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Mon.Not.R.Astron.Soc.000,1–10(2007) Printed2February2008 (MNLATEXstylefilev2.2) The Effect of Satellite Galaxies on Gravitational Lensing Flux Ratios ⋆ ⋆ E.M. Shin and N.W. Evans InstituteofAstronomy,UniversityofCambridge,MadingleyRoad,Cambridge,CB30HA,UnitedKingdom 2February2008 8 ABSTRACT 0 Gravitationallenseswithanomalousfluxratiosareoftencitedaspossibleevidencefordark 0 matter satellites predicted by simulations of hierarchical merging in cold dark matter cos- 2 mogonies.Weshowthatthefractionofquadswithanomalousfluxratiosdependsprimarily Jan othneitrheditsottraiblumtiaosns.aInfdds1p/a2ti∼ale1x0t0enktpocf,tthheensaftoerlliatems,oadnedrattheelycheallriapctitcearilstgiacllaexnygtwhistchalaeldin1/e2-ooff- sightvelocitydispersionof∼ 250kms−1,amassof∼ 3×109M inhighly-concentrated 6 ⊙ (Plummermodel)satellitesisneededfor20%ofquadrupletstoshowanomalousfluxratios, 1 risingto∼ 1.25×1010M⊙ for50%.Severaltimesthesemassesarerequiredifthesatellites ] have more extended Hernquist profiles. Compared to a typical elliptical, the flux ratios of h quadsformedbytypicaledge-ondiscgalaxieswithmaximumdiscsaresignificantlylesssus- p ceptibletochangesthroughsubstructure–threetimesthemassinsatellitegalaxiesisneeded - o toaffect50%ofthesystems. r In many of the lens systems with anomalous flux ratios, there is evidence for visible st satellites(e.g.,B2045+265orMG0414+0534).We showthatiftheanomalyisproducedby a substructurewithpropertiessimilar tothe simulations,thenopticallyidentifiedsubstructure [ shouldnotbepreponderantamonglenssystemswithanomalies.Thereseemtobetwopos- sibleresolutionsofthisdifficulty.First,insomecases,visiblesubstructuremaybeprojected 2 v withinorclosetotheEinsteinradiusandwronglyascribedastheculprit,whereasdarkmatter 8 substructureis causing the flux anomaly.Second, brightsatellites, in which baryoncooling 5 andcondensationhastakenplace,mayhavehighercentraldensitiesthandarksatellites,ren- 2 deringthemmoreefficientatcausingfluxanomalies. 2 . Keywords: gravitationallensing–darkmatter 1 0 8 0 : 1 INTRODUCTION dark substructures. They used 7 of the then available four-image v i lenssystemstoclaimdetectionofsubstructureamountingtoatotal X Theabundanceofsubstructureingalaxyhalosisemergingasakey massof0.6% 6%ofthelensgalaxymass.However,thisinter- testintheoriesofgalaxyassembly.InColdDarkMattercosmogo- − r pretationwaschallengedasfluxanomaliescouldarisefromalter- a nies,darkmatteroverdensitiescollapsetoformcuspedhalos,with nativesources,suchasabsorption,scattering,orscintillationbythe the smallest and least massive halos being the densest. The sim- interstellarmediumofthelensgalaxy,orbyhigherorderharmon- ulations of Klypinetal.(1999) and Mooreetal.(1999)predicted icsintheellipsoidalmodelsusedtofitlenssystems,orstellarmi- hundreds of small Galactic satellitehalos, in contrast to the nine crolensing (seee.g. Evans&Witt2003; Kochanek&Dalal 2004; thenknownsatellitesaroundtheMilkyWay.Efstathiou(1992)had Maoetal. 2004). In particular, the surface mass density in sub- already suggested that photoionisation may lengthen the cooling structure asjudged from simulations seems to be lower than that times of gas in haloes with low circular speeds. This effect sup- required by gravitational lensing, at least within the Einstein ra- presses the formation of satellite galaxies, but produces a large diuswhichprobesprimarilytheinnerpartsofhalos(e.g.,Maoetal. populationofentirelydarksatellites(seee.g.Kravtsovetal.2004; 2004).Itisalsohardtoreproducetheobservedstatisticsoncuspvi- Mooreetal.2006). olationswithsubstructure (Maccio` &Miranda2006).Thisargues Instronglensing,ithasbeenknownforsomeyearsthatsim- againstsubstructureasaprimarycauseofanomalousfluxratios. ple,smoothmodelsofgalaxylensesusuallyfittedtheimageposi- tionswell,butthefluxratiosoftheimagespoorly.Inaboldpaper, Nonetheless,thereisgoodevidenceinfavourofsubstructure, Dalal&Kochanek(2002)arguedthatthefluxanomaliesingravi- and, insomecases, visiblesubstructure can beidentified. Oneof tationallenssystemscouldbeinterpretedasevidenceforentirely thegravitationallenssystemswithafluxratioanomalyistheradio- loudquadrupleCLASSB2045+265discoveredbyFassnachtetal. (1999).RecentdeepHubbleSpaceTelescopeandKeckimagingof ⋆ E-mail:[email protected];[email protected] thissystembyMcKeanetal.(2007)haverevealedthepresenceof (cid:13)c 2007RAS atidallydisrupteddwarfgalaxyG2.Thismaybethecauseofthe Thecorresponding dimensionless deflection potential isthe solu- flux ratio anomaly, although caution is needed as modelling sug- tionofthePoissonEquation geststhat G2must be very highly flattened(q = 0.13). There is 1 alsoevidenceforvisiblestructure–possiblyasmallgalaxy–inthe κ(x,y)= 2ψ (2) 2∇ radio-loud quadruple MG0414+0534. Schechter&Moore (1993) andthedimensionlessbendingangleisα= ψ. alreadyarguedthattheperturbationcausedbythisobjectmayac- ∇ countfortherelativelypooragreementbetweentheobserveddata onthislensandthetheoreticalmodels.Thequadruplelenssystems 2.1.1 PrimaryLens CLASS B1608+656 (see e.g., Fassnachtetal. 1996) has a loose groupofgalaxiesatthesameredshiftasthemainlensinggalaxy, Weexaminetwomodelsforthemainlensgalaxy.Thefirstisap- aphenomenonthatalsooccursinthesix-imageradio-loudsystem propriate for an elliptical galaxy lens. It is a pseudo-isothermal B1359+154 (Rusinetal. 2001). It seems that visiblesubstructure elliptic deflection potential (see e.g. Kassiola&Kovner 1993; maywellberesponsibleforsomeofthefluxratioanomalies. Hunter&Evans2001;Evans&Hunter2002) In addition, there have been striking observational develop- ψ(x,y)=E r2+(1 ǫ)x2+(1+ǫ)y2 1/2 (3) ments nearer to home. The last two years have seen the discov- r c − ery of 10 faint, new Milky Way satellites in data from the Sloan wherer isadìmensionlesscoreradius(the´lengthscalebeingthe c DigitalSkySurvey(SDSS,seeWillmanetal.2005,Zuckeretal. arbitrarilychosenξ ),and 0 2006a,b;Belokurovetal.2006,2007).Itseemslikelythatapopu- E =σ2(ξ GΣ )−1 (4) lationofultra-faint,dwarfgalaxiesdoessurroundtheMilkyWay. r 0 cr These may be representatives of the “missing satellites”, as pre- is the dimensionless Einstein-ring radius of the singular isother- dictedbythesimulationsofKlypinetal.