Compton Scattering Notes | EduRev

: Compton Scattering Notes | EduRev

 Page 1


Astrophysics II, University of Turku 16 January 2009
Lecture 7 : Compton Scattering
7.1 INTRODUCTION
Thomsonscattering,orthescatteringofaphotonbyanelec-
tron at rest, strictly only applies at low photon energy,
i.e. when h?«mec
2
.
If the photon energy is comparable to or greater than
the electron energy, non-classical e?ects must be taken into
account, and the process is called Compton scattering.
A further interesting situation develops when the electron
is moving — in this case energy can be transferred to the
photon, and the process is called inverse Compton scat-
tering.Thislastprocessisanimportantmechanisminhigh
energy astrophysics.
7.2 THOMSON SCATTERING
In Thomson scattering, we have
dsT
dO
=
1
2
r
2
0
(1+cos
2
?) (7.1)
where sT is the Thomson cross-section, O the solid angle,
? is the angle of scattering, and r0 is the classical electron
radius,
r0 =
e
2
mec
2
. (7.2)
In Thomson scattering the incident photon and scatter
photon have the same wavelength or energy, so this scatter-
ing is also called coherent or elastic.
If we now move to photons of energy h?
>
~
mec
2
, the
scattering is modi?ed by the appearence of quantum e?ects,
through a change in the kinematics of the collision, and an
alteration of the cross-section.
7.3 COMPTON SCATTERING
Todothekinematicsofthecollisioncorrectlyathighphoton
energy, momentum and energy must be conserved.
Lettheincidentphotonhaveenergyh? andmomentum
h?/c, the scattered photon have energy h?
'
and momentum
h?
'
/c, and the electron (initially at rest) acquires energy E
and momentum pe. The scattering angle is ?.
Problem7.1 Showthattheenergyofthescatteredphoton
is given by
h?
'
=
h?
1+
h?
mec
2
(1-cos?)
(7.3)
In terms of wavelength, this reduces to
?
'
-? =?C(1-cos?) (7.4)
Figure 7.1. Compton scattering of an incident photon of energy
h? and momentum p to energy h?
'
and momentum p
'
. The elec-
tron is initially at rest and acquires energy E and momentum
pe.
where ? is the incident photon wavelength, ?
'
is the scat-
tered photon wavelength and ?C is the Compton wave-
length and is given by
?C =
h
mec
=0.02426
°
A. (7.5)
Problem 7.2 Prove Eqn 7.4.
The compton wavelength can be regarded as a wave-
length change ?? in the incident photon. Note that for
?»?C the change is negligible and we get back the Thom-
son scattering.
In full treatment of the problem yields the Klein-
Nishina formula for the scattering cross-section:
ds
dO
=
r
2
0
2

?
'
?

2

?
?
'
+
?
'
?
-sin
2
?

(7.6)
which can be shown to yield the following formulae for the
total cross-section (where x =
h?
mc
2
),
s˜sT

1-2x+
26x
2
5
+...

, for x« 1, (7.7)
and
s =
3
8
sT
1
x

ln(2x)+
1
2

, for x»1 (7.8)
for the non-relativistic and extremely relativistic cases. The
main e?ect is thus to reduce the cross-section at high pho-
ton energies, i.e. the scattering of the photons becomes less
e?cient.
7.4 INVERSE COMPTON: SCATTERING
FROM MOVING ELECTRONS
An important case arises when the electrons are no longer
considered to be at rest. In inverse scattering, energy is
transferred from the electrons to the photons, i.e. it is the
oppositeofComptonscattering,inwhichthephotonstrans-
fer energy to the electrons. Inverse Compton scattering can
Page 2


