A STUDY OF THE CORRELATION BETWEEN THE AMPLIFICATION OF THE Fe K� LINEAND THE X-RAY CONTINUUM OF QUASARS DUE TO MICROLENSING

L. Cˇ. Popovic,

1,2P. Jovanovic,

1,2E. Mediavilla,

3A. F. Zakharov,

4,5,6C. Abajas,

3J. A. Munoz,

3,7and G. Chartas

8

Received 2005 March 4; accepted 2005 October 4

ABSTRACT

The observed enhancement of the Fe K� line in three gravitationally lensed QSOs (MG J0414+0534, QSO2237+0305, and H1413+117) is interpreted in terms of microlensing, even when equivalent X-ray continuumamplification is not observed. In order to interpret these observations, first we studied the effects of microlensing onquasar spectra produced by a straight fold caustic crossing over a standard relativistic accretion disk. The diskemission was analyzed using the ray-tracing method, considering Schwarzschild and Kerr metrics. When theemission is separated into two regions (an inner disk corresponding to the Fe K� line and an outer annuluscorresponding to the continuum, or vice versa), we find microlensing events that enhance the Fe K� line withoutnoticeable amplification of the X-ray continuum, but only during a limited time interval. Continuum amplification isexpected if a complete microlensing event is monitored. Second, we studied a more realistic case of amplification bya caustic magnification pattern. In this case we could satisfactorily explain the observations if the Fe K� line isemitted from the innermost part of the accretion disk while the continuum is emitted from a larger region. We alsostudied the chromatic effects of microlensing, finding that the radial distribution of temperature in the accretion disk,combined with microlensing itself, can induce wavelength-dependent variability of �30% for microlenses with verysmall masses. All these results show that X-ray monitoring of gravitational lenses is a method well suited forstudying the innermost structure of active galactic nucleus accretion disks.

Subject headinggs: gravitational lensing — quasars: individual (H1413+117, MG J0414+0534, QSO 2237+0305) —X-rays: galaxies

1. INTRODUCTION

Recent observational and theoretical studies suggest thatgravitational microlensing can induce variability in the X-rayemission of lensed QSOs. Microlensing of the Fe K� line hasbeen reported in at least three macrolensed QSOs: MG J0414+0534 (Chartas et al. 2002), QSO 2237+0305 (Dai et al. 2003),and H1413+117 (Oshima et al. 2001b; Popovic et al. 2003b;Chartas et al. 2004).

The influence of microlensing in the X-ray emission has beenalso theoretically investigated.Mineshige et al. (2001) simulatedthe variation of the X-ray continuum due to microlensing,showing that the flux magnifications for the X-ray and opticalcontinuum emission regions are not significantly different dur-ing the microlensing event, while Yonehara et al. (1998, 1999)and Takahashi et al. (2001) found that simulated spectral var-iations caused by microlensing show different behavior, de-pending on photon energy. Also, microlensed light curves forthin accretion disks around Schwarzschild and Kerr black holeswere considered in Jaroszynski et al. (1992), and microlensinglight curves for the Fe K�were simulated by Jaroszynski (2002).On the other hand, the influence of microlensing in the Fe K�

spectral line shape was discussed in Popovic et al. (2001b,2003a, 2003b) and Chartas et al. (2002).9 Popovic et al. (2003a,2003b) showed that objects in a foreground galaxy with evenrelatively small masses can produce observable changes in theFe K� line flux much stronger than those expected for the UVand optical lines (Popovic et al. 2001a; Abajas et al. 2002; Lewis& Ibata 2004). In the optical spectra, microlensing inducedmagnification of broad UV lines (e.g., C iv and S iv/O iv) wasreported by Richards et al. (2004). Consequently, one can expectthat microlensing of the Fe K� line region will be more frequent.Observations of the X-ray continuum and the Fe K� line inmulti-imaged active galactic nuclei (AGNs) open new possi-bilities for the study of the unresolved X-ray–emitting structurein QSOs, particularly for high-redshift QSOs (Zakharov et al.2004; Dai et al. 2004).However, an explanation for the different behavior of the line

and continuum variability in the observed events should be givenin context of the microlensing hypothesis. Chartas et al. (2002)detected an increase of the Fe K� equivalent width in image B ofthe lensed QSO J0414+0534 that was not followed by the con-tinuum. Chartas et al. (2002) explained the nonenhancement ofthe continuum emission in the spectrum of image B by proposingthat the thermal emission region of the disk and the Compton up-scattered emission region of the hard X-ray source lie withinsmaller radii than the iron line reprocessing region. Analyzing theX-ray variability of QSO 2237+0305A, Dai et al. (2003) alsomeasured amplification of the FeK� line in component A ofQSO

1 Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia.2 Isaac Newton Institute of Chile, Yugoslavia Branch, and Universidad

Diego Portales, Chile.3 Instituto de Astrofisica de Canarias, 382005 La Laguna, Tenerife, Spain.4 Institute of Theoretical and Experimental Physics, 25, B. Cheremushkinskaya

st., Moscow, 117259, Russia.5 Astro Space Centre of Lebedev Physics Institute, Moscow, Russia.6 Isaac Newton Institute of Chile, Moscow Branch.7 Departamento de Astronomıa y Astrof ısica, Universidad de Valencia,

E-46100 Burjassot, Valencia, Spain.8 Astronomy and Astrophysics Department, Pennsylvania State University,

University Park, PA 16802.

9 Simulations of X-ray line profiles are presented in a number of papers;see, for example, Fabian (2001) and Zakharov & Repin (2002a, 2002b, 2002c)and references therein. In particular Zakharov et al. (2003) showed that infor-mation about the magnetic fieldmay be extracted fromX-ray line-shape analysis.Zakharov & Repin (2003a, 2003b) discussed signatures of X-ray line-shapes forhighly inclined accretion disks.

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The Astrophysical Journal, 637:620–630, 2006 February 1

# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

2237+0305, but not in the continuum. However, in this case theinterpretation was different. Dai et al. (2003) suggested that thelarger size of the continuum emission region (�1014 cm � 100Rg

for MBH ¼ 107 M�) with respect to the Fe K� emission region(�10Rg ) could explain this result. Finally, inH 1413+117, Chartaset al. (2004) found that the continuum and the Fe K� line wereenhanced by a different factor.

With the aim of discussing these results, we model here thebehavior of the X-ray continuum and the Fe K� line during amicrolensing event for different sizes of the continuum and theFe K� line emission regions.

2. MICROLENSING OF A COMPACT ACCRETION DISK

2.1. The Model

The assumption of a disk geometry for the distribution of theFe K� emitters is supported by the spectral shape of this line inAGNs (e.g., Nandra et al. (1997), where they have investigatedthe iron line properties of 18 Seyfert 1 galaxies). Regarding theX-ray continuum emission, it seems that it mainly arises from anaccretion disk. For instance, Fabian & Vaughan (2003) haveshown that the X-ray spectral variability of MCG�6�30�15can be modeled by a two-component model in which onecomponent varies and is a power law and the other component isconstant and is produced by very strong reflection from a rela-tivistic disk.

To study the effects of microlensing on a compact accretiondisk we use the ray-tracing method, considering only those pho-ton trajectories that reach the sky plane at a given observer’s an-gle �obs (see, e.g., Popovic et al. 2003b and references therein).The amplified brightness with amplification A(X, Y ) for thecontinuum is given by

IC(X; Y ;Eobs) ¼ IP Eobs; T (X; Y )½ �A(X; Y ); ð1Þ

and for the Fe K� line by

IL(X; Y ;Eobs) ¼ IP E0g (X; Y ); T (X; Y )½ �; � Eobs � E0g (X; Y )½ �A(X; Y ); ð2Þ

where T(X, Y ) is the temperature, X and Y are the impact pa-rameters that describe the apparent position of each point of theaccretion disk image on the celestial sphere as seen by an observerat infinity;E0 is the line transition energy (E

Fe K�0 ¼ 6:4 keV), and

g (X; Y ) ¼ Eobs/Eem is the energy shift due to relativistic effects(Eobs is the observed energy, and Eem is the emitted energy fromthe disk). Here we do not consider the cosmological redshift. Theemissivity of the disk is one of the important parameters thathas an influence on the continuum and line shapes. The observedcontinuum flux is very often fitted with one or two blackbodycomponents in the soft X-ray, in addition to a hard X-ray powerlaw (see, e.g., Page et al. 2004). The line shape, as well as thecontinuum distribution, strongly depends on emissivity law. Inthe standard Shakura-Sunyaev disk model (Shakura & Sunyaev1973), accretion occurs via an optically thick and geometricallythin disk. The effective optical depth in the disk is very high, andphotons are close to thermal equilibrium with electrons. Thesurface temperature is a function of disk parameters and results inthe multicolor blackbody spectrum. This component is thought toexplain the ‘‘blue bump’’ in AGNs and the soft X-ray emission ingalactic black holes. Although the standard model does not pre-dict the power-law X-ray emission observed in all sub-Eddingtonaccreting black holes, the power law for the X-ray emissivity in

AGNs is usually accepted (see, e.g., Nandra et al. 1999). But onecannot exclude other emissivity laws, such as blackbody or mod-ified blackbody emissivity laws. Therefore, here we use black-body,modified blackbody, and power emissivity laws for both theFe K� and continuum emission.

In the case of the blackbody radiation law, the disk emissivityis given as (e.g., Jaroszynski et al. 1992)

IP(X; Y ;E ) ¼ B½E; Ts(X; Y )�;

where

B E; Ts(X; Y )½ � ¼ 2E 3

h2c21

eE=kTs(X;Y ) � 1; ð3Þ

where c is the speed of light, h is the Planck constant, k is theBoltzmann constant, and Ts(X, Y ) is the surface temperature ofX-ray accretion disk.

