A weak lensing analysis of the Abell 383 cluster

Abstract

In this paper we use deep CFHT and SUBARU $uBVRIz$ archival images of the
Abell 383 cluster (z=0.187) to estimate its mass by weak lensing. To this end,
we first use simulated images to check the accuracy provided by our KSB
pipeline. Such simulations include both the STEP 1 and 2 simulations, and more
realistic simulations of the distortion of galaxy shapes by a cluster with a
Navarro-Frenk-White (NFW) profile. From such simulations we estimate the effect
of noise on shear measurement and derive the correction terms. The R-band image
is used to derive the mass by fitting the observed tangential shear profile
with a NFW mass profile. Photometric redshifts are computed from the uBVRIz
catalogs. Different methods for the foreground/background galaxy selection are
implemented, namely selection by magnitude, color and photometric redshifts,
and results are compared. In particular, we developed a semi-automatic
algorithm to select the foreground galaxies in the color-color diagram, based
on observed colors. Using color selection or photometric redshifts improves the
correction of dilution from foreground galaxies: this leads to higher signals
in the inner parts of the cluster. We obtain a cluster mass that is ~ 20%
higher than previous estimates, and is more consistent the mass expected from
X--ray data. The R-band luminosity function of the cluster is finally computed.

Full text

arXiv:1102.1837v1 [astro-ph.CO] 9 Feb 2011
Astronomy & Astrophysics
manuscript no. a383 c
ESO 2011
February 10, 2011
A weak lensing analysis of the Abell 383 cluster. ⋆
Zhuoyi Huang1, Mario Radovich2, Aniello Grado1, Emanuella Puddu1, Anna Romano3,
Luca Limatola1, and Liping Fu4,1
1INAF - Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131, Napoli, Italy
2INAF - Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, I-35122, Padova, Italy
3INAF - Osservatorio Astronomico di Roma, Monte Porzio, I-00185, Roma, Italy
4Key Lab for Astrophysics, Shanghai Normal University, 100 Guilin Road, 200234, Shanghai, China
received; accepted
ABSTRACT
Aims.
In this paper we use deep CFHT and SUBARU uBVRIz archival images of the Abell 383 cluster (z=0.187) to estimate its mass
by weak lensing.
Methods.
To this end, we first use simulated images to check the accuracy provided by our KSB pipeline. Such simulations include
both the STEP 1 and 2 simulations, and more realistic simulations of the distortion of galaxy shapes by a cluster with a Navarro-
Frenk-White (NFW) profile. From such simulations we estimate the effect of noise on shear measurement and derive the correction
terms. The R−band image is used to derive the mass by fitting the observed tangential shear profile with a NFW mass profile.
Photometric redshifts are computed from the uBV RIz catalogs. Different methods for the foreground/background galaxy selection are
implemented, namely selection by magnitude, color and photometric redshifts, andresults are compared. In particular, we developed
a semi-automatic algorithm to select the foreground galaxies in the color-color diagram, based on observed colors.
Results.
Using color selection or photometric redshifts improves the correction of dilution from foreground galaxies: this leads to
higher signals in the inner parts of the cluster. We obtain a cluster mass Mvir =7.5+2.7
−1.9×1014 M⊙: such value is ∼20% higher than
previous estimates, and ismore consistent the mass expected from X–ray data. The R-band luminosity function of the cluster is finally
computed, giving a total luminosity Ltot =(2.14 ±0.5) ×1012 L⊙and a mass to luminosity ratio M/L∼300M⊙/L⊙.
Key words. Galaxies: clusters: individual: Abell 383 – Galaxies: fundamental parameters – Cosmology: dark matter
1. Introduction
Weak gravitational lensing is a unique technique that allows to
probe the distribution of dark matter in the Universe. It mea-
sures the very small distortions in the shapes of faint back-
ground galaxies, due to foreground mass structures. This tech-
nique requires a very accurate measurement of the shape pa-
rameters as well as the removal of the systematic effects af-
fecting them. In addition, galaxies to be used in the weak lens-
ing analysis must be carefully selected so that they do not in-
clude a significant fraction of unlensed sources, with redshift
smaller than that of the lens. This would introduce a dilution
and therefore an underestimated signal, in particular towards the
cluster center (see Broadhurst et al. 2005): such effect may be
the reason of the under-prediction by weak lensing of the ob-
served Einstein radius, by a factor of ∼2.5 (Smith et al. 2001;
Bardeau et al. 2005). The optimal case would happen if photo-
metric redshifts were available. Even if for weak lensing high
accuracy in their estimate are not required for individual galax-
⋆Based on: data collected at Subaru Telescope (University of Tokyo)
and obtained from the SMOKA, which is operated by the Astronomy
Data Center, National Astronomical Observatory of Japan; observa-
tions obtained with MegaPrime/MegaCam, a joint project of CFHT and
CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which
is operated by the National Research Council (NRC) of Canada, the
Institute National des Sciences de l’Univers of the Centre National de
la Recherche Scientifique of France, and the University of Hawaii. This
work is based in part on data products produced at TERAPIX and the
Canadian Astronomy Data Centre as part of the Canada-France-Hawaii
Telescope Legacy Survey, a collaborative project of NRC and CNRS.
ies, on average we need at least σz/(1 +z)<0.1: this implies
having observations in several bands, spanning a good wave-
length range. If few bands are available, an uncontaminated
background sample can be obtained by selecting only galaxies
redder than the cluster red sequence (Broadhurst et al. 2005).
However, such method often does not allow to get a number
density of background sources high enough to allow an accu-
rate weak lensing measure. Including galaxies bluer than the red
sequence (Okabe et al. 2010) requires a careful selection of the
color offset, as bluer galaxies can be still contaminated by late–
types members of the cluster. Finally, if more than two bands
are available, Medezinski et al. (2010) discussed how to identify
cluster members and the foreground population as overdensities
in the color-color space.
