The complete catalogue of light curves in equal-mass binary microlensing

1.6 MB·English

by  Christine Liebig

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed2April2015 (MNLATEXstylefilev2.2)
The complete catalogue of light curves in equal-mass binary
microlensing
Christine Liebig,1(cid:63) Giuseppe D’Ago,2,3† Valerio Bozza,2,3‡ Martin Dominik,1,4§
5
1SUPA,SchoolofPhysics&Astronomy,NorthHaugh,UniversityofStAndrews,KY169SS,Scotland,UK
1
2DipartimentodiFisica“E.R.Caianiello”,Universita`diSalerno,ViaGiovanniPaoloII132,84084Fisciano(SA),Italy
0
3IstitutoNazionalediFisicaNucleare,SezionediNapoli,Italy
2
4RoyalSocietyUniversityResearchFellow

r
a
M
Accepted,30March2015.Received,12March2015;inoriginalform,23December2014.

1
3
ABSTRACT

] The light curves observed in microlensing events due to binary lenses span an extremely
P widevarietyofforms,characterisedbyU-shapedcausticcrossingsand/oradditionalsmoother
E
peaks. However, all peaks of the binary-lens light curve can be traced back to features of
. causticsofthelenssystem.Moreover,allpeakscanbecategorisedasoneofonlyfourtypes
h
p (cusp-grazing, cusp-crossing, fold-crossing or fold-grazing). This enables us to present the
- first complete map of the parameter space of the equal-mass case by identifying regions in
o
whichlightcurvesfeaturethesamenumberandnatureofpeaks.Wefindthatthetotalnumber
tr ofmorphologiesthatcanbeobtainedis73outof232differentregions.Thepartitionofthe
s parameter space so-obtained provides a new key to optimise modelling of observed events
a
[ throughacleverchoiceofinitialconditionsforfittingalgorithms.

2 Keywords: Gravitationallensing:micro–methods:numerical
v
9
1
2
2 1 INTRODUCTION Historically,itisnotuncommonformodellerstoexplorespe-
0 cific morphological traits of light curves to narrow down the pa-
Einstein(1936)showedthatthelightcurveofasourcemicrolensed
. rameter space to be searched, as has been done by authors such
1 by a single foreground compact object is given by an extremely
asMao&DiStefano(1995);Dominik&Hirshfeld(1996);DiSte-
0 simplesymmetricbell-shape,describedanalyticallybyaverycom-
fano&Perna(1997);Albrowetal.(1999a);Dominik(1999a);Han
5 pactformula,nowknownasPaczyn´skicurve(Paczynski1986).It
&Gaudi(2008),butliteraturethatsystematicallycoversthewhole
1 issomewhatfrustratingthatbyaddingjustanotherlensthecom-
: rangeofpossiblemorphologiesismorescarce.Themodellingof
v plexityofmicrolensingexplodessodramaticallythatafteralmost
observedmultiple-lensmicrolensinglightcurvesrequiresextensive
Xi 30yearsofactivetheoreticalandobservationalresearchacomplete computationofthemagnificationcurves.Muchefforthasbeenin-
classificationofallpossiblelightcurvemorphologiesisstillmiss-
vested into speeding up the modelling process, by improving the
r ingeveninthesimpleststaticcase!Thelackofacompleteknowl-
a parametrisation(Anetal.2002;Cassan2008;Bennett2010;Ben-
edge of the light curve zoology represents a considerable handi-
nett et al. 2012; Penny 2014), by employing neural networks to
capinthemodellingofrealmicrolensingevents.Infact,inorder
map light curve features to model light curves (Vermaak 2007).
tosetinitialconditionsforfitting,onemayfollowtworoutes:ei-
Ofcoursethisdevelopmenthappenedalongsideofsubstantialad-
therblindlyset-upadensegridintheparameterspaceoridentify
vancesinthecodeimplementationofexistingalgorithms.
good initial seeds with light curve morphologies close to the one
wewishtomodel.Thefirstapproachismoresystematicbutcanbe Mao&DiStefano(1995)discussedanewmethodformod-
timeconsumingandredundant;furthermore,itdoesnotguarantee elling binary microlensing events: the positions and amplitudes
thecompletenessoftheexplorationofallpossiblecorners,which of binary light curve extrema are compared to those stored in a
mayremainhiddeninthespacebetweenconsecutivepointsinthe pre-compiled(unblended,point-source)lightcurvelibrarytofind
grid.Thesecondapproachpromisestobemoreefficientintermsof promisingcandidateevents,whichinturnprovideinitialparameter
computingtimebutneedstobesupportedbyarobustandrigorous setsforamoreconventionalfittingprocedure.Thisapproachworks
theoreticalframeworkinordertobesafelypursued. wellformulti-peakevents,wherethesourcetrajectorypassesover
orclosetothebinarycaustics.
DiStefano&Perna(1997)developedthelibraryapproachfur-
therbydescribinganybinary-lenslightcurvebythesetofcoeffi-
(cid:63) [email protected]
cientsofChebyshevbasispolynomials.TheynotethattheCheby-
† [email protected]
‡ [email protected] shev expansion will never exactly match the microlensing light
§ [email protected] curve,becausetherewillbeextraextremaandinflectionpoints,but
(cid:13)c 0000RAS
2 C.Liebigetal.
anarbitrarilypreciseagreementcanbeachieved(limitedbycom- Consequently, the fundamental idea of this work is to iden-
putationalpower)byfurtherexpansion.Inthisway,amodelsearch tify the building blocks of microlensing light curves and develop
can be refined until the photometric precision of the data points a classification scheme that can be directly applied to observed
is matched. They find model parameter solutions to smooth and lightcurvesandthatallowsforasignificantnarrowingofthemod-
caustic-crossinglightcurvesbycomparingtheroughcharacteris- ellingparameterspace,while,unlikeanyotherapproach,guaran-
tics of the light curve (positions of extrema and inflection points teeingcompleteness.Wewanttogainagoodunderstandingofthe
andthemagnificationvaluesatthesepoints)withapre-computed range of possible light curves and how the identified morpholog-
lightcurvelibraryandthensearchingthenearbyenvironmentinthe icalclassesrelatetosubspacesofthemodellingparameterspace.
physicalparameterspacewithanincreasedsamplingdensityuntil As a first step, we focus on an in-depth study of the equal-mass
they find a match (or multiple matches) that satisfies the desired binarylens,whiletheframeworkdevelopedappliestothegeneral
precision.Inprinciple,thismethodisquitegoodatfindingdegen- case.OnreviewingthepropertiesofthisspecialcaseinSec.2,we
eratesolutionsandhigher-orderorevennon-microlensingparam- usetheopportunitytointroduceaconvenientnotationforcaustic
eterscaneasilybeintegrated,butagainitremainsunclearwhether elements.Sec.3introducesourmorphologyclassificationscheme,
allrelevantparameter-spaceregionshavecorrespondingentriesin whichisbasedonthefourfundamentalpeaktypesthatoccurinmi-
the library. The optimistic assertion that the “morphological fea- crolensing;wealsodiscussthepracticalities,suchasthelightcurve
tureschangeinawaythatisgradualandconsistentasthephysical simulation,thepeakcountingandtheidentificationofiso-maxima
parametersarechanged”(DiStefano&Perna1997)ismostlikely regionswithlightcurvemorphologies.InSec.4,wesummariseand
trueforsmoothlightcurves,butforcaustic-crossinglightcurves, discussthecurrentresultsofthisstudy,weleavesomefurthercon-
weknowthatverysmallchangesinthesourcetrajectorycanhave siderationstoSec.5,andstressitsfuturepotentialinSec.6,while
dramaticimplicationsforthenumberofextremaandtheirrelative thebulk ofthe contentisshown intabularand graphicalform in
positions. Table3andinFigures11to 18.
