Golden gravitational lensing systems from the Sloan Lens ACS Survey. I. SDSS J1538+5817: one lens for two sources
C. Grillo, T. Eichner, S. Seitz, R. Bender, M. Lombardi, R. Gobat, A. Bauer
aa r X i v : . [ a s t r o - ph . C O ] D ec Golden gravitational lensing systems from the Sloan Lens ACSSurvey. I. SDSS J1538+5817: one lens for two sources ∗ , ∗∗ C. Grillo , , T. Eichner , S. Seitz , , R. Bender , , M. Lombardi , , R. Gobat , andA. Bauer [email protected] ABSTRACT
We present a gravitational lensing and photometric study of the exceptional strong lensingsystem SDSS J1538+5817, identified by the Sloan Lens ACS survey. The lens is a luminouselliptical galaxy at redshift z l = 0 . z s = 0 . . +0 . − . × M ⊙ for the total mass projected within the Einstein radius of 2.5 kpc; (4) estimating a totalmass density profile slightly steeper than an isothermal one [ ρ ( r ) ∝ r − . +0 . − . ]. A fit of the SloanDigital Sky Survey multicolor photometry with composite stellar population models provides avalue of 20 +1 − × M ⊙ for the total mass of the galaxy in the form of stars and of 0 . +0 . − . for thefraction of projected luminous over total mass enclosed inside the Einstein radius. By combininglensing (total) and photometric (luminous) mass measurements, we differentiate the lens masscontent in terms of luminous and dark matter components. This two-component modeling,which is viable only in extraordinary systems like SDSS J1538+5817, leads to a description ofthe global properties of the galaxy dark matter halo. Extending these results to a larger numberof lens galaxies would improve considerably our understanding of galaxy formation and evolutionprocesses in the ΛCDM scenario. Subject headings: galaxies: elliptical and lenticular, cD − galaxies: individual (SDSS J1538+5817) − galaxies: structure − dark matter − gravitational lensing * Based on observations made with the NASA/ESAHubble Space Telescope, obtained from the data archiveat the Space Telescope Institute. STScI is operated by theassociation of Universities for Research in Astronomy, Inc.under the NASA contract NAS 5-26555. ** Based on observations made with the Nordic OpticalTelescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in theSpanish Observatorio del Roque de los Muchachos of theInstituto de Astrofisica de Canarias. Max-Planck-Institut f¨ur extraterrestrische Physik,Giessenbachstr., D-85748 Garching bei M¨unchen, Germany Universit¨ats-Sternwarte M¨unchen, Scheinerstr. 1, D-81679 M¨unchen, Germany European Southern Observatory, Karl-Schwarzschild- . Introduction Early-type galaxies host the majority of thebaryonic mass observed in galaxies in the Universe(e.g., Fukugita et al. 1998; Renzini 2006); hence,deciphering the processes that lead to their for-mation and the mechanisms that rule their subse-quent evolution is a key cosmological issue. For in-stance, it is still debated whether early-type galax-ies form at relatively high redshift ( z .
2) as a re-sult of a global starburst and then passively evolveto the present (e.g., Eggen et al. 1962; Larson1974; Arimoto & Yoshii 1987; Bressan et al. 1994)or whether they assemble from mutual disruptionof disks in merging events (e.g., Toomre 1977;White & Rees 1978). Information with whichto distinguish these scenarios lies in the char-acteristics of galaxy dark-matter halos. How-ever, the lack of suitable and easily interpretedkinematical tracers, such as HI in spirals, hasmade comprehensive studies on the dark mat-ter component in early-type galaxies rather dif-ficult (e.g., Bertin et al. 1992; Saglia et al. 1992;Thomas et al. 2007, 2009).In the last few years, strong gravitational lens-ing has allowed astrophysicists to make greatprogress in the understanding of the internalstructure of early-type galaxies. Through lens-ing, it has become possible to address in detailsome fundamental problems related to the mecha-nisms of formation of early-type galaxies, like thedetermination of the amount and distribution ofdark matter (e.g., Gavazzi et al. 2007; Grillo et al.2008c, 2009; Barnab`e et al. 2009) or the investi-gation of the total mass density profile and itsredshift evolution (e.g., Treu & Koopmans 2004;Koopmans et al. 2006). Several algorithms havebeen developed in order to fit the observationaldata of a strong gravitational lens system and,thus, to reconstruct the properties of a lens.Simplifying, a first difference between codes isthe use of a parametric model (e.g., gravlens ,Keeton 2001a,b; Seitz et al. 1998; Warren & Dye2003; Halkola et al. 2006; Rzepecki et al. 2007; Str. 2, D-85748 Garching bei M¨unchen, Germany University of Milan, Department of Physics, via Celoria16, I-20133 Milan, Italy CEA Saclay, DSM/DAPNIA/Service d’Astrophysique,F-91191 Gif-sur-Yvette Cedex, France http://redfive.rutgers.edu/ ∼ keeton/gravlens/ Grillo et al. 2008c) or a non-parametric model(e.g.,
PixeLens , Saha & Williams 2004; Koopmans2005; Suyu et al. 2009) to describe the mass dis-tribution of a lens. In the former case, the massdistribution of a lens is assumed to be accuratelydescribed by an analytical expression; the funda-mental scales of the model are determined by com-paring the observed and model-predicted proper-ties of the multiple images. In the latter case,a pixelated map or a multipole decompositionof the surface mass density of the lens is usu-ally estimated through a statistical analysis thatrequires, in addition to the observational informa-tion, some extra physically plausible constraints,called priors, on the surface mass density distri-bution (e.g., positive-definite and smooth) of thelens. On the one hand, parametric models providea great deal of freedom and complexity, but theydo not cover “naturally” all the possible realis-tic mass distributions (for instance, surface massdensity distributions with twisting isodensity con-tours); on the other hand, even if non-parametricmodels are more general, their number of degreesof freedom is often much larger than the con-straints and this can result in three-dimensionaldensity distributions that are dynamically unre-alistic or unstable. A viable solution to obtainphysically significant density distributions is toconsider a framework where the mass distribu-tion of the lens is reconstructed by combiningin a fully self-consistent way both gravitationallensing and stellar dynamics measurements (e.g.,Barnab`e & Koopmans 2007).Lensing codes are further distinguished bythe fact that the multiple images and the cor-responding sources can be modeled as point-like (e.g., gravlens ; PixeLens ; Seitz et al. 1998;Halkola et al. 2006) or extended (e.g., Warren & Dye2003; Koopmans 2005; Rzepecki et al. 2007;Grillo et al. 2008c; Suyu et al. 2009) objects. Inthe context of point-like algorithms, the best-fit model is defined as that which minimizes thechi-square between the measured positions of thecentroids of the images and the positions repro-duced by the model, weighted by the measure-ment uncertainties. Additional chi-square termsthat quantify the agreement between the observedand model-predicted relative fluxes and time de- gri image centered on the lensgalaxy.lays of the multiple images can also be included.For extended algorithms, the goodness of a modelis estimated by comparing on a pixelated gridthe image surface brightness morphology and dis-tribution which are observed to those which arereproduced by the model (after convolution withthe relevant point spread function).The relative positions of a multiply imaged sys-tem can sometimes be measured with an accuracyof a few milli-arcseconds (e.g., Patnaik & Kemball2001) and these positions represent the most im-portant constraints on the mass distribution of thelens. In fact, although the flux ratios of the mul-tiple images can be easily estimated and offer an-other important source of information, the sen-sitivity of the flux measurements to details suchas the dark matter substructure of the lens, theextinction in the interstellar medium of the lens,the microlensing effects of the stars present in thelens, and the time variability in the source de-crease their potential. Time delays can also helpto determine the mass distribution of a lens, buta statistically significant number of measurementsof this kind is just starting to become available.The projected total mass enclosed within acylinder of radius equal to the Einstein radiusof a lensing system can be accurately measuredby only fitting the observed positions of the mul-tiple images (e.g., Kochanek 1991; Grillo et al.2008c), whereas a detailed fit of the possible arcs Fig. 2.— SDSS spectrum obtained within an aper-ture of 3 ′′ diameter centered on the lens galaxy.and rings associated with an extended source isnecessary if the interest is in the properties ofboth lens and source (e.g., Swinbank et al. 2007;Vegetti & Koopmans 2009). By combining lens-ing and multiband photometric measurements, theamount of mass present in a lens galaxy in theforms of dark and visible matter can be deter-mined (e.g., Grillo et al. 2008a, 2009).In addition to projected total mass, lensinganalyses allow one to estimate also the total massdensity profile of lens galaxies (e.g., Rusin et al.2003; Rusin & Kochanek 2005). This can beachieved either by combining in a statistical waylensing and stellar dynamics data in a sampleof lens galaxies that are assumed to have a ho-mologous structure (e.g., Koopmans et al. 2006),by performing a joint lensing and extended stel-lar kinematics study in a single lens galaxy (e.g.,Barnab`e et al. 2009; Treu & Koopmans 2004), orby using lensing only in exceptional lensing sys-tems that show multiple images of different sourcesprobing wide angular and radial ranges of thelens mass distribution (e.g., Sykes et al. 1998; Nair1998).In this paper, we study the lensing system SDSSJ1538+5817, discovered by the Sloan Lens ACS(SLACS) survey . This system is particularly in- ′′ × ′′ field around the gravitational lensing system SDSS J1538+5817, before( on the left ) and after ( on the right ) the subtraction of an elliptical model, fitted on the luminosity profile ofthe lens galaxy. The images are obtained by combining the F606W
HST/WFPC2 and the
F814W
HST/ACSfilters.Table 1: The lens galaxy.
RA Dec zl q L θq L θe u g r i z (J2000) (J2000) (deg) ( ′′ ) (mag) (mag) (mag) (mag) (mag)15:38:12.92 +58:17:09.8 0.143 0.82 157.3 1.58 19 . ± .
06 18 . ± .
01 17 . ± .
01 16 . ± .
01 16 . ± . References – Bolton et al. 2008.Notes – Magnitudes are extinction-corrected modelMag (AB) from the SDSS.teresting because two different sources are lensed,one into an Einstein ring with four luminositypeaks and the other into two images, by an early-type galaxy that has an almost circular projectedlight distribution. The large number of images atvarious angular distances from the galaxy centerand the nearly perfect axisymmetric lensing con-figuration of the ring makes this system the ideallaboratory to disentangle the luminous and darkcomponents of the lens mass distribution.The paper is organized as follows. In Sect. 2,we describe the observational data for the complexstrong lensing system SDSS J1538+5817. We per-form parametric lensing analyses of this system inSect. 3. Then, in Sect. 4, we investigate the lu-minous and dark matter composition of the lensgalaxy. In Sect. 5, we summarize the results ob-tained in this study. Finally, in the Appendix, wemodel the lens mass distribution on a pixelated grid and compare these non-parametric results tothose from Sect. 3. Throughout this work we as-sume H = 70 km s − Mpc − , Ω m = 0 .
3, andΩ Λ = 0 .
7. In this model, 1 ′′ corresponds to alinear size of 2.51 kpc at the lens plane.
2. Observations
The SLACS survey was started in 2003 andaims at studying, from a lensing and dynamicsperspective, a statistically significant number ofgalaxies acting as strong lenses and located at red-shifts lower than 0.5. The candidate lenses werespectroscopically selected from the Sloan DigitalSky Survey (SDSS) database by identifying thoseobjects that show, in addition to the continuumand absorption lines of a possible lens galaxy at l u x Mg Na OI OI NIIH α NIIFe FeHeI H δ G H γ OIII H β OIII OIII
Fig. 5.— The NOT/ALFOSC 1D ( on the top ) and 2D ( on the bottom ) spectra of the lensing system SDSSJ1538+5817. In the wavelength range shown here, the most prominent spectral features at the redshift ofthe lens ( z l = 0 . z s = 0 . ′′ long-slit. − − − δ H β OIII Q Q D Fig. 6.— Intensity (in arbitrary units) of the three H δ , H β , and [OIII] λ D , Q , and Q im-ages.5edshift z l , one or multiple emission lines of ahypothetical source at a higher redshift z s . Themost promising candidates were then observed atleast once with the Hubble Space Telescope (HST)Advanced Camera for Survey (ACS) to confirmthe lens hypothesis (for further information, seeBolton et al. 2006, 2008). This procedure resultedin the sample of 63 “grade-A” strong gravitationallensing systems presented in Bolton et al. (2008).SDSS J1538 + 5817 is one of the lens galaxies dis-covered by the SLACS survey. The photomet-ric and spectroscopic observations taken by theSDSS are shown in Figs. 1 and 2. As describedabove, the redshifts of the lens galaxy and a source( z l = 0 .