In this paper we exploit deep uBVRIz images of the clus-
ter Abell 383, taken with the MEGACAM and SUPRIME cam-
era mounted on the 3.6m CFHT and 8m SUBARU telescopes
respectively, and publicly available. The mass of the cluster is
derived by weak lensing, and values obtained by different selec-
tion methods are compared. The properties of the cluster are re-
viewed in Sect.2. Data reduction is discussed in Sect.3. In Sect. 4
we describe the algorithm used for the shape measurement, and
some improvements for the removal of biases. The accuracy in
the mass estimate is derived by comparison with simulations. In
Sect. 5, we first summarize the different methods for the selec-
tion of the background galaxies from which the lensing signal
is measured. Such methods are applied to the case of Abell 383,
and the masses derived in such way are then compared. Finally,
in Sect. 6 we compare the mass derived in this paper with lit-
1

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
erature values, both by X–rays and weak lensing; a comparison
is also done with the mass expected for the R–band luminosity,
derived from the luminosity function of Abell 383.
A standard cosmology was adopted in this paper: ΩΛ=0.7,
ΩM=0.3, H0=70 km s−1Mpc−1, giving a scale of 2.92
kpc/arcsec at the redshift of Abell 383.
2. Abell 383
Abell 383 is an apparently well relaxed cluster of galaxies of
richness class 2 and of Bautz-Morgan type II-III (Abell et al.
1989), located at z =0.187 (Fetisova et al. 1993). It is domi-
nated by the central cD galaxy, a blue-core emission-line Bright
Cluster Galaxy (BCG), that is aligned with the X-ray peak
Smith et al. (2001). Abell 383 is one of the clusters of the
XBACs sample (X-ray-Brightest Abell-type Clusters), observed
in the ROSAT All-Sky Survey (RASS; Voges 1992): its X-ray
luminosity is 8.03 ×1044 erg sec−1in the 0.1-2.4 keV band and
its X-ray temperature is 7.5 keV (Ebeling et al. 1996). A small
core radius, a steep surface brightness profile and an inverted de-
projected temperature profile, show evidence of the presence of
a cooling flow, as supported by the strong emission lines in the
optical spectra of its BCG (Rizza et al. 1998).
An extensive study of this cluster was carried out by
Smith et al. (2001), in which lensing and X-ray properties were
analyzed on deep optical HST images and ROSAT HRI data,
respectively. A complex system of strong lensed features (a gi-
ant arc, two radial arcs in the center and numerous arclets) were
identified in its HST images, some of which are also visible in
the deep SUBARU data used here.
3. Data retrieval and reduction
The cluster Abell 383 was observed with the SUPRIME cam-
era mounted at the 8m SUBARU telescope: SUPRIME is
a ten CCDs mosaic, with a 34 ×27 arcmin2field of view
(Miyazaki et al. 2002). Data are publicly available in the BVRIz
filters, with total exposure times of 7800s (R), ∼6000s (B,V),
3600s (I) and 1500s (z); they were retrieved using the SMOKA1
Science Archive facility. Data were collected from seven differ-
ent runs, amounting to ∼55GB. Details about the observation
nights and exposure times for each band are given in Table 1. The
data reduction was done using the VST–Tube imaging pipeline,
specifically developed for the VLT Survey Telescope (VST,
Capaccioli et al. 2005) but adaptable to other existing or future
multi-CCD cameras (Grado et al. 2011) .
The field of Abell 383 was also observed in the u∗band with
the MEGACAM camera attached on the Canada-France Hawaii
Telescope (CFHT), with a total exposure time of 10541s. The
preprocessed images were retrieved from the CADC archive.
The basic reduction steps were performed for each frame,
namely overscan correction, flat fielding, correction of the ge-
ometric distortion due to the optics, and sky background sub-
traction. It is worth mentioning that to improve the photometric
accuracy, a sky superflat was used.
Geometric distortions were first removed from each expo-
sure, using the Scamp tool 2and taking the USNO-B1 as the as-
trometric reference catalog. The internal accuracy provided by
Scamp, as measured by positions of the same sources in differ-
ent exposures, is ∼0.05 arcsec. Different exposures were then
1http://smoka.nao.ac.jp/
2Stuff, Skymaker, SWarp and SExtractor are part of the
Astromatic software developed by E. Bertin, see www.astromatic.net
17 18 19 20
−0.4 0.0 0.2 0.4
R
zphot −zSDSS
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0 0.2 0.4 0.6
zSDSS
zphot
Fig.1. Density plots comparing the photometric redshifts in the
Abell 383 field available from the SDSS, and those here com-
puted from the uBVRIz photometry.
16 17 18 19 20 21 22 23
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
R
V−R
Fig.2. V−Rvs Rcolor plot: red and blue points are galaxies at
zphot =0.187±0.1 and classified as early and late-type, respec-
tively; dashed lines shows the ±1σlevels.
stacked together using SWarp. The coaddition was done in such
a way that all the images had the same scale and size.
The photometric calibration of the BVRI bands was per-
formed using the standard Stetson fields, observed in the same
nights of the data. In the case of the zband, we used a point-
ing also covered by the Stripe 82 scans in the Sloan Digital Sky
Survey (SDSS); the SDSS photometry of sources identified as
point-like was used to derive the zero point in the SUBARU im-
age.In the case of the MEGACAM-CFHT u∗band images, re-
duced images and photometric zero points are already available
from the CADC public archive, hence only astrometric calibra-
tion and stacking were required.
Table 2 summarizes the photometric properties of the fi-
nal coadded images (average FWHMs and limiting magnitudes
for point-like sources) for each band. All magnitudes were
converted to the AB system; magnitudes of sources classified
2

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
Table 1. Summary of observations done with the MEGACAM
(u) and SUPRIME (BVRIz) cameras, used in this paper.
