Strong Lensing as a Probe of the Mass Distribution Beyond the Einstein Radius. Mass & Light in SL2S J08544-0121, a Galaxy Group at z=0.35
Marceau Limousin, Eric Jullo, Johan Richard, Remi Cabanac, Sherry H. Suyu, Aleksi Halkola, Jean-Paul Kneib, Raphael Gavazzi, Genevieve Soucail
AAstronomy & Astrophysics manuscript no. draft c (cid:13)
ESO 2018October 29, 2018
Strong Lensing as a Probe of the Mass Distribution
Beyond the Einstein Radius
Mass & Light in SL2S J08544-0121, a Galaxy Group at z = . Marceau Limousin , , , Eric Jullo , , Johan Richard , , R´emi Cabanac ,Sherry H. Suyu , Aleksi Halkola , Jean-Paul Kneib , Raphael Gavazzi , & Genevi`eve Soucail (cid:63) Laboratoire d’Astrophysique de Marseille, UMR 6610, CNRS-Universit´e de Provence, 38 rue Fr´ed´eric Joliot-Curie, 13 388Marseille Cedex 13, France Laboratoire d’Astrophysique de Toulouse-Tarbes, Universit´e de Toulouse, CNRS, 57 avenue d’Azereix, 65 000 Tarbes, France Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Durham University, Physics and Astronomy Department, South Road, Durham DH3 1LE, UK Department of Astronomy, California Institute of Technology, 105-24, Pasadena, CA91125, USA Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany Excellence Cluster Universe, Technische Universit¨at M¨unchen, Boltzmannstr. 2, 85748 Garching, Germany CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France UPMC Universit´e Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France Laboratoire d’Astrophysique de Toulouse-Tarbes, Universit´e de Toulouse, CNRS, 14 avenue Edouard Belin, 31 400 Toulouse,FrancePreprint online version: October 29, 2018
ABSTRACT
Strong lensing has been employed extensively to obtain accurate mass measurements within the Einstein radius. In this article, weuse strong lensing to probe mass distributions beyond the Einstein radius. We consider SL2S J08544-0121, a galaxy group at redshift z = .
35 with a bimodal light distribution and with a strong lensing system located at one of the two luminosity peaks separated by ∼ (cid:48)(cid:48) . The main arc and the counter-image of the strong lensing system are located at ∼ (cid:48)(cid:48) and ∼ (cid:48)(cid:48) , respectively, from the lens galaxycentre. We find that a simple elliptical isothermal potential cannot satisfactorily reproduce the strong lensing observations. However,with a mass model for the group built from its light-distribution with a smoothing factor s and a mass-to-light ratio M / L, we obtainan accurate reproduction of the observations. We find M / L = ±
27 ( i band, solar units, not corrected for evolution) and s = (cid:48)(cid:48) ± σ confidence level). Moreover, we use weak lensing to estimate independently the mass of the group, and find a consistent M / L inthe range 66-146 (1- σ confidence level). This suggests that light is a good tracer of mass. Interestingly, this also shows that a stronglensing only analysis (on scales of ∼ (cid:48)(cid:48) ) can constrain the properties of nearby objects (on scales of ∼ (cid:48)(cid:48) ). We characterise thetype of perturbed strong lensing system allowing such an analysis: a non dominant strong lensing system used as a test particle toprobe the main potential. This kind of analysis needs to be validated with other systems since it could provide a quick way of probingthe mass distribution of clusters and groups. This is particularly relevant in the context of forthcoming wide field surveys, which willyield thousands of strong lenses, some of them being perturbed enough to pursue the analysis proposed in this paper.
Key words.
Gravitational lensing: strong lensing – Galaxies: groups –
1. Introduction
Gravitational lensing probes the mass distribution projectedalong the line of sight. When the surface mass density of a lensis larger than a critical threshold, i.e. in the strong lensing (SL)regime, the light from a background source galaxy is lensed into
Send o ff print requests to : [email protected] (cid:63) Based on observations obtained with MegaPrime / MegaCam, a jointproject of CFHT and CEA / DAPNIA, at the Canada-France-HawaiiTelescope (CFHT) which is operated by the National Research Council(NRC) of Canada, the Institut National des Sciences de l’Univers of theCentre National de la Recherche Scientifique (CNRS) of France, andthe University of Hawaii. This work is based in part on data productsproduced at TERAPIX and the Canadian Astronomy Data Centre aspart of the Canada-France-Hawaii Telescope Legacy Survey, a collab-orative project of NRC and CNRS. Also based on HST data, program10876 and Keck telescope data. multiple images. These multiple images provide strong observa-tional constraints on the projected mass distribution of the lenswithin the Einstein radius. Since the discovery of the first grav-itational arc in the galaxy cluster Abell 370 twenty years ago(Lynds & Petrosian 1986; Soucail et al. 1987; Richard et al.2010), strong lensing has been widely used to probe the massdistribution of structures at di ff erent scales: galaxies (see, e.g. the SLACS survey, Koopmans et al. 2006), galaxy clusters (see, e.g. Halkola et al. 2006) and recently galaxy groups (Cabanacet al. 2007; Limousin et al. 2009; Belokurov et al. 2009).
Because most of the galaxies in the Universe are part of largerstructures, either groups or clusters, so are many SL systems(see, e.g.
