Brightness temperature is the temperature a black body

Black body

A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...

 in thermal equilibrium

Thermal equilibrium

Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...

 with its surroundings would have to be to duplicate the observed intensity

Intensity (heat transfer)

In the field of heat transfer, intensity of radiation I is a measure of the distribution of radiant heat flux per unit area and solid angle, in a particular direction, defined according todq = I\, d\omega\, \cos \theta\, dAwhere...

 of a grey body object at a frequency .
This concept is extensively used in radio astronomy

Radio astronomy

Radio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The initial detection of radio waves from an astronomical object was made in the 1930s, when Karl Jansky observed radiation coming from the Milky Way. Subsequent observations have identified a number of...

 and planetary science

Planetary science

Planetary science is the scientific study of planets , moons, and planetary systems, in particular those of the Solar System and the processes that form them. It studies objects ranging in size from micrometeoroids to gas giants, aiming to determine their composition, dynamics, formation,...

.

For a black body, Planck's law gives:

where

• (aka. Brightness) is the amount of energy
  Energy
  In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
  
   per unit surface
  Surface
  In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
  
   per unit time
  Time
  Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
  
   per unit solid angle
  Solid angle
  The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
  
   emitted in the frequency range between and ;
• is the temperature
  Temperature
  Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
  
   of the black body;
• is Planck's constant;
• is frequency
  Frequency
  Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
  
  .
• is the speed of light
  Speed of light
  The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
  
  ; and
• is Boltzmann's constant.

For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity

Emissivity

The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...

 .
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: