A Theoretical Model of Lunar Optical Maturation: Effects of Submicroscopic Reduced Iron and Particle Size Variations

A Theoretical Model of Lunar Optical Maturation: Effects of Submicroscopic Reduced Iron and Particle Size Variations

Icarus 152, 275–281 (2001) doi:10.1006/icar.2001.6624, available online at http://www.idealibrary.com on A Theoretical Model of Lunar Optical Maturat...

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Icarus 152, 275–281 (2001) doi:10.1006/icar.2001.6624, available online at http://www.idealibrary.com on

A Theoretical Model of Lunar Optical Maturation: Effects of Submicroscopic Reduced Iron and Particle Size Variations L. V. Starukhina and Yu. G. Shkuratov Kharkov Astronomical Observatory, Kharkov, Ukraine, Sumskaya St., 35, Kharkov, 61022, Ukraine E-mail: [email protected] Received June 12, 2000; revised February 15, 2001; Posted online June 27, 2001

Diagrams, color index C(0.95/0.75 µm) versus albedo A(0.75 µm), are used to estimate composition and maturity degree of the lunar regolith. We simulate maturation curves on the diagrams with the model (Shkuratov, Yu. G., L. V. Starukhina, H. Hoffmann, and G. Arnold 1999. Icarus 137, 235–246). In simulation, decrease of characteristic particle size as well as increase of concentration of reduced iron were taken into account. For rather small changes of maturation parameters, almost linear A(0.75 µm)–C(0.95/0.75 µm) maturation trends are found which is in accordance with the method of determination of iron content and maturation degree proposed by Lucey, P. G., D. Blewett, and J. Johnson (1995. Science 268, 1150–1154). With increased changes in maturation parameters, the maturation trends are no longer linear and the branches with the opposite direction of optical maturation are obtained. This is in accordance with spectral measurements of particle size fractions of lunar soils. The maturation trends on A(0.75 µm)–C(0.95/0.75 µm) plots have in general no common origin. In most cases there is an area, where the trends can be focused, but the position of the area differs for different sets of data. Thus, Lucey’s assumption about the common origin of maturation trends is valid only as an approximation. Elongated clusters on A(0.75 µm)–C(0.95/0.75 µm) diagram for the lunar nearside (Shkuratov, Yu. G., V. G. Kaydash, and N. V. Opanasenko 1999. Icarus 137, 222–234) can be described by maturation trajectories corresponding to different Fe2+ abundance. The upper edge of the diagram, i.e., the greatest values of C(0.95/0.75 µm), can be determined by the processes of particle size decrease in maturation. °c 2001 Academic Press Key Words: Moon; surface; regoliths; spectroscopy.

INTRODUCTION

Information about composition of the lunar regolith can be derived from spectral measurements due to correlation between optical parameters and the abundance of the main lunar chromophore ions Fe2+ and Ti4+ (e.g., Pieters 1978). This derivation, however, is not a simple task because of alteration of the optical properties of regolith with time. The lunar surface material is exposed to space weathering which causes physical and chemical processes called maturation. Three processes accompanying maturation are usually considered: (1) formation of submicron

