The Lunar Opposition Effect: A Test of Alternative Models

ICARUS 128, 2–14 (1997)ARTICLE NO. IS975726

The Lunar Opposition Effect: A Test ofAlternative Models

Paul Helfenstein, Joseph Veverka, and John Hillier

Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853E-mail: [email protected]

Received April 26, 1994; revised February 5, 1997

scribe the scattering of light from rough, particulate sur-faces and for relating photometric behavior to physical andWe test the hypotheses that the lunar opposition effect is due

to shadow-hiding, coherent backscatter, or some combination of geological properties of the surface terrains. To extractthe two phenomena. Hapke’s photometric model is extended to maximum information from photometric modeling it isinclude M. I. Mishchenko’s (1993, Astrophys. J., 411, 351–361) important to verify the accuracy of the model at extremedescription of the coherent backscatter opposition effect. The geometries, especially at phase angles (a) that are eithermodel was fit simultaneously to V-filter (l 5 0.55mm) whole-

very large (a . 1208) or very small (a , 28).disk photometry of the Moon over many phase angles (28 ## aAt very small phase angles, photometric behavior of## 1438) and to disk-resolved observations of the lunar surface

particulate surfaces is dominated by the oppositionover a broad range of incidence, emission, and phase angles.The shape of the Moon’s opposition surge is accurately repre- surge—a rapid, nonlinear increase in brightness that occurssented by the combination of a narrow coherent-backscatter as the phase angle decreases to 08. Until recently, the oppo-peak whose effect is most strongly defined at a , 28 and a sition surge was considered to be well-understood andvery broad shadow-hiding peak, best defined over a , 208. The modeled in terms of interparticle ‘‘shadow-hiding’’ inrelative angular widths of the shadow-hiding and coherent

which shadows cast by particles upon one another disap-backscatter contributions can be used to estimate the grainpear near opposition because they are hidden from viewsizes most important for scattering at small phase angles. Scat-by the particles themselves. In the last few years anotherterers with sizes comparable to the wavelength of light are

required for coherent backscattering and are present in the mechanism, coherent backscatter (Hapke 1990, Mish-smallest size fraction of lunar soil, but are predicted only for chenko 1991, 1992a,b, 1993, Mishchenko and Dlugacha model in which coherent backscatter causes the narrow com- 1992, 1993, Hapke et al. 1993), has been proposed. Theponent and shadow-hiding the wide one. The amplitude of the

suggestion that coherent backscatter alone can explain theshadow-hiding opposition effect measures grain transparencylunar opposition effect remains to be verified. The coherentand our solution implies that the lunar regolith particles respon-backscatter opposition effect (CBOE) should be most im-sible for the shadow-hiding contribution behave like opaque

grains, consistent with complex structure of lunar regolith parti- portant for high-albedo regoliths because it depends uponcles at much larger scales than those which control the coherent multiple scattering of photons through and between scat-backscatter contribution. Most other airless planetary surfaces terers. That coherent backscatter is exclusively, or evenexhibit opposition surges whose broad components also appear significantly, responsible for the opposition surges for low-to be caused by shadow-hiding. We propose that the submicron

albedo surfaces, like that of the Moon, remains to be dem-sized grains that control coherent backscatter do not contributeonstrated. Initially, it was believed that coherent backscat-to the shadow-hiding opposition surge because the minute shad-ter could be important in natural particulate surfaces onlyows cast by these grains are illuminated by their own forward-

scattered light. We suggest that coherent backscatter cannot at very small phase angles (a , 38), however, circularbe responsible for the broad component of the Moon’s opposi- polarization measurements for lunar soil samples by Hapketion surge because such forward scattered light would illumi- et al. (1993) suggest that the process may contribute sig-nate all particle shadows and thereby eliminate the shadow- nificantly even at moderate phase angles. In addition,hiding contribution. 1997 Academic Press

Mishchenko (1992b) showed theoretically that the half-width of the coherent opposition peak can exceed a few de-grees.I. INTRODUCTION

In what follows, we modify Hapke’s traditional modelto take into account the effects of coherent backscatter onHapke’s (1981, 1984, 1986) radiative transfer model is

used routinely in lunar and planetary photometry to de- the opposition surge. We use the method of Helfenstein

20019-1035/97 $25.00Copyright 1997 by Academic PressAll rights of reproduction in any form reserved.

LUNAR OPPOSITION EFFECT 3

TABLE IAComponent Functions of the Photometric Model

Component function Description

Lommel–Seeliger coefficientg̃0

4e0

e0 1 e

P(a; g1 , g2 , f) 5 (1 2 f)PHG(a, g1) Particle phase function: A linear combination of two one-term Hen-1 fPHG(a, g2), yey-Greenstein functions PHG(a, g). PHG(a, g1) and PHG(a, g2) de-

scribe the backward and forward scattering lobes, respectivelywhere

PHG(a; g) 5(1 2 g2)

(1 1 2g cos a 1 g2)Ds

Describes the shape of the shadow-hiding opposition surgeB(a; B0 , h) 5

B0

1 1 (1/h) tan (a/2)

j(a; DaCB) Describes the shape of the coherent-backscatter opposition surge

M(e0 , e, a; g̃0 , g1 , g2 , f) Describes multiple scattering of light by anisotropically scattering parti-clesa