Date Band Exp. Time Total
23 Dec 2003 u∗8381s
21 Jan 2004 u∗2160s 10541s
09 Sept 2002 B2400s
08 Jan 2008 B2400s
09 Jan 2008 B1200s 6000s
02 Oct 2002 V4320s
01 Oct 2005 V1800s 6120s
12 Nov 2007 Rc2400s
08 Jan 2008 Rc5400s 7800s
02 Oct 2002 Ic3600s 3600s
10 Sept 2002 z′1500s 1500s
Table 2. Photometric properties of the coadded images; FWHM
(arcsec) and limiting magnitudes (in the AB system) were com-
puted for point-like sources. The signal to noise (SNR) is defined
as SNR=FLUX AUTO/FLUXERR AUTO.
Band FWHM mag (SNR=5) mag (SNR=10)
u∗1.3 26.1 25.3
B0.99 27.0 26.2
V0.95 26.6 25.7
Rc0.82 27.1 26.1
Ic0.97 25.0 24.2
z′0.74 24.7 23.9
as galaxies were corrected for Galactic extinction using the
Schlegel maps (Schlegel et al. 1998).
The weak lensing analysis was done on the R−band im-
age. The masking of reflection haloes and diffraction spikes
near bright stars are performed by ExAM, a code developed
for this purpose. In short, ExAM takes the SExtractor cata-
log as input, locates the stellar locus in the size-magnitude di-
agram (see Sect. 4.1), picks out stars with spike-like features
from the isophotal shape analysis, and outputs mask region file
that may be visualized in the DS9 software, and finally cre-
ates a mask image in FITS format. The reflection haloes are
masked by estimating the background contrast near the bright
stars, whose positions are obtained from the USNO–B1. The ef-
fective area available after removal of regions masked in such
way was 801 arcmin2. Catalogs for the other bands were ex-
tracted using SExtractor in dual–mode, where the R−band im-
age was used as the detection image.
Photometric redshifts were computed from uBVRIz pho-
tometry, using the ZEBRA code (Feldmann et al. 2006). This
software allows to define 6 basic templates (elliptical, Sbc,
Sbd, irregular and two starburst SEDs), and to compute log-
interpolations between each pair of adjacent templates. We first
applied to the BVz magnitudes the offset derived within the
COSMOS survey by Capak et al. (2007), that is +0.19 (B),
+0.04 (V), -0.04 (z). We then convolved the stellar spectra from
the Pickles’ library (Pickles 1985) with the transmission curves
used for each filter, and derived the offsets for the other filters,
that is: 0.0 (u), 0.0 (R), +0.05 (I). Fig. 3 shows the comparison
−0.5 0.5 1.5
2.5 2.0 1.5 1.0 0.5 0.0 −0.5
(B−V)AB
(u−B)AB
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Fig.3. Observed (red dots) and model colors (black dots) for
stars, after the offsets given in the text were applied. Model col-
ors were derived convolving the Pickles’ library of stellar spectra
with the filter transmission curves.
1000 2000 3000 4000
Redshift
N
0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
Fig.4. Distribution of the photometric redshifts computed from
the uBVRIz data for R<25 mag.
of the model colors with those derived for the stars in our cata-
logs, after the above offsets were applied. We also verified that
the offsets derived in this way are consistent with those obtained
by running ZEBRA in the so-called photometry-check mode,
that allows to compute magnitude offsets minimizing the aver-
age residuals of observed versus template magnitudes.
We then removed from the catalog those galaxies with a pho-
tometric redshift zph >3 and σz/(1 +z)>0.1, where σzwas
derived from the 68%–level errors computed in ZEBRA. The
distribution of the so-obtained photometric redshifts is displayed
in Fig. 4. The accuracy of the so obtained photometric redshifts
was derived from the comparison with the SDSS photometric
redshifts of galaxies in the SDSS (DR7), whose rms error is
∼0.025 for r′<20 mag. We derived (Fig. 1) a systematic off-
set ∆z/(1 +z)=0.003 and an rms error σ∆z/(1 +z)=0.07.
As a further check, we extracted from our catalog those galax-
ies classified by ZEBRA as early-type, with R<23 mag and
|zphot −0.187|=0.1. As expected, such galaxies define a red se-
3

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
quence (Fig.2): this was fitted as V−R=a+bR, with a=0.5,
b=−6×10−3.
4. Shape measurement
Ellipticities of galaxies were estimated using the KSB approach
(Luppino & Kaiser 1997): even if such algorithm does not allow
to achieve accurate measurements of very low shear signals, γ.
10−3, it is nevertheless adequate in the case of weak lensing by
clusters, as discussed e.g. by Gill et al. (2009) and Romano et al.
(2010).
In our KSB implementation, the SExtractor software was
modified to compute all the relevant quantities, namely the raw
ellipticity e, the smear polarizability Psm, and the shear polariz-
ability Psh. The centers of detected sources were measured using
the windowed centroids in SExtractor.
The KSB approach assumes that the PSF can be described
as the sum of an isotropic component (simulating the effect of
seeing) and an anisotropic part. The correction of the observed
ellipticity eobs for the anisotropic part is computed as:
eaniso =eobs −Psmp,(1)
where (starred terms indicate that they are derived from mea-
surement of stars): p=e∗
obs/Psm∗.(2)
The intrinsic ellipticity eof a galaxy, and the reduced shear, g=
γ/(1 −κ), are then related by:
eaniso =e+Pγg(3)
The term Pγ, introduced by Luppino & Kaiser (1997) as the pre–
seeing shear polarizability, describes the effect of seeing and is
defined to be:
Pγ=Psh −Psm Psh∗
Psm∗≡Psh −Psmq.(4)
The final output of the pipeline is the quantity eiso =
eaniso/Pγ, from which the average reduced shear, hgi=heisoi,
provided that the average intrinsic ellipticity vanishes, hei=0.
