Table Of ContentXXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004 1
Polarization signatures of strong gravity in black-hole accretion discs
Vladim´ır Karas
Astronomical Institute, Academy of Sciences, Prague, Czech Republic and
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic
Michal Dovˇciak
Astronomical Institute, Academy of Sciences, Prague, Czech Republic
Giorgio Matt
Dipartimento di Fisica, Universit`a degli Studi “Roma Tre”, Rome, Italy
We discuss the effects of strong gravity on the polarization properties of a black hole accretion
disc. The intrinsic polarization is computed taking into account light scattered on the disc surface
5
and using different approximations. The gravitational field of a black hole influences the Stokes
0
parametersofreflectedradiationpropagatingtoadistantobserver. Thelamp-postmodelisexplored
0
as an example of a specific geometrical arrangement relevant for AGNs. The degree and the angle
2
of polarization are computed as functions of the observer inclination angle, of the inner radius of
n the disc emitting region, and of other parameters of the model. The expected polarization should
a
be detectable by newgeneration polarimeters.
J
6
2 I. INTRODUCTION maybeessentialtodiscriminatebetweendifferentpos-
sible geometries of the source. The idea of using po-
1
larimetrytogainadditionalinformationaboutaccret-
v Accretion discs in central regions of active galactic
ing compact objects is not a new one.
9 nucleiaresubjecttostrongexternalilluminationorigi-
7 natingfromsomekindofcoronaandgivingrisetospe- In this context it was proposed by Rees [20] that
5 polarized X-rays are of high relevance. Pozdnyakov
cific spectral features in the X-ray band. In particu-
1 et al. [19] studied spectral profiles of iron X-ray lines
lar,theK-shelllinesofironarefoundtobeprominent
0 that result from multiple Compton scattering. Vari-
around 6–7 keV. It has been shown that the shape of
5
ous influences affecting polarization(due to magnetic
0 the intrinsic spectra must be further modified by the
fields, absorption as well as strong gravity) were ex-
/ strong gravitational field of the central mass, and so
h amined for black-hole accretion discs [1]. Temporal
X-ray spectroscopy could allow us to explore the in-
p variations of polarization were also discussed, in par-
nermost regions of accretion flows near supermassive
-
o black holes [10, 21]. Similar mechanisms operate also ticular the case of orbiting spots near a black hole
r in some Galactic black-hole candidates. [2, 4]. Furthermore, within general relativity frame-
t
s Recent XMM-Newton observations indicate the workthe polarizationhasbeenstudiedby variousau-
a thors: see Portsmouth & Bertschinger [18] for a very
: rather surprising result that relativistic iron lines are
v recentdiscussion. Withthepromiseofnewpolarimet-
not as common as previously believed; see Bianchi et
i ric detectors [5], quantitative examination of specific
X al. [3] and Yaqoob et al. [24] fort further references.
models becomes timely.
r This does not necessarily mean that the iron line is
a not produced in the innermost regions of accretion
discs but the situation is likely to be more complex
II. THE MODEL AND COMPUTATIONS
than in simple, steady scenarios. Some evidence for
thelineemissionarisingfromorbitingspotsispresent
in the time-resolved spectra of a few AGNs [9]. Even We have developed a new code for time-dependent
when clearly observed,relativistic lines behave differ- analysis of relativistic spectral features originating in
entlythanexpected. Thebestexampleisthepuzzling black-hole accretion discs [7, 8, 9]. Detailed track-
lack of correlation between the line and continuum ing of time-dependent spectral features (time-scale of
emission in MCG–6-30-15 [11], unexpected because severalkiloseconds)wouldbeparticularlyrelevantfor
the very broad line profile clearly indicates that the themodellingandinterpretationofvariableX-rayfea-
line originates in the innermost regions of the accre- tures. Thesehavebeenreportedina growingnumber
tion disc, hence very close to the illuminating source. ofactivegalacticnuclei andtentatively interpretedin
Miniutti et al. [15] have proposed a solution to this terms of reflection iron lines due to flares and spots
problem in terms of an illuminating source moving [7, 12, 22, 23, 24]. Narrow fluctuating profiles are of-
along the black-hole rotation axis or very close to it. tenredshiftedwithrespecttotherestenergyandthey
Polarimetric studies along with time-resolvedspec- may provide a powerful tool to measure the mass of
troscopy could provide additional information about central black holes in AGNs and Galactic black hole
accretion discs in the strong gravity regime, and this candidates [6, 16].
