[PPT] - Time lags and reverberation in the lamp-post geometry of the compact PowerPoint Presentation

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Time lags and reverberation in the lamp-post geometry of the compact corona illuminating a black-hole accretion disc

Michal Dovˇ ciak

Astronomical Institute Academy of Sciences of the Czech Republic, Prague Astrophysics group seminar, School of Physics, University of Bristol 13th March 2014

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Acknowledgement

StrongGravity (2013–1017) — EU research project funded under the Space theme of the 7th Framework Programme on Cooperation Title: Probing Strong Gravity by Black Holes Across the Range of Masses Institutes: AsU, CNRS, UNIROMA3, UCAM, CSIC, UCO, CAMK Webpages: http://stronggravity.eu/ http://cordis.europa.eu/projects/rcn/106556 en.html

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Active Galactic Nuclei – scheme

Urry C. M. & Padovani P . (1995) Unified Schemes for Radio-Loud Active Galactic Nuclei PASP , 107, 803

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Active Galactic Nuclei – X-ray spectrum

Fabian A.C. (2005) X-ray Reflections on AGN, in proceedings of “The X-ray Universe 2005”, El Escorial, Madrid, Spain, 26-30/9/2005

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References

◮ Blandford & McKee (1982) ApJ 255 419 → reverberation of BLR

Lo(v,t) =

∞

−∞ dt′ Lp(t′)Ψ(v,t −t′) ◮ Stella (1990) Nature 344 747 → time dependent Fe Kα shape (a = 0) ◮ Matt & Perola (1992) MNRAS 259 433

→ Fe Kα response and black hole mass estimate → t ∼ GM/c3 → time dependent light curve, centroid energy and line equivalent width (h = 6, 10; a = 0; θo)

◮ Campana & Stella (1995) MNRAS 272 585

→ line reverberation for a compact and extended source (a = 0)

◮ Reynolds, Young, Begelman & Fabian (1999) ApJ 514 164

→ fully relativistic line reverberation (h = 10; a = 0, 1) → more detailed reprocessing, off-axis flares → ionized lines for Schwarzschild case, outward and inward echo, reappearance of the broad relativistic line

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References

◮ Wilkins & Fabian (2013) MNRAS 430 247

→ fully relativistic, extended corona, propagation effects

◮ Cackett et al. (2014) MNRAS 438 2980

→ Fe Kα reverberation in lamp-post model

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Scheme of the lamp-post geometry

r

in

r

ut

δi δe

a

Ω

corona accretion disc black hole

bserver

∆Φ

h M

◮ central black hole – mass, spin ◮ compact corona with isotropic emission

→ height, photon index

◮ accretion disc

→ Keplerian, geometrically thin, optically thick → ionisation due to illumination (Lp, h, M, a, nH, qn)

◮ local re-processing in the disc

→ REFLIONX with different directional emissivity prescriptions

◮ relativistic effects:

→ Doppler and gravitational energy shift → light bending (lensing) → aberration (beaming) → light travel time

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Time delay

5 10 15 20 25 30 35 5 10 15 t [GM/c3] r [GM/c2] a = 1 height 3 6 15

−10 −5 5 10 x −10 −5 5 10 y

Time delay

(a = 1.0000 , θo = 60°)

−20 −10 10 20 −10 −5 5 10 x −10 −5 5 10 y

Total time delay

(a = 1.0000 , θo = 60°, h = 3)

8 16 24 32

◮ total light travel time includes the lamp-to-disc and disc-to-observer part ◮ first photons arrive from the region in front of the black hole

which is further out for higher source

◮ contours of the total time delay shows the ring of reflection that develops

into two rings when the echo reaches the vicinity of the black hole

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Light curve

0.05 0.1 0.15 0.2 0.25 10 20 30 40 Photon flux t [GM/c3] a = 0, h = 3, ∆t = 1 inclination 30° 60° 85° 0.5 1 1.5 2 2.5 10 20 30 40 Photon flux t [GM/c3] a = 1, h = 3, ∆t = 1 inclination 30° 60° 85° ◮ the flux for Schwarzschild BH is much smaller than for Kerr BH due to the hole

below ISCO (no inner ring in Schwarzschild case)

◮ the shape of the light curve differs substantially for different spins ◮ the “duration” of the echo is quite similar ◮ the higher the inclination the sooner first photons will be observed ◮ magnification due to lensing effect at high inclinations

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Dynamic spectrum

◮ signature of outer and inner

echo in dynamic spectra

◮ large amplification when

the two echos separate

◮ intrinsically narrow Kα line

can acquire weird shapes

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Transfer function for reverberation

Stationary emission from the accretion disc:

G = g µe ℓ g = E El µe = cosδe ℓ = dSo dS⊥

l

Ninc = gΓ

inc

dΩp dSinc Ψ0 = g GNinc M

F(E) =

r dr dϕ G(r,ϕ)Fl(r,ϕ,E/g)

Response to the on-axis primary emission:

F(E,t) =

dt′
r dr dϕ G(r,ϕ)×

Np(t′)Ninc(r)M(r,ϕ,E/g,t′ +tpd)δ([t −tdo]−[t′ +tpd])