S(i, e, a; u) Hapke’s (1984) macroscopic roughness correction

a We have used the exact solution for an anisotropic particle phase function of the form 1 1 b cos (u), whereu 5 f 2 a is the scattering angle and b the first-order Legendre coefficient (Chandrasekhar 1950). For this functionto represent the first-order Legendre expansion of our double Henyey–Greenstein particle phase function, thecoefficient b must be related to the model parameters g1 , g2 , and f by b 5 3 [(1 2 f)g1 1 fg2] (cf. Helfenstein etal. 1995a).

et al. (1994) to fit the parameters of this model simultane- model are: (1) we adapt the coherent backscatter modelof Mishchenko (1993) as described below, (2) we use aously to the lunar whole-disk phase curve out to large phase

angles (a 5 1438) and to surface-resolved observations double Henyey–Greenstein particle phase function,1 and(3) we account for the anisotropy of multiply scatteredcovering a large range of incidence (i), emission (e), and

phase angles. Our objective is to determine whether the light to first-order instead of using Hapke’s approximationof isotropically scattering particles.inclusion of the coherent backscatter process into the

model produces a significantly better fit to the data and, We assume that shadow-hiding (SH) and coherent back-scatter (CB) mechanisms may both be contributing to theif so, to evaluate whether the parameters implied are rea-

sonable from a geological point of view. Moon’s opposition effect. In Hapke’s (1986) model, theopposition surge is attributed entirely to classical shadow-

II. PHOTOMETRIC MODEL hiding described by a single function (here representedby [1 1 B(a; h, B0)]) that multiplies the singly scattered

We have used a modified form of Hapke’s (1981, contribution of reflected light.2 A detailed analytical de-1984, 1986) equation, scription of coherent backscatter that works within the

framework of Hapke’s model has yet to be developed. Inthe coherent backscatter model of Mishchenko (1993), ther(i, e, a) 5

g̃0

4e0

e0 1 e[[1 1 B(a; h, B0)] P(a; g1 , g2 , f)

total reflected intensity of light is the sum of the diffusely

1 M(e, e0 , a; g̃0 , g1 , g2 , f)] (1)1 Hapke (1984) used a two-term Legendre polynomial expression,[1 1 j(a; Da)] S(i, e, a; u),

P(a) 5 1 1 b cos (a) 1 c (1.5 cos2 (a) 2 0.5), where b and c are Legendrecoefficients. Most workers have followed the example of Veverka (1970)where e0 and e are, respectively, the effective cosines ofand adopted the single Henyey–Greenstein function (Henyey and

incidence (i) and emission (e) angles (after adjustment Greenstein, 1941; see Table 1A). The double Henyey–Greenstein func-for the effects of macroscopic surface roughness (Hapke tion (Kattawar 1975) has been used in Hapke’s model only recently

(Domingue et al. 1991, 1995, Domingue and Hapke 1992, Helfenstein et1984)), a is the phase angle, and r(i, e, a) is the radianceal. 1997a), although it has been used in atmospheric studies for many years.factor. Equation (1) incorporates a variety of constituent

2 In classical shadow-hiding, the opposition surge is caused by the factfunctions, each of which describes a different mechanismthat soil particles completely occult their own shadows at small phase

of light scattering and absorption by a rough particulate angles, but cast them more visibly upon each other with increasing phasesurface. These functions and their parameters are summa- angle. The surge occurs because of the contrast between dark interparticle

shadows and singly (Fresnel) reflected light from particle facets.rized in Table I. Our modifications to Hapke’s original

4 HELFENSTEIN, VEVERKA, AND HILLIER

TABLE IBSummary of Photometric Model Parameters

Parameter Property Description

g̃0 Average particle single-scattering Ratio of scattering and extinction efficiencies for average particlesa: g̃0 5 QS/QE 5

albedo QS/(QS 1 QA)Related to particle composition, size, and microstructure

h, DaCB Opposition surge angular width Relates angular width of the opposition surge in phase angle for shadow-hiding and co-herent backscatter opposition surges, respectively to properties like porosity and parti-cle size distribution

B0 Opposition surge amplitude Total strength of opposition surge for the shadow-hiding opposition surge. Related tocomposition and particle microstructure.

g1 , g2 , f Average particle phase function Directional scattering properties of average regolith particles, related to particle composi-tion and microstructure. Asymmetry factors 0 $ g1 $ 21 describes the backscatteringlobe and 0 # g2 # 11 describes the forward scattering lobe.

Partition coefficient f controls the relative contribution of each component.u Macroscopic roughness Average slope angle of subresolution-scale topographic roughness

a The efficiencies with which particles scatter and absorb light are defined relative to a circular cross section whose area, fr 2p , is the same as the

average geometric cross section of the particle. If sS , sA , and sE are respectively the scattering, absorption, and extinction cross sections of theparticle, then scattering efficiency QS 5 sS/fr 2

p , absorption efficiency QA 5 sA/fr 2p , and extinction efficiency QE 5 QA 1 QS 5 sE/fr 2

p .

scattered (ID) and coherently scattered (IC) radiation. TheJD(w, w0) 5

316f

cos2 f 13

8f[cos2 f Fxx;xx(0)

(4)coherent backscatter enhancement factor, z, is defined asthe ratio (ID 1 IC)/ID. With very little modification to

1 sin2 fFxx;yy(0)],Mishchenko’s (1994) coherent backscatter formulation,we can express the multiplicative enhancement factorz 5 [1 1 j(a; DaCB)]—a convenient form that uses one where f 5 w 2 w0 and the angular functions Fij;lm aremodel parameters (DaCB). The parameter DaCB is the an- given below (Eqs. (7)–(11)). For the coherent contribution,gular semi-half width of the coherent backscatter opposi- different approximations (also due to Ozrin 1992) are usedtion surge analogous to the h parameter of shadow-hid- depending upon the range of q. For 0 # q # 2,ing model.

