Asteroid Lightcurve Observations from 1981 to 1983

Icarus 142, 173–201 (1999) Article ID icar.1999.6181, available online at http://www.idealibrary.com on

Asteroid Lightcurve Observations from 1981 to 1983 A. W. Harris and J. W. Young Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 E-mail: [email protected]

E. Bowell Lowell Observatory, Flagstaff, Arizona 86001

and D. J. Tholen Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822 Received September 3, 1998; revised June 24, 1999

We present observations of 40 asteroids taken mostly from Table Mountain Observatory, mostly from 1982, but with a few observations from late 1981 and early 1983. Several new or substantially revised periods are reported. Perhaps most interesting are observations of two asteroids, 288 Glauke and 3288 Seleucus, which are in “tumbling,” or nonprincipal axis rotation, states. Glauke has the longest rotation period known, ∼1200 h. Absolute photometry was performed on most objects, allowing us to derive phase relations and absolute magnitudes. For objects where these results are superior to previous values, we have revised the IRAS albedos using our absolute magnitude (H ) values. °c 1999 Academic Press Key Words: asteroids; rotation; photometry.

INTRODUCTION

This paper is a continuation of our reports of photoelectric photometry of asteroids from Table Mountain Observatory (cf., Harris and Young 1989, Harris et al. 1992). We also include some observations from other observatories, mostly at Lowell Observatory and the University of Arizona, that were made in collaboration with our observations, or have otherwise escaped publication separately. Most of the observations were made during 1982, but we include here the analysis of observations of 433 Eros which were made from September 1981 through April 1982 (see also Harris et al. 1992) and observations of a few objects in early 1983, which were essentially a part of the 1982 “observing season.” In addition to the printed paper, we will provide a data file containing the original observations upon request to A.W.H. ([email protected]). The preferred mode is by FTP, although mail delivery of a floppy disk can

be arranged. The data files include all the data, even fragmentary and inconclusive data and any points which were deleted from final fits, so that other investigators can fully evaluate our work. We hope that by doing this, we can put into the public domain data which are by themselves insufficient to draw any conclusion, but which may prove useful when combined with additional observations. We have also submitted all of our data to be included in the next revision of the Asteroid Photometric Catalog (Lagerkvist et al. 1987), and the data files will be submitted to the NASA Planetary Data System, Small Bodies Node. Our primary objectives in this observing project were to determine unknown rotation periods, and to define mean phase curves with high precision. To do this, we endeavored to cover the longest possible span of time and phase angle, with only as much data as necessary to ensure unambiguous period determinations. In general, we attempted to observe each asteroid thoroughly during at least one observing run, near opposition, and less thoroughly either the month before or the month after, in order to establish the phase relation over a significant range of solar phase angle. The methods of data taking and reduction are described by Harris and Young (1983). In addition, we have employed the Fourier analysis procedure described by Harris et al. (1989a) to derive rotation periods, mean reduced magnitudes, and produce composite lightcurves. These procedures are also described by Harris and Lupishko (1989). The phase data are fit with the H–G magnitude relation, described by Bowell et al. (1989). In cases where our derived value of the absolute magnitude H differs significantly from that used in the IRAS Minor Planet Survey (Tedesco et al. 1992, hereafter IMPS), we recompute the albedo and diameter from the IMPS using the procedure of Harris and Harris (1997).

173 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.

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TABLES AND FIGURES

Table I is a summary of results, arranged in order by asteroid number (column 1). Column 2 lists the taxonomic classification as given by Tholen (1989) or Tedesco et al. (1989) in the Asteroids II Database (hereafter AIIdb). Classifications followed by an asterisk are inferred by us from other data, for example our own B–V observations, and are not from AIIdb (see comments under specific asteroids). Columns 3 and 4 list the diameter and albedo of the asteroid, from the IMPS. Albedos followed by an asterisk (and, implicitly, the diameters associated) are either from another source or rederived from our H values, as noted in the comments on specific objects. In cases noted as revised from IMPS using the new H value, the revision is done according to the recipe given by Harris and Harris (1997). Column 5, the “observation date,” is a date generally within the range of the observations, sometimes the opposition date, that can be taken as representative of the observation aspect and is quantified in terms of the longitude and latitude of the phase angle bisector (see Harris et al. 1984 for the definition and justification of use of the PAB), column 6. Column 7 is the range of solar phase angle observed, and column 8 is the mean phase angle of the observations, hαi. This is not just the median of the previous range, but is weighted according to the number and quality of the observations. Columns 9 and 10 contain the mean reduced magnitude at mean phase angle, V hαi (but see discussion of Table II regarding values in hbracketsi), and its formal uncertainty. Mean reduced magnitudes for each night of observation are obtained from the Fourier fitting procedure (Table II), as described by Harris et al. (1989a). These values are then used to obtain a least squares fit to the H–G magnitude relation, as described by Bowell et al. (1989). The fit values for the absolute magnitude, H , and slope parameter, G, and their formal uncertainties are given in columns 11–14. When the range of phase angle observed was insufficient to solve for G, we solved for H using a constrained value of G, representing the mean for that taxonomic class of object. Constrained values of G and their estimated uncertainties (from Harris and Young 1989) are enclosed in parentheses. One should keep in mind that due to uncertainties in the definition of the V magnitude band and various standard star systems, absolute errors (values of H or V hαi) are rarely reliable below a level of 0.01 magnitude, whereas relative measurements, e.g., over a month or two related to a common set of nearby stars) can result in magnitude levels of an asteroid which are calibrated in a relative sense to within a few thousandths of a magnitude, thus allowing a very precise definition of the phase relation (G value) even if the absolute brightness is less well defined. Column 15 is the full range of lightcurve variation, generally obtained by differencing the minimum from the maximum of the fitted Fourier function, and column 16 is the difference between the mean light level and the maximum light level, as defined by the fitted Fourier function. We do not tabulate error bars for these quantities, although σV hαi is perhaps a fair estimate the uncertainty of these quantities. Columns 17 and 18 are the

