Signatures of rotating binaries in micro-lensing experiments

1.5 MB·

by  A. A. Nucita

Mon.Not.R.Astron.Soc.000,1–??(2011) Printed27January2014 (MNLATEXstylefilev2.2)
Signatures of rotating binaries in micro-lensing experiments
A.A. Nucita1,2, M. Giordano1, F. De Paolis1,2, & G. Ingrosso1,2
4 1DepartmentofMathematicsandPhysics“E.DeGiorgi”,UniversityofSalento,ViaperArnesano,CP193,I-73100,Lecce,Italy
1 2INFN,Sez.diLecce,ViaperArnesano,CP193,I-73100,Lecce,Italy
0
2

n
Acceptedxxx;Receivedxxx;inoriginalformxxx
a
J

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2 ABSTRACT
Gravitationalmicro-lensingoffersapowerfulmethodtoprobeavarietyofbinarylenssystems

] asthebinarityofthelensintroducesintheeventlightcurvesdeviationsfromthetypical(sin-
R glelens)Paczyn´skibehaviour.Generally,astaticbinarylensisconsideredtofittheobserved
S light curve and, when the orbital motion is taken into account, an over-simplified model is
. usuallyemployed.Inthispaper,wetreatinarealisticwaythebinarylensmotionandfocus
h
onsimulatedeventswellfittedbyaPaczyn´skicurve.We showthat,mostoften,anaccurate
p
timing analysis of the residuals (calculated with respect to the best fit Paczyn´ski model) is
-
o sufficienttoinfertheorbitalperiodofthebinarylens.Itgoeswithoutsayingthattheindepen-
r dentlyestimatedperiodmaybe used to furtherconstrainthe orbitalparametersobtainedby
t
s thebestfittingprocedurethatoftengivesdegeneratesolutions.Wealsopresentapreliminary
a
analysisontheeventOGLE-2011-BLG-1127/MOA-2011-BLG-322whichwasrecognized
[
tobeduetoabinarylens.Theperiodanalysisresultsinaperiodicityof≃12dayswhichcon-

1 firmstheoscillationoftheobserveddataaroundthebestfitmodel.Theestimatedperiodicity
v is likely associated to an intrinsic variability of the source star, thus openingthe possibility
8 tousethistechniquetoinvestigateeithertheintrinsicvariabilityofthesourceandtheeffects
8
inducedbythebinarylensorbitalmotion.
2
6 Keywords: gravitationallensing:micro
.
1
0
4
1
1 INTRODUCTION it allows to detect exo-planets in a rather large range of masses,
:
v spanning from Jupiter-like planets down to few Earth-mass ob-
In thelast decades the gravitational micro-lensing technique, ini-
i jects(Bennett&Rhie1996;Bondetal.2004;Udalskietal.2005;
X tiallydevelopedtosearchforMassiveCompacthaloobjects(MA-
Beaulieuetal.2006;Gouldetal.2006;Gaudietal.2008;Bennett
r CHOs) in the Galactic halo and in the Galactic disk (Paczyn´ski
2009;Dominik2010;Gaudi2011).
a 1986; Alcocketal. 1993; Paczyn´ski 1996; Roulet&Mollerach
1997, 2002; Zakharov&Sazhin 1998), has been widely used to Micro-lensinghasalsobecomeapowerfultooltostudysev-
inferthepresenceofexo-planets1orbitingaroundthemainlensing eral aspects of stellar astrophysics. In fact, a sufficiently ampli-
stars(seee.g.thereviewsbyPerryman2000;Perrymanetal.2005; fied micro-lensing event gives the opportunity to investigate the
Bennett2009).Indeed, asshownbyMao&Paczyn´ski(1991)the source limb-darkening profile, i.e. the variation of intensity from
presenceofaplanetarounditshostingstarformsabinarylensthat the center of the disc up to its border, thus implying the possi-
can micro-lens a background source, inducing non-negligible de- bilitytotestamongdifferentatmospheremodels(Abeetal.2003;
viationswithrespecttotheusualsymmetricPaczyn´skilightcurve Gaudi&Gould1999).Thisopenstheuniquepossibilitytoinves-
(Witt&Mao1994;Gould1994;Alcocketal.1997).Inparticular, tigatetheGalacticbulgestaratmospheres, otherwisehardlytobe
thelight-curve analysis of highly magnified events is sensitiveto studiedduetotheirdistance.Besidesthebrightnessprofileofthe
thepresenceoflenscompanionswhenthebinarycomponentsare background source disc, highly magnified events with large radii
separatedbyadistanceoftheorderofEinsteinradiusRE associ- sourcesallowtomeasurethelensEinsteinradiusoncethephysical
atedtothewholelenssystem.Ingeneral,suchsignaturesarechar- radiusofthesourcestarisknownfromdifferentmethods.Morere-
acterizedbyshortdurationdeviationslastingfromafewhourstoa cently,ithasalsobeenproposed(see,e.g.,Ingrossoetal.2012and
fewdays(dependingontheparametersofthebinarylenssystem). referencestherein)that,whenfinitesizesourceeffectsbecomerele-
Asamatteroffact,themicro-lensingtechniqueissosensitivethat vant,acharacteristicpolarizationsignalmayariseinmicro-lensing
events.Thisisduetothedifferentialmagnificationinducedduring
thecrossingofthesourcestaroverthelens(eithersingleorbinary).
1 Atthetimeofwriting,thenumberofexo-planetsdetectedviathemicro- Thisproducesapolarizationsignalupto0.04%forlatetypestars
lensingmethodis24(seee.g.theextra-solarplanetencyclopediaavailable anduptoafewpercentforcoolgiants,dependingonthephysical
athttp://exoplanet.eu/). processesatthebasisofthepolarizationandtheatmosphereparam-
2 A.A. Nucita,M. Giordano,F.DePaolis,&G. Ingrosso
etersofthesourcestar.Suchasignalmaybeobservablewiththe 2 SIMULATINGAROTATINGBINARY
currently available technology. It was also shown (Ingrossoetal. MICRO-LENSINGEVENT
2013) that polarization measurements may help in disentangling
2.1 Thecomplexnotationforabinarywithpoint-likemasses
the particular degeneracy between the binary or planet lens solu-
tionwhichoccursinsomemicro-lensingevents. Itiswellknown(seee.g.Schneider,Ehlers&Falco1992)thatin
Notethatfinitesourceeffectsusuallyworkagainsttheappear- abinarymicro-lensingeventinvolvingapoint-likesourcethean-
anceofexo-planetsignaturesinmicro-lensinglightcurves,asthese gularpositionsoftheimages(x)andbinarylenscomponents(x
A
featurestendtobesmearedout.However,althoughthefinitesource andx )inthelensplanearerelatedtotheangularsourceposition
B
effectsaregenerallytiny,theycannotbeignoredwhenmodelinga (y)inthesourceplaneviathelensequation
lightcurve.Thisisparticularlytrueforlargeplanetarymassesorin
caseofbinarymicro-lensingeventscharacterizedbycausticcross- y=x−mA|xx−−xxAA|2 −mB|xx−−xxBB|2 , (1)
ing,i.e.whenthesourcepassesthroughfoldand/orcuspcaustics
(Schneider,Ehlers&Falco1992). where mA and mB are the masses of the binary lens compo-
Themainreasonforconsideringbinaryeventsisthatthebest nents normalized to the total system mass with mA > mB,
fit procedure to a micro-lensing light curve may allow, in princi- q = mB/mA and mA +mB = 1. All the positions are mea-
ple, to derive the parameters (the projected lens separation b and suredinunitsoftheangular EinsteinradiusθE ofthewholelens
system,i.e.
themassratioq)ofthelenssystem.Typically,astaticbinarylens
isconsideredandthisimpliestheminimizationofafunctionalde-
pendingonsevenfreeparameters.Theseare,inadditiontoband θE =r4Gc2M DODSLDSOL , (2)
q, the time t0 of closest approach to the lens system, the impact
parameter u0 (in units of the Einstein radius), the Einstein time being DOS and DOL (with DLS = DOS −DOL) the distance
from the observer to the source and lens, respectively. Here, G
oftheeventTE,theangleθthatthebackgroundsourcetrajectory
is the gravitational constant and c the speed of light. Eq. (1) can
forms with respect to the binary lens separation, and the source
beeasilygeneralizedinordertoaccount formultiplelensesasin
radius ρ∗. Considering in the fit procedure the orbital motion of
Kayser,Refsdal&Stabell(1986).Byusingcomplexnotation(see
the lenssystem (see e.g. Dominik 1997; Pennyetal. 2011a,b) is
Witt1990;Witt&Petters1993;Witt&Mao1995),eq.(1)canbe
extremelytimeconsumingsinceagoodmodelingofsuchmotion
easily rewritten in terms of complex variables. Let us define the
wouldimplysixadditionalparameters,i.e.thesemi-majoraxisa,
variablesz ≡ (x,y) = x+iy andζ ≡ (ξ,η) = ξ+iη,being
the orbit eccentricity e, the time of passage at periastron tp, the
(x,y)and(ξ,η)thedimensionlesscomponentsofthevectorxand
angle i between the normal to lens orbital plane and the line of
y,respectively.Withthissubstitutiononehas
sight,theorientationφaoftheorbitalplaneinthesky,andtheor-
bitingversus(eitherclockwiseorcounterclockwise).Asdiscussed ζ =z− mA − mB , (3)
inParketal.(2013), forthedetermination of thelensparameters z¯−z¯A z¯−z¯B
oneshouldalsoincludetherelativelens-sourceparallaxπE dueto wherethebaroverasymbolindicatestheoperatorofcomplexcon-
EarthmotionaroundtheSunwhichinvolvestwomoreparameters. jugation.
