Resolving Microlensing Events with Triggered VLBI

1.2 MB·

by  M. Karami

DRAFTVERSIONJANUARY25,2016
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
RESOLVINGMICROLENSINGEVENTSWITHTRIGGEREDVLBI
MANSOURKARAMI1,2,AVERYE.BRODERICK1,2,SOHRABRAHVAR4,1,MARKREID3,
1PerimeterInstituteforTheoreticalPhysics,31CarolineStreetNorth,Waterloo,ONN2L2Y5,Canada
2DepartmentofPhysicsandAstronomy,UniversityofWaterloo,200UniversityAvenueWest,Waterloo,ONN2L3G1,Canada
3Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
4SharifUniversityofTechnology,AzadiAve.Tehran,IranPOBox:11365-11155
DraftversionJanuary25,2016
6 ABSTRACT
1 Microlensing events provide a unique capacity to study the stellar remnant population of the Galaxy. Optical
0
microlensingsuffersfromanearcompletedegeneracybetweenthemass,thevelocityandthedistance. However,
2
a subpopulationof lensed stars, Mira variable stars, are also radio bright, exhibitingstrong SiO masers. These

n are sufficiently bright and compact to permit direct imaging using existing very long baseline interferometers
a such as the Very Long Baseline Array (VLBA). We show that these events are relatively common, occurring
J at a rate of 40 yr- 1 of which 1.3 yr- 1 are associated with Galactic black holes. Features in the associated
≈
1 images,e.g.,theEinsteinring,aresufficientlywellresolvedtofullyreconstructthelensproperties,enablingthe
2 measurementofmass,distance,andtangentialvelocityofthelensingobjecttoaprecisionbetterthan15%.Future
radio microlensing surveysconductedwith upcomingradio telescopes combinedwith modest improvementsin
] theVLBAcouldincreasetherateofGalacticblackholeeventstoroughly10yr- 1,sufficienttodoublethenumber
E
ofknownstellar massblackholesin acoupleyears, andpermittingthe constructionofdistributionfunctionsof
H stellarmassblackholeproperties.
. Keywords:gravitationallensing:micro—masers—radiolines:stars—stars: blackholes
h
p
-
o 1. INTRODUCTION magnifiedimage,typicallyincreasingbymorethananorderof
r magnitude. Secondly, the peculiar motions of objects within
st Gravitationallensingpresentsoneofahandfulofcurrentob- the Galaxy cause the source-lens system to evolve on time-
servational windows on the dark universe. The distortion of
a scalescomparableto
[ backgroundgalaxiesbygalaxyclustershasprovidedastriking θ
1 dimempoorntastnrtaltyi,ointhoafsgeennaebrlaeldrtehleatuivniitqyue(Breacrtoenlmstrauncnti2o0n1o0f)t.heMporore- tE ≡ |µS-EµL|, (2)
v jectedmassdensityoftheclusterdarkmatterhaloes(Refregier whereµS andµL arethesourceandlensapparentangularve-
1 2003; Paczynski1986). While the dynamicsof stars and gas locities,respectively.FortypicalvaluesintheGalaxytE ranges
0 probethedarkmatterhalowithin102kpcofthecentralgalaxy from days to months, dependingon the source mass and dis-
8 (Persicetal.1996),gravitationallensingcurrentlyprovidesthe tance of lens and source from the observer. The relationship
5 only means to detect and study the halo on larger scales. By betweentE andthelensstructureimpliesacharacteristiclight
0 combiningsuchmeasurements,ithasbecomepossibletomap curve,referredtoasamicrolensingevent.
.
1 the dark matter distribution in the local universe (Shan et al. The rarity of serendipitous lens-source alignments require
0 2014). the monitoringof large numbersof potential sources to iden-
6 Thedetailedstructureofagravitationallylensedimagemay tify candidate microlensing events. For this purpose optical
1 beseparatedintotheintrinsicstructureofthelensedobjectand surveysofdensestarfieldshavebeenundertakenbyanumber
v: a distortion imposed by the lens that depends upon the lens ofgroups,bothtoconstrainthecontributionofstellarremnants
i location and mass distribution. A typical scale is set by the totheGalacticdarkmatterbudget(Moniez2010)andmorere-
X Einsteinangle,thisisexactlytheangularscaleoftheEinstein cently to find extrasolar planets which results in short time-
r ringcastbyapointlens, scale features in the microlensing light curve (Gaudi 2012).
a These have successfully excluded remnants with masses be-
4GM(D - D ) tween 10- 7M –10M as a candidate for dark matter within
θ = S L , (1) ⊙ ⊙
E c2D D theGalactichalo(Lasserreetal.2000;Alcocketal.2001).
whereMisthetotalsourrcemassanSdLD andD arethesource Motivated by the potential to detect extra-solar planets, re-
S L
and lens distance, respectively. For galaxy clusters this can centyearshaveseentheinitiationofhigh-cadence,large-area
spanmanyarcminutes. surveys.Thesehavebeenenabledbythedevelopmentoflarge
CCDarraysfortelescopeswithwidefieldsofview.Asaresult,
GravitationallensingwithintheGalaxyalsooccurswhenthe
angularseparationof a backgroundstar and a foregroundob- the OGLE1 (Optical Gravitational Lensing Experiment) col-
ject becomes comparable to θ . This differs from the gravi- laborationrecorded 2 103 eventsin 2013, using the early
E ≈ ×
warningsystem(Udalskietal.1994)toalertthefollow-uptele-
tational lensing of clusters and galaxies in two important re-
spects. Firstly,theangularsizeofthelensedimagehasatyp- scopes for monitoring microlensing events with higher preci-
ical scale of θ 1(M/M )1/2 mas, well below the resolu- sion.TheKoreanMicrolensingTelescopeNetwork(KMTNet)
E ⊙
≈ willincreasingthenumberofeventsto6000peryearwitha10
tion of existing optical telescopes. As a result, direct optical
imaginghasnotbeenfeasible. Nevertheless,theconservation minutecadenceoperatingcontinuously(Poteetetal.2012).
of brightness, a consequence of Liouville’s theorem, implies
substantial amplificationsof the total flux associated with the 1http://ogle.astrouw.edu.pl/ogle4/ews/ews.html
2 KARAMI ET AL.
Reconstruction of microlensing events usually only deter- Because of the nonthermal nature of the maser process, the
minesthet .Asadirectconsequence,themassofthelens,and spotscanhavebrightnesstemperaturesashighas5 1010 K,
E
×
thedistancesandvelocitiesofthelensandsourcesufferfrom well above the thresholds for imaging with the VLBA (Reid
a fundamental degeneracy. Methods for breaking the mass- etal.2007). WithintheGalacticcenter,theinnerparsecofthe
distancedegeneracytypicallyrequireshigher-ordereffects. If Galactic bulge, the SiO masers associated with evolved stars
themicrolensingparallaxcanbemeasured–anasymmetryin have been imaged with the VLBA at angular resolutions of
thelightcurveinducedbythenon-uniformmotionoftheEarth 0.7 mas, limited by the interstellar electron scattering in that
–thelensdistanceandthemassofthelenscanbeconstrained region(Reidetal.2003,2007).OutsideoftheGalacticcenter,
(Gould 1992;Rahvaret al. 2003). Parallax effectcan also be itshouldbepossibletoimagethesewithresolutionsapproach-
employed by monitoring microlensing events from earth and ing0.3mas.