Kundic et al. 1997; Fassnacht & Lubin 2002; Faure a r X i v : . [ a s t r o - ph . C O ] S e p Limousin et al.: Strong Lensing Beyond the Einstein Radius et al. 2004; Morgan et al. 2005; Williams et al. 2006; Momchevaet al. 2006; Auger et al. 2007; Tu et al. 2008; Auger et al. 2008;Grillo et al. 2008; Treu et al. 2009; Inada et al. 2009). A massdistribution located at a small angular distance from a stronglens may induce measurable perturbations in the lensing sig-nal. Not taking this external perturbation into account can se-riously bias the results inferred from the SL modelling as shownby Keeton & Zabludo ff (2004): they found that if the environ-ment is neglected, SL modelling of double-image lenses largelyoverestimate both the ellipticity of the lens galaxy ( ∆ e / e ∼ . ∆ h / h ∼ . ∆ h / h ∼ . ff er-ent authors (see, e.g Keeton et al. 1997; Kochanek et al. 2001;Keeton & Zabludo ff a ff ected byexternal mass distributions, and people have tried to take thisbias into account in order to improve the SL modelling. As ob-servations become more and more accurate, we can expect tobe more and more sensitive to external mass distributions nearstrong lenses. In this article, we propose to exploit this externale ff ect by using the perturbations measured in SL modelling as probes of the external mass distribution . We first remind the reader of a previous attempt we made to lo-cally probe the potential of the galaxy cluster Abell 1689 with aperturbed SL system. In the core of galaxy cluster Abell 1689,Limousin et al. (2007) reported SL systems (“rings”) formedaround three elliptical galaxies located 100 (cid:48)(cid:48) away from the clus-ter centre, i.e. the transitional region between the strong andweak lensing regimes. These SL systems should be sensitiveto the external shear and convergence produced by their par-ent cluster (Kochanek & Blandford 1991). Based on simula-tions, Tu et al. (2008) showed that such strong lenses couldbe used to probe the cluster potential locally. They applied thismethod to the three rings discovered in Abell 1689, and foundthat solely modelling these three rings (i.e. without includingany other multiply-imaged systems that are also produced by thecluster) provides strong evidence for bimodality of the clustercore; it is not possible to model simultaneously the three ringsassuming a unimodal mass distribution for the cluster. This bi-modality confirms previous parametric SL studies of Abell 1689(Miralda-Escude & Babul 1995; Halkola et al. 2006; Limousinet al. 2007; Leonard et al. 2007; Saha et al. 2007; Okura et al.2008). More importantly, this result shows that SL features of1-2 (cid:48)(cid:48) -wide Einstein rings actually contain information on themass distribution of the parent cluster, i.e. on a much largerscale than their Einstein radii. In other words, this study sug-gests that strong lenses can be used to probe mass distributions beyond their Einstein radius. In this article, we further developthis idea on another perturbed SL system located in a galaxygroup, SL2S J08544-0121.All results are scaled to a flat, Λ CDM cosmology with Ω M = Ω Λ = =
70 km s − Mpc − .In this cosmology, 1 (cid:48)(cid:48) corresponds to a physical transverse dis-tance of 4.94 kpc at z = .
35. All images are aligned with the WCS coordinates, i.e. north is up and east is left. Magnitudes aregiven in the AB system. Luminosities are given for the i band, insolar units, not corrected for passive evolution. Ellipticities areexpressed as ( a − b ) / ( a + b ), and position angles are givencounterclockwise with respect to the west. Shear and conver-gence are computed for a source redshift of z s =
2. SL2S J08544-0121: Presentation & Data
SL2S J08544-0121 is part of the Strong Lensing Legacy Survey(SL2S, Cabanac et al. 2007) , which collects SL systems in theCanada France Hawaii Telescope Legacy Survey (CFHTLS).SL2S J08544-0121 is a galaxy group at z = .
35 presented byLimousin et al. (2009) which contains a SL system (Fig. 1).
SL2S J08544-0121 has been observed in five bands as part ofthe CFHTLS. The i -band data are used to build luminosity mapsfrom isophotal magnitudes of elliptical group members and toperform a weak-lensing analysis.The bottom panel of Fig. 1 shows a 10’ ×
10’ CFHTLS i -band image. The white cross gives the location of the stronglens. We draw luminosity isodensity contours of 10 , 3 × ,10 , 3 × and 10 L (cid:12) kpc − . The top-right panel also showsa CFHTLS 1-arcmin gri colour image centred on the lens. The strong lensing features detected from ground-based imageshave been observed with the
Hubble Space Telescope (HST).Observations were done in snapshot mode (C15, P.I. Kneib, ID10876) in three bands with the ACS camera (F475, F606, andF814). Figure 1 shows a colour image of the strong lens basedon these observations. We report two multiply-imaged systems:the first system is bright and forms a typical cusp configurationperturbed by a satellite galaxy (labelled Dwarf on Fig. 1). Thesecond system is a very faint arc located west of the lens at alarger radius. It is not possible to reliably identify individual im-ages on the faint arc. Moreover, given its faintness, spectroscopyis hopeless with current facilities, as the surface brightness isca. 31 mag arcsec − , and therefore it is not used in the followinganalysis. As can be appreciated on Fig. 1, the HST data bringssignificant amounts of additional information on the lensed fea-tures. We have used the Low Resolution Imager and Spectrograph(LRIS, Oke et al. 1995) on the Keck telescope to measure thespectroscopic redshift of both the lens and the brightest arc of theSL2S J08544-0121 system. On January 14 2007, we obtained300 seconds on the lensing galaxy and 4 exposures of 900 sec-onds each on the arc, using a 1.0 (cid:48)(cid:48) wide slit. A 600 lines mm − grism blazed at 4000 Å and a 400 lines mm − grating blazed at8500 Å were used in the blue and red channels of the instru-ment, both light paths being separated by a dichroic at 5600 Å.The corresponding dispersions are 0.6 / / / red channel. The resulting extractedspectra are shown in Fig. 2. The lens presents a typical ellipticalspectrum at z = . ± . imousin et al.: Strong Lensing Beyond the Einstein Radius 3 by ∼ z = . ± . The luminosity contours of SL2S J08544-0121 are elongated inthe east-west direction. The SL deflector is populated by a singlebright galaxy. Note (Fig. 1) that the innermost luminosity iso-density contour at 10 L (cid:12) kpc − encompasses the SL system butalso two bright galaxies located ∼ (cid:48)(cid:48) east from the SL system,making this light distribution bimodal. This is the only group ofthe sample presented by Limousin et al. (2009) for which theluminosity isodensity distribution is not clearly dominated bythe lens, making this configuration rather exceptional: the largeEinstein radius ( ∼ (cid:48)(cid:48) ) indicates a significant mass concentrationassociated with this lens, but the luminosity isodensity distribu-tion is actually bimodal. Fig. 2. up ) and of the bright arc ( low ). The greenshaded region masks the residuals of a strong OH atmospheric emission
3. Modelling the Lens
In this Section, we attempt to reproduce the SL multiple imagesusing a single elliptical isothermal potential centred on the brightgalaxy. All optimisations are performed in the image plane us-ing the L enstool software (Jullo et al. 2007). We quantify thegoodness of the fit by using the image plane RMS and the cor- responding χ . When necessary, we compare the fits using theBayesian Evidence. As explained in Section 2.2, we do not use the faint arc in theanalysis and focus on the bright multiply-imaged system. Thissystem is composed of 4 main images: 1.1, 1.2, 1.3 and 1.4.Additional images are produced by the satellite (dwarf) galaxy.These are not considered in the analysis because we do not wantto complicate the modelling by adding a sub-halo for the satel-lite galaxy. We also make the assumption that, given its smallsize, the satellite galaxy does not influence images 1.1 to 1.4.This assumption will be discussed in Section 8 and addressedfurther in a forthcoming publication. Since the merging arc com-posed of images 1.2 and 1.3 is well resolved, we can safely asso-ciate two other images on this arc, namely 2.2 and 2.3 (Fig. 1).Their counter-images expected near images 1.1 and 1.4 are notsafely identified, therefore we do not use them in the analysis.Indeed, parametric strong lensing analyses are highly sensitiveto misidentifications of images and we prefer to use only the im-ages we are confident in. This gives us a total of 8 observationalconstraints.