If ID(DaCB) and IC(a; DaCB) are, respectively, the diffuseJC(q, w, w0) 5

38f

hcos2 w cos2 w0Fxx;xx(q) 1 sin2 w sin2 w0Fyy;yyand coherent intensity of scattered light, then

1 [cos2 w sin2 w0 1 sin2 w cos2 w0]Fxy;yx(q) (5)j(a; Dacb) 5

IC(a; Dacb)ID(Dacb)

. (2)1 2 cos w cos w0 sin w sin w0[Fxy;xy(q) 2 Fxx;yy(q)]j.

As discussed in Mishchenko (1993), the scattered intensi- In the range q $ 4,ties can be computed from expressions for Stokes parame-ters (Van de Hulst 1957),

JC(q, w, w0) Q9

64q[cos2 w cos2 w0 1 Dk sin2 w sin2 w0

(6)I(q) 5

12

[J(q, 0, 0) 1 J(q, 0, f/2) 1 J(q, f/2, 0)(3) 1 cos w cos w0 sin w sin w0],

1 J(q, f/2, f/2)],and for 2 # q # 4, JC is obtained by nonlinear interpolation.Approximations for the functions Fij:lm are (Ozrin 1992)where angular parameter q P 0.56 (a/DaCB). The function

J(q, w, w0) models how the scattered light varies with thegeometry of polarized light, where w and w0 are, respec-

Fxy;xy(q) 112

> 0.85(1 1 0.99q2)1/8 (7)tively, the angles between the detected and incident polar-

izations and the scattering plane. In the Mishchenko model(due to Ozrin 1992), the diffuse component, JD , is indepen- Fxy;yx(q) 5

0.1251 1 0.57q2 (8)

dent of a and given by


N in Mishchenko’s original expression with the effectivenumber density of particles, Ne .Fxx;yy(q) > 1.07

(1 1 1.26q)2.3 (9)Equations (2) and (3) can be combined to eliminate N

and expressed in terms of other parameters in our photo-Fyy;yy(q) 2 Fxx;yy(q) 1

12

> 0.975(1 1 8q2)0.1 (10) metric model. By definition, se 5 Qe fr2

p , ss 5 Qsfr2p , and

g̃0 5 Qs/Qe . Also, for the double Henyey–Greensteinparticle phase function used in this study, kcos ul 5Fxx;xx(q) 2 Fyy;yy(q) > 0.95q2

1 1 5q2 . (11)(1 2 f)g1 1 fg2 (see Helfenstein et al. 1997a). Thus,

The angular half-widths of the shadow-hiding oppositionsurge (DaSH 5 2h; Hapke 1986) and of the coherent back- DaCB

DaSH5

«

2fg̃0[1 2 g1 2 (g2 2 g1) f ] Sl

rpD 1

Y(r). (15)

scatter opposition surge (DaCB) can be related to physicalproperties of the regolith and to each other. For theshadow-hiding opposition surge, Hapke (1986) gives the This expression provides a way of estimating the particlefollowing expression which relates DaSH to the mechanical size of the most dominant scatterers from Hapke parame-properties of the regolith at a depth where the slant-optical ters and from an estimate of plausible values for the func-path length is approximately unity: tion Y(r).

Correction for Angular Size of the SunDaSH 5 2h 5 NeserpY(r) 5 2

Nseln(1 2 F)rp

FY(r), (12)

In most applications it is practical to treat the Sun as apoint-source of illumination. For the Moon, the variations

where Ne 5 2N ln (1 2 F)/F is the effective number density in incidence and phase angles due to the finite angular sizeof particles (Hapke, 1986), N is the actual number density of the Sun (0.278) are important only for phase angles lessof particles at the surface, se is the mean particle extinction than about 0.258 and incidence angles greater than 89.758.cross section, and rp is the radius of a sphere whose geomet- Because we are interested in the detailed shape of theric cross section is the same as that of an average particle. opposition surge at very small phase angles (a , 18) weThe packing factor is F 5 NV, where V 5 4fr3

p/3 is the do take the angular size of the Sun into account. Let fsaverage volume of a particle and Y(r) is a function whose be the angular radius of the Sun measured as a polarform depends on the particle size distribution. A useful angle f from the center of the solar disk, and let z be thesimplification of Eq. 12 is obtained by noting that the azimuthal angle of any infinitesimal region on the solarporosity of the regolith, P 5 (1 2 F), and that for particu- disk. The mean reflectance of the scattering surface,late surfaces Qe P 1 (Hapke 1981, 1993)—an approxima- r(i, e, a), averaged over all such angles incident to thetion that is reasonably accurate for regolith porosities up infinitesimal regions covering the solar disk isto 80% (Mishchenko 1994). Using these definitions,

r(i, e, a) 51

f sin2 fsE2f

z50Efs

f50(16)

h P 238

ln (P)Y(r). (13)

r(i9, e, a9) sin f cos f df dz ,The half-width of the coherent backscatter oppositionsurge can be expressed as where i9 and a9 are angles of incidence and phase measured

from an arbitrary point (f, z) within the solar disk. If i0

and a0 represent incidence and phase angles, respectively,DaCB 5«l

2fltr5

«lNess(1 2 kcosul)2f

(14) for the Sun’s center-of-disk, then i9 and a9 can be calcu-lated from

5 2«lNssln(1 2 F)(1 2 kcosul)

