The Discovery of a Gravitationally Lensed Quasar at z = 6.51
Xiaohui Fan, Feige Wang, Jinyi Yang, Charles R. Keeton, Minghao Yue, Ann Zabludoff, Fuyan Bian, Marco Bonaglia, Iskren Y. Georgiev, Joseph F. Hennawi, Jiangtao Li, Ian D. McGreer, Rohan Naidu, Fabio Pacucci, Sebastian Rabien, David Thompson, Bram Venemans, Fabian Walter, Xue-Bing Wu, Ran Wang
DD RAFT VERSION O CTOBER
30, 2018Typeset using L A TEX twocolumn style in AASTeX62
The Discovery of a Gravitationally Lensed Quasar at z = 6 . X IAOHUI F AN , F EIGE W ANG , J INYI Y ANG , C HARLES
R. K
EETON , M INGHAO Y UE , A NN Z ABLUDOFF , F UYAN B IAN , M ARCO B ONAGLIA , I SKREN
Y. G
EORGIEV , J OSEPH
F. H
ENNAWI , J IANGTAO L I , I AN D. M C G REER , R OHAN N AIDU , F ABIO P ACUCCI , S EBASTIAN R ABIEN , D AVID T HOMPSON , B RAM V ENEMANS , F ABIAN W ALTER , R AN W ANG ,
12, 13
AND X UE -B ING W U
12, 131
Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721 Department of Physics, Broida Hall, University of California, Santa Barbara, CA 93106 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854 European Southern Observatory, Vitacura, Santiago 19, Chile Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, Florence, Italy Max Planck Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117, Heidelberg, Germany Department of Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109 Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Department of Physics, Yale University, New Haven, CT 06511 Max-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany Large Binocular Telescope Observatory,933 North Cherry Avenue, Tucson, AZ 85721 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Department of Astronomy, School of Physics, Peking University, Beijing 100871, China
ABSTRACTStrong gravitational lensing provides a powerful probe of the physical properties of quasars and their hostgalaxies. A high fraction of the most luminous high-redshift quasars was predicted to be lensed due to mag-nification bias. However, no multiple imaged quasar was found at z > in previous surveys. We report thediscovery of J043947.08+163415.7, a strongly lensed quasar at z = 6 . , the first such object detected at theepoch of reionization, and the brightest quasar yet known at z > . High-resolution HST imaging reveals amultiple imaged system with a maximum image separation θ ∼ . (cid:48)(cid:48) , best explained by a model of three quasarimages lensed by a low luminosity galaxy at z ∼ . , with a magnification factor of ∼ . The existence of thissource suggests that a significant population of strongly lensed, high redshift quasars could have been missedby previous surveys, as standard color selection techniques would fail when the quasar color is contaminated bythe lensing galaxy. Keywords: quasars: individual (J0439+1634) ; quasars: supermassive black holes; gravitational lensing: strong INTRODUCTIONLuminous quasars at z > ∼
150 quasars have been discovered at z >
6, with the high-est redshift at z = 7 . (Ba˜nados et al. 2018). Detections ofsuch objects indicate the existence of billion solar mass ( M (cid:12) )SMBHs merely a few hundred million years after the firststar formation in the Universe and provide the most stringent Corresponding author: Xiaohui [email protected] constraints on the theory of early SMBH formation (Volon-teri 2012).Much of our understanding of the nature of high-redshiftquasars assumes that their measured luminosities are intrin-sic to the quasars themselves. However gravitational lens-ing can substantially brighten quasar images. This effectis particularly important in flux-limited surveys, which aresensitive to the brightest sources; the resulting magnificationbias (Turner 1980) could cause a significant overestimationof the SMBH masses powering these objects. A large lens-ing fraction among the highest redshift luminous quasars haslong been predicted (Wyithe & Loeb 2002a; Comerford et al.2002) and was suggested as a solution to the difficulty informing billion M (cid:12) SMBHs in the early universe. However, a r X i v : . [ a s t r o - ph . GA ] O c t F AN ET AL .the two highest redshift known lensed quasars are at z ∼ . (McGreer et al. 2010; More et al. 2016), discovered in theSloan Digital Sky Survey (SDSS); no multiple imaged sys-tems were discovered at . (cid:48)(cid:48) resolution among the more than200 quasars at z = 4 − . observed in two HST programs(Richards et al. 2006; McGreer et al. 2014). The lack of thehigh-redshift lensed quasars has been a long-standing puzzle.The solution could be either a reduced magnification bias dueto a flat quasar luminosity function (Wyithe 2004) or a strongselection effect against lensed objects arising from the mor-phology or color criteria used in quasar surveys (Wyithe &Loeb 2002b).In our wide-area survey of luminous z ∼ quasars (Wanget al. 2017), we discovered an ultraluminous quasar UHSJ043947.08+163415.7 (hereafter J0439+1634) at z = 6 . .Subsequent Hubble Space Telescope ( HST ) imaging showsthat it is a multiple imaged gravitationally lensed quasar, themost distant strongly lensed quasar yet known. We presentthe initial discovery and followup imaging observations thatconfirm its lensing nature in §
2. In §
3, we present the lens-ing model in detail. In §
4, we discuss the possibility of alarge number of high-redshift lensed quasars missed in pre-vious surveys due to bias in color selection. We use a Λ CDMcosmology with Ω M = 0 . , Ω Λ = 0 . and H = 70 km s − . J0439+1634: A LENSED QUASAR AT Z=6.512.1.
