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found by calculating the mean specific intensity of the light scattered from a paraboloidal crater (see Hämeen-Anttila et al., 1965, for details), each element of which scatters according to Irvine's law.

Clearly, the j(a) calculated in this way for a surface with Q X 0 will be less than that found when Q = 0 at all phase angles a > 0. We will, in fact, have the following relationship:

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where 2(q, Q) is a macroscopic shadowing function that depends only on a and Q (and, of course, on the model of surface roughness) but not on f(o, D). Thus, the effective scattering law for the rough model planet may be

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where I(i, e, a) is given by equation (1) and 2(q, Q) can be determined by the calculation described above. For a macroscopically smooth planet (Q = 0), X(or, 0) = 1 for all o and

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The values of 2(q, Q) for this model, found using either a 36 or 100 point grid over the illuminated part of the planet and a 2500 point grid over each crater, are shown in figure 3. The numerical accuracy of these values is better than 1 percent. The results of figure 3 can now be used to study the effects of large-scale surface roughness on the photometric parameters of the model planet once f(o, D) is specified. Because the f(o, D) shown in figure 1 is very

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Figure 3.—The macroscopic shadowing function X(Q) versus phase angle o for various values of Q. The nature of X depends on the specific model of large-scale surface roughness used (in this case the surface is assumed to be covered with paraboloidal craters), but is independent of f(o, D). Note that beyond Q = 2, increasing the surface roughness produces little change in E. Values of E(Q) were calculated for o = 0°, 10°, 20°, 50°, 90°, 130°, and 170°; for all Q - 0, x (170°, Q) was found to be less than 0.001.


similar to that of the Moon, it is of interest to use it in these calculations. For this purpose, it may be extended linearly (on a magnitude scale) from a = 10° to a = 0°; that is, at 0.026 mag/deg, thus in effect neglecting any opposition effect. Values of the phase coefficient 3 (between o' = 10° and o. = 30°) and of

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for this model planet are shown in figure 4 as functions of the roughness parameter Q. The phase coefficient is seen to increase significantly as the surface gets rougher until about Q = 2; for larger values of Q the additional increase in 3 is slight. The phase integral, on the other hand, decreases appreciably as Q increases, but again a leveling off occurs beyond Q = 2. Note that the phase coefficient 3 of the disk integrated light is related to the laboratory phase coefficient slab, the slope of f(x, D) (on a magnitude scale), by the relation

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where 61s is the phase coefficient of a Lommel-Seeliger planet (that is, a planet with Q = 0 and f(o, D) = 1). Between a = 10° and a = 30°, 6is = 0.006

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Figure 4.—(a) The phase integral q of the model planet versus the surface roughness (represented by the parameter Q). The fifunction shown in figure 1 extrapolated to a = 0° as described in the text was used in this calculation. (b) The corresponding variation of the phase coefficient 3 measured between a = 10° and o = 30°.

mag/deg, Hence, because over the same interval of phase angles, slab = 0.026 mag/deg for the surface of figure 1, 3 = 0.032 mag/deg for Q = 0 in figure 3. For a scattering law of type (1), the geometric albedo p of the model planet is independent of Q. Thus, for the above model, it can be concluded that—

(1) Large-scale surface roughness has a strong effect on both the phase integral and the phase coefficient, but none on the geometric albedo.

(2) From equation (1) it follows that the phase coefficient is independent of the single particle albedo CŞ0, but the geometric albedo is not.

(3) Therefore, in view of conclusions (1) and (2), there can be, in general, no correlation between 3 and p.

(4) Within the framework of this model, 3 and q are independent of wavelength, unless p(a) has a wavelength dependence. But because it is assumed that the particles of the model surface are opaque and large compared to the wavelength, the wavelength dependence of q}(o) will be small.


Laboratory work with dark, microscopically complex surfaces (Halajian, 1965; Halajian and Spagnolo, 1966) is in accord with these conclusions. Even in the laboratory, when macroscopic shadowing is not important, no general correlation between glab and the surface reflectivity is found. Also, the observed wavelength dependence of slab is very small, but there is an interesting trend for glab to decrease slightly with increasing wavelength. Because the reflectivity of the samples used in this work tends to increase slightly with increasing wavelength, this suggests that the breakdown of the Irvine model is at least in part due to the increased importance of multiple scattering at longer wavelengths. Multiple scattering makes it easier for light to escape from the surface; this effect is relatively more important at large phase angles because it is then more difficult for singly scattered photons to escape from within the surface. Thus multiple scattering helps to get relatively more light out of the surface at large phase angles than near opposition. This tends to make phase coefficients smallest at those wavelengths at which multiple scattering is most important; that is, usually in the red portion of the spectrum. But for dark surfaces this effect is very small.

