1. Cyclotron & Synchrotron Radiation
Rybicki & Lightman
Chapter 6

2. Cyclotron and Synchrotron Radiation
Charged particles are accelerated by B-fields  radiation
“magnetobremsstrahlung”
Cyclotron Radiation
non-relativistic particles
frequency of emission = frequency of gyration
Synchrotron Radiation
relativistic particles
frequency of emission from a single particle  emission at a range of frequencies

3. Astronomical Examples:
(1) Galactic and extragalactic non-thermal radio and X-ray emission
Supernova remnants, radio galaxies, jets
(2) Transient solar events, Jovian radio emission
Synchrotron emission:
reveals presence of B-field, direction
Allows estimates of energy content of particles
Spectrum  energy distribution of electrons
Jet production in many different contexts

4. Equation of motion for a single electron:
Recall
4-momentum
Relativistic equation of motion
see Eqn
(1) or so

5. (2) Let
be divided into
Since
is a constant, and
is a constant,
is a constant

6. (3) Result: Helical motion
- uniform circular motion in plane
perpendicular to B field
- uniform velocity along the field line
(4) The frequency of rotation or gyration is
Remember so
(Larmor frequency)

7. Numerically, the Larmor frequency is
Radius of the orbit
Typical values:
small on cosmic scales

8. perpendicular, parallel acceleration in
Total Emitted Power
For single electron
Recall
perpendicular, parallel acceleration in
frame where the electron is instantaneously
at rest.
In our case, the acceleration is perpendicular to the velocity: So
and
classical
electron
radius
write

9. Average over an isotropic, mono-energetic velocity distribution
of electrons:
i.e. all electrons have the same velocity v, but random
pitch angle with respect to the B field,
Then So
per particle or

10. Write it another way
where
Thomson cross-section
magnetic energy density
For β1.

11. Life time of particle of energy E is

12. Spectrum of Synchrotron Radiation -- Qualitative Discussion
The spectrum of synchrotron radiation is related to the
Fourier transform of the time-varying electric field.
Because of beaming, the observer sees radiation only for
a short time, when the core of the beam (of half-width 1/γ)
is pointed at your line of sight:

13. The result is that E(t) is “pulsed”
i.e. you see a narrow pulse of E-field
 expect spectrum to be broad in frequency

14. It is straight-forward to show (R&L p. 169-173) that the
width of the pulse of E(t) is
where

15. Define CRITICAL FREQUENCYor
Spectrum is broad, cutting off at frequencies >> ωC

16. For the highly relativistic case, one can show that the
spectrum for a single particle:
Where F is a dimensionless function which looks like:

17. Transition from Cyclotron to Synchrotron Emission
observer

18. Slightly faster 

19. β ~ 1 Highly relativisticto
observer

20. Spectral Index for Power-Law Electron Distribution
Often, the observed spectra for synchrotron sources are
power laws
where s = spectral index
at least over some particular range of frequencies ω
Example: on the Rayleigh-Jeans tail of a blackbody spectrum
s = -2

21. A number of particle acceleration processes yield
a power-law energy distribution for the particles, particularly
at high velocities
e.g. “Fermi acceleration”
Maxwell-Boltzman distribution
“Non-thermal” tail of particle velocities v
Let N(E) = # particles per vol., with energies between E, E+dE
Power-law
p = spectral index
C = constant

22. Turns out that there is a VERY simple relation between
p = spectral index of particle energies
and s = spectral index of observed radiation

23. p = spectral index of particle energies
and s = spectral index of observed radiation
Since
can be written
(1)
Power/particle with energy
E, emitted at frequency ω
# particles /Vol.
with energy E
where E1 and E2 define the range over which the power law holds.

24. Equivalently, in terms of γ
(2)
(3)
where
Inserting (1) and (3) into (2),
change variables by letting
where

25. Then
can approximate x1  0, x2  ∞
Then the integral is ~constant with ω So
Relation between slope of power law of
radiation, s, and particle energy index, p.

26. Polarization of Synchrotron Radiation
First, consider a single radiating charge
 elliptically polarized radiation
Observer
The cone of radiation  projects onto an ellipse on the plane of
the sky
Major axis is perpendicular to the projection of B on the sky

27. Ensemble of emitters with different α 
emission cones from each side of line of sight cancel
 partial linear polarization
Frequency integrated polarization can be as high as 75%
For a power-law distribution of energies, per cent polarization
Linear polarization is perpendicular to direction of B

28. Synchrotron Self-Absorption
Photon interacts with a charge in a magnetic field and is
absorbed, giving up its energy to the charge
Can also have stimulated emission: a particle is induced to
emit more strongly in a direction and at a frequency at
which there are already photons present.
A straight-forward calculation involving Einstein A’s and B’s
(R&L pp )
yields the absorption coefficient for synchrotron self-absorption
for a power-law distribution of electrons

29. The Source function is simpler:
Independent of p
spectrum  dead give-away that synchrotron self-abs.
is what is going on
which is the Rayleigh-Jeans value

30. Summary:
For optically thin emission
For optically thick
 Low-frequency cut-off
Thick
Thin

31. Synchrotron Radio Sources
Map of sky at 408 MHz (20 cm).
Sources in Milky Way are pulsars, SNe.

32. Radio emission of M1 = Crab Nebula, from NRAO web site
The Crab Nebula, is the remnant of a supernova in 1054 AD, observed as a "guest star" by ancient Chinese astronomers. The nebula is roughly 10 light-years across, and it is at a distance of about 6,000 light years from earth. It is presently expanding at about 1000 km per second. The supernova explosion left behind a rapidly spinning neutron star, or a pulsar is this wind which energizes the nebula, and causes it to emit the radio waves which formed this image.
Radio emission of M1 = Crab Nebula,
from NRAO web site

33. IR
Optical
Radio
X-ray
(Chandra)

34. Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays
see Aharonian ApJ 614, 897
Synchrotron
Synchrotron
Self-Compton

35. Synchrotron Lifetimes, for Crab Nebula
Photon frequency
(Hz)
Electron Energy
U, (eV)
Electron lifetime
(Yr)
Radio (0.5 GHz)
5x108
3.0x108
109,000
Optical (6000A)
5x1014
3.0x1011
109
X-ray (4 keV)
1x1018
1.4x1013
2.4
Gamma Ray
1x1022
1.4x1015
0.024 = 9 days
Timescales
<< age of Crab
Pulsar is
Replenishing energy

36. Guess what this is an image of?

37. Extragalactic radio sources: Very isotropic distribution on the sky
6cm radio sources
right ascension
Milky Way
North Galactic Pole

38. Blowup of
North
Pole

39. VLA
Core of jets:
flat spectrum s=0 to .3
Extended lobes:
steep spectrum
s =

40. FR I vs. FR II On large scales (>15 kpc) radio sources divide into
Fanaroff-Riley Class I, II
(Fanaroff & Riley MNRAS P)
FRI: Low luminosity
edge dark
Ex.:Cen-A
FRII: High luminosity
hot spots on outer edge
Ex. Cygnus A

41. Lobes are polarized
 synchrotron emission with well-ordered B-fields
Polarization is perpendicular to B