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2 XXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004
θo=30°, rin=6, rout=400 θo=60°, rin=6, rout=400 θo=80°, rin=6, rout=400
85 60 60
Height Height Height
2 2 2
80 6 30 6 30 6
15 15 15
gle 75 100 gle 100 gle 100
an an 0 an 0
n n n
zatio 70 zatio -30 zatio -30
olari 65 olari olari
P P -60 P -60
60
-90 -90
55
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
θo=30°, rin=6, rout=400 θo=60°, rin=6, rout=400 θo=80°, rin=6, rout=400
0.5 0.5 0.5
reflected Height reflected Height reflected Height
2 2 2
0.4 6 0.4 6 0.4 6
15 15 15
ee 100 ee 100 ee 100
gr gr gr
de 0.3 de 0.3 de 0.3
n n n
o o o
ati ati ati
z 0.2 z 0.2 z 0.2
ari ari ari
ol ol ol
P P P
0.1 0.1 0.1
0 0 0
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
θo=30°, rin=6, rout=400 θo=60°, rin=6, rout=400 θo=80°, rin=6, rout=400
0.12 0.05 0.03
Height reflected+direct Height reflected+direct Height reflected+direct
2 2 2
6 0.04 6 6
gree 0.09 11050 gree 11050 gree 0.02 11050
de de 0.03 de
on 0.06 on on
ati ati ati
z z 0.02 z
ari ari ari 0.01
ol ol ol
P 0.03 P P
0.01
0 0 0
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
FIG.1: Theenergydependenceofpolarizationangle(toppanels)andpolarizationdegree(middlepanels)duetoreflected
radiationfordifferentobserver’sinclinationangles(θo =30◦,60◦ and80◦)andfordifferentheightsoftheprimarysource
(h=2,6,15and100). Polarizationdegreeforreflectedplusdirectradiationisalsoplotted(bottompanels). Theemission
comes from a disc within rin = 6 and rout = 400. Isotropic primary radiation with photon index Γ = 2 and angular
momentum of thecentral black hole a=0.9987 were assumed.
Since the reflecting medium has a disc-like geome- assumed processes that vary at a much slower pace
try, a substantial amount of linear polarization is ex- than the light-crossing time at the corresponding ra-
pected in the resulting spectrum because of Comp- dius). Theintrinsicpolarizationofemerginglightwas
ton scattering. Polarization properties of the disc computedlocally,assumingaplane-parallelscattering
emission are modified by the photon propagation in layerwhichwasilluminatedbylightradiatedfromthe
a gravitationalfield, providingadditionalinformation primary source.
onitsstructure. Herewecalculatetheobservedpolar-
ization of the reflected radiation assuming the lamp-
post model for the stationary power-law illuminating
source [14, 17]. We assumed a rotating (Kerr) black III. RESULTS
hole as the only source of the gravitationalfield, hav-
ing a common symmetry axis with an accretion disc.
The reflection component has been computed by a
The disc was also assumed to be stationary and we
Monte-Carlo code. The number of reflected photons
restricted ourselves to the time-averagedanalysis (we
is proportional to the incident flux NS(E ) arriving
i p
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XXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004 3
θo=30°, rin=rms, rout=400 θo=60°, rin=rms, rout=400 θo=80°, rin=rms, rout=400
90 60 60
Height Height Height
2 2 2
30
80 6 30 6 6
15 15 15
gle 100 gle 100 gle 0 100
n an 70 n an 0 n an -30
o o o
arizati 60 arizati -30 arizati -60
Pol Pol -60 Pol -90
50
-120
-90
40 -150
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
θo=30°, rin=rms, rout=400 θo=60°, rin=rms, rout=400 θo=80°, rin=rms, rout=400
0.4 0.16 0.12
reflected Height reflected Height reflected Height
2 2 2
6 6 6
ee 0.3 11050 ee 0.12 10105 ee 0.09 10105
gr gr gr
e e e
d d d
on 0.2 on 0.08 on 0.06
ati ati ati
z z z
ari ari ari
ol ol ol
P 0.1 P 0.04 P 0.03
0 0 0
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
θo=30°, rin=rms, rout=400 θo=60°, rin=rms, rout=400 θo=80°, rin=rms, rout=400
0.15 0.08 0.03
Height reflected+direct Height reflected+direct Height reflected+direct
2 2 2
0.12 6 6 6
gree 11050 gree 0.06 11050 gree 0.02 11050
de 0.09 de de
on on 0.04 on
ati ati ati
z 0.06 z z
ari ari ari 0.01
ol ol ol
P P 0.02 P
0.03
0 0 0
4 6 8 10 12 14 4 6 8 10 12 14 4 6 8 10 12 14
Energy [keV] Energy [keV] Energy [keV]
FIG. 2: Same as in theprevious figure butfor disc starting at rin=1.20.
