Moderate Galaxy-Galaxy Lensing

5.5 MB·

by  S. Mao

Mon.Not.R.Astron.Soc.000,000–000(2011) Printed11January2012 (MNLATEXstylefilev2.2)
Moderate Galaxy-Galaxy Lensing
Shude Mao1,2(cid:63), Jian Wang3, Martin C. Smith1,4
1NationalAstronomicalObservatories,ChineseAcademyofSciences,20ADatunRoad,Beijing100012,China
2JodrellBankCentreforAstrophysics,theUniversityofManchester,Manchester,M139PL,UK
3DepartmentofPhysicsandTsinghuaCenterforAstrophysics,TsinghuaUniversity,Beijing100084,China
4TheKavliInstituteforAstronomyandAstrophysics,PekingUniversity,YiHeYuanLu5,Beijing100871,China
2
1
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2 Accepted........Received.......;inoriginalform......

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a
ABSTRACT
J
We study moderate gravitational lensing where a background galaxy is magnified substan-
0
tially,butnotmultiplyimaged,byaninterveninggalaxy.Wefocusonthecasewhereboththe
1
lens and source are elliptical galaxies. The signatures of moderate lensing include isophotal

] distortionsandsystematicshiftsinthefundamentalplaneandKormendyrelation,whichcan
O potentiallybeusedtostatisticallydeterminethegalaxymassprofiles.Theseeffectsareillus-
C tratedusingMonteCarlosimulationsofgalaxypairswheretheforegroundgalaxyismodelled
asasingularisothermalspheremodelandobservationalparametersappropriatefortheLarge
.
h SynopticSurveyTelescope(LSST).Therangeinradiusprobedbymoderatelensingwillbe
p largerthanthatbystronglensing,andisintheinterestingregimewherethedensityslopemay
-
bechanging.
o
r
t Keywords: Gravitationallensing-galaxies:ellipticalandlenticular-darkmatter-Galaxies:
s
structure,Galaxies:formation
a
[

1
v
0 1 INTRODUCTION an elliptical galaxy is lensed by a foreground elliptical galaxy.
6 Many cases are expected to be found in future surveys (e.g. by
Gravitational lensing is usually divided into microlensing, strong
9 PANSTARRS1 andLSST2)wherehundredsofmillionsofgalax-
lensing,andweakgravitationallensing.Microlensingreferstothe
1 ies will be imaged. Many pairs of galaxies that are close to each
1. temporalchangeofmagnificationofabackgroundsourcelensedby otherbutatdifferentredshiftswillbediscovered.Mostofthesewill
an intervening object (usually stars). Strong lensing occurs when
0 notbemultiply-imaged.Atverylargeseparations,aweak-lensing
multiple images or strong distortions (e.g. giant arcs) of a back-
2 galaxy-galaxy analysis will be appropriate; at very small separa-
groundsourcearecausedbyaninterveninggalaxyorcluster.For
1 tions,multipleimagesmayform.InthispaperwecarryoutMonte
weaklensingthedistortionofabackgroundsourcebythelensing
: Carlosimulationsofmoderategalaxy-galaxylensingatintermedi-
v objectismuchmoresubtle.Forexample,acircularsourcewillbe
ateseparationstoillustratethesignatures,andwhatwecanlearn
i lensed into an ellipse with ellipticity of a few percent. Such sub-
X fromthese.
tledeviationshavetobeinferredstatisticallybyaveragingovera
r Thepaperisorganisedasfollows.InSection2,wedescribe
a large number of background galaxies. All these fields in gravita- the simple singular isothermal sphere model we use. In Section
tionallensinghavebeenextensivelystudied,withdiverseapplica-
3, we present the main results, including the predictions for the
tionsrangingfromcosmologytothedetectionofextrasolarplanets
optical depth, isophotal distortions, and systematic offsets in the
(forareviewonthesetopics,seeSchneideretal.2006andrefer-
fundamentalplaneandKormendyrelation.Wepresentasummary
encestherein).
anddiscussioninSection4.Throughoutthispaper,weadoptaflat
Inthiswork,weshallexploretheintermediateregimewhich
wetermas“moderategravitationallensing”.Inthiscase,themag- ΛCDMcosmologymodelwithΩm,0 = 0.27andΩΛ,0 = 0.73for
thematterandcosmologicalconstant,andtheHubbleconstantis
nification is still significant, but no multiple images occur. In the
writtenasH = 100hkms−1 Mpc−1 withh = 0.705(Komatsuet
contextofclustersofgalaxies,Williams&Lewis(1998)andFuta- 0
al.2009).