synodic rotation period and its formal uncertainty, obtained from the Fourier analysis routine. Rotation periods given in parentheses are taken from other sources, as noted in the remarks on individual objects, for the purpose of constructing composite lightcurves in cases where our data alone do not yield well defined periods. Column 19 is an estimate of the reliability of the period determination, as defined by Harris and Young (1983). A reliability of 1 is extremely tentative, and may be totally wrong; 2 corresponds to reasonably secure results, not likely wrong by more than a factor of two, but including cases of ambiguity (i.e., the true period may be half, double, 1.5 times, etc., the stated value). A reliability of 3 indicates a secure, unambiguous result. The final column, 20, lists the figures associated with each asteroid. All data are included in the machine readable data file, even if not included in figure form in the printed paper. In the next section, we give some remarks relating to specific objects. Table II contains the aspect data for each asteroid on each night of observation. On the header line, along with the asteroid name we list the color index, B–V, of the asteroid. Generally, we list B–V as tabulated in the AIIdb (Tedesco 1989), or for several objects, transformed from the Eight-Color Asteroid Survey (ECAS, Zellner et al. 1985). Generally, colors from these sources are more accurate than our measurements. Furthermore, in cases where we had literature values available, those values were used in the reduction procedure to define the color transformation, so in those cases, the results are not independent measures of the color index. However, in a few cases, literature values were not available, so our measures, while not highly accurate, can be considered independent measures and may be of some value. In those cases, we have listed our measured value of B–V, followed by an asterisk. In one case (392 Wilhelmina), we did not make any B observations and we could not find a literature value of B–V, so a value of 0.75 (listed in parentheses) was assumed for purposes of color transformations. For the main data entries, the first column is the date of observation, given to the nearest tenth of a day to the center of the observed lightcurve. All other entries in the table refer to this time. Columns 2 and 3 are the 1950 right ascension and declination of the asteroid, and columns 4 and 5 are the 1950 longitude and latitude of the phase angle bisector. Columns 6 and 7 are the distances in AU from the asteroid to the Sun and Earth, respectively, and column 8 is the solar phase angle. Column 9 is the predicted rate of change in mean magnitude, in mag/day, including distance and phase angle effects. This number is used to reduce all magnitude levels for the night to the geometry of the reference day fraction (cf., Eq. (9) in Harris and Lupishko 1989). Columns 10 and 11 are the observed reduced mean magnitude for the night and its formal error. Most values are solution values from the Fourier fitting procedure, as described by Harris et al. (1989a). Values enclosed in hbracketsi are means of the observations for the night, in cases where the period of rotation was not determined or known from other sources. In these cases, the number listed in column 11, also enclosed in hbracketsi, is the half-range of variation observed during the night. Values of V (α) enclosed in

Note. Classifications followed by an asterisk are not AIIdb values, but are inferred from our data or other sources as noted in comments on individual asteroids. Albedos followed by an asterisk are from other than the IMPS. See comments on the individual asteroid for details. (Magnitudes) and (sigmas) in parentheses are derived values using (nominal values) of H and G, with thier (sigmas). hMagnitudesi and hsigmasi in brackets are derived from nightly means and ± ranges observed, where no period was determined. Magnitudes followed by an asterisk refer to maximum light rather than mean level. (Periods) in parentheses are assumed values from other sources, not solution values from these data.

TABLE I Summary of Results

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TABLE II Aspect Data

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TABLE II—Continued

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TABLE II—Continued

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parentheses are mean levels which have been constrained to lie along the H–G magnitude relation, according to the value of G listed in Table I, with respect to neighboring values of the mean magnitude. The magnitudes for 1862 Apollo, flagged with asterisks, refer to maximum light. The same notations are used in Table I for magnitude results computed from these data. Columns 12, 13, and 14 give the V magnitude, B–V (see below), and identification of the comparison star used. As noted under the discussion of Table I, the largest uncertainty in the values of the mean magnitude (both the nightly values listed in Table II and

the solution values, V hαi and H , in Table I) lies in the reduction to an absolute magnitude scale. Thus, if one wanted to establish to a higher degree of accuracy any of the magnitudes tabulated, one could remeasure the comparison star(s) used to greater accuracy and adjust the tabulated magnitudes accordingly. In general, the magnitudes for a given asteroid are accurate to a few thousandths of a magnitude with respect to one another. This was accomplished by carefully observing the different comparison stars used for each object with respect to one anothor, as discussed by Harris and Young (1983). On the other hand, we paid

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less attention to reduction to a standard magnitude scale (in this case, the Johnson and Morgan V band), since errors of the order of one or two hundredths of a magnitude are nearly impossible to avoid, and the effort is unwarranted for our purposes. The star identification in column 14 is usually the SAO catalog number, but for stars not in that catalog, we give the 1950 coordinates in compressed form, e.g., 1613.4–1614 refers to star at right ascension 16h 13.4m and declination −16◦ 140 . Some entries refer to published data from others, which we have reanalyzed along with our data. These entries are included for the purpose of complete documentation of the machine readable data file, which contains the data from all of these nights. The machine readable files include Tables I and II. Regarding the matter of color transformations, we have attempted to measure the color index, B–V, of all of the asteroids and comparison stars used and apply a correction for the color difference between the two. The procedures used are discussed in Appendix I of Harris and Young (1983). For the 1982 data from TMO, we applied a correction of 0.031[(B–V)C − (B–V)A ]. In cases where the color index of the asteroid was not determined, a literature value was assumed, as noted above. In cases where the color index of the comparison star was not measured, reductions were carried out with an assumed value of the color index, (B–V)C = 0.75. In these cases, we have tabulated (0.75) in Column 13 of Table II to indicate that the value is assumed. The Lowell Observatory photometer has a negligible transformation to the Johnson and Morgan system, so that the color difference between the asteroid and comparison does not matter so much. Therefore, we have not tabulated the color indices of comparison stars used there. Two main types of figures are presented: lightcurves and phase relations. The lightcurves are composites constructed from the Fourier analysis. In most cases, we have attempted to separate the different nights with different symbols. In general, the time scale across the bottom was chosen to start at the reference time of the Fourier fit, which is chosen as 0h UT of the date nearest to the weighted mean time of the observations. Above the figure is given a value for the period, without an error estimate. This is, in all cases, the exact value of the period used to produce the composite and the value with which one could correctly reestablish the original time tags. It should be noted that the times plotted, and those given in the data file, are light time corrected from the times of observation to the times at the asteroid. The vertical scale is the relative magnitude, with the zero level corresponding to the mean light level as derived in the fitting procedure. The actual magnitude of any datum is thus the plotted level plus the value of V¯ (α) for the night, from Table II. While we have attempted to be precise in our definitions and plotting, we discourage extracting the original lightcurves from the plots and again remind the reader of the availability of the digital data files. In some cases of single nights of observations, or when a period could not be found, we present individual lightcurve figures, with the magnitude scale just the reduced magnitude as observed.

