber magnitudes (Adelman-McCarthy et al.
2007). Only four spectroscopically confirmed members
in the magnitude range r=13-16 redder than the red se-
quence are Virgo members; all four lie on the red se-
quence or blueward if the color is measured from the
model magnitudes.Nearly all Virgo members with
Fig.2.— Color-magnitude diagram for Virgo.
quence is clearly visible (solid line), although it is distorted at the
bright end, possibly due to known problems with SDSS photom-
etry of bright galaxies. Filled squares indicate spectroscopically
confirmed Virgo members, open squares indicate probable mem-
bers lacking reliable redshifts, and small dots indicate spectroscop-
ically confirmed background galaxies. Dashed lines indicate the
color-magnitude cuts we adopt for the red sequence of Virgo. Note
that again essentially all galaxies redder than the red-sequence with
spectra are background galaxies.
The red se-
r > 16 and redder than the red sequence are low sur-
face brightness galaxies and may have unreliable colors.
Because surface brightness correlates with absolute
magnitude, the faintest Virgo galaxies in SDSS may
be close to the surface brightness limit of the survey.
Blanton et al. (2005) studied the completeness of the
SDSS pipeline using simulated images of galaxies with
a wide range of apparent magnitudes and surface bright-
nesses. Figure 3 shows central surface brightness ver-
sus apparent magnitude for galaxies within Rp≤ 1Mpc
of M87.We define the central surface brightness as
µ0r= rPetro+2.123 to convert the fiber magnitudes into
mag arcsec−2(this definition assumes constant surface
brightness within the fiber). Galaxies in the background
of Virgo tend to have higher surface brightness at a fixed
apparent magnitude (Tolman 1930; Kurtz et al. 2007),
but the loci of Virgo galaxies and background galaxies
overlap. Galaxies from DR6 with z ≤ 0.01 but outside
of Virgo show a similar distribution, indicating that this
difference is not due to photometric issues specific to the
The dramatically different distributions of magni-
tude versus surface brightness for Virgo members and
background galaxies strongly support the use of sur-
face brightness as a membership classification (e.g.,
Binggeli et al. 1985; Conselice et al. 2002; Hilker et al.
2003; Mahdavi et al. 2005; Mieske et al. 2007).
SDSS spectra show the power of this classification.
Adopting µ0r = 0.9rPetro+ 6.2 to separate the two
populations (and excluding 34 very bright galaxies with
rfib < 16 that lie outside Figure 3), 65.2% of galaxies
with lower surface brightness are spectroscopically con-
firmed Virgo members. Virgo members comprise only
0.67% of galaxies with higher surface brightnesses. This
simple photometric cut removes 95.2% of the spectro-
4Rines & Geller
Fig. 3.— Central surface brightness versus apparent magnitude
for galaxies within 1 Mpc of M87. Filled (open) squares indicate
spectroscopically confirmed (probable) Virgo members. Small dots
indicate background galaxies. The straight line indicates an ap-
proximate division between Virgo members and background galax-
scopically confirmed background galaxies.
The power of this technique shows that there is
a tight correlation between absolute magnitude and
surface brightness for both
Andreon & Cuillandre 2002) and field galaxies (e.g.,
Blanton et al. 2005).This classification may be un-
usually clean for the Virgo cluster due to the large
deficit of galaxies in the immediate background of Virgo
(Ftaclas et al. 1984).
Figure 4 shows average surface brightness within the
Petrosian half-light radius versus apparent magnitude.
This is the definition of surface brightness used to con-
struct the SDSS spectroscopic target catalogs. Figure 4
shows the results of a completeness study of LSB galax-
ies performed by Blanton et al. (2005). There are signif-
icant numbers of Virgo members in the region where the
SDSS spectroscopic target catalog begins to become in-
complete. This incompleteness is mitigated by the inclu-
sion of large galaxies inserted by hand after the main tar-
get selection (Blanton et al. 2005) and by the procedures
we follow here to identify additional Virgo members be-
low the spectroscopic target limits (see Appendix).
Figure 3 provides a useful constraint on the abundance
of high surface brightness galaxies in clusters. In par-
ticular, a new class of high surface brightness galaxies
was discovered in the Fornax cluster (Drinkwater et al.
1999; Hilker et al. 1999).
tracompact dwarf galaxies (UCDs), usually are unre-
solved in ground-based imaging. The typical luminosi-
ties of UCDs place them between globular clusters and
compact elliptical galaxies like M32. Two groups have
found UCDs in the Virgo cluster (Ha¸ segan et al. 2005;
Jones et al. 2006), but they appear to be a relatively
rare type of galaxy. While many UCDs would be un-
resolved in ground-based SDSS imaging, the SDSS data
successfully recovers the compact Virgo members VCC
1313 and VCC 1627 (Trentham & Hodgkin 2002) as well
as a previously undiscovered UCD (classified as a galaxy
by the SDSS photometric pipeline and described by
cluster galaxies (e.g.,
These galaxies, termed ul-
Fig.4.— Average surface brightness versus apparent magni-
tude for galaxies within 1 Mpc of M87.
indicate spectroscopically confirmed (probable) Virgo members.