maseetal.(1998)haveconsideredaclassofgravitationallylensed,
highly magnified, yet morphologically regular images (originally
motivatedbyobservationsoftheobjectcB58,Yeeetal.1996;Seitz
etal.1998).Morerecently,Sonnenfeldetal.(2011)discussedthe
magnification effect in with a focus on how to remove the mass-
sheet degeneracy. In this paper, we shall explore the case where
1 http://pan-starrs.ifa.hawaii.edu/public/home.html
(cid:63) [email protected];[email protected] 2 http://www.lsst.org
(cid:13)c 2011RAS
2 Mao,WangandSmith
2 THELENSANDSOURCEMODEL galaxiesatredshift2.Belowthistruncationredshift,weassumea
constantcomovingnumberdensityoflenses,implyingthephysi-
Inthissection,wedescribethelensmassmodel,thesourcepop-
calvolumedensityevolvesasΦ (z ,σ)=Φ (σ)(1+z)3,where
ulationandthesurfacebrightnessmodelsofforegroundandback- G,L d G,L
thefactor(1+z)3accountsfortheexpansionoftheuniverse.Since
groundgalaxieswhichwewilluseforMonteCarlosimulationsof
mostoftheforeground(lensing)galaxiesarebelowredshift1,this
galaxypaircatalogues.
assumptionwillnothavemucheffectsonourresults.
2.1 Lensmassmodel
2.3 Ellipticalgalaxyluminosityfunction
Forsimplicity,wemodelthemassprofileofthelensinggalaxyas
Theluminosityfunction(LF)ofearly-typegalaxiesismodelledby
a singular isothermal sphere (SIS). This simple model is analyti-
theSchechter(1976)form
callytractable,andfitstheobservationaldatawell(e.g.,Koopmans
2009).Thesurfacemassdensitycanbemodelledas Φ (L)dL=Φ(cid:63)(cid:32) L (cid:33)αLexp(cid:32)− L (cid:33)dL. (6)
L L L L L
σ2 1 (cid:63) (cid:63) (cid:63)
Σ(ξ)= , (1)
2Gξ We adopt values from Choi et al. (2007) that were obtained us-
ing a galaxy sub-sample of the SDSS Data Release 5 with M −
where ξ is the (physical) distance in the lens plane, and σ is the 5log h = −20.23 ± 0.04, α = −0.527 ± 0.043, and Φ(cid:63)(cid:63) =
one-dimensional velocity dispersion. The angular Einstein radius 10 L L
0.71×10−2h−3Mpc−3,appropriatefortheSDSSr-band.
ofanSISisgivenby
Itismoreconvenientforourcalculationheretousethelumi-
σ2 D (cid:18) σ (cid:19)2 D nosityfunctionwithabsolutemagnitude M ratherthan L.Eq.(6)
θ =4π ds =1.(cid:48)(cid:48)4 ds, (2)
E c2 D 220kms−1 D canberewrittenas
s s
wherecisthespeedoflight,Dds istheangulardiameterdistance ΦL(M)dM=0.4ln10×Φ(cid:63)L10−0.4(M−M(cid:63))(αL+1)exp(−10−0.4(M−M(cid:63)))dM,(7)
between the lens and the source, and D is the angular diameter
s whereM isgivenbeloweq.(6).
distancetothesource. (cid:63)
Each survey has its own flux limit which corresponds to a
A point source at unlensed angular position 0 < θ < θ
s E magnitudelimitinaphotometricband(e.g.theSDSSrband),de-
from the galaxy produces two images at the angular positions
θ =θ ±θ .Themagnificationofeachimageandthetotalabso- notedbymr.Theabsolutelimitingmagnitude, Mr,ofagalaxyat
1,2 s E redshift z and galactic coordinates (l,b), can be constructed from
lutemagnificationaregivenby
theapparentmagnitudelimitm asfollows:
r
θ θ
µ1,2= θ1,2, µtotal=|µ1|+|µ2|=2θE, 0<θs<θE. (3) Mr =mr−DM(z)−Kr(z), (8)
s s
Noticethatthetotalmagnificationforthetwoimagesexceedstwo. whereDM(z)isthedistancemodulus,andKr(z)istheK-correction
A point source at angular separation θs > θE produces only (Hogg 1999). Here we use the Kr(z) given by Choi et al. (2007),
oneimageattheangularpositionθ = θ +θ ,withmagnification and we have ignored dust extinction by the Milky Way and the
s E
(θ +θ)/θ,givingavaluebetween1and2.Thusifasourceisafew foregroundgalaxy.
