Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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3.1 Definitions of the scattering and transmission functions 65
I(0, µ, ϕ)
I(τ
1
, −µ, ϕ)
S
0
τ
ττ
1
=0
=
Fig. 3.1 Schematic diagram of reflected and transmitted radiances at τ = 0 and
τ = τ
1
of a plane–parallel atmospheric layer.
where here and in the following 0 <µ≤ 1. Thus, I (0,µ,ϕ) denotes upward
directed radiation at the top of the atmospheric layer while I (τ
1
, −µ, ϕ) describes
downward directed radiation at τ = τ
1
. The reason that the factor 1/µ has been
introduced in the definitions (3.1) is to obtain symmetry of the S and T functions in
the variables (µ, ϕ) and (µ
0
,ϕ
0
) as required by Helmholtz’s principle of reciprocity
to be discussed soon.
It is noteworthy that the reflected and transmitted intensities include only the dif-
fuse radiation which has suffered one or more scattering processes, i.e. the directly
transmitted radiation S
0
exp(−τ
1
/µ
0
) is not included in (3.1). The radiances appear-
ing in (3.1) are solutions of the standard problem. The extended solution which
includes ground reflection will be taken up in a later section.
So far we have assumed that the atmospheric layer is illuminated by an incident
parallel beam. The solution for an arbitrary incident radiation field I
in
(0, −µ
,ϕ
)
with the same angular distribution at every point on the surface τ = 0 can also
be expressed in terms of the functions S and T . In this situation we would
have to integrate the incoming radiance over all directions of the incoming light
yielding
I (0,µ,ϕ) =
1
4πµ
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)I
in
(0, −µ
,ϕ
)dµ
dϕ
I (τ
1
, −µ, ϕ) =
1
4πµ
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)I
in
(0, −µ
,ϕ
)dµ
dϕ
(3.2)
66 Principles of invariance
By assuming that the incoming radiance is the solar radiation, I
in
may be
expressed in terms of the Dirac δ-functions as
I
in
(0, −µ, ϕ) = S
0
δ(µ − µ
0
)δ(ϕ − ϕ
0
) (3.3)
Note that δ(x) = δ(−x). Substitution of this equation into (3.2) yields, as it should,
the original equations (3.1).
Finally, we would like to point out that the layer thickness τ
1
has been explicitly
included in the definition of the S and T in (3.1) and elsewhere in order to emphasize
their dependence on the optical thickness of the atmospheric layer.
3.2 Diffuse reflection in a semi-infinite atmosphere
Consider a semi-infinite plane–parallel atmosphere being illuminated by a parallel
beam of radiation. In this case only the law of reflection will be of interest. Now
the emerging diffuse radiation at the top of the layer is written as
I (0,µ,ϕ) =
S
0
4πµ
S(µ, ϕ, µ
0
,ϕ
0
) (3.4)
omitting specific reference to the infinite optical thickness of the atmosphere.
The incident solar flux density S
0
at the upper boundary of the layer is reduced
according to Beer’s law, see (2.52). Thus at the optical depth τ we obtain the so-
called reduced parallel solar flux density S
0
exp(−τ/µ
0
). At this level in the down-
ward direction we not only have the reduced parallel solar radiation but also a diffuse
radiation field I (τ,−µ, ϕ). Both parts of the downward radiation will be reflected
by the atmospheric layer below τ so that the upward radiation at τ is given by
I (τ,µ,ϕ) =
1
4πµ
2π
0
1
0
S(µ, ϕ, µ
,ϕ
)I (τ,−µ
,ϕ
)dµ
dϕ
+
S
0
4πµ
exp
−
τ
µ
0
S(µ, ϕ, µ
0
,ϕ
0
)
(3.5)
Owing to the boundary condition
I (0, −µ, ϕ) = 0 (3.6)
at the top of the layer where τ = 0, (3.5) reduces to the first equation of (3.1) in
case of a finite optical thickness. Furthermore, by observing that for a semi-infinite
optical thickness the atmosphere below the level τ is still infinitely large, at
level τ the total downward radiation is reflected according to the same law of
diffuse reflection as (3.4). Thus equation (3.5) can be considered as a statement
3.2 Diffuse reflection in a semi-infinite atmosphere 67
of the invariance of S(µ, ϕ, µ
0
,ϕ
0
) to the addition and subtraction of layers in a
semi-infinite atmosphere.
