2 O 02MNRAS.336. .670S 4 2 2 43, 123 1 *E-mail: [email protected](MCS), [email protected](SM),edu.pl(MS),[email protected] (MK),[email protected](GP), [email protected] (PW),[email protected] (AU),[email protected]@astrouw.edu.pl(IS), [email protected] (KZ). Accepted 2002June24.Received21;inoriginalformFebruary11 UniversidaddeConcepcion,DepartamentoFísica,Casilla160-C,Concepción,Chile Los AlamosNationalLaboratory,MSD436,Alamos,NM87545,USA Mon. Not.R.Astron.Soc.336,670-684(2002) WarsawUniversityObservatory,Al.Ujazdowskie4,00-478Warszawa,Poland M. Kubiak,*G.Pietrzyñski,I.SoszyñskiandK.Zebrun Martin C.Smith,*'ShudeMao,P.Wozniak,A.Udalski,M.Szymañski, multipeak parallaxevent Optical gravitationalleasingexperiment:OGLE-1999-BUL-19-thefirst UniversityofManchester,JodrellBankObservatory,Macclesfield,CheshireSK119DL © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem -1 -1 binary-source signatureinadditiontotheseparallax-inducedmultiplepeaks.Withspectro- rather thanmicrolensingbyabinarysource.However,thiseventmaybeexhibitingslight parallax microlensingeventcanbeexactlyreproducedbyasuitablebinary-sourcemodel.A better fittedbyarotatingbinary-sourcemodel,althoughthisistobeexpectedsinceevery radius crossingtimeforthiseventisapproximately1yr,whichunusuallylong.Weshow We describeahighlyunusualmicrolensingevent,OGLE-1999-BUL-19.Unlikemoststan- bulge. that themajorityof/-bandblendingiscontributedbyasourceroughlyalignedwith mining thelensmass.Wealsoinvestigatednatureofblendingforthisevent,andfound the instancethathypothesisturnsouttobecorrect)simultaneouslyfitbothmodelsand multiple-peak parallaxeventeverobserved.Wealsofoundthatthecanbesomewhat This valueforvisthelowestvelocitysofarpublishedandwebelievethatthisfirst tive transversevelocityofthelensprojectedontoobserverplane(v^12.5=b1.1kms). that theadditionalpeaksinlightcurvecanbecausedbyverysmallvalueforrela- ABSTRACT kinematics anddynamics. ics andunderstandingtheoriginsoftheseeventsexcess oflongeventstowardthe lensed source.Thisimpliesthatmostofthe/-bandblendingiscausedbylightfromlensor face-on rotatingbinary-lensmodelwasalsoidentified,butthisprovidesasignificantlyworse dard microlensingevents,thiseventexhibitsmultiplepeaksinitslightcurve.TheEinstein Key words:gravitationallensing-binaries:generalGalaxy: bulge -Galaxy:centre very usefulforunderstandingthenatureofblends.Wealso suggestthataradialvelocity a binarycompaniontothesource.However,inVband,there appearstobeasecondblended obtain ameasurementofthelensmass.Furthermore,spectroscopicobservationscouldalso fit. Weconcludethatthemostlikelycauseforthismultipeakbehaviourisparallaxmicrolensing Since thisvalueissignificantlylessthanthespeedoforbitEartharoundSun survey ofallparallaxeventswillbeveryusefulforfurtherconstraining thelensingkinemat- source 0.35arcsecawayfromthelensedsource.HubbleSpace Telescopeobservationswillbe supply informationregardingthelensproperties,possiblyprovidinganotheravenuefordeter- scopic observationsitispossibletotestthis‘parallaxplusbinary-source’hypothesisand(in (n ~30kms),themotionofEarthinducesthesemultiplepeaksinlightcurve. 0 © 2002RAS 2 O 02MNRAS.336. .670S piece ofinformationisrequired,suchastheangularEinsteinra- partially liftsthelensdegeneracy,butisnotenoughtouniquely best parallaxmodel,whileinSection 4weexplorewhetherOGLE- paper isasfollows.InSection2we presenttheobservationaldata, per istopresentadetailedanalysis ofthisevent.Theoutlinethe black holesintheMilkyWay. be theonlymethodtoprovideacompletecensusofmassive the datareductionprocedureandour methodtoselecttheparallax multipeak signature.Sucheventshavebeenpredictedby in Wozniaketal.(2001),oneeventhasbeenuncoveredthatcanbe tified byaslightasymmetryinthelightcurvecausedmotion larly excitingbecausetheseblackholesmaybeoutsidethegaslayer intervening blackholes(seealsoAgoletal.2002).Thisisparticu- time andthekinematicsimplythatlensesareverylikelytobe lens tobedrawn.Sofarapproximately10microlensingparallax for thelenskinematics,whichallowsimportantconstraintson further informationtheparallaxeffectcanbecombinedwithamodel Delplancke, Górski&Richichi2001).However,intheabsenceof the availabilityofsuitableopticallong-baselineinterferometry(see lensed object,whichshouldbecomepossibleinthenearfuturewith this degeneracyproblemincluderesolvingthecomponentsof Hubble SpaceTelescope(HST),forexample].Otherapproachesto this instanceisverysmallandrequiresconfirmation[thiscanbe microlens propermotion.However,thismassdeterminationrelies to havemadeadeterminationofthemicrolensmassfordifferent lens-mass degeneracyandprovedtobeoneofthefirstevermea- in acaustic-crossingbinary-lensevent;thiscompletelybrokethe transverse velocity)intheeclipticplane.Thisadditionalconstraint low onetoderivetheprojectedEinsteinradius(orequivalently, these exoticmicrolensingevents(Gould1992).Theseal- the lensdistanceandmasscannotbeuniquelydetermined(see ter (Paczynski1986),theseeventshavealsodevelopeddiverse known. Inadditiontotheoriginalgoalsearchfordarkmat- At thetimeofwriting,morethan1000microlensingeventsare Gould, Miraida-Escude&Bahcall (1994).Thepurposeofthispa- very wellfittedbytheparallaxmodel.Thiseventshowsastriking et al.