E s s
Einstein radiiaway from theline of sight, itmay still experience The lensing probability averaged over all source redshifts is
substantial magnification and differential magnification, and have givenby(e.g.,Wyitheetal.2010)
observablesignatures. 1 (cid:88)NG 1 (cid:90) zs,(cid:63) dV
<τ>= τ(z)= dz C Ψ (M(z),z)τ(z), (9)
N i N dz G,S s s s
G i=1 G 0
2.2 Lensingprobability
whereN isthetotalnumberofbackgroundellipticalgalaxies,V
G C
Forabackgroundellipticalgalaxyatredshiftz,thelensingproba- isthecomovingvolume,andΨ (M(z),z)isthecomovingnum-
s G,S r s s
bility(opticaldepth)canbeobtainedasfollows berdensityofellipticalgalaxiesbrighterthanM(z)atredshiftz.
r s s
(cid:90) zi (cid:90) cdt NGcanbecalculatedas
τ(zs) = 0 dzd dσΦσ(zd,σ)σcr(σ) dzd, (4) N =(cid:90) zs,(cid:63)dzdVC Ψ (M(z),z). (10)
G dz G,S r s s
whereσ isthecross-sectionformoderatelensing,tisthecosmic 0
cr
time, and Φ (z ,σ) is the lens velocity dispersion function (e.g. andΨ (M(z),z)as
σ d G,S r s s
Turner, Ostriker & Gott 1984, Gott, Park & Lee 1989, Fukugita, (cid:90) Mr(zs)
Futamase&Kasai1990). Ψ (M(z),z)= Φ (M)dM, (11)
G,S r s s L
Thelensvelocitydispersionfunctionismodelledbyamodi- −∞
fiedSchechterfunction(Choietal.2007): whereΦ (M)isthebackgroundellipticalgalaxyluminosityfunc-
L
Φσ(σ)dσ=Φ(cid:63)σ(cid:32)σσ (cid:33)ασexp−(cid:32)σσ (cid:33)βσΓ(αβσ/β )dσσ, (5) trieodnsh(sifetezesqc.o7r)r,easnpdonMdirn(zgs)toistthheemabasgonliututedleimlimitiintgmmr.aTgonibtuedceoMnsrear-t
(cid:63) (cid:63) σ σ vativeandforsimplicity,weusetheLSSTsingle-visitdepthlimit
whereΦ(cid:63)σ=8.0×10−3h3Mpc−3,ασ=2.32±0.10,βσ=2.67±0.07, mr =24.7inouropticaldepthcalculation.
andσ =161±5kms−1.Welimitthevelocitydispersiontothein-
(cid:63)
terval70∼400kms−1(Loeb&Peebles2003).Mostmassiveearly-
2.4 Lensandsourcesurfacebrightnessprofiles
typegalaxieswerealreadyassembledatz<1;beyondthisredshift,
thedensityofearly-typegalaxiesdeclinessignificantly(e.g.,Ren- We model the background source (and the lens) as an elliptical
zini2006),andsowetruncatetheredshiftofearly-typebackground galaxywithadeVaucouleurssurfacebrightnessprofile:
(cid:13)c 2011RAS,MNRAS000,000–000
ModerateGalaxy-GalaxyLensing 3
I(R)=I exp(−7.67((R/R )1/4−1)), (12)
e e
where R is the two-dimensional radius, R is the effective radius
e
within which half of the light is enclosed and I is the surface
e
brightnessattheeffectiveradius.Wequotetheeffectiveradiusas
√
thegeometricmeanofthemajorandminoraxes,R = a b .The
e e e
total apparent magnitude m , average magnitude within effective
T
radius(cid:104)µ(cid:105) andeffectiveradiusR canberelatedby(Scodeggioet
e e
al.1998)
m =−2.5log (L )−48.6=(cid:104)µ(cid:105) −1.9954−5log (R ), (13)
T 10 T e 10 e
whereL isthetotalfluxoftheellipticalgalaxyandR ismeasured
T e
inarcsec.
