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Astronomy & Astrophysics manuscript no. 9731 c
ESO 2008
June 21, 2008
A weak-lensing analysis of the Abell 2163 cluster ?
M. Radovich1, E. Puddu1, A. Romano1, A. Grado1, and F. Getman2
1INAF - Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131, Napoli
2INAF - VSTceN, via Moiariello 16, I-80131, Napoli
received; accepted
ABSTRACT
Aims. We attempt to measure the main physical properties (mass, velocity dispersion, and total luminosity) of the cluster Abell 2163.
Methods. A weak-lensing analysis is applied to a deep, one-square-degree, r-band CFHT-Megacam image of the Abell 2163 field.
The observed shear is fitted with Single Isothermal Sphere and Navarro-Frenk-White models to obtain the velocity dispersion and the
mass, respectively; in addition, aperture densitometry is used to provide a mass estimate at different distances from the cluster centre.
The luminosity function is derived, which enables us to estimate the mass/luminosity ratio.
Results. Weak-lensing analyses of this cluster, on smaller scales, have produced results that conflict with each other. The mass and
velocity dispersion obtained in the present paper are compared and found to agree well with values computed by other authors from
X-ray and spectroscopic data.
Key words. Galaxies: clusters: individual Abell 2163 – Galaxies: fundamental parameters – Cosmology: dark matter
1. Introduction
Abell 2163 is a cluster of galaxies at z=0.203 of richness class 2
(Abell et al. 1989) and without any central cD galaxy (Fig. 1). It
is one of the hottest clusters known so far with an X-ray tempera-
ture of 14 keV and an X-ray luminosity of 6×1045 erg s−1, based
on Ginga satellite measurements (Arnaud et al. 1992). Elbaz
et al. (1995) used ROSAT/PSPC and GINGA data to map the gas
distribution; they showed that the gas extends to at least 4.6 Mpc
or 15 core radii and is elongated in the east-west direction; they
estimated a total mass (1.43±0.05)×1015 M(h=0.5) inside that
radius, which is 2.6 times higher than the total mass of Coma.
The corresponding gas mass fraction, 0.1 h−3/2, is typical of rich
clusters. The peak of the X-ray emission was found to be close
to a bright elliptical galaxy (α=16h15m49.0s,δ=−06◦0804100),
which was confirmed by later X-ray observations (Martini et al.
2007). Two faint gravitational arcs are visible close to this galaxy
(Fig. 1); the redshift of the source galaxies is zs∼0.73 (Miralda-
Escude & Babul 1995). The gas velocity dispersion is also very
high, σ=1680 km s−1(Arnaud et al. 1994); Martini et al. (2007)
derived a velocity dispersion of σ=1381±324 km/s from spec-
troscopic data.
ASCA observations of Abell 2163 (Markevitch et al. 1996)
measured a dramatic drop in the temperature at large radii: this
placed strong constraints on the total mass profile, assumed
to follow a simple parametric law (Markevitch et al. 1996).
Considerable gas temperature variations in the central 3-4 core
radii region were also found. The total mass derived inside
0.5h−1Mpc was (4.3±0.5) ×1014h−1M, while inside 1.5h−1
Mpc it was found to be (1.07 ±0.13) ×1015h−1M.
Send offprint requests to: M. Radovich: radovich@oacn.inaf.it
?Based on observations 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.
Fig. 1. r−band image of the Abell 2163 field. The zoomed image
shows the bright elliptical galaxy identified as the centre of the
cluster from X-ray maps: also visible are the faint gravitational
arcs.
Abell 2163 is remarkable also in the radio band: as first re-
ported by Herbig & Birkinshaw (1994), it shows a very extended
and powerful radio halo. Feretti et al. (2001) further investigated
the radio properties of the cluster. In addition to its size (∼2.9
Mpc), the halo is slightly elongated in the E-W direction; the
same elongation is also seen in the X-ray.
