Lens magnification by CL0024+1654 in the U and R band

19 Pages·1.1 MB·English

by  S. Dye

Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
Lens magnification by CL0024+1654 in the U and R band
S. Dye1, A.N. Taylor2, T.R. Greve3,4, O¨.E. Ro¨gnvaldsson5, E. van Kampen2, P. Jakobsson6, V.S.
6 6 3
Sigmundsson , E.H. Gudmundsson , J. Hjorth
2
0
0 1 Astrophysics Group, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, U.K.
2 2 InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,EdinburghEH93HJ,U.K.
3 Astronomical Observatory,Juliane Maries Vej30, DK-2100 Copenhagen Ø,Denmark
n
4 Department of Physics & Astronomy,University College London,Gower Street, London WC1E 6BT, UK
a
J 5 NORDITA,Blegdamsvej 17, DK-2100 Copenhagen Ø,Denmark
8 6 Science Institute, Dunhaga3, IS-107 Reykjavik,Iceland
2
Received Month XX,2002 / Accepted Month XX,2002

2
v Abstract. Weestimatethetotalmassdistributionofthegalaxy clusterCL0024+1654 from themeasured source
9 depletion due to lens magnification in the R band. Within a radius of 0.54h−1Mpc, a total projected mass of
39 (8.1±3.2)×1014h−1M⊙ (EdS)ismeasured.The1σ errorhereincludesshotnoise, sourceclustering, uncertainty
in background count normalisation and contamination from cluster and foreground galaxies. This corresponds
8
to a mass-to-light ratio of M/L = 470±180. We compute the luminosity function of CL0024+1654 in order
0 B
1 to estimate contamination of the background source counts from cluster galaxies. Three different magnification-
0 basedreconstruction methodsareemployed:1) Anestimator methodusingalocalcalculation oflensshear;2)A
/ non-local, self-consistent method applicable to axi-symmetric mass distributions; 3) A non-local, self-consistent
h
method for derivation of 2D mass maps. We have modified the standard single power-law slope number count
p
theory to incorporate a break and applied this to our observations. Fitting analytical magnification profiles of
-
o different cluster models to the observed number counts, we find that CL0024+1654 is best described either
r by a NFW model with scale radius r = 334±191h−1kpc and normalisation κ = 0.23±0.08 or a power-
t s s
s law profile with slope ξ = 0.61 ± 0.11, central surface mass density κ = 1.52 ±0.20 and assuming a core
0
a radius of r = 35h−1kpc. The NFW model predicts that the cumulative projected mass contained within a
v: radius R sccaolrees as M(<R)=2.9×1014(R/1′)1.3−0.5lg(R/1′)h−1M⊙. Finally, we have exploited the fact that flux
i magnification effectively enables us to probe deeper than the physical limiting magnitude of our observations in
X
searching for a change of slope in the U band number counts. We rule out both a total flattening of the counts
r with a break up to U ≤26.6 and a change of slope, reported by some studies, from dlogN/dm=0.4→0.15
a AB
up to U ≤26.4 with 95% confidence.
AB
Key words. Gravitational lensing - Galaxies: clusters: individual: CL0024+1654 - dark matter
1. Introduction The authors demonstrated that a concentration of large
cluster galaxies near the arc centre is necessary to cause
The lensing cluster CL0024+1654 ranks as one of the
this by perturbing the cluster cusp and were thus able
most highly studied clusters to date. Lying at a redshift
to constrain the potential of the cluster and the perturb-
of z = 0.39, early measurements of the cluster’s velocity
ing galaxies. A later analysis by Bonnet, Mellier & Fort
dispersionofσ 1300 100kms−1(Dressler,Schneider&
(1994) constrained the cluster’s mass profile more tightly
≃ ±
Gunn1985)suggestedaformidablemass.Thediscoveryof
with the first measurement of weak shear out to a ra-
a large gravitationallylensed arc from a blue background dius of 1.5h−1Mpc. Wallington, Kochanek & Koo (1995)
galaxybyKoo(1988)hassinceprovokedarangeofstudies
confirmed the perturbing galaxy hypothesis of Kassiola,
to estimate the cluster’s mass based on its lensing prop-
Kovner & Fort by fitting a smooth elliptical cluster po-
erties.
tential with two superimposed L∗ galaxy potentials near
The first lens inversion of CL0024+1654,by Kassiola, the centre of the arc.