2.5 Simulatingimagesofearly-typegalaxies
In an image of an early-type galaxy, in addition to the signal of
thesource,wealsohavethePoissonnoiseintheskybackground,
readoutnoiseoftheCCDsandthedarkcurrent.Thesignal-to-noise Figure1.Asimulatedlensedimageofabackgroundellipticalgalaxy,with
ratioisgivenby(McMasteretal.2008) a scale of 0.(cid:48)(cid:48)2 per pixel. The foreground lens is at redshift 0.3 and the
backgroundellipticalgalaxy(ontheright)isatredshift0.6.Thesizeofthe
S = (cid:113) NSt , (14) imageis30.(cid:48)(cid:48)0by25.(cid:48)(cid:48)6.Thedistancebetweenthetwogalaxycentresis
N N t+n(N t+N t+N2) 8.2arcsec.TheellipsesareobtainedbyrunningtheIRAFtaskELLIPSE
S sky DC R (see§3.2).Noticethesmallchangesinthepositionangleandtwistsinthe
whereN isthenumberofphotoelectronsfromthesourceperunit isophotes.
S
time,N istheintensityofskybackgroundinphotoelectronsper
sky
pixelperunittime, N isthedarkcurrentinelectronsperpixel
DC
perunittime,N isthereadoutnoiseinelectronsperpixel,nisthe
R
numberofpixelscoveredbyanimageofanearly-typegalaxyand
TheKormendy(1977)relationbetweenthesurfacebrightness
tistheexposuretime. andeffectiveradiusisgivenby(Bernardietal.2003a,asquotedin
Toproceedfurther,weneedtoadoptaconcretecaseforthe
Oohamaetal.2009)
parametersineq.(14).HerewerestrictourselvestoLSST.Thetele-
scopewillbesitedinCerroPacho´n,Chile,withexcellentmedian (cid:104)µ(cid:105) =2.04log R +18.7, (16)
e 10 e
free-air seeing of 0.(cid:48)(cid:48)7 in the r band. Correspondingly, the pixel
scaleischosentobe0.(cid:48)(cid:48)2.Thetelescopewillrepeatedlysurveythe where(cid:104)µ(cid:105) isthemeansurfacebrightness(inmag/arcsec2)within
e
sky covering an area of 20000 square degrees with a cadence of theeffectiveradius(R ,inunitsofkpc).Thescatterinthisrelation
e
≈ 7days.Theintegrationtime(t)foreachindividualframeis15 issubstantiallylargerthanthatinthefundamentalplane(see§3.4).
seconds.Weassumethatthereadoutnoiseisthesameforallexpo- ForLSST,themeanredshiftofthegalaxysampleisabout0.8
sures.ForthePoissonnoiseintheskybackground,readoutnoise (seetheLSSTsciencebookfordetails).However,tobespecificand
of the CCDs and the dark current, we use the mean values given somewhatconservative,weusealensredshiftof0.3andasource
bytheLSSTExposureTimeCalculator3.Usingthesenumbers,we redshiftof0.6.Inreality,galaxypairswillhavearedshiftdistribu-
verifiedthatthephotonnumbersweobtaindifferfromthosegiven tion,butwecaninprinciplebinthedataandstudythequantities
bytheLSSTExposureTimeCalculatorbylessthan3%. weareinterested,andsothissimplificationdoesnotchangethere-
Fig.1 shows an example of a simulated moderately lensed sultsofthepaper.Noticethatfortheopticaldepthcalculation,we
backgroundellipticalgalaxyover-plottedwiththebest-fitellipses stillassumethelensesandsourceshavearedshiftdistribution,as
fromtheIRAFtaskELLIPSE(formoredetails,see§3.2).Notice describedin§2.
thatwehavenotconvolvedtheimagewithseeing. For each random galaxy generated for redshift 0, we shift it
To simulate a population of early-type galaxies, we use the to redshift z by taking into account its evolution (Bernardi et al.
method given in Appendix A of Bernardi et al. (2003a). Briefly, 2003a)
eachgalaxyisrandomlyassignedanabsolutemagnitude, M,ef-
r
fective radius Re and velocity dispersion σ (V in the notation of Mr(z)=Mr(z=0)−Qz, (17)
Bernardi et al. 2003a), appropriate at redshift z = 0. This proce-
whereQ=0.85fortherband.
dure ensures the generated galaxies satisfy the observed correla-
We assume a total exposure time of 6000 seconds (i.e., 400
tionsamongobservedproperties,includingthefundamentalplane
exposures of 15 seconds). We further select only galaxies which
asgivenbyBernardietal.(2003b)
satisfy R > 0.(cid:48)(cid:48)7, as a way to crudely account for the effects of
e
log10Re=log10σ+0.2((cid:104)µ(cid:105)e−20.09). (15) seeing.Intotal,wegenerate11623pairsofforegroundandback-
ground galaxies which satisfy these conditions. The ellipticity of
We assign a typical error of 0.03dex in logσ and 0.02dex in
thebackgroundgalaxyisrandomlydrawnfromtheaxisratiodistri-
log R . The scatter in the fundamental plane is quite small, on
10 e butionasgiveninChoietal.(2007)(seetheirFig.13).Thelensed
theorderof≈0.08,andmaybemostlyduetomeasurementerrors.
background galaxy images are obtained using the lens equation.