spacesimultaneously. Spitzerhasbeenusedtothisend,mea- Hereweexploretheratesofsuitablemicrolensingeventsan-
suringmicrolensingparallaxandthusprovidingan additional ticipatedby currentand futuresurveysandthe precisionwith
constrain (Street et al. 2015). Alternatively the source finite- whichtheassociatedlensparameterscanbereconstructed.We
sizeeffectcouldbeusedtobreakthedegeneracy. Iftheangu- propose using optical microlensing surveys to trigger follow-
larimpactparameterissmallerthanthesourceangularsizethe up VLBA observationsbased on source locationin the color-
point source approximation is no longer valid and the finite- magnitude diagram, anticipated event duration, and the lack
size effect becomes important. By analyzing the light curve of a bright lens counterpart. The color-magnitude diagram
anadditionalrelationbetweenangularsourcesizeandangular (CMD) providesa proxyfor identifyingMiras, if notalready
Einsteinradiuscouldbefound. Inbothcases,thelensandthe identifiedduetointrinsicvariability. Massivelensesaremore
sourceapparentmotionsmustbesubsequentlymeasuredafter heavilyrepresentedinlong-durationevents,andthosewithout
theeventtoremovethesefromconsideration(see,e.g.,Gould obvious stellar counterparts are more likely to be associated
2014), which is only possible for stellar lenses and explicitly withmassivecompactobjects.
excludesthoseassociatedwithstellarremnants. In section 2 we describe a Monte Carlo simulation of mi-
Directly resolving the image would greatlysimplify the re- crolensingevents,discusstherelativeratesatwhichMirasare
construction of the lens parameters. The typical scales of expectedtobelensedbyvariousobjects,includingstellarrem-
microlensing events are well matched to those achievable by nants, and how event selection can be optimized for imaging
verylongbaselineinterferometry(VLBI)atcentimetrewave- based on optical properties. Section 3 discusses the absolute
lengths,e.g.,withtheVeryLongBaselineArray(VLBA).Un- ratesofradio-brightblackhole microlensingeventsof a vari-
fortunately,imagingmicrolenseswiththeVLBArequirescom- etyofpotentialsurveys. Section4describestheresultingim-
pactradiobrightsourceswith typicalbrightnesstemperatures agesforasimplemasermodelandestimatestheaccuracywith
in excess of 1010 K, well above the typical stellar tempera- which the lens properties can be reconstructed. The implica-
tures. However,theunprecedentedrateatwhichmicrolensing tions for constructing mass and distribution functions for the
eventsarebeingidentifiedbyopticalsurveysmakesitpossible Galacticremnantpopulationisdiscussedin5. Finally,conclu-
toleverageraresourcepropertiesforthispurpose. sionsarecollectedin6.
Mira variables provide a natural target for optically trig-
gered radio imaging microlensing experiments.2 Miras are
asymptotic giant branch (AGB) stars with month time-scale 2. STELLARREMNANTSINSIMULATED
pulsationslargelydeterminedbyconvectionintheirenvelopes MICROLENSINGSURVEYS
(Wood2006).WhiletheopticalluminositiesofMirascanvary
The rates of microlensing events associated with compact
over three orders of magnitude (Reid & Goldston 2002), the
remnants depends on the distribution of sources and lenses
typical peak luminosities are on the order of 103L , ensur-
⊙ within the Galaxy and the detection efficiency of current and
ing that they are among the brightest objects in microlensing
upcomingmicrolensingsurveys.Herewedescribesimulations
survey samples (Soszyn´ski et al. 2013). Assuming there are
ofanOGLE-likemicrolensingsurveyandestimatethenumber
6 105Mirastarsinthebulge(Matsunagaetal.2005),theop-
tic×aldepthofmicrolensingeventsshouldbeτ 2.35 10- 6in ofmicrolensingeventsweanticipatetobeassociatedwiththe
≃ × variousGalacticremnantpopulations. Todothis, weperform
directionofGalacticbulge(Sumietal.2003)andatypicaltime
mocksurveysassumingthemonitoredstellarfieldislocatedin
scaleofonemonthforthemicrolensingevents(i.e.<t > 1
E ≃ theGalacticbulge.
month),therateofeventswithhundredpercentdetectioneffi-
Necessary inputs are the mass, velocity, color, and mag-
ciencycanbeobtainedby
nitude distributions of potential sources and the mass and
N 2 τ
events = . (3) velocity distributions of potential lenses. For simulation of
NstarsTobs π<tE > microlensing events, we adapt the detection efficiency in
We expect to observe roughly 11 Mira-source microlensing terms of the Einstein crossing time of OGLE survey sources
eventsperyear. Withtypicalmassessimilarto1M⊙ Mirasare (Wyrzykowski et al. 2015). We describe the details of the
relativelynumerousdespitehavingAGBlifetimesofroughlya modelhere.
millionyears.
OfparticularimportancehereisthepresenceofSiOmasers
intheextendedatmospheresofMiras.Typicalsizesofthemas- 2.1. GalacticDistributionModelandStellarLens/Source
ingregionsare1AU,correspondingtoangularsizesof0.1mas PopulationModel
for distances characteristic of bulge stars (Reid et al. 2007).
Forgeneratingmicrolensingeventswe needthespatialdis-
tributionofstarsaswellastheirvelocities.
2Heyl(2011)exploreanumberofotherpotentialradio-brightmicrolensing
ForthestructureoftheGalaxy,weusethindiskmodeland
targets,includingthecontinuumemissionfromawideclassofgiants.Ofthese,
only theSiOmasers fromMiras andactive galactic nuclei havesufficiently standard bulge model from Binney & Tremaine (1987). The
highbrightnesstemperaturestobeamenabletoVLBIimaging. density distribution in disk is modeled in cylindrical coordi-
RESOLVING MICROLENSING EVENTSWITH TRIGGERED VLBI 3
11
Table1 Stellar
Assumedparametersofthegalacticmodel,The 10 White dwarf
modelispartlyadoptedfrom(Rahvaretal.2003). Black hole
9
Structure Parameter Value 8
Σ(M⊙pc- 2) 50 dM) 7
H(kpc) 0.325 N/
Disk Rd(kpc) 3.5 M d 6
velocity σrkms- 1 34. (0 5
sdiiosnpser- σσzθkkmmss--11 2280.. log1 4
MB(M⊙) 1.7×1010 3
a(kpc) 1.49
Bulge b(kpc) 0.58 2
c(kpc) 0.40
1
velocity σkms- 1 110 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Spheroid aρc0((kMp⊙c)pc- 3) 0.9320×.510- 5 Log10(M/ MO•)
velocity σkms- 1 120
RρHc⊙(kp(Mc)⊙pc- 3) 0.50.008 Fanigdubrleac1k.hoAlsessu(mdoetdtemd)aisnsthfuendcitsioknasndofbustlagres.T(shoelisdp),ikwehfeitaetudrewsaarftslo(wda(swhehdit)e,
Halo vMelionci6t0ykpcσ(k1m01s0-M1⊙) 25010 dnwanatrfm)aasnsdishingehar(lbylaicnkdehpoelne)demnatsosfesthcaotrroefspthoendprtoogreengitioorn.sTinhewsheicdhotnhoetrseumb--
stantiallyimpacttheresultshere.