The other main class of figures are plots of the phase data, as listed in Table II. The fitted H–G magnitude relation is plotted in each figure, with the solution value of the slope parameter, G, listed at the top. The listed G values are the exact values used to plot the H–G curve, not rounded to reflect the uncertainty in the determination. In cases where different plot symbols are used, they are generally keyed to the same symbols as used for each day in the composite lightcurve. We generally do not present phase plots consisting of only two phase points (or clusters of points at only two distinct phase angles) as they will be fit “perfectly,” and the resulting plots would not be very revealing. A third class of a few figures are “noise spectra” of the Fourier fit. The Fourier fit solution is linear with respect to the harmonic components (including the mean magnitude level which is the “zeroth-order” harmonic), so it is efficient to produce those solutions over a range of trial periods spanning the range of possible rotation period. A plot of the RMS fit residual vs period is in effect a “noise spectrum” of the fit, with minima corresponding to possible rotation periods. We scale the RMS residuals in units of the a priori error estimates of the observations, thus a value of 1.0 corresponds to a fit quality equal to the a priori estimated noise in the data. In the best of cases, only one possible period will emerge, with a fit noise level near 1.0. Sometimes no solutions measure up to this quality, and other times many solutions are possible. Such problematical cases are discussed under the comments on individual objects and the noise spectra presented for the reader’s evaluation. Variations from these standard formats are discussed under the remarks on individual objects. COMMENTS ON INDIVIDUAL ASTEROIDS

5 Astraea. We observed Astraea on only one night, January 8, 1983 (Fig. 1). The second and third observations of the

FIG. 1. Single night observations of 5 Astraea.

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FIG. 2. Composite lightcurve of 12 Victoria.

comparison star we deemed discordant with other observations taken near the same time, so we have adjusted the magnitudes of Astraea according to the first and fourth observations of the comparison with a nominal extinction correction and assigned considerably larger error estimates. Our curve is compatible with the period of 16.800 h reported elsewhere (e.g., De Angelis 1995). 12 Victoria. This asteroid has been well observed at several previous oppositions. See Dotto et al. (1995) for a summary and pole solution. We observed on three consecutive nights in October 1982. The amplitude of variation was low, and because of the short time span of the observations, neither the period nor the phase relation could be determined better than literature values, P = 8.661 h (Dotto et al. 1995), and G = 0.22 (H–G fit of data from Tempesti and Burchi (1969)). We thus constrained our data both horizontally and vertically according to these parameters to obtain the composite lightcurve presented in Fig. 2. The resulting lightcurve is noteworthy for its low amplitude of variation, only 0.08 mag, compared with a typical amplitude of 0.2 mag or more at other aspects. We thus infer that we were observing at an aspect quite close to polar. The mean magnitude level of our lightcurve is about 0.19 mag brighter than that inferred at the same phase angle from the Tempesti and Burchi observations, which further confirms that we were viewing from a more nearly polar aspect and also implies a substantial polar flattening (>10%) compared to the short equatorial dimension of Victoria. The pole solutions obtained by Dotto et al. do indeed lie near our observation longitude; however, they obtain an obliquity (polar latitude) about 50◦ away from our observing aspect. We suspect that our viewing geometry was more polar than that, so a reevaluation of the pole position including our lightcurve could be worthwhile. The diameter and albedo are slight revisions from the IMPS based on our H value.

FIG. 3. Single night observations of 15 Eunomia.

are better determined by previous data sets (see De Angelis (1993) for a recent summary of past observations and a pole and shape solution). The night-to-night magnitude levels were adjusted using a slope parameter G = 0.21, the mean value for M class asteroids. 19 Fortuna. We observed this asteroid on three nights in June 1981 (Harris et al. 1992). In 1982, Fortuna came to very low phase angle at opposition on 14 October, so we attempted observations to obtain a good phase relation. We were partially successful, obtaining observations on three consecutive nights in October including the night of lowest phase angle (0.30◦ ), and one more night in November at a phase angle of 15.83◦ . Additionally, a lightcurve was obtained by D. Thompson of Lowell Observatory on 15 September (R. Millis, pers. commun.), which is included in our composite lightcurve and phase relation plots (Figs. 5, 6). The Fourier fit to all the data is quite satisfactory, resulting in a very precise determination of the synodic period,

15 Eunomia. Only four data points were obtained on one night (Fig. 3). A change in magnitude of at least 0.5 occurred in the interval of 1.5 h of the observations, which is consistent with previous observations showing more than a half magnitude amplitude with a 6.083-h period and with pole solutions (De Angelis 1995) indicating that we were observing at a nearly equatorial aspect. 16 Psyche. The lightcurve in Fig. 4 was constructed with both the period and the phase relation constrained, since both

FIG. 4. Composite lightcurve of 16 Psyche.

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FIG. 5. Composite lightcurve of 19 Fortuna.

P = 7.4449 ± 0.0003 h, in excellent agreement with our previous result (Harris et al. 1992). A rather high-order Fourier fit was required to fully represent the very detailed Lowell lightcurve, which results in some overfitting in the less well sampled portion of the curve, which should be ignored. The overfitting does not significantly affect the period determination nor the phase relation. Fortuna has been observed on many previous oppositions (see De Angelis (1995) for a summary and pole solution). He and various others before him obtain a sidereal rotation period of 7.4432 h, with a direct sense of rotation. We question this result, because the synodic period of a directly rotating body in retrograde motion against the sky (i.e., at opposition) should be shorter than the sidereal period, if one considers only the geocentric position of the asteroid. The difference between our syn-

FIG. 6.

Phase relation of 19 Fortuna.

odic period and the sidereal period of De Angelis corresponds to a rotation of ∼0.26◦ /day, which is the right amount, but in the wrong direction. On the other hand, if one assumes that the sidereal–synodic period difference is more closely related to the rotation of the phase angle bisector (see Harris et al. (1984) for the definition and rationale of its use and Magnusson (1986) for its application to asteroid pole determination), then the sidereal– synodic period difference should be very small, since the phase angle bisector is nearly stationary during the months near opposition. In the case of Fortuna, one would expect a difference of only about 0.0004 hours between the sidereal and synodic periods (the sense of the difference and exact magnitude depending on the sense of rotation and pole position). Thus we suspect that there may be a cycle ambiguity problem with past pole solutions and suggest that a new solution is in order. Turning to the phase relation, we include in our fit the three magnitude levels observed in 1981 (plotted with + symbols). Although the position in the sky was very different, the fact that the amplitude of variation was almost exactly the same gives us some confidence that the aspect (sub-Earth latitude) of observation was not radically different. We performed a fit to the H–G phase relation using constant weighting, which yielded a fairly nominal slope parameter, although it appears (Fig. 6) that Fortuna exhibits somewhat less opposition effect than predicted by the H–G function. 22 Kalliope. Only two measurements of 22 Kalliope were taken, about 30 min apart, which differed by only 0.01 mag in level. No lightcurve is presented, although the two data points are included in the digital lightcurve file. 38 Leda. We observed Leda on only one night, January 8, 1983. At the time, the rotation period was unknown, but has subsequently been determined with high reliability by DeYoung and Schmidt (1996), P = 12.84 h. Our lightcurve (Fig. 7) is

FIG. 7. Single-night observations of 38 Leda.