Thefollowinganalysesarebasedonthissample,althoughweap-
3 http://dls.physics.ucdavis.edu:8080/etc43work/servlets/LsstEtc.html plyafurthercutintheisophoteshapeanalysis(seebelow).
(cid:13)c 2011RAS,MNRAS000,000–000
4 Mao,WangandSmith
Figure3.Theprobabilitydensitydistributionoftheeffectiveradiiofthe
Figure2.Theopticaldepth,τ,asafunctionoftheredshift,zs.Thecross- backgroundellipticalgalaxies,allatredshift0.6.Atthisredshift,1arcsec
sectionformoderatelensingistakenastheareabetween1θEto10θE. correspondsto4.7h−1kpc.
3 RESULTS
3.1 Opticaldepth
InFig.2,wepresenttheopticaldepthasafunctionofredshift.In
thisexercise,wetakethemoderatelensingcross-sectionforeach
galaxyasθ to10θ (correspondingtoµ=2and1.1).Notsurpris-
E E
inglytheopticaldepthincreaseswithincreasingredshift,reflecting
thefactthatthenumberofinterveninggalaxiesincreasesformore
distant background sources. The optical depth is about 0.0029 at
z = 0.5,0.016atz = 1.0,and0.062atz = 2.0.Theopticaldepth
ofaveragedoverallthebackgroundsourcesisabout0.0254.This
probabilitywillscalelinearlywiththecross-sectionweadopt.For
example, if the we take the cross-section from 2θ to 5θ (corre-
E E
spondingtoµ = 1.5and1.2)thentheprobabilitywillbereduced
byafactorofroughly5toabout0.005.Theprecisechoiceofcross-
sectionwilldependonthedepth,seeingetc.ofobservations.
Wehavealsocalculatedthemagnificationbias(Turneretal.
1984),whichcanbelargeformultiply-imagedquasarsorgalaxies.
Inourcase,wefindthemagnificationbiastobemodest,onlyabout
1.16.Themagnificationbiasissmallbecausethemagnificationin
ourcaseisbydefinitionmodest.
Aswehavementionedin§2.5,theredshiftofthebackground
Figure4.TheprobabilitydensitydistributionoftheangularEinsteinradius
sources is truncated at z = 2.0. With this limitation, the surface
θE ofthesimulatedlensingcases.
number density of early-type galaxies is about 10 per square ar-
cmin.ForLSST,itwillcarryoutasurveyof20000deg2ofthesky,
andabout700millionellipticalgalaxieswillbedetected.InFig.3, theobservedonefromtheCLASSsurvey,eventhoughthesource
wepresentthedistributionoftheeffectiveradiusofthebackground andlensdistributionstherearesomewhatdifferent(seeFig.11in
elliptical galaxy sources. The number of elliptical galaxies drops Browneetal.2003).
steeply for larger effective radii. About 54.1% of the background
sources have effective radii larger than 0.(cid:48)(cid:48)7, the median free-air
3.2 Isophotaldistortions
seeing at the site of LSST. Under this criterion, we estimate that
thereareabout9.5millionmoderategalaxylensingcasesthatwill Surface photometry is performed on the simulated lensed images
beobservedbyLSSTtothesingle-visitdepth(m =24.7mag). of the background elliptical galaxy by utilizing the IRAF task
r
ThepredictedangularEinsteinradiusisshowninFig.4.As ELLIPSE(Jedrzejewski1987).WhenrunningtheELLIPSEonthe
can be seen, the distribution peaks around 0.8 arcsec and has an lensed images, the geometric centre, elllipticy and position angle
extendedtailouttolargerseparations.Thedistributionissimilarto areallallowedtovaryfreely,withalogarithmicstepof0.05inthe
(cid:13)c 2011RAS,MNRAS000,000–000
ModerateGalaxy-GalaxyLensing 5
Figure 6. The probability density distributions of the apparent isophotal
axisratio(q),positionangle(ϕ,inunitsofdegrees)forthegalaxyshownin
Fig.1.