(2002),whichanalysedthe520microlensingeventspublished 2002). Threeoftheseeventsareparticularlyinteresting(Bennettet events havebeenfound(Alcocketal.1995;Mao1999;Bennett done byobtainingameasurementoftheastrometricparallaxusing event byutilizingmeasurementsofboththeparallaxeffectand dius. Thisisarareoccurrence,butstrikingexampleofthiscanbe determine themassofleasingobject;todothissomeother events arewelldescribedbythestandardshape(e.g.Paczyñski events fromthemicrolensingdata base. InSection3wepresentthe © 2002RAS,MNRAS 336,670-684 of theEartharoundSun.DuringsystematicsearchSmith any otherwavelengthssuchasX-rayandradio.Microlensingmay of thedisc,andhencehavenoaccretionsignaturesfordetectionin al. 2001;Maoet2002)becausethelongEinsteinradiuscrossing al. 2001;BondetMao2002;Smith,&Wozniak on theirphotometricmeasurementoftheparallaxeffect,whichin of microlensing. Section 2).Thisdegeneracyisoneofthemajorobstaclesinstudies applications (seePaczyñski1996forareview).Mostofthese surements ofthemicrolensmass.Alcocketal.(2001a)alsoclaim seen inAnetal.(2002),whereparallaxsignatureswereobserved standard shape.Theparallaxmicrolensingeventsareoneclassof 1 INTRODUCTION 1986). Unfortunately,fromthesestandardmicrolensinglightcurves, Fortunately, somemicrolensingeventsshowdeviationsfromthe Nearly allofthepublishedmicrolensingparallaxeventsareiden- © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem OGLE-1999-BUL-19 -thefirstmultipeakparallax671 hms 2 1 2 --1 -1 2 / 2 1 roughly 200-300framesperfieldduringobservingseasons1997- binary-lens models.Section5discussesthevariousapproachesthat parallax modelforthisuniqueevent, whileinSection4weexplore region. belled sc40_2895.ThepositionofthestarisRA=17510.76 refer tothiseventasOGLE-1999-BUL-19,althoughinthediffer- in thedrift-scanmode.Thepixelsizewas24jam,givingascale is operatedbytheCarnegieInstitutionofWashington.Thetele- telescope attheLasCampanasObservatory,Chile.Theobservatory the secondphaseofOGLEexperimentwith1.3-mWarsaw The observationspresentedinthispaperwerecarriedoutduring AND SELECTIONPROCEDURE 2 OBSERVATIONS,DATAREDUCTION http://www.astro.princeton.edu/~wozniak/dia/lens the alternatemodels. Reassuringly, thisconfirmedtheexistenceofahugespike,already to testthis,wesubsequentlyanalysedthedatafromthisseason. the parallaxmodelpredictsahugespikein2000season.Inorder this figureitisclearthatthestarlocatedinred-clump Fig. 1showsthecolour-magnitudediagramforstarswithin the lensedstarplusblend(s)isapproximately/^16.07.Theav- this cataloguewastoidentifypotentialparallaxmicrolensingevents, Wozniak (2000)forfurtherdetails.Theaimofthesearchthrough ing thedifferenceimageanalysistechnique,isavailableelectroni- teo &Saha1993)isavailablefromtheOGLEwebsite. were performedinthe‘medium’speedreadingmodewithagain 2048 xpixelcharge-coupleddevice(CCD)detectorworking discuss ourresults. can beutilizedtoprovideameasurementof(orstrongconstraints curve. Inthenextsectionwepresentbothbeststandardand erage V—Icolourofthecompositeisapproximately2.63±0.05. 268? 126and—9?637,respectively.Thetotal/-bandmagnitudeof / =357?077,&—3?147,andtheeclipticlongitudelatitudeare ence imageanalysiscatalogueofWozniaketal.(2001)itisla- cally andinterestedreadersarereferredtoWozniaketal.(2001) catalogue ofmicrolensingeventsthathadbeencompiledfromthe observing techniques,andthedophotphotometry(Schechter,Ma- of 7.1eADUandreadoutnoise~6.3.Detailsthein- of 0.417arcsecpixel.ObservationstheGalacticbulgefields on) thelensmass.Finally,inSection6,wesummarizeandfurther are showninFig.2.Intotal,there317datapointsthelight System. Thefour-seasondatafromthedifferenceimageanalysis a fieldofview5x5arcminaroundOGLE-1999-BUL-19.From and Dec=—33°0344.l(J2000).TheGalacticcoordinatesare System asOGLE-1999-BUL-19.Throughoutthispaperweshall and thedetailsofthisprocedurecanbefoundinSmithetal.(2002). 3-yr OGLE-IIbulgedata.Thiscatalogue,whichwasgeneratedus- ( 1997).ThemajorityoftheOGLE-IIframesweretakenin/band, strumentation setupcanbefoundinUdalski,Kubiak&Szymañski scope wasequippedwiththe‘first-generation’cameraaSITe seen (butunknowntothemodellers)byOGLEEarlyWarning (available on-line;Wozniaketal.2001).However,wenoticedthat http://www.astrouw.edu.pl/~ogle/ogle2/ews/1999/ews.html 1999. Udalskietal.(2000)givesfulldetailsofthestandardOGLE 1999-BUL-19 canbefittedbyalternaterotatingbinary-sourceand Initially, weanalysedjustthethree-seasondatafrom1997to1999 In fact,thiseventwasfirstdetectedbytheOGLEEarlyWarning OGLE-1999-BUL-19 wasidentifiedduringasearchthrough 2 O 02MNRAS.336. .670S 2 3 plane isgivenby nification), rtheEinsteinradiusprojectedontoobserverplane, The standardlightcurve,A{t)isgivenby(e.g.Paczyñski1986) microlens model.Inthismodelthe(point)source,lensand First, OGLE-1999-BUL-19isfittedwiththestandardsingle- 3 STANDARDANDPARALLAXMODELS the fullsequenceof/-bandsubframesforthisevent,finding Figure 1.Thecolour-magnitudediagramofthe5xarcminfieldaround dius crossingtime.TheEinsteinradiusprojectedontotheobserver where wistheimpactparameter(inunitsofEinsteinradius) difference imageanalysisdatathatwasusedtomodelthisevent. chart forthestar,dophotphotometry,andfullfour-season an asterisk.Thestarislocatedinthered-clumpregion. OGLE-1999-BUL-19. ThepositionofOGLE-1999-BUL-19ismarkedwith http://bulge.prmceton.edu/~ogle/ogle2/OGLE-1999-BUL-19.html with tobeingthetimeofclosestapproach(i.e.maximummag- and A(0 = observer areallassumedtomovewithconstantspatialvelocities. sight, alsoprojectedontotheobserverplane,andtEinsteinra- 672 v