Inverse Compton Radiation
Rybicki & Lightman Chapter 7
1 -- Review of the Compton Effect
2 – Review of Thomson Scattering
3 – Klein-Nishima Cross-section
4 – Inverse Compton Scattering
5 – The Compton Catastrophe
6 – Spectrum of IC; Kompaneets Eqn.; y parameter
7 – Synchrotron-Self-Compton
8 -- Sunyaev-Zel’dovich Effect
Inverse Compton
Low-energy photon
Higher-energy
photon
Eemitted ~ γ2 Eincident
Kyle & Skyler 2011
Thomson Scattering
hν << mc2
no electron recoil
RESULTS
(1) Eγ(before) = Eγ(after)
No change in photon energy: “coherent” or “elastic” scattering
(2) Thomson cross-section
T 
8 2
r0
3
(3) For unpolarized incident photons, cross-section per solid angle:
dT 1 2
 r0 1 cos2 
d 2
so...
Incident energy
flux, erg/s/cm2
d dP
s 

d d
Fraction of
incident flux
radiated into θ,φ
per solid angle
radiated power per unit solid
angle erg/s/ster
The Compton Effect
Allow for electron recoil
photon
photon
h

E   h 
e-
E  h 0
p0 
p 

h
0
electron: Kinetic energy after collision = T
momentum = p
(1) Conservation of momentum (x,y components)
p0  p cos  p cos
0  p sin   p sin 
Eliminate

p cos  p cos  p0
p sin    p sin 
by squaring and adding
p2  p02  p2  2 pp0 cos
(Eqn. A)
(2) Conservation of Energy:
or
square 
E0  E  T
T
p0  p  
c
T 
2


p  p  2 p0 p   
c
2
2
o
Subtract (B) from (A):
(Eqn. B)
2
T
p 2  2  2 p0 p1  cos 
c
Relativistic momentum for electron after collision 



p  mv  m0 v
E  mc 2  T  m0 c 2

E is conserved under L.T. so
E  pc  m 0c
2
2

2 2
T  m c   pc  m c 
2 2
0
pc
2
2 2
2
0
 T 2  2Tm0c 2
1 2
p  2 T  2m 0c 2T 
c
2
T
p 2  2  2m 0T  2m 0c( p0  p)
c
m0c p0  p  p0 p1 cos 
2
So
1 1
1


1 cos
p p0 m0c
Since p0  p 
T
c
h
1  cos 
   
m0c
h
  
1 cos 
m0c
Compton Scattering
After scattering, the photon energy has decreased.
o
h
 C  0.02426 A Compton Wa velength
mc
 ~ C
Arthur Compton
Note: Can re-write the Compton formula as
h
h '
1
h
1 cos 
2 
mc
where  ' = scattered frequency
 = incident frequency
The
scattered photon energy is shifted significantly as the
incident photon energy (h) becomes comparable to the rest
energy mc2 of the electron (=0.511 MeV).
Klein-Nishima Cross-Section
Account for quantum mechanical effects: important when
E ( photon) ~ E ( photon)
d 1 2  E f
 r0 
d 2  Ei



2


 Ei E f
2 

 sin  



 E f Ei

Upshot:
Cross-section becomes
smaller at large photon
energies
 Compton scattering
becomes inefficient at
high energies
Inverse Compton Scattering
Now consider the case where the electron is moving at
relativistic velocity, v
We assume that in the rest frame of the electron,
Thomson scattering holds, and then consider what happens
in the lab frame.
Ef
 i
Ef

i
Ei
photon
Ei
Lab Frame
Electron Rest Frame

From the formula for Doppler shifts,
we can write
   1   cos 

 
E f  E f  1  cos f 


E i  E i 1  cos i 
Now suppose the collision is elastic in the rest frame of the
electron, so that

Ei  Ef
For the moment, consider  i 

2
and  f 

2
Then
E f  Ef  and Ei  Ei
so the ratio
 Energyof photon  Energyof photon  Energyof photon
 

 

 beforescattering  :  beforescattering  :  afterscattering 
 in Lab frame
  in e - rest frame   in theLab frame 

 

 
 Ei : Ei : E f
1: :
2
This process converts a low-energy photon to a higher energy photon
by a factor of

2
A similar result holds if we instead assume Compton scattering in
the rest frame.
NOTES:
• This energy gain by the photon is the opposite of the energy loss
suffered during a Compton scattering event.
thus, the process is called
INVERSE COMPTON SCATTERING
• However, this name is slightly misleading because the process is not
the inverse of Compton or Thomson scattering, but is rather
ordinary Compton or Thomson scattering viewed in a frame
in which the electron is highly relativistic.
• One can think of the process as an interaction between
and electron gas and a “photon gas”
- if the electron gas is much “hotter” than the photon gas,
energy flows from the electrons to the photons
 Inverse Compton
- if the photon gas is hotter, energy flows from the photons
to the electrons
 Compton Heating
• Often γ >> 1, so this process can boost, say an IR photon,
(with λ=20 microns) to the very hard X-rays or
gamma rays
 Compton upscattered IR photons may be the
physical process which produces X-ray emission
in quasars
• Even though a particular electron loses a small fraction of
its energy in a single encounter, the cumulative effect of
many scatterings can be an important process of energy loss
– comparable to the energy loss due to synchrotron radiation.
Can show (pp. 199-201 in R&L) that
Ratio of
power emitted
by synch. to
power emitted by
Compton scattering
Psync
PCompton
magnetic field
energy density
UB

U photon
photon energy
density
The Compton Catastrophe
Let τ = the optical depth for Compton scattering
• In a situation where the electrons are relativistic, each photon
increases its energy by γ2 with each scattering event.
• Suppose there is a source of synchrotron photons with
luminosity = L(sync)
The luminosity in Compton scattered photons is
L(comp) ~
  Lsync
2
  Lsync
where K = some constant
However, these Compton photons will themselves be scattered,
carrying off another factor of
so
 2
of energy
Ltotal  Lsync 1    2   3  L

  if   1
You’d never see such a source, since the electrons would radiate
away their energy instantaneously.