2fF,

(17)cos i9 5 cos i0 cos f 2 sin i0 sin f cos z

where ltr is the mean free transport length of photons inandthe medium, ss is the particle scattering cross-section, and

« is an angular constant found to be approximately 0.56(Mishchenko 1992a,b, 1993). Here, u 5 f 2 a is the scatter- cos a9 5 (cos i0 cos e 1 sin i0 sin e cos c0) cos fing angle, and the asymmetry factor kcos ul is the first

1 (sin e sin c0) sin f sin z (18)moment of cos u with respect to P(a; g1 , g2 , f). In Eq.(14), we have followed Hapke’s (1986) advice and replaced 1 (cos i0 sin e cos c0 2 sin i0 cos e) sin f cos z ,


TABLE IIASources of Whole-Disk Data for this Study

Phase angle WavelengthsData source range (em) Comments

Rougier (1933) 2.48–144.58 0.445 Photoelectric photometryCombined photoelectric and early spacecraft photographic

Shevchenko (1982) 1.48–130.48 P0.55 photometryShorthill et al. (1969) 2.38–124.58 0.445 Video photometryLane and Irvine (1968) 6.68–118.68 0.548 (V) Photoelectric photometry

where the Shorthill et al. (1969) observations of ‘‘average’’ lunarmaterials. In addition, data of Pohn et al. (1969) and Whi-taker et al. (1969) are incorporated for the opposition surge

cos c0 5cos a0 2 cos i0 cos e

sin i0 sin e. (19) from 08 to 78.

IV. APPROACHEquation (19) is undefined when i0 5 0 and/or e 5 0. Wheni050 and e ? 0, then i9 5 f and cos a9 5 cos e cos f 1

To fit the photometric model simultaneously to whole-sin e sin f cos z. For the special case where i0 5 e 5disk and disk-resolved data, we minimize the rms residuala0 5 0, i9 5 a9 5 f. In practice, we use numerical integra-(x), defined such thattion to evaluate Eq. (16).

III. OBSERVATIONS AND DATA x2 51

3n On

j51[Mj 2 M(aj)]2

To determine the photometric parameters, we fit themodel simultaneously to two independent data types: the 1

6.253k Ok

j51[log10 rj 2 log10 skr(ij , ej , aj)]2

whole-disk phase curve and disk-resolved observations ofthe Moon. Whole-disk observations (summarized in Table

16.253,

O,

j51[log10 rj 2 log10 s,r(ij , ej , aj)]2,IIA) measure how the integral brightness of the lunar

disk varies with phase angle. Following the approach ofHelfenstein and Veverka (1987), we combined the visual

where the subscripts n, k, and , refer, respectively, to the(V-filter) observations of Lane and Irvine (1973), with ob-number of whole-disk phase curve, Shorthill et al. disk-servations of Rougier (1933), Shorthill et al. (1969), andresolved, and the Pohn/Whitaker disk-resolved data points.Shevchenko (1982). All data are rescaled to the absoluteMj is the whole-disk lunar magnitude at observed phasecalibration of Lane and Irvine. Disk-resolved observationsangle aj and M(aj) is the corresponding model whole-disk(Table IIB) measure how the brightness of lunar materialslunar magnitude. Likewise, rj is the disk-resolved radiancevaries with incidence, emission, and phase angles. We usefactor observed at photometric geometry (ij ej aj) and r(ij ej

aj) is the corresponding model radiance factor. Calibrationscale factors, sk 5 1.05 and s, 5 0.85 renormalize disk-

TABLE IIB resolved reflectances to be consistent with the absoluteDisk-Resolved Data for This Study

brightness scale of the whole-disk data.The technique used to find minimum rms residuals is anPhotometric

Source coverage Method improved version of that described by Helfenstein (1986).It employs repetetive trials that combine a grid search of

Shorthill et al. (1969) 1.98 # i # 89.68 Telescopic video images parameter space for model parameters that are least-well4.08 # e # 82.28

constrained by data with a gradient follower (at each grid-2.38 # a # 135.78searched increment) for the remaining well-constrainedWhitaker (1969) i 5 08 Apollo 8 photographs

08 # a # 5.88 parameters. We adopt the following strategy: We consider08 # e # 5.88 two end-member solutions in which the opposition surge

Pohn et al. (1969) i 5 08 Apollo 8 photographs is attributed only to shadow-hiding (SHO) and only to08 # a # 78

coherent backscatter (CBO), respectively. The first of08 # e # 78these cases is essentially the Hapke (1981, 1984, 1986)


TABLE IIIPhotometric Parameters Fits to Composite V-Filter Data

Shadow CoherentModel hiding only backscatter Solution 1a Solution 1b Solution 2a Solution 2b

parameter (SHO) only (CBO) (DaCB @ DaSH) (DaCB @ DaSH) (DaSH @ DaCB) (DaSH @ DaCB)

g̃0 0.31 0.43 0.278 6 0.007 0.169 0.279 6 0.002 0.167u 298 408 27.18 6 0.28 8.58 26.78 6 0.18 8.08

g1 20.33 20.42 20.252 6 0.007 20.240 20.235 6 0.005 20.226g2 0.52 0.52 0.621 6 0.001 [Not used] 0.638 6 0.012 [Not used]f 0.5 0.72 0.443 6 0.005 [Not used] 0.444 6 0.003 [Not used]

B0 1 [Not used] 0.84 6 0.11 0.84 1 1h 0.023 [Not used] 0.027 6 0.007 0.033 0.158 6 0.001 0.161

