5. Radiation transport
Reading: Shu, Vol.I, Ch.1 and Ch.3
5.1 The radiation transport equation
Being equipped with techniques to describe nonrelativistic matter we now turn our attention
to radiation.
We know that in many cirumstances we can use classical optics, i.e.
describe
radiation as freely propagating rays and waves or photons when interacting with matter. We
also know that when interactions with individual photons are concerned, such as the emission
or absorption of photons by atoms, a quantummechanical treatment is needed.
Let us Frst deFne a few quantities.
Assume a area element
dA
perpendicular to incoming
radiation. All rays through
dA
, whose direction is within the solid angle element
d
Ω, transport
the energy
dE
through
dA
in the time interval
dt
and frequency interval
dν
. We deFne
I
ν
=
dE
dA dt dν d
Ω
speciFc intensity
(5
.
1)
Averaging over solid angle yields
J
ν
=
1
4
π
I
I
ν
d
Ω
mean intensity
(5
.
2)
The energy density spectrum per solid angle element then is
u
ν
(Ω) =
dE
dV dν d
Ω
=
dE
c dt dA dν d
Ω
=
I
ν
c
(5
.
3)
And the total energy density spectrum
u
ν
=
I
u
ν
(Ω)
d
Ω =
I
I
ν
c
d
Ω =
4
π
c
J
ν
(5
.
4)
How do these quantities compare with the notion of a photon distribution function for spin
state
i
? Using
p
=
hν/c
we obtain
X
i
f
i
(
~x,~p, t
) =
X
i
dN
i
d
3
x d
3
p
=
u
ν
(Ω)
dν
E p
2
dp
=
I
ν
c
c
3
hν
(
hν
)
2
h
⇒
I
ν
=
h
4
ν
3
c
2
X
i
f
i
(
~x,~p, t
)
(5
.
5)
In thermodynamic equilibrium with matter the radiation Feld should be a blackbody or Planck
spectrum.
Planck
I
ν
= 2
h ν
3
c
2
1
exp
±
hν
kT
²

1
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentor because the blackbody emission is unpolarized
f
i
(
~x,~p, t
) =
1
h
3
1
exp
±
hν
kT
²

1
=
1
h
3
n
i
(
~x,~p, t
)
(5
.
6)
where
n
i
is called the photon occupation number.
If radiation passes through matter, its speciFc intensity may change. Energy may be added by
emission or taken from the radiation Feld by absorption processes.
Using
dV
=
dA ds
let us
deFne the spontaneous emission coe±cient as
j
ν
=
dE
dV dt dν d
Ω
=
dE
ds dA dt dν d
Ω
=
dI
ν
ds
(5
.
7)
²or the absorption let us visualize an ensemble of particles with density
n
, each of which blocks
the radiation over a area
σ
. The total absorbing area in a test volume then is
dA
a
=
n σ dA ds
.
The absorbed energy is
dE
=
I
ν
dA
a
d
Ω
dtdν
=

dI
ν
dA d
Ω
dtdν
⇒
dI
ν
=

nσ I
ν
ds
(5
.
8)
We now deFne the absorption coe±cient
α
ν
and the optical depth
τ
as
α
ν
=
nσ
τ
=
Z
s
s
0
α
ν
ds
dτ
=
α
ν
ds
(5
.
9)
In total we have derived the radiation transport equation without scattering
dI
ν
ds
=
j
ν

α
ν
I
ν
or
dI
ν
dτ
=
S
ν

I
ν
mit
S
ν
=
j
ν
α
ν
(5
.
10)
The quantity
S
ν
is called the source function. The radiation transport equation has the formal
solution
I
ν
(
τ
) =
I
ν
(0) exp(
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 RABE
 Physics, Thermodynamics, Photon, Radiation, spontaneous emission, dΩ Iν, radiation transport equation

Click to edit the document details