By analogy with the Rayleigh-Jeans approximation,
, which strictly
applies to long wavelengths, the mm-wave radio astronomers have introduced the
concept of ``radiation'' or ``effective'' temperatures, which scale linearly
with the detected power.
The noise power detected by the telescope is the sum of the power received by the
antenna,
, and of the noise generated by the receiver and
transmission lines,
.
Using Nyquist's relation
,
and
can be expressed in terms of the temperatures TA and
Trec of two resistors, located at the end of the transmission line, which would
yield noise powers equal to
and
,
respectively:
TA is called the ``antenna temperature'' and Trec the ``receiver
temperature''. TA becomes Tload when the receiver horn sees a load, instead
of the antenna, and Tgr when it sees the ground. It should be noted that
Tload and Tgr are not stricto sensus equal to the load and ground physical
temperatures, but are only ``Rayleigh-Jeans'' equivalent of these temperatures (they
are proportional to the radiated power). For ambient loads, they approach however
closely the physical temperature, since
K at
mm.
When observing with the antenna a source and an adjacent emission-free reference
field, one sees a change
in antenna temperature.
Because of the calibration method explained below, it is customary, in mm-wave
astronomy, to replace
, the source antenna temperature, by
, the source antenna temperature corrected for atmosphere absorption and
spillover losses. Both are related through:
where is the line-of-sight atmosphere opacity.
and
are the forward and beam efficiency factors, which denote the fractions of
the power radiated by the antenna on the sky and in the main beam, respectively
(they are typically of the order of 0.9 and 0.7).
The source equivalent ``radiation temperature'' TR (often improperly called
``brightness temperature'' and therefore denoted TMB when it is averaged over
the main beam) and
are related through
where
is the antenna power pattern. For a source
smaller than the main beam,
.
When observing a small astronomical source with an antenna temperature
K, located at an elevation el, one detects a signal
(of scale:
volt or counts per Kelvin):
This signal can be compared with the signals observed on the blank
sky (Tatm), close to the source, and to that observed on a hot load
(Tload):