The ellipticity calculation is done by using a window func-
tion in order to suppress the outer, noisy part of a galaxy: the
function is usually chosen to be Gaussian with size θ. The size
of the window function is commonly taken as the radius con-
taining 50% of the total flux of the galaxy (as given by e.g. the
FLUX RADIUS parameter in SExt ractor). In our case, we pro-
ceed as follows. We define a set of bins with θvarying between 2
and 10 pixels (sources with smaller and larger sizes are rejected
in our analysis), and a step of 0.5 pixel. For each bin we compute
eobs,Psh and Psm, and the ellipticity signal to noise ratio defined
by Eq. 16 in Erben et al. (2001):
SNe(θ)=RI(θ)W(|θ|)d2θ
σsky qRW2(|θ|)d2θ
.(5)
The optimal size of the window function, θmax, is then defined as
the value that maximizes SNe. Fig.5 shows the typical trend of
SNe, which was normalized for display purposes, as a function
of θ. It can be seen (Fig. 6) that, on average, there is a con-
stant offset between θmax and FLUX RADIUS. Below SNe ∼5,
FLUX RADIUS starts to decrease: this provides an estimate of
limit on SNe, below which the shape measurement is not mean-
ingful any more.
2 3 4 5 6 7
0.0 0.2 0.4 0.6 0.8 1.0
θ [pixels]
Normalized SNe
Fig.5. SNe is displayed as a function of the window function
size, θ, used to measure ellipticities. For display purposes, galax-
ies were selected to have the same value of θmax, and SNe was
normalized so that min(SNe)=0,max(SNe)=1. The vertical lines
indicate the average (solid) and standard deviation (dashed) of
FLUX RADIUS for the same galaxies.
0 10 20 30 40 50
0.0 0.1 0.2 0.3 0.4 0.5
SNe
(FLUX_RADIUS−θ) [pixels]
Fig.6. Running median of FLUX RADIUS-θmax as a function
of SNe. The vertical line shows the limit chosen for the selection
of background galaxies.
The terms pand q, derived from stars, must be evaluated
at each galaxy position: this is usually done fitting them by a
polynomial (see e.g. Radovich et al. 2008), whose order must be
chosen to fit the observed trend, without overfitting. The usage
of the window function introduces a calibration factor, which
is compensated by the Pγterm. This implies that stellar terms
must be computed and fitted with the same value of θused for
each galaxy (Hoekstra et al. 1998). An alternative approach, not
based on a constant (and somehow arbitrary) order polynomial,
is given e.g. by Generalized Additive Models: we found that
the implementation in R (function GAM in the mgcv library)
provides good results. Fig. 7 shows fitting and residuals of the
anisotropic PSF component: from the comparison between the
results obtained with polynomial and GAM fitting we see that
4

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
in the latter case we obtain lower residuals, in particular in the
borders of the image. To quantify the improvementcompared to
the usage of the polynomial, we obtain eaniso,1=(1±5)×10−4,
eaniso,2=(−2±9) ×10−4with a polynomial of order 3,
eaniso,1=(2 ±4) ×10−4,eaniso,2=(1 ±7) ×10−4with the
GAM algorithm. The values of the fitted terms pand qat the po-
sitions of the galaxies are predicted by GAM, that also provides
an estimate of the standard errors of the predictions, ∆pand ∆q.
From error propagation, the uncertainty on eiso was computed
as:
∆e2
iso =(∆eaniso/Pγ)2+(eaniso(Pγ)−2∆Pγ)2,(6)
where (∆eaniso)2=(∆eobs)2+(Psm∆p)2and (∆Pγ)2=(Psm∆q)2;
uncertainties on the measured values of Psm and Psh were not
considered.
For each galaxy, a weight is defined as:
w=1
∆e02+ ∆e2
iso
,(7)
where ∆e0∼0.3 is the typical intrinsic rms of galaxy elliptici-
ties.
4.1. Star-galaxy classification
Stars and galaxies were separated in the magnitude
(MAG AUTO) vs. size plot. Instead of using e.g.
FLUX RADIUS as the estimator of size, we used the quantity
δ=MU MAX-MAG AUTO, where MU MAX is the peak
surface brightness above background. Saturated stars were
found in the locus of sources with constant MU MAX; in the
δvs. MAG AUTO plot, stars are identified as sources in the
vertical branch. Sources with δlower than stars were classified
as spurious detections. In addition, we rejected those sources for
which δis ∼2σlarger than the median value. This is to exclude
from the sample of stars used to compute the PSF correction
terms, those sources for which the shape measurement may be
wrong due to close blended sources, noise, etc.
We further excluded those galaxies with w<1 or SNe <5,
for which the ellipticity measurement is not meaningful.
4.2. Error estimate in shear and mass measurement
In the case of faint galaxies used for the weak lensing analy-
sis, the ellipticity is underestimated due to noise. Such effect is
not included in the Pγterm, which can be only computed on
stars with high signal to noise ratio. Schrabback et al. (2010)
proposed the following parametrization for such bias, as a func-
tion of the signal to noise:
m=ek−em
em=a∗(SNe)b,(8)
where emand ekare the ellipticities before and after the correc-
tion respectively.Such parameters were derived using the STEP1
(Heymans et al. 2006) and STEP2 (Massey et al. 2007) simula-
tions, where both PSF and shear are constant for each simulated
image. We obtain a=−0.1 and b=−0.45, which corresponds to
a bias mchanging from ∼5% for SNe =5 to <2% for SNe >50.
After such correction was applied, we computed again the aver-
age shear from the STEP1 and STEP2 simulations, and obtained
a typical bias of ∼3% for SNe =5.
We then estimated the accuracy on the mass that can be ob-
tained from an image with the same noise and depth as in the
0 2000 6000 10000
0 2000 6000
Measured
X
Y
0 2000 6000 10000
0 2000 6000
Fitted
X
Y
0 2000 6000 10000
0 2000 6000
Corrected
X
Y
−0.10 0.00 0.05 0.10
−0.10 0.00 0.05 0.10
e1
e2
0 2000 6000 10000
0 2000 6000
Corrected
X
Y
−0.10 0.00 0.05 0.10
−0.10 0.00 0.05 0.10
e1
e2
Fig.7. PSF anisotropy correction derived with the GAM algo-
rithm: the first three panels show the ellipticity pattern (mea-
sured, fitted and residuals; X and Y are in pixels). The scale is
displayed by the arrows in the upper right part of each panel
(e=0.05). In the next panel, black dots are the measured val-
ues, green dots are after the correction; values rejected during
fitting are marked in red. The last row shows for comparison the
corrected ellipticities obtained using for the fit a polynomial of
order 3.