In principle, one can assume different distributions of the sur-face temperature along the disk. To obtain the X-ray continuumdistribution using equation (3), one can assume that Ts ¼ const:,taking that the black hole is a powerful X-ray source with aneffective temperature of 107–108 K. But, regarding the standarddisk model, it is expected that at least the surface temperature isradially dependent. Therefore, here we accept the radial distribu-tion of surface temperature given by (Shakura & Sunyaev 1973):

Ts(X; Y ) � r�3=2(X; Y ) 1� r�1=2(X; Y )h i4=5

K; ð4Þ

taking that an effective temperature is in an interval from 107 to108 K. The distribution of the temperature along the radius of thedisk used in this paper is given in the top panel of Figure 1, andthe corresponding shape of spectral energy distribution is shownin the bottom panels. In equation (4), r is the dimensionless pa-rameter, defined as

r(X; Y ) ¼ R(X; Y )

6Rg¼ 1

6

R(X; Y )c2

GM¼ M�

M

R(X; Y )

9 km;

where R(X, Y ) is the disk radius, expressed in gravitational radii(Rg ).

However, in the innermost part of the accretion disk thePlanck function cannot be used properly. Therefore, we also usethe standard (classical) Shakura-Sunyaev approach, where theemissivity law is described by a ‘‘modified’’ blackbody radiationlaw (eqs. [3.4] and [3.8] in Shakura & Sunyaev 1973; see alsothe discussion in Novikov&Thorne 1973; Shapiro & Teukolsky1983; Straumann 1984):

IP(E;X; Y ) / x3 exp (�x); ð5Þ

where x ¼ E/kT (X; Y ). Shalyapin et al. (2002) used similarexpressions to study microlensing in the optical continuum.Taking into account that the observed hard X-ray continuum hasa power-law–type spectral shape, we also assume that the time-independent intrinsic emissivity of the continuum is

I(E; r) � E��r��;

where, according to the investigation of observed X-ray spectra,� and � are taken to be 1.5 and 2.5 (see, e.g., Dovciak et al.

MICROLENSING OF Fe K� LINE AND X-RAY CONTINUUM 621

2004). For the Fe K� emission in this case we used the samecalculation as in Popovic et al. (2003a, 2003b).

We should note here that the disk may be considered to becomposed of a number of distinct parts with different physi-cal conditions (a radiation-pressure-dominant part, a matter-pressure-dominant part, etc.; see, e.g., Shakura & Sunyaev 1973).Consequently, in general one can expect that different partsof the disk can be described by different emissivity laws (e.g.,the blackbody law may be applied in outer part of the disk).Taking into account the huge number of parameters thatshould be considered in the case of a microlensed disk (see thenext section), we consider only one emissivity law for wholedisk.

The total observed flux for the continuum and the Fe K� lineis given as

F(E ) ¼Zimage

½IC(X; Y ;E )þ IL(X; Y ;E )� d�; ð6Þ

where d� is the solid angle subtended by the disk in the observer’ssky and the integral extends over the whole emitting region.

As one can see from equation (6) the total observed flux is asum of the continuum and the line fluxes; consequently, the am-

plification in the continuum and in the Fe K� line can be consid-ered separately as two independent components.On the other hand,the amplifications will depend on the sizes and geometry of thecontinuum and line-emitting regions. Consequently, we consideramplifications in the line and in the continuum separately.We would like to point out here that the aim of the paper is not

to create a perfect accretion disk model (taking into accountdifferent effects that can be present, e.g., opacity of the disk andspots in the disk), but only to illustrate the influence of micro-lensing on the continuum and the Fe K� line amplification anddemonstrate that this phenomenon could essentially changeour general conclusions. Therefore, we use the three emissivitylaws of the disk and the very important effect of strong gravi-tation (beaming and light-bending in Schwarzschild and Kerrmetrics).

2.2. Disk and Microlens Parameters

To apply the model one needs to define a number of param-eters that describe the emission region and the deflector. In prin-ciple, we should find constraints for the the size of the diskemission region, the disk inclination angle, the mass of the blackhole, the accretion rate, the relative amplification, the constant� (Chartas et al. 2002; Popovic et al. 2003b), the orientationof the caustic with respect to the rotation axis, and the direction

Fig. 1.—Top: The distribution of the temperature as a function of the radius along the direction of the disk rotation, given for two different values of angularmomentum a. Negative values of R correspond to the approaching side and positive values to the receding side of the disk. Bottom: Shapes of the continuum for thethree emissivity laws considered (normalized to the maximal value) for an accretion disk with other radius of 20Rg (left) and 80Rg (right). The other parameters of thedisk are given in x 2.2.1.

POPOVIC ET AL.622 Vol. 637

of the caustic crossing and microlens mass. In the followingsubsections we choose and discuss the parameters used in thecalculations.