from the primary source, and then specific Stokes parameters per energy bin,
i.e.∆i ,∆q ,∆u and∆v . The latterquantities are
dΩ o o o o
NiS(Ei)=NpΩ(Ep)dS p , (1) directly measurable, specifying the fluxes of photons
loc with a given polarization. One can write
whereNΩ(E )=N E−Γrepresentsanisotropicand
p p 0p p
steady power-law primary emission that is emitted
intothesolidangledΩpandeventuallyilluminatesthe ∆io(E,∆E) = N0Z dS Z dElociloc(Eloc)F , (3)
local area element dS on the disc (see refs. [7, 8, 9]
loc
for a more detailed description of computations and
∆q (E,∆E) = N dS dE q (E )cos2Ψ
for notation). Four Stokes parameters, Iν, Qν, Uν o 0Z Z loch loc loc
andVν,entirelydescribepolarizationpropertiesofthe −u (E )sin2Ψ F , (4)
loc loc
scattered light. One has to distinguish the quantities i
that are determined locally at the point of emission
∆u (E,∆E) = N dS dE q (E )sin2Ψ
on the disc surface (index ‘loc’) and those relevant to o 0Z Z loch loc loc
a distant observer (‘o’). We introduce specific Stokes
+u (E )cos2Ψ F, (5)
parameters, loc loc
i
iν ≡ IEν , qν ≡ QEν , uν ≡ UEν , vν ≡ VEν , (2) ∆vo(E,∆E) = N0Z dS Z dElocvloc(Eloc)F , (6)
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4 XXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004
r =6, r =400 r =r , r =400
in out in ms out
0.3 0.3
θ θ
o o
30° 30°
ee 60° ee 60°
egr 0.2 80° egr 0.2 80°
d d
n n
o o
ati ati
z z
ari 0.1 ari 0.1
ol ol
P P
0 0
200 200
e e
gl gl
n 160 n 160
a a
n n
atio 120 atio 120
z z
ari 80 ari 80
ol ol
P P
40 40
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Height Height
FIG. 3: Polarization degree and angle due to reflected radiation integrated over the whole surface of the disc and
propagated to thepoint of observation. Dependenceon height h is plotted. Left panel: rin =6; right panel: rin =1.20.
Inboth thepanels theenergy rangewas assumed 9−12 keV,thephotonindex of incident radiation Γ=2, theangular
momentum a=0.9987.
r =6, r =400 r =r , r =400
in out in ms out
0.4 0.4
Height Height
2 2
e e
e 0.3 8 e 0.3 8
gr gr
e 100 e 100
d d
n n
o 0.2 o 0.2
ati ati
z z
ari ari
ol 0.1 ol 0.1
P P
0 0
200 200
e e
ngl 160 ngl 160
a a
on 120 on 120
ati ati
ariz 80 ariz 80
Pol 40 Pol 40
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
µ µ
o o
FIG. 4: Polarization degree and angle as functions of µo (cosine of observer inclination, µo = 0 corresponds to the
edge-on view of thedisc). The same model is shown as in theprevious figure.
where F ≡F(r,ϕ)=g2lµ r is the transfer function, gree due to reflected and reflected plus direct radia-
e
g being the total energy shift between observed and tion. One can see that the polarization of reflected
emitted photons, l the lensing effect, µ the cosine of radiation can be as high as thirty percent for small
e
theemissionangle,andΨtheanglebywhichavector inclinations and small heights of the primary source.
rotateswhileitisparallellytransportedalongthelight Polarization of the reflected radiation depends only
geodesic. weakly on energy, except for the region close to the
InFigures1and2weshow,respectively,theenergy iron edge at ∼ 7.2keV. In order to compute observ-
dependence ofpolarizationangleand polarizationde- able characteristics one has to combine the primary
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XXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004 5
r =6, r =400 r =r , r =400
in out in ms out
0.12 θo=30° 3-6keV 0.12 θo=30° 3-6keV
6-9keV 6-9keV
0.09 9-12keV 0.09 9-12keV
12-15keV 12-15keV
0.06 0.06
0.03 0.03
0 0
ee 0.06 θo=60° ee 0.06 θo=60°
gr gr
e e
d d
n 0.04 n 0.04
o o
ati ati
z z
ari 0.02 ari 0.02
ol ol
P P
0 0
θ =80° θ =80°
o o
0.02 0.02
0.01 0.01
0 0
2 10 100 2 10 100
Height Height
FIG. 5: Net polarization degree of thetotal (primary plus reflected) signal as a function of h. Left panel: rin =6; right
panel: rin=1.20. The curvesare parametrized bythe corresponding energy range.