Fe0 grains in the regolith particles (e.g., Housley et al. 1973, Hapke et al. 1975, Morris 1976, 1978, 1980), (2) decrease of particle size due to micrometeorite bombardment of regolith (e.g., McKay et al. 1991), and (3) agglutination (e.g., Adams and McCord 1971). Processes (1) and (3) result in darkening of the lunar soils; the effect of particle size decrease is the opposite. The spectral characteristics depend on bulk abundance of the iron and titanium ions as well as on the maturation processes. Among them the optically main process is formation of the finely dispersed metal forms in regolith particles. In order to characterize maturation degree of the lunar regolith, the ratio Is /FeO (where Is and FeO are proportional to concentration of the finely dispersed metallic iron and bulk iron abundance, respectively) was introduced (Morris 1976, 1978, 1980). The effect of particle size on the albedo of powdered surface is well known (e.g., Adams and Filice 1967). Increase of the size results in attenuation of the light passing through the particles and decrease of the albedo of powdered surface. Darkening effect of agglutination (e.g., Adams and McCord 1971) is due mostly to abundant submicron grains of Fe0 on the surfaces and in the volume of aggregated particles. Therefore the optical effect of agglutinates can be included in the effects of reduced iron. As for the contribution of agglutinates to the average particle size, it should be emphasized that they are aggregates of smaller soil particles (mineral grains, glasses, and even older agglutinates). The size parameter of our model is the average size of scattering element (the distance between two successive scatterings), i.e., the size of subparticles of the agglutinates. This size is less than the average particle size, which widens the size range to the smaller values. Optical effects of maturation processes can be presented on the plot of color index C(0.95/0.75 µm) versus albedo A(0.75 µm). This allows us to develop techniques to deconvolve the composition and maturation effects on spectral properties of the lunar surface (Lucey et al. 1995, 1996, 1997, 1998a,b,c, Lucey and Blewett 1997, Blewett et al. 1997a,b, Shkuratov et al. 1999a, Giguere et al. 2000). In particular, Lucey et al. (1995, 1998a) suggested estimating iron content and maturity degree using a polar coordinate system defined on the coordinate plane A(0.75 µm)–C(0.95/0.75 µm), the polar angle and

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catrix of a particulate layer, respectively ρb = q · rb ρ f = q · r f + 1 − q.

(2)

The values rb and r f are back and forward components of the indicatrix of a particle, respectively, rb = Rb + Te Ti Re exp(−2τ )/2[1 − Ri exp(−τ )]

(3a)

r f = R f + Te Ti exp(−τ ) + Te Ti Re exp(−2τ )/2 × [1 − Ri exp(−τ )] τ = 4π kl/λ,

FIG. 1. The diagram A(0.75 µm)–C(0.95/0.75 µm) for lunar sites (Lucey et al. 1998a) recalibrated by Eqs. (19a, b) (symbols) and its simulation (lines).

radial coordinate being related to iron content and maturity degree, respectively. To choose the proper polar coordinate system, data on some lunar samples were taken. Lucey’s method is based on the assumption that maturation trajectories linearly extrapolated in the direction of high maturity are focused in a common point which can be the origin of the polar coordinates. On the A(0.75 µm)–C(0.95/0.75 µm) plane, maturity degree increases to the top left corner of the plane; this corner corresponds to darker material with less pronounced 0.95 µm absorption band. To illustrate the approach, in Fig. 1 the data for lunar sites adopted from Lucey et al. (1998a) are presented; the maturity trends corresponding to different Fe2+ content really tend approximately to a common point. It should be noted that the existence of the common origin on the diagram A(0.75 µm)–C(0.95/0.75 µm) has not yet been substantiated physically—it is just an empirical fact. Here we study this problem using our spectrophotometric model (Starukhina and Shkuratov 1996, Shkuratov et al. 1999b). Distinct from our previous studies (Starukhina and Shkuratov 1999), in the present study we take two opposite optical effects of maturation into account: surface darkening due to reduced iron and surface brightening due to particle size decrease.

A=

1 + ρb2 − ρ 2f 2ρb

s −

µ

(4)

Re = Rb + R f , Te = 1 − Re

(5)

Ti = Te /n 2 , Ri = 1 − Ti ,

(6)

where Rb and R f are the average backward and forward reflectance coefficients at a single scattering, Te is the average transmission coefficient for the light penetrating into a particle from outside, Ti is the average transmission coefficient for the light going in the opposite direction, and Ri is the average reflectance coefficient for the light inside the particle. For the coefficients Rb , Re , and Ri in the range of n from 1.4 to 2.0 these empirical approximations are satisfactory Rb ≈ 0.07n − 0.083 ≈ r0 + 0.055

(7)

Re ≈ 0.14n − 0.12

(8)

Ri ≈ 1.04 − 1/n ,

(9)

2

r0 = (n − 1)2 /(n + 1)2 being the Fresnel reflection coefficient at the normal incidence. The model is inversible; i.e., starting from the surface albedo and supposing the values of n and l known, we can solve Eq. (1) for the imaginary part κ of the refraction index of the particle material (Starukhina and Shkuratov 1996, Shkuratov et al. 1999b) κ = τ λ/4πl,

