There are numerous textbooks which derive the basic radiative transfer equations for the 21-cm line of neutral hydrogen, mostly following Wild (1952). For the sake of completeness and to avoid ambiguities due to a different choice of notation I will summarize the most significant formulae involved, following the notation used by Kulkarni and Heiles (1988).

The starting point is the equation of radiative transfer:

(1) |

where *I*() is the specific
intensity at frequency ,
*j*() the emissivity,
() the absorption
coefficient and *s* the distance along the line of sight. Let us
introduce the usual definitions for the optical depth
*d*() =
()*ds* and for the source function
*S*() =
*j*() /
(). In
thermodynamic equilibrium Kirchoff's law applies so that, using the
Rayleigh-Jeans approximation for the Planck law, the source function at
radio wavelengths can be written as:

(2) |

Here *c* is the speed of light in vacuum, *k* Boltzmann's
constant and *T*_{s} the spin temperature.
In an equivalent way one can define a brightness temperature
*T*_{B} which corresponds to the
temperature which a black body would have at frequency
if it were to emit the specific
intensity *I*(). Now we can
rewrite the equation of transfer as:

(3) |

This equation can be solved in the case of a homogeneous isolated HI
cloud embedded in
a background radiation field with a brightness temperature of
*T*_{bg}. Employing
*e*^{} as an
integrating factor one finds the following important relation:

(4) |

Let us assume that one can ignore the background radiation
(*T*_{bg} << *T*_{s}). In the
case of an optically thick cloud
(() >> 1) the formula for the brightness temperature
simply reduces to *T*_{B} = *T*_{s}, or the
measured brightness temperature is equal to the spin
temperature, or characteristic temperature, of the cloud. If the cloud
is optically thin
(() << 1) we find that *T*_{B} =
()*T*_{s}. It is important to realize that
() is directly
related to the absorption coefficient per H atom which is dependent on
atomic constants only. Then

(5) |

In this relation *N*_{H} is defined as the column density
of HI along the line of sight in
atoms cm^{-2} and the dependence on frequency has been changed
for dependence on velocity,
expressed in km s^{-1}. In general the profile of the HI line is
determined by its Doppler
width which in turn is dominated by the macroscopic motion of the
gas. Combining all
this, we find in the optically thin case the following useful relation:

(6) |

When reporting HI column densities and HI masses based on this formula, radio astronomers implicitly assume that the emission comes from an optically thin medium. However, HI absorption studies of the Galaxy have indicated that an appreciable fraction of the gas is optically thick so that column densities derived under the assumption of << 1 represent a lower limit to the true HI content.

Now let us look at what happens when *T*_{bg} >
*T*_{s}. In general an HI spectrum is
measured with respect to the background emission,

(7) |

If there is a strong background source present a spectrum along the line
of sight towards
this source will show an absorption feature. By measuring a spectrum
along a nearby
line of sight which presumably has an identical *emission*
spectrum, and assuming that
the brightness temperature, *T*_{src} of the source has
been determined separately, one can determine the optical depth profile

(8) |

Once the optical depth profile is known the spin temperature can be
derived directly via *T*_{s}(*v*) =
*T*_{B,off} /
(1 - *e*^{-(v)}). As explained in Kulkari and Heiles the spin
temperature
is in general a column-density weighted harmonic-mean
temperature. Therefore, and to
enable a meaningful comparison between different regions in the Galaxy,
a better parameter is the integrated optical depth which was introduced by
Crovisier (1981)
and which is defined as:

(9) |

Correspondingly one can define the mean spin temperature as:

(10) |

In the case that the spin temperature would drop as low as the 2.7 K cosmic
background temperature, no HI emission could be observed above the
background and clouds such as the giant HI ring in Leo detected by
Schneider et al. (1983),
but with even lower
densities, could go unobserved. However, it seems that this situation is
unlikely as pointed out by
Schneider et al. (1989)
for the M96 group and by
Corbelli and Salpeter
(1990),
who report on an extensive HI absorption line survey of High Velocity
Clouds (HVCs) and a
study of HI absorption towards bright extragalactic sources whose lines
of sight pass close
to galaxies. They find no evidence for HI column densities of about 5 ×
10^{18} atoms cm^{-2}
or larger which are hidden from detection in emission because their spin
temperature approaches the microwave background temperature (the
subthermal HI effect). Secondly,
they infer from their data a lower limit for *T*_{s} of
several hundred Kelvin for clouds with
column densities of the order of 10^{20} atoms cm^{-2}
with some evidence for cold HI being
present in directions with larger column densities (cf. the review by
J. Kenney in this volume for more information on this subject).