High Energy Astrophysics - Jodrell Bank Centre for Astrophysics

High Energy Astrophysics
Radiation Processes 1/4
Giampaolo Pisano
Jodrell Bank Centre for Astrophysics - University of Manchester
giampaolo.pisano@manchester.ac.uk - http://www.jb.man.ac.uk/~gp/
February 2012

Radiation Processes
- There are many radiation processes in Astrophysics:
- Bremsstrahlung (free-free emission)
- Compton Scattering
- Synchrotron emission
- Absorption processes
- Self-absorption
- Pair production
- Ionisation Losses
- Cherenkov radiation
- Nuclear interactions
Not discussed in this course

Thermal and non-thermal processes
- Thermal processes
- Black-body radiation
- Thermal Bremsstrahlung
- Non-thermal processes
- Non-thermal Bremsstrahlung
- Inverse Compton scattering
- Synchrotron radiation
Continuum radiation from particles with
energy spectrum not Maxwellian

Radiation Processes
- Thermal and non-thermal processes
- Radiation from accelerated charged particles
- Bremsstrahlung (Free-free emission)
- Compton Scattering
- Synchrotron emission
- Absorption processes
- Competing processes
References:
- Rybicki & Lightman - Chapter 1
- Rosswog, Bruggen - Par. 3.7
- Longair - Vol 1, Chapter 3

Radiation from accelerated charged particles 1/6
J.J. Thomson’s treatment : radiation origin in terms of electric field lines
Electric field
lines
(the standard complex derivation uses retarded potentials)
1) Stationary q in inertial frame at t=0
2) Small acceleration to ∆v in ∆t :
const v
3) After t, consider sphere radius r=ct :
q
∆v
c∆t
- Outside sphere:
field lines do not yet know that
charge has moved radial to O
- Inside sphere:
field lines radial in reference
frame of moving charge
∆v

Radiation from accelerated charged particles 2/6
E r Radial component
(Coulomb’s law)
E r
E ϑ Tangential component
E ϑ
ϑ
∆v
Emitted
at t=0
Emitted
after ∆t
r=ct
c∆t
Longair
Par 3.3.2
Watch very nice animation at:
http://www.ligo.caltech.edu/~tdcreigh/Radiation/
Eϑ ∆v
t sinϑ
=
E c∆t
r
→
⎛ ∆v
⎞⎛
t ⎞
Eϑ = Er
⎜ ⎟⎜
⎟sinϑ
⎝ ∆t
⎠⎝
c ⎠

Radiation from accelerated charged particles 3/6
E r Radial component
(Coulomb’s law)
E r
E ϑ
E ϑ Tangential component
( t=r/c )
ϑ
⎛ ∆v
⎞⎛ t ⎞
Eϑ = Er
⎜ ⎟⎜
⎟sinϑ
=
⎝ ∆
t
⎠⎝
c
⎠
q
4πε
r
&& r
r
2
2
0 c
sinϑ
∆v
Null E ϑ
&& p sinϑ
4πε
c r
Eϑ
=
2
0
-Tangential
electric field
p =
qr
- Dipole moment
r=ct
Max E ϑ
c∆t
Note: E ϑ α sinϑ /r
Pulse of radiation that propagates at c
Energy loss from accelerated charged particle

Radiation from accelerated charged particles 4/6
- The energy flux of the e.m. radiation pulse through dA in dt at distance r
is given by the Poynting vector:
2
2
⎛ dE ⎞
E
− ⎜ ⎟ = S = E × H ϑ ⎛ && p sinϑ
⎞ && p sin ϑ
= cε
0
=
2
2 3 2
⎝ dAdt ⎠
⎜
rad
4
⎟
Z0
⎝ πε
0c
r ⎠ 16π
ε
0c
r
2
2
⎛
⎜being
⎝
Z
0
=
µ
0
ε
0
, c =
1 ⎞
⎟
ε
0µ
0 ⎠
=
Energy loss rate α sin 2 ϑ
- The energy loss rate per solid angle dΩ and time dt at distance r is:
−
⎛
⎜
⎝
dE
dtd
⎞
⎟
Ω ⎠
rad
⎛
= −⎜
⎝
dE
dAdt
⎞
⎟
⎠
rad
r
2
=
Sr
2
=
&& p
2
sin
0
2
2
16π
ε c
ϑ
3
⎛
dA
⎜being dΩ
=
2
⎝
r
⎟
⎠
⎞
⎛ dE ⎞
− ⎜ ⎟
⎝ dtdΩ
⎠
rad
=
&& p
2
sin
0
2
2
16π
ε c
ϑ
3
- Energy loss per unit
time and solid angle

