Orbital motion effects in astrometric microlensing

1.2 MB·

by  Sedighe Sajadian

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed27January2014 (MNLATEXstylefilev2.2)
Orbital motion effects in astrometric microlensing
⋆
Sedighe Sajadian
SchoolofAstronomy,IPM(InstituteforResearchinFundamentalSciences),P.O.Box19395-5531,Tehran,Iran
4
1
0 27January2014
2

n
ABSTRACT
a
We investigate lens orbital motion in astrometric microlensing and its detectability. In mi-
J
crolensingevents,thelightcentroidshiftinthesourcetrajectory(theastrometrictrajectory)
4
falls offmuchmore slowly than the lightamplificationas the sourcedistance fromthe lens
2
positionincreases.Asaresult,perturbationsdevelopedwithtimesuchaslensorbitalmotion

canmakeconsiderabledeviationsinastrometrictrajectories.Therotationofthesourcetrajec-
]
M toryduetolensorbitalmotionproducesamoredetectableastrometricdeviationbecausethe
astrometriccross-sectionismuchlargerthanthephotometricone.Amongbinarymicrolens-
I
. ing events with detectable astrometric trajectories, those with stellar-mass black holes have
h
mostlikelydetectableastrometricsignaturesoforbitalmotion.Detectinglensorbitalmotion
p
in their astrometric trajectories helps to discover further secondary components around the
-
o primaryevenwithoutanyphotometricbinaritysignatureaswellasresolveclose/widedegen-
r eracy. For these binary microlensing events, we evaluate the efficiency of detecting orbital
t
s motion in astrometric trajectories and photometric light curves by performingMonte Carlo
a simulation.Weconcludethatastrometricefficiencyis87.3percentwhereasthephotometric
[ efficiencyis48.2percent.

1
1 INTRODUCTION ing high precision astrometry, which will be operational in near
v
6 future(Proftetal.2011).
1 Gravitational field of an object acts as a gravitational lens which During a microlensing event by measuring the astrometric
4 deviateslightpathofacollinearbackgroundsourceandproduces lensing and parallax effects, the mass of the deflector can be in-
6 distortedimages(Einstein1936).InGalacticscale,theangularsep- ferred without knowing the exact distances of the lens and the
1. aration of the images is of order of milli-acrsecond that is too source from the observer (Paczyn´ski1997; Miralda-Escde´1996).
0 smalltoberesolvedevenbythemostmoderntelescopes.Instead, Also the degeneracy in close/wide caustic-crossing binary mi-
4 the combined light of images received by an observer is mag- crolensingevents(Dominik1999)canberemovedbyastrometric
1 nified in comparison to the un-lensed source. This phenomenon measurements (Gould&Han2000; Chungetal.2009). The two
v: is called gravitational microlensing which was proposed as a configurationsofthelensandsourceproducedifferentastrometric
i methodtoprobedarkobjectsinGalactichalo,extrasolarplanets, trajectorieseventhoughtheymightpresentthesamelightcurves.
X studystellaratmosphere,etc.(Liebes1964;Chang&Refsdal1979; Hence, by measuring the astrometric trajectories this degeneracy
r Paczyn´ski1986a,b). canberemoved(Hanetal.1999).
a
One of features of gravitational microlensing is the devi- Generally, there are some anomalies in microlensing events
ation of the light centroid of the source star images from the which deviate the observed light curve and astrometric trajectory
source star position. For the case of a point-mass lens, the cen- of source star from the standard model. Some deviations owing
troidshiftofsourcestarimagestracesanellipseinthelensplane to these anomalies help obtain extra information about lens and
whilethesourcestarispassinganstraightforwardlineinthelens source and thus resolve degeneracy. One of them is the effect of
plane (Walker1995; Miyamoto&Yoshii1995; Høgetal.1995; lens orbital motion in a binary microlensing event. In this case
Jeongetal.1999).Incontrastwiththemagnificationfactorwhich two components in a gravitationally bound binary system which
isadimensionlessscalar,thelightcentroidshiftofthesourcestar act as lens rotate around their common center of mass. As a re-
images isa dimensional vector and itssizeis proportional toan- sult,theirorientationchangesasamicrolensingeventprogressand
gular Einsteinradius, i.e.theangular radiusof images ringwhen the resulting light curve is different from that due to a static bi-
observer, source and lens are completely aligned. For a stellar- narymicrolens.TheratiooftheEinsteincrossingtimetotheorbital
mass lens, the angular Einstein radius is a few hundreds micro- periodoflenssystemisconsidered asacriterionwhichindicates
arcsecond,toosmalltobeobserveddirectly.However,itsquantity theprobabilityofdetectingtheorbitalmotioninthemicrolensing
enhancesintwocases:(i)whenthelensmassishigh,e.g.astellar- lightcurve(Dominik1998;Iokaetal.1999).Measuringthiseffect
massblackholeasmicrolensand(ii)whenthelensisverycloseto givesinformationabout theorbitof microlensessystemwhichin
theobserver.Thelightcentroidshiftofsourcestarimagesinthese turn helps to resolve the close/wide degeneracy (Anetal.2002;
microlensingeventsismeasurablebyhigh-precision interferome- Gaudietal.2008; Dongetal.2009; Shinetal.2013). Detectabil-
terse.g. VeryLargeTelescope Interferometry(VLTI)ortheHST ityoforbitalmotioninlightcurvesofbinarymicrolensingevents
(Paczyn´ski1995).AnimportantfutureprojectisGAIA,perform- wasinvestigatedextensivelybyPennyetal.(2011)whereastherel-
c 0000RAS
(cid:13)
2 SedigheSajadian
evant deviation intheastrometrictrajectoryofsource starandits effectinmicrolensinglightcurvesinthesamewayasDominik’s
detectabilityarenotclearyet. approach.
