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of ordinary stars are fairly high (several thousands to several tens of thousand
Kelvin), their solid angles, which are inversely proportional to their squared
distances, are very small. For example, if we were to put the Sun at the
distance of the nearest star, which is about 3 light years, or 1 parsec, the
Sun’s angular diameter would be only ' 0.01 arcsecond! As a result, the
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flux density 5 × 10 Jy of the quiet Sun at 10 GHz (see Figure 2) would be
reduced down to 125 µJy at the distance of the nearest star. On the contrary,
the interstellar gas clouds, as cold as tens to hundreds of Kelvin, still clearly
shine in the radio sky, because their angular diameters are as large as minutes
to degrees of arc. For example, Figure 2 shows that the flux density of the
Orion Nebula is about 500 Jy at around 500 MHz.

9.5 Stefan–Boltzmann Law

The total intensity I(T) of the thermal radiation from a black body of tem-
perature T, over the entire frequency range, is given by
∞ ∞
2hν 1 1
Z Z 3
4
I(T) = I ν dν = dν = σT , (32)
c 2 e kT − 1 π
hν
0 0
where σ is the Stefan–Boltzmann constant:
5 4
2π k
−4
σ = = 5.6697 × 10 −8 W m −2 K .
2 3
15c h
Equation (32) can be derived using the integration formula:
∞
Z 2x 2n−1 B n
dx = , (33)
e 2πx − 1 2n
0
where B n is the n-th Bernoulli number, and B 2 = 1/30.
9.6 Total Blackbody Radiation from a Star or a Gas

Cloud

The power flux density S ? , at a surface of a blackbody, which is the power
over the entire frequency range through a cross section of unit area of the
surface (Figure 22), is given by the equation:

π
∞ ∞ 2 2π
Z Z Z Z
S ? = S ν dν = I ν cos θ sin θ dφ dθ dν
0 0 0 0
π
2π
2 Z Z
4
= I(T) cos θ sin θ dφ dθ = π I(T) = σT . (34)
0 0
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