R1/4 law(Haoetal.2006;Grahametal.2005).Inthesameplot,
thesedistributionsforseparationsbetween2.(cid:48)(cid:48)0and6.(cid:48)(cid:48)0andbe-
tween2.(cid:48)(cid:48)0and10.(cid:48)(cid:48)0arealsoshown.Asexpected,forbackground
sourceswithsmallerseparationsfromthelenscentre,thedistribu-
Figure5.CoefficientsoftheFourierseriesa3/a,a4/a,b3/aandb4/aob- tionsofthecoefficientsarebroaderbecausethelensingeffectsare
tainedbytheIRAFtaskELLIPSEforthegalaxyshowninFig.1.Thehori-
strongerinsuchcases.Noticethatintheaboveanalysis,werequire
zontallineineachpanelindicatesthevalueforaperfectellipse.
2r >1.5R ,thisconditionreducesthepairofgalaxiesfrom11623
s 50
to9094.
semi-majoraxis.Theisophotesofthelensedimagescanbemea- Both the a /a and b /a distributions show large deviations
3 3
suredindifferentways.HerewepresenttheoutputofthetaskEL- from those of normal elliptical galaxies as studied by Hao et al.
LIPSE.Adetaileddescriptionofthesemeasurementscanbefound (2006).ThisisfurtherillustratedinFig.8,hereweplotthetwo-
inHaoetal.(2006),weonlyrepeattheessentialshere.Theinten- dimensionaldistributionsofthesetwoquantitiesratherthanashis-
sityalongtheellipseisexpandedinFourierseries togramsfromprojection.Wecanseethattheunlensedgalaxiesare
(cid:88) mostlyatthecentre,whilethelensedgalaxiesclearlyshowmuch
I(θ)=I + (A cosnθ+B sinnθ), (18)
0 n n larger scatters than the unlensed ones. Thus they can be used as
where I istheintensityaveragedovertheellipse,and A and B effectiveindicatorswhethermoderategravitationallensinghasoc-
0 n n
arethehigherorderFouriercoefficients.Ifanisophoteisaperfect curredinapair.
ellipse,thenallthecoefficients,(A ,B ),n = 1,···,∞willbeex-
n n
actlyzero.Wewillusetwoquantitiesa andb extensively,which
n n
arerelatedtotheELLIPSEoutputsA andB by 3.3 Fundamentalplane
n n
a A Gravitationallensingpreservesthesurfacebrightnessofthesource
n = n, (19)
a γa becauseofLiouville’stheorem,butchangestheapparentsolidan-
gleofasource.Followingthis,anaturalexpectationisthatthefun-
whereγ=dI/dRisthelocalradialintensitygradient,andaisthe
damentalplaneofbackgroundellipticalsourcesshouldbeaffected
semi-majoraxislength.
bymoderategravitationallensing.
ForthegalaxyshowninFig.1,thecoefficientsoftheFourier
Toobtainthelensedandunlensedeffectiveradiiandsurface
series a /a,a /a,b /a and b /a as a function of the semi-major
3 4 3 4 brightness,R andR ,wefitthesurfacebrightnessprofileofthe
axisradiusareshowninFig.5.Thehorizontallineineachpanelis e,L e
lensedimagesofbackgroundellipticalgalaxieswiththeR1/4 law.
thevalueforaperfectellipse.Noticethatbotha /aandb /ashow
3 3 Inthismodel,wefixthecentreofthegalaxy,afterwhichthereare
significantdeviationsfromtheisophotesofaperfectellipse.
four remaining free parameters, {I ,R ,ϕ,q}. The χ2 function is
Forthesamegalaxy,wepresentthedistributionoftheappar- e e
givenby
entisophotalaxisratioq,positionangleϕinFig.6,.Thehorizontal
line presents the ellipticity (position angle) before lensing. There (cid:88)Np (I −I )2
are some small but systematic changes in these two quantities of χ2(Ie,Re,ϕ,q)= iσ2 i,0 . (20)
afewpercentfortheexampleshownhere.Thea /aandb /apa- i=1 i,0
4 4
rametersdescribewhetheranellipticalgalaxyisdiskyorboxy,but where N is the number of pixels of the lensed image. I is the
p i,0
wedidnotincorporatetheseintoourimagesimulations.Insteadwe photon number in the ith pixel, σ is the Poisson noise σ =
(cid:112) i,0 i,0
choosetofocusonthea3/aandb3/aparameters. Ii,0. The degree of freedom is Ndof = Np −4. We use the rou-
InFig.7,wepresentthedistributionofthecoefficientsofthe tinegsl multimin fminimizer nmsimplexintheGNUScientificLi-
Fourierseriesa3/a,a4/a,b3/aandb4/aforcaseswherethesepa- brary4 tofindtheχ2 minimumandobtaintheeffectiveradius;the
rationsbetweentheforegroundgalaxyandbackgroundgalaxyare meansurfacebrightnessisthenobtainedbyaveragingthesurface
betweenθE+Re,s ≈ 2.(cid:48)(cid:48)0and10.(cid:48)(cid:48)0,whereRe,s istheeffectivera- brightnesswithinRe.