Photometric selection and initial spectroscopy
J0439+1634 was selected by combining photometric datafrom the UKIRT Hemisphere Survey (UHS; Dye et al.(2018)) in the near-infrared J band, the Pan-STARRS1 sur-vey (PS-1; Chambers et al. (2016)) at optical wavelengths,and the
Wide-field Infrared Survey Explorer ( WISE ; Wrightet al. (2010)) archive in the mid-infrared. It was chosenas a high-redshift quasar candidate based on it having a z -band dropout signature with z AB = 19 . ± . , y Vega =17 . ± . , and a red z AB − y AB = 1 . ± . , alongwith a blue power-law continuum ( J Vega = 16 . ± . , y AB − J Vega = 1 . ± . ), and a photometric red-shift of z ∼ . . The object has a weak i -band detec-tion in PS1 ( i AB = 21 . ± . ), but is strongly de-tected in all bands in the Two Micron All Sky Survey(2MASS; Skrutskie et al. (2006)), at J Vega = 16 . ± . , H Vega = 15 . ± . , and J Vega = 15 . ± . , respec-tively, as well in all four WISE bands, with Vega magnitudesof . ± . , . ± . , . ± . and . ± . ,respectively, from W1 to W4.The initial identification spectrum, obtained on 6 Febru-ary, 2018, with the Binospec optical spectrograph (Fabricantet al. 2003) on the 6.5m MMT telescope, shows a promi-nent break consistent with a strong Ly α line at z ∼ . .Follow-up optical and near-infrared spectra were acquiredwith MMT/Binospec, the Low Resolution Imaging Spectro- graph (LRIS, Oke et al. 1995) on the 10m Keck-I Telescope,and the GNIRS instrument (Elias et al. 2006) on the 8.2mGemini-North Telescope. The combined optical-IR spectrumis shown in Figure 1. Strong MgII emission is detected byGNIRS, yielding a redshift of z = 6 . ± . .J0439+1634 is roughly 40% brighter than the luminous z = 6 . quasar SDSS J0100+2802 (Wu et al. 2015), makingit the brightest quasar known at z > . It is also the brightestsubmm quasar at z > ; it is detected by the SCUBA-2 in-strument (Holland et al. 2013) on the James Clerk MaxwellTelescope (JCMT) with a total flux of 26.2 ± µ m. However, its high luminosity is likely not intrinsic,but instead boosted via gravitational lensing. The opticalspectrum of J0439+1634 shows a faint, continuous trace at λ < ˚A, visible in the middle of the deepest region ofquasar Gunn-Peterson absorption at 8500 ˚A < z abs > ). This trace extends beyond the quasar Lyman Limit at λ < ˚A, blueward of the IGM transmisssion spikes inthe quasar Ly β region. No quasar continuum transmissionis expected at these wavelengths due to the extremely highIGM optical depth (Fan et al. 2006), indicating the presenceof a foreground object within the (cid:48)(cid:48) spectroscopic slit. Thelensing hypothesis is further supported by the presence of avery small quasar proximity zone (Figure 1) and an apparentsuper-Eddington accretion rate based on the Mg II measuredSMBH mass (Figure 2), both of which can be explained witha significant lensing magnification.2.2. High Resolution Imaging
J0439+1634 appears as an unresolved point source onarchival PS1 and UHS images (seeing of ∼ . (cid:48)(cid:48) ) and ondeeper near-infrared images taken with the Fourstar instru-ment (Persson et al. 2013) on the 6.5m Magellan-1 Telescope(seeing ∼ . (cid:48)(cid:48) ). To test the lensing hypothesis, we obtaineda high resolution K-band image using the Advanced Rayleighguided Ground layer adaptive Optics System (ARGOS; Ra-bien et al. 2018) on the × . m Large Binocular Telescope,with a ground-layer AO corrected FWHM of . (cid:48)(cid:48) . Thisimage (Figure 3A), taken with the LUCI (Buschkamp et al.2012) instrument, marginally resolves J0439+1634 beyondthe PSF (FWHM = . (cid:48)(cid:48) ± . (cid:48)(cid:48) ).Even more revealing are the high resolution observa-tions of J0439+1634 with the Advanced Camera for Sur-veys (ACS) on the HST , taken on 3 April, 2018, using twointermediate band ( ∆ λ ∼ ˚A) ramp filters (Figure 1).The FR782N observation is centered at 7700 ˚A, fully cov-ers the quasar Ly β emission, and is the shortest wavelengthat which quasar emission is still detectable, thus provid-ing the highest possible spatial resolution of . (cid:48)(cid:48) . TheFR853N observation is centered at 8750 ˚A, within the Gunn-Peterson trough, and images only the foreground galaxy. The“galaxy+quasar” FR782N image (Figure 3B) clearly resolves ENSED Q UASAR AT Z =6.51 3