11
natesbyadoubleexponentialfunctions, Stellar
10 White dwarf
ρ (R,z)= Σ exp - (R- R⊙) exp - |z| , (4) 9 Black hole
D
2H (cid:18) Rd (cid:19) (cid:18) H (cid:19) M) 8
whereΣisthecolumndensityofthediskattheSunposition, N/d 7
H the height scale and Rd the length scale of the disk. The M d 6
distribution of the lens transverse velocity with respect to the (0 5
1
lineofsightisestablishedfromthesun’spropermotionwithin g
thethindiskandfromthelocalvelocitydistributions(seeTable lo 4
1). 3
The bar is described in a Cartesian frame positioned at the 2
galactic center with the major axis x tilted by φ= 45◦ with
1
respecttotheGalacticcenter-Sunline.Thebardensityisgiven -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
by Log (M/ M)
10 O•
M
ρB= B e- rs2/2
6.57πabc Figure2. Assumedmassfunctionofstars(solid),whitedwarfs(dashed),and
r4= x 2+ y 2 2+z4 , (5) banladckhighhol(ebsla(dckothteodl)e)inmtahseseSspchoerrroeisdp.onTdhetosrpeigkieonfesaitnurweshiacthlothwer(ewmhnitaendtwmaarsfs)
s a b c4 is nearly independent of that of the progenitor. These do not substantially
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21) impacttheresultshere.
whereM isthebulgemass,anda,bandcarethescalelength
B
factors. The dispersion velocity of bulge stars is 110 kms- 1.
2.2. RemnantPopulationModel
ForthedensityoftheSpheroidstructurewetakethefollowing
function(Robinetal.2003), In addition to the known stellar populations, gravitational
lenscandidatesincludetheGalacticpopulationofcompactob-
ρ (a /8.5kpc)- 2.44 a a jects. Inprinciple,thesemaybecomprisedoftheremnantsof
ρspher= ρ0(ac/8.5kpc)- 2.44 a≤ac (6) stellarevolutionandprimordialobjects. Wemaketheconser-
0 c
(cid:26) ≥ vativeassumptionthatthelatterareabsent,andconsideronly
a2=R2+z2/(0.76)2,a ,ρ andspheroidvelocitydispersionare theremnantsofmassivestars.
c c
givenintable(1). The end pointof stellar evolutiondependsprimarilyon the
Combining the density distribution of Galaxy, kinematics, massoftheprogenitor. We considertwo classesof remnants,
mass function and stellar population of stars, we can gener- whitedwarfsandblackholes. Weignoretheintermediateneu-
ate microlensingevents in terms of t , M, D, D , θ and π, tron stars, which we expect to contribute marginally to the
E l l s E
whereπ is theparallaxparameterthatis describedinthe Ap- high-masstailofthewhitedwarfpopulationandotherwisebe
pendix. We use the detection efficiency of OGLE survey in indistinguishable from them using microlensing observations
terms of Einstein crossing time (i.e. ǫ(t )) to select the ob- alone. Hence in practice the remnants are effectively white
E
served microlensing events in our simulation (Wyrzykowski dwarfs/neutronstarsandblackholes.
et al. 2015). A general overviewabout the simulation of mi- EncodedintheremnantpopulationistheGalactichigh-mass
crolensingeventscanbefoundinRahvar(2015). starformationhistory.Oncetherelationshipbetweenthezero-
age main sequence progenitormass (M ) and the final rem-
MS
nantmass(M ,whereRmaybeWDorBH)isspecificed,the
R
4 KARAMI ET AL.
massfunction3(φ(M ))isgivensolelyintermsofthestarfor- surveys.Theobservationof pulsarswith agesless than 3 Gyr,
R
mation rate (Λ(t))and initialmass functionof mainsequence indicatesthepulsarbirthvelocityofabout400kms- 1 (Hobbs
stars(Φ(M )): etal.2005).
MS
Λ(t- T )Φ(M )
φ(MR)= dt MS MS , (7) 2.3. MockObservationStatistics
dlog M /dlog M
Z 10 R 10 MS Inordertosimulatethemicrolensingevents,wechoosethe
whereTMS 10(MMS/M⊙)- 1.5Gyristhelifetimeofamainse- source stars from a Hipparcos like color-magnitude distribu-
≈
quencestar withmassMMS associatedwith theremnantmass tion, distributed proportional to the density of matter. After
MR,andtheintegrationisovertheentireGalactichistory. Ef- selectingthesourcestarwithagivenabsolutecolorandmag-
fectively, this is simply the number of remnants formed over nitude, the position of star in the color–magnitude diagram
thehistoryofGalacticstarformation,excludingthosestarsthat changes due to the distance modulus and interstellar extinc-
remainonthemainsequence.Forthisweadoptthestarforma- tion. Those stars brighter than the limiting magnitude of the
tion rate from Hartwick (2004), which peaks roughly10 Gyr microlensing survey (which is taken about 19 in this simula-
ago, and the “Disk and Young Clusters” stellar initial mass tion) are selected for the observation. It should be noted that
function (IMF(M)) from Chabrier (2003), though employing theresultsarenotsensitivetothelimitingmagnitudeandcould
the“Universal”IMFfromKroupa(2001)makesnodiscernible beappliedtomicrolensingsurveyswithdifferentlimitingmag-
difference. nitudes.
We relate MWD and MMS via the piecewise continuous ex- We note that all the microlensing events generated in this
pression presentedin (Salaris et al. 2009), obtained for white simulationarenotobservableinreality.Dependingonduration
dwarfsinopenclustersbycomparingthewhitedwarfagesin- ofobservation,cadenceofphotometricdataandnon-observing
ferredbycoolingmodelsandclusteragefromisochronefitting: nights due to bad weather and technical failures, only a frac-
tion of eventsas a function of Einstein crossing time, t , can
E
0 mMS 0.5588 be observed. We adapt the detection efficiency function (i.e.
≤
0.01m +0.5418 0.5588<m 1.7 ǫ(t ))forselectingtheobservedeventsfromOGLEpublished
MS MS E
M /M =0.134m +0.331 1.7<m 4≤ (8) function(Wyrzykowskietal.2015).Detaileddescriptionabout
WD ⊙ 0.047mMMSS+0.679 4≤mMSM<S≤8 g(2r0av1i5ta).tionalmicrolensing formalism can be found in Rahvar
0 8 m ,
wheremMS MMS/M⊙. ≤ MS wiEthacthheevaepnptriospcrioantestrpurcotpeedrbtiyesfi,rssutbcsheoqouseinngtlyaraanradnodmomsoulernces
≡ withtheappropriateproperties,andfinallyapplyingtheOGLE
TherelationshipbetweenM andM ishighlydependent
BH MS
detectionefficiency.Thatis,
upontheuncertainevolutionofmassivestarsandthedetailsof
corecollapsesupernovae.Asaresult,above30M itisunclear
⊙ 1. Sourcestarselection:
whattheslopeoftheM -M relationshipis(seeFigure9of
BH MS
Fryer et al. 2012). Here we make a moderately conservative (a) Asourcestarpositionisselectedwithaprobability
assumption that the resulting black hole mass is independent distribution given by the stellar density and geo-
ofM ,setting metricfactors,i.e.,
MS
dP
0 mMS≤8 dDS ∝ρ(DS)D2S, (10)
MBH/M⊙=0.3mMS 8<mMS≤33 (9) (b) The source star stellar type is randomly chosen
10 33<mMS, accordingtoHipparcosColor-Magnitudedistribu-
tion.
which lies in the middle of the permitted range presented in

Fryeretal.(2012)andisroughlyconsistentwithnarrowblack (c) Thestellarcomponent(bulge,disk,spheroid)isse-
hole mass range inferred from X-ray binaries by (Özel et al. lectedwithweightsequaltotheirdensityatDS.