(1999)andMooreetal. malspherewithline-of-sightvelocitydispersionσ corresponding (1999),ortheymaybeapopulationoftidaldwarfgalaxiesoreven tother = 0, ǫ = 0case.Therearetwocriticalcurves:asmall c starclusters(seee.g.,Belokurovetal.2007).Suchultra-faintob- inner one which maps to a ‘radial’ caustic and an outer ‘tangen- jects are only detectable nearby, and so would be – to all intents tial’one whichmaps toan astroidcaustic. At theranges of ǫwe andpurposes–darkatthetypicalredshiftsofstronglenses. consider, theastroid causticiswhollywithintheouter caustic. A All this suggests that it is worth re-examining the effects of pointsourcehasoneimageifitisoutsidebothcaustics,threeifit substructureonstronglenses.Inthispaper,weanswerthefollow- isinsidetheoutercaustic,andfiveifitisinsidetheastroidcaus- ingquestions.Givenwhatweknowaboutthesatellitepopulations, tic.Tripletsandquintuplets,however,areeffectivelydoubletsand how frequently might we expect anomalous flux ratios for ellip- quadruplets, because one of the multiple images is a highly de- tical and spiral galaxies? If luminous satellite galaxies represent magnifiedcentralimagewithinthesmallinner criticalcurve. For the bright and massive end of a predominantly faint population concreteness, we consider the elliptical potential to have (unless of objects, how frequently might weexpect toattributeflux ratio otherwisespecified)avelocitydispersionσ = 250kms−1 anda anomaliestovisibleobjects?Three(B2045+265,MG0414+0534, coreradiusof100pc,assuggested byKassiola&Kovner (1993) B1608+656) outofthesampleofsixquadrupletsoriginallyiden- andEvans&Hunter(2002).Werestricttheellipticityparameterto tified by Dalal&Kochanek (2002) as anomalous have optically besmallerthanǫ 0.2,otherwisethecorrespondingsurfacemass identifiedcompanions thatarepossiblecauses.Attheoutset,this density becomes≈dumbbell-shaped, which is inappropriate for el- seemssurprisinglyhigh,ifthesubstructureispredominantlydark. lipticalgalaxies(seeKassiola&Kovner1993). The paper is arranged asfollows. In 2, models of elliptical Thesecondmodelisappropriateforaspiralgalaxylens.Itisa § andspiralgalaxiesarebrieflyintroduced,togetherwiththeirsatel- three-componentmodelwidelyusedingalacticastronomy(seee.g. litegalaxypopulations.In 3,thefrequencywithwhichanomalous Dinescuetal.1999,Fellhaueretal.2006)asamodelfortheMilky § fluxratiosoccur isshown todepend primarilyonthescalelength Way. It has a Hernquist (1990) bulge, a Miyamoto-Nagai (1975) ofthesatellitedistribution,themassmodelusedfor thesatellites discandacoredisothermalhalo.TheHernquistbulgehas3Dmass andthetotalmassinsubstructure,whiledependingonlyweaklyon distribution howthemassisdistributedbetweensatellites.Thesimulationsre- M r portedin 4givetheexpectedfractionofanomalousfluxratiosfor ρ(rˆ)= b b , (5) bothellipt§icalsandspirallenses,togetherwiththetypicalnumbers 2π rˆ(rˆ+rb)3 causedbyhighmassandluminoussatellitegalaxies. whererîsthesphericalpolarradiusandrb acoreradius.Thatof thehalois ρ ρ(rˆ)= c . (6) 1+rˆ2/r2 h 2 METHODOLOGY where r is the core radius and ρ the central density. The mass h c 2.1 MassModels distributionoftheMiyamoto-Nagaidisciscomplicated,butthede- flectionpotentialintheedge-oncaseissimple: Forconvenience, wefollowSchneideretal.(1992)bydefiningin the lens plane the dimensionless distance, dimensionless surface ψ = 1m log x2+(a+ b2+y2)2 , (7) massdensityandcriticalsurfacemassdensity d 2 d h p i Rˆ Σ(ξ R) c2D where md = Md/(πΣcrξ02) is the dimensionless mass and a R= , κ(R)= 0 , Σ = s , (1) andbcontroltheshapeofthedistribution.Wenormalizeourdisc ξ Σ cr 4πGDD 0 cr l ls galaxylenstotheMilkyWay,accordingtotheparametersgivenin withD , D, D beingthedistancestothesource,lens,andbe- Shin&Evans(2007).Discgalaxiesgiverisetothreemaindifferent s l ls tweenlensandsource, andξ anarbitraryscalelengthwhichre- classesof multiple-image configurations (see e.g.Mo¨ller&Blain 0 lates the scaled (uncapped) coordinates to the unscaled (capped). 1998): ‘core triplets’ (in effect, doublets), ‘disc triplets’ (where 2 images straddle the plane of the disc) and quintuplets (in effect, 0.03 quadruplets).Thesmall7-imagingbutterflycuspinthecausticof this edge-on Milky Way is ignored here (see e.g. Shin&Evans ) 1 2007). –c p Wechoosetheredshiftofthelenstobe0.46andthatofthe k 0.02 ( sourcetobe2.15.Thesearethemedianredshiftsofknown4-image ) R lenssystems(seetheCASTLESwebsite),omittingthoseknownto ( n havemorethanone mainlensandthose without known lensand 1 – sourceredshifts.WeuseaflatΛCDMcosmologywithΩm =0.27, N 0.01 Ω =0.73,H =71kms−1Mpc−1. R Λ 0 (cid:652) 2 2.1.2 PlummerModelSatellites 0 10 20 30 40 50 60 70 The Plummer model is often fitted to observed dwarf spheroidal galaxies (see e.g. McConnachie&Irwin 2006, Wilkinsonetal R (kpc) 2002).Wegiveoursatellitesdensities Figure1.Probabilitydensityofsatelliteswithrespecttothelens-planepo- κ(r )=κ(k) 1+λ2r2 −2 (8) larradius R.Thesolidline is ford = 100kpc,thedashed line for k 0 k k 1/2 d =50kpc. 1/2 where r = ` x2+y2´, x and y being Cartesian coordinates k k k k k withtheiroriginatthecentreofthekthsatellitegalaxy,κ(k)isthe p 0 centraldensityofthatgalaxy,andλk = ξ0/rp(k) whererp(k)isthe M(k) (1+rˆ /r(k))−3, rˆ<rˆ(k) Plummermodelscaleradius.Themassofanysatelliteis ρ(rˆk)= 2πrs(k)2rˆk k s t (13) ( 0, rˆ>rˆ(k), M(k) =Σ κ(k)πr(k)2. (9) t p cr 0 p whererˆ istheradialdistancefromthecentreofthekthsatellite, Wegivea‘typical’107M⊙Plummersatelliteascaleradius(equal M(k)arketheuntruncatedHernquistmassesandr(k)areHernquist to its half-light radius) of 140pc, which is the median half-light s scalelengths. radiusofknownMilkyWaysatellitespheroidal galaxies(seee.g. Thescalelengthofeachsatelliteischosensothatthelocation Belokurovetal. 