Microlensing of the X-ray, UV and optical emission regions of quasars: Simulations of the time-scales and amplitude variations of microlensing events

0.96 MB·English

by  P. Jovanović

Mon.Not.R.Astron.Soc.000,1–11(2007) Printed3February2008 (MNLATEXstylefilev2.2)
Microlensing of the X-ray, UV and optical emission regions
of quasars: Simulations of the time-scales and amplitude
variations of microlensing events
8
0 P. Jovanovi´c1, A. F. Zakharov2,3,4, L. Cˇ. Popovi´c1, T. Petrovi´c1
0 1Astronomical Observatory,Volgina 7, 11160 Belgrade, Serbia
2 2Institute of Theoretical and Experimental Physics, 25, B.Cheremushkinskaya, 25, Moscow, 117259, Russia
n 3Bogoliubov Laboratory for Theoretical Physics, JINR, 141980 Dubna, Russia
a 4Centerof Advanced Mathematics and Physics, National Universityof Science and Technology, Rawalpindi, Pakistan
J

9
2 Accepted 2007Received2007;inoriginalform2007

]
h
ABSTRACT
p
We consider the influence of microlensing on different spectral bands of lensed QSOs.
-
o We assumedthat the emitting X-ray,UV and opticalregionsare differentin size, but
r that the continuum emission in these spectral bands is originating from an accretion
t
s disc.Estimationsofthetime scalesformicrolensingandfluxamplificationindifferent
a bands are given. We found that the microlensing duration should be shorter in the
[
X-ray(severalmonths)thaninUV/opticalemittingregion(severalyears).Thisresult

1 indicates that monitoring of the X-ray variations in lensed QSOs that show a ’flux
v anomaly’ can clarify the source of this anomaly.
3
7 Key words: accretion, accretion discs – gravitational lensing – galaxies: active –
4 ultraviolet: galaxies – X-rays: galaxies
4
.
1
0
8 1 INTRODUCTION J1131-1231. Inparticular,theyfounddiscrepanciesbetween
0
the X-ray and optical flux ratio anomalies. Such anomalies
: Recent observational and theoretical studies suggest that
v in the different spectral band flux ratios can be attributed
gravitational microlensing can induce variability not only
i to micro- or milli-lensing in the massive lensing halo. In
X in optical light, but also in the X-ray emission of lensed
the case of milli-lensing one can infer the nature of sub-
r QSOs (Chartas et al. 2002, 2004; Dai et al. 2003, 2004; structure in the lensing galaxy, which can be connected to
a Jovanovi´c2006;Popovi´c et al.2001,2003a,b,2006a,b).Vari-
ColdDarkMatter(CDM)structures(seee.g.Dobler&Kee-
ability studies of QSOs indicate that the size of the X-
ton 2006). Besides microlensing, there are several mecha-
ray emitting region is significantly smaller (∼ several light
nismswhich can producefluxanomalies, such asextinction
hours),thantheopticalandUVemittingregions(∼several
and intrinsic variability. These anomalies were discussed in
light days).
Popovi´c & Chartas(2005)wheretheauthorsgaveamethod
Gravitational lensing is achromatic (the deflection an- that can aid in distinguishing between variations produced
gleofalight raydoesnotdependonitswavelength),butit by microlensing from ones resulting from other effects. In
isclear thatifthegeometries of theemittingregions atdif- thispaperwediscussconsequencesofvariationsduetograv-
ferentwavelengthsaredifferentthenchromaticeffectscould itationalmicrolensing,inwhichthedifferentgeometriesand
occur. For example, if the microlens is a binary star or if dimensions of emitting regions of different spectral bands
themicrolensedsourceisextended(Griest & Hu1992,1993; are considered.
Bogdanov and Cherepashchuk 1995a,b; Zakharov 1997;
The influence of microlensing on QSO spectra emit-
Zakharov & Sazhin 1998; Popovi´c & Chartas 2005) differ-
ted from their accretion discs in the range from the X-ray
entamplificationsindifferentspectralbandscanbepresent.
to the optical spectral band is analyzed. Moreover, assum-
Studies aiming to determine the influence of microlensing
ing different sizes of the emitting regions, we investigate
on the spectra of lensed QSOs need to take into account
the microlensing time scales for those regions, as well as a
the complex structure of the QSO central emitting region
time dependent response in amplification of different spec-
(Popovi´c & Chartas 2005). Since the sizes of the emitting
tral bands due to microlensing. Also, we give the estimates
regions are wavelength dependent, microlensing by stars in
of microlensing timescales for a sample of lensed QSOs.
the lens galaxy may lead to a wavelength dependent mag-
nification. For example, Blackburne et al. (2006) reported InSection2,wedescribeourmodelofthequasaremit-
such a ’flux anomaly’ in quadruply imaged quasar 1RXS ting regions and a model of the micro-lens. In Section 3 we
2 P. Jovanovi´c, A.F. Zakharov, L.Cˇ. Popovi´c, T. Petrovi´c
discuss the time scales of microlensing and in Section 4 we point of the accretion disc image on the celestial sphere as
present our results. Finally, in Section 5, we draw our con- seen byan observer at infinity,E is theobserved energy,
obs
clusions. I is an emissivity function.
P
InthestandardShakura-Sunyaevdiscmodel(1973),ac-
cretion occurs via an optically thick and geometrically thin
2 A MODEL OF QSO EMITTING REGIONS disc.Theeffectiveopticaldepthinthediscisveryhighand
AND MICROLENS photonsareclosetothethermalequilibriumwithelectrons.
