Role of emission angular directionality
in spin determination of accreting black holes
with broad iron lines
Jiřı́ Svoboda (Astronomical Institute of AS CR, Prague)
Michal Dovčiak & Vladimı́r Karas (Prague),
René W. Goosmann (Strasbourg)
Probing Strong Gravity near Black Holes, Prague - Carolinum, 16/02/2010
Light rays and wave fronts
• vacuum curved space-time: light rays are not “straight”
• different emission angle at different positions on the disc
Emission angle from a black hole accretion disc
↓ direction to the observer
Emission directionality
• intensity of the reflection radiation from an accretion disc:
Ie (re , µe , Ee ) ≡ R(re ) M(µe , re , Ee ) E(Ee )
• emission directionality M(µe , re , Ee ) = dependence of the intensity on
the emission angle (µe = cos θe )
• usually, a unique profile is assumed:
→ limb darkening M(µe ) = 1 + 2.06 µe (Laor 91)
→ limb brightening M(µe ) = ln(1 + µ−1
e ) (Haardt 93)
• limb brightening suggested in simulations of X-ray irradiated disc
atmospheres (Ghisellini et al. 94, Życki & Czerny 94, ..., Goosmann et al. 06)
Emission directionality in cold disc simulation
• we applied the Monte-Carlo radiative transfer code NOAR (Dumont 00)
to calculate reprocessing model for neutral or weakly ionized matter
(primary radiation: power law with photon index Γ = 1.9)
2.2
2 - 6 keV
6 - 15 keV
15 - 100 keV
2 - 100 keV
2
M(µe)
1.8
limb darkening
1.6
1.4
limb brightening
1.2
isotropic
1
0.8
reprocessing
0.6
0
0.2
0.4
0.6
µe
0.8
1
arbitrary flux [cts cm-2 s-1 keV-1]
Line profiles for different directionalities
limb brightening
limb darkening
isotropic
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
Energy [keV]
6
7
• parameters of the line: a = 0.9982, q = 3, θo = 30 deg, E0 = 6.4 keV,
rin = rms (a) = 1.23 rg , rout = 400 rg , KY model (Dovčiak et al. 04)
• stronger red wing with limb brightening
• steeper line profile with limb darkening
(see also Beckwith & Done 04, Niedźwiecki & Życki 08,...)
Role of directionality in spin determination
• the aim of our research is to investigate how sensitive the spin
measurements are to the assumptions about the directionality
• current X-ray data
→ XMM-Newton observation of MCG–6-30-15 (2001/07/31 - 08/05)
Constraining spin in MCG–6-30-15
• broad iron line detected (Fabian et al. 02, Ballantyne et al. 03, Vaughan &
Fabian 04,...), we re-analyzed the data and applied KY model with two
cases of directionality (Svoboda et al. 09):
190
χ2
185
180
E=6.60(1)keV
i=32(1)deg
q1=3.7(1)
q2=2.1(1)
rb=18(1)rg
limb
darkening
190
185
χ2
limb brightening
180
2σ
2σ
175
E=6.67(1)keV
i=27(1)deg
q1=5.3(1)
q2=2.8(1)
rb=4.9(2)rg
175
0.8
0.85
0.9
0.95
1
spin a/M
0.8
0.85
0.9
0.95
spin a/M
• high value of spin required in both cases
• main difference: values of the parameters related to the radial part of
the intensity given by R(re ) = re−q1 (re ≤ rb ) , R(re ) = re−q2 (re > rb )
1
Role of directionality in spin determination II
• the aim of our research is to investigate how sensitive the spin
measurements are to the assumptions about the directionality
• current X-ray data
→ XMM-Newton observation of MCG–6-30-15 (2001/07/31 - 08/05)
• artificial data of high resolution next generation X-ray missions
→ simulation of IXO data
→ seed model of the simulated data: POWERLAW + KYL2CR
(relativistic reflection model with the directionality of the
reprocessing model from the NOAR numerical computations)
→ generated data fitted with the model equipped with a simple
formula for the directionality (limb darkening, isotropic, limb
brightening)
χ2 test-fit analysis on simulated data
• fiducial values: af = 0.7, if = 30 deg (q = 3, rin = rms , rout = 400 rg )
limb brightening
isotropic
limb darkening
400
400
400
χ
χ
χ
2
500
2
500
2
500
300
300
300
200
200
200
0.55
0.6 0.65 0.7 0.75