THE MODEL

The model (Starukhina and Shkuratov 1996, Shkuratov et al. 1999b) enables us to calculate reflectance (albedo) A of a particulate surface as a function of the wavelength λ. The parameters of the model are: the real n and the imaginary κ parts of particle refractive index, the volume fraction q filled by the particles, and the average size l of the regolith particles. The following expressions have been used for modeling 1 + ρb2 − ρ 2f 2ρb

¶2 − 1,

(1)

where ρb and ρ f are back and forward components of the indi-

(3b)

where the optical density τ is calculated as s ¸ µ ¶2 · c b b + , − τ = −ln a a a where a = Te Ti (y Ri + qTe ), q b = y Rb Ri + Te2 (1 + Ti ) − Te (1 − q Rb ), 2 c = 2y Rb − 2Te (1 − q Rb ) + qTe2 , y = (1 − A)2 /2A.

(10)

(11)

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The model allows us to estimate the change in albedo of a particulate surface when particle size and/or composition are modified: the values of κ are calculated from the surface albedo A; then κ and/or l can be changed and the surface reflectance A is recalculated for new values of κ and/or l. For instance, to take the presence of fine (much smaller than the wavelength) absorbing inclusions of Fe0 into account, we modify values of κ as (Starukhina et al. 1994, Shkuratov et al. 1999b) κeff (λ) = κ(λ) + c · κFe (λ),

(12)

where the contribution of Fe0 grains per unit volume fractions is κFe (λ) = 1.5 · n · Im{[ε(λ) − 1])/[ε(λ) + 2]},

Possible directions of maturation trajectories on the A–C plane are determined by the condition dl/dc ≤ 0; i.e, for two points (A0 , C0 ) and (A1 , C1 ) 1c(A0 , C0 , A1 , C1 , l0 ) > 0, l1 /l0 ≤ 1.

(17)

If both conditions (17) are satisfied, there are maturation trajectories directed from a point (A0 , C0 ) to a point (A1 , C1 ) on the A–C plane. Using Eqs. (15) and (16) it can be shown that 1c(A0 , C0 , A1 , C1 , l0 ) = −1c(A1 , C1 , A0 , C0 , l1 ); i.e., calculated 1c changes its sign with exchange of the order between the points 0 and 1.

(13)

ε(λ) being the ratio of the complex dielectric constant of Fe0 to that of a regolith particle at a given wavelength. The volume fraction c of such grains is very small: c < 10−3 (Morris 1980). For calculations, the optical constants of Fe0 measured by Johnson and Cristy (1974) were used. To find a new reflectance of the surface, we substitute another value of the size l and κeff instead of κ into formulas (1)–(4). Equation (12) allows us to find the difference 1c between the volume fractions of fine-grained reduced iron for any two points 0 and 1 with coordinates (A0 , C0 ) and (A1 , C1 ) on the A(0.75 µm)–C(0.95/0.75 µm) plane, if the change of the coordinate is due only to the above-mentioned maturation processes. Denote the average sizes of the particles corresponding to the points 0 and 1 by l0 and l1 , respectively, and the wavelengths λb = 0.75 µm, λr = 0.95 µm. Multiplying Eq. (12) written at λb and λr by 4πl1 /λb and 4πl1 /λr , respectively, we obtain the system of equations for maturation parameters 1c and l1 /l0 , the optical densities τ defined by formula (4) being the coefficients of the system

RESULTS

The above-described model (Eqs. (1)–(13)) has been already used for simulation of Fe0 effects on the spectra of lunar soils (Starukhina et al. 1994) and terrestrial minerals (Starukhina and Shkuratov 1998). In our previous simulation of Lucey’s trends (Starukhina and Shkuratov 1999) only change of Fe0 concentration with maturity degree was taken into account. In this work we consider also the effect of the decrease of particle size. Decrease of particle size results in brightening of the surface that is opposite to the darkening due to reduced iron. General View of Maturation Trajectories Examples of maturation trajectories on A(0.75 µm)– C(0.95/0.75 µm) plane are shown in Fig. 2. Starting points of the curves are at the bottom of the plot where C(0.95/0.75 µm) < 1, which corresponds to immature soils. The curves start at different values of A(0.75 µm), i.e., at different Fe2+

