# Harvard Astronomy 201b

## Sample Post: Definitions of Temperature

In Uncategorized on January 22, 2011 at 4:03 am

A list of the more common uses of the term “temperature”, and their interrelationships

The Radex manual gives a nice, concise description of these terms

## Kinetic Temperature

The “normal” temperature, that appears in the ideal gas law. The average translational kinetic energy of particles.

## Excitation Temperature

A property of a pair of energy levels, equal to the temperature of a system in thermodynamic equilibrium that would produce the same population ratio.Defined by the following equation

$\frac{n2}{n1} = \frac{g2}{g1} {\rm exp}\big[\frac{E_1 - E_2}{k T}\big]$

The excitation temperature is equal to the kinetic temperature in the limit of thermodynamic equilibrium.

## Brightness Temperature

A property of a body radiating at a specific intensity. Equal to the temperature of an ideal blackbody that would emit at the same intensity.

$I_\nu = \frac{2 h\nu^3}{c^2} \frac{1}{{\rm exp}\big[ \frac{h \nu}{k T_B} \big] - 1}$

Equal to the Rayleigh Jeans temperature at long wavelengths.

## Rayleigh-Jeans Temperature

A property of an object radiating at a given specific intensity. Equal to the temperature of a Rayleigh-Jeans emitter that would emit at the same intensity

$I_\nu = \frac{2kT \nu^2}{c^2}$

Converges to the brightness temperature at long wavelengths.

Synonymous with the Rayleigh Jeans temperature.

## Antenna Temperature

The power received by an antenna, expressed as a temperature

$P^{source} = kT_A \Delta \nu$

or, expressed in terms of source flux,

$T_A = \frac{A}{2k} F_\nu \eta_A$

Here, \eta_A  is the efficiency of the antenna (the fraction of radiation detected by the telescope), and A is the telescope area.

## Main Beam Temperature

The antenna temperature, corrected for the telescope efficiency

$T_{mb} = T_A / \eta_{A}$

The main beam temperature and (Rayleigh Jeans) radiation temperature are equal when a source is resolved (i.e. it fills the telescope beam). Otherwise,

$T_{mb} = T_r \frac{\Omega_{\rm source}}{\Omega_{\rm beam}}$