2007), and vary r as M . This mass-radius p p ofthepeakcircularvelocityofthemodel(whichisr )varieswith scaling is consistent with those found in N-body simulations, s p themassboundwithinthetidalradiusM Mr2/(r2+r2)as wherethesizeofsatellitesscalesasapowerofthemass.Wealso t ≡ t t s putalowerlimitonrp of70pc,thesmallestknownhalf-lightra- rs=10−4 Mt/M⊙ kpc (14) diusofaMilky-Waydwarfspheroidal(Belokurovetal.2007). inagreemenptwiththenumericalsimulationsof,forexample,Die- mand,Kuhlen&Madau(2007). 2.1.3 HernquistModelSatellites IfthetidalradiusofanyHernquistsatelliteislessthanitsscale radius,werejectitasbeingtoostronglytidallydisruptedtobeap- We also examine a second density profile for the satellites, proximated by this mass profile. For example, Metcalf&Madau motivated by numerical simulations. The NFW density pro- (2001)findthatNFWclumpslosetheirmassrapidlyforr < r . file(Navarroetal.1996) t s Itisbeyondthescopeofthispapertoconsiderlensingeffectsdue rˆ −1 rˆ −2 tothetidally-strippedmatterfromdwarfgalaxies. ρ =ρ 1+ , (10) NFW s r r „ s« „ s« isfoundtobeagoodfittocolddarkmattersubhalos.Ratherthan usingtheNFW,whichfallsoffasrˆ−3 atlargeradiiandtherefore 2.1.4 SpatialDistributionofSatellites hasformallyinfinitetotalmass,weusetheHernquistdensitypro- file,whichissimilartotheNFWprofile,buthasanrˆ−4asymptotic The distribution of satellite galaxies is taken as spherically sym- density decay. Our Hernquist satellitesareless concentrated than metric.Explicitly, weassume that thenumber densityin thelens thePlummers,andwemakealowestorderestimateoftheeffects planeis oftidalstrippingbytruncatingthemattidalradiirˆ(k) definedby t n(R) (R2+r2)−m/2. (15) thecondition ∝ d ρ¯(k)(rˆ(k))=ρ¯(rˆ) (11) wherem>1determinestheasymptoticfall-offofthedistribution, t andr isacentralsofteningparameter.Wechoosem=5/2sothat d where rîs the radial distance of the kth satellitefrom the centre thethree-dimensionalnumberdensity,obtainedbyAbeldeprojec- of the main galaxy, ρ¯(k)(rˆt(k)) its mean mass density, and ρ¯(rˆ) tionof(15),fallsofflike(distance)−3.5,similartothebehaviour the mean mass density due to the main galaxy enclosed withina observed in the Milky Way (see e.g., Wilkinson&Evans 1999). sphereofradiusrˆwhenthepseudo-isothermal ellipticalpotential Wechoosetwodifferentr ,sothathalfthesatellitesarewithina d isapproximatedbyasingularisothermalsphere,thatis, (sphericalpolar)radiusd of100kpcor50kpcrespectivelyof 1/2 ρ¯(rˆ)= 4πrˆ3 −1 rˆ4πrˆ′2 σ2 drˆ′ = 3σ2 rˆ−2. (12) Nthoetceetnhtarte,oatfltehaesmtaasinjugdagleadxyf.roTmhetdhiestcraisbeutoiofnthseaMreiplkloyttWedayin,wFiege.x1-. 3 2πGrˆ′2 2πG „ « Z0 pecthalfthesatellitestoliewithin100kpc,andsod1/2 ∼100kpc ThetruncatedHernquistprofileisthen isperhapsthemorerealistic. 3 ratherthanN or M .ThesolidlineinFig.2suggeststhatsatel- cted 100 litesofmass109M⊙p areonly50%moreefficientatalteringflux affe 80 ratios than those of mass 107M⊙. The Plummer scale radius rp, ms if set differently, can affect the results. For example, fixing rp to ste 60 140pcforallMpcausesFifalloffrapidlybelowMp =107M⊙, y asthecentraldensityofthesatellitesdecreases. s of 40 Fig. 2 can be understood qualitatively as follows. The mag- e ag nification of an image is changed by > 5% – and theflux ratios ent 20 of the lens system affected – if the image is within a ‘radius of c Per 0 influence’reff ofasatellite.Here,reff dependsonthelocationand magnificationoftheimage,aswellasthemassofthesatellite.(r 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 eff increaseswiththemassofthesatelliteandthemagnificationofthe LogLo10g 1(0M [aMsps /p Mer(cid:130) d]warf) image. Note that there can be an ambiguity inr . Asa satellite eff movesclosertotheimage,theimagemaybefirstslightlybright- Figure2.TheproportionsF4ofquad(circles)andF2ofdoublet(squares) ened, before dimming rapidly as the satellite approaches, or first lenssystemsthathaveafluxratiochangedby>5%byPlummersatellite slightlydimmedbeforebrighteningrapidly.So,foralargeenough galaxies, asafunctionofthemasspersatellite galaxy.Thetotalmassin satellite, there may be two ranges in which the flux changes by satellites,Msat,isfixedat5×109M⊙andthescalelengthofthedistribu- tionatd = 100kpc(solidpointsandline)andd =50kpc(open > 5%.Forthepurposeofourroughargumenthere,however,we 1/2 1/2 points anddotted line). Thelens isapseudo-isothermal elliptic potential ignorethiscomplication.)Foragivenconfiguration,theprobability withǫ = 0.1.NoteespeciallythatF4 isonlyweaklydependentonhow thatthefluxratioischangedappreciablybysatellitesdependsnot themassissharedbetweensatellites.Theerrorbarsare2σofabinomial only onthe number of satellitesand their probability distribution distribution. in space, but on the individual values of r . Even so, ensemble eff averaging over many imageconfigurations of the same type(e.g. quads)yieldsanr thatdependsonlyonthemassofthesatellite. 2.2 NumericalDetails eff Thefractionsofquadsanddoubletsthatareaffectedbysatel- Foragivensourceposition,thelensequationcanbesolvednumer- litesdependonthefractionalareaA ofthelensplanethatlies sat ically by first triangulating the image plane, mapping the grid to withinthecirclesofinfluence.Thatis, thesourceplanetotriangulatetheimagepositions,andthenusing 1 F (1 A )4 F 4A 6A2 , amulti-dimensionalNewton-Raphsonprocedure. − 4 ≈ − sat ⇒ 4 ≈ sat− sat Thesatellitegalaxieshavesmalleffectsonthepositionsofthe 1 F (1 A )2 F 2A A2 , (16) − 2 ≈ − sat ⇒ 2 ≈ sat− sat imagesand oncriticalcurves andcaustics. Their effectismainly where r ,A are different for quads and doublets. If A is ontheratiosbetweenimagefluxes.Whenasatellitegalaxyissuf- eff sat sat small,thecirclesofinfluencedonotoverlap,so ficiently near an image, it changes the magnification of that image. Onlyif theseparation betweenthesatelliteandtheimage is N stillsmallerdoesimagesplittingoccur.Inthispaper,wefocuson Asat ∝ πre(iff)2. (17) the effect of the satellites on the flux ratios of lenses rather than i=1 X theimagemultiplicity.Whenimagessplit,theyarenearnewcriti- ThegentlyincreasingF inFig.2overarangeofM reflectsa i p