As a result of the large number of parameters involved2, when a Todeterminetheimagepositionandmagnification,onehasto
bestfitprocedureisattempted,sometricksarerequiredinorderto takethecomplexconjugateofeq.(3)andsubstitutetheexpression
makethefitconvergingtoreliableresults. forz¯backinit.Aftersomealgebra,onegetsafifth-orderpolyno-
Inthispaper,wewillconcentrateonbinarysystemswithor- mialinz,i.e.p(z)= 5i=0cizi =0withcoefficientscidepend-
bitalparametersforwhichtheresultingmicro-lensing lightcurve ingonb,andq,whosePsolutionsgivetheimagepositions.Here,b
isvery close toa Paczyn´ski curve (i.e.a planetary case). Indeed, represents the separation of the two binary lens components in a
oneexpectsthat thepresence ofplanetsrotatingaroundthehost- referenceframe(Oξηκ)withorigininthemiddlepointandξaxis
ingstarshouldcause,mostoften,onlysmallperturbations(seealso oriented from the star to the planet and so that ξA = −b/2 and
Bozza1999)tothePaczyn´skilightcurveassociatedtoanequiva- ξB = +b/2, i.e. the star and its companion are always on the ξ
lentsinglelensevent.Weconsidertwocases:1)thebinarysystem (real)axis.Theverticalaxisκisorientedinordertohavearight-
orbitalperiodP islowerthanthetypicalEinsteintimeTE ofthe handed reference frame. A (fixed) reference frame CM ξ′η′κ′,
event,and2)P iscomparableorlargerthanTE.Inthelatter,we withorigin inthecenter of massand axesξ′ and η′ parallel toξ
requiredthatthelightcurveislongenoughtocontainatleastthree andη,willbealwaysusedinthefollowing.
fullcycles.Thenweshowedthatanaccuratetiminganalysisofthe Thelensingphenomenonseparatesthesourcestarimageinto
residuals(calculatedwithrespecttothePaczyn´skimodel)issuffi- severalpieces(NI)andconservesthesourcebrightness(see,e.g.,
cienttoinferP. Schneider,Ehlers&Falco 1992). Hence, for a point-like source,
Theplanofthepaperisasfollows:inSection2,wedescribe thesummationoverthemagnificationofeachoftheimagesgives
themodeladoptedforsimulatingamicro-lensinglightcurvetaking thetotalmagnification(Witt&Mao1995)
intoaccounttheorbitalmotionofthebinarysystemandthefinite
sourcesizeeffects.InSection3,forselectedsimulatedevents,we A0(y)= NI πi , (4)
apply thetiminganalysis tobest fitresidualsand, finally,inSec- Xi (cid:18)detJ(xi)(cid:19)
tion 4, wediscuss the advantages of our procedure and present a
whereπi istheparityofthei-thimageandthedeterminantofthe
preliminaryanalysisontheOGLE-2011-BLG-1127/MOA-2011-
Jacobianis
BLG-322event,andaddressfutureperspectives.
∂ξ ∂ξ
detJ =1− . (5)
∂z∂z
2 Notethatinthemostgeneralcaseoneshouldalsoconsiderthebaseline The point-source magnification A0(y) diverges if any of the im-
magnitudeandtheblendingparameter. agesxjappearonthecriticalcurveofthelens,i.e.thecurvealong
Signaturesofrotatingbinariesin micro-lensingexperiments 3
whichdetJ vanishes.Thiscurve,oncemappedbackinthesource whereyc isthecenterofthebackgroundsourcestaronitstrajec-
plane via the lens equation in eq. (1) or, equivalently, in eq. (3), toryatagiventimetandthesurfaceintegralisextendedoverthe
gives the caustic curves. As shown by Witt (1990), the critical wholesurfaceΣS =πρ2∗.
curvescanbeobtainedbysolving Wenumericallysolve5 thepreviousintegralwhenthesource
(or part of it) is close enough (or crosses) the caustic curves as-
mA + mB =expiψ , (6) sociated tothebinary lens system, i.e. when theIRSmethod has
(z¯−z¯A)2 (z¯−z¯B)2
to be employed (R 6 0.05 RE). Otherwise, for any source po-
with real phase ψ varying from 0 to 2π. The previous equation sitionforwhichthecomplexformalism(Witt1990)isagoodap-
correspondstoafourth-ordercomplexpolynomialinz,therootsof proximation,weusethehexadecapoletechnique(Gould2008).The
which3 lieonthecriticalcurves.Byvarying ψ andsearchingfor hexadecapolemethodsubstantiallyconsistsincalculatingthemag-
therootsofeq.(6),onefinallygetsthecausticcurvesviathelens nification in 13 locations of the source disk: one position coinci-
equation4. dentwiththestarcenter-themonopoleapproximation-,eightpo-
sitionsonthesourcelimb-withradiusρ∗-,andfourpositionsona
ringofradiusρ∗/2.Asalastremark,weremindthatweassumed
2.2 TheInverseRayShooting(IRS)method the source surface brightness to be described by the linear limb-
darkeningprofile(Choietal.2012)
Thecomplexmethoddescribedabovefailswhenthesourceisvery
closetothelenscaustics.Insuchconditions,amoreaccurate(but 3
S (y)= 1−Γ 1− y , (8)
moretimeconsuming)approachisgivenbytheIRStechnique(see λ (cid:20) λ(cid:18) 2 (cid:19)(cid:21)
e.g. Schneider,Ehlers&Falco (1992); Kayser,Refsdal&Stabell
wheretheparameterΓ dependsonwavelength,spectraltype,sur-
(1986);Wambsganss(1997)).NotethattheIRSmethodisageneral λ
facegravityandmetallicityofthesourcestar.Inthefollowing,we
technique,thusallowingonetoconsiderallthoseeffectswhichcan
assumeΓ =0.5asatypicalvalue(Claret2000).
notbeeasilyimplementedwhengeneratingasyntheticlightcurve. λ
Themethodconsistsinsolvingnumericallythelensequationineq.
