We have calculated the scattering of light by media consisting of densely packed spherical particles by applying geometric optics and Monte Carlo simulation. We found that when the packing density is increased, dark surfaces are clearly brightened, especially near grazing emergence and incidence. Also, the transmission through a finite layer is reduced with opaque particles but not with transparent ones. The results indicate that previous models that use low-density approximations (Lommel–Seeliger, Hapke, Lumme–Bowell, etc.) are not accurate for typical regoliths.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Plane A(μ0) and Spherical A* Albedos of Densely Packed Mediaa
Density
A(1)
A(0.5)
A(0.1)
A*
Remarks
0.0
0.03
0.05
0.08
0.04
0.05
0.034
0.053
0.16
0.055
0.1
0.044
0.054
0.18
0.065
0.2
0.048
0.069
0.19
0.078
0.3
0.052
0.081
0.20
0.088
0.4
0.062
0.12
0.21
0.10
0.4
0.060
10% size variations
0.4
0.059
40% size variations
0.4
0.060
0.067
0.16
0.066
rough particle (see text)
0.1
0.041
0.045
0.10
0.047
rough particle
Absorbing, monodisperse, homogeneous spheres, except as indicated. The refractive index is n = 1.55 + i, the single-scattering albedo
= 0.2, and the size parameter is x = 500.
Table 2
Scattering by a Finite Layer with Normal Incidencea
n
τ0
D
Reflected
Transmitted
Absorbed
1.33
3.75
0.1
31
69
0
1.33
0.4
31
69
0
+10−5i
0.1
27
61
12
+10−5i
0.4
27
61
12
+10−4i
0.1
9
27
63
+10−4i
0.4
10
24
66
+10−3i
0.1
0.6
1.5
97
+10−3i
0.4
0.9
0.8
98
+10−2i
0.1
0.4
0.8
98
+10−2i
0.4
0.7
0.7
99
1.33
1.5
0.1
12
88
0
1.33
0.4
11
89
0
+10−5i
0.1
11
85
4
+10−5i
0.4
10
86
4
+10−4i
0.1
6
65
29
+10−4i
0.4
6
62
32
+10−3i
0.1
0.6
24
76
+10−3i
0.4
0.9
9
90
+10−2i
0.1
0.4
19
81
+10−2i
0.4
0.7
5
94
Perfectly Reflecting
∞ + ∞i
1.5
0.0
47
54
0
0.1
55
45
0
0.4
72
28
0
Given the refractive index n, the original sparse-medium optical thickness τ0, and the volume density D, we show the percentages of reflected, transmitted, and absorbed fluxes integrated over a hemisphere.
Table 3
Direct Transmissiona as a Function of Volume Density D and Classical Sparse-Medium Optical Thickness τ0
D
τ0
V1
V2
V3
VEspo
VHap
Vsimu
V1−k
0.1
0.5
0.61
0.58
0.58
0.57
0.59
0.66
0.59
0.2
0.5
0.61
0.55
0.54
0.54
0.58
0.52
0.57
0.4
0.5
0.61
0.54
0.53
0.43
0.53
0.42
0.47
0.1
1.5
0.22
0.19
0.19
0.19
0.21
0.16
0.21
0.4
1.5
0.22
0.13
0.10
0.08
0.15
0.06
0.14
0.1
3.75
0.024
0.015
0.014
0.016
0.019
0.015
0.020
0.4
3.75
0.024
0.0059
0.0036
0.0019
0.0084
0.007
0.0076
The probability that a ray path is not intercepted by the particles. Given are the standard exponential extinction V1 or the first-order cumulant, the second- and third-order cumulant expansion results, and the extinction corrected by Esposito’s 1/(1 − D) factor, by Hapke’s −ln(1 − D)/D factor, by our simulation, and by experimental results of Ishimaru and Kuga24 normalized to agree with other results at the low-density limit (see text).
Tables (3)
Table 1
Plane A(μ0) and Spherical A* Albedos of Densely Packed Mediaa
Density
A(1)
A(0.5)
A(0.1)
A*
Remarks
0.0
0.03
0.05
0.08
0.04
0.05
0.034
0.053
0.16
0.055
0.1
0.044
0.054
0.18
0.065
0.2
0.048
0.069
0.19
0.078
0.3
0.052
0.081
0.20
0.088
0.4
0.062
0.12
0.21
0.10
0.4
0.060
10% size variations
0.4
0.059
40% size variations
0.4
0.060
0.067
0.16
0.066
rough particle (see text)
0.1
0.041
0.045
0.10
0.047
rough particle
Absorbing, monodisperse, homogeneous spheres, except as indicated. The refractive index is n = 1.55 + i, the single-scattering albedo
= 0.2, and the size parameter is x = 500.
Table 2
Scattering by a Finite Layer with Normal Incidencea
n
τ0
D
Reflected
Transmitted
Absorbed
1.33
3.75
0.1
31
69
0
1.33
0.4
31
69
0
+10−5i
0.1
27
61
12
+10−5i
0.4
27
61
12
+10−4i
0.1
9
27
63
+10−4i
0.4
10
24
66
+10−3i
0.1
0.6
1.5
97
+10−3i
0.4
0.9
0.8
98
+10−2i
0.1
0.4
0.8
98
+10−2i
0.4
0.7
0.7
99
1.33
1.5
0.1
12
88
0
1.33
0.4
11
89
0
+10−5i
0.1
11
85
4
+10−5i
0.4
10
86
4
+10−4i
0.1
6
65
29
+10−4i
0.4
6
62
32
+10−3i
0.1
0.6
24
76
+10−3i
0.4
0.9
9
90
+10−2i
0.1
0.4
19
81
+10−2i
0.4
0.7
5
94
Perfectly Reflecting
∞ + ∞i
1.5
0.0
47
54
0
0.1
55
45
0
0.4
72
28
0
Given the refractive index n, the original sparse-medium optical thickness τ0, and the volume density D, we show the percentages of reflected, transmitted, and absorbed fluxes integrated over a hemisphere.
Table 3
Direct Transmissiona as a Function of Volume Density D and Classical Sparse-Medium Optical Thickness τ0
D
τ0
V1
V2
V3
VEspo
VHap
Vsimu
V1−k
0.1
0.5
0.61
0.58
0.58
0.57
0.59
0.66
0.59
0.2
0.5
0.61
0.55
0.54
0.54
0.58
0.52
0.57
0.4
0.5
0.61
0.54
0.53
0.43
0.53
0.42
0.47
0.1
1.5
0.22
0.19
0.19
0.19
0.21
0.16
0.21
0.4
1.5
0.22
0.13
0.10
0.08
0.15
0.06
0.14
0.1
3.75
0.024
0.015
0.014
0.016
0.019
0.015
0.020
0.4
3.75
0.024
0.0059
0.0036
0.0019
0.0084
0.007
0.0076
The probability that a ray path is not intercepted by the particles. Given are the standard exponential extinction V1 or the first-order cumulant, the second- and third-order cumulant expansion results, and the extinction corrected by Esposito’s 1/(1 − D) factor, by Hapke’s −ln(1 − D)/D factor, by our simulation, and by experimental results of Ishimaru and Kuga24 normalized to agree with other results at the low-density limit (see text).
Add to BibSonomy