Gravitational Lensing by Ring-Like Structures

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by  Ethan Lake

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:[email protected] callkeygravitationallensequationsandpresentthesolutionstothe
† Contacte-mail:[email protected] 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|<a)areundeflected.Thelatterresult
I
tweentheobserverandthelens,theobserverandthesource,and
canbeeasilyinferredbyconsideringtwochordspassingthrough
thelensandthesource,respectively.ThemassMisthetotalmass
the impact point with an infinitesimally small opening angle be-
ofthelenssystemweconsider.
tweenthem–thedeflectionanglescausedbythetwoarcelements
Thegenerallensequationcanbewrittenas
ontheringboundedbythetwochordsarethesamebutintheoppo-
r =r −α(r). (2) sitedirections,leadingtonullcontributiontothedeflection.Thus,
S I I
thelensingequationreads
Thenormalizeddeflectionangleαisfoundbyintegratingthecon-
r
tributionovertheprojectedmassdistributionofthelens, r =r − I Θ(|r|−a), (9)
S I |r|2 I
α(rI)= π1(cid:90) d2r(cid:48)Iκ(r(cid:48)I)(cid:12)(cid:12)(cid:12)rrII−−rr(cid:48)I(cid:48)I(cid:12)(cid:12)(cid:12)2, (3) weqhuearteioΘn(ixs)IwisritthteenHineaavivsiedcetosrtefpormfu,ncittiroend.uAcelsthtoougahscthaelarleenqsiunag-
tion(byreplacingr andr withr andr,respectively)giventhe
whereκ(r)isthedimensionlesssurfacemassdensityofthelens, S I S I
I symmetryoftheface-onring/beltcasediscussedinthissubsection.
κ(r)= Σ(rI). (4) Forsuchascalarequation,whilethesourcepositionrS ispositive
I Σcr bydefinition,theimagepositionrI canbeeitherpositiveornega-
ThecriticalsurfacedensityΣcrisdefinedas tive,correspondingtoimageslocatedonthesameside(rI > 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|<a).Theleftandrightdashedcurvesarethesolutionfor
1 I 2 I I i
imagesisobtainedbyµ = (cid:80) |µ|,whereirunsoverallimages. rays deflected by the whole belt (i.e. equivalent to putting all the
tot i i
Thesignofthemagnificationreflectstheparityoftheimage.That massofthebeltattheorigin),whichisthesolutionforrayspassing
is, µ > 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 <r <a thatpassthroughthebelt,ifthebeltiswide(low
i I o
Schneideretal.(1992)andPettersetal.(2001). surfacedensity),∂r /∂r >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<ul0tipfoleriamfiangietse(reainthgeeromf=rI.3Tohremtr=an5si)tifoonr PS1=(cid:12)(cid:12)(cid:12)(cid:12)ai− (1+1q)ai(cid:12)(cid:12)(cid:12)(cid:12) and PS2=(cid:12)(cid:12)(cid:12)(cid:12)ao− a1o(cid:12)(cid:12)(cid:12)(cid:12). (15)
betweenthesetworegimesoccursatthecriticalbeltwidth We can also derive the condition for pseudo-caustics that change
2 1 the image multiplicity, as opposed to those that just change the
(∆a) = = , (12)
cr a +a a sizeofimages.Asbefore,thisisdonebyfindingtheparametersat
i o c
MNRAS000,1–??(2016)
4 E.LakeandZ.Zheng
2.0
1.5
2 images
rS1.0
4 images
0.5
0.0
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
r
I
Figure2. Left:Solutionstothelensequationforabelt+pointmasslens.Thebeltisface-on,withmeanradiusac =1.25andwidth∆a=0.1.Thebeltto
pointmassratioissettoq=1.0.Thetwoinner(outer)blackdashedcurvescorrespondtothesolutionsfromjustthepointmass(fromthetotalbelt+point
masssystemactingasapointmass).Thebluesolidcurvesarethephysicallyrealizedsolutionsforthebelt+pointmasscase.Thepresenceofthebeltcreates
aregioninthesourceplanewherefourimagescanbecreated.NegativerIcorrespondstoimageslocatedontheoppositesideofthelensasthesource.Right:
MagnificationforthelensintheleftplotasafunctionofthedistancerS ofthe(point)sourcefromtheorigin.Thestep-likejumpinmagnificationmarks
thesourcepositionscorrespondingtothepseudo-caustics(seethetext),andtheplateauboundedbythetwostepscorrespondstosourcepositionswherefour
imagesareformed.