τ (A1 ) = [τ (A0 ) + 4π1c · l0 κ Fe (λb )/λb ] · l1 /l0 , (14a) τ (A1 · C1 ) = [τ (A0 · C0 ) + 4π1c · l0 κ Fe (λr )/λr )] · l1 /l0 . (14b) The solution to the system (14a, b) is 1c(A0 , C0 , A1 , C1 , l0 ) =

τ (A0 C0 )τ (A1 ) − τ (A0 )τ (A1 C1 ) 1 , κ Fe (λb ) κ Fe (λr ) 4π · l0 τ (A1 C1 ) − τ (A1 ) λb λr

(15)

where the optical density τ as a function of albedo can be calculated with formula (11). The system (14a, b) can also be solved for the size ratio l1 /l0 l1 = l0

κ Fe (λb ) κ Fe (λr ) − τ (A1 ) λb λr . κ Fe (λb ) κ Fe (λr ) τ (A0 C0 ) − τ (A0 ) λb λr τ (A1 C1 )

(16)

FIG. 2. Maturation trajectories on A(0.75 µm)–C(0.95/0.75 µm) plot. Dotted lines are simulation results of the increase of Fe0 content without changing particle size. Solid lines correspond to dl/dc = −2.2 · 105 µm (see formula (18)). Approximations of tangent lines are also shown (dashed lines).

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abundance (which is not determined here). The initial particle size l0 = 100 µm is rather typical for immature lunar soils (McKay et al. 1991). In first approximation, we consider the accompanied decrease of l with increasing c as a linear function l(c) = l0 + (dl/dc) · c.

(18)

The order of magnitude of |dl/dc| can be estimated as l0 /cmax ≈ 100 µm/10−3 = 105 µm. For the solid curves shown in Fig. 2 we took dl/dc = −2.2 · 105 µm. Their high-albedo ends correspond to particle sizes 18 µm for curve 1 and 23 µm for curves 2 and 3. Difference in Fe0 volume fractions c between the ends of the curves is 3.75 · 10−4 for curve 1 and 3.5 · 10−4 for curves 2 and 3, that is about 1/3 of the total range of c(cmax ≈ 10−3 ) for fine-grained Fe0 (Morris 1980). The maturation trends are quasi-linear in their initial (low color index) parts, with almost the same slope and direction as observed by Lucey et al. (1995, 1998a). Linear extrapolations of the initial parts are also given in Fig. 2 by dashed lines. The values of dl/dc were chosen so that the these tangent (more exactly, secant) lines can be focused on a small area. Thus, the assumption of Lucey et al. (1995, 1998a) about the common origin of maturation trends is valid for some sets of data. On the other hand, if other values of dl/dc are taken or dl/dc is considered as a function of regolith composition, a system of maturation trends without a focusing area is obtained. The existence and the position of common origin strongly depends on the points where tangent or secant lines are drawn; i.e., there is no unique point which could be the common origin for all sets of maturation trends. Most of the trends can be focused, but the intersection area is usually localized beyond the domain obtained by Lucey et al. (1995, 1998a). To illustrate this, we simulate variations of Fe0 content without changing particle size (dotted lines in Fig. 2). For this limit case the maturation trends are almost parallel in their initial part. Another limit case is shown in Fig. 3: the parameter l varies without changing Fe0 content. Figure 3 presents a system of 1c isolines and can be used to determine the difference in Fe0 content between any two points of A–C plane (the volume fraction 1c of Fe0 is related to its weight percentage by 1cwt.% = 102 · 1c · ρFe /ρreg ≈ 285 · 1c, where ρFe and ρreg are the densities of iron and regolith material, respectively). The coordinates of the reference point (see Eq. (15)) are A0 = 0.05 and C0 = 1, where l0 = 100 µm; particle sizes decrease by a factor of 5 from the left to the right ends of the curves. Our model predicts also that maturation can change its direction in high-color-index parts of maturation trajectories (see solid lines in Fig. 2). These branches are just due to the particle size effect accompanying maturation. It is interesting to note that for curve 3 in Fig. 2 both tangent lines corresponding to low-color-index and high-color-index branches are focused on a common point. In case of dl/dc = 0, the slope inversion of maturation curves takes place too (see dotted lines in Fig. 2). After reaching the maxima at c = 5 · 10−4 , the curves

