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ϕ
0
). Actually
in the cloudless atmosphere, the intensity of solar direct radiation is essentially
greater than the intensity of scattered radiation. In this case, the direction of
solar radiation is only one, the intensity depends only on the altitude, and the
transfer equation (1.35) transforms to the following:
ϑ
0
,
it conforms to the direction of the direct radiation spreading (
dI
(
z
)
dz
ϑ
0
=
α
cos
(
z
)
I
(
z
) .
(1.37)
ϑ
0
>
0 in (1.37). Differential equation (1.37) together
with boundary condition
I
Markthatitisalwayscos
=
I
(
z
∞
), where
z
∞
is the altitude of the top of
the atmosphere (the level above which it is possible to neglect the interaction
between solar radiation and atmosphere) is elementary solved that leads to:
⎛
⎞
z
1
cos
=
⎝
α
(
z
)
dz
⎠
.
I
(
z
)
I
(
z
∞
)exp
ϑ
0
z
∞
It is convenient to rewrite this solution as:
=
I
(
z
∞
)exp
⎛
⎞
z
∞
1
cos
(
z
)
dz
⎝
−
α
⎠
.
I
(
z
)
(1.38)
ϑ
0
z
This relation illustrates the exponential decrease of the intensity in the extinct
medium and it is called Beer's Law.
Introduce the dimensionless value:
z
∞
(
z
)
dz
,
τ
=
α
(
z
)
(1.39)
z
that is called
the optical depth
oftheatmosphereataltitude
z
.Itsimportant
particular case is
the optical thickness
of the whole atmosphere:
z
∞
τ
0
=
α
(
z
)
dz
.
(1.40)
0
Then Beer's Law is written as:
=
τ
|
ϑ
0
) .
I
(
z
)
I
(
z
∞
) exp(−
(
z
)
cos
(1.41)
As it follows from definitions (1.39) and (1.40) and from “summarizing rules”
(1.23), the analogous rules are correct for the optical deepness and for the
optical thickness:
M
M
τ
=
1
τ
i
(
z
),
τ
0
=
1
τ
0,
i
.
(
z
)
=
=
i
i
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