Night,DiStefano&Schwamb(2008)makeabroaddistinc-
tionbetweensmoothlightcurvesandcaustic-crossinglightcurves,
buttheclassificationisnotbasedonthelightcurveitself,buton 2 MICROLENSINGOFEQUAL-MASSBINARY
thesourcetrajectoryanditsclosenesstothecaustics,i.e.theknown SYSTEMS
simulationparameters,nottheobservabledata.Theycometothe
2.1 Parametrisation
conclusionthattheratioofsmoothly-perturbedtocaustic-crossing
binary-lens light curves is rather low in survey detections, which GravitationalmicrolensingischaracterisedbytheangularEinstein
canpartlybeexplainedbythefactthatcaustic-crossingpeaksstand radius
outunambiguously,whereassmoothperturbationsoftencanhavea (cid:114)
4GM D –D
rangeofcompetingexplanations(suchasbinarysources,parallax θ = S L , (1)
effects,orbitalmotion). E c2 DLDS
InBozzaetal.(2012),adetailedmorphologicalassessmentis where M is the total mass of the (foreground) lens object,
usedforthemodellingofOGLE-2008-BLG-510andfurthermore while D and D denote the lens and source distances from the
L S
thegroundworkislaidforareal-timebinaryeventmodellingcode observer.Inthecourseofamicrolensingevent,theseparationbe-
(further based on Bozza (2001) and Bozza (2010)). The code re- tweeneachpairofimagesisoftheorderofθ ,whichislessthan
E
liesonawidechoiceofstartingconditions(“seeds”)fromwhere amilliarcsecondfortypicalobservedeventswiththesourceinthe
asearchforlocalχ2 minimaiscarriedout.Thechoiceofseedsis bulge(D ∼ 8kpc),andthelensbeingamainsequencestarhalf-
S
basedonthemorphologyofthebinarycaustics,withtheassump- waytothesource(D ∼4kpc,M∼0.3M )
L (cid:12)
tion that binary-lens light curves sampled from a given region of If one assumes uniform, rectilinear relative proper motion µ
theparameterspacelieonasmoothslopeoftheχ2 landscapeas betweenthelensandsource,themagnificationduetoasinglepoint
long as the morphology of the light curves does not change. The lensisdescribedbyonlythreeparameters:u ,theclosestangular
0
morphologyisunderstood,inthiscase,asagivenpeaksequenceof impact of the source to the centre of mass expressed in units of
causticcrossingsandgrazings,withanynewlycreatedordestroyed θ , the Einstein radius crossing time t ≡ θ /µ, and the time of
E E E
peakleadingtoachangeinmorphology. closestapproacht ofthesourcetothecentreofmassofthelens
0
Our intention, with the present work, is to take the move system, which is typically used to fix the epoch of observations,
from the path traced by Mao & Di Stefano (1995) and Di Ste- but is irrelevant for the light curve shape. Beyond the single lens
fano & Perna (1997) and achieve the first complete classification parameters,weneedthebinarymassratioq=m /m ,wherem is
2 1 1
of light curves in the binary microlensing problem. By studying theprimarymassofthebinarylenssystemandm thesecondary
2
peak-number plots, we can separate groupings of light curves in massinunitsofthetotalmassofthesystemM,theangularsepa-
the binary-lens parameter space. We are not concerned with di- rationofthebinarycomponentssinunitsofθ ,theangularsource
E
rectlyestablishinglightcurvemodels,butwewanttoensurethat star radius ρ still in units of θ , and the angle α, between the di-
E
weclassifyallpossiblelightcurves.Wethenwanttoimproveour rectionvectorfromtheprimarytothesecondaryandthedirection
understandingoftherelationsbetweentheparameterspaceandthe ofthesourcerelativetothelens,seealsoFig.1.Weassumeuni-
lightcurves. form,rectilinearrelativepropermotionbetweensourceandlensfor
Thevarietyofmicrolensinglightcurvescanseemoverwhelm- thesimulationsandignorehigher-ordereffects.Theobservedlight
ing,butthetrainedeyerecognisesfamiliarpatternsandtranslates curveisthesumofthesourcefluxF ,amplifiedbythemicrolens-
S
thembacktotheparameterspace.Infact,theshapeofamicrolens- ingeffectA(t;u ,t ,t ,q,s,ρ,α),andtheblendfluxF contributed
0 E 0 B
inglightcurvedoesfollowcertainrules:notanyarbitrarycurvecan byunresolvedsources
beinterpretedasamicrolensinglightcurve.Specifically,thelim-
F(t)=F A(t)+F . (2)
ited topologies of the binary lens magnification maps allow only S B
foralimitedrangeoflightcurvemorphologies. For the purposes of this morphological study, F and F have no
S B
(cid:13)c 0000RAS,MNRAS000,000–000
Thecatalogueofbinarymicrolensinglightcurves 3
top
u >0
0
B b B
α t1 t t2
Lens Lens a a
1 binaryaxis 2 ts1 ts2
secondary caustic
C
centreofmass ts
Source C
tp
primary caustic a a
Figure1.Ourdefinitionofu0andα.Theimpactparameteru0ispositive, 1 tp1 tp2 2
whensourceandlens(centreofmass)passeachotherontheright-hand A A
1 2
sideasprojectedontheplaneofsky.αistheanglebetweenthebinaryaxis a a
(pointingfromprimarytosecondarymass)andthesourcetrajectory. bp1 bp2
C
bp
impactastheyjustrepresentamultiplicativeandanadditivefactor
respectively. C
bs
Our parametrisation is equivalent to the convention detailed a a secondary caustic
in Skowron et al. (2011, Appendix A), except that we regard the bs1 bs2
sourceratherthanthelenssystemasmoving,resultinginadiffer- B b B
enceofαhere = αSkowron–π.Achangebyπjustmeansthesource b1 b b2
istravellinginthe0oppositedirectiononthesametrajectorywhich bottom
doesnotaffectthemorphologyofthelightcurve,inotherwordsit
isatimereversalofthelightcurve.Moreontheparameterspace
symmetriesinSec.3.3. Figure2.Causticfeaturenotationoftheclose-separationbinarylens.