All of this evidence indicates that the cluster is unrelaxed and
has experienced a recent or is part of an ongoing merger of two
large clusters (Elbaz et al. 1995; Feretti et al. 2004). This was
confirmed by Maurogordato et al. (2008), who interpreted the
properties of Abell 2163 in terms of a recent merger, in which
the main component is positioned in the EW direction and a fur-
arXiv:0804.4652v3 [astro-ph] 21 Jun 2008
2 M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster
ther northern subcluster (Abell 2163-B) is related to the same
complex. They used optical and spectroscopic data to compute,
in addition, the virial mass, Mvir =(3.8±0.4) ×1015 Mand the
gas velocity dispersion, ∼1400 km s−1.
Squires et al. (1997) first performed a weak-lensing analysis
of Abell 2163 using a 2048 ×2048 CCD at the prime focus of
the Canada-France Hawaii Telescope (CFHT). They mapped the
dark matter distribution up to 70(∼1h−1Mpc); the mass map
showed two peaks, one close to the elliptical galaxy, the other at
30W. The mass obtained by weak lensing alone was a factor of
∼2 lower than that derived from X-ray data: they interpreted the
discrepancy in mass measurement as the result of an extension of
the mass distribution, beyond the edges of the CCD frame; tak-
ing this effect into account, a reasonable agreement is achieved
between the mass determined by X-ray and weak lensing. A fit
of the shear profile with that expected for a singular isothermal
model provided a velocity dispersion measurement of σ=740
km s−1, which was lower than the expected value σ > 1000 km
s−1.
Cypriano et al. (2004) completed a weak-lensing analysis of
Abell 2163 using FORS1 at the VLT in imaging mode. They
measured a higher velocity dispersion of σ=1021 ±146 km
s−1. They explained their disagreement with Squires et al. (1997)
by the fact that those authors used a bright cut (V>22 mag,
I>20.5 mag) for the selection of background galaxies, whereas
they chose R>23.3 mag.
Wide-field cameras, such as the ESO Wide-Field Imager
(WFI) with a field of view of 340×330, and the Megacam cam-
era mounted at the CFHT (∼1 square degree), are particularly
well suited for the weak-lensing study of clusters because they
enable the clusters to be imaged well beyond their radial extent.
We use public archive data of Abell 2163, acquired using the
Megacam camera, to complete a revised weak-lensing analysis
of this cluster and derive the luminosity function of the cluster
galaxies.
This paper is organized as follows. Section 2 describes the
data and steps followed in the reduction. The weak-lensing anal-
ysis and determination of mass are discussed in Sect. 3. Finally,
the cluster luminosity function is derived and the mass to lumi-
nosity ratio is computed in Sect. 4.
We adopt H0=70 km s−1Mpc−1,ΩΛ=0.7, Ωm=0.3,
which corresponds to a linear scale of 3.34 kpc/00 at the redshift
of Abell 2163.
2. Observations and data reduction
Abell 2163 was observed in 2005 with the Megacam camera at
the 3.6m Canada-France Hawaii Telescope in the r-band, with a
total exposure time of 2.7hr. The prereduced (bias and flat-field
corrected) images were retrieved from the Canadian Astronomy
Data Centre archive1. Before coadding the different exposures,
it was necessary to remove the effect of distortions produced
by the optics and by the telescope. This was completed us-
ing the AstromC package, which is a porting to C++ of the
Astrometrix package described in Radovich et al. (2004); we
refer the reader to this paper for further details. For each image,
an astrometric solution was computed, assuming the USNO-A2
catalog as the astrometric reference and taking into account the
positions of the same sources in the other exposures. The ab-
solute accuracy of the astrometric solutions with respect to the
USNO-A2 is limited to its nominal accuracy, ∼0.300; the internal
accuracy of the same sources detected in different images is far
1http://www4.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/cadc/
Fig. 2. Selection of galaxies used for the lensing analysis (area
with oblique lines) and of the stars used for the PSF correction
(area with horizontal lines) in the CFHT-Megacam r-band im-
age; rhis the half-light radius.