In order to obtain the required integral equation for S, we differentiate (3.5) with
respect to τ , set τ = 0 and find
d
dτ
I (τ,µ,ϕ)
τ =0
=
1
4πµ
2π
0
1
0
S(µ, ϕ, µ
,ϕ
)
d
dτ
I (τ,−µ
,ϕ
)
τ =0
dµ
dϕ
−
S
0
4πµµ
0
S(µ, ϕ, µ
0
,ϕ
0
) (3.7)
The derivatives occurring in this equation can be evaluated with the help of the RTE
in the form
µ
d
dτ
I (τ,µ,ϕ) = I (τ,µ,ϕ) − J(τ, µ, ϕ) (3.8)
where J is the source function which, according to equation (2.53), may be written
as
J (τ,µ,ϕ) =
ω
0
4π
2π
0
1
−1
P(µ, ϕ, µ
,ϕ
)I (τ,µ
,ϕ
)dµ
dϕ
+
ω
0
4π
S
0
exp
−
τ
µ
0
P(µ, ϕ, −µ
0
,ϕ
0
) (3.9)
The two differential expressions appearing in (3.7) follow from (3.8) and are given
by
d
dτ
I (τ,µ,ϕ)
τ =0
=
1
µ
[
I (0,µ,ϕ) − J (0,µ,ϕ)
]
d
dτ
I (τ,−µ, ϕ)
τ =0
=
1
µ
J (0, −µ, ϕ)
(3.10)
where use was made of the boundary condition (3.6). Substituting (3.10) into (3.7)
yields
I (0,µ,ϕ) − J (0,µ,ϕ) =
1
4π
2π
0
1
0
S(µ, ϕ, µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
−
S
0
4πµ
0
S(µ, ϕ, µ
0
,ϕ
0
) (3.11)
With the help of (3.4), equation (3.11) can be rewritten as
S
0
4π
1
µ
+
1
µ
0
S(µ, ϕ, µ
0
,ϕ
0
) =
1
4π
2π
0
1
0
S(µ, ϕ, µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
+ J (0,µ,ϕ) (3.12)
68 Principles of invariance
In order to replace J (0,µ,ϕ) on the right-hand side of (3.12), we again use (3.9).
To make the mathematical operation more obvious, we first rewrite the integral term
as
1
−1
P(µ, ϕ, µ
,ϕ
)I (τ,µ
,ϕ
)dµ
=
1
0
P(µ, ϕ, µ
,ϕ
)I (τ,µ
,ϕ
)dµ
+
1
0
P(µ, ϕ, −µ
,ϕ
)I (τ,−µ
,ϕ
)dµ
(3.13)
At τ = 0, due to the upper boundary condition, the second integral in (3.13) vanishes
so that
J (0,µ,ϕ) =
ω
0
S
0
(4π)
2
2π
0
1
0
P(µ, ϕ, µ
,ϕ
)S(µ
,ϕ
,µ
0
,ϕ
0
)
dµ
µ
dϕ
+
ω
0
4π
S
0
P(µ, ϕ, −µ
0
,ϕ
0
) (3.14)
Introducing this expression into (3.12) finally yields
1
µ
+
1
µ
0
S(µ, ϕ, µ
0
,ϕ
0
)
=
ω
0
(4π)
2
2π
0
1
0
2π
0
1
0
P(−µ
,ϕ
,µ
,ϕ
)S(µ
,ϕ
,µ
0
,ϕ
0
)S(µ, ϕ, µ
,ϕ
)
×
dµ
µ
dϕ
dµ
µ
dϕ
+
ω
0
4π
2π
0
1
0
P(−µ
,ϕ
, −µ
0
,ϕ
0
)S(µ, ϕ, µ
,ϕ
)
dµ
µ
dϕ
+
ω
0
4π
2π
0
1
0
P(µ, ϕ, µ
,ϕ
)S(µ
,ϕ
,µ
0
,ϕ
0
)
dµ
µ
dϕ
+ ω
0
P(µ, ϕ, −µ
0
,ϕ
0
)
(3.15)
which is the desired integral equation for S. This integral equation is nonlinear,
but for simple phase functions there exist methods to evaluate it without excessive
numerical difficulties.
