- Email: [email protected]
Shadow-Hiding Opposition Surge for a Two-Layer Surface
ICARUS
128, 15–27 (1997) IS975724
ARTICLE NO.
Shadow-Hiding Opposition Surge for a Two-Layer Surface John K. Hillier MS 183-501, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109 E-mail: [email protected] Received December 15, 1995; revised February 18, 1997
roughness and a shadow-hiding opposition surge (a nonlinear increase in brightness as opposition is approached observed on virtually every airless Solar System body). It has recently been suggested that a second mechanism, coherent backscatter, may be responsible for the extremely narrow opposition surges observed on several Solar System bodies (Hapke 1990, Mischenko 1992a,b, Mischenko and Dlugach 1992) including some outer planet satellites (Domingue et al. 1991, Thompson and Lockwood 1992, Goguen et al. 1989), asteroids (Harris et al. 1989), and the Moon (Nozette et al. 1994). In this mechanism, light rays which travel identical but reversed paths in a surface will emerge in phase and interfere constructively leading to up to a factor of two increase in brightness at opposition. A review of recent work on coherent backscatter is given by Muinonen (1993). However, this mechanism has not been included in the model. The shadow-hiding explanation for the opposition surge was first proposed by Seeliger (1887, 1895) to explain his observations of the phenomena in Saturn’s rings. It has since been discussed by a number of researchers and several models for the shadow-hiding mechanism (e.g., Irvine 1966, Lumme and Bowell 1981, Hapke 1986) exist. One way of looking at the shadowhiding opposition surge is that if an incident ray can penetrate to a certain point in the surface it will, in essence, have preselected a preferential escape route back along or near the direction the incident ray came from (at or near zero degrees phase). Indeed, at precisely zero degrees phase, emergent rays will automatically escape (if it did not hit anything going in, it must be able to escape back along precisely the same path). Another way of visualizing the shadow-hiding opposition surge is that at zero phase particles will hide their own shadows but the shadows quickly become visible as one moves away from opposition. Here I extend the shadow-hiding model of Hapke (1986) for use with a two-layer surface. It should be noted that, as an extension of Hapke’s (1981, 1984, 1986) photometric model, the model I propose here includes similar approximations (e.g., particles large compared to the wavelength, diffraction unimportant for a densely packed surface) and thus potentially suffers from the same limitations as have
A shadow hiding opposition surge, based on B. Hapke (1986, Icarus 67, 264–280), is derived for a surface consisting of an optically finite veneer of material over an underlying substrate. As done by Hapke in his photometric model, it is assumed that standard radiative transfer theory with correction for macroscopic roughness and a shadow hiding opposition surge provides an adequate description of the two-layered surface. The scattering properties of the upper layer are calculated using the doubling technique while Hapke’s photometric model is used to describe the underlying layer. An adaptation of the doubling technique is used to calculate the scattering from the combined system. The model should have wide applicability in the Solar System and has been used with success to describe the polar regions of Ganymede (J. Hillier et al., 1996, Icarus 124, 308–317) as well as the equatorial regions of Triton (J. Hillier and J. Veverka, in preparation). Using this model, it is shown that extremely narrow opposition surges, as observed on many Solar System bodies, might be explained by a shadowhiding model even for relatively compact surfaces if the particle size decreases toward the surface and may provide an alternative explanation to the coherent backscatter mechanism for the observed surges. 1997 Academic Press
1. INTRODUCTION
The surfaces of several bodies in the Solar System appear to be overlain by thin veneers of material. Examples include variations in albedo features on Mars which have been attributed to the deposition or erosion of layers of dust (Sagan et al. 1973, Guiness et al. 1982), many regions of Io which may be overlain by thin volcanic deposits or SO2 frosts which may condense on the night side, the polar regions of Ganymede (which appear to be overlain by thin frost deposits), and the equatorial regions of Triton whose scattering properties have led to the suggestion that they are overlain by a thin veneer of frost (Lee et al. 1992, Helfenstein et al. 1992, Hillier et al. 1994). In this paper, I discuss a two-layer surface photometric model which can be used to describe such regions. The model is based on standard radiative transfer theory, with (following Hapke 1984, 1986) corrections for macroscopic 15
0019-1035/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
16
JOHN HILLIER
been suggested for Hapke’s model (Peltoniemi and Lumme 1992, Mischenko 1994). The model has been used with success to describe the polar regions of Ganymede (Hillier et al., 1996) as well as the equatorial regions of Triton (Hillier and Veverka, in preparation). 2. PHOTOMETRIC MODEL
The model (Fig. 1) assumes that the optically active region of the surface consists of two distinct layers. The doubling technique of Hansen (1969) is used to compute the scattering behavior of the upper layer while Hapke’s (1981, 1984, 1986) photometric model is used to describe the optically thick lower layer. Following Hansen (1969), the total scattering from the combined system is given by I(i, e, a) S(tu ; e, w; e0 , w0) e2(tu / e1tu / e0) 5 1 S0(tu ; e, w; e0 , w0) F 4e 4e
EE
1
e2tu / e0 4fe
3
1 S0(tu ; e9, w9; e0 , w0) dw9 de9 4e9
1
e2tu / e 4fe
1
2f
0
0
EE 1
2f
0
0
T(tu ; e, w; e9, w9) FIG. 1. Schematic representation of two-layer model used in this study.
S0(tu ; e, w; e9, w9)
(1)
1 T(tu ; e9, w9; e0 , w0) dw9 de9 3 4e9
EE EE
1
1 4fe
3
1 S0(tu ; e0, w0, e9, w9) 4fe0
3
T(tu ; e9, w9; e0 , w0) dw0 de0 dw9 de9, 4e9
1
2f
1
2f
0
0
0
0
T(tu ; e, w; e0, w0)
where S0(tu ; e, w; e0 , w0) ;
O y
n51,3 · ? ?
Sn(tu ; e, w; e0 , w0),
(2)
S1(e, w; e0 , w0) 5 4eRH(i, e, a(e, w; e0 , w0)),
(3)
H
J
and
Sn(tu ; e, w; e0 , w0) 5
1 4f
EE 1
2f
0
0
S1(e, w; e9, w9), n odd
S(tu ; e, w; e9, w9), n even
3 Sn21(tu ; e9, w9; e0 , w0) dw9
de9 . e9 (4)
i, e, and a are the incidence, emission, and phase angles, respectively, w is the azimuth angle, e 5 cos(e), e0 5 cos(i), fF is the incident normal flux, S and T are the scattering and transmission functions, respectively, for the upper layer of optical depth tu (as defined by Chandrasekhar 1960) and RH is Hapke’s (1981, 1984, 1986) reflectance function. As with Hapke’s model, the model assumes that the scattering can be adequately described by standard radiative transfer theory with corrections for macroscopic roughness and a shadow hiding opposition surge. Indeed, except for the fact that multiple scattering is fully accounted for in the upper layer of my model, my model yields the same results as Hapke’s in the limiting case of a single layer (either an optically thick upper layer or when the upper and lower layer properties are identical). The model parameters used to describe the scattering properties of the surface are shown in Table I. These are the optical depth of the overlying layer, Hapke’s (1984) macroscopic roughness parameter, u, and, for each layer, the single particle scattering albedo, g˜ 0 , one or more parameters to describe the particle phase function, and the opposition surge amplitude and width parameters, B0 and h (as defined by Hapke (1986)). An additional parameter, the ratio of the upper to lower layer particle radii, rr , is required in the calculation of the opposition surge. Hapke’s (1984) correction for macroscopic roughness can be applied to any bidirectional reflectance function for
17
TWO LAYER OPPOSITION SURGE
TABLE I Parameters for the 2-Layer Surface Photometric Model tu u¯ g ˜ 0u , g˜ 0l Pu(a), Pl(a) B0u , B0l hu , hl rr
determined below, reduce to Hapke’s (1986) B(a) term in the appropriate limit:
Optical depth of the upper layer Average topographic slope angle Single particle scattering albedo for the upper and lower layer One or more parameters to describe the single particle phase function for the upper and lower layer. Amplitude of opposition surge (both layers) Width of opposition surge (both layers) Ratio of upper to lower layer particle radii
a smooth surface. Therefore, the reflectance after correction for macroscopic roughness is given by, r2R(i, e, a) 5 r2S(i9, e9, a)Sh(i, e, a),
(5)
where r2s is the reflectance predicted for a smooth surface, i9 and e9 are the effective incidence and emission angles, respectively, after correction for the tilt as defined by Hapke, and Sh is Hapke’s shadowing function. A derivation of the correction for the shadow-hiding opposition surge, based on Hapke (1986), for a two-layered surface is the subject of the next section.
lim Bu(i, e, a, tu) 5 B(a)
tuRy
(9)
lim Bl(i, e, a, tu) 5 B(a).
t u R0
3.1. Opposition Surge in Overlying Layer For the overlying layer, the derivation of Hapke (1986) is applicable if we limit the integration of the intensity to the optical depth of the overlying layer (instead of infinity for the one layer case). Therefore, we can use the results of Hapke’s Eqs. (29) and (30) without modification. The single scattered radiance (including the opposition surge) is given by
I(i, e, a) 5
Fg˜ 0u 4
pu(a)
E
tu
0
e2(t1t9)/ kel
dt , e
(10)
where
t9 5
1 zl
E
z1zl
z
t(z9) dz9,
(11)
3. DERIVATION OF TWO-LAYER OPPOSITION SURGE
Following Hapke (1986) I assume that the shadow hiding opposition surge affects only the single scattering component. Let us write the total reflectance as r2s 5 rno 1 Bu(i, e, a, tu)rsu 1 Bl(i, e, a, tu)rsl ,
(6)
where i, e, and a are the incidence, emission, and phase angles, respectively, rno is the calculated reflectance without the opposition surge, and rsu(i, e, a) 5
Isu(i, e, a) g˜ 0u e0 5 pu(a) (1 2 e2(tu / e01tu / e)) F 4 e0 1 e (7)
and Isl(i, e, a) g˜ 0l e0 rsl(i, e, a) 5 5 pl(a) e2(tu / e01tu / e) F 4 e0 1 e
and t is the optical depth. zl and kel are given by Hapke’s Eqs. (22) and (23), z1 P kelkrl cot(a/2)
(12)
kel 5 2e0e/(e0 1 e),
(13)
and krl is the effective particle radius defined in terms of the average particle cross section ksl: krl 5
!ksfl.