Inthiswork,westudylensorbitalmotioninastrometricmi- Letusconsideragravitationallyboundsystemasmicrolens.
crolensing and itsdetectability. In microlensing events, theastro- Under the gravitational effect, the secondary component with re-
metrictrajectoryofsourcestartendstozeromuchmoreslowlythan specttothefirstonetracesoutanellipseoneofwhosefocusisat
thelightamplificationasthesourcedistancefromthelensposition thecenterofmassposition.Twocomponentsoftheirrelativedis-
rises.Asaresult,perturbationsenlargedwithtimesuchaslensor- tancearegivenby:
bitalmotioncanproduceconsiderabledeviationsintheastrometric
x(ξ) = a(cosξ ε)
trajectories.Sincetheastrometriccross-sectionismuchlargerthan −
the photometric one, so the rotation in the source trajectory ow- y(ξ) = a 1 ε2sinξ, (1)
−
ing to the orbital motion is more probable to be detected in the
wherexandypaxesarealongthesemi-majorandsemi-minoraxes,
astrometric trajectory than the photometric light curve. However,
ε is the eccentricity, a isthe semi-major axis and ξ is a periodic
dependingonthesizeoftheangularEinsteinradiusthiseffectcan
functionwithtimewhichforasmalleccentricityanduptoitsfirst
bedetected.Wefindthat,amongbinarymicrolensingeventswith
orderisestimatedas:
detectableastrometrictrajectories,thosewithrotatingstellar-mass
black holes have most likely detectable astrometric signatures of ξ(t)=2πt−tp +2εsin(2πt−tp), (2)
orbitalmotion.Forthesebinarymicrolensingevents,weevaluate P P
theefficiencyofdetectingorbitalmotioninastrometrictrajectories where tp is the time of arriving at the perihelion point of orbit,
andphotometric light curvesbyperforming Monte Carlosimula- P is the orbital period of the lenses motion. To study the orbital
tion.Detectinglensorbitalmotionintheirastrometrictrajectories motion of lenses on microlensing light curves, weshould project
helpstodiscoverevensecondarycomponentswithnophotometric theorbitplaneof lensesintotheskyplane.Inthatcase,weneed
binaritysignaturesaswellasresolveclose/widedegeneracy. totwoconsecutiverotationangles:β aroundx-axisandγ around
Thispaperisorganizedasfollows:Insection(2)theeffectof y-axis.Theprojectedcomponentsoftherelativepositionvectorof
orbitalmotioninbinarymicrolensingeventsareexplained.Wenext thesecondonewithrespecttothefirst,intheskyplanenormalized
describeastrometricpropertiesof microlensingeventsandfinally totheEinsteinradiusare:
weinvestigate theorbital motioneffect on theastrometric trajec-
x (t) = ρ[cosγ(cosξ(t) ε)+
tories.InthenextsectionbypreformingaMonteCarlosimulation 1 −
westudytheastrometricandphotometricefficienciesfordetecting +sinβsinγ 1 ε2sinξ(t)]
−
lensorbitalmotion.Insection(4)weexplaintheresults.
x (t) = ρcosβ 1 pε2sinξ(t), (3)
2
−
where ρ isthe normpalized semi-major axis tothe Einstein radius
(Dominik1998). Hence, lens orbital motion causes the projected
2 ORBITALMOTIONINASTROMETRIC
distance between two lenses in the lens plane changes with time
MICROLENSING
whichisgivenby:
Inthissectionwefirststudytheorbitalmotioneffectinbinarymi-
crolensing events. Havingreviewed theastrometric microlensing, d(t)= x1(t)2+x2(t)2. (4)
weinvestigatethemicrolensingeventswithdetectableastrometric On thepother hand, the binary axis with respect to the projected
shiftsinsource trajectories. Finally,westudy whether theorbital source trajectory rotates with angle θl(t) = tan−1(x2/x1). So,
motioneffectintheastrometrictrajectoriesisdetectable.
duringamicrolensingeventwithbinaryrotatinglenses,thesource
trajectory rotates with respect to the binary axis with angle θ
l
−
aroundthelineofsighttowardstheobserver.However,thisrotation
2.1 Orbitalmotioneffectofbinarymicrolenes
isnotalwaysdetectableintheobserverreferenceframe.
Dominik(1998)firststudiedtheeffectoflensorbitalmotioninbi- Thesignatureoflensorbitalmotioninthemicrolensinglight
narymicrolensingeventsandconcludedthatinthemostmicrolens- curves is not always observable and its Detectability depends
ing events this effect is ignorable, but in some long-duration mi- strongly on the ratio of the lensing characteristic time to the or-
crolensingeventscanprobablybeobserved.Orbitalmotioneffects bitalperiodoflensesmotion.Dominik(1998)consideredtheEin-
ontheplanetarysignalsinhigh-magnificationmicrolensingevents steincrossingtimeasthecharacteristictimeofabinarymicrolens-
wasinvestigatedbyRattenburyetal.(2002).Recently,Pennyetal. ingeventtoevaluatetheorbitalmotiondetectability.However,the
(2012) indicated how fraction of binary and planetary microlens- orbitalmotiondetectionefficiencyforbinarymicrolensingevents
ingeventsexhibittheorbitalmotioneffectsintheirlightcurvesby withoutcaustic-crossing featuresistoosmallandoforder ofone
performingMonteCarlosimulationandinvestigatedthosefactors per cent (Pennyetal.2011). As a result, we consider the effec-
whichchangethisfraction.Untilnow,severalobservedmicrolens- tive time scale for detecting signatures of lens orbital motion as
ing events have shown the signatures of lens orbital motion. In t˜E = ∆(q,d)tE instead of the Einstein crossing time, where
somecases,thesesignatureshaveallowedtoinfersomeoforbital ∆(q,d) refers to the average over the ratios of the lengths of
parametersandremovetheclose/widedegeneracy(Anetal.2002; the source trajectories providing the sources are inside the caus-
Gaudietal.2008;Dongetal.2009; Shinetal.2013). Sometimes, ticcurvetothetotallengthofthesourcetrajectories.Weassume
there are several degenerate best-fitting solutions for orbital mo- thatthereareseveralparallelsourcetrajectoriesoverthelensplane
tion modeling. Three systems with complete orbital solutions withthesamelengthsi.e.Lsothattheycoverthecausticcurve.In
have been found, e.g. the microlensing event OGLE-2011-BLG- thiscase,∆(q,d)isgivenby:
0417 (Shinetal.2011; Shinetal.2012). However, one of these
N N
solutions can actually be checked with radial velocity method 1 1
∆(q,d)= ( li/N)= liδ= c, (5)
(Gouldetal.2013). In the following we study the orbital motion L NLδ S
i i
X X
c 0000RAS,MNRAS000,000–000
(cid:13)
Orbitalmotioneffects inastrometricmicrolensing 3
0.5
u0=0.1
u0=0.4
1 10-1 0.4 u0=0.8
u0=1.2
u0=2.0
0.3 u0=5.0
0.5 10-2
M (solar mass) 0 10-3 δθθ/c,yE 00..12
log
-0.5 10-4 0
-0.1
-1 10-5 -0.4 -0.2 0 0.2 0.4
-1 -0.5 0 0.5 1 δθc,x/θE
log s(Au)
Figure 1. The map of the detectability factor f in the lenses parameter Figure2.Thetrajectory ofthe lightcentroid shiftnormalized totheθE
spacecontainingthelensestotalmassMandthesemi-majoraxisoflenses assumingapoint-like sourcestarispassinganstraightforward lineinthe
orbit s. The black dashed and solid lines represent the branch values of gravitational fieldofapoint-masslensforvariousamountsofimpactpa-
theprojected distance between twolenses normalized totheEinstein ra- rameter.Thestarsignsindicatethemostdeviationinthelightcentroidof
diusbetween3differenttopologiesofcaustic:close,intermediateandwide imageswithrespecttothesourcepositionwhichhappensatu=√2.
