MNRAS000,1–??(2016) Preprint8November2016 CompiledusingMNRASLATEXstylefilev3.0 Gravitational Lensing by Ring-Like Structures Ethan Lake1(cid:63) and Zheng Zheng1† 1DepartmentofPhysicsandAstronomy,UniversityofUtah,115South1400East,SaltLakeCity,UT84112,USA 8November2016 ABSTRACT 6 Westudyaclassofgravitationallensingsystemsconsistingofaninclinedring/belt,withand 1 0 withoutanaddedpointmassatthecentre.Weshowthatacommonfeatureofsuchsystemsare 2 so-called“pseudo-caustics”,acrosswhichthemagnificationofapointsourcechangesdiscon- tinuouslyandyetremainsfinite.Suchamagnificationchangecanbeassociatedwitheithera v changeinimagemultiplicityorasuddenchangeinthesizeofalensedimage.Theexistence o of pseudo-caustics and the complex interplay between them and the formal caustics (which N correspondtopointsofinfinitemagnification)canleadtointerestingconsequences,suchas 5 truncated or open caustics and a non-conservation of total image parity. The origin of the pseudo-caustics is found to be the non-differentiability of the solutions to the lens equation ] across the ring/belt boundaries, with the pseudo-caustics corresponding to ring/belt bound- P ariesmappedintothesourceplane.Weprovideafewillustrativeexamplestounderstandthe E pseudo-causticfeatures,andinaseparatepaperconsideraspecificastronomicalapplication . h ofourresultstostudymicrolensingbyextrasolarasteroidbelts. p Keywords: gravitationallensing:strong—gravitationallensing:micro—asteroids:general - o r t s a [ 1 INTRODUCTION 1987; Evans & Wilkinson 1998; Rhie 2010) and variants thereof (Wang & Turner 1997; Shin & Evans 2008; Tessore & Metcalf 2 Gravitational lensing has developed from an interesting novelty 2015). These systems possess features called “pseudo-caustics”, v predictedbygeneralrelativity(Einstein1936)toanessentialtool where the number of images of a source can change without the 1 inmodernastrophysics.Ithasbeenappliedwithgreatsuccessto 5 sourcecrossingacaustic. (amongotherthings)detectextrasolarplanets(e.g.Mao&Paczyn- 0 Inthispaper,weexaminelensingsystemsconsistingofcircu- ski1991;Bondetal.2004;Batistaetal.2014),estimatethemasses 3 larringsandbeltsatvariousinclinations,withorwithoutapoint ofgalaxyclusters(e.g.Tysonetal.1998;Bradacˇetal.2009;Hoek- 0 massatthecentre.Wefindthatthesesystemspossessmoregeneral straetal.2013),anddeterminethelarge-scaledistributionofmatter 1. intheUniverse(e.g.Wittmanetal.2000;Vikrametal.2015).In typesofpseudo-caustics,characterizedbyadiscontinuousbutfi- 0 nitechangeinthesourceplanemagnification.Theyareeitherasso- addition,thetheoreticalstudyofgravitationallensingsystemsisof 6 ciatedwithachangeinimagemultiplicityorwithasuddenchange interest in mathematics, namely in catastrophe theory and Morse 1 in the size of an image. These pseudo-caustics produce complex theory (e.g. Erdl & Schneider 1993; Petters 1995; Petters et al. : andinterestingmagnificationpatterns,despitetherelativesimplic- v 2001). ity of the lensing systems in question. We aim to obtain a basic i Onecommongoalofexamininglensingsystemsfromathe- X understandingofsuchlensingsystems.Althoughwemainlyfocus oretical standpoint is the characterization of the magnification of r onthelimitingcaseofasharpbeltwithadiscontinuousmassdis- a source at a given position from the lens mapping. In particu- a tribution, we also analyse smooth mass distributions for a better lar, there exist singularities in the mapping which form caustics understandingoftheoriginofthepseudo-causticsandtheeffectof (curvesorpoints)inthesourceplane,wherethemagnificationis the smoothing on the pseudo-caustics. We show that the pseudo- formally infinite. Studying the behaviour of these caustics allows causticsarisefromdiscontinuitiesinthelensingmassdistribution. forthedeterminationofcertainpropertiesofthelensmapping,and Although our interest here is mainly theoretical and phe- providesinformationaboutthenumberofimagesasourcegener- nomenological, these kind of lensing systems do exist in astron- ates.Normallythenumberofimagesofasourcechangeswhen,and omy,witharelevantexamplebeingtheasteroidbeltaroundastar. onlywhen,thesourcecrossesacausticcurve,withthetotalparity Inarelatedpaper(Lake,Zheng,&Dong2016),aidedbytheresults of the number of images conserved throughout the source plane. presentedhere,westudythemicrolensingsignaturesofextrasolar However,somelensmodelshavebeenfoundtoviolatethisprinci- asteroid belts and explore the possibility of detection with future ple,namelysingularisothermalspheresandellipsoids(e.g.Kovner surveys. Thispaperisorganizedasfollows.InSection2,webrieflyre- (cid:63) Contacte-mail:lake@physics.utah.edu callkeygravitationallensequationsandpresentthesolutionstothe † Contacte-mail:zhengzheng@astro.utah.edu lensingsystemsweconsider.InSection3,weexaminethemagni- (cid:13)c 2016TheAuthors 2 E.LakeandZ.Zheng ficationpatternsandpseudo-causticstructuresforeachofthesys- foralensingsystemcomposedofathincircularbeltwithorwithout temsinconsiderationanddiscusstheoriginofpseudo-caustics.We apointmassatthecentre.Wefirstshowtheface-oncasetobuild summarizeourresultsanddiscussimplicationsinSection4.Sev- anintuitiveunderstandingandthengeneralizetoinclinedbelts.For eralmoretechnicalresultsarepresentedinacollectionofappen- simplicity, we assume the belt to have a uniform surface density. dices. Wechoosethecentreofthebeltinprojectionastheoriginofthe sourceandlensplanes,andinthecaseoftheinclinedbelt,weset thedirectionsofthebelt’smajorandminoraxesinthelensplane asxˆ andyˆ ,respectively. 