where
x ¼ zb=2:
(4.12)
And the absorption function A n becomes rather simple for small values x
A n ¼ 2x
ffiffiffi
p
p
¼ 2
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
Sau d
=
p
:
(4.13)
The Eq. 4.13 determines the region of the square root low introduced with the
Eq. 4.6.
4.3 The Statistic Molecule Band Model (Goody’s Model)
Goody R.M. studied in 1952 the water vapor rotational band and has found that the
random position of spectral lines is the only feature of the absorption lines spectral
distribution in spectral intervals Dn wider than 25 cm
À1 . It allows the possibility of
analytically calculating the absorption coefficient basing on the random distribution of
absorption lines, characterized by known statistical properties (distribution functions).
Let the spectral interval Dn contain n lines spaced at average distance d, and it be
true Dn ¼ nd. Assume the discrete uniform distribution of the line location within
the spectral interval Dn. Introduce the function P(S i ) determining the probability of
i-th line possessing the intensity S i , and it is normalized in according with:
ð
1
0
PðSÞdS ¼ 1:
(4.14)
Then the mean function value over the interval Dn is found with averaging the
absorption coefficient over all intensities and all line locations:
P n ¼ P n ¼
1
Dn
ð Þ
ð
Dn
dn 1 :::
ð
Dn
dn n Â
ð
1
0
PðS 1 Þ e
Àk1 u dS 1 :::
ð
1
0
PðS n Þ e
Àk n u dS n ;
(4.15)
where k n is the absorption coefficient of n-th line. All integrals are similar that
leads to:
P n ¼
1
Dn
ð
dn
ð
1
0
PðSÞ e
Àk u dS
2
4
3
5
n
¼ 1 À
1
Dn
ð
dn
ð
1
0
PðSÞ ð1 À e
Àk u
ÞdS
2
4
3
5
n
:
(4.16)
42
4 Calculating Transmission Functions with Modeling Absorption Bands