In this subsection, we describe the properties of the light dis-tribution of the bright galaxy populating the strong lensing de-flector. We use the IRAF task ellipse to measure the shape of itsisophotes. We find an ellipticity of e = .
206 and a position angleof 39 ± (cid:48)(cid:48) from the centre. A closer inspection of thegalaxy centre clearly reveals a double core even though the outerisophotes are elliptical. The above measurements therefore cor-respond to the superposition of the light from each component.The spectrum of the galaxy presented in Fig. 2 does not showfeatures of another galaxy at a di ff erent lens redshift along theline of sight. Given the similar colours of these two components,the bright galaxy may be the result of a recent merger. The sizes of the multiple images have been estimated using theIRAF task imexamine . They range from 0.11 (cid:48)(cid:48) to 0.15 (cid:48)(cid:48) , witha mean of 0.13 (cid:48)(cid:48) . Therefore, the positional uncertainty is set to0.13”. We note that this positional uncertainty may seem largefor HST data but we stress that we are not in the case of point-like objects (quasar lensing) where the astrometric precision canreach 0.01 (cid:48)(cid:48) . In our case, the images are extended and the depthof the snapshot observations does not allow us to resolve betterthe conjugated points with imexamine . We note that large posi-tional uncertainties are often used in the case of extended images(see, e.g.
Oguri 2010).
The lens potential is parametrised by a dual Pseudo IsothermalElliptical Mass Distribution (dPIE, see El´ıasd´ottir et al. 2007).The 3D density distribution of the dPIE is: ρ ( r ) = ρ (1 + r / r )(1 + r / r ) ; r cut > r core . (1)This distribution represents a spherical system with scale radius r cut , core radius r core and central density ρ . Limousin et al.: Strong Lensing Beyond the Einstein Radius
Fig. 1.
Group SL2S J08544-0121 at z spec = . Upper Left: composite CFHTLS gri colour image (1 arcmin = ×
297 kpc ). Upper Right: composite HST / ACS F814W-F606W-F475W colour image (24 (cid:48)(cid:48) × (cid:48)(cid:48) = ×
118 kpc ). We show the proposed multiple-image identification.The dwarf galaxy and the main extra image it produces is labelled. Lower:
CFHTLS i band (10 (cid:48) × (cid:48) = × ). Luminosity isodensitycontours of 10 , 3 10 , 10 , 3 10 and 10 L (cid:12) kpc − are drawn (continuous black line), and the white cross shows the location of the SL system.imousin et al.: Strong Lensing Beyond the Einstein Radius 5 This profile is formally the same as the Pseudo IsothermalMass Distribution (PIEMD) described in Limousin et al. (2005).Its scale radius is set to 250 kpc, i.e. larger than the range wherethe observational constraints are found. Allowing r core to varyproduces models with core radii much smaller than the range ofradii over which we have observational constraints. Therefore,we can set r core =
0, and the dPIE profile are close to isothermalin the range of interest. The remaining free parameters of thedPIE profile are:- the halo centre position (X,Y), which is allowed to varywithin 3 (cid:48)(cid:48) of the light distribution centre- the halo ellipticity e , which is forced to be smaller than 0.6,as suggested by numerical simulations (Jing & Suto 2002)- its position angle θ , which is allowed to vary between 0 and180 degrees- The fiducial velocity dispersion which is allowed to varybetween 200 and 900 km / s.We emphasise that this fiducial velocity dispersion is not theSpherical Isothermal Sphere velocity dispersion. It is usuallysmaller, and we refer the reader to El´ıasd´ottir et al. (2007) fora self-contained description of the dPIE profile. Results of the optimisation are given in Table 1. This first op-timisation results in a poor fit to the data, with the RMS errorof image positions ∼ . (cid:48)(cid:48) in the image plane (i.e. significantlylarger than the assumed positional uncertainty of 0.13 (cid:48)(cid:48) ) and areduced χ of 29. The halo position is found to coincide with thelight distribution centre within error bars. The halo ellipticity isat the upper bound of the input prior, and the position angle isequal to ∼
18 deg. Only when we allow the halo ellipticity toreach values as high as 0.9 are we able to reproduce the obser-vational constraints (RMS equals to 0.06 (cid:48)(cid:48) for e = . θ ∼ ff set from the light distributioncentre by one arcsecond.We conclude that a single potential does not satisfactorilyreproduce the observational constraints. We have used the lensmodelling method based on Halkola et al. (2006, 2008) in par-allel to our method, and found that the observational constraintsused in this work require an external shear component in order tobe properly reproduced. In the rest of the paper, we include thecontribution of the external mass distribution in the lens mod-elling.
4. An External Mass Perturbation Based on theLight Distribution: Does Light traces Mass?
The large scale properties of SL2S J08544-0121 shown onFig. 1, together with the failed modelling attempted in the pre-vious Section, suggest the need to take into account an externalmass perturbation. In order to test the hypothesis that light tracesmass, this external perturbation will be mapped from the knownlight distribution properties.
