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other direction is perpendicular to the line of sight at the model's center, about which a rotation causes a change in aspect (the angle between the rotation axis and the line of sight). The light source can be moved horizontally to change the phase. For each of the models, up to 27 lightcurves were produced by varying the aspect (90°, 60°, 35°), the obliquity (90°, 50°, 15°) and the phase (20°, 40°, 60°). The average probable error estimate of all angle measurements is +1°. Figure 3 illustrates all the lightcurves obtained from a smooth-surfaced, long, cylindrical model with rounded ends. One end and part of one side were artificially darkened with graphite powder to produce the apparent reflectivity differences between primary and secondary features seen in the Geographos lightcurves (Dunlap and Gehrels, 1971). Ignoring these differences, the lightcurves illustrate in general the effects of changing the aspect, phase, and obliquity. Several characteristics of the lightcurves can be identified that are used later in making comparisons of models:
(1) Amplitude: the height of the curve from minimum to maximum (The estimated probable error of the amplitudes is +0.01 mag.)
(2) Shape of minima: sharp, flat, and/or asymmetric
(3) Width of minima at half amplitude
90° I 6O" I 35°
Figure 3.—Model lightcurves in a 3 by 3 by 3 matrix. The model was a cylinder with hemispherical ends having one end darkened and an overall length-to-width ratio of about 4. The lightcurve at 90° obliquity, 90° aspect, and 20° phase (top left) has an amplitude of 2.12 mag.
(4) Time shifts of maxima or minima relative to the observation at 90° aspect,90° obliquity, 20° phase (5) Lightcurve inversions: maxima become minima and vice versa (time shift is 90°) (6) Primary and secondary maxima and minima Looking horizontally from left to right in figure 3, one sees the changes produced by decreasing the aspect; most noticeable is the decrease in amplitude and the time shifts (leading to two lightcurve inversions and two partial inversions at the top right of the figure). The inversions are understood roughly as occurring when the illuminated part of the “true” maxima has a smaller area (as seen by the detector) than the illuminated part of the “true” minima. Looking vertically, one sees sometimes a noticeable change in amplitude with obliquity and sometimes changes in asymmetry. Looking diagonally (in groups of three), one sees the changes due to phase—usually small changes in amplitude with some asymmetries and time shifts.
Figure 4 is the set of nine amplitude-aspect curves for the lightcurves from figure 3 (using secondary amplitudes to avoid reflectivity effects). The turnup in the curves at 90° obliquity and 40° and 60° phase is associated with
Figure 4.—Amplitude-aspect curves obtained from the secondary amplitudes of figure 3.
lightcurve inversions. (See fig. 3). Curves for the other models are similar but not exactly the same as these. It is clear, however, that there is no unique amplitude-aspect function for this or any of the models studied. Therefore, it is not possible, in general, to determine a rotation axis precisely by using a single amplitude-aspect function. Of course, approximations can be made; and they may be better if the phase angles are always small. However, the amplitudeaspect function is model dependent in an as-yet-unknown way.
COMPARISONS OF MODELS
Table I is a brief summary of the results of five comparisons of models. To see how differences in the shape affect the observed light variation, each model was compared with one having a different shape; finally, the lightcurves made at the same orientations were examined for differences in the characteristics described earlier. The changes in the light variation usually depend not only on the shape of the model, but also on aspect, obliquity, and phase. We cannot, for example, look at a single asteroid lightcurve and deduce the shape of the asteroid. Therefore, before comparisons can be made with actual observations, the orientation of the rotation axis in space must be known precisely (~ + 1"). Probably the weakest point in our present method for obtaining the rotation axis (see Taylor”) is in accounting for differential time shifts in the maxima (or minima) that depend on aspect, obliquity, and phase. It may be possible to utilize the time shifts from the models to improve our determination of rotation axes. We notice also in table I that the presence of a third body axis is clearly evident in the change in brightness of the maxima as the aspect changes.
COMPARISON WITH TELESCOPIC OBSERVATIONS
Figure 5 shows the August 31, 1969, lightcurve of Geographos and also the average of the two model lightcurves from figure 3 that are closest to the calculated orientation of Geographos if its pole is at No = 113°, 60 = 84°.
In the laboratory we were modeling direct rotation, but Geographos' rotation is retrograde. Making the necessary corrections to the lightcurve and moving the dark side to again follow the dark end, the model will reproduce the asymmetries in the minima; but the difference in the widths of the minima is still unexplained. We are currently developing a computer method of using the model data along with observed amplitudes to determine a pole, but results are not yet available. Other asteroids with large amplitudes (433 Eros, 624 Hektor, and possibly 15 Eunomia, 39 Laetitia, and 44 Nysa) might also be used for comparisons.
The extreme smoothness (<0.004 mag) of all the model lightcurves is not usually seen in asteroid lightcurves, although asteroids with large light
TABLE I.—Comparisons of Models
Model Observed changes in light variation relative to the reference model” compared with referencea Amplitude Shape of minima Time shifts Other . Same shape; cross- Larger by up to a factor Narrower, sharper, more Some larger shifts in Lightcurve inversion is sectional area ratio of 2 asymmetry maxima possible at smaller phase
. One end pointed (conical); some brighter spots; 5 percent smaller area ratio . Elongation along a second body axis 0.7 times the major axis; 3 percent larger area ratio at 90° aspect
Primary and secondary
Smaller, except at 90° aspect
2.3 times larger . Two tangent spheres of equal radii; 5 percent larger area ratio . Both ends pointed (conical); ~ 5 percent larger area ratio
Complicated changes with
Some smaller shifts in max-
No significant change
Occasionally the minima
*The reference model is a cylinder with hemispherical ends and a cross-sectional area ratio Amax/Amin of 1.9. ese changes usually depend on the aspect, phase, and obliquity and therefore do not characterize all the lightcurves.
Figure 5.-Comparison of model to telescopic observations: • Geographos, August 31, 1969; o model whose lightcurves are in figure 3.
variations appear to have somewhat smoother lightcurves. Two models with rougher surfaces (about 5 and 20 percent deviations from an average dimension) were also observed. The roughest model produced the only deviation: a smooth tertiary hump seen in certain orientations near the minimum of light. A few lightcurves were also made using a very irregularly shaped and somewhat porous rock, and they show several small features with some indication of deviations from smoothness. More work needs to be done in modeling the surface texture, which is apparently the major source of small features in the lightcurves.
There is no unique amplitude-aspect function for any of the models studied, and this method should not be used for precise determinations of rotation axes. The differences in the amplitude-aspect function as well as in other characteristics of the lightcurves are partly due to the shape, but the orientation of the rotation axis must be precisely known—by photometric astrometry (see Taylor”)—before the shape of an asteroid may be confidently determined.
This work is supported by a grant from the National Geographic Society. I am also indebted to several of my colleagues for their help with the observations and especially to M. Howes who has helped extensively with the reductions.