Astrophysics II, University of Turku 16 January 2009
Lecture 7 : Compton Scattering
7.1 INTRODUCTION
Thomsonscattering,orthescatteringofaphotonbyanelec-
tron at rest, strictly only applies at low photon energy,
i.e. when h?«mec
2
.
If the photon energy is comparable to or greater than
the electron energy, non-classical e?ects must be taken into
account, and the process is called Compton scattering.
A further interesting situation develops when the electron
is moving — in this case energy can be transferred to the
photon, and the process is called inverse Compton scat-
tering.Thislastprocessisanimportantmechanisminhigh
energy astrophysics.
7.2 THOMSON SCATTERING
In Thomson scattering, we have
dsT
dO
=
1
2
r
2
0
(1+cos
2
?) (7.1)
where sT is the Thomson cross-section, O the solid angle,
? is the angle of scattering, and r0 is the classical electron
radius,
r0 =
e
2
mec
2
. (7.2)
In Thomson scattering the incident photon and scatter
photon have the same wavelength or energy, so this scatter-
ing is also called coherent or elastic.
If we now move to photons of energy h?
>
~
mec
2
, the
scattering is modi?ed by the appearence of quantum e?ects,
through a change in the kinematics of the collision, and an
alteration of the cross-section.
7.3 COMPTON SCATTERING
Todothekinematicsofthecollisioncorrectlyathighphoton
energy, momentum and energy must be conserved.
Lettheincidentphotonhaveenergyh? andmomentum
h?/c, the scattered photon have energy h?
'
and momentum
h?
'
/c, and the electron (initially at rest) acquires energy E
and momentum pe. The scattering angle is ?.
Problem7.1 Showthattheenergyofthescatteredphoton
is given by
h?
'
=
h?
1+
h?
mec
2
(1-cos?)
(7.3)
In terms of wavelength, this reduces to
?
'
-? =?C(1-cos?) (7.4)
Figure 7.1. Compton scattering of an incident photon of energy
h? and momentum p to energy h?
'
and momentum p
'
. The elec-
tron is initially at rest and acquires energy E and momentum
pe.
where ? is the incident photon wavelength, ?
'
is the scat-
tered photon wavelength and ?C is the Compton wave-
length and is given by
?C =
h
mec
=0.02426
°
A. (7.5)
Problem 7.2 Prove Eqn 7.4.
The compton wavelength can be regarded as a wave-
length change ?? in the incident photon. Note that for
?»?C the change is negligible and we get back the Thom-
son scattering.
In full treatment of the problem yields the Klein-
Nishina formula for the scattering cross-section:
ds
dO
=
r
2
0
2

?
'
?

2

?
?
'
+
?
'
?
-sin
2
?

(7.6)
which can be shown to yield the following formulae for the
total cross-section (where x =
h?
mc
2
),
s˜sT

1-2x+
26x
2
5
+...

, for x« 1, (7.7)
and
s =
3
8
sT
1
x

ln(2x)+
1
2

, for x»1 (7.8)
for the non-relativistic and extremely relativistic cases. The
main e?ect is thus to reduce the cross-section at high pho-
ton energies, i.e. the scattering of the photons becomes less
e?cient.
7.4 INVERSE COMPTON: SCATTERING
FROM MOVING ELECTRONS
An important case arises when the electrons are no longer
considered to be at rest. In inverse scattering, energy is
transferred from the electrons to the photons, i.e. it is the
oppositeofComptonscattering,inwhichthephotonstrans-
fer energy to the electrons. Inverse Compton scattering can
2
produce substantial ?uxes of photons in the optical to X-
ray region. Analysis shows that the mean frequency of the
photons after the collision increases by a factor ?
2
, so that
high frequency radio photons in collisions with relativistic
electrons for which ? is of order 10
3
to 10
4
can beboosted
in the UV and X-ray regions. There is a practical limit to
the amount of boosting possible beyond the Thomson limit
(h? ˜ ?mc
2
), which can be seen from the conservation of
energy
h?
'
=?mc
2
+h?. (7.9)
Scattered photon energies are thus limited to ?mc
2
.
The power emitted in the case of an isotropic distribu-
tion of photons is
PComp =
4
3
scU
rad
?
2
ß
2
(7.10)
whereU
rad
istheradiationenergydensityofthephoton?eld
(before scattering).
Note how similiar this is the power due to synchrotron
emission
P
Synch
=
4
3
scUB?
2
ß
2
(7.11)
where UB is the energy density of the magnetic ?eld. Thus
P
Synch
PComp
=
UB
U
rad
. (7.12)
The losses due to synchrotron and Compton processes
are in the ratio of the magnetic ?eld energy density to the
photon ?eld energy density, and is independent of ?.
The scattered photons may be produced in the source
throughsynchrotronradiation,andifthesearebosstedthen
theresultantphotonsarecalledSynchrotronSelfComp-
ton.
7.5 COMPTONISATION
If the spectrum of a source is primarily determined by
Compton processes it is termedComptonised. In this case
the plasma must be thin enough that other processes, such
as bremsstrahlung, do not dominate the spectrum instead.
The hotter the gas, the more chance of Comptonisation.
Some examples of astrophysical sources in which comp-
tonisation is important are:
• hot gas near binary X-ray sources
• hot plasma in clusters of galaxies
• hot plasma near center of active galactic nuclei
• primordial gas cooling after the Big Bang
7.5.1 Non-relativistic Comptonisation
We consider non-relativistic electrons and photons with en-
ergy h?«mec
2
. From Eqn 7.4 one can show that the rela-
tive change in the photon energy ?E/E is given by
?E
E
=
h?
mec
2
(1-cos?) (7.13)
Problem 7.3 Verify Eqn 7.13
In the electron frame, the scattering is Thomson, and
therefore symmetric around the incident direction, so that
?E
E
=
h?
mec
2
, for h?«mec
2
. (7.14)
This is the average energy increase of the electron for
low photon input energies, h?«mec
2
.
Now consider high energy photons. In this case the
power produced per scattering is given by Eqn. 7.10. For
non-relativistic electrons, ?˜ 1, so for electron velocity v,
PComp =
4
3
sc