whichr ceasestobecomeamonotonicallyincreasingfunctionof belt,i.e.|r| > 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 <ao,cr,wehave∂rS/∂rI <0inthispartofthesolution(see
a = 2q +a2, ifa >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),meaningthatthissolutionisnotphysicallyrealized.The
I i whichbecomesanellipsewhenseeninprojection.Theellipseis
middle right dashed curve corresponds to rays passing interior to
parametrizedbythesetofvectorsinthelens(image)plane,which
the belt and deflected only by the point mass. For |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<isRaptpoeraSrs>”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 <a ,andthesurfacedensityofthebeltisnothigh
o o,cr
andouteredgesofthebeltmappedontothesourceplaneaccord- enoughtogenerateadditionalimages,similartothecaseindicated
MNRAS000,1–??(2016)
10 E.LakeandZ.Zheng
Figure5. Magnificationandimagemultiplicitymapsforaring+pointlens.Theringandpointmasshaveequalmasses.Theringhasradiusa=0.75inthe
topplotanda=1.25inthebottomplot.AsinFigure3thereexistpseudo-caustics associatedwithchangesintheimagemultiplicity.SeeFigure4forthe
correspondingpseudo-causticsassociatedwiththecaseswithi(cid:62)30◦.
bythebluecurveinFigure1.However,weseearing-likestruc- a = 0.75 case). In particular, at edge on, the three cases from
turepresentinthemagnificationmap,withnocorrespondentim- a = 0.75 to a = 1.25 are similar to the triple point lensing sys-
agemultiplicitychange.Thisring-likestructurecorrespondstorays temgoingfromclosetowardintermediateseparations.Theoverall
passingthroughthebelt,experiencingdeflectionfromboththebelt features for the case with half mass inside and half mass outside
andthecentrepointmass.Thisleadstothechangeintheslopeof theEinsteinringradiusareinbetweenthea = 0.75anda = 1.25
ther -r relation(asillustratedinFigure1).Thismeansthatoneof cases.
S I
theimagesbecomeslargerinsize,resultinginanoverallincreasein
magnification(dr2/dr2).Thisexampleshowsthatpseudo-caustics
I S
neednottobeassociatedwithchangesintheimagemultiplicity– 3.2.3 ExamplesofImagesfromtheRing+PointSystem
theycanalsocorrespondtodiscontinuouschangesinthesizeofan
In Figure 7, we provide examples of the sorts of images formed
image.
forsourceslensedbyring+pointmasssystems.Thetopplotshows
When the belt inclines, a narrow m = 4 annular region the case of a face-on ring with a = 1.25. The left panel shows
boundedbypseudo-causticsappearsthatevolvesintoafan-shaped theimagemultiplicitymapwiththeredandbluecirclesrepresent-
regionati=60◦.Whenthebeltisseenedge-on,themagnification ingtwosourcesinthesourceplane,andthecorrespondingimages
mapissimilartothecaseofthea=0.75edge-onring(rightpan- are shown in the image plane in the right panel. The red source
elsinthetopplotofFigure5),asbothapproximatelyapproachthe ismappedtotwoimagesbythecentrepointmass,whiletheblue
close-separatedtriplepointlensingsystem. sourceinsidetheregionboundedbythepseudo-causticsisableto
For the belt with radius centred at a = 1.25 (middle plot of passraysoutsidethering(blackcircle)andtakeadvantageofthe
Figure 6), the magnification pattern shares some broad similari- ring’slensingpotentialtoformthreeimages,withthethirdappear-
tieswiththatofthea = 1.25ringcase(bottomplotofFigure5). ingjustoutsidethering.
The main noticeable differences are that at low inclination the Thebottomplotconsidersaslightlymorecomplicatedcase,
belt+pointcasehasanarrowerregionboundedbypseudo-caustics withtheringinclinedtoi=60◦.Eventhoughtheimagemultiplic-
andthatintheface-oncase,thereisnoimagenumberchangewhen itymapintheleftpanellookscomplicated,thewayinwhicheach
crossingthepseudo-caustics(forthesamereasonasintheabove sourceislensediseasytounderstand.Theredsourceislensedby
MNRAS000,1–??(2016)