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Abell 611: I. Weak lensing analysis with LBC
ARTICLE in ASTRONOMY AND ASTROPHYSICS · FEBRUARY 2010
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Astronomy & Astrophysics manuscript no. A611˙Romano˙aph
February 3, 2010
Abell 611
I. Weak lensing analysis with LBC
arXiv:1002.0727v1 [astro-ph.CO] 3 Feb 2010
A. Romano1,2 , L. Fu3,12 , F. Giordano4,2 , R. Maoli1 , P. Martini5 , M. Radovich3,14 , R. Scaramella2 , V.
Antonuccio-Delogu6,13 , A. Donnarumma7,8,9 , S. Ettori8,9 , K. Kuijken10,3 , M. Meneghetti8,9 , L. Moscardini7,9 , S.
Paulin-Henriksson11,6 , E. Giallongo2 , R. Ragazzoni14 , A. Baruffolo14 , A. DiPaola2 , E. Diolaiti8 , J. Farinato14 , A.
Fontana2 , S. Gallozzi2 , A. Grazian2 , J. Hill16 , F. Pedichini2 , R. Speziali2 , R. Smareglia15 and V. Testa2
1
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Dipartimento di Fisica, Universitá La Sapienza, piazzale A. Moro 2, I-00185 Roma, Italy
e-mail: Anna.Romano@roma1.infn.it
INAF- Osservatorio Astronomico di Roma, via Frascati 33, 00044 Monte Porzio Catone (Roma), Italy
INAF- Osservatorio Astronomico di Napoli, via Moiariello 16, 80131 Napoli, Italy
Dipartimento di Fisica, Universitá Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy
Department of Astronomy and Center for Cosmology and Astroparticle Physics, The Ohio State University, Columbus, OH 43210,
USA
INAF- Osservatorio Astrofisico di Catania, Via Santa Sofia 78, I-95123 Catania, Italy
Dipartimento di Astronomia, Universitá di Bologna, via Ranzani 1, 40127 Bologna, Italy
INAF- Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy
INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy
Leiden Observatory, PO Box 9513, 2300 RA Leiden, The Netherlands
Service d’Astrophysique, CEA Saclay, Batiment 709, 91191 Gif-sur-Yvette Cedex, France
Key Lab for Astrophysics, Shanghai Normal University, 100 Guilin Road, 200234, Shanghai, China
Astrophysics, Department of Physics, University of Oxford, Oxford, UK
INAF- Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, I-35122 Padova, Italy
INAF- Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34131 Trieste, Italy
Large Binocular Telescope Observatory, University of Arizona, 933 N Cherry Avenue, 85721-0065 Tucson, Arizona
Received ..., 200?; ... 200?
ABSTRACT
Aims. The Large Binocular Cameras (LBC) are two twin wide field cameras (FOV ∼ 23′ × 25′ ) mounted at the prime foci of the 8.4m
Large Binocular Telescope (LBT). We performed a weak lensing analysis of the z = 0.288 cluster Abell 611 on g-band data obtained
by the blue-optimized Large Binocular Camera in order to estimate the cluster mass.
Methods. Due to the complexity of the PSF of LBC, we decided to use two different approaches, KSB and Shapelets, to measure the
shape of background galaxies and to derive the shear signal produced by the cluster. Then we estimated the cluster mass with both
aperture densitometry and parametric model fits.
Results. The combination of the large aperture of the telescope and the wide field of view allowed us to map a region well beyond
the expected virial radius of the cluster and to get a high surface density of background galaxies (23 galaxies/arcmin2 ). This made
possible to estimate an accurate mass for Abell 611. We find that the mass within 1.5 Mpc is: (8 ± 3) × 1014 M⊙ from the aperture mass
technique and (5 ± 1) × 1014 M⊙ using the model fitting by a NFW mass density profile, for both Shapelets and KSB methods. This
analysis demonstrates that LBC is a powerful instrument for weak gravitational lensing studies.
Key words. Cosmology: dark matter; Galaxies: clusters: individual; Gravitational lensing
1. Introduction
According to the hierarchical model of structure formation, clusters of galaxies are the most massive objects in the universe
and the cluster mass function is a powerful probe of cosmological parameters (e.g. Evrard 1989; Eke et al. 1998; Henry 2000;
Allen et al. 2004; Vikhlinin et al. 2009). In addition, the ratio between the cluster gas mass, as estimated with X-ray observations, and the total mass in a galaxy cluster provides stringent
constraints on the total matter density. Specifically, the apparent
evolution of the gas fraction with redshift can be used to estimate the contribution of the dark energy component to the cosmic density (e.g Allen et al. 2008; Ettori et al. 2009). The use of
clusters as a cosmological probe therefore requires reliable mass
estimates.
Several techniques are commonly used to estimate masses for
galaxy clusters: the X-ray luminosity or temperature of the hot
intracluster gas, the Sunyaev-Zel’dovich effect, the number of
bright galaxies in a cluster, and the velocity dispersion of the
cluster galaxies. The disadvantage of all of these methods is that
they are indirect and require significant assumptions about the
dynamical state of the cluster. Gravitational lensing, in contrast,
is only sensitive to the amount of mass along the line of sight and
allows reconstruction of the projected cluster mass regardless
its composition or dynamical behavior (e.g. Kaiser & Squires
1993). The only direct method to estimate cluster masses is
therefore via measurement of the distortion (shear) of the shapes
of background galaxies that are weakly lensed by the gravita-
2
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Fig. 1. A three-color image (2′ × 1.6′ ) of Abell 611 obtained
with LBC observations in the u-, g-, and r-bands. A giant arc
is clearly visible close the brightest cluster galaxy (α= 08h 00m
57s,δ=+36d 03’ 23”).
tional potential of the cluster.
This distortion is very small and lensed galaxies are usually at
high redshift. Observational studies to measure weak gravitational lensing by clusters require deep images in order to detect
these faint sources and to obtain a high number density of background galaxies. Moreover this kind of analysis requires very
high quality images in order to measure the shape of the lensed
sources with high precision: good seeing (< 1′′ ) conditions and a
high signal-to-noise ratio (typically > 10) are needed. Wide-field
images are also required to obtain a statistical measure of the tangential shear as function of distance from the cluster center so
that the projected mass measured by weak lensing includes essentially all of the mass of the cluster (Clowe & Schneider 2001,
2002).