2.2 Caustics Moving from point to extended sources, the singularities of
thelensmapareregularisedbyanintegrationoverthefiniteangu-
In the theoretical treatment of multiple lens systems, caustics are
lar disk of the source. As Schneider & Weiß (1986, Fig. 9) have
singularlineswherethefluxofapointsourceisinfinitelymagni-
shown,anincreaseinangularsourcesizeleadstodecreasedpeak
fied. Schneider & Weiß (1986) have shown that there are exactly
magnification,abroadeningofthepeakwidthandadisplacement
threedistinctcaustictopologiesforthecaseofanequal-massbi-
ofthepeak,whichmeansthatthemaximumwilloccurlaterwhena
nary lens. Erdl & Schneider (1993) confirmed this to be true for
largersourceentersacaustic(andearlierattheexit).Magnification
arbitrary mass ratios. They also noted the transition points in the
mapsofextendedsourcesfeatureclosedhigh-magnificationlines
binarylensseparationwherethecaustictopologychangesdepend-
that can be easily recognised as originating from the smoothing-
ingonthelensmassratioq(alsocf.Dominik(1999b)).Acaustic
outofcaustics.Thesehighmagnificationlinesasymptoticallyap-
enterstheclose-separationtopologydomainwhens<s ,
c proachthemathematicalcausticsasthesourcesizeshrinkstozero.
1 (cid:32)1–s4 (cid:33)3 Asanaside,introducingathirdlenscanleadtoexceedingly
m m = c, (3) more complicated caustic structures (Rhie 2002). Daneˇk & Hey-
1 2 s8 3
c rovsky´ (2015b,a) have set out to explore the full range of triple-
andwillshowthewide-separationtopologywhens <s, lens caustic topologies. To quote just one very specific example,
w
s2w=(cid:16)3√m1+ 3√m2.(cid:17)3, (4) tahneglceawseitohfttwhoreeeqmuaalsmseasspseossiattio(n1e–dµ)a/t2tahnedtiaptshoirfdamnaesqsuailtaµte,rbaolatsrtis-
10differentcaustictopologies.Manyofthosecanbefoundinother
Thethreetopologies(close,intermediate,wide)areshownin
triple-lensscenarios,butthelistoftenisnowhereclosetocovering
Figs. 2-4 for representative choices of the separation parameter.
thewholerangepossible.
These figures also contain the labels of the notation to be intro-
ducedanddiscussedinSe√ction2.3.Anisolatedpairoflensesclose
to each other (i.e. s < 2/2 for q = 1) result in three caustics
2.3 Notationforcausticelements
(Fig.2):onediamondshapedatthecentreofmass,andtwosmall,
triangular, secondary caustics set off from the binary axis. If the Allextremaofabinary-lenslightcurvecanbetracedbacktofea-
angularseparationofthetwolensesisoftheorderofoneEinstein turesofthecausticofthelenssystem.Wehavedevelopeda“short-
radius, there will be only one central, relatively large, six-cusped hand” notation for these features, sketched in Figures 2, 3 and 4
caustic,seeFig.3.Fort√heequal-massbinarylens“oftheorderof” andlistedinTable1.Inthisstudy,weuseanddepictthisshorthand
meanstheexactrange 2/2<s<2.Ifthetwolensesarefarfrom onlyfortheequal-massbinarylens,butwepointoutitsuniversal
eachother(s > 2),twodiamondshapedcausticsclosetothetrue applicabilitytobinary-lenscausticsofanymassratio.
positionofthelensesresult.Werecollectthatcausticlinesareal- We denote folds of a caustic with a lower-case letter and
waysconcaveinthecoplanarbinarylenscaserelevantforGalactic cusps with an upper-case letter. Local maxima arise either when
microlensingapplications.Petters,Levine&Wambsganss(2001) the source trajectory approaches or crosses a fold or a cusp. We
go into more mathematical detail in describing caustics through discussthepeaktypesindetailinSec.3.1.
singularitytheoryofdifferentiablemaps. Werecallthatthemagnificationofapointsourcedivergesas
(cid:13)c 0000RAS,MNRAS000,000–000
4 C.Liebigetal.
top
Notation Meaning
a,b fold
A,B,C,D cusp
1 nearerbinarymass1
Bt1 bt Bt2 2 nearerbinarymass2
t “above”binaryaxis(i.e.leftofbinaryvector)
b “below”binaryaxis(i.e.rightofbinaryvector)
a a p primarycaustic(inclose-separationcase)
t1 t2 s secondarycaustic(inclose-separationcase)
[a...;[A... causticentry(viafold;viacusp)
...a];...A] causticexit(viafold;viacusp)
1 A A 2 [...a...] foldgrazing(alwaysinside(oron)causticforbinarycase)
1 2 ...A... cuspgrazing(alwaysoutside(oron)causticforbinarycase)
Table 1. Caustic feature notation, also illustrated in the sketches in Fig-
a a ures2,3and4.
b1 b2
takesplace,whenthetwolensesaresetfurtherapart,exceptthatin
b thiscasethe“b”-foldswillmergetoformthenew“D”-cusps.
Bb1 b Bb2 In the close topology, the closer the two lenses are posi-
tioned,thefurtherthetwotriangular,secondarycausticswillmove
out from the axis and they will continually decrease in size and
bottom
strength, whereas the central caustic only decreases in size but
gains in strength, until the binary lens becomes indistinguishable
fromasinglelensfors→ 0.
Figure 3. Caustic feature notation of the intermediate-separation binary
Conversely,inthewidetopology,thetwoarrow-shapedcaus-
lens.
tics become more and more symmetric towards a diamond shape
and decrease in size, until for s → ∞ the B-cusps point perpen-
top diculartotheaxisandtheD-cuspsbecomemoreequalinstrength
totheA-cusps.Ultimatelythetwocausticsshrinktotwopoints,at
whichstagetwoindependentsinglelenseswillbeobservedrather
thanonebinarysystem.
B B
t1 t2 Allpeaksarisingfromfeaturesclosertoorfacingthelenson
1 at1 bt1 bt2 at2 2 theleftsidearefurnishedwithanindex“1”,whereasthosenearer
A D D A therightsidelensareindexed“ ”.Wealsowanttodistinguishthe
1 a b 1 2 b a 2 symmetric caustic features, whi2ch are mirrored across the binary
b1 b1 b2 b2
axis.Quitearbitrarily,wedenotethemwith“”ortop,iftheyare
B B t
b1 b2 on the left-hand side of the binary axis (looking from primary to
bottom secondary) and “ ” or bottom, if they lie on the right-hand side.
b
Figures2,3and4betterillustratethe“logic”behindthischoice.