R−band SUBARU image. To this end we dropped the assump-
tion on constant shear, and produced more realistic simulations:
the effect on galaxy shapes by weak lensing from a galaxy cluster
was produced using the SHUFF code, that will be described in a
separate paper (Huang et al., in preparation). To summarize, the
code takes as input a catalog of galaxies produced by the Stuff
tool; it computes the shear produced by a standard mass profile
(e.g. Navarro-Frenk-White, NFW hereafter) and applies it to the
ellipticities of the galaxies behind the cluster. Such catalog is
then used in the SkyMaker software, configured with the tele-
scope parameters suitable for the SUBARU telescope and with
the exposure time of the R−band image, producing a simulated
image; the background rms of such image was set to be as close
as possible to that of the real image. We considered for the lens
5

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
Table 3. Masses derived by simulations with NFW model fitting
(M3D) and aperture densitometry (M2D). Input masses used for
each simulation are given in the first column (Min).
Min M3D M2D/M3D
1014 M⊙1014 M⊙
0.316 0.31 ±0.13 1.3±0.9
1.00 1.01 ±0.24 1.4±0.4
3.16 3.12 ±0.36 1.4±0.3
10.0 10.1±0.5 1.3±0.1
a range of masses at log Mvir/M⊙=13.5,14,14.5,15.0, and a
NFW mass profile with cvir =6. Each simulation was repeated
50 times for each mass value, randomly changing the morphol-
ogy, position and redshift of the galaxies.
For each of these images, we run our lensing pipeline with
the same configuration used for the real data. The density of
background galaxies used for the lensing analysis was ∼20
gals arcmin−2. The fit of the mass was done as described in
Radovich et al. (2008) and Romano et al. (2010): the expres-
sions for the radial dependence of tangential shear γTderived
by Bartelmann (1996) and Wright & Brainerd (2000) were used,
and the NFW parameters (Mvir,cvir) were derived using a max-
imum likelihood approach. In addition, the 2D projected mass
can be derived in a non-parametric way by aperture densitome-
try, where the mass profile of the cluster is computed by the ζ
statistics (Fahlman et al. 1994; Clowe et al. 1998):
ζ(θ1)=¯κ(θ≤θ1)−¯κ(θ2< θ ≤θout)=2Zθ2
θ1
hγTidlnθ(9)
+2
1−(θ2/θout)2Zθout
θ2
hγTidlnθ.
The mass is estimated as Map(θ1)=πθ2
1ζ(θ1)Σcrit, and θout is
chosen so that ¯κ(θ2, θout)∼0.
The average errors on mass estimate obtained in such way
are displayed in Table 3, showing that masses can be estimated
within an uncertainty of <20% for M≥1014M⊙. Such accuracy
only includes the contribute due to shape measurement and mass
fitting method, but it does not include the uncertainty due to the
selection of the lensed galaxies.
Finally, the masses derived by aperture densitometry are ∼
1.3 higher than those obtained by mass fitting: this is in agree-
ment with Okabe et al. (2010), who find M2D/M3D =1.34 for
virial overdensity.
5. Results
One of the most critical source of systematic errors, which can
lead to an underestimation of the true WL signal, is dilution of
the distortion due to the contamination of the background galaxy
catalog by unlensed foreground and cluster member galaxies
(see e.g. Broadhurst et al. 2005). The dilution effect increases as
the cluster-centric distance decreases because the number den-
sity of cluster galaxies that contaminate the faint galaxy catalog
is expected to roughly follow the underlying density profile of
the cluster. Thus, correcting for the dilution effect is important
to obtain unbiased, accurate constraints on the cluster parameters
and mass profile.
As discussed by Broadhurst et al. (2005); Okabe et al.
(2010); Oguri et al. (2010), the selection of background galaxies
to be used for the weak lensing analysis can be done taking those
galaxies redder than the cluster red sequence. However, such
selection produces a low number density (10 galaxies/arcmin2
in our case), and correspondingly high uncertainties in the de-
rived parameters. In the following, we compare the results ob-
tained by different methods. We first assumed that no informa-
tion on the redshift is available, and that photometry from only
one band is available (magnitude cut), or from more than two
bands (color selection). Finally, we included in our analysis the
photometric redshifts. The density of background galaxies is ∼
25-30 galaxies/arcmin2, see Table 4.
In order to derive the mass, we need to know the critical
surface density:
Σcrit =c2
4πGDs
DlDls =c2
4πG1
Dlβ,(10)
Dls,Ds, and Dlbeing the angular distances between lens and
source, observer and source, and observer and lens respectively.
This quantity should be computed for each lensed galaxy. As
the reliability of photometric redshifts for the faint background
galaxies is not well known, we prefer to adopt the single sheet
approximation, where all background galaxies are assumed to
lie at the same redshift, defined as β(zs)=hβ(z)i. In the case
of the selection based only on magnitude or colors, such value
was derived from the COSMOS catalog of photometric redshifts
(Capak et al. 2007), to which the same cuts used for the Abell
383 catalog are applied. Later on, we computed β(zs) from the
photometric redshifts themselves, and compared such two val-
ues.The mass was computed both by fitting a NFW profile
(M3D =Mvir), and by aperture densitometry (M2D). In the case
of the NFW fit, in addition to a 2-parameters fit (virial mass Mvir
and concentration cvir), we also show (MNFW) the results ob-
tained using the relation proposed by Bullock et al. (2001) be-
tween Mvir,cvir and the cluster redshift zcl:
cvir =K
1+zcl Mvir
M⋆!α
,(11)
with M⋆=1.5×1013/h M⊙,K=9, α=−0.13.
