Bremsstrahlung
Rybicki & Lightman Chapter 5
Bremsstrahlung
“Free-free Emission”
“Braking” Radiation
Radiation due to acceleration of charged particle by the Coulomb
field of another charge.
Relevant for
(i) Collisions between unlike particles: changing dipole  emission
e-e-, p-p interactions have no net dipole moment
(ii) e- - ions dominate: acc(e-) > acc(ions) because m(e-) << m(ions)
recall P~m-2  ion-ion brems is negligible
Method of Attack:
(1) emission from single epick rest frame of ion
calculate dipole radiation
correct for quantum effects (Gaunt factor)
(2) Emission from collection of e thermal bremsstrahlung
or non-thermal bremsstrahlung
(3) Relativistic bremsstrahlung (Virtual Quanta)
A qualitative picture
Emission from Single-Speed Electrons
e-
b
v
R
Electron moves past ion, assumed
to be stationary.
b= “impact parameter”
Ze
ion
- Suppose the deviation of the e- path is negligible
 small-angle scattering
The dipole moment
the encounter.


d

e
Ris a function of time during
- Recall that for dipole radiation
where
ˆ()
d
2
energy
dW 8 4 ˆ


d ( )
3
frequency d
3c
is the Fourier Transform of

d
After some straight-forward algebra, (R&L pp. 156 – 157), one can
derive
dW(b)
d
in terms of impact parameter, b.
Now, suppose you have a bunch of electrons, all with the same
speed, v, which interact with a bunch of ions.
Let ni = ion density
(# ions/vol.)
ne = electron density (# electrons / vol)
The # of electrons incident on one ion is
n

#
v
electro
area/
e
# e-s /Vol
d/t

around one ion, in terms of b
Area

2
b
db
So total emission/time/Vol/freq is
dW
dW(b)
 ne ni 2v 
bdb
d dV dt
d

Again, evaluating the integral is discussed in detail in
R&L p. 157-158.
We quote the result 

Energy per volume per frequency per time due to bremsstrahlung
for electrons, all with same velocity v.
dW
16e 6 1
2

n
n
Z
g ff (v, )
e i
3
2
ddVdt 3 3c m v
ne elec trons/
vol.
ni ions/vol
Z#ofc harges
inion
gff (v,)Gaunt
factor
Gaunt factors are quantum mechanical corrections
 function of e- energy, frequency
Gaunt factors are tabulated (more later)
Naturally, in most situations, you never have electrons with just
one velocity v.
Maxwell-Boltzmann Distribution  Thermal Bremsstrahlung
Average the single speed expression for dW/dwdtdV
over the Maxwell-Boltzmann distribution with temperature T:
dW
(
v
,
)2 m
v


v
exp

d
v



d

dVdt
2
kT
dW



2
dVdtd


m
v
2

 
d
v
vexp


kT
2

2
The result, with


d

2
d


dW
2
e
2




T
Z
n
n
eg

3
dVdtd
3
mc
3
km


5
6
where
1
/
2

1
/
2
2 
h
/
kT
e
i
ff
g

velo
ver
Ga
fa
ff
In cgs units, we can write the emission coefficient
 
dW

38
2 
1
/
2

h

/
kT


6
.
8

1
Z
n
0
n
T
e
g
d Vd
e
i
ff
td
ff
Free-free emission coefficient
ergs /s /cm3 /Hz
Integrate over frequency:
dW
2 e 2kT  2

 Z n e n i gB
3 
dVdt 3hm c  3km 
5
where

In cgs:

6
1/ 2
gB  frequency average of t he
velocity averaged Gaunt factor

ff
dW
27 2
1/ 2

1.4 10 Z ne niT gB
dVdt
Ergs sec-1 cm-3
g
T
,
)
ff(
The Gaunt factors
- Analytical approximations exist to evaluate them
- Tables exist you can look up
- For most situations,

h
g
~
1
for
10

1
ff
kT

4
so just take
g
1
.2
ff
Handy table, from Tucker: Radiation Processes in Astrophysics
ff
Important Characteristics of Thermal Bremsstrahlung Emissivity  
(1) Usually optically thin. Then
 ff
(2)
 ff is ~ constant with hν at low frequencies h


kT
(3)
 ff
falls of exponentially at
h~
 kT
Examples:
Important in hot plasmas where the gas is mostly ionized, so
that bound-free emission can be neglected.
T (oK)
Obs. of
 ff
Solar flare
107 (~ 1keV)
radio flat
X-ray  exponential
H II region
105
radio flat
Orion
104
radio-flat
Sco X-1
108
optical-flat
X-ray  flat/exp.
Coma Cluster ICM
108
X-ray  flat/exp.
Bremsstrahlung (free-free) absorption
Brems emission
photon
e-
eion
photon
e-
collateral
Inverse Bremss.
free-free abs.
Recall the emission coefficient, jν, is related to the absorption
coefficient αν for a thermal gas:
 ff
j

B
(
T
)



is isotropic, so

4
j

ff
ff
and thus
6
 2 
4e
ff
2 1/ 2 3
h / kT
 
g ff

 ne ni Z T  1 e

3m hc3km 
1/ 2
in cgs:
ff  3.7 108 ne ni Z 2T1/ 2 3 1 eh / kT g ff
Important Characteristics of
(1)
ff
h


kT(e.g. X-rays)



8 2

1
/
2

3
e
i
ff

3
.
7

1
n
n
Z
0
T
g
ff

Because of T
term, 
is very small unless ne is very large.

1
/2 
3
ff
in X-rays, thermal bremsstrahlung emission can be
treated as optically thin
(except in stellar interiors)
(2)
h


kT


e.g. Radio: Rayleigh Jeans holds

2

3
/
2

2
e
i
ff

0
.
018
n
n
Z
T
g
ff
Absorption can be important, even for low ne
in the radio regime.
From Bradt’s book: BB spectrum is optically thick limit of
Thermal Bremss.
HII Regions, showing free-free absorption in their radio spectra:
R&L Problem 5.2
Spherical source of X-rays, radius R
distance L=10 kpc
flux F= 10 -8 erg cm-2 s-1
(a) What is T? Assume optically thin, thermal bremsstrahlung.
Turn-over in the spectrum at log hν (keV) ~ 2
E
9o
T
max

10
K
k
(b) Assume the cloud is in hydrostatic equilibrium around a
central mass, M.
Find M, and the density of the cloud, ρ
3
14

R
ff
F
 2


4

L3
Vol. emission coeff.
1/r2
Vol.
1 4 R 3
27 1/ 2
2
F
1.4

10
T
n
n
Z
gB 

e i
2
4 L
3

- Since T=109 K, the gas is completely ionized
- Assume it is pure hydrogen, so ni = ne, then
n
n
in
e
2
H
ρ=mass density, g/cm3
 
47 2



3
.
6

10

m
 H
2
Z=1 since pure hydrogen
gB  1.2

20
2
1
/
2
3

2
(1)
F

2
.
0

10
T
R
L
- Hydrostatic equilibrium  another constraint upon ρ, R
Virial Theorem:

For T=109 K
 K.E.  grav.energy
2  
 

particle
particle

 

GMm
H
3
kT

R
M

(2)


R

5

10
cm
M
 s un

8

3
/
2
- Eqn (1) & (2) 
M





4

10
LF


M
sun
 

261
/
2
Substituting L=10 kpc, F=10-8 erg cm-2 s-1

3
/
2
M





4

10
g
cm


M
 sun


7
3