(DaSH) (2.68) (3.18 6 0.48) (3.88) (18.18 6 0.18) (18.48)DaCB [Not used] 1.78 238 6 88 248 2.08 6 0.58 2.38

RMS error 0.00317 0.00537 0.00266 0.00375 0.00260 0.00371

model currently in widespread use. We also consider two (double Henyey-Greenstein function). The rms residualfor Solutions 2a is about 2% lower than that for Solutionmore sophisticated models in which shadow-hiding and

coherent backscatter act together. In the first of these 1a, a difference probably too small to be statistically sig-nificant.(hereafter called Solution 1), the shadow-hiding opposition

effect is narrow in comparison to the coherent backscatter- The solid curve in Fig. 1 shows how rms errors variedas we performed a grid search of possible solutions ining opposition effect (DaSH ,,, DaCB). Conversely, in the

second (Solution 2), the coherent backscatter opposition which we increased DaCB incrementally while allowing theremaining parameters to adjust to optimal values at eacheffect is much narrower than the shadow hiding opposition

effect (DaCB ,,, DaSH). increment. Solution 2a is characterized by a well-definedminimum that is clearly separated from the shallow mini-To assess the sensitivity of our model to different particle

phase functions, we have divided each of Solutions 1 and2 into two parts (labeled ‘‘a’’ and ‘‘b,’’ respectively). InSolutions ‘‘a,’’ we adopt a double Henyey–Greenstein par-ticle phase function which allows the simultaneous repre-sentation of strong forward and backward scattering. ForSolutions ‘‘b,’’ we substitute a more commonly used singleHenyey–Greenstein particle phase function to show thatour major conclusions are not dependent critically uponthe specific representation of the phase function.

V. RESULTS

Optimal Solutions

Our models contain up to eight model parameters (Ta-bles IB and Table III). Following the approach outlinedin Section IV, we obtained the least-squares fits to ourfour candidate models summarized in Table III. Of the fitslisted in Table III, the CBO yields the largest rms error.

FIG. 1. Result of an incremental parameter search on DaCB . A well-The rms error for the SHO is 41% smaller. Not surprisingly, defined narrow minimum at Solution 2a and a less-distinct broad mini-the most sophisticated models (Solutions 1a and 2a), each mum at Solution 1a are shown. The minimum rms error that defines

Solution 2a is slightly better than that which defines Solution 1a. Dashedhaving an additional model parameter, offer improvementslines show the margin of rms errors over which we consider the minimaof 50 and 52%, respectively, over the CBO model and 14to be adequately defined. For Solution 1a, the rms error intervaland 18% improvements, respectively, over the SHO model.Drms 5 0.000014 corresponds to the parameter interval 148 # DaCB #

The rms residuals for Solutions 1b and 2b (single Henyey– 298. For Solution 2a, the same rms error interval corresponds to 1.68 #Greenstein function) are, respectively, 40 and 43% poorer, DaCB # 2.58. We have used this margin in rms residual to compute the

uncertainties for Solutions 1a and 2a reported in Table III.respectively, than the corresponding Solutions 1a and 2a


FIG. 2. Composite V-filter phase curve data for the Moon and predicted phase curves from two different Hapke models (Solutions 1a and 2a).(a) Complete phase curve out to a 5 1808. (b) Details of (a) showing whole-disk phase coverage 08 # a # 408. (c) Details of the narrow componentof the opposition surge revealed by disk-resolved observations of Whitaker (1969) and Pohn et al. (1969) from Apollo 8 photographs of the commandmodule shadow point on the Moon’s surface. Also shown are the two Hapke model fits (see text for discussion).

mum of Solution 1a by a prominent maximum. The mini- the narrow component of the opposition surge, especiallyat a # 18. Solution 2a offers only a 0.3% improvement overmum corresponding to Solution 1a is very broad and cannot

be distinguished clearly from immediately surrounding so- Solution 1a in its ability to model the disk-resolved data, a5% improvement in its ability to model the whole-disk phaselutions. In contrast, the rms error minimum for Solution

2a is very narrow. Thus both in terms of the depth of the curve data, but a 65% improvement in its ability to modelthe near-opposition data. The goodness of fit of Solution 2aminimum and in terms of its narrowness, we conclude that

Solution 2a is to be preferred over Solution 1a. to disk-resolved lunar data is shown in Fig. 3, where we havebinned (in increments of incidence, emission, and phaseSolutions 1a and 2a differ primarily in terms of their oppo-

sition effect parameters: h, B0, and DaCB. The fits of Solu- angles) the averaged ratios of observed radiance factors rel-ative to those computed from Solution 2a for correspondingtions 1a and 2a to the whole-disk lunar data are compared

in Fig. 2: the two solutions cannot be distinguished at phase i, e, and a. Systematic deviations of average ratios from unitywith photometric geometry, as might be expected from aangles greater than 28. Figure 2c indicates that the two solu-

tions differ mainly in Solution 2a’s better ability to model poor fit, are not evident.


FIG. 3. Fit of Solution 2a to disk resolved data of Shorthill et al. at many angles of (a) phase, (b) incidence, and (c) emission. Vertical axis isthe ratio of observed radiance factors to radiance factors predicted from Table III photometric parameters. All data (N 5 363) were binned in 2.58

increments of i, e, or a and averaged. Error bars are 1 SD. No systematic misfit with photometric angle is evident, indicating that a good fit hasbeen obtained. Note that the angular extent of the Sun was accounted for in these results.