Inthespecialcaseofanequal-massbinaryunderexamination,
Figure4.Causticfeaturenotationofthewide-separationbinarylens. we have a second symmetry axis through the centre of mass, i.e.
throughthemidpointbetweenthetwolensesandperpendicularto
the binary axis. This does not affect the choice of notation. The
chosen caustic feature notation scheme covers all scenarios with
fc,on/δyifonehitsthecuspalongitsaxisandasfc,off/δy2/3 ifone two point lenses, including mass ratios very different from unity.
hitsitoff-axis(Schneider,Ehlers&Falco1992).Forfoldcrossing,
ThenotationschemeissummarisedinTable1.
themagnificationdivergesasf /δy1/2whenapproachingthesingu-
f
lar line from the inside. Following these basic analytic formulae,
allcuspsarestronglymagnifyingcomparedtotheirimmediatesur-
3 CLASSIFICATIONSCHEMEANDMETHODOLOGY
roundings. The strength of magnification varies considerably be-
tweenonepointonafoldlineandanotherdependingonthefactor Having revisited the basic structure of equal-mass binary lenses
f ,whichbecomesweakeraswemoveoffthebinarylensaxis. and having established an alphanumeric notation to identify ev-
f
Itisthecuspsclosesttothebinaryaxisthatarethestrongestin ery fold and cusp in each of the three topologies, we now move
comparison.Intheequal-massbinarycase,regardlessofthespe- to the classification of microlensing light curves. First, we define
cific topology, the points of maximum magnification are the two a light curve morphology based solely on observable features of
“A”-cusps on the binary axis, followed by those parts of the “a”- lightcurves(Section3.1).Byspanningthewholeparameterspace
foldsclosesttotheaxis. ofbinarymicrolensing,wesimulatelightcurves(Section3.2)and
The four off-axis cusps (“B”) in the intermediate case, cf. assign them to the corresponding morphology class. In this way,
Fig. 3, can be traced across different separations. When the two wecanidentifyeveryregionintheparameterspaceinwhichthe
lensesaremovedclosertogether,thea-foldswilleventuallymerge samemorphologyarisesastheresultoftheencounterofadetermi-
and split the single caustic line into three separate caustics. The natesequenceofcausticfeaturesbythesourcealongitstrajectory
newlycreatedcuspsaredenotedby“C”.Asimilarmetamorphosis (Section3.3).
(cid:13)c 0000RAS,MNRAS000,000–000
Thecatalogueofbinarymicrolensinglightcurves 5
3.1 Thefourpeaktypesinmicrolensing • afoldgrazingcanonlytakeplaceinsideacaustic
• acuspgrazingcanonlytakeplaceoutsideacaustic
Given that the most obvious characteristic of microlensing light
• due to the concave curvature, a straight caustic-crossing tra-
curves is the sequence and shape of their local extrema, this se-
jectorymustexitbyafold(orcusp)differentfromtheonethrough
quenceprovidesanaturaltaxonomickeyforourlightcurveclas-
whichitentered(Cassanetal.2010)
sification scheme. We propose that a class of light curves can be
identifiedbythecommonsequenceofpeaktypes.Wethenrecog- Allbinarymicrolensinglightcurves(intheparameterspace
nisethatanymicrolensinglight-curvemaximumiscreatedbyone consideredinthisstudy)adheretotheserules,butjustconforming
offourbasicmechanisms.Wediscussthefourpeaktypesindetail totheserulesdoesnotguaranteeamicrolensinglightcurvesince
below,butinshortsummarytheyare: thepossiblecaustictopologiesarelimited(Erdl&Schneider1993).
It is well known that similar light curve morphologies may
(i) acuspgrazing(C),
ariseincompletelydifferentsituations,withsourcetrajectoriesin-
(ii) acausticfoldentry(F-)orexit(-F),
teractingwithdifferentcuspsorfoldsindifferenttopologies.Such
(iii) acuspentry(C-)orexit(-C),
disconnectedregionscanbeidentifiedbyspecifyingthefoldsand
(iv) afoldgrazing(-F-). cuspsinvolvedusingthenotationintroducedinSec.2.3.Thenthe
symbolsidentifyingasequenceofpeaksconformingtoaspecific
Now,indetail:(i)thecuspgrazing,C:Thepeakthatariseswhen
morphologyclass(e.g.F-FC),canbereplacedbythecorrespond-
thesourcepassesoutsidethecausticbutcloseenoughtooneofthe
ingcausticelementsinvolved(e.g.[a b]B ).Sincethefoldsand
cuspstopassoverthelobeofincreasedmagnification,isa“cusp t1 t t2
cusp symbols already carry subscripts, in order to generate more
grazing”.Weunambiguouslycallalightcurve“cusp-grazing”,if
reader-friendly sequences, we indicate the caustic entry and exits
the source trajectory is outside the caustic pre and post-peak and
bysquarebracketsandsuppressthebarforthegrazings.Soafold
onlyasinglepeakresults.ThePaczyn´skicurvecanbeunderstood
entry is “[a...”, a cusp entry “[A...”, with the exit being “...]”. A
asagrazingofthepointcaustic(orinfinite-ordercuspoid)ofthe
foldgrazingis“[...a...]”andacuspgrazingisgivenby“A”.These
singlelens.ThenamePaczyn´skicurveshouldbereservedforsin-
notationsdetailingthecausticfeaturesinvolvedinthelightcurve
glelenslightcurvesonly,butinthelimitswhereabinarylensre-
morphologysequencearealsosummarisedinTable1.Wewilluse
semblesasinglelens,whenthesourcedoesnotpassclosetothe
thesyntheticnotation(e.g.F-FC)foridentifyingalightcurvemor-
caustics or when the caustics are very small relative to the solid
phologyclassirrespectiveofitspossibleinterpretationsintermsof
angleofthesource,asingle-peakedlightcurvewillresult.Wedo
notregisteranymorphologicaldifferencetothecuspgrazinginthe sourcetrajectoriesandcausticsinvolved,andthedetailednotation
([a b]B inthisexample)toidentifytheiso-maximaregion(s)in
narrowsense. t1 t t2
(ii)thefoldentry/exit,F-/-F:Whenthesourceentersonacaus- theparameterspacegivingrisetothatspecificmorphology.
To see an example of a light curve classification “at work”,
ticfold,thiscreatesaverydistinctlyshapedcurve(cf.Schneider,
considerthelightcurveinFig.11(h)whereweseea(symmetric)
Ehlers&Falco(1992);Gaudi&Petters(2002)),withasteep,al-
cusp entry (C-) paired with an (asymmetric) fold exit (-F) and a
mostverticalrisefollowedbyamoreparabolicfall,whichdoesnot
post-causticgrazingofthecusplobe(C),
descendaslowasthecaustic-exteriormagnification.Themorphol-
ogyismirroredinthefoldexit.Apairoffoldentryandexitpeaks C-F C .