Observed Wavelength ( Å ) f ( e r g s c m Å ) Rest-frame Wavelength ( Å ) | Ly | N V | Si IV | C IV | C III] | Mg II
Observed Wavelength ( Å ) f ( e r g s c m Å ) (b) Observed Wavelength ( Å ) f ( e r g s c m Å ) (a) Figure 1. Combined optical and near-infrared spectrum of the lensed quasar J0439+1634 at z = 6.51. The optical portion of the spectrumis from the Binospec instrument on the 6.5m MMT telescope and the LRIS instrument on the 10m Keck-I telescope. The near-infrared portionof the spectrum is from the GNIRS instrument on the 8.2m Gemini North Telescope. The proximity zone around the quasar is denoted by thegrey shaded area blueward of Ly α ; its size ( R p = 3 . ± . ) is > × smaller than for other luminous quasars at z ∼ . (Wu et al.2015; Mazzucchelli et al. 2017), suggesting that the intrinsic ionizing flux is much lower. Insert (a) shows the spectrum in the Ly α forestregion. A faint continuum is clearly detected in the darkest region of the quasar Gunn-Peterson trough, suggesting the presence of a foregroundgalaxy. Orange dashed lines are the traces of the HST /ACS ramp filters used to image the lensing galaxy and quasar (images shown in Figure3).
Insert (b) shows the Mg II region of the quasar spectrum. The red line is the best fit spectrum when including a power-law continuum,Balmer continuum, and Mg II+Fe II emission. The best-fit redshift based on Mg II is . ± . . The best-fit FWHM of the MgII line is2924 ±
188 km s − , yielding a SMBH mass (McLure & Dunlop 2004) of (4.93 ± . × M (cid:12) before correction for lensing magnification. Figure 2. Distribution of quasar bolometric luminosities and SMBH masses estimated from Mg II emission . The open red star representsJ0439+1634 without lensing correction; the filled red star represents the same object after applying a lensing magnification correction factorof 51 × (from the fiducial lensing model in Table 1). The green circle represents SDSS J0100+2922 at z = 6 . (Wu et al. 2015), the bluesquare SDSS J1148+5251 at z = 6 . (Fan et al. 2003), and the magenta circles ULAS J1120+0641 at z = 7 . (Mortlock et al. 2011) andULAS J1342+0928 at z = 7 . (Ba˜nados et al. 2018). Black dots denote other z (cid:38) quasars (Wu et al. 2015; Mazzucchelli et al. 2017). Theblack contours and grey dots show SDSS low redshift quasars (Shen et al. 2011) (with broad absorption line quasars excluded). The error barsrepresent the 1 σ measurement errors. For comparison, the dashed lines illustrate fractions of the Eddington luminosity. F AN ET AL .the system into multiple components: there are at least twopoint sources separated by . (cid:48)(cid:48) and a faint, extended source ∼ . (cid:48)(cid:48) to the east, which we interpret as the lensing galaxy.The “galaxy-only” FR853N image (Figure 3C) shows onlythe lensing galaxy, best fit with an exponential profile, anellipticity of ∼ . , and an effective radius of ∼ . (cid:48)(cid:48) .2.3. Properties of the lensing galaxy
We use the best-fit galaxy position and shape parametersfrom the FR853N image to derive the lensing galaxy flux inthe two