2.2.1. Accretion Disk Parameters

For the disk inclination we adopt the averaged values givenby Nandra et al. (1997) from a study of the Fe K� line profilesof 18 Seyfert 1 galaxies: i ¼ 35�. The inner radius, Rin , cannotbe smaller than the radius of the marginally stable orbit, Rms ,that corresponds to Rms ¼ 6Rg (gravitational radii Rg ¼ GM /c2,where G is gravitational constant,M is the mass of central blackhole, and c is the velocity of light) in the Schwarzschild metricand to Rms ¼ 1:23Rg in the case of the Kerr metric with angu-lar momentum parameter a ¼ 0:998. To select the outer radius,Rout , we take into account previous investigations of the X-rayvariability that support very compact X-ray–emitting disks. Inparticular, Oshima et al. (2001a) infer from the observed vari-ation in the lensed blazar PKS 1830�211 a size of the X-ray con-tinuum emission region of �3 ; 1014 cm, which is in agreementwith the estimation for QSO 2237+03050 given by Dai et al.(2003). So, considering a range of black hole masses of 107–109M�we can conclude that the X-ray emission is coming froma compact region of the order of 10Rg–100 Rg. This range ofsizes is also acceptable for the Fe K� emission region (see, e.g.,Nandra et al. 1997, 1999).

To explore the suitability of the various hypotheses explain-ing the lack of adequate response of the X-ray continuum to themicrolensing events detected in the Fe K� line (see x 1), weconsider several combinations of disk sizes for the emitters ofboth the continuum and the line: (1) the inner and outer radii ofboth emission regions are the same,Rin ¼ Rms andRout ¼ 20 Rg;(2) the inner radius is the same, Rin ¼ Rms, but the outer radius ofthe X-ray continuum disk is smaller,Rout ¼ 20Rg, than the radiusof the line emission disk, Rout ¼ 80Rg; (3) the continuum emis-sion disk has radii Rin ¼ Rms and Rout ¼ 20Rg, and the line emis-sion disk hasRin ¼ 20Rg andRout ¼ 80Rg (the continuum emissiontakes place in an inner part of disk surrounded by an annulusof Fe K� emission); (4) the continuum emission disk has radiiRin¼ 20Rg andRout ¼ 80Rg, and the line emission disk hasRin ¼Rms and Rout ¼ 20Rg (the Fe K� emission is located in the innerdisk and the continuum emission in the outer annulus).

We adopt the central object mass from Bian & Zhao (2002).We assume a black hole of mass M8 ¼ 108 M�. We use thisvalue in order to determine the effective temperature distribu-tion. This value is in agreement withWang et al. (2003), where itwas found that the majority of QSOs have black hole masses inthe range of 108–109 M� .

It is difficult to discuss the validity of different emissivitylaws for demonstrating the X-ray emission (in the line as well asin the continuum), but sometimes, as for example in the case of

Fig. 2.—Left:Microlensing map of QSO 2237+0305A image with 16 ERR (177,372Rg) on a side (Abajas et al. 2005). Right: A small part (square in the left panel )of the microlensing pattern, compared to a face-on accretion disk. The assumed outer radius of the disk is Rout ¼ 1000Rg.

TABLE 1

Projected ERR for Different Deflector Masses for the Three Lensed QSOs where Microlensing of the Fe K� Line Is Suspected

Object zs zl 1 ; 10�4 M� 1 ; 10�3 M� 1 ; 10�2 M� 1 ; 10�1 M� 1 M�

MG J0414+0534......................................... 2.64 0.96 20.3 64.2 203.1 642.3 2031.1

QSO 2237+0305......................................... 1.69 0.04 11.2 35.4 112.1 354.5 1121.0

QSO H1413+117 ........................................ 2.56 1.00 19.8 62.5 197.7 625.2 1977.0

Notes.—Expressed in gravitational radii. The three QSOs are J0414+0534 (Chartas et al. 2002), QSO H1413+117 (Oshima et al. 2001b; Chartas et al. 2004), andQSO 2237+0305 (Dai et al. 2003). The values used for the cosmological constants are H0 ¼ 50 km s�1 Mpc�1 and �0 ¼ 1. The black hole mass is assumed to be108 M�.

MICROLENSING OF Fe K� LINE AND X-RAY CONTINUUM 623No. 2, 2006

blackbody emissivity law, the emissivity at X-ray wavelengthscan be extremely small comparedwith, for example, opticalwave-lengths, and X-ray photons are emitted from a quite small re-gion. In Figure 1 (bottom), we present the continuum shapesfor different emissivity laws used in the calculation (maxi-mum of each is normalized to one). The shapes of the contin-uum were calculated for different dimensions of the disk. Asone can see from the figure (bottom), the shape of the contin-uum strongly depends not only on emissivity law, but also ondisk dimensions.