(3)byshootinglightraysbackwardsfromtheobserverthroughthe
2.4 Thelenssystemorbitalmotion
lensplaneuptothesourceplane.Therayswhicharecollectedin
thesourceplane,i.e.thoseforwhichthetwosidesofthelensequa- For a binary system constituted by two masses mA and mB or-
tiondifferatleastbyachosentolerance,andthedensityoftherays bitingaroundthecommoncenterofmass,thereducedmassµ =
at a particular location in the source plane is proportional to the mAmB/(mA+mB)movesonanellipse,withsemi-majoraxisa
magnificationatthispoint.Alsointhiscasethemagnificationmap andeccentricitye.Theorbitequation6is(seee.g.Smart1980)
dependsonthemassratioq andtheprojectedbinarylenssepara-
tionb.Asonecaneasilyverify,thecomplexandIRSmethodsgive r(ψ)=a(1−ecosψ), (9)
comparable results beyond a distance R (in Einstein radii in the
wherer(ψ)istheradialdistancefromthecenterofmassandψthe
sourceplane) fromthefoldandcuspcaustics.Hence, weusethe
trueanomaly,respectively.Thetrueanomalyisrelatedtothemean
IRSmethodforanysourcedistancelessthanR,andthecomplex
anomalybytheKeplerequation
method otherwise. We verified that a good compromise between
thecalculationspeedandthequalityoftheoutputsimulatedlight
φ≡ω(t−tp)=ψ−esinψ, (10)
curvewasobtainedbychoosingR=0.05RE.
beingtagenericinstantoftimeduringtheorbitalmotion,tp the
timeofthepericenterpassageandω theangularfrequencygiven,
2.3 Finitesourceeffects asusual,by
For real astrophysical source stars, the point-like source approx-
imation described above breaks down where the magnification 2π GM
ω≡ = , (11)
A0(y)variesnonlinearlyonthescalescomparabletothesource P r a3
radiusρ∗.Inthissituation, themagnification isobtained by inte- where P is the keplerian orbital period. Since we are giving the
grating A0(y) in eq. (4) - weighted with the surface brightness semi-majoraxisainunitsoftheEinsteinradius,theorbitalperiod
S(y)-overthesourcediskanddividingbytheunamplifiedsource
P resultstobegiveninEinsteintime.
flux.Inparticular,
WethensolvedtheKeplerequationbyusingaseriesexpan-
A∗(yc)= RΣSA0(ycS+(yy′′))dS2(yy′′)d2y′ , (7) sionofBesselfunctions(seee.g.Watson1966).Inparticular,itcan
ΣS
R
5 ThenumericalintegrationisperformedbyusingtheCuhremethodim-
plementedinHahn(2005).
3 Inthepresentworkweusethenewandrobustalgorithmtosolvepoly- 6 Inthiswork,weassumethatthebinarylensmotionoccurswithrespect
nomialequationspresentedinSkowron&Gould(2012)andoptimizedfor toaright-handedreferenceframe(CM XYZ)withorigininoneofthe
themicro-lensingcase.Thecodeisafewtimesfasterthansimilarmethods ellipsefoci.ThetrajectoryisintheXY plane,beingtheZaxisorthogonal
designedformoregeneralpurposes. toit.Ingeneral, theorbitofthereduced massisalsocharacterized bya
4 Forabinarysystemofpoint-likemasses,thecausticcurveisconstituted semi-major axis forming aphase angle φa (measured counterclockwise)
either by a single continuous curve or two (or three) separate paths de- withtheX axis.Finally,theorbitalplaneisseenbythedistantobserver
pendingonthebinaryparametersbandq(Schneider,Ehlers&Falco1992). withaninclinationangleibetweentheZ axisandthelineofsight.Note
Foracompletepictureofthecausticzoo,werefertoPejcha&Heyrovsky´ thatthreerotations andatraslation (depending ontimet)arerequiredin
(2009). ordertoswitchbetweentheCMXYZandOξηκreferenceframes.