binariesrespectively.
2.2 Astrometricpropertiesofmicrolensing
Inmicrolensingevents,thelightcentroidvectorofsourcestarim-
whereliisthelengthoftheportionofithsourcetrajectoryprovid- ages does not coincide withthesource position. Thisastrometric
ingthesourceisinsidethecausticcurve,N isthenumber ofthe shift in source star position changes with time as a microlensing
sourcetrajectories,δ = L/N isthenormaldistancebetweentwo eventprogresses.Forapoint-masslens,thecentroidshiftvectorof
consecutivesourcetrajectories,distheprojecteddistancebetween sourcestarimagesisgivenby:
twolenses normalized tothe Einsteinradiusand q istheratioof
tihseeqleunasltmoathsesersa.tiAococfotrhdeinagretaoinetqeuraiotirotno(th5e),ciatuisstciclecaurrvtheatto∆an(qa,rdea) δθc = µ1θµ11++µµ22θ2 −uθE = u2θ+E 2u, (7)
equaltoL2 i.e. c.Byconsideringthisfactor,thedetectabilityof where θi and µi are the position and magnification factor of ith
lensorbitalmotiSoncanbecharacterizedbythefactorf: image,u=pxˆ+u0yˆisthevectoroftheprojectedangularposition
ofthesourcestarwithrespecttothelensnormalizedbytheangular
f = t˜E. (6) EinsteinradiusofthelensθE inwhichp=(t−t0)/tE ,t0isthe
P time of the closest approach, tE is the Einstein crossing time, xˆ
andyˆaretheunitvectorsinthedirectionsparallelwithandnormal
Themapofdetectabilityfactorf isshowninFigure(1)inthepa-
tothedirectionofthelens-sourcetransversemotion.Theangular
rameterspaceoflensescontainingthetotalmassoflensesM and
Einsteinradiusoflensisgivenby:
semi-major axiss. Inthis figurethe black dashed and solidlines
represent the threshold values between the close-intermediate (or
M π (mas)
resonance) and intermediate-wide binaries respectively. For cal- θE =√κMlπrel =300µas l rel , (8)
culating the inner area of caustics, we first determine the caustic s0.3M⊙r 0.036
pointswhichhavesosmalldeterminantofJacobianmatrix.Then, whereM isthelensmass,κ= 4G andπ =1Au( 1 1 )
aftersortingtheconsecutivepointsonthecausticlinewecalculate where Dll and Ds are the lensc2aAnud sourcreeldistancesDflro−mDtshe
the interior area of caustics numerically. The detectability factor observer. Usually, a microlensing parallax is defined as πE =
has the highest amount for the intermediate microlensing events πrel/θE whichcanbemeasuredfromthephotometricevent.
whichlocateneartotheclose-intermediatethresholdline(dashed Thecomponentsofthiscentroidshiftvectoraregivenby:
line).However,Gaudi(2012)intuitivelypointedoutthatthereso-
p
nancemicrolensingeventsaccordingtotheirlargesizesandcross- δθc,x(u0,p) = u2+p2+2θE,
sections are more sensitive to the small changes in d owing to 0
u
lensorbitalmotion.Accordingtothisfigure,theorbitalmotionde- δθc,y(u0,p) = u2+p02+2θE. (9)
tectabilityinmicrolensingeventswithmoremassivelenseswhich 0
have the same caustic curve, is larger than those with less mas- Theshiftinthelightcentroidtrajectoryofthesourcestarimages
sive lenses. For this plot, we consider q = 1, vt = 175km/s, owingtoapoint-masslenstracesanellipsewhilethesourcestar
Dl = 4KpcandDs = 8Kpc.However, thesmalleramountsof ispassinganstraightforwardlineinthelensplane,plottedinFig-
q just move two threshold lines towards each other so that they ure(2).BydefiningX = δθc,x andY = δθc,y − 2(uu020θ+E2),these
mostlyformcloseandwidebinarieswhilethedetectabilityfactor coordinatessatisfythefollowingequation:
ismaximumforintermediatebinariessimilartotheplottedcase.
