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A Study of the Correlation between the Amplification of the Fe Kalpha Line and the X-Ray Continuum of Quasars due to Microlensing

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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´,
1,2
P. Jovanovic´,
1,2
E. Mediavilla,
3
A. F. Zakharov,
4,5,6
C. Abajas,
3
J. A. Mun˜oz,
3,7
and G. Chartas
8
Recei
v
ed 2005 March 4; accepted 2005 October 4
ABSTRACTThe 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 disk emission 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 without noticeable 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,combinedwithmicrolensingitself,caninducewavelength-dependentvariabilityof
30%formicrolenseswithverysmall masses. All these results show that X-ray monitoring of gravitational lenses is a method well suited for studying the innermost structure of active galactic nucleus accretion disks.
Subject headin
g
g s:
gravitationallensing — quasars:individual(H1413+117, MGJ0414+0534, QSO 2237+0305) — X-rays: galaxies1. INTRODUCTIONRecent observational and theoretical studies suggest that gravitational microlensing can induce variability in the X-rayemission of lensed QSOs. Microlensing of the Fe K
line has been 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 beenalsotheoreticallyinvestigated.Mineshigeetal.(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 for thin accretion disks around Schwarzschild and Kerr black holeswere considered in Jaroszyn´ski et al. (1992), and microlensinglightcurvesfortheFeK
weresimulatedbyJaroszyn´ski(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 UVandopticallines(Popovic´ etal.2001a;Abajasetal.2002;Lewis& Ibata 2004). In the optical spectra, microlensing inducedmagniﬁcation of broad UV lines (e.g., C
iv
and S
iv
/O
iv
) wasreportedbyRichardsetal.(2004).Consequently,onecanexpect thatmicrolensingoftheFeK
lineregionwillbemorefrequent.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 lineand continuum variability intheobservedevents shouldbe givenin context of the microlensing hypothesis. Chartas et al. (2002)detected an increase of the Fe K
equivalent width in image B of the lensed QSO J0414+0534 that was not followed by the con-tinuum. Chartas et al. (2002) explained the nonenhancement of thecontinuumemissioninthe spectrumofimageBbyproposingthat the thermal emission region of the disk and the Compton up-scattered emission region of the hard X-ray source lie withinsmallerradiithantheironlinereprocessingregion.AnalyzingtheX-ray variability of QSO 2237+0305A, Dai et al. (2003) alsomeasuredampliﬁcationoftheFeK
lineincomponentAofQSO
1
Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia.
2
Isaac Newton Institute of Chile, Yugoslavia Branch, and UniversidadDiego Portales, Chile.
3
Instituto de Astroﬁsica de Canarias, 382005 La Laguna, Tenerife, Spain.
4
Institute of Theoretical and Experimental Physics, 25, B. Cheremushkinskayast., Moscow, 117259, Russia.
5
Astro Space Centre of Lebedev Physics Institute, Moscow, Russia.
6
Isaac Newton Institute of Chile, Moscow Branch.
7
Departamento de Astronomı´a y Astrofı´sica, 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-mationaboutthemagneticﬁeldmaybeextractedfromX-rayline-shapeanalysis.Zakharov & Repin (2003a, 2003b) discussed signatures of X-ray line-shapes for highly inclined accretion disks.
