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```					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
B ~5x10-6G                  larmor  2 14 rad s-1

For an electron having g ~ 104 the width of the pulse is

t 
1                     1
                    1010 s
g 2 larmour       10   2 14
pulse                         4 2


g
The time between pulses is ~                103 s
 larmour



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

NdE
ECE
()
dE () 
 
P  Ergs/s
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:

)

 
p2
(1/
For optically thin emission   I 
For optically thick

 
I S

52
/

 Low-frequency cut-off

( )/2
p1

Thick

Thin

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

Milky Way
Interstellar
Cosmic Ray
Energy spectrum

energy spectrum
has p~2.4

Synchrotron has
s~0.7

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.                    LG  4 ´ 10 20 m       Scale
Rotation time
l0  3 ´ 1018 m    Scale                t G  2 ´ 10 8 years
Eddy turnover time
t 0  10 7 years
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”

B2
Recall:     UB                  Energy density of magnetic field
8

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?

p
Assume power-law electron energy distribution   N(E)  KE
between energies Emin and E max

which produces synchrotron radiation between frequencies min and max

E(m ax)      
Ue       EN(E )dE               Energy density of electrons
E (m in)

 (m ax)
L          L d          Synchrotron radiation luminosity
                    (m in)


p
Substitute   N(E)  KE                               and

(dE /dt)  B E                  2        2
Energy per electron from synch.

                       K
E(m ax)

E      1 p
dE        E   2 p
E(m ax)

Ue
        E(m in)
E(m ax)                            E(m in)
E(m ax)
L                                            3 p
KB   2
   E 2 p
dE       B2 E
E(m in)
E(m in)
Approximate each electron emits at energy E, and

So   E max  B 1/ 2       and         E min  B 1/ 2   and

B   B1 p / 2  B3 / 2
2 p
1/ 2
Ue

               L  2 B 1/ 2 3 p B 2 B 3 / 2 p / 2
B

So
3 / 2
Ue  B

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

So total energy        U  U e  U ion  U B
 (1 )U e  U B
 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                         “Equipartition”
field energy   3

So given an optically thin synchrotron source of luminosity L,
numerical formulae on Condon & Ramson web site
Assume equipartition, and then compute B

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
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
this image.

Radio emission of M1 = Crab Nebula,
from NRAO web site
IR
Optical

(Chandra)
Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays
see Aharonian+ 2004 ApJ 614, 897

Synchrotron
Synchrotron
Self-Compton

Photon         Electron   Electron
(Hz)            U, (eV)    (Yr)
GHz)
Optical           5x1014          3.0x1011   109
(6000A)
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
Timescales

for  =       ,B in teslas                      << age of Crab
Pulsar is
2
Replenishing energy
Guess what this is an image of ?
Extragalactic radio sources: Very isotropic distribution on the sky

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)

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)

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