Radiation from accelerated charged particles 5/6
- Integrating over the solid angle:
⎛
− ⎜
⎝
dE
dt
⎞
⎟
⎠
rad
= −
∫
⎛ dE ⎞
⎜ ⎟
⎝ dtdΩ
⎠
rad
dΩ
2
2
&& p sin ϑ
= ∫ dΩ
2 3
16π
ε c
0
2
&& p
2
= ∫ sin ϑdΩ
2 3
16π
ε c
0
⎛
⎜but
⎝
8
∫ sin ϑdΩ =
3
2 π
⎞
⎟
⎠
⎛ dE
− ⎜
⎝ dt
⎞
⎟
⎠
rad
=
q
2
&& r
2
=
&& p
3
3
6πε
0c
6πε
0c
2
- Larmor’s Formula
p = qr
- Dipole
moment
- Power emitted by a non-relativistic accelerated charge
- Proportional to the square of acceleration and charge
- Valid for any form of acceleration
- ‘Proper’ acceleration: loss rate measured in particle rest frame

Radiation from accelerated charged particles 6/6
Radiation intensity distribution
Polarisation orientation
−
dE
dtdΩ
=
&& p
2
sin
0
2
2
16π
ε c
ϑ
3
&& p sinϑ
4πε
c r
Eϑ
=
2
0
I α sin 2 ϑ
ϑ
E α sinϑ
- Intensity dipolar distribution:
- Null along acceleration vector
- Max at right angles
- Radiation polarised with electric
field lying along acceleration
vector as projected on the sphere

Radiation Processes
- Thermal and non-thermal processes
- Radiation from accelerated charged particles
- Bremsstrahlung (Free-free emission)
- Compton Scattering
- Synchrotron emission
- Absorption processes
- Competing processes
References:
- Longair - Vol 1 - Chapter 3
- Bradt - Chapter 5

- What is Bremsstrahlung emission ?
Bremsstrahlung radiation 1/2
An accelerated charged particle
emits e.m. radiation
e -
γ
Acceleration of e - in the
electrostatic field of a nucleus
Nucleus: +Ze
Bremsstrahlung:
Radiation emitted in the encounter between an e - and a nucleus
(it means ‘braking radiation’)
- Also called Free-free emission because the radiation corresponds
to transitions between unbound states in nucleus field

Bremsstrahlung in Astrophysics
Whenever there is hot ionised gas in the Universe it emits
Free-Free radiation or Bremsstrahlung
Examples:
Radio emission:
- HII regions: compact regions of ionised hydrogen at T~10 4 K
X-ray emission:
- Binary X-ray sources at T~10 7 K
Diffuse X-ray emission:
- Hot intergalactic gas in cluster of galaxies at T~10 8 K

Bremsstrahlung radiation 2/2
- We are going to study three different cases:
- Single electron
- Thermal Bremsstrahlung
- Relativistic Bremsstrahlung (mention)
References:
- Longair - Vol 1 - Chapter 3
- Bradt - Chapter 5

Single electron Bremsstrahlung 1/7
- Radiation associated with the acceleration of a single electron
in the electrostatic field of ions and nuclei
- Under most circumstances the accurate treatment requires QED,
only in the low frequency limit a classical description is appropriate
Classical derivation steps
- To compute the spectrum of the emitted radiation the starting point
is the Larmor’s formula:
⎛ dE
− ⎜
⎝ dt
⎞
⎟
⎠
rad
=
q
2
&& r
0
2
6πε
c
3
We need to calculate the electron acceleration in its reference frame