143 and z s = 0 . F814W and
F606W filters ofthe HST/ACS and Wide Field Planetary Cam-era 2 (WFPC2) respectively, we model the lu-minosity distribution of the lens galaxy and sub-tract the best-fit model from the images. In de-tail, a model for the lens galaxy is constructedby using an iterative procedure: first, presum-able background source images are masked andisophotal contours of the lens galaxy are derivedfor surface brightness levels separated by 0.1 magarcsec − . Then, all isophote contours (even ifpartially masked) are fitted by ellipses followingthe method of Bender & Moellenhoff (1987). Thisprovides five parameters (center coordinates, ma-jor and minor axis, and position angle) per sur-face brightness level. The resulting table of theseparameters and associated surface brightnesses isemployed to calculate a smooth elliptical modelfor the lens galaxy. This model is subtracted fromthe image, leaving as residuals only the imagesTable 2: Astrometric and photometric measure-ments for the multiple images. x a y a z s δ x,y f δ f d a ( ′′ ) ( ′′ ) ( ′′ ) ( ′′ ) D D − − Q Q − Q − − Q − With respect to the galaxy center. of the background source galaxy. The residualsare used to check and improve the masking ofthe source galaxy. This procedure is repeatedtwice until the final model for the lens galaxyis obtained. In addition, images taken in differ-ent wavelength bands provide color informationfor lens and source which is of additional help toidentify and separate lens and source components.Color composite images of the strong lensing sys-tem and the residuals after the lens galaxy sub-traction are shown in Fig. 3.The excellent angular resolution of the HST al-lows us to identify two systems of multiple images(for labels, see Fig. 7): a double ( D and D ) anda quad ( Q , Q , Q , and Q ). The images of thetwo systems have different colors, but the averagedistances from the galaxy center of D and D and Q , Q , Q , and Q are consistent within theerrors. This fact would imply approximately thesame redshift for the two sources, if the lens to-tal mass were close to an isothermal distribution.In addition, the absence in the SDSS spectrum ofevident emission lines at a possible third redshiftsupports the hypothesis that the two sources areat the same distance to the observer.Since the precise knowledge of the redshift ofthe two sources plays a crucial role in the deter-mination of the total mass distribution of the lensgalaxy, we decided to perform additional spectro-scopic measurements to understand whether theemission lines observed in the SDSS spectrum areassociated to one or both of the lensed sources.The data were obtained on June 25, 2009 as aFast-Track Observing Program (P38-428) with theAndalucia Faint Object Spectrograph and Camera(ALFOSC) at the 2.5-m Nordic Optical Telescope(NOT) on La Palma (Spain). We positioned a 1 ′′ -wide long-slit centered in Q and passing through D and Q , as shown in Fig. 4. We used ALFOSCwith the 8 grism, that covers a wavelength rangebetween 5825 and 8350 ˚A with a dispersion of 1.3˚A per pixel. In good atmospheric conditions (see-ing between 0.7 and 1 ′′ ) and in the same obser-vational configuration, we obtained six exposuresof 24 minutes each, resulting in a total integrationtime of 2.4 hrs.In Fig. 5, we show the wavelength-calibrated1D and 2D spectra. We identify several prominentabsorption lines at redshift 0.143 and at least sixsecure emission lines (H δ , G, H γ , H β , [OIII] λ λ δ , H β , and [OIII] λ Q and Q ), at angular positions consistent with thoseof the three images D , Q , and Q . The mea-surement of the same emission lines at the sameobserved wavelengths proves in a conclusive waythat the two sources D and Q are equally distantfrom the observer. We remark that the intensityvalues of the emission lines shown in Fig. 6 aredifferently contaminated by the lens galaxy flux.In Table 1, we summarize the photometric andspectroscopic properties of the lens galaxy: the co-ordinates (RA, Dec, z l ), the minor to major axisratio ( q L ) and its position angle ( θ q L , degrees eastof north), and the SDSS multiband magnitudes ( u , g , r , i , z ). In Table 2, we report the coordinatesof the multiple images ( x , y , z s ) and the adoptedposition uncertainty on the first two coordinates( δ x,y ), the relative flux of the double system com-ponents ( f ) and the respective error ( δ f ), and thedistance of the images to the galaxy center ( d ).
3. Strong gravitational lensing
We address parametric (Sect. 3.1) point-likemodeling of the strong gravitational lensing sys-tem. We focus mainly on projected total massand total mass density profile measurements. Acomparison with the results obtained from non-parametric models is provided in the Appendix.
Gravlens (Keeton 2001a) is a publicly-availablelensing software that, starting from the measuredobservables of a strong lensing system, recon-structs the properties of a lens in terms of anadopted model that is defined by some relevant pa-rameters. By using this code, we perform a para-metric analysis in which we describe the total massdistribution of the lens galaxy in terms of either anelliptical de Vaucouleurs model (deV), or a singu- lar isothermal ellipsoid (SIE) model, or a singularpower law ellipsoid (PL) model (for further detailson the model definitions, see e.g. Keeton 2001b).Both a deV and an SIE model are characterized byfive parameters: a length scale ˜ b (corresponding tothe value of the Einstein angle θ Ein in the circularlimit), the two coordinates of the center ( x l , y l ),the minor to major axis ratio q , and its positionangle θ q . For the deV model, we fix the value ofthe effective angle ( θ e ) to that shown in Table 1. APL model is more general than an SIE model. Inparticular, the former requires as an additional pa-rameter the value of the exponent γ of the threedimensional density distribution ρ ( r ) ∝ r − γ (anSIE model is retrieved by setting γ equal to 2).The convergence κ ( x, y ) of a PL model, definedas the surface mass density of the model dividedby the critical surface mass density of the studiedlensing system (for definitions, see Schneider et al.1992), depends on the previous parameters as fol-lows κ ( x, y ) ∝ ˜ b γ − (cid:16) x + y q (cid:17) γ − . (1)Due to the normalization used in the code, gravlens provides values of a length scale b thatare related to the values of ˜ b by a function f ( · ) ofthe axis ratio q : b = ˜ b f ( q ) . (2)Varying the parameters of the two adoptedmass models and the positions of the sources[( x D , y D ); ( x Q , y Q )], we minimize a chi-square χ function. This function compares first only the ob-served (see Table 2) and predicted positions of themultiple images [deV(nf), SIE (nf), and PL (nf)models] and then also the measured (see Table2) and reconstructed fluxes of the double system[deV (wf), SIE (wf), and PL (wf) models]. In thelatter case, the flux of the source imaged twiceis an additional free parameter of the models. Inour lensing analysis, we decide to neglect the fluxconstraints relative the quad system because thepresence of the Einstein ring prevents us fromseparating accurately the individual components.For the multiple images, we assume position un-certainties equal to the size of one pixel of ACS(0.05 ′′ ) and flux uncertainties as reported in Ta-ble 2 and determined by considering the different7able 3: The best-fit (minimum χ ) parameters of the different models.Model b x l y l q θ q θ e γ χ d.o.f.