(cid:124)(cid:123)(cid:122)(cid:125)(cid:124)(cid:123)(cid:122)(cid:125)
giverisetothefamiliardoublecausticcrossingsignature.
caustic cusp
(iii)thecuspentry/exit,C-/-C:Ifthecausticisenteredorexited traversal lobe
alongacusp,thepeakwillhaveamoresymmetricshape,because The detailed sequence specifying the folds and cusps involved in
thelobeoutsidethecausticandthecloseproximityofthefoldlines thislightcurveis[A1at2]A2.
ontheinsideofthecausticattenuatethegradientofthepassageon Fig.12(d)givesaniceexamplewithaclear-cutfoldentry(F-),
bothsides.Thefactthatthemagnificationinthecausticinterioris followedbyasecondpeakstillinsidethecaustic,whichcanonly
increasedcanhelptodistinguishitfromacusp-grazing1. beaninnerfoldapproach(-F-),afoldexit(-F)andfollowedbya
(iv)the“interiorfoldapproach”orfoldgrazing,-F-:Thistype finalcusplobegrazing(C),soweclassifyitas
ofpeakoccursinsidethecaustic,whilethesourcetrajectorypasses
F-F-F C .
closetoacausticfold.Duetotheconcavityofthecausticlines,the (cid:124)(cid:123)(cid:122)(cid:125)(cid:124)(cid:123)(cid:122)(cid:125)
fold-grazingpeakwillonlybeobservedifitisaninteriorapproach. trcaavuesrtsical cloubspe
A special case is the peak that occurs when two or more caustic The detailed sequence specifying the folds and cusps involved in
linesarecloseenoughorstrongenoughtoraisethemagnification thislightcurveis[a a a ]B .
b1 t1 t2 t2
ofanextendedareabetweenthem,givingrisetoapeakthatcannot Keepingthecausticgeometryfixedanddisplacingthesource
bedirectlyattributedtoonesinglefold. trajectory, we can appreciate the changes in the light curve mor-
These“buildingblocks”ofmicrolensinglightcurvescanbe phology,withpeaksmergingordisappearingwhileotherpeaksap-
sequenced,subjecttoafewrules: pearorseparateintwo.Thesetransitionmorphologiesneedsome
furtherattentioninordertobeassignedtospecificclasseswithout
• acausticentrymustbefollowedbyacausticexit2
ambiguities.
• acausticexitcannotoccur,ifthecaustichasnotbeenentered
In this respect, consider the case of Fig. 5, representing the
before
morphingfromtwofoldcrossingsF-FtoacuspgrazingC.When
the extended source trajectory cuts a cusp nearly perpendicularly
to its axis, the light curve features a transition morphology with
1 Mao,Witt&An(2013)haverecentlyshownthatthisisnotnecessarily
asinglepeakprecededandfollowedbyderivativediscontinuities,
thecaseforamulti-planarlensdistribution.
2 Forn>3lensesthenumberofentriesandexitsmaybeunequalascaustic typicaloffoldcrossings(trajectoriesST2andST3inFig.5).Intro-
linescanbeintersectingandnesting.Forn=2,onecausticentrymustbe ducinganewintermediate“cuspcutting”classwouldnotbevery
followedbyonecausticexit,beforeanothercausticentrycanoccur. useful,sincethedetectionofthetwodiscontinuitiesatthebaseof
(cid:13)c 0000RAS,MNRAS000,000–000
6 C.Liebigetal.
ρ=0.01 ρ=0.03
ST1 ST2
ρ=0.05 ρ=0.07
ST3 ST4
Figure6.Classificationinthecaseofabeak-to-beakmetamorphosis.The
magnificationcurvesresultfromthesamesourcetrajectory,butwithdif-
ferentsourcesizes(asindicatedbythewhitecircles).Thesmallestsource
producesanunambiguousfold-grazing,asthecentralpeakoccursinside
thecaustic(C-F¯-C).Interestingly,thelargersourcescreateacentralpairof
Figure5.Comparisonoffold-crossingandcusp-grazingsourcetrajectories
peaksinstead,thusleadingustoclassifythelightcurvesasC-CC-C.This
(ST1to4)andresultinglightcurves.Theangularsourcesizeisindicated
mightseemcounterintuitive,beforeoneconsiderstheconvolutedmagnifi-
bythewhitecircles.Fromlefttoright,thelightcurvemorphologyevolves
cationpattern,whereitbecomesclearthatalargersourceshiftstheposi-
from a double fold crossing F-F for ST 1 to a cusp grazing C (ST 2, 3
tionofthebeak-to-beakfoldmerger–therebycausingthecaustictopology
and4).Whereexactlythistransitionoccursdependsontheangularsource
changetooccuratasmallerseparationcomparedtothesmallersource.
size;withasmallersource,ST2wouldalsoleadtoadouble-peakedfold
crossing.
secanttothefold.Adoptingthesameconventionasbefore,weex-
tendthe“foldgrazing”classtoincludethetransitionpeakoccur-
thepeakcouldneverbeunambiguouslyassessedinrealobserva-
ringwhenthesourcemovestangentiallytothefold,aslongasthe
tions. Only a very detailed analysis of the light curve would dis-
peakremainssingle.
tinguishacusp-cuttingfromacusp-grazingtrajectory.Keepingin
Transitionmorphologiescanbemorecomplicatedthanthese
mindthatthepurposeofourstudyistoidentifyregionsinthepa-
two cases illustrated above and may also involve changes in the
rameter space that may give rise to independent seeds for model
caustic topologies. In Fig. 6, we have a fold-grazing source tra-
searches, we assign these cusp-cutting peaks to the broader cusp jectory C-F¯-C, across a caustic that is close to the limits of
grazingclass,extendingitsdefinitionbyincludingalltrajectories
the intermediate-to-wide transition, which morphs into a cusp
forwhichthecuspcuttingdoesnotgiverisetotwofold-crossing exit/entrypair,C-CC-C,withanincreasedsourcesize.
peaks with a dip in between. In some sense, this statement is al-
In summary, all sorts of transitions can be safely treated by
readycontainedintheabovedefinition,inwhichwerequiredthat
adoptingtheextendeddefinitionsofcuspgrazingandfoldgrazing
thesourceisoutsidethecausticpre-andpost-peakandonlyone
classes just described, including the transition peaks before they
peakoccurs.Thisspecificexampleshouldhelpavoidinganycon-
splitintotwo.Nowwearereadytoapplyourclassificationscheme
fusion.
toarbitrarilycomplicatedlightcurveswithoutfacinganymoream-
Itfollowsthatthepeakclassificationdoesnotjustdependon
biguities.