Assuming a step function distribution function we have, following Hapke (1986), z $ 0: ne(z) 5 0, (above the surface)
(8)
are the single scattered reflectance for the upper and lower layers (ignoring the opposition surge), respectively. The second and third terms in Eq. (6) represent the additional reflectance from the opposition surge in the upper and lower layers, respectively. The functions Bu and Bl , to be
(14)
z , 0: ne(z) 5 ne 5 constant,
(15)
where ne is the effective particle density (Hapke’s Eq. (12)). The optical depth as a function of depth into the surface is then given by
t(z) 5 2nekslz.
(16)
18
JOHN HILLIER
From Eq. (11) we then have (Hapke, Eq. (31)) z $ 0: t9 5 0 2zl , z , 0: t9 5 t 2 /(2tl)
(17)
z # 2zl : t9 5 t 2 tl /2, where
tl 5 nekslzl 5 2kelhu cot(a/2),
(18)
hu ; nekslkrl/2
(19)
and
is Hapke’s (1986) width parameter (for the upper layer). Substitution into Eq. (10) yields Case 1: zl . zupper layer (or tl . tu): I(i, e, a) 5
Fg˜ 0u 4
E
tu
pu(a)
0
e2(t91t9
2
/2tl)/ kel
dt9 , e
(20a)
Case 2: zl , zupper layer (or tl , tu): Fg˜ 0u
I(i, e, a) 5
4 1
pu(a)
FE
tl
0
e2(t91t9
2
/2tl)/ kel
dt9 e
(20b)
G
e0 h2etl /2kele22t / kel 1 e23tl /2kelj . e 1 e0
where, following Hapke, I have multiplied the expression for Bu by his B0 parameter. Bu can then be found by numerical integration of the integrals in Eq. (22).
From the definition of Bu , we have
3.2. Opposition Surge in Underlying Layer
I(i, e, a) 5 Isu(i, e, a)[1 1 Bu],
(21)
where, Isu(i, e, a) is the radiance for single scattering ignoring the opposition surge (Eq. (7)). After some algebraic manipulation, we have, for tl . tu ,
Bu 5 B0u
E
1
tu
0
e2(t91[t9
2
(dt9/e)
tan(a /2)]/4hukel)/ kel
(e0 /(e0 1 e))(1 2 e
2[tu / e01tu / e]
)
2
2 1 , (22a)
and, for tl , tu ,
Bu 5 B0u 1
E
3
tl
0
e2(t91[t9
2
tan(a /2)]/4hukel)/ kel
(e0 /(e 1 e0))(1 2 e
FIG. 2. Schematic diagram showing a cross section through the scattering plane of the overlap of the incident and emergent probability cylinders. In order to simplify the figure, the figure is shown for the same sized particles in both layers. In general, however, the cylinder radii used for each layer will be different. The overlap between the cylinders is denoted by the filled in regions (the darker region that contributed by the lower layer, Ul , and the lighter that contributed by the upper layer, Uu , see text). Note that in the case shown the overlap extends above the surface. However, in general the overlap may end before the surface is reached or be contained entirely in the lower layer.
(dt9/e)
2[tu / e01tu / e]
)
(22b)
4
h2e hu cot(a /2)e22tu / kel 1 e23hu cot(a /2)j 21 , (1 2 e2[tu / e01tu / e])
The probability that an incident ray will penetrate to a given point in the surface without being scattered is just the probability that no particle has its center in the cylinder of radius equal to the effective particle radius, krl, whose axis connects the point with the source. Similarly, the probability that a ray can escape from a point in the surface is just the probability that no particle has its center in the cylinder of radius krl and whose axis connects the point and the observer. However, as illustrated in Fig. 2, portions of these two cylinders overlap. It is this overlap, unaccounted for in standard radiative transfer theory, which leads to the opposition surge. Following Hapke (1986), let us define the optical depth associated with this overlap, equal to the integral of ne over the volume of overlap, to be tc . Then, the total radiance from singly scattered light is given by I(i, e, a) 5
Fg˜ 0l Pl(a) 4
E
y
tu
e2(t / e1t / e02tc)
dt , e
(23)
19
TWO LAYER OPPOSITION SURGE
where the factor of 2tc in the exponent is the correction for the opposition effect. By definition, Bl is given by Bl(i, e, a, tu) 5 B0l
S
D
I(i, e, a) 21 , ISl(i, e, a)
z(t) 5
t 5 tu 2 neuksluz ⇒
(25) z . 0,
where z is the depth in the surface, ne is the effective particle number density, ksl the extinction cross section, and the subscripts u or l refer to the upper or lower layers, respectively. For a step function distribution, ne is constant (at least within each layer) and thus the quantity which must be subtracted from the total optical depth is (26)
where tcl and tcu are the contributions to tc from the overlap in the lower and upper layers and Ul and Uu are the shared volume between the incident and emergent cylinders in the lower and upper layer, respectively. Following Hapke, the overlap between the incident and emergent cylinders extends to a distance of zl , zll P kelkrll cot(a/2)
(27)
zlu P kelkrlu cot(a/2), where, as above, the subscripts denote the lower or upper layer above the scattering location. The total shared volume is (Hapke’s Eqs. (24) and (26)) Uc P
kslkrl cot(a/2) 1 5 kslzl , 2 2kel
(29)
where zl 5 zll or zlu are used to calculate Uc or Ua for the desired layer. 3.2.1. Lower layer contribution (tcl). If z 1 zll , 0 (or t 2 tu , tll), then the total shared volume occurs in the lower layer and we have
tcl 5 nelUcl 5
nelksllzll 2kel
5
tl l
,
(30)
SD
(31)
2kel
tll ; nelksllzll 5 2hlkel cot
a , 2
and hl is Hapke’s (1986) width parameter. If z 1 zll . 0 (or t 2 tu . tll), then the shared volume above z9 5 0 (i.e., above the lower layer) does not contribute to the opposition surge and we have
1 (tu 2 t) z(t) 5 neukslu
tc 5 tcl 1 tcu 5 nel 3 Ul 1 neu 3 Uu ,
D
z9 2 z 2 , zl
where
z,0
1 (tu 2 t) nel ksll
S
Ua(z9) P Uc 1 2
(24)
where, following Hapke, I have multiplied the expression for Bl by his B0 parameter (the subscript l denoting the lower layer), and ISl is the singly scattered radiance ignoring the opposition surge (Eq. (8)). Therefore, all that is left to finish the derivation is to obtain an expression for tc with which Eq. (23) can be numerically integrated. Let us define the boundary between the upper and lower layers to be the z 5 0 point, then
t 5 tu 2 nel ksllz ⇒
while that lying above any given plane at z9 between z and z 1 zl is
(28)
S F GD
tcl 5 nel(Ucl 2 Ual(0)) 5 nelUcl 1 2 1 1
z zll
2
.
(32)
Some algebraic manipulation and substituting in for Ucl from Eq. (28) yields
tcl 5 nel
S
D
ksll z2 22z 2 . 2kel zll
(33)
Finally, substituting in for z and zll from Eqs. (26) and (28) yields the desired result:
tc l 5
H
J
t 2 tu)2 1 (t 2 tu) 2 . kel 2tll
(34)
3.2.2. Overlying layer contribution (tcu). If z 1 zlu , 0 then there is no overlap between the incident and emergent cylinders in the upper layer and we have
tcu 5 0.
(35)
If z 1 zlu . 0 and z 1 zlu , zu ;
tu , neukslu
(36)
20
JOHN HILLIER
then the overlap between the incident and emergent cylinders ends before reaching the top of the upper layer and we have
tcu 5 neu 3 Ua(0),
tcu 5 neu 3 Uc 5
S D S D
neuksluzlu 2kel
2
11
5
neuksluzlu 2kel
S D z 11 zlu
tcu 5 (38)
2
tlu 2kel
tlu
S
D
z z2 1 2 , kel zlu 2z lu
(39)
where
tlu ; neuksluzlu 5 2kelhu cot(a/2).
(40)
tcu 5
2kel
1
S
tl u
D
tl u 2kel
1
tl u
krlu zlu 5 . krll zll
(42)
tlu , texp
D
tu 2 t (tu 2 t)2 1 . kel tll rr 2(tll rr)2
(43)
Finally, if z 1 zlu . zu then the surface is reached before the overlap between the incident and emergent cylinders ends and we have
tcu 5 neu 3 (Ua(0) 2 Ua(zu)) 5 n eu Uc 5 neu Uc
t1u , texp 2 tu , tlu . texp 2 tu
S
SF G F H JG D SF G FS D G D 11
z zlu
2
11
z zlu
2
2 12
zu 2 z zlu
2
zu z 2 zlu zlu
11
S F S
G D D
tu 1 1 ( tu 2 t) t 2u 2 2 kel tlu tll rr 2t lu
(46)
(47)
where hu and hl are the opposition surge width parameters for the upper and lower layers, respectively. With this modification the boundary conditions can be expressed in terms of the optical depth as
We have the desired result
tcu 5
(45)
(41)
Making the definition rr ;
tlu
texp 2 tu neukslu hu 1 5 5 , t 2 tu nelksll hl rr
tu 2 t (tu 2 t) 1 . kel nelksllzlu 2(nelksllzlu)2 2
.