( ′′ ) ( ′′ ) ( ′′ ) (deg) ( ′′ )deV (nf) 1.97 − − − − − − χ minimizations on simulated data sets. Theseare obtained by extracting the image positionsand fluxes from Gaussian distributions centeredon the measured values and with standard devi-ations equal to the observational errors reportedin Table 2. In addition, starting from the sets ofoptimized parameter values, we estimate the totalprojected mass M totlen ( ≤ R i ) enclosed within sevendifferent circular apertures of radii R i . The firstthree radii are chosen as the projected distancesfrom the lens galaxy center of the inner image ofthe double system, the “average” Einstein circle,and the outer image of the double system. Theremaining four radii are given by the values of themidpoints of the three segments defined by theprevious three points and a further point at thesame distance from the outer double image as thefirst point is from the second one [i.e., (0.26, 0.52,0.75, 0.98, 1.28, 1.58, 1.84) ′′ ].The best-fit (minimum chi-square) parameterand χ values of the different models are sum-marized in Table 3. For all the models, wefind that the best-fit χ values are smaller thanthe corresponding number of degrees of freedom(d.o.f.). This implies that the reconstructed po-sitions of the images are angularly very close tothe measured positions of Table 2. In this sec-tion, we concentrate on the results given by theone-component PL and, as a comparison with theresults of previous studies, simpler SIE models andonly in the next section we will address the two- component mass decomposition.In Fig. 7, for the best-fit SIE (nf) model weshow the reconstructed positions of the sourcesand the caustics, the observed and reconstructedpositions of the images and the critical curves,and the Fermat potential (for definition, seeSchneider et al. 1992) with its stationary points.The inclusion of the fluxes of the double systemdoes not change significantly the best-fit parame-ters of the models. Moreover, the projected totalmass estimates, which are presented below, arenot particularly sensitive to the flux constraints.For these reasons, in the following we will mainlyconcentrate on the properties of the models thatomit the additional source of information comingfrom the fluxes of the double system, i.e., the SIE(nf) and PL (nf) models. We note that the best-fit b values are on the order of 1 ′′ , the typical distanceof an image of the quad system from the center ofthe lens (see Tables 2 and 3). The best-fit valuesof the lens center and ellipticity show that thelens mass distribution is well centered and alignedwith the galaxy light distribution (see Tables 1and 3). In particular, we remark that the totalsurface mass of the lens is well approximated byan axisymmetric distribution. The best-fit valuesof the parameter γ of the PL models suggest thatthe lens total density profile is slightly steeperthan an isothermal one. We estimate maximumtime delays of approximately 30 and 3 days forthe double and quad systems, respectively (seeTable 4 and Fig. 7). Finally, as far as the po-sitions of the sources are concerned, the modelpredicted angular distance of the two sources isbetween approximately 0.5 and 0.7 ′′ (see Table 5),8orresponding, respectively, to 3.2 and 4.5 kpc ata redshift of 0.531.In Figs. 8 and 9, we plot, respectively, the jointprobability distributions of the SIE (nf) and PL(nf) model parameters, with the 68% and 95%confidence regions and the 68% confidence inter-vals. These intervals are determined by excludingfrom the 2000 χ minimizations the 320 smallestand the 320 largest values for each model param-eter. We have checked that the error estimatesdetermined in this way are unbiased and equiva-lent to the uncertainties provided by a full Markovchain Monte Carlo analysis. The comparison Figs.8 and 9 shows clearly that adding the exponent ofthe total mass distribution among the parametersincreases their degeneracies, hence their error es-timates. The probability distribution of the posi-tion angle θ q is bimodal, with a secondary peaklocated nearly 90 ◦ away from the primary one,found at approximately 150 ◦ . From the last col-umn of plots in Fig. 8, we see that the secondarypeak is included only in the 95% CL regions, and,from the last panel of the same figure, we notethat the low values of θ q are associated with al-most circular models (i.e., q ≃ ◦ of the lensmass distribution (supposing the positions of thesources are fixed).The degeneracies between b , q , and γ are con-nected to their relations defined in Eqs. (1) andTable 4: The model-predicted time delays for thebest-fit model parameters given in Table 3.Model ∆ t D , ∆ t Q , ∆ t Q , ∆ t Q , (days) (days) (days) (days)SIE (nf) 24.8 0.66 0.70 2.88PL (nf) 33.5 0.82 0.87 3.26 (2). In particular, the strong anti-correlation be-tween the value of the length scale and the steep-ness (see the fifth panel of Fig. 9) is caused bythe fact that the Einstein ring of a circular lensmodel defines a region on the image plane withinwhich the average value of the convergence κ isequal to one. In order for this equality to be ap-proximately valid inside the average circle definedby the positions of our quad system, from Eq. (1)and by holding the value of q fixed, it follows thata higher value of b requires a lower value of γ , andvice versa.The previous considerations on the almostmodel-independent average value of κ inside theEinstein ring can also be translated in terms of to-tal mass estimates within the same ring. Distinctmodels, defined by different parameters, that canreproduce well an approximately complete Ein-stein ring, provide total mass measurements in-side this typical aperture that differ by only a fewpercent. This is shown in Fig. 10. There, weplot the median values and the 68% confidenceintervals (obtained by excluding from the 2000 χ minimizations the 320 smallest and the 320largest mass estimates) of the lens projected totalmass within the Einstein ring and measure valuesof 8 . +0 . − . × M ⊙ for an SIE (nf) model and8 . +0 . − . × M ⊙ for a PL (nf) model. We noticethat the median values of the 2000 Monte-Carlocumulative total mass estimates do not necessar-ily follow a global PL model, but they have inprinciple more freedom. In fact, even if the totalmass values of each of the 2000 models do follow apower law model precisely at all radii, the medianvalues shown in Fig. 10 and used in the followingfor the luminous and dark matter decompositionare more general and do not provide the samevalue of the steepness γ at each radial position.In general, for the two different models the totalmass estimates, that are measured within variousapertures (approximately between 1 and 4 kpcTable 5: The model-predicted source positions forthe best-fit model parameters given in Table 3.Model ( x D , y D ) ( x Q , y Q ) d D,Q ( ′′ , ′′ ) ( ′′ , ′′ ) ( ′′ )SIE (nf) (0.27,0.46) (0.00,0.00) 0.53PL (nf) (0.39,0.60) (0.02, − .
