The multiple quasar Q2237+0305 under a microlensing caustic

0.32 MB·English

by  T. Anguita

Astronomy&Astrophysicsmanuscriptno.8221 c ESO2008
(cid:13)
February2,2008
⋆
The multiple quasar Q2237+0305 under a microlensing caustic
T.Anguita1,R.W.Schmidt1,E.L.Turner2,J.Wambsganss1,R.L.Webster3,K.A.Loomis4,D.Long4,andR.
McMillan4
1 AstronomischesRechen-Institut,Zentrumfu¨rAstronomiederUniversita¨tHeidelberg,Mo¨nchhofstrasse12-14,69120Heidelberg,
Germany.
8
2 PrincetonUniversityObservatory,PeytonHall,Princeton,NJ08544,USA
0
3 SchoolofPhysics,TheUniversityofMelbourne,Parkville,Victoria3052,Australia
0
4 ApachePointObservatory,2001ApachePointRd,SunspotNM88349,USA
2
n February2,2008
a
J ABSTRACT

5
Weusethehighmagnificationeventseeninthe1999OGLEcampaignlightcurveofimageCofthequadruplyimagedgravitational
1
lens Q2237+0305 to study the structure of the quasar engine. We have obtained g’- and r’-band photometry at the Apache Point

Observatory3.5mtelescopewherewefindthattheeventhasasmalleramplitudeinther’-bandthanintheg’-andOGLEV-bands.
]
h Bycomparingthelightcurveswithmicrolensingsimulationsweobtainconstraintsonthesizesofthequasarregionscontributingto
p theg’-andr’-bandflux.Assumingthatmostofthesurfacemassdensityinthecentralkiloparsecofthelensinggalaxyisduetostars
o- 1an0d15b√yMm/o0d.1eMlingctmhe,wsohuerrceeMwiitshthaeGmaueassniamnicprroolfielnes,inwgemobastasi,nanfodrathreatiGoaσuss/iaσnw=id1th.215.+200.45×.W10i1th5√thMe/li0m.1iMts⊙oncmthe.vσelgo′ci.ty7o.9f6th×e
str laenndsiangragtiaolaσxy/f⊙rσom=Gi1l-.M45e+r0i.n90o(eatlallv.a(2lu0e0s5)atas68oupreorncelyntprcioonr,fiwdeenocbet)a.inA0d.d6i0ti×ro′n1a0l1gl5′y,√fMro/m0.1o−M0u.1r⊙5mcimcr.oleσngs′in.g1s.i5m7u×l1a0ti1o5n√sMw/e0fi.1nMd⊙thcamt,
a r′ g′ 0.25
during the chromatic micro−lensing event observed, the continuum emittingregion of the quasar crossed acaustic at 72 percent
[ ≥
confidence.

2
Keywords.gravitationallensing–quasars:individual:Q2237+0305–cosmology:observations–accretiondisks
v
5
6
2 1. Introduction Ostensenetal. 1996; Alcaldeetal. 2002; Schmidtetal. 2002).
4 The one that has the longest and best sampled data is that
1. OQn2e237o+f03th0e5. bTehsits qstuuadsiaerdwqausadsaisrcolveenrseidngbysythsteemCsentyeertfoisr of the OpticalGravitationalLensingExperiment(OGLE)team
1 (Woz´niaketal. 2000; Udalskietal. 2006). They have moni-
Astrophysics (CfA) redshift survey (Huchraetal. 1985). The
7 tored Q2237+0305 since 1997 delivering the most complete
systemismadeupofabarredspiralgalaxy(z=0.0394)andfour
0 lightcurvesforthissysteminwhichmanymicrolensingevents
imagesofaquasarataredshiftofz=1.695thatappearinanearly
: can be seen. These data have been used for various studies
v symmetricconfigurationaroundthecoreofthespiralgalaxyat
on the system: Wyitheetal. (2000a,b,d,e,f); Yonehara (2001);
Xi adistanceofapproximately1”fromthecenter. Shalyapinetal. (2002) and Kochanek(2004) used this data to-
Q2237+0305is an idealsystem to study quasar microlens-
r getherwithmicrolensingsimulationsnotonlytoobtainlimitson
a ing. At the position of the quasar images the optical depth thepropertiesofthequasarsuchassizeandtransversalvelocity,
due to microlensing by the stars is high (Kayseretal. 1986;
butalsoonthemassofthemicrolensingobjects.