Finally, for use in Eq. (23), it is necessary to express the boundaries between the different regimes (z 1 zlu , 0; z 1 zlu . 0 and z 1 zlu , zu ; z 1 zlu . zu) in terms of the optical rather than the physical depth. In general, the physical distance traveled in each layer for a given optical depth will be different. The optical depth expected at depth z assuming that we were still in the upper layer, texp , is related to the actual optical depth, t, in the lower layer by the expression
Substituting in for z (,0) from Eq. (25) we have
tl u
2
tu tu (tu 2 t) 5 11 2 . kel tll rr 2tlu
2z z 1 , zlu z 2lu
1
zu z zu 2 zlu zlu zlu
Substituting in for Uc , z, zu , and zlu from Eqs. (28), (25), (36), and (27) we have the desired result:
2
or
tc u 5
S F G F GD
tcu 5 neu Uc 2 1 1
(37)
where Ua(0) is the shared volume in the upper layer (which begins at z 5 0). Substituting in for Ua(0) we have z 11 zlu
After some algebraic manipulation we have
2
(44)
2
.
tlu . texp ,
regime 1 (tcu given by Eq. (35)),
(48a)
regime 2 (tcu given by Eq. (43)), (48b) regime 3 (tcu given by Eq. (46)).
(48c)
4. SUMMARY
In summary, a photometric model for describing twolayered surfaces, based on standard radiative transfer theory with corrections for macroscopic roughness (Hapke, 1984) and a shadow-hiding opposition surge, is developed. The scattering properties of the optically finite upper layer are calculated using the doubling procedure of Hansen (1969), while Hapke’s (1981, 1984, 1986) photometric model, ignoring the opposition surge and macroscopic roughness, is used to describe the underlying surface. The scattering from the combined system, R2 , again ignoring
21
TWO LAYER OPPOSITION SURGE
the opposition surge and macroscopic roughness is calculated using an adaptation of the adding technique of Hansen (1969), R2 5
S(tu ; e, w; e0 , w0) e2(tu / e1tu / e0) 1 S0(tu ; e, w; e0 , w0) 4e 4e 1
e2tu / e0 4fe
EE 1
2f
0
0
where Sh is Hapke’s (1984) shadowing function, and i9 and e9 are calculated using procedures outlined in Hapke (1984). The second and third terms in the expression in parenthesis are the corrections for the opposition effect. Rus and Rls are the reflectance from single scattering for the upper and lower layers, respectively,
T(tu ; e, w; e9, w9)
Rus(i, e, a) 5
1 S0(tu ; e9, w9; e0 , w0) dw9 de9 3 4e9
EE
1
e2tu / e 4fe
3
1 T(tu ; e9, w9; e0 , w0) dw9 de9 4e9
1 1 4fe
1
2f
0
0
EE EE 2f
1
2f
0
0
0
0
1 S0(tu ; e0, w0, e9, w9) 4fe0
3
T(tu ; e9, w9; e0 , w0) dw0 de0 dw9 de9, 4e9
Rls(i, e, a) 5
g˜ 0l 4
pl(a)
O y
n51,3 · ? ?
Sn(tu ; e, w; e0 , w0),
(50)
1
2f
0
0
H
S1(e, w; e9, w9), n odd
J
S(tu ; e, w; e9, w9), n even
3 Sn21(tu ; e9, w9; e0 , w0) dw9
Bu 5 B0u
(51)
and
EE
(55)
de9 . e9
i, e, and a are the incidence, emission, and phase angles, respectively, w is the azimuth angle, e 5 cos(e), e0 5 cos(i), fF is the incident normal flux, S and T are the scattering and transmission functions for the upper layer of optical depth tu (as defined by Chandrasekhar 1960), and Rh is Hapke’s (1981, 1984, 1986) reflectance function. The scattering from the system after correction for macroscopic roughness and the shadow hiding opposition surge is then given by R(i, e, a) 5 (R2(i9, e9, a) 1 Bu(i9, e9, a)Rus(i9, e9, a)
(53)
E
1
tu
e2(t91[t9
0
2
(dt9/e)
tan(a /2)]/4hukel)/ kel
(e0 /(e0 1 e))(1 2 e
2[tu / e01tu / e]
)
2
2 1 , (56a)
if tu , tlu and
Bu 5 B0u 1
(52)
1 Bl(i9, e9, a)Rls(i9, e9, a))Sh(i, e, a),
e0 e2(tu / e01tu / e), e0 1 e
where i, e, a are the incidence, emission, and phase angle, respectively, e0 5 cos(i), e 5 cos(e), g˜ 0 is the single particle scattering albedo, p(a) is the single particle phase function (the subscripts u and l denoting the upper and lower layers, respectively), and tu is the upper layer optical depth. Bu is given by
S1(e, w; e0 , w0) 5 4eRH(i, e, a(e, w; e0 , w0)),
1 4f
e0 (1 2 e2(tu / e01tu / e)) (54) e0 1 e
(49)
where
Sn(tu ; e, w; e0 , w0) 5
pu(a)
T(tu ; e, w; e0, w0)
3
S0(tu ; e, w; e0 , w0) ;
4
and
S0(tu ; e, w; e9, w9)
1
g˜ 0u
3
E
tl
0
e2(t91[t9
2
tan(a /2)]/4hukel)/ kel
(dt9/e)
(e0 /(e 1 e0))(1 2 e2[tu / e01tu / e])
h2e hu cot(a /2)e22tu / kel (1 2 e
1e
j
23hu cot(a /2)
2[tu / e01tu / e]
)
21
(56b)
4
for tu . tlu , where B0u and hu are Hapke’s (1986) opposition surge amplitude and width parameters, respectively, for the upper layer and
tlu 5 2kelhu cot(a/2),
(57)
where
kel 5
2ee0 . e 1 e0
(58)
22
JOHN HILLIER
FIG. 3. Predicted scattering at 08 phase for a surface overlain by a thin forward scattering layer. Significant limb darkening is seen at moderate optical depths.
Finally Bl is given by Bl(i, e, a, tu) 5 B0l
S
D
R(i, e, a) 21 , Rls(i, e, a)
hl is Hapke’s opposition surge width parameter for the lower layer, and (59)
where B0l is Hapke’s opposition surge amplitude parameter for the lower layer, Rls is given by Eq. (55) and R(i, e, a) 5
g˜ 0l pl(a) 4
E
tcu 5 y
tu
e2(t / e1t / e02tc)
dt . e
(60)
Equation (60) can be solved numerically given tc . tc is given by
tc 5 tcl 1 tcu ,
tlu
2kel
1
S
D
tlu tu 2 t (tu 2 t)2 1 , kel tll rr 2(tll rr)2
tlu . texp 2 tu tlu , texp
S
tlu . texp ,
D
(tu 2 t) tu tu 11 2 , kel tll rr 2tlu
where rr 5 ru /rl is the ratio of the radii of the upper and lower layer particles and
texp 5 tu 1
5H
tll 2kel
(t 2 tu) $ tll
,
J
(t 2 tu)2 1 (t 2 tu) 2 , kel 2tll
(64)
(61)
where
tcl 5
5
tlu , texp 2 tu
0,
hu l (t 2 tu). hl rr
(65)
5. DISCUSSION
(62) (t 2 tu) , tll ,
where
tll 5 2kelhl cot(a/2),
(63)
Having derived the opposition surge for a two-layer surface, we now proceed to examine the effects of a second layer on a surface’s scattering properties. The effect of adding a thin forward scattering layer is shown in Fig. 3. As long as the upper layer is not too thick, the addition of the forward scattering layer tends to increase the degree of limb darkening seen at low phase angles. This effect is
TWO LAYER OPPOSITION SURGE
23
FIG. 4. Predicted scattering at 08 phase for a high albedo surface overlain by a layer of dark particles. Significant limb darkening is seen at moderate optical depths.
similar to that found for Triton’s thin forward scattering hazes (Hillier and Veverka 1994), which is not surprising: the upper layer behaves very much like an overlying haze layer. A similar effect is seen for dark particles placed over a brighter substrate (Fig. 4). In both cases the cause is clear: the larger slant path optical depths seen toward the limb and terminator tend to increase the contribution of the darker upper layer, decreasing the predicted brightness. As expected, the reverse (more backscattering or brighter particles in the upper layer) tends to lead to limb brightening (Fig. 5). As the upper layer becomes optically thick, the optically active portion of the surface approaches the single layer limit and the pronounced increase in limb darkening (or brightening) disappears. The effects of varying the upper layer opposition surge parameters as compared to the lower layer parameters are shown in Figs. 6–8. Lowering the upper layer opposition surge amplitude parameter leads to significant limb darkening near opposition for moderate optical depths (Fig. 6). This is similar to the effect seen for a forward scattering or dark upper layer (Figs. 3 and 4). In each case, the limb darkening is due to the reduced reflectance in the upper layer. However, changes in B0 substantially affect only the low phase data while changes to the single scattering albedo or phase function can be seen at higher phase angles as well. As with a forward scattering or lower albedo layer, the substantial limb darkening seen disappears as the upper layer becomes optically thick. As expected, decreasing the upper layer opposition surge width parameter tends to decrease the width of the overall opposition surge (Fig. 7). Perhaps the most interesting behavior is exhibited when
the particle size is allowed to vary between the layers. Decreasing the size of the upper layer particles decreases the width of the opposition surge, in some cases substantially (Fig. 8). Physically, though the full opposition effect is still seen right at opposition, the smaller particles in the upper layer lead to smaller ‘‘holes’’ for the light to escape through from the lower layer—these holes are filled in much more quickly as the phase angle is increased than otherwise would be the case leading to a quick reduction in the opposition effect. The width of the opposition surge appears to be essentially independent of the albedo or phase function of the surface, as would be expected if the effect is due to a mechanical property of the surface. Such narrow opposition surges have been observed on several Solar System bodies including the Moon (Nozette et al., 1994), icy outer planet satellites (Domingue et al. 1991, Goguen et al. 1989), and some asteroids (Harris et al. 1989). The fact that such narrow opposition surges require extremely (and probably unreasonably) porous surfaces in standard shadow-hiding models has led to the suggestion that the shadow-hiding model is deficient and another mechanism such as coherent backscatter (Hapke 1990, Mischenko 1992a,b, Mischenko and Dlugach 1992) may be responsible for the observed surges. However, the above results suggest that shadow-hiding could lead to such narrow surges even for relatively compacted surfaces if the particle size decreases toward the surface and thus coherent backscatter may not need to be invoked. Being based on Hapke’s shadow-hiding model, the opposition model presented here will yield essentially identical results to his model in the single layer limit (optically
24
JOHN HILLIER
FIG. 5. Predicted scattering at 08 phase for a surface overlain by a thin backscattering layer. As expected, significant limb brightening is seen at moderate optical depths. Similar results are observed for a dark surface overlain by bright particles.