v
c

2
. (7.15)
Thenumberofscatteredphotonspersecondisthenum-
berofphotonsencounteredpersecondbyanelectron,which
is given by the photon number density, N
phot
, the photon
velocity c and the Thomson cross-section sT
N
phot
csT. (7.16)
The photon number density is just
N
phot
=
U
rad
h?
(7.17)
so we have
sTN
phot
c =
sTU
rad
c
h?
. (7.18)
Comparison of Eqns. 7.15 and 7.18 readily shows that the
energy gain of the photons per collision must be
?E
E
=
4
3

v
c

2
for h?»mec
2
. (7.19)
Let’s summarise what we have so far:
• For h?«mec
2
, the electrons gain energy. (Eqn 7.14).
• For h?»mec
2
, the photons gain energy. (Eqn 7.19).
7.5.2 Thermal electrons
To make things practical, let’s consider a thermal distribu-
tion of electrons, with temperature Te. We have
3
2
kTe =
1
2
mev
2
(7.20)
where v is the typical electron velocity. Eqn 7.19 can thus
be written
?E
E
=
4kTe
mec
2
for h?«kTe. (7.21)
If we now combine the results of Eqns 7.21 and 7.14 we can
derive a simple equation for the energy gain/loss for both
the high and low frequency regimes
?E
E
=
1
mec
2
(4kTe-h?). (7.22)
Therefore, for
• h? = 4kTe, there is no energy exchange
• h? > 4kTe, electrons gain energy
• h? < 4kTe, photons gain energy
Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are
“hotter” than the photons.
Page 3


Astrophysics II, University of Turku 16 January 2009
Lecture 7 : Compton Scattering
7.1 INTRODUCTION
Thomsonscattering,orthescatteringofaphotonbyanelec-
tron at rest, strictly only applies at low photon energy,
i.e. when h?«mec
2
.
If the photon energy is comparable to or greater than
the electron energy, non-classical e?ects must be taken into
account, and the process is called Compton scattering.
A further interesting situation develops when the electron
is moving — in this case energy can be transferred to the
photon, and the process is called inverse Compton scat-
tering.Thislastprocessisanimportantmechanisminhigh
energy astrophysics.
7.2 THOMSON SCATTERING
In Thomson scattering, we have
dsT
dO
=
1
2
r
2
0
(1+cos
2
?) (7.1)
where sT is the Thomson cross-section, O the solid angle,
? is the angle of scattering, and r0 is the classical electron
radius,
r0 =
e
2
mec
2
. (7.2)
In Thomson scattering the incident photon and scatter
photon have the same wavelength or energy, so this scatter-
ing is also called coherent or elastic.
If we now move to photons of energy h?
>
~
mec
2
, the
scattering is modi?ed by the appearence of quantum e?ects,
through a change in the kinematics of the collision, and an
alteration of the cross-section.
7.3 COMPTON SCATTERING
Todothekinematicsofthecollisioncorrectlyathighphoton
energy, momentum and energy must be conserved.
Lettheincidentphotonhaveenergyh? andmomentum
h?/c, the scattered photon have energy h?
'
and momentum
h?
'
/c, and the electron (initially at rest) acquires energy E
and momentum pe. The scattering angle is ?.
Problem7.1 Showthattheenergyofthescatteredphoton
is given by
h?
'
=
h?
1+
h?
mec
2
(1-cos?)
(7.3)
In terms of wavelength, this reduces to
?
'
-? =?C(1-cos?) (7.4)
Figure 7.1. Compton scattering of an incident photon of energy
h? and momentum p to energy h?
'
and momentum p
'
. The elec-
tron is initially at rest and acquires energy E and momentum
pe.
where ? is the incident photon wavelength, ?
'
is the scat-
tered photon wavelength and ?C is the Compton wave-
length and is given by
?C =
h
mec
=0.02426
°
A. (7.5)
Problem 7.2 Prove Eqn 7.4.
The compton wavelength can be regarded as a wave-
length change ?? in the incident photon. Note that for
?»?C the change is negligible and we get back the Thom-
son scattering.
In full treatment of the problem yields the Klein-
Nishina formula for the scattering cross-section:
ds
dO
=
r
2
0
2

?
'
?

2

?
?
'
+
?
'
?
-sin
2
?

(7.6)
which can be shown to yield the following formulae for the
total cross-section (where x =
h?
mc
2
),
s˜sT

1-2x+
26x
2
5
+...

, for x« 1, (7.7)
and
s =
3
8
sT
1
x

ln(2x)+
1
2

, for x»1 (7.8)
for the non-relativistic and extremely relativistic cases. The
main e?ect is thus to reduce the cross-section at high pho-
ton energies, i.e. the scattering of the photons becomes less
e?cient.
7.4 INVERSE COMPTON: SCATTERING
FROM MOVING ELECTRONS
An important case arises when the electrons are no longer
considered to be at rest. In inverse scattering, energy is
transferred from the electrons to the photons, i.e. it is the
oppositeofComptonscattering,inwhichthephotonstrans-
fer energy to the electrons. Inverse Compton scattering can
2
produce substantial ?uxes of photons in the optical to X-
ray region. Analysis shows that the mean frequency of the
photons after the collision increases by a factor ?
2
, so that
high frequency radio photons in collisions with relativistic
electrons for which ? is of order 10
3
to 10
4
can beboosted
in the UV and X-ray regions. There is a practical limit to
the amount of boosting possible beyond the Thomson limit
(h? ˜ ?mc
2
), which can be seen from the conservation of
energy
h?
'
=?mc
2
+h?. (7.9)
Scattered photon energies are thus limited to ?mc
2
.
The power emitted in the case of an isotropic distribu-
tion of photons is
PComp =
4
3
scU
rad
?
2
ß
2
(7.10)
whereU
rad
istheradiationenergydensityofthephoton?eld
(before scattering).
Note how similiar this is the power due to synchrotron
emission
P
Synch
=
4
3
scUB?
2
ß
2
(7.11)
where UB is the energy density of the magnetic ?eld. Thus
P
Synch
PComp
=
UB
U
rad
. (7.12)
The losses due to synchrotron and Compton processes
are in the ratio of the magnetic ?eld energy density to the
photon ?eld energy density, and is independent of ?.
The scattered photons may be produced in the source
throughsynchrotronradiation,andifthesearebosstedthen
theresultantphotonsarecalledSynchrotronSelfComp-
ton.
7.5 COMPTONISATION
If the spectrum of a source is primarily determined by
Compton processes it is termedComptonised. In this case
the plasma must be thin enough that other processes, such
as bremsstrahlung, do not dominate the spectrum instead.
The hotter the gas, the more chance of Comptonisation.
Some examples of astrophysical sources in which comp-
tonisation is important are:
• hot gas near binary X-ray sources
• hot plasma in clusters of galaxies
• hot plasma near center of active galactic nuclei
• primordial gas cooling after the Big Bang
7.5.1 Non-relativistic Comptonisation
We consider non-relativistic electrons and photons with en-
ergy h?«mec
2
. From Eqn 7.4 one can show that the rela-
tive change in the photon energy ?E/E is given by
?E
E
=
h?
mec
2
(1-cos?) (7.13)
Problem 7.3 Verify Eqn 7.13
In the electron frame, the scattering is Thomson, and
therefore symmetric around the incident direction, so that
?E
E
=
h?
mec
2
, for h?«mec
2
. (7.14)
This is the average energy increase of the electron for
low photon input energies, h?«mec
2
.
Now consider high energy photons. In this case the
power produced per scattering is given by Eqn. 7.10. For
non-relativistic electrons, ?˜ 1, so for electron velocity v,
PComp =
4
3
sc

