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PASJ: Publ. Astron. Soc. Japan 61, 833–872, 2009 August 25
c 2009. Astronomical Society of Japan.
Subaru Weak-Lensing Survey II:
Multi-Object Spectroscopy and Cluster Masses
Takashi HAMANA, Satoshi MIYAZAKI, and Nobunari KASHIKAWA
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588
Richard S. ELLIS
California Institute of Technology, 105-24 Astronomy, Pasadena, CA 91125, USA
Richard J. MASSEY
Institute for Astronomy, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
Alexandre REFREGIER
Service d’Astrophysique CEA Saclay, Bat. 709 F-91191 Gif sur Yvette, France
and
James E. TAYLOR
University of Waterloo, Department of Physics and Astronomy, Waterloo, Ontario N2L 3G1, Canada
(Received 2008 August 28; accepted 2009 May 15)
Abstract
We present the first results of a multi-object spectroscopic campaign to follow up cluster candidates located via
weak lensing. Our main goals are to search for spatial concentrations of galaxies that are plausible optical counterparts of the weak-lensing signals, and to determine the cluster redshifts from those of member galaxies. Around
each of 36 targeted cluster candidates, we obtained 15–32 galaxy redshifts. For 28 of these targets, we confirmed
a secure cluster identification, with more than five spectroscopic galaxies within a velocity of ˙3000 km s1. This
includes three cases where two clusters at different redshifts are projected along the same line-of-sight. In 6 of
the 8 unconfirmed targets, we found multiple small galaxy concentrations at different redshifts, each containing at
least three spectroscopic galaxies. The weak-lensing signal around those systems was thus probably created by the
projection of groups or small clusters along the same line-of-sight. In both of the remaining two targets, a single
small galaxy concentration was found. In some candidate super-cluster systems, we found additional evidence of
filaments connecting the main density peak to an additional nearby structure. For a subsample of our most cleanly
measured clusters, we investigated the statistical relation between their weak-lensing mass (MNFW, SIS) and the
velocity dispersion of their member galaxies (v), comparing our sample with optically and X-ray selected samples
from the literature. Our lensing-selected clusters are consistent with v = SIS, with a similar scatter to that of
optically and X-ray selected clusters. We also derived an empirical relation between the cluster mass and the galaxy
velocity dispersion, M200E(z) = 11.0  1014  (v=1000 km s1)3:0 h1Mˇ, which is in reasonable agreement with
predictions of N -body simulations in the ΛCDM cosmology.
Key words: cosmology: dark matter — cosmology: large-scale structure of universe — cosmology:
observations — galaxies: clusters: general
1. Introduction
The development of weak-lensing techniques, coupled with
deep panoramic imaging surveys, has enabled us to locate clusters of galaxies via the gravitational distortions of background
galaxies’ shapes. Since the first spectroscopically confirmed
discovery of a shear-selected cluster by Wittman et al. (2001),
there has been rapid progress toward a large weak-lensing
selected cluster catalogue. Miyazaki et al. (2002) first reported
the detection of several significant shear-selected cluster candidates in an untargeted 2.1 deg2 field. Hetterscheidt et al.
(2005) found 5 cluster candidates in 50 randomly selected VLT
FORS1 fields (0.64 deg2 in total), all of which are associated
with an overdensity of galaxies. Wittman et al. (2006) reported
8 candidates in the 8.6 deg2 Deep Lens Survey. Gavazzi and
Soucail (2007) found 14 cluster candidates in the 4 deg2 CFHT
Legacy Survey (Deep), of which nine have optical or X-ray
counterparts, and are thus secure clusters.
The first sizable sample of weak-lensing shear-selected
cluster candidates was presented by Miyazaki et al. (2007;
hereafter P1). Their sample was obtained solely via peak
finding in weak-lensing density maps, and included 100 significant peaks in a 16.7 deg2 survey area.
Before such a sample is used for statistical cosmological
or astronomical analyses, two additional follow-up observations are required. Firstly, each cluster candidate should be
confirmed by independent observations, since a fraction of
lensing peaks could be false positives from e.g., a chance
tangential alignment of the galaxies’ intrinsic ellipticities
(White et al. 2002; Hamana et al. 2004; Hennawi & Spergel
2005). Secondly, the redshifts of confirmed clusters need
to be determined in order to derive their physical quantities,
including mass.
We conducted a multi-object spectroscopic (MOS)
campaign that accomplishes both goals. We measured the
redshifts of a few tens of galaxies within an expected cluster
834 T. Hamana et al. [Vol. 61,
scale radius (or core radius, typically a few arcmins), and
searched for spatial concentrations that are plausible optical
counterparts of the weak-lensing signals. Once a galaxy
overdensity is found, it is easy to determine the cluster redshift
from the redshifts of member galaxies. It is important to note
that cluster confirmation based on prominent galaxy concentrations would not be very effective for very high mass-to-light
ratio (M=L), galaxy-poor systems. Although our methodology
can confirm normal or galaxy-rich clusters, the absence of
a galaxy concentration in our fairly sparsely-sampled data
therefore does not necessarily prove that a weak-lensing signal
is false.
In addition to our primary goals, MOS observations provide
several useful by-products. If redshifts can be obtained for
sufficient galaxies in a cluster, their line-of-sight velocity
dispersion provides an estimate of the cluster’s dynamical
mass. MOS observations can also detect multiple structures
along the same line of sight. Because of the relatively broad
redshift window function of gravitational lensing, physically
unrelated structures in the same line of sight may contribute to
a single peak in a weak-lensing density map, resulting in an
overestimation of the cluster mass (White et al. 2002). It will
therefore be important to quantify and properly account for
such projections when computing statistics of cluster masses
from weak-lensing observations.
In this paper, we present results of cluster confirmations and
cluster redshifts. We discuss the detailed weak-lensing properties of each system and, for a clean subset of our clusters,
examine statistical relations between the weak-lensing masses
and dynamical masses. A statistical analysis of the entire
sample, taking into account additional selection effects, will
be presented in A. Green et al. (2009, in preparation).
This paper is organized as follows. In section 2, we discuss
our selection of cluster candidates. In section 3, we describe
our new observations, data reduction, and measurements of
galaxy redshifts. In section 4, we identify optical counterparts to cluster candidates, and discuss our measurements of
their velocity dispersions and dynamical masses. In section 5,
we analyze the weak-lensing signal of confirmed clusters. In
section 6, we consider cluster scaling relations within our
sample. In section 7, we summarize our results. In appendix 1,
we calculate the gravitational lensing shear profile of a truncated NFW model. Detailed discussions of each system,
including comparisons of the dynamic and lensing masses,
follow in appendices 2 (for clusters we have observed) and 3
(for observations taken from the literature).
Throughout this paper, we adopt a flat ΛCDM cosmology
with a matter density of Ωm = 0.3, a cosmological constant of
ΩΛ = 0.7, and a Hubble constant of H0 = 100hkm s1 Mpc1,
with h = 0.7.
2. Cluster Candidates
Targets for our spectroscopic follow-up campaign were
selected from amongst the Subaru weak-lensing survey’s shearselected cluster candidates. We refer readers to P1 for
details of the Subaru weak-lensing survey, including the image
acquisition and analysis, the creation of weak-lensing density
maps, and the selection of cluster candidates. Briefly, RC -band
imaging data were obtained in thirteen fields (except for the
COSMOS field where i 0-band data was used). Each survey
area was 1.07–2.8 deg2, and the total area was 21.82 deg2.
Weak-lensing density maps were computed with a pixel size
of 1500  1500, using the Kaiser and Squires (1993) inversion
algorithm, with a Gaussian smoothing filter of G = 10. Noise
maps were also created using the same algorithm, but from
the root-mean-square (RMS) of 100 realizations in which the
orientations of the measured shears were randomized. Maps
of the signal-to-noise ratio (S=N ) were created by dividing
the density map by the noise RMS map. Positive peaks were
searched for in the S=N maps, and peaks with S=N greater
than a threshold value were defined as candidates of massive
halos. P1 took a threshold value of S=N = 3.69. After
carefully avoiding areas close to bright stars or field boundaries, where a weak-lensing density map could be affected by
a lack of background lensed galaxies, 100 significant detections remained in a 16.7 deg2 area (see table 2 of P1 for their
weak-lensing and optical properties).
We have carried out multi-object spectroscopic follow-up
observations of 36 cluster candidates (see table 1 for our
targets’ optical and weak-lensing properties, galaxy number
densities, and peak  values;  is defined in subsection 5.1).
Our target selection differs slightly from that of P1 because, in
the planning stages of this follow-up program, we were still
evaluating the optimal criteria for reliable cluster selection.
The current selection was based on both the peak  values and
a visual inspection of the optical images. The peak  value
was evaluated with two filter scales (10 and 20). Since the
Gaussian filter acts as a matched filter, large nearby systems
can be detected with a higher S=N in the 20 filter. Due to
a visual inspection, our target could be biased toward optically rich clusters. Of our 36 targets, 24 are listed in the P1
catalog. The remaining 12 targets are either below the P1 S=N
threshold with a single filter scale, or in discarded survey areas
(close to bright stars or field boundaries).
3. Spectroscopic Observations and Data Reduction
3.1. Spectroscopic Observations
We used the Subaru telescope’s Faint Object Camera and
Spectrograph (FOCAS: Kashikawa et al. 2002) in MultiObject Spectroscopy (MOS) mode. Each cluster candidate was
observed with one 60 diameter slit mask, on which we placed
25–38 slits. We used a 150 mm1 grating and an SY47 order
sorting filter, resulting in wavelength coverage between 4700
and 9400 Å, with a pixel resolution of 2.8 Å pixel1. The slit
width was 0.800 for all cases, which corresponds to a spectral
resolution power of R  250, or Δ  30 Å at 8000 Å.
We conducted FOCAS observations in 2004 May 16,
December 9, 17, 2005 June 1–2, July 31–August 1,
December 9 and 23–24. Observing conditions were not always
good: some targets were observed under a cloudy/cirrus sky.
We took two or three exposures per mask. The total exposure
times are listed in table 1; they were 30–70 min, depending
on the apparent magnitude of the targeted galaxies and the
observing conditions.
Within each field, target galaxies were selected by their
apparent RC magnitude (with higher priorities for brighter
No. 4] Multi-Object Spectroscopy and Cluster Masses 835
Table 1. Summary of spectroscopic observations.
