Synchrotron Radiation,
continued
Rybicki & Lightman Chapter 6
Also Course notes for “Essential Radio Astronomy” at NRAO,
Condon & Ransom
http://www.cv.nrao.edu/course/astr534/ERA.shtml
Synchrotron Theory: Summary of Results
1. Synchrotron = relativisitic electrons & B-field
 Very spiky E(t) because of beaming
 larmor  2 14 rad s-1
B ~5x10-6G
For an electron having g ~ 104 the width of the pulse is
t
pulse 

1
g 2 larmour

1
10   2 14
4 2
1010 s
The time between pulses is ~
g
 larmour
 10 3 s

E(t)
t
2. Spectrum of single energy electron:
Critical frequency
x=  / c
E= electron
energy
Each electron of energy E contributes
to the spectrum at x=1
3. Spectral index related to energy index of electron energy distribution
p = spectral index of particle energies
and s = spectral index of observed radiation

s
P
(

)


N
(
E
)
dE

CE
dE
Ergs/s

p
p1
s
2
(Optically thin synchrotron)
4. For optically thin synchrotron, the slope of the spectrum must always
be greater than -1/3 because the low-frequency spectrum is a superposition
of single spectra, and F(x) ~ -1/3
Single electron spectrum
s > -1/3
Summary:
For optically thin emission
For optically thick


(
p

1
)
/
2
I






5
/
2
I

S



 Low-frequency cut-off
(
1
)/2
p
Thick
Thin
Synchrotron Radio Sources
Map of sky at 408 MHz (20 cm).
Sources in Milky Way are pulsars, Sne;
Diffuse radio spectrumGalactic B-field + cosmic rays
Milky Way magnetic field ~ 5 microGauss, along spiral arms
measured via Zeeman splitting of OH masers
pulsar dispersion measures
polarization of starlight by dust aligned in B-field
c.f. Earth’s Magnetic field: 500,000 microGauss
Spectrum of Cosmic Rays in ISM of the Milky Way has
p~2.4
Spectrum of synchrotron radiation s~0.7
Milky Way
Interstellar
Cosmic Ray
Energy spectrum
energy spectrum
has p~2.4
Synchrotron has
s~0.7
Casadel & Bindli 2004
ApJ 612, 262
M51 Polarization derived from
Synchrotron (6 cm).
Beck 2000
Coherent structure, B-field along arms
Cowley 2011
Milky Way B-field: Theory
Vertical Field in Center B ~1mG
Horizontal
Field in Disc B
~ 3G
V
V
Supernova explosion.
l0  3 ´ 1018 m
t 0  10 7 years
Scale
Eddy turnover time
LG  4 ´ 10 20 m
t G  2 ´ 10 8 years
Scale
Rotation time
6. Dynamo  amplification of primordial seed magnetic field
E. Parker: Galactic Dynamo
Differentital rotation & convection or SN explosions --> loops
Loops align with existing B-field
Net result is amplification
Zirker,
The Magnetic
Universe
7. Minimum Energy and Equipartition
Synchrotron spectrum spectral index  electron energy index, but
not B-field
B-fields often estimated by assuming “equipartition”
Recall:
B2
UB 
8
Energy density of magnetic field
What can we say about the minimum energy in relativistic particles
and magnetic fields that is required to produce a synchroton source
of a given luminosity?
What is UE?
Assume power-law electron energy distribution
N(E)  KE
p
between energies Emin and E max
which produces synchrotron radiation between frequencies min and max
E(max)

 EN(E)dE
Ue 
Energy density of electrons
E(min)
 (max)


L
 L d
 (min)
Synchrotron radiation luminosity
Substitute
N(E)  KE
p
(dE /dt)  B E
2

and
2
Energy per electron from synch.
E(max)
Ue

L
K
KB
E
1 p
dE
E(min)
E(max)
2
2 p
E
E(min)

dE
E
2 p
E(max)
E(min)
E(max)
B2 E
3 p
E(min)
Approximate each electron emits at energy E, and
So
E max  B 1/ 2
and
E min  B 1/ 2
and
B 

Ue
B 1 p / 2
3 / 2



B
L B 2 B 1/ 2 3 p B 2 B 3 / 2 p / 2
1/ 2 2 p

So

Ue  B
3 / 2
Need total energy density in particles: electrons plus ions
Let
U ions

Ue
Don’t usually know what is, but ~ 40 for cosmic rays near Earth

U  U e  U ion  U B
 (1 )U e  U B
So total energy
 B 3 / 2

 B2
So there is a B for which U is minimum


Find minimum U by taking dU/dB and setting = 0
Result: Get minimum energy when
particle energy 4
 1
field energy
3
“Equipartition”
So given an optically thin synchrotron source of luminosity L,
Assume equipartition, and then compute B

numerical formulae on Condon & Ramson web site
Physically plausible: B field cannot have U>>U(particles) and still have
Coherent structures
Large extragalactic jets have an enormous amount of particle energy as
It is, so putting more energy into particles makes theory more difficult
Crab Nebula
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
IR
Optical
Radio
X-ray
(Chandra)
Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays
see Aharonian+ 2004 ApJ 614, 897
Synchrotron
Synchrotron
Self-Compton
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
5.16 1
t 2
electron decay time,sec.
B g
for  =

2
,B in teslas
Timescales
<< age of Crab
Pulsar is
Replenishing energy
Guess what this is an image of ?
Extragalactic radio sources: Very isotropic distribution on the sky
6cm radio sources
right ascension
Milky Way
North Galactic Pole
Blowup of
North
Pole
VLA
Core of jets:
flat spectrum s=0 to .3
Extended lobes:
steep spectrum
s = 0.7-1.2
www.jb.man.ac.uk/atlas/dragns.html
DRAGNS: Double-Lobed Radio-loud Active Galactic Nuclei
Cen A, Full moon and CSIRO radio
Observatory
Radio lobes are ~ million light years
across
APOD April 13, 2011
FR I vs. FR II
On large scales (>15 kpc)
radio sources divide into
Fanaroff-Riley Class I, II
(Fanaroff & Riley 1974
MNRAS 167 31P)
FRI: Low luminosity
edge dark
Ex.:Cen-A
FRII: High luminosity
hot spots on outer edge
Ex. Cygnus A
Lobes are polarized
 synchrotron emission with well-ordered B-fields
Polarization is perpendicular to B
8. Synchrotron spectra steepen with age
Energy radiated by electrons
E
2
So high energy electrons lose their energy faster than low energy electrons
Spectrum steepens
At high freqencies:

Typically:
Cores have “flat spectra” s~0.5
Outer lobes have “steep spectra” s ~ 1.5-2
Real spectra can be complex: non-uniform B-fields, geometries
(Kellerman & Owen 1988)