2010). (d) The source star velocity is the sum of the bulk
Theresultingstellarandremnantmassfunctionsforthedisk, Galactic motion for the chosen stellar component
bulge and spheroid are shown in Figures 1, 2. The spikes in atthechosensourcelocationandarandomcontri-
thelowandhighmassendsofthewhitedwarfandblackhole butionpulledfromaGaussiandistributionwiththe
mass functionsare associated with regionsin which the rem- appropriatedispersion(listedinTable1).
nantmassare nearlyindependentofthe progenitormass, and
2. Lensselection:
contain a finite number of objects. For both the disk and the
bulge there is a breakin the stellar mass functionwherestars
(a) Alenspositionisselectedfromthestellardensity,
producedduringthe star formationpeakare leavingthe main
modifiedbythelensingcrosssection,i.e.,
sequence,roughly1M .
⊙
Because neither white dwarfs nor black holes are expected dP D (D - D )
L S L
to experience strong kicks at formation, we assume the same ρ(D ) . (11)
velocitydispersionfortheremnantsasthesurroundingstellar dDL ∝ L s D2S
population.Thisisnotablynotthecaseforneutronstars,which
(b) A lens component (bulge, disk, spheroid) is ran-
oftenexperiencesufficientkicksto be launchedintothe halo,
domly chosen according to the local densities of
diluting their density in the disk and therefore their represen-
eachatD .
tation in the lens sample for currentand planed microlensing L
(c) A lens component type (star, white dwarf, black
3Wedefinethemassfunctiontobethenumberofobjectsperlogarithmic hole) is randomly chosen according to the local
decade,i.e.,dN/dlog10M. fractionofeachforthegivencomponent.
RESOLVING MICROLENSING EVENTSWITH TRIGGERED VLBI 5
10000 4000
Black hole
3000
Non remnant
White dwarf 2000
Non remnant events
Total
1000
1000
0
s
ent 300
v
of e 100 200 White dwarf events
mber nts 10 00
Nu eve 112500
10 of 90
er 60 Black hole events
mb 30
u 0
N 0 2 4 6 8 10 12 14 16 18
1
-1 0 1 2 3 4 5 Heliocenteric distance [kpc]
Log(t /days)
e
Figure3. DistributionofmicrolensingeventsinEinsteincrossingtimefrom Figure4. Distributionoflenspositionsingalaxyforvariouslenspopulations.
Monte-Carlosimulation. ThedetectionefficiencyofOGLEIVsurveyisap- Contributionsofthedisk(solid)andbulge(dashed)areshownseparately;the
pliedduringeventselection. Thefractionofblackholeeventstotheoverall spheroidcontributionisnegligible.
eventsis0.035andweightedheavilytowardslongtE.
(d) Alensmassisselectedaccordingtothetotalmass
functionofthechosencomponentmodifiedbythe 14
approximatelensing cross section; for a fixed ob-
serving time the probability of detecting a lens is
θ M1/2. 16
E
∝ ∝
(e) AlensvelocityisthesumofthebulkGalacticmo- g]
tionforthechosencomponentanda randomcon- ma 18
tribution pulled from a Gaussian distribution with I [
theappropriatedispersion(listedinTable1).
20
3. Detectionefficiencycut:Giventhelensmass,sourceand
lensvelocities,andsourceandlenspositionst iscom-
E 22
puted,andtheeventisacceptedwithaprobabilitysetby
theOGLEdetectionefficiency.
-2 0 2 4 6 8
Theresultisasetoflensingeventswithknownintrinsicprop- V-I [mag]
erties(sourceandlensmasses,distances,velocities,andtypes)
Figure5. Color-MagnitudediagramforOGLEIIIMicrolensingevents. Half
selectedusingarealisticmicrolensingsurveybiases. ofthestarsintheredboxarelikelytobeMiravariables. Thesmallblueline
Figure3 showsthedistributioninEinsteincrossingtimeof insidetheredboxshowshowmuchaMIIIgiantwillmoveinHRdiagramas
simulatedevents.Sinceinthesimulationweflaglensesthatare aresultofblendingwithasolartype(G2V)lensstarlocatedhalfwaybetween
thesourceandobserver.
either main sequenceor remnantsand, we can classify lenses
based on lens type. Here we compare the overall microlens- veys necessarily probe the lens population throughoutthe in-
ingeventswiththoseproducedbyblackholesspecifically;the terveningdisk. Thisremainstrueindependentofthelenstype.
fractionofeventswithblackholelensesis0.035.Thisfraction
iscomparabletotheestimateofGould(2000),whichfinds0.01 3. RADIO-BRIGHTMICROLENSINGEVENTRATES
assuminga100%detectionefficiency,andthusneglectingthe
higherdetectionefficiencyofMiraevents. Keytotheradioimagingofmicrolensingeventsistheiden-
Another important point is the average t for the overall tification of radio-brightlensed sources. In the near term this
E
eventsandblackholeeventsare38daysand127days,respec- requires the identification of candidate events based on their
tively. Thisisunsurprisingsincethedurationofeventsscales optical properties alone. Here we discuss the rates implied
with lens mass as M1/2, and is an immediate consequenceof by the previous section employing surveys that exploit opti-
the fact that the average black hole mass is roughly an order cal counterparts of radio bright sources (i.e., Mira variables)
andthepotentialratesfromfutureradiomicrolensingsurveys.
of magnitude larger than that of the overall sample. Hence,
Thisisultimatelydependentonthepopulationanddistribution
black hole lensing events are typically more than three times
ofcandidatesources.
longerthanthetypicalmicrolensingevent,withimportantcon-
sequencesforidentificationandparameterestimation.