Name Field (No.) peak ng texpk Obs. date# Nspec Nc
(arcmin2) (s)
SL J0217.30524 SXDS 0.041 12 (7.2) 2  1200 + 900 2005/12/23 24 1
SL J0217.60530 SXDS 0.041 11 (7.2) 3  900 2005/12/23 19 0(1)
SL J0217.90452 SXDS 0.061 12 (6.2) 2  1200 + 900 2005/12/9 21 0(1)
SL J0218.00444 SXDS 0.047 13 (6.6) 3  1200 2005/12/9 19 0(2)
SL J0219.60453 SXDS 0.060 12 (6.9) 2  1200 + 900 2005/12/23 25 1
SL J0222.80416 XMM-LSS (23) 0.060 12 (6.7) 2  1500 + 1200 2005/12/24 27 1
SL J0224.40449 XMM-LSS (02) 0.065 14 (6.0) 2  1200 + 900 2005/12/23 25 1
SL J0224.50414 XMM-LSS (12) 0.047 15 (6.7) 3  900 2004/12/9 25 1
SL J0225.30441 XMM-LSS (15) 0.086 7.1(6.0) 3  1200 2005/12/24 26 1
SL J0225.40414 XMM-LSS (22) 0.083 14 (6.9) 3  1200 2004/12/9 20 1
SL J0225.70312 XMM-LSS (01) 0.114 8.1(7.1) 3  1200 2005/12/24 25 1
SL J0228.10450 XMM-LSS (16) 0.082 13 (6.1) 3  1200 2005/12/24 25 1
SL J0850.5+4512 Lynx (08) 0.099 11 (6.2) 3  900 2005/12/17 26 1
SL J1000.7+0137 COSMOS (00) 0.092 14 (7.0) 3  900 2004/12/9 32 1
SL J1001.2+0135 COSMOS 0.081 13 (8.8) 2  900 2004/12/9 32 2
SL J1002.9+0131 COSMOS 0.047 18 (8.8) 2  1800 2004/5/16 21 0(2)
SL J1047.3+5700 Lockman Hole (05) 0.101 18 (6.9) 3  1200 2005/12/23 27 2
SL J1048.1+5730 Lockman Hole (15) 0.077 10 (7.8) 3  1200 2005/12/23 27 1
SL J1049.4+5655 Lockman Hole (06) 0.087 14 (6.9) 3  1200 2005/12/23 26 1
SL J1051.5+5646 Lockman Hole (03) 0.082 8.4(8.6) 2  1200 + 600 2005/6/2 15 0(2)
SL J1052.0+5659 Lockman Hole 0.085 7.1(8.6) 2  1200 2005/6/2 24 0(2)
SL J1052.5+5731 Lockman Hole 0.075 11 (8.6) 2  1200 + 600 2005/6/2 17 0(2)
SL J1057.5+5759 Lockman Hole (00) 0.091 23 (6.3) 3  1200 2004/12/9 32 1
SL J1135.6+3009 GD140 (00) 0.117 10 (6.7) 2  900 2004/12/17 17 1
SL J1201.70331 PG1159-035 (05) 0.070 16 (6.6) 2  1800 2004/5/16 18 1
SL J1204.40351 PG1159-035 0.079 14 (6.4) 3  900 2004/12/17 20 1
SL J1334.3+3728 13hr field (00) 0.097 24 (7.1) 2  1800 + 420 2004/5/16 31 1
SL J1335.7+3731 13hr field (01) 0.074 16 (7.1) 3  900 2005/6/1 23 1
SL J1337.7+3800 13hr field (04) 0.082 14 (7.3) 3  1200 2005/6/1 27 1
SL J1601.6+4245 GTO 2 deg2 0.107 14 (6.2) 3  1200 2005/7/31 29 2
SL J1602.8+4335 GTO 2 deg2 (00) 0.099 22 (7.0) 3  900 2005/6/1 22 1
SL J1605.4+4244 GTO 2 deg2 (09) 0.067 11 (7.8) 3  1200 2005/7/31 20 1
SL J1607.9+4338 GTO 2 deg2 0.079 11 (6.5) 3  1200 2005/8/1 21 1
SL J1634.1+5639 CM DRA (06) 0.104 5.1(6.3) 3  900 2005/6/1 16 1
SL J1639.9+5708 CM DRA (04) 0.044 12 (7.3) 3  1200 2005/8/1 24 0(2)
SL J1647.7+3455 DEEP16 (00) 0.070 9.7(5.0) 3  1200 2004/12/9 32 1
 Cluster name in the IAU convention.
 Field and catalogue number given in P1 (in parentheses, if listed).
 The peak  value.
 The number density of galaxies with 18 < mag < 23 (Vega mag) within 10 of the  peak, and the mean value over the same Suprime-Cam
FOV in parentheses. Here the luminosity is defined by the SExtractor MAG AUTO in the RC -band, except for the COSMOS field where
i 0-band data is used.
k Exposure time.
#
The date of the observation.
The number of galaxies for which a spectroscopic redshift was obtained.
 The number of clusters identified (see section 4 for our cluster identification criteria) and, when no clusters were identified, the number of
small galaxy concentrations in parentheses.
galaxies) and color information when it was available. To
obtain multicolor imaging, we searched the Subaru archive,
or took pre-imaging data for the MOS mask design with the
IC -band filter. When color information was available, we gave
a higher priority to galaxies of a color consistent with any early
type red-sequences evident in the color–magnitude diagram.
To avoid possible selection effects, we set the color range for
this increased priority relatively broad (˙0.5 mag in color from
a possible red-sequence). Finally, we visually inspected the
selected galaxies so as to avoid obviously nearby galaxies.
836 T. Hamana et al. [Vol. 61,
3.2. Data Reduction
The data were reduced with the standard FOCAS data reduction package FOCASRED,1 which operates in IDL and IRAF.
After bias subtraction and flat-fielding, each slit-let spectrum
was extracted, then wavelength- and flux-calibrated. Nightsky lines within the spectra themselves were used to define the
wavelength scale. We then carried out skysubtraction using
a 2nd order Chebyshev function. Individual spectra of 420–
1800 s exposure time were combined using the imcombine
IRAF task. The final wavelength determination accuracy was
a few angstroms.
3.3. Redshift Measurement
Redshifts were determined by centroiding multiple emission
and/or absorption lines. The statistical error of the redshift
measurement was less than 2  104. In our samples, there
are 18 galaxies whose redshifts were also obtained by Sloan
Digital Sky Survey (SDSS). The differences between SDSS
and our measurements are less than 4  103. We thus
conclude that any systematic errors in our redshift measurement are very small.
We adopted a simple spectral classification of galaxies,
following Cohen et al. (1999), and refer readers to their paper
for details. Briefly, “E” (emission) denotes a galaxy in
which emission lines dominate the spectrum; “A” (absorption)
denotes a galaxy where no emission lines are detected; and
“C ” (composite) is an intermediate case, where both emission
and absorption (usually [O II] 3727 and H + K) are seen.
Note that the [O II] 3727 emission line is blueward of our
wavelength coverage for galaxies at z . 0.2, so the discrimination between C and A was somewhat ambiguous for such
nearby galaxies.
We successfully obtained the redshifts of 15–32 galaxies
near each cluster candidate. All of the results2 (the spatial and
redshift distributions) are presented in figures 8–43.
4. Dynamical Masses
We next searched for galaxy concentrations in the redshift
data. With only information on the line-of-sight velocity
and the sky coordinates of relatively sparse samples, it was
often difficult to judge whether galaxies that appear clustered
were really gravitationally bound. We adopted a quantitative criterion for galaxy cluster identification that there be
more than five galaxies within ˙3000 km s1. As shown in
table 2, 31 galaxy concentrations satisfied this condition. Three
cluster candidates (SL J1001.2+0135, SL J1047.3+5700, and
SL J1601.6+4245) contained two galaxy concentrations. The
velocity distribution of galaxies within or near each concentration is presented in figure 1. Note that there may be additional
galaxy clusters that remain unconfirmed by our MOS observation because of our relatively sparse sampling.
We then computed the velocity dispersion of galaxies within
the 31 detected clusters. To do this, we used the ROSTAT
1 The Subaru data reduction manual is available from hhttp://www.naoj.org/
Observing/DataReduction/i.
2 Machine-readable tables are available from hhttp://th.nao.ac.jp/˜hamanatk/
SLmosz.dati.
Table 2. Summary of dynamical analysis of galaxy groups/clusters.
Name N  zc v Note
(km s1)
SL J0217.30524 8 0.4339+0:00100:0041 958+350224 PS
SL J0219.60453 11 0.3322+0:00080:0005 404+20378 PS
SL J0222.80416 6 0.3219+0:00360:0035 970+1649 PS
SL J0224.40449 10 0.4945+0:00060:0002 271+355202 PS
SL J0224.50414 12 0.2627+0:00030:0004 268+22351 FB
SL J0225.30441 7 0.2642+0:00130:0006 530+622148 PS
SL J0225.40414 8 0.1419+0:00030:0007 400+411113 PS
SL J0225.70312 15 0.1395+0:00060:0010 739+15086 CS
SL J0228.10450 13 0.2948+0:00060:0006 447+8252 NS
SL J0850.5+4512 15 0.1935+0:00070:0009 650
+115
53 CS
SL J1000.7+0137 14 0.2166+0:00020:0006 729
+526
439 CS
SL J1001.2+0135A 11 0.2205+0:00240:0026 1382
+337
160 PS
SL J1001.2+0135B 11 0.3657+0:00120:0013 931
+315
170 PS
SL J1047.3+5700A 6 0.2427+0:00130:0002 412
+205
133 PS
SL J1047.3+5700B 10 0.3045+0:00070:0016 691
+143
95 PS
SL J1048.1+5730 9 0.3173+0:00080:0008 448
+138
59 PS
SL J1049.4+5655 6 0.4210+0:00040:0004 276
+112
71 PS
SL J1057.5+5759 18 0.6011+0:00210:0021 1552
+320
204 FB
SL J1135.6+3009 15 0.2078+0:00080:0012 893
+365
145 CS
SL J1201.70331 8 0.5218+0:00420:0019 1221+425279 PS
SL J1204.40351 14 0.2609+0:00080:0005 568+37494 CS
SL J1334.3+3728 21 0.3012+0:00150:0017 1443
+199
164 NS
SL J1335.7+3731 14 0.4070+0:00400:0017 1064
+166
88 CS
SL J1337.7+3800 16 0.1798+0:00090:0004 783
+285
248 CS
SL J1601.6+4245A 7 0.2075+0:00040:0007 285
+97
36 PS
SL J1601.6+4245B 8 0.4702+0:00150:0014 965
+694
285 PS
SL J1602.8+4335 15 0.4156+0:00050:0018 675
+589
265 CS
SL J1605.4+4244 6 0.2233+0:00140:0068 1525
+382
188 PS
SL J1607.9+4338 9 0.3109+0:00040:0004 273
+66
40 PS
SL J1634.1+5639 13 0.2377+0:00170:0021 1402
+334
121 CS
SL J1647.7+3455 12 0.2592+0:00100:0008 673
+158
99 CS
 Number of spectroscopic galaxies within a velocity space of ˙3000 km s1.
 The cluster redshift.
 The velocity dispersion.
 PS indicates poor statistics in the estimation of velocity dispersion (N < 12),
NS the existence of a neighbor system, FB proximity (< 3 0) to the SuprimeCam field boundary, and CS membership of the clean sample (see subsection 6.1).
routine, an implementation of the robust bi-weight algorithm
by Beers, Flynn, and Gebhardt (1990). We recorded the
bi-weight estimators for the cluster redshift (zc) and velocity
dispersion (v), computing their errors by a bootstrap technique. The results are summarized in table 2.
No. 4] Multi-Object Spectroscopy and Cluster Masses 837
Fig. 1. Distribution of spectroscopically confirmed member galaxies (located within ˙3000 km s1) in redshift (lower scale) and velocity (upper scale)
space.
5. Weak-Lensing Analysis
5.1. Basic Equations
Let us first summarize several expressions useful for weaklensing analysis. We closely follow the notation of Bartelmann
and Schneider (2001). For more details, see an excellent review
and references therein.
The aim of weak-lensing mass reconstruction is to measure
the dimensionless surface mass density, frequently called the
gravitational-lensing density,
 =
Σ
QΣcr.zl/
; (1)
where Σ is the two-dimensional projected mass density, and
QΣcr.zl/ is the source-weighted critical density,
QΣcr.zl/= c
2
4 G
1
Dl R1
zl dzs ns.zs/Ds=DlsR1
0
dzs ns.zs/
; (2)
where ns.z/ is the redshift distribution of source galaxies,
and Dl, Ds, Dls are the angular diameter distances from the
838 T. Hamana et al. [Vol. 61,
Fig. 1. (Continued)
observer to a lens, from the observer to a source, and from the
lens to a source, respectively.