thick upper layer or same parameters in both layers). The model has been used with success to fit Clementine observations of the lunar opposition surge (Buratti et al. 1996). They found that models with h as high as 0.12 (for a particle size ratio of 0.14) can fit the observations though a simple Hapke model with h 5 0.040 6 0.004 between the previously determined values of 0.05–0.07 (Hapke 1986, Hel-
fenstein and Veverka 1987) and the recent value of 0.023 found by Helfenstein et al. (1997) for their shadow-hiding only solution can fit the observations as well. This shows that significantly higher values of h (suggesting lower porosities) can indeed be obtained for a real surface if the particle size decreases toward the surface. While Hapke’s model predicts no dependence on the opposition surge
FIG. 6. Predicted scattering at 08 phase assuming a low upper layer opposition surge amplitude. Other parameters are identical between the layers.
TWO LAYER OPPOSITION SURGE
25
FIG. 7. Scattering from a two-layer surface with lower opposition surge width parameter in the upper layer. Other parameters are identical between the layers. In this figure, emission angle is held constant at 08 while the phase and incidence angles vary. As expected, the opposition surge width tends to decrease as the upper layer optical depth is increased.
with incidence or emission angle, the model presented here does predict such a dependence for a finite upper layer (as the slant path optical depth of the upper layer varies with incidence and emission angle), as illustrated in Fig. 9. Interestingly, the effect of lowering the particle ratio
parameter, leading to a larger opposition surge away from the center of the disk, is the opposite of that predicted by the model of Lumme and Bowell (1981) which predicts that the opposition surge should disappear at the limb. This provides a diagnostic which may be
FIG. 8. Effect of varying the particle size ratio on the opposition surge. All other parameters are identical between the layers. As can be seen, an upper layer consisting of small particles leads to a decrease in the width of the opposition surge.
26
JOHN HILLIER
FIG. 9. Comparison of opposition model predictions across the planetary disk. Shown are results along the mirror meridian at a 5 58. An optical depth of 1.0 is assumed and, except for the particle size, all parameters are identical between the layers. The function B shown on the ordinate can be directly compared with Hapke’s B(a) function (and is essentially equivalent to it when rr 5 1). While Hapke’s model predicts no dependence on incidence or emission angle (as seen in the rr 5 1 plot), the model proposed here does suggest a significant dependence.
useful in distinguishing between the models. Unfortunately, the Clementine observations are almost all nadir pointing, making them ill suited for this purpose. ACKNOWLEDGMENTS John Hillier acknowledges support from NASA Grant NAGW-3837. Portions of this work were performed while the author held a National Research Council-JPL Research Associateship. The author thanks Audouin Dollfus and an anonymous reviewer for constructive reviews.
Hapke, B. W. 1990. Coherent backscatter and the radar characteristics of outer planet satellites. Icarus 88, 407–417. Harris, A. W., J. W. Young, L. Contreiras, T. Dockweiler, L. Belkora, H. Salo, W. D. Harris, E. Bowell, M. Poutanen, R. P. Binzel, D. J. Tholen, and S. Wang 1989. Phase relations of high albedo asteroids: The unusual opposition brightening of 44 Nysa and 64 Angelina. Icarus 81, 365–374. Helfenstein, P., and J. Veverka 1987. Photometric properties of lunar terrains derived from Hapke’s equation. Icarus 72, 342–357. Helfenstein, P., J. Veverka, D. McCarthy, P. Lee, and J. Hillier 1992. Large quasi-circular features beneath frost on Triton. Science 255, 824–826.
REFERENCES
Helfenstein, P., J. Veverka, and J. Hillier 1997. The lunar opposition effect: A test of alternative models. Icarus 128, 2–14.
Buratti, B. J., J. K. Hillier, and M. Wang 1996. The lunar opposition surge: Observations by Clementine. Icarus 124, 490–499.
Hillier, J., P. Helfenstein, and J. Veverka 1996. Latitude variations of the polar caps on Ganymede. Icarus 124, 308–317.
Chandrasekhar, S. 1960. Radiative Transfer. Dover, New York. Domingue, D. L., B. W. Hapke, G. W. Lockwood, and D. T. Thompson 1991. Europa’s phase curve: Implications for surface structure. Icarus 90, 30–42. Goguen, J. D., H. B. Hammel, and R. H. Brown 1989. V Photometry of Titania, Oberon, and Triton. Icarus 77, 239–247. Guiness, E. A., C. E. Leff, and R. E. Arvidson 1982. Two Mars years of surface changes seen at the Viking landing sites. J. Geophys. Res. 87, 10051–10058. Hansen, J. E. 1969. Radiative transfer by doubling very thin layers. Astrophys. J. 155, 565–573. Hapke, B. 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 86, 3039–3054. Hapke, B. 1984. Bidirectional reflectance spectroscopy. 3. Correction for macroscopic roughness. Icarus 59, 41–59. Hapke, B. 1986. Bidirectional reflectance spectroscopy. 4. The extinction coefficient and the opposition effect. Icarus 67, 264–280.
Hillier, J., and J. Veverka 1994. Photometric properties of Triton hazes. Icarus 109, 284–295. Hillier, J., J. Veverka, P. Helfenstein, and P. Lee 1994. Photometric diversity of terrains on Triton. Icarus 109, 296–312. Irvine, W. 1966. The shadowing effect in diffuse reflectance. J. Geophys. Res. 71, 2931–2937. Lee, P., P. Helfenstein, J. Veverka, and D. McCarthy 1992. Anomalous scattering region on Triton. Icarus 99, 82–97. Lumme, K., and E. Bowell 1981. Radiative transfer in the surfaces of atmosphereless bodies. I. Theory. Astron. J. 86, 1694–1704. Mischenko, M. I. 1992a. Enhanced backscattering of polarized light from discrete random media: Calculations in exactly the backscattering direction. J. Opt. Soc. Am. 9, 978–982. Mischenko, M. I. 1992b. The angular width of the coherent backscatter opposition effect: An application to icy outer planet satellites. Astrophys. Space Sci. 194, 327–333.
TWO LAYER OPPOSITION SURGE Mischenko, M. I. 1994. Asymmetric parameters of the phase function for densely packed scattering grains. J. Quant. Spectrosc. Radiat. Transfer 52(1), 95–110. Mischenko, M. I., and J. M. Dlugach 1992. The amplitude of the opposition effect due to weak localization of photons in discrete disordered media. Astrophys. Space Sci. 189, 151–154. Muinonen, K. 1993. Coherent backscattering by Solar System dust particles. In IAU Symposium 160, Asteroids, Comets, Meteors (A. Milani, M. DiMartino, and A. Cellino, Eds.), pp. 271–296. Kluwer, Dordrecht, The Netherlands. Nozette, S., P. Rustan, L. P. Pleasance, D. M. Horan, P. Regeon, E. M. Shoemaker, P. D. Spudis, C. H. Acton, D. N. Baker, J. E. Blamont, B. J. Buratti, M. P. Corson, M. E. Davies, T. C. Duxbury, E. M. Eliason, B. M. Jakosky, J. F. Kordas, I. T. Lewis, C. L. Lichtenberg, P. G. Lucey, E. Malaret, M. A. Massie, J. H. Resnick, C. J. Rollins, H. S. Park, A. S. McEwen, R. E. Priest, C. M. Pieters, R. A. Reisse, M. S. Robinson, R. A. Simpson, D. E. Smith, T. C. Sorenson, R. W. Vorder
27
Bruegge, and M. T. Zuber 1994. The Clementine mission to the Moon: Scientific overview. Science 266, 1835–1839. Peltoniemi, J. I., and K. Lumme 1992. Light scattering by closely packed particulate media. J. Opt. Soc. Am. A 9, 1320–1326. Sagan, C., J. Veverka, P. Fox, R. Dubisch, R. French, P. Gierasch, J. Lederberg, E. Levinthal, R. Tucker, B. Eross, and J. B. Pollack 1973. Variable features on Mars: 2. Mariner 9 global results. J. Geophys. Res. 78, 4163–4196. Seeliger, H. 1887. Zur Theorie der Beleuchtung der Grossen Planeten Insbesondere des Saturn. Abhandl. Bayer. Akad. Wiss. Math.-Naturw. Kl. II 16, 405–516. Seeliger, H. 1895. Theorie der Beleuchtung Staubformiger Kosmischen Masses Insbesondere des Saturinges. Abhandl. Bayer. Akad. Wiss. Math.-Naturw. Kl. II 18, 1–72. Thompson, D. T., and G. W. Lockwood 1992. Photoelectric photometry of Europa and Callisto 1976–1991. J. Geophys. Res. 97, 14761–14772.