thesourcetrajectoryrelativetothelenspositions,butequallyonthe
angularsourcesizerelativetothecausticsize.I.e.agivensource
trajectory(e.g.ST2inFig.5)canyieldanF-Fmorphologyfora
3.2 Lightcurvesimulationandprocessing
smallersourceandaCmorphologyforalargersource,whereasfor
agivensourcesizeST1canresultinanF-Fpair,butST2willonly In order to achieve a complete classification of binary lens light
showasinglepeakandbeclassifiedascusp-grazingC. curve morphologies, we process simulated light curves. We then
Anothersituationalmostcomplementarytothepreviousone considerlightcurvesgroupedintheparameterspacebytheirnum-
occurswhenafoldgrazingmorphsintotwofoldcrossingsasthe berofmaxima.Theparameterspacewewanttocoveristheequal-
source trajectory changes from fully internal to tangent and then mass lens (q = 1), the separation s across all topologies and the
(cid:13)c 0000RAS,MNRAS000,000–000
Thecatalogueofbinarymicrolensinglightcurves 7
sourcetrajectoryparameters0≤α<2πandu asfarasnewmor- givesthesymmetryofatimereversal(wherethesignofu hasto
0 0
phologiescanbeexpectedtooccur.Weuseanextendedsourcewith changeaccordingtotheconvention,becausethesourcenowpasses
angular radius 0.002θ . For each light curve we record the num- thelensontheotherside).Wecanalsocombinethetwo,
E
berofpeaksandvisualisetheresultsinpeak-numberplots(overα
(u ,α)⇔(u ,π–α). (7)
and u ). The resulting iso-maxima regions are examined with re- 0 0
0
gardtothecontainedlightcurvemorphologies.Broadlyspeaking, Forthespecialcaseoftheequal-massbinary,wealsohaveaperfect
aniso-maximaregion,coveringa“bundle”ofneighbouringsource degeneracy
trajectories,correspondstoaspecificsequenceofcausticfeatures.
Onestepupintheclassificationhierarchy,differentiso-maximare- u0⇔–u0, (8)
gionsarecollectedinmorphologyclasses(asintroducedinSec.3).
i.e.theplotisaxis-symmetricinu .
0
Thefixedsourcesizeusedinourinvestigationissmallenoughto
We note that whenever one moves from one iso-maxima re-
probethecausticsofanequal-massbinarylensindetailbutlarge
giontoaneighbouringone,themorphologyofthelightcurvepeaks
enoughtoletcuspcrossingsoccurinfiniteregionsoftheparameter
changes–naturally,becausetheborderwillbeoversteppedwhen-
space.Differentchoiceswillcauseaslightshiftoftheboundaries
everapeakiscreatedordestroyed.Werecordthecausticfeature
oftheiso-maximaregions(cf.Figs.5and6).Thispointisfurther
sequenceforthelightcurvesofeachregion,seeTable2,andre-
discussedinSec.5.
alisethatinagivenquadrant,therearenotwoiso-maximaregions
In our examination of the equal-mass binary lens case, we
withthesamenumberofpeaksthatcontainthesamesequenceof
simulatemicrolensinglightcurvesforall(relevant)volumesofthe
causticfeatures.
(s,α,u )parameterspace.Wesimulatethelightcurveswithinverse
0 Wethenmapthecausticfeaturestothebroaderpeaktypology,
rayshooting,usingasoftwarelibrarywrittenin2010byMarnach3.
therebyreducingthecomplexityofthelightcurvedescriptionand
Assumingstaticlenses,thismeanswecancomputemagnification enablingustocollatedifferentregionsinmoregeneralmorphology
mapsforevery(q,s)set,foldthemwiththesourcestarprofilewith
classes.
a radius ρ and then extract a large number of light curves differ-
inginαandu atvirtuallynocomputationalcost.Duringthepeak
0
counting, numerical noise can create artificial peaks and troughs,
4 RESULTS
especially for source trajectories that run at a small angle to fold
lines.Toavoidthese,werequireaminimaldifferencebetweenthe Focussingontheequal-massbinarylens,weanalysedpeak-number
maximumandtheminimaoneithersideof5%ofthenearestlocal plotsspanningallthreecaustictopologiesandthetwotransitioning
minimumvalue,beforeatrough-peak-troughoccurrenceiscounted cases:close(s = 0.5,0.65),close-to-intermediate(s = 0.7),inter-
asapeak.Becauseofthisthreshold,sometimestruepeakswillbe mediate(s = 0.85,1.0,1.5),intermediate-to-wide(s = 2.05)and
disregardedinthemaximacountingalgorithm.Butthisisunlikely wide(s=2.5).AsdiscussedinSec.3.2,weweremotivatedtouse
to make us miss a whole iso-maxima region, as generally the re- anextendedsourcewithanangularradius,ρ = 0.002(inunitsof
gionboundary(wheretheformerlydisregardedpeakbecomessig- θ )andworkwithapeakthresholdof5%abovethenearestmin-
E
nificant)willonlybeslightlyshiftedinthe(u0,α)plane. ima.
Withinthepeak-numberplots,weknowthelightcurvecom-
position in each (substantial) iso-maxima region, i.e. we know
3.3 Iso-maximaregions
whichsequenceofcausticfeaturesproducestheobservablepeaks
Perexaminedseparation,weplotthenumberoflocalmaximaper ofalllightcurvesinthatregion.Wenotethatitismostlyabijec-
light curve over α, u of its source trajectory, see Fig. 7. In the tivemapping,withonlyveryfewregionscontainingmorethanone
0
resultingplot,wecanidentifyandisolateregionsofauniformpeak kindofcausticfeaturesequences.Innocase,dotwounconnected
number,whichwecalliso-maximaregions. regionssharethesamecausticfeaturesequence.
Duetotheinherentsymmetries,wecanrestrictourselvesto The light curves (and iso-maxima regions) are collected in
the first quadrant, 0 < u , 0 < α < π/2. Beyond the trivial pe- morphologyclasses,whereeachpeakismorphologicallyclassified
0
riodicityofαwithperiod2π,thereareseveralsymmetriesinthe asoneofthefollowing:cusp-grazing,cusp-crossing,fold-crossing
two-dimensional(u ,α)space.Generally,forabinarylens, orfold-grazing.Asubstantialsubsetofmorphologyclassescanbe
0
foundinallexaminedseparationsettings.Otherclassesonlyappear
(u0,α)⇔(–u0,–α) (5) whenahigherorlowerseparationleadstomulti-caustictopologies,
whereasthespecificexampleofadoublefoldgrazingisnecessarily
isanexactdegeneracy,whichiscausedbytheintrinsicsymmetry
limitedtotheintermediatecausticcases.
ofthebinarylensacrossthebinaryaxis.Skowronetal.(2009,Ap-
The extreme variety of binary microlensing phenomenology
pendixA)argues(andthishasbeencommonpracticeforawhile,
can be appreciated by summarising the results of our search in a
seee.g.Albrowetal.(1999b))thatmodelsforstaticbinariesshould
few numbers. We have found 73 different light curve morpholo-
beexpressedintherangeu≥0and0≤α<2π,withtheexception
ofcasesthatdisplayparallaxeffectwheretheapparentsourcepo- giesaccordingtoourclassificationbasedonthesequenceofpeaks.
