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of energy)
F
(
t
) according to (Sobolev 1972; Hulst 1980; Minin 1988) (often
it is specified as net spectral energy flux) as a factor of the proportionality
of radiation energy
dE
λ
incoming within a particular infinitesimal interval
λ
λ
λ
]andtime[
t
,
t
+
dt
]tothesurface
dS
of wavelength [
,
+
d
from
the all
λ
,
dS
si.e.:
directions
to values
dt
,
d
dE
dt d
=
F
(
t
)
.
(1.3)
λ
λ
dS
Adduce the “physical” definition of the irradiance that is often used instead of
the “formal” one expressed by (1.3). Radiation energy incoming per unit area
per unit time, per unit wavelength is called a radiation flux or irradiance. This
definition corresponds correctly to (1.3) provided the meaning that energy
is equivalent to the difference of incoming and emitted energy and uses the
differential scale of area, time and wavelength. Proceeding from this interpre-
tation, we will further use the term
energy
as a synonym of the
flux
implying
the value of energy incoming per unit area, time and wavelength.
To characterize the direction of incoming radiation to the element
dS
in
ϑ
ϕ
addition to the angle
,whichiscountedoff
as an angle between the projection of the vector
r
to the plane
dS
and any
direction on this plane (0
,introducetheazimuthangle
≤
ϕ
≤
2
π
). That is to say in fact that we are using the
spherical coordinates system. Energy
dE
incoming to the surface
dS
from all
directions is expressed in terms of energy from a concrete direction
dE
(
ϑ
ϕ
,
)
as:
dE
=
ϑ
ϕ
Ω
dE
(
,
)
d
,
Ω
=
π
4
where the integration is accomplished over the whole sphere. Using the well-
known expression for an element of the solid angle in the spherical coordinates
d
Ω
=
ϕ
ϑ
ϑ
d
sin
d
we will get:
π
π
2
dE
=
ϕ
ϑ
ϕ
ϑ
ϑ
d
dE
(
,
) sin
d
.
0
0
After the substituting of this expression to (1.3) with accounting (1.2) we will
get the formula to express the irradiance:
π
π
2
=
ϕ
ϑ
ϕ
ϑ
ϑ
ϑ
F
(
t
)
d
I
(
,
,
t
) cos
sin
d
.
(1.4)
λ
λ
0
0
ϑ
ϕ
λ
In addition to direction (
and time
t
the solar radiance in the
atmosphere depends on placement of the element
dS
.Owingtothesphericity
of the Earth and its atmosphere, it is convenient to put the position of this ele-
ment in the spherical coordinate systemwith its beginning in the Earth's center.
,
), wavelength
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