lower (∼0.0100), which enables us to optimize the Point Spread
Function (PSF) of the final coadded image. AstromC for each
exposure computed an offset to the zero point to take into ac-
count changes e.g. in the transparency of the atmosphere, relative
to one exposure that was taken as reference. Photometric zero
points were given already in the header of the images; Table 1
summarizes the photometric parameters. All images were resam-
pled according to the astrometric solution and coadded together
using the SWarp software developed by E. Bertin2. Finally, cat-
alogs of sources were extracted using SExtractor. Galaxies and
stars were selected by the analysis of the rhversus magnitude
diagram, where rhis the half-light radius (Fig. 2). The coadded
image was inspected to search for regions with spikes and halos
around bright stars. Such regions were masked on the image with
DS93and sources inside them were discarded from the catalog.
In addition, we did not use the outer part of the image, where
the PSF rapidly degraded. The residual available area is 1775
arcmin2.
We note that Abell 2163 is located in a region of high
Galactic extinction: from the maps by Schlegel et al. (1998), us-
ing the dust getval code4we obtain 0.27 <E(B −V) <0.43 in
the field, with an average value of E(B −V) =0.35. Such change
is significantly higher than the typical uncertainty in E(B −V)
(∼16%, Schlegel et al. 1998): it is therefore more appropriate to
correct the magnitude of each galaxy for the extinction value at
its position, rather than using the same average value.
3. Weak-lensing analysis
Weak-lensing relies on the accurate measurement of the aver-
age distortion produced by a distribution of matter on the shape
of background galaxies. As the distortion is small, the removal
of systematic effects, in particular the effect of the PSF both
from the telescope and from the atmosphere, is of crucial impor-
tance. Most of published weak-lensing results have adopted the
so–called KSB approach proposed by Kaiser et al. (1995) and
Luppino & Kaiser (1997). We summarize the main points here
2http://terapix.iap.fr/
3http://hea-www.harvard.edu/RD/ds9/
4http://www.astro.princeton.edu/∼schlegel/dust/
M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster 3
Table 1. Photometric calibration terms and magnitude limits computed for point-like sources at different signal to noise ratios in
the CFHT-Megacam r-band. Aris the average Galactic extinction, for E(B−V)=0.35. Megacam zero points are defined so that
magnitudes are already on the AB system, and are given here such that the airmass is 0.
Zero point Color term Extinction coeff.ArmAB(σ=3) mAB (σ=5) mAB(σ=10)
26.1 0.00×(g-r) 0.10 0.92 27.0 26.4 25.5
Fig. 3. PSF correction: the first three panels show the spatial pat-
tern of the observed, fitted and residual ellipticities of stars: a
scaling factor was applied for display purposes; Xand Yare
the pixel coordinates in the image. The last panel shows the ob-
served versus corrected ellipticities.
and refer to e.g. Kaiser et al. (1995), Luppino & Kaiser (1997),
and Hoekstra et al. (1998) for more detailed discussions.
In the KSB approach, for each source the following quanti-
ties are computed from the moments of the intensity distribution:
the observed ellipticity e, the smear polarizability Psm, and the
shear polarizability Psh. It is assumed that the PSF can be de-
scribed as the sum of an isotropic component (simulating the
effect of seeing) and an anisotropic part. The intrinsic ellipticity
esof a galaxy is related to its observed one, eobs, and to the shear,
γ, by:
eobs =es+Pγγ+Psm p.(1)
The term pdescribes the effect of the PSF anisotropy (starred
terms indicate that they are derived from measurement of stars):
p=e∗
obs/Psm∗.(2)
It is necessary to fit this quantity as it changes with the position
in the image, using e.g. a polynomial such that it can be extrap-
olated to the position of the galaxy. In our case, we find that a
polynomial of order 2 describes the data well.