In the last decade substantial progress has been made with weak
lensing studies thanks to the advent of wide-field data with linear
detectors, the development of sophisticated algorithms for shape
measurements (e.g. Kaiser et al. 1995, Bernstein & Jarvis 2002,
Refregier 2003, Kuijken 2006), and the availability of multiband photometry which provides information about the redshift
distribution of the lensed sources (Ilbert et al. 2006).
Here we describe the results of a weak lensing analysis of the
Abell 611 cluster. This analysis is based on images obtained with
the Large Binocular Camera (LBC), which are a pair of prime
focus cameras mounted on the two 8.4m diameter mirrors of the
Large Binocular Telescope (LBT). Each LBC has a 23′ × 25′
field of view (FOV) and, combined with the collecting area of
LBT, is a very powerful instrument for weak lensing studies.
Abell 611 is a rich cluster at redshift z = 0.288
(Struble & Herbert 1999) that appears relaxed in X-ray data, has
a regular morphology, and the brightest cluster galaxy (BCG)
is coincident with the center of X-ray emission (Donnarumma
et al. in preparation). A giant arc due to strong lensing is also
clearly visible close to the BCG (Fig. 1).
In this work we describe a weak lensing analysis to estimate the
mass of Abell 611 from a deep g-band LBC image whose field
of view extends well beyond the expected virial radius of the
cluster. We compare the mass estimated from gravitational lensing with previous lensing results and with other mass estimates
available in the literature that were derived by secondary tech-
niques. In particular we compared our weak lensing results with
new mass measurements obtained by X-ray analysis of Chandra
data provided to us by Donnarumma et al. (in preparation).
Mass measurements by weak lensing do not need any assumption about the geometry of the cluster; however, assumptions are
required to compare projected lensing masses with other mass
estimates. Thus projection effects have to be taken into account
during this kind of analysis, in particular the true triaxiality of the
halo (De Filippis et al. 2005, Gavazzi 2005) and the presence of
unrelated structures along the line of sight (Metzler et al. 2001,
Hoekstra 2007) can be sources of noise or bias on the projected
mass measurements.
The paper is organized as follows. In the first sections we
describe the data (Section 2) used for a weak lensing analysis of Abell 611, the catalog extraction of the background
sources (Section 3), and the selection of candidate cluster galaxies (Section 4). The two different pipelines used to extract the
shear signal from the images are described in Section 5, and their
results are compared in Section 5.3. Finally, both shear maps
are used to estimate the mass of the cluster with different techniques (Section 6). The results are summarized and discussed in
the Section 7.
Throughout this paper we adopt H0 = 70 km s−1 M pc−1 , Ωm =0.3,
and ΩΛ =0.7. At the distance of Abell 611, 11′ radius corresponds to a projected physical distance of nearly 3 Mpc.
2. Observations and Data Reduction
Abell 611 was observed in March 2007, during the Science
Demonstration Time (SDT) for the blue-optimized Large
Binocular Camera (LBC), which is one of the two LBCs built
for the prime foci of the LBT. The LBC focal plane consists
of four CCDs (2048 x 4608 pixels, pixel scale 13.5 µm, gain
∼ 2 e− /ADU, readout-noise ∼ 11 e− ). The CCDs are arranged
so that three of the chips are butted along their long edges and
the fourth chip is rotated counterclockwise by 90 degrees and
centered along the top of the other CCDs (see Fig.2). The field
of view is equivalent to 23′ × 25′ and provides images with a
sampling of 0.225′′/pixel. Because each LBC is mounted on a
swing arm over the primary mirror, the support structure lacks
the symmetry of most prime focus instruments. Moreover LBC
PSFs are dominated by optical aberrations from mis-alignments,
which can cause PSFs to not have bilateral symmetry. This is
a potential complication for the weak lensing analysis and we
discuss this point further below. More details about the characteristics of LBC are given in Giallongo et al. (2008).
The observations, collected in optimal seeing conditions
(FWHM ∼ 0.6′′ ), consisted of several sets of exposures of 5
minutes each in a wide u-band, SDSS g- and r-band filters. The
total exposure time was 1 hour in g, 15 minutes in r, and 30 minutes in u. Unfortunally some u-band observations were not usable, so the real total exposure time used for this band is 20 minutes. For the present work we used the deep, g-band data for the
weak-lensing analysis and the u- and r-band data to select cluster
galaxies. Each image was dithered by 5 arcseconds in order to
remove bad pixels, rows, columns, and satellite tracks. This offset is not large enough to fill the gaps between the CCDs, but the
analysis plan was to treat CCDs separately due to expected PSF
discontinuities at the chip boundaries. The offsets were therefore
kept small to maximize depth and uniformity. The images were
reduced by the LBT pipeline1 implemented at INAF-OAR. The
flat-field correction was done using both a twilight flat-field and
1
http://lbc.oa-roma.inaf.it
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
3
28
26
g [mag]
24
22
20
18
5
10
15
20
rh [pixels]
Fig. 2. g-band image of the full field of LBC, centered on Abell
611. The box marks the region of 5000 pixels we used for the
analysis.
Filter
Nexp
g
r
u
12
3
4
exptime
(s)
3600
900
1200
mag
(5σ)
28.0
26.0
27.2
mag
(20σ)
26.0
24.2
25.0
zero point
28.4
27.7
27.1
Table 1. Exposure times, limiting magnitudes for point-like
sources and zero points (AB) for the observations in each band.
a superflat obtained during the night. Moreover a geometric distortion correction was performed in order to normalize the pixel
size, which showed differences across the CCDs due to field distortions in the optics. The astrometric solution was computed
using the ASTROMC package (Radovich et al. 2008). This solution was then used to resample and coadd the images using the
SWarp2 software.
Standard fields for photometric calibration were not observed
during the SDT. So we used the values of zero points for each
band (Tab. 1) given by Giallongo et al. (2008). Table 1 also
shows the limiting magnitudes estimated from the faintest pointlike objects detected at the 5σ and 20σ level.
3. Catalog extraction
The detection of sources was performed using the SExtractor
package (Bertin & Arnouts 1996). Regions of the image presenting potential problems, such as spikes and halos around bright
stars, were masked by visual inspection; sources inside such regions were discarded from the final catalog. In addition, we removed sources located at the borders of each CCD, where the
SNR was lower due to the small dither offset. Finally, a very
bright star dominates one of the CCDs. We therefore decided to
limit our analysis to a box (displayed in Fig.2) with a size of
5000 pixels (corresponding to ∼ 18.7′) centered on Abell 611.