Y
Inthefollowing,weexplaintheastrometricpropertiesofmi- X2+( )2 =a2, (10)
b
crolensing and seek binary microlensing events which have de-
whereb= u0 anda= θE .Thisellipseiscalledtheas-
tectable astrometric trajectories as well as very likely detectable √u20+2 2√u20+2
orbitalmotionsignatures. trometricellipse(Walker1995;Jeongetal.1999).Theratioofthe
c 0000RAS,MNRAS000,000–000
(cid:13)
4 SedigheSajadian
M(M ) Dl(Kpc) Ds(Kpc) RE(Au) θE(millias) µl(′′/yr) Vt(Km/s) tE(day) s(Au) P(day)
⊙
( ) >5 6.5 8.0 >7.0 >1.1 0.006 175 >70 4 61300
A ∼ ∼ ∼ ∼ ∼
( ) 0.3 60.1 8.0 60.5 >4.9 >0.4 185 65 4 5000
B ∼ ∼ ∼ ∼ ∼
Table1.Thepossiblerangesofphysicalparametersoftwosetsofbinarymicrolensingeventswithmeasurableshiftsintheastrometriclightcentroidofsource
starimages:( )gravitational microlensing events withbinarystellar-mass blackholes and( )thosewithhighpropermotionstarslocated indistances
A B
smallerthan100parsecfromthesun.
axes of this ellipse is a function of impact parameter, so that for
4
2.6 3 3.6
thelargeimpact parameter thisellipseconverts toa circlewhose (a) 2
radius decreases by increasing impact parameter and for small 2.4 R)E 01 2.4
Y( -1
amounts, itbecomes astraight line(seeFigure2) (Walker1995). 2.2 -2
-3 1.2
sswionhcifcreohmabasa=swtreo2lθlmuE0aestratinhcdedwpataearawclaleanxgsemitmetpahsleyuraoenmbgtuealniantrtuEh0einflsretoenmisnmprahadsoistuoscmafnertorbmye, A 11 ..268 -4-4-3-2-1x( R0E )1 2 3 4 - 01.2 δθ(mas)c,y
inferred. In this figure the points with the maximum amounts of
the centroid shift (i.e. u = √2) were shown with star symbols. 1.4 -2.4
1.2
bθFEeoir/nuuglw≫enhse√erde2abstyhtheaespimzoeiangotn-fmifithacseastailosetnnrosfmafcoetrotrruicocfeant1prootiiendntd-slshikitfeotsf1oalu+lrsc2oe/ffsutaa4sr. mag) 0 . 011 -3 -2 -1 δθ 0c,x(mas 1) 2 -343..56 mas)
Hence,thecentroidshiftofsourcestarim≫agesfallsoffmuchmore res( 1.5 res(
-0.1 0
slowlythanthelightamplificationofsourcewhilethesourcedis- -2 -1 0 1 2 3 -6 -4 -2 0 2 4 6
t(tE) t(tE)
tancefromthelensrises(Dominik&Sahu2000).Asaresult,the
astrometriccross-sectionislargerthanthephotometriconeinmi- 2.4 4 1.6
2
crolenInsincgonetvraesnttsw.ith the magnification factor which is a dimen- 2.2 (b) Y(R)E - 02 0.8
2
sionlessscalar,thelightcentroidshiftofthesourcestarimagesis -4
astediinmreandsiiuosnaglivveenctboyraenqduaittisosniz(e8)i.sTphroepaonrgtiuolnaarlEtointshteeiannrgaudliaursEoifna- A 11..68 -4 -2x( R0E) 2 4 0 δθ(mas)c,y
K-orM-dwarfstarasthemostprobablelenswithM 0.3M
l ∼ ⊙ 1.4 -0.8
locatedintheGalacticdiskD 6.5Kpc,intheobservationsto-
l
whuanrddsretdhemGicarola-catriccsebcuolngde(i.see.eDeqs∼ua∼tio8n.08K)pwch,icishoisftoorodesrmoafllatofebwe 01. .112 δθc,x(mas) -1.6
-1 0 1 3
olabrsEerivnesdte.inHorawdeivuesr,enfohrantwceos:se(ts)ofwmheincrothleenlseinnsgmevaesnstsisthhiegahnagnud- res(mag) 0 1.5 res(mas)
( )whenthelensissoclosetoAtheobserver.Thesetwoclassesof -0.1-2 -1 0 1 2 3 -6 -4 -2 0 2 4 6 0
mBicrolensing events with detectable astrometric trajectories have t(tE) t(tE)
differentpropertiesowingtodifferentamountsoftheEinsteinra- Figure3.Twotypicalbinarymicrolensingeventswithrotatingstellar-mass
dius.Intable(1)webrieflyexpressthepossiblerangesofamounts blackholes.Ineachsubfigure,thelightcurves(leftpanels),astrometrictra-
oftheirphysicalparameters. jectories(rightpanels)andthesourcetrajectorieswithrespecttothecaustic
curve(insetsintheleft-handpanels)withoutandwithconsideringtheef-
Theset( )ofmicrolensingeventswhicharecharacterizedin
A fectoflens orbital motionareshownwithgreen dashed andbluedotted
thesecondrowoftable(1)arelong-durationmicrolensingevents
linesrespectively.Thepoint-masslensmodelsareshownbyredsolidlines.
withnolightfromthelens.Theseeventswithdetectableastromet-
Thephotometricandastrometricresidualswithrespecttothestaticbinary
rictrajectoriesarebest candidates toindicate the massof stellar-
modelareplotted withgraydottedlines. Therelevant parameters canbe
massblackholeslocatedintheGalacticdiskthroughthephotomet- foundinTable(2).
ricandastrometricdataaswellastheparallaxeffect(Sahu2011a).