4 A.A. Nucita,M. Giordano,F.DePaolis,&G. Ingrosso
1.5 1.5
1
1 1
0.5
0.5 0.5 0.5
0 0 0 0
η η
−0.5 −0.5 −0.5 −0.5
−1 −1 −1 −1
−−1.25 −1 −05. 0 05. 1 −−1.25 −1 −05. 0 05. 1
5 5
4 4
A∗ 3 A∗ 3
2 2
1 1
0.001 0.001
uals 0 uals 0
d d
esi esi
r r
−0.001 −0.001
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2
ξ ξ
3 2 1 0 −1 −2 −3 3 2 1 0 −1 −2 −3
t/TE t/TE
Figure1. Leftside:asnapshotatagenerictimetofthepositionofthesourceandoftherotatinglensesisgiven(upperpanel)togetherwithazoom(insets)
aroundtheassociatedcentralcausticcurve.Here,thegravitationallensisassumedtobeabinarysystemrotatingcounterclockwisewithparametersa=0.2,
q=1×10−3,e=0,andi=φa=0◦sothattheorbitalperiodPSim=0.25TE.Thebackgroundsource(withradiusρ∗=0.03)ismovingwithimpact
parameteru0 = 0.18(asmeasuredfromthebinarycenterofmass),atanangleθ ≃ 138◦withrespecttotheξaxis(i.e.fromrighttoleft).Inthemiddle
panel,wegivethemagnificationlightcurve(redline)togetherwiththebestfitPaczyn´skimodel(greenline),herepracticallysuperimposed.Theresiduals
betweenthesimulateddataandthebest-fitmodelarealsoshown.InthebottompaneltheLomb-Scargleperiodogramisgivenwiththedashedlineindicating
theperiod(PEst)recoveredbytheanalysis(seediscussioninthetext).Rightside:thesameasbeforebutfordifferentvaluesoftheeccentricity,inclination
andphaseangles(e=0.5,i=45◦,andφa=0◦)ofthebinarysystem.Forclarity,theticklabelsoftheinsetsweremultipliedby104(left)and105(right),
respectively.
beeasilyshownthat formalism,theradialcoordinatesofthetwolensesontheirorbits
are
+∞
2
ψ=φ+ Jn(ne)sin(nφ), (12)
n
nX=1
rA(ψ)=−r(ψ)µ/mA, rB(ψ)=r(ψ)µ/mB. (13)
where Jn(ne) is the Bessel function of order n, and the mean
anomaly φ contains the time dependence. Note that a few terms
intheprevioussummationareusuallyrequiredtogetthesolution Ofcourse,theseparationb(whichentersinthecomplexandIRS
oftheKeplerequationwithinafewpercentofaccuracy.Hence,for methodsdescribedpreviously)betweenthetwolenscomponentsis
any time t, the previous equation gives the true anomaly ψ with nowtimedependentanditisgivenbytheprojectionontothelens
thereducedmasspositionobtainedbymeansofeq.(9).Withthis planeofthevectorrA−rB.
Signaturesofrotatingbinariesin micro-lensingexperiments 5
1.5 2 0.03
1 0.03 0.02
0.5 0.02 0.01
η −0.05 −00..00110 η 0 −−00..00120
−1 −0.02 −2 −0.03
−1−.25 −0.03−003. −002. −001. 0 001. 002. 003. −003. −002. −001. 0 001. 002. 003.
3.5
3
3
2.5
A∗ 2.25 A∗ 2
1.5 1.5
1
1
0.1 0.1
0.08
residuals −0000....000006422 residuals 0.005
−0.04 −0.05
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −3 −2 −1 0 1 2 3
ξ ξ
3 2 1 0 −1 −2 −3 5 4 3 2 1 0 −1 −2 −3 −4 −5
t/TE t/TE
Figure2.Leftside:micro-lensinglightcurveforabinarysystemwitha=0.23,q=0.8,e=0andi=φa=0◦.Thesourceandtrajectoryparametersare
thesameasinthecasedescribedinFig.1.ThesystemischaracterizedbyanorbitalperiodPSim ≃0.33TE,andtheeventlastsfor≃6TE.Rightside:by
usingthesameorbitalparameters,wefixedtheorbitalperiodto≃2TE andanobservationtimeof≃10TE.Thelarge(shortduration)spikeclearlyvisible
intheresiduallightcurveisduetoaplanetarycausticcrossingeventwhichoccurredduringthesourcetransit.ThebottompanelsshowtheLomb-Scargle
periodogramsasdiscussedinthetext.
3 THEOUTPUTSYNTHETICLIGHTCURVEANDTHE the distances in units of the Einstein radius, we simulated a bi-
TIMINGANALYSISONTHERESIDUALS nary system rotatingcounterclockwise withparameters a = 0.2,
q = 1×10−3, e = 0, and i = φa = 0◦. The corresponding
TheformalismintroducedinthepreviousSectionallowsustoob- binary system has an orbital period PSim = 0.25TE, being TE
tain simulated micro-lensing light curves taking into account the the Einstein time of the event. The purple ellipse represents the
orbitalmotionofthebinarylens.Here,weconcentrate onbinary planet trajectory as projected in the plane of the sky. The back-
systemscharacterizedbyorbitalperiodsP lowerthanthetypical groundsource(withradiusρ∗ =0.03)ismovingattheimpactpa-
Einstein time TE of the event and by orbital parameters b and q rameteru0 =0.2(asmeasuredfromthebinarycenterofmass)at
givingrisetoeventlightcurvesclosetothePaczyn´skibehaviour. anangleθ≃138◦withrespecttotheξaxis.Withtheassumedbi-
In Fig. 1 (left side), we give a snapshot at a generic time t narysystemandmicro-lensingparameters,weexpecttohaveonly
of theposition ofthesource, therotatinglenssystemand theas- minordeviations(aplanetarycase)withrespecttothesinglelens
sociated caustic curves (inset) in the source plane (upper panel). light curve. Indeed, in the middle panel, we show the magnifica-
Intheadditionalmaterialavailableon-line,wepresentamovieof tionlightcurve(redline)resultingfromeq.(7)towhichthebestfit
themicro-lensingeventcorrespondingtoFig.1.Bymeasuringall lightcurve(greenline)obtainedwithaPaczyn´skimodelissuper-
6 A.A. Nucita,M. Giordano,F.DePaolis,&G. Ingrosso
imposed.Asonecansee,thetwocurvesarealmostindistinguish- 4
ableinthiscase.AsitisclearfromthebottompanelofFig.1,the 3
3
residualsbetweenthesimulatedlightcurveandthebestfitmodel 2 2
areaslowasafewpartsperthousandbutclearlyshowaperiodic 1 1
0 0
behaviour (here, for simplicity, we are ignoring the effect of any η −1 −1
noiseinthedatastrain).Asaguidefortheeye,wealsoshowase- −2 −2
−3 −3
riesofverticalsolidlinesseparatedintimebyoneorbitalperiodP.