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∗.(3)
As discussed by Hoekstra et al. (1998), the quantity Psh∗
Psm∗should
be computed with the same weight function used for the galaxy
to be corrected. For this reason, the first step is to compute its
Fig. 4. S-maps obtained by aperture densitometry and convolu-
tion with up: a Gaussian filter function (size: 5 arcmin); down:
the filter function proposed by Schirmer (2004). The contour
levels are plotted at S=(3,4,5,6,7), where Sis defined in
Sec. 3.1.1. The small box indicates the position of the elliptical
galaxy with arcs. The dashed contours show for comparison the
density distribution of cluster galaxies (see also Maurogordato
et al. 2008).
value using weight functions of size drawn from a sequence of
bins in the half-light radius rh. In many cases, Psh∗
Psm∗can be as-
sumed to be constant across the image and be computed from
the average of the values derived from the stars in the field. We
preferred to fit the quantity for the Megacam image, considering
its size, as a function in addition of the coordinates (x, y), us-
ing a polynomial of order 2. For each galaxy of size rh, we then
assumed the coefficients computed in the closest bin to finally
derive the value of Psh∗
Psm∗.
4 M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster
1 2 3
1
2
3
4
5
r (Mpc)
M(<r) [1015 Msun]
Fig. 5. Mass profile obtained by aperture densitometry.
The implementation of the KSB procedure is completed us-
ing a modified version of Nick Kaiser’s IMCAT tools, kindly
provided to us by T. Erben (see Hetterscheidt et al. 2007); these
tools enable measurement of the quantities relevant to the lens-
ing analysis, starting from catalogs obtained using SExtractor.
The package also enables us to separate stars and galaxies in the
rh-mag space and compute the PSF correction coefficients Pγ,
p. In addition, we introduced the possibility to fit Pγversus the
coordinates (x,y), as explained above, and for each galaxy used
the values of both Pγand pcomputed in the closest bin of rh.
Stars were selected in the range 17.5mag <r<21mag,
0.3700 <rh<0.4500, providing 2400 stars usable to derive the
quantities needed for the PSF correction. As discussed above,
these quantities were fitted with a polynomial both for PSF
anisotropy and seeing correction: we verified that the behaviour
of the PSF across the CCDs enabled a single polinomial func-
tion to be used for the entire image (Fig. 3). Galaxies used for
shear measurement were selected using the following criteria:
Pγ>0.25, νmax >5 , rh>0.4500, 23mag <r<25mag, and
ellipticities smaller than one. We finally obtained approximately
17000 galaxies, which implied that the average density of galax-
ies in the catalog was ∼8 galaxies/arcmin2.
The uncertainty in ellipticities was computed as in Hoekstra
et al. (2000):
w=1
σ2
γ
=Pγ2
Pγ2σ2
e0+∆e2,(4)
where D∆e2E1/2was the uncertainty in the measured ellipticity,
σe0∼0.3 was the typical intrinsic rms of galaxy ellipticities.
3.1. Mass derivation
Weak lensing measures the reduced shear g =γ
(1−κ). The con-
vergence κis defined by κ= Σ/Σcrit, where Σis the surface mass
density and Σcrit is the critical surface density:
Σcrit =c2
4πG
Ds
DlDls
=c2
4πG
1
Dlβ,(5)
Dls ,Ds, and Dlbeing the angular distances between lens and
source, observer and source, and observer and lens respectively.
In the weak lensing approximation, κ1, so that g∼γ.
However, the measured value of κincludes an unknown additive
constant (the so-called mass-sheet degeneracy): this degeneracy
200 400 600 800 1000
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
θ (arcsec)
γt
1000 2000 3000
r (kpc)
Fig. 6. The tangential component of the shear (dark points) is
displayed as a function of the distance from the assumed cen-
tre of the cluster. The lines show the result of the best-fit to the
unbinned data using: a NFW profile with cvir given by Eq. 17
(solid line), and a SIS profile (dashed line). Also shown is the
radial component of the shear (diamonds), which is expected to
be null in the absence of systematic errors.
can be solved by assuming either that κvanishes at the borders
of the image, or a particular mass profile for which the expected
shear is known. Both approaches are used here.