In summary, Solutions 1a and 2a provide the best fits to the fit over our best RMS residual from Table III (Solution2a). In the context of Hapke’s (1986) model, derived valuesthe lunar data. Solutions 1a and 2a are distinct and Solution

2a provides a small improvement in the fit. As Fig. 1 shows, of B0 greater than unity from earlier studies are physicallymeaningless and most likely confirm that the shadow-hid-it is defined by a sharper minimum than is Solution 1a. As

demonstrated in the following section, a compelling prefer- ing model alone is inadequate to account fully for observedopposition effects. Whitaker (1969) first noted that theence for Solution 2a can made on the basis of the physical

interpretation of the opposition surge parameters. In either total amplitude of the Moon’s opposition surge (includingthe narrow component at a , 28) exceeds theoretical limitssolution, the Moon’s opposition surge is best described as a

narrow (28–38 wide) angular peak at very small phase angles for shadow-hiding. By modeling the simultaneous contri-butions from coherent backscatter and shadow-hiding,that transitions to a broader (188–238 wide) surge at a . 28.

Together, Solutions 1a and 2a yield the following average however, we have achieved optimal fits to a very complexphotometric data set (including a narrow component ofparameters: g̃0 5 0.28, u 5 278, f 5 0.44, 20.24 # g1 # 20.25,

0.62 # g2 # 0.64, and 0.8 # B0 # 1.0. the opposition surge seen by Whitaker) without demandingthat model parameters exceed their admissible bounds.A significant difference between our study and earlier

works is that we have restricted the shadow-hiding opposi-tion surge amplitude parameter, B0, to be within its physi- VI. DISCUSSIONcally admissible range of values. Many earlier workers (seeTable IV) obtained best fits of Hapke’s (1986) model to As discussed earlier, Solutions 1a and 2a fit our lunar

data set almost equally well. A choice between them canlunar and planetary data by relaxing the requirement that0.0 # B0 # 1.0. As shown in Table IV and Fig, 4, values be made on the basis of the realism of physical properties

that each solution implies for the lunar surface. Our valuesof B0 . 1 were especially important for obtaining optimummodel fits for relatively low-albedo objects like the Moon, of g̃0 , u, g1 , g2 , and f above are in good agreement with

values obtained in Helfenstein et al. (1996b) who providesMercury, Phobos and Deimos, dark asteroids, Callisto, andUmbriel. Because B0 is exactly at the upper physical limit a thorough discusion of their physical interpretation.

Coherent backscatter requires a significant contributionof unity in Solution 2a, we conducted an additional gridsearch (Fig. 5a) in which we allowed B0 to exceed unity; of light from scatterers whose sizes are too small for scatter-

ing to be described by geometric optics (i.e., particles can-however, we found only marginal (2%) improvements in


TABLE IV rosity, P, Eq. (13) can be rearranged to solve for a corre-Survey of Opposition Surge Amplitudes for sponding range of values for the unknown function Y(r).

Solar System Objects A lower limit of P P 0.45 may be taken from in situestimates of the bearing strength of lunar soil (Carrier etObject g̃0P(08) B0 Studyal. 1972, 1973, 1974, Houston et al. 1972). The state of

Mercury 0.276 2.40 Veverka et al. (1988) compaction of lunar soil likely varies with depth and may0.378 1.85 Bowell et al. (1989) be considerably higher at the topmost surface than the

Moon bearing strength implies. We conservatively take P 5 0.80Whole disk 0.353 2.01 Helfenstein and Veverka (1987)

as a plausible upper limit of regolith soil porosity (it makes0.329 1.89 Helfenstein et al. (1996a)little difference to our conclusion if a larger upper limit isDark terrains 0.181 2.81 Helfenstein and Veverka (1987)

Average terrains 0.425 1.84 assumed). This range of 0.45 # P # 0.80 and values of hBright terrains 0.564 1.83 from Solutions 1a (Table III) yield a corresponding range

Mars 0.629 1.00 Arvidson et al. (1989)of Y(r) values listed in Table V. The range of particle radiiPhobos 0.113 4.0 Simonelli et al. (1996)(rp) corresponding to the derived interval of Y(r) can be0.108 5.8 Efford and Wilson (1989)

Deimos 0.202 1.65 Thomas et al. (1996) derived using Eq. (5), Table III values of photometricAsteroids parameters g̃0 , g1 , g2 , f, DaCB , and DaSH , and the effective

1 Ceres 0.222 1.58 Helfenstein and Veverka (1989) wavelength of the photometric observations (l 5 0.55 em).4 Vesta 1.061 1.03Table V reveals that Solution 2a predicts the presence of24 Themis 0.187 1.6 Bowell et al. (1989)scatterers with diameters 0.1 # 2rp # 0.4 em, near the44 Nysa 2.256 0.6

69 Hesperia 0.599 0.94 wavelength of light used to detect them (l 5 0.55 em)—a82 Alkmene 0.452 1.4 size range consistent with the ‘‘weak localization’’ nature133 Cyrene 0.741 1.19

of coherent backscatter. Particles in this size range are419 Aurelia 2.040 0.47smaller than the mean size of regolith grains (40 em;243 Ida 0.646 1.53 Helfenstein et al. (1996c)

951 Gaspra 0.646 1.63 Helfenstein et al. (1994) McKay et al. 1974), but typical of the fine grain fraction1862 Apollo 0.814 0.98 Bowell et al. (1989) of the lunar soil (Gold et al. 1970, Zook and McCoy 1991).