Kayser&Refsdal1989;Wambsganssetal.1990).Furthermore
TheOGLE team hasmonitoredthis quasarwith a singleV
the expected time-delay between the images of this quasar is
bandfilter. However,Wambsganss&Paczynski(1991) showed
of the order of a day or less (Schneideretal. 1988; Rixetal.
that color variations should be seen in quasar microlensing
1992;Wambsganss&Paczynski1994).Asmicrolensingevents
events and how they would provide additional information on
forthissystemareoftheorderofmonthsoryears,itiseasyto
the background source. We know from thermal accretion disk
distinguishintrinsic variabilityfrommicrolensing.Anotherbig
modelsthatemissioninthecentralregionsofaquasarshouldbe
advantage is the fact that the quasar is 10 times farther from
∼ bluerthanintheouterregions(e.g.,Shakura&Sunyaev1973).
usthanthelensinggalaxy.Thisleadstoalargeprojectedveloc-
As an approximation we can assume the central engine of the
ityofthestarsinthesourceplaneandthusashorttimescaleof
quasartobeoftheorderof1000Schwarzschildradii:
caustic events.Notlongafterits discoveryin1985,Irwinetal.
(1989)observedthefirstmicrolensingsignatureinthisquasar.
The quasar has been monitored for almost two decades
M
bydifferentsurveys(e.g.,Corriganetal. 1991; Penetal. 1993; R =1000 r =2.8 1016 BH cm (1)
ad × s × 108M
Send offprint requests to: T. Anguita, E-mail: [email protected] ⊙!
heidelberg.de
⋆ BasedonobservationsobtainedwiththeApachePointObservatory where rs is the Schwarzschild radius, Rad is the radius of the
3.5-metertelescope,whichisownedandoperatedbytheAstrophysical accretion disk and MBH is the mass of the black hole. We can
ResearchConsortium. compare this value with the Einstein Radius (R ) - the length
E
2 T.Anguitaetal.:Aquasarunderamicrolensingcaustic
TheAPOdataweretakenwiththe3.5mtelescopeatApache
Point Observatory using the Seaver Prototype Imaging camera
(SPIcam) in both the SDSS g’ and r’ filters. The SPIcam has
a 2048 2048pixels CCD and a minimumpixel scale of 0.141
×
”/pixel, however, for the observations we use, the pixels were
binnedto0.282”/pixel.
2.1.StandardCCDReduction
For the standard CCD reduction we use a combination of the
2237+0305 astronomicalimagereductionsoftwarepackageanddata
STD Star
language.We applybiascorrection,createonemasterflat-field
image for each night and each filter and calibrate the science
frameswiththese.Cosmicraysareeliminatedfromthescience
framesusingmedianfilteringoutsidethesources.
Inordertoremovetheremainingcosmicraysandimperfec-
30 arcsec
tionsoftheCCD aswellastoimprovethesignaltonoiseratio
Fig.1.APOg’imageofthequasarsystemQ2237+0305. oftheimages,wecombinetheframescorrespondingtothesame
night, excluding those with asymmetric point spread functions
(PSFs).Intheendweobtainacleanhighqualityimageforeach
scaleinquasarmicrolensing-whichcanbegiveninthesource nighting’andr’filters.
planeby(Schneideretal.1992):
2.2.Photometry
4GMD D M
R = s ls =5.68 1016 cm (2)
E r c2 Dl × s0.1M The photometry for this system is made difficult because
⊙ the quasar images are seen through the core of the lens-
where we have replacedthe valuesof the distances D, D and ing galaxy. Our first attempt, following the procedure as in
l s
D to the ones corresponding to Q2237+0305. The accretion Woz´niaketal. (2000), was to use an imagesubtractionmethod
ls
diskscalesarethuscomparabletothoseoftheEinsteinringsof (Alard&Lupton 1998). However our field of view lacks the
stars in the lensing galaxy. This shows that differential magni- high number of bright stars required in order to run a suc-
fication of the different emission regions of the quasar can be cessful correlation and PSF characterization for the procedure.
observed. Thisiswhyweuse(Pengetal.2002)instead.isa
Some multi-band observations of this system have been galaxy/pointsourcefittingalgorithmthatfits2-Dparameterized
carried out (Corriganetal. 1991; Vakuliketal. 1997, 2004; imagecomponentsdirectlytotheimages.
Koptelovaetal. 2005), howevermostof themhave eitherlarge
gaps between the observations, or no obvious microlensing
2.2.1. model
event.
A very interesting event, that has been one of the bases ForthegalaxyweusethemodelshownbySchmidt(1996)(see
forpredictions(Wyitheetal.2000b)andstudy(Yonehara2001; also Trott&Webster 2002) from HST F7815LP band data, in
Shalyapinetal. 2002; Vakuliketal. 2004), is the highmagnifi- whichthelensinggalaxyisparameterizedwithadeVaucouleurs
cationeventofimageCintheyear1999asobservedbyOGLE. profileforthebulgeandanexponentialprofileforthedisk.The
Inthispaperwepresentdataofthisparticulareventobtained bulge’sdeVaucouleursprofileisconstrainedtoa4.1”(3.1kpc)
in two filters at Apache Point Observatory (APO). We analyze scale length,a0.31ellipticityanda 77 positionangle.Forthe
◦
the event using both the OGLE light curve and the APO data exponentialprofileweusea11.3”(8.6kpc)scalelength,a0.5el-
and study the implications for the size of the quasar emission lipticityanda77 positionangle.Theseprofilesareconstrained
◦
regionusingmicrolensingsimulations.Weuseaflatcosmology to have the exactsame central position within the image and a
withΩm =0.3andH0 =70km s−1Mpc−1. fixedmagnitudedifference.
