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Synchrotron Radiation
Emission from relativistic electrons gyrating round magnetic field lines. This is the dominant continuum emission mechanism in quasars and radio-galaxies (cores, jets and lobes) and from supernovae. The main observable differences from Bremsstrahlung are:-

Synchrotron radiation is polarised.

e



%$ !

Different spectrum ( - optically thick - optically thin

#" ! ?? ?? ? ? ?? Å

'&'( '&'('&'('&'(

&( &(&(&(



The emitters are not in thermal equilibrium ). (c.f. Bremsstrahlung (c.f. Bremsstrahlung )

)

B


Cyclotron radiation

with

.

The radiation is emitted in a dipole pattern and is intrinsically polarised.

B

v

r

g

a g

VU TS RQ PI HG F 3@ B DC E B 9A 4

21 W0 E 3 C E

) ) 98 7 65 4 3 21 0 )

)

For non-relativistic electrons can calculate gyro-frequency .

Spectrum is just line emission at this frequency. Can calculate radiated power from Larmor's equation.

dipole pattern


Mildly Relativistic Electrons
Even for , relativistic beaming changes emitted spectrum significantly. Depar ture from dipole pattern and spectrum no longer a single emission line.

gf !e d c 'b a` Y

X h


Ultra-Relativistic Electrons: Synchrotron Radiation

B e

v

v||

v_ |

Relativistic equation of motion

Electron moves in helix about

V R~ }| {z y p wp r xs vu tv 9s

qp o

In frame

in which electron at rest, the power radiated is

h

vector precesses around cone of semi-angle

h

Pitch angle,

given by

ut s s D ?y xw v
with .

w gf

Tn k h ml k H ji

?y w

6

ed

pq pipipqpipqpi pq

q iiqiqi q r r r r

. The electron's velocity .


? ? Ñ ÔÆ Å Ó ? Á Ä Ã Á Â À
: , and substituting gives

Ï Á RÒ Ñ

Á RÐ É ÁÏ ÊÉ Â È Î ÍÌ Ë ÇÆ Å '? W? ? ? ?? Å ?

?

? ?? ?Å ? ?? } { 9? e ? ?? ?Å H D P V T R P H ? T
Lorentz transorming back into the lab frame



Also assuming that

and

Averaging over a random set of

gives

.






Spectrum

Observer sees pulse star ting at A and ending at B

Radiation from B arrives at

Pulse length since

Pulse is

Expect a frequency spread

Define

Sychrotron source emits most power at

åù ö õ ö ß ? ÿþ ô Ö õ ý ?ü ?û ú Dß ö ø Ö ôø Ü Dù ö eß ö ø Ö õ 2÷ ö Ö õ ö ç ô òî ñ ð óï ô õ ô ð óï ô í ì Þá é Ø ë ò xñ î {ð ï í tì Þ é Ø ë áà Ü êÛ Ú é èß Ú qÜ Ú ç á åà Þ äã Ý â áà Ü Íß Ú Þ Ý Ü Û Ú

Radiation from A arrives at

9Ö 6Ù Ø
A

Ø

For semi-angle

×Ö

Õ Õ Õ Õ

the radiation is beamed into a narrow cone of about the velocity vector:

L

B R

Õ

Õ

æ

shor ter than the gyro period

.

.

.



Emission from a Non-Thermal Emsemble
In general the emitters are not in thermal equilibrium. We can model their energy distribution with a power law

Each electron emits a power


The energy of the electron is related to
)

by

6

!

U

So a syncrotron source has a spectral index optically thin .

6? ? ?

)





?

!

A8 6

?? ) ? a? ? 0 ? ? x? ? ?? ? ? ?

TQ ÅS G



)

?

RQ PI HG

??

!

E

98 F6 5 ) 4



?

?

? YX ?



?

(

CB



CB

DC B

The emissivity

is

!3

21



?

??

!

A8 6

R? ?

)

?





of which emerges at frequency
'

& %$ ?# "! ? ?? ?? ?? ?

? ? Å ?  ? V? ? ?

? {? a? ? ? ? ? ? x? ? ?? ? ?
( ( ! )! )

? ?

(!

@

!

98 76 5 ) 4

(

0

? ? ?

(

?

(

C

WV

(

?

?



?

'

CB

?

? ? ?

, most

.

.

?

if it is


Synchrotron Self-Absorption
As well as synchrotron emission also get synchrotron absorbtion.
a ` ` ` ` ` a `

Electron are not in thermal equilibrium choff 's Law.

Cannot use Kirin an

.


.

x Å r

h



In R-J region the specific intensity

so

r 9 2 y



i

a D y r i

h

y

Electron radiates principally at
r A y i c b

sr q

2p i

hg

f

e

The mean electron energy is to .
c b c xw v u t

. Equate this .

dc b

However we can assign a brightness temperature optically thick case.

r y


The "Minimum Energy" Argument
Consider a sychrotron source and find the energies contained in the magnetic field and in the par ticles . .
h P t H t ef t s s ed x w w v ?s %u r t qp o nm xl kj i h gd

for

For an optically thin source we have found that where
Å s w

. This is a minimum for , i.e. if there is equipatition of the energy. Making that assumption will allow an estimate of to be made.
?d ef ?d ? gf h ?? Y? 7?



P t

|

s



|

Å

??





w

w

f

|

s





w

gives


.

? %? ? t

h





2

h

Combining


and

d

?



t h | s |

s Å





w

t

R

.

t

~ }| z s {z y

h

gf



Evolution of Spectrum
Par ticles accelerated in shocks to high energy. Initial (injected) spectrum will then change as electrons loose energy through radiation.
ÃÅ Á Ã ?Ä "Ã Â Á ÅÀ ? P? ? Æ

Loss rate of energy

.

Most energetic electrons, emitting at hightest frequency, loose energy fastest. Leads to "knee" in spectrum - location gives estimate of age of electrons since acceleration.




Sychrotron Radiation

B e

Emission as result of relativistic electrons gyrating round magnetic field lines
É É É

Emitters not in thermal equilibrium Polarised

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