diusforthesource.Foreachgalaxy,thesecoefficientsarethemean Fig.9 shows the distribution of differences in effective radii
valuewithintheregionbetween2r and1.5R ,wherer ≈0.(cid:48)(cid:48)7is
s 50 s
themedianseeingradiusandR isthePetrosianhalf-lightradius.
50
ForSDSS,R50 = 0.71Re forearly-typegalaxiesdescribedbythe 4 http://www.gnu.org/software/gsl/
(cid:13)c 2011RAS,MNRAS000,000–000
6 Mao,WangandSmith
Figure 7. The probability density distributions of the coefficients of the
Figure 9. The probability density distribution of differences in effective
Fourier series a3/a,a4/a,b3/a and b4/a obtained by the IRAF task radii between the lensed and unlensed sources in our simulated sample.
ELLIPSE.Theblacksolidhistogramineachpanelshowsthedistribution
Noticethesystematicshifttoalargereffectiveradius.
ofthecoefficientsforanearbysampleofearly-typegalaxiesfromHaoet
al.(2006).Thegreenlineisformoderatelylensedgalaxypairswithinsep-
arationbetween2to6arcsecwhiletheredsolidlineisforgalaxiesfrom2
to10arcsec.
Figure10.Theprobabilitydensitydistributionofdifferencesinmeansur-
face brightness within effective radius between the lensed and unlensed
sourcesinoursimulatedsample.
Figure8.Thescatteredplotforthesimulatedgalaxypairsintheplanea3/a
vs.b3/a.Thereddotsareforunlensedgalaxieswhiletheblackonesarefor meansurfacebrightnesseswithintheeffectiveradiusbetweenthe
lensedones.Intotal,thereare9094pairsofgalaxies. lensed and unlensed sources, defined as ∆(cid:104)µ(cid:105) = (cid:104)µ(cid:105) − (cid:104)µ(cid:105) ,
e e,L e
where (cid:104)µ(cid:105) and (cid:104)µ(cid:105) are the mean surface brightness within the
e,L e
lensed and unlensed effective radius of the lensed images. The
betweenthelensedandunlensedsources,definedas∆log yR = meanandmediandifferencesare−0.0003and−0.0003,whichare
10 e
log R −log R .Themeanandmedianvaluesof∆log R are muchsmallerthanthosefortheeffectiveradius,i.e.,gravitational
10 e,L 10 e 10 e
0.03and0.03respectively.Thesevalueswilldependonthesepara- lensingdoesnotaffectthemeansurfacebrightnesswithintheef-
tionrangewetakeforthegalaxypair,herewehaveselectedpairs fectiveradius.
withseparationsbetween∼2to10arcsec. Thefundamentalplaneofthelensedellipticalgalaxiesispre-
Similarly,Fig.10showsthedistributionofdifferencesinthe sentedinFig.11.Theblacksolidlineshowsthefundamentalplane
(cid:13)c 2011RAS,MNRAS000,000–000
ModerateGalaxy-GalaxyLensing 7
Figure11.Fundamentalplaneofthelensedellipticalgalaxies.Theblack Figure12.Kormendyrelationoftheellipticalgalaxies.Theblackdashed
solidlineshowsthefundamentalplaneforthebackgroundellipticalgalax- line shows the Kormendy relation of the unlensed background elliptical
ies.Thereddotsareforthelensedellipticalgalaxieswiththebestregression galaxies(blackdots).Thereddashedlinepresentstherelationofthelensed
lineshowsasthereddashedline.Thelensgalaxiesareatredshiftzl =0.3 ellipticalgalaxies(reddots).Inthisfigure,thelensgalaxiesareatredshift
andthesourcegalaxiesatzs=0.6. z=0.3andthesourcegalaxiesatz=0.6.