Kovner & Fort (1992), noted a violation of the ‘length
theorem’(Kovner1990)thatthelengthofthemiddleseg- By parameterising the source and lens models and fit-
ment of the arc should equal the sum of the other two. ting to six images of the lensed source galaxy,Kochanski,
Dell’ Antonio & Tyson (1996) again showed that the
Sendoffprintrequeststo:SimonDye,e-mail:[email protected] mass profile of CL0024+1654is consistent with a smooth
2 Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band
isothermaldistribution.Furthermore,theyfoundthatthe ing results of R01. We take the observations from R01
cluster mass traces light fairly well out to a radius of but create new, more reliable R and U band object cat-
0.5h−1Mpc and were able to rule out the existence of alogues. Using these, we re-calculate the depletion signal
any significant substructure larger than 15h−1kpc in the in both bands and use this to re-fit the isothermal lens
central region. Using HST images of the cluster, Tyson, model of R01. Secondly, we wish to extend the analysis
Kochanski&Dell’Antonio(1998,TKDhereafter)isolated of R01: 1) We fit a power-law profile and a NFW model
eight well-resolved images of the blue background galaxy to the depletion profiles. 2) We transform the measured
to construct a high resolution mass map of the cluster. depletion into mass estimates using three recently devel-
Theirparametricinversionconcludedthatmorethan98% oped magnification reconstruction methods; the local es-
ofthemassconcentrationexcludingthatcontributedfrom timator method of van Kampen (1998) applied by Taylor
discretegalaxieswasrepresentedbyasmoothdistribution et al. (1998, T98 hereafter), the non-local axi-symmetric
of mass with a shallower profile than isothermal. solution of T98 for reconstruction of radial mass profiles
The most recent lensing analysis of CL0024+1654 to and the non-local method of Dye & Taylor (1998) for de-
date is that by Broadhurst et al. (2000, B00 hereafter) termining 2D mass maps. 3) We investigate the relation-
who provide the first measurement of the redshift of the ship between mass, light and galaxy number density in
bluebackgroundgalaxyatz =1.675.Thisredshiftbreaks the cluster. 4) We quantify contamination of our back-
themass-redshiftdegeneracypresentinallmassestimates ground source sample by cluster and foreground objects
oftheclusterthusfar.TheirfitofNFWprofiles(Navarro, andincorporatethisintoouranalyses.5)Finally,weapply
Frenk & White, 1997)to the eight brightestcluster mem- twomethodstosearchforachangeinslopeintheU band
bers is found to be an adequate model to explain the po- fieldgalaxynumbercountsbyexploitingthefactthatlens
sitions of the five main lensed images.This, they suggest, magnificationenablesustoeffectivelyseedeeperthanthe
highlights the possibility that sub-structure has not been physical magnitude limit imposed by the observations.
erased in the cluster. The following section briefly describes data acquisi-
The first of only two measurements of the cluster’s tion, reduction, object extraction and mask generation.
lens magnification to currently exist was that investi- Contaminationofthesourcecountsfromclusterandfore-
gated by Fort, Mellier & Dantel-Fort (1997). This was ground galaxies is estimated. Section 3 details the recon-
thefirstdetectionofdepletioninbackgroundgalaxynum- struction methods used in this paper. In Section 4, we
ber counts due to geometrical magnification as predicted consider the properties of the R band background galaxy
by Broadhurst, Taylor & Peacock (1995, BTP hereafter). population and fit cluster mass models to the measured
Rather than reconstruct cluster mass, this work concen- number count profile. This is necessary for determination
trated on characterising the radial distribution of critical of the mass results presented in Section 5. In Section 6,
lines to infer the redshift range of the background popu- we investigate the U band galaxy population and test for
lations in B and I. Using this data, van Kampen (1998) the existence ofabreakinthe numbercounts.Finally,we
produced an estimate of the mass of CL0024+1654 from discuss and summarise our findings in Section 7.
lens magnification.
The only other magnification analysis of the cluster
published so far is that of Ro¨gnvaldssonet al. (2001; R01
hereafter)using R andU bandobservations.Their choice 2. The data
to observe in U was inspired by the findings of Williams
Dataacquisitionandreductionaredescribedinfulldetail
et al. (1996) which suggested a flattening in the U band
in R01. Here, we highlight the key deviations in the gen-
number count slope at faint magnitudes. Such a break is
eration of the object catalogues of this paper from those
reported to occur at U 25.5 26 with a change of
AB
≃ − in R01:
slope of dlogN/dm 0.4 0.15. Given suitably deep
≃ →
U band imaging, this should therefore manifest itself as
a depletion in the number density of galaxies observed in – A Gaussian smoothing kernel with a FWHM equal to
the presence of lens magnification (see Section 3.2). R01 theseeingwasusedtoextractobjects(seeSection2.2).
claimed to have detected depletion in U implying that a This compares with the narrower smoothing kernel of
break in the slope must be present. FWHM equal to slightly larger than half the seeing
Further investigationhassincehighlightedconcernre- used by R01 (note the incorrect statement that their
garding the reliability of faint objects extracted by R01 smoothingkernelisequaltothe seeing).Matchingthe
from the R and U band observations. While this is not a width ofthe smoothing kernelto the seeing allowsop-
large concern for R where the depletion signal is strong, timal extraction in the sense that spurious detections
the claim of detection of depletion in U and hence the due to background noise are kept as small as possible
reported break in the number count slope is strongly af- whilemaintainingasufficientlylargedetectionsuccess
fected.Are-examinationofthesefindingsisthereforenec- rate. This creates the most noticeable difference be-
essary. tween this dataset and that of R01; our U and R cat-
Themotivationdrivingthepaperpresentedhereissev- alogues contain fewer faint objects (our total paired
eralfold.Firstly,there is the needto re-evaluatethe exist- object catalogue contains 29% fewer objects).
Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band 3
– Galactic extinction corrections have been applied to between both bands, a linear co-ordinate transformation
both the U and R band, brightening all objects (see was calculated and applied to map the U band catalogue
Section 2.1). onto the R band. Objects were paired within a positional
– A more thorough determination of the background tolerance equal to the seeing and yielded a total of 875
noise has been obtained. The mean RMS background objects excluding stars. Those objects with star/galaxy
variation has been calculated within a circular aper- classification indices larger than 0.95 were assumed to be
ture of diameter equal to the seeing FWHM. The re- stars and excluded from the analysis below.
sult of this is that the ‘3σ’ detection limits calculated
inR01fromPoissionstatisticsarefainterthanthosein
2.3. Object selection
thispaperby1.3maginU and1.7maginRincluding
the galactic extinction correction. Separationofthebackgroundgalaxiesfromtheforeground
– A linear co-ordinate transformation has been applied and cluster galaxies must be achieved before lens magni-
to the U band object positions resulting in a more ac- fication can be evaluated. Segregation of the cluster and
curate mapping to theR band.This has improvedthe foreground objects is important to enable estimation of
position coincidence matching between both bands. their background sky obscuration which affects binned
number counts.
