Page 27 - 83 basic knowledge of astronomy

Let the intensity I ν change by dI ν due to absorption and emission as
the radiation passes through an infinitesimal distance dl (Figure 23). The

s
I (s, l) I (s, l + dl)
ν
ν
dl

Figure 23: Radiative transfer through a cylinder of infinitesimal length.

variation can be described as a sum of two contributions: one due to the
absorption −κ ν I ν dl, and one due to the emission ν dl, where κ ν and ν are
the following frequency–dependent coefficients:
−1
: opacity m ,
κ ν
−1
: emissivity W m −3 Hz −1 sr .
ν
We then obtain the radiative transfer equation in the following form:
dI ν
= −κ ν I ν + ν . (36)
dl
10.2 Derivation of the Radiative Transfer Equation from

Einstein’s Elementary Quantum Theory of Radi-
ation

Equation (36) can also be derived from equation (15) obtained in the discus-
sion of Einstein’s elementary quantum theory of radiation:
dn p (s, ν)
n
m
m
= (n n β − n m β )u ν (s) + n n α ,
dt n m n
where n p is the number density of photons per unit solid angle and per unit
frequency bandwidth, u ν is the spectral energy density per unit solid angle,
n m and n n are the number densities of the particles in states Z m and Z n ,
n
m
and α , β n m and β are Einstein’s differential coefficients. In fact, using the
m
n
relations:
u ν (s) = hνn p (s, ν),
I ν (s) = cu ν (s),
dl = cdt ,

we can transform equation (15) to:
hν
dI ν n m m
= −(n m β − n n β ) I ν + hνn n α . (37)
dl m n c n
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