The first step is to build luminosity maps of SL2S J08544-0121from which we will derive the external mass perturbation prop-erties. To identify group members, we select all galaxies having linked to the central density by: σ = G πρ r r ( r cut − r core )( r cut + r core ) , a r − i colour di ff erence smaller than 0.15 magnitudes from thebright galaxy deflector (Limousin et al. 2009). Because we wantto describe the perturbation of the galaxy group on the SL sys-tem, this luminosity map should not take into account the lightcoming from the galaxies populating the SL deflector. Therefore,we select all group members apart from the bright galaxy popu-lating the deflector and the associated satellite galaxy . This par-tial group luminosity is referred to as L ext hereafter. From thiscatalogue, we generate smoothed luminosity maps, and hencethe mass maps, assuming mass follows light. An important in-gredient of this procedure is the smoothing scale of the lumi-nosity maps. Since this influences the properties of the derivedmass maps, we adopt the smoothing scale as a free parameter fordescribing the external mass perturbation.We use the following smoothing scheme: the 10’ × i -band image is divided into cells of size c pixels,which translates into c × (cid:48)(cid:48) (since the pixel size equals0.186 (cid:48)(cid:48) ). We compute the rest-frame i -band luminosity L of eachgalaxy located in a given cell with L = ( M (cid:12) − M + DM + k) / . (2)where M is the i -band isophotal magnitude of the galaxy, M (cid:12) isthe solar absolute magnitude in the i band, DM is the distancemodulus, and k the k-correction factor that is estimated fromelliptical templates by Bruzual & Charlot (2003) using single-burst stellar formation models. Then we sum up the luminositiesof all galaxies in each cell to get the total luminosity for the cell.The resulting luminosity isodensity map is then convolved witha Gaussian kernel of width w . This gives an angular smoothingscale s that equals c × × w (cid:48)(cid:48) . Figure 3 shows three lumi-nosity maps, where we distribute the same total luminosity L ext for three smoothing scales. We draw luminosity isodensity con-tours of 10 and 10 L (cid:12) arcsec − . One can appreciate how thesmoothing scale s influences the shape of the luminosity isoden-sity contours. Once a luminosity map with a given smoothing scale s is ob-tained, we assume a constant mass-to-light ratio M ext / L ext to con-vert it into a mass map. This M ext / L ext is the second free param-eter describing the perturbation produced by the galaxy group.Since we have excluded the bright galaxy populating the deflec-tor when building the luminosity map, this mass map can be con-sidered as the external mass perturbation. Therefore we refer tothe mass contained in this map as M ext . Then, we use the algo-rithm developed by Jullo & Kneib (2009) to transform this massmap into a grid of analytic circular dPIE potentials, supportedby L enstool .We model the mass distribution of SL2S J08544-0121 witha 5 (cid:48) hexagonal grid of dPIE potentials. In order to build an adap-tive grid where the resolution follows the 2-D mass density, werecursively split the input mass map into equilateral triangles un-til the mean surface mass density per triangle is lower than 10 M (cid:12) arcsec − . Then we place a dPIE potential at each node ofthe grid with the following parameters: core radii r core are setto the local grid resolution and cut-o ff radii r cut = × r core . InJullo & Kneib (2009), we found that such values of r cut ensureda smooth and extended density profile. We estimate the dPIEcentral velocity dispersions σ i by inverting the equation σ i = M i , j Σ j , (3)where Σ j is the surface mass density at the grid nodes location,and M i , j is a mapping matrix whose coe ffi cients depend of the Limousin et al.: Strong Lensing Beyond the Einstein Radius
Table 1.
Parameters of the lens inferred from two optimisations:
First line:
A single halo models the lens potential.
Second line:
We add an externalmass perturbation on top of the halo lens potential. Coordinates are given in arcseconds with respect to the centre of the galaxy deflector. e is thecentral halo ellipticity. Error bars correspond to 1 σ confidence levels.Model δ ( x ) δ ( y ) e θ σ (km s − ) RMS χ log(Evidence) Prior1 Lens -0.24 ± ± + . − . ± ± (cid:48)(cid:48) / e < ± ± ± ± ± (cid:48)(cid:48) e < Fig. 3.
Three luminosity maps of the same luminosity L ext for di ff erent smoothing scales as indicated on each panel. We draw in white luminos-ity isodensity contours of 10 and 10 L (cid:12) arcsec − . The smoothing scale s influences the shape of the luminosity isodensity contours, and byconstruction the shape of the resulting mass distribution, hence its lensing properties. dPIE core and cut-o ff radii (see Jullo & Kneib 2009). In orderto prevent negative σ i , we invert equation 3 by a mean-squareminimisation technique. The density threshold (for splitting thecells into triangles) controls the grid resolution, and might thusbe considered as an important parameter. However, we have triedto use smaller thresholds, down to 10 M (cid:12) arcsec − , and the re-sults were unchanged. Therefore, we keep the 10 M (cid:12) arcsec − threshold because the corresponding mass maps require lessmass clumps. We also force the algorithm to stop after 4 lev-els of splitting. On average, a grid cell contains about 200 dPIEpotentials. We now model the SL system, taking into account the exter-nal mass perturbation parametrised by a smoothing scale s anda mass-to-light ratio M ext / L ext . We generated mass maps withsmoothing scales s ranging from 1 to 40 (cid:48)(cid:48) in steps of 2.5 (cid:48)(cid:48) andmass-to-light ratios from 10 to 190 in steps of 20. Each massmap is then included in the modelling of the SL system. Thismodelling is performed in the image plane. We note that thesetwo extra parameters describing the external mass perturbationare not treated the same way as the five parameters of the de-flecting halo. For each set of parameters ( s , M ext / L ext ), we opti-mise the 5 parameters of the halo. Parameters for the strong lensdeflector are the same as in Section 3.4. For each set of param-eters ( s , M ext / L ext ), we quantify the goodness of the SL mod-elling using the image plane RMS, the corresponding χ and theBayesian Evidence.Since our goal is to constrain the galaxy group as a whole,in the following we use M / L corresponding to the total mass-to-light ratio of the group; i.e. M (L) is the sum of the external mass(luminosity) perturbation and the mass (luminosity) of the lens. We checked that degeneracies of each mass component near thelens are small. For the range of parameters ( s , M ext / L ext ) inves-tigated in this work, we compute M ext / M lens in a circle of radius10 (cid:48)(cid:48) centred on the lens. This ratio falls between 10 − and 10 − .Total masses and luminosities are computed within a regionof 10’ ×
10’ centred on the lens. At the redshift of the group, itcorresponds to ∼ × (Fig. 1).