The scattering function S and the transmission function T have the important
property of being symmetric in the pair of variables (µ, ϕ) and (µ
0
,ϕ
0
). In order
to recognize this, we first refer to the definition of the phase function in the form
(2.68). Recalling the symmetry relations of the associated Legendre polynomials
as stated in (2.59) we immediately recognize the validity of the relations
P(µ, ϕ, µ
,ϕ
) = P(µ
,ϕ
,µ,ϕ)
P(−µ, ϕ, −µ
,ϕ
) = P(µ, ϕ, µ
,ϕ
)
P(µ, ϕ, −µ
,ϕ
) = P(−µ, ϕ, µ
,ϕ
) = P(µ
,ϕ
, −µ, ϕ)
(3.16)
3.2 Diffuse reflection in a semi-infinite atmosphere 69
It is convenient to introduce the function
˜
f (µ, ϕ, µ
,ϕ
) which is obtained from
f (µ, ϕ, µ
,ϕ
) by transposing the variables (µ, ϕ) and (µ
,ϕ
)
˜
f (µ, ϕ, µ
,ϕ
) = f (µ
,ϕ
,µ,ϕ) (3.17)
Using this notation, the symmetry properties of P(µ, ϕ, µ
,ϕ
) may be expressed
by means of
˜
P(µ, ϕ, µ
,ϕ
) = P(µ
,ϕ
,µ,ϕ),
˜
P(µ, ϕ, −µ
,ϕ
) = P(µ
,ϕ
, −µ, ϕ)
(3.18)
Transposing the variables (µ, ϕ) and (µ
0
,ϕ
0
) in (3.15) applying the relations (3.16),
we observe that
˜
S satisfies the same equation as S. Thus if S is a solution of (3.15) so
is
˜
S. However, it does not follow that S is necessarily symmetrical in the variables
(µ, ϕ) and (µ
0
,ϕ
0
). We omit the complete proof of the symmetry relations but
simply assume that the scattering and the transmission functions are symmetric, i.e.
˜
S(µ, ϕ, µ
0
,ϕ
0
) = S(µ, ϕ, µ
0
,ϕ
0
),
˜
T (µ, ϕ, µ
0
,ϕ
0
) = T (µ, ϕ, µ
0
,ϕ
0
)
(3.19)
Verbally, the scattering and the transmission functions are unaltered when the
directions of incidence and emergence are interchanged. This is the principle of
invariance as applied to the functions S and T . A rigorous proof of (3.19) may be
found in Chandrasekhar (1960).
We will now briefly consider the simple case of isotropic radiation which, accord-
ing to (2.13), is given by P = 1. Owing to the axial symmetry of the radiation field,
(3.15) reduces to
1
µ
+
1
µ
0
S(µ, µ
0
) =
ω
0
4
1
0
1
0
S(µ
,µ
0
)S(µ, µ
)
dµ
µ
dµ
µ
+
ω
0
2
1
0
S(µ, µ
)
dµ
µ
+
ω
0
2
1
0
S(µ
,µ
0
)
dµ
µ
+ ω
0
(3.20)
which can be also be written in the form
1
µ
+
1
µ
0
S(µ, µ
0
) = ω
0
1 +
1
2
1
0
S(µ, µ
)
dµ
µ
1 +
1
2
1
0
S(µ
,µ
0
)
dµ
µ
(3.21)
Since S(µ, µ
) is symmetric in the variables (µ, µ
), on the right-hand side of this
equation the two terms in parentheses must be the values of µ and µ
0
of the same
70 Principles of invariance
function. It is customary to denote this function by the symbol H (µ), as defined by
H(µ) = 1 +
1
2
1
0
S(µ, µ
)
dµ
µ
= 1 +
1
2
1
0
S(µ
,µ)
dµ
µ
(3.22)
For the conservative case we obtain
1
0
H(µ)dµ = 2,
1
0
µH (µ)dµ =
2
√
3
(3.23)
With (3.22) equation (3.21) assumes the more simple form
1
µ
+
1
µ
0
S(µ, µ
0
) = ω
0
H(µ)H (µ
0
) (3.24)
By substituting S back into equation (3.22), we find the following expression for
the so-called H-function which is a nonlinear integral equation
H(µ) = 1 +
ω
0
µ
2
H(µ)
1
0
H(µ
)
µ + µ
dµ
(3.25)
The H -function plays an important role in some parts of radiative tansfer theory.
