brightness temperature

Brightness temperature is the temperature at which a blackbody in thermal equilibrium with its surroundings would have to be in order to duplicate the observed intensity of an object at a frequency ν. This is a useful concept only for radiation that obeys the Rayleigh-Jeans Law, and it is extensively used in radio astronomy and planetary science.

For a blackbody, the Planck distribution gives:

where

• is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between ν and ν+dν;
• is the temperature of the black body;
• is Planck's constant;
• is the speed of light; and
• is Boltzmann's constant.

In the Rayleigh-Jeans limit of low frequency, we find:

This can be rewritten to define the brightness temperature as:

Brightness temperature is a useful diagnostic for temperature measurement if the astronomical source is a blackbody and we are in the Rayleigh-Jeans regime. It is not useful if the source is non-thermal and/or we are in the high frequency limit.

If the Planck distribution is reintroduced into the expression for brightness temperature we find:

Failed to parse (unknown function\text): {T_b=\frac{h \nu}{k (\text{Exp}[h \nu /k T]-1)}}

So for the Sun, where the temperature may be estimated to be 6000K, we can plot the brightness temperature against wavelength.

See also

Compare with color temperature.

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