Thesurfacetemperatureisafunctionofdiscparametersand
2.1 A model of the QSO emitting regions
resultsinthemulticolorblack bodyspectrum.Thiscompo-
In our models we adopt a disc geometry for the emit- nentisthoughttoexplain the’bluebump’in AGNand the
ting regions of Active Galactic Nuclei (AGN) since the soft X-ray emission in galactic black holes. Although the
most widely accepted paradigm for AGN includes a super- standard Shakura-Sunyaevdisc model does not predict the
massive black hole fed by an accretion disc. Fabian et al. power-lawX-rayemissionobservedinallsub-Eddingtonac-
(1989)calculated spectrallineprofilesforradiation emitted creting black holes, the power law for the X-ray emissivity
from the inner parts of accretion discs and later on such in AGN is usually adopted (see e.g. Nandra et al. (1999)).
features of Fe Kα lines were discovered by Tanaka et al. Butonecannotexcludeotherapproximationsforemissivity
(1995) in Japanese ASCA satellite data for Seyfert galaxy laws, such as black-body or modified black-body emissivity
MGC-6-30-15. Moreover, the assumption of a disc geome- laws. Moreover, we will assume that the emissivity law is
try for the distribution of the emitters in the central part the same through the whole disc. Therefore we used the
is supported by the spectral shape of the Fe Kα line in black-body radiation law. The disc emissivity is given as
AGN (e.g. Nandra et al. (1997), see also results of simula- (e.g. Jaroszynski (1992)):
tions (Zakharov & Repin 1999, 2002; Zakharov et al. 2003;
Zakharov & Repin 2003a,b,c, 2004, 2005, 2006; Zakharov
I (X,Y;E)=B[E,T (X,Y)],
P s
2004, 2007)). On the other hand, very often a bump in the
UV band is present in the spectra of AGN, that indicates where
that the UV and optical continuum originates in an accre- 2E3 1
tion disk. B(E,Ts(X,Y))= h2c2eE/kTs(X,Y)−1, (2)
We should note here that probably most of the X-ray
where c is the speed of light, h is the Planck constant,
emission in the 1–10 keV energy range originates from in-
k is the Boltzmann constant and T (X,Y) is the sur-
verse Compton scattering of photons from the disc by elec- s
face temperature. Concerning the standard accretion disc
trons in a tenuous hot corona. Proposed geometries of the
(Shakura& Sunayev 1973), here we assumed that (see
hot corona of AGN include a spherical corona sandwiching
Popovi´c et al. (2006a)):
the disc and a patchy corona made of a few compact re-
gions covering a small fraction of the disc (see e.g. Malzac T (X,Y)∼r−3/2(X,Y)(1−r−1/2(X,Y))4/5K, (3)
s
(2007)). On the other hand, it is known that part of the
accretion disc that emits in the 1–10 keV rest-frame band taking that an effective temperature in the innermost part
(e.g. the region that emits the continuum Compton reflec- (where X-ray is originated) is in an interval from 107 K to
tion component and the fluorescent emission lines) is very 108 K.
compactandmaycontributetoX-rayvariabilityinthisen-
ergy range. In order to study the microlensing time scales,
2.2 A model for microlensing
one should consider the dimensions of the X-ray emitting
region which are very important for the integral flux vari- Toexplaintheobservedmicrolensing eventsinquasars,one
ations due to microlensing. The geometry of the emitting can use different microlensing models. The simplest ap-
regions adopted in microlensing models will affect the sim- proximation is a point-like microlens, where microlensing
ulated spectra of the continuum and spectral line emission is caused by some compact isolated object (e.g. by a star).
(Popovi´c & Chartas2005;Jovanovi´c2006).Inthispaperwe SuchamicrolensischaracterizedbyitsEinsteinRingRadius
assumethattheAGNemissionfromtheopticaltotheX-ray 4GmDD
in the lens plane: ERR = l ls or by the corre-
band originates from different parts of the accretion disc. r c2 D
s
Also we assume that we have a stratification in the D
sponding projection in the source plane: R = sERR =
disc, where the innermost part emits the X-ray radiation E D
l
and outer parts the UV and optical continuum emission. 4GmD D
s ls, where G is the gravitational constant, c is
To study the effects of microlensing on a compact accre- r c2 D
l
tion disc we use the ray-tracing method, considering only the speed of light, m is the microlens mass and Dl, Ds
thosephotontrajectoriesthatreachtheskyplaneatagiven and Dls are the cosmological angular distances between
observer angle (see, e.g. Popovi´c et al. 2003a,b and refer- observer-lens, observer-source and lens-source, respectively.
ences therein). The amplified brightness with amplification Inmost cases we can not simply consider that microlensing
A(X,Y) for thecontinuum is given by is caused by an isolated compact object but we must take
into account that the micro-deflector is located in an ex-
tended object (typically, the lens galaxy). Therefore, when
I (X,Y;E )=I (E ,T(X,Y))·A(X,Y), (1)
C obs P obs the size of the Einstein ring radius projection R of the
E
whereT(X,Y)isthetemperature,X andY aretheimpact microlens is larger than the size of the accretion disc and
parameters which describe the apparent position of each when a number of microlenses form a caustic net, one can
Microlensing of the X-ray, UV and optical emission regions of quasars 3
Figure1.MagnificationmapoftheQSO2237+0305Aimage(left)andofa”typical”lenssystem(right).Thewhitesolidlinesrepresent
threeanalyzedpaths ofanaccretiondisccenter:horizontal (y=0),diagonal(y=−x)andvertical(x=0).