For Abell 2163, Σcrit =9.69 ×1013β−1Marcmin−2. In our
case, we were unable to assign a redshift to the source galax-
ies; we however assumed the single-sheet approximation, which
implies that the background galaxies lie at the same redshift
(King & Schneider 2001). To compute the value of the redshift,
we used the publicly-available photometric redshifts obtained by
Ilbert et al. (2006) for the VVDS F02 field with Megacam pho-
tometric data. We applied the same cuts adopted here for the
r-band magnitude data and assumed a Gamma probability dis-
tribution (Gavazzi et al. 2004):
n(z)=za−1exp(−z/zs)
Γ(a)za
s
,(6)
We found that a=2.04 and zs=0.57 were the best-fit parame-
ters, and Med(z) =0.98. We therefore adopted a median redshift
of z∼1, which provided hβi=0.58.
3.1.1. Mass aperture maps
Figure 4 displays the S-map introduced by Schirmer et al.
(2004), that is:
Map =Piet,iwiQ(|θi−θ0|)
Piwi
(7)
σ2
Map =Pie2
t,iw2
iQ2(|θi−θ0|)
2(Piwi)2,(8)
where et,iare tangential components of the lensed-galaxy ellip-
ticities computed by considering to be the centre the position in
the grid, wito be the weights defined in Eq. 4, and Qthe filter
function discussed below. The ratio S=Map/σMap , defined as
the S-statistics by Schirmer et al. (2004), provides a direct esti-
mate of the signal-to-noise ratio of the halo detection.
M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster 5
Table 2. Best-fit values obtained from the fit of the shear from the NFW model (c=4.09). For comparison, the lower rows give the
masses computed at the same radii from the best-fit SIS model (σv=1139+53
−56 km/s, θe=2200) and from aperture mass densitometry
(A.D.).
rvir Mvir r200 M200 r500 M500 r2500 M2500
(kpc) (1014 M) (kpc) (1014 M) (kpc) (1014 M) (kpc) (1014 M)
NFW 3012 ±170 22 ±4 2362 ±130 18 ±3 1514 ±90 12 ±2 623 ±40 4.2±0.7
SIS 18 ±5 14 ±4 9 ±2 4 ±1
A.D. 38 ±10 33 ±7 20 ±3 5.8±0.8
For the window function, we tested two possible forms: a
Gaussian function and a function close to a NFW profile.
The Gaussian window function is defined by:
Q(|θ−θ0|)=1
πθ2
c
exp −(θ−θ0)2
θ2
c!,(9)
where θ0and θcare the centre and size of the aperture.
Schirmer (2004) proposed a filter function, whose behaviour
is close to that expected from a NFW profile:
Q(x)=1+ea−bx +e−c+d x−1tanh(x/xc)
πθ2
c(x/xc),(10)
where x=(θ−θ0)/θc, and we adopted the following parameters:
a=6, b=150, c=47, d=50, xc=0.15 (Hetterscheidt et al.
2005).
In both cases, we found consistently that (i) the peak of the
lensing signal was coincident with the position of the bright el-
liptical galaxy with arcs (BCG hereafter), confirming that this
was in fact the centre of the mass distribution as indicated by
the X-ray maps; and (ii) the weak-lensing signal is elongated
in the E-W direction. A comparison with Fig. 5 and Fig. 9 in
Maurogordato et al. (2008) indicates that the mass distribution
follows the density distribution of early-type cluster galaxies
(the A1 and A2 substructures).