2.2.2. Microlens Model and Parameters

Different types of caustics can be used to explain the observedmicrolensing events in quasars. Moreover, for the exact eventone can model the caustic shape to obtain different parameters

(see, e.g., Abajas et al. 2005 and Kochanek 2004 for the case ofQ2237+0305). In order to apply an appropriate microlens model,first we consider a standard microlensing magnification pat-tern (Fig. 2, left) for the Q2237+0305A image with 16 Einsteinring radii (ERRs) on a side and � ¼ 0:36, k ¼ 0:40, and �c ¼ 0.The mass of microlens is taken to be 1 M�. The simulationwas made employing ray-shooting techniques that send raysfrom the observer through the lens to the source plane (Kayseret al. 1986; Schneider &Weiss 1987; Wambsganss et al 1990a,1990b).We assume a flat cosmological model with� ¼ 0:3 andH0 ¼ 70 km s�1 Mpc�1.In Figure 2 we present a comparison between the projected

magnification map in the source plane and an accretion disk witha size of 1000Rg (the circle in the right panel of Fig. 2). Takinginto account the small dimensions of the X-ray emission region

Fig. 3.—Simulations of the behavior of the Fe K� line and continuum variations due to microlensing by a caustic in the cases of Schwarzschild (left) and Kerr (right)metrics. The parameters of the caustic are A0 ¼ 1, � ¼ 1, and ERR ¼ 50Rg. In the first and second rows we present the caustic crossing perpendicular to the rotating axisfor � ¼ �1, respectively. In the third and fourth rows we show the caustic crossing along the rotation axis with � ¼ �1, respectively. The radii of the continuum and theFe K� line emission accretion disks are the same: Rin ¼ Rms and Rout ¼ 20Rg (where Rg ¼ GM /c2). The unperturbed and normalized emission correspond to solid anddashed lines, respectively. The relative intensity ranges from 0 to 7 (Y-axis), and the energy interval ranges from 0.1 to 10 keV (X-axis).

Fig. 4.—Same as in Fig. 3, but Rout ¼ 80Rg for the Fe K� line.

POPOVIC ET AL.624 Vol. 637

(several tens times Rg), the approximation of a straight foldcaustic can be assumed for this pattern. We explored the generalbehavior of the total continuum and Fe K� flux amplificationdue to microlensing. First we used the straight fold caustic ap-proximation. (see eqs. [5]–[8] in Popovic et al. 2003b). We alsoconsidered an example assuming a caustic magnification patternfor the Q2237+0305A image produced by a population of low-mass microlens in x 3.3.

In Table 1 we give the projected ERRs for the lensed QSOswhere amplification of the Fe K� line (MG J0414+0534

[Chartas et al. 2002], QSO H1413+117 [Oshima et al. 2001b;Chartas et al. 2004], and QSO 2237+0305 [Dai et al. 2003]) hasbeen observed. The ERRs (expressed in gravitational radii) arecomputed for different deflector masses and for a black holemass of 108 M�. We found that even deflectors with small masshave ERR with sizes from several tens to several hundredsof gravitational radii. To obtain a qualitative understanding ofthe influence of the microlens mass on our results we considermicrolens masses that correspond to ERR values equal to 50Rg

(see Figs. 3–7). Qualitatively the shape of the flux amplificationwill not be changed if we consider massive deflectors (e.g.,ERR ¼ 2000Rg ; see Fig. 8). Even if we apply simple assump-tion of the straight fold caustic, we should define the caustic param-eters A0 and �. These values can be considered to be differentfor different microlensing events. Higher values of A0 and � willcause higher amplification. Here we would like to demonstratecorrelation between the line and flux amplification due to mi-crolensing, and we adopt values considered in more detail byChartas et al. (2002), A0 ¼ 1 and � ¼ 1 (Witt et al. 1993). Herewe considered three directions of caustic crossing; parallel(Y ¼ 0) and perpendicular (X ¼ 0) to the rotation axis, as wellas in a direction inclined 45

�with respect to the rotation axis

(X ¼ Y ).

3. RESULTS AND DISCUSSION

3.1. Continuum and Line Profile Variability

In Figures 3 and 4 we present the variations of the total X-rayemission spectra (continuum + Fe K� line) during a straight foldcaustic crossing (A0 ¼ 1, � ¼ 1, and ERR ¼ 50Rg ). The radialdependence of the emissivity is related to the blackbody radia-tion law (see x 2). In Figure 3 the sizes of the continuum and lineemission regions are the same, Rin ¼ Rms and Rout ¼ 20Rg. InFigure 4 the Fe K� line emitting disk is larger, Rout ¼ 80Rg. We

Fig. 5.—Variations of the Fe K� line and X-ray continuum flux due to causticcrossing (ERR ¼ 50Rg) perpendicular to the rotation axis (X ¼ 0) for differentpositions of the caustic in the Y-axis (Schwarzschild metric). Panels correspondto (a) blackbody law and case 2 of x 2.2.1, (b) modified blackbody law and case 2,(c) blackbody law and case 3, and (d ) modified blackbody law and case 3. Theflux axis ranges from 1 to 1.7.

Fig. 6.—Variations of the Fe K� line and X-ray continuum flux during a caustic crossing (ERR ¼ 50Rg) for the case 4 of x 2.2.1. We consider three differentdirections of caustic crossing (X ¼ 0, Y ¼ 0, and X ¼ Y ; see x 2.2.2). The left and right panels correspond to the blackbody and modified blackbody laws, respectively.From top to bottom, the three first panels on both the left and the right correspond to Schwarzschild metric and the last three to Kerr metric. The flux axis ranges from1 to 1.7.