describethemicrolensingintermsofthecrossingofthedisc ofcausticcrossingperpendiculartotherotatingaxisκ=+1
by a straight fold caustic (Schneider et al. 1992). The am- fordirectionofcausticmotionfrom−Y to+Y,otherwiseit
plification at a point of an extended source (accretion disc) is −1. A microlensing event where acaustic crosses overan
close to the caustic is given by Chang & Refsdal (1984) (a emissionregioncanbedescribedinthefollowingway:before
moregeneralexpressionforamagnificationnearacusptype thecaustic reaches theemission region, theamplification is
singularity is given by Schneider& Weiss (1992); Zakharov equaltoA becausetheHeavisiedfunctionofequation(4)is
0
(1995)): zero. Just as thecaustic begins tocross theemittingregion
the amplification rises rapidly and then decays gradually
r
A(X,Y)=A0+Krκ(cξa−ustξicc) ·H(κ(ξ−ξc)), (4) towaMrdosrAeo0vears,tfhoretshoeurscpeecmifiocveesveanwtaoynferocmantmheodcealutshtiec-cfaoulds-.
where A is the amplification outside the caustic and K = tic shape to obtain different parameters (see e.g. Abajas
0
A β is the caustic amplification factor, where β is constant et al. 2005, Kochanek 2004 for the case of Q2237+0305).
0
of order of unity (e.g. Witt et al. 1993). The ”caustic size” In order to apply an appropriate microlens model, ad-
r is the distance in a direction perpendicular to the ditionally we will consider a standard microlensing mag-
caustic
causticforwhichthecausticamplificationis1,andtherefore nification pattern for the Q2237+0305A image (Fig. 1
thisparameterdefinesatypicallinearscaleforthecausticin left). For generating this map we used the ray-shooting
the direction perpendicular to thecaustic. ξ is the distance method (Kayseret al. 1986; Schneider& Weiss 1986, 1987;
perpendiculartothecausticingravitational radiiunitsand Wambsganss et al. 1990a,b). In this method the input pa-
ξ istheminimumdistancefromthedisccentertothecaus- rameters are the average surface mass density σ, shear γ
c
tic. Thus, andwidth of themicrolensing magnification map expressed
in units of the Einstein ring radius (defined for one solar
ξc = Xc2+Yc2, (5) mass in thelens plane).
p
First, wegenerate arandom star fieldin thelensplane
withuseoftheparameterσ.Afterthat,wesolvethePoisson
tgα= Yc, (6) equation∇2ψ=2σ inthelensplanenumerically,sowecan
X
c determine the lens potential ψ in every point of the grid in
and the lens plane. To solve the Poisson equation numerically
(X−X )tgφ+Y −Y one hasto write its finitedifference form:
ξ=ξ + c c , (7)
c
tg2φ+1
where φ=α+π/2. H(κ(ξ−pξ )) is the Heaviside function, ψi+1,j+ψi−1,j+ψi,j+1+ψi,j−1−4ψi,j =2σi,j. (8)
c
H(κ(ξ −ξ )) = 1, for κ(ξ −ξ ) > 0, otherwise it is 0. κ
c c
is ±1, it depends on the direction of caustic motion; if the Here we used the standard 5-point formula for the two-
direction of the caustic motion is from approaching side of dimensionalLaplacian.Nextstepisinversionoftheequation
thediscκ=−1,otherwiseitis+1.Also,inthespecialcase (8) usingFouriertransform. Aftersome transformations we
4 P. Jovanovi´c, A.F. Zakharov, L.Cˇ. Popovi´c, T. Petrovi´c
obtain: (2004)), r = 2Gm is the Schwarzschild radius correspond-
s c2
σˆ
ψˆ= mn , (9) ing to microlens mass m, h75 = H0/((75 km/sec)/Mpc) is
2(cosmπ +cos nπ −2) dimensionless Hubbleconstant.
N1 N2
The corresponding angular scale is
where N and N are dimensions of the grid in the lens
1 2
(Treyer& Wambsganss 2004)
plane. Now, using the finite difference technique, we can
compute the deflection angle α~ = ∇ψ in each point of the R m
θ = E ≈2.2·10−6 h−0.5 arcsec, (11)
grid in thelens plane.After computingdeflection angle, we 0 Ds rM⊙ 75
canmaptheregulargridofpointsinthelensplane,vialens
equation, onto the source plane. These light rays are then Using the length scale (10) and velocity scale (say
collected in pixels in the source plane, and the number of v⊥ ∼600km/secasTreyer& Wambsganss(2004)did),one
raysinonepixelisproportionaltothemagnificationdueto couldcalculatethestandardtimescalecorrespondingtothe
microlensing at this point in thesource plane. scale tocross the projected Einstein radius
Typically, for calculations of microlensing in gravita-
R m
tionally lensed systems one can consider cases where di- tE =(1+zl)v⊥E ≈25rM⊙v6−010h−750.5 years, (12)
mensionless surface mass density σ is some fraction of 1
e.g. 0.2, 0.4, 0.6, 0.8 (Treyer& Wambsganss 2004), cases wherearelativetransversevelocityv600 =v⊥/(600km/sec).
without external shear (γ = 0) and cases with γ = σ for ThetimescaletE,correspondingtoapoint-masslensandto
isothermal sphere model for the lensing galaxy. In this ar- asmallsource(comparedtotheprojectedEinsteinradiusof
ticle we assume the values of these parameters within the thelens),couldbeused ifmicrolenses aredistributedfreely
range generally adopted by other authors, in particular by at cosmological distances and if each Einstein angle is lo-
Treyer& Wambsganss (2004). catedfarenoughfromanotherone.However,theestimation
Microlensing magnification pattern for the (12) gives long and most likely overestimated time scales
Q2237+0305A image (Fig. 1 left) with 16 R on a especially for gravitationally lensed systems. Thus we must
E
side(whereR ≈5867R )iscalculatedusingthefollowing apply another microlens model to estimate time scales.