01) 0.729 x l (arcsec)0.91.01.11.21.31.4 b ( a r c s ec ) b − x l joint probability distribution % % -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 y l (arcsec)0.91.01.11.21.31.4 b ( a r c s ec ) b − y l joint probability distribution % % q b ( a r c s ec ) b − q joint probability distribution % %
60 80 100 120 140 160 θ q (deg)0.91.01.11.21.31.4 b ( a r c s ec ) b −θ q joint probability distribution % % % -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 y l (arcsec)-0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l − y l joint probability distribution % % q -0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l − q joint probability distribution % %
60 80 100 120 140 160 θ q (deg)-0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l −θ q joint probability distribution % % % q -0.15-0.10-0.05-0.000.050.100.15 y l ( a r c s ec ) y l − q joint probability distribution % %
60 80 100 120 140 160 θ q (deg)-0.15-0.10-0.05-0.000.050.100.15 y l ( a r c s ec ) y l −θ q joint probability distribution % % %
60 80 100 120 140 160 θ q (deg)0.60.70.80.91.0 q q −θ q joint probability distribution % % Fig. 8.— Estimates of the errors and correlations in the parameters for an SIE (nf) model. Results of the χ minimizations on 2000 Monte-Carlo simulated data sets. Thick bars on the co-ordinate axes and contourlevels on the planes represent, respectively, the 68% confidence intervals and the 68% and 95% confidenceregions. For each model parameter, the 68% confidence interval is determined by excluding from the 2000 χ minimizations the 320 smallest and the 320 largest values.from the lens center), are consistent, given theerrors. We remark that fixing the exponent of the total mass profile (i.e., γ equal to 2 for the SIEmodels) result in significant smaller uncertainties10 x l (arcsec)0.00.51.01.52.0 b ( a r c s ec ) b − x l joint probability distribution % % -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 y l (arcsec)0.00.51.01.52.0 b ( a r c s ec ) b − y l joint probability distribution % % q b ( a r c s ec ) b − q joint probability distribution % % %
60 80 100 120 140 160 θ q (deg)0.00.51.01.52.0 b ( a r c s ec ) b −θ q joint probability distribution % % % % γ b ( a r c s ec ) b −γ joint probability distribution % % -0.15 -0.10 -0.05 -0.00 0.05 0.10 0.15 y l (arcsec)-0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l − y l joint probability distribution % % q -0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l − q joint probability distribution % % %
60 80 100 120 140 160 θ q (deg)-0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l −θ q joint probability distribution % % % γ -0.15-0.10-0.05-0.000.050.100.15 x l ( a r c s ec ) x l −γ joint probability distribution % % % q -0.15-0.10-0.05-0.000.050.100.15 y l ( a r c s ec ) y l − q joint probability distribution % % %
60 80 100 120 140 160 θ q (deg)-0.15-0.10-0.05-0.000.050.100.15 y l ( a r c s ec ) y l −θ q joint probability distribution % % % γ -0.15-0.10-0.05-0.000.050.100.15 y l ( a r c s ec ) y l −γ joint probability distribution % % %
60 80 100 120 140 160 θ q (deg)0.60.70.80.91.0 q q −θ q joint probability distribution % % % % γ q q −γ joint probability distribution % % % γ θ q ( d e g ) θ q −γ joint probability distribution % % % % % Fig. 9.— Estimates of the errors and correlations in the parameters for a PL (nf) model. Results of the χ minimizations on 2000 Monte-Carlo simulated data sets. Thick bars on the co-ordinate axes and contourlevels on the planes represent, respectively, the 68% confidence intervals and the 68% and 95% confidenceregions. For each model parameter, the 68% confidence interval is determined by excluding from the 2000 χ minimizations the 320 smallest and the 320 largest values.in the total mass values. As mentioned above, bymodeling also the fluxes of the double system wefind total mass measurements that are consistent within 1 σ with the estimates obtained by fittingthe image positions only.11 ource plane -1.0 -0.5 0.0 0.5 1.0 x s (arcsec)-1.0-0.50.00.51.0 y s ( a r c s ec ) DQ Image plane -2 -1 0 1 2 x i (arcsec)-2-1012 y i ( a r c s ec ) D D Q Q Q Q -2 -1 0 1 2 x i (arcsec)-2-1012 y i ( a r c s ec ) Fermat potential (double) -2 -1 0 1 2 x i (arcsec)-2-1012 y i ( a r c s ec ) Fermat potential (quad)
Fig. 7.— Best-fit SIE (nf) model.
Top left : sourceplane with caustics. The predicted source posi-tions of the double and quad systems are repre-sented by a plus and a cross symbol, respectively.
Top right : image plane with critical curves. Theobserved and predicted image positions of the dou-ble (diamond and plus symbols, respectively) andquad (square and cross symbols, respectively) sys-tems are shown.
Bottom left : contour levels of theFermat potential for the double system. The im-ages are one minimum (D ) and one saddle point(D ). Bottom right : contour levels of the Fermatpotential for the quad system. The images aretwo minima (Q and Q ) and two saddle points(Q and Q ).We generalize our result by emphasizing thatthe adoption of an isothermal model for stronglenses often provides a good fit of the observedimages, but the errors on the projected mass es-timates may be considerably underestimated al-ready at projected distances from the center of thelens that differ from the Einstein radius by half itsvalue. This fact has non-negligible consequenceson the inferred properties of a lens dark matterdistribution (see below).We notice that the value of 205 ±
13 km s − for the central stellar velocity dispersion σ , whichis determined by rescaling the value of the SDSSspectroscopic stellar velocity dispersion measured Total projected mass R (kpc)051015 M ( < R ) ( M O • ) SIE (nf)SIE (wf)
Total projected mass R (kpc)051015 M ( < R ) ( M O • ) SIE (nf)PL (nf)
Fig. 10.— Comparison of the projected total massestimates for the SIE (nf), SIE (wf), and PL (nf)models. For each aperture, the 1 σ confidenceintervals are determined from 2000 Monte-Carlosimulations by excluding the 320 smallest and the320 largest mass estimates.The arrows show the projected distances of theobserved multiple images from the lens center.inside an aperture of 1.5 ′′ [ σ = (189 ±
12) km s − ]to an aperture of radius equal to θ e /
8, is consis-tent, within the errors, with the value of 215 ± σ SIE .We remark that the best-fit parameters of ourSIE point-like models are consistent, given the er-rors, with the best-fit parameters of the SIE ex-tended model measured by Bolton et al. (2008).We also note that previous studies (Kochanek1993, 1994; Treu et al. 2006; Grillo et al. 2008b)agree on finding that the central stellar velocitydispersion of early-type galaxies is a good estima-tor of the velocity dispersion of a one-componentisothermal model.