HST bands and LBT K-band: AB ˚ A = 22 . ± . , AB ˚ A = 22 . ± . and K Vega = 18 . ± . .The non-detection in the blue channel of the Keck/LRISspectrum yields an upper limit of g AB > for the galaxy.We estimate the synthetic PS-1 g, r, i band magnitudes ofthe lensing galaxy using the spectrum of J0439+1634 (Fig-ure 1), which shows the trace of the lensing galaxy spectrumin the quasar Gunn-Peterson trough. We scale the spectrumby matching it to the HST /FR853N band magnitude, whichdoes not include quasar flux. We choose a wavelength rangefree of quasar flux, between 8600 and 8900 ˚A in the Gunn-Peterson trough, and blueward of the Lyman limit ( < g , r and i band AB magnitudes are estimated to be . ± . , . ± . , . ± . , respectively.Based on these photometric data and after applying theGalactic extinction correction (Cardelli et al. 1989), we es-timate the photometric redshift using the EAZY (Brammeret al. 2008) code. The peak value of the p(z) probability dis-tribution is z peak = 0.67, and the 1 σ confidence intervalfrom the probability distribution is . ≤ z ≤ . . Withthe LeP hare code (Arnouts et al. 2002; Ilbert et al. 2006)and a set of 12 template galaxies using Bruzual & Charlot(2003) models, we find a best-fit stellar mass of 10 . M (cid:12) .Deeper photometry is needed to further improve the photo-metric redshift and stellar mass determinations. LENSING MODELA purely photometric fit of the
HST /ACS FR782N data us-ing only two quasar images has a significant residual, sug-gesting a more complex lensing configuration. We fit a sin-gular isothermal ellipsoid lensing model, fixing the lens po-sition and ellipticity ( e = 0 . ) to match the observed galaxyin the FR853N image, while varying the lens mass and posi-tion angle along with the source position to reproduce theobserved configuration (Keeton 2001). We vary the Ein-stein ring radius and position angle of the galaxy along withthe position of the source. For each set of parameters, wesolve the lens equation to predict the positions of the im-ages, place copies of the HST
PSF at those positions, and compare with the FR782N image to compute a χ goodnessof fit. We then use Markov Chain Monte Carlo methods tosample the parameter space. The resulting model is depictedin Figure 3 and the parameters are summarized in Table 1.To interpret the Einstein radius, we assume the galaxy is athin rotating disk such that the projected ellipticity reflectsthe inclination, and we compute the corresponding circularvelocity (Keeton & Kochanek 1998). A three-image modelis preferred (Figure 3D), with a best-fit Einstein radius of θ E = 0 . (cid:48)(cid:48) ± . (cid:48)(cid:48) , corresponding to a circular velocityof v c = 160 +8 − kms − and a high total magnification of . ± . . In this model, the separation of the two brighterimages is only . (cid:48)(cid:48) , unresolved even by HST .We estimate the observed optical luminosity at rest-frame3000 ˚A to be (4 . ± . × erg s − by fitting thecalibrated spectrum. Applying an empirical factor (Shenet al. 2011) to convert the luminosity at 3000 ˚A to the bolo-metric luminosity gives L bol = 2 . × ergs s − =5 . × L (cid:12) . After correction for magnification factor of51.3, the bolometric luminosity of J0439+1634 is reduced to . × L (cid:12) , and the SMBH mass to . ± . × M (cid:12) .This corresponds to an Eddington ratio of . ± . .However, this model seems to underpredict the flux of thefaintest quasar image. It is not clear whether the discrepancyis due to limitations in the current data (e.g., in the HST
PSFmodel) or to fundamental problems with this class of lensmodels. As an alternative, we consider the possibility thatthe lens galaxy could actually lie between the quasar imagesand be blended with them. In this scenario, the galaxy lightdetected in the
HST image could be offset from the mass cen-troid, due perhaps to strong dust obscuration. For example,if the lensing galaxy is seen mostly edge-on, then we mighthave detected only the part of the galaxy with the highest sur-face brightness along the disk. The smallest residuals are ob-tained for a highly inclined galaxy with projected ellipticity e = 0 . , which produces four images and a total magnifica-tion of . ± . (see Figure 4 and Table 1). The impliedcircular velocity v c = 88 +4 − km s − is quite low, compara-ble to that of the Large Magellanic Cloud. The orientation isconsistent with the hypothesis that the observed galaxy lightis from part of the disk. It also possible that the nearby galaxyis not related to the lensing. In this