3.1.2. Aperture densitometry
We first estimate the mass profile of the cluster by computing the
ζstatistics (Fahlman et al. 1994; Clowe et al. 1998):
ζ(θ1)=¯κ(θ≤θ1)−¯κ(θ2< θ ≤θmax)=2Zθ2
θ1hγTidln θ(11)
+2
1−(θ2/θmax)2Zθmax
θ2hγTidln θ
The quantity Map (θ1)=πθ2
1ζ(θ1)Σcrit provides a lower limit
to the mass inside the radius θ1, unless ¯κ(θ2, θmax )=0. This for-
mulation of the ζstatistics is particularly convenient because it
enables a choice of control-annulus size (θ2,θmax) that satisfies
this condition reasonably well; in addition, the mass computed
inside a given aperture is independent of the mass profile of the
cluster (Clowe et al. 1998). Clowe et al. (2004) discussed how
the mass estimated by aperture densitometry is affected by as-
phericity and projected substructures in clusters, as in the case
of Abell 2163: they found that the error was less than 5%.
The cluster X-ray emission was detected out to a cluster-
centric radius of 2.2h−1Mpc (Squires et al. 1997), which corre-
sponds to ∼90000. We took advantage of the large available area
and chose θ2=130000,θmax ∼150000, which provided ∼3000
sources in the control annulus. The mass profile is displayed in
Fig. 5; the mass values computed at different radii are shown in
Table 2.
3.1.3. Parametric models
We consider a Singular Isothermal Sphere (SIS) and a Navarro-
Frenk-White (NFW) mass profile, for which the expected shear
can be expressed analytically. The fitting of the models is com-
pleted by minimizing the log-likelihood function (Schneider
et al. 2000):
lγ=
Nγ
X
1=1"|i−g(θi)|2
σ2[g(θi)] +2 ln σ[g(θi)]#,(12)
with σ[g(θi)] =(1 −g(θi)2)σe.
In the case of a SIS profile, the shear is related to the velocity
dispersion σby:
γt(θ)=2π
θ
σ2
c2
Dls
Ds
=θE
θ(13)
For the Navarro-Frenk-White (NFW) model, the mass profile
is (Wright & Brainerd 2000):
ρ(r)=δcρc
(r/rs)(1 +r/rs)2,(14)
where ρc=3H2(z)/(8πG) is the critical density of the universe
at the cluster redshift; rsis a characteristic radius related to the
virial radius by the concentration parameter cvir =rvir/rs;δcis a
characteristic overdensity of the halo:
δc=∆vir
3
c3
ln(1 +c)−c/(1 +c),(15)
∆vir ∼(18π2+82(ΩM(z)−1) −39(ΩM(z)−1)2)/ΩM(z).
The mass of the halo is:
Mvir =4
3π∆virρcr3
vir (16)
Bullock et al. (2001) used simulations of clusters to show
that the virial mass and the concentration are linked by the rela-
tion:
cvir =K
1+z Mvir
M?!α
,(17)
with M?=1.5×1013h−1M,K=9, α=−0.13.
Comerford & Natarajan (2007) computed the values of K
and αfitting Eq. 17 to the values of virial mass and concentra-
tion measured in a sample of 100 clusters; they adopted M?=
1.3×1013h−1Mand found K=14.5±6.4, α=−0.15 ±0.13,
which provides values of the concentration which are approxi-
mately 1.6 higher than obtained using the relation proposed by
Bullock et al. (2001). Given the large uncertainty in the value
of K, we preferred to adopt the values of Bullock et al. (2001).
We used the expression of the shear γt(r) derived by Bartelmann
6 M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster
Fig. 7. The r-band LF of Abell 2163 compared with other deter-
minations in the literature.
Table 3. Best-fit parameters and errors. a,b, and cdescribe the
shape of galaxy counts, whereas α,m?, and Φ?describe the
shape of the cluster LF. a,b,c, and Φ?are in units of deg−2.
The last two columns show the half width of the 68% and 95%
confidence range, with two degrees of freedom.