2.1. Acquisition and completeness
CL0024+1624wasobservedintheCousinsU andCousins
Rbandswitha6′ 6′fieldofviewusingtheNordicOptical
×
Telescope in August 1999. A total integration time of 37
ksec and seeing 1.1′′ was obtained in the U band com-
pared with 8.7 ksec and 1.0′′ in the R. Following R01, we
useABmagnitudesthroughoutthispaperconvertingfrom
the Cousins U and R magnitudes with an offset of +0.71
magand+0.20magrespectively(Fukugitaetal.1995).In
addition,wehavecorrectedthedataforgalacticextinction
amounting to 0.15 mag and 0.31 mag in the R and U
− −
bandrespectively(Schlegeletal.1998).Thelimitingmag-
nitudescorrespondingtoasignal-to-noiseratioof3within
a seeing disk are U = 25.7 mag and R = 25.8 mag,
AB AB
with the noise level taken as the mean RMS of the back-
ground. All photometry was performed using SExtractor
(Bertin&Arnouts1996;seeSection2.2).Magnitudeswere
measuredas either totalor correctedisophotaldepending
on the proximity of neighbouring objects. This is man-
aged automatically via SExtractor’s MAG BEST magnitude
definition.
Completeness was estimated from the detectability of
synthetic objects of varying brightness added to the im-
ages. Further details can be found in R01. In re-applying
thisprocesstothedataofthispaper,wefindthatthecom-
pleteness at the 3σ detection limit is 84% at U = 25.7
AB
mag and 81% at R =25.8 mag.
AB
2.2. Object extraction
Objects were extracted from the final reduced images us-
ing SExtractor. With a detection threshold of 1σ above Fig.1. Colour-magnitude diagrams for all 875 matched ob-
background and a Gaussian filtering kernel of FWHM jects. Top and bottom shows selection of mask objects in the
equal to the seeing, catalogues of all objects with at least U and R bandrespectively bythecriteria (U−R)AB>3and
U <24, R <23.5 (dashedlines).The3σ detection limits
10connectingpixelsbrighterthanthethresholdweregen- AB AB
U = 25.7 and R = 25.8 are shown in both plots by the
erated. A total of 1887 objects in the R band and 1122 AB AB
dot-dashed lines.
objects inthe U bandweredetected. These totalsinclude
stars but exclude objects classified by SExtractor as hav-
ing saturated pixels, being truncated or possessing cor- Figure1showsthecolour-magnitudeplotof(U R)
AB
−
rupt isophotal data. By matching the 30 brightest stars versusthe U andR magnitudesfor the 875matched
AB AB
4 Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band
objects. By matching our sample with the redshift mea-
surements of the field of CL0024+1654 by Czoske et al.
(2001b), we have found that a large fraction of cluster
members can be immediately discarded by removing ob-
jects with (U R) >3. The noticeable lack of objects
AB
−
seen in the vicinity of (U R) 3 reflects a mini-
AB
− ≃
muminthebi-modal(U R)distributionofclustergalax-
−
ies identified in the Czoske et al. sample by the criterion
0.388<z <0.405.Applying afurtherselectionR <22
AB
to avoid incompleteness in both the Czoske et al. sample
and our data, we find that 65% of identified cluster ob-
jects lie at (U R) >3. In addition, we find that 12%
AB
−
of objects brighter than R = 22 with (U R) > 3
AB AB
−
are foregroundgalaxies and only 2% of objects in this se-
lectionarebackgroundobjects.Insummary,the selection
(U R) > 3 very efficiently removes cluster galaxies,
AB
−
at least up to R =22, discarding a very small fraction
AB
of background sources.
Fig.3. Distribution of light (surface flux density L con-
In Figure 2, we plot the number density of cluster
verted to the restframe B band and expressed in units of
galaxies defined as objects with colour (U − R)AB > 3 1013L⊙Mpc−2h2)fromclustergalaxiesselectedeitherby(U−
or by their redshift for (U −R)AB < 3. Figure 4 shows R)AB > 3 or redshift. Contours are separated by equal flux
how the number counts of galaxies meeting these crite- intervals.
ria vary with magnitude in the U and R. In Figure 3 we
plot the distribution of light from these selected cluster
galaxies. The surface flux density shown in this plot is Our foreground and cluster objects are therefore cho-
scaledtotherestframeB bandforeasycomparisonofthe sen as a combination of objects identified in the Czoske
mass-to-light ratio we calculate later with other authors. et al. sample with z < 0.405 as well as those satisfying
This scaling was carried out using the K and colour cor- (U R) > 3 and an additional R < 23.5 for the R
AB AB
−
rections presented in Fukugita et al. (1995). Both distri- band sample or an additional U <24.0 for the U band
AB
butionsarecomparedwiththereconstructedclustermass sample.ThechoiceofRandU limitsherearechosentobe
map in Section 5.1. The fact that both distributions are optimalin the sense thatthey selectas many background
relatively concentrated around the known cluster centre objects as possible while preventing too high a degree of
indicates that we have reliable selection criteria for the contamination from cluster and foreground objects (see
(bright) cluster members. Section 2.4). Extracted parameters such as size, elliptic-
ity and orientation of the cluster and foreground objects
are used to generate an obscuration mask as Section 2.5
details.