E g
parameters: σ = 0.36 and γ = 0.4 (see Popovi´c et al. For a simple caustic model, such as one that considers
(2006a),Fig.2),themassofmicrolensistakentobe0.3M⊙ astraightfoldcaustic1,therearetwotimescalesdepending
and we assume a flat cosmological model with Ω=0.3 and either on the ”caustic size” (rcaustic) or the source radius
H0=75 km s−1Mpc−1. (Rsource).Inthecasewhensourceradiusislargeroratleast
We also calculated microlensing magnification pattern close to the ”caustic size” (Rsource & rcaustic), the relevant
fora”typical”lenssystem(Fig.1right),wheretheredshifts time scale is the ”crossing time” (Treyer& Wambsganss
ofmicrolensandsourceare:z =0.5andz =2.Inthiscase, 2004):
l s
themicrolens parameters are taken arbitrary: σ=0.45 and
R
γ16=R0.3×an16dRthe,swizheeorfeoRbtai≈ne3d10m7icRrol.ensingpatternisalso tcross = (1+zl)v⊥(Dsosu/rcDel)
E E E g D −1
≈ 0.69 R v−1 s h−0.5 years (13)
15 600„D « 75
l
3 TYPICAL TIME SCALES FOR ≈ 251 R15 v6−010 h−750.5 days,
MICROLENSING
where D and D correspond to z = 0.5 and z = 2, re-
l s l s
Typical scales for microlensing are discussed not only in spectively and R15 =Rsource/1015 cm. As a matter of fact,
books on gravitational lensing (Schneider,Ehlers & Falco thevelocity perpendicular to thestraight fold caustic char-
1992; Zakharov 1997; Petters, Levine& Wambsganss acterizes the time scale and it is equal to v⊥sinβ where
2001), but in recent papers also (see, for example, β is the angle between the caustic and the velocity v⊥ in
Treyer& Wambsganss (2004)). In this paper we dis- thelens plane, but in our rough estimates we can omit fac-
cuss microlenses located in gravitational macrolenses tor sinβ which is about unity. However, if the source ra-
(stars in lensing galaxies), since optical depth for mi- dius Rsource is much smaller than the ”caustic size” rcaustic
crolensing is then the highest (Wyithe& Turner 2002a,b; (Rsource ≪ rcaustic), one could use the ”caustic time”, i.e.
Zakharov, Popovi´c & Jovanovi´c 2004, 2005) in comparison the time when the source is located in the area near the
with other possible locations of gravitational microlenses, caustic:
as for example stars situated in galactic clusters and t = (1+z) rcaustic
extragalactic dark halos (Tadros, Warren & Hewett 1998; caustic l v⊥(Ds/Dl)
Totani 2003; Inoue& Chiba 2003). D −1
Assuming the concordance cosmological model with ≈ 0.69 r15 v6−010„Ds« h−750.5 years (14)
Ω = 1, Ω = 0.3 and Ω = 0.7 we recall that typ- l
icatoltlength smcaalteterfor microlensinΛg is (Treyer& Wambsganss ≈ 251 r15 v6−010 h−750.5 days,
2004):
R = 2r · DsDls ≈3.2·1016 m h−0.5 cm, (10) 1 Weusethefollowingapproximationfortheextramagnification
E r s Dl rM⊙ 75 near the caustic: µ = rrξca−usξtcic, ξ > ξc where ξ is the perpen-
where”typical”microlensandsourcesredshiftsareassumed dicular direction to the caustic fold (it is obtained from Eq. (4)
to be zl = 0.5,zs = 2. (similar to Treyer& Wambsganss assumingthatfactorK isaboutunity).
Microlensing of the X-ray, UV and optical emission regions of quasars 5
Time (in years) Time (in years)
-20 -15 -10 -5 0 5 10 15 20 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
25 25
Optical band Optical band
UV band UV band
X band X band
20 20
mplification 15 mplification 15
Flux a 10 Flux a 10
5 5
0 0
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
Distance from the center of accretion disc (in Rg) Distance from the center of accretion disc (in Rg)
Figure 2. The variations of normalized total continuum flux in Figure 4. The same as in Fig. 2. in the case when time scale
optical (3500–7000 ˚A), UV (1000–3500 ˚A) and X (1.24–12.4 ˚A corresponds to zl = 0.04, zs = 1.69 (i.e. to Q2237+0305 lens
i.e. 1–10 KeV) band due to microlensing by a caustic crossing system).
alongy=−xdirectioninthecaseofSchwarzschildmetric.Time
scale corresponds to ”typical” redshifts of microlens and source:
zl = 0.5 and zs = 2. The parameters of the caustic are: A0=1, ized total flux in the X-ray, UV and optical spectral bands
β=1, κ = +1 and its ”size” is 9000 Rg. Negative distances and (seeFigs.2-4)bymeasuringthetimefromthebeginningto
times correspond to the approaching side, and positive to the thepeak of themagnification event
recedingsideofaccretiondisc.Inthiscase,duetoκ=+1,caustic
(ii) bycalculatingthecaustictimes(t )accordingto
caustic
motion is from the receding towards the approaching side (i.e.
equation (14)
fromtherighttotheleft).Thesourcemassis108 M⊙.Theradii
(iii) usinglightcurves(seeFigs.5and6)producedwhen
of optical emitting region are: Rin = 100 Rg, Rout = 2000 Rg,
the source crosses over a magnification pattern. Rise times
forUV emitting region: Rin =100 Rg, Rout =1000 Rg and for
Xemittingregion:Rin=Rms,Rout=80Rg. of high magnification events (tHME) are then measured as
thetimeintervalsbetweenthebeginningandthemaximum
of the corresponding microlensing events (for more details,
Time (in years) see thenext section).