IthasbeenfiveyearssincetheHSTstartedobservingsomeoflong-
durationmicrolensingeventswithnolightfromthelens.Because
noisolatedblackholehasbeendiscoveredtodate,sotheseevents Thesecond set ( ) whicharecharacterized inthethirdrow
B
verylikelyshowbinaritysignatures(Sahu2011b).Thebinarymi- of table (1) contains gravitational microlensing events in which
crolensingeventsduetostellar-massblackholeslocatedinGalac- the lens is so close to the sun, e.g. closer than 100 parsec from
tic disk with the total mass of lenses larger than 5M and semi thesunposition.Thesestarsarebestcandidatesfortheastromet-
⊙
major axis of order of 4Au which have the orbital period larger ricmicrolensing through GAIA mission for accurately indicating
than1300days,arecharacterizedbytheEinsteinradiuslargerthan their mass(Proftetal.2011). Owing to the small distance of the
7.8AuandtheEinsteincrossingtimelargerthan80days.Therefore, lens from the sun, their proper motion is so high and larger than
mostoftheseeventsarecloseandafewofthemareintermediate 0.5arcsecondinyear.Thesemicrolensingeventsarecharacterized
orwidebinarymicrolensingeventsinwhichtheorbitalmotionef- by the Einstein radius smaller than 0.5Au and Einstein crossing
fectoflensesisnotignorableduetotheconsiderableratioofthe timesmallerthan5days.Letusassumethatthelensisabinarysys-
Einsteincrossingtimetotheorbitalperiodandthefinitesizeeffect tem.Inthatcase,mostofthesebinarymicrolensingeventsarewide
istoosmallowingtothelargeEinsteinradius. andafewofthemareintermediateorclosemicrolensingeventsin
c 0000RAS,MNRAS000,000–000
(cid:13)
Orbitalmotioneffects inastrometricmicrolensing 5
Ml(M ) q s(Au) d(RE) tE(day) P(day) β◦ γ◦ α◦ u0 vt(km/s)
⊙
(a) 8.5 0.06 2.5 0.33 48.6 478.9 5.3 85.7 101.7 0.43 271.6
−
(b) 13.7 0.09 3.9 0.28 276.7 734.8 23.1 69.3 71.6 0.45 88.6
−
(c) 15.4 0.12 9.5 0.64 233.1 2572.1 40.3 82.7 233.4 0.77 110.0
−
Table2.TheparametersusedtomakethemicrolensinglightcurvesshowninFigure(3)andFigure(4).
whichtheorbitalmotioneffectoflensesisignorableduetothetoo 1.7 6 1.6
smallratioof theEinsteincrossing timetotheorbital period and (c) 4
the finite-lens effect is probably considerable owing to the small 1.6 R)E 02
Y( -2 0.8
Einsteinradius.Notethat,thefinite-sourceeffectismostlyignor- 1.5 -4
atobltehebHeseconaucuresc,eebositfnatarhrfeyrosmmmiactlrhloealmeonbossuiennrgtveoerfviet.hnee.tsxrew(l=aittihvDerlod/tiaDsttisan)ngc6estoe0lf.l0ath1r-e2m5lea.nsss A 11..34 -6-6-4-2x( R0E) 2 4 6 0 δθ(mas)c,y
blackholeslocatedintheGalacticdiskwhichhavedetectableas- 1.2 -0.8
trometricshiftsinthesourcestartrajectorieshavemostlikelyde-
1.1
ttieoctaobflethoerbEiitnasltmeionticornosssiignngattuirmese(tcoasteheAo)r.bIintatlhepseerieovdenists,htihghe.raIf- 0. 11 -1 δθc ,0x(mas) 1 2 -31.6
tmheertreicissnigoncaatuursetioc-fcororbsistianlgmfeoatitounreoirnetvheenirolifgthhtecsuercvoensd,athryeplehnostois- res(mag) 0 1.5 res(mas)
-0.1 0
too small tobe detected whereas theastrometricsignature owing -2 -1 0 1 2 3 -6 -4 -2 0 2 4 6
t(tE) t(tE)
totheorbitalmotioncanbedetectedintheastrometrictrajectories
which can show the existence of a secondary component. In this Figure4.Asimulatedmicrolensinglightcurveandastrometrictrajectoryof
workwepayattentiontotheseeventsowingtomuchprobabilityof sourcestar.DatapointstakenbysurveytelescopesandtheHSTareshown
withvioletpointsandblackstars.Theparametersusedtomakethisevent
detectingastrometricsignaturesoforbitalmotion.
canbefoundintable(2).
2.3 Astrometricmicrolensingwithrotatingstellar-mass
blackholes
Here,westudytheastrometricmicrolensingduetorotatingstellar- Theorbitalmotionoflensesmakestheastrometrictrajectoryrotate
mass black holes. Asmentioned, lensorbital motion changes the inthesamewayasthesourcetrajectory.
lenses orientation. As a result, the magnification pattern changes Inthesubfigure(b)someoscillatoryfashionscanbeseenin
withtimeowingtothetimevariationoftheprojecteddistancebe- the astrometric trajectory which is owing to lens orbital motion.
tween two lenses as well as the straightforward source trajectory Indeed, whenever the source trajectorydoes not crossthe caustic
rotateswithrespect tothebinary axis.However, sourcestar does curve, the astrometric centroid shift can be estimated as the case
not always receive the gravitational effect of the secondary com- of point-mass lens, i.e. δθc′ = u′u2+′ 2 where u′ = x′s2+ys′2,
ponent, e.g.when theratioof thelensmassesissosmall.Inthat x′s and ys′ arethecomponents of angular source-lenspdistance by
case, the source trajectory does not change in the observer refer- consideringtheeffectoflensorbitalmotion:
enceframe.Consideringthispoint,weletthelensesrotatearound
theircommoncenterofmasswhentheastrometrictrajectorydevi- x′s = 1−ε2sinξ(t)[u0(cosβcosα−sinαsinβsinγ)
atesmorethan 5microarcsecondsfromtheastrometricellipsedue + pp(t)(cosβsinα+sinβsinγcosα)]
toapoint-masslenslocatedatthecenter ofmasspositionwhose
+ cosξ(t)cosγ[p(t)cosα u sinα] p(t)ε
0
massisequal tothetotal massof thelenses. However,whenthis − −
differencebecomessmallerthanthethresholdamount,westopthe ys′ = 1−ε2sinξ(t)[u0(cosβsinα+cosαsinβsinγ)
lensesrotationand letthe sourcetrajectoryslowlyreturn backto + pp(t)( cosβcosα+sinβsinγsinα)]
−
itsstraightforwardtrace. + cosξ(t)cosγ[u cosα+p(t)sinα] u εcosγ, (11)
0 0
In Figure (3) we represent two examples of microlensing −
eventswithrotatingstellar-massblackholes.Ineachsubfigure,the whereαistheanglebetweenthesourcetrajectoryandthebinary
lightcurves(leftpanels)andastrometrictrajectories(rightpanels) axis.Byconsideringtheorbitalmotionsomeoscillatoryfashions,
withoutandwithconsidering theeffectoflensorbitalmotionare termscontainingsinξandcosξ,willappearintheastrometrictra-
shown with green dashed and blue dotted lines respectively. The jectorieswhichdependingontheratiooftheEinsteincrossingtime
point-mass lens models are shown by red solid lines. The photo- totheorbitalperiodtheyorsomepartsofthemcanbedetected.