calcurvescreatedbythesatellitegalaxyandarehighlymagnified, dependencer2 Mpforp&1.Ford =50kpc,thepropor- eff ∝ p 1/2 whichcertainlyresultsinafluxratiochange. tionalincreaseinF4between,say,107M⊙and109M⊙(seedotted Wegeneratepositionsofsourcesrandomlytofind1000five- lineinFig.2)islessthanford = 100kpcbecauseofgreater 1/2 imageand1000three-imagesystems.Foreachsystem,theimage overlapping(A increasesmoreslowlythan N πr(i)2).Dou- positions and magnifications are found numerically, both for the sat i=1 eff bletsaremuchlessstronglyaffectedthanquadsnotonlybecause maingalaxyaloneandwithsatellitegalaxiesatpositionsrandomly P thereareonlytworather thanfour imageswhosefluxescouldbe generatedaccordingtothedistribution(15)(orits3Ddeprojection, affected(hencethe2A ratherthan4A termin(16)),but,more sat sat for the purposes of finding tidal radii). We count the number of importantly,becausetheirimagesaretypicallyofmuchlowermag- systems where the satellite galaxies change the ratio of any two nification. imagefluxesby5%ormore(discountingtheusuallyunobservable Letusnow allowthetotalmassM insatellitegalaxiesto sat centralimages). vary. Since F are not quite independent of how M is appor- i sat tionedbetweensatellites,weneedtoallowfordifferentindividual satellitemasses.WedrawthemfromanM−2distributionwithcut- 3 THETOTALMASSINSATELLITES offsat5.0 106M⊙and5.0 109M⊙,andvaryN tovaryMsat. × × (Cutoffmassesofbelow 5.0 106 weretoocomputationallyex- 3.1 PlummerSatellites × pensivebecausemanymoresatelliteswouldhavebeenneededfor Forthemoment,letusfixthetotalmassinsatellitegalaxiesM the same total masses.) The masses of the satellites (along with sat as 5 109M⊙ and share it equally among a varying number N theirpositions)areregeneratedforeachnewsourceposition.The × of satellites. The lens galaxy is an ǫ = 0.1 pseudo-isothermal resultsareshowninFig.3,whereF areplottedasafunctionM i p ellipticalpotential,and thesatellitesarespatiallydistributedwith onalog-logscale.Thescalings(16)and(17)canbeseen:F ini- i d =100kpc.TheresultsareshowninFig.2,whichillustrates tiallyincrease.linearlywithM (asA N M fora 1/2 sat sat ∝ ∝ sat that, over a large range of M or N, the proportion F of quads givensatellitemassfunction),untiltheoverlapsbetweencirclesof p 4 withfluxratiosaffectedbythesatellitesisonlyweaklydependent influencecannolongerbeneglected,afterwhichF approachunity i onhowmassisapportionedbetweenthesatellites.Thatis,thepro- asymptotically.TheproportionsF gofrom20%to80%overabout i portion F is mainly sensitive to the total mass M = NM oneorderofmagnitudeofM (thiscanbeseenevenmoreclearly 4 sat p sat 4 100 50 d e s affected ms affect 40 ge of system 10 e of syste 2300 a g Percent ercenta 10 P 0 1 6.5 7.0 7.5 8.0 8.5 9.0 9.5 8.5 9.0 9.5 10.0 10.5 11.0 11.5 LogL1o0 g(T1o0 t[aMl msaats /s Min (cid:130)dw(cid:3)]arfs) LLoogg101 0[[MMtt // MM(cid:130) ]] Figure 3. The proportions Fi of quads and doublets (upper and lower Figure4.AsFig.2,butwithaninitialtotalmassinHernquistprofilesatel- curves)as functions ofthetotal massMsat inPlummersatellites drawn lites fixed at 1.0×1010M⊙. Fi are plotted against the average tidally fromtheM−2 distribution, onalog-log scale. Here, d = 100kpc; boundmasspertruncatedHernquistsatellite. 1/2 similarbehaviourholdsforotherd (notillustrated).Thedottedlinesare 1/2 ofunitgradient:whenthepercentageofaffectedsystemsislow,Fiscales as.Msat. 100 d e ct fromFig.6inthenextsection). AtallMsat,fluxratiosofquads affe aremuchmorelikelytobeaffectedthanthoseofdoublets. s m e st sy 10 of 3.2 HernquistSatellites e g a nt SincetheboundmassMt ofaHernquistprofilesatellitedepends, erce through(11)and(14)onitsinitialmassM anditsdistancefrom P themaingalaxy(generatedrandomlyinthissimulation),wecannot 1 repeat 3.1exactly.Wecanonlyfixthetotalinitialmass,apportion- 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 ingite§qually betweenthevarying number of satellitesN. More- Log10L (oTogt1a0l [tMidasla mt /a Mss (cid:130)in(cid:3)] dwarfs) massivesatellites,whicharethereforelessconcentrated,aremore pronetotidaldisruption:ford =100kpc,atotalinitialmassof Figure 5. The proportions Fi of quads and doublets (upper and lower 1/2 1.0 1010M⊙resultsinatotaltidallyboundmassof 7 109M⊙ curves) as functions of tidal mass Msat in Hernquist dwarfs, d1/2 = × ∼ × 100kpc.Thedottedlinesareofunitgradient. ifsharedequallybetween1000dwarfsofinitialmass107M⊙,and 6 109M⊙ ifsharedbetween10ofinitialmass109M⊙.For ∼d1/2×=50kpcthesefallto∼6.3×109M⊙and∼5.7×109M⊙ asignificantprobabilityofflux-ratiochanges.WefirstusePlummer respectively. Even more significantly, the most massive satellites model satellites, drawn from the M−2 mass function, and again cannot stay bound close-in to the main galaxy (whereas smaller varythenumberofsatellitestovarythetotalmass.Theresultsare satellitescan),decreasingtheirchancesoflyingnearthecentreof plottedinFigure6. thelensinprojection,wheretheimagestypicallyare.Thisisdra- Forthemoderatelyellipticcaseǫ=0.1,massesof 1.25 pmoarttiicoanlloyfisllyussttermatsedwiinthFailgt.er4e,dwflhuexreraftoirosd1o/n2ly=va1ri0e0skwpecakthlyewprioth- 1010M⊙ are needed for 50% of quadruplets to show a∼nomalou×s flux ratios if d = 100kpc, decreasing to 5 109 for aNre,mwuhcehremasofroeredffi1c/i2en=ta5t0chkapncgisnagtefllluitxeesswthiathnmMatss.ive5o×ne1s.0H7Mow⊙- d1/2 = 50kpc1./T2he lengthscale of the satellite d∼istrib×ution d1/2 hasalargeeffectonbothF andF .Asexpected,thehigherthe ever,thisresultdependsontheassumptionthatasatellitewhichis 4 2 probabilitydensityofsatellitesintheinnerpartsofthelensplane highlytidallydisrupted(r < r )canbeignored,onthegrounds t s (where the images lie), the greater the proportion of anomalous thatitsmasssurfacedensityissodiffusethatithaslittleeffecton flux ratio systems. Comparing left and right panels, we see that thelensingfluxes. theeffectofsatellitesonfluxratiosofquadrupletsdecreasesasǫ Drawing initial masses from the same M−2 distribution as increases – and the quadruplet cross-section increases, while the before,andvaryingN