-20 -15 -10 -5 0 5 10 15 20
11
Optical band
10 UV band
X band
9
8
mplification 567 4In oRrdEerStUoLeTxpSloAreNdDiffeDreInStCcaUseSsSoIfOmNicrolensing and eval-
Flux a 4 uatetimescalesfordifferentspectralbands,firstwenumer-
3 ically simulate the crossing of a straight-fold caustic with
2
parameters A = 1, β = 1, κ = +1 and r = 9000 R
1 0 caustic g
0 overan accretion disc with an inclination angle 35◦, that is
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
Distance from the center of accretion disc (in Rg) stratified intothree parts:
(i) The innermost part that emits X-ray continuum
Figure 3.ThesameasinFig.2.butforrcaustic=2000Rg. (1.24 ˚A–12.4 ˚A or1–10 keV).Theinnerradius istakento
be R =R (where R is the radius of the marginally
inn ms ms
stable orbit: R = 6 R in the Schwarzschild metric) and
where r =r /1015 cm. ms g
15 caustic outer radius is R = 80 R (where R = GM/c2 is the
Therefore,t couldbeusedasalowerlimitfortypical out g g
cross gravitational radius for a black hole with mass M).
time scales in the case of a simple caustic microlens model.
(ii) An UV emitting part of thedisc (contribute to the
From equations (13) and (14) it is clear that one cannot
emission from 1000 ˚A – 3500˚A), with R = 100 R and
unambiguously infer thesource size R from variability inn g
source R =1000 R .
measurementsalone,withoutmakingsomefurtherassump- out g
(iii) An outer optical part of the disc with R =100
tions.Ingeneral,however,weexpectthatt corresponds inn
cross
R and R =2000 R that emits in the wavelength band
to smaller amplitude variations than t , since in the g out g
caustic from 3500 ˚A until7000 ˚A.
first case only a fraction of a source is significantly ampli-
Having in mind that the aims of this investigation are
fiedbyacaustic(duetoassumptionthatR &r ),
source caustic
to study the microlensing time scales for different emitting
while in the second case it is likely that the entire source
regionsandtimedependentresponseofamplificationindif-
could be strongly affected by caustic amplification (due to
ferent spectral bands, we considered microlensing magnifi-
assumption that R ≪r ).
source caustic
cation patterns only for image A of Q2237+0305 and for
Inthispaper,weestimatedthemicrolensingtimescales
a ”typical” lens system. Our intention was not to create a
fortheX-ray,UVandoptical emittingregionsoftheaccre-
completemicrolensing modelfor aspecificlenssystem, and
tion disc using thefollowing three methods:
thereforewedidnotanalyzethedifferencesbetweenimages
(i) by converting the distance scales of microlensing (as for instance the time delay between them). The vari-
eventstothecorrespondingtimescalesaccordingtothefor- ations in the total flux in the different spectral bands are
mula (13) in which R is replaced by thedistance from given in Figs. 2–6. In Figs. 2 and 3 the simulations for a
source
the center of accretion disc. Caustic rise times (t ) are typical lens system with z = 0.5 and z = 2 and for two
HME l s
then derived from the simulated variations of the normal- different”causticsizes”aregiven.AsonecanseefromFigs.
6 P. Jovanovi´c, A.F. Zakharov, L.Cˇ. Popovi´c, T. Petrovi´c
Table 1.Theestimated timescales (inyears)formicrolensingof the X-ray,UV andoptical emissionregionforlensedQSOsobserved
by Chandra X-ray Observatory (Daietal. 2004). The calculated caustic times tcaustic are obtained according to formula (14) for the
following values of the cosmological constants: H0 = 75 km s−1Mpc−1 and Ω0 = 0.3. The caustic rise times tHME are derived from
causticcrossingsimulations(seeFigs2–4).Theblackholemassisassumedtobe108M⊙.
Object zs zl X-ray UV optical
t t t t t t
caustic HME caustic HME caustic HME
HS0818+1227 3.115 0.39 0.572 0.660 7.147 7.070 14.293 15.160
RXJ0911.4+0551 2.800 0.77 0.976 1.120 12.200 12.080 24.399 25.880
LBQS1009-0252 2.740 0.88 1.077 1.240 13.468 13.330 26.935 28.570
HE1104-1805 2.303 0.73 0.918 1.050 11.479 11.370 22.957 24.350
PG1115+080 1.720 0.31 0.451 0.520 5.634 5.570 11.269 11.950
HE2149-2745 2.033 0.50 0.675 0.780 8.436 8.350 16.871 17.890
Q2237+0305 1.695 0.04 0.066 0.080 0.828 0.820 1.655 1.760
Table2.Averagerisetimes(<tHME >)ofhighmagnificationevents,theiraveragenumber(<Ncaustic>0)perunitlength(RE)and
theiraveragenumber(<Ncaustic>y)peryearinthelightcurvesofQ2237+0305A (Fig.5)and”typical”lenssystem(Fig.6).