128, 15–27 (1997) IS975724
ARTICLE NO.
Shadow-Hiding Opposition Surge for a Two-Layer Surface John K. Hillier MS 183-501, Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109 E-mail: [email protected] Received December 15, 1995; revised February 18, 1997
roughness and a shadow-hiding opposition surge (a nonlinear increase in brightness as opposition is approached observed on virtually every airless Solar System body). It has recently been suggested that a second mechanism, coherent backscatter, may be responsible for the extremely narrow opposition surges observed on several Solar System bodies (Hapke 1990, Mischenko 1992a,b, Mischenko and Dlugach 1992) including some outer planet satellites (Domingue et al. 1991, Thompson and Lockwood 1992, Goguen et al. 1989), asteroids (Harris et al. 1989), and the Moon (Nozette et al. 1994). In this mechanism, light rays which travel identical but reversed paths in a surface will emerge in phase and interfere constructively leading to up to a factor of two increase in brightness at opposition. A review of recent work on coherent backscatter is given by Muinonen (1993). However, this mechanism has not been included in the model. The shadow-hiding explanation for the opposition surge was first proposed by Seeliger (1887, 1895) to explain his observations of the phenomena in Saturn’s rings. It has since been discussed by a number of researchers and several models for the shadow-hiding mechanism (e.g., Irvine 1966, Lumme and Bowell 1981, Hapke 1986) exist. One way of looking at the shadowhiding opposition surge is that if an incident ray can penetrate to a certain point in the surface it will, in essence, have preselected a preferential escape route back along or near the direction the incident ray came from (at or near zero degrees phase). Indeed, at precisely zero degrees phase, emergent rays will automatically escape (if it did not hit anything going in, it must be able to escape back along precisely the same path). Another way of visualizing the shadow-hiding opposition surge is that at zero phase particles will hide their own shadows but the shadows quickly become visible as one moves away from opposition. Here I extend the shadow-hiding model of Hapke (1986) for use with a two-layer surface. It should be noted that, as an extension of Hapke’s (1981, 1984, 1986) photometric model, the model I propose here includes similar approximations (e.g., particles large compared to the wavelength, diffraction unimportant for a densely packed surface) and thus potentially suffers from the same limitations as have
A shadow hiding opposition surge, based on B. Hapke (1986, Icarus 67, 264–280), is derived for a surface consisting of an optically finite veneer of material over an underlying substrate. As done by Hapke in his photometric model, it is assumed that standard radiative transfer theory with correction for macroscopic roughness and a shadow hiding opposition surge provides an adequate description of the two-layered surface. The scattering properties of the upper layer are calculated using the doubling technique while Hapke’s photometric model is used to describe the underlying layer. An adaptation of the doubling technique is used to calculate the scattering from the combined system. The model should have wide applicability in the Solar System and has been used with success to describe the polar regions of Ganymede (J. Hillier et al., 1996, Icarus 124, 308–317) as well as the equatorial regions of Triton (J. Hillier and J. Veverka, in preparation). Using this model, it is shown that extremely narrow opposition surges, as observed on many Solar System bodies, might be explained by a shadowhiding model even for relatively compact surfaces if the particle size decreases toward the surface and may provide an alternative explanation to the coherent backscatter mechanism for the observed surges. 1997 Academic Press
1. INTRODUCTION
The surfaces of several bodies in the Solar System appear to be overlain by thin veneers of material. Examples include variations in albedo features on Mars which have been attributed to the deposition or erosion of layers of dust (Sagan et al. 1973, Guiness et al. 1982), many regions of Io which may be overlain by thin volcanic deposits or SO2 frosts which may condense on the night side, the polar regions of Ganymede (which appear to be overlain by thin frost deposits), and the equatorial regions of Triton whose scattering properties have led to the suggestion that they are overlain by a thin veneer of frost (Lee et al. 1992, Helfenstein et al. 1992, Hillier et al. 1994). In this paper, I discuss a two-layer surface photometric model which can be used to describe such regions. The model is based on standard radiative transfer theory, with (following Hapke 1984, 1986) corrections for macroscopic 15
0019-1035/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
16
JOHN HILLIER
been suggested for Hapke’s model (Peltoniemi and Lumme 1992, Mischenko 1994). The model has been used with success to describe the polar regions of Ganymede (Hillier et al., 1996) as well as the equatorial regions of Triton (Hillier and Veverka, in preparation). 2. PHOTOMETRIC MODEL
The model (Fig. 1) assumes that the optically active region of the surface consists of two distinct layers. The doubling technique of Hansen (1969) is used to compute the scattering behavior of the upper layer while Hapke’s (1981, 1984, 1986) photometric model is used to describe the optically thick lower layer. Following Hansen (1969), the total scattering from the combined system is given by I(i, e, a) S(tu ; e, w; e0 , w0) e2(tu / e1tu / e0) 5 1 S0(tu ; e, w; e0 , w0) F 4e 4e
EE
1
e2tu / e0 4fe
3
1 S0(tu ; e9, w9; e0 , w0) dw9 de9 4e9
1
e2tu / e 4fe
1
2f
0
0
EE 1
2f
0
0
T(tu ; e, w; e9, w9) FIG. 1. Schematic representation of two-layer model used in this study.
S0(tu ; e, w; e9, w9)
(1)
1 T(tu ; e9, w9; e0 , w0) dw9 de9 3 4e9
EE EE
1
1 4fe
3
1 S0(tu ; e0, w0, e9, w9) 4fe0
3
T(tu ; e9, w9; e0 , w0) dw0 de0 dw9 de9, 4e9
1
2f
1
2f
0
0
0
0
T(tu ; e, w; e0, w0)
where S0(tu ; e, w; e0 , w0) ;
O y
n51,3 · ? ?
Sn(tu ; e, w; e0 , w0),
(2)
S1(e, w; e0 , w0) 5 4eRH(i, e, a(e, w; e0 , w0)),
(3)
H
J
and
Sn(tu ; e, w; e0 , w0) 5
1 4f
EE 1
2f
0
0
S1(e, w; e9, w9), n odd
S(tu ; e, w; e9, w9), n even
3 Sn21(tu ; e9, w9; e0 , w0) dw9
de9 . e9 (4)
i, e, and a are the incidence, emission, and phase angles, respectively, w is the azimuth angle, e 5 cos(e), e0 5 cos(i), fF is the incident normal flux, S and T are the scattering and transmission functions, respectively, for the upper layer of optical depth tu (as defined by Chandrasekhar 1960) and RH is Hapke’s (1981, 1984, 1986) reflectance function. As with Hapke’s model, the model assumes that the scattering can be adequately described by standard radiative transfer theory with corrections for macroscopic roughness and a shadow hiding opposition surge. Indeed, except for the fact that multiple scattering is fully accounted for in the upper layer of my model, my model yields the same results as Hapke’s in the limiting case of a single layer (either an optically thick upper layer or when the upper and lower layer properties are identical). The model parameters used to describe the scattering properties of the surface are shown in Table I. These are the optical depth of the overlying layer, Hapke’s (1984) macroscopic roughness parameter, u, and, for each layer, the single particle scattering albedo, g˜ 0 , one or more parameters to describe the particle phase function, and the opposition surge amplitude and width parameters, B0 and h (as defined by Hapke (1986)). An additional parameter, the ratio of the upper to lower layer particle radii, rr , is required in the calculation of the opposition surge. Hapke’s (1984) correction for macroscopic roughness can be applied to any bidirectional reflectance function for
17
TWO LAYER OPPOSITION SURGE
TABLE I Parameters for the 2-Layer Surface Photometric Model tu u¯ g ˜ 0u , g˜ 0l Pu(a), Pl(a) B0u , B0l hu , hl rr
determined below, reduce to Hapke’s (1986) B(a) term in the appropriate limit:
Optical depth of the upper layer Average topographic slope angle Single particle scattering albedo for the upper and lower layer One or more parameters to describe the single particle phase function for the upper and lower layer. Amplitude of opposition surge (both layers) Width of opposition surge (both layers) Ratio of upper to lower layer particle radii
a smooth surface. Therefore, the reflectance after correction for macroscopic roughness is given by, r2R(i, e, a) 5 r2S(i9, e9, a)Sh(i, e, a),
(5)
where r2s is the reflectance predicted for a smooth surface, i9 and e9 are the effective incidence and emission angles, respectively, after correction for the tilt as defined by Hapke, and Sh is Hapke’s shadowing function. A derivation of the correction for the shadow-hiding opposition surge, based on Hapke (1986), for a two-layered surface is the subject of the next section.
lim Bu(i, e, a, tu) 5 B(a)
tuRy
(9)
lim Bl(i, e, a, tu) 5 B(a).
t u R0
3.1. Opposition Surge in Overlying Layer For the overlying layer, the derivation of Hapke (1986) is applicable if we limit the integration of the intensity to the optical depth of the overlying layer (instead of infinity for the one layer case). Therefore, we can use the results of Hapke’s Eqs. (29) and (30) without modification. The single scattered radiance (including the opposition surge) is given by
I(i, e, a) 5
Fg˜ 0u 4
pu(a)
E
tu
0
e2(t1t9)/ kel
dt , e
(10)
where
t9 5
1 zl
E
z1zl
z
t(z9) dz9,
(11)
3. DERIVATION OF TWO-LAYER OPPOSITION SURGE
Following Hapke (1986) I assume that the shadow hiding opposition surge affects only the single scattering component. Let us write the total reflectance as r2s 5 rno 1 Bu(i, e, a, tu)rsu 1 Bl(i, e, a, tu)rsl ,
(6)
where i, e, and a are the incidence, emission, and phase angles, respectively, rno is the calculated reflectance without the opposition surge, and rsu(i, e, a) 5
Isu(i, e, a) g˜ 0u e0 5 pu(a) (1 2 e2(tu / e01tu / e)) F 4 e0 1 e (7)
and Isl(i, e, a) g˜ 0l e0 rsl(i, e, a) 5 5 pl(a) e2(tu / e01tu / e) F 4 e0 1 e
and t is the optical depth. zl and kel are given by Hapke’s Eqs. (22) and (23), z1 P kelkrl cot(a/2)
(12)
kel 5 2e0e/(e0 1 e),
(13)
and krl is the effective particle radius defined in terms of the average particle cross section ksl: krl 5
!ksfl.