A table of the H -function can be found, for example, in Chandrasekhar (1960).
3.3 Chandrasekhar’s four statements of the principles of invariance
Let us consider a plane–parallel atmospheric layer of total optical thickness τ
1
so
that 0 ≤ τ ≤ τ
1
. We will now state the following four principles.
(1) The radiance I (τ,µ,ϕ) in the upward direction at any level τ results from the reflection
of the reduced incident flux density S
0
exp(−τ/µ
0
) and the diffuse downward radiation
I (τ,−µ
,ϕ
) incident on the surface τ , reflected by the atmosphere below τ having the
optical thickness (τ
1
− τ ). Hence we may write
I (τ,µ,ϕ) =
1
4πµ
2π
0
1
0
S(τ
1
− τ,µ, ϕ,µ
,ϕ
)I (τ, −µ
,ϕ
)dµ
dϕ
+
S
0
4πµ
exp
−
τ
µ
0
S(τ
1
− τ,µ, ϕ,µ
0
,ϕ
0
)
(3.26)
(2) The radiance I (τ,−µ, ϕ) in the downward direction at any level τ results from the
transmission of the incident flux density by the atmosphere of optical thickness τ ,
above the surface, and the reflection by this same surface
1
of the diffuse radiation
1
The formulation ‘reflection by this same surface’ sounds as if some sort of a mirror in horizontal position is
placed in the atmosphere reflecting the radiation. Chandrasekhar simply implies that the atmospheric sublayer
above the surface acts as if it were an imperfect mirror.
3.3 Chandrasekhar’s four statements of principles of invariance 71
I (τ,µ,ϕ) incident on it from below, that is
I (τ,−µ, ϕ) =
1
4πµ
2π
0
1
0
S(τ,µ,ϕ,µ
,ϕ
)I (τ, µ
,ϕ
)dµ
dϕ
+
S
0
4πµ
T (τ,µ, ϕ,µ
0
,ϕ
0
)
(3.27)
(3) The diffuse reflection of the incident light by the entire atmospheric layer of optical
thickness τ
1
is equivalent to the reflection by the part of the atmosphere above the
level τ and the transmission by the same part of the atmosphere of the diffuse radiation
I (τ,µ
,ϕ
) incident on the surface τ from below. The corresponding equation reads
S
0
4πµ
S(τ
1
,µ,ϕ,µ
0
,ϕ
0
) =
1
4πµ
2π
0
1
0
T (τ,µ, ϕ,µ
,ϕ
)I (τ, µ
,ϕ
)dµ
dϕ
+
S
0
4πµ
S(τ,µ,ϕ,µ
0
,ϕ
0
) + I (τ,µ, ϕ)exp
−
τ
µ
(3.28)
The last term on the right-hand side refers to the direct transmission of the diffuse
radiance I (τ,µ, ϕ) which is already in the direction (µ, ϕ).
(4) The diffuse transmission of the incident light by the entire atmospheric layer of optical
thickness τ
1
is equivalent to the transmission of the reduced incident flux density
S
0
exp(−τ/µ
0
) and the diffuse radiation I (τ,−µ
,ϕ
) incident on the surface τ and
transmitted by the atmosphere of optical thickness (τ
1
− τ ) below τ . This principle is
expressed by
S
0
4πµ
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
) =
1
4πµ
2π
0
1
0
T (τ
1
− τ,µ, ϕ,µ
,ϕ
)I (τ, −µ
,ϕ
)dµ
dϕ
+
S
0
4πµ
exp
−
τ
µ
0
T (τ
1
− τ,µ, ϕ,µ
0
,ϕ
0
)
+ I (τ,−µ, ϕ)exp
−
τ
1
− τ
µ
(3.29)
The last term on the right-hand side refers to the direct transmission of the diffuse
radiance I (τ,−µ, ϕ), already in the direction (−µ, ϕ).