2 SOLUTIONSTOTHELENSEQUATION I I Inallthegeometriesweconsiderinthispaper,abackgroundsource islensedbyamasslyingbetweenthesourceandtheobserver.We 2.1 Face-onSystems denotetheangularpositionintheimageandsourceplaneas r = I Asafirststep,weconsiderauniformthinringseenface-on,centred (x,y)andr =(x ,y ),respectively.Wewriteallangularvectors I I S S S attheorigin,withradiusa(inunitsoftheEinsteinringradius).This inunitsoftheEinsteinringradius, lensinggeometryhasbeenbrieflystudiedbefore(Schneideretal. (cid:114) 4GM D 1992,p.247),andherewepresentamoredetailedanalysis. θE = c2 DLLDSS, (1) Rayspassingoutsidethering(|rI|>a)experienceadeflection as if the ring were a point mass located at the origin, while rays where D , D , and D are the (angular diameter) distances be- L S LS passingthroughthering(|r| 0)or oppositeside(r <0)ofthelensasthesource. I Σ ≡ c2DS . (5) The generalization from a thin ring to a belt of finite width cr 4πGDLDLS isstraightforward.Onlythemassinsidethecircleintersectingthe For a circularly symmetric lens, Σ is equal to the mean surface impactpointcontributestothedeflection,asifitwereapointmass cr density inside the Einstein ring radius. The surface density κ is atthecentre.Forabeltofconstantsurfacedensitywithinnerand connectedtotheprojectedgravitationallensingpotentialψ(i.e.de- outerradiiaiandao,wehave flectionpotential)throughPoisson’sequation, r (cid:32)r2−a2(cid:33) r =r − I R I i , (10) ∇2rIψ(rI)=2κ(rI). (6) S I |rI|2 a2o−a2i The magnification of the image at the image position r can be whereR(x)isamodifiedrampfunctiondefinedas I cµo(rmI)pu=teddeat1(sA), (7) R(x)= 10x iiifff0xx>(cid:54)<10x.,(cid:54)1, (11) wheredet(A)isthedeterminantofthelensingJacobianA ,asym- Thesolutiontoequation(10)isplottedinFigure1forafixed ij metricmatrixformedfromthederivativesofthelensingdeflection belt mass with different values of the belt width ∆a = a −a. o i α=α xˆ +α yˆ , With a mean radius a = (a +a )/2 = 0.8, the belt has width x I y I c i o ranging from ∆a = 0.1 (dark red) to ∆a = 1.4 (dark blue). The ∂α A =δ − i, (8) middledashedlinecorrespondsto r = r,whichisthesolution ij ij ∂x S I j for the undeflected rays passing interior to the inner edge of the with x = x and x = y. The total magnification for multiple belt(|r| 0 (µ < 0) corresponds to positive (negative) parity of exteriortotheouteredgeofthebelt(|r|>a ).Werefertothetwo i i I o the i-th image, where the lens mapping is orientation-preserving solutionsastheinteriorandexteriorsolution,respectively.Forrays (orientation-reversing). For derivations of these results, see e.g. witha 0.However,ifthebeltisnarrow(high S I Fortheinvestigationinthispaper,wederivethelensequation surfacedensity),∂r /∂r < 0fora < r < a .Thisisbecauseas S I i I o MNRAS000,1–??(2016) GravitationalLensingbyRing-LikeStructures 3 whichInisthdeerciavseedobfyase>ttin1g,w∂reSfi/∂nrdI|mrI=(ra+i )==01. forsmallr nearthe o S S originofthesourceplane.If∆a>(∆a) ,m(r )=1forallr ,asr cr S S S isastrictlymonotonicallyincreasingfunctionofr.If∆a<(∆a) , I cr thebeltlensleadstoanannulusinthesourceplanewherem = 3. Inthelimitofthethinring,thecentralimage(fromrayspassing throughthebelt)disappears,sinceitssizeiszero.Theremaining twoimagescorrespondtoundeflectedrayspassinginteriortothe ringandrayspassingexteriortotheringthatarethendeflectedto theobserver. If a < 1, we always have ∆a < (∆a) , and as a result o cr m(r ) > 1atsomepointinthesourceplane.Intuitively,ifa < 1, S o theentiremassofthebeltlieswithinitsownEinsteinradius(with thebeltshrinkingtoapointmassastheextremecase),andsothe beltcanformmorethanoneimageforsmallr .Wefindthatm=5 S neartheorigininthesourceplaneandm=3inanannularregion surrounding the central m = 5 region, which becomes an m = 1 regionatlarger . S Alltheabovefeaturescanbeunderstoodpreciselybyconsid- eringthebehaviorofthetotalsource-planemagnificationµ(r )of S thelensmapping.Basedonthesolutionofthebelt-onlycase,we seethatasasourcemovesfromr = 0tor (cid:29) 1,µ(r )changes S S S discontinuouslyastheimages(orimpactpoints)crossthebelt.The changeiscausedbythediscontinuityin∂r /∂r at|r| = a ora , Figure1. Solutionstothelensequationforaface-onbeltwithdifferent S I I i o wacuoi)rd/vt2ehss=.sAh0ol.lw8b(teihnletsushnoailtuvsteiooefnqtuhtoaeltEhmeinalssestneasininndrgieneagqcurhaahdtiaiousnsa)i.fmTtheheaenboerulattdewirue(srbeolafrceakpc)la=dcae(sdahibe+yd wnfoahrrircaohwwcbioderelrtebswpeilotthnwd∆siathto<∆aa(∆cha>a)ncr(g∆oerai)ancrs.thhTaehrpneuclmohcabinegrineofitnhiemthsaeogiuemrscamegepflosairnzeea apointmass(withrS =rI−1/rI),whiletheinner(grey)dashedcurveshows associated with the discontinuous change in µ(rS) are defined as thesolutionifthebelthadzeromass,andissimplygivenbyrS =rI (i.e. pseudo-caustics,whichwewilldefinemorepreciselyin§3.Since undeflectedrays).Thesolidcurvesshowthephysicallyrealizedsolutions, jumpsinµ(r )occurwhentheimageofasourcecrossestheinner S withthewidthofthebeltincreasingfromredtoblue.Notethatinthelimit or outer edge of the belt, the pseudo-caustics can be found sim- ofaninfinitesimallythinringthedownward-slopingsectionsdisappear(or ply by mapping circles with radius a and a in the lens (image) i o becomevertical)andthecorrespondingimagesvanish,sincetheybecome plane into the source plane. From this and equation (10), we see zero-sized.Beltswithhighenoughsurfacedensities(redcurves)canform thattheselensingsystemsalwayspossesstwoconcentriccircular 5images,withtwolocatedontheoppositesideofthelensasthesource pseudo-caustics,locatedatsourceradii (withrI <0) (cid:12) (cid:12) (cid:12) 1(cid:12) PS1=ai and PS2=(cid:12)(cid:12)(cid:12)ao− a (cid:12)(cid:12)(cid:12). (13) r increases,thehighsurfacedensitymakesasubstantialincrease o I in the mass inside the radius of the impact point, which causes a Forbeltswith∆a<(∆a) ,whichcanformregionswithm>1,the cr large deflection and allows rays from small r to be deflected to sourceradiir = PS andr = PS definetheboundariesofthe S S 1 S 2 theobserver.Theinteriorandexteriorsolutionsareconnectedby differentimagemultiplicityregionsmentionedpreviously. rays passing through the belt, which complete the final solution. Wenowintroduceapointmasstoourlensingsystem,located In Figure 1, the number of images formed by a point source at a at the centre of the face-on belt. We denote the mass ratio of the radiusofr inthesourceplaneisfoundbycountingthenumberof belt to the point mass as q = M /M and normalize the an- S belt point intersectionpointsofthesolutioncurvewithahorizontallineatr . gular positions by the the Einstein ring radius for the combined S Inthegeneralcase,thenumberofimagesthatcanformwith belt+pointsystem.Thishastheeffectofaddinganextraterm(the a face-on belt lens depends on the configuration of the belt. The lensingcontributionfromthepointmass)tothelensequation,with lensing equation (10) is linear, quadratic, and quadratic for rays equation(10)becoming passinginteriortotheinneredgeofthebelt,throughthebelt,and 1 r q r (cid:32)r2−a2(cid:33) exteriortotheouteredgeofthebelt,respectively.Sowecanhave r =r − I − I R I i . (14) fivesolutionsintotal.Sincenotallthesolutionsarephysicalgiven S I 1+q|rI|2 1+q|rI|2 a2o−a2i theaboveboundaryconditions,weexpectthemaximumnumberof Justlikethebelt-onlylens,thebelt+pointmasslenspossesses imagesformedbythebelttobem =5 max pseudo-caustics which correspond to the inner and outer bound- Let m(r ) denote the number of images formed by a point S aries of the belt mapped into the source plane. In this case, the sourceatalocationr inthesourceplane.Wefindthatingeneral, S pseudo-caustics take the form of two concentric rings at source- widebeltswithlowsurfacedensityhavem(r )=m(r )=1forall S S planeradiiof r ,and∂r /∂r > 0forallr.Fornarrowbeltswithhighsurface S S I I (cid:12) (cid:12) (cid:12) (cid:12) dsmenasliltireSs,,wwiethca∂nrSh/a∂vreIm a .Asawhole,thesolidbluecurvesrepresentthe S I o r.Givenaninnerradiusa,wedenote(a ) asthemaximumouter physicallyrealizedsolution.Thetransitionfromtheinteriorsolu- I i o cr radius of the belt that allows for image-forming pseudo-caustics. tiontotheexteriorsolutionoccurswhentheimagesliewithinthe Weobtain rangeofthebelt,i.e.whena < |r| < a .Sincethebeltisnarrow i I o (cid:115) andao 1; Figure1foracomparison). o,cr 1+q+1/a2 i o,cr The right panel of Figure 2 shows the total magnification a i 1 (cid:20) (cid:18) point source experiences when lensed by the face-on belt+point ao,cr = (cid:112)2q+2 a2i(1+q)+q−1+ a4i(1+q)2 psylasnteem.A,atsboathfusnmctailolnanodfdlairsgtaenrce,rmS =fro2m(steheethoerigleinftipnantheelosfouFricge- +(q−1)2+2a2q(3q2+4q+1)(cid:19)1/2(cid:21)1/2, ifa (cid:54)1. ure2),whileintheregionboundSedbythetwopseudo-caustics(see i o,cr equation(15);whichcorrespondsto69/130(cid:54)r (cid:54)47/60forthe S (16) casehere),wefindm = 4.Inthemagnificationmap,asthenum- In the belt-only limit (q → +∞), it can be shown that the above berofimageschangesfromtwotofour,thereisasharpincrease inthemagnification(aboutafactorof2.5).Notethatinthiscase, equationsreducetoequation(12). the source does not need to cross a formal caustic (defined as a Thesolutiontoequation(14)isplottedintheleftpanelofFig- ure2foraface-onbeltwitha = 1.2, a = 1.3,andq = 1.The curve in the source plane where |µ(rS)| = ∞ for a point source) i o for the number of images to change. The associated loci across two middle curves (with the left one solid blue and the right one which the magnification is finite but changes discontinuously are bluetransitioningtodashedblack)correspondtothepossibleso- thepseudo-causticsmentionedintheintroduction.Theexistenceof lutionstothelensingequationforrayspassinginteriortothebelt, pseudo-caustics implies that we cannot rely solely on the formal whereonlythecentralpointmassisresponsibleforthedeflection. causticsinordertounderstandtheselensingsystems. The two dashed curves at large |r| are the possible solutions for I rayspassingexteriortothebelt,wherethedeflectioncomesfrom thetotalmassofthesystem.Thesolutionsincludeimageslocated atbothpositiveandnegativer,whichcorrespondtoimageswith I positiveandnegativeparities,andarelocatedonthesamesideof 2.2 InclinedSystems the lens as the source and on the opposite side, respectively. The We now allow our system to be inclined, with the normal of the farleftdashedcurveincludesraysdeflectedfromboththebeltand planeofthebeltinclinedatanangleirelativetothelineofsight. the point mass, but the rays in fact pass interior to the belt (i.e. We again start with the case of a uniform ring with radius a, |r| a, de- I i satisfy flectionfromthebeltstartstocontributeandsothissolutionisnot allowed.Similarly,fortherightmostdashedcurve,theallowedso- x2 y2 lutionshouldhaveanimagepositionbeyondtheouterradiusofthe + =1, (17) a2 b2 MNRAS000,1–??