. : http://www.mrao.cam.ac.uk/~kjbg1/lectures/lect3.pdf
: Fri Oct 17 15:32:24 2003
: Tue Oct 2 03:14:57 2012
: IBM-866
:
Free-Free or Bremsstrahlung Radiation
Electrons in a plasma are accelerated by encounters with massive ions. This is the dominant continuum emission mechanism in thermal plasmas.

Examples :









An impor tant coolant for plasmas at high temperature

Radio emission from HII regions Radio emission from ionised winds and jets X-ray emission from clusters of galaxies

e
-

b

Z

+


Calculation of Bremsstrahlung Spectrum
Impor tant ingredients:

When a charged par ticle accelerates it emits radiation (Larmor's formula). Acceleration is a function of , and . Acceleration as a function of time intensity spectrum via the Fourier Transform (Parseval's theorem). Integrate over (exact details tricky gives rise to the Gaunt Factor, which is a function of ). Include term for collision rate (depends on number densities and of electrons and ions respectively). Integrate over . Assume plasma in thermal equilibrium Maxwellian distribution of .

is the emissivity, the emitted frequency. This is related to the cient (the emitted power per unit unit solid angle) by

power per unit volume per unit spontaneous emission coeffivolume per unit frequency per .

&P xD w )v )u D t

3 S y 4 3 2 1

s

with the result having the units

rh q c 2 p fi h UEg fe dc Sb a `Y EX WV UT (' S$ &% $ H RH !Q &P D

.









Consider one par ticle at a specific

# " !







IH GF ED CB "A @9 87 6 5 4 3 2 1







(' $

3 2 1

)

&% $

0

and .



Simple Example: Hydrogen Plasma
A common case is that of an optically thin hydrogen plasma, so and . Because the plasma is optically thin, the total emitted specific intensity is propor tional to the emissivity integrated along the line of sight.

This is propor tional to process.

as we would expect for a collisonal

The integral is called the emission measure, and is often written in units of .

Total Emissivity
Integrate over frequency to get the total emissivity:

If we set correct result.

~ }o )| h {

This has units of

zy w x ! vu ts r qp o n #n m

.

we will probably be within 20% of the

Ee d ! ) (

g lk )j ih g





Ee d f



y wx

( &


Free-Free Absorption

Now wish to find how much an observer receives. These two are not equal because free-free absorbtion occurs.

1

If optically thick, spectrum is effectively blackbody.

1

In the Rayleigh-Jeans region



If optically thin, spectrum is as calculated before ( mately flat until turnover).

E x

Can now find optical depth



in units of

.

appoxi-

S q

# 2 S q t 8 } @ G E C l f E q S E r &



p



Kirchoff 's Law :



Find how much absorbed as a function of frequency i.e. (= fraction of intensity lost per unit distance)

& }

2 U



Have calculated how much radiation emitted.


Example: HII regions around OB stars
Plasma Cloud

uv

I



The uv-photons from OB stars photoionises the gas surrounding them. The resulting plasma has a temperature of K. around The optical depth in the R-J limit is given by

In this regime

. e.g.

GHz for Orion.

i G





l p p



At low like spectrum. At high spectrum, Turnover when





R # i E r q p z

i

l W )





(

G S q 2



x C







. Blackbody "Flat"


Example: X-ray emission from clusters of galaxies
Gas in clusters of galaxies at temperatures of . Therefore Bremsstrahlung emission extends into X-rays.

Cluster gas also gives rise to the SunyaevZel'dovich effect .

Can combine SZ and X-ray data to get and the line of sight depth. If assume that line of sight depth is equal to distance across cluster, can then calculate Hubble's constant.

&% $ ("





'

Bolometric (total) X-ray luminosity

&% $ #" !





X-ray flux density

&% $



Estimate

from location of "knee" in spectrum. . .

C? ?

? x?



Very low gas density, thin. Cluster core radius

, so emission optically

kpc.

? }? C? ?

(



? ? 8

21 0)

? ?