2
developed by E. Bertin, http://terapix.iap.fr
Fig. 3. Magnitude (g) vs. half-light radius (rh ) plane. Green zone
shows the background galaxies selected for the lensing analysis
(23 < g < 26, rh > 1.8 pixels) and pink zone the unsaturated
stars selected for correction of PSF anisotropy (20 < g < 23,
1.4 < rh < 1.8 pixels). See §3 for further details.
Starting from this box of 350 arcmin2 , the effective area used for
the analysis was ∼ 290 arcmin2 after removing all the masked
regions (30% due to bright stars, 70% due to regions between
adjacent CCDs with no data or low S/N).
The separation between stars and galaxies was performed in the
mag − rh plane, where magnitudes (mag) and half-light radii (rh )
were obtained from the MAG AUTO and FLUX RADIUS parameters computed by SExtractor. Unsaturated stars were selected on the vertical branch (see Fig. 3) in the range 20 < g < 23
mag and 1.4 < rh < 1.8 pixels. In this way we obtained 302 stars
for the PSF correction, with a SNR> 200 for the faintest ones.
For the lensing analysis, only background galaxies located at
redshifts larger than z = 0.288, the redshift of the cluster, should
be used. Unfortunately, in our case the number of available bands
does not allow us to estimate accurate photometric redshifts of
these faint galaxies. The selection of the background galaxies
was therefore done by choosing an adequate cut in apparent
magnitude.
The choice of the upper magnitude limit was based on the galaxy
redshift distribution obtained by Ilbert et al. (2006) from the
Canada-France-Hawaii Telescope Legacy Survey (CFHTLS),
which also used the SDSS photometric system. Taking into
account the accuracy of the photometric redshifts (3%) of
Ilbert et al. (2006), the approximations due to the different bands
they used compared to ours and the assumption that we have the
same galaxy distribution in our field, we chose 0.4 as redshift reference value to perform the magnitude cut. Galaxies with z ≤ 0.4
were assumed to belong to the cluster or be foreground galaxies.
This reference value was chosen to be larger than cluster redshift, in order to reduce the contamination of foreground galaxies as much as possible, taking into account the approximations
discussed above.
Figure 4 shows the fraction of the total CFHTLS sources at
z ≤ 0.4 (upper panel) and the fraction of background galaxies
at z > 0.4 (bottom panel) as a function of the apparent magnitude cut. From this figure we conclude that a magnitude cut
at g > 23 is a good compromise to minimize the contamination
from likely foreground and cluster galaxies (∼ 10%) and to maximize the number density of background galaxies (∼ 98%). The
faint magnitude cut was chosen at g < 26, that is the magnitude
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
1.5
0.0
0.5
1.0
g−r
2.0
2.5
3.0
4
17
18
19
20
21
22
23
r
Fig. 4. The fraction of likely foreground and cluster galaxies
(top-panel) and background galaxies (bottom-panel) from the
CFHTLS versus magnitude cut in g-band. The dotted vertical
line is the lower limit of the magnitude cut adopted to select
background galaxies. See §3 for details.
limit where we have a signal-to-noise ratio SNR > 10 for the
sources, where SNR is defined as FLUX/FLUX ERR as measured by SExtractor. The final catalog contained 8134 background galaxies.
4. Candidate cluster members
The candidate cluster members were selected from the simultaneous usage of u-, g- and r- band photometry. To this end
catalogs were extracted from these images, running SExtractor
in dual-mode with the g-band image as detection image. Only
sources detected in all three bands were used.
We then applied an algorithm (Fu et al., in preparation) similar to the C4 Clustering Algorithm (Miller et al. 2005). The algorithm is based on the assumption that galaxies in a cluster should
have similar colors and locate together in space. It evaluates the
probability of each galaxy to be field-like: hence candidate cluster galaxies are those for which this probability is below a given
threshold, as outlined below.
1. Each galaxy was set in a four-dimensional space of α, δ,
u − g and g − r. For each galaxy (named “target galaxy”),
we counted the number of neighbors within the fourdimensional box, Ntarget . The angular size of the box was set
to 1 h−1 Mpc (∼ 5.5′ at the redshift of Abell 611). The sizes
of the boxes in two color dimensions were determined as
q
δmn = σ2mn (stat) + σ2mn (sys),
(1)
where σmn (stat) is the observed error for two magnitudes
(m, n), and σmn (sys) is the intrinsic scatter of the color m − n.
For LBC A611 data, δug , δgr are 0.49 and 0.31 respectively.
2. This four-dimensional box was placed on 100 randomly
chosen galaxy positions and at each position we counted
the number of neighbors. These randomized number counts
Fig. 5. Colour-magnitude plot of the galaxies in the Abell 611
field. Red points are the candidate cluster galaxies selected by
the C4 method. Also displayed (solid line) is the results of the
biweight regression fit.
constructed a distribution of counts for the target fourdimensional box. This distribution is represented by the median value Nmedian of the randomization counts.
3. The probability p of the target galaxy being field-like was
derived by comparing the target galaxy count Ntarget to the
distribution of randomization values Nmedian .
4. The distribution of p values was derived by repeating the
above steps for all galaxies. We ranked the p values from
smallest to highest and derived the value after which p starts
to rise significantly. In this way we identified ∼ 150 galaxies
at r < 23 mag as the candidate cluster members. We further
removed outliers in the g − r vs u − g diagram, leaving 125
candidate members.
Starting from these galaxies, we fitted the g − r vs. r Red
Sequence (g − r = a + b · r) using a biweight regression method
+0.02
(Fig. 5), and obtained: a = 2.39+0.61
−0.45 , b = −0.04−0.03 .
5. Weak lensing analysis
Weak lensing is based on the measurement of the coherent distortion of the shapes of background galaxies produced by a distribution of matter. This distortion is very small and it is unmeasurable for any single background galaxy because the galaxy
intrinsic ellipticity is not known. The dispersion
√ of intrinsic ellipticity is a source of noise which is ∝ σǫ / N. A statistical
approach is therefore required, where the distortion can be measured for a large number of sources in order to bring down that
noise. This requires a careful treatment of systematic effects, as
the shapes of galaxies may also be affected by contributions to
the point spread function (PSF) by both the telescope and the
atmosphere. In the last decade several methods have been developed for this kind of analysis. The most popular is the KSB approach, originally proposed by Kaiser et al. (1995) and improved
by Luppino & Kaiser (1997) and Hoekstra et al. (1998). Several
different implementations of this method exist in the literature
and have been used in a number of KSB analysis pipelines.