FIG. 3. Lines of equal volume fractions of the iron reduced in maturation on A(0.75 µm)–C(0.95/0.75 µm) plane. The figures on the lines indicate the difference in c (multiplied by 1000) between the points of the A–C plane and the point with coordinatesA = 0.05, C = 1.

turn down to the low-albedo part of the diagram and at c > 10−3 approach the values of C(0.95/0.75 µm) = 1 and A(0.75 µm) = 0.02. According to Eq. (15), the values of 1c on isolines in Fig. 3 depend on a choice of l0 defined above as the particle size. In fact, it should be less than particle size, because, strictly speaking, it is the distance between two successive scatterings. As l0 is known within the order of magnitude, scaling laws valid for all calculated maturation trends should be pointed out. Since l(c) is approximately proportional to l0 and 1c ∝ 1/l0 , the derivative |dl/dc| is proportional to (l0 )2 . So the same set of maturation trends is obtained at another value l1 of the initial size, if dl/dc is multiplied by (l1 /l0 )2 . Lunar Samples To test our model, we used spectral data for particle size separates of lunar soils in the visible and NIR ranges (Pieters et al. 1993). The measurements have shown that the size fractions differ in their effect on the spectra of bulk samples. Fine fractions with particle size <45 µm have spectra most close to those of the bulk samples. Owing to their higher surface/volume ratio, the fine separates (<45 µm) are more sensitive to exposure to space environment, than the separates with larger particle size. The increase of the red slopes of the spectra with decrease of particle size proves increasing content of reduced iron, so the data of Pieters et al. (1993) illustrate both increase of Fe0 content and decrease of particle size. The data are plotted with symbols on A(0.75 µm)–C(0.95/0.75 µm) diagram in Fig. 4. Note that the sign of the slopes of the trends differs from that observed by Lucey et al. (1995, 1998a). The points corresponding to each sample can be connected by a simulated curve with reasonable parameters. Using our model we calculated the values of κ at 0.75 µm and 0.95 µm for the

LUNAR OPTICAL MATURATION MODEL

279

Lucey et al. (1998a) in a recalibrated form (see Fig. 1), changing A(0.75 µm) and A(0.95 µm) according to the equations (Shkuratov et al. 2000) A(0.75 µm)new = {[b02 − 4a0 · (q0 − A(0.75 µm)Lucey )]1/2 − b0 }/2a0 , (19a) A(0.95 µm)new = {[b12 − 4a1 · (q1 − A(0.95 µm)Lucey ]1/2) − b1 }/2a1 , (19b)

FIG. 4. The diagram A(0.75 µm)–C(0.95/0.75 µm) for particle size separates of the lunar samples measured by Pieters et al. (1993) (points) and its simulation (curves).