spin a/M [GM/c]
0.8
0.55
0.6 0.65 0.7 0.75
spin a/M [GM/c]
0.8
0.55
0.6 0.65 0.7 0.75
spin a/M [GM/c]
0.8
Min: 202.4; Levels = 204.7, 207.0, 211.6
Min: 185.1; Levels = 187.4, 189.7, 194.3
30.3
30
29.7
inclination i [deg]
30.3
30.3
inclination i [deg]
inclination i [deg]
Min: 193.5; Levels = 195.8, 198.1, 202.7
30
29.7
29.4
29.4
0.6 0.63 0.66 0.69
spin a/M [GM/c]
0.72
29.7
29.4
29.1
29.1
0.57
30
0.66
0.69
0.72
spin a/M [GM/c]
0.75
0.66
0.69
0.72
0.75
spin a/M [GM/c]
0.78
Svoboda et al., A&A (2009), 507, 1
χ2 test-fit analysis on simulated data
• fiducial values: af = 0.998, if = 60 deg (q = 3, rin = rms , rout = 400 rg )
limb brightening
isotropic
limb darkening
450
500
250
375
2
χ
2
300
χ
χ
2
450
200
400
225
150
0.99
Min: 182.0; Levels = 184.3, 186.6, 191.2
60.6
inclination i [deg]
inclination i [deg]
60.6
1
60.3
60
59.7
0.98
0.99
spin a/M [GM/c]
1
0.992 0.994 0.996 0.998
spin a/M [GM/c]
1
0.994
Min: 149.0; Levels = 151.2, 153.5, 158.1
60.3
60
59.7
0.997
0.998
0.999
spin a/M [GM/c]
0.996
0.998
spin a/M [GM/c]
1
Min: 369.5; Levels = 371.7, 374.0, 378.6
inclination i [deg]
150
0.97 0.975 0.98 0.985 0.99 0.995
spin a/M [GM/c]
350
1
60.3
60
59.7
0.998
0.999
spin a/M [GM/c]
Svoboda et al., A&A (2009), 507, 1
1
χ2 test-fit analysis on simulated data
• fiducial values: af = 0.998, if = 60 deg (q = 3, rin = rms , rout = 400 rg )
limb brightening
isotropic
limb darkening
450
500
250
375
2
χ
2
300
χ
χ
2
450
200
400
225
150
0.99
χ2/ ν = 1.25
60.6
inclination i [deg]
60.6
inclination i [deg]
1
60.3
60
59.7
0.98
0.99
spin a/M [GM/c]
1
0.992 0.994 0.996 0.998
spin a/M [GM/c]
1
χ2/ ν = 1.02
60.3
60
59.7
0.997
0.994
inclination i [deg]
150
0.97 0.975 0.98 0.985 0.99 0.995
spin a/M [GM/c]
350
0.998
0.999
spin a/M [GM/c]
1
0.996
0.998
spin a/M [GM/c]
1
χ2/ ν = 2.53
60.3
60
59.7
0.998
0.999
spin a/M [GM/c]
Svoboda et al., A&A (2009), 507, 1
1
Conclusions
• black hole spin measurements depend on the employed definition of the
angular distribution of the disc emission
• our numerical ‘cold disc’ simulations show effect of the limb brightening,
but not as strong as I(µe ) ≈ ln(1 + µ−1
e )
• from the considered directionality formulae, isotropic angular emissivity
reproduces the simulated data to the best precision
• limb darkening seems to be not suitable approach of the emission
directionality in the case of reflection on a black hole accretion disc
and using it leads to:
→ overestimated spin values (around 10 – 20 % in rin )
→ steeper radial dependence of the intensity (q)
→ higher χ2 values (with the higher resolution X-ray missions)
Conclusions
• black hole spin measurements depend on the employed definition of the
angular distribution of the disc emission
• our numerical ‘cold disc’ simulations show effect of the limb brightening,
but not as strong as I(µe ) ≈ ln(1 + µ−1
e )
• from the considered directionality formulae, isotropic angular emissivity
reproduces the simulated data to the best precision
• limb darkening seems to be not suitable approach of the emission
directionality in the case of reflection on a black hole accretion disc
and using it leads to:
→ overestimated spin values (around 10 – 20 % in rin )
→ steeper radial dependence of the intensity (q)
→ higher χ2 values (with the higher resolution X-ray missions)
Thank you very much for your attention!
Contours a vs. q
• af = 0.7, θf = 60 deg
• the simulated data: powerlaw + kyl2cr, IXO res.
• Left: brightening – middle: isotropic – right: darkening
-2.8
Min: 272.9; Levels = 275.2, 277.5, 282.1
-2.8
-3
q
-3
q
Min: 146.2; Levels = 148.5, 150.8, 155.4
-3.2
-3.2
-3.4
-3.4
0.6
0.65
0.7
0.75
0.8
spin a/M [GM/c]
0.85
0.9
-3.2
-3.4
-3.6
-3.6
Min: 214.7; Levels = 217.0, 219.4, 224.0
-3
q
-2.8
-3.6
0.6
0.65
0.7
0.75
0.8
spin a/M [GM/c]
0.85
0.9
0.6
0.65
0.7
0.75
0.8
spin a/M [GM/c]
0.85
0.9