Our weak-lensing mass reconstruction used the Fourier
space relation between  and the gravitational-lensing shear,
(Kaiser & Squires 1993). Note that the directly observable
quantity is not
, but the reduced shear,
g =
1  : (3)
We adopted the weak-lensing approximation, g '
, because
most of our signal lies in the wings of clusters, where  . 0.1
(see figures 8–43).
5.2. Source Redshift Distribution
For the redshift distribution of source galaxies, we adopted
the conventional parametric model,
ns.z/=
ˇ
zΓŒ.1+ ˛/=ˇ

z z
exp
"


z z ˇ#
: (4)
No. 4] Multi-Object Spectroscopy and Cluster Masses 839
The mean redshift is given by hzsi = zΓŒ.2+ ˛/=ˇ =ΓŒ.1+
˛/=ˇ . Since our galaxy catalog [RC < 25.5 mag (Vega)]
was much deeper than any sizable galaxy catalog with spectroscopic redshifts, the redshift distribution of our galaxies is
uncertain. We took a fiducial model of hzsi = 1, ˛ = 1.5, and
ˇ = 1, and evaluated two errors arising from uncertainty in the
model parameters. The first error, due to uncertainly in hzsi,
we denote as zs. We evaluated this in the standard manner
as zs = d QΣcr=.dzsızs/, and considered an uncertainty of
ızs = 0.2. We found zs ' 0.2  zl for zl < 0.35 and
zs ' 0.07 for zl > 0.35. The second error, due to uncertainly
in ˛ and ˇ, we denote as ns. In order to evaluate this, we
considered two models of (˛, ˇ) = (2, 1.5) and (3, 2). The
relative differences in QΣcr from the fiducial model were found
to be about 10% or less at the redshifts of our clusters. We
therefore decided to take ns = 0.1  QΣcr. These errors were
properly propagated to the errors in the weak-lensing mass and
the SIS velocity dispersion (see below).
5.3. Aperture Densitometry
One convenient way to compute the mass of galaxy clusters exploits the relation between  and the tangential shear,
t (Squires & Kaiser 1996),
h ti./= 1
2
d N
d ln
; (5)
where h
ti is the azimuthal average of the tangential shear, and
N is the averaged  over a circular aperture with radius  . From
this, one can obtain an expression for the average projected
density within a radius of 1, N( < 1), subtracted from that
within an annulus 2 <  < 3, N(2 <  < 3),
.1;2;3/= N. < 1/ N.2 <  < 3/
= 2
Z 2
1
d lnh
ti+ 2
1 22=23
Z 3
2
d lnh
ti:
(6)
This is the so-called -statistic. In our computation of , we
adopted the weak-lensing approximation
' g, which should
be valid in all but the very inner part of clusters (r . 100 kpc
or  . 0:05). It is then straightforward to relate with the aperture mass,
M.< /= 2 .;2;3/ QΣcr: (7)
This is the mass within an aperture  minus an unknown mass.
While the value of this additional mass is difficult to evaluate, it presumably tends to zero if 2 is sufficiently large to
be hardly affected by the cluster mass. We took 2 = 100
and 3 = 200. The statistical error in was estimated from
the rms of 100 recalculations of , each time randomizing the
orientation of galaxy ellipticities. This error was propagated
to M (< ) in the standard manner. Results of aperture mass
profiles for our clean sample (defined in subsection 6.1) are
presented in figures 8–43.
5.4. Lens Models of Clusters
An alternative way to estimate the cluster mass from weaklensing data is to fit the shear profile to an analytical model of
a galaxy cluster. One merit of this method over the -statistic
is that it is free from the uncertainty in the additive mass (see
subsection 5.3). It is also of fundamental importance to check
that a model prediction agrees with the observed shear profile,
to provide a direct observational test on the theoretical model.
5.4.1. NFW model
Currently, the most successful model is that proposed by
Navarro, Frenk, and White (1996, 1997, NFW hereafter). We
adopted a truncated NFW model, in which the density profile
is truncated at the virial radius,
nfw.x/=
(
s
r=Œrs.1+r=rs/2
; for r < rvir
0 otherwise,
(8)
where rs and rvir are the scale radius and virial radius respectively. It is convenient to introduce the concentration parameter cnfw = rvir=rs. Bullock et al. (2001) found from N -body
simulations that the concentration parameter is related to the
halo mass as
cnfw.M;z/ =
c 1+ z

M
1014h1Mˇ
0:13
; (9)
where c ' 8 for the ΛCDM model. By definition, the mass
enclosed within a sphere of radius rvir gives the virial mass,
Mvir = 4
sr3s

log .1+ cnfw/ cnfw
1+ cnfw

: (10)
The virial mass is also defined from the spherical top-hat
collapse model as
Mvir =
4
3
ıvir.z/ N
.z/r3vir; (11)
where ıvir is the threshold over-density for spherical collapse
[see Nakamura and Suto (1997) and Henry (2000) for useful
fitting functions]. Using equations (10) and (11), one can
express
s in terms of ıvir(z) and cnfw. Introducing ıs =
s= N
1,
one finds
ıs =
ıvir
3
c3nfw
log.1+ cnfw/ cnfw=.1+ cnfw/ : (12)
Thus, the density profile of the NFW halo can be characterized by two parameters: the virial mass, Mvir, and the concentration parameter, cnfw. In the following analyses we fixed
the mass-concentration relation to the empirical result, equation (9). This is necessary because, with high S=N measurements of the tangential shear in only a limited angular range
(typically 10 <  < 40), only a poor constraint on the concentration parameter is obtained. See appendix 1 for details.
Cluster masses are also often defined by M200, which is the
mass contained within a radius of r200, where the mean mass
density of the halo is equal to 200-times the critical density at
the redshift of the cluster. In order to allow for a direct comparison with other works, we computed M200 while assuming
the NFW profile.
Analytical expressions for weak-lensing convergence and
shear behind an NFW profile are given in Takada and Jain
(2003a, 2003b). Note that those expressions differ from those
by Bartelmann (1996) and Wright and Brainerd (2000). The
latter are for a non-truncated NFW halo, and give a larger
surface mass density than ours, due to the infinite extent of the
840 T. Hamana et al. [Vol. 61,
mass. In appendix 1, we illustrate how the shear and projected
mass profile depend on the virial mass and scale radius.
In order to evaluate the cluster mass from the measured shear
profile, we performed a maximum likelihood analysis. We
computed the 2 function,
2.Mvir/=
X
i .
modeli 
obsi /2
2
;i ; (13)
where the summation runs over radial bins. The best-fit cluster
mass and the 1 confidence interval were evaluated from the
2 function in a standard manner.
5.4.2. SIS model
The singular isothermal sphere (SIS) model has frequently
been adopted as a lens model, because of its simplicity. One
convenient feature of this model is that it provides an estimate
of the velocity dispersion, which is a useful measure of the
gravitational potential and allows a direct comparison with the
observed velocity dispersion in galaxies.
The density profile of the SIS model is
SIS.r/=
2SIS
2 G
1
r2 ; (14)
where SIS is the SIS velocity dispersion. The mass within
a radius r is
MSIS.< r/=
22SIS
G
r: (15)
The shear profile is
./=
1
QΣcr
2SIS
2G
1
Dl

sis
.=10/
; (16)
where  is the angular separation from the lens center, and
we define
sis as the amplitude of the shear profile at  = 10.
We evaluated
sis by fitting the measured shear over the range
10 <  < 40. We used this range for three reasons: (1) for
most clusters, the shear signal was measured with a good S=N
throughout this interval, but degraded outside it. (2) at larger
radii, nearby structures (either physically related or unrelated
to the main cluster) could contribute to the shear signal. (3) at
smaller radii, the reduced shear, g, and the dilution effect due to
the cluster member galaxies could have a non-negligible effect
on the shear signal (Broadhurst et al. 2004). The measured
sis
values are summarised in table 3.
By combining equations for the definition of virial mass (11)
and the mass in an SIS (15), one finds
MSIS.< rvir/ =

6
6SIS
G3ıvir.zl/ N
.z/
1=2
' 6:6 1014.h1Mˇ/C 
 SIS
1000kms1
3
;
(17)
where
C 

1
Ωm
1
.1+ zl/3
200
ıvir.zl/
1=2
: (18)
In the notation of Bryan and Norman (1998), C is written as
C = (200=Δc)1=2E(z)1, where Δc is the critical overdensity
of the spherical collapse, and
E.z/=
H.z/
H0
=

Ωm.1+ z/3+ΩΛ
1=2
(19)
for a Λ-flat cosmology. A reasonably accurate fitting function
of C for a Λ-cosmology (Ωm= 0:3, ΩΛ= 0:7) over the redshift
range 0 < zl < 0.8 is found to be C ' 1.44=(1 + zl)1:08, and in
the case of Einstein–de Sitter cosmology (Ωm = 1, ΩΛ = 0), it
reduces to C =
p 200=178=(1 + zl)1:5 ' 1.06=(1 + zl)1:5.
5.5. Results
The results of our weak-lensing analyses are compiled in
table 3. A detailed discussion of each system is presented in
appendix 2, including a comparison of the weak-lensing and
optical properties, and the results of MOS observations.
6. Cluster Scaling Relations
6.1. Clean Sample of Clusters
We now examine statistical relations between the dynamical mass estimator (v) and the weak-lensing mass estimators (SIS and MNFW). In this paper, we restrict our analysis
to clusters with the very cleanest measurements; a statistical
treatment of the entire sample, taking into account all selection
effects, will follow in A. Green et al. (2009, in preparation).
We define a clean (sub)-sample of clusters (“CS” in table 2)
as those whose velocity dispersion was evaluated from at least
12 spectroscopic member galaxies, and whose weak-lensing
mass estimation could not have been affected by proximity to
either a neighboring system (“NS”) or a field boundary (“FB”).
Ten clusters satisfy these criteria.
We additionally included two clusters in our survey
area, whose velocity dispersions had been measured by
other authors. Observations by Willis et al. (2005) of
SL J0221.70345 (z = 0.43, v = 821+9274 km s1 from
39 galaxy redshifts) and SL J0228.40425 (z = 0.43,
v = 694+20491 km s
1 from 13 redshifts) satisfy the same stringent selection criteria as mentioned above. We examined the
weak-lensing properties of those clusters in the same manner
as described in section 5; the results are described fully
in appendix 3.
6.2. Velocity Dispersions
The relation between the cluster galaxies’ velocity dispersion (v) and the velocity dispersion parameter of the best-fit
SIS model (SIS) has been measured for various clusters, in the
context of their dynamical status (Irgens et al. 2002; Hoekstra
et al. 2002; Hoekstra 2007; Milvang-Jensen et al. 2008).
Results for our sample are presented in figure 2 (with a comparison to optically selected clusters) and figure 3 (with a comparison to X-ray selected clusters). The optically selected catalog
of Milvang-Jensen et al. (2008) contains less massive clusters,
at higher redshifts (Nz = 0.58) than our sample ( Nz = 0.28).
Conversely, the wide, but shallow, Einstein Medium Sensitivity
Survey (EMSS: Gioia et al. 1990) and X-ray Brightest Abell
Cluster Survey (XBACS: Ebeling et al. 1996) include the most
massive clusters at more modest redshifts. The mean redshift
of the catalog by Cypriano et al. (2004) is Nz = 0.13 and that by
Hoekstra (2007) is Nz = 0.31.
