Page 18 - 83 basic knowledge of astronomy

n n Z n E n

n m Z m E m

Figure 18: Number densities of particles in the two energy states.

On the other hand, the number density of photons in the same solid angle
dΩ, and the same bandwidth dν, absorbed by the Z m → Z n transition during
the time interval dt, is equal to:
n
n m df ab = n m β u ν dν dΩ dt.
m
The differential changes in the number density of photons per unit solid
angle around -s, and per unit bandwidth around ν mn = (E n − E m )/h, in the
course of the absorption and emission is now given by:
n
m
m
dn p (s, ν) dν dΩ = {−n m β u ν + n n [α + β u ν (s)]}dν dΩ dt.
m
n
n
Therefore, the time variation of the number density of photons is described
by the following equation (Figure 19):
dn p (s, ν)
m
n
m
= (n n β − n m β )u ν (s) + n n α . (15)
dt n m n
n dΩ
s n
nm
n (s, ν) at t n (s, ν) at t + dt
p
p

Figure 19: Time variation of the number density of photons.

9 Blackbody Radiation

We now consider the case when the radiation and matter are in thermody-
namic equilibrium. Then, we have the following:

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