tovarythetotalboundmassM ,weob- sat meanmagnificationofquadsfalls.TheproportionofdoubletsF , tainFig.5,whichistheanalogueofFig.3.TheHernquistprofile 2 however,isnotnoticeablyaffectedbytheellipticityǫ. satellites,whicharemoreextendedanddiffusethanthePlummer Changingtheredshiftsoflensandsourceaffectstheseresults models,arelessefficientatalteringfluxes. onlythroughΣ ,onwhichE ineq(4)andκ(k)ineq(8)depend. cr r 0 F fortwosetsofredshiftsareplottedinFig.7,ford =50kpc i 1/2 andǫ=0.1.Weseethattheresultsarenotverysensitivetodiffer- 4 ASTROPHYSICALAPPLICATIONS entredshiftsordifferentΣ .Theresultsarealsonotverysensitive cr to the core radius of the pseudo-isothermal elliptic potential, al- 4.1 EllipticalGalaxyLenses thoughtheyareaffectedbyitsvelocitydispersionσ,asshownin Letusconsiderseveraldifferentsetsofparametersofellipticalpo- Fig.8.WerecallthatthevelocitydispersioncontrolstheEinstein tentials(3)andaskwhatmassinsatellitesisrequiredfortheretobe radiusE = σ2(ξ GΣ )−1,withmoremassivelensgalaxiesdi- r 0 cr 5 100 d(cid:72)1 /2= =0 . 11 00kpc 100 d(cid:72)1 /2= =0 . 2100kpc ed 100 ms affected 6800 ms affected 6800 ms affect 80 Percentage of syste 2400 Percentage of syste 2400 ntage of syste 246000 0 0 e 8.5 9.0 Lo9g.510 [ M10sa.0t / M10(cid:130).5] 11.0 11.5 8.5 9.0 Lo9g.510 [ M10sa.t0 / M10(cid:130).5] 11.0 11.5 Perc 0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 Log10L [oMg1sa0t /( M )(cid:130)] 100 d1/2 = 50kpc 100 d1/2 = 50kpc (cid:72) = 0.1 (cid:72) = 0.2 Percentage of systems affected 24680000 Percentage of systems affected 24680000 σFmr(edidoogs=dtutheerildef2tssl80ainzt.0eelPlk)l=.eimtTrec0hseg.e−n4atr61laeagmax(enidasedaisnso,zhifnsaegsds=ypalsai2tnfre.uaem1mn)5,sc,et2tiwfeo5orni0rsthtkaohrmfrfleeuteǫhsx−ed=ir1ftafo0teit(.oras1elsonlamtiandfvadfeseldliscon1tceei/ni)d2t,ysb3=ady0tie0s1lPpl0kielt0umresmski,ospm−nfcoes1.rr: lFoadefiifspgtttsouroei0trbfua8e.uld5etao6mico-.9aihn.Ps0solpshoLtahaoMt9ngs.se51e0ros al[m a.fM1tc0asF.ha0lti /ainMer1(alP(cid:130)0cl.c5]il iputrectm1ril1cie.ms0stpiec1for1ot.5sedrcnwqatuialaerallfdeswsn,,gfistothhqruveoa0laf8rlr.ie5dipsot1uif/c9os.2i0rtpyLadaos9orgǫ.5a1ui,0 mn[b wMd1lee0ishat.ctt0e is/al rM)ets1e0.(cid:130)adt.5]shTeihfn1ue1s.n0talhcteeet1ni1losl.t5iontipess Percentage of systems affected 12468000000 d(cid:72)1 /2= =0 . 11 00kpc Percentage of systems affected 12468000000 d(cid:72)1 /2= =0 . 2100kpc 0 0 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 100 Log10 [ Msat / M(cid:130)] Log10 [ Msat / M(cid:130)] d e ct e aff 80 ems 60 100 d(cid:72)1 /2= =0 . 15 0kpc 100 d(cid:72)1 /2= =0 . 250kpc Percentage of syst 240008.5 9.0 9.5 10.0 10.5 11.0 11.5 Percentage of systems affected 24680000 Percentage of systems affected 24680000 Log10L [oMg1sa0t /( M )(cid:130)] 08.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 08.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 Log10 [ Msat /M(cid:130)] Log10 [ Msat / M(cid:130)] Figure7.ProportionofsystemswithfluxratiosaffectedbyPlummermodel Figure9.EffectofHernquistsatellitesonfluxratios(analoguetoFig.6). satellite galaxies, asafunction oftotal massinsatellites, fortwosetsof redshifts.Filledpointsandsolidline:zl=0.46,zs=2.15(corresponding toΣcr=2.04×109M⊙kpc−2).Openpointsanddottedline:zl=0.21, initial masses of the Hernquist dwarfs are drawn from the same zs=1.22(correspondingtoΣcr=3.05×109M⊙kpc−2).Thefirstset M−2distribution,andFiareplottedagainstthetotaltidalmassin is the median lens- and source-redshifts of known 4-image systems; the satellites.Fortheǫ = 0.1ellipticalpotential,ifd = 100kpc, 1/2 secondzlandzsareonestandarddeviationbelowthemedian. some4.5 1010M⊙insatellitesisneededfor50%ofquadruplet × fluxratiostobealtered(overthreetimesthemassinPlummersatel- lutingtheeffectofthesatellitesonfluxratios.Apseudo-isothermal litesneeded),droppingonlyto2×1010ford1/2 =50kpc(about ellipticpotentialwithσ=300kms−1requires 2timesasmuch eighttimesthemassinPlummersatellitesneeded)becauseofthe massinsatellitegalaxiesasaσ=200kms−1m∼odelforF =0.5, increasedtidalstrippingofcloser-insatellites. 4 or 50%of thequads tohaveanomalous fluxratios.However, the Einstein radius of the main galaxy goes as σ2, and the projected 4.2 SpiralGalaxyLenses mass within the Einstein ring as σ4, so a given relative mass in satellitecompanionsismorelikelytoaffectfluxratiosinmoremas- Nowletuschangethelensgalaxytoanedge-onspiral,usingthe siveellipticallenses. modelbasedoneqns(5)-(7).PlotsofF asfunctionsofM ,for i sat Whenthelessconcentrated,tidally-strippedHernquistprofile Plummermodelsatellites,areplottedinFig.10assolidlinesalong- satellites are used instead of Plummer models, the effect on flux sidethoseforapseudo-isothermalellipticpotentialasdottedlines. ratiosissignificantlyweaker:Fig.9istheanalogueofFig.6.The The upper panel shows the model in which the disc is maximal 6 This makes sense, as early-type galaxies are more massive than 100 ed late-type. If instead we were to carry out the comparison using s affect 80 σcom=pavr0a/b√le2,,thsoenthtahtetheellikpitniceamlawtiocupldrohpaevrteieasolofwtheer mveolodceiltsywdeirse- em 60 persion andso thediscrepancy between theeffectsof spiralsand yst ellipticalswouldbeincreased(seeFigure8). s of 40 e g nta 20 4.3 VisibleSubstructure e c er Some flux anomalies appear to be attributable to single dwarf P 0 galaxies. The most obvious examples are the Sixth Ob- 8.5 9 9.5 10 10.5 11 11.5 Log10[ToLtoagl 1m0 a[Msss aitn / dMw(cid:130)a]rfs / M ] jCeLctASinSBM2G004451+42+60553(4Mc(KScehanecehttearl.