Rather than perform our analyses on the associated
object catalogue (ie. containing only objects with paired
U and R mags) which would miss faint objects detected
in only either the U or R band, we take our background
source list for each band from the individual complete
U and R catalogues. To obtain our source list, we re-
move objects from each complete catalogue identified as
cluster members andforegroundobjects using the colour-
magnitude and redshift criteria above.
ForourU bandbreakanalysisinSection6,removalof
contaminantscausesaslightdilemma.Althoughwewould
like our U band sample to be contamination free so that
any lensing signal present is maximised, we also wish to
remain consistent with existing U band studies to pre-
vent our search for the break from being biased in some
way.It turns out that since the Czoske et al. redshift sur-
veyextendstoonlyrelativelyshallowU bandmagnitudes,
we only identify and remove 5 foreground objects which
Fig.2.Numberdensity(n=number/squarearcmin)ofcluster fall within the colour-magnitude selection criteria for our
galaxies selected either by(U−R)AB>3 or redshift. background sample. We have therefore introduced only a
Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band 5
Figure 4 plots the number counts of both these samples
asafunctionofmagnitude.NoticehowtheU bandcounts
aresignificantlysteeperthaninR.Allsubsequentmagnifi-
cationanalysiswillbeperformedonthesetwocatalogues.
2.4. Contamination of background source samples
We discussed in the previous section that 65% of clus-
∼
ter galaxies lie at (U R) > 3 for R < 22. For the
AB AB
−
R band sample, this does not provide a useful means of
removing cluster galaxies as the background source selec-
tionlimitofR 23.5preventsinclusionofobjectswith
AB
≥
(U R) >3 anywaydue to the U band detectionlimit
AB
−
of U =25.7. By including objects from the full R band
AB
catalogue,faintobjectswith(U R) >3willinevitably
AB
−
fall into the backgroundR sample.Additionally, the clus-
ter galaxies which lie at (U R) <3 will contaminate
AB
−
bothU andRsamplesalongwithanyforegroundgalaxies.
This contamination must be quantified.
2.4.1. Foreground galaxy contamination
To estimate the foreground galaxy contamination ex-
pected within our chosen magnitude ranges, we use the
luminosityfunctionsmeasuredbytheCNOC2fieldgalaxy
redshift survey (Lin et al. 1999). The CNOC2 survey is
ideal for our purposes being the largest intermediate red-
shiftsurveytodatewithmulticolourUBVRI photometry.
The number of objects N which exist within the ap-
parent magnitude range m < m < m and the redshift
1 2
range z <z <z can be estimated as
1 2
Fig.4. Number counts as a function of magnitude for the U
band(top)andRband(bottom).Plottedarecountsofgalaxies z=z2 M2(z)
N = dV(z) φ(M)dM (1)
inthebackgroundsample(solid)andclustergalaxies(dashed).
Errors account for shot noise only. Zz=z1 ZM1(z)
wheredV(z)isthecomovingvolumeelement,φ(M)isthe
luminosityfunctionandtheabsolutemagnitudesM (z)&
1
negligible inconsistency with other U band number count M (z)correspondtotheapparentmagnitudesm andm
2 1 2
studiesandyethaveremovedallwecanintermsofknown ataredshiftz.Thefractionofobjectswhichliecloserthan
foreground objects. We know that there are more fore- a redshift z in a sample of galaxies observed within the
f
ground objects in our background sample than we have magnitude range m <m<m is therefore given by,
1 2
been able to remove (see Section 2.4.1) so we are forced
N(m <m<m ,z <z )
to allow for these as a contamination error in our lensing 1 2 f . (2)
N(m <m<m )
analysis.As farasremovalofobjects with(U R) >3 1 2
AB
−
is concerned, this simply removes the majority of surplus The denominator here is the total number of galaxies
cluster galaxieswhich aren’tpresentin published number within m < m < m integrated over all redshifts.
1 2
count studies anyway. Using this equation, we estimate the fraction of galax-
In the R band, this does not pose a problem. We do ies within our U and R background samples lying at
not have to ensure that our R band data is consistent z < 0.405, allowing for an early+intermediate mixed
withanyothersamplesinceourdepletionisnormalisedto galaxy K-correction (Fukugita et al. 1995) for the mag-
counts near the edge of the field rather than independent nitudes in equation (1).
measurements.Removalofasmanyforegroundandcluster We use the early+intermediate (Cousins) R and
objects as possible is therefore the ideal. As with the U (Cousins) U luminosity functions from the CNOC2 sur-
band sample, we can not achieve this fully so must resort vey for calculation of N in equation (1). These predict
to treating the remainder as a source of error in the mass that the contamination due to foreground galaxies is 3%
reconstructions. in the U band sample and 2% in the R band assuming
After removal of cluster and foreground objects from an EdS cosmology. For the case Ω = 0.3, Λ = 0.7, these
our full U and R catalogues, we find a total of 863 back- fractions drop only very slightly. We find that this result
ground galaxies remaining in the U and 1367 in the R. is insensitive to the choice of K-correctionmix.
6 Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band
2.4.2. Cluster contamination and the cluster
luminosity function
To determine the cluster contaminationfraction, we need
to be able to predict the number of cluster members ex-
pected within the magnitude range spanned by the se-
lection magnitude and the detection limit in both bands.
Thisrequiresknowledgeoftheclusterluminosityfunction
(CLF) for CL0024+1654.
We fit our own CLF to the cluster counts determined
by matching with the Czoske et al. cluster members (see
Section 2.3) in both the U and R. We use the likelihood
method of Sandage Tammann & Yahil (1979) and check
theconsistencyofourresultswithseveralpublishedCLFs.
In the R band, we compare with three recent studies of
the composite CLF (CCLF): Paolillo et al. (2001) who
construct a CCLF from 39 Abell clusters, Piranomonte
et al. (2000) whose CCLF is constructed from 86 Abell
clusters and Garilliet al. (1999)who use 65 Abell and X-
rayselectedclusterstocalculateaCCLF.Allthreestudies
use observations in the Gunn r filter so we must apply a
K-colour-correctionfortheabsolutemagnitudeconversion
M M .Since 60 70%ofgalaxieswithinclusters
RAB → r ∼ −
are E/S0 types (Dressler et al. 1997; see also Smail et al.
1997 for specific case of CL0024+1654),we calculate this
correction using an elliptical galaxy spectrum taken from
Kinney et al (1996).
IntheU band,publicationsonCLFsarerareandthere
arenoknownU bandCCLFstodate.ApplyingK-colour-
corrections to convert redder-band magnitudes to the U
bandwhenthegalaxytypemixisnotknownaccuratelyis
notreliable.Wethereforechoosethemostheavilystudied Fig.5. Luminosity functions of CL0024+1654 galaxies. Top:
clusterintheU band,theComacluster,andcompareour U band Schechter LF from Beijersbergen et al. (2001) for the
fit with the CLF from the recent study by Beijersbergen Coma cluster (dashes) and the best fit LF determined in this
etal.(2001).InasimilarmannertotheRband,weapply work (solid). Bottom: r band CCLFs for rich clusters from
Paolillo et al. (2001, dots), Piranomonte et al. (2000, dashes),
a correctionto take absolute U magnitudes atz =0.39
AB
Garillietal.(1999,dot-dashes)andthebestfitfromthiswork
to rest-frame absolute Cousins U magnitudes.
(solid).
Figure 5 plots the various CLFs/CCLFs, all of which
are described by a Schechter function (Schechter 1976)
with parameters given in Table 1. For the r band CCLF to obtaina morefairlevelofcontamination.We calculate
fromeachstudy,wehavetakenSchechterparameterscor- the number of cluster galaxies as
responding to the rich subsample of clusters for consis-
tency with CL0024+1654. All luminosity functions are Mb
normalised to our cluster counts which are superimposed nc =A φ∗(M∗,α) dMφ(M;M∗,α) (3)
inbothplots.Absolutemagnitudesassumeq0 =h0 =0.5. * ZMa +M∗,α
Forcorrectnormalisation,wemustensurethattheCzoske
where the angular brackets denote averaging over each
et al. matched sample for each band does not suffer
realisation of M∗ and α weighted by the corresponding
from incompleteness. We therefore limit the matched U
probability from the CLF fit. A is a normalisation con-
band counts to the magnitude range 21.8 < U < 24.0
AB stant calculated as the inverse of the sum of these proba-
( 22.4 < M < 20.2) and the matched R band counts
t−o 19.9<RU <−22.1 ( 23.3<M < 21.1). bilities. The absolute magnitudes Ma and Mb correspond
AB − r − to the apparent magnitude ranges 23.5<RAB <25.5 for
IntegratingovereachoftheCLFs/CCLFs,weareable r or 24.0 < U < 25.5 for U . These ranges are chosen
AB c
to calculate the number of cluster galaxies expected. For to extend slightly less deep than the 3σ detection limits,
the published luminosity functions presented here, this is allowing comparison with a more complete background
a straight-forwardintegral.Forour ownfitted CLFs how- sample to give a more accurate estimation of contamina-
ever,we incorporate the uncertainty arising from their fit tion.
Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band 7
curately(forthispurposeatleast)byanisothermalsphere
CLF/CCLF Band M∗ α C (%) (see Section 4.3). The radial dependence of the error on
Paolillo CCLF r -22.2 -1.01 15 the surface mass density, κ, can thus be written
Piranomonte CCLF r -22.0 -0.91 13
Garilli CCLF r -22.1 -0.60 5 k
σ (r)= (4)
This work CLF r -21.9 -0.70 19 κ 2nr(β 1)µβ(r)
−
Beijersbergen CLF U -19.4 -1.54 8
c
This work CLF U -21.5 -1.41 9 where n is the backgroundnumber density in the absence
c
oflensing,µ(r)isthemagnificationgivenbyequation(24)
Table 1. Schechter parameters of the CLF/CCLFs used
and β is the number count slope (see Section 3.1). We
for estimation of cluster contamination, C, expressed as
haveapproximatedthenumberdensityprofileoftheclus-
the percentage of objects expected to be cluster members
ter galaxies as k/r where k is set by normalising to the
within 23.5<R <25.5 for r or 24.0<U <25.5 for
AB AB
contamination fractions given earlier in this section.