Single electron Bremsstrahlung 2/7
- The acceleration of the electron during its collision with a nucleus
depends on the charge +Ze and the collision parameter b :
γ
e -
a
b
+Ze
- In the derivation it is necessary to consider the:
- Nucleus field experienced by the relativistic moving electron
- Resolve the acceleration in components parallel and perpendicular to v
- Fourier Transform the acceleration: a(t) a(ω)
- Investigate how the loss rate is transformed between reference systems

Single electron Bremsstrahlung 3/7
- Electric and magnetic fields from a uniformly moving relativistic charge q
b
E
y
B
vt
z
v
q
x
E
E y
E x
Parallel
Component (x)
Perpendicular
Component (y)
t
Rybicki
Lightman
Par 4.6
- Time dependence of the fields from a particle of uniform relativistic velocity
- Note: the acceleration on a test particle is parallel to the total electric field

Single electron Bremsstrahlung 4/7
- The final result, that we will not derive, is the following:
- Intensity spectrum from a single collision
e - - nucleus with collision parameter b:
I(
ω)
∝
2
Z
e
2
m
e
6
2
ω
2
γ v
3
⎡
1 2
⎛
ω
b
⎞ 2
⎛
ω
b ⎞⎤
⎢ K0
⎜ ⎟ + K ⎜ ⎟
2
1 ⎥
⎣γ
⎝ γv
⎠ ⎝ γv
⎠⎦
ω= angular frequency
K 0
and K 1
: modified Bessel functions
of order 0 and 1
Parallel Perpendicular
component component

Single electron Bremsstrahlung 5/7
- Plotting separately the two terms:
from J.D.Jackson
Classical
Electrodynamics
(1975)
Perpendicular
Parallel
- Single Electron
Bremsstrahlung Spectrum
- Perpendicular impulse:
- Greater intensity
- Significant at low frequencies even in the non relativistic case (γ=1)
- Parallel component:
- In the relativistic case it suffers a factor 1/γ 2
Perpendicular impulse dominant contribution in Bremsstrahlung

Single electron Bremsstrahlung 6/7
High frequency:
ω >> γv/b
I(
ω)
∝
2
Z e
m
2
e
6
2
ω
2
γ v
3
⎡ 1
⎢ 2
⎣γ
⎤
+ 1⎥
e
⎦
2ωb
−
γv
Exponential cut-off at high frequencies
Reasons:
1) Cut-off must exist: the electron cannot radiate a photon with
energy greater than its kinetic energy
2) Origin cut-off depends on the duration of the collision:
Major interaction
within ± b
b
b
b
τ ≈
2b
2π
πγv
γv
→ ωcutoff
≈ = ≈
γv
τ b b

Single electron Bremsstrahlung 7/7
2 6
Z e
2 2
m b v
Low frequency: ω >b
Reason:
b
- In the low frequency range the momentum impulse is a delta function:
Duration collision

Bremsstrahlung radiation
- We are going to study three different cases:
- Single electron
- Thermal Bremsstrahlung
- Relativistic Bremsstrahlung
References:
- Longair - Vol 1 - Chapter 3
- Bradt - Chapter 5

Thermal Bremsstrahlung 1/6
- Emission from an hot plasma cloud
e -
γ
+Ze
Single electron emission
Thermal Bremsstrahlung emission
The emission of photons by an hot plasma cloud is due to
‘near’ collisions between electrons and ions

Thermal Bremsstrahlung 2/6
- Plasma physical condition assumptions
- Completely ionised plasma
- Gas with sufficiently high T:
Collisions energetic and frequent enough to keep gas ionised
Depending on the density, to avoid recombination
- Plasma in thermal equilibrium: e - and ions same average kinetic energies
- Ions considered as stationary (in H plasma, e - ~40x faster than protons)
- Non-relativistic speed:
v

Thermal Bremsstrahlung 3/6
- Free-free emission of a hot ionised gas at temperature T
- The velocities of non-relativistic electrons in a ionised gas can be
described by a Maxwellian distribution:
N e (v)
( T,m e )
Maxwell
distribution
v
N
e
(v) dv
=
4π
N
e
⎛ me
⎜
⎝ 2πkT
⎞
⎟
⎠
3 2
v
2
⎛
exp
⎜ −
⎝
mev
2kT
2
⎟ ⎞
dv
⎠
( N e (v): e - number density
in velocity interval dv)
- Notice the exponential cut-off for high velocities

