Page 31 - 83 basic knowledge of astronomy

B ν (T) at temperature T. Such a case is called “local thermodynamic
equilibrium”, denoted by “LTE”.
In LTE, the radiative transfer equation reduces to

dI ν
= −κ ν [I ν − B ν ] . (45)
dl
Then, introducing the “optical depth” τ ν , which satisfies

dτ ν = −κ ν dl, (46)

we can further transform equation (45) to

dI ν
= I ν − B ν (T) . (47)
dτ ν

Suppose we have an emitting and absorbing gas medium confined within
a finite linear extent l 0 , and we consider the radiation coming from the
outside (l < 0, see Figure 25). The optical depth τ ν is chosen to be
equal to τ ν (0) at l = 0, and to 0 at l = l 0 .
The solutions to equation (47) are:

Background Medium with finite κ ν and ε ν
Radiation dl

I (0) I (l )
ν
ν 0
s
I (s, l) I (s, l + dl)
ν
ν
κ = 0
ν
ε = 0
ν
d τ ν
0 l l 0

l
τ (0) τ ν τ ( )= 0
ν
ν
0
Figure 25: Radiative transfer in LTE and optical depth.

within the medium (0 ≤ l ≤ l 0 ):

τ ν (0)
Z
0
0
I ν (l) = I ν (0)e τ ν −τ ν (0) + e τ ν B ν (T(τ ))e −τ 0 dτ , (48)
τ ν
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