thelenstransversevelocityrelativetoobserver-sourcelineof r(t) = E 0 E 3 The dataforthiseventcanbefoundon-line.Thissiteincludes 2 2 t-k u^Ju +4 4GMDx 2 c(l —x)’ s ÍE w +2 M. C.Smithetal. © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem 2 u(t) =\Ju'q-fir(t), (3) (2) (1) 2 4 25 4 5 reference image(/).Allthefluxeshereareinunitsof10ADUand be writtenas properties ist.Thismeansthatforagivenvalueofonecannot x =D\/Distheratioofdistancetolensand behaviour shownbythedata.The/perdegreeoffreedomisgreater Best-fitting parameters(andtheirerrors)arefoundbyminimizing need thefiveparametersgivenabove:/,A/(or)uq,toandt. the baselinefluxoflensedstarplusfromlensand/or is thedifferencebetweenbaselineflux(/)andof infer v,Mandvuniquely,evenifthesourcedistanceisknown. measurable quantitythatholdsanyinformationconcerningthelens to thesource.Equations(l)-(3)showwell-knownlensdegen- m/(0=m/,-2.51og[/(0//R], where /=1048.01andm/, f, theEinsteinradiusprojectedontoobserverplaneandi/u meters torepresentthelenstrajectory.Wedescribetrajectory Earth willhavelesseffectthaniftheprojectedEinsteinradiusis to thisprojectedEinsteinradius.Therefore,iftheEin- important quantityisthemagnitudeofEarth’smotionrelative length-scale onwhichthemagnificationiscalculated,andso magnitude oftheparallaxdeviationsissizeEinsteinra- to beespeciallyprominentforeventswheretherelativetransverse the lightcurve,andthiscanbecomeimportantforeventswithtime- The centripetalaccelerationoftheEarthinducesaperturbationon the modelthatobservermoveswithaconstantspatialvelocity. its orbitaroundtheSun.Thisinvalidatesstandardassumptionof than 50,andsothismodelcanbediscountedunequivocally. Table 1.Clearly,thismodelcomprehensivelyfailstoreproducethe the dottedlineinFig.2,andfitparametersarepresented the usualxusingminuitprograminCERNlibrary. et al.2001).Tofitthis/-banddatawiththestandardmodel,we can bebrighterthanthetruebaselineimage(/r>/o).ForOGLE- does nothavetobezeroorevenpositiveasthereferenceimage given inWozniaketal.(2001).Thisvalue/iscomposedofboth can beconvertedinto/-bandmagnitudesusingthetransformation where /isthebaselinefluxoflensedsourceandA/=— eracy instandardmicrolensinglightcurves.Inthismodeltheonly where Misthelensmass,Ddistancetosourceand http://wwwinfo.cern.ch/asd/cernlib/ comparable to1au. dius projectedontotheobserverplane.Thisradiusdetermines velocity ofthesystemissmall(i.e.comparabletoorbitalspeed days, duringwhichtimetheEarthwillhavemovedsubstantiallyin can beseenfromthelightcurve,tisinregionofafewhundred any unlensedblendedstar(s),ifpresent.NotethatingeneralA/ an angleintheeclipticplanedescribing theorientationoflens advocated byGould(2000),andusethetwoadditionalparameters: of thelensineclipticplane,followingnaturalformalism of theEarth),sincethismeansthatorbitEarthhasamore allax effect,sincethedurationofeventisparticularlylong:as stein radiusissignificantlygreaterthan1authenthemotionof significant effectonthetrajectory.Anotherfactorthataffects scales greaterthanafewmonths.Parallaxdeviationsareexpected /(0 =/[A(0-l]+A/,(4) 15.86 arethereferencefluxand/-band magnitude,respectively. 1999-BUL-19, thereferenceimagefluxis/=1048.01(Wozniak R E s s0E 0 R10 R E 0 s0R s E s R The fluxdifferenceobtainedfromimageanalysiscan The descriptionoftheparallaxeffectrequirestwoadditionalpara- The nextlogicalstepistoattemptafitthatincorporatesthepar- The best-fittinglightcurveforthestandardmodelisshownby e 2002RAS,MNRAS 336,670-684 2 O 02MNRAS.336. .670S 2 parameters areagainfoundbyminimizingx-Notethatthedefini- parameters havebeendeterminedthenthetrajectoryoflensin two parametersnolongerhavestraightforward intuitivemeanings trajectory forpositivevaluesofmq. Unlikethestandardmodel,these now correspondstotheclosestapproach ofthelenstrajectoryto tions ofMoandtoareslightlydifferent fromthestandardmodel:uq to searchforanydegeneracyintheparameterspace.Thebest-fitting ues ofrandÿarearbitrarilychosenforanumbercombinations model). Thelightcurvecanthenbecalculatedfromthisseparation. the quantitythatisanalogoustouqparameterfromstandard to bemadeoftheseparationbetweenlensandobserver(i.e. the eclipticplaneiscompletelydetermined,allowingacalculation including afigureillustratingthegeometryofsituation,canbe trajectory (givenbytheanglebetweenheliocentriceclipticv- the twounusualbumpsinthislightcurve,clearlyshowinghowsuccessfulparallaxmodelisfittingthesefeatures.Thetoppanelshowsresidual flux magnitudes usingthetransformationgiveninSection3.Thedottedandsolidlinesareforbeststandardparallaxfits,respectively.insets detail Figure 2.The/-bandlightcurveforOGLE-1999-BUL-19fromdifferenceimageanalysis,withfluxgiveninunitsof10ADU,whichcanbeconverted into guesses fortheparameters/,A/,uq,toandt.However,initialval- found inSoszyñskietal.