4. Luminous and dark matter
We combine the surface brightness distributionmeasurement obtained from the HST images (seeSect. 2) with the multicolor photometric obser-vations of the SDSS (see Table 1) and the pro-jected total mass estimates determined from thelens modeling (see Sect. 3) to study the amountand distribution of luminous and dark matter inthe lens galaxy.First, we compare in Fig. 11 the surface bright-ness and the total surface mass [for the SIE (nf)model] isodensity contours of the best-fit modelsdescribed in the previous section. We use two im-12 x l (arcsec)-2-1012 y l ( a r c s ec ) Surface brightness -2 -1 0 1 2 x l (arcsec)-2-1012 y l ( a r c s ec ) SIE (nf) surface mass density
Fig. 11.— Light and mass distributions. Isoden-sity contours of the best-fit surface brightness ( onthe left ) and parametric [SIE (nf), on the right ] to-tal surface mass profiles. The observed image po-sitions of the double (diamond) and quad (square)systems are shown.ages with the same area and pixel size, normalizethe images to the sum of the values of all theirpixels, and plot the same contour levels in bothimages. We observe that the distributions of lightand total mass from the best-fit SIE (nf) modelare nearly axisymmetric, but the former is slightlymore concentrated than the latter. This can beinferred by looking at the positions of the innerand outer contour levels. The position angle ofthe surface brightness and total surface mass dis-tributions are consistent within the errors. Thus,we conclude that the light distribution is approx-imately circular symmetric in projection and it isa good tracer of the total mass distribution.Next, we fit the lens spectral energy distribu-tion (SED), consisting of the SDSS ugriz mag-nitudes (see Table 1), with a three-parameterBruzual & Charlot composite stellar population(CSP) model computed by adopting a Salpeterinitial mass function (IMF) and solar metallic-ity (for further details, see Grillo et al. 2009).The best-fit model, shown in Fig. 12, provides aphotometric (luminous) mass M ∗ phot of the lensof 20 +1 − × M ⊙ . We then estimate the valueof the mass in the form of stars M ∗ phot( ≤ R ),at a projected distance R from the center of thelens, by multiplying M ∗ phot by an aperture fac-tor f ap( ≤ R ), that represents the fraction of lightmeasured within a circular aperture of radius R divided by the total light of the galaxy. The quan- Fig. 12.— SED and best-fit model of the lensgalaxy SDSS J1538+5817. The observed total fluxdensities, measured in the u , g , r , i , and z pass-bands, and their 1 σ errors are represented by cir-cles and error bars. The best-fit is obtained byusing Bruzual & Charlot 2003 models. On thebottom, the best-fit values of the age ( T ), the char-acteristic time of the SFH ( τ ), and the luminousmass ( M ∗ phot ) are shown.tities introduced above are explicitly defined as M ∗ phot( ≤ R ) = M ∗ phot f ap( ≤ R ) (3)and f ap( ≤ R ) = R R I ( ˜ R ) ˜ R d ˜ R R ∞ I ( ˜ R ) ˜ R d ˜ R , (4)where I ( R ) is the de Vaucouleurs profile I ( R ) = I exp " − . (cid:18) RR e (cid:19) , (5)with R e equal to D ol θ e . As discussed above,the circular symmetry of the light distribution as-sumed in the previous three equations is a plausi-ble approximation for the lens surface brightness.In Fig. 13, we plot at different radii the projectedtotal and luminous mass estimates obtained fromthe best-fit SIE (nf) and PL (nf) models of Sect.3 and the best-fit SED model.In the same figure, we show the fraction of pro-jected mass in the form of stars f ∗ ( ≤ R ) := M ∗ phot ( ≤ R ) M totlens ( ≤ R ) (6)13 otal and luminous projected mass R (kpc)051015 M ( < R ) ( M O • ) SIE (nf)Phot
Total and luminous projected mass R (kpc)051015 M ( < R ) ( M O • ) PL (nf)Phot
Luminous over total mass fraction R (kpc)0.20.40.60.81.01.21.4 f ∗ ( < R ) SIE (nf)PL (nf)
Mass-to-light ratio R (kpc)024681012 M L B − ( < R ) ( M O • L O • , B - ) SIE (nf)PL (nf) FPPhot
Fig. 13.— Comparison of the projected total massestimates of the SIE (nf) ( on the top left ) and PL(nf) ( on the top right ) models with the luminousmass measurements from the best-fit SED model.The fraction of projected mass in the form of stars( on the bottom left ) and the total and stellar mass-to-light ratios ( on the bottom right ) are obtainedby using lensing and photometric information. Ineach panel, the arrows show the projected dis-tances of the observed multiple images from thelens center and the curves the 1 σ confidence inter-vals.and the total mass-to-light ratio M tot L B ( ≤ R ) := M totlens( ≤ R ) L B ( ≤ R ) (7)plotted versus the projected radius R . We com-pare this last quantity with the values of the stel-lar mass-to-light ratio estimated from the best-fit SED model ( M ∗ phot L − B = 5 . +0 . − . M ⊙ L − ⊙ ,B )and the evolution of the Fundamental Plane[ M ∗ FP L − B = (6 . ± . M ⊙ L − ⊙ ,B ] (for more in-formation, see Grillo et al. 2009).The need for a dark component to be added tothe luminous one to reproduce the total mass mea-surements of the SIE (nf) models is suggested bylooking at the outer galaxy regions probed by lens-ing. Due to the larger error bars, the evidence onthe presence of dark matter is reduced if the totalmass estimates obtained from the PL (nf) models are considered. According to all the lensing mod-els (i.e., deV, SIE, and PL), a value of 0 . +0 . − . forthe fraction of projected mass in the form of starsover total is estimated at a projected distance fromthe galaxy center of approximately 2.5 kpc, and at4 kpc from the galaxy center a value of one for thesame quantity is excluded by the SIE (nf) mod-els at more than 3 σ level. Moreover, at the samedistance, the value of the total mass-to-light ratiodetermined from the SIE (nf) mass measurementsis not consistent with the value of the mass-to-light ratio of the luminous component estimatedfrom the galaxy SED modeling. Between 1 and 4kpc, the same decrease of f ∗ ( ≤ R ) and deviationof M tot L − B from M ∗ phot L − B are also indicatedby the values of the PL (nf) mass estimates, butthese results are not highly significant because ofthe large