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rial. This is consistent with the pole solution given by Taylor et al. (1988) or more recently by De Angelis (1995). It is noteworthy that our observed light levels, both mean and maximum, are considerably brighter than reported by Taylor et al. This is consistent with an interpretation that Eugenia is substantially flatter in polar dimension than its equatorial dimensions and that we were observing at a more polar aspect than Taylor et al. in 1969 and 1984. This supports the estimates of De Angelis (and others, which he summarizes in his paper) that the axis ratio b/c of Eugenia is ∼1.5; that is, its polar profile is more elongate than its equatorial profile. The IRAS observations used to determine albedo were taken during the 1983 apparition (opposition in July), when Eugenia was closest to polar aspect, according to all pole solutions. Thus the amplitude of variation was probably even less than we observed, but the mean light level was likely brighter than was assumed in computing the IMPS albedo of 0.040. Thus it is likely that the albedo is somewhat higher, perhaps as much as 0.050. FIG. 8.

Composite lightcurve of 45 Eugenia.

quite consistent with that period, indeed a Fourier analysis of the half-period yields a best fit value for P/2 of 6.42 h, in perfect agreement. The amplitude appears to be ∼0.1 mag, not far different from that observed by DeYoung and Schmidt. 45 Eugenia. We observed this asteroid on four nights in April, 1982. It was also observed on March 13, 1982, by Debehogne et al. (1983) at quite low phase angle. We have digitized those observations and included them in our composite lightcurve and phase relation (Figs. 8, 9). The period is consistent with previous observations, and the amplitude (0.18 mag) is about half the maximum value reported from other apparitions, indicating a viewing aspect midway between polar and equato-

FIG. 9. Phase relation of 45 Eugenia.

46 Hestia. This asteroid was observed on four nights in 1982, at almost exactly the same aspect as was observed in 1978 by Scaltriti et al. (1981). The 1982 observations are insufficient to obtain an independent rotation period, so we formed a composite constrained by the Scaltriti et al. value (21.04 h) and further adjusted the magnitude level on December 14 by +0.067 mag with respect to the level on December 19, corresponding to a phase relation slope parameter G = 0.12, which is the solution value obtained from the larger span of data. This was done because of the minimal overlap of coverage on December 14 when composited against the other nights. The resulting lightcurve (Fig. 10) is in general agreement with the Scaltriti et al. lightcurve, but appears to have a few bad data points and is unsatisfyingly irregular in form. Thus we encourage further observations to confirm the rotation period. Fig. 11 is a plot of the phase relation, where we have added in the phase data from

FIG. 10. Composite lightcurve of 46 Hestia.

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FIG. 11. Phase relation of 46 Hestia.

the 1978 observations (plotted with + symbols). The magnitude levels appear consistent, although there is some indication that 46 Hestia is deficient in opposition effect compared to the H–G model fit. 50 Virginia. This asteroid was observed on two nights from Lowell Observatory. The individual lightcurves are shown in Fig. 12. There seems to be some indication of a period of ∼9.25 h, although the result is very insecure. Other published lightcurves are similarly uncertain. Harris and Young (1980) suggest a period longer than 24 h, Holliday (1996a) finds a period of 17.88 h, and Shevchenko et al. (1997) derive a period of 14.31 h, all with amplitudes of variation ∼0.1 mag.

FIG. 13. Composite lightcurve of 93 Minerva.

et al. also took lightcurve observations on seven nights, one the same as one of ours. We have included their data in our solution and present the combined results here. We obtain a rotation period of 5.9823 h, in good agreement with earlier results (e.g., Debehogne et al. 1982, Harris and Young 1989). The resulting composite lightcurve is presented in Fig. 13. As noted by Millis et al., the magnitude levels measured in October, considerably before opposition, and in March, far after opposition, do not fit the phase relation followed by the data near opposition. In Fig. 14, we plot all the phase data and the H–G fit to only the data from December 1992 through January 1993. The February point fits fairly well, but was excluded because it is a single observation. We also plot (but exclude from the fit) the magnitude

93 Minerva. We observed this asteroid on three nights in December 1982 and January 1983, in support of an occultation which occurred on 22 November (Millis et al. 1985). Millis

FIG. 12. Individual nights’ observations of 50 Virginia.

FIG. 14. Phase relation of 93 Minerva. See text for description of the 1980 and 1981 data, plus fitting procedure.

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levels observed in 1980 (Debehogne et al.) and in 1981 (Harris and Young). The vertical bars are not error bars, but instead represent the range of variation observed at these aspects. Note that 93 Minerva was ∼0.2 mag fainter in 1981 than in 1982–1983 and ∼0.2 mag brighter in 1980 than in 1982–1983. At no aspect did it exhibit more than ∼0.1 mag amplitude of variation. We can thus conclude that the 1981 aspect was nearly equatorial while the 1980 aspect must have been nearly polar and that Minerva has a very substantial polar flattening, perhaps as much as a 2 : 1 axis ratio, but at the same time has a quite regular equatorial profile, with only ∼10% variation as indicated by the low amplitude of variation at that aspect. This extreme polar flattening is no doubt the reason that the magnitude levels far off opposition, and hence at different aspect, fail to fall along the phase curve closer to opposition. We can further infer a spin axis orientation of high obliquity and pointing toward a longitude about 90◦ from the 1981 aspect (λ = 318◦ ). Thus we infer a pole orientation of (λ, β) = (230◦ , 0◦ ), with an uncertainty of ∼30◦ . We make no guess of the sense of rotation. 127 Johanna. We observed on only one night, following up a rumor that this asteroid might have a rotation period of less than 1 h. Our single lightcurve (Fig. 15), which appears to cover about one half of a normal rotation cycle, is sufficient to refute the rumor and indicates an ordinary period of ∼11 h. No other lightcurves appear in the published literature. 130 Elektra. This asteroid has been observed extensively, e.g., see De Angelis (1995) for a recent summary, including a pole solution. We present two nights’ observations obtained by T. Dockweiler using the Lowell Observatory 42-in. telescope. Only relative magnitudes are available. The two lightcurves cover all rotation phase and yield a period solution of 5.231 ± 0.005 h. The accepted sidereal period is 5.225 h, in good agreement with our period determination. Figure 16 is the resulting composite.