metric and astrometric residuals with respect to the static binary Whentheanglebetweentheorbitalmotionplaneoflensesand
modelareplottedwithgraydottedlines.Theparametersofthese skyplaneisabout π/2i.e.γ 0andβ π/2,therotationof
∼ ∼ ∼
microlensingeventscanbefoundinTable(2).Weusethegeneral- binaryaxisissosmall.Inthatcase,theorbitalmotionsignaturein
izedversionoftheadaptivecontouringalgorithm(Dominik2007) theastrometrictrajectoriesisignorableespeciallywhenthereisno
forplottingastrometrictrajectoriesinbinarymicrolensingevents. caustic-crossingfeature.
Even though, theorbital motion effect isnot obvious in thelight Depending onthesizeoftheangular Einsteinradiusthisef-
curves, this effect can be detected in the astrometric trajectories. fectcanbeseenintheastrometrictrajectory.Ifthereisnocaustic-
c 0000RAS,MNRAS000,000–000
(cid:13)
6 SedigheSajadian
crossing feature,theprobability of detecting thephotometricsig-
natureoforbitalmotionissosmall.
0.09
0.08
3 MONTECARLOSIMULATION
In this section we perform a Monte Carlo simulation to obtain 0.07
quantitativelydetectabilityoflensorbitalmotioninastrometricand
0.06
photometric microlensing with rotating stellar-mass black holes.
In the first step, we produce an ensemble of binary microlens- )max
d(t0.05
iinngg teovetnhtespwhyitshicsatleldlaisrt-rmibaustsiobnlaocfkphaoralemsetaesrsm. Tichroenle,ncsoesrreascpcoonrdd-- dN/event0.04
inglightcurvesandastrometrictrajectoriesaregeneratedaccord-
ing to the real data points from an ensemble of observatories in 0.03
themicrolensingexperiment.Byconsideringadetectabilitycrite-
0.02
rion, weinvestigatewhether thesignatures of lensorbital motion
canbeseeninobservationsofmicrolensinglightcurvesandastro-
0.01
metrictrajectories.Ourcriterionfordetectabilityoforbitalmotion
is∆χ2 = χ2OM −χ2SB > ∆χ2th where χ2OM and χ2SB are the 0 0 1 2 3 4 5 6 7
χ2softheknownrotatingbinarymicrolensingmodelandthestatic tmax(tE)
model with the same binary parameters respectively. We assume
thatfromfittingprocessandsearchingallparameterspacethebest- Figure 5. Distribution functions of the time, with respect to the time of
fittedsolutionistheknownbinarysolution.Ouraimistocompare theclosestapproach andintheunitoftheEinstein crossingtime,ofthe
photometricefficiencyofdetectinglensorbitalmotionwiththeas- maximumphotometric(greenhistogram)andastrometric(redhistogram)
trometricone.Hence,wecalculate∆χ2forsimulatedlightcurves deviations duetolensorbitalmotionforbinarymicrolensing eventswith
andastrometrictrajectoriesindependently. detectableeffectsoflensorbitalmotionintheirlightcurvesandastrometric
Here,weexplaindistributionfunctionsofthelensandsource trajectoriesrespectively.
parameters. For lens parameters we take the mass of the stellar-
mass black hole as the primary from the following distribution
function(Farretal.2011): m⋆ isthemassofthesourcestarintheunitofthesolarmassand
M α = 0.3for0.01 6 m⋆ 6 0.08,α = 1.3for0.08 6 m⋆ 6 0.5
P(Ml)∝exp(−Mol), (12) Aanndotαhe=rpa2r.a3m5eftoerrwme⋆ne>ed0i.n5o(uKrrsoimupualaettioanl.i1s9t9h3e;raKdriouuspoafs2o0u0r1c)e.
whereP(Ml)=dN/dMlandMo =4.7M intherangeofMl star. This parameter will be used in generating the light curve
⊙ ∈
[4.5,25]M . The mass ratio of the secondary to the primary is of microlensing events with the finite size effect. For the main-
⊙
drawnfromthedistributionfunction(Duquennoy&Mayor1991): sequence stars,the relationbetween themass and radiusisgiven
(q q )2 byR⋆ =m0⋆.8whereallparametersnormalizedtothesun’svalue.
ρ(q) exp( − 0 ) (13) Wedidnotconsidergiantsourcestars.Forindicatingthetrajectory
∝ − 2σ2
q ofthesourcewithrespecttothebinarylens,weidentifythispath
whereρ(q)=dN/dqintherangeofq [0.01,1],q =0.23and withanimpactparameterandorientationdefinedbyananglebe-
0
∈
σq = 0.42. Wechoose the semi-major axis s of thebinary orbit tweenthetrajectoryofsourcestarandthebinaryaxis.Weletthis
fromtheO¨piklawwherethedistributionfunctionfortheprimary- anglechangeintherangeof[0,2π].Theimpactparameteristaken
secondary distance is proportional to ρ(s) = dN/ds s−1 uniformlyintherangeofu0 [0,1]andthecorresponding time
(O¨pik1924) in the range of s [0.6,30]Au. The loca∝tion of foru issetzero. ∈
0
∈
lenses from the observer is calculated from the probability func- Thevelocitiesofthelensandthesourcestararetakenfromthe
tionofmicrolensingdetectiondΓ/dx ρ(x) x(1 x),where combinationoftheglobalanddispersionvelocitiesoftheGalactic
∝ −
x = Dl/Ds changes in the range of x ∈ [0,p1] and ρ(x) is the diskandbulge(Rahaletal.2009;Binney&Tremaine1987).The
stellar density of thin disk, chosen from Rahal et al. (2009). The relative velocities of the source-lens is determined by projecting
time of arriving at the perihelion point of orbit tp is chosen uni- thevelocityofthesourcestarintothelensplane.