v
c

2
. (7.15)
Thenumberofscatteredphotonspersecondisthenum-
berofphotonsencounteredpersecondbyanelectron,which
is given by the photon number density, N
phot
, the photon
velocity c and the Thomson cross-section sT
N
phot
csT. (7.16)
The photon number density is just
N
phot
=
U
rad
h?
(7.17)
so we have
sTN
phot
c =
sTU
rad
c
h?
. (7.18)
Comparison of Eqns. 7.15 and 7.18 readily shows that the
energy gain of the photons per collision must be
?E
E
=
4
3

v
c

2
for h?»mec
2
. (7.19)
Let’s summarise what we have so far:
• For h?«mec
2
, the electrons gain energy. (Eqn 7.14).
• For h?»mec
2
, the photons gain energy. (Eqn 7.19).
7.5.2 Thermal electrons
To make things practical, let’s consider a thermal distribu-
tion of electrons, with temperature Te. We have
3
2
kTe =
1
2
mev
2
(7.20)
where v is the typical electron velocity. Eqn 7.19 can thus
be written
?E
E
=
4kTe
mec
2
for h?«kTe. (7.21)
If we now combine the results of Eqns 7.21 and 7.14 we can
derive a simple equation for the energy gain/loss for both
the high and low frequency regimes
?E
E
=
1
mec
2
(4kTe-h?). (7.22)
Therefore, for
• h? = 4kTe, there is no energy exchange
• h? > 4kTe, electrons gain energy
• h? < 4kTe, photons gain energy
Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are
“hotter” than the photons.
Lecture 7 : Compton Scattering 3
7.6 THE COMPTON PARAMETER
Nowconsideraplasmacloudofelectrondensityne andchar-
acteristicsizeD.Theopticaldepthte toComptonscattering
is just
te =nesTD. (7.23)
If te»1, the cloud is optically thick, and a large num-
ber of scatterings are required for a photon to escape from
the cloud. The path taken by the photon to get out of the
cloud is termed arandom walk. Figure 7.2 illustrates two
possible paths for a photon executing a random walk from
the center of an optically thick region until it reaches the
edge and escapes.
Suppose the photon typically moves a distance l before
being scattered. In order to escape from the cloud, the pho-
ton must move a distance D where
D =
v
Nl (7.24)
where N is the number of scatterings.
Problem 7.4 Consider a photon emitted in an in?nite,
homogeneous scattering region. Let it travel a distance r1
before the ?rst scatter, a distancer2 before the second scat-
ter, and so on. The displacement of the photon R after N
scatterings
R =r1 +r2 +r3 +...r
N
(7.25)
Show that the mean total displacement D = |R| is of
order
D =
v
Nl (7.26)
where l is the mean free path. The optical thickness t of the
region is of order R/l. Show that the number of scatterings
is given by
N ˜t
2
. (7.27)
The mean free path l is given by
l =
1
nesT
. (7.28)
After one scattering, the initial energy E is increased by
?E, where
?E
E
= 1+
4kTe
mec
2
(7.29)
and so after N scatterings, the energy is E
'
where
E
'
E
=

1+
4kTe
mec
2

N
. (7.30)
We now de?ne y, the called the Compton y-parameter, as
y = (Number of scatterings)×(Energy gain/scattering).
7.6.1 Weak Comptonisation
Problem 7.5 By assumption, 4kTe « mec
2
. Use this to
show that
Figure 7.2. Compton scattering of a photon in a mildly optically
thick region. The photon begins at the central dot and executes
a random walk until it reaches the edge of the cloud and escapes.
A shorter and a longer random walk are shown.
E
'
E
=e
4y
. (7.31)
This is the energy gain for weak Comptonisation,
in which the photon energy remains small relative to the
electron energy.
7.6.2 Strong Comptonisation
In strong Comptonisation, the photon energy is increased
until the electron and photon energy distributions approach
equilibrium, i.e. there is further no net energy gain by
one population to the other. In this case the photons are
“heated” to such a temperature that
h? =4kTe. (7.32)
Lettheopticalthicknessnecessaryforthistohappenbe
t. If the medium is optically thick, then a large number of
scatterings are required for photons to escape. The number
of scatterings N is given by Eqn 7.27, and it thus follows
from Eqn 7.31 that
4kTe
h?0
= exp
h
4
kTe
mec
2
t
2
i
. (7.33)
In this case the medium will be strongly Comptonised,
and the photon spectrum will approach an equilibrium form
given by
Page 4