FIG. 16. Composite lightcurve of 130 Elektra.

nights in 1983, 1 month after our observations. We have combined our one lightcurve with theirs to obtain a more definitive result. A reexamination of the Goguen et al. lightcurves suggests that an uncertainty of ∼0.1 h can be allowed for their period determination. We thus separated the composite lightcurve given by Weidenschilling et al. into individual nightly lightcurves, and examined their data plus ours for a range of periods from 20.7 to 21.1 h. A very robust solution value of 20.991 ± 0.003 h was found (Fig. 17), corresponding to a match in coverage of the principal maximum on January 8 and February 20. Our curve does not match the other maximum observed in February, so a period ambiguity would require one full cycle error between January and February, or a period ∼0.4 h greater or less than the solution shown in Fig. 17. Thus neighboring solutions are disallowed by

139 Juewa. We observed this asteroid on only one night, in an attempt to resolve the ambiguity in the period previously reported. Goguen et al. (1976) obtained a value of 20.9 h, but Debehogne and Zappal`a (1980) proposed a value of twice that, based on a single long lightcurve with a near-constant slope. Weidenschilling et al. (1990) reported observations on three

FIG. 15.

Single-night observations of 127 Johanna.

FIG. 17. Composite lightcurve of 139 Juewa.

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FIG. 20. Composite lightcurve of 160 Una.

from the fit. The mean magnitude at mean phase angle is almost exactly coincident with the magnitude levels observed in 1979 and 1980 (Harris and Young 1989), as is the range of variation and the general shape of the lightcurve. FIG. 18. Phase relation of 139 Juewa.

the series of lightcurves presented by Goguen et al., and we can consider our solution unique. Although the two maxima of our composite lightcurve are quite unequal, a period twice as long seems unlikely. The phase data (Fig. 18) are very well fit by the H–G relation, with a G value modestly higher than average for a dark asteroid. The diameter and albedo listed in Table I are minor revisions from the IMPS, based on our value of H . 146 Lucina. This asteroid was observed by us in 1979 and 1980 (Harris and Young 1983, 1989). Both of those sets of observations spanned larger ranges of time and phase angle; thus the present composite lightcurve (Fig. 19) has been constrained both in rotation period and in phase relation. The fit is satisfactory, except for some points at the end of the night of April 23, taken at greater than two air masses, which have been deleted

160 Una. We observed Una on three nights in October 1982, using the same comparison star as was used for 19 Fortuna, which was close by. A Fourier scan of all periods from 3 h to >12 hours revealed only harmonics of the solution P = 5.55 h, corresponding to 2, 3, or 4 pairs of extrema per rotation cycle. While we cannot rule out the longer periods, we select the shortest one with two pairs of extrema per rotation cycle as being most probable (Fig. 20). The resulting phase data are very well fit by the H–G relation, yielding a well-determined and very ordinary value for G, in spite of the very small range of phase covered. Una was observed by Di Martino et al. (1994) at almost exactly the same place in the sky in October 1991. Their lightcurve is similar to ours with a very slightly larger amplitude and a somewhat more regular shape. Their derived period is consistent with ours. Unfortunately they report only relative photometry, so it is not possible to use their data to extend the phase angle range of our H–G solution. 166 Rhodope. We observed on only two nights in October. No other lightcurve observations have been published, and it was not observed by IRAS. A Fourier scan of periods from 3 to 40 h revealed possible solutions at periods of 7.87, 11.8, and 23.7 h, with a less likely possibility at 9.6 h. After examining all of the possible composites, we favor the period of 7.87 h (Fig. 21), although the other values cannot be fully ruled out,

FIG. 19.

Composite lightcurve of 146 Lucina.

FIG. 21. Composite lightcurve of 166 Rhodope.

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FIG. 22. Composite lightcurve of 187 Lamberta.

and there is some possibility that none of these values is correct; thus we assign the result a reliability of only 1. The solution presented was obtained without constraining the magnitude levels, even though the phase angle on the two nights was nearly identical. It is somewhat reassuring to note that even though the magnitude levels were not constrained, the free solution yielded the expected levels within 0.001 mag, thus lending some further support to this solution. 187 Lamberta. We observed Lamberta on six nights in April 1982. Hainaut-Rouelle et al. (1995) recently published observations from April 1991, at almost the same aspect. The period and amplitude of variation we obtain (Fig. 22) is in excellent agreement with their results. The diameter and albedo listed in Table I is a revision from the IMPS value based on our H value.

the magnitude levels adjusted to the same phase angle using a nominal phase relation (G = 0.09). The figure is not intended as a “composite lightcurve,” but only to indicate the relative magnitude level on the two nights. Medea has also been observed by Di Martino et al. (1995), Holliday (1996b), and C. Blanco (unpublished, pers. commun.). The two published papers give periods of 10.12 and 10.25 h, respectively, both with amplitudes of variation of ∼0.12 mag. Blanco suggests a period of 18.174 h. None of these period values lead to an overlap of rotation phase coverage by our two lightcurves, so all that can be said is that our data are consistent with any of these periods, but do not serve as a significant confirmation. It is noteworthy that the Di Martino observations suggest an absolute magnitude H ≈ 8.50, Holliday’s observations suggest H ≈ 8.95, our observations yield H = 8.18 ± 0.06 (assuming G = 0.09); and the IMPS lists H = 8.28 (assuming G = 0.09). This range of H values is hard to reconcile with the small but nearly equal amplitude of variation observed at the various aspects. Our value and the IMPS one differ mainly due to the different G values assumed and are thus quite concordant. 230 Athamantis. We observed this asteroid from September 1982 to January 1983. The rotation period is so close to commensurate with Earth’s that only one part of the lightcurve is visible from a given location all season. Thus we requested Zappal`a and Di Martino in Torino, Italy, to observe for us, which they did on January 9–10, 1983. The resulting composite lightcurve (Fig. 24) yields a period very close to 24 h and, thanks to the Torino lightcurve, covers more than half of the rotation phase, eliminating any ambiguity in the period. There is some discordance in the fit near the right-hand end, due mainly to the September data. It seems likely that the shape of the curve changed from its appearance in September to a somewhat different form in December and later, due to changing aspect perhaps. The phase data (Fig. 25) are extremely well fit by the H–G relation, with a very ordinary value of G.

212 Medea. We obtained only two nights of data on this asteroid, in September 1982. Our results are inconclusive. In Fig. 23, we plot the 2 days’ data on the same UT time scale, with

FIG. 23. Individual nights’ observations of 212 Medea. Magnitude levels on September 23 are adjusted for the phase angle on September 22 (8.09◦ ) and plotted on the same time scale, to show the consistency of intrinsic brightness night to night. This is not intended to represent a composite lightcurve.

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FIG. 24. Composite lightcurve of 230 Athamantis.

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FIG. 25. Phase relation of 230 Athamantis.

266 Aline. Two consecutive nights of observation in September 1982 failed to yield any conclusive sign of variation. Figure 26 is a plot of the 2 days’ data in the same style as in Fig. 23. The asteroid appeared a bit brighter on the second night, even though the phase angle was increasing slightly, but the difference is only barely above the scatter in the data. Thus the period is either very long, or the amplitude of variation very low, or both. No other lightcurve observations appear in the literature. The diameter and albedo listed in Table I are revisions from the IMPS, based on our value of H . 288 Glauke. This asteroid was observed for seven nights over a 9-day interval, during which time it faded in brightness at a nearly uniform rate by 0.4 mag, even after correcting for distance and a nominal phase correction. Other observers were alerted to this bizarre behavior, and beginning the next dark run it was monitored on nine nights, and the next month, quite far from opposition, it was observed on another two nights, yielding a data span of nearly 60 days. This, it turned out, was only barely long enough to cover one full rotation cycle, indicating that 288 Glauke is by far the slowest spinning asteroid known.

FIG. 26. as Fig. 23.

Individual nights’ observations of 266 Aline, in the same format

FIG. 27. Lightcurve of 288 Glauke, plotted with different phase relation corrections.

In addition to the observations reported here, Glauke was observed on May 15, 21, and 23 by Zellner et al. (1985) as a part of the Eight-Color Asteroid Survey. We have checked those observations and find that they agree with those reported here within a few hundredths of a magnitude in each case. Since those observations have already been published, and would not contribute additional days to our coverage, we have not included them in the present analysis. Our set of observations cannot be uniquely composited, due to the coupling between the phase angle variation and the rotational variation. In Fig. 27, we plot the observed magnitudes with no phase angle correction, that is V (α), and the same data points adjusted to V (0) values for three assumed values of the slope parameter G: 0.12, 0.23, and 0.34. These slope parameters are the mean value for S class asteroids and values ±1 σ from that mean. Since 288 Glauke never varied by more than ∼0.05 mag in the course of a night, we plot only one normal point for each night, which is a weighted average of all observations for the night. The key to deriving the period is the placement of the last two observations, which overlap the linear string of seven observations in different ways for each G value. For the G = 0.12 curve, the points at both 135 and 140 days overlap the magnitude range of the first seven points and are of a different slope; thus that solution is inconsistent with at least one observation. For the other two choices of G, the magnitude on day 140 is fainter than the last one in the string of April observations, so either curve can be composited without contradiction by selecting the period which places the point at 135 days at the appropriate magnitude level on the string of April observations. For G = 0.23, we get a period of 1150 h. For G = 0.34, we obtain a period of 1110 h. The best solution for G = 0.12 (discordant with one or both of the last two points, as noted above) has a period of 1210 h.

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FIG. 28. Photographic photometry of Glauke in 1983.

We can turn to observations at later oppositions for further insight. Bowell (pers. commun.) observed Glauke photographically in August/September 1983. Photometric reductions were performed, which yield typical deviations from photoelectric observations of only ∼0.1 mag. Figure 28 is a plot of those data. The phase angle ranged from 2.5◦ to 6.1◦ ; thus the uncertainty in G contributes no significant error in reducing the magnitudes even to zero phase, so we plot V (0), for an assumed value of G = 0.23. The magnitudes are quite consistent with the maximum level we observed in 1982, and with the exception of one of the last two points, they are consistent with the very long period. At the next opposition, in 1984, Binzel (1987) observed it from September 24 to November 27, covering just slightly more than a full rotation. Since his observations covered the month before as well as the month after opposition, the overlap in rotational phase occurs at almost exactly the same phase angle, so the uncertainty in G value of the phase relation does not affect the period determination. Overlaying the first two and last three data points of his curve (Fig. A10 in Binzel 1987) we can obtain a period of 54 ± 1 days, or 1296 ± 24 h, which is inconsistent with the period we derived from the 1982 observations, no matter what G value is chosen. The most likely resolution of this discrepancy is that 288 Glauke is not in a principalaxis rotation state, but instead is “tumbling” in a general state of three-axis rotational motion. Such rotational state has been confirmed by radar observations for 4179 Toutatis (Hudson and Ostro 1995) and is indicated by photometric lightcurve observations of Toutatis (Spencer et al. 1995), 253 Mathilde (Mottola et al. 1995) and 3288 Seleucus (this paper, below). Harris (1994) estimates a time scale for damping of non-principal-axis rotation for Glauke of ∼300 billion years; thus one would not expect any significant damping to have occurred for as long as Glauke has been spinning so slowly. What one can say is that Glauke and the other very slowly spinning tumbling asteroids were not slowed from much faster spins by some simple drag mechanism over a very long time span. If this were the case, we would expect the tumbling component of spin to have been damped while the asteroids were spinning much faster, thus leaving them in states of principal-axis rotation, even though the present damping time scale is very long. Thus we must con-