The shear profiles obtained from the different methods dis-
cussed below are displayed in Fig. 8; also displayed are the av-
erage values of tangential and radial components of shear, com-
puted in bins selected to contain at least 200 galaxies, and cen-
tered on the BCG: this is also where the peak of the X–ray emis-
sion is located (Rizza et al. 1998). In order to check the possible
error introduced by a wrong center, we considered a grid around
the position of the BCG, with a step of 2 arcsec: for each posi-
tion in the grid, we took it as the center, performed the fit and
derived the mass. Within 30 arcsec, we obtain that the rms is
σ(Mvir)<5%. The NFW parameters obtained by model fitting,
and the reduced χ2computed from the binned average tangential
shear, are given in Table 4.
The magnitude cut is the simplest approach as it only re-
quires photometry from the same band in which the lensing mea-
surement is done. Taking galaxies in the range 23 <R<26 mag
(a), produces a sample dominated by faint background galaxies,
but the inner regions of the cluster may still present an unknown
contamination by cluster galaxies.
To improve the selection, we proceeded as follows. The locus
of foreground galaxies was first found, allowing a better separa-
tion of different galaxy populations (see Medezinski et al. 2010,
and references therein), compared e.g. to methods based on the
selection of only red galaxies. Here we considered two colors
selections, namely B−zvs. V−z(b) and B−Vvs. V−I(c),
6

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
100 200 300 400 500 600 700
−0.05 0.00 0.05 0.10 0.15
θ (arcsec)
γ
mnfw
nfw
a
500 1000 1500 2000
r (kpc)
100 200 300 400 500 600 700
−0.05 0.00 0.05 0.10 0.15
θ (arcsec)
γ
mnfw
nfw
b
500 1000 1500 2000
r (kpc)
100 200 300 400 500 600 700
−0.05 0.00 0.05 0.10 0.15
θ (arcsec)
γ
mnfw
nfw
c
500 1000 1500 2000
r (kpc)
100 200 300 400 500 600 700
−0.05 0.00 0.05 0.10 0.15
θ (arcsec)
γ
mnfw
nfw
d
500 1000 1500 2000
r (kpc)
Fig.8. Shear profiles obtained with the different selection methods (see Table 4, where the parameters derived for each model are
given). The fitting was done using the maximum likelihood approach. Binned points are shown for display purposes only. In panel
a, the curves obtained by the two models (nfw/mnfw) overlap.
with 21 <R<26 mag. For intermediate redshifts (z∼0.2),
foreground and background objects are well separated in such
two colors diagrams. We explored the possibility to find the
best selection criteria based only on the observed colors, with-
out any information on the redshift of the galaxies. To this end,
we developed a semi-automatic procedure, implemented in the
R language (R Development Core Team 2010). We first selected
bright (R<21 mag) galaxies, which are expected to be mainly at
zph <0.2. A kernel density estimate, obtained by the kde2d pack-
age in R (Venables & Ripley 2002) was then applied to these
points, giving the plots displayed in Fig.9: the normalization was
done so that the value of the maximum density in the binned data
was equal to one. The boundary of the foreground galaxy region
was then defined by the points within the same density level l
(e.g., l=0.2). Such region was converted to a polygon using the
splancs3package in R, that also allows to select for a given cata-
log those sources whose colors lie inside or outside the polygon.
A comparison with the model colors obtained in ZEBRA from
the convolution of the spectral templates with filter transmission
curves, shows that colors inside the area selected in such way
are consistent with those expected for galaxies at redshift <0.2.
Galaxies classified as foreground in such way were therefore ex-
cluded from the weak lensing analysis.
We finally used photometric redshifts (d), both for the selec-
tion of background galaxies, defined as those with 21 <R<26
mag, 0.3<zph <3, and to compute the average value of β: we
3http://www.maths.lancs.ac.uk/˜rowlings/Splancs
obtain in such way β(zs)=0.74, in good agreement with the
value obtained from the COSMOS catalog with the same mag-
nitude and redshift selection, β(zs)=0.73.
The effect of the different selections on the residual pres-
ence of cluster galaxies is displayed in Fig. 11, showing the den-
sity of background galaxies computed in different annuli around
the cluster: a clear increase of the density in the inner regions
is visible in case a, which indicates that magnitude selection
alone does not allow to completely remove the contamination by
cluster galaxies. Such contamination is greatly reduced by color
selection, and the optimal result is given by photometric red-
shifts, as expected. As a further check, we also found for each
method that the tangential shear signals of the rejected ’fore-
ground/cluster’ galaxies average out. In the following discus-
sion, we take as reference the results from case d, which is very
close to cin terms of uncertainties on fitted parameters, density
of background galaxies and residuals in the radial component of
the shear.
It was pointed out in Hoekstra (2003) and Hoekstra et al.
(2010) that large-scale structures along the line of sight provide
a source of uncertainty on cluster masses derived by weak lens-
ing, which is usually ignored and increases as a larger radius
(θmax) is used in the fitting. The uncertainty introduced by such
component on the mass estimate can be ∼10-20% for a cluster
with M=1015M⊙at z∼0.2, θmax =10 arcmin as in our case
(see Fig. 6 and 7 in Hoekstra (2003)), which is comparable to
the uncertainties derived in the fitting.
7

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
Table 4. Best-fit NFW parameters: for each selection method, in the upper row both Mvir and cvir were taken as free parameters,
in the lower row the (Mvir,cvir,zcl) relation (Bullock et al. 2001) was used. The reduced χ2was computed from the binned values
displayed in Fig. 8. Also given are the values of M200: the larger uncertainties are due to the fact they were derived from the fitted
values (Mvir,cvir). The values of M2D before and after applying the factor 1.34 (Sect.4.2) are displayed in the upper and lower rows,
respectively.