formlyintherangeoftp [t0 P,t0+P].Theprojectionangles Weignorethemicrolensingeventsinwhichtheminimumdis-
∈ −
toindicatetheorientationoforbitoflenseswithrespecttothesky tancebetweentwolensesandalsominimumimpactparameterare
plane, i.e.β and γ, aretaken uniformly inthe range of [ π,π]. notoneorderofmagnitudelargerthantheSchwarzschildradiusof
−2 2
Wetaketheeccentricityofthelenses’orbituniformlyintherange theprimarylens.BecausetheydonotobeyKeplerianlaws.
ofε [0,0.15]. Now, we generate data points over the light curves and as-
∈
For the source, we take the coordinate toward the Galactic trometrictrajectories.Weassumethatmicrolensingeventsareob-
bulge(l,b), distributionof thematter instandard Galacticmodel servedwiththeHST,OGLEandMOAsurveys.Thetimeinterval
and generate the distribution of the source stars according to the betweendatapointsandthephotometricuncertaintiesforeachdata
Besanconmodel(Dweketal.1995;Robinetal.2003).Weassign pointtakenbysurveytelescopesaredrawnfromthearchiveofthe
thethe absolute, apparent color and magnitude of thesource star microlensinglightcurves.TheHSTstartobservingtheseeventsaf-
inthesameapproachwhichisexplainedin(Sajadianetal.2013). terthatthelightcurvereachestoagiventhresholdofmagnification
Themassof source star istaken fromthe Kroupamass function, (i.e. 3 ).WesupposethattheHSTismonitoringthesemicrolens-
ξ = dN/dm ∝ m−⋆α,intherangeofm⋆ ∈ [0.3,3]M⊙,where inge√ve5ntswithcadenceoneday.Theotherfactorinsimulationof
c 0000RAS,MNRAS000,000–000
(cid:13)
Orbitalmotioneffects inastrometricmicrolensing 7
1 1 0.9 1
0.8
0.8 0.8 0.7 0.8
0.6 0.6 00..56 0.6
0.4 0.4 0.4 0.4
0.3
0.2 0.2 0.2 0.2
0.1
0 0 0 0
5 10 15 20 25 -2 -1.5 -1 -0.5 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ml(Mo) log10(q) u0 x
1 1
1 0.9
0.8 0.8 0.8
0.75 0.7
0.6 0.6 0.6
0.5
0.5
0.4 0.4 0.4
0.3
0.2 0.25 0.2 0.2
0.1
0 0 0 0
-1 log10(d(RE))0 close resonance wide -90 -60 -30 β0 30 60 90 -90 -60 -30 0γ 30 60 90
Figure6.Theastrometric(solidline)andphotometric (dashedline)effi- Figure7.Theastrometric(solidline)andphotometric (dashedline)effi-
cienciesofdetectinglensorbitalmotionfordifferentparametersoflenses cienciesofdetectinglensorbitalmotionfordifferentparametersofsource
anddifferenttopologyofcaustics(thelastpanel). starandlensestrajectories.
torieshelpstodiscover further secondary components around the
thelightcurveistheexposuretimeforeachdatapointtakenbythe primarylensaswellasresolveclose/widedegeneracy.
HST.Fromtheexposuretimewecancalculatethetheerrorbarfor InFigure(5)weplotthedistributionfunctionsofthetimeof
eachdatapoint.Wecantunetheexposuretimeinsuchawaythat themaximumphotometric(greenhistogram)andastrometric(red
wehaveuniformerror barsthought out thelight curve. Here,we histogram)deviationsduetolensorbitalmotion,measuredwithre-
demandthephotometricaccuracyforeachdatapointtakenbythe specttothetimeoftheclosestapproachandintheunitoftheEin-
HSTabout0.4percentwhichischosenaccordingtotherecorded steincrossingtimei.e.tmax,forbinarymicrolensingeventswith
exposuretimesforthoselong-duration microlensingeventsbeing detectable effects of lens orbital motion in their light curves and
observedbytheHST. astrometric trajectories respectively. According to this figure, the
Astrometricobservations of sourcestarpositionaredone by orbital motion signature in the astrometric trajectory can be seen
the HST with the astrometric accuracy of σas = 200 micro- atsometimeoftenafteroneEinsteincrossingtimewithrespectto
arcsecond. The simulated astrometric data points are shifted ac- the time of the closest approach whereas this signal in the light
cording to the astrometric accuracy of the HST by a Gaussian curvecanbedetectedatsometimealmostbeforeit.Inwideorin-
function. A sample of the simulated light curve and astrometric termediatebinaryeventswithdetectableastrometricsignaturesof
trajectory of source star is shown inFigure (4). Inthis figurethe orbitalmotion,themaximumdeviationhappensatverylatetime,
datatakenbysurveytelescopesandtheHSTareshownwithvio- e.g.t t0 >6tE,whichhavemadesomesosmallpeaksinFigure
−
letpointsandblackstars.Theparametersusedtomakethisevent (5).
arebrought inthefourthrowoftable(2).Thetime-variationrate Inordertostudythesensitivityofdetectingorbitalmotionin
of the astrometric centroid shift vectors is not constant. As a re- astrometrictrajectoriesofsourcestarandphotometriclightcurves
sult, in some places over the astrometric trajectory the HST data ontheparametersofthemodel,weplottheastrometric(solidline)
pointsdonotuniformlysetdespitesimilarcadences(seeFigure4). andphotometric(dashedline)detectionefficienciesintermsofthe
Inthisevent,theorbitalmotioneffectisobviousintheastrometric relevant parameters of the binary lens and source in Figures (6)
trajectoryaswellasthephotometriclightcurve. and(7).Weignoretheirrelevantparametersthatdonotchangethe
Our criterion for detectability of orbital motion is ∆χ2 = efficiency function. The detection efficiency function in terms of
χ2 χ2 > ∆χ2 .Weconsider∆χ2 =250forbothphoto- thebinarylensandsourceparametersisgivenasfollows.