power-law continuum with the reflected component. metry. The change in angle is due to gravitation for
Thepolarizationdegreeoftheresultingsignaldepends which we assumed a rapidly rotating black hole. The
onthe ratiobetween the two components and alsoon two panels of this figure correspond to different loca-
theenergyrangeofanobservation. Theoveralldegree tions of the inner disc edge: r = 6 and r = 1.20,
in in
ofpolarizationincreaseswithenergy(seebottompan- respectively. The curves are strongly sensitive to r
in
els in Figs. 1–2) due to the fact that the intensity of and h, while the dependence on r is weak for a
out
radiation from the primary source diminishes, the in- large disc (here r = 400). Sensitivity to r is par-
out in
tensityofthereflectedradiationincreaseswithenergy ticularlyappealingifonerememberspracticaldifficul-
(in the energy range 3−15keV)and the polarization ties in estimating r by fitting spectra. The effect is
in
of the reflected light alone stays roughly constant. clearly visible even for h∼20. Graphs corresponding
to r = 6 and a = 0.9987 resemble the non-rotating
In our computations we assumed that the irradi- in
case(a=0)quite closelybecause draggingeffects are
ating source emits isotropically and its light is af-
most prominent near horizon.
fected only by gravitational redshift and lensing, ac-
Figure 4 shows the polarization degree and angle
cording to the source location at z = h on axis.
as functions of the observer’s inclination. Again, by
This results in a dilution of primary light by factor
∼g2(h,θ )l (h,θ ), where g2 = 1 −2h/(a2 +h2) is comparingthetwocasesofdifferentrinonecanclearly
h o h o h
recognizethatthepolarizationissensitivetodetailsof
square of the redshift of primary photons reaching
theflowneartheinnerdiscboundary. Finally,depen-
directly the observer, l is the corresponding lensing
h
dence of the polarization degree of overall radiation
factor. Here, the redshift is the dominant relativistic
(primary plus reflected) on the height of the primary
term, while lensing of primary photons is a few per-
sourceandtheobserverinclinationindifferentenergy
centatmostandcanbesafelyignored. Anisotropyof
ranges is shown in Figures 5–6.
primary radiation may further attenuate or amplify
the polarization degree of the final signal, while the
polarizationangle is almostindependent of this influ-
ence as long as the primary light is itself unpolarized. IV. CONCLUSIONS
The polarization of scattered light is also shown in
Fig. 3, where we plot the polarization degree and the We examined strong-gravity polarization features
changeofthepolarizationangleasfunctionsofh. No- in X-rays reflected from accretion discs. In order to
tice that in the Newtonian case only polarization an- compute directly observable characteristics one has
◦ ◦
glesof0 or90 wouldbeexpectedforreasonsofsym- to combine the primary continuum with the reflected
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6 XXII Texas Symposium on Relativistic Astrophysics, Stanford University, December 2004
r =6, r =400 r =r , r =400
in out in ms out
0.15 0.15
3-6keV h=2 3-6keV h=2
0.12 0.12
6-9keV 6-9keV
0.09 9-12keV 0.09 9-12keV
12-15keV 12-15keV
0.06 0.06
0.03 0.03
0 0
e h=8 e h=8
e e
gr 0.09 gr 0.09
e e
d d
n n
o 0.06 o 0.06
ati ati
z z
ari 0.03 ari 0.03
ol ol
P P
0 0
h=100 h=100
0.09 0.09
0.06 0.06
0.03 0.03
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
µ µ
o o
FIG.6: Netpolarizationdegreeofthetotal(primaryplusreflected)signalasafunctionofµo. Thesamemodelisshown
as in theprevious figure.
component. Polarization degree of the resulting sig- and spectroscopy, a tool for exploring the properties
nal depends on mutual proportion of the two com- of the accretion flows in the vicinity of black holes.
ponents and the energy range of observation. Po- The authors gratefully acknowledge support from
larization properties represent the scattering mech- Czech Science Foundation grants 205/03/0902 (VK)
anism, source geometry as well as the gravitational and 205/05/P525 (MD), and from the Grant
field structure acting on reflected photons. New gen- Agency of the Academy of Sciences (IAA300030510).
erationphotoelectricpolarimetersinthefocalplaneof GM acknowledges financial support from Agenzia
largeareaoptics, suchas those foreseenfor Xeus, can Spaziale Italiana (ASI) and Ministero dell’Istruzione,
probe polarization degree of the order of one percent dell’Universit`a e della Ricerca (MIUR), under grant
inbrightAGN,makingpolarimetry,alongwithtiming cofin–03–02–23.
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