Only a few outliers in figures 2 and 3 are inconsistent with v = SIS. The main outlier from our sample,
No. 4] Multi-Object Spectroscopy and Cluster Masses 841
Table 3. Summary of weak-lensing analyses.
Name zc
sis SIS MNFW M200 M500k rvir#
(km s1) (1014h1Mˇ) (comoving Mpch1)
SL J0217.30524 0.4339 0.048 723+439451 1.95+1:440:95 1.71+1:260:83 1.20+0:880:58 1.3
SL J0219.60453 0.3322 0.072 789+383409 2.75+1:251:02 2.37+1:080:88 1.67+0:760:62 1.5
SL J0222.80416 0.3219 0.058 695+388408 2.09+0:090:40 1.81+0:080:35 1.28+0:060:24 1.3
SL J0224.40449 0.4945 0.071 946+437477 4.79+0:831:42 4.19+0:731:24 2.85+0:490:84 1.8
SL J0224.50414 0.2627 0.066 679+310340 1.78+0:730:59 1.53+0:630:51 1.09+0:450:36 1.3
SL J0225.30441 0.2642 0.087 800+405425 3.02+1:321:04 2.58+1:130:89 1.82+0:800:63 1.5
SL J0225.40414 0.1415 0.059 573+273293 1.86+0:450:45 1.57+0:380:38 1.14+0:270:27 1.2
SL J0225.70312 0.1395 0.112 790+356379 2.95+0:830:79 2.48+0:700:66 1.77+0:500:47 1.4
SL J0228.10450 0.2948 0.102 898+523527 4.37+1:671:86 3.73+1:431:59 2.60+0:991:11 1.7
SL J0850.5+4512 0.1935 0.068 650+361371 1.95
+1:17
0:77 1.66
+0:99
0:65 1.19
+0:71
0:47 1.3
SL J1000.7+0137 0.2166 0.091 775+332347 2.69
+1:03
0:32 2.29
+0:88
0:27 1.63
+0:62
0:19 1.4
SL J1048.1+5730 0.3173 0.069 758+441435 2.51
+1:13
1:03 2.16
+0:97
0:89 1.53
+0:69
0:63 1.4
SL J1049.4+5655 0.4210 0.091 983+502525 3.72
+3:28
0:70 3.23
+2:85
0:61 2.23
+1:97
0:42 1.6
SL J1057.5+5759 0.6011 0.087 1194+523544 8.71
+2:58
1:56 7.65
+2:27
1:37 5.05
+1:50
0:90 2.2
SL J1135.6+3009 0.2078 0.100 804+344373 4.17
+0:69
1:06 3.52
+0:58
0:90 2.48
+0:41
0:63 1.6
SL J1201.70331 0.5219 0.087 1085+527530 7.24+2:972:46 6.32+2:592:15 4.23+1:741:44 2.1
SL J1204.40351 0.2609 0.098 844+427460 3.55+1:641:28 3.03+1:401:09 2.13+0:980:77 1.6
SL J1334.3+3728 0.3006 0.123 991+454482 4.79
+1:67
1:35 4.09
+1:43
1:15 2.84
+0:99
0:80 1.8
SL J1335.7+3731 0.4070 0.093 978+452493 6.61
+2:18
1:92 5.70
+1:88
1:66 3.88
+1:28
1:13 2.0
SL J1337.7+3800 0.1798 0.071 655+313346 1.62
+0:69
0:42 1.38
+0:59
0:36 1.00
+0:42
0:26 1.2
SL J1602.8+4335 0.4155 0.087 954+418418 4.79
+1:20
1:05 4.15
+1:04
0:91 2.85
+0:71
0:62 1.8
SL J1605.4+4244 0.2233 0.066 665+319342 1.82
+0:79
0:63 1.55
+0:67
0:54 1.12
+0:48
0:39 1.2
SL J1607.9+4338 0.3109 0.063 718+418447 1.74
+0:04
0:74 1.50
+0:03
0:64 1.07
+0:02
0:46 1.3
SL J1634.1+5639 0.2377 0.076 724+402404 2.09
+1:01
0:87 1.79
+0:86
0:74 1.28
+0:62
0:53 1.3
SL J1647.7+3455 0.2592 0.079 759+551560 4.90
+2:48
1:89 4.16
+2:11
1:60 2.90
+1:47
1:12 1.8
 The amplitude of tangential shear profile at 10, when fitted with an SIS model (see sub-subsection 5.4.2).
 The SIS velocity dispersion parameter.
 The virial mass estimated by fitting the radial shear profile with an NFW model.
 The M200 estimated adopting the NFW model.k The M500 estimated adopting the NFW model.
#
The virial radius computed from the NFW mass using the relation equation (11).
SL J1634.1+5639, has v  2SIS  1400 km s1, but the
velocity distribution of the spectroscopic member galaxies is
also strongly skewed (see figure 1). The reason for this is
currently not clear, but is discussed further in appendix 2.34.
Overall, despite variations in the samples’ range of cluster
masses and redshifts, the scatter in the v–SIS relation is
remarkably similar for all four catalogs. Thus, as far as the relation between v and SIS is concerned, no strong selection bias
is identified between the various cluster detection techniques.
It is worth noting that the density (and shear) profile of a real
cluster is typically not a single power law. The best-fit SIS
model, and the value of SIS, may therefore depend on the
specific fitting method, and the range over which data are fit.
Our above finding, that v ' SIS, may therefore be somewhat
method-dependent. A corollary of this issue is that it might
also be possible to minimize scatter in the v–SIS relation by
optimizing the fitting method used to obtain SIS. We have not
attempted to do this.
6.3. Velocity Dispersion versus Mass
We next examine the relation between the velocity dispersion of a cluster’s galaxies (v) and its weak-lensing mass.
Since the NFW virial masses and aperture masses agree for all
12 cleanly-measured clusters, we adopt Mvir = MNFW as our
sole weak-lensing mass estimate. Figure 4 shows our results,
and compares them to measurements of X-ray selected clusters by Hoekstra (2007). It is important to note that weaklensing measurements do not depend upon the dynamical status
of the clusters. Motivated by an SIS model prediction [equations (17) and (18)], we adopt a functional form of the scaling
842 T. Hamana et al. [Vol. 61,
Fig. 2. Comparison between the velocity dispersion measured from
galaxy redshifts (v ) and the velocity dispersion parameter of the
weak-lensing SIS model (SIS). Filled circles show our clean sample
(see text). Filled triangles show two clusters whose v was measured
by Willis et al. (2005) (see appendix 3). Open squares show optically
selected clusters from Milvang-Jensen al. (2008; note that clusters
with structures possibly affecting the velocity dispersion estimate are
excluded, see section 8 of their paper for details).
relation, Mvir / Mvp=(1 + zc)1:08. For pure SIS clusters
in a ΛCDM cosmology, the parameters would be p = 3 and
M = 9.5  1014 h1Mˇ [see equation (17) and below]. This
simplistic toy model is shown as a dotted line, and already
provides a reasonable approximation to the observed Mvir–v
data. In order to find the best-fit model parameters (M and p)
we performed a maximum likelihood analysis while taking into
account not only the errors in Mvir, but also in v . To do
this, we made an assumption that the error in v follows the
Gaussian distribution, and calculated the likelihood function
L.M;p/=
X
i 1p 2 v;i
Z
dvexp
"
 . Nv;i  v/
2
22v;i
#
Li .M;p;v/; (20)
where the summation runs over the cluster sample, Nv;i and
v;i denote the measured velocity dispersion and its 1 error,
respectively, and
Li .M;p;v/ / exp
(
 ŒMi M
model.M;p;v/ 2
22M;i
)
;
(21)
where Mi and M;i denote the measured cluster mass and its
1  error, respectively. To obtain a constraint on a parameter (M or p), we marginalized the likelihood function over
the other parameter. The best-fit model (excluding the outlier
SL J1634.1+5639) is found to be Mvir.1+zc/1:08 = (15.1+1:71:2)
Fig. 3. Same as figure 2, but compared with X-ray selected clusters
from Cypriano et al. (2004; open squares) and Hoekstra (2007; open
triangles).
 1014  (v=1000 km s1)3:0+0:50:4 h1Mˇ. The best-fit powerindex is thus consistent with the SIS prediction of 3, but the
normalization is higher.
A more sophisticated prediction of the cluster M –v
relation, using N -body simulations, was obtained by Evrard
et al. (2008). They found M200E(z) = 9.358  1014
 (v=1000 km s1)2:975h1Mˇ, and argued that it is insensitive to cosmological parameters in a variety of CDM models.
To aid in a comparison, we estimated M200 from our measurements of MNFW by assuming that every cluster had an NFW
density profile. These masses are listed in table 3 and are
shown in figure 5, together with those of Hoekstra (2007).
We performed the same maximum likelihood analysis as
mentioned above, and the best-fit model (excluding the outlier
SL J1634.1+5639) was found to be M200E(z) = (11.0+1:21:0)
 1014  (v=1000 km s1)3:0+0:40:4 h1Mˇ. In figure 5, the bestfit power-law model is plotted as a dashed line and Evrard
et al.’s prediction is overlaid as a dotted line.
An interpretation of this apparent consistency is not trivial,
because Evrard et al. evaluated v from simulated dark matter
particles instead of galaxies. However, since both galaxies
and cold dark-matter particles may be safely regarded as being
collisionless particles in the cluster potential, it is reasonable to
assume that they have approximately the same velocity dispersion outside a central region in which the effect of dynamical friction is important (Okamoto & Habe 1999). If this
is the case, our findings indicate that the dynamical structure of galaxy clusters is indeed consistent with that expected
in the standard CDM paradigm of structure formation. It
will be interesting to compare our observations with simulations of cluster evolution that also incorporate mechanisms for
galaxy formation.
No. 4] Multi-Object Spectroscopy and Cluster Masses 843
Fig. 4. Comparison between the velocity dispersion measured from
galaxy redshifts (v ) and the virial mass measured from weak lensing
(MNFW). In order to account for redshift evolution in the relation, cluster masses were multiplied by a factor of (1 + zc)1:08 [as
motivated by equations (17) and (18)]. The filled circles show clusters in this study; the two filled triangles show clusters in this study
whose v were measured by Willis et al. (2005) (see appendix 3),
and the open circles show the sample of Hoekstra (2007). The dotted
line shows the prediction of a pure SIS model, Mvir  (1 + zc)1:08
= 9.5  1014  .v=1000 km s1)3 h1 Mˇ. The dashed line shows
the best-fit empirical relation, Mvir  (1 + zc)1:08 = 15.1  1014
 (v=1000 km s1)3:0 h1Mˇ.
7. Summary and Discussions
We have presented the results of a multi-object spectroscopic campaign to target 36 cluster candidates located
by the Subaru weak-lensing survey (Miyazaki et al. 2007).
We obtained the redshifts of 15–32 galaxies within a few
arcminutes of each cluster candidate. Our primary goals
were to search for a spatial concentration of galaxies as an
optical counterpart of each weak-lensing density peak, and to
determine the cluster redshifts. We found 31 galaxy concentrations containing more than five spectroscopic galaxies
within a velocity of ˙3000 km s1, and determined their
redshifts. These included 25 detections of isolated clusters,
and three systems (SL J1000.7.3+0137, SL J1047.3+5700,
SL J1601.6+4245) in which two galaxy clusters at different
redshifts are projected along the same line-of-sight. This demonstrates that spectroscopic follow-up of weak-lensing cluster
candidates is a reliable way not only to identify their optical
counterparts, but also to distinguish superposed systems.
