Page 22 - 83 basic knowledge of astronomy

9.2 Wien’s Law

The peak frequency of the Planck spectrum at a given temperature T is

10
ν max (in Hz) = 5.8789 × 10 T (in K). (31)
One can easily derive this law by equating to zero the derivative of the
intensity I ν with respect to frequency ν, in equation (27).
This, Wien’s law, explains why stars with higher temperatures appear
bluer, and those with lower temperatures appear redder. Conversely, as-
tronomers infer one of the most important physical parameters of stars, the
surface temperature, from measuring their color.

Can we observe the peak of the thermal blackbody spectrum in
the radio region?
• No hope for thermal radiation from a stellar surface, or from a fairly
warm interstellar cloud.

• For submillimeter waves at around 500 GHz (which are still regarded
as radio waves), we can see the peak if the temperature of the medium
is below 10 K (T ≤ 10 K).

• For example, the cosmic background radiation with T ' 2.7 K has its
peak at around 170 GHz.

9.3 Spectral Indices of Thermal Continuum Radio Sources

We now understand why the spetral indicies α (S ν ∝ ν −α ) of thermal contin-
uum radio sources like the Moon, the quiet Sun, and the left half portion of
the spectrum of Orion nebula, are all close to −2 (α ' −2). This is because
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thermal continuum sources mostly show Rayleigh–Jeans spectra, I ν = 2ν 2 kT,
c
in the radio region.

9.4 Stars are Faint and Gas Clouds are Bright in the
Radio Sky

According to the Planck spectrum, the intensity of a hotter black body is
always stronger than the intensity of a colder body, for any frequency range.
On the other hand, the flux density S ν , which we directly detect with our
radio telescopes, is proportional to the product of the intensity I ν and the
solid angle of the radio source Ω (S ν ∝ I ν Ω). While the surface temperatures

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