case, the true lens galaxyis too faint for detection here, could lie between the quasarimages, and be relatively round. We therefore test a thirdmodel with ellipticity e = 0 . , which produces just two im-ages that have a total magnification of . +1 . − . .This modelhas a modest circular velocity of v c = 121 +6 − km s − .We consider the fiducial three-image model to be the mostlikely lensing configuration, because it naturally places thecenter of the lensing galaxy at the position of the detectedgalaxy image in the two HST bands. However, further ob-servations are needed to clearly distinguish between the dif-
ENSED Q UASAR AT Z =6.51 5
LBT/LUCI K
G BA HST/ACS FR782N G HST/ACS FR853N Fiducial Model
Figure 3. High-resolution images of the strongly lensed quasar J0439+1634 and the best-fit three-image lensing model. A:
LBT/ARGOSLUCI image in the K-band. With ground layer AO correction, the FWHM of the PSF is . (cid:48)(cid:48) . The quasar image has a FWHM of . (cid:48)(cid:48) ± . (cid:48)(cid:48) .The contours show the image core elongated in the North-South direction as well as excess light towards the East, consistent with the highresolution HST imaging. B: HST /ACS WFC image with the FR782N ramp filter centered at 7700 ˚A, covering the quasar Ly β emission. This“galaxy+quasar” image is resolved into at least two point-like components (A and B) and a faint extended source towards the East (G). C: HST /ACS WFC image with the FR853N ramp filter centered at 8750 ˚A, covering the deepest part of the quasar Gunn-Peterson trough. This“galaxy only” image is used to determine the location and shape parameters of the lensing galaxy. D: Best-fit three image lensing model to the
HST /ACS FR782N image, using the lens location and shape derived from the FR853N image. White crosses show the locations of the best-fitquasar images and blue lines show lensing critical curves of the fiducial lensing model. Red lines show the lensing caustics in the source plane.In this model, the total magnification is . ± . and the Einstein radius is . (cid:48)(cid:48) , which corresponds to a circular velocity of v c = 160 +8 − km s − assuming a lens redshift z = 0 . +0 . − . . F AN ET AL . Fiducial Model 4-Image Model 2-Image ModelResidual Residual Residual
Figure 4. Fiducial and Alternative Lens Models: Fits (top row) and residuals (bottom row) of the
HST /ACS FR782N image.
As in Figure3, white crosses are locations of quasar images, and blue and red lines represent the lensing critical curves and caustics, respectively. ferent models. Images that are deeper than the current
HST observation could fully characterize the lensing galaxy, whileobservations with higher spatial resolution (possible onlywith
JWST or ALMA) would reveal whether there are two,three, or four image components. DISCUSSIONThe probability that a luminous quasars is gravitationallylensed with magnification factor µ > at z ∼ ranges from ∼ , if the bright end of the quasar luminosity function is Φ( L ) ∝ L − . (Jiang et al. 2016), to ∼ , if the quasarluminosity function is as steep as Φ( L ) ∝ L − . (Yang et al.2016). Yet J0439+1634 is the first strongly lensed quasar dis-covered at z > among the several hundred quasars knownat this redshift. A reexamination of the color selection usedin previous high-redshift quasar surveys suggests a strong se-lection bias against lensed quasars.Selecting z (cid:38) quasars requires either a non-detection(Wang et al. 2017; Jiang et al. 2016) or a strong drop inthe dropout band below the quasar Lyman break (Mazzuc-chelli et al. 2017). The presence of a lensing galaxy, how-ever, introduces flux into the dropout bands when the im-age is not fully resolved. Most lensing galaxies are expectedto be massive galaxies at z ∼ . - 1.5 and to have de- tectable r - or i -band flux in the SDSS or PS-1 survey. Forexample, among the 62 lensed z < quasars in the SDSSsample (Inada et al. 2012) with measurements of the lensinggalaxy, the faintest lens