Assuming a step function distribution function we have, following Hapke (1986), z $ 0: ne(z) 5 0, (above the surface)
(8)
are the single scattered reflectance for the upper and lower layers (ignoring the opposition surge), respectively. The second and third terms in Eq. (6) represent the additional reflectance from the opposition surge in the upper and lower layers, respectively. The functions Bu and Bl , to be
(14)
z , 0: ne(z) 5 ne 5 constant,
(15)
where ne is the effective particle density (Hapke’s Eq. (12)). The optical depth as a function of depth into the surface is then given by
t(z) 5 2nekslz.
(16)
18
JOHN HILLIER
From Eq. (11) we then have (Hapke, Eq. (31)) z $ 0: t9 5 0 2zl , z , 0: t9 5 t 2 /(2tl)
(17)
z # 2zl : t9 5 t 2 tl /2, where
tl 5 nekslzl 5 2kelhu cot(a/2),
(18)
hu ; nekslkrl/2
(19)
and
is Hapke’s (1986) width parameter (for the upper layer). Substitution into Eq. (10) yields Case 1: zl . zupper layer (or tl . tu): I(i, e, a) 5
Fg˜ 0u 4
E
tu
pu(a)
0
e2(t91t9
2
/2tl)/ kel
dt9 , e
(20a)
Case 2: zl , zupper layer (or tl , tu): Fg˜ 0u
I(i, e, a) 5
4 1
pu(a)
FE
tl
0
e2(t91t9
2
/2tl)/ kel
dt9 e
(20b)
G
e0 h2etl /2kele22t / kel 1 e23tl /2kelj . e 1 e0
where, following Hapke, I have multiplied the expression for Bu by his B0 parameter. Bu can then be found by numerical integration of the integrals in Eq. (22).
From the definition of Bu , we have
3.2. Opposition Surge in Underlying Layer
I(i, e, a) 5 Isu(i, e, a)[1 1 Bu],
(21)
where, Isu(i, e, a) is the radiance for single scattering ignoring the opposition surge (Eq. (7)). After some algebraic manipulation, we have, for tl . tu ,
Bu 5 B0u
E
1
tu
0
e2(t91[t9
2
(dt9/e)
tan(a /2)]/4hukel)/ kel
(e0 /(e0 1 e))(1 2 e
2[tu / e01tu / e]
)
2
2 1 , (22a)
and, for tl , tu ,
Bu 5 B0u 1
E
3
tl
0
e2(t91[t9
2
tan(a /2)]/4hukel)/ kel
(e0 /(e 1 e0))(1 2 e
FIG. 2. Schematic diagram showing a cross section through the scattering plane of the overlap of the incident and emergent probability cylinders. In order to simplify the figure, the figure is shown for the same sized particles in both layers. In general, however, the cylinder radii used for each layer will be different. The overlap between the cylinders is denoted by the filled in regions (the darker region that contributed by the lower layer, Ul , and the lighter that contributed by the upper layer, Uu , see text). Note that in the case shown the overlap extends above the surface. However, in general the overlap may end before the surface is reached or be contained entirely in the lower layer.
(dt9/e)
2[tu / e01tu / e]
)
(22b)
4
h2e hu cot(a /2)e22tu / kel 1 e23hu cot(a /2)j 21 , (1 2 e2[tu / e01tu / e])
The probability that an incident ray will penetrate to a given point in the surface without being scattered is just the probability that no particle has its center in the cylinder of radius equal to the effective particle radius, krl, whose axis connects the point with the source. Similarly, the probability that a ray can escape from a point in the surface is just the probability that no particle has its center in the cylinder of radius krl and whose axis connects the point and the observer. However, as illustrated in Fig. 2, portions of these two cylinders overlap. It is this overlap, unaccounted for in standard radiative transfer theory, which leads to the opposition surge. Following Hapke (1986), let us define the optical depth associated with this overlap, equal to the integral of ne over the volume of overlap, to be tc . Then, the total radiance from singly scattered light is given by I(i, e, a) 5
Fg˜ 0l Pl(a) 4
E
y
tu
e2(t / e1t / e02tc)
dt , e
(23)
19
TWO LAYER OPPOSITION SURGE
where the factor of 2tc in the exponent is the correction for the opposition effect. By definition, Bl is given by Bl(i, e, a, tu) 5 B0l
S
D
I(i, e, a) 21 , ISl(i, e, a)
z(t) 5
t 5 tu 2 neuksluz ⇒
(25) z . 0,
where z is the depth in the surface, ne is the effective particle number density, ksl the extinction cross section, and the subscripts u or l refer to the upper or lower layers, respectively. For a step function distribution, ne is constant (at least within each layer) and thus the quantity which must be subtracted from the total optical depth is (26)
where tcl and tcu are the contributions to tc from the overlap in the lower and upper layers and Ul and Uu are the shared volume between the incident and emergent cylinders in the lower and upper layer, respectively. Following Hapke, the overlap between the incident and emergent cylinders extends to a distance of zl , zll P kelkrll cot(a/2)
(27)
zlu P kelkrlu cot(a/2), where, as above, the subscripts denote the lower or upper layer above the scattering location. The total shared volume is (Hapke’s Eqs. (24) and (26)) Uc P
kslkrl cot(a/2) 1 5 kslzl , 2 2kel
(29)
where zl 5 zll or zlu are used to calculate Uc or Ua for the desired layer. 3.2.1. Lower layer contribution (tcl). If z 1 zll , 0 (or t 2 tu , tll), then the total shared volume occurs in the lower layer and we have
tcl 5 nelUcl 5
nelksllzll 2kel
5
tl l
,
(30)
SD
(31)
2kel
tll ; nelksllzll 5 2hlkel cot
a , 2
and hl is Hapke’s (1986) width parameter. If z 1 zll . 0 (or t 2 tu . tll), then the shared volume above z9 5 0 (i.e., above the lower layer) does not contribute to the opposition surge and we have
1 (tu 2 t) z(t) 5 neukslu
tc 5 tcl 1 tcu 5 nel 3 Ul 1 neu 3 Uu ,
D
z9 2 z 2 , zl
where
z,0
1 (tu 2 t) nel ksll
S
Ua(z9) P Uc 1 2
(24)
where, following Hapke, I have multiplied the expression for Bl by his B0 parameter (the subscript l denoting the lower layer), and ISl is the singly scattered radiance ignoring the opposition surge (Eq. (8)). Therefore, all that is left to finish the derivation is to obtain an expression for tc with which Eq. (23) can be numerically integrated. Let us define the boundary between the upper and lower layers to be the z 5 0 point, then
t 5 tu 2 nel ksllz ⇒
while that lying above any given plane at z9 between z and z 1 zl is
(28)
S F GD
tcl 5 nel(Ucl 2 Ual(0)) 5 nelUcl 1 2 1 1
z zll
2
.
(32)
Some algebraic manipulation and substituting in for Ucl from Eq. (28) yields
tcl 5 nel
S
D
ksll z2 22z 2 . 2kel zll
(33)
Finally, substituting in for z and zll from Eqs. (26) and (28) yields the desired result:
tc l 5
H
J
t 2 tu)2 1 (t 2 tu) 2 . kel 2tll
(34)
3.2.2. Overlying layer contribution (tcu). If z 1 zlu , 0 then there is no overlap between the incident and emergent cylinders in the upper layer and we have
tcu 5 0.
(35)
If z 1 zlu . 0 and z 1 zlu , zu ;
tu , neukslu
(36)
20
JOHN HILLIER
then the overlap between the incident and emergent cylinders ends before reaching the top of the upper layer and we have
tcu 5 neu 3 Ua(0),
tcu 5 neu 3 Uc 5
S D S D
neuksluzlu 2kel
2
11
5
neuksluzlu 2kel
S D z 11 zlu
tcu 5 (38)
2
tlu 2kel
tlu
S
D
z z2 1 2 , kel zlu 2z lu
(39)
where
tlu ; neuksluzlu 5 2kelhu cot(a/2).
(40)
tcu 5
2kel
1
S
tl u
D
tl u 2kel
1
tl u
krlu zlu 5 . krll zll
(42)
tlu , texp
D
tu 2 t (tu 2 t)2 1 . kel tll rr 2(tll rr)2
(43)
Finally, if z 1 zlu . zu then the surface is reached before the overlap between the incident and emergent cylinders ends and we have
tcu 5 neu 3 (Ua(0) 2 Ua(zu)) 5 n eu Uc 5 neu Uc
t1u , texp 2 tu , tlu . texp 2 tu
S
SF G F H JG D SF G FS D G D 11
z zlu
2
11
z zlu
2
2 12
zu 2 z zlu
2
zu z 2 zlu zlu
11
S F S
G D D
tu 1 1 ( tu 2 t) t 2u 2 2 kel tlu tll rr 2t lu
(46)
(47)
where hu and hl are the opposition surge width parameters for the upper and lower layers, respectively. With this modification the boundary conditions can be expressed in terms of the optical depth as
We have the desired result
tcu 5
(45)
(41)
Making the definition rr ;
tlu
texp 2 tu neukslu hu 1 5 5 , t 2 tu nelksll hl rr
tu 2 t (tu 2 t) 1 . kel nelksllzlu 2(nelksllzlu)2 2
.