OM − SB th th
metricandastrometricobservations.FromtheMonteCarlosimu- (i)Thefirstparameteristheprimarylensmass,M.Here,the
l
lationweobtainthat80.2,16.9and2.9percentoftotalsimulated astrometricefficiencyofdetectinglensorbitalmotionriseswithin-
events are close, intermediate and wide binaries and the average creasingtheprimarylensmass.Thephysicalinterpretationofthis
detectionefficiency< ǫOM >fortheorbitalmotiondetectionin featureisthattheangular Einsteinradiusandasaresult ofitthe
theastrometrictrajectoriesandtheamplificationlightcurvesofbi- astrometricsignal risewithincreasing theprimarylensmass. On
narymicrolensingeventswithstellar-massblackholesare87.3and thephotometric detection efficiency therearetwo factorsthat ef-
48.2percent,respectively.Asaresult,intheseeventslensorbital fectinversely.WithincreasingtheprimarylensmasstheEinstein
motioncanbeseenmostlikelyinastrometrictrajectorieswhereas radiusriseswhichdecreasesthenormalizeddistancebetweentwo
photometricsignaturesoforbitalmotionareoftentoosmalltobe lenscomponentsandthecausticsize.Ontheotherhand,theratio
detected.Detectinglensorbitalmotionintheirastrometrictrajec- oftheEinsteincrossingtimetotheorbitalperiodoflensesmotion
c 0000RAS,MNRAS000,000–000
(cid:13)
8 SedigheSajadian
increases.However,thefirsteffectisdominant.Because,thepho- this signal in the light curve can be detected at sometime almost
tometric signature of orbital motion can more likely be detected beforeit.
whilethesourceiscrossingthecaustic. Binary microlensing events with rotating stellar-mass black
(ii) The second parameter is the ratio of the lens masses, q. holes which have detectable astrometric shifts in the source star
Theratioofthelensmasseshasageometricaleffectontheshapeof trajectorieshavemost likelydetectable orbital motion signatures.
thecausticlines,whereincreasingittowardsthesymmetricshape Intheseevents,theratiooftheEinsteincrossingtimetotheorbital
maximizesthedetectionefficiency.Indeed, thesizeofthecentral periodishigh.Although,orbitalmotioneffectintheirlightcurves
causticscalesasq(Gaudi2012). is too small especially when there is no caustic-crossing feature,
(iii)Thethirdparameter istheprojectedSemi-majoraxisof but the astrometric signature owing to the orbital motion can be
lensorbit normalized tothe Einsteinradius. The photometric de- detectedattheendpartsofastrometrictrajectories.Inaddition,the
tectionefficiencyhasthemaximumamountaround1RE inwhich Hubble Space Telescope (HST) is observing some long-duration
thecausticcurveshavethemaximumsize.However,theastromet- microlensing events which are likely to be owing to stellar-mass
ric detection efficiency has the highest amount for close binaries blackholes,followingtheapprovedproposalbySahuetal.(2010).
withthelargeratiooftheEinsteincrossingtimetotheorbitalpe- Hence,weexpectthatifthereisasecondarylens,theastrometric
riod. signatureofitsorbitalmotioncanmuchprobablybeobserved.
(iv)Differenttopologiesofcausticcurve:Theastrometricand ByperformingMonteCarlosimulationweevaluatedtheeffi-
photometricdetectionefficienciesaremaximizedincloseandinter- ciencyofdetectingorbitalmotioninastrometricandphotometric
mediatebinariesrespectively.Indeed,thesizeofthecentralcaus- microlensingwithbinarystellar-massblackholes.Weconsidered
ticwhichindicatestheprobabilityofcausticcrossingandthede- ∆χ2 > 250 as the detectability criterion and concluded that as-
tectability of lens orbital motion photometrically has the highest trometric efficiency is 87.3 per cent whereas the photometric ef-
amount for the intermediate microlensing events (see Figure 1). ficiency is 48.2 per cent. From total simulated events 80.2, 16.9
Notethatmostsimulatedeventsareclosebinaries. and 2.9 per cent were close, intermediate and wide binaries re-
(v)Impactparameter,u ,showninFigure(7):Decreasingthe spectively.Also,themoremassivelenses,themoredetectablesig-
0
impactparameterincreasesthephotometricdetectionefficiency.A natures of orbital motion. Detecting orbital motion in binary mi-
smallerimpactparameterfromthecenterofthelensconfiguration crolensingeventshelpsnotonlytodetectfurthersecondary com-
risestheprobabilityofthecausticcrossing.However,theastromet- ponentsevenwithoutanyphotometricbinaritysignaturebutalsoto
ricdetectionefficiencydoesnotdependontheimpactparameter. resolveclose-widedegeneracyaswell.
(vi) Thenext parameter isx = Dl/Ds therelativedistance Acknowledgment I am especially thankful to Sohrab Rah-
ofthelenstothesourcestarfromtheobserver. Theangular Ein- var for his encouragement, useful discussions and reading of the
steinradius andastrometric signalsincrease by decreasing x.On manuscript.IwouldliketothankMartinDominikforgeneralizing
theotherhand,theEinsteinradiushasthehighestamountaround hisadaptivecontouringalgorithmtocalculatetheastrometricshifts
x=0.5.Whenxtendsto0.5,byincreasingtheEinsteinradiusthe inbinarymicrolensingevents,MatthewPennyforpointingoutan
normalizeddistancebetweentwolenscomponentsandtheproba- importanterror,AndyGouldandCheonghoHanforcarefulread-
bilityofthecausticcrossingdecrease. ingandcommentingonthemanuscript.Finally,Ithankthereferee
(vii)and(viii)Inclinationanglesoforbitalplaneoflenseswith forusefulcommentsandsuggestionswhichcertainlyimprovedthe
respect to the sky plane, β and γ. The astrometricdetection effi- manuscript.
ciencyismaximizedinβ 0andγ π/2inwhichtherotation
∼ ∼±
angleofthesourcetrajectorywithrespecttothebinaryaxisowing
tolensorbitalmotion,θ,hasthemaximumamount.
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