More recently, Refregier (2003) and Massey & Refregier (2005)
have proposed a new method based on Shapelets. Several available pipelines have also used this approach to measure the shear
5.1. KSB method
We used the weak lensing pipeline described in Radovich et al.
(2008) to compute the quantities relevant to the lensing analysis.
This pipeline implements the KSB approach using a modified
version of the IMCAT3 tools that was provided to us by T. Erben
(Erben et al. 2001 and Hetterscheidt et al. 2007).
In the KSB approach stars and galaxies are parametrized according to the weighted quadrupole moments of the intensity distribution using a Gaussian weight function whose scale length is
the size of the source (the formalism is described in Kaiser et al.
1995). The main assumption of this approach is that the PSF can
be described as the sum of a large isotropic component (seeing)
and small anisotropic part. In this way the observed ellipticity
eobs can be related to the intrinsic source ellipticity e s and shear
γ by the relation:
eobs = e s + Pγ γ + P sm p,
(2)
where P sm is the smear polarizability tensor and Pγ is the preseeing polarizability tensor, which is related to the shear polarizability tensor P sh and P sm by Eq. 5 (see also Hoekstra et al.
1998). The quantity p characterizes the anisotropy of the PSF
and is estimated from stars, which have zero intrinsic ellipticity:
p∗ =
e∗obs
.
P sm∗
(3)
If we average over a large number of sources, assuming a random orientation of the unlensed galaxies, we expect he s i = 0 and
so
eiso
γ = h γ i,
(4)
P
3
http://www.ifa.hawai.edu/∼kaiser/imcat
10
9
Psh Psm
8
7
ccd 1
ccd 2
ccd 3
ccd 4
6
signal in various ways (e.g. Massey & Refregier 2005, Kuijken
2006).
One of the key differences between these two approaches is the
treatment of the PSF. The KSB method assumes that the PSF
can be written as a convolution of a very compact anisotropic
kernel with a more extended, circular function. These two terms
are expressed in terms of the quadrupole moments of the surface brightness. As summarized below, the isotropic component
is subtracted from the measured ellipticity and the anisotropic
component is subtracted from a responsivity term. In contrast,
in the Shapelets approach there is no assumption about the PSF
shape. Individual galaxy images are decomposed into a complete orthonormal basis set consisting of Hermite (or Laguerre)
Polynomials and the PSF correction is performed through deconvolution.
Since the PSF of LBC presents a significant deviation from symmetry, the analysis of Abell 611 presents a very good opportunity to compare the results produced by these two methods. For
this comparison we started with the same initial catalogs of stars
and galaxies for both pipelines. Specifically, we only considered
sources with a SExtractor FLAG < 4, which removes sources
that are possibly blended. As the subsequent steps performed by
each algorithm are different, the same galaxy may be rejected
by one algorithm but not by the other. The result is that different output catalogs are produced by the KSB and Shapelets
pipelines. To have a homogeneous comparison, we therefore finally selected only the sources common to both output catalogs.
Sources with an unphysical ellipticity |e| > 1 were also removed
from the final catalogs.
5
11
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
2
3
4
5
6
rh (pixels)
Fig. 7. P sh∗ /P sm∗ values computed in different bins of rh , for each
CCD. These values are lower for the central CCD (#2) where the
quality of PSF is better.
where eiso = eobs − P sm p∗ is the ellipticity corrected for
anisotropic distortions.
We computed the contribution due to the PSF anisotropy (Eq.
3) from the selected stars. This quantity changes with position
in the image, so we needed to fit it on each CCD in order to
extrapolate its value at the position of the galaxy we want to
correct. In our case we performed this fitting on each CCD and
a second-order polynomial fit was sufficient.
Fig. 6 displays the spatial pattern of the ellipticities for the stars
in all CCDs, before and after the PSF correction.
After that we computed Pγ for each source:
Pγ = P sh − P sm
P sh∗
.
P sm∗
(5)
As the PSF correction was done separately on each CCD, we
sh∗
decided not to fit PPsm∗ as a function of the coordinates and instead
we took an average value.
Stellar ellipticities should be computed using the same
weight function used for galaxies (see Hoekstra et al. 1998), so
we considered a sequence of bins in rh and for each galaxy we
selected the PSF correction terms computed in the closest rh bin.
sh∗
Fig. 7 shows the PPsm∗ values computed in different bins of rh , for
each CCD.
We weighted the shear contribution from each galaxy according
to:
Pγ2
,
(6)
w = γ2 2
P σe0 + h∆e2 i
as in Hoekstra et al. (2000), where σe0 ∼ 0.3 is the intrinsic rms
of galaxy ellipticities, h∆e2 i1/2 is the uncertainty in the measured
ellipticity, where the average is computed on both components
of ellipticity for each galaxy (see eq. A8, A9 in Hoekstra et al.
(2000)).
A crucial point in this kind of study is the selection of the galaxies to use for the shear analysis, as the contamination of foreground galaxies can dilute the lensing signal and lead to an underestimate of the mass.
6
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Fig. 6. Removal of PSF anisotropy in the KSB approach. The observed (top-left), fitted (top-right) and residual (bottom-left) ellipticities of stars for all CCDs. The observed (black points) versus corrected (green points) ellipticities are also shown (bottom-right).
Since the PSF degrades somewhat at the borders of the image,
we limited our analysis to an 18.7′ × 18.7′ box centered around
the BCG, as discussed in Section 3. After that we filtered the
source catalog using the following criteria: Pγ > 0.25, SNR
> 10, rh > 1.8 pixels, 23 < g < 26, ellipticities smaller than
one, obtaining a surface density of ∼ 25 galaxies/arcmin2.
The cut Pγ > 0.25 allowed us to discard sources that appeared
too circular (e.g. stars incorrectly classified as galaxies). We considered only galaxies with SNR > 10 in order to avoid noisy objects, which can be a source of error in the computation of the
shear signal. Finally, we used the magnitude cut 23 < g < 26
to select background galaxies, whose choice was previously explained in Section 3.
5.2. Shapelets
Another approach for weak lensing analysis is the use of
Shapelets, which are basis functions constructed from twodimensional Hermite polynomials weighted by a Gaussian. The
translation, magnification, rotation and shear of astronomical
images can be expressed as matrices acting on Shapelets coefficients. The advantage of Shapelets is that a galaxy image can be
described in reverse order: pixelation, convolution with the PSF,
and finally distortion by shear. Shapelets have a free scale radius β which is the size of the Gaussian core of the functions. Its
truncated expansion describes deviations from a Gaussian over
a particular range of spatial scales, which widens with order N.