−4 −3−2−1 01 2 3
Intherightsideofthefigure,wesimulatedamicro-lensingevent
due to a binary system with the same mass and semi-major axis
asbefore,butdifferentvaluesofeccentricity,inclinationandphase 3
angleB(eefo=re0d.5is,cius=sin4g5◦th,eanrdesφulats=ob0t◦ai)n.edwiththeperiodanalysis A∗ 2.25
ontheresiduals,wenotethat,duetothefactthatthesimulatedlight 1.5
curveandthebest-fitPaczyn´skimodeldonothavethepeakatthe 1
sametime(andthisisageneralcharacteristicofsuchevents),the 0.1
residualsshow anasymmetry withrespect tot0 andanumber of als 0.05
extra peaks (especially concentrated around the event maximum) du
thatarenotcorrelatedtotherealbinarysystemperiod.Aswewill resi 0
discusslater,thisintroducesspuriouspeaksintheperiodanalysis −0.05
that might make impossible to recover the simulated period. In- −4 −2 0 2 4
deed, any algorithm used for theperiod search looks for features ξ
inphaseandfailsaroundthemicro-lensingmaximumtogivethe
6 4 2 0 −2 −4 −6
rightperiod.
t/TE
We tested different methods to extract the period from the
residual light curve for several simulated events: theFast Fourier
Transform -FFT- (Pressetal. 2007); the Phase Dispersion Min-
imization -PDM- (Stellingwerf 1978); the Lomb-Scargle peri-
odogram(Lomb1976;Scargle1982);itsgeneralizedformgivenin
Zechmeister&Kuerster (2009) and the epoch folding technique.
All the algorithms give consistent results, but here we prefer to
presentthoseobtainedviatheclassicalLomb-Scarglemethod(the
generalizedversiondidnotimprovedtheperioddetection)because
itsstatisticalbehaviouriswellknownandthetechniquecanbeeas-
ilyimplemented.
Themethodrequirestospecifytheminimum(νmin)andmax-
imum (νmax) frequencies to be searched for in the input signal,
together with the frequency step ∆ν. We chose to set these pa-
rameters depending on the observational data set, i.e. νmin =
1/(3Tobs),νmax = 1/(2δt),oversampling byafactor 10,being
T thedurationoftheobservationandδttheassociatedtimestep.
obs
Notethatbyusingtheminimumfrequencyνminweareimplicitly
requiring to have at least three full cycles per observational win- Figure3.Theexpectedmicro-lensinglightcurveforabinarysystemwith
dow. orbitalparametersa= 0.23,q = 0.8,e= 0.5,i = 45◦,andφa =0◦,
The blind application of the Lomb-Scargle method to the assumingtheorbitalperiodPSim≃2TE andatotalobservationduration
residual light curves resulted in the periodograms shown in Fig. of14TE.Forclarity,theticklabelsoftheinsetweremultipliedby103.
1,wherethedashedverticallines(alwaysatperiodPSim ≃2000)
mark the periodicity detected in the signal. Note that the Lomb-
Scargleperiodogram(aswellastheotherperiodsearchalgorithms) The second case we discuss is a rotating binary similar to
doesnot giveexactlythesimulatedperiodsincethefrequency of that considered in Pennyetal. (2011b) with orbital parameters
the periodic signal isaffected by the relative source-lens motion. a = 0.23, q = 0.8, e = 0, and i = φa = 0◦ rotating with a
Asdiscussedabove,thecentralpartoftheresiduallightcurvehas period PSim ≃ 0.33TE. Sincethesystem isface-on and theec-
spuriouspeaksthatdisturbthetiminganalysissincetheyintroduce centricityisnull,theprojecteddistancedoesnotchangeduringthe
additionalpoweratfrequenciesdifferentfromthefundamentalone, micro-lensingevent,thusimplyingthatthecausticssimplyrotate
i.eν =2π/PSim. withoutanydeformation. Thisparticulargeometry appearsinthe
We have found that the best results in the period search are periodogramasapeakalwaysathalfofthesimulatedperiod(see
obtainedbyremovingacentralregionintheresidualcurve,around theleftbottompanelofFig.2)asaconsequenceoftheNorth-South
the event peak. Without this cut, the Lomb-Scargle periodogram symmetryinthecausticplane.Intherightsideofthesamefigure,
mayreturnotherpeaksinadditiontothatcorrespondingtothesim- wesimulateabinarylens event withthesame orbitalparameters
ulatedone.Itisremarkablethat,inallthecaseswehaveconsidered, but with PSim ≃ 2TE and an observational window of ≃ 6TE.