Astrophysics II, University of Turku 16 January 2009
Lecture 7 : Compton Scattering
7.1 INTRODUCTION
Thomsonscattering,orthescatteringofaphotonbyanelec-
tron at rest, strictly only applies at low photon energy,
i.e. when h?«mec
2
.
If the photon energy is comparable to or greater than
the electron energy, non-classical e?ects must be taken into
account, and the process is called Compton scattering.
A further interesting situation develops when the electron
is moving — in this case energy can be transferred to the
photon, and the process is called inverse Compton scat-
tering.Thislastprocessisanimportantmechanisminhigh
energy astrophysics.
7.2 THOMSON SCATTERING
In Thomson scattering, we have
dsT
dO
=
1
2
r
2
0
(1+cos
2
?) (7.1)
where sT is the Thomson cross-section, O the solid angle,
? is the angle of scattering, and r0 is the classical electron
radius,
r0 =
e
2
mec
2
. (7.2)
In Thomson scattering the incident photon and scatter
photon have the same wavelength or energy, so this scatter-
ing is also called coherent or elastic.
If we now move to photons of energy h?
>
~
mec
2
, the
scattering is modi?ed by the appearence of quantum e?ects,
through a change in the kinematics of the collision, and an
alteration of the cross-section.
7.3 COMPTON SCATTERING
Todothekinematicsofthecollisioncorrectlyathighphoton
energy, momentum and energy must be conserved.
Lettheincidentphotonhaveenergyh? andmomentum
h?/c, the scattered photon have energy h?
'
and momentum
h?
'
/c, and the electron (initially at rest) acquires energy E
and momentum pe. The scattering angle is ?.
Problem7.1 Showthattheenergyofthescatteredphoton
is given by
h?
'
=
h?
1+
h?
mec
2
(1-cos?)
(7.3)
In terms of wavelength, this reduces to
?
'
-? =?C(1-cos?) (7.4)
Figure 7.1. Compton scattering of an incident photon of energy
h? and momentum p to energy h?
'
and momentum p
'
. The elec-
tron is initially at rest and acquires energy E and momentum
pe.
where ? is the incident photon wavelength, ?
'
is the scat-
tered photon wavelength and ?C is the Compton wave-
length and is given by
?C =
h
mec
=0.02426
°
A. (7.5)
Problem 7.2 Prove Eqn 7.4.
The compton wavelength can be regarded as a wave-
length change ?? in the incident photon. Note that for
?»?C the change is negligible and we get back the Thom-
son scattering.
In full treatment of the problem yields the Klein-
Nishina formula for the scattering cross-section:
ds
dO
=
r
2
0
2

?
'
?

2

?
?
'
+
?
'
?
-sin
2
?

(7.6)
which can be shown to yield the following formulae for the
total cross-section (where x =
h?
mc
2
),
s˜sT

1-2x+
26x
2
5
+...