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clude that whatever is responsible for the slow rotation is not a dissipative process or that it acted very quickly “in the beginning.” The explanation for these “tumbling asteroids” is indeed a mystery. Glauke was observed once more, in 1986, by Weidenschilling et al. (1990). They observed on five consecutive nights, January 17 to 21, as Glauke passed through very low phase angle. By subtracting the magnitude levels before opposition from those after at the same phase angle, one can infer a nearly constant fading of ∼0.03 mag/day and a rather steep opposition surge, somewhat like that observed for some other high albedo asteroids (Harris et al. 1989b). Based on the magnitude levels observed at other oppositions, they were probably not observing near maximum light; thus the night-to-night fading is very modest, e.g., compared to our observations in April 1982 where the slope was more than twice as great. 375 Ursula. We observed this asteroid in conjunction with a campaign to observe a stellar occultation by Ursula on 15 December, 1982 (Millis et al. 1984). We observed on four nights in November, December, and January. In addition, Millis et al. observed it from Lowell Observatory on a total of six nights from October to March. We have combined their observations into our analysis. A preliminary inspection of the individual lightcurves gave very little indication of any variation. Taking the mean light levels each night we constructed a very ordinary looking phase curve, with all points agreeing within a few hundredths of a magnitude of a fit to the H–G relation, which is further evidence for almost no lightcurve amplitude. Schober (1987) published lightcurves from three consecutive nights in August 1981, obtaining a composite lightcurve with a period of 16.83 ± 0.07 hours and an amplitude of 0.17 magnitude. The position in the sky in August 1981 was almost exactly 90◦ away from the position during our observations, so it is plausible that he was observing a nearly equatorial aspect, and we were observing a nearly polar one. Guided by his period determination, we did a fourth-order Fourier scan of periods from 16.5 to 17.5 h (Fig. 29). Within the range of periods allowed by the Schober composite, values of 16.819, 16.898, or 17.110 appear possible. We constructed best-fit composites for each of these values and found only the middle one to be satisfactory. It turned out that the fourth harmonic for this period was so small that the third-order fit was statistically superior, with a period of 16.900 ± 0.003 h. We present this composite in Fig. 30. The amplitude of variation is only ∼0.06 mag, or 1/3 that observed by Schober, supporting the conclusion that we were observing at a nearly pole-on aspect. It can be noted that the moment of occultation, November 15 at 11.2 h UT (Earth received time, not light time corrected), corresponds to a time on our composite lightcurve (light time corrected) of 4.8 h UT on 23 November, or just past a minimum in the lightcurve. On examining the mean magnitude levels from the Fourier fit, it was apparent that all of the Table Mountain values were systematically brighter than the Lowell Observatory ones by several hundredths of a magnitude. Thus we did separate H–G fits of the

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FIG. 29. Noise spectrum of 375 Ursula, showing several possible periods. Schober (1987) suggested the 16.82-h period, but the 16.9-h period is much more plausible. See text for discussion.

two sets of magnitudes, and found remarkably that the solution values of G for each were essentially identical, 0.077 ± 0.03 for TMO and 0.078 ± 0.07 for Lowell. The H values obtained are 7.21 ± 0.02 and 7.27 ± 0.02, respectively. In Fig. 31 we present the phase relation data along with both solution lines. We also plot the maximum magnitude levels observed by Schober (1987) and recently by Hainaut-Rouelle et al. (1995). Note that our H–G solutions, if referenced to maximum light level, would be about 0.03 mag higher on the plot (about half the spacing between the two curves). The Hainaut-Rouelle observations were taken at a sky position about 170◦ away from our aspect, presumably also close to polar, which is confirmed by the level of maximum light and their observed amplitude of variation of about 0.05 mag. It

FIG. 30. Composite lightcurve of 375 Ursula.

FIG. 31. Phase relation of 375 Ursula. The two H–G solution curves are for the Lowell and TMO data separately. They both have the same G value.

can be noted that their two lightcurves are consistent with the 16.9-h period, but like our data, do not yield a unique solution. Finally, the magnitude level observed by Schober is distinctly fainter than that seen in our observations, by at least 0.1 mag, further supporting the conclusion that he was observing at a nearly equatorial aspect, and suggesting a polar flattening of ∼10% compared to the short equatorial axis. Thus we can infer a triaxial shape of Ursula of about 1.0 : 1.1 : 1.25. The diameter listed in Table I is the occultation diameter given by Millis et al. (1984), and the albedo is derived from that diameter and our value of H . 392 Wilhelmina. We observed on only two consecutive nights in October. A noise spectrum spanning the period range from 3 to 30 h revealed possible solutions of 8.5 and 13 h, plus longer harmonics. We present a composite (Fig. 32) with the shorter period, but assign the result a reliability of only 1. The light levels on the two nights resulting from this fit are plausible and within a few thousandths of a magnitude of a phase relation with a value of G = 0.09, appropriate for a dark asteroid. The IMPS lists an albedo of 0.059, based on a somewhat fainter H

FIG. 32. Composite lightcurve of 392 Wilhelmina.

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value. We have computed a slightly higher albedo, 0.065, based on our H value. 423 Diotima. This asteroid has a long history of observations and misinterpretations. We obtained only three data points on two nights in 1982, but were motivated to reanalyze most of the other data to try to resolve the ambiguity in the rotation period (with the result that we present three figures from only three data points, which must be some sort of a record). Diotima was also observed in 1982 by Schober (1983) and Di Martino and Cacciatori (1984a). Zappal`a et al. (1985) reanalyzed the combined set of data in a paper with the theme of the difficulty of deriving unambiguous periods. Despite that emphasis, it must be noted that the periods proposed in all three papers are wrong. The most convincing period was that obtained by Zappal`a et al., 4.622 h; however, Dotto et al. (1995) present additional data suggesting a period of 4.755 h, and Hainaut-Rouelle et al. (1995) present data from which they suggest three possible periods, 4.784, 9.563, or 11.953 h, favoring the 9.563 one which is twice the Dotto et al. period. We began by reanalyzing all the 1982 data. Figure 33 is a Fourier scan of periods in the range 4.5–5.0 h, which show a sequence of very nearly equal minima in the RMS residual, thus indicating that the period determination from these data is very nonunique. Both the 4.620-h and the 4.775-h values are clearly seen as equally possible. The 11.953-h period is not a possibility for the 1982 data. We next reanalyzed the Hainaut-Rouelle et al. Data. The noise spectrum of that data set (Fig. 34) clearly rules out the 4.620-h period, as well as the other periods that appear in Fig. 33, leaving only 4.775 as a possible period. Figure 35 is the plot of all of the 1982 data composited with that period. We still cannot rule out a period of double this value, although we note that the double period would result in a lightcurve with four sets of extrema and furthermore would not improve the most significant discordance in the fit, in the range

FIG. 33. Noise spectrum of 423 Diotima; 1982 data from Schober (1983) and Di Martino and Cacciatori (1984a), plus this paper.