Line reverberation:

F(E,t) =

dt′ Np(t′)
r dr dϕ Ψ0(r,ϕ)δ(E −gErest)δ([t −t′]−[tpd +tdo]
∆t

)

Transfer function → response to a flare [Np(t′) = δ(t′)]:

Ψ(E,t) =

∑

g = E/Erest tpd +tdo = t

Ψ0 r Erest

∂(g,∆t)

∂(r,ϕ)

−1

F(E,t) =

dt′ Np(t′)Ψ(E,t −t′)

∂(g,∆t) ∂(r,ϕ) = ∂g ∂r ∂(∆t) ∂ϕ − ∂g ∂ϕ ∂(∆t) ∂r = 0 ⇒ ∇g ∦ ∇(∆t)

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Caustics – Schwarzschild case

◮ the black curves show the points where the energy shift contours are

tangent to the time delay ones

◮ contour of ISCO in energy-time plane is shown by the blue curve ◮ the correspondent points A, B, C, D and E are shown in each plot for

better understanding

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Caustics – extreme Kerr case

◮ the black curves show the points where the energy shift contours are

tangent to the time delay ones

◮ contour of ISCO in energy-time plane is shown by the blue curve ◮ the correspondent points A, B, C, D and E are shown in each plot for

better understanding

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Caustics

θo = 5°, h = 3

−10 −5 5 10 x −10 −5 5 10 y

5 10 15 20 25 30 t 0.0 0.5 1.0 1.5 g

◮ plots of infinite magnification in the x-y (top) and g-t (bottom) planes ◮ the plots for Schwarzschild case (red) above ISCO are very similar to the

extreme Kerr case (blue)

◮ the shape of these regions change with inclination

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Dynamic spectrum – narrow spectral line

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Dynamic spectrum – neutral disc

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Dynamic spectrum – ionised disc

E2 ×F(E)

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Definition of the phase lag

Frefl(E,t) = Np(t)∗ψ(E,t) ⇒ ˆ Frefl(E,f) = ˆ Np(f). ˆ ψ(E,f) where ˆ ψ(E,f) = A(E,f)eiφ(E,f) if Np(t) = cos(2πft) and ˆ ψ(E) = A(E)eiφ(E) then Frefl(E,t) = A(E)cos{2πf[t +τ(E)]} where τ(E) ≡ φ(E) 2πf F(E,t) ∼ Np(t)∗(ψr(E,t)+δ(t)) ⇒ ˆ F(E,f) ∼ ˆ Np(f).( ˆ ψr(E,f)+1) and tanφtot(E,f) = Ar(E,f)sinφr(E,f) 1+Ar(E,f)cosφr(E,f)

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Parameter values and integrated spectrum

100 1000 0.1 1 10 E2 x F(E) E [keV] a = 1, h = 3, θo = 30° reflected primary

M = 108M⊙ a = 1 (0) θo = 30◦ (60◦) h = 3 (1.5, 6, 15, 30) Lp = 0.001LEdd Γ = 2 (1.5, 3) nH = 0.1 (0.01, 50, 5, 0.2)×1015cm−3 qn = −2 (0, −5, −3) Energy bands: soft excess: 0.3−0.8 keV primary: 1−3 keV iron line: 3−9 keV Compton hump: 15−40 keV

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Phase lag dependence on geometry

0.02 0.04 0.06 0.08 0.1 0.12 10 20 30 40 50 relative photon flux t [ks] a = 1, θo = 30° height [GM/c2] 1.5 3 6 15 30 0.05 0.1 0.15 0.2 0.25 10 20 30 40 50 relative photon flux t [ks] a = 1, θo = 60° height [GM/c2] 1.5 3 6 15 30

300
250
200
150
100
50

50 1 10 100 phase lag [s] f [µHz] a = 1, θo = 30° height [GM/c2] 1.5 3 6 15 30

160
140
120
100
80
60
40
20

20 1 10 100 phase lag [s] f [µHz] a = 1, θo = 60° height [GM/c2] 1.5 3 6 15 30

◮ reflected photon flux

decreases with height

◮ primary flux increases with

hight

◮ the delay of response is

increasing with height

◮ the “duration” of the

response is longer

◮ the phase lag increases

with height, it depends mainly on the “average” response time and magnitude of relative photon flux

◮ the phase lag null points

are shifted to lower frequencies for higher heights due to longer timescales of response

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Phase lag dependence on geometry

0.02 0.04 0.06 0.08 0.1 0.12 10 20 30 40 50 relative photon flux t [ks] a = 1, θo = 30° height [GM/c2] 1.5 3 6 15 30 0.05 0.1 0.15 0.2 0.25 10 20 30 40 50 relative photon flux t [ks] a = 1, θo = 60° height [GM/c2] 1.5 3 6 15 30

