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Figure 2. —The most probable albedo combinations for diffuse A and geometrical B reflectivities. Both scales have been normalized in terms of the average normal albedo of the asteroid. The broken line is the locus of points for ao = 0.5. The numbered curves enclose the allowed albedo combinations as defined by the number of Fourier terms used to approximate the observed lightcurve. (a) 4 Vesta, April 10, 1967. (b) 39 Laetitia, December 1955.
n = 3 and greater that we can verify the presence of diffusely or geometrically reflecting areas. The lightcurves used in the present analysis are those taken by Gehrels (1967) for 4 Vesta and by van Houten-Groeneveld and van Houten (1958) for 39 Laetitia. These are typical examples of single maximum and double maximum lightcurves that also satisfy the requirement of being observed within the equatorial plane of the asteroid. This is indeed true for 4 Vesta, which is observed in the equatorial plane, although for 39 Laetitia the inclination may be as large as 20°. Also, it should be noted that the phase angle for 4 Vesta was -18° and that of 39 Laetitia was about the same. Although the present calculations are specifically applicable for a zero phase angle, the derived albedo limits should not be significantly affected unless the form of the assumed reflectivity laws changes significantly with phase. The results of Fourier analyzing the 4 Vesta and 39 Laetitia lightcurves are summarized in table I. For convenience of comparison, the Fourier coefficients of the sin nqo and cos nqo terms have been combined and are normalized with respect to Co, the mean normal albedo of the asteroid. Also included is the root-mean-square difference between the observed points of the lightcurve and the nth Fourier representation. The difference between the single maximum and double maximum lightcurves is illustrated by the prominence of the even-order terms in the 39 Laetitia lightcurve and the odd-order terms in the 4 Vesta lightcurve. TABLE I.—Fourier Analysis of Lightcurves
4 Vesta 39 Laetitia n (C; + D})*sco rms/C0 (C; + D})*sco rms/C0 0. . . . . . . . . . . . • 1.000 0.042 1.000 0.139 ! . . . . . . . . . . . . - .056 .01.1 .025 .137 2 . . . . . . . . . . . . . .005 .01.1 .203 .022 3. . . . . . . . . . . . . .007 .009 .009 .020 4. . . . . . . . . . . . . .002 .009 .022 .016 5 . . . . . . . . . . . . . .005 .008 .012 .015 6 . . . . . . . . . . . . . .000 .008 .018 .010
Clearly, the even-order terms for 39 Laetitia may be interpreted as being due to the projected area of an irregular object. This leaves a 5 to 10 percent light variation due to the odd-order terms, which may be associated with either a spotted surface or Lambert law reflectivity. However, it appears likely that if 39 Laetitia is observed closer to its equatorial plane, the size of these terms will be somewhat reduced. For 4 Vesta, interpreting the 1.4 and 1.0 percent light variation contributed by the n = 3 and n = 5 terms as significant would indicate the definite presence of a Lambert law contribution to the surface reflectivity. The n = 5 term seems to suggest bright diffusely reflecting spots on a darker geometrically reflecting surface. However, the limited accuracy of such high-order terms does not necessarily preclude a model with dark spots on a bright geometrically
reflecting surface. REFERENCES
Gehrels, T. 1967, Minor Planets. I. The Rotation of Vesta. Astron. J. 72, 929–938.
VEVERKA: In your analysis you assume a surface composed of areas which scatter either “geometrically” (a cos ?) or “diffusely” (a cos” Y) according to Lambert's law. For asteroids, this is an invalid assumption. For intricate surfaces, in which multiple scattering is not dominant, the first part of the assumption is not too bad at small phase angles, but still, strictly speaking, you can only have “geometric” scattering at opposition and nowhere else. (See, for example, Irvine, 1966.)
However, you can never have Lambert scattering on such a surface at visible wavelengths. At visible wavelengths, even quasi-Lambert scatterers are rare and usually consist of extremely bright patches in which multiple scattering is dominant (for example, snow or MgO). There is no evidence that such areas occur in asteroids, and much evidence that they do not occur (for example, deep negative branches in the polarization curves).
You are therefore trying to force a fit using two generally inappropriate scattering laws, and a probably incorrect shape (a spherical asteroid). Thus the calculation, although interesting, cannot have much application to asteroids.
JOHNSON: The lack of color variation seems to indicate that in many cases the asteroid variation is due to shape rather than spots. Is it possible to make the model yield shape information as well as spot distribution?