uncertainties.Finally, by taking advantage of the total massmeasurements available at different distances fromthe center of the lens (not only in the vicinity ofthe Einstein angle, as in the majority of the knownlensing systems), we decide to investigate the darkmatter component in greater detail. To make pos-sible a direct comparison of our results with thoseobtained from dynamical analyses or cosmologicalsimulations, we consider two-component models inwhich the luminous ρ L ( r ) and dark ρ D ( r ) matterdensity distributions are parametrized by ρ L ( r ) = (3 − γ L ) M L r L πr γ L ( r + r L ) − γ L ρ D ( r ) = (3 − γ D ) M D r D πr γ D ( r + r D ) − γ D , (8)where M L / D is the total mass, r L / D a break ra-dius, and γ L / D the inner density slope of the lumi-nous and dark matter distributions. The densityprofiles of Eq. (8) are projected along the line-of-sight to give the corresponding surface mass den-sity profiles Σ L / D ( R ):Σ L ( R ) = 2 Z ∞ R ρ L ( r ) r d r √ r − R Σ D ( R ) = 2 Z ∞ R ρ D ( r ) r d r √ r − R , (9)which, once integrated, result in the following cu-mulative mass distributions M L / D ( ≤ R ): M L ( ≤ R ) = Z R Σ L ( R ) 2 πR d R D ( ≤ R ) = Z R Σ D ( R ) 2 πR d R . (10)The total density ρ T ( r ), surface mass densityΣ T ( R ), and cumulative mass M T ( R ) distributionsare defined as the sum of the luminous and darkcontributions ρ T ( r ) = ρ L ( r ) + ρ D ( r ) , Σ T ( R ) = Σ L ( R ) + Σ D ( R ) ,M T ( ≤ R ) = M L ( ≤ R ) + M D ( ≤ R ) . (11)We notice again that the circular approximationis plausible for this particular lens.The luminous quantities introduced in theabove equations are completely determined fromthe photometric observations. In fact, for the lu-minous component we have estimated the totalmass M L by modeling the SED and, to obtain asurface brightness profile close to a de Vaucoleursprofile, we assume a Hernquist (1990; γ L = 1 and r L = R e / . γ L = 2 and r L = R e / . M D , the break radius r D ,and the inner density slope γ D can assume valuesincluded between 0.1 and 100 times M L , 0.1 and10 ′′ , and 0.5 and 2.5, respectively. The first twointervals are divided logarithmically into 31 and21 points respectively, the last one linearly into 21points. The best-fit dark matter profile is foundby minimizing the following chi-square function: χ ( M D , r D , γ D ) = X i =2 M totlens ( ≤ R i ) − M T ( ≤ R i ) σ M totlens ( ≤ R i ) . (12)In order to estimate the errors in the best-fit pa-rameters, we perform 500 Monte-Carlo simula-tions varying the total mass of the luminous com-ponent according to the corresponding measure-ment errors and the luminous break radius by as-suming a realistic 10% uncertainty.In Fig. 14, we show the luminous and dark massdecomposition obtained from the best-fit (mini-mum chi-square) model and in Fig. 15 the pa-rameter joint probability distributions. We decideto plot the best-fit dark matter model obtainedby assuming a Jaffe profile (no significative differ-ences are present if a Hernquist profile is adopted)for the luminous component and considering the projected total mass estimates coming from thePL (nf) models. The confidence levels on the pa-rameter space of the dark matter component areexpressed in terms of the luminous mass fraction f L = M L / ( M L + M D ), i.e., the mass in the formof stars to the total mass of the galaxy, the ratioof the dark to luminous break radius r D /r L , and γ D . Luminous and dark matter density profile r (kpc)10 -4 -3 -2 -1 ρ ( r ) ( M O • p c − ) ρ L ( r )ρ D ( r )ρ T ( r ) r −2 Fig. 14.— Best-fit (minimum chi-square) lu-minous and dark matter decomposition, deter-mined by assuming a Jaffe profile for the three-dimensional luminous density and projected totalmass measurements as estimated from the PL (nf)model. The arrows show the projected distancesof the observed multiple images from the lens cen-ter.We find a best-fit χ value of 0.8 with two de-grees of freedom (derived from the total mass mea-surements at the five central radii fitted by three-parametric models). We measure that the valuesof the dark matter density overcome those of theluminous matter density at radii larger than ap-proximately 1.5 times the effective radius of thegalaxy ( R e = 4.0 kpc). As in the previous sections,a three-dimensional total density profile close butnot exactly equal to a function decreasing as 1 /r (i.e., an isothermal profile) is found. We note thatthe uncertainties in the dark matter parametersdetermined by using the projected total mass esti-15 .5 1.0 1.5 2.0 2.5 γ D f L f L −γ D joint probability distribution % r D / r L f L f L − r D / r L joint probability distribution % r D / r L γ D γ D − r D / r L joint probability distribution % γ D f L f L −γ D joint probability distribution % r D / r L f L f L − r D / r L joint probability distribution % r D / r L γ D γ D − r D / r L joint probability distribution % Fig. 15.— Estimates of the errors and correla-tions in the parameters related to the dark mattercomponent: the luminous mass fraction f L , thedark to luminous break radius ratio r D /r L , andthe dark matter inner density slope γ D . The pro-jected total mass measurements of the PL (nf) ( onthe left ) and SIE (nf) ( on the right ) models areused. The small squares on the three left panelsshow the best-fit parameters corresponding to thedark matter density profile represented in Fig. 14.mates of the PL (nf) models are significantly largerthan those coming from the measurements of theSIE (nf) models. This is a consequence of the dif-ferent error sizes of the two sets of projected totalmass estimates. For the same reason, as alreadydiscussed looking at Fig. 13, large values of f L areexcluded at a 95% CL only if the lens three dimen-sional total density profile is fixed to be isother-mal. We observe that the dark matter componentis in any case more diffused than the luminous one.In fact, r D /r L is larger than 2 at more than a 95% CL. Given the assumed parametrization, we alsofind that the dark matter density profile ρ D ( r ) isprobably shallow in the inner galactic regions. Thevalue of γ D is indeed lower than 0.7 at a 68% CL.