It is important to note that equations (3.26)–(3.29) are sufficient to uniquely
determine the radiation field in terms of the scattering and the transmission functions
S and T for plane–parallel atmospheres of finite optical thickness.
The four principles of invariance listed in the previous section will now be used
to derive a set of four integral equations for the scattering and the transmission
functions. These will be obtained by differentiating equations (3.26)–(3.29) with
72 Principles of invariance
respect to τ and then passing (3.26) and (3.29) to the limit τ = 0 and (3.27) and
(3.28) to the limit τ = τ
1
. Furthermore, we will make use of the boundary conditions
I (0, −µ, ϕ) = 0, I (τ
1
,µ,ϕ) = 0 (3.30)
which apply to the standard problem. The resulting equations are
d
dτ
I (τ,µ,ϕ)
τ =0
=
1
4πµ
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)
d
dτ
I (τ,−µ
,ϕ
)
τ =0
dµ
dϕ
+
S
0
4πµ
−
1
µ
0
S(τ
1
,µ,ϕ,µ
0
,ϕ
0
)
+
d
dτ
S(τ
1
− τ,µ, ϕ,µ
0
,ϕ
0
)
τ =0
d
dτ
I (τ,−µ, ϕ)
τ =τ
1
=
1
4πµ
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)
d
dτ
I (τ,µ
,ϕ
)
τ =τ
1
dµ
dϕ
+
S
0
4πµ
d
dτ
T (τ,µ, ϕ,µ
0
,ϕ
0
)
τ =τ
1
0 =
1
4πµ
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)
d
dτ
I (τ,µ
,ϕ
)
τ =τ
1
dµ
dϕ
+
S
0
4πµ
d
dτ
S(τ,µ,ϕ,µ
0
,ϕ
0
)
τ =τ
1
+
d
dτ
I (τ,µ,ϕ)
τ=τ
1
exp
−
τ
1
µ
0 =
1
4πµ
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)
d
dτ
I (τ,−µ
,ϕ
)
τ =0
dµ
dϕ
+
S
0
4πµ
−
1
µ
0
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
)+
d
dτ
T (τ
1
− τ,µ, ϕ,µ
0
,ϕ
0
)
τ =0
+
d
dτ
I (τ,−µ, ϕ)
τ =0
exp
−
τ
1
µ
(3.31)
The derivatives dI/dτ appearing in these equations can be replaced with
the help of the RTE in the form (3.8) by employing the basic definitions (3.1) and
the boundary conditions (3.30). Thus we obtain
d
dτ
I (τ,µ,ϕ)
τ =0
=
1
µ
S
0
4πµ
S(τ
1
,µ,ϕ,µ
0
,ϕ
0
) − J(0,µ,ϕ)
d
dτ
I (τ,−µ, ϕ)
τ =0
=
1
µ
J (0, −µ, ϕ)
d
dτ
I (τ,µ,ϕ)
τ =τ
1
=−
1
µ
J (τ
1
,µ,ϕ)
d
dτ
I (τ,−µ, ϕ)
τ =τ
1
=−
1
µ
S
0
4πµ
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
) − J(τ
1
, −µ, ϕ)
(3.32)
3.3 Chandrasekhar’s four statements of principles of invariance 73
Substituting (3.32) into (3.31) results in the four integral equations
(a)
S
0
4π
1
µ
+
1
µ
0
S(τ
1
,µ,ϕ,µ
0
,ϕ
0
) −
d
dτ
S(τ
1
− τ,µ,ϕ,µ
0
,ϕ
0
)
τ =0
=
1
4π
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
+ J(0,µ,ϕ)
(b)
S
0
4π
1
µ
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
) +
d
dτ
T (τ,µ,ϕ,µ
0
,ϕ
0
)
τ =τ
1
=
1
4π
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)J (τ
1
,µ
,ϕ
)
dµ
µ
dϕ
+ J(τ
1
, −µ, ϕ)
(c)
S
0
4π
d
dτ
S(τ,µ,ϕ,µ
0
,ϕ
0
)
τ =τ
1
(3.33)
=
1
4π
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)J (τ
1
,µ
,ϕ
)
dµ
µ
dϕ
+ J(τ
1
,µ,ϕ)exp
−
τ
1
