Post on 03-Sep-2018
1.Concept of emissivity profiles
• Why should I care?
2.Determination from observed spectra
3.Test with XSPEC model
4.Theoretical predictions – ray tracing
Outline
2
• Parameter in XSPEC models
• “Vary it to obtain best fit for model”
Emissivity Profile
3
XSPEC12>model laor
Input parameter value, delta, min, bot, top, and max values for ... 6.4 0.05 0 0 100 1001:laor:lineE>6.4 3 -‐0.1 -‐10 -‐10 10 102:laor:Index>
• Parameter in XSPEC models
• “Vary it to obtain best fit for model”
Emissivity Profile
3
XSPEC12>model laor
Input parameter value, delta, min, bot, top, and max values for ... 6.4 0.05 0 0 100 1001:laor:lineE>6.4 3 -‐0.1 -‐10 -‐10 10 102:laor:Index>
• BUT - physical interpretation...
‘Lamppost Model’
4
PLC
RDC
Hard X-ray source in coronaIC scattering of seed photons
Reflection from accretion disc. Atomic lines imprinted (reflionx)
• Reflected power per unit area from disc.
• Flux received at point on disc falls off with distance from X-ray source.
Emissivity Profile
5
• Reflected power per unit area from disc.
• Flux received at point on disc falls off with distance from X-ray source.
Emissivity Profile
5
F ∝ 1d2
=1
r2 + h2
d
h
• e.g. Euclidean space
r 1e-05
0.0001
0.001
0.01
0.1
1 10 100
• Depends on
• Source location/height
• Source extent
• Source/disc geometry
Emissivity Profile
6
• Constrain these from observed emissivity profiles?
• Typically assume (broken) power law emissivity profile.
• Can we determine emissivity profile from observed spectrum?
Emissivity Profiles from Spectra
8
105
11.
52
2.5
3
ratio
Energy (keV)
data/model
drw 5 May 2010 16:38
• Gravitational redshift
• Doppler shift
Broadened Emission Lines
9
F0(ν0) =
Ie(re,ν0
g)T (re, g)dgredre
I(re, ν) = δ(ν − νe)(re)
• Consider the line as sum of contributions from successive radii.
• Number of photons from each annulus
• A projected to observer.
Emissivity from Broad Lines
10
F0(ν0) =
T (re, g)redre(re)
N(r) ∝ A(r)(r)
• Model spectrum
• For K-line (3-10 keV), only dominant features are power law and reflected line.
• Power law, inclination and reflionx parameters from best fit to whole spectrum (Zoghbi+09).
• Fit for norm of each reflionx component.
• Divide by projected area of each annulus (transfer fn).
Emissivity from Broad Lines
11
powerlaw +
kdblur⊗ reflionx
10 3
0.01
norm
aliz
ed c
ount
s s
1 keV
1
data and folded model
105
0.8
1
1.2
ratio
Energy (keV)drw 20 May 201
χ2 = 255.48
χ2 / NDoF = 1.1255
12
1H0707-495
• kdblur3 model
• As kdblur2
• Convolve with relativistic broad line profile but with twice-broken power law emissivity.
• Fit emissivity params.
Test Result with XSPEC Model
15
kdblur2 kdblur3
Index 1 4.82 7.72
R break 1 6.85RG 5.53RG
Index 2 2.09 8.46x10-5
R break 2 — 34.73RG
Index 3 — 3.38
χ2 287.52 272.39
χ2 / Ndof 1.11 1.07
3-10keV : powerlaw + kdblur⊗reflionx
Theoretical PredictionsRaytracing from the source to the accretion disc...
Building on work of Miniutti et al, Suebsuwong, Malzac et al in light of observations.
Develop a formalism for analysis...
xa + Γabcx
bxc = 0 gabxaxb = 0
• Light rays follow null geodesics in Kerr spacetime around (rotating) black hole.