300
250
200
150
100
50

50 1 10 100 phase lag [s] f [µHz] a = 1, θo = 30° height [GM/c2] 1.5 3 6 15 30

160
140
120
100
80
60
40
20

20 1 10 100 phase lag [s] f [µHz] a = 1, θo = 60° height [GM/c2] 1.5 3 6 15 30

◮ relative photon flux and the

phase lag increase with inclination for low heights

◮ the delay and duration of

response do not change much with the inclination and thus the phase lag null points frequencies change

nly slightly

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Phase lag dependence on spin and energy band

0.005 0.01 0.015 0.02 5 10 15 20 relative photon flux t [ks] a = 0, h = 3, θo = 60° energy band 0.3-0.8 keV 1-3 keV 3-9 keV 15-40 keV 0.05 0.1 0.15 0.2 5 10 15 20 relative photon flux t [ks] a = 1, h = 3, θo = 60° energy band 0.3-0.8 keV 1-3 keV 3-9 keV 15-40 keV

50
45
40
35
30
25
20
15
10
5

5 1 10 100 phase lag [s] f [µHz] a = 0, h = 3, θo = 60° energy band 0.3-0.8 keV 3-9 keV 15-40 keV

100
80
60
40
20

20 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 60° energy band 0.3-0.8 keV 3-9 keV 15-40 keV

◮ the relative flux in the

energy band where primary dominates may in some cases be larger than that in Kα and Compton hump energy bands

◮ the magnitude of the

phase lag in different energy bands differs (in extreme Kerr case the larger lag in SE is due to larger ionisation near BH)

◮ the magnitude of the

phase lag is smaller in Schwarzschild case due to the hole in the disc under the ISCO

◮ the null points of the phase

lag change only slightly with energy and spin

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Ionisation

1 10 100 1000 10000 1 10 100 ξ r [GM/c2]

15
10
5

5 10 15 20 25 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 30° ionisation 1 2 3 4 5

◮ the phase lag in Kα band is shown ◮ the reflection component of the spectra are steeper for higher ionisation ◮ the magnitude of the phase lag depend on ionisation ◮ the null points of the phase lag does not change with the ionisation

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Directionality and photon index

80
70
60
50
40
30
20
10

10 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 30° directionality darkening isotropic brightening

120
100
80
60
40
20

20 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 30° Np:Nr 0.5 1 2

80
70
60
50
40
30
20
10

10 20 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 30° photon index 1.5 2 3

◮ the phase lag in SE band is shown ◮ the magnitude of the phase lag changes in all three cases ◮ the null points of the phase lag does not change with different directionility

dependences or power-law photon index

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Phase lag energy dependence

for low f: Ar(E,f) ≃ AE(E)Af(f) φr(E,f) ≃ φr(f) and ∆τ(E,f) ≃ 1 2πf atan [Ar(E,f)−Ar(E0,f)] sinφr(f) 1+[Ar(E,f)+Ar(E0,f)] cosφr(f)+Ar(E,f)Ar(E0,f) and for f such that φr(f) = ±π 2 : ∆τ(E,f) ≃ 1 2πf [Ar(E,f)−Ar(E0,f)]

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Phase lag energy dependence

100
90
80
70
60
50
40
30
20
10

10 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 30° energy band 0.3-0.8 keV 3-9 keV 15-40 keV

100
80
60
40
20

20 1 10 100 phase lag [s] f [µHz] a = 1, h = 3, θo = 60° energy band 0.3-0.8 keV 3-9 keV 15-40 keV 0.001 0.01 0.1 0.1 1 10 phase lag [s] E [keV] a = 1, h = 3, θo = 30° f = 40µHz f = 130µHz E2 x F(E) 0.001 0.01 0.1 0.1 1 10 phase lag [s] E [keV] a = 1, h = 3, θo = 60° f = 40µHz f = 120µHz E2 x F(E)

◮ the energy dependence of

the phase lag follows the spectral shape perfectly at particular frequencies

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Phase lag energy dependence

70
60
50
40
30
20
10

10 1 10 100 phase lag [s] f [µHz] a = 0, h = 3, θo = 30° energy band 0.3-0.8 keV 3-9 keV 15-40 keV

50
45
40
35
30
25
20
15
10
5

5 1 10 100 phase lag [s] f [µHz] a = 0, h = 3, θo = 60° energy band 0.3-0.8 keV 3-9 keV 15-40 keV 0.0001 0.001 0.01 0.1 1 10 phase lag [s] E [keV] a = 0, h = 3, θo = 30° f = 30µHz f = 110µHz E2 x F(E)

0.001

0.001 0.002 0.003 0.004 0.005 0.006 0.1 1 10 phase lag [s] E [keV] a = 0, h = 3, θo = 60° f = 30µHz f = 100µHz E2 x F(E)

◮ if the second phase lag

maximum is too small the phase lag energy dependence does not follow the spectral shape that well

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Summary

◮ two aspects of reverberation – in the timing and frequency domains ◮ the response of the disc peaks in the vicinity of the black hole ◮ the phase lag is used to get information on the system properties ◮ the frequency dependence of the phase lag is mainly due to geometry

(height of the corona)

◮ the magnitude of the phase lag depends on many details of the model

(height, spin, ionisation, unisotropy, energy, ...)

◮ extended corona