Method Mvir cvir rvir M200 χ2/d.o.f. M2D Density
1014 M⊙arcsec 1014 M⊙1014 M⊙gal/arcmin2
a 7.783.34
−2.21 4.731.99
−1.49 69888
−73 6.447.94
−3.23 1.13 12.3 ±2.98 30
7.801.65
−1.52 4.710.13
−0.12 69946
−49 6.461.98
−1.68 1.13 9.20 ±2.22
b 7.612.94
−2.04 5.772.33
−1.69 69380
−69 6.426.85
−3.21 0.40 8.14 ±3.15 25
9.001.85
−1.69 4.620.13
−0.11 73347
−49 7.442.20
−1.86 0.70 6.07 ±2.35
c 7.512.90
−2.00 5.402.13
−1.52 69079
−68 6.306.15
−3.08 0.87 10.5 ±3.03 28
8.381.69
−1.57 4.670.13
−0.11 71645
−48 6.942.01
−1.73 0.61 7.82 ±2.26
d 7.542.66
−1.91 5.682.11
−1.60 691.0573
−64 6.356.45
−3.02 1.1 10.2 ±2.94 25
8.731.75
−1.60 4.640.12
−0.11 725.7746
−47 7.222.09
−1.77 1.90 7.59 ±2.19
Also displayed in Table 4 are the projected masses computed
by aperture densitometry from Eq. 10, at a distance from the
cluster center r=rvir, and θ2=900′′,θout =1000′′ . A good
agreement is found between these masses and the values com-
puted from parametric fits, if we take into account the expected
ratio M2D/M3D =1.34 (see Sect. 4.2).
From the catalog based on photometric redshifts selec-
tion (case d), we finally derived the S-map introduced by
Schirmer et al. (2004), that is: S=Map/σMap , where
Map =Piet,iwiQ(|θi−θ0|)
Piwi(12)
σ2
Map =Pie2
iw2
iQ2(|θi−θ0|)
2(Piwi)2,
The map was obtained by defining a grid of points along the
image; the tangential components et,iof the lensed galaxy ellip-
ticities were computed taking as center each point in such grid.
The weight wiwas defined in Eq. 7, and Qis a Gaussian function
as in Radovich et al. (2008):
Q(|θ−θ0|)=1
πθ2
sexp −(θ−θ0)2
θ2
s!,(13)
where θ0and θsare the center and size of the aperture (θs∼
1.5 arcmin). The S-map is displayed in Fig. 10, showing a quite
circular mass distribution centered on the BCG.
6. Discussion
Several mass measurements of this cluster are available in liter-
ature, based on different data and/or methods. Schmidt & Allen
(2007) used Chandra data and modeled the dark mat-
ter halo by a generalized NFW profile, obtaining a mass
value Mvir =9.16+1.89
−1.85 ×1014M⊙and a concentration value
Table 5. Luminosity function parameters, and uncertainties. Φ⋆
is in units of deg−2.
Best-fit Standard error
α-1.06 0.07
m⋆18.32 0.39
Φ⋆/1031.273 0.374
cvir =5.08+0.55
−1.03.
A weak lensing analysis of Abell 383 was done by
Bardeau et al. (2007) using CFH12K data in the B,R,Ifilters.
For the shape measurements they used a Bayesian method
implemented into the IM2SHAPE software. To retrieve the
weak lensing signal they selected the background galaxies
as those within 21.6<R<24.9 mag and (R−I)&0.7,
obtaining a number density of ∼10 gal arcmin−2. Their fit of
the shear profile by a NFW profile gave as result a mass of
M200 =4.19 ±1.46 ×h−1
70 1014M⊙at r200 =1.32 ±0.17h−1
70 Mpc
and a concentration value of c=2.62 ±0.69.
Another weak lensing mass estimate of Abell 383 from
CHF12K data was obtained by Hoekstra (2007) using two bands,
B(7200 s) and R(4800 s). The background sample was se-
lected by a magnitude cut 21 <R<24.5, from which cluster
red sequence galaxies were discarded. The remaining contam-
ination was estimated from the stacking of several clusters by
assuming that fraction of cluster galaxies fgc was a function of
radius ∝r−1. This function was used to correct the tangential
shear measurements. As discussed in Okabe et al. (2010), this
kind of calibration method does not allow to perform an unbi-
ased cluster-by-cluster correction. Assuming a NFW profile, the
fitted virial mass of Abell 383 was Mvir =2.81.6
−1.5×h−11014M⊙.
8

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
0.0 0.5 1.0 1.5
0.5 1.0 1.5 2.0 2.5 3.0
V−z
B−z
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
B−V
V−I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2
Fig.9. Color selection of foreground galaxies (zphot <0.2). The
contour in red is the density level chosen for the selection; the
points display the model colors computed by ZEBRA for this
redshift range.
Abell 383 belongs to the clusters sample selected for the
Local Cluster Substructure Survey (LoCuSS) project (P.I. G.
Smith). Within this project, a weak lensing analysis of this clus-
ter has been recently performed by Okabe et al. (2010) using
SUBARU data in two filters, i′(36 minutes) and V(30 min-
utes). In addition to a magnitude cut 22 <i<26 mag, they used
the color information to select galaxies redder and bluer than the
cluster red sequence. Looking at the trend of the lensing signal
as function of the color offset, they selected the sample where
the dilution were minimized, obtaining a background sample of
∼34 gal arcmin−2for the computation of tangential shear pro-
file of the cluster. The fit of this profile by a NFW model did
not give an acceptable fit: the virial mass computed assuming
a NFW profile was Mvir =3.62+1.15
−0.86 ×h−11014M⊙with a high
concentration parameter cvir =8.87+5.22
−3.05. The same authors also
derived the projected mass, obtaining M2D=8.69 ×h−11014 M⊙
at the virial radius.
Scaling such mass values to h=0.7, we derive Mvir ∼
(4 −5) ×1014 M⊙. This value is still consistent within uncer-
tainties with the value derived in the present analysis (Mvir =
(7.52.7
−1.9×1014 M⊙,cvir =5.72.1
−1.6, case d). In our case, we obtain
Fig.10. Weak lensing S-map showing the mass distribution de-
rived by weak lensing; overlaid is the central region of the Abell
383 field.
