Brightness temperature

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Brightness temperature is the temperature a black body in thermal equilibrium with its surroundings would have to be to duplicate the observed intensity of a grey body object at a frequency .[1] This concept is extensively used in radio astronomy and planetary science.

The brightness temperature is not a temperature as ordinarily understood. It characterizes radiation, and depending on the mechanism of radiation can differ considerably from the physical temperature of a radiating body (though it is theoretically possible to construct a device which will heat up by a source of radiation with some brightness temperature to the actual temperature equal to brightness temperature).[2] Nonthermal sources can have very high brightness temperatures. In pulsars the brightness temperature can reach 1026 K. For the radiation of a typical helium–neon laser with a power of 60 mW and a coherence length of 20 cm, focused in a spot with a diameter of 10 µm, the brightness temperature will be nearly 14×109 K.[citation needed]


For a black body, Planck's law gives:[2][3]

where

(the Intensity or Brightness) is the amount of energy emitted per unit surface area per unit time per unit solid angle and in the frequency range between and ; is the temperature of the black body; is Planck's constant; is frequency; is the speed of light; and is Boltzmann's constant.

For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity . That makes the reciprocal of the brightness temperature:

At low frequency and high temperatures, when , we can use the Rayleigh–Jeans law:[3]

so that the brightness temperature can be simply written as:

In general, the brightness temperature is a function of , and only in the case of blackbody radiation it is the same at all frequencies. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation.

Calculating by frequency[edit]

The brightness temperature of a source with known spectral radiance can be expressed as:[4]

When we can use the Rayleigh–Jeans law:

For narrowband radiation with very low relative spectral linewidth and known radiance we can calculate the brightness temperature as:

Calculating by wavelength[edit]

Spectral radiance of black-body radiation is expressed by wavelength as:[5]

So, the brightness temperature can be calculated as:

For long-wave radiation the brightness temperature is:

For almost monochromatic radiation, the brightness temperature can be expressed by the radiance and the coherence length :

See also[edit]

References[edit]

  1. ^ Brightness Temperature https://www.astro.cf.ac.uk/observatory/radiotelescope/background/?page=brightness\title= Brightness Temperature Check |url= value (help).  Missing or empty |title= (help)
  2. ^ a b Rybicki, George B., Lightman, Alan P., (2004) Radiative Processes in Astrophysics, ISBN 978-0-471-82759-7
  3. ^ a b "Blackbody Radiation". 
  4. ^ Jean-Pierre Macquart. "Radiative Processes in Astrophysics" (PDF). 
  5. ^ "Blackbody radiation. Main Laws. Brightness temperature" (PDF).