fractions corresponding to the lower symbols on A(0.75 µm)– C(0.95/0.75 µm) plots in Fig. 4. Then these values were changed by formula (12) to take the increase of Fe0 volume fraction into account. The obtained κeff were used to calculate new values of A(0.95 µm) and A(0.75 µm). Decreasing l together with 4 increasing c as l(c) = l0 (1 − a)4·10 c , where a = 0.19, 0.0275, 0.125, 0.054, and 0.06 for Apollo 16 and 11 and Luna 20, 24, and 16, respectively, we obtained fitting curves shown in Fig. 4. Lower points of each trend in Fig. 4 correspond to coarse fractions 45–94 µm for Luna 16, Luna 20, and Luna 24 soils, to the fraction 25–45 µm for Apollo 11, and 75–250 µm for Apollo 16 samples. For calculations the initial particle sizes of 90, 80, 70, 35, and 80 µm, respectively, were taken. Particle sizes for the upper points of the trends (fines with spectra close to the bulk samples) are <45 µm for Luna 16, Luna 20, and Luna 24 soils, <25 µm for Apollo 11, and 20–45 µm for Apollo 16 soils. On the calculated curves these points correspond to sizes 43, 30, 32, 22, and 35 µm, respectively. Differences in Fe0 volume concentrations 1c between the lower and the upper points are 3.2 · 10−4 (Luna 16), 2 · 10−4 (Luna 20), 3.8 · 10−4 (Luna 24), 4.4 · 10−4 (Apollo 11), and 1.1 · 10−4 (Apollo 16). These are reasonable values. The values of 1c and final sizes are in accordance with Eqs. (15) and (16). Simulation of Lucey’s Plot for Lunar Sites With the procedure described above we tried to reproduce the A(0.75 µm)–C(0.95/0.75 µm) the diagram obtained by Lucey et al. (1998a) using Clementine data for small craters and their mature surroundings near six sample-returned lunar sites. Recently, it was found that there is a significant difference between photometric scales of the Clementine and Earth-based observations of the Moon (see, e.g., Shkuratov et al. 2000). Clementine albedos are systematically higher (by a factor of about 2.5) than telescope determinations. Therefore, we used the diagram of

where a0 = 1.84, b0 = 1.03, q0 = 0.01, a1 = 1.94, b1 = 0.962, and q1 = 0.016. This recalibration decreases the values of Clementine reflectance at 0.75 and 0.95 µm approximately by the same factor, so that the color remains almost unchanged. To reproduce the diagram, we took the initial values of A(0.75 µm) and A(0.95 µm), corrected by Eqs. (19a, b) for the “immature” low-color ends of the trends by Lucey et al. (1998a), and simulated the addition of fine-grained Fe0 in regolith particles together with a decrease of particle size. The 4 trends were reproduced, at l(c) = l0 (1 − a)4·10 c , where a = 0.034, 0.033, 0.02, 0.024, 0, and 0 for Apollo 12, Apollo 11, Luna 16, Apollo 14, Luna 20, and Apollo 16 landing sites, respectively. As most of the trends start at C(0.95/0.75 µm) > 1, which corresponds to rather mature soils, the value l0 = 70 µm was taken for the initial particle size. In Fig. 1 Lucey’s trends for different lunar sites are shown by points, and their simulations are shown by lines. Changes in Fe0 volume concentrations between the ends of Lucey’s trends are 2 · 10−4 (Apollo 12), 2.86 · 10−4 (Apollo 11), 1.65 · 10−4 (Luna 16), 1.48 · 10−4 (Apollo 14), 8.5 · 10−5 (Luna 20), and 7 · 10−5 (Apollo 16), which agrees with formula (15). The values of the final particle sizes for Lucey’s trends turned out to be 53 µm (Apollo 12), 48 µm (Apollo 11), 60 µm (Luna 16 and Apollo 14), and 70 µm (Luna 20 and Apollo 16). Constant particle size for the Apollo 16 trend may be due to saturation of the parameter l for soils of high maturity (average Is /FeO for Apollo 16 samples is about 83). Constant l for the Luna 20 trend may result from different chemical composition for some points within the observed assemblage. Besides, if chemical composition for some points on the Luna 20 and Apollo 16 trends is the same, there is a possibility that some parts of the points for the two highland sites correspond to a common trend. The trajectory from Luna 20 to Apollo 16 data is in accordance with the maturity degrees for these landing sites: Is /FeO for Apollo 16 is twice as great as that for Luna 20 (Stakheev and Lavrukhina 1979). In Fig. 1 a trajectory from Luna 20 to Apollo 16 set of data is shown (cf. curve 3 in Fig. 2). For this trajectory, l(c) = l0 − 5 · 10−4 c − 3 · 109 c2 (l(c) and l0 are in microns). Such an exotic trend meets the conditions (17); i.e., dl/dc < 0 in each point of the curve. It shows that, in principle, what is thought to be different maturation trends can be different branches of the same trend. Only the data on chemical composition can prove the trend type. For example, different FeO and TiO2 content in Luna 20 and Apollo 16 soils