Best-fit 68% 95%
α-0.88 0.09 0.15
m?18.66 0.23 0.40
Φ?/1033.8
a3.19
b0.36
c-0.022
(1996); the minimization in Eq. 12 was completed using the
MINUIT package. Figure 6 shows the results of the fit and, for
comparison, the binned values of the tangential and radial com-
ponents of the shear: these are consistent with zero, as expected
in the absence of systematic effects. Table 2 shows the masses
obtained by model fitting (SIS and NFW), as well as those ob-
tained by aperture mass densitometry at different distances from
the BCG, which was assumed to be the centre of the cluster. In
addition to Mvir , the masses obtained for ρ/ρc=200,500,2500
and the corresponding radii r200,r500 and r2500 , are also dis-
played. The value of the virial mass, Mvir =(22 ±4) ×1014 M,
confirms Abell 2163 as a massive cluster compared to other clus-
ters (Comerford & Natarajan 2007).
4. Luminosity function
To compute the total r−band luminosity of the cluster and hence
the M/Lratio, we first derived its luminosity function (LF here-
after). The r−band magnitudes were corrected for Galactic red-
dening as explained in Sect. 2; no k-correction was applied be-
cause it is negligible at the redshift of Abell 2163, according to
Yasuda et al. (2001).
Fig. 8. Galaxy counts derived from the control field (empty
red circles), compared to literature. The triangles represent the
deep counts from the CFH12K-VIRMOS field (McCracken et al.
2003) corrected to the rAB according to Fukugita et al. (1995);
the stars mark the counts from the SDSS commissioning data
(Yasuda et al. 2001). The error bars of this work take into ac-
count only the Poissonian errors ( √n/area), whereas for the liter-
ature counts the errors are not displayed because they are smaller
than the point dimensions.
Fig. 9. Galaxy counts in the Abell 2163 line of sight (circles) and
in the control field (triangles: for display purposes, an offset in
rwas applied to these points). The lines show the result of the
joint fit. The errors bars were computed as described in the text.
We defined the cluster region to be the circular area encom-
passed within r200 (∼0.2 deg, see Table 2) and centred on the
BCG. Our control field of galaxies was assumed to be those
falling into the outer side (0.36 deg2) of a squared region cen-
tered on the BCG of area about 0.25 deg2.
With this choice, we are confident that we minimize the con-
tamination of cluster galaxies and take into account background
non-uniformities in the angular scale of the cluster. Figure 8
shows that the r−band galaxy counts in the control field are con-
sistent with those found in the literature.
The LF was computed by fitting the galaxy counts in the
cluster and control-field areas: we adopted the rigorous approach
M. Radovich et al.: A weak-lensing analysis of the Abell 2163 cluster 7
introduced by Andreon et al. (2005), which allows us to include,
at the same time, in the likelihood function to be minimized,
the contribution of both background and cluster galaxies. As the
model for the counts of the cluster field, we used the sum of a
power-law (the background contribution in the cluster area) and
a Schechter (1976) function, normalized to the cluster area Ωcl:
pcl,i= ΩclΦ?100.4(α+1)(m?−m)exp(−100.4(m?−m)) (18)
+Ωcl10a+b(m−20)+c(m−20)2.
For the control field, this reduces to the power-law only, nor-
malized to the background area Ωbkg:
pbkg,i= Ωbkg ∗10a+b(m−20)+c(m−20)2,(19)
where Φ?,α, and m?are the conventional Schechter parameters
as usually defined; a,b, and cdescribe the shape of the galaxy
counts in the reference-field direction; and the value of 20 was
chosen for numerical convenience.
Best-fitting parameters (see Table 3) were determined simul-
taneously by using a conventional routine of minimization on the
unbinned distributions. The data were binned for display pur-
pose only in Fig. 9, which shows the binned galaxy counts in
the control field (empty triangles) and cluster (empty circles) ar-
eas; the joint fit to the unbinned data sets is also overplotted.
Error bars are calculated to be √n/Ω. To check the effect of the
uncertainty in the position-dependent extinction correction (see
Sect. 2), we computed the same parameters in a set of catalogs
for which the extinction correction of each galaxy was randomly
modified within ±15%, the expected uncertainty in E(B−V)
(Schlegel et al. 1998): the rms uncertainty in the parameters de-
rived in this way is negligible compared to the uncertainties in
the fitting. Figure 7 displays the derived LF compared with se-
lected determinations from the literature, which have been con-
verted to our cosmology.