For the analysis of the LBC data of Abell 611 we used the
Shapelets pipeline developed by Kuijken (2006), starting with
the same stars and galaxies used in the KSB method. Each star
was first fit with a circular Gaussian and the median radius was
computed. This radius was then multiplied by a factor of 1.3 for
the Shapelets fits, which was found by Kuijken (2006) to work
well for a range of model PSFs up to Shapelets order N = 8.
Then we obtained a Shapelets description of the PSF for each
star. In order to estimate the PSF model at the position of each
galaxy, the Shapelets coefficients were interpolated by a fourthorder polynomial on the whole image frame (Fig. 8). The residual of the PSF model fitting is shown in Fig. 9.
The ellipticity of each source is then determined by least-squares
fitting a model, which is expressed as the shear applied to a circular source to fit the object optimally. The extension order of
Shapelets for galaxies is taken the same as for stars. Performing
the least-squares fit, the minimum of χ2 can be found in a few
Levenberg-Marquardt iterations (Press et al. 1986). The errors
of Shapelets coefficients for each source are derived from the
photon noise, and these can be propagated through in the χ2
function. Thus the error of shear measurement σγ is calculated
from the covariance matrix which is given by the second partial
derivatives of χ2 at the best fit.
The shear contribution from each galaxy is weighted according
to:
σ2
w = 2 e0 2 ,
(7)
σe0 + σγ
which combines the error in the shear measurements σγ and the
intrinsic scatter σe0 . The last quantity was computed from the
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
1673
3280
4886
Y=
891
2475
4058
Y=
891
6432
X=
7
X=
602
1130
2845
3915
5077
6332
Fig. 8. Shapelets PSF models interpolated at different positions on each CCD. The distribution of these models corresponds to their
actual placement on the CCD mosaic, CCD1 to CCD3 from left to right in the bottom panel and CCD4 in the top. Contours are the
representations of PSF shape decomposed by Shapelets. The X and Y values correspond to the pixel position.
5.3. Comparison between KSB and Shapelets ellipticities
After matching the two output catalogs, our final background
galaxy sample has a surface density of ∼ 23 galaxies/arcmin2.
In weak lensing studies, the background number density
of ground-based telescope usually is around 15 to 20
galaxies/arcmin2 (e.g. Paulin-Henriksson et al. 2007). It depends
not only on the size of telescope, the exposure time, the color filter, seeing condition, but also on galaxy selection criteria.
The maps of the PSF correction computed by both methods show
a good-quality PSF in the central regions that then degrades further from the center of the field. Nevertheless the final correction
is good with fluctuations in the PSF anisotropy less than 1%.
Figure 10 compares the first component of the shear γ1 measured by the KSB and Shapelets methods. It shows good agreement in the range of −0.5 < γ1 < 0.5. Some scatter is present
1
0.5
γ1 (Shapelets)
distribution of ellipticity components of galaxies.
Finally, we removed galaxies with SNR < 10 and that failed the
Shapelets expansion and radial profile cuts defined by Kuijken
(2006). This eliminated galaxies that were not well detected or
measured, providing a number density of ∼ 26 galaxies/arcmin2.
Further details about the selection criteria are available in
Kuijken (2006).
0
−0.5
−1
−1
−0.5
0
γ1 (KSB)
0.5
1
Fig. 10. Comparison between the first component of the shear
from the Shapelets and KSB methods (results with the second
components are very similar).
8
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Fig. 9. The star at the position near the center of each CCD are taken as an example to show the residual of PSF model fitting (left
column). The real PSF shape and fitted PSF model are shown in the middle and right columns.
1.5
1.4
1.3
1.2
1.1
w
for very elongated galaxies, but the fraction of these galaxies is
less than 5% in the final, common catalog. As shown in Fig. 11,
these strongly elongated galaxies are down-weighted nearly by
a factor of 2 compared to small ellipticity galaxies, so that they
do not affect the final mass measurements of the cluster (as described further below). A similar behavior is seen for the second
component γ2 . These matched catalogs were used to compute
estimates of the cluster mass as described in the next sections.
1
0.9
6. Mass Measurements
0.8
The relationship between the shear γ and the surface mass density is:
Z
1
γ(θ) =
d2 θ′ D(θ − θ′ )κ(θ′ ),
(8)
π R2
where: D
c2 D s
4πG Dl Dls
−1
= (θ1 −iθ
2 , κ ≡ Σ/Σcr is the convergence,
2)
c2 1
4πG Dl β is the critical density of the cluster, and
Σcr =
D s , Dl
=
and Dls are source-observer, lens-observer and lens-source distances respectively. While weak lensing actually measures the
γ
, for κ ≪ 1, g ∼ γ.
reduced shear g = 1−κ
An optimal approach for computing the mass requires knowledge of the redshift of each background galaxy. As we do not
have this information, we assumed that the background sources
all lie at the same redshift according to the single sheet approximation (King & Schneider 2001). An estimate of the redshift
value to use for the weak lensing analysis was computed using the first release of photometric redshifts available for the D1
0.7
0.6
0.5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
|e1|
1
Fig. 11. The plot of the normalized galaxy weight as function
of the absolute first component of ellipticity for KSB measurement. The applied normalization was Σwi ×δe1 = 1, where δe1 is
the constant bin width of e1 . The Shapelets measurement shows
similar down-weight for elongated ellipticity galaxies.
deep field of the CFHTLS, adopting the magnitude cut in the gband chosen here for the background galaxies selection (23 <
In Figure 12 we plot the so-called S-maps (Schirmer et al. 2004)
for these data. S-maps are computed as the ratio S = Map /σ Map
where:
Σi et,i wi Q(|θi − θ0 |)
(10)
Map =
Σi wi
Σi e2t,i w2i Q2 (|θi − θ0 |)
,
(11)
σ2Map =
2(Σi wi )2
For this calculation the image is considered as a grid of points,
et,i are the tangential components of the ellipticities of the lensed
galaxies, which are computed by considering the center of each
point of the grid, wi is the weight as defined in Equation 6, and Q
is a window function, chosen to be a Gaussian function defined
by:
!
1
(θ − θ0 )2
Q(|θ − θ0 |) = 2 exp −
(12)
πθc
θc2
where θ0 and θc are the center and the size of the aperture. The
ratio S = Map /σ Map provides an estimate of the SNR ratio of
the dark matter halo detection. S-maps are discussed further by
Schirmer et al. (2004).