The quasarimagesare parameterizedas pointsources.The
relative separation between them is fixed in the  code to
2. ApachePointobservatoryData,Observations&
the valuesobtainedby Blantonetal. (1998) from UV data and
Datareduction
the separation between the group of images and the center of
thebulgeisfixedtothevaluesobtainedwithHSTF7815LPfil-
Forthisstudyweusetwodifferentdatasets.Thefirstoneisthat
of the OGLE1 team lightcurvesfor this quasar(Woz´niaketal. ter data (HST proposalID 3799):∆RA = 0.075′′ and ∆DEC =
−
0.937 .
2000;Udalskietal.2006)forwhichweselectthedatabetween ′′
Julian days2451290(April21,1999)and 2451539(December ThePSFweusetoconvolvethedifferentparameterizations
26,1999),whichcomprises83datapoints.Theresultisadense is taken from the bright star (STD) shown in Fig. 1. After fit-
light curve for the event we analyze. The second data set is ting the different profiles, the brightness of each componentis
APO two-band monitoring data for Q2237+0305.We selected measured.Weruntheroutineovereachnightineachone
the nights between May 17 1999 (Julian day: 24511316) and ofthetwo bandsavailable.Theresidualsfromthisfittingtech-
January 8 2000 (Julian day: 24511551) which also track the niquearelow,usuallyoftheorderof5percent.Inordertoob-
event,comprising8nightswithdatapoints. tain error bars on the magnitude measurements, we create 500
Monte-Carlorealizationsoftheobservedimagesanddetermine
1 http://bulge.princeton.edu/ ogle/ogle2/huchra.html the measurement uncertainty from the scatter of the  re-
∼
T.Anguitaetal.:Aquasarunderamicrolensingcaustic 3
Table 2. Localmacro-lensingparametersfor quasar imagesB
andCfromSchmidtetal.(1998).κisthescaledsurfacedensity,
γistheshear,µ isthetotalmagnification.
tot
Image κ γ µ
tot
B 0.36 0.42 4.29
C 0.69 0.71 2.45
(Lewis&Irwin 1996)). Approximately 1011 rays are shot, de-
flected in the lens plane and collected in a 10,000 by 10,000
pixels(equivalentto50Einsteinradiiby50Einsteinradii)array
in the source plane. The values of surface density κ and shear
γ are taken from Schmidtetal. (1998) (Table 2). The conver-
genceκisseparatedintoacompactmatterdistributionκ anda
⋆
smoothmatterdistributionκ .Eventhoughweexploredifferent
c
stellarfractions,asthequasarimagesarelocatedwithinthecen-
tralkiloparsecofthelensinggalaxy,wefocusourworkmainly
onthe κ⋆ =1stellarfraction.
κ
AsshowninSect.1,inquasarmicrolensingthesourcesize
isanimportantparameterbecauseitiscomparablewithtypical
causticscales(e.g,Kayseretal.1986).Inordertostudyasam-
ple ofdifferentsourcesizes, theraw magnificationpatternsare
convolvedwithasetofprofiles.Mortonsonetal.(2005)showed
that microlensing fluctuations are relatively insensitive to the
source shape, so it is of little consequence whether we choose
astandardmodelaccretiondisk(Shakura&Sunyaev1973)ora
Gaussianprofile.ForsimplicitywechooseaGaussianprofilefor
thesurfacebrightnesswheretheextentofthesourceisdescribed
bythevarianceσ.Thefullwidthhalfmaximum(FWHM)isde-
finedas2.35σ.
WevarytheFWHMofaGaussianprofilefrom2to120pix-
els(whichcorrespondsto0.01E and0.60E ,respectively)for
R R
quasarimageBandfrom2to216pixels(whichcorrespondsto
Fig.2. Top panel: Black squares show the APO g’-band light 0.01ERand1.08ER,respectively)forquasarimageC(theaddi-
curve(relativetothefaintestdatapoint)ofthehighmagnifica- tionalmagnificationpatternsforimageCareusedforthecolor
tioneventseeninimageCofQ2237+0305in1999.Forcompar- curvefittingexplainedinSection5).Forpatternsbelow12pix-
ison, in lightgray,the OGLE V-bandlight curvesamplingthis elsthestepsizeis2pixels,andforpatternsabove12pixelsthe
eventis shown. Middle panel: APO r’-bandlight curve for the stepsizeis4pixels.Wefindthatthislinearsamplinggivesmore
event(blacksquares)andtheOGLEV-bandlightcurveplotted accurate results when interpolating compared to a logarithmic
inlightgray.Bottompanel:APOg’-r’colorcurve.Thedashed sampling.
lineisshowntoguidetheeye. For a specific convolved pattern we can now extract light
curves.Bydefiningatrack(pathofthesourceinthesourceplane
magnificationpattern)withastartingpoint,directionandveloc-
sults. The g’- and r’-band light curves for image C are shown ity, we extract the pixel counts in the positions of the pattern
in Fig. 2. The measurements for images A and C are given in defined by this track using bi-linear interpolation. The values
Table1.NoreliablephotometryisobtainedforimagesBandD arethenscaledwiththemagnificationvaluesofTable2foreach
becausetheyweretoofaint. imageandconvertedintomagnitudes.