Laboratory work such as that referred to above (Halajian, 1965; Halajian and Spagnolo, 1966) shows conclusively that no mineralogical information is contained in phase coefficients; at best one can distinguish materials in which multiple scattering is dominant from those in which it is negligible. In addition, this work shows that away from opposition (o: X 10°) phase coefficients contain no information about whether a surface is particulate. For example, as already noted, both particulate samples of volcanic cinders and solid samples of furnace slag reproduce the lunar photometric function in V equally well as phase angles larger than a few degrees.


A serious complication in interpreting phase coefficients is that many asteroids are not even approximately spherical. What can be said about the brightness variations with phase of an irregular asteroid whose aspect changes with time? Clearly, as the aspect changes, so will the importance of large-scale shadowing.

Consider the following idealized example of an ellipsoidal asteroid. Two of the semiaxes are equal to A, and the third is equal to B > A. The asteroid rotates about one of the short axes. Two extreme cases may occur: (1) the asteroid is viewed pole-on and the light fluctuations are minimum and (2) the rotation axis of the asteroid is perpendicular to the line of sight and the light variations are maximum. Also, suppose that a spherical planet of the same material and surface macrostructure has a phase coefficient 3,phere.

In case (2), at maximum light, the situation is identical to case (1) and

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The inequality follows from the fact that on the ellipsoid, at maximum light, the average i and e are effectively smaller than on the sphere, and the effects of shadowing are therefore less important. However, at minimum light, the average i and e are effectively larger than in the case of a sphere and therefore shadowing is more important. Hence

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Usually, in case (2), 3 would be determined by using the mean magnitude of the lightcurve, so that

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Therefore, it is possible to predict that for an irregular asteroid whose aspect changes with time and whose surface is macroscopically rough:

(1) The apparent 3 is largest when the amplitude of the lightcurve is
(2) If the aspect of an asteroid stays approximately constant during an
opposition, then the phase coefficient determined from the minima
of the lightcurve should be larger than that determined from the
maxima; that is
Smin > Pmax

Thus, to even meaningfully define a phase coefficient for a very irregular asteroid whose aspect changes significantly with time may require a long series of accurate observations. CONCLUSIONS

In summary, the situation appears bleak. One cannot expect to derive the geometric albedos of asteroids from their phase coefficients. The contrary claim by Widorn (1964) and others is largely based on a fortuitous empirical relationship obtained by plotting 3 against p for the Moon and some of the large planets. Jupiter and Venus are intrinsically bright (large p) and have cloud decks in which multiple scattering is important (low 6). Mercury and the Moon are intrinsically dark (low p) and have rough dark surfaces (high 6). Thus one can arrive at the unfounded conclusion that 3 must always be inversely correlated with p, which in the case of dark surfaces certainly need not be true. Because the degree of surface roughness (Q in the above model) of any particular asteroid is not known, one cannot convert an observed phase coefficient 3 in its laboratory counterpart slab. Furthermore, even if this were possible, little diagnostic information could be obtained from Blab. (See the previous discussion of slab.) In addition, for very irregular asteroids with rough surfaces it may be difficult to even define a meaningful phase coefficient (as discussed in the preceding section). Fortunately, there are some asteroids, Ceres and Flora, for example, that are almost spherical, so that at least this complication does not arise. Flora has a phase coefficient similar to that of the Moon: 0.028 mag/deg in V (Veverka, 1971). If it is composed of photometrically similar material, its surface roughness must also be similar. If it is rougher than the Moon, its surface material must be less backscattering, and vice versa. The phase coefficient of Ceres, 0.050 mag/deg in V (Ahmad, 1954), is considerably larger than that of the Moon. This is probably not entirely due to surface roughness because, as figure 4(b) shows, for lunarlike materials it is difficult to increase 3 beyond 0.05 mag/deg by increasing surface roughness. This suggests that the surface material of Ceres is intrinsically more backscattering than that of the Moon. According to the above model the color dependence of asteroid phase coefficients should be small. This does seem to be the case. For Vesta, for example, 3, −0.0253 mag/deg, 3B = 0.0264 mag/deg, and 3U = 0.0291 mag/deg (Gehrels, 1967). Because the reflectivity of Vesta increases with increasing wavelength in the UBV region of the spectrum, this slight decrease in 3 with increasing wavelength may perhaps be attributed to the increased importance of the small, multiply scattered component at long wavelengths, as suggested above. If this is true, then the wavelength dependence of asteroid phase coefficients mostly contains information about the wavelength dependence of the surface reflectivity, information that can be obtained more easily from a single spectral reflectivity measurement.

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