(2001)andwillnotberepeatedhere(see goodness oftheparallaxfit.ThelargescatterinearlypartfourthseasondataisexplainedSection3text. © 2002RAS,MNRAS 336,670-684 Uq isdefinedsuchthattheSunlies ontheleft-handsideoflens Sun intheeclipticplane,andtois the timeofthisclosestapproach. also Alcocketal.1995;Dominik1998a).Oncethesetwoadditional axis andthenormaltotrajectory).Thedetailsofthisprocedure, (the observeddatapointssubtractedbytheparallaxmodel).Thefactthattherearenosignificantsystematicdeviationsfromhorizontalaxisindicate the E sE The best-fittingvaluesofthestandardmodelaretakenasinitial 2 P 1628.43±0.21372.0±3.3-0.469±0.004706.0+^3-188.01.13.1369^}?2.684+¡];¡]g590.1/310 5 1628.146±0.044166.9±1.60.1075±0.0014783±11-124.66±0.88-17912/312 Model toA:(d)uqfAf\f/?e(au)x/dof in Section3. Table 1.Thebeststandardmodel(firstrow)andtheparallax(secondforOGLE-1999-BUL-19.parametersareexplained s © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem OGLE-1999-BUL-19 -thefirstmultipeakparallax JD -2450000 2 2 2 2 parallax model.Anyproblemswith theparallaxfitwouldmanifest plane andhencedoesnotmatchthepeakoflightcurve. ple, theclosestapproachoflenstrajectoryineclipticplane because ofgeometricprojectionsandtheparallaxeffect.Forexam- This xvalueof590fortheparallax fitcorrespondstoaxperde- from aslightunderpredictionofthe fluxinthesecondseason, themselves assystematicdeviations fromthehorizontalaxis;apart Fig. 2,whichshowsthedifferencebetweendatapointsand insets onthebottompanelofFig.2),resultinginawhollyinfeasible two smallbumpsoneithersideofthemainpeak(shownin under 18000.Thisimprovementcanclearlybeseenfromthelight in thexvalue.Thenewfithasaxvalueof590for310degrees gree offreedom1.90,aformally unacceptablevalue.However, flux consistentlymatchesthevalue predictedbytheparallaxmodel. curve plottedinFig.2.Thestandardfitisunabletoreproducethe does not,ingeneral,correspondtotheclosestapproachlens both bumpsaresuitablyfitted. of freedom(dof),comparedwiththevaluestandardfitjust X value.However,theparallaxmodelhasnosuchdifficultiesand The accuracyoftheparallaxfitishighlightedintoppanel The inclusionoftheparallaxeffectresultsinadramaticreduction 673 2 O 02MNRAS.336. .670S 2 parallax modelpredictsthatthelensedstarcontributes82percent reduces thesizeofphotometricerrorbars.Thetotal/-bandbase- parallax deviationsareespeciallypronounced.Anotheraspectof parameters, andmostnotablythosedescribingthelenstrajectory residual plot,whenconvertedfrommagnitudesintocounts,shows reduction (i.e.differenceimageanalysis),wemadeaparallaxfitto possible explanationmightbethelargerthanaverageextinctionin part oftheBUL_SC40field.Theerrorbarsmaybeunderestimated pressed incountsshouldriseasthesquarerootofmagnifiedflux. large as1.4M©. then jri=0.125mas,andthiswould implyalensmassofaround lens islyinghalf-wayin-between(i.e.A^8kpcand4kpc), lens-source distance(seeSoszyñskietal.2001;Gould2000), light curvethiscanonlybeexpressedasafunctionoftherelative From theseanexpressionforthelensmasscanbedetermined, f =2.68±0.23au,t372.03.3d.(5) than 10(A^10.2). line magnitudeforthiseventisapproximately16.07mag,andthe nature oftheparameterswasbrightnessevent,sincethis the lightcurvethatprobablycontributedtotightlyconstrained is basedsolelyonthedifferenceimageanalysisdata. marginally largeroverallscatter.Therefore,thefollowinganalysis the regularphotometryobtainedwithdophot.Thecorresponding kernels andpointspreadfunctions.Toensurethattheseobserved this field,contributingtoalowersignal-to-noiseratiointhefitted in thisregion,whichthenresultanartificiallylargexvalue.A not fullyreflectthephotometricuncertaintyforbrightstarsinthis the rmsscatterisapproximately1percentinpresentdata,fully in Wozniak2000).Nearthepeakofeventat/^13.8mag, II dataareaccountedforbyasimplerescalingoftheerrorbars(fig.3 in mostground-basedCCDimaging.MostoftheseeffectsOGLE- However, therearealsoadditionalsourcesofnoiseforbrighterstars from thetoppanelofFig.2itisclearthatmostexcessscatter clump starinthebulge,soitsdistanceisapproximately8kpc.If field. Theproblemisthattheaveragescalingcurveforallfieldsmay consistent withthescatterforothercomparablybrightobjectsinthis was stronglymagnified.Obviously,thePoissonphotonnoiseex- 0.11 Mq.However,ifthelensisvery closetous,sayA~550pc ative lens-sourceparallax,jri.Fig.1suggeststhatthestarisared- M =—M— although becauseofthedegeneracyinherentinmicrolensing of thelensedstaraloneis16.3mag.Thewasalsoquite of this,i.e.thereisonlyslightblendingandthebaselinemagnitude all thesamefeaturesthathavebeendiscussedabove,butwitha occurs nearthebeginningoffourthseason,whensourceflux strongly magnifiedatthepeak,risingtoamagnificationofgreater (wo andi/r),areverytightlyconstrained,owingtothefactthat sented inTable1.Itcanbeseenfromthistablethatallofthe systematics arenotanartefactofourmethodphotometricdata 674 M.C.Smithetal. server planesuffersfromnosuchdegeneracy andcanbedetermined (which correspondsto7ri>1.6mas), thenthislensmasscanbeas re E max As canbeseenfromthisequation,thelensmassdependsonrel- re ie The modelparametersforthebest-fittingparallaxarepre- The valueofthetransversevelocity projectedontotheob- The parametersthathelpindeterminingthelenspropertiesare: _ au 4G - © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem A Ä 1 0.088 M© 2.68 au O.lmas rel (6) -1 1 bottom panelshowsamoretypicalparallaxevent,sc33_4505(seeSec- the toppanelofFig.3,andhighlynon-linearnaturecanclearly motion ofthelensasinnormalmicrolensingevents.Thetrajectory Because ofthis,asthelenstrajectorypassesEarthchangein is muchlessthantheorbitalspeedofEarth(u®^30kms). low velocityarediscussedinSection6. This velocityisexceptionalinthatitthelowesteverrecordedfor uniquely, this toamoretypicalparallaxmicrolensing event,sc33_4505,highlightsthe linear, leadingtotheexceptionallydramatic parallaxsignature.Comparing that thetrajectoryincaseofevent OGLE-1999-BUL-19 ishighlynon- verses fromrighttoleft,asdepictedby