As shown in Fig. 2, the APO g’-band light curve shows a
very similar behaviour to the OGLE V-band light curve, how-
ever, the APO r’-band light curve shows a lower amplitude at 4. Lightcurvefitting
the brightness peak. Thus, the resultant g’-r’ color curve, does
not appear flat but shows a chromatic variation during the mi- 4.1.OGLEVlightcurve
crolensingevent.
Since quasars vary intrinsically, we need to separate this vari-
ation from microlensing.We thereforestudy the difference be-
tweentwolightcurvesextractedfrommicrolensingpatternsfor
3. MicrolensingSimulations
different images to eliminate this effect and allow for an ad-
Inordertocomparetheobserveddatawithsimulationswegen- ditional magnitude difference (the time delay is negligible, see
erate source-plane magnification patterns for quasar images C Sect.1).
andBusingtheinverserayshootingmethod(Wambsganssetal. Using the V band OGLE light curves sampling this event
1990; Wambsganss 1999). We assume identical masses for in both images C and B of the quasar (Woz´niaketal. 2000;
all the microlenses distributed in the lens plane (the results Udalskietal.2006)weconstructthedifferencelightcurveC-B.
do not depend on the mass distribution of the microlenses Simulatedlightcurvesforeachimageareextractedusingtracks
4 T.Anguitaetal.:Aquasarunderamicrolensingcaustic
Table1. APOlightcurvesforquasarimagesAandC.MagnitudesareshownrelativetothevalueofthelastdatapointofimageC
intheg’filter.
JulianDay A C
+2451000 g’ r’ g’ r’ FWHM
∆[mag] err[mag] ∆[mag] err[mag] ∆[mag] err[mag] ∆[mag] err[mag] [arcsec]
1315 -0.822 0.016 -0.675 0.012 -0.533 0.016 -0.381 0.013 1.6
1334 -0.826 0.056 -0.680 0.021 -0.554 0.056 -0.393 0.021 1.4
1373 -0.925 0.017 -0.729 0.013 -0.672 0.017 -0.445 0.013 1.2
1385 -0.971 0.017 -0.782 0.015 -0.635 0.017 -0.411 0.015 1.0
1400 -0.912 0.019 -0.757 0.014 -0.487 0.019 -0.298 0.014 1.1
1531 -1.148 0.024 -0.887 0.015 -0.013 0.026 0.067 0.015 1.1
1541 -1.158 0.015 -0.909 0.015 -0.037 0.016 0.016 0.018 1.1
1551 -1.137 0.018 -0.916 0.014 0.000 0.024 0.094 0.015 1.2
over microlensing patterns created independently for each im- tracksfor30differentsourcesizeswithbest-fittingvelocityand
age and convolvedwith a Gaussian profile with the same size. magnitudeoffset).
Byrepeatedlightcurveextractionwithvariationoftheparame-
ters,thebestfittotheseareobtainedfromthecomparisonwith
4.2.APOColorCurve
theOGLEC-Bdifferencelightcurve.Whatisactuallyfittedare
theparametersthatdefinethetrackfromwhichbothlightcurves Fig.2showsthattheOGLEV-bandandtheAPOg’-bandlight
areobtained.Thistrackisconstrainedtohaveidenticaldirection curves are very similar. Since the microlensing effect depends
andvelocityinboththepatternsCandB,butthestartingpoint strongly on the source size, we assume, for simplicity, that the
can be differentamong the differentpatterns. It is importantto OGLEV-bandcontinuumregionisofsimilarsizeasthatofthe
remarkthatthe directionis set to be the same in both patterns, APO’sg’-band.HerewedeterminetheratiooftheAPOr’-band
takingintoconsiderationthatthesheardirectionbetweenimages regionandthe g’-/OGLEV-bandregionsthatfollowsfromthe
CandDisapproximatelyperpendicular(Witt&Mao1994). colorcurveinFig.2.