(2016) GravitationalLensingbyRing-LikeStructures 5 whereaisthering’ssemi-majoraxisandb = acosiisthering’s get a relatively simple analytic classification for the two limiting semi-minor axis. Even for arbitrary inclinations, rays passing in- cases of the inclination i – we have addressed the face-on i = 0 teriortotheringareremarkably stillundeflected(Chandrasekhar caseintheprevioussection,andpresentananalysisoftheedge-on 1969;Bray1984). i=π/2caseinAppendixA. Tofindthedeflectionforanimpactpointr intheimageplane Equipped with the solutions of the deflection angle and the I thatliesoutsidethering(i.e.theexteriorsolution),weconsideran lensequationfortheinclinedsystemandusingourunderstanding ellipsepassingthrough r andconfocaltotheprojectedring.The oftheface-onsystem,weturnnowtoaninvestigationofthegeneral I confocalellipseisexpressedas lensingpropertiesofbelt+pointmasssystemsasafunctionofthe inclination. x2 y2 + =1, (18) a(cid:48)2 b(cid:48)2 wheretheprimedsemi-axesaredefinedbya(cid:48)2 =a2+λandb(cid:48)2 = 3 LENSINGPROPERTIESOFBELT+POINTMASS b2+λ,withλfoundbysubstitutingrI =(xI,yI)intoequation(18) SYSTEMS:CAUSTICSANDPSEUDO-CAUSTICS andtakingthepositivesolution. Thedeflectionangleα(r) = α xˆ +α yˆ fromtheringcan Tostudythelensingpropertiesofbelt+pointlensingsystems,we r I 1 I 2 I be found analytically using results from ellipsoidal potential the- employtheinverseray-shootingtechnique(e.g.Schneider&Weiss ory (Bourassa & Kantowski 1975), which are solved nicely in 1986).Lightraysaredrawnuniformlyfromtheimageplaneand Schramm(1990).For r = x xˆ +y yˆ ,thedeflectionanglefrom mapped back to the source plane according to the lens equation. I I I I I theringis Themagnificationateachpositioninthesourceplaneiscomputed (cid:18) x y (cid:19) as the ratio of the density of rays at this angular position to that α(r;a,i)= p(cid:48)2 I + I , (19) intheimageplane,whichisalsothedensityofraysinthesource r I a(cid:48)3b(cid:48) a(cid:48)b(cid:48)3 planeintheabsenceofthelens. wherep(cid:48)=(cid:16)x2/a(cid:48)4+y2/b(cid:48)4(cid:17)−1/2istheperpendiculardistancefrom Of interest in analysing the properties of lensing systems I I the centreof theconfocal ellipseto the tangentline at r and we is the behaviour of caustic curves, which are loci in the source I alsousethenotationx =x xˆ andy =y yˆ forsimplicity. plane where the Jacobian matrix of the lens mapping is singular ToobtainanexpreIssionI fIorthedIeflecItioIncausedbyabelt,we (det(A) = 0), and where the magnification µ(rS) is formally infi- makeuseofmoreresultsinellipsoidalpotentialtheory.Consider nite. The caustic curves correspond to the singular points of the acircularuniformdiscinprojection,whichappearsasanelliptical lens mapping. For smooth lens mappings, we may invoke Sard’s discwithconstantnormalizedsurfacedensityκ ,semi-majoraxis theoremtoshowthatthesetofcausticcurvesmusthavemeasure 0 a,andinclinationi.Thenormalizeddeflectionangleatapointr in zero, which means that they must consist of collections of one- I theimageplaneis(Schramm1990) dimensionalcurvesandisolatedpoints. Forallbutthemostsimplelensinggeometries,findinganalyt- (cid:32) (cid:33) α (r;a,i)=2πa˜b˜κ 1 xI + 1 yI , (20) icalexpressionsforthecausticcurvesisdifficult,sincethemapping d I 0 a(cid:48)+b(cid:48) a(cid:48) a(cid:48)+b(cid:48) b(cid:48) r (cid:55)→ r ishighlynon-linear.Whiletheinverse-rayshootingtech- I S whereforpointsinteriortothedisc,weseta(cid:48)2=a˜2=x2+y2/cos2i niqueenablesustofindhighmagnificationregions,itislimitedby andb(cid:48) = b˜ = a(cid:48)cosi.Forpointsexteriortothedisk,wIeseIta˜ = a theresolutionofthemapping.Tobetterlocatethecausticcurves, andb˜ = bandobtaina(cid:48) andb(cid:48) bysubstituting(x,y)intoequa- weuseequation(7)totrackdownpointswherethemagnification I I isabovesomethresholdvalue(weadopt|µ(r )| > 1000).InAp- tion(18). S pendixB,weprovideanalyticexpressionsforthemagnificationin Tocalculatethedeflectionangleofabeltofinnerandouter eachofthesystemsweconsider. semi-majoraxesa anda ,wesuperimposeadiscwithsemi-major axisa ,inclinationii,andosurfacedensityκ withadiscofeffective Forisolatedlensingsystems1withlensingpotentialsψ(rI)that o 0 are smooth for all r away from a finite number of point singu- surfacedensity−κ ,semi-majoraxisa,andinclinationi.Theto- I 0 i larities, several results from singularity theory can be applied to taldeflectionfromthetwosuperimposeddiscsisjustthatfroma constrainthetypesofimagesthatsourcescanform.Inparticular, uniformbeltwithinnerandouterradiia