LACIS: The even terms of the spot distribution function h(p) can be directly associated with the shape of the object. By assuming a constant albedo over the surface, a rough estimate for the shape is given by R(p) = 1 — h(p). For 39 Laetitia, this relation indicates an oblong object with a length-to-width ratio of approximately 3:2.
KUIPER: When we started our precision photometry of asteroids at McDonald about 1949, we found that the rule was to have two maxima and two minima in the lightcurve. It was concluded that the light variation was primarily due to shape. Variation in surface reflectivity could contribute something, but when the variation is 0.3 mag or more, the main effect must be due to shape.
LACIS: Inverting the lightcurve in terms of a spotted sphere gives us little more than a geometrical model that is capable of reproducing the observed light variation. However, in the case where the observations are made in the equatorial plane of the asteroid, we can infer the type of reflectivity law from the strength of the different Fourier terms present in the observed lightcurve. At opposition, the geometrical reflectivity is a special case of the Lommel-Seeliger law. The assumed Lambert law reflectivity could conceivably refer to a more specular type of reflection law. It is just that the presence of odd Fourier terms (n = 3, 5, ...) cannot be accounted for in terms of geometrical reflectivity alone.
Also, it may be of interest to note that there is a systematic increase in the odd Fourier terms and a decrease in the even-order terms as the observing point moves away from the equatorial plane of the asteroid. This may be helpful in locating the orientation of the rotational axis.
Irvine, W. M. 1966, The Shadow Effect in Diffuse Radiation. J. Geophys. Res. 71,2931. LABORATORY WORK ON THE SHAPES OF ASTEROIDS
J. L. DUWLAP
Photometric lightcurves of about 50 asteroids have been obtained over the past 20 yr, yet very little is known about the shape of these objects. Perhaps 100 lightcurves (including photographic ones) of 433 Eros have been obtained with amplitudes up to 1.5 mag. Some authors' have attempted to calculate the dimensions of Eros assuming it to be a three-axis ellipsoid. The most recent determination (35 km, 16 km, 7 km) was given by Roach and Stoddard in 1938. In the case of 624 Hektor, amplitudes up to 1.1 mag were observed on lightcurves in which the primary and secondary maxima differed by less than 0.04 mag. Van Houten (1963) noted that for lightcurve amplitudes greater than 0.2 mag, the two maxima were about the same level and differed by 0.04 mag on the average. This is an indication of the small effect of reflectivity differences between the opposite sides. Assuming then that the light variation of Hektor is due almost entirely to shape, Dunlap and Gehrels (1969) used a cylindrical model with rounded ends to calculate a length of 110 km and a diameter of 40 km. The nearly constant (+0.02 mag) absolute magnitude of the maxima ruled out a third axis being significantly different from the second. More recent lightcurves of 1620 Geographos have been obtained with amplitudes up to 2.0 mag (Dunlap and Gehrels, 1971). If all of this variation is caused by shape, Geographos might be nearly six times longer than wide! However, the 0.1 mag difference between maxima (and an even larger difference in the minima) suggests a possible reflectivity effect that appears to reduce the length-to-width ratio to about 4. It was decided to make a laboratory investigation of the lightcurves of models to clarify our understanding of light variations caused by shape and perhaps enable us to find a particular shape that would reproduce the observations of Geographos. The work is still in progress, but we already have obtained some interesting results.
PRODUCTION OF MODEL LIGHTCURVES
Figure 1 illustrates some of the first of 12 models that have been observed. Each model was made with a Styrofoam center covered with a thin layer of Plasticene and finally dusted with powdered rock. The model was turned about
its shortest body axis by a stepping motor, and integrations were usually made every 3° (or 5°) over 240° (or 360°) of rotation using a photometer as used at the telescopes (Coyne and Gehrels, 1967). Figure 2 defines the geometry of the observations. The model's rotation axis can be oriented in space around two perpendicular directions. One is the line of sight, a rotation about which causes a change in asterocentric obliquity.” The
Figure 2.-The geometry of the laboratory observations. (Note that obliquity as defined here is not the same as the obliquity defined in the glossary.)
*Throughout this paper, obliquity refers to the dihedral angle between the plane determined by the line of sight and the axis of rotation, and by the plane perpendicular to the scattering plane and containing the line of sight. (See fig. 2.)