Small dots indicate background galaxies. The straight line indi-
cates an approximate division between Virgo members and back-
ground galaxies. The four curves indicate from top, 50%, 75%,
90%, and 95% completeness contours for the SDSS imaging pipeline
(Blanton et al. 2005).
Filled (open) squares
Chilingarian & Mamon 2007).
The SDSS spectroscopy demonstrates conclusively
that the Virgo cluster contains very few high surface
brightness galaxies that are resolved in ground-based
imaging.Estimating the total number of stellar-like
UCDs in Virgo is observationally expensive, requiring
spectroscopy of all stellar-like objects (Jones et al. 2006)
or HST imaging (Ha¸ segan et al. 2005). Existing studies
indicate that UCDs are not sufficiently common to sig-
nificantly affect the luminosity function of Virgo cluster
We use the 479 Hectospec redshifts for A2199 along
with 207 redshifts from SDSS and the literature to de-
termine the luminosity function. Of these 686 galaxies,
351 are members of A2199. In Virgo, we find 484 definite
or probable members (including 5 with r≥17.77) out of a
total of 3971 galaxies within 1 Mpc of M87 and r<17.77.
The luminosity functions for both clusters suggest rel-
atively shallow faint-end slopes, α = −1.13 ± 0.07 for
A2199 and α = −1.28 ± 0.06 for Virgo.
3.1.1. Membership Fractions and Composition of Cluster
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 galax-
ies are ≤0.05 mag redder than the cutoff (five are ≤0.02
mag redder), suggesting that our cutoff might be too re-
strictive. Inspection of the two remaining galaxies reveals
that their g−r colors are likely overestimated due to de-
blending problems in 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
Faint End of the Luminosity Function5
Fig. 5.— Top panels: Fraction of spectroscopically observed
galaxies in A2199 that are cluster members as a function of appar-
ent magnitude (left) or projected radius (right). The lines indicate
the fractions for galaxies on the red sequence (RS), redder than
the red sequence (“Red”; also shown by isolated points), and bluer
than the red sequence (“Blue”). Bottom panels show the fraction
of the cluster population in each of these categories as a function
of apparent magnitude (left) or projected radius (right). The blue
fraction increases with either increasing apparent magnitude or in-
creasing clustrocentric radius.
are background galaxies. The exceptions are either de-
blending problems or low surface brightness galaxies for
which accurate colors are difficult to obtain (§2.3). 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 mem-
bership 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.
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 pop-
ulation at all magnitudes. Because most of the member
galaxies in the “very red” population lie close to the cut-
off, the fraction of “very red” cluster members is almost
certainly overestimated. Similarly, the membership frac-
tion of the very red population should be regarded as an
The right-hand panels of Figure 5 show these fractions
as functions of projected clustrocentric radius. As ex-
pected, the membership fractions generally decline with
radius, and the fraction of blue galaxies increases with ra-
dius (and hence decreasing density, e.g., Abraham et al.
1996; Balogh et al. 2004; Tanaka et al. 2004; Rines et al.
Fig. 6.— Luminosity function of A2199 determined using spectra
from MMT/Hectospec and SDSS DR6 (thick red solid line). Er-
rorbars indicate Poissionian uncertainties. The blue line shows the
results of a correction for low surface brightness galaxies (§4.1.5).
The dashed line at 17<r<20 shows spectroscopically confirmed
members from our Hectospec data. The rising dotted line indi-
cates the cluster luminosity function of Popesso et al. (2006) and
the dash-dotted line indicates the field LF of Blanton et al. (2005).
The thick solid line at faint magnitudes shows an extrapolation
of the LF assuming membership fractions for the color bins of
fRS = fblue = 0.3 and fred = 0 (Figure 5). The short-dashed
lines at faint magnitudes show the extrapolated LF assuming (up-
per) all LSB galaxies are members or (lower) that fmemis the same
as in Virgo at the comparable absolute magnitude (Figure 7).
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.
3.1.2. The Luminosity Function of A2199
Based on Figures 1 and 5, we estimate the total lumi-
nosity function in 0.5 mag bins by applying the member-
ship fraction of spectroscopically observed galaxies to the
three color populations. Figure 6 shows the luminosity
function of A2199. The upper solid line shows the counts
of galaxies in 0.5 mag bins when restricted to the “red se-
quence” and “blue” cuts defined above. The raw counts
of galaxies on or bluer than the red sequence show a sig-
nificant upturn at Mr=-17, similar to the upturn seen by
Popesso et al. (2006).
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
where M∗is the characteristic magnitude of the LF