µ
(d)
S
0
4π
1
µ
0
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
) −
d
dτ
T (τ
1
− τ,µ,ϕ,µ
0
,ϕ
0
)
τ =0
=
1
4π
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
+ J(0, −µ, ϕ)exp
−
τ
1
µ
The validity of
(a)
d
dτ
S(τ,µ,ϕ,µ
0
,ϕ
0
)
τ =τ
1
=−
d
dτ
S(τ
1
− τ, µ, ϕ,µ
0
,ϕ
0
)
τ =0
(b)
d
dτ
T (τ,µ,ϕ,µ
0
,ϕ
0
)
τ =τ
1
=−
d
dτ
T (τ
1
− τ, µ, ϕ,µ
0
,ϕ
0
)
τ =0
(3.34)
may be easily verified. Multiplying (3.33b) by −1/µ
0
and (3.33d) by 1/µ and
adding the results, with the help of (3.34b) we obtain an expression for the derivative
of the transmission function with respect to τ analogously to (3.33c)
S
0
4π
1
µ
−
1
µ
0
d
dτ
T (τ,µ, ϕ,µ
0
,ϕ
0
)
τ =τ
1
=
1
4πµ
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
−
1
4πµ
0
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)J (τ
1
,µ
,ϕ
)
dµ
µ
dϕ
−
1
µ
0
J (τ
1
, −µ, ϕ) +
1
µ
J (0, −µ, ϕ)exp
−
τ
1
µ
(3.35)
74 Principles of invariance
Finally, in (3.33a,b) the derivatives of S and T with respect to τ may be replaced
by means of (3.33c,d) yielding
S
0
4π
1
µ
+
1
µ
0
S(τ
1
,µ, ϕ,µ
0
,ϕ
0
)=
1
4π
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
−
1
4π
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)J (τ
1
,µ
,ϕ
)
dµ
µ
dϕ
+ J(0,µ,ϕ) − J(τ
1
,µ,ϕ)exp
−
τ
1
µ
S
0
4π
1
µ
−
1
µ
0
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
)=
1
4π
2π
0
1
0
S(τ
1
,µ,ϕ,µ
,ϕ
)J (τ
1
,µ
,ϕ
)
dµ
µ
dϕ
−
1
4π
2π
0
1
0
T (τ
1
,µ,ϕ,µ
,ϕ
)J (0, −µ
,ϕ
)
dµ
µ
dϕ
+ J(τ
1
, −µ, ϕ) − J (0, −µ, ϕ)exp
−
τ
1
µ
(3.36)
Inspection shows that these equations still contain the source functions J at
the boundaries τ = 0 and τ = τ
1
. These may be eliminated with the help of
equations (3.1), (3.9) and the boundary conditions (3.30). Without difficulties we
obtain the required expressions
J (0,µ,ϕ) =
ω
0
S
0
(4π)
2
2π
0
1
0
S(τ
1
,µ
,ϕ
,µ
0
,ϕ
0
) P(µ, ϕ, µ
,ϕ
)
dµ
µ
dϕ
+
ω
0
4π
S
0
P(µ, ϕ, −µ
0
,ϕ
0
)
J (τ
1
,µ,ϕ) =
ω
0
S
0
(4π)
2
2π
0
1
0
T (τ
1
,µ
,ϕ
,µ
0
,ϕ
0
)P(µ, ϕ, −µ
,ϕ
)
dµ
µ
dϕ
+
ω
0
4π
S
0
exp
−
τ
1
µ
0
P(µ, ϕ, −µ
0
,ϕ
0
)
(3.37)
Equations (3.36) together with (3.37) are the desired integral relations for S
and T in analogy to equation (3.15) for S in case of a semi-infinite atmosphere.
These integral relations describe the problem of diffuse reflection and transmission
of plane–parallel atmospheres of finite optical thickness. As Chandrasekhar points
out, (3.36) may be regarded as the expression of the invariance of the laws of diffuse
reflection and transmission to the addition (or removal) of layers of arbitrary optical
thickness to (or from) the atmosphere at the top and the simultaneous removal (or
addition) of layers of equal optical thickness from (or to) the atmosphere at the base
of the layer.