FortheminimizationmethodweuseaLevenberg-Marquardt While theparametersthatdefinethetracksare fixedbythe
leastsquaresroutinein2.Thisroutineisaχ2-basedminimiza- procedureshownintheprevioussection,thetwoparametersthat
tionbaseduponMINPACK-1(More´etal.1980).Asweareus- influence the shape of the color curve in this step are the ratio
ing68percentvaluesforourerrors,theχ2 issimplycalculated betweenthesizesofthetworegionsandtheintrinsiccolorg’-r’
as: ofthequasar(c ).Thesearetheonlyparametersallowedtovary
0
duringthisstep. Asbefore,we usethe patternsconvolvedwith
χ2 =DOF yi− f(xi) 2 (3) Gaussianprofilesupto120pixelsFWHMforimageCtoobtain
σerr theg’-bandlightcurve.Toobtainther’-bandlightcurveweuse
Xi=0 i ! thefullsetofpatternsforimageC.Duringthefittingprocesswe
whereDOFisthenumberofdegreesoffreedom,yisthemea- interpolatethemagnitudevaluesoftheextractedlightcurvesin
suredvalue, xistheindependentvariablemodelandσerr isthe ordertoobtaincontinuousvaluesforthesourcesizeratio.After
one sigma error of the particular measurement. In the case in this second fitting procedure,the χ2 values of the track library
whichwefittheOGLEdataforthisevent,thenumberofdegrees areupdated:χ2 =χ2OGLE +χ2APO.
offreedomis75.y arethedifferencesbetweentheOGLElight
i
curvesforimagesCandB, σerr aretheuncertaintiesmeasured
i 5. Considerations
by the OGLE team, x are Julian days and f(x) are the differ-
i i
encebetweenofthetwo lightcurvesextractedfromthe source 5.1.Degeneracies
planemagnificationpatternsforimagesCandBconvolvedwith
an identical source profile. We also include a parameter which As source size and microlens mass are degenerate,we express
we call m that accounts for the magnitude offset between the the size values in Einstein radii or scaled by a stellar mass as
0
images. showninequation(2).Anotherimportantdegeneracyistheone
As the size of each one of the convolved patterns is very between the size of the source and the transversal velocity we
largecomparedtothetimescaleofthechosenevent(oftheor- fit.Ifwefixthepathvelocityofasourceandincreasethesizeof
derofseveralhundredtimes),andconsideringthefactthatitis it,itslightcurvewillgetbroader.Conversely,ifwefixasource
relatively easy to find good fits because of the huge parameter size anddecreasethevelocity,thelightcurvewillalsobecome
spacethatsuchlargepatternsgive,weneedtodetermineahigh broaderbecauseit takeslongertime forthe sourceto crossthe
number of tracks for each pattern. To obtain a statistical sam- magnificationregion.
ple,wesearchfor10,000tracksforeachconsideredsourcesize.
The first guesses foreach one of the tracksare distributed uni-
5.2.Velocityconsiderations
formly.However,duringthefittingprocessthestartingpointsof
the tracksare constrainedto stay within a square of 200pixels Thevelocityonemeasuresforamicrolensingeventiscomposed
sidelength around the random initial value in order to force a of velocities of the observer, the lenses and the source. The
sampling of the whole pattern. After this minimizationis done Earth’s motion relative to the microwave background is likely
we have a track library containing 3 105 fitted tracks (10,000 tobealmostparalleltothedirectionofthequasarQ2237+0305
×
(Witt&Mao 1994) so the velocity of the observer is negligi-
2 http://cow.physics.wisc.edu/ craigm/idl/ blecomparedtotheothervelocityvalues.Weassumethephys-
∼
T.Anguitaetal.:Aquasarunderamicrolensingcaustic 5
(Shakura&Sunyaev 1973). Kochaneketal. (2006) show that
theradiiR whereradiationwithwavelengthλisemittedscale
λ
asλ43.Thus:
4
f = Rλ1 = λ1 3 . (5)
R λ
λ2 2!
FortheobserveddatafromAPO,thetransmissioncurveforthe
SPIcam g’ and r’ filters have central wavelengths of 4700Å
∼
Pattern Image C Pattern Image B and 6400Å,respectively.Using these valuesand equation(5)
∼
weobtainapredictedthin-diskratiobetweenther’-andg’-band
sourcesizesof 1.5.
∼
6. StatisticalAnalysis
There are different ways to approach quasar microlensing
studies. It can be by analytical model fitting of the light
curves (e.g., Yonehara 2001), studying the structure func-
tion (e.g., Lewis&Irwin 1996), doing statistics of parame-
ter variationsovertime intervals(e.g.,Schmidt&Wambsganss
1998;Wambsganssetal.2000;Gil-Merinoetal.2005),obtain-
ing probability distributions with light curve derivatives (e.g.,
Wyitheetal.2000c)orBayesiananalysis(e.g.,Kochanek2004).
SimilartoKochanek(2004),weconstructaprobabilitydistribu-
tionforthequantitiesofinterest(V/g’-bandsourcesize,source
sizeratioandtransversalvelocity)fromourtracklibrary.
