Page 28 - 83 basic knowledge of astronomy

Therefore, if we denote

n
m hν
κ ν = (n m β − n n β ) c : absorption and induced emission,
m
n
ν = hνn n α m : spontaneous emission, (38)
n
equation (37) is reduced to the radiative transfer equation (36).
Role of the induced emission
In the resulting radiative transfer equation:

dI ν
= −κ ν I ν + ν ,
dl
the opacity κ ν now contains not only the contribution of simple absorption
but also that of induced emission, as we see in equation (38). According to
equation (21) obtained by Einstein:

m
n
g m β = g n β ,
n
m
we can now express the opacity κ ν as
g m hν
n
κ ν = (n m − n n )β m . (39)
g n c
If we consider the simple case where the statistical weights of the two states
are equal to each other (g m = g n ), the above equation (39) reduces to

hν
κ ν = (n m − n n )β n . (40)
m
c
It is worthy to note that the opacity takes on a negative value when the
number density of the particles in the higher energy level is larger than that
at the lower level (n n > n m ).

10.3 The Simplest Solutions of Radiative Transfer Equa-
tion

Let us solve the radiative transfer equation (36):

dI ν
= −κ ν I ν + ν ,
dl
under the following simple conditions.

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