Page 20 - 83 basic knowledge of astronomy

Henceforth, we omit the s dependence in u ν , since the RHS of equation
(23) does not depend on any specific direction, as expected from the
isotropic nature of thermal radiation.

2. The energy density per unit solid angle u ν must follow the well-known
Rayleigh–Jeans radiation law:
2ν 2
u ν = kT, (24)
c 3
in the classical limit hν kT. Therefore, we must require in equation
(23)
α m = 2hν 3 . (25)
n
β n m c 3

The above discussions thus lead to Planck’s formula of blackbody radiation:

2hν 3 1
u ν = . (26)
hν
c 3 e kT − 1
For the intensity I ν , we obtain
2hν 3 1
I ν = cu ν = , (27)
hν
c 2 e kT − 1
(see Figure 20). For the energy density U ν , we have

8πhν 1
I 3
U ν = u ν dΩ = 4πu ν = . (28)
hν
c 3 e kT − 1

9.1 Two Extreme Cases of the Planck Spectrum

hν
• Rayleigh–Jeans region (hν kT, and hence e kT ' 1 + hν ):
kT
2ν 2
I ν = kT. (29)
c 2

Note that thermal radiation in the radio frequency range is mostly in
the Rayleigh–Jeans region.

• Wien region (hν kT):

2hν 3 hν
I ν = e − kT . (30)
c 2

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