&20M07o)o.rIen1b9o9th3)theasnedcaGs2es,ina singlepieceofsubstructuregivesasubstantialimprovementinthe 100 ed fitofasmoothmodelofthelensgalaxy.Aninterestingquestionis affect 80 –weifeflxupxecrtattioosaeneomaavliiseisblaerecoduunetetorpsaurbt?strAutcttuhreet–yphiocwaloreftdesnhimftisgohft s em 60 thelenses,onlytheverylargestdwarfgalaxiescanbedetectedwith yst ground-basedtelescopes,ofcourse. s of 40 This question can be answered by simulations in which the age satellitesare divided into twopopulations –N1 dark satellitesof ercent 20 laonwdflmuaxs-sraMtio1cahnadngNes2abrreigshotugsahtteflolirtethseotfwhoigphopmualastsioMns2se≫parMat1ely– P 0 andtogether.Somesampleresultsofthesimulationsareshownin 8.5 9 9.5 10 10.5 11 11.5 Table1fortheusualǫ = 0.1pseudo-isothermal ellipticpotential Log10[ToLtoagl 1m0 a[Msss aitn / dMw(cid:130)a]rfs / M ] withd = 100kpc.Notethat,forthePlummersatellites,when 1/2 theratioofthetotalmassinpopulation1tototalmassinpopula- Figure10.TheupperpanelshowsFiasfunctionsofthetotalmassinPlum- tion2is1:1,thenthenumberofsystemswithanomalousfluxratios mersatellites,foranedge-onMilkyWaywithmaximaldisc(solidlinesand causedbypopulation1ascomparedtopopulation2isroughlyin filledsymbols)andapseudo-isothermal elliptic potential forcomparison theratioof2:3,anda3:2massratiogivesaroughly1:1effect.For (dottedlinesandopensymbols)withσ = 250kms−1.Thelowerpanel theHernquistsatellites,a1:1massratiogives1:1effect.Thiscon- showsthesamequantities butforasub-maximaldisc. Bothgalaxies are atzl = 0.46,lensingsourcesatzs = 2.15.d1/2 = 50kpc.Quadsare sfiprhmesrothidealobisseornvlaytions5o0f%§3mtohraeteaffi1c0i9eMnt⊙atPpleurmtumrbeirn-gmfloduexlesdwthaarnf shownascircles,doubletsassquares,anddisctripletsastriangles. ∼ a 107M⊙ one, and more-massive Hernquist dwarfs are no more efficientthanlessmassiveones. SofromTable1,weseethatthe andprovidesmostoftherotational supportintheinner parts,the mostmassivesatellitesdonotcontributeverydisproportionatelyto lowerpanelshowsamodelinwhichthediscissub-maximal(see anomalousfluxratios. Shin&Evans2007foradetaileddiscussionofthelensingproper- Theneat correspondence between ratiosisnotseen for10:1 tiesofthesemodels).Therearethreesolidlinesinthepanels,asthe massratiosbecauseF increasessub-linearlywithmass:e.g.asin- 4 resultsaredivided according tothefiveimage(ineffect,quadru- gle109M⊙ inPlummersatellitechangesfluxesin12%ofquads, plet),coretriplet(ineffect,doublet)anddisctripletmorphologies. butfourofthemchangefluxesin28%,not48%,ofquads.How- Compared to a typical elliptical, the flux ratios of maximal ever, it is confirmed that 85% of systems with affected fluxes ∼ disclensquadsaresignificantlylesssusceptibletochangesthrough stillhaveaffectedfluxesifthe10%ofmassinmassivesatellitesis substructure – almost three times the mass in satellite galaxies removed. is needed to affect half the flux ratios. Spiral doublets, however, The fraction of observed anomalous flux ratio systems with are slightly more susceptible. This is as expected: the greater visible (and therefore high mass) culprits is actually quite high. the asymmetry of the matter distribution, the larger the 4-image Ofthesixfour-imagelenssystemsproposedbyDalal&Kochanek cross-section,thelowerthetypicalmagnificationofquadsandthe (2002), three (B2045+265, MG0414+0534, B1608+656) have smaller the effect of satellite galaxies on the 4-image flux ratios. identified,visiblesubstructurethatmaycausethefluxperturbation. (Sodisctriplets,withtypicalmagnificationsinbetweenquadsand Ifthisdatumistakenatfacevalue,itsuggeststhatabout halfthe doublets,arealsoinbetweeninsusceptibilitytofluxchanges.)In- massinsubstructureisindwarfgalaxieslargeenoughtobeopti- deed,wedonotexpectmanyanomalousfluxratiosindiscgalaxy callyidentified.Simulations,however,tendtofindthatthesatellite lensestobecausedbythesatellitegalaxiesintheirhaloesunless masses behave more like an M−2 distribution. This implies that these satellitestotal at least & 1010M⊙ inmass, that is, 10% thereisequalmassinequaldecades,andthereforethateachdecade ≈ oftheluminousgalaxymass!(Andthatassumesthesatellitesare isresponsibleforthecausingroughlythesamenumberoffluxra- distributedwithd1/2 = 50kpc.)Onmovingtothesub-maximal tioanomalies(the109M⊙decadecausingonly50%morethanthe case, the 4-image and disc triplet cross sections are substantially 107M⊙decade). reduced,andsothenumbersreturntowardstheirvaluesintheel- One resolution of this difficulty is to postulate that the visi- lipticalgalaxycase. ble dwarf has been mistakenly designated as the culprit and that Notethatwehavecomparedtheeffectsofatypicalspiralwith theanomalousfluxratioisreallyproducedbyanotherdarksatel- asymptoticcircular speed of v 220kms−1 toatypical ellip- lite. A hint that this may sometimes be the case is given by the 0 tical with line-of-sight velocity ≈dispersion of σ = 250kms−1. unrealisticallylargeflatteningdeduced forsatelliteG2inCLASS 7 Table1.TheproportionF4ofquadswithanomalousfluxratiosforfiverunsofthetwo-populationmodel.Here,thesatellitepopulationshaved1/2=100kpc and the elliptical lensing galaxy has the standard parameters. Population 1 satellites have mass 1.0×107M⊙, while population 2 satellites have mass 1.0×109M⊙.(ForHernquistsatellites,theseareinitialmasses.)ThePlummermodelPopulation2contributestoF4by∼ 50%morethanpopulation1 satellites,whilstthetwoHernquist-modelpopulationscontributeequally. Satellite N1M1 N2M2 F4 F4,1 F4,2 N1M1:N2M2 F4,1:F4,2 Profile (M⊙) (M⊙) (Bothpops) (Pop1only) (Pop2only) Plummer 1.0×1010 1.0×109 43.2% 37.3% 12.2% 10:1 3.1:1 Plummer 5.0×109 5.0×109 45.6% 21.5% 34.5% 1:1 1:1.6 Plummer 6.0×109 4.0×109 44.3% 23.4% 28.1% 3:2 1:1.2 Hernquist 2.0×1010 2.0×109 28.4% 25.1% 5.1% 10:1 4.9:1 Hernquist 1.2×1010 8.0×109 22.8% 16.3% 10.6% 3:2 3.1:2 Hernquist 1.0×1010 1.0×1010 20.9% 13.2% 12.1% 1:1 1.1:1 B2045+265 using model-fittinginMcKeanetal. (2007). Inother Table2.The(average)tidalmassMsat inHernquistsatellites ofuntrun- words,itmaysimplybeachanceeffectthatmanyanomalousflux catedmass109M⊙neededtoaffectfluxesin50%offour-imagesystems, ratiosystemsappeartohavevisibleobjectsatorneartheEinstein for various q. The elliptical galaxy has the standard parameters and the radius.TheprobabilitythatalargedwarfliesclosetotheEinstein scalelengthofthespatialdistributionofsatellitesisd =100kpc. 1/2 radius is easily computed from eq (15). When d = 100kpc, 1/2 thenthereisa10% probabilityofalargedwarflyingwithintwo Satellite N Msat Einstein radii and 3% within one. This still does not seem large