1 2 3 4 5 6 7
15 20 25 30 35
r (arcmin)
ρ
no sel
a
b
c
d
Fig.11. Density (gal/arcmin2) of background galaxies used for
the lensing analysis, as a function of the distance from the clus-
ter, for the different selection methods here considered. The case
with no selection is also displayed for comparison.
however a better agreement with both the values of Mvir and cvir
given by the X-ray data, and a better consistency between para-
metric and non-parametric mass estimates, once the projection
factor is taken into account. This may be due either to how the
selection of background/foreground galaxies was done, or to a
higher accuracy in the shape measurement as a combination of
the deeper image and calibration of the bias due to SNR (Sect. 4).
9

Zhuoyi Huang et al.: A weak lensing analysis of the Abell 383 cluster.
Fig.12. R-band LF of the galaxies in Abell 383. Data points are
derived by binning the data in magnitude bins of 0.5 mag and
error bars by Poissonian errors; the curve traces the best-fitting
LF and the shaded area marks the model uncertainty, obtained
by a bootstrap technique.
Finally, we computed the luminosity function (LF hereafter)
in the R-band and derived the total luminosity by integrating
the fitted Schechter (Schechter 1976) function as described in
Radovich etal. (2008). In order to obtain the cluster LF, that is
the number of galaxies per unit luminosity and volume belong-
ing to the cluster, we need to remove from our catalog all the
background and foreground galaxies. Usually, this is done by
statistically subtracting the galaxy counts in a control field from
galaxy counts in the cluster direction. Here we take advantage
of the selection on the color-color diagrams described in Sect. 5,
and extract a catalog that includes cluster, foreground and resid-
ual background galaxies. The last two components were further
removed by the statistical subtraction, where we defined as clus-
ter area the circular region around the cluster center of radius
r=10.2 arcmin (1.3Mpc); the control field was instead defined as
the area outside the circle of radius r=15.3 arcmin. For the best-
fitting procedure to the Schechter function, we adopted conven-
tional routines of minimization on the binned distributions. Best-
fitting parameters are listed in Table 5. The R-band total lumi-
nosity, calculated as the Schechter integral, is Ltot =(2.14 ±0.5)
×1012 L⊙. The errors were estimated by the propagation of the
68%-confidence-errors of each parameter.
For comparison, we then used the relation in Popesso et al.
(2007) between M200 and the optical luminosity, Lop, to see
whether the mass obtained in this paper is consistent with the
value expected for that luminosity. According to this relation,
the mass expected for such luminosity is M200 =(4.73 ±1.3)×
1014 M⊙, in good agreement with the value derived by our weak
lensing analysis, M200 ∼6.3×1014 M⊙(case d), corresponding
to M/L∼300M⊙/L⊙.
7. Conclusions
We have computed the cluster Abell 383 mass by weak lens-
ing, using a deep R-band image taken with the Suprime cam-
era on the Subaru telescope. Catalogs extracted from combined
CFHT+SUBARU uBVRIz images were used to derive photo-
metric redshifts, and improve the weak lensing analysis. The
data were reduced using a pipeline developed in–house, that was
specifically designed for wide-field imaging data. The elliptic-
ities, from which the shear signal was derived, were measured
using a pipeline based on the KSB approach. We discussed some
aspects that may improve the results, namely the size of the win-
dow used to suppress the noise from the outer part of the galax-
ies, the selection of a limit on SNR below which the measure-
ment ellipticity is not accurate enough, and a weighting scheme
where uncertainties on spatial fitting of the PSF correction terms
were taken into account.
The accuracy on the mass estimate by weak lensing available
with our KSB pipeline was first derived on simulated images
which were built in such a way to mimic as closely as possi-
ble the background noise and the depth of the real image. From
these simulations we conclude that the mass can be measured
with an uncertainty ∼5-10% for log M/M⊙≥14.5. Such ac-
curacy takes into account the measurement errors of the ellip-
ticity, but not the errors due to e.g. the foreground/background
galaxy separation, that may introduce an underestimate of the
mass. The impact of such selection was evaluated by comparing
three methods for the foreground/background galaxy separation,
namely: magnitude cut in one band, color selection and usage of
photometric redshifts. All methods gave consistent estimates of
the total virial mass, but from the shear profile it can be seen that
a dilution of the signal in the inner regions is still present, in the
case of a simple magnitude cut. Color selection and photometric
redshifts provide better results, even if the accuracy of the pho-
tometric redshifts is not high due to the few available bands. The
virial mass of Abell 383 here obtained by NFW model fitting is
in agreement with the value obtained from the non-parametric
mass estimate, that is Mvir ∼7×1014 M⊙. Other previous weak
lensing analyses give Mvir ∼(4−5)×1014 M⊙: the value found in
this paper seems more in agreement with the value found by X-
ray data, and we also have a better agreement between paramet-
ric and non-parametric estimates, compared to e.g. Okabe et al.
(2010).
Finally, we estimated the R-band LF of Abell 383, and de-
rived the total R-band luminosity of the cluster: starting from this
value and using the relation between mass and luminosity found
for clusters by Popesso et al. (2007), we conclude that the mass
derived by weak lensing is consistent with the value expected for
this luminosity.
8. Acknowledgments
L.F., Z.H. and M.R. acknowledge the support of the European
Commission Programme 6-th framework, Marie Curie Training
and Research Network “DUEL”, contract number MRTN-
CT-2006-036133. L.F. was partly supported by the Chinese
National Science Foundation Nos. 10878003 & 10778725, 973
Program No. 2007CB 815402, Shanghai Science Foundations
and Leading Academic Discipline Project of Shanghai Normal
University (DZL805), and Chen Guang project with No.
10CG46 of Shanghai Municipal Education Commission and
Shanghai Education Development Foundation. A.R. acknowl-
edges support from the Italian Space Agency (ASI) contract
Euclid-IC I/031/10/0. We are grateful to the referee for the useful
comments that improved this paper.
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