Q2237+0305 ”Typical”lens
Discpath Sp.band <tHME > <Ncaustic>0 <Ncaustic>y <tHME > <Ncaustic>0 <Ncaustic>y
X 0.37 0.95 0.20 3.94 1.63 0.06
y=0 UV 1.29 0.59 0.12 10.96 0.78 0.03
Optical 2.96 0.51 0.10 15.81 0.62 0.02
X 0.57 0.52 0.11 2.77 1.65 0.06
y=−x UV 1.92 0.36 0.07 6.62 0.49 0.02
Optical 3.98 0.26 0.05 17.04 0.38 0.01
X 0.68 1.61 0.33 4.52 1.40 0.05
x=0 UV 1.38 0.88 0.18 13.25 0.54 0.02
Optical 2.96 0.44 0.09 31.62 0.31 0.01
2 and 3, the microlensing time scales are different for dif- event beyond the maximum of the amplification could not
ferent regions, and the durations of variations in the X-ray beaccuratelydeterminedbecauseoftheasymptoticdecrease
are on order of several months to a few years, but in the ofthemagnificationcurvebeyondthepeak.Estimatedtime
UV/optical emission regions they are on order of several scales for different spectral bands are given in Table 1. As
years. Also, as one can see in Figs. 2 and 3, the time scales one can see from Table 1, the microlensing time scales are
do not depend on ”caustic size” which, on the other hand, significantly smaller for the X-ray than for the UV/optical
affectsonly themaximal amplifications in all threespectral bands.
bands.TheresultscorrespondingtothelenssystemofQSO
Unamplifiedand amplified brightnessprofilesof theX-
2237+0305 (z = 0.04, z = 1.69) are given in Figs. 4 and
l s ray emitting region, corresponding to the highest peak in
5.Weconsideredthestraight-foldcaustic(Fig.4)andami-
Fig. 5 (top right) are presented in Fig. 7. As one can see in
crolensingpatternfortheimageAofQSO2237+0305 (Fig.
Fig. 7, the assumed brightness profile of the source is very
5).AsonecanseefromFigs.4and5,ahigheramplification
complex due to applied ray tracing method, which allows
intheX-raycontinuumthanintheUV/opticalisexpected.
us to obtain an image of the entire disc, not only its one
Inthiscase,thecorrespondingtimescalesaremuchshorter
dimensional profile. Therefore, we could not use a simple
and they arefrom afew daysupto a few monthsfor X-ray
sourceprofileforthecalculation ofmicrolensing timescales
and a few years for UV/optical spectral bands. The similar
(asitwasdonebyWitt et al.(1995)).Instead,weestimated
conclusion arises when we compare the latter results with
the frequency of high amplification events (HMEs), i.e. the
thosefora”typical”lenssysteminthecaseofmicrolensing
number of such events per unit length, directly from the
pattern (Fig. 6).
lightcurvespresentedinFigs5and6.Formodelswithnon-
We also estimated time scales for seven lensed QSOs zeroshear,thisfrequencydependsonthedirectionofmotion
which have been observed in the X-ray band (Dai et al. andforbothcalculated maps(Q2237+0305A and”typical”
2004).Foreachspectralbandtwoestimatesaremade:caus- lens) we counted the number of high magnification events
tic time (t ) - obtained from equation (14) and caustic along the following paths (see Fig. 1): i) horizontal (y =0)
caustic
rise time (t ) - obtained from caustic crossing simula- in the direction from −x to x, ii) diagonal (y = −x) in
HME
tions (see Figs. 2 –4). Inthesecond case, we measured the direction from −x to x and iii) vertical (x = 0) in direc-
time from the beginning of the microlensing event until it tion from y to −y. For each map the lengths of the hori-
reaches its maximum (i.e. in thedirection from theright to zontal and vertical paths in the source plane are the same
the left in Figs. 2 – 4). The duration of the magnification (13.636 R for Q2237+0305A and 12.875 R for ”typical”
E E
Microlensing of the X-ray, UV and optical emission regions of quasars 7
Time (in years)
Time (in years)
-40 -30 -20 -10 0 10 20 30 40
-30 -20 -10 0 10 20 30 4.5
4 X
X UV
UV 4 OPT
3.5 OPT
3.5
3
3
n
plification 2 .25 mplificatio 2.5
m a 2
Flux a 1.5 Flux 1.5
1 1
0.5 0.5
0 0
-40000 -20000 0 20000 40000 -60000 -40000 -20000 0 20000 40000 60000
Distance from the center of accretion disc (in Rg) Distance from the center of accretion disc (in Rg)
Time (in years)
-30 -20 -10 0 10 20 30
4
X
UV
OPT
3.5
3
n
o
ati
fic
pli 2.5
m
a
ux
Fl 2
1.5
1
-40000 -20000 0 20000 40000
Distance from the center of accretion disc (in Rg)
Figure 5. Variations inthe X-ray(solid), UV (dashed) and optical (dotted) spectral bands corresponding to the horizontal (top left),
diagonal(topright)andvertical(bottom) pathinthemagnificationmapoftheQSO2237+0305A (Fig.1left).
lens),aswellasarethecorrespondingcrossing times(66.22 7 in UV and 5 in optical band and iii) vertical path: 22 in
yearsforQ2237+0305Aand337.32yearsfor”typical”lens). X-ray,12 in UV and 6 in optical band. In case of ”typical”
ForQ2237+0305Athelengthofthediagonalpathinsource lens these numbers are: i) horizontal path: 21 in X-ray, 10
planeis19.284R andthecrossingtimeis93.62years,while in UVand8 inoptical band,ii) diagonal path:30in X-ray,
E
thecorresponding length in the”typical”case is 18.208 R 9 in UV and 7 in optical band and iii) vertical path: 18 in
E
and the crossing time is 477.04 years. X-ray,7 in UV and 4 in optical band.
