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

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 13 th March 2014

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

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

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

References ◮ Blandford & McKee (1982) ApJ 255 419 → reverberation of BLR � ∞ − ∞ dt ′ L p ( t ′ )Ψ( v , t − t ′ ) L o ( v , t ) = ◮ Stella (1990) Nature 344 747 → time dependent Fe K α shape ( a = 0) ◮ Matt & Perola (1992) MNRAS 259 433 → Fe K α response and black hole mass estimate → t ∼ GM / c 3 → time dependent light curve, centroid energy and line equivalent width ( h = 6 , 10; a = 0; θ o ) ◮ 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

References ◮ Wilkins & Fabian (2013) MNRAS 430 247 → fully relativistic, extended corona, propagation effects ◮ Cackett et al. (2014) MNRAS 438 2980 → Fe K α reverberation in lamp-post model

Scheme of the lamp-post geometry ◮ central black hole – mass, spin ◮ compact corona with isotropic emission → height, photon index ◮ accretion disc → Keplerian, geometrically thin, optically thick → ionisation due to illumination observer ( L p , h , M , a , n H , q n ) a ◮ local re-processing in the disc → REFLIONX with different directional corona emissivity prescriptions ◮ relativistic effects: → Doppler and gravitational energy shift h → light bending (lensing) δ i δ e black hole → aberration (beaming) → light travel time M ∆Φ r in Ω r accretion disc out

Time delay Time delay Total time delay ( a = 1.0000 , θ o = 60° ) ( a = 1.0000 , θ o = 60°, h = 3 ) a = 1 −20 −10 0 10 20 0 8 16 24 32 35 height 10 10 15 30 6 3 25 5 5 t [GM/c 3 ] 20 0 0 y y 15 10 −5 −5 5 −10 −10 0 0 5 10 15 −10 −5 0 5 10 −10 −5 0 5 10 r [GM/c 2 ] x x ◮ 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

Light curve a = 0, h = 3, ∆ t = 1 a = 1, h = 3, ∆ t = 1 0.25 2.5 inclination inclination 85 ° 85 ° 0.2 60 ° 2 60 ° 30 ° 30 ° Photon flux Photon flux 0.15 1.5 0.1 1 0.05 0.5 0 0 0 10 20 30 40 0 10 20 30 40 t [GM/c 3 ] t [GM/c 3 ] ◮ 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

Dynamic spectrum ◮ signature of outer and inner echo in dynamic spectra ◮ large amplification when the two echos separate ◮ intrinsically narrow K α line can acquire weird shapes

Transfer function for reverberation Stationary emission from the accretion disc: g µ e ℓ G = � E F ( E ) = r dr d ϕ G ( r , ϕ ) F l ( r , ϕ , E / g ) g = E l µ e = cos δ e Response to the on-axis primary emission: dS o ℓ = � � dS ⊥ dt ′ l F ( E , t ) = r dr d ϕ G ( r , ϕ ) × d Ω p N p ( t ′ ) N inc ( r ) M ( r , ϕ , E / g , t ′ + t pd ) δ ([ t − t do ] − [ t ′ + t pd ]) g Γ N inc = inc dS inc Line reverberation: � � dt ′ N p ( t ′ ) r dr d ϕ Ψ 0 ( r , ϕ ) δ ( E − gE rest ) δ ([ t − t ′ ] − [ t pd + t do ] F ( E , t ) = ) Ψ 0 = g GN inc M � �� � ∆ t Transfer function → response to a flare [ N p ( t ′ ) = δ ( t ′ ) ]: � � − 1 � r ∂ ( g , ∆ t ) � � dt ′ N p ( t ′ )Ψ( E , t − t ′ ) ∑ Ψ( E , t ) = Ψ 0 F ( E , t ) = � � E rest � ∂ ( r , ϕ ) � g = E / E rest t pd + t do = t ∂ ( g , ∆ t ) ∂ ( r , ϕ ) = ∂ g ∂ (∆ t ) − ∂ g ∂ (∆ t ) � = 0 ⇒ ∇ g ∦ ∇ (∆ t ) ∂ r ∂ϕ ∂ϕ ∂ r

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

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

Caustics θ o = 5°, h = 3 10 5 0 y −5 −10 −10 −5 0 5 10 x 1.5 1.0 g 0.5 0.0 0 5 10 15 20 25 30 t ◮ 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