thearrows.Thisfiguredemonstrates Fig. 2andTable1.Thesoliddotsindicatethelenspositionat50-dintervals, not necessarilycorrespondtothebest-fittingstandardlightcurvegivenin magnification fortheevent.Thestraightdashedlinerepresentsequiv- Earth correspondstotheuvaluegiveninequation(1)anddetermines tion 3andSmithetal.2002).Thisseparationbetweenthelens The toppaneldepictsthetrajectoryforOGLE-1999-BUL-19,whilst location ofthelenswithrespecttoEarth(denotedbysmallcross). in theobserverplanerelativetoobserver-sourcelineofsight,i.e. Figure 3.Thetrajectoryofthelens(givenbybest-fittingparallaxmodel) counted forwhenitisnotedthattheprojectedlensvelocity,f, axes aregiveninunitsofthisEinstein radius.Forbotheventsthelenstra- dotted circlerepresentsthesizeofEinsteinradiusevent,and and 650dforOGLE-1999-BUL-19sc33_4505,respectively.Thelarge and theopendotscorrespondtoapproximatepeaktimes,i.e.t=1650 a standard‘constant-velocity’lightcurve).Itshouldbenotedthatthisdoes alent trajectorywithoutaccountingfortheorbitalmotionofEarth(i.e. of thelensrelativetoobserver-sourcelinesightisshownin a publishedparallaxmicrolensingevent.Possiblecausesforsuch spectacular natureofthetrajectoryfor OGLE-1999-BUL-19. separation isdominatedbythemotionofEarth,ratherthan v =—12.5±1.1kms“.(7) The exaggeratednatureoftheparallaxsignaturescanbeac- e 2002RAS,MNRAS 336,670-684 x x 2 O 02MNRAS.336. .670S -1 back towardthelensresultinginasharpriseflux.Thisperiod be contrastedwiththebehaviourduring6monthsdirectlypre- be seen.Duringthe6monthsbeforepointofclosestapproach by anyotherphenomena,suchasa rotatingbinarysourceor It isnecessarytotestwhetherthisunusual behaviourcouldbecaused 4.1 Binary-sourcemodel 4 ALTERNATEMODELS much smootherandsmallerinmagnitude.Theoriginsofthe tion ismuchlessclearforthesc33_4505event;oscillationsare form, andfromthisoscillationonecanunmistakablyidentifythe Earth inducesanannualoscillationontothisstandardhyperbolic ing lightcurve(shownasadashedline);theorbitalmotionof BUL-19 event(givenintheupperleft-handpanel)clearlyvaries how thisseparationvarieswithtimeishelpfultoelucidatethesitu- factor thatleadstotheparallaxeffectbeingmorepronouncedin BUL-19 theEarthcompletesnearlytwoorbitsinthistime.Another has completedapproximatelyoneorbit,whereasforOGLE-1999- that ittakesthelenstocrossitsprojectedEinsteindiameterEarth for OGLE-1999-BUL-19.Thisisbecausethevelocityoflens Fig. 3tobemuchcloserthestandardlinearapproximationthan ing eventstowardtheGalacticbulge(seeSmithetal.2002).This the parallaxsearchthroughOGLE-IIdatabaseofmicrolens- for themicrolensingeventsc33_4505whichwasdiscoveredduring trajectory iscomparedwiththatofanother,moretypical,parallax- trough aroundt=1800d. however, inthesubsequent6-monthperiodEarthbeginstomove Earth movesawayfromthelens,resultinginasharpdeclineflux; the 6monthsdirectlyfollowingpointofclosestapproach in the12monthsfollowingpointofclosestapproach:during respectively, totheunusualdecliningandrisingsectionsoflight tory broughtitbacktowardsthelens.Thesetworegionscorrespond, the Earthswungawayfromlensandincreasedseparation, velocity issolow,asthecaseinOGLE-1999-BUL-19,motion correspond tothepointsatwhichrateofchangeseparation, discussed above,anditsinfluencecanclearlybeseeninFig.3. dius, r,ismuchsmallerthanforsc33_4505.Thiseffecthasbeen OGLE-1999-BUL-19 isthatthevalueofprojectedEinsteinra- event hadunexceptionalvaluesoffandv(6.37au57km curve whichoccurbeforethemainpeak,i.e.parallax-induced ceding thisregion(i.e.approximately1250,between have sixmoreparametersthatdescribethebinaryandsource to bedegenerate(Mao&diStefano1995;Dominik1999).For it isoftendifficulttolocatetheglobalminimum.Furthermore,for Mao &Paczyñski1991).Asaresultofthesenewfeatures,thex may containsharprisesandfallscausedbycausticcrossings(e.g. model. Inthissection,wewillexplorewhetherOGLE-1999-BUL- In theprevioussubsections,wehavefittedlightcurveofOGLE- 4.2 Binary-lensmodel this resultstrengthenstheargumentthatbinary-sourcemodelis when non-face-onorbitsandblendingarealsoconsidered. does notmatchthedatapointsforfirstthreeseasons,including were performed).Fig.5showsthebestfitthatwasfound.Thetotal from anumberofinitialguesses(althoughnoexhaustivesearches first andsecondlenses,respectively.Notethattcorrespondstothe Ai =+m],whereandarethemassesof eccentricity, e;thetimeofperihelion,tp;andmassratio OGLE-1999-BUL-19, themultipeakbehaviourisreminiscentof weak binariesandill-sampledlightcurves,thesolutionsareknown all ofthelight. at thetimeofperihelion;periodbinaryorbit,T; of sight.Otherthanthefiveparametersinstandardmodel,we orbit isface-on,i.e.theorbitalplaneperpendiculartoline of rotatingbinaries(Dominik1998a)whereweassumethebinary allax modelsarealwayssmooth,thoseproducedbybinarylenses source model.Muchofthexactuallyarisesfromfailure X is1522.4.TheparametersaregiveninTable4. surface inthemultidimensionalspaceisevenmorecomplex,and simply reproducingamirror-imageoftheparallaxmodel. 678 M.C.Smithetal. 19 canbefittedbyarotatingbinary-lensmodel. 