Download Lecture 4

Astrophysical Radiation
Processes
4: Relativistic effects II
Dr. J. Hatchell, Physics 407, [email protected]
Course structure
1.  Radiation basics. Radiative transfer.
2.  Accelerated charges produce radiation. Larmor formula.
Acceleration in electric and magnetic fields – non-relativistic
bremsstrahlung and gyrotron radiation.
3.  Relativistic modifications I. Doppler shift and photon
momentum. Thomson, Compton and inverse Compton
scattering.
4.  Relativisitic modifications II. Emission and arrival times.
Superluminal motion and relativistic beaming. Gyrotron,
cyclotron and synchrotron beaming. Acceleration in particle
rest frame.
5.  Bremsstrahlung and synchrotron spectra.
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
PHY3145 Topics in Theoretical Physics
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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Motion in the particle’s rest frame
Aim: Calculate motion in particle’s rest frame (usually so that we
can apply Larmor formula)
Method: Lorentz Transform position, fields etc. from observer’s
frame S to comoving frame S’ (standard relativity convention)
S y
S’ y’
v
L.T.
v
x
Observer’s frame
x’
Particle’s rest frame
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
Example:
Synchrotron particle motion and acceleration
in rest frame
Aims: i.  Relativistic motion of charged particle in magnetic field; ii.  Acceleration in particle rest frame (needed for Larmor
formula)
iii.  Total power
Method:
i.  Lorentz force in lab. frame
ii.  Lorentz transform B into instantaneous rest frame. Use
Lorentz force to calculate acceleration.
iii.  Larmor’s formula in particle rest frame
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Examples
•  Acceleration in E-field: Bremsstrahlung (relativistic) - HOMEWORK
•  Acceleration in B-field: Cyclotron/Synchrotron
2
Synchrotron motion (continued…):
B
i. Relativistic motion of charged particle
in magnetic field
θ
Use Lorentz force in lab. frame
a
γv ma = qvB sin θ
r
v
Equate with centripetal acceleration
v⊥ 2
v 2 sin2 θ
qvB sin θ
=
=
r
r
γv m
Angular velocity – relativistic gyrofrequency
θ is the pitch angle
(angle between B
and v)
ωr =
v⊥
qB
ωg
=
=
r
γv m
γv
Synchrotron acceleration (continued…):
z
B
Instantaneously take v along x and B in x-z plane
y
v
B = (Bx , 0, Bz ) = (B cos θ, 0, B sin θ)
x
E = (0, 0, 0)
S
ii. Acceleration in charge rest frame. Lorentz transform E and B
and apply Lorentz force remembering that in rest frame v ! = 0
Ex! =
Ey!
Ez!
=
=
Ex
γ(Ey − vBz )
γ(Ez + vBy )
=0
Bx! =
Bx
= −γvB sin θ
By! =
γ(By + v/c2 Ez )
=0
=
γ(Bz − v/c Ey )
= γB sin θ
=0
Bz!
= B cos θ
2
!
γv! ma! = q(v! × B! + E! ) only has non-zero terms in E
a! =
Dr J. Hatchell
γv qvB sin θ
m
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
S
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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iii) Emitted power. Use Larmor’s formula to get the
emitted power in the particle rest frame
Total emitted power in observer’s frame is the same as in
charge rest frame (dW/dt Lorentz invariant): Gyroradiation (non-relativistic)
Power varies as sin2 (ωgt) so electric field varies as sin(ωgt). All
radiation emitted at gyrofrequency of the charged particle.
(NB independent of
velocity)
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Emitted power in rest and observer’s frame
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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Results:
S’ y’
v
u’
L.T.
θ’
S y
u
θ
x
Observer’s frame
x’
Aberration formula
For photons u’=c
also
Relativistic beaming
Aim: How does the viewed power pattern of emission from an
accelerated particle change when the particle moves relativistically
with respect to an observer?
Quick estimate: put θ’ = π/2 in aberration formula.
S’
v
Particle rest frame
Dr J. Hatchell
S
a
Observer’s frame
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Relativistic aberration
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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v
a
a
Rest frame
Lab frame
Gyroradiation (non-relativistic)
Power varies as sin2 (ωgt) so electric field varies as sin(ωgt). All
radiation emitted at gyrofrequency of the charged particle.
(NB independent of
velocity)
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
2b (ii) Relativistic beaming
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
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Aim: Cyclotron frequency spectrum
Method: Consider increasing narrowing of pulse due to beaming
(qualitatively).
Result (without derivation): Frequency spectrum consists of a
series of harmonics
ωl =
lωg
γ(1 − (v! /c) cos θ)
θ is angle
of B to
LOS
Doppler factor for motion parallel to B
simplifying to ωl =
lωg
γ
for motion perpendicular to B
X-ray pulsar cyclotron harmonics
X-ray pulsar V0332+53
Image: Tsygankov et al. 2006 MNRAS 371, 19
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Cyclotron (mildly relativistic)
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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amplitude
time
cos(ωg t)
cos(ωg t) + a1 cos(2ωg t)
7
!
al cos(lωg t)
l=1
Gyrofrequency
only
Gyrofrequency +
first harmonic
l=1,2
Gyrofrequency +
first six harmonics
l=1,2,3,4,5,6
Synchrotron beaming (γ >> 1)
Aim: Understand (qualitatively) pulse length of received power and
therefore frequency of received radiation
Method: Combine relativistic beaming (Lecture I) with relativistic
motion around circle.
Result (see lectures):
duration of pulse
S
A
Dr J. Hatchell
B
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Cyclotron harmonics: demonstration
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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Maximum frequency component
log(P)
log(ν)
Exact
result
Diamond Synchrotron
High power X-ray / UV synchrotron
radiation used experimentally to probe
properties of materials.
Images: www.diamond.ac.uk
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
Synchrotron spectrum for single charge
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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i. Acceleration of electron and total power
Bremsstrahlung = free-free radiation = radiation by an unbound
charged particle due to acceleration by another charged particle
S y S’ y’
eE
Ze
v
b
x’
x
Relativistic accelerations
Total radiated power from a single electron-ion encounter
Dr J. Hatchell
3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell
3145 Topics in Theoretical Physics Astrophysical Radiation Processes
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