, for x« 1, (7.7)
and
s =
3
8
sT
1
x

ln(2x)+
1
2

, for x»1 (7.8)
for the non-relativistic and extremely relativistic cases. The
main e?ect is thus to reduce the cross-section at high pho-
ton energies, i.e. the scattering of the photons becomes less
e?cient.
7.4 INVERSE COMPTON: SCATTERING
FROM MOVING ELECTRONS
An important case arises when the electrons are no longer
considered to be at rest. In inverse scattering, energy is
transferred from the electrons to the photons, i.e. it is the
oppositeofComptonscattering,inwhichthephotonstrans-
fer energy to the electrons. Inverse Compton scattering can
2
produce substantial ?uxes of photons in the optical to X-
ray region. Analysis shows that the mean frequency of the
photons after the collision increases by a factor ?
2
, so that
high frequency radio photons in collisions with relativistic
electrons for which ? is of order 10
3
to 10
4
can beboosted
in the UV and X-ray regions. There is a practical limit to
the amount of boosting possible beyond the Thomson limit
(h? ˜ ?mc
2
), which can be seen from the conservation of
energy
h?
'
=?mc
2
+h?. (7.9)
Scattered photon energies are thus limited to ?mc
2
.
The power emitted in the case of an isotropic distribu-
tion of photons is
PComp =
4
3
scU
rad
?
2
ß
2
(7.10)
whereU
rad
istheradiationenergydensityofthephoton?eld
(before scattering).
Note how similiar this is the power due to synchrotron
emission
P
Synch
=
4
3
scUB?
2
ß
2
(7.11)
where UB is the energy density of the magnetic ?eld. Thus
P
Synch
PComp
=
UB
U
rad
. (7.12)
The losses due to synchrotron and Compton processes
are in the ratio of the magnetic ?eld energy density to the
photon ?eld energy density, and is independent of ?.
The scattered photons may be produced in the source
throughsynchrotronradiation,andifthesearebosstedthen
theresultantphotonsarecalledSynchrotronSelfComp-
ton.
7.5 COMPTONISATION
If the spectrum of a source is primarily determined by
Compton processes it is termedComptonised. In this case
the plasma must be thin enough that other processes, such
as bremsstrahlung, do not dominate the spectrum instead.
The hotter the gas, the more chance of Comptonisation.
Some examples of astrophysical sources in which comp-
tonisation is important are:
• hot gas near binary X-ray sources
• hot plasma in clusters of galaxies
• hot plasma near center of active galactic nuclei
• primordial gas cooling after the Big Bang
7.5.1 Non-relativistic Comptonisation
We consider non-relativistic electrons and photons with en-
ergy h?«mec
2
. From Eqn 7.4 one can show that the rela-
tive change in the photon energy ?E/E is given by
?E
E
=
h?
mec
2
(1-cos?) (7.13)
Problem 7.3 Verify Eqn 7.13
In the electron frame, the scattering is Thomson, and
therefore symmetric around the incident direction, so that
?E
E
=
h?
mec
2
, for h?«mec
2
. (7.14)
This is the average energy increase of the electron for
low photon input energies, h?«mec
2
.
Now consider high energy photons. In this case the
power produced per scattering is given by Eqn. 7.10. For
non-relativistic electrons, ?˜ 1, so for electron velocity v,
PComp =
4
3
sc

v
c

2
. (7.15)
Thenumberofscatteredphotonspersecondisthenum-
berofphotonsencounteredpersecondbyanelectron,which
is given by the photon number density, N
phot
, the photon
velocity c and the Thomson cross-section sT
N
phot
csT. (7.16)
The photon number density is just
N
phot
=
U
rad
h?
(7.17)
so we have
sTN
phot
c =
sTU
rad
c
h?
. (7.18)
Comparison of Eqns. 7.15 and 7.18 readily shows that the
energy gain of the photons per collision must be
?E
E
=
4
3

v
c

2
for h?»mec
2
. (7.19)
Let’s summarise what we have so far:
• For h?«mec
2
, the electrons gain energy. (Eqn 7.14).
• For h?»mec
2
, the photons gain energy. (Eqn 7.19).
7.5.2 Thermal electrons
To make things practical, let’s consider a thermal distribu-
tion of electrons, with temperature Te. We have
3
2
kTe =
1
2
mev
2
(7.20)
where v is the typical electron velocity. Eqn 7.19 can thus
be written
?E
E
=
4kTe
mec
2
for h?«kTe. (7.21)
If we now combine the results of Eqns 7.21 and 7.14 we can
derive a simple equation for the energy gain/loss for both
the high and low frequency regimes
?E
E
=
1
mec
2
(4kTe-h?). (7.22)
Therefore, for
• h? = 4kTe, there is no energy exchange
• h? > 4kTe, electrons gain energy
• h? < 4kTe, photons gain energy
Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are
“hotter” than the photons.
Lecture 7 : Compton Scattering 3
7.6 THE COMPTON PARAMETER
Nowconsideraplasmacloudofelectrondensityne andchar-
acteristicsizeD.Theopticaldepthte toComptonscattering
is just
te =nesTD. (7.23)
If te»1, the cloud is optically thick, and a large num-
ber of scatterings are required for a photon to escape from
the cloud. The path taken by the photon to get out of the
cloud is termed arandom walk. Figure 7.2 illustrates two
possible paths for a photon executing a random walk from
the center of an optically thick region until it reaches the
edge and escapes.
Suppose the photon typically moves a distance l before
being scattered. In order to escape from the cloud, the pho-
ton must move a distance D where
D =
v
Nl (7.24)
where N is the number of scatterings.
Problem 7.4 Consider a photon emitted in an in?nite,
homogeneous scattering region. Let it travel a distance r1
before the ?rst scatter, a distancer2 before the second scat-
ter, and so on. The displacement of the photon R after N
scatterings
R =r1 +r2 +r3 +...r
N
(7.25)
Show that the mean total displacement D = |R| is of
order
D =
v
Nl (7.26)
where l is the mean free path. The optical thickness t of the
region is of order R/l. Show that the number of scatterings
is given by
N ˜t
2
. (7.27)
The mean free path l is given by
l =
1
nesT
. (7.28)
After one scattering, the initial energy E is increased by
?E, where
?E
E
= 1+
4kTe
mec
2
(7.29)
and so after N scatterings, the energy is E
'
where
E
'
E
=