5. Summary and conclusions
By means of HST/ACS and WFPC2 imagingand NOT/ALFOSC spectroscopy, we have estab-lished that SDSS J1538+5817 is a rare lensing sys-tem composed of a luminous elliptical galaxy, lo-cated at redshift z l = 0 . z s = 0 . − Parametric models predict image positionsthat match closely the observed lensing ge-ometry, and describe lens total mass distri-butions that are almost circular in projec-tion, moderately steeper than an isothermalprofile, and well aligned with the lens lightdistribution. − The value of the total mass projected withinthe Einstein circle of radius 2.5 kpc is slightlylarger than 8 × M ⊙ and approximately10% of this mass is in the form of dark mat-ter. − In the inner galactic regions, the galaxydark-matter density distribution is shallowerand more diffuse than the luminous one. Theformer starts exceeding the latter at a dis-tance of roughly 6 kpc from the galaxy cen-ter, corresponding to 1.5 times the value ofthe luminous effective radius.16e conclude by remarking that strong gra-vitational lens systems with configurations com-parable to or more complex than that of SDSSJ1538+5817 are excellent laboratories to study thedistribution of luminous and dark matter in early-type galaxies. However, to achieve realistic resultson the dark matter component, it is essential toverify the commonly accepted isothermality of thetotal mass distribution at a higher level than doneso far. Strong lensing systems with an Einsteinradius significantly larger than the effective radiusof the lens galaxy would be invaluable to deter-mine the dark matter properties of the halos ofearly-type galaxies.We acknowledge the support of the EuropeanDUEL Research Training Network, TransregionalCollaborative Research Centre TRR 33, and Clus-ter of Excellence for Fundamental Physics and theuse of data from the accurate SDSS database. Wethank Piero Rosati for useful suggestions and theNOT staff for carrying out our observations in ser-vice mode. We are grateful to the NOT Scien-tific Association for awarding some observing timesolely on the basis of scientific merit and support-ing the NOT Summer School where CG gainedsome observational experience with the ALFOSCspectroscopic data. 17 . Non-parametric models
PixeLens (Saha & Williams 2004) is a non-parametric lensing program that generates an ensemble ofmodels consistent with the observed data of a lensing system. Each model is composed of a pixelated surfacemass density map of the lens, the reconstructed position of the source, and, optionally, an estimate of thevalue of the Hubble parameter. These results are obtained by using the observed positions of the multipleimages (ordered by arrival time, even if time delays are not known), the redshifts of the lens and the source,and some priors based on previous knowledge of general galaxy mass distribution (for further details, seeSaha & Williams 1997; Coles 2008). Interestingly,
PixeLens has been employed to measure the value of theHubble parameter from samples of strong lensing systems with measured time delays (e.g., Saha & Williams2006; Coles 2008).We model here the surface mass density of the lens on a symmetric circular grid of 2 ′′ radius divided into20 pixels. We consider 400 models with fixed cosmological values and with decreasing total projected massprofiles [i.e., Σ( R ) ∝ R − α , where α > Total projected mass R (kpc)051015 M ( < R ) ( M O • ) NP (nf)PL (nf) -2 -1 0 1 2 x l (arcsec)-2-1012 y l ( a r c s ec ) NP (nf) surface mass density γ probability distribution γ β=+0.25β=0 β=−0.25 Fig. 16.—
Left:
Comparison of the total projected mass estimates, with 1 σ confidence intervals, fromparametric [PL (nf)] and non-parametric [NP (nf)] modeling. The arrows show the projected distancesof the observed multiple images from the lens center. Middle:
Isodensity contours of the best-fit non-parametric [NP (nf)] total surface mass profile. The observed image positions of the double (diamond)and quad (square) systems are shown.
Right:
Marginal probability distribution (histogram) of the three-dimensional total density exponent γ from non-parametric modeling. The thick bar on the x -axis showsthe 1 σ confidence interval. The same probability distribution as obtained by combining strong lensing andstellar dynamics measurements is represented by the smooth curves.The cumulative total projected mass, the total surface mass density profile of the average model, and themarginalized probability distribution of the three-dimensional total density exponent γ are shown in Fig.16. We measure a value of the total mass projected within the Einstein radius of 8 . +0 . − . × M ⊙ , at a68% CL. At the same confidence level, we estimate a value of γ included between 1.62 and 2.87. We observethat the contour levels of the non-parametric total surface mass show non-negligible values of ellipticity inthe inner regions. The differences between the surface brightness of Fig. 11 and total surface mass of Fig. 16are significant within the area defined by the Einstein radius. This is not surprising since here the totalsurface mass density distribution is almost completely unconstrained by the lensing observables. This is theequivalent of Gauss’ law in gravitational lensing (see Kochanek 2004). We notice that these differences areless evident outside the Einstein ring, where the positions of the multiple images limit the freedom of the non-parametric models in determining the lens total mass distribution. In Fig. 16, we also show for comparisonthe mass estimates obtained in the equivalent parametric modeling [PL (nf)] and the probability distributionof the density exponent that is expected by combining strong lensing and stellar dynamics measurements. Indetail, the combined lensing and dynamical probability distribution for γ is obtained by using the followingexpression c π θ Ein σ ˜ r ( z l , z s ; Ω m , Ω Λ ) = (cid:18) θ Ein θ e (cid:19) − γ g ( γ, δ, β ) (A1)18hat relates through the spherical Jeans equations the values of the central stellar velocity dispersion σ , Ein-stein angle θ Ein , effective angle θ e , exponent of the three-dimensional luminosity density profile δ , anisotropyparameter of the stellar velocity ellipsoid β , and ratio of angular diameter distances between observer-sourceand lens-source ˜ r ( z l , z s ; Ω m , Ω Λ ) [ g ( γ, δ, β ) is a numerical factor that depends on the three cited quantities;for definitions and further details, see Koopmans 2005]. In the plots of Fig.16, we fix δ equal to 2 and choosetwo values of β ( − .
25 and +0 .
25) representative of small tangential and radial orbit anisotropy. By doublingthe size of the grid but keeping the same size of the pixels, we have checked that the choice of a circular gridwith a radius of 2 ′′ to reconstruct the total surface mass density distribution of our not perfectly circularlens galaxy does not introduce any artificial shear component and does not affect significantly the results.According to these results and looking at Fig. 16, we can conclude that the two independent parametricand non-parametric analyses are in general consistent, within the errors, as far as total projected massand three-dimensional total density exponent measurements are concerned, but small differences and someconsiderations are worth noticing.The projected total mass estimates obtained with PixeLens are systematically larger than those obtainedwith gravlens . This can be caused by a combination of the mass-sheet degeneracy (see Falco et al. 1985;Schneider & Seitz 1995) and the prior on the positive definiteness of every pixel of the grid of the totalsurface mass density. Among all the arbitrary constants that can be added to the convergence κ , leavingthough the image positions unchanged, those which provide a negative value of κ somewhere on the grid areexcluded, a priori , from the non-parametric lensing analysis. This fact may bias the projected total massmeasurements to slightly larger values.As far as γ is concerned, the larger uncertainty coming from the non-parametric reconstruction withrespect to the parametric one is probably just a consequence of the more general allowed models. A biastowards small values of γ may be associated to the prior present in PixeLens that constrains the value of κ on one pixel of the grid to be lower than twice the average value of the neighboring pixels. For large valuesof γ , two adjacent pixels located in the central region of the lens may have very different values of κ , hencethese models may not be included in the statistical ensemble.Finally, we remark on the overall agreement between the lensing only and lensing plus dynamics probabilitydistributions of γ . We notice, though, that lensing alone does not reach the precision needed to distinguishamong models with different values of the stellar anisotropy parameter β .19 EFERENCES
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