620
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 thelargersizeofthecontinuumemissionregion(
10
14
cm
100
R
g
for
M
BH
¼
10
7
M
) with respect to the Fe K
emission region(
10
R
g
)couldexplainthisresult.Finally,inH1413+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 the behavior 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 continuumemission, itseems that itmainly 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 onecomponentvaries and isapower law and the other componentisconstant 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
I
C
(
X
;
Y
;
E
obs
)
¼
I
P
E
obs
;
T
(
X
;
Y
)
½
A
(
X
;
Y
)
;
ð
1
Þ
and for the Fe K
line by
I
L
(
X
;
Y
;
E
obs
)
¼
I
P
E
0
g
(
X
;
Y
)
;
T
(
X
;
Y
)
½
;
E
obs
E
0
g
(
X
;
Y
)
½
A
(
X
;
Y
)
;
ð
2
Þ
where
T
(
X
,
Y
) is the temperature,
X
and
Y
are the impact pa-rameters that describe the apparent position of each point of theaccretiondiskimageonthecelestialsphereasseenbyanobserver atinﬁnity;
E
0
isthelinetransitionenergy(
E
Fe K
0
¼
6
:
4keV),and
g
(
X
;
Y
)
¼
E
obs
/
E
em
is the energy shift due to relativistic effects(
E
obs
is the observed energy, and
E
em
is the emitted energy fromthe disk).Here wedo not considerthecosmological redshift. Theemissivity of the disk is one of the important parameters that has 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 power law (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, and photons are close to thermal equilibrium with electrons. Thesurface temperature isafunctionofdiskparametersandresultsinthemulticolorblackbodyspectrum.Thiscomponentisthoughttoexplainthe ‘‘bluebump’’ inAGNsandthesoftX-rayemissioningalactic black holes. Although the standard model does not pre-dictthe power-lawX-rayemissionobservedinallsub-Eddingtonaccreting black holes, the power law for the X-ray emissivity inAGNs 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,modiﬁedblackbody,andpoweremissivitylawsforboththeFe K
and continuum emission.In the case of the blackbody radiation law, the disk emissivityis given as (e.g., Jaroszyn´ski et al. 1992)
I
P
(
X
;
Y
;
E
)
¼
B
½
E
;
T
s
(
X
;
Y
)
;
where
B E
;
T
s
(
X
;
Y
)
½ ¼
2
E
3
h
2
c
2
1
e
E
=
kT
s
(
X
;
Y
)
1
;
ð
3
Þ
where
c
is the speed of light,
h
is the Planck constant,
k
is theBoltzmann constant, and
T
s
(
X
,
Y
) is the surface temperature of X-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
T
s
¼
const
:
,taking that the black hole is a powerful X-ray source with aneffective temperature of 10
7
–10
8
K. But, regarding the standarddisk model, it is expected that at least the surface temperature isradiallydependent.Therefore,hereweaccepttheradialdistribu-tionofsurfacetemperaturegivenby(Shakura&Sunyaev1973):
T
s
(
X
;
Y
)
r
3
=
2
(
X
;
Y
) 1
r
1
=
2
(
X
;
Y
)
h i
4
=
5
K
;
ð
4
Þ
taking that an effective temperature is in an interval from 10
7
to10
8
K. The distribution ofthe temperature along the radius of thedisk used in this paper is given in the top panel of Figure 1, andthe corresponding shapeofspectralenergydistributionis shownin the bottom panels. In equation (4),
r
is the dimensionless pa-rameter, deﬁned as
r
(
X
;
Y
)
¼
R
(
X
;
Y
)6
R
g
¼
16
R
(
X
;
Y
)
c
2
GM
¼
M
M R
(
X
;
Y
)9 km
;
where
R
(
X
,
Y
)isthediskradius,expressedingravitationalradii(
R
g
).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 theemissivitylawisdescribedbya‘‘modiﬁed’’blackbodyradiationlaw (eqs. [3.4] and [3.8] in Shakura & Sunyaev 1973; see alsothediscussioninNovikov&Thorne1973;Shapiro&Teukolsky1983; Straumann 1984):
I
P
(
E
;
X
;
Y
)
/
x
3
exp (
x
)
;
ð
5
Þ
where
x
¼
E
/
kT
(
X
;
Y
). Shalyapin et al. (2002) used similar expressions 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., Dovcˇiak 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-dominantpart,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 that should 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
)
¼
Z
image
½
I
C
(
X
;
Y
;
E
)
þ
I
L
(
X
;
Y
;
E
)
d
;
ð
6
Þ
where
d
isthesolidanglesubtendedbythediskintheobserver’ssky and the integral extends over the whole emitting region.As one can see from equation (6) the total observed ﬂux is asumof the continuum and the line ﬂuxes; consequently,the am- pliﬁcationinthecontinuumandintheFeK
linecanbeconsid-eredseparatelyastwoindependentcomponents.Ontheotherhand,the ampliﬁcations will depend on the sizes and geometry of thecontinuum and line-emitting regions. Consequently, we consider ampliﬁcations in the line and in the continuum separately.Wewouldlike to point outhere that the aim of thepaper isnot to create a perfect accretion disk model (taking into account different 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 Kerr metrics).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 disk emission region, the disk inclination angle, the mass of the black hole, 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 angular momentum