itissufficienttoremoveaverysmallregionaroundthemaximum Inspite of the fact that there areonly afew full cycles, theperi-
magnificationwithsizeoftheorderofafractionoftheorbitalpe- odogramgivesagainhalfoftheexpectedperiod.
riodPSiminordertogetthetrueperiodicity(indicatedasPEst). Wenotethatthegeometryofthebinarylensdescribedinthe
Signaturesofrotatingbinariesin micro-lensingexperiments 7
twoexamplesofFig.2isratherunlikelytoappearinrealobserva- assolid,dottedanddashedlines,respectively.Asonecannote,the
tionsandso,inFig.3,weconsideramorerealisticbinaryconfig- periodsat≃4daysand≃30daysarenotsignificant.
urationwithe = 0.5andi = 45◦,beingalltheotherparameters Note also that interpreting the ≃ 30 days feature as a true
unchanged.Asonecansee,inthiscasetheLomb-Scargleanalysis periodicitycouldbeproblematicsinceitiscomparabletotheevent
allows us to recover the right period even if the associated peri- EinsteintimeTE ≃23.4days.Inaddition,atimeresolvedperiod
odogrampeakisquietbroad. analysisdoesnotreturnastableresult.Asamatteroffact,wenote
thatthelightcurveoftheeventOGLE-2011-BLG-1127showsan
oscillationaroundthebestfit(static)model,asitisapparentinFig.
1of Shvartzvaldetal. (2013),withatimescaleof ≃ 15 days. If
4 RESULTANDDISCUSSIONS
the bump-like structure observed ≃ 50 days after the event peak
We have considered the orbital binary lens motion in simulated isreallypresentalongthewholelightcurve(asalsomentionedin
micro-lensingeventsthathavelight-curvessimilartothePaczyn´ski Shvartzvaldetal.2013)thenouranalysisisreturningaperiodicity
behaviourandhaveshownthatinmostcasesatiminganalysisof relatedtothesourceortothebinarylensmotion.
theresidualsallowsonetorecoverthebinaryperiodP.Sincethe
simulatedlightcurveandthebest-fitPaczyn´skimodeldonothave,
ingeneral,thepeakatthesametime,theresidualsshowanasym-
REFERENCES
metrywithrespecttot0 andanumber ofextrapeaksthatarenot
correlatedtotherealbinarysystemperiod.Thisintroducesspuri- Abe,F.,etal.,2003,A&A,411,L493.
ous features in the analysis that can be overcome by excluding a Alcock,C.,etal.,1993,Nature,365,621.
central region of the light curve whose size is reduced smoothly Alcock,C.,etal.,1997,ApJ,491,436.
untilthetrueperiodislost. Beaulieu,J.-P.,etal.2006,Nature,439,437.
The importance of this procedure is that of inferring the or- Bennett, D. P., 2009, Exoplanets. Mason J., editor. Springer,
bital period of the binary lens without performing a numerically Berlin;2009.
heavymulti-dimensionalfitprocedurewhich,generally,donotgive Bennett,D.P.&Rhie,S.H.,1996,ApJ,472,660.
unique solutions. Here, by considering Paczyn´ski-like events, we Bond,I.A.,etal.,2004,ApJ,606,L155.
showed that the procedure is robust enough to recover the simu- Bozza,V.,1999,ApJ,348,311.
lated orbital period, provided that the event duration contains at Claret,A.,2000,A&A,363,1081.
leastthreefullcycles. Choi,J.-Y.,Shin,I.-G.,Park,S.-Y.etal.,2012,ApJ751,41.
Asystematicsearchforsuchsignaturesonreal(archival)ob- Dominik,M.,1997,GalacticMicrolensingBeyondTheStandard
served micro-lensing events will be presented elsewhere. Here, Model,Ph.D.ThesisatUniversityofDortmund.
we show a preliminary analysis on the event OGLE-2011-BLG- Dominik,M.,2010,Gen.Rel.Grav.,42,2075.
1127/MOA-2011-BLG-322(Shvartzvaldetal.2013)possiblydue Gaudi,B.S.&Gould,A.,1999,ApJ,513619.
to a binary lens. By using the publicly available OGLE data GaudiB.S.,etal.,2008,Science,319,927.
(Udalskietal.2003,http://ogle.astrouw.edu.pl/)wehavefittedthe Gaudi,B.S.,Exoplanets.SeagerS.,editor.Tucson:Univ.Arizona
datawithasimplePaczyn´skimodel(seeFig.4,panela)andana- Press;2011.p.79.
lyzedtheresiduals(Fig.4,panelb)byusingeitherthegeneralized Gould,A.,1994,ApJ,421,L71.
Lomb-Scargle and the discrete fast Fourier transform techniques Gould,A.,etal.,2006,ApJ,644,L37.
(DFFT treated via the CLEAN algorithm of Robertsetal 1987). Gould,A.,2008,ApJ,681,1593.
Thesearethebest-performingperiodsearchmethodsforthiskind Hahn,T.,2005,ComputerPhysicsCommunications,168,2,78.
of data. Both techniques return the presence of a period at about Horne,J.H.,&Baliunas,S.L.,1986,ApJ,302,757.