3.1. MirasasMicrolensingSources
Figure4showsthedistributionoflenslocationsinoursim-
ulated survey for each of the Galactic components and lens Mirasprovideanaturalclassofopticallybrightsourceswith
types. Events are dominatedby lenses within the disk (65%) strongradioemissionintheformofcircumstellarSiOmasers.
with the remainder from the bulge (35%); the spheroid con- In most microlensing surveys Miras have already been iden-
tributesnegligibly. Of noteisthatthelensdistributionwithin tified due to their variable nature. Where they have not, they
the disk is nearly evenly distributed between the bulge and a are readily identified in the optical color magnitude diagram.
heliocentricdistanceof2kpc,implyingthatmicrolensingsur- (Figure5). Importantly,thereislittlechanceforsourceconfu-
6 KARAMI ET AL.
sion–blendingofsourceandlensstarsdoesnotmoveobjects togetherwiththeboxtoselecttypeMIIIstars. Theseareafew
appreciablywithintheregionpopulatedbyMiras. orders of magnitude brighter than M-dwarfs, increasing their
The variable nature of Miras provide an immediate imped- representationinmicrolensingsurveys.Anadvantageofusing
iment to their use as microlensing surveytargets; typical Mi- bright stars is that blending by background and lens stars do
ras vary with periodsof 1 yr, I-bandvariabilityamplitudes notaffectthebrightnessofthesourcestarsignificantly(Figure
≈
between 1.4 mag and 3 mag (Soszyn´ski et al. 2013), and V- 5). Moreover,fromthelensequation,andestimatingthemass
band variability amplitudes typically two–three times larger of the lens star, we can calculate the contributionof blending
(Kholopov 1990). However, black hole microlensing events andcorrectthe position of sourcestar in CMD (Murakiet al.
can be easily distinguished from the intrinsic variability as a 2011;Miyakeetal.2011).
result of three importantdifferences. All of these may be ex- WithintheOGLEIIIcataloguetherearecurrently6528Mi-
ploitedinpartduetothehighluminosityofMiras,andthere- ras (Soszyn´ski et al. 2013). The larger number of Miras in-
fore the high photometric accuracy with which they may be ferred in the bulge suggest that this can be readily enlarged
monitored. by surveying the entire bulge region. Estimates based on the
Firstly,microlensinglightcurveshaveawell-understoodand OGLEII catalogue suggest as many as 6 105 Miras in
≈ ×
uniform structure set by the lens-source geometry and nature GalacticBulge(Matsunagaetal.2005).Thelargeluminosities
of gravitational lensing, characterized by a divergent rise as ofMirasimpliesanearunitydetectionefficiency,andthusas
the source approachesthe lens caustics (moderated by the fi- discussedaroundequation(3),weanticipateatotalbulge-Mira
nitesourcesize). Thisdiffersstarklyfromthesmoothintrinsic microlensingeventrate of11yr- 1. Therefore,monitoringthe
variationsofMiras. Thuscarefullightcurvefittingshouldbe GalacticBulgealonewillproduce 0.4blackholemicrolens-
≈
abletorapidlydistinguishbetweenmicrolensingandintrinsic ing events per year associated with a radio-brightsource, en-
variability. ablingradioimaging.
Second,theamplitudeofthefluxvariationduringmicrolens-
3.3. CurrentandNear-futureOpticalSurveyMicrolensing
ing events is typically larger than that due to intrinsic vari-
Rates
ability. Typicalmagnificationsareoforder450%,whilehigh-
magnificationeventscanexceedan1000%. Thesecorrespond At least two future microlensingsurveysare underway,ex-
to magnitudevariationsof 1.5 mag and 2.5 mag, comparable ceedingOGLEIIIinthreeways: highercadence,deepermag-
the that caused by intrinsic variations in I-band. Hence, the nitudelimits, andincreasedskycoverage. OGLEIV,inoper-
microlensing signal is an order unity modification of the un- ation since 2010, has a bulge field of 330 deg2, 3.3 times
≈
derlying source variability in I-band. At longer wavelengths larger than that of OGLE III. The resulting OGLE IV EWS
theintrinsicvariabilitydecreasesfurther,onaveragevariations eventrate is three times larger than that of OGLE III;OGLE
inthemagnitudeat1.25µmare20%ofthoseatopticalwave- IIIEWSdetectedabout650eventsin2008whereasOGLEIV
lengths(Smithetal.2002)4. EWS detected almost 2000 events in 2013 and over 2000 in
Thirdly, the flux variations due to microlensing are achro- 2014. The forthcoming The Korean Microlensing Telescope
matic. Againthisisinstarkcontrasttotheintrinsicvariability Network (KMTNet) consists of three dedicated 1.6 m tele-
of Miras, which varies dramatically with wavelength. Thus, scopes located in Australia, South Africa, and South Amer-
multi-band monitoring should be readily able to disentangle ica,eachwith4deg2 fieldofview. Thus,KMTNetanticipates
thetwovariabilitycomponents. detecting 2300 microlensing events per year with constant
≈
cadenceofalmost10min(Hendersonetal.2014).
3.2. BulgeMirasMicrolensingRates The high luminosity of Miras and the long time-scales of
blackholemicrolensingeventsimplythatthefirsttwo,higher
TheOGLEIIIsurveymonitoredapproximately150million
cadence and deeper magnitude limits, will at best produce
starsinGalacticbulgefrom2001to2009insearchforgravita-
modestimprovementsintheMiramicrolensingrate. Highca-
tionalmicrolensingeventsandprovideduswiththelargestcat-
dences will vastly over-sample black hole microlensing light
alogueofmicrolensingeventsavailable.OGLEusedtheEarly
curves, and thus while important for planet searches are un-
WarningSystem(EWS)todetectongoingmicrolensingevents
likelytobehelpfulforthestudyofGalacticremnants.
(Udalski et al. 1994). EWS uses Difference Image Analysis
EvenwithintheextinctedregionneartheGalacticcenterMi-
photometryandbyanalyzingallthestarsinthefieldflagsthe
rasremainvisibleduetotheirhighintrinsicluminosity. Since
potentialmicrolensingevents (Alard & Lupton1998). In do-
opticallydimstarsdonotproducestrongradioemission(Stel-
ing soitcurrentlyidentifiesandfiltersoutvariablestars. The
lar photospheres of dwarf stars are essentially undetectable
flaggedstarsgothroughseveraltestsincludingavisualinspec-
withcurrentradiotelescopes),expandingthesurveysampleto
tion of the lightcurveand after satisfying all the criteria they
lessluminousobjectsprovideslittlevalueforthestudyofblack
areannouncedasamicrolensingcandidate.
holes. Nevertheless,thedeepermagnitudelimitsoffuturesur-
WeusedOGLEIIImicrolensingalertsdata(2002-2009)to
veyswillpermittheinclusionofregionsofhighextinction,i.e.,
find therelativenumberofMira-typecandidates,demonstrat-
darkareasofthebulge,insurveyfields(Schultheisetal.2009),
ing their high detection efficiency (thoughnote that all Miras
marginallyincreasingthenumberMirasthatcanbemonitored.
willhavebeenexcluded).Ofallthe4052events2015hadboth
Thismaybeamelioratedbyoperatingmicrolensingsurveysin
V and I magnitudesavailable which were used for ouranaly-
theinfraredaswell.
sis. The events which only have I bandmeasurementswould
Much more important is the expanded sky coverage. We
nothaveasubstantialcontributiontoMIIIstars,sincetheyare
have already assumed that the entire Galactic bulge will be
mainly faint stars. Figure 5 shows the CMD for these events
monitoredinourpreviousrateestimate. Furtherincreasingthe
numberof monitoredMiras substantially in an opticalsurvey
4 Thereareinstancesforwhichnear-infraredvariabilityisneverthelesssiz-
necessarily requires expanding the surveys beyond the bulge.
able,e.g.,RCas,wheretheamplitudeofvariationsat1.25µmareaslargeas
1.42mag. However,thevariabilitycontinuestodroprapidlywithincreasing Moderatemodificationstotheobservingstrategiesofexisting
wavelength. optical surveyspresent an obviousway in which radio-bright
RESOLVING MICROLENSING EVENTSWITH TRIGGERED VLBI 7
microlensing event rates can be substantially increased. Be- (e.g.,Miras)or primarilyextragalacticobjects. Thedetailsof
causeMirasareintrinsicallyluminous,theyliewellabovethe thelensparameterestimationareonlyweaklydependentonthe
detectionthresholdofallongoingmicrolensingsurveyslocated distance to the radio source; placing sources at extragalactic
anywherewithintheGalaxy. Therefore,therelativepaucityof distancesdecreasestheEinsteinanglebyatmost30%.