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 20
R
g
(
left
) and 80
R
g
(
ri
g
ht
). 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 given by Nandra et al. (1997) from a study of the Fe K
line proﬁlesof 18 Seyfert 1 galaxies:
i
¼
35
. The inner radius,
R
in
, cannot be smaller than the radius of the marginally stable orbit,
R
ms
,that corresponds to
R
ms
¼
6
R
g
(gravitationalradii
R
g
¼
GM
/
c
2
,where
G
is gravitational constant,
M
is the mass of central black hole, and
c
is the velocity of light) in the Schwarzschild metricand to
R
ms
¼
1
:
23
R
g
in the case of the Kerr metric with angu-lar momentum parameter
a
¼
0
:
998. To select the outer radius,
R
out
, we take into account previous investigations of the X-rayvariability that support very compact X-ray–emitting disks. In particular, Oshima et al. (2001a) infer from the observed vari-ationinthelensedblazarPKS1830
211asizeoftheX-raycon-tinuum emission region of
3
;
10
14
cm, which is in agreement with the estimation for QSO 2237+03050 given by Dai et al.(2003). So, considering a range of black hole masses of 10
7
– 10
9
M
we can conclude that the X-ray emission is coming froma compact region of the order of 10
R
g
–100
R
g
. This range of sizes 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 of both the continuum and the line: (1) the inner and outer radii of bothemissionregionsarethesame,
R
in
¼
R
ms
and
R
out
¼
20
R
g
;(2)theinnerradiusisthesame,
R
in
¼
R
ms
,buttheouterradiusof theX-raycontinuumdiskissmaller,
R
out
¼
20
R
g
,thantheradiusof the line emission disk,
R
out
¼
80
R
g
; (3) the continuum emis-siondiskhasradii
R
in
¼
R
ms
and
R
out
¼
20
R
g
,andthelineemis-siondiskhas
R
in
¼
20
R
g
and
R
out
¼
80
R
g
(thecontinuumemissiontakes place in an inner part of disk surrounded by an annulusof Fe K
emission); (4) the continuum emission disk has radii
R
in
¼
20
R
g
and
R
out
¼
80
R
g
,andthelineemissiondiskhas
R
in
¼
R
ms
and
R
out
¼
20
R
g
(the Fe K
emission is located in the inner disk 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
M
8
¼
10
8
M
. We use thisvalue in order to determine the effective temperature distribu-tion. Thisvalue isinagreement withWang etal.(2003), where it was found that the majority of QSOs have black hole masses inthe range of 10
8
–10
9
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 (177,372
R
g
) on a side (Abajas et al. 2005).
Ri
g
ht:
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
R
out
¼
1000
R
g
.TABLE
1
Projected ERR for Different Deflector Masses for the Three Lensed QSOs where Microlensing of the F
e
K
Line Is Suspected
Object
z
s
z
l
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 2031.1QSO 2237+0305......................................... 1.69 0.04 11.2 35.4 112.1 354.5 1121.0QSO 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
H
0
¼
50 km s
1
Mpc
1
and
0
¼
1. The black hole mass is assumed to be10
8
M
.
MICROLENSING OF Fe K
LINE AND X-RAY CONTINUUM 623
No. 2, 2006
blackbody emissivity law, the emissivity at X-ray wavelengthscanbeextremelysmallcomparedwith,forexample,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
Differenttypes ofcaustics canbe usedto explain theobservedmicrolensing events in quasars. Moreover, for the exact event one can model the caustic shape to obtain different parameters(see, e.g., Abajas et al. 2005 and Kochanek 2004 for the case of Q2237+0305).Inordertoapplyanappropriatemicrolensmodel,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 (Kayser et al. 1986; Schneider & Weiss 1987; Wambsganss et al 1990a,1990b).Weassumeaﬂatcosmologicalmodelwith
¼
0
:
3and
H
0
¼
70 km s
1
Mpc
1
.In Figure 2 we present a comparison between the projectedmagniﬁcationmapinthesourceplaneandanaccretiondiskwitha size of 1000
R
g
(the circle in the right panel of Fig. 2). Takinginto account the small dimensions of the X-ray emission region
Fig.
3.—Simulationsof thebehavior of the FeK
line andcontinuum variations due to microlensing by acausticin the casesof Schwarzschild (
left
)and Kerr (
ri
g
ht
)metrics.Theparametersofthecausticare
A
0
¼
1,
¼
1,andERR
¼
50
R
g
.Intheﬁrstandsecondrowswepresentthecausticcrossingperpendiculartotherotatingaxisfor
¼
1,respectively.Inthethirdandfourthrowsweshowthecausticcrossingalongtherotationaxiswith
¼
1,respectively.TheradiiofthecontinuumandtheFe K
line emission accretion disks are the same:
R
in
¼
R
ms
and
R
out
¼
20
R
g
(where
R
g
¼
GM
/
c
2
). 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
R
out
¼
80
R
g
for the Fe K
line.
POPOVIC´ET AL.624 Vol. 637

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