5. Results: Properties of SL2S J08544-0121
For a certain range of parameters characterising the externalmass perturbation we obtain excellent fits to the observed con-straints. We present first the best-fit model, and then the derivedconstraints on the galaxy group properties. We emphasise thatwhat we achieve here is to constrain the properties of the galaxygroup as a whole (on scales of 100 (cid:48)(cid:48) ) based on a local SL analysis only (on scales of 10 (cid:48)(cid:48) ). The modelling results are given in Table 1. The best-fit modelhas a total mass-to-light ratio of (cid:39)
75 ( i band, solar units, notcorrected for evolution) and a smoothing scale of (cid:39) (cid:48)(cid:48) (Fig. 4).The RMS error between observed and modelled image positionsin the image plane is 0.05 (cid:48)(cid:48) , yielding a reduced χ of 0.96. Thisis a significant improvement compared to the modelling withoutexternal mass perturbation, which had RMS = (cid:48)(cid:48) . To com-pare quantitatively the two models (i.e. mass models with andwithout the inclusion of external perturbations), we compute theBayesian evidence values of the two models. The evidence takesinto account the additional complexity of the new model with theextra parameters for the external perturbations. The di ff erence inthe evidence of the two models, which is the relative probabil- imousin et al.: Strong Lensing Beyond the Einstein Radius 7 Fig. 4.
Results: constraints on the galaxy group mass-to-light ratio M / Land smoothing scale s that characterises the size of dark matter clumps.Vertical dashed lines mark the lower and upper bounds of the constraint(1 σ error bars) on the group mass-to-light ratio from an independentweak-lensing analysis (Section 7). ity of the models given the data (assuming the two models areequally probable a priori), is 2 × . The data therefore rankthe perturbed model much higher than the simple model.We find the position of the halo to coincide with the centre ofthe light distribution. The modelled position angle of the halo is21.5 deg. Comparing this value to the position angle of the lightdistribution is complicated due to the bimodal light distributionof the bright galaxy (Section 3.2). In particular, a merger willa ff ect the light and mass distributions so that agreement may notnecessarily be expected. χ di ff erences between models with di ff erent s and M / L valuesare translated into confidence levels, which are drawn on Fig. 4.Considering the 2 σ contour, we find M / L = ±
27 ( i band, solarunits, not corrected for evolution) and s = (cid:48)(cid:48) ± We are able to reproduce accurately the observational constraintswhen considering an external mass perturbation drawn from thelight distribution. Because our SL analysis is sensitive to themass, this finding is consistent with the hypothesis that light is agood tracer of mass. We note, however, that we have not demon-strated the uniqueness of the smoothed light model.
6. Local Effect of Large Scale Perturbation
In this Section, we propose to explain why a local
SL analysisis able to constrain global properties of the galaxy group hostingthe lens. First, we investigate the impact of the external perturba-tion on the local SL modelling (i.e. on the local image positions).
The lensing properties of a mass distribution are commonlyparametrised by a shear γ and a convergence κ (see, e.g. Schneider et al. 1992, for the definition of these quantities). Here we estimate the mean shear and convergence experienced locallyby the lens for each set of parameters ( s , M ext / L ext ) by averaging γ and κ generated by the perturbation within a 7 (cid:48)(cid:48) square encom-passing all the multiple images.Figure 5 shows κ and γ maps generated by an external per-turbation of fixed mass (5.7 × M (cid:12) ) for di ff erent values of s (reported on each panel). These maps have been generated for asource redshift of 1.268. Red crosses indicate the lens position.One can appreciate how the experienced shear and convergenceare correlated with the smoothing scale. Figure 6 shows contours of κ and γ overlaid on the results fromFig 4. We see that the constraints inferred from the SL analysisdo not follow κ contours but do follow γ contours. In particu-lar, the best model generates a shear of ∼ ∼ s ), the smaller thegenerated shear. Therefore, the smoother the mass distribution,the higher the total mass in order to generate a given shear level. Figure 6 suggests that the observational constraints require lo-cally an external shear component of ∼ ff (2004), that the shear approximation fails to captureall of the environmental e ff ects . In other words, the shape of thecontours for the constraints follows the curves of constant shear.However, the contours do close, which means that the constraintsare sensitive to more than the shear, most probably to higher or-der terms beyond the shear that are naturally provided by themodelling proposed in this work.In addition, we estimate the shear experienced by images1.1, 1.2, 1.3 and 1.4. To do so, we consider all models fallingin the 1 σ contour. For each model, we compute the shear ex-perienced by the images, and from these numbers, we estimatethe mean shear and the associated standard deviation. The cor-responding shear values for the images listed above are 0.075,0.074, 0.073 and 0.073 respectively, with a typical uncertaintyof 0.01. Therefore, each image does experience the same shearvalue within the error bars. We investigate further the di ff erencesbetween our approach and a constant external shear approach inthe Appendix.
7. Mass-to-Light Ratio from Weak Lensing
We have presented constraints on the mass-to-light ratio ofgalaxy group SL2S J08544-0121 based on a local SL analy-sis. In this Section, we aim to constrain the mass-to-light ratioof SL2S J08544-0121 from an independent weak lensing (WL)analysis, which is intrinsically more sensitive to the projectedmass distribution on large scales. The goal is to check whetherthe M / L ratios inferred from the two techniques are consistent.For a detailed description of our WL methodology seeLimousin et al. (2009). Here we give a brief reminder. We se-lect as background sources all galaxies with i -band magnitudesin the range 21.5-24. The resulting galaxy number density is 13.5 Limousin et al.: Strong Lensing Beyond the Einstein Radius
Fig. 5.
