# Brightness temperature

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 $\nu$. This concept is extensively used in radio astronomy and planetary science.

For a black body, Planck's law gives:

$I_\nu = \frac{2 h\nu^{3}}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}$

where

$I_\nu$ (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 $\nu$ and $\nu + d\nu$; $T$ is the temperature of the black body; $h$ is Planck's constant; $\nu$ is frequency; $c$ is the speed of light; and $k$ is Boltzmann's constant.

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:

$T_b^{-1} = \frac{k}{h\nu}\, \text{ln}\left[1 + \frac{e^{\frac{h\nu}{kT}}-1}{\epsilon}\right]$

At low frequency and high temperatures, when $h\nu \ll kT$, we can use the Rayleigh–Jeans law:

$I_{\nu} = \frac{2 \nu^2k T}{c^2}$

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:

$T_b=\frac{h\nu}{k} \ln^{-1}\left( 1 + \frac{2h\nu^3}{I_{\nu}c^2} \right)$

When $h\nu \ll kT$ we can use the Rayleigh–Jeans law:

$T_b=\frac{I_{\nu}c^2}{2k\nu^2}$

For narrowband radiation with very low relative spectral linewidth $\Delta\nu \ll \nu$ and known radiance $I$ we can calculate the brightness temperature as:

$T_b=\frac{I c^2}{2k\nu^2\Delta\nu}$

## Calculating by wavelength

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

$I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1}$

So, the brightness temperature can be calculated as:

$T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right)$

For long-wave radiation $hc/\lambda \ll kT$ the brightness temperature is:

$T_b=\frac{I_{\lambda}\lambda^4}{2kc}$

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

$T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} }$

It should be noted that 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 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 14×109Lua error: Unmatched close-bracket at pattern character 67..