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Finally, I would like to stress again that typical asteroid phase coefficients (0.025 to 0.035 mag/deg) cannot be interpreted unambiguously. This is because the observed phase coefficient may depend as much on the photometric properties of an individual surface element f(o, D) as on the degree of large-scale roughness 2(o, Q). (See preceding discussions of these functions.) If only disk integrated measurements of the scattered light are available, these two effects cannot be separated. In spite of this, there does seem to be some point in looking for objects with unusual phase coefficients, such as Ceres.
In some cases, it should be possible to estimate the relative surface roughness of two quasi-spherical asteroids by combining photometric and polarimetric observations. For example, if the two asteroids have almost identical polarization curves, but quite different phase coefficients, it is likely that the asteroid with the larger phase coefficient has a macroscopically rougher surface.
I wish to thank C. Sagan and F. L. Whipple for their advice and generous assistance. The numerical calculations described in this paper were carried out at the Smithsonian Astrophysical Observatory, and I am grateful to the Smithsonian Foundation for support during this phase of the project. This work was also supported in part by NASA NGR 33-010-082.
Ahmad, I. I. 1954, Photometric Studies of Asteroids. IV. The Light Curves of Ceres, Hebe,
HAPKE: It is very likely true, as Veverka has stated, that one could not make a phase function as steep as Ceres' with lunar soil roughened in some fashion. However, the following conceptual model shows that it is possible to construct a surface with a phase function as steep as one wishes. Imagine a body with a relatively smooth surface; such a body would have a phase function something like that corresponding to Lambert's law. Now replace the smooth surface by large dark particles. Since the phase function of the body is given by the product of the phase function of the particles and their shadowing function, the body phase function will be steepened. Next, replace the particles by clumps of smaller particles. Now the phase function of the body involves a product: the phase function of the smaller particles times that of the clumps times the shadowing function. The body phase function is even steeper. This procedure can be followed until the diffraction limit for casting sharp shadows is reached. Such a hierarchy of clumps of material similar to lunar soil may not be a completely unreasonable model in view of the very low surface gravity on asteroids.
ASTEROID POLARIMETRY: A PROGRESS REPORT
This is a brief report on the present status of asteroid polarimetry. A detailed paper is in preparation jointly with T. Gehrels. The first polarization measurements of asteroids were made by Lyot (1934), who photographically determined the polarization curves of Ceres and Vesta. These curves are reproduced by Dollfus' (1961). Unfortunately, because of the low sensitivity of the photographic method, they do not agree very well with recent photoelectric measurements. The first photoelectric polarization measurements of asteroids were made by Provin (1955) (details of this work are given by Dollfus," 1961), and in recent years this work has been extended by Gehrels (unpublished) and by Veverka (1970). To date, fairly complete polarization curves for about a dozen asteroids have been obtained, and at least partial data are available for twice that number. Invariably, all asteroids for which sufficient data have been accumulated show lunarlike polarization curves with well-developed negative branches. This suggests that asteroid surfaces generally consist of a microscopically intricate, porous material in which multiple scattering is not dominant (Lyot, 1929). But, although these curves have similar shapes, they can differ considerably in detail (fig. 1). It is clear from figure 1 that, for example, the polarization curves of Ceres and Vesta are significantly different from each other, and from that of the Moon. It is therefore reasonable to suppose that some information about the composition and texture of asteroid surfaces can be obtained by matching these curves in the laboratory. Such work is now in progress using powdered meteorite surfaces. Asteroid polarization curves are also phenomenologically useful because from them one can estimate the absolute reflectivities of the surfaces. For dark, texturally complex surfaces, h, the slope of the linear part of the polarization curve beyond the inversion angle, seems to be inversely correlated with surface reflectivity. This relationship has been exploited by KenKnight et al. (1967), Widorn (1967), Gehrels et al. (1970), and others. Veverka (1971a) gives a short discussion of this method. Presently, the drawback is that this relationship has been calibrated adequately only for lunar regions and for
Figure 1.-A comparison of the polarization curves of the Moon and four asteroids. The lunar curve is from Lyot (1929). The asteroid curves are based on previous measurements as follows: Ceres and Pallas from Provin (1955; details given in Dollfus, 1961), Gehrels (unpublished), and Veverka (1970); Vesta from Gehrels (unpublished) and Veverka (1970); and Metis from Veverka (1970). The measurements by Gehrels were made either at 0.52 or 0.56 um; all others were made in integrated light. Note that from Earth few asteroids can be observed at phase angles larger than 30°.
possible lunar materials. But in view of the results of McCord, Adams, and Johnson? (1970) it may be more appropriate to calibrate the relationship by using pulverized meteorite surfaces. Nevertheless, the calibration now available is probably adequate to yield reasonable estimates of asteroid reflectivities. For example, one finds in this way 0.25 + 0.07 for the reflectivity of Vesta (Veverka, 1971a) and 0.16 + 0.03 for that of Flora (Veverka, 1971b), both in visible light. An interesting possibility being investigated in the laboratory is that for dark, microscopically intricate surfaces, the minimum of the negative branch of the polarization curve Pmin may be a crude indicator of surface reflectivity. Data for volcanic cinders and ashes by Lyot (1929) suggest that this may be the case (the deeper the negative branch, the lower the reflectivity). If such a relationship could be established, even approximately, it would provide a means of estimating the absolute reflectivity of an asteroid from a single measurement at a phase angle near 10°. This would be of special importance in the case of Trojan asteroids because they cannot be observed at sufficiently large phase angles to determine h, and there is at present no reliable way of estimating their reflectivities. (See, for example, Dunlap and Gehrels, 1969.) Furthermore, Trojans are faint and difficult to locate, making the determination of a complete polarization-phase curve a formidable task.
*See p. 51.
It has been suggested (Gehrels and Teska, 1963) that simultaneous polarization and brightness measurements can be used to decide conclusively to what extent the short-period brightness fluctuations of asteroids are due to changes in surface reflectivity, rather than to changes in the projected surface area. The idea is to observe at large phase angles where the polarization should be positive and inversely correlated to surface reflectivity (Gehrels, 1970). Such observations are difficult because at large phase angles asteroids are not well placed in the sky, and can be observed for only a few hours, making it impossible, in general, to follow a complete rotation during any one night. So far, no polarization-brightness variations related to rotation have been established, but only two asteroids have been observed simultaneously in brightness and in polarization: Pallas (Gehrels, unpublished) and Eunomia (Veverka, 1970).
I wish to thank T. Gehrels, W. Liller, C. Sagan, and F. L. Whipple for helpful discussions. Some of the observations reported in this paper were made at Harvard. The Harvard polarimeter project is supported by AFOAR contract no. F19–628-68-C-0228. This work was supported in part by NASA NGR 33-010-082.
Dollfus, A. 1961, Polarization Studies of Planets. Planets and Satellites (eds., G. P. Kuiper