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Fig. 8.1
Intensity across a planetary disk of uniform albedo for a phase angle of
α
=
70
◦
for
macroscopic roughness parameters
θ
of (
from the left
)0
◦
,10
◦
,20
◦
, and 30
◦
, computed based on
the reflectance function introduced by Hapke (
1984
). The other parameters of the Hapke model
have been chosen according to solution 1 obtained for the Moon by Warell (
2004
)
try parameter. However, it is commonly more favourable, as discussed by Warell
(
2004
), to adopt the double Henyey-Greenstein formulation
b
2
b
2
1
+
c
1
−
1
−
c
1
−
p
2HG
(α)
=
b
2
)
3
/
2
+
(8.3)
b
2
)
3
/
2
2
(
1
−
2
b
cos
α
+
2
(
1
+
2
b
cos
α
+
introduced by McGuire and Hapke (
1995
), modelling a backward and a forward
scattered lobe. In (
8.3
), the width of the two lobes is denoted by the value of
b
,
which may lie in between 0 and 1. The parameter
c
denotes the relative strengths of
the two lobes (McGuire and Hapke,
1995
).
The function
H(x)
in (
8.1
) takes into account multiple scattering processes. It
is tabulated by Chandrasekhar (
1950
), and a first-order approximation is given by
Hapke (
1981
) according to
1
+
2
x
H(x)
≈
2
x
√
1
−
w
.
(8.4)
1
+
An improved, second-order approximation of
H(x)
is introduced by Hapke (
2002
),
which is relevant essentially for high-albedo surfaces. Only very small differences,
however, are observed compared to the formulation in (
8.4
) for low-albedo surfaces
like that of the Moon.
An extension of (
8.1
) is introduced by Hapke (
1984
) to take into account the
roughness of the surface on spatial scales between the submillimetre range and the
resolution limit of the imaging device. It is shown by Helfenstein (
1988
), based on
the analysis of close-range stereo images of the lunar surface acquired during the
Apollo missions, that the strongest influence on the reflectance behaviour comes
from surface roughness on the submillimetre and millimetre scales. The full form of
the reflectance function including the macroscopic roughness
θ
is given by Hapke
(
1984
). For illustration, modelled planetary disks are shown in Fig.
8.1
for values of
θ
between 0
◦
and 30
◦
. The other parameters of the reflectance functions are set to
w
3
.
1 according to Warell (
2004
)
(cf. Table 1 in that study, Moon solution 1 therein). For the angular scattering func-
tion, the formulation according to (
8.3
) has been used. Figure
8.1
illustrates that
the Hapke model yields an unrealistically bright rim around the limb of the plane-
tary disk for
=
0
.
168,
b
=
0
.
21,
c
=
0
.
7,
h
=
0
.
11, and
B
0
=
θ
0
◦
, i.e. when the macroscopic surface roughness is not taken into
=
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