1+
4kTe
mec
2

N
. (7.30)
We now de?ne y, the called the Compton y-parameter, as
y = (Number of scatterings)×(Energy gain/scattering).
7.6.1 Weak Comptonisation
Problem 7.5 By assumption, 4kTe « mec
2
. Use this to
show that
Figure 7.2. Compton scattering of a photon in a mildly optically
thick region. The photon begins at the central dot and executes
a random walk until it reaches the edge of the cloud and escapes.
A shorter and a longer random walk are shown.
E
'
E
=e
4y
. (7.31)
This is the energy gain for weak Comptonisation,
in which the photon energy remains small relative to the
electron energy.
7.6.2 Strong Comptonisation
In strong Comptonisation, the photon energy is increased
until the electron and photon energy distributions approach
equilibrium, i.e. there is further no net energy gain by
one population to the other. In this case the photons are
“heated” to such a temperature that
h? =4kTe. (7.32)
Lettheopticalthicknessnecessaryforthistohappenbe
t. If the medium is optically thick, then a large number of
scatterings are required for photons to escape. The number
of scatterings N is given by Eqn 7.27, and it thus follows
from Eqn 7.31 that
4kTe
h?0
= exp
h
4
kTe
mec
2
t
2
i
. (7.33)
In this case the medium will be strongly Comptonised,
and the photon spectrum will approach an equilibrium form
given by
4
Figure 7.3. Comptonisation of photons for various values of the
potential, µ. From top to bottom the curves show the spectrum
for µ = 0,0.1,0.3.1.0,3.0 and 10.0. Note that µ = 0 is the pure
black-body curve.
u? =
8ph?
3
c
3
h
exp

h?
kT
+µ
i
-1
. (7.34)
This form, for µ = 0, is just the Planck (or blackbody)
spectrum. When the equilibrium state involves two species
(photons and electrons), then we can introduce a chemical
potential between them, µ. The e?ect on the Planck spec-
trum is illustrated in ?gure 7.3 for various potentials µ.
For µ» 1, the spectrum at high frequency approaches
the Wien spectrum with an extra factor of e
-µ
,
u? =
8ph?
3
c
3
e
-µ
e
-
h?
kT
(7.35)
and at low frequency
u? ??
3
. (7.36)
Figure 7.4 illustrates strong comptonisation of a
bremsstrahlung spectrum in an optically thick, non-
relativistic medium. The bremsstrahlung spectrum dom-
inates at low frequency and shows a characteristic self-
absorption region (I? ? ?
2
) and a ?at region (I? ? ?
0
).
At higher frequency, photons have been multiply scattered
via the Compton process so that a Wien spectrum forms
(I? ??
3
e
-h?/kT
).
7.7 KOMPANEETS EQUATION
In the previous section we derived forms for the spectrum
when the amount of Comptonisation in a non-relativistic
mediumisweakandstrong.Whataboutintermediatecases?
The equation which describes these cases was derived by
Kompaneets in 1949. Its derivation is non-trivial. The equa-
tioninvolvestheevolutionofthedistributionofthephotons
in phase space, n,
?n
?y
=
1
x
2
?
?x
h
x
4

n+n
2
+
?n
?x
i
(7.37)
Figure7.4.SaturatedComptonisationofabremsstrahlungspec-
trum. See text for details.
where
x =
h?
kT
(7.38)
and the compton y-parameter is generalised to an integral
along the photon path
y =
Z
kTe
mec
2
sTnedl (7.39)
and where the terms in the square brackets represent
• the increase/decrease in photon numbers in frequency
space, (the
?n
?x
term)
• cooling of photons/electron recoil, (the n term)
• and the cooling due to induced Compton scattering,
(the n
2
term)
In general, solutions to the Kompaneets equation must
be found numerically, although there are some useful limit-
ing cases which can be solved analytically.
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