Shear (upper panels) and convergence (lower panels) maps generated by the external perturbation and experienced by the SL system whosecentre is given by the red cross. The total mass is fixed to the same value (5.7 10 M (cid:12) ) in all panels. Panel sizes are 600 ×
600 square arcseconds,and the smoothing scales s vary as indicated on each panel. White contours correspond to shear levels of 0.1 and convergence levels of 0.1 and0.2. One can appreciate how the shear and convergence generated by the group are correlated with the smoothing scale. arcmin − . The completeness magnitude in this band is 23.91 andthe seeing is ∼ (cid:48)(cid:48) . A Bayesian method, implemented in theI m shape software (Bridle et al. 2002), is used to fit the shapeparameters of the faint background galaxies and to correct forthe PSF smearing. From the catalogue of background galaxies,Limousin et al. (2009) performed a one-dimensional WL analy-sis. They fit a Singular Isothermal Sphere (SIS) model to the re-duced shear signal between 150 kpc and 1.2 Mpc from the groupcentre, finding an Einstein radius of 5.4 ± (cid:48)(cid:48) . In order to re-late the strength of the WL signal to a physical velocity disper-sion characterising the group potential, Limousin et al. (2009)estimate the mean geometrical factor using the photometric red-shift catalogue from the T0004 release of the CFHTLS-Deepsurvey (Ienna & Pell´o 2006). They find σ SIS = + − km s − .This translates into a total mass within the considered square of5.3 ± M (cid:12) . Because the total luminosity is 5 10 L (cid:12) , wefind a mass-to-light ratio of 106 ±
40 ( i band, solar units, not cor-rected for passive evolution).This is comparable to the M / L constrained by SL only . Thegood agreement (Fig. 4) between the two methods gives supportto the analysis based on SL only. http: // / users / roser / CFHTLS T0004 /
8. Discussion
An external mass perturbation derived from the light of the groupmembers allows us to fit accurately the observed constraints.Because the observed constraints are sensitive to mass ratherthan light, this suggests that light is a good tracer of mass. Wenote that this result brings forth an e ffi cient way of taking intoaccount an external mass perturbation in SL modelling. Indeed,this perturbation is fully described with only two parameters, themass-to-light ratio and the smoothing scale. In contrast, describ-ing this perturbation parametrically using a mass clump wouldrequire at least three parameters (position and velocity disper-sion), unless independent data constrain one or more of theseparameters (see, e.g. Tu et al. 2009, where X-ray observationsallow one to constrain the group centre).
Why does our SL analysis allow us to infer properties on thewhole galaxy group? We claim that this is due to the perturbedstate of the SL system of SL2S J08544-0121. Most of the per-turbed signal of the multiply imaged system comes from image1.4, because it is located farther from the lens centre ( ∼ (cid:48)(cid:48) ) thanimages 1.1, 1.2 and 1.3 ( ∼ (cid:48)(cid:48) ). If we remove image 1.4 from theset of observational constraints, we are able to fit very well theremaining images without considering any external mass pertur- imousin et al.: Strong Lensing Beyond the Einstein Radius 9 Fig. 6.
As in Fig. 4, constraints obtained on the galaxy group as a whole derived from the local
SL analysis are shown as dashed contours. Solidlines corresponds to lines of constant κ (left) and γ (right) generated by the external mass perturbation and experienced locally by the lens, theirvalues are labelled on each line. bation (the lens being modelled as in Section 3.4.) In that case,we get RMS = (cid:48)(cid:48) and a reduced χ equals to 0.03. Therefore,ignoring image 1.4 prevents us from putting any constraints onthe external mass perturbation, i.e. the host galaxy group. Thisshows that image 1.4 yields the constraints presented in thiswork. This finding will help us diagnose the type of the SL sys-tems to which our new analysis technique can be applied (seeSection 8.5).We note that the SL analysis presented here is very simplesince we just conjugate a couple of images with each other. Inparticular, we do not use the constraints coming from the wholeEinstein ring. More sophisticated methods fully take into ac-count arc surface brightness constraints (see, e.g. Warren & Dye2003; Suyu et al. 2006; Barnab`e & Koopmans 2007). We areaware that we ignore some information that could allow us toput stronger constraints on the galaxy group. On the other hand,the basic level of the SL analysis done here emphasises evenmore the prospects of this method.
We have assumed that the satellite galaxy does not produce sig-nificant shear on the images used as constraints. However, onecould argue that neglecting this satellite galaxy e ff ectively pro-duces the claimed constraints from the SL analysis. This is notlikely – due to the location of the satellite galaxy with respect tothe multiple images (Fig. 1), the satellite galaxy may produce amarginal shear on on images 1.1, 1.2 and 1.3, but is unlikely tohave any significant influence on image 1.4, the image that yieldsmost of the constraints. Indeed, the distance between the satellitegalaxy and image 1.4 is ∼ (cid:48)(cid:48) . We note that we do not quantifythe bias that could result under our working assumption.Besides, a paper focusing on the properties of the satellitegalaxy (Suyu & Halkola, submitted to A&A) shows that evenwith the satellite galaxy included in the lens model, an externalshear of roughly the same magnitude is needed to fit the observedconstraints. The dPIE scale radius is where the logarithmic slope of the 3Ddensity profile smoothly decreases from -2 to -4. The scale ra-dius of the lens is set to 250 kpc in the present analysis. We havealso done a complete analysis for a scale radius of 400 kpc as asanity check and found that the results inferred for the group donot change significantly. To understand why, we superimposedcritical lines of the best-fit parameters of Table 1, for a sourceredshift of 1.268 (without external perturbation), and the criticallines of the best-fit parameters of Section 5.1 (with external massperturbation). We find that the external mass perturbation gener-ates a critical line shift of 1.3 (cid:48)(cid:48) . In parallel, we investigated thecritical lines shifts between various scale radii; increasing from250 kpc to 400 kpc and decreasing from 250 kpc to 100 kpc. Theshifts are 0.12 (cid:48)(cid:48) , an order of magnitude smaller than the shift dueto the external mass perturbation.
We propose to characterise the kind of perturbed SL systemsone should target in order to perform analyses similar to the onepresented in this work. From the ring test done in Abell 1689(Tu et al. 2008, see Section 1.3) and the analysis presented inthis paper, we hint at the need for a non-dominant SL system tobe used as a test particle for probing the main potential.
This is linked to the global geometry of the structure hostingthe SL system: to be a ff ected by a perturbation, the SL systemshould not be at the centre of the structure. Indeed, if the lensstudied in this paper would have dominated the whole group po-tential, image 1.4 would have been located at a similar distancefrom the lens centre as images 1.1, 1.2 and 1.3.The Cosmic Horseshoe (Belokurov et al. 2007; Dye et al.2008) illustrates this point: it is an almost complete Einstein ringof radius 5 (cid:48)(cid:48) containing a luminous red galaxy in its centre. Asrevealed by the SDSS photometry, this galaxy is the brightestobject in the group of ∼
26 members and it dominates the grouplight distribution. No external shear is required in the model ofthe Cosmic Horseshoe SL system, which is already suggested bythe nearly perfect circle outlined by the ring.