U .Notethatthecontaminationfractionscalculatedfrom
c
Inallreconstructionsofκfoundlaterinthispaper,we
the CLFs of this work are weighted averages(see text).
add the error given by equation (4) in quadrature to the
sum of the other errors, including the error due to fore-
groundgalaxycontamination.Theeffectofcontamination
Withintheappropriatemagnituderangesgivenabove, onthesearchfortheU bandbreakisdiscussedseparately
the last column of Table 1 lists the predicted number of in Section 6.
cluster galaxies as a percentage of the total number of
galaxiesdetected.InR,theestimatedcontaminationfrom
2.5. Obscuration masks
thepublishedluminosityfunctionsrangesbetween5%and
15%.Theweightedaveragepredictedcontaminationfrom Toaccountforobscurationofthebackgroundskybyclus-
our own R band CLF is higher at 19%. In the U band, ter members and foreground objects, we create a mask.
the contamination levels are not as high since the known Without giving consideration to obscuration, the ratio of
Czoske cluster galaxies have been discounted in the cal- observed objects to expected objects used later in our
culation. We find that the Coma CLF predicts that 8% analysiswouldbebiasedtolowervalues.Thisisespecially
of objects in our background sample are cluster objects, true ofbins nearthe cluster centre whererelativelyheavy
compared with our CLF which gives a weighted average obscuration occurs due to large central cluster galaxies.
of 9%. Figure 6 shows the obscuration mask produced us-
The total contamination predicted from both fore- ing the SExtracted parameters of mask objects selected
groundandclustergalaxiesthereforehasarelativelywide in Section 2.3. The top half shows the U band mask
spread in the R band of 7% to 21% comparedto between (greyellipses),U bandbackgroundsourcepositions(black
11% and 12% in U. Unfortunately, none of the published crosses) and the position of the annular bins used in
CLFs/CCLFs presented here extend sufficiently deep to Section4.3andSection6tobincountsradially.Thelower
be able to reliably predict the number of cluster objects halfofFigure6showsthecorrespondingRbandmaskand
in CL0024+1654 at the faint end of our sample. In addi- background sources with the grid used in Section 5.1 for
tion, some CLF studies (eg. Driver et al. 1994; Wilson et 2D binning. Note the horizontal phase of the bins with
al.1997)indicatethepossibilityofsteeperfaintendslopes respecttothefieldofviewtoallowplacementofbin(7,5)
whichwouldincreasethefractionofclustercontaminants. directly over the cluster centre, aligning it with the area
Inlightofthis,wethereforeassumethegenerouscontami- encompassed by the critical line.
nationlevelspredictedfromourfittedCLFsintheanalysis
which follows in this paper.
3. Mass reconstruction theory
Clearly,thecontaminationisdominatedbythecluster
members.Thismeansthatthevariationincontamination In this section, we outline the process used for the deter-
acrossthefieldofviewisgovernedbytheirdensityprofile. mination of magnification from number counts. We then
Since this increases toward the centre of the cluster and detail the three different methods used to transformfrom
since the background number counts fall off toward the magnification to surface mass density.
centre where the magnification is strongest (see Section
4.3) the observed number density of objects in the back-
3.1. Lens magnification: Single power-law number
groundsampleisaffectedmostbycontaminationatsmall
counts
radii. This distribution will have a strong fall-off as one
moves away from the cluster centre. Fortunately, this re-
A background population of galaxies whose integrated
duces the impact of the contamination on the cumulative
number counts follow the standard power-law n(< m)
mass measurements (Section 5.3) at large radii where the 100.4βm will be observed under a lens magnification fa∝c-
strongest mass contribution comes from. tor µ to have the number count n′(< m) given by (BTP,
Anapproximatecontaminationprofilecanbeobtained
T98);
by assuming that the cluster mass distribution and hence
thenumberdensityofgalaxiesisdescribedsufficientlyac- n′ =nµβ−1(1+δ )=λ(1+δ ). (5)
nl nl
8 Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band
ground probed by our small field of view and is positive-
definite.
The Poisson-lognormal distribution can be defined as
a compound distribution formed from a Gaussian distri-
bution G of a linearised field and a Poission distribution
D as
∞
P [nλ(µ)]= dδD[nλeδ−σ2/2]G(δ). (6)
LN
| |
−∞
Z
Here, δ is the linear density fluctuation which relates to
the non-linear fluctuation as 1+δ =eδ−σ2/2 with σ the
nl
linear clustering variance. Following the method of T98
but applied to the R band correlation function of Hudon
&Lilly(1996),wecalculatethenon-linearclusteringvari-
ance. Averaging the angular correlation function over a
circular area of radius θ gives,
σ2 =1.5 10−2z−1.8(θ/1′)−0.8. (7)
nl ×
The linear clustering variance is calculated from the non-
linear variance using σ2 =ln(1+σ2 ).
nl
Equation (6) gives the probability distribution for the
lens magnification in a given bin containing n galaxies.
This directly gives the most probable magnification for
thatbinwithitsassociatederrortakenasthewidthofthe
distribution. The integral in equation (6) must be evalu-
ated numerically.