Finally, for use in Eq. (23), it is necessary to express the boundaries between the different regimes (z 1 zlu , 0; z 1 zlu . 0 and z 1 zlu , zu ; z 1 zlu . zu) in terms of the optical rather than the physical depth. In general, the physical distance traveled in each layer for a given optical depth will be different. The optical depth expected at depth z assuming that we were still in the upper layer, texp , is related to the actual optical depth, t, in the lower layer by the expression
Substituting in for z (,0) from Eq. (25) we have
tl u
2
tu tu (tu 2 t) 5 11 2 . kel tll rr 2tlu
2z z 1 , zlu z 2lu
1
zu z zu 2 zlu zlu zlu
Substituting in for Uc , z, zu , and zlu from Eqs. (28), (25), (36), and (27) we have the desired result:
2
or
tc u 5
S F G F GD
tcu 5 neu Uc 2 1 1
(37)
where Ua(0) is the shared volume in the upper layer (which begins at z 5 0). Substituting in for Ua(0) we have z 11 zlu
After some algebraic manipulation we have
2
(44)
2
.
tlu . texp ,
regime 1 (tcu given by Eq. (35)),
(48a)
regime 2 (tcu given by Eq. (43)), (48b) regime 3 (tcu given by Eq. (46)).
(48c)
4. SUMMARY
In summary, a photometric model for describing twolayered surfaces, based on standard radiative transfer theory with corrections for macroscopic roughness (Hapke, 1984) and a shadow-hiding opposition surge, is developed. The scattering properties of the optically finite upper layer are calculated using the doubling procedure of Hansen (1969), while Hapke’s (1981, 1984, 1986) photometric model, ignoring the opposition surge and macroscopic roughness, is used to describe the underlying surface. The scattering from the combined system, R2 , again ignoring
21
TWO LAYER OPPOSITION SURGE
the opposition surge and macroscopic roughness is calculated using an adaptation of the adding technique of Hansen (1969), R2 5
S(tu ; e, w; e0 , w0) e2(tu / e1tu / e0) 1 S0(tu ; e, w; e0 , w0) 4e 4e 1
e2tu / e0 4fe
EE 1
2f
0
0
where Sh is Hapke’s (1984) shadowing function, and i9 and e9 are calculated using procedures outlined in Hapke (1984). The second and third terms in the expression in parenthesis are the corrections for the opposition effect. Rus and Rls are the reflectance from single scattering for the upper and lower layers, respectively,
T(tu ; e, w; e9, w9)
Rus(i, e, a) 5
1 S0(tu ; e9, w9; e0 , w0) dw9 de9 3 4e9
EE
1
e2tu / e 4fe
3
1 T(tu ; e9, w9; e0 , w0) dw9 de9 4e9
1 1 4fe
1
2f
0
0
EE EE 2f
1
2f
0
0
0
0
1 S0(tu ; e0, w0, e9, w9) 4fe0
3
T(tu ; e9, w9; e0 , w0) dw0 de0 dw9 de9, 4e9
Rls(i, e, a) 5
g˜ 0l 4
pl(a)
O y
n51,3 · ? ?
Sn(tu ; e, w; e0 , w0),
(50)
1
2f
0
0
H
S1(e, w; e9, w9), n odd
J
S(tu ; e, w; e9, w9), n even
3 Sn21(tu ; e9, w9; e0 , w0) dw9
Bu 5 B0u
(51)
and
EE
(55)
de9 . e9
i, e, and a are the incidence, emission, and phase angles, respectively, w is the azimuth angle, e 5 cos(e), e0 5 cos(i), fF is the incident normal flux, S and T are the scattering and transmission functions for the upper layer of optical depth tu (as defined by Chandrasekhar 1960), and Rh is Hapke’s (1981, 1984, 1986) reflectance function. The scattering from the system after correction for macroscopic roughness and the shadow hiding opposition surge is then given by R(i, e, a) 5 (R2(i9, e9, a) 1 Bu(i9, e9, a)Rus(i9, e9, a)
(53)
E
1
tu
e2(t91[t9
0
2
(dt9/e)
tan(a /2)]/4hukel)/ kel
(e0 /(e0 1 e))(1 2 e
2[tu / e01tu / e]
)
2
2 1 , (56a)
if tu , tlu and
Bu 5 B0u 1
(52)
1 Bl(i9, e9, a)Rls(i9, e9, a))Sh(i, e, a),
e0 e2(tu / e01tu / e), e0 1 e
where i, e, a are the incidence, emission, and phase angle, respectively, e0 5 cos(i), e 5 cos(e), g˜ 0 is the single particle scattering albedo, p(a) is the single particle phase function (the subscripts u and l denoting the upper and lower layers, respectively), and tu is the upper layer optical depth. Bu is given by
S1(e, w; e0 , w0) 5 4eRH(i, e, a(e, w; e0 , w0)),
1 4f
e0 (1 2 e2(tu / e01tu / e)) (54) e0 1 e
(49)
where
Sn(tu ; e, w; e0 , w0) 5
pu(a)
T(tu ; e, w; e0, w0)
3
S0(tu ; e, w; e0 , w0) ;
4
and
S0(tu ; e, w; e9, w9)
1
g˜ 0u
3
E
tl
0
e2(t91[t9
2
tan(a /2)]/4hukel)/ kel
(dt9/e)
(e0 /(e 1 e0))(1 2 e2[tu / e01tu / e])
h2e hu cot(a /2)e22tu / kel (1 2 e
1e
j
23hu cot(a /2)
2[tu / e01tu / e]
)
21
(56b)
4
for tu . tlu , where B0u and hu are Hapke’s (1986) opposition surge amplitude and width parameters, respectively, for the upper layer and
tlu 5 2kelhu cot(a/2),
(57)
where
kel 5
2ee0 . e 1 e0
(58)
22
JOHN HILLIER
FIG. 3. Predicted scattering at 08 phase for a surface overlain by a thin forward scattering layer. Significant limb darkening is seen at moderate optical depths.
Finally Bl is given by Bl(i, e, a, tu) 5 B0l
S
D
R(i, e, a) 21 , Rls(i, e, a)
hl is Hapke’s opposition surge width parameter for the lower layer, and (59)
where B0l is Hapke’s opposition surge amplitude parameter for the lower layer, Rls is given by Eq. (55) and R(i, e, a) 5
g˜ 0l pl(a) 4
E
tcu 5 y
tu
e2(t / e1t / e02tc)
dt . e
(60)
Equation (60) can be solved numerically given tc . tc is given by
tc 5 tcl 1 tcu ,
tlu
2kel
1
S
D
tlu tu 2 t (tu 2 t)2 1 , kel tll rr 2(tll rr)2
tlu . texp 2 tu tlu , texp
S
tlu . texp ,
D
(tu 2 t) tu tu 11 2 , kel tll rr 2tlu
where rr 5 ru /rl is the ratio of the radii of the upper and lower layer particles and
texp 5 tu 1
5H
tll 2kel
(t 2 tu) $ tll
,
J
(t 2 tu)2 1 (t 2 tu) 2 , kel 2tll
(64)
(61)
where
tcl 5
5
tlu , texp 2 tu
0,
hu l (t 2 tu). hl rr
(65)
5. DISCUSSION
(62) (t 2 tu) , tll ,
where
tll 5 2kelhl cot(a/2),
(63)
Having derived the opposition surge for a two-layer surface, we now proceed to examine the effects of a second layer on a surface’s scattering properties. The effect of adding a thin forward scattering layer is shown in Fig. 3. As long as the upper layer is not too thick, the addition of the forward scattering layer tends to increase the degree of limb darkening seen at low phase angles. This effect is
TWO LAYER OPPOSITION SURGE
23
FIG. 4. Predicted scattering at 08 phase for a high albedo surface overlain by a layer of dark particles. Significant limb darkening is seen at moderate optical depths.
similar to that found for Triton’s thin forward scattering hazes (Hillier and Veverka 1994), which is not surprising: the upper layer behaves very much like an overlying haze layer. A similar effect is seen for dark particles placed over a brighter substrate (Fig. 4). In both cases the cause is clear: the larger slant path optical depths seen toward the limb and terminator tend to increase the contribution of the darker upper layer, decreasing the predicted brightness. As expected, the reverse (more backscattering or brighter particles in the upper layer) tends to lead to limb brightening (Fig. 5). As the upper layer becomes optically thick, the optically active portion of the surface approaches the single layer limit and the pronounced increase in limb darkening (or brightening) disappears. The effects of varying the upper layer opposition surge parameters as compared to the lower layer parameters are shown in Figs. 6–8. Lowering the upper layer opposition surge amplitude parameter leads to significant limb darkening near opposition for moderate optical depths (Fig. 6). This is similar to the effect seen for a forward scattering or dark upper layer (Figs. 3 and 4). In each case, the limb darkening is due to the reduced reflectance in the upper layer. However, changes in B0 substantially affect only the low phase data while changes to the single scattering albedo or phase function can be seen at higher phase angles as well. As with a forward scattering or lower albedo layer, the substantial limb darkening seen disappears as the upper layer becomes optically thick. As expected, decreasing the upper layer opposition surge width parameter tends to decrease the width of the overall opposition surge (Fig. 7). Perhaps the most interesting behavior is exhibited when
the particle size is allowed to vary between the layers. Decreasing the size of the upper layer particles decreases the width of the opposition surge, in some cases substantially (Fig. 8). Physically, though the full opposition effect is still seen right at opposition, the smaller particles in the upper layer lead to smaller ‘‘holes’’ for the light to escape through from the lower layer—these holes are filled in much more quickly as the phase angle is increased than otherwise would be the case leading to a quick reduction in the opposition effect. The width of the opposition surge appears to be essentially independent of the albedo or phase function of the surface, as would be expected if the effect is due to a mechanical property of the surface. Such narrow opposition surges have been observed on several Solar System bodies including the Moon (Nozette et al., 1994), icy outer planet satellites (Domingue et al. 1991, Goguen et al. 1989), and some asteroids (Harris et al. 1989). The fact that such narrow opposition surges require extremely (and probably unreasonably) porous surfaces in standard shadow-hiding models has led to the suggestion that the shadow-hiding model is deficient and another mechanism such as coherent backscatter (Hapke 1990, Mischenko 1992a,b, Mischenko and Dlugach 1992) may be responsible for the observed surges. However, the above results suggest that shadow-hiding could lead to such narrow surges even for relatively compacted surfaces if the particle size decreases toward the surface and thus coherent backscatter may not need to be invoked. Being based on Hapke’s shadow-hiding model, the opposition model presented here will yield essentially identical results to his model in the single layer limit (optically
24
JOHN HILLIER
FIG. 5. Predicted scattering at 08 phase for a surface overlain by a thin backscattering layer. As expected, significant limb brightening is seen at moderate optical depths. Similar results are observed for a dark surface overlain by bright particles.
