A plane wave propagating along the z direction in the atmosphere can be
represented by its electric field vector :
The real part, the true refraction index, n= c/vp (where vp is the phase velocity of
the wave), is often expressed in terms of the refractivity,
The imaginary part of the refraction index, nj causes an
exponential attenuation of the wave amplitude, and is related to the
power absorption coefficient
by:
At a frequency close to a rotational transition
,
the optical depth is:
The line profile is given to a good
approximation by the well known Van Vleck and
Weisskopf collisional profile (here, multiplied by , [Townes & Schawlow 1975] p.342):
At the centre of the line, the second term of Eq.8.26 becomes negligible, and
Note that the density in Eq.8.24 is , whereas that in
Eq.8.27 is
: the absorption coefficient at the centre of the line
is proportional to
. It is independent of the total air pressure, as
long as this ratio (hence RH) stays constant. This is not the case, of course,
away from the centre, since
: as the density drops, the lines
become narrower and narrower. In the far wings of the line, the second term in
Eq.8.26 and the contribution from the wings of other lines cannot anymore
be neglected. In fact, a better fit to the water emission in the far wings is
reached if Eq.8.26 is replaced by another collisional line shape, called
the ``kinetic profile''
The opacity and width of the main H216O lines are large:
and
GHz, for the fundamental line in normal conditions of p and
T, and for
g.m-3 and ww= 1 mm (dry weather). These lines
dominate most of the millimetre and submillimetre atmospheric attenuation;
deviations from theoretical line shapes in the far wings (typically 1/10th of
intensity) are accounted for by an empirical continuum. The rare isotopomer
H218O, a few hundred times less abundant than H216O, makes a negligible
contribution (see the discussion by [Waters 1976])