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TABLE I.—Asteroid Characteristics–Concluded
B – V. U - B, Rotation Amplitude Number of Lightcurve
*The numbers are in brackets in the References.
his table II. The B- V and U- B columns are the colors at 5° phase; the period is synodic unless the sidereal period is known. Amplitude ranges and the number of oppositions observed are indicated. The hours column refers to the number of hours of good lightcurves obtained. Massalia was observed for the expressed purpose of determining magnitudephase relations at small phase angles, and the opposition effect was discovered: a sharp increase in brightness from 7° phase on toward 0° phase (Gehrels, 1956). Figure 6 illustrates. the phase relations along with the opposition effect as they appeared with Lydia (Taylor, Gehrels, and Silvester, 1971). With Lydia, when B and U were plotted as a function of phase, it appeared that the opposition effect was independent of wavelength. Also, the opposition effect for Massalia, Vesta, and Lydia appeared the same, as is illustrated in figure 7 (Taylor, Gehrels, and Silvester, 1971). Certain asteroids should perhaps be reevaluated in view of our present knowledge of the opposition effect. Three examples are as follows:
(1) The absolute magnitude and phase coefficient of 9 Metis was determined using five observations, four of which were under 6” phase (Groeneveld and Kuiper, 1954b).
(2) Observations of Iris near 23° phase yielded consistent V(1,0) values but a later run at 4° phase was 0.2 mag brighter than expected (van Houten-Groeneveld and van Houten, 1958).
(3) V(1,0) was found to be different by 0.01 mag for 12 Victoria before and after opposition. For Victoria it was assumed that the opposition effect started at phase angles less than 5°. In the linear plot after opposition, three of four data points lie in the region of 5°
Figure 6.—Phase functions of Lydia. Ordinates: top curve, the observed magnitudes (V on the UBV system) reduced to unit distances from the Sun and Earth; middle curve, the B – V colors; bottom curve, the U – B colors. Open circles are before opposition and filled circles are after opposition.
to 6° phase (Tempesti and Burchi, 1969). If the assumption is made that the opposition effect starts at 7° to 8° phase, I feel there may be a unique V(1,0).
SENSE OF ROTATION AND POLE DETERMINATIONS
To determine the sense of rotation of Eunomia, Groeneveld and Kuiper (1954a) assumed the ecliptic latitude of the pole to be at 90° and used the relation
where N is the number of cycles, P is the period, At, is time interval corrected for light time, AL is the difference in ecliptic longitudes, the plus sign is used for retrograde rotation, and the minus sign for direct rotation. A retrograde solution gave smaller residuals in P for the two intervals attempted. “Clearly . . . some uncertainty may be introduced . . . by the assumptions made concerning the position of the rotational axis” (Groeneveld and Kuiper, 1954a). The van Houtens with additional data and using the same method confirmed retrograde rotation; they present a concise and informative discussion of limitations in their “Concluding Remarks” (van HoutenGroeneveld and van Houten, 1958). The sense of rotation of Vesta was determined by using many intervals of both increasing and decreasing longitudes, and by assuming that the latitude of the pole was high (Gehrels, 1967a). Groeneveld and Kuiper (1954a) presented a pole determination method and claim 10 percent precision if optimum conditions are met: axis fixed in space and an observation at each of the stationary points of two successive oppositions. With 39 Laetitia, a formula was developed to adjust the period because of the relative motions of Earth and the asteroid. The formula depends on knowing the pole orientation. An amplitude-aspect relation derived by Stobbe (1940) and Beyer (1953) for Eros, scaled down, was applied to determine an estimate of the pole. The investigators admittedly had limited precision (van Houten-Groeneveld and van Houten, 1958). The poles of eight asteroids were calculated by combining two techniques: a sine relation between aspect and amplitudes, and a cosine relation between absolute magnitude changes with respect to aspect (Gehrels and Owings, 1962). It is not clearly established whether a sine relation is proper for those asteroids studied. With his work on Vesta, Gehrels developed what is now known as “photometric astrometry.” The method is basically the same as that used by Groeneveld and Kuiper for finding the sense of rotation of Eunomia. The main difference is that Gehrels did not restrict his analysis to a 90° orientation of the pole. He considered the asteroid-centric longitude changes between observations for various pole possibilities. Those differences were applied as corrections to the number of cycles for each interval. By attempting different orientations, he sought minimum residuals from the mean sidereal period of each trial. His method is “... independent of any assumptions regarding the shape of the asteroid and the relationship between amplitude and the aspect.” He also introduced a phase shift to correct for the displacement of the center of light on the apparent disk due to the effects of phase. Gehrels compared his data with earlier observations, over 20 000 cycles, to improve the precision (Gehrels, 1967a); that part of the analysis will, however, have to be redone, as is planned for a future paper, because the additional cycle correction for each orbital revolution was omitted. In the Hektor analysis (Dunlap and Gehrels, 1969), photometric astrometry was used, but it was difficult to determine the number of cycles. There were only a few observations over long intervals. As an aid, the relation AN = +N(Pyn - Pid)|Poid was used, where Pyn is the synodic period and Pid is the sidereal period. Figure 8 shows how the apparent number of cycles AN is changed as a function of longitude for four different pole orientations. That figure assumes the asteroid is on the ecliptic, as was the case with Hektor. Figure 8 could not be used for Icarus because the asteroid was not on the ecliptic. Figure 9 shows how the apparent number of cycles are affected if the asteroid is 20° above the ecliptic. The entire photometric astrometry routine, including the problems of cycles, was computerized before the Icarus analysis. The concept of light centers was also introduced: the center of the projection of the illuminated part of the disk, as seen from Earth, assuming uniform reflectivity and a spherical shape. The light center is on the great circle through the subsolar and sub-Earth points. The purpose for light centers is basically the same as for Gehrels’ phase shift (Gehrels et al., 1970). In conclusion, it is clearly seen that additional work is needed to improve the quality and the extent of the sample in table I. I feel that high priority should be given to improving pole determinations. For this purpose, high