anda : i o theimagemultiplicitym(r )canonlyeverchangebytwo.Thiscan S α (r;a,a ,i)=α (r;a ,i)−α (r;a,i). (21) beseenbyapplyingthePoincare-Hopfindextheorem,whichstates b I i o d I o d I i (cid:80) thatthesum sgn(µ)isaconstantforallnon-causticsourceloca- Here,κ isdefinedasκ =1/[π(a b −ab)]forthebeltsothatthe i i 0 0 o o i i tionsr ,andimpliesthatimagescanonlybecreated/annihilated totaldimensionlessmassofthebeltisunity.Itiseasytoverifythat S in opposite-parity pairs (Burke 1981; McKenzie 1985). For mass thebeltdoesnotdeflectraysthatpasswithintheinneredgeofthe distributionsκ(r)withnopointsingularities,thisimpliesthatthe belt. I imagemultiplicitym(r )mustbeoddforallr ,sinceweexpecta Withthedeflectionanglefromthebelt,thelensequationfor S S sourceatr toformonlyone,undeflectedimagewithpositivepar- thegeneralcaseofaninclinedbelt+pointsystemthenbecomes S ityinthelimit|r |→∞.Moregenerally,ifψ(r)possessesgpoint S I r =r − 1 rI − q α (r;a,a ,i), (22) singularities,thetotalimagemultiplicityisalwaysodd(even)ifg S I 1+q|r|2 1+q b I i o iseven(odd),withasourceat|r | → ∞formingoneundeflected I S positive-parity image and g negative-parity images located at the where q is the belt-to-point mass ratio. The second term on the positionsofeachsingularity(Pettersetal.2001). right-handsidecorrespondstothedeflectionfromthecentrepoint mass. Taking the limit of q → +∞ recovers the equation for the Additionally,forisolatedlenseswithsmoothψ(rI),theimage belt-onlycase. Whilecomputingthedeflectionforsystemswitharbitraryin- 1 AlensingsystemisisolatedifthetimedelayfunctionTrS(rI)= 12|rI− clinationisstraightforward,analyticallysolvingthelensingequa- rS|2−ψ(rI),whichisthesumofthegeometricandgravitationaltimedelays, tionbecomesmuchmorecumbersome.However,itispossibleto satisfiesTrS(rI)→∞as|rI|→∞forallfinite,non-causticrS. MNRAS000,1–??(2016) 6 E.LakeandZ.Zheng Figure3. Magnificationandimagemultiplicitymapsforaring-onlylens.Thetopplotshowsaringofradiusa=0.75acrossarangeofinclinations.Thetop rowshowsthemagnificationµinthesourceplane,withthebottomrowshowingthenumberofimagesasourceatagivenpositionproduces.Thebottomplot isthesameastheupperplot,butwiththeringradiuschangedtoa=1.25.Notethatinbothcasestherearelociwithfinitechangeinmagnificationandwith correspondingchangeinthenumberofimages,whicharepseudo-causticsdiscussedinthetext. multiplicity m(r ) can be constrained to change when, and only allowfortheexistenceofregionsinthesourceplaneacrosswhich S when,thesourcecrossesacaustic(Schneideretal.1992).Thiscan theimagemultiplicitycanchangebyone,violatingtheinvariance beseenbyconsideringthemapr →m(r ),whichassociateseach oftheimageparity. S S pointrS inthesourceplanewiththetotalnumberofimagesformed In the literature, curves in the source plane across which byapointsourcelocatedatrS.Sincem(rS)isanintegeritcannot the image multiplicity changes by one are often called “pseudo- change continuously, and hence for smooth lens mappings m(rS) caustics” (or “cuts” in e.g. Kovner 1987). Several previous stud- canonlychangewhenthesourcecrossesacaustic.Becauseofthis, ieshavedealtwithpseudo-caustics,primarilyinthestudyofvari- thecausticsetmustconsistofclosedcurves(togetherwithpossible ants of singular isothermal spheres and ellipsoids (e.g. Wang & isolatedpointcaustics).Indeed,iftherewereopencausticcurves,a Turner 1997; Evans & Wilkinson 1998; Keeton et al. 2000; Rhie closedloopcouldbeconstructedinthesourceplanethathadonly 2010), with which the ring+point lens shares several features. In oneintersectionwithacausticcurve.Asourcetravelingalongthis mostofthesemodels,thepseudo-causticsappearaslargesmooth loop would then arrive back at its starting point with a different circles or ellipses. For example, the singular isothermal sphere imagemultiplicitythanitstartedwith,whichisimpossible. modelisdefinedbythelensingpotentialψ(r) = R|r|(implying I I Theseconstraintsonimageformationcanberelaxedinsys- κ(rI) ∝ 1/|rI|),andpossessesacircularpseudo-causticatrS = R, tems, like the ones considered in this paper, whose lensing po- wherethenumberofimageschangesbyone.Asasourcetraveling dteintitoianls.ψM(roI)stsaretilsefiveasntwlyeafokreruds,iffthereesneticaobnilsittryaionrtssicnagnulbaeritryelacxoend- fimroamgerS“d”iRntcorothsesessinthgeulpasreituydoat-craIu=sti0c.,Tahniesgiastisvime-iplaarrittoy if ψ(r) is not twice-differentiable (C2). By Poisson’s equation whathappenstotheimagesofsourcescrossingthecircularpseudo- I ∇2ψ(r)=2κ(r),thisisguaranteedifκ(r)isdiscontinuous,which caustics in the face-on ring examples considered earlier, when a ocrcIursIforthemIajorityofthelensinggeomI etriesconsideredinthis negative-parity image “disappears behind the ring” and becomes paper(althoughweconsiderthecaseofaC∞ massdistributionin zero-sized. Sec.3.3).Aswasalreadyshownintheface-onringcaseandwillbe In this paper, we adopt a slightly more general definition of studiedinmoredetailbelow,discontinuousmassdistributionsmay pseudo-caustics, which is better suited for discussing the lensing MNRAS000,1–??