To summarise, we should look for multiply imaged systemswhere one of the images is found at a larger radial distance thanthe other images of the SL system.
We have shown that the modelling based on strong lensing only provides strong hints for a bimodal mass distribution: the firstmass component is clearly associated with the strong lensing de-flector, and the second one that perturbs the strong lensing con-figuration seems, to first order, to be associated with the secondlight peak of the bimodal light map. This suggests a dynamicallyyoung structure in the process of formation. A spectroscopic sur-vey of the group further supports this hypothesis: we measuredredshifts for 36 galaxies along the direction of SL2S J08544-0121 by using spectroscopic data acquired with FORS2 at theESO Very Large Telescope (VLT), and confirmed the presenceof a high concentration of galaxies at z ∼ .
35 (Mu˜noz et al.,in preparation). A careful analysis of the redshift distributionof galaxies around this peak reveals two close structures witha radial velocity di ff erence of V r = − . This result isin agreement with the interpretation of our strong lensing only analysis.
9. Conclusion
We propose a method to constrain the dark matter distributionof galaxy groups and clusters. Exploiting information containedin perturbed SL systems, we use the SL geometry to probe themain potential of the host structure responsible for that perturbedstate.We show that the SL only constraints on the mass-to-lightratio of SL2S J08544-0121 are in good agreement with WLconstraints obtained independently, supporting the reliability ofthe proposed method. Moreover, the SL only analysis providesstrong hints for a bimodal mass distribution, which is confirmedby the spectroscopic survey of galaxy group members.We advocate the need for a dedicated search of perturbed SLsystems in the HST archive in order to test and validate furtherthis method, which is particularly promising in the light of futurelarge surveys that will yield thousands of SL systems, some ofthem being perturbed enough to perform the test presented inthis paper.
Acknowledgement
ML acknowledges Bernard Fort, Masamune Oguri & PhilMarshall for related discussions. ML acknowledges the anony-mous referee for a detailed report, and Christopher Kochanek forinsightful comments on the submitted version of this paper. MLacknowledges the Centre National d’Etudes Spatiales (CNES)and the Centre National de la Recherche Scientifique (CNRS)for their support. ML est b´en´eficiaire d’une bourse d’accueil dela Ville de Marseille. The Dark Cosmology Centre is fundedby the Danish National Research Foundation. We thank theDanish Centre for Scientific Computing at the University ofCopenhagen for providing a generous amount of time on itssupercomputing facility. EJ is supported by the NPP, adminis-tered by Oak Ridge Associated Universities through a contractwith NASA. Part of this work was carried out at Jet PropulsionLaboratories, California Institute of Technology under a con-tract with NASA. JR acknowledges support from an EU Marie-Curie fellowship. SHS is supported in part through the Deutsche Forschungsgemeinschaft (DFG) under project SCHN 342 / References
Auger, M. W., Fassnacht, C. D., Abrahamse, A. L., Lubin, L. M., & Squires,G. K. 2007, AJ, 134, 668Auger, M. W., Fassnacht, C. D., Wong, K. C., et al. 2008, ApJ, 673, 778Barnab`e, M. & Koopmans, L. V. E. 2007, ApJ, 666, 726Belokurov, V., Evans, N. W., Hewett, P. C., et al. 2009, MNRAS, 392, 104Belokurov, V., Evans, N. W., Moiseev, A., et al. 2007, ApJ, 671, L9Bridle, S., Kneib, J.-P., Bardeau, S., & Gull, S. 2002, in The shapes of galax-ies and their dark halos, Proceedings of the Yale Cosmology Workshop ,New Haven, Connecticut, USA, 28-30 May 2001. Edited by PriyamvadaNatarajan., ed. P. Natarajan, 38– + Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000Cabanac, R. A., Alard, C., Dantel-Fort, M., et al. 2007, A&A, 461, 813Dye, S., Evans, N. W., Belokurov, V., Warren, S. J., & Hewett, P. 2008, MNRAS,388, 384Dye, S., Smail, I., Swinbank, A. M., Ebeling, H., & Edge, A. C. 2007, MNRAS,379, 308El´ıasd´ottir, ´A., Limousin, M., Richard, J., et al. 2007, ArXiv e-prints, 710Fassnacht, C. D. & Lubin, L. M. 2002, AJ, 123, 627Faure, C., Alloin, D., Kneib, J. P., & Courbin, F. 2004, A&A, 428, 741Grillo, C., Lombardi, M., Rosati, P., et al. 2008, A&A, 486, 45Halkola, A., Hildebrandt, H., Schrabback, T., et al. 2008, A&A, 481, 65Halkola, A., Seitz, S., & Pannella, M. 2006, MNRAS, 372, 1425Ienna, F. & Pell´o, R. 2006, in SF2A-2006: Semaine de l’AstrophysiqueFrancaise, ed. D. Barret, F. Casoli, G. Lagache, A. Lecavelier, & L. Pagani,347– + Inada, N., Oguri, M., Shin, M., et al. 2009, AJ, 137, 4118Jing, Y. P. & Suto, Y. 2002, ApJ, 574, 538Jullo, E. & Kneib, J. 2009, MNRAS, 395, 1319Jullo, E., Kneib, J.-P., Limousin, M., et al. 2007, New Journal of Physics, 9, 447Keeton, C. R., Kochanek, C. S., & Seljak, U. 1997, ApJ, 482, 604Keeton, C. R. & Zabludo ff , A. I. 2004, ApJ, 612, 660Kochanek, C. S. & Blandford, R. D. 1991, ApJ, 375, 492Kochanek, C. S., Keeton, C. R., & McLeod, B. A. 2001, ApJ, 547, 50Koopmans, L. V. E., Treu, T., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006,ApJ, 649, 599Kundic, T., Hogg, D. W., Blandford, R. D., et al. 1997, AJ, 114, 2276Leonard, A., Goldberg, D. M., Haaga, J. L., & Massey, R. 2007, ApJ, 666, 51Limousin, M., Cabanac, R., Gavazzi, R., et al. 2009, A&A, 502, 445Limousin, M., Kneib, J.