Each one of our tracks is a fit to the light and color curves
Fig.3. Exampletrackfor a 0.05 ER (10 pixel)FWHM g’-band and therefore has a χ2 value attached to it. We therefore infer
source size and a source size ratio σr/σg = 1.48. In the top information from the whole ensemble of models: the track li-
′ ′
panels the fitted track is shown in a 1.0ER 1.0ER section of brary.Usingthe standardapproachforensembleanalysis(e.g.,
×
themicrolensingpatterns.Inthemiddlepanelweplotthefitted Sambridge1999),weassignalikelihoodestimatortoeachtrack
differencelightcurve(solidline)andtheobservedOGLEdiffer- t: p(t) exp(χ2(ti)).Usingthislikelihoodestimatorthestatisti-
ence light curve.In the lower panel the best-fitting color curve i i ∝ 2
calweightforeachtrackcanbewrittenas:
(solidline)andtheAPOg’-r’colorcurveareshown.
p(t)
w(t)= i (6)
i
n(t)
ical velocities of the lens and the source of the same order of i
magnitude, thus, the main contribution to the total velocity is wherew(t), p(t)andn(t)are,respectively,theweight,thelike-
i i i
the velocity of the lens due to the fact that the source veloc- lihoodobtainedfromtheχ2 andthedensityparameterforeach
ity is weighted by 3.5 percent compared to the lens velocity trackt.Thedensityparameterdescribesthelocaltrackdensity
∼ i
(Kayseretal.1986). inamultidimensionalgridinwhicheachdimensioncorresponds
The velocity of the lens has two components: one of toaparameterofinterest.Thedensityisnormalizedsothatthe
the bulk velocity of the galaxy vb and the velocity disper- sumofallthedensityparametersinthegridequalsone.Bysum-
sion of the stars in the galaxy (microlenses) vµ. Since the ming over the statistical weights, we can calculate probability
time span of the event that we fit is 200 days, the sig- distributionhistogramsforalltheparametersofinterest.
∼
nificance of individual stellar motions in the fitting should Gil-Merinoetal. (2005) (from now on G-M05) found an
be fairly low. Moreover, Kundic&Wambsganss (1993) and upper limit of 625 km (90 percent confidence) on the effective
Wambsganss&Kundic (1995) show that, in a statistical sense, transversevelocityosfthelensinggalaxyinQ2237+0305.They
theoveralleffectoftheindividualstellarmotionscanbeapprox- performedmicrolensingsimulationsassumingmicrolenseswith
imatedbyanartificiallyincreasedvelocity(afactor 1.3)ofthe M = 0.1M forthreedifferentsourcesizesyieldingsimilarre-
≈
sourceacrossthepattern. sults. We u⊙se their probability distribution for the velocity ob-
The scale we use forfittingthe velocityis pixelsperJulian tainedwiththelargestsourcesizevalue(Fig.5inG-M05)ina
day.Usingequation(2)wecanseethatthisturnsinto: furtheranalysisstep,wherewefactoritintoourownprobability
distributionsbyimportancesampling.Inotherwords,we scale
1 pix = 0.005 ER =32870 M km (4) theprobabilityoffindingatrackbasedontheG-M05prior,and
jd jd s0.1M s thenreobtainthebest-fittingvalueforeachparameterandconfi-
⊙ denceregions.
in the source plane. Using the appropriate velocity scaling
(Kayseretal.1986),wefind:1 pix 3000 km inthelensplane.
jd ∼ s 7. ResultsandDiscussion
As described in the previous section, we obtain limits on the
5.3.SizeConsiderations
sourcesizeandtransversevelocityfromtwosetsofprobability
In a standard thin accretion disk model of a quasar the black distributions. The first corresponds to the parameters obtained
hole is surrounded by a thermally radiating accretion disk without using any prior, and a second one in which we apply
6 T.Anguitaetal.:Aquasarunderamicrolensingcaustic
Table 3. Resultsandconfidencelimitsforg’-bandsourcesize, ratiobetweenr’-andg’-bandsourcesize andtransversevelocity
(projectedtothelensplane).
NoPrior WithG-M05Prior
SourceSizeσ Ratio Velocity(*) SourceSizeσ Ratio Velocity
g′ g′
×10−2RE 1q0105.1M[Mc⊙m] σσgr′′ q[km0.1M/Ms]⊙ ×10−2RE 1q0105.1M[Mc⊙m] σσgr′′ q[km0.1M/Ms]⊙
× ×
8.10+5.90 4.60+3.36 1.25+0.45 2930 2.34+0.43 1.33+0.24 1.45+0.90 682+227
5.99 3.40 0.15 1.28 0.73 0.25 379
− − − − − − −
(*)ThevelocityobtainedwithouttheG-M05priorisshownwithouterrorbarsaswedonotobtainlimitsforit.