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makes us ascribe the points for proper sites to different trends, but different trends do not prove different chemical composition. However, such a deep bend on maturation trajectories are rather limit case. Simulation of IR Albedo-Color Diagram of the Moon IR albedo-color diagrams of the Moon can be analyzed in terms of maturation trends. We use here the A(0.75 µm)– C(0.95/0.75 µm) diagram for the lunar nearside published in (Shkuratov et al. 1999a). The lower edge of the diagram (see Fig. 5) with the least values of C(0.95/0.75 µm) corresponding to the least mature soils was chosen as an initial line of maturation trajectories. Starting from the line C(0.95/0.75 µm) = 0.96, we calculated the trajectories. The albedo A(0.75 µm) of the initial points for the curves shown in Fig. 5 is 0.043 + 0.02 · j, where j is a curve number from left to right. The last two curves were cut from bottom and top as there are not enough points in those regions of the lunar diagram. For the size of immature particles (starting points of the curves) l0 = 100 µm was taken. To fit the directions of most elongated clusters of the lunar diagram, l should decrease with increasing c. For Fig. 5 the function l(c) = l0 − 1.6 · 105 · c · (0.95 + 0.05 · j) was used, where j is the curve number counted from left to right of the diagram. Note again that l and c change within reasonable limits. The difference in c between the top and bottom edges of the lunar diagram is from 3 · 10−4 to 4 · 10−4 . It corresponds to about 1/3 of the total range of the values of c for fine-grained Fe0

measured by Morris (1980). This 1/3 is not too low, because in the upper regolith layer of a thickness of about 1 mm, which contributes to light scattering, completely immature soils can hardly be expected, so the c range can be less than the total one. Another interesting feature of the simulated A(0.75 µm)– C(0.95/0.75 µm) diagram is the existence of natural top edge. This edge is formed by an envelope line of the maturation trajectories. The top edge is reached at particle sizes from 40 µm (left corner) to 60 µm (right corner), i.e., at values typical for mature soils (McKay et al. 1991). Note again that the opposite directions of maturation trends are not typical; they just can occur in case of wider (but still possible) ranges of parameters. CONCLUSIONS

1. Quasi-linear maturation trends for lunar soils on the A(0.75 µm)–C(0.95/0.75 µm) plot are verified theoretically for small changes of the maturation parameters (volume fraction c of Fe0 and particle size l). 2. For greater but still possible changes of the maturation parameters, beside the branches where A(0.75 µm) decreases and C(0.95/0.75 µm) increases in maturation, the branches with the opposite behavior of C(0.95/0.75 µm) and A(0.75 µm) were obtained. 3. The maturation trends on A(0.75 µm)–C(0.95/0.75 µm) diagrams cannot be extrapolated to a unique origin common for all sets of trends. However, for most cases there is an area where the trends can be focused. Thus, the assumption of Lucey et al. (1995, 1998a) about the common origin of maturation trends is confirmed only in a first approximation. 4. To reproduce the spectral data on lunar sites (Lucey et al. 1998a) and samples (Pieters et al. 1993), besides from the increase of Fe0 content, the decrease of particle size in maturation should be taken into account. 5. Elongated clusters on the telescope lunar A(0.75 µm)– C(0.95/0.75 µm) diagram (Shkuratov et al. 1999a) can be described as maturation trajectories for soils with different Fe2+ abundance; on each trajectory Fe0 content increases together with decrease of particle size of the soil. 6. The upper edge of the telescope lunar A(0.75 µm)–C(0.95/ 0.75 µm) diagram, i.e., the greatest color value, can be controlled by processes of particle size decrease in maturation. ACKNOWLEDGMENTS We thank Prof. Paul G. Lucey for very fruitful remarks. We are grateful to V. G. Kaydash and M. A. Kreslavsky for help in manuscript preparation.

FIG. 5. The diagram A(0.75 µm)–C(0.95/0.75 µm) for the lunar nearside of Shkuratov et al. (1999a) and its simulation. Darker shades denote higher frequency of the corresponding A–C combinations. The initial particle size corresponding to the bottom ends of the curves is 100 µm. The top edge of the diagram corresponds to particle sizes from 40 µm (left corner) to 60 µm (right corner).

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