1999-BUL-19 withaparallaxmodelandrotatingbinary-source E eri 2 While thelightcurvesproducedbybinary-sourceorpar- The xismuchworsethanthebestparallaxmodelandbinary- We havesearchedforthebest-fitface-onrotatingbinarystarting Clearly, thisfitisnotacceptable.Itpossible,however,thatan 2 2 complexity ofthexsurface. Table 3.Thebest-fittingparametersfortheellipticalbinary-sourcemodel,withAifixedat1andT0.areexplainedin to tE(d)U{)/A/aß{à)Tep(fixed)£oAi0xdof Section 4.1andintheAppendix(AppendicesA1A2).Theerrorshavebeenomittedsincetheywerefoundtobemisleading,owing 1627.8 428-0.41557-193-4.350.123670.0100.329123.5601-1.24544.84305 s 2 2 have beenomittedbecauseofthecomplexity ofthexsurface. Table 4.Thebest-fittingparametersfor theface-onrotatingbinary-lensmodel.TheparametersareexplainedinSection 4.2.Theerrors to t(d)UQf Af AiPwindow is(V—/)cio=1.114mag.Therefore,thereddeningfor dereddened colourofthecentrered-clumpregionforBaade’s determined fromitsdereddenedcolourandmagnitude.Thecolour- constraint wasfoundforp*: corporated intoourmodel;atthe2crconfidencelevelfollowing curve. However,ascanbeseenfromFig.2,thepeakisreason- example, Gould1994;Nemiroff&WickramasingheWitt 5.3 Finitesourcesize ÖE =—>66.8nas. (14) © 2002RAS,MNRAS 336,670-684 and hencethefollowingconstraints onthelensmassanddistance p* (equation9)todeterminetheangular Einsteinradiusofthelens, & Brett1988).Oncethishasbeendone,avaluefor0*,theangular Aid =IA9E(V-I)d^2.14mag.(11) of theangularsourcesize,acorrespondingconstraintcanbede- p. <0.119.(9) ably well-fittedbythepoint-sourceparallaxmodel.Nosignificant and atsuchlargemagnificationsitbecomesnecessarytoconsider Section 3,thepeakmagnificationforthiseventisapproximately10, surface brightness-colourrelation(equation4fromAlbrowetal. size ofthesource,canbecalculatedfromfollowingempirical (11). Thiswouldimplythatthesourcehasanintrinsicbrightnessof same reddeningandextinctionascalculatedinequations(10) (V —/)ci,obs^2.55mag.FromPaczynskietal.(1999),theintrinsic (e.g. Albrowetal.2000),theangularsizeofsourcecanbe size ofthesourceinunitsangularEinsteinradiuslens. 7 ^14.15magandanintrinsiccolourof(V—7)^1.19mag, E(V -7)=(V/)—^1.44mag.(10) re Ss 0 Si0Si0 clclobscl0 When thisiscombinedwithanotherindependentdetermination This effectshouldbemostprominentaroundthepeakoflight This valueforcanbecombinedwiththeaboveconstrainton This colourcanthenbeconvertedfrom(V—7)^1.19mag Si0 P* © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem => 0.025mas. > 0.022Mq, 681 (16) (15) 2 O 02MNRAS.336. .670S -1 1 -1 be anunrelatedstaror,ifthelensiscloseenoughtoobserver, proximately 0.83pixels(~0.35arcsec).Thismisalignedblendcould blended sources.TheV-and/-banddatatogetherimplythatthereare be aworthwhileeffortsinceradialvelocitieswillprovidefurther red-clump starinthebulge(seeFig.1;seealsoBennettetal.(2001) prediction caneasilybediscountedbyacoupleofmeasurements bulge-bulge lensing,sinceinthisinstancevaluesofñ<100kms peaks, onecangainanindicationoftheirprevalencebystudyingthe- peak eventsoccur,ifindeedtheyarecausedbytheparallaxeffect. reveal periodicshiftsintheradialvelocity.Iflightisfrom that wearecurrentlyundertaking. Thisisimportantbecausesuch the prevalenceofsuchmultiplepeak events,andthisissomething unlikely accordingtotheirfindings. Onewouldneedtoperform ities comparabletoOGLE-1999-BUL-19 alsoappeartobehighly hand, valuesofvareexpectedtobemuchsmaller,althoughveloc- that itisextremelyunlikelysucheventscouldbecausedby the lensmass. These observationsmaybeparticularlyvaluableforfurtherunder- for thelensbecauseofrelativedifferencebetweenradial lens, thencross-correlationsofthespectramaystillshowevidence Furthermore, tounderstandthenatureofalignedblending,it microlensing lightcurve).However,theOGLE-IIdataarenotsuffi- then theseparationwouldbearound400au,whichtoo the lensisatadistanceof1.3kpc,aswassuggestedinSection5.2, this couldbeawide-separationcompaniontothelens(althoughif is alignedwiththelensedsourcewhileothermisalignedbyap- two blendedsourceswithintheseeingdiscoflensedsource,one light ontherelated(unsolved)problemofnatureexcess information onthesourcekinematics.Suchasurveywillalsoshed tain theradialvelocitiesofOGLE-1999-BUL-19.Aspectroscopic is producedwhenthelensandsourcemovemoreorlessradially for otherblackholecandidates).Usually,alowtransversevelocity true physicalsolution. further evidencethatthis‘pure’binary-sourcemodelmaynotbea the limitplacedon0byfinitesourcesizeconsiderations,providing these predictionsfromthe‘pure’binary-sourcemodelmayviolate velocities ofthesourceandlens(Mao,Reetz&Lennon1998). will beveryusefultoobtainspectrafromtheground.Ifaligned cient todecipherthesetwocomponents.TheAdvancedCamerafor great aseparationforthiscompaniontohaveanyinfluenceonthe to bethecaseforOGLE-1999-BUL-19,assourceseemsa caused byadisc-disclensingevent.However,thisdoesnotappear et al.(2002)pointedoutthatsuchlow-transversevelocitiesmaybe with 1kms“precision.However,aswasnotedinSection5.3, velocity variationspredictedbythisbinary-sourcemodelshouldbe events caneasilybeoverlookedwhen compilingmicrolensingcata- detailed MonteCarlosimulations to gainafirmerunderstandingof are stronglydisfavoured.Forlensesresidinginthedisc,onother oretical distributionsoftheprojectedvelocity,ñ(e.g.Han&Gould Since anespeciallylowvelocityisrequiredtoproducesuchmultiple Surveys onHSTistheidealinstrumenttoresolvesuchcomponents. of longevents,whichwasfirstnoticedbyHan&Gould(1996). approximately 5kmswithaperiodof1yr,andthereforethis standing thelensedsystem,includingapotentialdeterminationof source isfromabinarycompanion,thenspectroscopywill survey ofallparallaxevents,includingOGLE-1999-BUL-19,will (Mao &Paczynski1996).Therefore,itwillbeveryimportanttoob- velocity ñ,whichisthelowestseeninanylenscandidate.Smith 682 M.C.Smithetal. 