Brightness temperature or radiance temperature is the temperature of a black body in thermal equilibrium with its surroundings, in order to duplicate the observed intensity of a grey body object at a frequency. This concept is used in radio astronomy, planetary science and materials science. The brightness temperature of a surface is typically determined by an optical measurement, for example using a pyrometer, with the intention of determining the real temperature. As detailed below, the real temperature of a surface can in some cases be calculated by dividing the brightness temperature by the emissivity of the surface. Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. At high frequencies and lowtemperatures, the conversion must proceed through Planck's law. 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. 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. For a black body, Planck's law gives: where 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: 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
The brightness temperature of a source with known spectral radiance can be expressed as: 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
Spectral radiance of black-body radiation is expressed by wavelength as: 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 :