the G-M05 prior on the velocity of the lens galaxy. For each Table 4. Probabilitesof the imageC highmagnificationevent
probability distribution of a particular parameter we select the being produced by the source crossing a caustic and of the g’
68 percent confidence levels. This is done by making horizon- and r’-bandregion touchinga caustic. The table shows the ob-
talcutsstartingfromthehighestprobabilityandscreeningdown tained values for the probability both using and not using the
until 68 percent of the cumulative probability is reached. The Gil-Merinoetal.(2005)velocityprior.
distribution,thebest-fittingvalues(ortheaverageofthe68per-
cent confidence region where no single best-fitting solution is NoPrior WithG-M05Prior
found)andthe68percentconfidencelimitsfortherelevantpa- cross g’ r’ cross g’ r’
rameters are shown in Fig. 4 and Table 3. After applying the 0.75 0.92 0.97 0.72 0.89 0.94
G-M05prioronly 1000trackscarry99percentofthestatisti-
∼
calweight.Thismakestheprobabilityhistogramforthesource
ratiosparselypopulatedforσ /σ > 2.Therefore,werequire
r′ g′ Yonehara 2001; Shalyapinetal. 2002; Kochanek 2004). Using
atleast70ofthese1000trackstobepartofeachbininthehis-
different methods it was found that this event is likely to have
togram.
been produced by the source passing near a cusp. We investi-
Without any velocity prior (see Fig. 4), we have more fast
gatethisforthetracksinourlibrarybycomputingthelocation
tracks than slow tracks. No limit on the transversal velocity
of the caustics for each of the magnificationpatterns using the
can be determined. This mostly is due to the degeneracy be-
analytical method by Witt (1990, 1991) (combination of both
tween source size and transverse velocity (see Section 5.1).
ray shooting simulations and analytical caustics are shown in
Using the G-M05 prior, we obtain a best-fitting velocity of
Wambsganssetal.1992).
682+227 M km/s(projectedintothelensplane).Note,how- For each track we determine whether either the center, the
379 0.1M
ever−,thaqtthed⊙istributionshowsalongertailtowardsthehigher center±σg′ or the center±σr′ of the source touchesa caustic at
anytime.Ouranalysisagreeswiththepreviousresultswhenwe
velocitieswhencomparedtotheG-M05probabilitydistribution
use the OGLE light curve to fit for this event. However, to re-
because the steepness of the event fitted prefers fast tracks or
produce our APO color dataset, tracks that cross a caustic are
smallsourcesizes.
favored (see Table 4). We find that a source with a radius as
TheaverageGaussianwidthweobtainontheOGLEV/APO
big as σ touched a caustic with 97 percent and 94 percent
g’ source size without setting the velocity prior is σg′ = of confidre′nce without and with the G-M05 prior, respectively.
8.10+55..9909 × 10−2RE or 4.60+33..3460 ×1015 0.1MM cm. These limits Furthermore, the source, regardless of dimensions, crossed a
agree− with those obtained −by Yoneharaq(200⊙1) and Kochanek caustic with a 75 percent and 72 percent of confidence with-
(2004) (who used a logarithmic prior on the transversal ve- outandwiththeG-M05prior,respectively(i.e.thecenterofthe
locity). When we impose the velocity constraints of G-M05 sourcetouchedacaustic).
we obtain an OGLE V/APO g’ source size with a Gaussian Theprobabilitiesofthesourcetouchingacausticatdifferent
width of σ = 2.34+0.43 10 2R which is equivalent to radiiare biggerin the case in which we do notuse the G-M05
g′ −1.28 × − E prior as seen in detail in Table 4. This is due to the fact that a
1.33+0.24 1015 M cm.Thismakestheupperlimittighterby
0.73× 0.1M higher velocity makes a track cover more distance through the
a fac−tor of fiveqcompa⊙red to the previous result, placing it just patternandthusismorelikelytoencounteracaustic.
below the 68 percent confidence limit obtained by Kochanek
(2004), very close to their resolution limit. For m we obtain
0
8. Conclusions
1.75 . m . 0.25 without and 0.75 . m . 1.25 with the
0 0
− −
G-M05prior,respectively(seeSection4.1). We present two band (g’ and r’) APO data covering the high
The source size ratio between the r’- and g’-band emit- magnification event seen in image C of the quadruple quasar
ting regions of the quasar is σr′/σg′ = 1.25+00..4155 without, and Q2237+0305 in the year 1999. We find that the amplitude
σ /σ = 1.45+0.90withtheG-M05prior,resp−ectively,coupled of the brightness peak is more pronounced in the g’-band
r′ g′ 0.25
with an intrins−ic color parameter: c = 0.18 0.05 in both than in the r’-band. This is also consistent with the observa-
0
− ±
cases (see Section 4.2). As shown in Fig. 4, both source size tions by Vakuliketal. (2004) (see also the recent analysis by
ratio distributions are similar. The measurements are close to Koptelovaetal.2007).