We compare our LF with Garilli et al. (1999) and Paolillo
et al. (2001), which both used data calibrated to the Thuan &
Gunn photometric system; according to Fukugita et al. (1995),
the offset between this magnitude system and the one used by us
is negligible.
Our determination of LF agrees well with Garilli et al.
(1999), within the 68% confidence level (see Fig. 7), and has a
value of M?that is consistent with that of Paolillo et al. (2001).
The r−band total luminosity was calculated to Ltot =
L?φ?Γ(2+α). The transformation from absolute magnitudes M?
to absolute luminosity L?, in units of solar luminosities, was
performed using the solar absolute magnitude, obtained using
the color-transformation equation from the Johnson-Morgan-
Cousins system to the SDSS system of Fukugita et al. (1995).
The errors were estimated by the propagation of the 68%-
confidence-errors of each parameter. In this way we found that
Ltot =(80 ±2) ×1011 L, which corresponds to M200 /Ltot ∼230
(M200 =1.8×1015 M, Table 2). Popesso et al. (2007) found
a relation between M200 and the r−band luminosity in 217
clusters selected from the Sloan Digital Sky Survey (see their
Eq. 6): according to this relation, the luminosity expected for
M200 =1.8×1015 Mwas L=(73 ±10) ×1011 L, in excellent
agreement with the observed value.
5. Conclusions
For the galaxy cluster Abell 2163, we have shown that by the
usage of wide-field imaging it is possible to achieve far bet-
ter agreement than before, between mass and velocity disper-
sion measured using weak-lensing and those derived for exam-
ple from X-ray data.
The dispersion velocity here measured, σ=1139+52
−55 km s−1,
agrees well with those derived by X-ray and spectroscopic data,
as found by Cypriano et al. (2004), whereas it was underesti-
mated in the previous analysis by Squires et al. (1997).
On the other hand, the comparison with the masses obtained
from X-ray measurements (Markevitch et al. 1996) shows that
at r∼2 Mpc, MX∼(1.5±0.2) ×1015 M(h=0.7); at the
same distance, we obtain from the NFW fit (Sect. 3.1.3) Mwl ∼
(1.6±0.3) ×1015 M. We therefore agree with Squires et al.
(1997) about the consistency of the mass obtained by weak lens-
ing and X-rays: no correction factor is required in our case due
to the larger field of view. We also find a substantial agreement
between our estimate of the virial mass, Mvir =(2.2±0.4) ×1015
Musing the NFW fit, and the value Mvir =(3.8±0.4) ×1015
Mobtained by Maurogordato et al. (2008) from optical and
spectroscopic data; in addition, as noted by these authors, their
estimate of the virial mass could be overestimated by 25%. Our
weak-lensing analysis also confirms that the mass distribution is
extended along the E-W direction, in agreement with that ob-
served in optical, radio and X-ray data (see e.g. Maurogordato
et al. 2008).
Finally, the r−band total cluster luminosity within r200, de-
rived from the luminosity function, gives Ltot /M200 =240. The
observed luminosity is in very good agreement with that ex-
pected for the mass measured by weak lensing, according to the
L−Mrelation proposed by Popesso et al. (2007).
Acknowledgements. We warmly thank Thomas Erben for having provided us the
software for the KSB analysis. E. Puddu thanks S. Andreon for useful sugges-
tions and comments about the LF determination. We are grateful to the referee
for his comments, which improved the paper. This research is based on observa-
tions made with the Canada-France Hawaii Telescope obtained using the facili-
ties of the Canadian Astronomy Data Centre operated by the National Research
Council of Canada with the support of the Canadian Space Agency. This research
was partly based on the grant PRIN INAF 2005.
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List of Objects
‘Abell 2163’ on page 1
