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µ
0
to cosine
µ
that characterizes an arbitrary direction and specifying the
∆τ
, we obtain the following due to the method of the continuous
random values simulating:
free pass as
∆τ
=
µ
ln
β
τ
:
=
τ
+
∆τ
−
,
.
(2.7)
µ
Mark that, firstly, (2.7) is correctedboth for the photonmoving downward (
>
τ
decreases).
Secondly,deriving(2.7)wedon'tneedtheexplicitformoftheprobability
density of the photon free path function, however we will need it later. As
follows from the above-mentioned relations the probability of the photon path
within the range of 0 to
τ
increases) and for the photonmoving upward (
µ
0and
<
0and
τ
τ
=
τ|µ
0
). After differentiating with
is
P
(
)
1−exp(−
τ
respect to
the following is obtained:
1
|
µ
|
ρ
∆τ
)
=
∆τ
|µ
).
(
exp(−
(2.8)
τ
)(di-
rectly to the terminology - the probability of the quantum surviving). Thus if
β
≤
ω
0
(
ω
0
(
The probability of the photon scattering in the atmosphere is
τ
),thenthephotonscatteringisoccurringintheoppositecasethe
absorptionishappening,i.e.attheendofthetrajectory.Thecosineofthe
scattering angle
χ
φ
are to be obtained in
the scattering case. As the phase function does not depend on the azimuth it
is uniformly distributed within the interval [0, 2
and the azimuth of the scattering
π
φ
=
πβ
]thatgives
2
.The
χ
density of the probability of the scattering to the angle with cosine
is phase
χ
function
x
(
) [according to the definition (1.2) in Sect. 1.1]. As this value is
specified in the look-up tables, based on simulating rule (2.6) and repeating
literally the reasons for (2.1)-(2.5) with accounting of 1
|
2 factor in normalizing
relation (1.18) we obtain:
χ
k
)−
x
2
(
τ
,
τ
,
χ
k
)+2
∆
k
(2
β
τ
))
χ
=
χ
k
+
x
(
−
X
k
(
,
(2.9)
∆
k
=
i
−1
τ
)
1
1
τ
,
χ
j
+1
)+
x
(
τ
,
χ
j
)](
χ
j
−
χ
j
+1
), number
k
is derived
where
X
i
(
2
[
x
(
=
j
τ
)
≤
2
β
τ
), value
β
from condition
X
k
(
<X
k
+1
(
is the same as in (2.9) and:
.
τ
,
χ
k
+1
)−
x
(
τ
,
χ
k
)
β
τ
)
x
(
χ
=
χ
k
−
2
−
X
k
(
∆
k
=
∆
k
=
0p vid s
χ
k
−
χ
k
+1
τ
,
χ
k
)
x
(
τ
)and
x
(
τ
,
χ
j
)itisconvenienttoconstructthe
Owing to linear relation
X
j
(
=
=
preliminary table
X
j
(
z
i
), where
j
1,...,
M
,
i
1,...,
N
from the initial data
τ
) with (2.3) and (2.5). After
scattering, the photon needs to determine the next direction of its motion. This
not complicated problem is reduced to the solving of the spherical triangles
(Stepanov 1948). Specifying the direction of the photon before the scattering
and to use it for the interpolation of function
X
j
(
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