(2016) GravitationalLensingbyRing-LikeStructures 7 systems we consider. We define pseudo-caustics to be curves in above. In the face-on case (leftmost panel), from small to large thesourceplaneacrosswhichthemagnificationµ(r )changesdis- radii, we see a circular m = 3 region, an annular m = 2 region, S continuously, and yet remains finite. This definition is more gen- and finally the m = 1 region, divided by the two pseudo-caustic eralthanthedefinitionintermsofimagemultiplicities—besides curves.Examiningourearlierresultsonface-onsystemsexplains including loci in the source plane where the number of images that the ring can have m = 3 for small r because a < 1. Ap- S changes by one, it also includes curves across which the size of plyingequation(13)inthelimita = a = aallowsustoderive i o animagejumpssharply,whichwillberelevantinwhatfollows. the radii of the pseudo-caustic circles as PS = a = 0.75 and 1 The pseudo-caustics are found to possess a broad variety of PS =a−1−a≈0.58.Thedifferentimagemultiplicityregionscan 2 morphologiesacrosstheparameterspaceofthelensingsystemswe alsobeunderstoodbyplottingthesolutionstothelensequationin consider.Theformalcausticscorrespondtothedegeneracyofthe awaysimilartoFigures1and2.Therearethreepossiblesolutions, lensingJacobian(implyingµ(r )→∞),andasaconsequencethey one(from r = r)correspondingtotheundeflectedlightpassing S S I areusuallycusped.Bycontrast,thepseudo-causticsaredefinedto interiortothering(theinteriorsolution)andtheothertwo(from alwayshavefiniteµ,andtheyalwaysappeartobesmooth.Inmany r = r(1−1/|r|2)) corresponding to the deflection by a central S I I cases,wewillseethattheexistenceofpseudo-causticscanleadto pointwiththesamemassasthering(theexteriorsolutions).The thepresenceofopencaustics. physicallyrealizedsolutionsaredeterminedbytheringradius.For It is tempting to attribute the presence of pseudo-caustics in example,at|r |(cid:29)1onlyoneoftheexteriorsolutionsisphysical, S thethinringcasetothediscontinuityofthelensmappingbetween givingrisetothem=1region. the inner-ring and outer-ring solutions. However, the belt lensing The solution for the m = 3 region has one image from rays systemsweconsideralsoexhibitawidevarietyofpseudo-caustic passinginteriortothering(henceundeflected),sothecorrespond- features,eventhoughthedeflectionangleαb(rI;ai,ao,i)iscontinu- ingregionintheimageplanemustbeinteriortothering,whichcan ousthroughouttheentirelensplane.Rather,wewillseethatitisthe beusedtotracktheinclinationofthering.Indeed,astheringbe- non-differentiabilityofthedeflectionangle(andhencethediscon- comesmoreinclined,theregionbecomesmoreelliptical(topplot tinuityofκ(rI))thatcausestheformationofthepseudo-caustics. of Fig.3). As the ring inclines, a m = 5 diamond-shaped region Since the pseudo-caustics represent loci where µ is non- forms near the centre and a vertical structure of increased mag- differentiable,theycaninprinciplebefoundbyfindingpointsin nification appears. Since the projected mass of the inclined ring thesourceplaneacrosswhichthemagnificationisdiscontinuous. becomesmoreandmoreconcentratedtowardthetwoendsofthe However, the explicit analytic expression for |∇rSµ| is very un- majoraxis,thenormalcausticfeatureexhibitssomesimilarityto wieldy. In practice, we find the pseudo-caustics by using our in- thatfromanequal-massbinarylensingsystemwiththetwopoint verse ray-shooting results to numerically compute the magnifica- massessymmetricallyplacedonthemajoraxis,withthepositions tionmapinthesourceplaneandtheassociatedimagemultiplicity approaching ±a as the ring becomes edge-on. That is, as the in- map,whichthenallowsustofindthepseudo-causticsbyinspec- clinationincreases,thecausticfeaturesmimicthoseofclose-and tion. intermediate-separationequal-massbinarypointmass(e.g.Schnei- Aswasthecaseforface-onringsandbelts,wewillseethat der&Weiss1986;Liebigetal.2015)andbinaryisothermalsphere thepseudo-causticscorrespondtotheboundariesofthebelt/ringin (Shin & Evans 2008) lenses. Additionally, the combination of a theimageplanemappedtothesourceplane,whichallowsustode- largecircularpseudo-causticandacentralasteroidalcausticseen riveananalyticexpressionforthepseudo-caustics(seeAppendix inthei = 30◦ panelissimilartothecausticstructuresofsingular C).Thus,forthemodelsweconsider,wecanassociatediscontinu- isothermalspherepotentialsandvariantsthereof(seee.g.Figure1 itiesinκ(rI)withdiscontinuitiesinµ(rS),whichdefinethepseudo- ofRhie2010andFigure5ofKovner1987). caustics. We demonstrate this by explicitly mapping the belt/ring Thecasewitha=1.25(bottomplot)isfairlydifferent.When boundariesintothesourceplane,andcheckingthattheycorrespond seen face-on, the ring is no longer concentrated enough to form toregionsinwhichµ(rS)isfiniteanddiscontinuous.Wenotethat multipleimagesnearthecentre,sincetheentiremassdistribution thisisnotageneralproofthatdiscontinuitiesinκ(rI)areinaone- of the ring lies outside of its own Einstein radius. The possible to-onecorrespondencewith discontinuitiesinµ(rS),andthat this physical solution to the lens equation includes that from the un- correspondencemaynotholdforothertypesoflensingsystems. deflectedraysandthatcorrespondingtothecaseofapoint-mass lens(theonewiththeimageoutsidetheEinsteinringradius),with theformerandlatterapplyingtosmallandlargeradii,respectively. 