MICROLENSING OF Fe K� LINE AND X-RAY CONTINUUM 625No. 2, 2006

consider both metrics, Schwarzschild and Kerr. We simulate thecaustic crossing perpendicular to (first and second rows) andalong (third and fourth rows) the rotation axis in both directions;� ¼ �1, respectively.

3.2. Integrated Flux Variability

In Figures 5–8 we present the variation of the integrated flux(normalized to the integrated flux in the absence of microlens-ing) for both the X-ray continuum and the Fe K� line duringstraight fold caustic crossings (considering that the amplificationoutside the caustic A0 ¼ 1, as well as � ¼ 1 and ERR ¼ 50Rg;see figure legends). We considered all three emissivity laws (seex 2.1). Figures 5a and 5b correspond to case 2 of x 2.2.1, Figures5c, 5d, and 7 (left) to case 3, and Figures 6 and 7 (right) to case 4.(Case 1 is the same for the line and continuum; it corresponds tothe continuum variation in Figs 5a and 5b.) In Figure 7 wepresent cases 3 and 4, where a power-law emission is taken intoaccount. In Figure 8 a very favorable case of high ERR and in-clination is considered for case 3 of x 2.2.1. Notice that Fe K�microlensing events were observed in BAL QSOs, which mayhave highly inclined accretion disks; i � 75

�.

As one can see from Figure 8 (as was also shown in Popovicet al. 2003a, 2003b), the amplified component is mainly verynarrow in comparison with the undeformed line. This result is inagreement with the observations of Chartas et al. (2002, 2004)

and Dai et al. (2003) and supports the conclusions of these au-thors that enhancement of the Fe K� line observed in only oneof the images of a lensed quasar was caused by microlensing.Some interesting results can be inferred from our explora-

tory work. (1) When both the line and continuum disk have thesame inner radius, the differences in outer radius cannot causesignificant differences in the total line and the continuum fluxvariation (see Figs. 5a and 5b) for the considered emission laws.(2) When we separate the emission considering an inner diskcontributing to the continuum and an outer annulus contribut-ing to the Fe K� line (or vice versa) we find significant differ-ences between the continuum and the line amplification (seeFigs. 5c, 5d, and 6–8). (3) The results are qualitatively simi-lar for all considered emissivity laws (see Figs. 5–8). Finally,(4) interchanging between Schwarzschild and Kerr metrics in-duces only slight differences in normalized flux amplification(see Figs. 5–7).Figures 5–8 were intended to explore the two scenarios sug-

gested by Dai et al. (2003) and Chartas et al. (2002) to explainthe non observed associated enhancement of the X-ray contin-uum in objects with a microlensed Fe K� line. This behavior canbe expected in case 3 when the microlens crosses the outer partof the disk (Figs. 5c and 5d and the right panel of Fig. 7) and incase 4 when the microlens crosses the inner part of the disk (seeFig. 6 and the left panel of Fig. 7). However, in none of the

Fig. 7.—Same as in Fig. 5, but for the case of power-law radiation; the left panel corresponds to case 3 and the right to case 4 of x 2.2.1.

Fig. 8.—Variations of the X-ray continuum and the Fe K� line flux for a highly inclined disk (i ¼ 75�) due to microlensing by a caustic with ERR ¼ 2000Rg.Blackbody (left) and modified blackbody (right) are considered. The geometry corresponds to case 3 of x 2.2.1. The flux axis ranges from 1 to 4.

POPOVIC ET AL.626 Vol. 637

figures does the continuum remain strictly constant during acomplete Fe K�microlensing event. In the most favorable case(the inner Fe K� disk plus an outer continuum annulus; Figs. 6and 7), we achieve a significant and relatively quick change ofthe FeK� emissionwhile the continuum experiences only a slowincrease. This behavior could well approximate a nonvaryingcontinuum, but only if we consider observations in a temporalwindow that fall on the peak of the microlensing event in theFe K� line. In this case the continuum of the microlensed imageexperiences an (slowly changing or almost constant) amplifica-tion with respect to the continua of the other images, but prac-tically it is indistinguishable from the amplifications due to globalmacrolensing.

3.3. Microlensing by a Caustic Magnification Pattern:An Example for Q2237+0305A Image

Here we consider a situation where a low-mass population ofmicrolenses (smaller than one solar mass) can form pattern struc-tures (see Table 1) that are comparable with the size of the X-rayaccretion disk. Moreover, the black hole mass of the lensedquasar may be of the order of 109–1010 M�, taking that Rg �MBH, the pattern structure of low-mass microlenses are compa-rable with a X-ray disk size of several dozen Rg. Therefore, herewe consider that the black hole mass of the lensed quasar is109 M�.