3.2. Lens magnification: Dual power-law number
counts
In Section 6.1, we discuss evidence which suggests the
presence of a break in the U band number count slope
at faint magnitudes. Section 4.1 also highlights the possi-
bility ofa breakin the R band counts.To accountfor the
effect this has on the magnification calculated from num-
bercountdepletion,wemodifyequation(5)toincorporate
a second slope applicable beyond some break magnitude
m .
b
Writing the observed differential number counts as a
dual power-law,
Fig.6.Top:U bandmask(greyellipses)withsourcepositions
a100.4β1m , m<m
(black crosses) and annular bins used in Section 4.3 and 6. n(m)= b (8)
b100.4β2m , m m
Bottom: Corresponding R band mask with grid used for 2D b
(cid:26) ≥
maps in Section 5.1. Observed arcs lie along the outer heavy
where the normalisation coefficients a and b are con-
dashed circle in both plots. The inner dashed circle shows the
strained by continuity, the integrated number counts up
critical line determined from the isothermal sphere model fit
to theR band numbercounts(see Section 4.3). to some limiting magnitude ml > mb for µ 1 can be
≥
expressed as
The quantity δnl accounts for perturbations in n due to n′(<ml) = µ−1[n1(<mb)+
non-linearclusteringandwedefineλastheexpectednum- n (<m )µβ2 n (<m ) . (9)
2 l 2 b
ber of sources in the absence of clustering. We model the −
fluctuationδ withalognormaldistributionandcombine Here, n and n denote number count(cid:3)s integrated up to
nl 1 2
this with the additional uncertainty due to shot noise to a limiting magnitude over a constant slope of β and β
1 2
give a Poisson-lognormal distribution (eg. Coles & Jones respectively. The clustering term which features in equa-
1991;BTP).UnlikeaGaussiandistribution,thelognormal tion (5) has been omitted here for clarity but is used in
distributionaccountsfornon-linearclusteringoftheback- our analysis later.
Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band 9
If the limiting magnitude of observations matches the 3.4. Self-consistent axi-symmetric mass estimator
break magnitude then
Thesecondmethodwhichweusetoestimateclustermass
istheso-calledself-consistentaxi-symmetricmassestima-
β
1
n2(<ml)= n1(<ml). (10) tor introduced by T98. The non-local nature of this esti-
β
2
matorallowsthe magnificationequation(12)to be solved
Equation (9) can thus be simplified to foraself-consistentκandγradialprofile.Althoughthees-
timatorisvalidonlyforaxi-symmetricmassdistributions,
β it can be applied to data binned in any self-similar set of
n′ =nµ−1 1+ 1 µβ2 1 , (µ 1) (11)
β − ≥ contours centred on the peak of the mass distribution.
(cid:20) 2 (cid:21)
(cid:0) (cid:1) Inagivenannularbini,the shearcanbe expressedas
where n=n (<m ) here,is the unlensed surface number
1 l
γ = κ κ (16)
count density observed, for example, at the edge of the i i i
| − |
field of view. If β = β and hence there is no break,
1 2 where κ is the convergence in the bin and κ is the con-
i i
equation (11) becomes equation (5). For the case when
vergence averaged over the area interior to and including
µ<1, equation (5) must be reverted to.
the bin. Substituting this into equation (12) gives for the
magnification in bin i
3.3. Local mass estimator
µ−1 =(1 κ )(1 2κ +κ ). (17)
P i − i − i i
The simplest means of arriving at a mass estimate given
Dividing κ into two terms, one for bin i and the other,
a measurement of magnification is by assuming a local i
relationship between the convergence and shear. Such a which we denote ηi−1, for all interior bins, so that
relationship can be derived once a model is chosen to de- 2
scribe the mass distribution. Since magnificationdepends κi =ηi−1+ i+1κi (18)
on the convergence,κ, and shear, γ, as
allows rearrangementof equation (17) to give
µ= (1 κ)2 γ2 −1, (12) (i+1)
− − κi = i+1 (i 1)ηi−1
4i { − − −
(cid:12) (cid:12)
athnids (cid:12)ghievnecseaadlliorwecstκly(cid:12)tionvbeerteisbtliemraetlaedti.on for µ in terms of κ S[(i−1−(i+1)ηi−1)2+4iPµ−i 1]1/2}. (19)
For simplicity, we use the ‘parabolic estimator’ sug- Theparities and havethesamefunctionasinSection
P S
gested by the cluster simulations of van Kampen (1998) 3.3.
which relates γ to κ via Using equation (19), κ is calculated iteratively. The
only freedom is choice of η which as equations (16) and
0
γ = 1 c κ/c . (13) (18) show, is γ , the shear in the first bin. To avoid non-
| − | 1
physical solutions, γ2 µ−1 must be enforced leaving
As in T98,pwe adopt the value c=0.7 corresponding to a onlyasensiblerange1of≥vaPlue1s.AsT98discuss,theoverall
profile which lies between that of a homogeneous sheet of mass and shear profile obtained with the self-consistent
matter(κ=constant)andanisothermalprofile(κ r−1, axi-symmetric estimatorprovesto have only a minor sen-
∝
r the distance fromthe clustercentre).The magnification sitivity to the choice of γ .
1
can therefore be written
µ−1 =[(κ c)(κ 1/c)], (14) 3.5. Self-consistent 2D mass estimator
P − −
The finalreconstructionmethod we apply inthis paper is
where = 1 accounts for image parity on either side of
P ± that discussedby Dye & Taylor(1998).By pixellising the
the critical line implied by the modulus in equation (12).
imageinto arectangulargridofpixels,the componentsof
Rearranging this equation for κ gives
the shear in any pixel n can be expressed as
κ= 1 (c2+1) (c2+1)2 4c2(1 µ−1) . (15) γin =Dimnκm, i=1,2 (20)
2c −S − −P
(cid:16) p (cid:17) where summationoverindex m is implied for allN pixels
The second parity is due to the parabolic nature of
S onthegrid.ThematricesD1 andD2 aresimplegeometri-
equation(14)whichpermitsasecondcriticallineandthus
cal functions of the different combinations of positions of
higher values of κ. Since there is no evidence for the ex-
pixels m and n (see Dye & Taylor 1998 for details).
istence of a double critical line in CL0024+1654, we will
Writingequation(12)initspixellisedformandsubsti-
adopt =+1 throughout this paper.