Mirasdoessuggestanalternativestrategy: broadandshallow Moreimportantistheover-allnumberofsources,whichdi-
insteadofnarrowanddeep. rectly translates into the predicted rate enhancement. These,
IncludingtheGalacticdiskisexpectedtogrowtheMirasam- in turn, depend on the flux limit and wavelength of interest,
ple by a factor of 6-7 assuming the stellar populationdoes both set by the VLBA. For a 10M black hole lens multiple
⊙
≈
notvarywidelybetweenthebulgeandthedisk. Hence,weex- imagecomponentsareeasilyvisiblebytheVLBAforfrequen-
pect 4 106MirasintotalwithinMilkyway. Unfortunately, cies above 3 GHz, providing a natural upper limit of 10 cm
≈ ×
themicrolensingopticaldepthinthediskisroughlyathirdof on the observation wavelength. For 10 min integration times
thatofthebulgeasaresultoftheformer’ssmallerstellarden- with500MHzbandwidthstheVLBAfluxlimitshouldbenear
sity. Takingintoaccountthesmalleropticaldepthin thedisk 20µJyand40µJyat22GHzand43GHz,respectively;where
asurveythatmonitorstheentireGalacticplanetoamagnitude requiredweadoptafluxlimitsof30µJy.5 Large-amplification
limit of 15 would capture radio-brightmicrolensing events at eventswill reduce the effective flux limit further. As we will
a rate of 37 yr- 1, corresponding to a radio-bright black hole seebelow,eitherimprovementsinthesensitivityoftheVLBA
microlensingevents at a rate of 1.3 yr- 1. This is sufficientto ornewradio-brightsourceclasseswillbenecessarytoleverage
doublethe numberof knownGalactic black holeswithin less aradiomicrolensingsurvey.
thantwodecades. Radio-brightactivegalacticnuclei(AGN)provideanatural
Importantly, these survey strategies need not be exclusive. class of sources, being both numerous, and more importantly
Since the objects of primaryinterest for radio imaging, black compact. Conveniently, they are also typically flat-spectrum
holes,resultinlong-durationmicrolensingevents,evensparse radio sources, meaning that neither the source identification
timesamplingissufficient(ontheorderofonceperweek).The northesubsequentmicrolensingsurveyneedbe performedat
OGLEIVsurveyhasalreadyadoptedsuchastrategy,monitor- the same wavelengths at which the events are ultimately im-
ingtheentireGalacticdiskwithacadenceof1-2daysandthe aged. Thus,toestimatethenumberofpotentialsourcesabove
bulgewithacadence.1hr(Udalskietal.2015). the VLBA fluxlimit we employthe NRAO VLA Sky Survey
at 1.4 GHz reportedin Condonet al. (1998)(thoughsee also
3.4. Far-futureOpticalMicrolensingSurveys Condon 1984;Condon et al. 2012) finding roughly 1.6 107
TheLSST(LargeSynopticSurveyTelescope)isa8.4meter and 1.5 108 radio-brightAGN in ellipticals and spiral×s, re-
×
telescope equipped with a wide field camera with 9.6 square spectively,withfluxesaboveabove30µJy.
degreesfieldofview.Itwillprovideadeepsurveyofsouthern Thesearedistributedisotropically,andthusalargefraction
sky(over20000squaredegrees)giving1000exposuresofeach of AGN will be visible at high Galactic latitude with corre-
patchduring10yearsofobservations.Theoptimalcadenceof spondinglysmaller lensing opticaldepth. As a result, the av-
thesurveyfordifferentareasoftheskyisyettobedetermined. erage lensing opticaldepthfor an AGN surveyto thatfor the
The single exposure depth in r-filter will be 24.5 mag and opticalsurveysthatmonitortheGalacticbulgealoneisapprox-
≈
it can yet go deeper ( 26.5 mag) by co-adding the images. imately0.01. Thus,despitehavingnearly290timesthenum-
≈
Being a ground base survey the angular resolution is limited ber of targets, a radio microlensing survey of objects above
byseeing(0”.7)andultimatelybyitspixelsize (0”.2)(Ivezic 30 µJy would produce a black hole microlensing event rate
etal.2008). 1yr- 1,aboutthesameasthatfromMira-basedopticalsurveys.
Themeancadenceof3- 4daysismuchshorterthanthetyp- To obtain an event rate of 10 yr- 1, doubling the number
icaltime-scalesofthelongdurationblackholelensingevents. of know black holes in two years, requires a flux limit of
FurthermorethedeepsurveycombinedwiththefactthatMira 3 µJy. This is would require correspondingimprovementsin
variablesarebrightstarsguaranteesthatmostoftheMiralens- theVLBA.However,inprinciplethismaybeachievedviaan
ing eventsin the Galaxy,LMC, andSMCwillbe detectedby extended integration time combined with an expanded band-
LSST.Thus,forfindingblackholelensingeventsamenableto width. Currently, 4 GHz bandwidths supported by 16 Gbps
radio imaging, the LSST presents a similar capability to cur- recorders are planned stations participating in millimetre-
rent generation microlensing surveys operated with modified wavelength VLBI observations (Whitney et al. 2012), and
observationstrategy. could be deployed to VLBA stations for use at centimetre
wavelengths. Combined with 2 hr integration times, these
3.5. AFutureRadio-ContinuumMicrolensingSurvey
reach the 3 µJy flux limit needed. Below 3 µJy the number
Were a microlensingsurveyperformedin the radiodirectly ofsourcesscaleapproximatelyasN S- 0.5,andthusfurther
>S
∝
the need to identify optically-brightradio counterpartswould growthinthenumberofsourcesisaslowfunctionoffluxlimit.
beeliminatedaltogether,resultinginthefindingofeventsthat
are amenable to radio imaging with a near 100% efficiency.
Moreover such a radio survey would permit monitoring the
muchmorenumerouscompactcontinuumradiosources. With 4. RADIOVLBIEVENTRECONSTRUCTION
the adventofa numberofrapid-surveyradiotelescopes(e.g.,
theSquareKilometreArrayTaylor2007,CanadianHydrogen Motivated by the prospect of substantial number of opti-
Intensity MappingExperimentBandura et al. 2014, etc.), en- cal microlensing events amenable to radio imaging, we now
abling rapid nearly all-sky surveys, high-cadence radio tran- present illustrative examples of what these may look like for
sientsearcheswillbecomecommon.Aradiomicrolensingsur- typicaleventparameters,anddiscusstheprecisionwithwhich
veyisanaturalbyproductofthese.