thick upper layer or same parameters in both layers). The model has been used with success to fit Clementine observations of the lunar opposition surge (Buratti et al. 1996). They found that models with h as high as 0.12 (for a particle size ratio of 0.14) can fit the observations though a simple Hapke model with h 5 0.040 6 0.004 between the previously determined values of 0.05–0.07 (Hapke 1986, Hel-
fenstein and Veverka 1987) and the recent value of 0.023 found by Helfenstein et al. (1997) for their shadow-hiding only solution can fit the observations as well. This shows that significantly higher values of h (suggesting lower porosities) can indeed be obtained for a real surface if the particle size decreases toward the surface. While Hapke’s model predicts no dependence on the opposition surge
FIG. 6. Predicted scattering at 08 phase assuming a low upper layer opposition surge amplitude. Other parameters are identical between the layers.
TWO LAYER OPPOSITION SURGE
25
FIG. 7. Scattering from a two-layer surface with lower opposition surge width parameter in the upper layer. Other parameters are identical between the layers. In this figure, emission angle is held constant at 08 while the phase and incidence angles vary. As expected, the opposition surge width tends to decrease as the upper layer optical depth is increased.
with incidence or emission angle, the model presented here does predict such a dependence for a finite upper layer (as the slant path optical depth of the upper layer varies with incidence and emission angle), as illustrated in Fig. 9. Interestingly, the effect of lowering the particle ratio
parameter, leading to a larger opposition surge away from the center of the disk, is the opposite of that predicted by the model of Lumme and Bowell (1981) which predicts that the opposition surge should disappear at the limb. This provides a diagnostic which may be
FIG. 8. Effect of varying the particle size ratio on the opposition surge. All other parameters are identical between the layers. As can be seen, an upper layer consisting of small particles leads to a decrease in the width of the opposition surge.
26
JOHN HILLIER
FIG. 9. Comparison of opposition model predictions across the planetary disk. Shown are results along the mirror meridian at a 5 58. An optical depth of 1.0 is assumed and, except for the particle size, all parameters are identical between the layers. The function B shown on the ordinate can be directly compared with Hapke’s B(a) function (and is essentially equivalent to it when rr 5 1). While Hapke’s model predicts no dependence on incidence or emission angle (as seen in the rr 5 1 plot), the model proposed here does suggest a significant dependence.
useful in distinguishing between the models. Unfortunately, the Clementine observations are almost all nadir pointing, making them ill suited for this purpose. ACKNOWLEDGMENTS John Hillier acknowledges support from NASA Grant NAGW-3837. Portions of this work were performed while the author held a National Research Council-JPL Research Associateship. The author thanks Audouin Dollfus and an anonymous reviewer for constructive reviews.
Hapke, B. W. 1990. Coherent backscatter and the radar characteristics of outer planet satellites. Icarus 88, 407–417. Harris, A. W., J. W. Young, L. Contreiras, T. Dockweiler, L. Belkora, H. Salo, W. D. Harris, E. Bowell, M. Poutanen, R. P. Binzel, D. J. Tholen, and S. Wang 1989. Phase relations of high albedo asteroids: The unusual opposition brightening of 44 Nysa and 64 Angelina. Icarus 81, 365–374. Helfenstein, P., and J. Veverka 1987. Photometric properties of lunar terrains derived from Hapke’s equation. Icarus 72, 342–357. Helfenstein, P., J. Veverka, D. McCarthy, P. Lee, and J. Hillier 1992. Large quasi-circular features beneath frost on Triton. Science 255, 824–826.
REFERENCES
Helfenstein, P., J. Veverka, and J. Hillier 1997. The lunar opposition effect: A test of alternative models. Icarus 128, 2–14.
Buratti, B. J., J. K. Hillier, and M. Wang 1996. The lunar opposition surge: Observations by Clementine. Icarus 124, 490–499.
Hillier, J., P. Helfenstein, and J. Veverka 1996. Latitude variations of the polar caps on Ganymede. Icarus 124, 308–317.
Chandrasekhar, S. 1960. Radiative Transfer. Dover, New York. Domingue, D. L., B. W. Hapke, G. W. Lockwood, and D. T. Thompson 1991. Europa’s phase curve: Implications for surface structure. Icarus 90, 30–42. Goguen, J. D., H. B. Hammel, and R. H. Brown 1989. V Photometry of Titania, Oberon, and Triton. Icarus 77, 239–247. Guiness, E. A., C. E. Leff, and R. E. Arvidson 1982. Two Mars years of surface changes seen at the Viking landing sites. J. Geophys. Res. 87, 10051–10058. Hansen, J. E. 1969. Radiative transfer by doubling very thin layers. Astrophys. J. 155, 565–573. Hapke, B. 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 86, 3039–3054. Hapke, B. 1984. Bidirectional reflectance spectroscopy. 3. Correction for macroscopic roughness. Icarus 59, 41–59. Hapke, B. 1986. Bidirectional reflectance spectroscopy. 4. The extinction coefficient and the opposition effect. Icarus 67, 264–280.
Hillier, J., and J. Veverka 1994. Photometric properties of Triton hazes. Icarus 109, 284–295. Hillier, J., J. Veverka, P. Helfenstein, and P. Lee 1994. Photometric diversity of terrains on Triton. Icarus 109, 296–312. Irvine, W. 1966. The shadowing effect in diffuse reflectance. J. Geophys. Res. 71, 2931–2937. Lee, P., P. Helfenstein, J. Veverka, and D. McCarthy 1992. Anomalous scattering region on Triton. Icarus 99, 82–97. Lumme, K., and E. Bowell 1981. Radiative transfer in the surfaces of atmosphereless bodies. I. Theory. Astron. J. 86, 1694–1704. Mischenko, M. I. 1992a. Enhanced backscattering of polarized light from discrete random media: Calculations in exactly the backscattering direction. J. Opt. Soc. Am. 9, 978–982. Mischenko, M. I. 1992b. The angular width of the coherent backscatter opposition effect: An application to icy outer planet satellites. Astrophys. Space Sci. 194, 327–333.
TWO LAYER OPPOSITION SURGE Mischenko, M. I. 1994. Asymmetric parameters of the phase function for densely packed scattering grains. J. Quant. Spectrosc. Radiat. Transfer 52(1), 95–110. Mischenko, M. I., and J. M. Dlugach 1992. The amplitude of the opposition effect due to weak localization of photons in discrete disordered media. Astrophys. Space Sci. 189, 151–154. Muinonen, K. 1993. Coherent backscattering by Solar System dust particles. In IAU Symposium 160, Asteroids, Comets, Meteors (A. Milani, M. DiMartino, and A. Cellino, Eds.), pp. 271–296. Kluwer, Dordrecht, The Netherlands. Nozette, S., P. Rustan, L. P. Pleasance, D. M. Horan, P. Regeon, E. M. Shoemaker, P. D. Spudis, C. H. Acton, D. N. Baker, J. E. Blamont, B. J. Buratti, M. P. Corson, M. E. Davies, T. C. Duxbury, E. M. Eliason, B. M. Jakosky, J. F. Kordas, I. T. Lewis, C. L. Lichtenberg, P. G. Lucey, E. Malaret, M. A. Massie, J. H. Resnick, C. J. Rollins, H. S. Park, A. S. McEwen, R. E. Priest, C. M. Pieters, R. A. Reisse, M. S. Robinson, R. A. Simpson, D. E. Smith, T. C. Sorenson, R. W. Vorder
27
Bruegge, and M. T. Zuber 1994. The Clementine mission to the Moon: Scientific overview. Science 266, 1835–1839. Peltoniemi, J. I., and K. Lumme 1992. Light scattering by closely packed particulate media. J. Opt. Soc. Am. A 9, 1320–1326. Sagan, C., J. Veverka, P. Fox, R. Dubisch, R. French, P. Gierasch, J. Lederberg, E. Levinthal, R. Tucker, B. Eross, and J. B. Pollack 1973. Variable features on Mars: 2. Mariner 9 global results. J. Geophys. Res. 78, 4163–4196. Seeliger, H. 1887. Zur Theorie der Beleuchtung der Grossen Planeten Insbesondere des Saturn. Abhandl. Bayer. Akad. Wiss. Math.-Naturw. Kl. II 16, 405–516. Seeliger, H. 1895. Theorie der Beleuchtung Staubformiger Kosmischen Masses Insbesondere des Saturinges. Abhandl. Bayer. Akad. Wiss. Math.-Naturw. Kl. II 18, 1–72. Thompson, D. T., and G. W. Lockwood 1992. Photoelectric photometry of Europa and Callisto 1976–1991. J. Geophys. Res. 97, 14761–14772.