Geoscience Reference
In-Depth Information
Therefore, it is possible to specify the optical thickness of the molecular scat-
tering, the optical thickness of the aerosol absorption etc.
According to the condition accepted in Sect. 1.1 we are considering solar
radiation incoming to the plane atmosphere top as an incident solar parallel
flux
F
0
from direction (
ϕ
0
). Then, deducing the intensity through delta-
function (1.10) and substituting it to the formula of the link between the flux
and intensity (1.5) it is possible to obtain Beer's Law for the solar irradiance
incoming to the horizontal surface at the level
z
:
ϑ
0
,
=
ϑ
0
exp(−
τ
|
ϑ
0
) .
F
d
(
z
)
F
0
cos
(
z
)
cos
(1.42)
In particular, it is accomplished for the solar direct irradiance at the bottom of
the atmosphere
8
:
=
ϑ
0
exp(−
τ
0
|
ϑ
0
) .
F
d
(0)
F
0
cos
cos
(1.43)
Returntothegeneralcaseofthetransferequationwithtakingintoaccount
scattering (1.35). Accomplish the transformation to the dimensionless param-
eters in the transfer equation for convenience of further analysis. In accordance
with the optical thickness definition (1.39) the function
τ
(
z
) is monotonically
α
(
z
)
>
0. In this case
decreasing with altitude that follows from condition
τ
there is an inverse function
z
(
) that is also decreasing monotonically. Using
the formal substitution of function
z
(
τ
)rewritethetransferequationandpass
τ
from vertical coordinate
to coordinate
z
, moreover, the boundary condition
is at the top of the atmosphere
τ
=
τ
=
τ
0
,andthedirection
0andatthebottom
τ
τ
=
α
of axis
is opposite to axis
z
. It follows from the definition (1.39):
d
−
(
z
)
dz
.
µ
=
ϑ
Specify
cos
and pass from the zenith angle to its cosine (the formal
ϑ
=
µ
ϑ
ϑ
=
µ
substitution
arccos
with taking into account sin
d
−
d
). Finally,
α
τ
divide both parts of the equation to value
(
), and instead (1.35) obtain the
following equation:
π
2
1
τ
µ
ϕ
)+
ω
0
(
τ
dI
(
,
,
)
)
µ
=
τ
µ
ϕ
ϕ
τ
χ
τ
µ
,
ϕ
)
d
µ
,
−
I
(
,
,
d
x
(
,
)
I
(
,
(1.44)
τ
π
d
4
0
−1
where
=
σ
τ
σ
τ
(
)
(
)
ω
0
(
τ
)
=
)
,
(1.45)
α
τ
σ
τ
κ
τ
(
(
)+
(
)
α
=
8
Point out that according to Beer's Law the radiance in vacuum (
0) does not change (the
same conclusion follows immediately from the radiance definition). It contradicts to the everyday
identification of radiance as a brightness of the luminous object. Actually, it is well known that the
viewing brightness of stars decreases with the increasing of distance. It is evident that as the star is
further, then the solid angle, in which the radiation incomes to a receiver (an eye, a telescope objective),
is smaller, hence energy perceived by the instrument is smaller too. Just this energy is often identified
with the brightness (and it is called radiance sometimes), although in accordance to definition (1.1) it is
necessary to normalize it to the solid angle. Thus, the essence of the contradiction is incorrect using of
the term “radiance”. In astronomy, the notion equivalent to radiance (1.1) is the absolute star quantity
(magnitude).
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