For modeling of the caustic magnification pattern for imageQ2237+0305Awe used the same values for the convergence andexternal shear as presented in Figure 2, but for a low-mass pop-ulation, taking that the mass of the deflectors are randomly dis-tributed in an interval ranging from 0.1 to 0.6 M�, with a meanvalue of mh i ¼ 0:35 M�. Also, the positions of the lenses weredistributed randomly in a rectangular region in the lens plane,significantly larger than the considered region in the source plane.Now, 1 ERR projected in the source plane corresponds to

ERR(M ) ¼ ERR(M�)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM

M�

M8

MBH

s;

where ERR(M�) ¼ 0:054 pc is the projected ERR for a solarmass deflector, M is the mean mass of the deflectors, and MBH

is the blackhole mass. Taking the mean deflector mass as mh i ¼0:35 M� and Rg ¼ 9:547:10�6 MBH/M8 pc, we modeled a caus-

tic magnification pattern of 1 ERR ; 2 ERR, that corresponds to334:63Rg ; 669:26Rg in the source plane for a black hole mass ofMBH ¼ 109 M� (Fig. 9). For numerical reasons, the microlensmagnification map is given in pixels, 1000 ; 2000 (1 pixel ¼0:33463Rg in source plane). As one can see from Figure 9, themicrolensing pattern structures are comparable with a compactX-ray accretion disk.

In our previous modeling based on the straight fold causticapproximation the lack of a correlation between the continuumand Fe K� line is expected only if the line and X-ray continuumregion are separated. Recent investigations of the Fe K� lineprofile from active galaxies show that the line should be emittedfrom the innermost part of the accretion disk. In particular,Ballantyne & Fabian (2005) found that in BLRG 4C+74.26 theouter radius of the relativistic iron line should be within 10Rg.Consequently, here we assume that the Fe K� line is formed inthe innermost part of the disk (Rin ¼ Rms and Rout ¼ 20Rg) andthat the continuum (emitted in the energy range between 0.1 and10 keV) is mainly originated from a larger region (Rin ¼ 20Rg

and Rout ¼ 100Rg).10 On the other hand, from the straight fold

caustic modeling we conclude that the correlation between thetotal line and continuum flux due to microlensing is not very dif-ferent for different emissivity laws. Consequently, here we usedthe blackbody emissivity law.A disk (Schwarzschildmetric) withan inclination of 35

�is considered.

To explore the line and X-ray continuum variation we movedthe disk center along the microlensing map as shown in Figure 9(going from 0 to 2000 pixels, from left to right). In Figure 10 wepresent the corresponding total line and X-ray continuum fluxvariation. As one can see from Figure 10, there is a global cor-relation between the total line and continuum flux during thecomplete path. However, the total continuum flux variation issmooth and has a monotonic change, while the total line fluxvaries very strongly and randomly.

In fact, during some portion of the microlensing of theemission regions by the magnification pattern, we found the totalFe K� line flux changes, while the continuum flux remainsnearly constant (e.g., the position of the disk center between1000 and 1200 pixels). This and the shapes of the line andcontinuum total flux amplification indicate that the observedmicrolensing amplification of the Fe K� in three lensed quasarsmay be explained if the line is originated in the innermost part ofthe disk and the X-ray continuum in a larger region. Also, it

10 Note that here, taking the continuum disk size from Rin ¼ 20Rg toRout ¼ 100Rg, we neglected the contribution of the innermost part emission(from Rms to 20Rg) to the total continuum flux only in the energy interval from0.1 to 10 keV. It does not mean that there is no the continuum emission.

Fig. 9.—Microlensingmap ofQSO2237+0305A imagewith 1 ERR ; 2 ERR(1000 pixel ; 2000 pixel ¼ 334:63Rg ; 669:26Rg) on a side and scheme of theprojected diskwith outer radiusRout ¼ 20Rg and 100Rg for the Fe K� line and theX-ray continuum, respectively. The straight line presents the path of the centerof the disk (the left side of the pattern corresponds to 0 pixels).

Fig. 10.—Amplification of the Fe K� line and the X-ray continuum totalflux for different positions of the center on the microlensing map of QSO2237+0305A image (see Fig. 9).

MICROLENSING OF Fe K� LINE AND X-RAY CONTINUUM 627No. 2, 2006

seems that the contribution of the continuum emitted from theinnermost part of the disk (within 10Rg) to the total continuum(in the energy interval from 0.1 to 10 keV) flux is not significant.Further observations are needed to provide more data that mightbe compared with our theoretical results.

3.4. Wavelength Dependence of the X-RayContinuum Amplification

The influence of gravitational microlensing on the spectra oflensed QSOs was discussed in several papers (see Popovic &Chartas 2005 and references therein). Mainly, the color indexwas calculated as an indicator of the microlensing (see, e.g.,Wambsganss & Paczynski 1991; Wyithe et al. 2000), as wellas amplified flux behaviors (see Yonehara et al. 1998, 1999;Takahashi et al. 2001) of a disk, but the exact shape of the am-plification as a function of wavelength (or energy) for a partlymicrolensed disk has not been calculated. Here, taking into ac-count that the emitters at different radii in the accretion disk havedifferent temperatures (see Fig. 1) and make different contribu-tions to the observed continuum flux at a given wavelength, wecalculated the amplification as a function of observed energies.During a caustic crossing microlensing effects would depend on