Thesemorphologiesarisein232independentregionsoftheparam-
sitioncanappearonbothsidesofthelens.Wegenerallysubscribe
eterspace.Thesimplestmorphologiescanbeobtainedinmanydif-
tothisview,nonethelessitisinstructiveto,atleastonce,visualise
ferentways.Forexample,thesimplestcausticcrossinglightcurve
the“full”parameterspace,seeSec.4.
class,F-F,canbefoundin7disconnectedvolumesoftheparam-
Sinceweareinterestedinthemorphologyonly,
eter space. If we add shoulders to this caustic crossing, with the
(u ,α)⇔(–u ,α+π), (6) classicalsequenceCF-FC,wefind9differentvolumes.Weem-
0 0
phasise the fact that thanks to our thorough investigation we are
abletoquantifytheexactnumberofindependentphysicalmodels
3 Publishedathttps://github.com/smarnach/luckylensing. that can qualitatively reproduce an observable light curve for the
(cid:13)c 0000RAS,MNRAS000,000–000
8 C.Liebigetal.
Figure7.Plotofthenumberofmaximaperlightcurveinthefirstquadrantofthe(u0,α)parameterspacefortheequal-massbinarylensatseparations=1.0
(intermediatecaustictopology):whitemeansthelightcurvehasasinglepeak,darkbluemeanssixpeaks,highervaluesarenumericalnoiseinthisinstance.
Eachlabelledregionrepresentsagroupingofsourcetrajectoriesandcorrespondinglightcurvesthatfollowaspecificcausticfeaturesequence,seeTable2.
Rarelyaretworegionswiththesamenumberofpeaksdirectlyconnected(cf.IIIbandIIIgabove).
first time! More complicated morphologies with multiple caustic asdiscussedinSec.3.Apairoffoldcrossingscanbemergedinto
crossings are rarer and appear in fewer regions of the parameter asinglepeak,awholecausticcanbecrossedandappearasasin-
space. A microlensing light curve for an equal-mass binary lens glepeak,butaslongasthesolidangleofthesourceareaissmall
canhaveupto10peaks,ifthesourcemovesneartheverticalaxis relativetothecausticextent,theabsolutesizewillnotchangethe
ofacloseconfiguration. numberofdistinctmorphologiesthatcanbestudied.Forthestud-
iedmassratioq=1,wecanaffordtouseamoderatelylargesource
thatreducesthenumericalnoiseinoursamples.Meaningfulstud-
iesofplanetarymassratiosq(cid:46)10–3,requireasmallersourcesize.
5 FURTHERCONSIDERATIONS
WepointoutthatnotallofthepeaktypesofSec.3.1canbesim-
Source size A hypothetical point source is often useful in theo- ulatedwithapointsource:thecuspcrossingcanonlyoccur,ifthe
reticalstudiesofthebehaviourofgravitationallenses,butbecause pointsourceentersthecausticexactlyovertheinfinitesimalcusp
wewanttoexaminetherangeofreal,observablelightcurvemor- point.Theprobabilityforthisoccurrenceisthereforezero.
phologies,weuseanextendedsourcesizeof0.002θ foroursim-
E
ulations.Thesourcesizedoesinfluencetheshapeofalightcurve, Twopeakswillgenerallymergeintoone,iftheirangularsep-
(cid:13)c 0000RAS,MNRAS000,000–000
Thecatalogueofbinarymicrolensinglightcurves 9
Regionlabel Causticfeaturesequence Morphologyclass
Ia A1orBt1 C¯
IIa [A1A2] C-C
IIb [at1at2] F-F
IIc Bt1Bt2 C-C
IId [ab1at2] F-F
IIe [at1bt] F-F
IIf [ab1bt] F-F
IIg A1Bt1 C-C
IIh [bbbt] F-F
IIi [bbbt] F-F
IIIa [A1at2]A2 C-FC¯
IIIb A1[at1at2] C¯ F-F
IIIc [at1btat2] F-F¯-F
IIId A1[ab1at2] C¯ F-F
IIIe [at1bt]Bt2 F-FC¯
IIIf [ab1at1at2] F-F¯-F
IIIg A1[at1Bt2] C¯ F-C
IIIh A1[at1bt] C¯ F-F
IIIi [ab1at2]Bt2 F-FC¯
IIIj [ab1at1Bt2] F-F¯-C
IIIk [ab1bt]Bt2 F-FC¯
IIIl [ab1at1bt] F-F¯-F
IIIm Bb1[ab1bt] C¯ F-F
IIIn [bbab1bt] F-F¯-F
IIIo [ab1at1]Bt1 F-FC¯
IIIp Bb1A1B1 C¯ C¯ C¯
IVa A1[at1at2]A2 C¯ F-FC¯
IVb [at1bt][btat2] F-FF-F
IVc A1[ab1at2]A2 C¯ F-FC¯
IVd A1[at1bt]Bt2 C¯ F-FC¯
IVe [ab1at1][at1at2] F-FF-F
IVf A1[at1at2]Bt2 C¯ F-FC¯
IVg [ab1at1at2]Bt2 F-F¯-FC¯
IVh [ab1at1][at1Bt2] F-FF-C
IVi [ab1at1bt]Bt2 F-F¯-FC¯
IVj Bb1[ab1at2]Bt2 C¯ F-FC¯
IVk [ab1at1][at1bt] F-FF-F
IVl [bbab1][ab1bt] F-FF-F
IVm Bb1[ab1at1bt] C¯ F-F¯-F
IVn [bbab1at1bt] F-F¯-F¯-F
IVo Bb1[ab1at1]Bt1 C¯ F-F¯-F
Va [ab1at1][at1at2]Bt2 F-FF-FC¯
Vb [ab1at1][at1bt]Bt2 F-FF-FC¯
Vc Bb1[ab1at1][at1bt] C¯ F-FF-F
Vd [bbab1][ab1at1bt] F-FF-F¯-F
Ve Bb1[ab1at2][at2bt] C¯ F-FF-F
VIa [bbab1][ab1at1][at1bt] F-FF-FF-F
Table2.Causticfeaturesequencesfortheiso-maximaregionsinFig.7(q= Figure8.Comparisonofpeak-numberplotsresultingfromdifferentsource
1.0,s=1.0).Eachsequenceisuniquetoitsregion,whilethemorphology sizes(fromtoptobottom:ρ = 0.005,0.01,0.02θE),scaleandrangesas
classesgenerallyspanseveralindependentregions. inFig.7:x-axis:0 ≤ u0 ≤ 1.0,y-axis:0.0 ≤ α ≤ π/2.Thechangein
iso-maximaregionsissubtle,butnoticeable.Thesmallestsourcenotonly
leads to more iso-maxima regions, but also to more numerical artefacts.
