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and outside the medium (l 0 ≤ l):

τ ν (0)
Z
0
0
I ν (l) = I ν (l 0 ) = I ν (0)e −τ ν (0) + B ν (T(τ ))e −τ 0 dτ , (49)
0
where I ν (0) is the incoming background radiation.

5. LTE and isothermal medium
0
If the medium is isothermal (T(τ ) = T = const), the solutions are:
within the medium (0 ≤ l ≤ l 0 ):

I ν (l) = I ν (0)e τ ν −τ ν (0) + B ν (T)(1 − e τ ν −τ ν (0) ), (50)

and outside the medium (l 0 ≤ l):

I ν (l) = I ν (l 0 ) = I ν (0)e −τ ν (0) + B ν (T)(1 − e −τ ν (0) ) . (51)

Note that I ν → B ν (T) when τ ν (0) → ∞ in the above equations. This
means that thermal radiation becomes blackbody radiation re-
flecting the temperature of the medium when, and only when,
the medium is completely opaque.

If we denote the intensity I ν in terms of the brightness temperature
T B , and adopt the Rayleigh–Jeans approximation for B ν (T):

2ν 2 2ν 2
I ν = kT B and B ν (T) = kT ,
c 2 c 2

the equation (51) for the solution of the radiative transfer equation
outside the medium can be described by

T B (l) = T B (0)e −τ ν (0) + T(1 − e −τ ν (0) ) . (52)

10.4 What is LTE?

How can the Kirchoff law ν /κ ν = B ν (T) be satisfied, even though the inten-
sity of the radiation is not blackbody?
The opacity and emissivity are expressed in terms of Einstein’s differential
coefficients, as we saw in equation (38)

hν
m
n
κ ν = (n m β − n n β ) ,
m
n
c
m
ν = hνn n α .
n
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