used to produce the background elliptical galaxies (black dots)
while the red symbols present the lensed elliptical galaxies. The WealsoexploretheoffsetsintheKormendyrelationandfun-
bestregressionlineforthelensedsampleisshownasthedashed damentalplaneofthelensedellipticalgalaxies,andattributethese
redline.Onaverageanupwardoffsetofabout0.03isintroduced tothemagnificationeffectoftheeffectiveradii,whilethemeansur-
bygravitationallensing. face brightnesses within the radii are nearly unaffected. The sys-
tematic offsets are on the order of 0.03 in logR . To observe the
e
shiftinthefundamentalplane,weneedtodeterminethevelocity
3.4 Kormendyrelation
dispersionspectroscopicallywhilefortheKormendyrelation,we
Toderivethesystematicshiftsinthefundamentalplane,inprinci- only need the source redshift (e.g., from photometric redshift) to
pleweneedtomeasurethevelocitydispersionfromspectroscopic obtainthephysicaleffectiveradius,sotheobservationaldemandis
observations. This will be very time consuming given the large somewhatlower.
number of early-type galaxies available from LSST. However, it Forstronglensing,itispossibletomodelanindividuallensto
iseasiertomeasuretheKormendy(1977)relationbetweenthesur- extractinformationaboutthelens.Formoderatelensing,thiswill
face brightness and effective radius, which can be obtained from bedifficult,anditsapplicationwillbemostlystatistical.Sincethe
imagingdataandanapproximatephotometricredshift.InFig.12, scatterinthefundamentalplaneissmall(0.08inlog10Re),a0.03
weshowtheKormendyrelation,ofthelensedandunlensedellip- shiftcaninprinciplebedetecedwithafewthousandgalaxies.For
tical galaxies. On average, an upward offset of about 0.03 in the theMonteCarlosimulationswediscussedin§2,wefirstfittheun-
KormendyRelationisseeninFig.11.Asmightbeexpected,this lensedsamplewithalinearmodelandthenrenormalisetheerror
shiftissimilartothatinthefundamentalplane.Thisoffsetismuch barssothatthetotalχ2perdegreeoffreedomisunity.Wethenfit
smallerthanthescatterorthogonaltotheKormendyrelation(about thesamemodeltothelensedsample.Iftherewerenosystematic
0.166)andthescatterinthedirectionoflogR (about0.182). shifts,thentheresultingχ2 shouldfollowroughlyanormaldistri-
e butionwithmeanequaltoχ2 =N =N−2andadispersionof
√ mean dof
σ = 2N ,whereNisthenumberofpairsofgalaxies.Anysig-
χ dof
nificantdeviationwillindicatethenooffsetmodelisunsatisfactory.
4 CONCLUSIONANDDISCUSSION
Foroursimulatedsample,wefindtheχ2forthenooffsetmodelis
In this paper we have investigated the moderate lensing of back- significantlyhigherthanthisexpectation,andtheexcessχ2 above
groundellipticalgalaxiesbyinterveningellipticalgalaxies.Wefind χ2 isontheorderof4.2σ and6.6σ for1600and5000pairs
mean χ χ
thetheopticaldepthformoderatelensingisontheorderof∼1%. ofgalaxies.FortheKormendyrelation,thecorrespondingsignif-
We have also performed image simulations based on the design icancelevelsare3.4σ and6.3σ ,somewhatlowerthanthatin
chi chi
specificationsofLSSTandobtainedrealisticlensedimagesofthe thefundamentalplane,whichisnotsurprisinggiventhelargerscat-
backgroundellipticalgalaxies.Thedistortionsofthelensedimages tersinthisrelation.
havebeenquantifiedwiththeIRAFtaskELLIPSE.Wefindmoder- Inpractice,therewillbeseveralcomplicationssincethelens
atelylensedgalaxiescanbepotentiallydifferentiatedfromnormal andsourceredshiftsmayonlybeavailablephotometrically.How-
galaxiesasoutliersinthecoefficientsofa /a,b /aetc. ever,iftherearenosystematicerrors,wecaninprinciplebinthe
3 3
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ACKNOWLEDGMENTS
Sersic, J.-L. 1968, Atlas de Galaxias Australes (Co rdoba: Obs.
We thank Drs. Cheng Li, Hu Zhan and Zuhui Fan for helpful Astron.)
discussions. JW and SM acknowledge the Chinese Academy of WilliamsL.L.R.,LewisG.F.,1998,MNRAS,294,299
Sciences and NSFC (grants 10821061 and 11033003) for finan- WyitheJ.S.B.,OhS.P.,PindorB.,2010,arXiv:1004.2081v1
cialsupport.MCSacknowledgesfinancialsupportfromthePeking WyitheJ.S.B.,WinnJ.N.,RusinD.,2003,ApJ,583,58
University One Hundred Talent Fund (985) and NSFC grants WyitheJ.S.B.,TurnerE.L.,SpergelD.N.,2001,ApJ,555,504
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