Thestringentrequirementsonsourcesize(.1mas)restricts 5 We assume system-equivalent flux densities are roughly 500 Jy and
the potentialradio microlensingsurveysource targetsto stars 1000Jyat22GHzand43GHz,respectively.
8 KARAMI ET AL.
theeventparametersmaybereconstructed. Over the duration of a lensing event, the position of the
source and lens are assumed to evolve with a fixed velocity:
4.1. ModelsofResolvedMasers
SiO masers at 43 GHz ( 7 mm) have routinely been x =x +v t and x =x +v t, (13)
S s,0 S L L,0 L
≈ ≈
detected around late-type giants in the central parsec of the
respectively. In practice only the transverse motion is impor-
Galaxy (Reid et al. 2003). Located within the extended at-
tant. Incontrast,theobserverposition,i.e.,thatoftheEarth,is
mospheres of Miras, these are typically within 8 AU of their
orbital,andthusincludestheorbitalacceleration:
parent, correspondingto an astrometricoffsetof 1 mas in the
Galactic center. Note that this is comparable to the Einstein x =x + dtΩ r . (14)
⊕ ⊕,0 ⊕ ⊕
angle, θ , and therefore the lensing of the maser emission is ×
E Z
distinctfromthatof opticalemission ofthe star dueto the fi- Thelensgeometryisthenfullydefinedbythedistances
nitesizeeffectofsourcestar(Witt&Mao1994). D x - x and D x - x , (15)
Themasingspotsareresolvedbothbecauseoftheirintrinsic L≡| L ⊕| S≡| S ⊕|
structureandasaresultofaninterstellarscatteringscreenthat andthetransverseangulardisplacement
scatter-broadens images of the Galactic center (Backer 1978; (x - x ) (x - x ) (x - x )(x - x )
β S ⊕ - S ⊕ L ⊕ L ⊕ . (16)
Bower et al. 2006; Gwinn et al. 2014). Because of the latter ≡ D D · D D
S S L L
effect,evenpointsourcesexhibitanextendedsourcestructure Neglecting the acceleration in the Earth’s motion and the
withafull-widthhalf-max(FWHM)of 0.7mas,effectively line-of-sight motion of the source and lens permits a simpli-
≈
limitinglongestbaselinesthatcanbeemployedbytheVLBA. ficationofβ toa linearfunctionoftime, thoughherewe will
However,thisscatteringishighlylocalized,occurringonlyfor makeuseofthemoregeneralexpressioninequation(16).
sources in the immediate vicinity of the Galactic center, and Forapoint-masslenstheobservedpositiononthesky,θ is
unlikely to limit the resolution attainable by efforts to image givenbythethin-lensequation. Usuallythisisexpressedasa
MirasthroughouttheGalacticbulge. potentiallymulti-valuedfunctionoftheprojectedsourceposi-
TypicalintrinsicsizesfornearbySiOindividualmaserspots tion. However, we need only to identify the projected source
are 1 AU, corresponding to 0.1 mas at 8 kpc distance, positionintermsoftheobservedpositionθ,givenby
≈ ≈
andthereforealsounlikelytosignificantlylimittheresolution
of VLBA observations. We ignore the impact of intervening θ2
scatter broadening and model the intrinsic emission from an β=θ 1- E . (17)
individualspotby a Gaussian intensity profile with a FWHM θ 2!
| |
of0.1mas: Since surface brightness is conserved, this then immediately
Iint(β)=I0e- |β|2/2σs2pot, (12) definesthelensedimageintermsoftheintrinsic
whereσ =0.1mas/√8ln2=0.042mas.
spot I (θ)=I [β(θ)]. (18)
lens int
Inpractice,SiOmasersfromMiravariablesoftenformarc-
like structures that are dominated by a handful of individual Finally,weconvolvethelensedimagewitharealistic,ifnot
masingspots. Forblackholelensestheangularsizesofthese pessimistic,beam.Forthisweassumea43GHzbeam,B(θ)is
structuresaretypicallysmallerthanθ ,andthusmultiplespots ananisotropicGaussian,consistentwiththebeaminLuetal.
E
are likely tobe stronglylensed simultaneously. Forless mas- (2011): semi-minor axis of 0.5 mas and semi-major axis of
sivelensestheringangularangularscaleandθ maybemore 1.4mas, oriented12◦ EastofNorth. Thelargeaspectratiois
E
similar. Nevertheless, because the emission between maser aresultofacombinationofthelowdeclinationoftheGalactic
spotsremainsincoherent6. theresultinglensedimageisalin- centerandtheNorthAmericanlocationoftheVLBAantennae.
ear superpositionofa numberof individualspots. Thus, here Inpractice,thisresolutionislowerthanmaybeachievedbythe
weconsiderthesimplerproblemofasinglemasingspottoas- VLBAat43GHz; in Reidetal.(2003)theinterstellarscatter
sessthesizeoftheconstraintsthatcanbeplacedonthelensin broadeninglimitedthesizeofthearraythatcouldbeeffectively
principle. employed. OutsideoftheGalacticcenteritshouldbepossible
Net velocity offsets between the star and masing spots are toincreasetheresolutionbyafactorof2–3,thoughweadopt
systematicallyincorporatedintotheradiosourcevelocitiesand the Lu et al. (2011) beam. The resulting observed intensity
thusdonotpresentanadditionalsystematicuncertainty.How- distributionisthen
ever, the masing spotscan also evolvein size and luminosity.
I (θ)= d2θ′B(θ- θ′)I (θ′), (19)
Bothareunlikelytoproducesubstantialcomplicationssinceit obs lens
doesn’timpacttheseparationofthelenseimages(seeSection where prior to the convZolution the angular positions are con-
4.4). While thelatter cancomplicatethedeterminationofthe vertedintoequatorialcoordinates.
radiolightcurve,thismaybeamelioratedviatheopticallight All that remains is to specify the relative positions, veloci-
curve. ties,andmassofthesourceandlens. Wedothisforahandful
ofillustrativeexamplesinthefollowingsections.
4.2. GeneratingMockImages
Time sequencesof mock imagesare generatedvia a multi- 4.3. ExampleLensedMaserImages
stepprocess: beginningwiththespecificationofthepositions
As fiducial source parameters we assume D = 8 kpc and
of the source, lens, and observer, the mapping of the source a proper velocity of 120 kms- 1, consistent wSith sources in
totheimageplaneviathethin-lensequation,andconvolution
Galactic bulge. The fiducial lens parameters are D =4 kpc
witharealisticbeam. andapropervelocityof30kms- 1,consistentwiththLevelocity
6Thisisdistinctfrom,e.g.,forpulsars,whichproducemanycoherentspots dispersion in the Galactic disk. The relative positions of the
duetoscintillationwithintheinterstellarmedium,andthereforerequirecareful source and lens are chosen so that minimum impact parame-
considerationofwave-opticseffects(Penetal.2014). ter is 0.3 mas oriented in declination. These result in typical
RESOLVING MICROLENSING EVENTSWITH TRIGGERED VLBI 9
images that are illustrative of the general situation. All that of magnitude for high signal-to-noise detections (Reid et al.