HMEs are asymmetric peaks in the light curves which
depend not only on microlens parameters but also on the Theaveragenumberofcausticcrossingsperunitlength
sizes of emitting regions in the following sense: the larger (< N > ) and per year (< N > ) are given
caustic 0 caustic y
emitting regions are expected to produce smoother light in Table 2. This table also contains the average rise times
curves and more symmetric peaks (Witt et al. 1995). Con- (< t >) in all three spectral bands, derived from the
HME
sequently, it can be expected that the majority of HMEs rise times (t ) of individual HMEs which are presented
HME
shouldbedetectedinX-raylightcurves,lessoftheminUV in Figs. 8 and 9 in the form of histograms. These results,
and the smallest number in optical light curves. Therefore, as expected, show that the rise times are the shortest and
weisolatedonlyclearlyasymmetricpeaksinalllightcurves the frequency of caustic crossings is the highest in the X-
andmeasuredtheirrisetimest astheintervalsbetween ray spectral band in comparison to the other two spectral
HME
the beginning and the maximum of the corresponding mi- bands. One can also see from Tables 1 and 2 that in the
crolensingevents.InthecaseofQ2237+0305Awefoundthe case of Q2237+0305A, the average rise times of HMEs for
followingnumberofHMEs:i)horizontalpath:13inX-ray,8 allthreespectralbands(obtainedfrommicrolensmagnifica-
in UVand7 in optical band,ii) diagonal path:10 in X-ray, tion pattern simulations) are longer than both, caustic rise
8 P. Jovanovi´c, A.F. Zakharov, L.Cˇ. Popovi´c, T. Petrovi´c
Time (in years)
Time (in years)
-200 -100 0 100 200
-150 -100 -50 0 50 100 150 4
4 X
X UV
UV OPT
OPT 3.5
3.5
3
3
n
n o
plificatio 2.5 mplificati 2.5
Flux am 2 Flux a 2
1.5 1.5
1 1
0.5 0.5
-20000 -10000 0 10000 20000 -30000 -20000 -10000 0 10000 20000 30000
Distance from the center of accretion disc (in Rg) Distance from the center of accretion disc (in Rg)
Time (in years)
-150 -100 -50 0 50 100 150
3.5
X
UV
OPT
3
2.5
n
o
ati
fic
pli 2
m
a
ux
Fl 1.5
1
0.5
-20000 -10000 0 10000 20000
Distance from the center of accretion disc (in Rg)
Figure 6.ThesameasinFig.5butforthemagnificationmapofa”typical”lenssystem(Fig.1right).
0.025 0.025
Intensity (in arbitrary units)00..0000..01005512 Intensity (in arbitrary units)00..0000..01005512
0 0
−50 50 −50 50
−40 40 −40 40
−30 30 −30 30
−20 20 −20 20
−10 10 −10 10
0 0 0 0
10 −10 10 −10
20 −20 20 −20
30 −30 30 −30
40 −40 40 −40
50 −50 Y (in Rg) 50 −50 Y (in Rg)
X (in Rg) X (in Rg)
Figure 7.Unamplified(left)andamplified(right)brightnessprofileoftheX-rayemittingregion,correspondingtothehighestpeakin
Fig.5(topright).Theprofilesareobtainedusingtheraytracingmethod(see,e.g.,Popovi´cetal.2003a,bandreferencestherein).
Microlensing of the X-ray, UV and optical emission regions of quasars 9
6 6
s) s)
ear 4 ear 4
y y
n n
(iE2 (iE2
M M
H H
t t
0 0
−20 −20
−10 Opt. −10 0 Opt.
0 10 UV 10 20 UV
20 X 30 X
Time (in years) 30 Time (in years)
s) 4
ar
e
y
n
(iE2
M
H
t
0
−30
−20
−10 Opt.
0
10 UV
20 X
Time (in years) 30
Figure8.Risetimes(tHME)ofhighmagnificationeventsforallthreespectralbandsinthelightcurvesofQ2237+0305A(Fig.5).Top
leftpanel correspondstothehorizontal,toprighttothediagonal andbottom totheverticalpathofaccretiondisc.
30
40
s) s)
ear 20 ear 30
y y
(in HME10 (in HME1200
t t
0 0
−150 −200
−100 −50 Opt. −150 −100 −50 Opt.
0 UV 0 50 UV
50 100
Time (in years) 100 X Time (in years) 150 200 X
ars) 40
e
y
n
(iE20
M
H
t
0
−150
−100
−50 Opt.
0
50 UV
100 X
Time (in years)
Figure 9.ThesameasinFig8.butforthelightcurvesof”typical”lenssystem(Fig.6)
10 P. Jovanovi´c, A.F. Zakharov, L.Cˇ. Popovi´c, T. Petrovi´c
times(obtainedfrom causticsimulations) andcaustictimes Dai, X., Chartas, G., Agol, E., Bautz, M. W. & Garmire,
(calculated from equation (14)). G.P. 2003, ApJ, 589, 100
Microlensing can result in flux anomalies in the sense Dai, X., Chartas, G., Eracleous, M., Garmire, G. P. 2004,
thatdifferentimagefluxratiosareobservedindifferentspec- ApJ,605, 45
tralbands(Popovi´c & Chartas2005;Popovi´c et al.2006b). Dobler, G. and Keeton, C.R. 2006, MNRAS,365, 1243
As shown in Figs 2–6. the amplification in the X-ray band Fabian,A.C.,Rees,M.J.,Stella,L.&White,N.E.1989,
is larger and lasts shorter than it does in the UV and opti- MNRAS,238, 729
cal bands. Consequently, monitoring of lensed QSOs in the Griest, K. & Hu,W. 1992, ApJ,397, 362
X-ray and UV/optical bands can clarify whether the flux Griest, K. & Hu,W. 1993, ApJ,407, 440
anomalyisproducedbyCDMclouds,massiveblackholesor InoueK.T. & Chiba M. 2003, ApJ, 591, L83
globularclusters(millilensing)orstarsinforegroundgalaxy Jovanovi´c, P. 2006, PASP,118, 656
(microlensing). Jaroszyn´ski, M., Wambsganss, J.W., Paczyn´ski, B. 1992,
ApJ,396, L65.