the location of the emitters and, consequently, would induce awavelength dependence in the amplification. This dependencecan be clearly appreciated in the spectra of Figures 3, 4, and 8. InFigures 11 and 12 we present the amplification as a function ofthe observed energies of an accretion diskwith the characteristicsgiven in x 2.2.1, with inner radius Rin ¼ Rms and outer radiusRout ¼ 30Rg, assuming caustic crossing along the rotation axis(X ). We have considered the blackbody (Figs. 11a and 11b) andthe modified blackbody (Fig. 12) emissivity laws for both theSchwarzschild and Kerr metrics.As can be seen in Figures 11 and 12, the amplification is

different for different observed energies. The amplification ishigher for the hard X-ray continuumwhen the caustic crosses thecentral part of the disk (see Fig. 12). Depending mainly on thecaustic location and on the emissivity law selected, the differ-ence of the amplification in the energy range studied by us canbe significant (e.g., �20% for very small mass microlenses,ERR ¼ 50Rg; see Figs. 11 and 12). This effect could induce anoticeable wavelength-dependent variability of the X-ray con-tinuum spectrum during a microlensing event (as much as 30%),providing a tool to study the innermost regions of accretiondisks.

Fig. 11.—Microlensing amplification as a function of emitted energies (the chromatic effect of microlensing). The calculations were performed for causticcrossing along the rotation axis (Y ¼ 0) for the following positions on X-axis: (a) X ¼ 20Rg; (b) X ¼ 10Rg; (c) X ¼ 0Rg; (d ) X ¼ �10Rg; and (e) X ¼ �20Rg . Wetake ERR ¼ 50Rg. The radii of the disk are Rin ¼ Rms, Rout ¼ 30Rg. The blackbody emissivity law in both the Schwarzschild (left) and Kerr metrics (right) is used incalculation.

Fig. 12.—Same as in Fig. 11, but for the ‘‘modified’’ blackbody law in the Schwarzschild (left) and Kerr (right) metrics.

POPOVIC ET AL.628 Vol. 637

4. CONCLUSIONS

We have developed a model of microlensing by a straight foldcaustic of a standard accretion disk in order to discuss the ob-served enhancement of the Fe K� line in the absence of corre-sponding continuum amplification found in three lensed QSOs.Here we summarize several interesting results inferred from ourstraight fold caustic simulations.

1. As expected, both the Fe K� and the continuum may ex-perience significant amplification by a microlensing event (evenfor microlenses of very small mass). Thus, the absence of ade-quate continuum amplification in the observed Fe K� micro-lensed QSOs should be related to the structure of the accretiondisk and/or the geometry of the event.

2. Extending the outer radius of the distribution of Fe K�emitters does not result in any significant changes in our results.This is due to the radial dependence of the emissivity as expressedin the standard accretion disk model (Shakura & Sunyaev 1973),which concentrates the emission near the center of the black hole,making negligible the contribution from the outer parts to theintegrated flux. In principle, we could consider other less steepemissivity laws that make the outer parts of the disk more impor-tant; however, previous studies (e.g., Nandra et al. 1999) supportthe hypothesis of a strong emissivity gradient.

3. Segregation of the emitters allows us to reproduce theFe K� enhancement without equivalent continuum amplifica-tion if the continuum emission region lies interior to the Fe K�emission region or vice versa but only during limited time in-tervals. In fact, in none of the simulations does the continuumremain constant during a complete Fe K�microlensing event. Inthe case of an inner Fe K� disk plus an outer continuum annu-lus a significant change of the Fe K� emission is obtained whilethe continuum experiences only a very shallow gradient. Thisbehavior could well approximate the detected nonvarying con-

tinuum for observations covering only the peak of the Fe K�microlensing event.

4. We have also studied a more realistic case of microlensingby a caustic magnification pattern assuming a population of low-mass deflectors. In this case we can successfully reproduce theobserved lack of correlation between the X-ray continuum andFe K� emission amplification only if the line and continuumemission regions are separated.

An extreme case of segregation is that of the two-componentmodel for the continuum suggested in several papers (see, e.g.,Fabian & Vaughan 2003; Page et al. 2004) if the contribution ofthe disk component to the X-ray continuum is weak enough.

5. We have studied the chromatic effects of microlensing inthe X-ray continuum and find that the dependence with wave-length of the amplification can induce, even for small massmicrodeflectors (ERR ¼ 50Rg), chromatic variability of about30% in the observed energy range (from 0.1 to 10 keV) during amicrolensing event.

Further observations of the lensed quasars in the X-ray areneeded to confirm these results. In any case, monitoring gravi-tational lenses may help us to understand the physics of theinnermost part of the relativistic accretion disks.

This work is a part of the projects ‘‘Astrophysical Spectros-copy of Extragalactic Objects,’’ supported by the Ministry ofScience of Serbia, and P6/88, ‘‘Relativistic and Theoretical As-trophysics,’’ supported by the IAC. L. C. P. was supported byAlexander von Humboldt Foundation through the program forforeign scholars. J. A. M. is a Ramon y Cajal Fellow from theMCyT of Spain. Also, we would like to thank the anonymousreferee for very useful comments.

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