-P., & Natarajan, P. 2005, MNRAS, 356, 309Limousin, M., Richard, J., Jullo, E., et al. 2007, ApJ, 668, 643Lynds, R. & Petrosian, V. 1986, in Bulletin of the American AstronomicalSociety, 1014– + Miralda-Escude, J. & Babul, A. 1995, ApJ, 449, 18Momcheva, I., Williams, K., Keeton, C., & Zabludo ff , A. 2006, ApJ, 641, 169Morgan, N. D., Kochanek, C. S., Pevunova, O., & Schechter, P. L. 2005, AJ, 129,2531Oguri, M. 2010, ArXiv e-printsOguri, M., Inada, N., Blackburne, J. A., et al. 2008, MNRAS, 391, 1973Oguri, M., Keeton, C. R., & Dalal, N. 2005, MNRAS, 364, 1451Oke, J. B., Cohen, J. G., Carr, M., et al. 1995, PASP, 107, 375Okura, Y., Umetsu, K., & Futamase, T. 2008, ApJ, 680, 1Richard, J., Kneib, J., Limousin, M., Edge, A., & Jullo, E. 2010, MNRAS, 402,L44Saha, P., Williams, L. L. R., & Ferreras, I. 2007, ApJ, 663, 29Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses (Berlin:Springer-Verlag)Soucail, G., Fort, B., Mellier, Y., & Picat, J. P. 1987, A&A, 172, L14Suyu, S. H., Marshall, P. J., Hobson, M. P., & Blandford, R. D. 2006, MNRAS,371, 983Treu, T., Gavazzi, R., Gorecki, A., et al. 2009, ApJ, 690, 670Tu, H., Gavazzi, R., Limousin, M., et al. 2009, A&A, 501, 475Tu, H., Limousin, M., Fort, B., et al. 2008, MNRAS, 1169Warren, S. J. & Dye, S. 2003, ApJ, 590, 673Williams, K. A., Momcheva, I., Keeton, C. R., Zabludo ff , A. I., & Leh´ar, J. 2006,ApJ, 646, 85 imousin et al.: Strong Lensing Beyond the Einstein Radius 11 Appendix A: Taking an External Mass PerturbationInto Account: Comparison with OtherApproaches
We have proposed in this article a way of taking into account anexternal mass perturbation in a strong lensing (SL) modelling.Here we try other possible and more conventional approaches:i) a constant external shear profile and ii) a Singular IsothermalSphere centred on the second high luminosity peak, which, byconstruction, is the main mass concentration perturbing the SLin the method proposed in this work.
A.1. A Constant External Shear
Although unphysical (any mass distribution will not generateshear only but also convergence), the external shear modelis widely used and is often a good approximation. Here weaddress the modelling of the SL system with a constant ex-ternal shear component parametrised by a position angle anda strength ( γ Kst ). This modelling is performed in the imageplane. Parameters of the potential describing the lens are setas in Section 3.4. The external shear strength is allowed tovary between 0 and 0.3. The upper limit corresponds to a verystrong shear value: for comparison, the massive galaxy clusterAbell 1689 produces an average shear value of 0.23 at 50 (cid:48)(cid:48) awayfrom its centre.We are able to get a very good fit, with χ <
1. The bestmodel corresponds to a circular halo for the lens ( e = . = -0.36 (cid:48)(cid:48) , Y = (cid:48)(cid:48) ), making itsposition angle (95 degrees) irrelevant. The lens fiducial veloc-ity dispersion equals to 450 ± − (1 σ ). The external shearis described by γ Kst = ∼ enstool software does explore the parameter space us-ing a MCMC sampler (Jullo et al. 2007). Therefore, we canuse these MCMC realisations in order to investigate the degen-eracies between the di ff erent parameters. The following figureshave been generated this way.Figure A.1 shows that there is a strong degeneracy between e and γ Kst . We see that the solution derived in Section 5.1 (i.e.an external shear of ∼ ∼ σ contour. On the other hand, the position angleof the external shear is very well constrained to be ∼
20 degrees.This position angle points towards the second high luminositylight clump. This suggests that, to first order, the external massperturbation is dominated by this component. We note that thebest-fit model needs an external shear of order 0.18, which is apretty unlikely value in our case because it is comparable to whatwould be experienced at ∼ (cid:48)(cid:48) from the centre of Abell 1689. A.2. An SIS profile
The first order mass perturbation is associated to a second peakof high luminosity. We put an SIS mass distribution at the lo-cation of this second luminosity peak (X,Y = -53,10 (cid:48)(cid:48) wrt thelens). We allow its velocity dispersion to vary up to 800 km s − ,an upper limit motivated by the WL analysis of the full group(Section 7), and do the SL modelling with parameters set as inSection 3.4. We are able to get a very good fit, with χ <
1. Thelens halo is centred on the bright galaxy. Its ellipticity equals0.43 + . − . and its position angle 27 ± ± − (1 σ ). The externalshear and convergence generated by the SIS profile at the lo- Fig. A.1.
Results of a constant external shear model.
Top: degeneracybetween the halo ellipticity e and the strength of the external shear γ Kst . Bottom: posterior probability distribution for the position angle of theexternal shear. cation of the multiple images are equal by definition, between0.07 and 0.10.We show on Fig. A.2 the degeneracies between the lens haloellipticity and the SIS profile velocity dispersion, related to thestrength of the external shear experienced by the multiple im-ages. We see that the solution derived in Section 5.1 (i.e. an ex-ternal shear of ∼ σ SIS ∼
700 km s − andan ellipticity of ∼ σ contour. A.3. Discussion
In each cases investigated in this Appendix, we find that the solu-tion we have derived in Section 5.1 using our original method isconsistent with solutions derived with more conventional meth-ods.We note that conventional methods exhibit strong degenera-cies between the lens halo ellipticity and the strength of the ex-ternal shear. These degeneracies are smaller in the case of theSIS profile (lens ellipticity is constrained between 0.3 and 0.6)compared to the case of a constant shear profile (ellipticity un-
Fig. A.2.
Degeneracies between the lens halo ellipticity and the SIS pro-file velocity dispersion, related to the strength of the external shear ex-perienced by the multiple images. constrained between the allowed priors: 0 and 0.6). The maindi ff erence between the SIS profile and the constant shear profileis that the SIS profile generates both shear and convergence.With respect to the lens itself, we note that all fitted fidu-cial velocity dispersions are consistent, whatever the methodused to take into account the external mass perturbation. Theyfall between 433 km s − and 458 km s − . This translates into aprojected mass computed in a radius of 10 (cid:48)(cid:48) between 0.93 and1.04 × M (cid:12)(cid:12)