We computed these maps using shear catalogs obtained from
both the KSB and Shapelets pipelines. In both cases (see Fig.
12) the maps show that the lensing signal is peaked around the
BCG, confirming that this is indeed the center of the mass distribution. The mass distribution also appears quite regular, which
is in agreement with what is indicated by the X-ray maps. In
Figure 13 S-map contours are overlaid on the r-band luminosityweighted density distribution of the red sequence galaxies of
Abell 611, selected in Section 4, showing that the mass distribution follows that of the red cluster galaxies.
6.2. Aperture densitometry
In order to trace the mass profile of the cluster, we computed the
ζ-statistic described in Clowe et al. (1998) and Fahlman et al.
(1994):
Z θ2
hγT id ln θ
(13)
ζ(θ1 ) = κ̄(θ ≤ θ1 ) − κ̄(θ2 < θ ≤ θmax ) = 2
θ1
+
2
1 − (θ2 /θmax )2
Z
θmax
θ2
hγT id ln θ,
36.07
36.06
δ
36.03
36.04
36.05
4
6.1. S-Maps
3.5
The convergence κ gives an estimate of the surface mass
density apart from an unknown additive constant – the so-called
mass-sheet degeneracy. We tried to solve this degeneracy using
two different approaches: assuming either that κ vanishes at the
borders of the image or a particular mass profile whose expected
shear profile is known.
4.5
We computed this factor using the CFHTLS catalog of photometric redshifts and obtained hβ2 i/hβi2 = 1.167.
5
g < 26 mag; see Section 3) and assuming a Gamma probability
distribution (Gavazzi et al. 2004). This yielded a median redshift
z = 1.05 and for our analysis we assume all of the background
galaxies lie at z ∼ 1, corresponding to hβi = hDls /D s i ∼ 0.65.
As discussed by Hoekstra (2007), the single-sheet approximation results in an overestimate of the shear by a factor
" 2
#
hβ i
1+
− 1 κ.
(9)
hβi2
9
36.08
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
120.28
120.26
120.24
120.22
120.20
α
Fig. 13. r-band luminosity-weighted density distribution of red
sequence galaxies of Abell 611. The overlaid contour (black
lines) is the S-map (computed and discussed in Section 6.1)
showing the SNR of the shear signal around the cluster obtained from the Shapelets analysis. The levels are plotted between σmin = 3.5 and σmax = 5.
Mass measurements are computed within different apertures of
increasing radius using a control annulus far from the center of
the distribution (the BCG). Map = πθ12 ζ(θ1 )Σcr is the mass within
the last aperture and provides a lower limit on the mass, unless
the value in the control annulus is equal to zero. This method
allows us to choose the size of the annulus that satisfies the desired condition; moreover it has the advantage that this mass
computation does not depend on the mass profile of the cluster (Clowe et al. 1998).
We chose 30′′ ≤ θ ≤ 500′′ and θmax = 600′′ , which yielded
projected masses within ∼ 1500 kpc of 7.7 ± 3.3 × 1014 M⊙ and
8.4 ± 3.8 × 1014 M⊙ using the KSB and Shapelets shear catalogs,
respectively.
Unfortunately we could not extend our analysis further from the
center of the cluster because of the presence of a very bright star
in the field that made the outer regions unusable.
Since weak lensing is sensitive to the total mass along the line
of sight, the observed aperture mass is the sum of the mass of
the cluster and any contribution from other, uncorrelated structures along the line of sight. This contribution is assumed to be
negligible in the central regions of the cluster, which are much
denser, and become more relevant in its outer regions. As discussed by Hoekstra (2001), the effect of this contribution does
not introduce any bias, but it does add a source of noise to the
lensing mass. Aperture densitometry is more affected by this uncertainty than parametric methods because it is sensitive to the
lensing signal at large radii. Nevertheless, for observations of
rich clusters at intermediate redshifts, this uncertainty is fairly
small because the bulk of the background sources are at much
higher redshifts than the cluster.
6.3. Model fitting
The model fitting approach for estimating the mass of a cluster
consists of assuming a particular analytic mass density profile
for which calculate the expected shear and then fitting the observed shear with the model by minimizing the log-likelihood
10
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Fig. 12. S-maps obtained from the KSB (left panel) and the Shapelets (right panel) analysis. The levels are plotted between σmin =
3.5 and σmax = 5. They are overplotted on a g-band greyscale image (∼ 4 arcmin) of the center of the field of Abell 611.
function (Schneider et al. 2000):
lγ =
Nγ "
X
|ǫi − g(θi )|2
i=1
σ2 [g(θi )]
#
+ 2 ln σ[g(θi )] ,
(14)
with σ[g(θi )] = (1 − g(θi )2 )σe .
In this analysis we assumed both a Singular
Isothermal Sphere (SIS) and a Navarro-Frenk-White
(Navarro, Frenk & White 1997) model.
In the SIS model the density profile depends on one parameter,
the velocity dispersion σ:
ρS IS (r) =
σ2 1
.
2πG r2
(15)
In this profile the shear is found to be related to σ by:
γT (θ) =
2π σ2 Dls θE
= .
θ c2 D s
θ
(16)
(e.g. Bartelmann & Schneider 2001).
The mass density profile predicted by the Navarro-Frenk-White
model (hereafter NFW) is:
ρNFW (r) =
δc ρc
(r/r s )(1 + r/r s )2
(17)
where ρc = 3H 2 (z)/(8πG) is the critical density of the universe at
the cluster redshift, r s is a scale radius related to the virial radius
by means of the concentration parameter cvir = rvir /r s and δc is
a characteristic overdensity of the halo:
δc =
∆vir
c3
.
3 ln(1 + c) − c/(1 + c)
(18)
where ∆vir is the virial overdensity, approximated by ∆vir ∼
(18π2 +82(Ω M (z)−1)−39(Ω M (z)−1)2 )/Ω M (z) using the spherical
collapse model (Bryan & Norman 1998) for flat cosmologies.
The mass of the halo is:
Mvir =
4
3
π∆vir ρm rvir
.
3
(19)
where ρm = ρc · Ω M (z) is the mean density at the cluster
redshift. We solved for the mass of the cluster with the expression for the shear γT (r) derived by Bartelmann (1996) and
Wright & Brainerd (2000) and minimized Equation 14 with the
MINUIT package.