Profile (M⊙) enoughtoexplaintheeffect,butcautionisneededastheremaybe Hernquist(dark) 100 5.6×1010 othersuppliesofsubstructurealongthelineofsightforsomeofthe Hernquist(10×concentration) 20 1.5×1010 lenses ingroups and clusters, likeB1359+154. It isworthnoting Hernquist(100×concentration) 12 1.1×1010 thatsimulations(seee.g.Zentneretal.2005)cansometimesyield highlyanisotropic distributionofsubstructure insimulatedhalos: theprojectedsubhalomasswithin10kpccanvarybyfactorof10 dependingonviewingangle.Inthiscase,sightlineswhichproject rs=q10−4 Mt/M⊙, (18) massive satellites onto small radius are also much more likely to whereq = 1p, 1/√10, or 0.1,correspondingtocentraldensities projectother satellitesontosmallradius.Thismaymeanthatour of 1, 10 and 100 times that of the dark (uncontracted) Hernquist computedprobabilitiesof3-10%maybeonthelowside. profile.Thenumberofsatellitesofinitialmass109M⊙ neededto Another possibleresolutionofthedifficultyisthatluminous affectfluxratiosin50%offour-imagesystems(allotherparame- satellites, the baryons in which have cooled and condensed, may tersbeingthesameasinTable1)isshowninTable2.Almostfour bemuchmorecentrallyconcentratedthandarksatellites.Inother timesthetidalmassinmassivedarksatellitesisrequired,compared words, it is possible that luminous satellites would have a much tomassiveluminoussatelliteswith10timesthecentraldensity,to larger effect on flux ratios than their dark brethren because they affectthesameproportionoffluxratios.Raisingthecentraldensity arestructurallydifferent and morecompact. Ofcourse, theeffect afurtherfactorof10hasasmallereffect.Theincreasingeffecton ofbaryonsondarkhaloesissubjecttoconsiderableuncertainties. fluxesseenwithdecreasingr isamplifiedbytheextraresistance s This effect must be small for dwarfs like Draco, with a mass-to- totidaldisruptionofthemore-concentratedsatellites. lightratioof> 350(Kleynaetal.2001),althoughitmaybemore Table3showstheresultsoftwo-populationmodels,inwhich significant for satellites like the Large Magellanic Cloud with a population 1 are dark satellites of initial mass 107M⊙ and pop- mass-to-lightratioof∼5(Alves2004).Itisalsoknownthatsemi- ulation 2 are bright satellites of initial mass 109M⊙, with 10 or analyticcalculationsofgalaxyformationleadtotoomanycompact, 100timesthecentraldensityoftheirdarkbrethren.Thefractionof luminoussatellites,ascomparedtowhatisseenaroundtheMilky anomalousfluxratiosystemscausedbybrightsubstructureisnow Way(seee.g.,Koposovetal.2007).Thebrightsatellitespredicted impressivelyhigh–forexample,thesecondlineofthetabletells by semi-analytic theories are much too concentrated, suggesting usthat42%ofsystemshaveanomalousfluxratios,ofwhich28% thatthiseffectisoverplayedinthemodelling. remainanomalouswhenthedarkpopulationisremoved. Inother Nonetheless, the effect certainly exists at some level and is words,overhalfoftheanomalousfluxratiosystemsarecaused,at worthinvestigating.Baryoncondensationmayincreasethecentral leastinpart,bybrightsatellites.Thisisclosetothestatisticsonob- densitybyafactorofbetween4and160(Gnedin&Zhao2002),al- servedsystems–althoughcautionisneededaslargecompression thoughthelargernumbersareprobablymoreappropriateforgalax- factorslike10or100maywellcausetheimportanceofthiseffect iesliketheMilkyWayratherthansatellitegalaxies.Wechangethe tobeoverestimatedforsatellitegalaxies. lengthscaler oftheHernquistmodeltomimictheresultofbaryon s condensationandquantifytheextraeffect,comparedtodarkHern- quist satellites,that highly-concentrated luminous satellitescould 5 CONCLUSIONS have on flux ratios. (The Plummer satellites are already so com- pactthatchangingthePlummerscaleradiusmakesnoappreciable It remains unclear whether dark matter satellites and substruc- differenceevenwhenthecentraldensityisraisedbyafactor100.) ture are responsible for anomalous flux ratios in strong lensing. TheHernquistdensitylaw(13)meansthatthecentraldensitygoes Dalal&Kochanek (2002) originally studied a sample of 7 radio- asr−2,sowemodifythescalelength-to-massrelation(14)to loudfour-image lenssystemsand claimed evidence of anomalies s 8 Table3.TheproportionF4ofquadswithanomalousfluxratiosforthreerunsofthetwopopulationmodel,withthesameparametersasinTable1butwhere population2satelliteshaveincreasedcentraldensity.Asthecompressionincreases(qdecreases),thetidallyboundmassofpopulation2satellitesincreases, buttheireffectonfluxesincreasesdisproportionately,outstrippingtheeffectofthediffuselow-masspopulation1satellites.Thedifferencecausedbyafactor 10increaseincentraldensityismuchlargerthanthedifferencecausedbyanextrasteptoa100-foldincrease. Satellite N1 N2 Mspaotp1 Mspaotp2 F4 F4,1 F4,2 Profile (M⊙) (M⊙) (M⊙) (M⊙) (Bothpops) (Pop1only) (Pop2only) Hernquist 1200 8 7.4×109 4.5×109 23% 16% 11% Hernquist(Pop2with10×concentration) 1200 8 7.4×109 6.0×109 42% 16% 28% Hernquist(Pop2with100×concentration) 1200 8 7.4×109 7.2×109 48% 16% 36% in 6of them. Thisseemingly suggests that anomalous fluxratios the visible substructure may have been wrongly identified as the areverycommon.Here,wehavecarriedoutatheoreticalstudyof cause, whereasdarksubstructuremaybethetrueculprit.Alarge thefrequencyoffluxratioanomaliesasafunctionoflensinggalaxy dwarfgalaxymaybychancebeprojectedclosetotheEinsteinra- anddarkmattersubstructureparameters. dius,whereasunrelateddarksubstructuremaybethemajorcause Thelikelihoodthatsatellitesaffectfluxratiosinstronglenses of the anomaly. Second, visible satellites may be more concen- dependsontheirmassprofile.Here,weconsideredcompactPlum- tratedthantheirdarkcousins,aphysicaleffectthatmaynaturally merspheresanddiffuse,tidallystrippedHernquistprofilesforour arise from baryon condensation. Compression factors causing an satellites,withthesatellitesizescalingasapowerofmass.Asthe enhancementofthecentraldensitybyafactorof10inbrightsatel- Hernquist satellitesaremoreextended thanthePlummermodels, litesseemtobeampletogiveasatisfactoryexplanationoftheob- they therefore have a smaller effect on image fluxes – typically served statistics. Nonetheless, such high compression factors are about 3 times the mass is needed to generate the same numbers probablyimplausibleexceptforthelargestsatellitegalaxies.This ofanomalousfluxratios. seemstobeinaccordwiththeresultsofMaccio` etal.