AlsocompareFig.7,whereρ=0.002θE.
aration is smaller than the angular source diameter (disregarding
limb-darkeningeffects).Inoursimulationsthesourcehasadiame-
terof4×10–3θ ,i.e.peakswithin4×10–4t ofeachotherwouldbe Errormargin Whileweaimforcompleteness,duetothenumer-
E E
missed.Weworkwiththeassumptionthatalargersourcesizecan icalnatureofourstudywehavetoignoreverysmallsub-regions
onlyleadtoasmallernumberofidentifiedmorphologies.Thishas of the studied parameter space and therefore might have missed
beenvisuallydemonstratedforq = 1.0,s = 1.0inFig.8.Liebig out on a particular light curve morphology. Within this space we
(2014)alsodocumentstheentiretyofmorphologicalclassesfora haveexaminedalliso-maximaregionslargerthan10by10pixel,
source radius of 10–2θ and they are a subset of the morphology i.e.102×1/(u -sampling)× π/(α-sampling),meaningthatwithina
E 0 2
presentedhere. givenEinsteinradiusandwithoursamplingof1600,theprobabil-
(cid:13)c 0000RAS,MNRAS000,000–000
10 C.Liebigetal.
or fold-grazing. In order to achieve this complete classification,
we have developed a general notation scheme for the features of
binary-lenscaustics.Ourtool,plotsofpeaknumberoveru andα,
0
servestoprovideinsightintothethemicrolensingparameterspace.
Before this work, statements of the diversity of binary mi-
crolensing light curves were only made on reasonable but vague
arguments.Withourdetailedstudyweareabletoassignnumbers
toallspecificcasesandopenthewaytomorequantitativestudies
ofbinarymicrolensing.
Apartfromthepuretaxonomicalaspects,whichareveryin-
terestingfromthetheoreticalpointofview,Table3standsoutasa
verypowerfultoolformodellerstorelateanobservedlightcurveto
allpossibleregionsoftheparameterspaceinwhichthislightcurve
canbefound.Thiscapabilitywouldhelptheconstructionofmore
fail-safealgorithmsthatwillguaranteeafullexplorationofthemi-
crolensingparameterspace.Inpractice,seedsforfittingalgorithms
canbeplacedinthemiddleofeachiso-maximaregionsoastoen-
sureafullexplorationofallpossiblecases.Amongthecurrently
runningplatformsformodellingusingthisprincipleforsettingini-
tial conditions, we mention RTModel (Bozza 2010; Bozza et al.
2012).Theinclusionofourcatalogueinthetemplatelibrarycon-
Figure9.Extendedplotoftheiso-maximaregionsfors = 1.0toillus- sultedbyRTModelwouldfurtherdiminishthechancesofmissing
trateexistingsymmetriesandtheseamlesscontinuationofiso-maximare- anyparticularregionintheparameterspace.
gionsbeyondthefirstquadrant.Wehavemarkedacausticfeatureregion:
Anotherinteresting aspectthat canbe furtherinvestigated is
thegreenoutlineframestheareawheretheat1 foldgivesrisetoalight the probability of the occurrence of a given morphology. Having
curvepeak,morespecificallythetopandbottomregionscontainthefold
tracedtheiso-maximaregionsintheparameterspace,itshouldnot
entry [at1...] whereas the middle region contains the fold exit [...at1]. be difficult to translate the volumes of the iso-maxima regions in
Thegreenshademarksareaswheretheat1 foldiscrossedtwice(requir-
ing [...at1][at1...] to be part of the light curve). Moving to a slightly probabilitiesnormalisedbyaphysicallymotivatedmeasure.Inthis
smalleru0fromtheshadedarea,thelightcurveswilldisplaythefoldgraz- way,wewouldbeabletoquantifyhow“rare”orcommonamor-
ing[...at1...]. phologyis.Thecurrentiso-maximaplotswouldalreadysufficefor
probabilities at fixed lens separations. However, for a more com-
pleteandusefulstudy,oneshouldmovethroughdifferentlenssep-
itytoobservethatparticularlightcurvemorphologyissmallerthan arationswithamuchsmallersamplingstep,soastocharacterise
(cid:46)1/16000. theshapesandthevolumesoftheregionsinamoreaccurateway.
Furthermore, the final result would depend on the assumed prior
Caustic feature regions It is highly instructive to consider the distribution function for the separations of binary systems. Sum-
completeparameterspacevolumethatcorrespondstoapeakcre- mingup,thestudyoftherelativefrequenciesofthedifferentmor-
atedbyaparticularcausticfeature.Thiscausticfeatureregionwill phologyclassesiscertainlyoneofthemostinterestingdirections
coverseveraliso-maximaregions,wherethelightcurvesshowone openedupbyourwork,whichdeservesthegreatestattentionand
ortwopeaksduetothisparticularcausticelement,seeanexample anadequatespaceindedicatedfutureworks.
inFig.9.Ifthisinformationcontentcouldbeproperlyharnessed, Wehaveonlyverybrieflymentionedtheexistenceofcaustic-
itwouldprovideanimmediatekeyforthemappingofthelenssys- feature regions as “meta regions” to the iso-maxima regions, i.e.
temtothelightcurveandfromthelightcurvemorphologytothe thecombinationofalliso-maximaregionscontainingonespecific,
caustic. caustic-relatedpeak.Unfortunately,wehavenotyetfoundagood
Forcaustic-crossingpointsources,thisproblemoftenreduces way to extract and preserve the information about these meta re-
to registering the intersection points of the straight source trajec- gions,butinfacttheycanprovideamorefundamentalunderstand-
torywiththecaustic,whichisamathematicallywell-definedprob- ingoftheparameterspace,sinceiso-maximaregionsarebasically
lem. However, for non-caustic crossing peaks (i.e. cusp and fold just“stacks”ofcaustic-featureregions.Incontrasttoiso-maxima
approaches),causticlinesdonotprovidesufficientinformation.It regions,caustic-featureregionsaresmoothstructuresand,likethe
wouldbenecessarytostudythemagnificationmaparoundcaustics causticstheyarederivedfrom,theychangecontinuouslyoverthe
in order to pin down the position of the maximum magnification parameterspace.Iftheirboundariescouldbeanalyticallyderived
alongthesourcetrajectoryandthenassignthispositiontoanearby fromthecausticlines,anelegantautomaticclassificationcouldbe
caustic fold or cusp for classification. This approach appears too achieved.
complextoimplementinpractice. Finally, we must remember that our work is limited to the
equal-mass static lens case. Higher order effect such as parallax
andorbitalmotionwoulddramaticallyincreasethecomplexityof
the classification, adding very few new morphologies (at least in
6 CONCLUSIONANDFUTUREPROSPECTS
reasonable physical cases) and would mainly distort existing iso-
We have compiled an unprecedented catalogue of microlensing maximaregions.Theonlyreallyrelevantandhumanlyachievable
light curve morphologies for the equal-mass binary lens. We re- upgradeofourcatalogueshouldincludeavariablemassratio.
alisedthatallpeaksinmicrolensinglightcurvescanbeclassified While previous literature has shown only up to three pairs
in just four categories: cusp-grazing, cusp-crossing, fold-crossing ofcausticcrossingsforasinglemicrolensingcaustic(e.g.Cassan
(cid:13)c 0000RAS,MNRAS000,000–000