Table 2 shows the best-fit parameters and the mass values derived by model fitting. The NFW profile was used keeping
both the concentration and mass as free parameters (marked as
NFW), and by using the relation between cvir and Mvir proposed
M α
K
vir
,
by Bullock et al. (2001) (hereafter MNFW): cvir = 1+z
M∗
13
where M∗ = 1.5 × 10 /h M⊙ , K = 9 and α = −0.13.
These values were computed assuming spherical symmetry for
the cluster halo. The effect of departures from spherical symmetry (e.g. triaxial halos) on the determination of the total cluster mass have been studied by several authors (e.g. Gavazzi
2005, De Filippis et al. 2005). De Filippis et al. (2005) showed
that these effects are negligible when the mass is computed at
large distances from the cluster center, although they are important at small radii. The same authors tried to recover a threedimensional reconstruction of Abell 611 through a combined
analysis of X-ray and Sunyaev-Zel’dovich observations and concluded that the cluster was approximately spherical, supporting
our symmetry assumption.
In Figure 14 the results of fitting by a NFW model (dashed
lines), a constrained (M)NFW (solid lines) model (Bullock et al.
2001) and a SIS profile (dotted lines) are displayed, for the KSB
(left panels) and Shapelets (right panels) pipeline, respectively.
The black lines represent the best fit to the unbinned data as function of distance from the center of the cluster. Average values
of tangential (black) and radial (red) components of the shear,
computed in logarithmic scale bins, are also plotted. The latter
components are expected to be zero in the absence of systematics errors.
In Fig. 15 the confidence contours for the NFW profile are plotted in a plane r s vs c: the levels show confidence at 68% and
90%, starting with the innermost one.
The goodness of each fit (χ2 ) and its probability (Q) are also
listed in Table 2.
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Parameter
M200 (1014 )
r200 (kpc)
cvir
σcl (km/s)
red.χ2
Q
MNFW
+1.4
5.3−1.2
1513+119
−123
4.51
1.46
0.21
KSB
NFW
+4.7
5.6−2.7
1545+345
−306
+5.6
3.9−2.1
2.02
0.11
SIS
MNFW
+0.8
5.3−0.8
1516+77
−78
4.50
778+26
−27
1.94
0.10
1.62
0.17
Shapelets
NFW
+2.2
5.9−1.7
1570+177
−170
+2.2
3.7−1.3
2.12
0.10
11
SIS
781+26
−27
2.31
0.06
Table 2. Mass values computed for M200 by model fitting using a SIS profile, a NFW profile and a constrained NFW (MNFW),
according with Bullock et al. (2001). Best fit parameters and reduced χ2 are listed for each fit performed using both KSB and
Shapelets shear catalogs, together with the goodness of fit probability Q.
r (kpc)
r (kpc)
1500
2000
2500
500
1000
1500
2000
2500
0.15
1000
0.15
500
γt
0.05
0.00
−0.05
−0.05
0.00
γt
0.05
0.10
mnfw
nfw
sis
0.10
mnfw
nfw
sis
100
200
300
400
500
600
θ (arcsec)
100
200
300
400
500
600
θ (arcsec)
Fig. 14. Results of fitting by a NFW model (dashed lines), a constrained (M)NFW (solid lines) model (Bullock et al. 2001) and a
SIS profile (dotted lines), with the KSB (left panel) and Shapelets (right panel) pipeline. Average values of tangential (black) and
radial (red) components of the shear, computed in logarithmic scale bins, are also plotted. Cluster masses were computed at r200 ,
estimated to be ∼ 350 arcsec.
7. Discussion
We have conducted a weak lensing analysis of Abell 611 with
deep g-band images from the LBC. Due to the complexity of
the LBC PSF, we decided to use both the KSB and Shapelets
methods to measure galaxy shapes and extract the shear signal.
KSB parametrizes the sources using their weighted quadrupole
moments and is based on a simplified hypothesis of a nearly
circular PSF. In contrast, Shapelets uses a decomposition of the
images into Gaussian-weighted Hermite polynomials and does
not make any assumption about the best PSF model. Due to the
large collecting area of LBT and the wide field of the LBC, we
were able to extract a high number density of ∼ 25 background
galaxies per arcmin2 over a wide field of 19 × 19 arcmin2 and
this allowed us to perform an accurate weak lensing analysis.
The two shear catalogs, derived by the KSB and Shapelets
pipelines, were matched and common sources (with a number
density of ∼ 23 galaxies/arcmin2) were used to estimate mass
measurements for Abell 611 by two different weak lensing
techniques: aperture densitometry and a parametric model
fitting. In both approaches we assumed that the BCG was the
center of the mass distribution, which is supported by S-Maps
(see Fig. 12, Section 6.1).
The projected mass values obtained by aperture densitometry
within a radius of ∼ 1500 kpc are: 7.7 ± 3.3 × 1014 M⊙ and
8.4 ± 3.8 × 1014 M⊙ , using KSB and Shapelets, respectively.
These estimations are model independent (Clowe et al. 1998),
but they are affected by large uncertainties. As discussed in
Section 6.2, the contribution of uncorrelated structures along the
line of sight can be source of noise for aperture measurements,
although they can be decreased by averaging the results for several clusters (Hoekstra 2001) or corrected by using photometric
redshifts of the sources, if available.
Table 2 shows the results of fitting the observed shear with a
parametric model. We assumed both a SIS and a NFW mass
density profile. We first fitted a NFW model leaving both c and
r s as free parameters: as displayed in Fig. 15, the uncertainty
on these two parameters provided by the fit is high and does
not allow to put a strong constrain on the concentration.
Nevertheless, the best-fit value (c ∼ 4) is in good agreement
with the one obtained when the Bullock et al. (2001) relation is
adopted (c = 4.5). For the NFW fits the quoted masses are M200 ,
within the radius r200 where the density is 200 times the critical
density. The estimated value of r200 for Abell 611 is ∼ 1.5 Mpc
(5.8′ ).