has i AB = 21 . . On the other hand,the J0439+1634 lens is among the faintest lensing galaxiesknown, with i AB = 22 . . The faintness of this lens, com-bined with the high apparent luminosity of the lensed quasar,minimizes the impact of lensing galaxy flux to the overallunresolved quasar+lens color used in candidate selection. Ifthe lens were brighter by even 0.5 mag, J0439+1634 wouldnot have been selected as a high-redshift quasar candidateby our color selection criteria (Wang et al. 2017), suggestingthat previous surveys have potentially missed the majority oflensed quasars at the highest redshifts due to their stringentdropout criteria. Thus a full modeling of quasar+lens colorsand selection procedure modifications are needed to cover themajority of the high-redshift lensed quasar population.A statistical study of strong lensing properties using theMillennium Simulation (Hilbert et al. 2008) shows that for asource at z = 5 . , only 5% of the lensing optical depth is pro-vided by galaxies with a halo mass lower than × M (cid:12) ,comparable to J0439+1634’s lensing galaxy ( v c = 160 kms − ) in the fiducial three-image lensing model. This im-plies up to ∼ lensed high-redshift quasars could have ENSED Q UASAR AT Z =6.51 7been missed in our survey due to contamination from lens-ing galaxy light. If these lensed quasars do exist, it wouldsignificantly impact the measurement of the quasar luminos-ity function, especially at the brightest end (Wyithe & Loeb2002b). Benefiting from the boosted flux, an object such asJ0439+1634 is a powerful probe of the physical properties ofquasars and their host galaxies as well as serving as an idealbackground source for studying high redshift metal absorp-tion lines and early IGM chemical enrichment.We acknowledge the support of the staff at the MMT, Mag-ellan, LBT, Keck and Gemini Telescopes, and thank the Di-rectors of LBTO, Gemini Observatory, JCMT, and STScI for granting us Director Discretionary time for follow up obser-vations of this object. X.F., J.Y., M.Y. and I.D.M. acknowl-edge support from US NSF Grant AST-1515115, NASAADAP Grant NNX17AF28G and HST-GO-13644 grant fromthe Space Telescope Science Institute. C.K. acknowledgessupport from US NSF grant AST-1716585. A.I.Z. acknowl-edges support from NSF grant AST-1211874. F.P. acknowl-edges support from the NASA Chandra award No. AR8-19021A and from the Yale Keck program No. Y144. B.P.V.and F.W. acknowledge funding through the ERC grants “Cos-mic Dawn” and “Cosmic Gas”. R.W. and X.-B.W acknowl-edge support from NSFC grant No. 11533001.
Facilities:
UHS, WISE, PS1, MMT, Magellan, LBT,Gemini, Keck, JCMT, HSTREFERENCES
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Fiducial 3-image model Alternate 4-image model Alternate 2-image modelImage 1 (∆ RA , ∆ Dec ) ≡ (0 ,
0) (0 ,
0) (0 , µ = 5 . ± . µ = 1 . µ = 3 . +0 . − . Image 2 ( − . , − . − . , − . − . , − . µ = 21 . ± . µ = 5 . ± . µ = − . +0 . − . Image 3 ( − . , − . − . , − . — µ = − . ± . µ = − . ± . Image 4 — (0 . , − . — µ = − . ± . Source (0 . , . − . , − . − . , − . µ tot = 51 . ± . µ tot = 10 . ± . µ tot = 23 . +1 . − . Lens (0 . , . − . , − . − . , − . θ E = 0 . ± . (cid:48)(cid:48) θ E = 0 . ± . (cid:48)(cid:48) θ E = 0 . ± . (cid:48)(cid:48) v c = 160 +8 − km s − v c = 88 +4 − km s − v c = 121 +6 − km s − e = 0 . , PA = 103 . ± . e = 0 . , PA = 101 . ± . e = 0 . , PA = 112 . +6 ..
0) (0 , µ = 5 . ± . µ = 1 . µ = 3 . +0 . − . Image 2 ( − . , − . − . , − . − . , − . µ = 21 . ± . µ = 5 . ± . µ = − . +0 . − . Image 3 ( − . , − . − . , − . — µ = − . ± . µ = − . ± . Image 4 — (0 . , − . — µ = − . ± . Source (0 . , . − . , − . − . , − . µ tot = 51 . ± . µ tot = 10 . ± . µ tot = 23 . +1 . − . Lens (0 . , . − . , − . − . , − . θ E = 0 . ± . (cid:48)(cid:48) θ E = 0 . ± . (cid:48)(cid:48) θ E = 0 . ± . (cid:48)(cid:48) v c = 160 +8 − km s − v c = 88 +4 − km s − v c = 121 +6 − km s − e = 0 . , PA = 103 . ± . e = 0 . , PA = 101 . ± . e = 0 . , PA = 112 . +6 .. − ..