(2006),who Theprobabilitythatstronglensingfluxratiosareaffectedby foundthatincludingbaryonsinnumericalsimulationsdidnothelp satellites is crucially dependent on their spatial distribution. The in reconciling simulation results with the statistics of anomalous characteristiclengthscaled hasalargeeffectonthefractionof fluxratios. 1/2 lenseswithaffectedfluxratios.Ourspatialdistributionsofsatellite Finally,weremarkthatthelikelihood thatfluxratiosareaf- galaxiesareinspiredbytheobservationaldataontheMilkyWay, fecteddepends on theellipticityof themainlens galaxy, but this for which the satellitenumber density falls off as rˆ−3.5 inthree- dependenceismuchstrongerinquadsthandoublets.Thisisacon- dimensionswithalengthscaleofd 100kpc.Forsuchdistri- sequenceofhighmagnificationimagesbeingmoreeasilyaffected butions,mostsatellitesaretoofaro1u/t2to∼affectthefluxesofimages. by a dwarf galaxy than low magnification ones. Quads are more Forexample,evenwithPlummersatellites,amassof 3 109M⊙ highly magnified than doublets (the disc triplets of spirals are in isneededfor20%ofquadrupletstoshowanomalous∼flux×ratiosfor between),andchangingtheellipticityofthemaingalaxychanges atypicalellipticalgalaxy, risingto 1.25 1010M⊙ for50%. thetypicalmagnificationofquadsmorethanitchangesthatofdou- Lensesthatareedge-onspiralgalax∼ieswith×maximumdiscs(like blets.Generally,thegreatertheellipticitythelesstheeffectofsatel- theMilkyWay)aremoreresistanttofluxchangesbysatellites,so litegalaxies. For example, a given mass Msat of dwarf satellites themassinsatellitesandsubstructurehastoberoughlyafactorof around a typical edge-on maximum-disc spiral galaxy is signifi- 3 times as great for the same proportion of quads to be affected. cantlylesslikelytochange imagefluxratiosthanMsat around a Toobtainanythingliketheapparentabundanceofanomalousflux typicalellipticalgalaxy,eventhoughthespiralislessmassivethan ratios,thenthescalelengthofthesubstructurehastobedifferentto theelliptical. whatisknownfortheMilkyWaysatellites. Whetherthefluxratiosinalenssystemareaffectedbysatel- litesissensitivetothetotalmassinsatellites,butmoreweaklyde- ACKNOWLEDGEMENTS pendentofhowthismassisapportionedbetweenthem.ForPlum- We thank the referee for some helpful comments. EMS thanks mermodelsatellites,theprobabilitythatagivensatellitechangesa the Commonwealth Scholarship Commission and the Cambridge fluxratioincreaseswithitsmassonlyslightlyfasterthanlinearly, CommonwealthTrustfortheawardofaStudentship.NWEthanks atleastwhenitsmassisbetween∼ 5×106M⊙ and∼ 109M⊙. NealJacksonforsomeinsightfuldiscussions.Thisworkhasbeen For example, satellites of mass ∼ 109M⊙ are only responsible supportedbytheANGLESnetwork. for the causing 50% more flux ratio anomalies than those of ∼ mass 107M⊙. For Hernquist model satellites, more massive ∼ ones seem no more efficient (per unit mass) at changing fluxes; REFERENCES indeed,moremassivedwarfs,beingmorepronetotidaldisruption, mightevenbelessefficientthanlighter,morecompactones.One AlvesD.R.,2004,ApJ,601,L151 interesting consequence is that, if matter in dark satellites is not BelokurovV.etal.,2006,ApJ,647,L111 predominantly inthemost massive ones, then thecontributionof BelokurovV.etal.,2007,ApJ,654,897 the most massive satellitegalaxies toflux ratioanomalies should Bradac˘M.,SchneiderP.,LombardiM.,SteinmetzM.,Koopmans notbepredominant. L.V.E.,NavarroJ.F.,2004,A&A,423,797 Inthelight ofthis, thefact that somanyanomalous fluxra- DalalN.,KochanekC.S.2002,ApJ,572,25 tios systems have optically identified substructure seems at out- Dinescu D.I., van Altena W.F., Girard T.M., Lo´pez C.E., 1999, set surprising. There seem tobe two possible explanations. First, AJ,117,1792 9 Diemand J., Kuhlen M., Madau P. 2007, ApJ, in press, arXiv:astro-ph/0703337 EfstathiouG.1992,MNRAS,256,43P EvansN.W.,HunterC.2002,ApJ,575,68 EvansN.W.,WittH.J.,2003,MNRAS,345,1351 Fassnacht,C.D.,Womble,D.S.,Neugebauer,G.,Browne,I.W.A., Readhead, A.C.S., Matthews, K., & Pearson, T.J. 1996, ApJ, 460,L103 FassnachtC.D.,etal.,1999,AJ,117,658 FellhauerM.,etal.,2006,ApJ,651,167 GnedinO.Y.,ZhaoH.,2002MNRAS,333,299 HernquistL.,1990,ApJ,356,359 HunterC.,EvansN.W.,2001,ApJ,554,1227 JohnstonK.V.,SpergelD.N.,HernquistL.,1995,ApJ,451,598 KassiolaA.,KovnerI.,1993,ApJ,417,450 Kleyna J.T., Wilkinson M.I., Evans N.W., Gilmore G.F., 2001, ApJ,563,L115 Klypin,A.,Kravtsov,A.V.,Valenzuela,O.,&Prada,F.1999,ApJ, 522,82 KochanekC.S.,DalalN.2004,ApJ,610,69 KoposovS.,etal.2007,ApJ,submitted,(arXiv:0706.2687) Kravtsov, A.V., Gnedin, O.Y., & Klypin, A.A. 2004, ApJ, 609, 482 Maccio`,A.V.,MirandaM.2006,MNRAS,368,599 Maccio`, A.V., Moore, B.,Stadel,J.,&Diemand, J.2006, MN- RAS,366,1529 MaoS.,SchneiderP.,1998,MNRAS,295,587 MaoS.,JingY.,OstrikerJ.P.,WellerJ.2004,ApJ,604,L5 McConnachieA.W.,IrwinM.J.,2006MNRAS365,1263 McKean J.P., Koopmans L.V.E., Flack C.E., Fassnacht C.D., Thompson D., Mathews K., Blandford R., Readhead A.C.S., SoiferB.,2007,MNRAS,378,109 MetcalfR.B.,MadauP,2001,ApJ,563,9 MiyamotoM.,NagaiR.,1975,PASJ,27,533 Mo¨llerO.,BlainA.W.,1998,MNRAS,299,845 Moore,B.,Ghigna,S.,Governato,F.,Lake,G.,Quinn,T.,Stadel, J.,&Tozzi,P.1999,ApJ,524,L19 MooreB.,DiemandJ.,MadauP.,ZempM.,StadelJ.2006,MN- RAS,368,563 Navarro,J.F.,Frenk,C.S.,White,S.D.M.1996,ApJ,462,563 Rusin,D.,etal.2001,ApJ,557,594 SchechterP.L.,MooreC.B.1993,AJ,105,1 Schneider P., Ehlers J., Falco E.E., 1992, Gravitational Lenses, Springer-Verlag,NewYork ShinE.M.,EvansN.W.,2007,MNRAS,374,142 Wilkinson,M.I.,&Evans,N.W.,1999,MNRAS,310,645 Wilkinson M.I., Klenya J, Evans N.W.,GilmoreG., 2002, MN- RAS330,778 WillmanB.,etal.,2005,ApJ626,L85 Zentner, A.R., Kravtsov, A.V., Gnedin, O.Y., & Klypin, A.A., 2005,ApJ629,219 Zucker,D.B.,etal.,2006a,ApJ,643,L103 Zucker,D.B.,etal.,2006b,ApJ,650,L41 10