We have therefore identified secure optical counterparts of
the weak-lensing signal in 28 out of 36 targets. In 6 of the
8 unconfirmed cluster candidates, we found multiple small
galaxy concentrations at different redshifts (each containing at
least 3 spectroscopic galaxies). This suggests that the weaklensing signal in those cases may arise from the projection
of small clusters along the same line-of-sight. However, it is
Fig. 5. Same as figure 4, but with M200 instead of virial mass MNFW,
for a comparison with numerical simulations. The dotted line shows
a prediction from N -body simulations M200E(z) = 9.358  1014
 (v=1000 km s1)2:975h1 Mˇ (Evrard et al. 2008). The dashed
line shows the best-fit empirical relation M200E(z) = 11.0  1014
 (v=1000 km s1)3:0 h1Mˇ.
also possible that a real, massive cluster is responsible for the
weak-lensing density peak, but was missed by our relatively
sparse MOS observations. This is also the case for the final
two unconfirmed candidates, where only a single small galaxy
concentration was identified. In order to obtain a firm confirmation of the optical counterpart of such unconfirmed candidates, denser spectroscopic observations would be required.
We measured the mass of single cluster systems with known
redshifts using two weak-lensing methods: aperture densitometry and by fitting the shear profile to an NFW model. In most
cases, the two mass estimators agree well: providing observational support for the NFW model. In the few clusters where
the mass estimators did not agree, the weak-lensing  signal
clearly deviated from spherical symmetry. This could account
for the disagreement. It was also found, by eye, that the aperture mass profile of some clusters did not flatten even at a large
radius of   100. This can be accounted for by the mass contribution from surrounding structures. We found some candidates
of super-cluster systems, whose weak-lensing mass reconstructions show evidence of filamentary structure connecting the
main cluster to surrounding systems.
We investigated statistical relations between clusters’
weak-lensing properties (SIS and Mvir or M200) and the
velocity dispersion of their member galaxies (v), comparing
our results to optically and X-ray selected cluster samples
from the literature. Although our clean sample contained
only 12 clusters, we found our clusters to be consistent with
v = SIS, with a scatter as large as that of optically and
X-ray selected samples. Therefore, as far as the relation
between v and SIS is concerned, no strong bias between the
cluster selection techniques was identified. We also derived
844 T. Hamana et al. [Vol. 61,
an empirical relation between the cluster virial mass and the
galaxy velocity dispersion: Mvir(1 + zc)1:08 = (15.1+1:71:2)
 1014  (v=1000 km s1)3:0+0:50:4 h1Mˇ. The derived Mvir–SIS
relation is similar to theoretical expectations from the SIS
model, equation (17). It is important to note that, unlike
the SIS model assumption, real cluster shear profiles (and
density profiles) are not single power-laws, so this result may
depend upon details of the fitting technique. For a comparison
with numerical simulations, we also derived the M200–v
relation, and found M200E(z) = .11:0+1:21:0/  1014 
(v=1000 km s1)3:0
+0:4
0:4 h1 Mˇ. This is in good agreement
with predictions by Evrard et al. (2008), demonstrating that
the dynamical structure of galaxy clusters is similar to that
expected in the standard CDM paradigm of structure formation.
We are very grateful to Subaru astronomers: Y. Ohyama,
K. Aoki, and T. Hattori for their dedicated support of the
FOCAS observing. Numerical computations presented in
this paper were carried out on computer system at CfCA
(Center for Computational Astrophysics) of the National
Astronomical Observatory Japan. Data reduction and analysis
were in part carried out on a general common-use computer
system at ADAC (Astronomical Data Analysis Center) of the
National Astronomical Observatory of Japan. This research
was supported in part by the Grants-in-Aid from MonbuKagakusho and Japan Society of Promotion of Science: Project
number 15340065 (TH&SM) and 21740202 (TH).
Appendix 1. Weak-Lensing Properties of the Truncated
NFW model
Here, we present a reduced shear profile of a truncated NFW
model (sub-subsection 5.4.1), to illustrate the dependence on
the model parameters. In figures 6 and 7, we plot the reduced
shear (upper panel) and the projected mass within an aperture
of  (lower panel) for various values of the model parameters,
Mvir and cnfw. Since we adopt the truncated model [equation (8)],  becomes zero outside the virial radius, so the aperture mass flattens and the reduced shear (where g =
) scales
as / 2.
As shown in figure 6, changes in Mvir alter the amplitude of
the shear profile, but leave the overall shape almost unchanged.
The slight discontinuity at the virial radius in the reduced shear
profile is a numerical artifact caused by a discontinuity there in
. Figure 7 illustrates that changes in the concentration parameter alter the slope of the shear profile, but not the amplitude
at the virial radius. The higher is the concentration, the steeper
does the slope become, as expected.
In principle, a measurement of the weak-lensing shear
profile over a broad angular range therefore allows simultaneous constraints on both the cluster mass and the concentration parameter (i.e., the mass would be determined mainly by
the shear amplitude near the virial radius, and the concentration parameter by the slope at  < vir). However, in our case,
the angular range over which tangential shear is measured
with a good S=N is rather narrow (typically 10 <  < 40). This
especially prevents us from obtaining a tight constraint on the
Fig. 6. The top panel shows the reduced shear profile of a truncated
NFW model, for different virial masses (5 , 2 , 1 , and 0.5
 1014 h1 Mˇ from upper to lower). The bottom panel shows the
projected mass within an aperture  . In all cases, the concentration
parameter is cnfw = 5, and the lens and source redshifts are zl = 0.4
and zs = 1 (a single source plane approximation is employed).
Fig. 7. Same as figure 6, but for different concentration parameters
(cnfw = 15, 10, 5, and 3, from upper to lower). In all cases, the virial
mass is Mvir = 5  1014 h1Mˇ.
concentration parameter, which requires measurements over
a broad angular range. Instead of treating both the virial mass
and the concentration parameter as free parameters, we therefore
decided to adopt an empirically observed relation between the
concentration parameter and the mass, equation (9).
No. 4] Multi-Object Spectroscopy and Cluster Masses 845
Fig. 8. SL J0217.30524. Left panel: RC -band image with the weak-lensing density overplotted as contours (starting from  = 0.04 and in increments
of Δ = 0.01). Galaxies with measured redshifts are marked with circles, and labeled with their target IDs and redshifts (in parentheses). In the electric
version, red circles represent absorption galaxies, blue circles represent emission, and green circles represent composite galaxies (see subsection 3.3 and
Cohen et al. 1999 for details). Right panels: Cone diagrams showing the 3D locations of galaxies. The horizontal axis shows the radial comoving
distance. On the vertical axes, x and y correspond to the RA and Dec directions respectively.
Fig. 9. Same as figure 8, but for SL J0217.60530.
846 T. Hamana et al. [Vol. 61,
Fig. 10. Same as figure 8, but for SL J0217.90452.
Fig. 11. Same as figure 8, but for SL J0218.00444.
No. 4] Multi-Object Spectroscopy and Cluster Masses 847
Fig. 12. Same as figure 8, but for SL J0219.60453.
Fig. 13. Same as figure 8, but for SL J0222.80416.
848 T. Hamana et al. [Vol. 61,
Fig. 14. Same as figure 8, but for SL J0224.40449.
Fig. 15. Same as figure 8, but for SL J0224.50414.
No. 4] Multi-Object Spectroscopy and Cluster Masses 849
Fig. 16. Same as figure 8, but for SL J0225.30441.
Fig. 17. Same as figure 8, but for SL J0225.40414.
850 T. Hamana et al. [Vol. 61,
Fig. 18. Sl J0225.70312. Top panels are the same as figure 8. Bottom-left panel: The gray scale shows the weak-lensing  map (over an extended
area), with gray (red in online edition) contours starting from  = 0.04 and in increments of Δ = 0.02. White contours show the smoothed number
density of galaxies (18 < mag < 23), starting from ng = 10 arcmin2 and in increments of 2 arcmin2 . Bottom-right panels: (Upper plot) the measured
weak-lensing tangential shear profile gt =
t=.1 /, with the best-fit SIS model (dashed line) and NFW model (solid line, plotted up to the virial
radius). (Lower plot) the aperture mass profile M (<  ), computed from the tangential shear. The diamond shows the virial mass and virial radius both
derived from the best-fit NFW model.
Appendix 2. Properties of Individual Targets
Here, we describe each target’s weak-lensing and optical
properties, and discuss the full results of the MOS
observations. Quantitative summaries of these data can be
found in tables 1, 2, and 3.
Figures 8–43 also present the data in a uniform format for an
easy visual comparison. Each figure is laid out as follows. The
left or top-left panel shows an optical image of the cluster core,
overlaid with contours reproducing the weak-lensing density.
The contours start from  = 0.04 and increase in increments
of Δ = 0.01. Galaxies with successfully measured redshifts
are marked with circles, their target IDs, and their redshifts (in
parentheses). The colors in the electric version correspond to
the galaxies’ observed spectral types. The right or top-right
panels show the positions of those galaxies in cone diagrams.
The horizontal axis corresponds to radial comoving distance,
and the vertical axes to sky directions, with x and y standing
for RA and Dec respectively. For clusters defined as a clean
sample (see subsection 6.1), two more plots are presented as
follows. The bottom-left panel reproduces the weak-lensing
density map on a larger scale to show any nearby structure.
No. 4] Multi-Object Spectroscopy and Cluster Masses 851
Fig. 19. Same as figure 8, but for SL J0228.10450.
Overlaid on the gray scale map are gray (red in online edition)
contours, starting from  = 0.04 and increasing in increments
of Δ = 0.02. White contours show instead the smoothed
number density of galaxies with 18 < RC < 23 (18 < i 0
< 23 for the COSMOS field), starting from ng = 1002 and
in increments of Δng = 202. The bottom-right panels show
the weak-lensing tangential shear profile (upper plot) and aperture mass profile (lower plot). Black dots with error bars show
measured data. The dashed line shows the best-fit SIS model,
and the solid line shows the best-fit NFW model, plotted up to
the virial radius. The diamond shows the virial mass of this
best-fit NFW model.
A.2.1. SL J0217.30524
This is not listed in the P1 catalogue because the peak 
S=N does not exceed P1’s threshold. As observed in figure 8,
the weak-lensing  peak is well correlated with a galaxy overdensity. In the redshift data, there exists one galaxy concentration that passes our cluster criterion, with 8 members at
zc = 0.43. There is also a small concentration at z = 0.31.
The velocity dispersion of the galaxy cluster is consistent with
the SIS velocity dispersion. The NFW cluster mass agrees with
the aperture mass at the corresponding virial radius. However,
the aperture mass keeps increasing even outside of the expected
virial radius. This is likely due to the mass associated with two
 over-densities located a few arcminutes to the west and the
north–east of the cluster center.
A.2.2. SL J0217.60530
This is not listed in the P1 catalogue because the peak  S=N
is below the threshold. The weak-lensing density peak appears
to be isolated, and is correlated with a galaxy over-density. In
the redshift data, there is no galaxy concentration passing our
criterion of five galaxies within a velocity of ˙3000 km s1,
but there is a small group of three galaxies at z = 0.43. With
only this information, it is currently not clear whether the
weak-lensing shear signal comes solely from the halo of the
small galaxy concentration, or whether there are other mass
concentrations along the same line-of-sight. Since no galaxy
concentration that passes our cluster criterion was found, no
weak-lensing mass estimation or radial profiles are displayed.