The weak lensing mass measurements obtained from both
the KSB and Shapelets shear catalogs are in agreement,
within uncertainties: by using a (M)NFW profile we obtain
14
+0.8
14
M200 ∼ 5.3+1.4
−1.2 × 10 M⊙ and 5.3−0.8 × 10 M⊙ , respectively. The
smaller uncertainties of Shapelets results show that this method
12
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
rs (kpc)
0
200
400
600
rs (kpc)
800
1000
1200
1400
0
8
8
6
6
200
400
600
800
1000
1200
1400
c
10
c
10
4
4
2
2
50
100
150
200
250
300
rs (arcsec)
50
100
150
200
250
300
rs (arcsec)
Fig. 15. The confidence contours for NFW profile are plotted in the (r s ,c) plane, for KSB (left panel) and Shapelets (right panel),
respectively. The confidence levels are at 68% and 90%, starting with the innermost one. The triangle shows the best-fit position.
provide a higher accuracy than KSB. Moreover, the goodness
of the fits (χ2 ) in Table 2 shows that both Shapelets and KSB
provide a best-fit with higher probablity (Q) by using a NFW
mass density profile than a SIS profile.
Abell 611 was previously targeted for a weak lensing study
by Dahle (2006), who used V and I observations from several
facilities to target a large number clusters (see Dahle et al. 2002
for more details). They used the KSB (Kaiser et al. 1995) shear
estimator as described in Kaiser (2000) to derive the shear signal
from the images. The mass of the cluster was then derived by
fitting the observed shear with a NFW mass density profile
(Navarro, Frenk & White 1997). They assumed a concentration
parameter as predicted by Bullock et al. (2001) and obtained a
mass value of M180 = (5.21 ± 3.47) × 1014h−1 M⊙ within r180 , the
radius within which the density is 180 times the critical density.
A more recent mass estimate of Abell 611 is presented in
Pedersen & Dahle (2007), who used the data collected in Dahle
(2006) and obtained a value of M500 = 3.83 ± 2.89 × 1014 h−1 M⊙
within r500 , the radius within which the density is 500 times
the critical density. In these papers the authors extrapolated
the NFW profile out to r500 because the data were insufficient to extend this far in projection from the cluster center. For
their work on Abell 611 r f it /r500 = 0.59 (note r500 = 0.66×r200 ).
Our weak lensing estimates for the mass of Abell 611 are
in agreement with the previous results of Dahle (2006) and
Pedersen & Dahle (2007), but the depth and the larger area
covered by LBC data allowed us to perform a more accurate
analysis.
A recent weak lensing analysis of Abell 611 has recently
been done by Okabe et al. (2009, submitted to PASJ) using
Subaru/Suprime-Cam observations in two filters (i’ and V). The
authors used the color (V − i′ ) information to select the background galaxies to use for their cluster lensing analysis, getting
a galaxy number density of ∼ 21 galaxies/arcmin2, with a SNR
∼ 10. By fitting the mass density profile of the cluster using a
14
NFW model, they found a mass value M200 = 5.13+1.16
−1.00 ×10 M⊙
+1.73
and a cvir = 4.14−1.21 (private communication), which are in
good agreement with our results.
A new mass estimation of Abell 611 was performed by
Newman et al. (2009) over wide range of cluster-centric
distance (from ∼ 3 kpc to 3.25 Mpc) by combining weak,
strong and kinematic analysis of the cluster, based on Subaru,
HST and Keck data, respectively. They found a mass value
14
M200 = 6.2+0.7
−0.5 × 10 M⊙ with c = 6.95 ± 0.41 by using a NFW
model fitting, in agreement with our results. We note that such a
large value of c cannot be rejected by our NFW fits, due to the
large uncertainties on NFW parameters (see Tab.2 and Fig.15).
Finally, we also compared the mass values obtained by our
weak lensing analysis to X-ray estimates of M200 available
in literature. Schmidt & Allen (2007) analyzed Chandra data
of several clusters and modelled their total mass profile (dark
plus luminous matter) using a NFW profile. They found for
Abell 611 a scale radius r s = 0.32+0.10
−0.20 Mpc and a concentration parameter c = 5.39+1.60
,
which
provide a mass value
−1.51
M200 ∼ 8 × 1014 M⊙ at r200 ∼ 1.7 Mpc, in agreement, within
the uncertainties, with the results obtained by us. A more
recent X-ray analysis of Chandra observations of Abell 611
has been performed by Donnarumma et al. (in preparation).
They obtained a value of M200 = 1.11 ± 0.21 × 1015 M⊙ at
+0.87
r200 ∼ 1900 kpc, with r s = 407+120
−86 kpc and c = 4.76−0.78 . Their
15
projected mass at r200 is 1.28 ± 0.24 × 10 M⊙ . Such value is
in agreement, within the statistical uncertainties, with the mass
measured by aperture densitometry, but higher than the value
estimated by the parametric model. Additional information on
the mass will be derived from a strong lensing analysis of Abell
611 (Donnarumma et al. in preparation).
This work shows that LBT is a powerful instrument for weak
lensing studies, but we want to stress that the data here analyzed
did not allow us to use the full capabilities of the telescope. The
presence of bright saturated stars hampered to use the whole
field of the camera for the analysis of Abell 611. In addition,
the analysis of weak lensing is expected to be improved by the
usage of the Red Channel in LBC, which was not yet available
during these observations. The present results are nevertheless
important to demonstrate the capabilities of LBC for weak
lensing; we therefore plan to extend to other clusters such
analysis, now using the Red Channel.
A. Romano et al.: Abell 611: I Weak lensing analysis with LBC
Acknowledgements. Observations were obtained with the Large Binocular
Telescope at Mt. Graham, Arizona, under the Commissioning and Science
Demonstration phase of the Blue Channel of the Large Binocular Camera. The
LBT is an international collaboration among institutions in the United States,
Italy and Germany. LBT Corporation partners are: The University of Arizona on
behalf of the Arizona university system; Istituto Nazionale di Astrofisica, Italy;
LBT Beteiligungsgesellschaft, Germany, representing the Max-Planck Society,
the Astrophysical Institute Potsdam, and Heidelberg University; The Ohio State
University, and The Research Corporation, on behalf of The University of Notre
Dame, University of Minnesota and University of Virginia.
This paper makes use of photometric redshifts produced jointly by Terapix and
the VVDS team.
Part of the data analysis in this paper was done using the R software
(http://www.R-project.org).
We thank the anonymous referee for useful comments and suggestions which
improved the presentation of this work. AR acknowledges the financial support
from contract ASI-COFIS I/016/07/0. LF, KK and MR acknowledge the support
of the European Commission Programme 6-th framework, Marie Curie Training
and Research Network DUEL, contract number MRTN-CT-2006-036133. LF is
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). AD, SE, LM, MM acknowledge the financial contribution from
contracts ASI-INAF I/023/05/0 and I/088/06/0. SPH is supported by the P2I
program, contract number 102759.
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