the theoretical value (f 1.5) estimated in Section 5.3 from the ByusingthisdatatogetherwiththewellknownOGLEdata
∼
Shakura&Sunyaev(1973)thermalprofile. (Woz´niaketal.2000;Udalskietal.2006)andcombiningitwith
For a long time there has been discussion on the nature microlensingsimulations,wehavebeenabletoobtainlimitson
of the image C event in year 1999 (e.g., Wyitheetal. 2000b; thesizeofregionsofthequasar’scentralengineemittinginthese
T.Anguitaetal.:Aquasarunderamicrolensingcaustic 7
σban/dσs:G=au1s.s2i5a+n0w.45i.dthσg′ =4.60+−33..3460×1015√M/0.1M⊙cmand SThroatlty,aCp.inM,V.&.NW.,eGbostiecro,eRch.eLa.,2L0.0J2.,,AMlNcaRldAeS,,D3.,3e4t,a6l2.12002,ApJ,579,127
r′ g′ 0.15 Udalski,A.,Szymanski,M.K.,Kubiak,M.,etal.2006,ActaAstronomica,56,
Because of−the degeneracy between source size and trans-
293
verse velocity we use a prior on the velocity obtained from Vakulik,V.G.,Dudinov,V.N.,Zheleznyak,A.P.,etal.1997,Astronomische
the work of Gil-Merinoetal. (2005) in order to improve the Nachrichten,318,73
results: Gaussian width σ = 1.33+0.24 1015√M/0.1M Vakulik,V.G.,Schild,R.E.,Dudinov,V.N.,etal.2004,A&A,420,447
cm and σ /σ = 1.45+0g.′90. Both v−a0l.u73es×for the ratio be⊙- Wambsganss,J.1999,JournalofComputationalandAppliedMathematics,109,
r′ g′ 0.25 353
tween the source sizes a−re close to the ratio obtained for Wambsganss,J.&Kundic,T.1995,ApJ,450,19
a face-on Shakura&Sunyaev (1973) accretion disk (f 1.5). Wambsganss,J.&Paczynski,B.1991,AJ,102,864
Recentstudies(e.g.,Poindexteretal.2007;Morganetal.∼2007; Wambsganss,J.&Paczynski,B.1994,AJ,108,1156
Wambsganss,J.,Paczynski,B.,&Schneider,P.1990,ApJ,358,L33
Pooleyetal.2007)suggestmicrolensingyieldsaslightlybigger
Wambsganss,J.,Schmidt,R.W.,Colley,W.,Kundic´,T.,&Turner,E.L.2000,
value for the ratio than that obtained analytically with the thin A&A,362,L37
diskmodel(seeSection5.2),alsoinagreementwithourresults. Wambsganss,J.,Witt,H.J.,&Schneider,P.1992,A&A,258,591
We alsoshowthatthiseventwasprobablyproducedbythe Witt,H.J.1990,A&A,236,311
Witt,H.-J.1991,PhDThesis,Universita¨tHamburg
sourcedirectlyinteractingwithacausticasweobtainprobabili-
Witt,H.J.&Mao,S.1994,ApJ,429,66
tiesof97percentand94percentthatther’-bandemittingregion
Woz´niak,P.R.,Udalski,A.,Szyman´ski,M.,etal.2000,ApJ,540,L65
touchesacausticwithoutandwiththeG-M05prior,respectively, Wyithe,J.S.B.,Webster,R.L.,&Turner,E.L.2000a,MNRAS,318,762
andof75percentand72percentthatthesourcecenter(regard- Wyithe,J.S.B.,Webster,R.L.,&Turner,E.L.2000b,MNRAS,318,1120
lessofsize)crossedacausticwithoutandwiththeG-M05prior, Wyithe,J.S.B.,Webster,R.L.,&Turner,E.L.2000c,MNRAS,312,843
Wyithe,J.S.B.,Webster,R.L.,&Turner,E.L.2000d,MNRAS,315,337
respectively.
Wyithe,J.S.B.,Webster,R.L.,Turner,E.L.,&Agol,E.2000e,MNRAS,318,
1105
Acknowledgements. TAwouldliketoacknowledgesupportfromtheEuropean Wyithe,J.S.B.,Webster,R.L.,Turner,E.L.,&Mortlock,D.J.2000f,MNRAS,
Community’s Sixth Framework Marie Curie Research Training Network 315,62
Programme, Contract No. MRTN-CT-2004-505183 “ANGLES” and the Yonehara,A.2001,ApJ,548,L127
InternationalMaxPlanckResearchSchoolforAstronomyandCosmicPhysics
attheUniversityofHeidelberg.
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8 T.Anguitaetal.:Aquasarunderamicrolensingcaustic
Fig.4.Probabilityhistogramsforg’ orV sourcesize, r’/g’sourcesize ratio andtransversevelocity(projectedto the lensplane),
respectively.Thedottedline showsthe best-fittingvalues(or averageof the 68percentconfidenceregionif no single best-fitting
valueisfound).Theerrorbarsdescribethe68percentconfidencelimits.