0.908 0.90 Helfenstein and Veverka (1989)In contrast, Solution 1a implies particle diameters ofJovian moons0.01 # 2rp # 0.04 em—particles that are 14 to 55 timesIo 2.494 0.80 McEwen et al. (1988)

EuropaLeading side 3.770 0.50 Domingue et al. (1991)Trailing side 2.761 0.50

GanymedeDark terrain 0.953 0.69 Helfenstein (1986)Bright terrain 1.493 0.69

Callisto 0.816 1.50 Helfenstein et al. (1996a)Saturnian moons

Mimas 1.702 0.67 Verbiscer and Veverka (1992)Enceladus 3.850 0.21 Verbiscer and Veverka (1994)Rhea

Whole-disk 2.201 0.66 Verbiscer and Veverka (1989)Global-average 2.541 0.66Leading side 2.936 0.43 Domingue et al. (1995)Trailing side 2.914 0.43

Uranian moonsTitania 1.185 0.65 Helfenstein et al. (1988)Oberon 1.021 1.09 Helfenstein et al. (1989)Umbriel 0.523 2.01 Helfenstein et al. (1991)

not be much larger than a wavelength of light) and too FIG. 4. Log–Log plot of B0 vs g̃0P(08) from the data of Table IV.The straight line is a least-squares fit B0 5 1.083 [g̃0P(08)]20.629, whichlarge for scattering to be described by quantum physicalyields a correlation coefficient of 92%. The tendency of B0 to decreaseoptics alone (particles cannot be much smaller than a wave-with increasing particle albedo is consistent with the predictions of thelength of light). Equations (13) and (15) can be used toshadow-hiding model of the opposition surge, but is opposite to the

estimate the the particle radii, rp , predicted from Solutions predictions of the coherent backscatter model. This result suggests that1a and 2a. Given the best-fit values of h from Table III the broad component of the opposition surge for most planetary surfaces

is due to shadow-hiding.and reasonable range of assumptions about lunar soil po-


FIG. 5. (a) Results of an incremental parameter search on B0 for which it was allowed to exceed its physical upper limit of unity. rms residualsnear 0.0255 (about 2% lower than that for Solution 2a are found for many values of B0 . (b) Coupling of B0 to the total amount of backscatteredlight at zero phase, g̃0P(08) for the corresponding solutions represented in (a). (c) Coupling of the angular width of the shadow-hiding oppositionsurge, DaSH , to B0 for the solutions represented in (a). As B0 increases, the angular width of the shadow-hiding opposition surge increases tocompensate. (d) Coupling of the shape of the backscattering lobe of the particle phase function (represented by g1) to B0 values from solutionsshown in (a). As B0 increases, the backscattering lobe of the particle phase function becomes flatter at small phase angles to compensate.

smaller than the wavelength of light and inconsistent with average particle backscatter, g̃0P(08). This follows becausein the case of shadow-hiding B0 should decrease with in-the physical basis of coherent backscatter.creasing g̃0P(08), whereas the opposite is true for coherentFurther support for Solution 2a and for the view thatbackscatter, as we now demonstrate.shadow-hiding is responsible for the broad (rather than

Hapke (1986) defines the total amplitude of the shadownarrow) component of the lunar opposition surge comeshiding opposition effect B0 as the fraction S(0)/g̃0P(08),from published fits of Hapke’s (1986) model to the widewhere P(08) is the particle phase function evaluated atvariety of planetary surfaces (Table IV; Fig. 4). Few of thea 5 08, S(0) is the contribution only from first-surfacephotometric data sets represented by Table IV containreflections (for euhedral crystal grains, S(0) is the Fresnelsufficient observations at phase angles small enough toreflectance), and g̃0P(08) is the total amount of backscat-characterize adequately the narrow component of the op-tered light from a typical regolith particle. The fractionalposition surge but they are sufficient to characterize thecontribution of first-surface reflected light should be largerbroad component. We can test whether the broad compo-for opaque particles than for more tansparent grains be-nents are due to shadow-hiding or coherent backscattercause in the latter case, light can penetrate more deeplyby considering that the derived values of B0 vary withinto the grain and be backscattered from internal crystaldefects. Because opaque grains have smaller albedos thansimilarly sized more transparent grains, B0 should be larger

TABLE V for lower-albedo particles than for higher-albedo grains.Estimated Mean Particle Sizes Corresponding to DifferentIf the broad component is due to shadow-hiding, thenParameter Solutions and Assumed PorositiesB0 should decrease with increasing g̃0P(08). In contrast,coherent backscatter is a mechanism that depends on theParticle

Y(r) radius availability of multiply scattered photons and should thusDaCB/DaSH (0.45 # P # 0.80) rp(em) be more intense for high-albedo than for low-albedo rego-

liths. If the broad component is due to coherent backscat-Solution 1a 7.68 0.088 # Y(r) # 0.32 0.02 $ rp $ 0.005

ter, then B0 should increase with increasing g̃0P(08).Solution 2a 0.109 0.053 # Y(r) # 1.88 0.2 $ rp $ 0.06Figure 4 shows an overall trend for B0 to decrease with


increasing g̃0P(08), as expected for the shadow-hiding but phase function for the lunar regolith is significantly moreforward scattering than expected for perfectly opaquecontrasting to the expectations for coherent backscatter.

Similar trends have been noted in laboratory studies of particles (Hapke 1963), but less forward scattering thanfor small transparent grains (cf. McGuire and Hapkethe opposition surge in lunar soil samples (cf. Hapke et al.