A.2.3. SL J0217.90452
This is not listed in the P1 catalogue because the peak 
S=N is below the threshold. The weak-lensing  map shows
a bimodal feature, with a second peak located at 20 to the west
of the main peak. There is a galaxy over-density near the main
peak. No galaxy concentration in the redshift data is sufficiently rich to qualify as a cluster under our criterion, but there
is a small group of four galaxies at z = 0.19. With only the
current information, it is not clear whether the weak-lensing
shear signal comes from the halo of the small galaxy concentration alone, or from additional concentrations along the same
or an adjacent line-of-sight. Since no galaxy concentration
passing our cluster criterion was found, we did not make the
weak-lensing mass estimation.
A.2.4. SL J0218.00444
This is not listed in the P1 catalogue because the peak  S=N
does not exceed P1’s threshold. The weak-lensing density
distribution is elongated along the south–east to north–west
direction. There is a galaxy over-density overlapping with the
weak-lensing  peak, but elongated perpendicular to this. In
the redshift data, no galaxy concentration passes our cluster
criterion, but there are two small concentrations at z = 0.37
(five galaxies) and at z = 0.31 (four galaxies). Since no
852 T. Hamana et al. [Vol. 61,
Fig. 20. Same as figure 18, but for SL J0850.5+4512.
dominant galaxy concentration was found, we have not estimated a weak-lensing mass. Note that in the vicinity of this
target there is a known galaxy cluster, identified in optical–near
infrared imaging and with an estimated photometric redshift
of zp = 0.71˙ 0.03 (van Breukelen et al. 2006, their ID 5,
RA = 34:h50, Dec = 4:ı72). This cluster is located 20 to the
north–west and is within the elongated over-density region. It
is therefore likely that  excess consists of the chance projection of several halos located at different redshifts along adjacent lines-of-sight.
A.2.5. SL J0219.60453
This is not listed in the P1 catalogue because the peak  S=N
is below the threshold. As shown in figure 12, the weak-lensing
 peak is elongated in the east–west direction, with a smaller,
secondary peak 30 east of the cluster center. An extended
galaxy over-density overlaps with both  peaks. In the redshift
data, a galaxy concentration with 11 members at zc = 0.33
passes our cluster criterion. There is also a small concentration
of three galaxies at z = 0.3. The measured velocity dispersion
of the galaxy cluster is smaller than the SIS velocity dispersion,
but within a 1 error. The NFW cluster mass is slightly larger
than the aperture mass at the corresponding virial radius. This
could be due to a contaminated measurement of the tangential shear. Looking more closely at the aperture mass profile
shows a flattening at  = 10–20, followed by a second subsequent increase at larger radii up to 40. Mass associated with
the secondary peak may account for that second rise.
No. 4] Multi-Object Spectroscopy and Cluster Masses 853
Fig. 21. Same as figure 18, but for SL J1000.7+0137.
A.2.6. SL J0222.80416
The weak-lensing density distribution looks relaxed, and
correlates well with the galaxy over-density. In the redshift
data, one concentration of six galaxies at zc = 0.32 passes
our cluster criterion. In addition, there are small concentrations at z = 0.435 (4 galaxies) and at z = 0.227 (3 galaxies).
The velocity dispersion of the galaxy cluster is larger than the
SIS velocity dispersion, though they are within a 1  error.
The NFW cluster mass agrees with the aperture mass at the
corresponding virial radius. However, the aperture mass keeps
increasing at outer radii of the expected virial radius. This is
due to the mass associated with the structure located at about
40–80 to the east–south from the cluster center where no associated galaxy excess is observed.
A.2.7. SL J0224.40449
This is a galaxy cluster previously identified by weaklensing shear (Cl-02 of Gavazzi & Soucail 2007). The redshift
was photometrically estimated to be zp = 0.497 (Gavazzi &
Soucail 2007) but has not been spectroscopically obtained.
The weak-lensing density distribution is elongated in the
north–south direction. The elongation of the galaxy distribution is less pronounced, but in the same direction. In the
redshift data, there is one strong concentration of ten, mainly
absorption galaxies at zc = 0.49. There may also be a small
concentration at z = 0.32. The velocity dispersion of the
galaxy cluster is found to be smaller than the SIS velocity
dispersion. The reason for this discrepancy is not clear, but
one possibility is the small number of member galaxies used
854 T. Hamana et al. [Vol. 61,
Fig. 22. Same as figure 8, but for SL J1001.2+0135.
Fig. 23. Same as figure 8, but for SL J1002.9+0131.
No. 4] Multi-Object Spectroscopy and Cluster Masses 855
Fig. 24. Same as figure 8, but for SL J1047.3+5700.
Fig. 25. Same as figure 8, but for SL J1048.1+5730.
856 T. Hamana et al. [Vol. 61,
Fig. 26. Same as figure 8, but for SL J1049.4+5655.
Fig. 27. Same as figure 8, but for SL J1051.5+5646.
No. 4] Multi-Object Spectroscopy and Cluster Masses 857
Fig. 28. Same as figure 8, but for SL J1052.0+5659.
Fig. 29. Same as figure 8, but for SL J1052.5+5731.
858 T. Hamana et al. [Vol. 61,
Fig. 30. Same as figure 8, but for SL J1057.5+5759.
for estimating the velocity dispersion. The NFW cluster
mass agrees with the aperture mass at the corresponding virial
radius. The aperture mass flattens at scales larger than the virial
radius, and no substructure is observed in the weak-lensing
density map.
A.2.8. SL J0224.50414
This is a known cluster, XLSS J022433.8041405 (also
XLSSC-044), first identified via X-ray emission by Pierre
et al. (2006, the spectroscopic redshift they obtained is
z = 0.26). The weak-lensing density distribution shows irregular morphology, with a second peak about 30 east of the first
peak. There is an apparent galaxy over-density that largely
overlaps with the first peak. In the redshift data, there is one
strong galaxy concentration at zc = 0.26 with 12 members,
dominated by absorption galaxies. In addition, there is a small
concentration at z = 0.316 (4 galaxies). The velocity dispersion of the galaxy cluster is smaller than the SIS velocity
dispersion, though they are within a 1  error. The NFW
cluster mass agrees with the aperture mass at the corresponding
virial radius. The aperture mass profile increases erratically at
larger radii, probably reflecting contributions to the signal from
nearby structures. The weak-lensing mass of this cluster was
also measured by Bergé et al. (2008) and found to be M200 =
7.2+2:31:7  1013h1Mˇ, which is smaller than our measurement
(M200 = 1.53+0:630:51  1014h1Mˇ). The reason for this discrepancy is currently unclear.
A.2.9. SL J0225.30441
This is a known cluster, XLSS J022524.5044042 (also
XLSSC-025), first identified via X-ray emission by Pierre et al.
(2006, the spectroscopic redshift they obtained is z = 0.26).
This was also identified by weak-lensing shear (Cl-05 of
Gavazzi & Soucail 2007). The weak-lensing density distribution is slightly elongated in the north–west to south–east
direction, with an additional small extension to the north–east.
Interestingly, the center of the galaxy over-density is at the
position of the small extension, and a possible cD galaxy is also
found there (target ID 1). At zc = 0.26, there is a concentration
of seven (mainly absorption) galaxies, including the possible
cD galaxy. In addition, there are nearby concentrations of three
galaxies at both z = 0.21 and z = 0.46. The velocity dispersion
of the galaxy cluster is consistent with the best-fit SIS velocity
dispersion parameter, and the NFW cluster mass is consistent
with the aperture mass at the virial radius.
A.2.10. SL J0225.40414
This is a known cluster, XLSS J022530.6041419 (also
XLSSC-041), first identified via X-ray emission by Pierre et al.
(2006, the spectroscopic redshift they obtained is z = 0.14).
This was also identified by weak-lensing shear (Cl-14 of
Gavazzi & Soucail 2007). The weak-lensing density distribution is slightly elongated in the north–south direction. Two
additional peaks lie in the same direction: one 40 to the north
and another 70 to the south. There is a filamentary-like structure
connecting the three peaks. The northern clump was previously
identified by weak-lensing shear (Cl-04 of Gavazzi & Soucail
2007), but the spectroscopic redshift has not been obtained.
There are galaxy over-densities corresponding to
(but slightly offset from) all three weak-lensing peaks.
Interestingly, the galaxy over-density associated with the
central peak is elongated in a perpendicular direction to the
weak-lensing density. In the redshift data, there is a concentration of eight absorption galaxies at zc = 0.14. The velocity
dispersion of the galaxy cluster is consistent with the SIS
velocity dispersion. The tangential shear profile is not well
No. 4] Multi-Object Spectroscopy and Cluster Masses 859
Fig. 31. Same as figure 18, but for SL J1135.6+3009.
fit by an NFW model, with an excess at large radii due to the
surrounding structures observed in the weak-lensing density
map. Similarly, the aperture mass profile does not flatten at
scales even as large as  = 100. The mass contribution from
the surrounding structures may also account for this. It was
therefore difficult to define the boundary of this galaxy cluster,
in which to calculate the total mass. The weak-lensing mass
of this cluster was also measured by Bergé et al. (2008), and
found to be M200 = 4.9+1:61:2  1013 h1Mˇ, which is smaller
than our measurement (M200 = 1.57+0:380:38  1014h1Mˇ). The
reason of this discrepancy is currently unclear.
A.2.11. SL J0225.70312
This is a known cluster, XLSS J022540.6031121 (also
XLSSC-050), first identified via X-ray emission by Pacaud
et al. (2007, the spectroscopic redshift they obtained is
z = 0.14). This cluster has one of the strongest weak-lensing
signals (peak = 0.114) of our catalog. The weak-lensing
density distribution appears to be relaxed, except for an extension to the north–east. The galaxy distribution correlates with
the weak-lensing density but is off-centered towards the extension. In the redshift data, there is an apparent galaxy concentration at zc = 0.14, with 15 members dominated by absorption galaxies. The velocity dispersion of the galaxy cluster
is in good agreement with the SIS velocity dispersion. The
NFW cluster mass is consistent with the aperture mass at
the largest radius.
860 T. Hamana et al. [Vol. 61,
Fig. 32. Same as figure 8, but for SL J1201.70331.
A.2.12. SL J0228.10450
This is a known cluster, XLSS J022803.4045103 (also
XLSSC-027), first identified via X-ray emission by Pacaud
et al. (2007, the spectroscopic redshift they obtained is
z = 0.29). This was also identified by weak-lensing shear
(Cl-14 of Gavazzi & Soucail 2007). We found a strong concentration of 13 absorption galaxies at zc = 0.29. The weaklensing density distribution shows an irregular morphology.
The low-level feature to the south may be edge effects due to
a bright star mask. However, a second cluster, 70 (16.5 h1
comoving Mpc) to the west and at the same redshift is real.
This was listed as SL J0227.70450 in P1 and was first identified from XMM-Newton data by Pierre et al. (2006), where it
was named XLSS J022739.9045129 (also XLSSC-022). The
weak-lensing map shows a filamentary structure connecting the
two clusters, which appear to form a super cluster system.
There is an apparent galaxy over-density that largely overlaps with the weak-lensing high-density region. There is
a possible cD galaxy (the target ID 3, z = 0.294) slightly
south–east of the weak-lensing density peak. The velocity
dispersion of the galaxy cluster is found to be smaller than the
SIS velocity, though they are within a 1  error. The NFW
cluster mass agrees with the aperture mass. However, the
boundary of the cluster is very uncertain because of the filament. The aperture mass profile does not show flattening at
large scales of  = 100.
