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Brightness temperature or radiance temperature is a measure of the intensity of electromagnetic energy coming from a source. In particular, it is the temperature at which a black body would have to be in order to duplicate the observed intensity of a grey body object at a frequency \nu. This concept is used in
radio astronomy Radio astronomy is a subfield of astronomy that studies Astronomical object, celestial objects using radio waves. It started in 1933, when Karl Jansky at Bell Telephone Laboratories reported radiation coming from the Milky Way. Subsequent observat ...
,
planetary science Planetary science (or more rarely, planetology) is the scientific study of planets (including Earth), celestial bodies (such as moons, asteroids, comets) and planetary systems (in particular those of the Solar System) and the processes of ...
,
materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...
and
climatology Climatology (from Greek , ''klima'', "slope"; and , '' -logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. Climate concerns the atmospher ...
. The brightness temperature provides "a more physically recognizable way to describe intensity". When the electromagnetic radiation observed is thermal radiation emitted by an object simply by virtue of its temperature, then the actual temperature of the object will always be equal to or higher than the brightness temperature. Since the emissivity is limited by 1, the brightness temperature is a lower bound of the object’s actual temperature. For radiation emitted by a non-thermal source such as a pulsar, synchrotron, maser, or a laser, the brightness temperature may be far higher than the actual temperature of the source. In this case, the brightness temperature is simply a measure of the intensity of the radiation as it would be measured at the origin of that radiation. In some applications, the brightness temperature of a surface is 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 (short wavelengths) and low temperatures, 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 (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). Nonthermal sources can have very high brightness temperatures. In pulsars the brightness temperature can reach 1030 K. For the radiation of a helium–neon laser with a power of 1 mW, a frequency spread Δf = 1 GHz, an output aperture of 1 mm, and a beam dispersion half-angle of 0.56 mrad, the brightness temperature would be . For a black body, Planck's law gives:Rybicki, George B., Lightman, Alan P., (2004) ''Radiative Processes in Astrophysics'', I_\nu = \frac \frac where I_\nu (the Intensity or Brightness) is the amount of
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
emitted per unit
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
per unit time per unit solid angle and in the frequency range between \nu and \nu + d\nu; T is the
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
of the black body; h is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
; \nu is
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
; c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
; and k is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
. For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity \epsilon. That makes the reciprocal of the brightness temperature: T_b^ = \frac\, \text\left + \frac\right/math> At low frequency and high temperatures, when h\nu \ll kT, we can use the Rayleigh–Jeans law: I_ = \frac so that the brightness temperature can be simply written as: T_b=\epsilon T\, In general, the brightness temperature is a function of \nu, 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: T_b=\frac \ln^\left( 1 + \frac \right) When h\nu \ll kT we can use the Rayleigh–Jeans law: T_b=\frac For narrowband radiation with very low relative spectral linewidth \Delta\nu \ll \nu and known radiance I we can calculate the brightness temperature as: T_b=\frac


Calculating by wavelength

Spectral radiance of black-body radiation is expressed by wavelength as: I_=\frac\frac So, the brightness temperature can be calculated as: T_b=\frac \ln^\left(1 + \frac \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: T_b = \frac For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: T_b = \frac


In oceanography

In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature and roughness (e.g. from wind-driven waves) of the water.


References

{{reflist Temperature Radio astronomy Planetary science