1995, 1996) and in multispectral data for the lunar surface 1995, Mishchenko 1994).In summary, we find that grain sizes predicted by Solu-(Helfenstein et al. 1997b).

The mutual consistency of opposition surge parameters tion 1a are inconsistent with the theory of coherent back-scatter and the known size distribution of lunar soil grains.provides yet another way of distinguishing the realism of

Solutions 1a and 2a. The presence of nearly opaque parti- Solution 1a is also inconsistent with the simultaneous pres-ence of coherent backscatter and shadow hiding. In con-cles implied by B0 5 1.0 (Solution 2a) and B0 5 0.84

(Solution 1a) would seem to be inconsistent with the small trast, Solution 2a predicts grain sizes that are known tooccur in lunar soil and that are consistent with coherentparticle sizes needed for coherent backscatter and with the

moderate forward scattering implied by our best-fit particle backscatter theory. Solution 2a also provides a mechanismby which coherent backscatter and shadow-hiding can oc-phase function parameters (cf. Helfenstein et al. 1997a).

This apparent inconsistency can be reconciled by consider- cur simultaneously in the lunar regolith. Therefore, weadopt Solution 2a as a more reasonable choice and con-ing how regolith structure at different spatial scales con-

trols the light scattered into different photometric geome- clude that coherent backscatter controls the narrow peakof the lunar opposition surge, but that the broader segmenttries. The sub-micron sized grains responsible for coherent

backscatter must exhibit significant forward scattering at is definitely due to shadow-hiding. This choice is supportedby the fact that observed amplitudes of opposition surgesl 5 0.55 em, even if they are made out of an opaque

material (such as metal) because diffraction can occur for Solar-System objects tend to be larger for darker sur-faces than for brighter ones.around particles at the topmost surface of the regolith

where grains are exposed to free space. Forward scatteredFurther Tests of Coherent Backscatter and

light from the sub-micron sized grains illuminates the mi-Shadow-Hiding Theories

nute shadows that they cast and therefore they do notcontribute to the shadow-hiding opposition effect. The Measurements of circular polarization from lunar soil

samples have been offered as unequivocal proof that mostshadow-hiding opposition effect must be controlled bymuch larger particles in the regolith which the smaller of the lunar opposition surge is caused by coherent back-

scatter and not by shadow-hiding (Hapke et al. 1993). Thatparticles coat.Hapke (1996) has shown that comparatively large, struc- conclusion, however, is founded upon a qualitative under-

standing of how circular polarization is expected to varyturally complex particles such as lunar agglutinates andmulticomponent grains, especially those with opaque inclu- with phase angle if the opposition effect is due either to

shadow-hiding or to coherent backscatter. Shadow-hidingsions, have phase functions that are strongly backscatter-ing. Such particles, which are much larger than the minute is so fundamental a physical process that it is difficult to

imagine how it could not significantly contribute to thegrains that cause coherent backscatter, may be nearlyopaque at size scales that are important for shadow-hiding. Moon’s opposition surge. The present study indicates that

the Moon’s opposition surge at visual wavelengths (l 5Since the median size for lunar regolith grains is 40 em(McKay et al. 1974), shadows cast by clumps of these grains 0.55 em) is best explained by strong contributions from

both shadow-hiding and coherent backscatter.are two to three orders of magnitude larger in size thanthose cast by the sub-micron sized particles responsible Information about the roles of coherent backscatter and

shadow-hiding can be derived from studying how photo-for coherent backscatter. Our Solution 2a value of B0 5 1.0is thus consistent with the known presence of structurally metric properties of regolith materials vary with spectral

wavelength. Equation (15) shows, for example, that thecomplex lunar regolith grains with diameters much largerthan a wavelength of visible light. angular width of the coherent backscatter opposition effect

should increase relative to that of the shadow hiding com-Solution 1a is automatically excluded as a reasonablemechanism because if the broad component of the lunar ponent with increasing wavelength. In addition, the ampli-

tude of the shadow-hiding opposition surge should de-opposition surge were due to coherent backscatter, thenall particle shadows would be illuminated by forward- crease with increasing particle albedo as discussed earlier.

Because the albedos of lunar materials vary with wave-scattered light and there would be no detectable shadow-hiding contribution to the opposition surge (B0 P 0). length, multi-spectral data, like that returned from Clem-

entine and Galileo spacecraft, as well as spectrophotomet-Solution 2a is the only proposed mechanism which ac-counts for the simultaneous presence of opposition effects ric studies of lunar soil samples provide and additional

ways of distinguishing the coherent backscatter fromdue to shadow-hiding and coherent backscatter. Theabove hypothesis also explains why the average particle shadow-hiding opposition effects.


Thus far, our conclusion is supported by recent labora- plitudes are consistent with the presence of a broadshadow-hiding contribution.tory studies of lunar soil samples (Hapke et al. 1996)

as well as by newly analyzed Clementine multi-spectralobservations of the lunar surface (Buratti et al. 1996).Hapke et al. (1996) and Buratti et al. (1996) have both ACKNOWLEDGEMENTSfound no wavelength dependence on the angular width

This study was supported by NASA Grants NSG 2084 and NSG 4986.of the opposition surge of lunar soils, consistent with theWe thank Mary Roth for providing assistance with manuscript prepara-contribution expected from shadow-hiding. In addition,tion. M. Mishchenko and B. Hapke provided careful reviews.Hapke et al. (1996) found that the circular polarization

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The Lunar Opposition Effect: A Test of Alternative Models

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