A.2.13. SL J0850.5 4512
This is a known cluster first identified from its galaxy
overdensity (NSC J85029+451141: Gal et al. 2003), but no
spectroscopic redshift has been obtained. The weak-lensing
density distribution appears to be relaxed. There is a possible
associated substructure north–east from the cluster. The galaxy
distribution agrees well with the weak-lensing density map.
Two possible cD galaxies (target IDs 1 and 31) are located
very close to the weak-lensing density peak. In the redshift
data, there exists one strong galaxy concentration at zc = 0.19
with 15 members dominated by the absorption galaxies. The
velocity dispersion of the galaxy cluster is found to be in good
agreement with the SIS velocity dispersion. The NFW cluster
mass agrees with the aperture mass.
A.2.14. SL J1000.7 0137
This is a known cluster first identified from its galaxy
concentration (NSC J100047+013912: Gal et al. 2003), and
later via X-ray emission by Finoguenov et al. (2007, their ID
67; the photometric redshift they obtained is zp = 0.22), but
the spectroscopic redshift had not been determined. The weaklensing density distribution shows an irregular morphology.
There is an over-density of galaxies, but its peak is about
20 north of the weak-lensing density peak. About 60 east of
the cluster center there is another weak-lensing density peak,
which is our target, SL J1001.2+0135, described in subsection A2.15. A smaller, third peak lies at a similar distance to
the east. In the redshift data, we find a strong galaxy concentration at zc = 0.22, with 14 members dominated by absorption
galaxies. Note that we find galaxy cluster SL J1001.2+0135 at
the same redshift. There are additional small concentrations of
galaxies at z = 0.34 and z = 0.52. The separation between the
two main clusters is 11h1 comoving Mpc (at z = 0.22), and
a filamentary structure connecting the two clusters is observed
in the weak-lensing density map. Thus, it is likely that they
form a super cluster system.
The velocity dispersion of the galaxy cluster is in good
No. 4] Multi-Object Spectroscopy and Cluster Masses 861
Fig. 33. Same as figure 18, but for SL J1204.40351.
agreement with the SIS velocity dispersion. The NFW cluster
mass is slightly larger than the aperture mass. It is likely that
the spherical NFW model does not give a good description of
this cluster, because of asymmetry in the density distribution.
A.2.15. SL J1001.2 0135
This is not listed in the P1 catalogue because the peak
 S=N does not exceed the required threshold. Note
that an extended X-ray source discovered by XMM-Newton
(Finoguenov et al. 2007, their ID 54; RA = 150:h33413, Dec
= 1:ı60301) lies about 30 east of the weak-lensing density
peak, and SL J1001.2 is a known optically selected cluster
(Gal et al. 2003, NSC J100113+013335, RA = 150:h30812,
Dec = 1:ı55967, zp = 0.242). The weak-lensing  map appears
to be elongated, with a filamentary structure connecting this
cluster to SL J0850.5+4512 (subsection A2.14).
In the redshift data, there are two strong galaxy
concentrations at z = 0.22 (11 members; we name it
SL J1001.2+0135A) and z = 0.37 (11 members; we name
it SL J1001.2+0135B). The velocity dispersions are SIS
 1380 km s1 (A) and 930 km s1 (B). Since we do not have
enough information to de-project the weak-lensing density into
two components (e.g., accurate photometric redshifts of faint
galaxies, e.g., Massey et al. 2007), we are unable to separately
estimate the weak-lensing mass of each cluster.
862 T. Hamana et al. [Vol. 61,
Fig. 34. Same as figure 8, but for SL J1334.3+3728.
A.2.16. SL J1002.9 0131
This is not listed in the P1 catalogue, because the  peak
is located close to the field edge and thus outside the secure
survey area. There is a galaxy over-density correlated with the
weak-lensing  peak. In the redshift data, there is no galaxy
concentration passing our cluster criterion, but there are two
small concentrations at z = 0.37 (3 galaxies) and at z = 0.66
(3 galaxies). Since no galaxy concentration that passes our
cluster criterion was found, we did not estimate the weaklensing mass.
Note that very near this target is a known X-ray cluster with
an estimated photometric redshift of zp = 0.75 (Finoguenov
et al. 2007, their ID 9; RA = 150:h75121, Dec = 1:ı52793).
Since the lensing efficiency of such a high redshift cluster is
low (see figure 3 of Hamana et al. 2004), it is unlikely to
be solely responsible for the  peak. Therefore, one possible
explanation of the observed  excess is a chance projection of
several halos located at different redshifts in adjacent lines-ofsight.
A.2.17. SL J1047.3 5700
The weak-lensing density distribution is elongated in the
north–west to south–east direction. The galaxy over-density
correlates with the  map very well. However, the redshift
information reveals that the weak-lensing and galaxy overdensities arise from not one, but two clusters, located at
different redshifts along the same line-of-sight. The foreground
cluster is at z = 0.24 with 6 members (v = 412 km s1);
the background cluster is at z = 0.30 with 10 members
(v = 619 km s1). We call them SL J1047.3+5700A and
SL J1047.3+5700B respectively. Since we do not have
enough information to de-project the weak-lensing density into
two components (e.g., accurate photometric redshifts of faint
galaxies, e.g., Massey et al. 2007), we can not make separate
mass estimates.
A.2.18. SL J1048.1 5730
The weak-lensing density distribution is slightly elongated.
There is a separate weak-lensing density peak at about 30
south–east of the cluster center, which was not listed in the
P1 catalogue because its peak  S=N (as opposed to the illustrated ) is lower than the required threshold. There is a galaxy
over-density that largely overlaps with the  peak. The redshift
data contain one strong galaxy concentration at zc = 0.31, with
9 members of mainly absorption type. There is also a small
concentration at z = 0.36. The velocity dispersion of the
galaxy cluster is found to be smaller than the SIS velocity
dispersion, but within a 1  error. The aperture mass profile
shows a jump at  ' 3:03, probably due to the south–east
peak. The NFW cluster mass agrees with the aperture mass
within  < 30.
A.2.19. SL J1049.4 5655
The weak-lensing density distribution is elongated from the
north–east to the south–west, with a second peak about 30
north–east of the cluster center. The distribution of galaxies
is more isotropic. In the redshift data, there is one galaxy
concentration at zc = 0.42 (6 members) that passes our
cluster criterion, plus small groups at z = 0.24 (4 galaxies),
z = 0.31 (4 galaxies), and possibly z = 0.59 (3 galaxies). The
velocity dispersion of the galaxies is smaller than the bestfit SIS velocity dispersion. This discrepancy is probably due
to a combination of line-of-sight projections and the small
number of redshifts used to compute v .
No. 4] Multi-Object Spectroscopy and Cluster Masses 863
Fig. 35. Same as figure 18, but for SL J1335.7+3731.
A.2.20. SL J1051.5 5646
The weak-lensing density distribution is slightly elongated,
and there is a second peak (catalogued as SL J1051.6+5647)
40 to the north–east, for which we have not obtained galaxy
spectra. There is no galaxy concentration sufficiently rich to
fulfill our cluster criterion in the redshift data, but two small
groups lie at z = 0.33 and z = 0.35. Note that galaxies with
target IDs 2, 3, and 5 are found to be very nearby. Also
note that the bright galaxy located at the peak of the  map
(SDSS SpecObjID 255522745545129984) is a nearby galaxy
at z = 0.047, and a second bright galaxy at RA = 162:h87,
Dec = 56:ı82 (SDSS SpecObjID 267344926176444416) is
also at z = 0.46. Since no galaxy concentration passing our
cluster criterion was found, we did not estimate the weaklensing mass.
A.2.21. SL J1052.0 5659
This is not listed in the P1 catalogue because its peak  S=N
is below the required threshold. The weak-lensing density
distribution appears to be very relaxed. A galaxy over-density
is present, but its peak is off-center, about 20 south of the 
peak. In the redshift data, there is no galaxy concentration
passing our cluster criterion, but there are two small concentrations at z = 0.34 (4 galaxies) and z = 0.52 (4 galaxies).
Since no rich galaxy concentration was found, we did not make
a weak-lensing mass estimation.
864 T. Hamana et al. [Vol. 61,
Fig. 36. Same as figure 18, but for SL J1337.7+3800.
A.2.22. SL J1052.5 5731
This is not listed in the P1 catalogue because its peak  S=N
is below the required threshold. The weak-lensing density
distribution shows an irregular morphology. A projected
galaxy over-density overlaps with the  peak, but in the redshift
data there is no galaxy concentration passing our cluster criterion. There are two small concentrations around z = 0.34
(5 galaxies) and z = 0.61 (3 galaxies). Since no dominant
galaxy concentration was found, we did not calculate the weaklensing mass.
Note that this region also contains two X-ray cluster
candidates (Kolokotronis et al. 2006): SEXCLAS-12 (RA
= 163:h159, Dec = 57:ı514, at photometric redshift zp = 0.61)
and SEXCLAS-13 (RA = 163:h226, Dec = 57:ı536; zp = 0.58).
The sky position of SEXCLAS-12 is very close to the  peak
( 10) and its estimated photometric redshift is very similar
to the redshift of our small galaxy concentration. It is therefore likely that the galaxy concentration at z = 0.61 is an
optical counterpart of X-ray cluster candidate SEXCLAS-12
(Kolokotronis et al. 2006). The observed  peak appears to
consist of a chance projection of halos of galaxy clusters at
different redshifts.
A.2.23. SL J1057.5 5759
The weak-lensing density distribution appears to be relaxed,
and is coincident with a very prominent overdensity of
galaxies. 18 (mainly absorption) galaxies are found at
zc = 0.60. Their velocity dispersion is larger than the velocity
No. 4] Multi-Object Spectroscopy and Cluster Masses 865
Fig. 37. Same as figure 8, but for SL J1601.6+4245.
dispersion parameter of the best-fit SIS model, but within a 1
error. The NFW cluster mass agrees with the aperture mass
at the corresponding virial radius. The aperture mass does not
flatten at large radii. The reason for this is not currently clear,
although measurements beyond 50 are noisy because the cluster
is near the edge of a field.
A.2.24. SL J1135.6 3009
The weak-lensing density distribution appears to be very
isolated but with an extension to the north. A prominent overdensity of bright galaxies includes a cD galaxy precisely at the
 peak position. The redshift data reveal a concentration of
15 (mainly absorption) galaxies at zc = 0.21. The velocity
dispersion of the galaxy cluster is consistent with the best-fit
SIS parameter and the NFW cluster mass is consistent with the
aperture mass.
A.2.25. SL J1201.70331
The weak-lensing density distribution appears to be relaxed.
There is a clear galaxy over-density that coincides with the
 peak. The redshift data contain a galaxy concentration at
zc = 0.52 with 8 members dominated by absorption galaxies.
The velocity dispersion of the galaxy cluster is consistent with
the SIS velocity dispersion. The NFW cluster mass is larger
than the aperture mass at the corresponding virial radius. This
small disagreement is likely to be due to poor measurements of
the shear near the edge of a Subaru field. High S=N measurements were obtained only at scales between 10 <  < 3:03.
A.2.26. SL J1204.40351
This is not listed in the P1 catalogue because the peak  S=N
is below that threshold. However, it is a known cluster, first
identified by its extended X-ray emission (RX J1204.30350:
Vikhlinin et al. 1998) and later confirmed with optical data
(OC5 12040351: Donahue et al. 2002). The spectroscopic
redshift of this cluster is z = 0.261 (Mullis et al. 2003).