Dynamic spectrum – narrow spectral line

Dynamic spectrum – neutral disc

Dynamic spectrum – ionised disc E 2 × F ( E )

Definition of the phase lag F refl ( E , f ) = ˆ ˆ N p ( f ) . ˆ F refl ( E , t ) = N p ( t ) ∗ ψ ( E , t ) ⇒ ψ ( E , f ) where ψ ( E , f ) = A ( E , f ) e i φ ( E , f ) ˆ if ψ ( E ) = A ( E ) e i φ ( E ) N p ( t ) = cos ( 2 π ft ) and ˆ then τ ( E ) ≡ φ ( E ) F refl ( E , t ) = A ( E ) cos { 2 π f [ t + τ ( E )] } where 2 π f F ( E , f ) ∼ ˆ ˆ F ( E , t ) ∼ N p ( t ) ∗ ( ψ r ( E , t )+ δ ( t )) ⇒ N p ( f ) . ( ˆ ψ r ( E , f )+ 1 ) and A r ( E , f ) sin φ r ( E , f ) tan φ tot ( E , f ) = 1 + A r ( E , f ) cos φ r ( E , f )

Parameter values and integrated spectrum a = 1, h = 3, θ o = 30 ° 10 8 M ⊙ M = reflected primary a = 1 ( 0 ) 1000 30 ◦ ( 60 ◦ ) θ o = E 2 x F(E) h = 3 ( 1 . 5 , 6 , 15 , 30 ) L p = 0 . 001 L Edd Γ = 2 ( 1 . 5 , 3 ) 0 . 1 ( 0 . 01 , 50 , 5 , 0 . 2 ) × 10 15 cm − 3 n H = 100 q n = − 2 ( 0 , − 5 , − 3 ) 0.1 1 10 E [keV] Energy bands: soft excess: 0 . 3 − 0 . 8 keV primary: 1 − 3 keV iron line: 3 − 9 keV Compton hump: 15 − 40 keV

Phase lag dependence on geometry ◮ reflected photon flux a = 1, θ o = 30 ° a = 1, θ o = 60 ° 0.12 0.25 decreases with height height [GM/c 2 ] height [GM/c 2 ] 1.5 1.5 0.1 ◮ primary flux increases with 3 3 0.2 6 6 relative photon flux relative photon flux 15 15 0.08 hight 30 30 0.15 0.06 ◮ the delay of response is 0.1 increasing with height 0.04 0.05 0.02 ◮ the “duration” of the response is longer 0 0 0 10 20 30 40 50 0 10 20 30 40 50 t [ks] t [ks] ◮ the phase lag increases a = 1, θ o = 30 ° a = 1, θ o = 60 ° with height, it depends 50 20 mainly on the “average” 0 0 -20 response time and -50 -40 phase lag [s] phase lag [s] magnitude of relative -100 -60 photon flux -80 -150 height [GM/c 2 ] height [GM/c 2 ] -100 ◮ the phase lag null points -200 1.5 1.5 3 3 -120 6 6 -250 are shifted to lower 15 -140 15 30 30 -300 -160 frequencies for higher 1 10 100 1 10 100 f [ µ Hz] f [ µ Hz] heights due to longer timescales of response

Phase lag dependence on geometry ◮ relative photon flux and the a = 1, θ o = 30 ° a = 1, θ o = 60 ° 0.12 0.25 phase lag increase with height [GM/c 2 ] height [GM/c 2 ] 1.5 1.5 inclination for low heights 0.1 3 3 0.2 6 6 relative photon flux relative photon flux 15 15 ◮ the delay and duration of 0.08 30 30 0.15 response do not change 0.06 0.1 much with the inclination 0.04 and thus the phase lag null 0.05 0.02 points frequencies change 0 0 only slightly 0 10 20 30 40 50 0 10 20 30 40 50 t [ks] t [ks] a = 1, θ o = 30 ° a = 1, θ o = 60 ° 50 20 0 0 -20 -50 -40 phase lag [s] phase lag [s] -100 -60 -80 -150 height [GM/c 2 ] height [GM/c 2 ] -100 -200 1.5 1.5 3 3 -120 6 6 -250 15 -140 15 30 30 -300 -160 1 10 100 1 10 100 f [ µ Hz] f [ µ Hz]