Kayser, R., Refsdal, S.& Stabell, R.1986, A&A,166, 36
Malzac, J. 2007, Mem. S.A.It. 78, 382
5 CONCLUSION
Nandra, K., George, I.M., Mushotzky, R.F., Turner, T.J.
In this paper we calculated microlensing time scales of dif- & Yaqoob,T. 1997, ApJ, 477, 602.
ferent emitting regions. Using a model of an accretion disc Nandra, K., George, I. M., Mushotzky, R. F., Turner, T.
(in the center of lensed QSOs) that emits in the X-ray and J., Yaqoob, T. 1999, ApJ523, 17
UV/optical spectral bands, we calculated the variations in Oshima, T., Mitsuda, K., Fujimoto R., Iyomoto N., Futa-
thecontinuumfluxcausedbyastraight-foldcausticcrossing moto K., et al. 2001, ApJ,563, L103
an accretion disc. We also simulated crossings of accretion Petters,A.O.Levine,H.&Wambsganss, J.2001, Singular
discs over microlensing magnification patterns for the case Theory and Gravitational Lensing, (Boston, Birkh¨auser)
of image Aof Q2237+0305 and for a”typical” lens system. Popovi´c, L. Cˇ., Chartas, G. 2005, MNRAS,357, 135
From thesesimulations we concluded thefollowing: Popovi´c, L.Cˇ., Jovanovi´c, P., Mediavilla, E.G., Zakharov,
(i)onecanexpectthattheX-rayradiationismoream- A.F.,Abajas,C.,Mun˜oz,J.A.&Chartas,G.2006a,ApJ,
plifiedthanUV/opticalradiationduetomicrolensingwhich 637, 620
can inducetheso called ’flux anomaly’ of lensed QSOs. Popovi´c,L.Cˇ.,Jovanovi´c,P.,Petrovi´c,T.,Shalyapin,V.N.
(ii) the typical microlensing time scales for the X- 2006b, AN,10, 981
ray band are on order of several months, while for the Popovi´c, L.Cˇ., Mediavilla, E.G., Jovanovi´c, P. & Mun˜oz,
UV/opticaltheyareonorderofseveralyears(althoughthe J.A. 2003, A&A,398, 975
time scales obtained from microlensing magnification pat- Popovi´c, L.Cˇ., Jovanovi´c, P., Mediavilla, E.G. & Mun˜oz,
ternsimulationsarelongerincomparison tothoseobtained J.A. 2003b, Astron. Astrophys.Transactions, 22, 719
from caustic simulations). Popovi´c, L., Cˇ., Mediavilla, E.G., Mun˜oz J., Dimitrijevi´c,
(iii) monitoring of the X-ray emission of lensed QSOs M.S. & Jovanovi´c, P. 2001, SerAJ, 164, 73
can reveal the nature of ’flux anomaly’ observed in some Shakura,N.I. & Sunayev,R.A.1973, A&A,24, 337
lensed QSOs. Schneider, P., Ehlers, J., Falco, E.E. 1992, Gravitational
Allresultsobtainedinthisworkindicatethatmonitor- Lenses, (Springer, Berlin)
ing the X-ray emission of lensed QSOs is useful not only Schneider,P. & Weiss, A. 1986, A&A,164, 237
to discuss the nature of the ’flux anomaly’, but also can be Schneider,P. & Weiss, A. 1987, A&A,171, 49
used for constraining thesize of theemitting region. Schneider,P. & Weiss, A. 1992, A&A,260, 1
Totani, T. 2003, ApJ,586, 735
Tadros, H., Warren S. & Hewett, P. 1998, New Rev., 42,
115
ACKNOWLEDGMENTS
Tanaka,Y.,Nandra,K.,Fabian,A.C.,Inoue,H.,Otani,C.,
This work is a part of the project (146002) ”Astrophysical Dotani,T.,Hayashida,K.,Iwasawa,K.,Kii,T.,Kunieda,
Spectroscopy of Extragalactic Objects” supported by the H.,Makino, F., Matsuoka, M. 1995, Nat, 375, 659
Ministry of Science of Serbia. The authors would like to Treyer, M. & Wambsganss, J. 2004, A&A,416, 19
thank theanonymousreferee for veryuseful comments. Wambsganss,J.,Paczyn´ski,B.&Katz,N.1990a,ApJ,352,
407
Wambsganss,J.,Schneider,P.&Paczyn´ski,B.1990b,ApJ,
358, L33
REFERENCES
Witt, H.J., Kayser, R.,Refsdal, S. 1993, A&A 268, 501
Bogdanov, M.B. and Cherepashchuk,A.M.1995a, Astron. Witt, H.J., Mao, S., Schechter,P.1995, ApJ, 443, 18
Lett., 21, 505 Wyithe, J.S.B. & Turner,E.L. 2002a, ApJ, 567, 18
Bogdanov,M.B. andCherepashchuk,A.M.1995b,Astron. Wyithe, J.S.B. & Turner,E.L. 2002b, ApJ, 575, 650
Rep.,39, 779 Zakharov, A.F. 1994, MNRAS,269, 283
Chang, K., Refsdal, S.1984, A&A,132, 168 Zakharov, A.F. 1995, A&A,293, 1
Chartas, G., Agol, E., Eracleous, M., Garmire, G., Bautz, Zakharov, A.F. 1997, Gravitational lenses and microlens-
M. W., Morgan, N. D. 2002, ApJ, 568, 509 ing, (Janus-K,Moscow)
Chartas,G.,Eracleous,M.,Agol,E.,Gallagher,S.C.2004, Zakharov,A.F. 2004, in Proc. of the22nd SummerSchool
ApJ,606, 78 andInternationalSymposiumon”ThePhysicsofIonized