≃12days(seeFig.4,panelscandd)7. Kayser,R.,Refsdal,S.&Stabell,R.,1986,A&A,166,36.
It seems that thisperiodic feature isremarkably stablesince Ingrosso,G.,etal.,2012,MNRAS,426,1496.
italwaysappearswhenatimeresolvedperiodogramisperformed, Ingrosso, G., et al., 2013, Physica Scripta. 2013. preprint
i.e.whentheperiodsearchalgorithmisappliedondifferentparts arXiv:1310.5866.
oftheresiduallightcurve,providedthateachpartcontainsasuf- Lomb,N.R.,1976,Ap&SS,39,447.
ficiently large number of cycles. It is therefore possible that the Mao,S.,&Paczyn´ski,B.,1991,ApJ,374,L37.
periodicityat≃ 12daysisassociatedtoanintrinsicvariabilityof Paczyn´ski,B.,1986,ApJ,304,1.
the source star as if it is in a binary system or intrinsically vari- Paczyn´ski,B.,1996,ARA&A,34,419.
able(seee.g.Wyrzykowskietal.2006)orcontaminatedbyaclose Park,H.,etal.,2013,eprintarXiv:1306.3744.
variable star. Note also that some period search techniques, such Pejcha,O.,&Heyrovsky´,D.,2009,ApJ,690,1772.
asthecleanedDFFT(Fig.4,panelsc)andthegeneralizedLomb- Penny,M.T.,Mao,S.,&Kerins,E.,2011,MNRAS,412,607.
Scargleperiodogram,givesignificantpoweralsoat≃ 4daysand Penny,M.T.,Kerins,E.,&Mao,S.,2011,MNRAS,417,2216.
≃ 30days. Sincewetesttrialperiodsinagivenrange,itispos- Perryman,M.,2000,Rep.Prog.Phys.,63,1209.
sibletoevaluate thesignificance of each feature compared tothe Perryman, M., et al., 2005, Report by the ESA–ESO Working
poweratallotherfrequencies.Hence,followingHorne&Baliunas grouponExtra-SolarPlanets,astro-ph/0506163.
(1986) (see also Zechmeister&Kuerster 2009), we evaluate the Press,W.H..Teukolsky,S.A.,Vetterling,W.T.&Flannery,B.P.,
falsealarmprobabilityandobtainedthesignificancelevelsat68%, Numerical Recipes. The Art of Scientific Computing, 3rd edn.
90%,and99%whichareshownatthebottom-rightpanelofFig.4 Cambridge:CambridgeUniversityPress,2007.
Roberts,D.H.,Lehar,J.,&Dreher,J.W.,1987,AJ,93,968.
Roulet,E.&Mollerach,S.,1997,Phys.Rep.,279,2.
7 Herewenotethattheregionaroundtheeventpeakhasnotbeenremoved Roulet,E.&Mollerach,S.,GravitationalLensingandMicrolens-
sincealreadypoorofdata. ing.Singapore:WorldScientific,2002.
8 A.A. Nucita,M. Giordano,F.DePaolis,&G. Ingrosso
Figure4.TheeventOGLE-2011-BLG-1127lightcurveintheIbandandthebestfitPaczyn´skimodelisshowninpanela),togetherwiththeassociated
residuals(panelb).Inpanelsc)andd),weshowtheperiodogramsobtainedviatheCLEANedDFFTandgeneralizedLomb-Scarglemethods,respectively
(seetextfordetails).FortheLomb-Scargleperiodogram,weevaluatedthefalsealarmprobabilityandobtainedtheconfidencelevelsat68%,90%,and99%
shownintheright-bottompanelasthesolid,dottedanddashedhorizontallines,respectively.
Scargle,J.D.,1982,ApJ,263,835.
Schneider, P., Ehlers, J. & Falco E. E., , Gravitational Lenses.
Berlin:Springer,1992.
Shvartzvald,Y.,etal.,2013,preprint(arXiv:1310.0008).
Skowron,J.,&Gould,A.,2012,preprint(astro-ph/1203.1034).
Smart, W., M., Textbook on Spherical Astronomy, Cambridge:
CambridgeUniversityPress,1980.
Stellingwerf,R.F.,1978,ApJ,224,953.
Udalski,A.,etal.,2003,ActaAstronomica,53,4.
Udalski,A.,etal.,2005,ApJ628,L109.
Wambsganss,J.,1997,MNRAS,284,172.
Watson,G.N.,ATreatiseontheTheoryofBesselFunctions,Cam-
dridge:CambridgeUniversityPress,1966.
Witt,H.J.,1990,A&A,236,311.
Witt,H.J.,&Mao,S.,1994,ApJ,430,505.
Witt,H.J.,&Mao,S.,1995,ApJ,447,L105.
Witt,H.J.,&Petters,A.O.,1993,JournalofMathematicalPhsy-
ics,34,4093.
Wyrzykowski,L.,etal.,2006,ActaAstronomica,56,145.
ZakharovA.F.&Sazhin,M.V.,1998,SovietPhysicsUspekhi,41,
945.
Zechmeister,M.&Kuerster,M.,2009,A&A,496,577.