remainsistospecifythemassofthelens. 2003). Thus,together,theseyieldahigh-precisionestimateof
Thefirstcaseweconsideristhatofa10M blackhole,for θ :
⊙ E
whichθE =3.2mas. A time-sequenceofimagesforourfidu- ∆θ 1/2
cial case is shown in Figure 6, coveringnearly a year. Since θE = √2 Amax A2max- 1- (A2max- 1)
the minimum impactparameter is small in comparisonto θE, (cid:20) q (cid:21) (22)
multiple images are present. Because θ is also considerably ∆θ 1
E 1- ,
largerthanthe intrinsicsourceandbeam sizes, these are well ≈ 2 8A2
resolved,suggestingthatdetectingandinterpretingthefeatures (cid:18) max(cid:19)
ofstronglensingbyblackholeswillbestraightforward.Inpar- where the latter expression is for large Amax, and a fractional
ticular,adirectmeasurementofθ ispossiblesimplybyfitting precisionof
E
thespotseparations,thoughwedefertohowwellthismaybe σθ2E σ∆2θ + σA2max . (23)
doneinpracticetoSection4.4. θ2 ≈ ∆θ2 4A6
The same qualitative conclusions hold for a 1M lens, in- E max
⊙
dicative of a white dwarf, shown in Figure 7.7 Despite the When the uncertainty is dominated by the astrometric uncer-
smaller Einstein angle (θE = 1.0 mas), multiple images are tainty,thisimpliesthatfora10M⊙ blackholeθE canbemea-
resolved, again implying that θ can be directly measured. suredtowithin 1%.
E ≈
Where these differ most dramatically is near peak magnifica- Even without detailed event modeling, θE immediately en-
tion (center panel in Figures 6 and 7); the more massive lens ablesthedeterminationoftherelativeangularlens-sourceve-
produces correspondinglylarger distortions in the image as a locityfromequation(2):
result of the better resolution of the Einstein ring. The mi- ω - ω = θEuˆ, (24)
crolensingeventisalsosignificantlyshorter. L S t
E
The effect of the accelerated observer frame, due to the whereuˆ is the asymptoticdirectionof the vectorbetweenthe
Earth’s orbital motion, is evident in the asymmetric entrance first and second images. The source proper motions can be
and exist from the microlensing event. However, this effect directly measured by follow-up observations; the angular ve-
may be confusedwith featuresarisingfrom the impactof the locitiesofindividualmaserspotshavebeenmeasuredwithan
asymmetricbeamandarbitraryeventorientation. ThusinFig- accuracybetterthan1masyr- 1. Aswiththesourcepositions,
ures6and7wehavealsoshownbytheredcontourstheimages thisislimitedbytheelectronscatteringReidetal.(2003),and
withtheEarth’sorbitalmotionneglected.Fromtheseitisclear thusforsourcesbeyondthecentralpcitshouldbepossibleto
thatasymmetryinthetemporalevolutionoftheimagestructure measurethisnearlyanorderofmagnitudebetter,givingatyp-
atearlyandlatetimesisarobustindicatorofparallax. icalaccuracyof0.1masyr- 1, correspondingtoaphysicalve-
locityof4kms- 1atthedistanceoftheGalacticcenter. Witha
4.4. ParameterEstimation typicallensapparentvelocityof30kms- 1,comparabletothe
The primary parameters of interest are those of the lens: velocity dispersion within the disk, this induces to a roughly
mass, velocity, distance. Given only the magnification light 10%uncertaintyintheinferredωL.
curvethesearedegenerate,constrainedonlybytheeventtime- The remaining degeneracy with DL may be broken via the
scaleandpeakmagnification.8 Asaconsequence,allthatcan measurementof aparallaxfromthe microlensingeventitself.
berecoveredeveninprinciplearetheprojectedimpactparame- The sources of most interest also are expected to have the
terinunitesofθE,tE,andthetimeofmaximummagnification. longesttE,andthusencompassasubstantialfractionofayear.
However,imageslikethoseinFigures6and7introduceatleast As madeexplicitin Figures6 and7, longtE permitmeasure-
twokeyadditionalobservables:θ andalensingparallax. mentsoftheimpactoftheEarth’sorbitalmotion,i.e.,aparal-
E
The peak magnification occurs at the projected closest ap- lax. Thiscomesintwoforms.
proach, for which the projected impact parameter is β and First, the light curve itself is asymmetric as a result of the
min
themagnificationis modifiedevolutionof the Earth-lens-sourcealignment(Smith
etal.2003).Figure8showstheeffectofparallaxonlightcurve
β2 +2θ2 ofa10M blackholelensingevent. SincetheEarthacceler-
A = min E . (20) ⊙
max atessubstantiallyduringthisperiod,theadditionalcomponent
β β2 +4θ2
min min E isdistinguishablefromtheotherwiseunknownbutessentially
q fixedvelocitiesofthesourceandlens,resultinginanasymme-
ThedegreetowhichA maybemeasureddependsuponthe
max tryinthemagnificationlightcurve. Whilethemaseremission
quality of the photometry that can be performed, and thus is
andopticalstellaremissionarenotspatiallycoincident,theoff-
likelytobelimitedbysystematicerrorsintheradiofluxcali-
setinthedistanceismuchsmallerthanD ,andthusD canbe
bration(fluxcalibrationaccuracyforthe VLBAandtheVLA S L
reconstructed from either the optical or infrared light curves.
at43GHzwouldberoughly10%). Atthistimetheseparation
Typicaluncertaintiesof%10inradiofluxcalibrationmaypre-
betweenthecentroidoftheprimaryandsecondaryimagesis
cludetheuseofradiolightcurveforthispurpose.SinceMiras
∆θ= β2 +4θ2, (21) areamongthebrighteststarsinthemicrolensingsurveyfields,
min E
theymaybegoodcandidatesforobtainingD inthisway,as-
whichmaybemeasureddireqctlyfromtheimagestoaprecision L
sumingtheunderlyingvariabilitycanbeadequatelymodeled.
thatexceedstheinterferometricbeamwidthbyuptoanorder
Second,asdescribedintheprevioussection,theunderlying
impactonthelensedimagesthemselvesisdirectlymeasurable.
7Recallthatasolar-typestarwouldnotmodifythecolorofthecombined
Notethatthisisessentially thesame effect,theasymmetryin
lens-sourcesystemsufficientlytoplaceitoutsideoftheallowedregioninthe
CMD. themagnificationlightcurvearisesfromtheasymmetryinthe
8Thismaybeimprovedsubstantiallythroughtheexploitationoftheimpact evolvingimage structure. However,unlike the parallaxeffect
ofparallaxonthelightcurve(Gould1992;Rahvaretal.2003). inthelightcurve,theimagestructureisinsensitivetotheMira
10 KARAMI ET AL.
Figure6. Example10M⊙blackholemicrolensingevent.Impactparameterof0.3mas,relativelens-sourcevelocityof150kms- 1,1AUFWHMmaser,andradio
beamtypicalofGalacticcenterobservationswiththeVLBA(semi-minor/majoraxesof0.5masand1.4mas,respectively,withapositionangleof12◦eastofnorth).
RedcontoursshowimagesignoringtheorbitalmotionoftheEarth,whileblackcontoursincludetheparallax.Forreference,thebluedottedlineshowstheEinstein
ring.