Thermal Bremsstrahlung 4/6
- It is necessary to integrate the single-electron formula:
I(
ω)
∝
2
Z e
m
2
e
6
2
ω
2
γ v
3
⎡ 1
⎢ 2
⎣γ
K
2
0
⎛ ωb
⎜
⎝ γv
⎞
⎟ +
⎠
K
2
1
⎛ ωb
⎞⎤
⎜ ⎟⎥
⎝ γv
⎠⎦
Over all the possible:
γv/b
- Velocities v
- Impact parameters b (b max & b min )
b
We jump directly to the final results

Thermal Bremsstrahlung 5/6
- The calculations lead to the following:
2
Z NNe ⎛ hω
⎞
I(
ω,
T ) ∝ exp⎜
− ⎟ g(
ω,
T )
T ⎝ kT ⎠
- Plasma spectral emissivity
High frequency
Low frequency
- N: nuclei/ionised atoms number density
- g(ω,T): ‘correction’ Gaunt factor depends on electron velocity
hω >>
kT
hω =
- Exponential cut-off at that reflects the Maxwell distribution tail
Cut-off can be used to determine T plasma
hω

Thermal Bremsstrahlung 6/6
- In summary:
- Spectrum of Thermal Bremsstrahlung
log(I(ω))
τ >>1
I(ω)
∝ const
τ

Thermal Bremsstrahlung in Astrophysics 1/3
- X-ray emission from Cluster of Galaxies
Abell 2029
Optical - HST
X-ray - Chandra
www-xray.ast.cam.ac.uk
- These clusters can contain hundreds or thousands of galaxies
- Largest and most massive objects in the Universe
- Large amount of intergalactic material falling towards the centre
of the cluster reaching very high temperatures

Thermal Bremsstrahlung in Astrophysics 2/3
- X-ray emission in the Perseus Cluster of Galaxies
Optical
X-ray
apod.nasa.gov
- Diffuse Thermal Bremsstrahlung X-ray emission due to hot
intergalactic plasma at T~10 8 K

Thermal Bremsstrahlung in Astrophysics 3/3
- X-ray spectrum of the Perseus Cluster of Galaxies
- Continuum emission from thermal
Bremsstrahlung of hot intergalactic
gas @ T=7.5 x 10 7 K
- Radiation thermal nature confirmed
by highly ionised Fe +25 lines
- Temperature estimate:
from R.Mushotzky
X-ray Astronomy (1980)
hω = hν ~ kT
T
→
h
~ ν ~
k
3
−19
5×
10 × 1.6×
10
~ ~ 6×
10
−23
1.38×
10
T
5KeV
k
7
K

Bremsstrahlung radiation
- We are going to study three different cases:
- Single electron
- Thermal Bremsstrahlung
- Relativistic Bremsstrahlung
References:
- Longair - Vol 1, Chapter 3

Relativistic Bremsstrahlung
- Relativistic particles interacting with ionised gas
Example:
γ-ray emission of the interstellar medium (ISM) caused by the interaction
of relativistic particles with the ionised ISM
- In this case, the energy distribution of the ultra-relativistic cosmic ray
electrons has a power law form (non-thermal):
N e
( E)
dE
∝
E
−α
dE
- Electrons energy
distribution
N(E)dE:# e- per unit
volume in (E,E+dE)
The final spectrum is obtained integrating over the energies

Relativistic Bremsstrahlung in Astrophysics
- Low energy gamma-ray emission of the interstellar medium (ISM)
CGRO / EGRET - 30-50 MeV
- Caused by the interaction of relativistic particles with the ionised ISM

Radiation Processes
- Thermal and non-thermal processes
- Radiation from accelerated charged particles
- Bremsstrahlung (Free-free emission)
- Compton Scattering
- Synchrotron emission
- Absorption processes
- Competing processes