Ray Tracing
18
ds2 = c2
1− 2µr
ρ2
dt2 +
4µacr sin2 θ
ρ2dtdϕ− ρ2
∆dr2 − ρ2dθ2
−
r2 + a2 +2µa2r sin2 θ
ρ2
sin2 θdϕ2
• Solve geodesic equations of the null (photon) geodesics...
Derivatives w.r.t. affine parameter, σ
Geodesic Equations
19
t =(r2 + a2 cos2 θ)(r2 + a2) + 2µa2r sin2 θ
k − 2µar
c h
r21 + a2 cos2 θ
r2 − 2µr
(r2 + a2) + 2µa2r sin2 θ
ϕ =2µacrk sin2 θ + (r2 + a2 cos2 θ − 2µr)h
(r2 + a2)(r2 + a2 cos2 θ − 2µr) sin2 θ + 2µa2r sin4 θ
θ2 =Q + (kca cos θ − h cot θ)(kca cos θ + h cot θ)
ρ4
r2 =∆ρ2
kc2t− hϕ− ρ2θ2 − 2
• So, given starting position of a photon and its constants of motion (initial direction), can propagate it.
• Affine parameter step variable as required:
(and similar in θ, ϕ – take the smallest, with limit).
Ray Tracing
20
r(σ + dσ) = r(σ) + rdσ
dσ = r−rH
r
τ
e(a) · e
(b) = η(a)(b)
• Source frame (flat)
• Photons at equal intervals in cos α and β
• Calculate h, Q from α and β (set k=1)
• Isotropic Point Source
• Equal power radiated into equal solid angle, in source frame.
The Source
21
dΩ = d(cos α)dβ
α
βei
e’i
θϕ
r
dΩ’
-10
-5
0
5
10
-10-5
0 5
10
0
2
4
6
8
10
"geodesic_x.dat"
• Kerr Black Hole, a = 0.998
• 6RG
24
Frame dragging
-4
-2
0
2
4
-4-2 0 2 4
0
1
2
3
4
5
"geodesic_x.dat"
• Kerr Black Hole, a = 0.998
• 3RG, θ = π / 4
26
• Trace rays until they hit disc (or you get bored).
• Disc divided into radial bins.
• Count photons in bin.
• Emissivity – divide by area of annulus (rdr from definition of T!!!)
27
Emissivity Profiles from Ray Tracing
• Develop modular ray tracing code in Fortran 95 and C++.
• Re-use same library for ray tracing in any context.
• Parallelised
• For each point source, divide parameter space (cos α) equally between cluster nodes – each traces a set of rays.
• Master node collects and sums radial bins.
The Code
28
0.001
0.01
0.1
1
10
100
1000
10000
100000
1 10 100 1000
h = 3Rgh = 6Rg
h = 10Rgh = 20Rgh = 50Rg
Axial Source
29
50RG: α~2.7
3RG: α~3.4
50RG: α~3
0.001
0.01
0.1
1
10
100
1000
10000
100000
1 10 100 1000
a = 0a = 0.998
Axial Source – Schwarzschild
30
3RG
Lesser effect at greater source height.
0.001
0.01
0.1
1
10
100
1000
10000
100000
1 10 100 1000
th=pi/4, r=3Rgth=pi/4, r=6Rg
Ring Source
31
Symmetry... equivalent to
• Stationary sources do not reproduce steep emissivity profile in centre.
• Stationary ring source a bit unphysical!
• Moving source (e.g. ‘co-rotating’ with disc).
• Beaming of emission into ‘forward’ direction
• Increased emission onto inner disc.
• Coding this...
Moving Source
32
• Determination of emissivity profile from observed spectrum of 1H0707-495.
• Twice-broken power law.
• Fits with kdblur3 model.
• Beginnings of ray tracing simulations/theoretical profiles.
• Can reproduce index at large r, but not steep profile for inner disc.
Conclusions
33