This led people to conclude that one must have k<1, i.e.

LComp
Lsync

U photon
UB
1
for any viable model for a synchrotron source.
However, two effects cause L(total) to converge:
1 – As the photons attain higher and higher energies, the average
scattering will result in a smaller and smaller energy loss to the
electron
 eventually the photons will “cool” and the electrons will “heat”
2 – The relevant cross-section for scattering becomes the full
Klein-Nishima formula which is less than the Thomson cross-section,
so Compton scattering becomes less effective as the photon
energy increases.
Inverse Compton Spectra
Refn: Longair: High energy
Astrophysics
• So far, we’ve only considered overall energetics. Obviously, we
also want to compute the spectrum of the radiation, to compare
with observations.
• The spectrum will depend upon
(1) the incident energy spectrum of the photons
(2) the energy distribution of the electrons
and in general, one has to solve the equation of radiative transfer,
to see how many scattering events each photon suffers
• The approach is to
(1) compute what happens when a photon of particular energy
m c2
Eo scatters off an electron of particular energy
(2) average over the photon and electron energy distributions
What is the IC spectrum of a single electron?
Straight-forward calculation shows that the spectrum of
the IC radiation produced by scattering a photon of frequency
0 off an electron with energy  is
Very peaked, even more peaked that synchrotron F(x) function
So the spectrum of IC
(1) Scattering off a power-law distribution of electron energies,
N(E) ~E-p results in a power law for photon energies
with spectral index
s
p 1
2
Identical to the spectral index for synchrotron emission.
(2) Repeated scatterings off a non-power law distribution can
also yield a power law spectrum.
(3) Repeated scatterings off non-relativistic electrons can be
calculated using the “Kompaneets Equation”
Repeated scatterings off non-relativistic electrons , each
with a small energy change, can be calculated using the
“Kompaneets Equation”
Approach: Consider evolution of phase space density of photons
proceeds with time
n(x,p) d3xd3p
The result is a differential equation for n which must be solved
numerically
Kompaneets Equation
f(p) = maxwellian distribution
n( )
d
3
 c  d pd  f ( p)n()(1 n( ))  f (p)n(w)(1 n())
t
d
Photon scatters from frequency w’ to w
Photon scatters from frequency w to w’
1+n(w) term: n(w) increased probability due to stimulated emission
n 1   4 
n 
2
 2 x n  n  
y x x  
x 
kTe
dy 
2 T N e cdt
mec

h
x
kTe


(4) Compton y-parameter
 averagefractional   mean # 

 

y   energy changeper    of

 scattering
  scatterings 

 

When y>>1: Spectrum is altered  “saturated” comptonization
When y < 1: spectrum is not changed much, “coherent scattering”
Synchrotron-Self-Compton Radiation
When a synchrotron source is sufficiently compact
that the synchrotron photons can Inverse Compton
scatter off the relativistic electrons
 emergent spectrum is called “synchrotron self-Compton”
of SSC
Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays
see Aharonian+ 2004 ApJ 614, 897
Synchrotron
Synchrotron
Self-Compton
Spectrum of MKN 501 (Seyfert Galaxy)
Showing synchrotron peak and SSC peak
Galactic Center Hinton & Arahonian 2006
TeV
Gamma rays
Sunyaev-Zel’dovich Effect
CMB photons inverse
Compton scatter off of hot,
X-ray emitting intra-cluster
medium in clusters of
galaxies
 temperature distortion
in CMB
 Cosmological
parameters


Holzapfel et al. 1997, LaRoque et al. 2002

T
k
T
SZE
B
e

f
()
n
dl
e
2T

T
m
c
CMB
e


T

n
T
dl

n
T
D
d
CMB
e
e
e
e
A


• Assume a spherical
cluster
c
• D
 
f(

,z
)
A
H
0
• Need ne and Te
Observations
• X-rays (false color)
– Gives us Te
– Temperature profile
allows us to
calculate Λee
– Allow us to model ne
• Radio (contours)
– Gives us dθ
– ΔT
Bonamente et al. 2006
Observations

2
2
S

n

d
l

n

D
d
X
e
e
e
e
e
A
e



S

n

d
l

n

D
d




T

n
T
dl

n
T
D
d
CMB
e
e
e
e
A


2
2
X e
e ee
e
A
e
Result:
2
(

T
)

1
CMB
ee
D

A
2
S
T

Xe
Carlstrom, Holder, Reese 2002
c
D
 
f(

,z
)
A
H
0
Constraining ΩM
• Can constrain ΩM by determining the
gas-mass fraction
• fgas can be measured directly from SZE
–

T
S ZE
fg a
s
2
T
e
mgas
fgas
mtotal
Measure
d
From
BBN

b

fg a

f
fb 
sb 

M
Growth of Structure
• SZE is z independent
• Can use SZE to find clusters to high z
• This can be used as a probe of cluster abundance,
which is a probe of large scale structure