3.1 Ring-OnlySystems In between, there is an annular region where both apply, and the Toproceed,wefirstturnourattentiontothesimplestcase–that boundaries give the two pseudo-caustic curves, which are circles of a ring-only lensing system. Figure 3 shows magnification and ofradiiPS = a = 1.25andPS = a−a−1 = 0.45.Asthering 1 2 imagemultiplicitymapsforarangeofringparameters. becomesinclined,thisannularregionpinchesdownaboutthema- ThetopplotinFigure3showsaringwithradiusa = 0.75, jor axis, and two folded regions of high magnification appear at whileinthebottomplottheringhasa=1.25.Inbothcases,thein- the inner parts of the annulus. As the inclination continues to in- clinationoftheringincreasesfromlefttoright,goingfromface-on crease, regions with higher image multiplicities start to form. As toedge-on.Thetoprowsofeachplotshowmagnification,whilethe withthea = 0.75case,thenormalcausticssharesimilaritieswith bottompanelsindicatethenumberofimagesforasource,m(r ),at thosefromanintermediate-separationequal-massbinarylens.For S thegivenlocationinthesourceplane.Wecomputem(r )bysweep- bothvaluesofa,weseetheappearanceofanarrowhigh-mregion S ingovertheimageplane,identifyingthesetofallpoints{(xI,yI)} locatedalongthexS axis.Forapointsourcetheadditionalimages thatmaptowithinacircleofradius(cid:15) (cid:28)1centredatrS,andthen formedinthisregionoccurwhentheyS-coordinateofthesource countthenumberofpath-connectedregionsintheset{(x,y)}. is strictly zero, and so the corresponding high-m region has zero I I Inthea = 0.75case(topplot),weseemanyboundariesbe- thickness. tweenregionswithdifferentvaluesofm,wherethemagnification Wenowturnourattentiontothering+pointmasssystemand hasafinitejump.Thesearethepseudo-causticcurvesmentioned provide an illustrative understanding of the pseudo-caustics and MNRAS000,1–??(2016) 8 E.LakeandZ.Zheng open caustics. Finally, we present the results for the belt+point formoutsideoftheprojectedringarenotphysical.Thereforethe masssystem. part of the critical curve outside the ring (indicated by thin blue lines)isnotallowed.Theallowedlens-planeareafortheinterior solutionbecomessmallerastheringbecomesmoreinclined.The 3.2 Ring/Belt+PointMassSystems corresponding caustic is still the point at the origin of the source plane. 3.2.1 Ring+PointMassSystem Fortheexteriorsolution,thesituationisreversed.Thepartof Withapointmassplacedatthecentreofthering/belt,thesolutions thecriticalcurveinsidethering(indicatedbythinredlines)isnot becomemorecomplex.Examiningthecriticalcurvesintheimage allowed.Inafewcases(e.g.witha = 0.75andi = 60◦),thefour planeandthecorrespondingcausticcurvesinthesourceplaneis smallringlet-likecriticalcurvesareinsidetheringandhenceare helpful for us to understand the lensing properties, especially the not allowed, which leads to the disappearance of the “planetary” pseudo-causticsandopencaustics. caustics.Forthecasewitha = 1.25andi = 60◦,theallowedpart ThethickblackcurvesinFigure4arethecriticalcurves(top ofthecriticalcurveformsfourseparatedarclets,asaresultofthe plots)andthecorrespondingcausticcurves(bottomplots)forthe truncationimposedbythering.Theyleadtofourseparatedcaustic ring+point mass system. To understand these curves, it is useful curves, and the truncation in the critical curve makes the caustic to present the full sets of critical and caustic curves for the inte- curvebecomeopen. rior and exterior solutions, which are shown as the blue and red Since it is the projected position of the ring that determines curves,respectively.Fortheinteriorsolution,theterm“full”refers which solution is valid, the pseudo-caustics can be obtained by to the curves determined from the centre point mass, as if the mappingtheprojectedringontothesourceplane,accordingtothe ring did not exist. For the exterior solution, “full” means that we interiorandexteriorsolution,respectively.Theinner(outer)bound- plainlyapplytheexteriorsolutioneveniftherayspassinteriorto ary of the mapped ring is shown as purple (green) curves in the thering.Oreffectively,weshrinktheprojectedringconfocallyto sourceplanethatresultfrommappingtheprojectedringwiththe a degenerate ellipse with zero minor axis so that the exterior so- interior(exterior)solutioninthecaustic-curvepanelsofFigure4. lution can be extended to small radii. We investigate which parts Formal caustics can become open as they intersect the mapped of the critical curves are allowed to form the final critical curves ring.Sincethemappingpropertieschangediscontinuouslyasthe (thickblack)giventheconfigurationofthesystem,anddeducethe impact point (image) crosses the ring, the curves of the mapped pseudo-caustics and open caustics. For the system shown in Fig- ringarejustthepseudo-causticsmentionedbefore.Noticethateach ure 4, the ring and point mass have equal masses, with the ring pseudo-causticissmooth,sincethelensmappingiscontinuousjust havingaradiusofeithera = 0.75ora = 1.25aslabelledinthe withinandjustoutsidethering(whichalsoholdsforthethickbelt panels.Theinclinationoftheringincreasesfrom30◦ to90◦ from case).Sincethemagnificationisfiniteatthepseudo-caustics,mag- lefttoright.ThescalesarenormalizedtotheEinsteinringradius nificationandimagemultiplicitymapsmustbeconstructedtostudy