The weak-lensing density distribution shows an irregular
morphology. The galaxy over-density is clearly observed, and
its peak position is very close to the  peak. A concentration
of 14 galaxies is indeed seen at zc = 0.261, dominated by
the absorption spectral types. Our measured velocity dispersion is consistent with the SIS velocity dispersion. The NFW
cluster mass is slightly larger than the aperture mass at the
corresponding virial radius. This small disagreement is likely
to be due to deviations from spherical symmetry apparent in
the  map. In this case, the NFW model would not be a good
description of the cluster density distribution.
A.2.27. SL J1334.3 3728
This is a known cluster, first identified by galaxy counts
(NSC J133424+372822: Gal et al. 2003), but a spectroscopic redshift has only been obtained for the cD galaxy
(RA = 203:h60, Dec = 37:ı48, z = 0.305; SDSS SpecObjID
591610245776670720). This is located very close to the  peak
(RA = 203:h60, Dec = 37:ı48), whose redshift is z = 0.305
(SDSS, SpecObjID is 591610245776670720).
The weak-lensing density distribution appears to be very
irregular, with elongations to the north–west and south–east,
as well as a neighbouring structure about 20 north–east of
the  peak. However, the distribution of galaxies is centered
neatly on only the main  peak. The galaxy redshifts reveal
21 members of a cluster at zc = 0.30, most of which are
absorption galaxies. Upon further inspection, there is also
a significant trend for southern (northern) galaxies to be at
lower (higher) redshifts, which may imply an ongoing merger
of two clusters. The measured velocity dispersion is larger than
866 T. Hamana et al. [Vol. 61,
Fig. 38. Same as figure 18, but for SL J1602.8+4335.
the best-fit SIS velocity dispersion parameter, but is consistent
within a 1 error. The NFW cluster mass agrees with the aperture mass at the corresponding virial radius. However, the aperture mass profile does not flatten, even at  = 100. This is probably due to the mass of neighbouring structures, and it is very
difficult to define the boundary of the cluster.
A.2.28. SL J1335.7 3731
The weak-lensing density distribution appears to be very
irregular, with two distinct main peaks aligned in the east–west
direction, and an additional structure between them extending
towards the north. Only the central structure was listed in P1,
as the S=N in peak  (as opposed to the  values shown) is
below the required threshold. The galaxy over-density closely
follows this elongated structure. A well-defined concentration
of 14 galaxies is located at zc = 0.41. Interestingly, more than
half of them have emission or composite type spectra. There
is also a small group of 3 galaxies at z = 0.20. The velocity
dispersion of the galaxy cluster is in reasonable agreement with
the SIS velocity dispersion parameter. The NFW cluster mass
is consistent with the aperture mass at the corresponding virial
radius. However, it is unlikely that a spherical NFW model is
a good description of this cluster, because of the asymmetry
apparent in the weak-lensing density map.
A.2.29. SL J1337.7 3800
The weak-lensing density distribution appears to be relaxed.
There is a galaxy over-density that coincides with the  peak,
No. 4] Multi-Object Spectroscopy and Cluster Masses 867
Fig. 39. Same as figure 8, but for SL J1605.4+4244.
Fig. 40. Same as figure 8, but for SL J1607.9+4338.
including a cD galaxy very close to the center. In the redshift
data, there is a prominent galaxy concentration at zc = 0:16
with 16 members dominated by absorption galaxies. The
velocity dispersion of the galaxy cluster is consistent with
the SIS velocity dispersion. The NFW cluster mass is in
good agreement with the aperture mass at the corresponding
virial radius.
A.2.30. SL J1601.6 4245
This is not listed in the P1 catalogue because the peak 
S=N is below the P1’s threshold. The weak-lensing density
distribution looks relaxed, except for an additional filament
extending to the north–west. A prominent over-density of
galaxies coincides with the  peak. This includes a cD
galaxy at RA = 240:h3934, Dec = 42:ı75902, whose spectrum
868 T. Hamana et al. [Vol. 61,
Fig. 41. Same as figure 18, but for SL J1634.1+5639.
was obtained by SDSS, and was found to be z = 0.208
(SpecObjID 375714238640947200). Spectra of another two
galaxies in this field were obtained by SDSS: SpecObjID =
375714238619975680 at RA = 240:h34840, Dec = 43:ı73718,
z = 0.208 and SpecObjID = 375714238666113024 at RA
= 240:h45011, Dec = 42:ı79000, z = 0.292.
Our MOS observations in fact reveal a projection of several
clusters at different redshifts along the same line-of-sight.
A foreground cluster (SL J1601.6+4245A) is at z = 0.208,
with 7 spectroscopically confirmed member galaxies, and
a background cluster (SL J1601.6+4245B) at z = 0.47 with
8 members. We also found a small group of 5 new galaxies at
z = 0.29. Therefore, SL J1601.6+4245A has 2+ 7 spectroscopically confirmed cluster members including a cD galaxy,
and the small galaxy concentration has 1 + 5 spectroscopic
members. We therefore conclude that the observed  excess is
caused by a chance projection of at least three galaxy clusters
located at different redshifts. Since we do not have sufficient
information to de-project the weak-lensing density into components, we can not compute their weak-lensing masses.
A.2.31. SL J1602.8 4335
The weak-lensing density distribution appears to be relaxed,
with an additional low-level filamentary structure running from
north to south. There is a galaxy over-density, including a cD
galaxy, that coincides with the  peak. In the redshift data,
a strong concentration at zc = 0.42 of 15 members is dominated
by absorption galaxies. The velocity dispersion of the galaxy
No. 4] Multi-Object Spectroscopy and Cluster Masses 869
Fig. 42. Same as figure 8, but for SL J1639.9+5708.
cluster is consistent with the SIS velocity dispersion. The NFW
cluster mass is in good agreement with the aperture mass at
the corresponding virial radius. No conclusive explanation has
been found for the north–south filament.
A.2.32. SL J1605.4 4244
The weak-lensing density distribution appears to be relaxed.
There is a galaxy over-density that largely overlaps with the 
peak. In the redshift data, there exists one galaxy concentration at zc = 0.22 with 6 members. The velocity dispersion is
significantly larger than the SIS velocity dispersion, probably
on account of the low number of observed member galaxies.
The aperture mass profile behaves irregularly at larger radii.
This may be noise due to a shortage of source galaxies, which
are hidden by masks around nearby bright stars.
A.2.33. SL J1607.9 4338
This is not listed in the P1 catalogue because the peak 
S=N is below the P1 threshold. This is a known, optically
selected cluster (GHO 1606+4346: Gunn et al. 1986), but no
spectroscopic redshift has been obtained. The weak-lensing
density distribution appears to be elongated towards the north
and south–west, with local  maxima about 30 from the target
center in both directions. The south–west peak has the highest
 and is listed in the P1 catalogue (GTO 2 deg2 #04). A galaxy
over-density overlaps the central and northern  peaks. Our
redshift data reveal a galaxy concentration at zc = 0.31, passing
our cluster criterion with 9 members. This is dominated by
absorption galaxies. In addition, there is a small concentration
of five galaxies at z = 0.25. The velocity dispersion of the
galaxy cluster is smaller than the SIS velocity dispersion. The
reason for this is not clear, but is probably due to poor statistics
from the small number of spectroscopically confirmed cluster
members. The NFW cluster mass is consistent with the aperture mass at the corresponding virial radius.
A.2.34. SL J1634.1 5639
The weak-lensing density distribution appears to be relaxed.
The clustering of bright galaxies is apparent, but the number
density of galaxies with 18 < RC < 23 is lower than the
surrounding mean density. The redshift data reveal a single
concentration of 13 galaxies at zc = 0.4. The northern galaxies
tend to have absorption spectra but, interestingly, the southern
galaxies have emission spectra. The velocity dispersion of
the galaxy cluster is significantly larger than the SIS velocity
dispersion. As shown in figure 1, the velocity distribution of
spectroscopic members appears to be strongly skewed toward
the bluer side. This skewness may account for the large
measurement of dispersion. The reason for the large skewness
is currently not clear; dynamical activity of the cluster may be
involved, although there are poor statistics to constrain higher
moments. The NFW cluster mass is in good agreement with
the aperture mass at the corresponding virial radius.
A.2.35. SL J1639.9 5708
The weak-lensing density distribution is elongated in the
north–south direction. The corresponding galaxy over-density
is clearly found. In the redshift data, there is no galaxy concentration passing our cluster criterion, but there are two small
concentrations at z = 0.2 and z = 0.63. Since no sufficiently
rich concentration of galaxies was found, we did not calculate
the weak-lensing mass.
A.2.36. SL J1647.7 3455
The weak-lensing density distribution is elongated in the
north–south direction. A prominent overdensity of bright
870 T. Hamana et al. [Vol. 61,
Fig. 43. Same as figure 18, but for SL J1647.7+3455.
galaxies includes one cluster of 12 galaxies at zc = 0.26. This
is dominated by absorption galaxies. There are small additional groups of galaxies at z = 0.41 and z = 0.47. The
velocity dispersion of the main galaxy cluster is in reasonable
agreement with the SIS velocity dispersion. The NFW cluster
mass is consistent with the aperture mass. However, the aperture mass profile does not flatten as expected at larger radii.
This may be accounted for by the mass of a second cluster
located about 60 south–west of the cluster. It is also likely
that the spherical NFW model does not accurately describe
this cluster, as asymmetry is clearly visible in the weak-lensing
density map.
Appendix 3. Properties of Previously Studied Clusters
When studying cluster scaling relations in section 6, we
included two cluster candidates from our weak-lensing survey
that had already been spectroscopically verified and whose
velocity dispersions were previously known. Data from the
literature were exposed to the same selection criteria used to
make our clean sample (see subsection 6.1). Observations
of clusters SL J0221.70345 and SL J0228.40425 by
Willis et al. (2005, where they are named XLSSC 006 and
XLSSC 012), both satisfy our conditions.
These two clusters were originally identified via their
X-ray emission. Later spectroscopic observations by
No. 4] Multi-Object Spectroscopy and Cluster Masses 871
Table 4. Summary of weak-lensing analyses.
Name XMM-LSS ID zc
sis SIS MNFW M200 M500k rvir#
(km s1) ( 1014h1Mˇ) (comoving Mpc h1)
SL J0221.70345 XLSSC 006 0.429 0.079 926+406438 5.01+1:371:39 4.35+1:191:21 2.97+0:810:82 1.7
SL J0228.40425 XLSSC 012 0.433 0.065 839+449448 3.24+1:461:19 2.82+1:271:04 1.95+0:880:72 1.6
 The amplitude of the tangential shear profile at 10, when fitted with an SIS model (see sub-subsection 5.4.2).
 The best-fit SIS velocity dispersion parameter.
 The virial mass estimated by fitting the radial shear profile with an NFW model.
 The M200 computed assuming the NFW profile.k The M500 computed assuming the NFW profile.
#
The virial radius computed from the NFW mass using the relation equation (11).
Fig. 44. Same as the bottom two panels of figure 18, but for SL J0222.80416.
Fig. 45. Same as the bottom two panels of figure 18, but for SL J0222.80416.
872 T. Hamana et al.
Willis et al. (2005) revealed galaxy velocity dispersions for
SL J0221.70345 of v = 821+9274 km s1 (computed from
39 galaxy redshifts) and for SL J0228.4042 of 694+20491 km s1
(from 13 galaxy redshifts).
In P1, the two clusters are listed as XMM-LSS-00 and
XMM-LSS-21. We measured the weak-lensing properties of
these clusters using the method described in section 5, and
summarize our results in table 4. Weak-lensing mass maps,
galaxy density maps, tangential shear profiles, and aperture
mass profiles are presented in figures 44 and 45.
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