S tuting the expression for shear in equation (20) gives the
Sincetheparabolicestimatorrequiresonlylocalκand
vector equation
γ,itfindsapplicationtobothradialand2Dmagnification
distributions in this paper. 1 2κ+κGκt µ−1 =0 (21)
− −P
10 Dyeet al.: Lensmagnification by CL0024+1654 in the U and R band
where isagaintheimageparityfromSection3.3,µ−1 is contaminants,any bias as a result of normalisingout this
P
the N-dimensional vector of pixellised inverse magnifica- effect falls within our error budget. We therefore expect
tionvalues,κt isthetransposeofthevectorκofpixellised only a small bias to remain from the underestimation of
convergencevaluesand1isthe vector(1,1,1,...).Gisan background counts at large radii.
N N N matrix with the elements As far as the number count slopes are concerned, the
× ×
R band slope determined by Hogg et al. (1997)over21<
Gpqn =δpnδqn−D1pnD1qn−D2pnD2qn (22) RAB < 25 is βR = 0.83, in agreement with Smail et al.
(1995) who measure β = 0.80 over the same magnitude
where δ is the Kr¨onecker delta, and summation occurs R
ij range. This is consistent with the slopes in the V and I
only over indices p and q in equation (21).
fromtheHubbleDeepFielddataofPertetal.(1998)with
We solve equation (21) for κ using a hybrid Powell
β 0.9 and β 0.8 over23<(V,I) <26. However,
method provided by the NAG routine C05PCF. V ≃ I ≃ AB
over the fainter magnitude range 26 < (V,I) < 29,
AB
they measure a shallower slope of β <0.5 and find the
V,I
4. Cluster Model constraints from Number trend that flattening of the number count slope is more
Counts pronounced in shorter wavelength bands. The fact that
this is seen at all in the I band therefore suggests that
BeforeapplicationofthetheorypresentedinSection3,the flattening most probably also occurs in R at these faint
properties of the background population must be under- magnitudes. This is fortified by the more recent R band
stood. Our initial attention is turned toward the R band countsofMetcalfeetal.(2001)fromtheWilliamHerschel
population of backgroundgalaxies since this proves to be DeepField,whichsuggestaslopeofβ 0.6fainterthan
R
the mostsuitableforthe applicationoflensmagnification R 26. ≃
AB
owingtoitsshallowernumbercountslope.Inthissection, O≃ur R band observations tantalisingly extend to ap-
weusetheradiallybinnednumbercountsofthesampleto proximately the depth where the apparent break occurs.
fit an isothermal, power-law and NFW profile. We defer We comparein Section4.3and 5.3the difference between
discussion of the U band sample properties until Section the results obtained using the single and dual slope mod-
6. els.Intheanalysishereafter,forthesingleslopemodel,we
take β = 0.80 and for the dual slope model, β = 0.80
R R1
and β =0.5 with a break magnitude of R=26.
4.1. R band sample characteristics R2
The characteristics of the R band background source
4.2. Magnification profiles
galaxy population must be constrained before we can ap-
ply our lensing analysis to the source counts. Specifically, In the case of lens magnification, the least biased method
the unlensed surface number density nR and the number of determining the best fit mass model is to fit depletion
count slope must be determined. curvestothenumbercountprofile.Inthisway,thelensing
T98 estimate the unlensed surface number density signalisusedinitspurestformbeforepotentialbiasesare
from deep number counts of field galaxies observed inde- introduced by calculation of the κ profile.
pendently oftheir work.We optfor the alternativechoice To fit the depletion profile from a given mass model,
here of taking nR from our field where the lens effect of its lens magnification must be determined. We choose to
theclusterisexpectedtobe small.Thiseliminatespoten- fit an isothermal sphere, a power-law mass model and a
tial biasing of the reconstructions which arise from differ- NFW profile, all of which have analytical forms for their
ing completeness characteristics between the lensed field magnification. In the case of the isothermal sphere, the
and the reference unlensed field (Gray et al. 2000). This massprofileiscompletelydeterminedbyitscriticalradius
is particularly important for the U band break search in r ,
crit
Section 6.2. In an annular region centred on the cluster
r
and bounded by the radial limits 120′′ < r < 180′′, we κ(r)= , (23)
measure n = 43 2 arcmin−2 in the R. This value is 2rcrit
R
±
assumed in our subsequent analysis. giving a magnification of
Ideally, this normalising annulus should be further
away from the centre of the cluster than we have chosen. µ(r)= 1 rcrit/r−1. (24)
| − |
However,aswediscussinSection5,theedgeofourfieldis
The power-law model we choose is that of Schneider,
somewhat noisy due to the presence of some bright stars.
Ehlers& Falco(1993)whichgivesa smooth,non-singular
This causes a drop in the number counts at larger radii
surfacemassdensitydistributionwithaκprofilegivenby
so that normalising here would force a large over density
of counts at medium radii and hence negative mass. An 1+ξx2
κ(x)=κ . (25)
alternative explanation is that such an overdensity would 0(1+x2)2−ξ
be caused by contaminating cluster galaxies. In fact, it
is most likely that both effects jointly contribute. Since Here κ is the peak surface mass density at the centre, ξ
0
our reconstructionsinclude the uncertaintydue to cluster is the power-lawslope andx=r/r withr the core
core core