ofthecombinedring+pointmasssystem. them. The “full” critical√curve for the interior solution is simply Figure5showsthemagnificationandimagemultiplicitymaps a circle with radius 1/ 2, the Einstein ring for the centre point forthering+pointsystemsshowninFigure4,withq=1.Clearly, mass(sincewenormalizebytheEinsteinringradiusforthetotal themainfeaturesinthemagnificationmapcorrespondcloselywith ring+pointmass,whichistwicethepointmasshere).Thecorre- theformalcausticsandpseudo-causticsinFigure4. spondingcausticisthepointattheorigin.Fortheexteriorsolution, the“full”criticalcurveisalsoacircle(theEinsteinringfortheto- For a = 0.75 (top plot), when seen face-on, we see two talring+pointmass)forthesystematface-on.Astheringbecomes pseudo-caustics,similartothering-onlycase.However,theregions moreinclined,itsmassdistributionapproachesmoretowardabi- insidethepseudo-causticshavehigherimagemultiplicitiesthanthe nary lens with the two equal point masses symmetrically placed ring-onlycase.Thereasonisthatthepresenceofthecentrepoint onthex-axisandonthetwosidesofthecentrepointmasstoward massallowsthemaximumnumberofsolutionstobefour,opposed (±a,0).Thewholesystemmimicstheonecomposedofthreepoint tothreeasinthering-onlycase.Usingourearlierresultsforface- masses (Daneˇk & Heyrovský 2015). For the a = 0.75 case, be- on belt+point mass systems and taking ai = ao = a, we can de- sides a big ring-like critical curve, four small ringlet-like critical rivetheradiiofthepseudo-causticcirclestobePS1 ≈ 0.11and curvesareformed.Thebigandsmallcurvescorrespondtothecen- PS2 ≈ 0.82.Astheinclinationincreases,thethree-imageregion traldiamond-shapedcausticandthefourplanetarycuspcaustics, becomes larger and more oblate. Interestingly, two lobes of de- respectively.Asawhole,athighinclination,thecriticalandcaustic creasedimagemultiplicityappearwithinthem = 3region.When curvessharesimilaritieswiththeclose-separationtriplepointlens- seenedge-on,theseregionsbecomelargerandpushoutthem=3 ingsystem.Forthea=1.25case,theanalysisissimilarandathigh regionstolargerradii. inclinationwefindthesystemtobesimilartotheintermediate-or InthebottomplotofFigure5,theringisenlargedtohavea wide-separationtriplepointlens. semi-majoraxisofa = 1.25.Thepseudo-causticsinthiscaseap- Withthe“full”criticalandcausticcurvesinplaceforboththe peardistinctfromthea=0.75case.Whenseenface-on,weseean interiorandexteriorsolutions,wenowanalysewhichpartsofthe m = 3annularregionboundedbythetwopseudo-caustics.When curvesarephysicallyrealizedbyconsideringthevalidityoftheso- theringinclines,theannulusfoldsdownoveritself,creatingcom- lutions.UnlikeinSection3.1whereweexplainthingsinthesource plicatedoverlappingfeatures(e.g.inthepanelwithi = 60◦)from plane,wefinditismoreintuitivetoworkintheimageplane,since theinterplaybetweenthepseudo-causticsandtheformalcaustics. the validity of the solutions is determined by the image position Whenseenedge-on,m = 1regionsandsmallm = 7regionsare (whetheritisinteriororexteriortothering)andsincetheprojected formed.Althoughedge-ona=0.75anda=1.25ring+pointmass ringisalsonaturallydescribedintheimageplane. systemsareverysimilargeometrically,theyapproximatelycorre- Thedashedblackcurveineachcritical-curvepanelofFigure4 spondtoclose-andintermediate-separationtriplepointmasslens- delineatestheprojectedring.Fortheinteriorsolution,imagesthat ingsystemsandhencethemagnificationmapsaredifferent. MNRAS000,1–??(2016) GravitationalLensingbyRing-LikeStructures 9 Figure4. Criticalcurves(topplot)andcausticcurves(bottomplot)foraring+pointlens.Theringandthepointmasshaveequalmasses.Theradiusand inclinationoftheringarelabelledineachpanel.Ineachcritical-curvepanel,theblue(red)thincurvesconnectedwiththethicksolidonesformthe“full” criticalcurvesfromtheinterior(exterior)solution.Thedashedcurvedelineatestheprojectedringintheimage(lens)plane.Thecriticalcurvesfromtheinterior (exterior)solutionbecomesinvalidforthepartsoutside(inside)thering.Thefinalallowedcriticalcurvesareinthickblack.Thecorrespondingcausticsare denotedwiththesamecolourandlinetypesinthecaustic-curvepanels.Theringismappedontothesourceplaneaccordingtotheinterior(purple)andexterior (green)solution,respectively.Thesemappedringcurvesformthepseudo-caustics.Seethetextfordetail. 3.2.2 Belt+PointMassSystems ingtotheinteriorandexteriorsolutions,respectively.Theexplicit expressionforthepseudo-causticscanbefoundinAppendixC. Figure6showsthemagnificationandimagemultiplicitymaps We now widen the ring to a belt, and study the resulting effects forbeltswithdifferentmeanradii.With∆a = 0.3,thebeltradius onthemorphologyofthepseudo-causticsandthegeneralmagni- iscentredatac = 0.75inthetopplotandac = 1.25inthemiddle ficationpattern.Thelensingsolutionsareaffectedbytheprojected plot.Inthebottomplotitissituatedsothathalfofitsmassliesat massdensity(i.e.thewidthofthebelt),asseeninFigure1.Forthe a>1andtheotherhalfliesata<1.Inallplots,wetakeq=1. followingexamples,wefixthebeltwidthtobe∆a = 0.3.Inthe Fora=0.75,weonlyseetwoimagesformedfortheface-on belt+pointmasscase,thepseudo-causticscorrespondtotheinner casesincea

Gravitational Lensing by Ring-Like Structures

Similar Gravitational Lensing by Ring-Like Structures