1995). ThedistributionsproducedbyHan&Gould(1995)suggest E We haveinvestigatedthephotometryandastrometryof It wouldbeinterestingtoknowhowfrequentlythesemultiple One strikingpredictionoftheparallaxmodelisprojected © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem ber ofmarginalmicrolensingeventsthatwereuncovered.Detailed been mistakenforavariablesource. Warsaw TelescopeattheLasCampanasObservatoryofCarnegie This workwasbasedonobservationsobtainedwiththe1.3-m ACKNOWLEDGMENTS is expectedintheOGLEexperiments.Wearecurrentlyperforming Monte Carlosimulationsareneededtocheckwhetherasimilarrate Wozniak etal.(2001).TheseincludeOGLE-1999-BUL-19,along found amongthe520microlensingeventsincatalogueof last forwellover5yearsintotal.Therefore,iftheobservationshad For example,thevariabilityforOGLE-1999-BUL-19isexpectedto long timetoreachaconstantbaseline(owingtheexceptionally microlensing lightcurveandalsobecausetheytakeanunusually logues sincetheirvariabilitydifferssignificantlyfromthestandard Duquennoy A.,MayorM.,1991,A&A,248,485 Dominik M.,1999,A&A,341,943 Dominik M„1998b,A&A,333,893 Dominik M.,1998a,A&A,329,361 Derue F.etal.,1999,A&A,351,87 Delplancke F.,GórskiK.M.,RichichiA.,2001,A&A,375,701 Bond LA.etal.(theMOACollaboration),2001,MNRAS,327,868 Bessell M.S.,BrettJ.M.,1988,PASP,100,1134 Bennett D.P.etal.,2001,ApJ,submitted(astro-ph/0109467) Becker A.C.,2000,PhDthesis,Univ.Washington An J.H.etal.,2002,ApJ,572,521 Alcock C.etal.,2001b,ApJ,552,259 Alcock C.etal.,2001a,Nat,414,617 Alcock C.etal.,2000,ApJ,541,734 Alcock C.etal.,1995,ApJ,454,L125 Albrow M.D.etal.,2000,Api,534,894 Alard C.,MaoS.,GuibertJ.,1995,A&A,300,L17 Agol E.,KamionkowskiM.,KoopmansL.,BlandfordR.,2002,preprint NSF grantAST-9820314. tion 5.MCSacknowledgesthefinancialsupportofaPPARCstu- for apromptandcomprehensivereportthatweimplementedinSec- Institution ofWashington.WethankBohdanPaczyñskifordiscus- cent. However,asdiscussedinSmithetal.(2002),thereareanum- 2002). Soitappearsthattheparallaxrateisapproximately1per with twoeventsthathavebeenstudiedinpreviouspapers,namely Griest K.,HuW.,1992,ApJ,397,362 (erratumApJ407,440) Gould A.,Miralda-EscudeJ.,Bahcall J.N., ApJ,1994,423,L105 Gould A.,AnJ.H.,2002,ApJ,565,1381 Gould A.,2000,ApJ,542,785 Gould A.,1998,ApJ,506,785 Gould A.,1994,ApJ,421,L71 Gould A.,1992,ApJ,392,442 Girardi L„SalarisM.,2001,MNRAS,323,109 Cox A.,2000,Allen’sAstrophysicalQuantities,4thedn.Springer-Verlag, REFERENCES dentship. PWwassupportedbytheLaboratoryDirectedResearch also supportedbythePolishgrantKBN2P03D01418and & Developmentfunds(XIEMprogramatLANL).Thisworkwas such astudy,theresultsofwhichwillbereportedelsewhere. sc33_4505 (Smithetal.2002)andOGLE-1999-BUL-32(Mao started oncetheeventwasunderway,thiscouldhaveeasily small valueofvthatisrequiredtoproducesuchmultiplepeaks). sions andencouragement.WearedeeplyindebtedtoAndyGould So faratleastthreeconvincingparallaxcandidateshavebeen New York (astro-ph/0203257) e 2002RAS,MNRAS 336,670-684 2 O 02MNRAS.336. .670S (1)2) (1)2) (1)2) Wozniak R,2000,ActaAstron.,50,421 Witt H.J.,MaoS.,1994,ApJ,430,505 plane. respectively. Inthisprescriptionthex-axisisdefinedsuchthatsource mass fractionofthefirstsource,A4,i.e.A4=ra/|ra+m], the firstsource,JF,i.e.JF=/[+hwhereand the Einsteinradiusprojectedontosourceplane;fluxratioof to thefivefromstandardmodel.Themotionoftwosources The descriptionofthiscircular-orbitbinary-sourcemodelrequires A1 Circularorbits A completeformalismcanalsobefoundinDominik(1998a). The generalizationtoellipticalorbitsisdiscussedinAppendixA2. this modelisfirstsimplifiedbyconsideringonlycircularorbits. the parameterspacewouldprovetobedemandingcomputationally, A comprehensivedescriptionofthebinary-sourcemotionshould MODEL APPENDIX A:ROTATINGBINARY-SOURCE Udalski A.,ZebruñK.,SzymañskiM.,KubiakM.,PietrzyñskiG.,Soszyñski Udalski A.,KubiakM.,Szymañski1997,ActaAstron.,47,319 Paczyñski B.,UdalskiA.,SzymañskiM.,KubiakPietrzyñskiG., Paczyñski B.,1998,ApJ,494,L23 Paczyñski B.,1996,ARAA,34,419 Paczyñski B.,1986,ApJ,304,1 Nemiroff R.J.,WickramasingheW.A.D.T.,1994,ApJ,424,L21 Miyamoto M.,YoshiY,1995,AJ,110,1427 Mao S.etah,2002,MNRAS,329,349 Mao S.,ReetzJ.,LennonD.J.,1998,A&A,338,56 Mao S.,PaczyñskiB.,1996,ApJ,473,57 Mao S„PaczyñskiB„1991,ApJ,374,L37 Mao S.,diStefanoR.,1995,ApJ,440,22 Mao S.,1999,A&A,350,L19 Landau L.D.,LifshitzE.M.,1960,Mechanics,Volume1ofCourseThe- Hosokawa M.,OhnishiK.,FukushimaT.,Takeuti1993,A&A,278,L27 Hpg E.,NovikovLD.,PolnarevA.G.,1995,A&A,294,287 Han C.,GouldA.,1997,ApJ,480,196 Han C,GouldA.,1996,ApJ,467,540 Han C„GouldA.,1995,ApJ,447,53 Han C,2001,MNRAS,328,611 Han C,JeongY.,1998,MNRAS,301,231 where mandarethemassesoffirstsecondsource, consider ellipticalorbits.However,sinceanexhaustivesearchof Wozniak P.R.,UdalskiA.,SzymañskiM.,KubiakPietrzyñskiG., Walker M.A.,1995,ApJ,453,37 circular andthereforerotationallyinvariant. Unlessotherwisestated, © 2002RAS,MNRAS 336,670-684 and giveninunitsoftheEinsteinradius projectedontothesource all ofthedistancesinthissection are describedintheorbitalplane a parametertoaccountfortheorbital phasebecausetheorbitsare are thefluxesfromfirstandsecondsource,respectively; orbit, T\thebinarysourceseparation,p,whichisgiveninunitsof are describedusingthefollowingfourparameters:periodof