8–8
Compton Scattering
E’,
Thomson scattering: initial and final
wavelength identical.
But: in reality: light consists of photons
=⇒ Scattering: photon changes direction
=⇒ Momentum change
=⇒ Energy change!
This is quantum picture =⇒ Compton
scattering.
p’
θ
E, p
T
Energy/wavelength change (see handout):
E
E
E =
∼E 1−
(1 − cos θ)
E
mec2
1 + me c2 (1 − cos θ)
h
(1 − cos θ)
λ0 − λ =
me c
0
(8.14)
(8.15)
where h/mec = 2.426 × 10−12 m (Compton wavelength).
Averaging over θ, for E mec:
∆E
E
≈−
E
me c2
(8.16)
E.g., at 6.4 keV, ∆E ≈ 0.2 keV.
1
Compton Scattering
8–13
Amplification factor, I
In electron frame of rest,
∆E
E
=−
E
mec2
(8.16)
Assuming a thermal (Maxwell) distribution of electrons (i.e., they’re not at rest),
one can show that the relative energy change is given by
∆E 4kT − E
=
=A
E
mec2
(8.48)
where A is the Compton amplification factor.
Thus:
E . 4kTe =⇒ Photons gain energy, gas cools down.
E & 4kTe =⇒ Photons loose energy, gas heats up.
Thermal Comptonization
1
8–14
Amplification factor, II
In reality, photons will scatter more than once before leaving the hot electron
medium.
The total relative energy change of photons by traversal of a hot (E kTe)
medium with electron density ne and size ` is then approximately
rel. energy change
(rel. energy change y) =
scattering
× (# scatterings)
(8.49)
The number of scatterings is max(τe, τe2), where τe = neσT` (“optical depth”),
such that
y =
4kTe
me
c2
max(τe, τe2)
(8.50)
“Compton y -Parameter”
2
Thermal Comptonization
8–15
Spectral shape
Photon spectra can be found by analytically solving the “Kompaneets equation”, but this is very
difficult.
Approximate spectral shape from the following arguments:
After k scatterings, the energy of a photon with initial energy Ei is approximately
Ek = EiAk
(8.51)
But the probability to undergo k scatterings in a cloud with optical depth τe is pk (τe) = τek
(follows from theory of random walks, note that the mean free path is ` = 1/τe ).
Therefore, if there are N (Ei) photons initially, then the number of photons emerging at energy Ek
is
−α
ln τe
Ek
k
with
α=−
(8.52)
N (Ek ) ∼ N (Ei )A ∼ N (Ei)
Ei
ln A
Comptonization produces power-law spectra.
General solution: Possible via the Monte Carlo method.
Thermal Comptonization
3
Generate Photon
(E,w,r,a)
Generate τ from
exp−distribution
Propagate Photon
Bin photon in
Yes
spectrum
No
Escaped?
Simulate collision
8–17
Spectral shape
10 6
τ=5
kT=200 keV
E N(E) [keV cm −2 s−1 keV −1 ]
10 5
10 4
5
4
10 3
3
2
1
10 2
10 1
10 0
0.01
0.10
1.00
10.00
100.00
1000.00
E [keV]
Monte Carlo simulation shows: Spectrum is =⇒ Power law with exponential
cutoff (here: with additional “Wien hump”, see next slide)
Thermal Comptonization
11
8–18
Spectral shape
10 2
τe=10
E2 N(E) [keV2cm-2 s-1 keV-1 ]
10 1
10 0
τe=5
-1
10
τe=2
τe=1
-2
10
τe=0.05
y 1: pure power-law.
y < 1: power-law with
τe=0.5
10-3
10-4
0.1
1.0
10.0
E [keV]
100.0
1000.0
Sphere with kTe = 0.7me c2 (∼ 360 keV), seed photons come from
center of sphere.
exponential cut-off
y 1: “Saturated
Comptonization”.
Saturated Comptonization has
never been observed.
Thermal Comptonization
12
9–1
Application of Comptonization
9–2
Galactic Black Holes
Typical BHC Spectra
102
E N(E) [keV cm-2 s-1 keV-1]
GS 2000+25, x10
100
Cyg X-1
10-2
GRO J0422+32, x0.1
10-4
TTM
10-6
0.1
OSSE
BBXRT
1.0
HEXE
10.0
E [keV]
100.0
1000.0
(Cyg X-1: Wilms et al., 1996, ; GRO J0422+32, GS2000+25: Sunyaev et al.,
1993, Kroeger [priv. comm.])
X-ray spectra of
galactic black hole
candidates can be
well explained by
thermal
Comptonization in a
plasma with
kT ∼ 150 keV and
with y ∼ 1.
1
Observations
9–3
INTEGRAL/RXTE, I
Observations
2
9–4
χ
keV (keV cm−2 s−1 keV−1)
INTEGRAL/RXTE, III
Fit of a model to RXTE/INTEGRAL
data from the galactic black
hole Cygnus X-1
1.0
kTsoft = 1.21 keV,
τe = 1.09,
kTe ∼ 100 keV
0.1
4
2
0
−2
−4
10
Note presence of a Compton
reflection hump (evidence of
close vicinity of hot electrons
and only mildly ionised
material)
100
Energy [keV]
Fritz, et al., 2005, in prep.
4
Observations
9–5
AGN
NGC 4151 and PG 1416-129
10-2
Photon flux [ph/cm2 s keV]
10-3
10-4
10-5
10-6
10-7
ROSAT
1
GINGA
10
Energy [keV]
OSSE
HEXE
100
(PG 1416−129: de Kool et al., 1994, Williams et al., 1992, Staubert & Maisack, 1996; NGC 4151: Maisack 1991, 1993)
Spectral shape of AGN
very similar to galactic
Black Holes =⇒ Same
physical mechanism
(=Comptonization)
responsible!
Note: NGC 4151 not corrected for interstellar absorption.
Observations
5