20

40

60

80

23 46

44 DECLINATION (B1950)

42

40

38

36

11 53 00

52 45 30 RIGHT ASCENSION (B1950)

15

Figure 1: The cluster of galaxies A1413. Greyscale is X-rays from ROSAT PSPC. Contours are the SZ effect from Ryle Telescope


3

on H ml kj i (d hg 8 xu d f 9 9P

i ed 9 P

T v s ` Q Q ( 8 u d X e H e Wt s

f

In this case the optical depth is a function of distance from the star . Need to integrate along line of sight where .

WB @

&I H S9 R8 QP 9 5 &I H GF 0FB E 9 &8 7 65

6 D CB A@ 3 4

assuming

Example: Ionised winds from stars

If wind speed constant

in wind is constant, in R-J region and

d

cb a` P YX WF F U V

The flux from the wind

9 T s ` (` ` e X CB @ u X & y w u d X b B (` ` ` e X QB @ u X & y xw vu d X e H e Wt s D &r H &r H B q 6 B p
;

Si 8 e 9 8 P e

using substitution

A full analysis will allow calculation of the mass loss rate.

p i ed 9 P e D

my xw cv u n 6 6Bp 4

t sr q 3

D Bq 3 3 4 D 97 3

.

.

9 f hg 9 e

6 ` eX B @

T

3



Free-Free or Bremsstrahlung Radiation

e
-

Emission as result of collisions between charged par ticles, usually electrons and ions.

Emitters in thermal equilibrium

Unpolarised

e.g.

-- HII regions -- X-ray emission from clusters of galaxies -- Ionised winds from stars

{}{}{}{} }}}} {~{~{~{~ ~~~~
Z
+

{z{|{z{|{z{| z|z|z|


Appendix: Derivation of Bremsstrahlung Spectrum
Radiation from single accelerating electron Larmor's formula gives the power from an electron as a function of acceleration

.

An electron in a harmonic field acceleration

undergoes an





# v 2





& ( &

Y

This follows from Parseval's theorem (that and from the symmetry proper ty that





Y









2



#



we can write the total energy emitted as where the spectral density is





2 #



Y

#

If we introduce the fourier transform of

(









&





#

Integrating over solid angle gives

( Q ! x h Q
)

#

x Q Q





2



#

& 2 2










Now the incident flux in the wave is just , so writing the radiated power as a cross section so that the power radiated is flux, we find

Collision of single electron at one speed Consider an electron travelling at speed colliding with an ion with impact parameter . Assuming the deviation from a straight line is small:

The frequency spectrum is given by

? ? ? W ? ?

?

We define a characteristic interaction time is the impact parameter.

h m Q x Q

2 ? ? ? &? ?

W A 2

m Q











m G m W ? ? Q ? ? ? ?

, where


Collision of many electrons at one speed Suppose we have number densities and for the electrons and ions. The collison rate per unit volume between impact parameters and is then

and the total emitted power per unit volume per unit (angular) frequency is found by integration to be

The upper limit is set by the condition that we expect no emission beyond a frequency , so that . There are two limits on the lower limit to . The straight line assumption breaks down when the par ticle kinetic energy is smaller than the potential energy for a given : this gives

fe



and most of the energy is emitted at a frequency

0 )

54 1

d cV 7U S

ba `Y S X WV U TS







32 1



Thus for frequencies up to some value frequency spectrum



, we have a flat

' % " 3

RQ PI H G' $ & $% " $ F 54 A1 @2 1

9 B8 54 A1 @2 1

(' $ & $% #" ! 0 9 8 76

ha `Y (gS

D ED C
? ? ?

?

.


The second is the quantum limit set by uncer tainty:

We define the Gaunt Factor to encode all the uncer tainties in the above analysis:

Thermal Free-Free emission If the par ticles obey a Maxwellian distribution, we have a probability density , and we can perform the necessary average to obtain the total specific emissivity:

where

The Gaunt Factor Maxwellian distribution.

has now been averaged over the

9 B v } 3 9| { z

)x y

s

9 R q m x y ~w W} # 9| { z

ut

p

bt `s r e Wd r i

x y m w s 5v su (t 5k 5s Ai @j i rq p

m

xy

o 5n m l P

v w

w q

u bt `s r 3q p i 5k i j i ) u )x y yx ) x y w m l m @ sv w o n q uh g u f u g