a totalof12parameters,i.e.furthersevenparametersinaddition Stanek K.Z.,1996,ApJ,460,L37 Soszyñski I.etah,2001,ApJ,552,731 Smith M.C.,MaoS.,WozniakR,2002,MNRAS,332,962 Schechter P.L.,MateoM.,SahaA.,1993,PASP,105,1342 1 liesonthe(positive)x-axisati= i-Thereisnoneedtoinclude 0 h, WozniakR,2000,ActaAstron.,50,1 oretical Physics.Pergamon,Oxford Soszyñski h,ZebruñK.,2001,ActaAstron.,51,175 Soszyñski I.,WozniakR,ZebruñK.,1999,ActaAstron.,49,319 © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem OGLE-1999-BUL-19 -thefirstmultipeakparallax683 r (1) 0) (1)2) (2) (ï) u>() (z) (?) i(0=Uo plane, ©,whichismeasuredfromthex-axistowardsdirection plane (alongitudeintheorbitalplane,a,andalatitudemeasured be determined,andthepositionvectorsfortwosources,r the centreofmasslyingonleft-handsidelenstrajectory. the parallaxmodel,apositivevalueofuqischosentocorrespond minimum separationofthelenstrajectoryfromcentremass the fivefromstandardmodel(t,tuq,fA/),thisgives is themassratiooffirstsource,asdefinedabove. Einstein radius,thisquantityr^g) isanalogoustotheseparation line ofsight.Thisrequirestheline-of-sightvectortoobserver, tain r^g),thecomponentofthisseparationperpendicularto this vectorneedstobeprojectedonthesourceplaneob- the lenspositiontosourceposition,i.e.<>r(i)=r—rig), to determinetheseparationbetweensourceandlensforboth their paper). the parallaxmodelinSoszyñskietal.(2001)(cf.equation16 This isanalogoustothedescriptionoflenstrajectorygivenfor the lensplane,andatthispointonlycircularorbitsareconsidered method, thelensmotionisdefinedinorbitalplane,ratherthan his studyofrotatingbinarysources.However,unlikeDominik’s lensed sources[i.e./=+]. Also, /nowcorrespondstothecombinedbaselinefluxoftwo complete setofparameters.However,uqnowcorrespondstothe quired: two,aandß,todeterminetheorientationoforbital f = r, aregivenby(see,forexample,Landau&Lifshitz1960), ñ =(cosßcosa,sina,sinß), andisgivenby where rg)isthepositionofsourceiandrig) ^E,;A©) =/ where r{t)= and ismeasuredinunitsoftheprojectedEinsteinradius.Aswith of thesourcesystem.Thisseparationisdefinedinorbitalplane, of motionthelens.Whenaboveparametersareaddedto observer); andanangletodescribethelenstrajectoryinorbital out oftheorbitalplane,ß,describingline-of-sightvectorto Since alloftheabovedistancesare givenintermsoftheprojected ¿/{'(O =Sr(t)-[ár'í)-ñ]ñ. (A3) Since <5rg)correspondstotheseparationinorbitalplane, of thelens,asdefinedinequations(Al)and(A2),respectively. and sources individually,<5r(¿).Thisissimplygivenbythevectorfrom (see above). 0Es s s In thisequation,istheusualKroneckerdeltafunctionandA4 In additiontotheseparameters,afurtherthreeanglesarere- The aboveparametersallowthemotionofbinarysourcesto The lenspositionintheorbitalplaneisgivenby This methodissimilartothatemployedbyDominik(1998a)in Once thesequantitieshavebeendeterminedthenthenextstepis y(t -k), 27t = (8-A4)r(0, U 72 y— cos©J f sin0\ y! -cos(/3)cosg©a) i =1,2. + r,(®)Tt) Ep where r(i) = y sin0 f cos©\ losing) y ip cos(f)\ (A2) (Al) 2 O 02MNRAS.336. .670S (l {l (1)2 be significant.ThisisbecauseforOGLE-1999-BUL-19thesource binary natureofthesource.IncaseLMCself-lensingevents parameter ufromthestandardandparallaxmodels(seeSection3). the Einsteinradiusprojectedontoobserverplaneissmaller,and is closertous(mostprobablylocatedintheGalacticbulge),hence into accounttheparallaxmotionofobserverinadditionto This completestheprescriptionforbinary-sourcemodel.How- independently. Therefore,thetotalfluxisgivenby the twofluxesisjustifiedsincebothsourcesareabletobetreated independently bysubstitutingu\t)=lár^COIintotheusualmi- The magnificationofeachsource,A\t),canthenbecalculated ever, itmustbenotedthatamorecompletemodelshouldalsotake erence image(thisparameter,A/,wasintroducedinequation4). difference betweenthetotalbaselinefluxandofref- can thenbeaddedtogethertogivethetotalflux.Thisadditionof was employedinthepreviousmodels(equation4).Thesetwofluxes each sourcetobecalculatedindividually,usingtheequationthat crolensing equation(equation1).Thisallowsthelensedfluxfrom so theparallaxeffectisexpectedtobelarger(seeSection3). (e.g. Alcocketal.2001b)thiseffectshouldbenegligible(seeGould 684 M.C.Smithetal. fit) ={[A(0-1][AV)(1F)}f+A/, 1998), butforOGLE-1999-BUL-19onewouldexpectthiseffectto s(A4) where fisthebaselinefluxoflensedsourcesandA/ s © Royal Astronomical Society •Provided bytheNASA Astrophysics DataSystem p nowdescribestheorbitalsemimajoraxis,inunitsofEinstein radius projectedontothesourceplane. required: theeccentricity,e;andphaseoforbit,£,whichis in thisequationareasdescribedAppendixAl.Therestofthe Except foreand§(whichweredefinedabove),alloftheterms The parametrizationofthisequationisnowmorecomplicated,with nary sources.Thisequationshouldbereplacedbyitselliptical into equation(Al),whichdescribesthetrajectoryofbi- A2 Ellipticalorbits This paperhasbeentypesetfromaTpX/ETpXfilepreparedbytheauthor. ? -esing)=271(^4)+?(>-<’sing,,).(A6) £ beinggivenby XV) =(Si,-M)v(t).wherer(i)(~^). counterpart, defined astheanglebetweenx-axisandpositionoffirst analysis isidenticaltothatforcircularorbits(AppendixAl). When consideringellipticalorbits,twoadditionalparametersare source at¿=i-Also,sincetheorbitsareelliptical,parameter 0 0 0 The onlydifferencebetweenellipticalandcircularorbitscomes e 2002RAS,MNRAS 336,670-684 2 \j3Vl -esin(£p (A5)