FIG. 34. Noise spectrum of 423 Diotima, 1992 data from Hainaut-Rouelle et al. (1995). This spectrum plus that in Fig. 33 indicate that only the 4.775-h period is acceptable.

near 3h UT, since the discordant data would plot the same on the double-length lightcurve. Thus we adopt the 4.7748 ± 0.0005-h solution as the most likely one, although we assign a reliability of 2 to the result, recognizing that the double period is not fully ruled out. In addition to the various data sets mentioned above, Gil Hutton (1990) has also published observations from 1987, which he composited with the 4.620-h period. Both he and Dotto et al. attempt pole solutions, but neither use data from all of the oppositions that are now available. A new pole solution would appear to be worthwhile. Although our period is based almost entirely on others’ data, we do obtain a new H value from our three observations which were at very low phase angle, thus we are able to offer a revised diameter and albedo based on the IMPS and our new H value.

FIG. 35. Composite lightcurve of 423 Diotima.

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FIG. 38. Single-night lightcurve of 433 Eros in October 1981. FIG. 36. Composite lightcurve of 433 Eros, from the single night 81/09/03, but covering the full rotation cycle.

433 Eros. We present here observations of Eros from September 1981 to April 1982. During that time Eros moved more than 90◦ across the sky, and the amplitude of the lightcurve changed from a minimum of only a few hundredths of a magnitude (late September) to more than one full magnitude (January/February). To illustrate the range of variations, we present the lightcurves in 11 figures (36 to 46). Several of the lightcurves were obtained at the Lowell Observatory, as indicated in Table II. The lightcurve on December 18 (Fig. 40) was obtained by D. Tholen at the University of Arizona 61-in. (1.54-m) telescope. The observations were made in the B band, differentially from the star SAO 037504, which was observed to have a B magnitude of 9.918. On the subsequent two nights, a few observations were made, also by Tholen with the same telescope, as a part of the Eight-Color Asteroid Survey (Zellner et al. 1985). Those observations have been composited onto the December 18 data as shown in the figure. The magnitude scale has been shifted to the V band, using the color index B–V = 0.928, derived from the 8-color observations on the other two nights. Table II lists the mean magnitude levels for each night of observation, which are mainly useful for establishing the zero point of the magnitude scales and the magnitude offsets used to construct the composites. Because of the large change in amplitude, the mean light levels are not well suited for determining a phase relation of Eros. Instead we follow past observers in using the levels of maximum light. In Table III we list the maximum light levels which we observed in 1981–1982, and also similar data extracted from previously published data for the 1974–1975 apparition, and one point observed in 1972. Even though the two maxima are sometimes of differing brightness, the magnitude difference between

FIG. 37. Composite lightcurve of 433 Eros from late in September 1981, showing almost zero amplitude of variation.

them changes with changing aspect during an apparition, and even changes sign; that is, what one would assign “M1” as the brightest maximum early in the apparition may in fact become the fainter of the two maxima later on. In 1973–1974, the magnitude difference, M2–M1 (as defined by Millis), ranged from −0.08 to +0.07. In 1982, the greatest difference we observed was 0.16 (on February 19), but during most of the apparition it was much less. Since most geometric models that are used to fit amplitude data are symmetric and predict no difference in light level, we list in Table III the magnitude which is the mean of the two maxima. In cases where only one maximum was observed, we have adjusted that value up or down according to a predicted difference in the two levels, based on observations before and after the date in question. We have done this for the previously published data as well as for our own, so all of the light levels listed are new determinations, rather than just a listing from the literature. Figure 47 is a plot of all of these data, with the H–G model fit indicated. It is noteworthy that all of the points fall rather well along a single curve, even though they are taken at three different apparitions with somewhat different viewing aspects. This confirms earlier results (e.g., Dunlap 1976), that Eros is essentially biaxial; that is its polar dimension is not much different than its shortest equatorial dimension, so the maximum light level at equatorial aspect is about the same as at polar aspect. This is dramatically illustrated by comparing our composite lightcurve of September 22 and 26, which has an almost unmeasurable amplitude of not more than a few hundredths of a magnitude, with the lightcurve observed by Scaltriti and Zappala (1976) on December 11–12, 1974, which has an amplitude of 1.3 mag. Both lightcurves were taken at a solar phase angle of 35◦ . The 1981 lightcurve must be at nearly polar aspect, and the 1974 one very nearly equatorial, yet the

FIG. 39. Composite lightcurve of 433 Eros from early December 1981.

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FIG. 40. Single-night lightcurve from December 18, 1981, plus some data points taken for ECAS the next two nights, observed by Tholen at University of Arizona 61-in. telescope.

maximum light levels, corrected for the very small differences in solar phase angle, differ by less than 0.05 mag. Not only that, but the difference is in the sense that would imply a slightly larger polar dimension than the smallest equatorial one (that is, the light level at polar aspect is slightly fainter than the maximum at equatorial aspect), which is dynamically impossible for a principal-axis rotator; thus it seems likely that the polar dimension and the shortest equatorial dimension are equal, within the measurement accuracy. The H–G fit of the averaged maximum light data as tabulated and plotted yields a plausible value of the slope parameter G for an S class asteroid. Harris (the other one, 1998) has applied a new thermal model to the radiometric observations of Lebofsky and Rieke (1979) and preliminary values of H and G from our present data (10.47 and 0.32) to obtain an effective diameter of Eros of 23.6 km and an albedo of 0.20. Unfortunately, the final values of H and G that we obtain from the present data are slightly different from the values we sent to the other Harris, so our values of diameter and albedo listed in Table I are slight revisions from his values. Preliminary size and albedo determinations from the NEAR flyby in December 1998 (Veverka et al. 1999) suggest that Eros is somewhat smaller and higher albedo than the above estimates. The near-identical polar and short-equatorial axes of Eros raises an interesting dynamical possibility. For such a body, a “polar motion” in the direction of the short equatorial axis would

FIG. 41. Composite lightcurve of 433 Eros, Jan. 24–25, 1982.

FIG. 42. Single-night lightcurve of 433 Eros, Jan. 31, 1982, defining maximum and minimum light levels.

have almost no restoring force; thus it could have a very large amplitude, or even circulate. Black et al. (1999) show that the precession frequency ν of such a polar wander would be · ν=

(C − B)(C − A) AB

¸1/2 ω3 ,

where ω3 is the spin frequency. Similarly, they show that the polar precessional motion, viewed in the body coordinate frame, would be an ellipsoidal trajectory with axis ratio · ¸ |ω2 | (C − A)A 1/2 , = |ω1 | (C − B)B

FIG. 43. Single-night lightcurve of 433 Eros, Feb. 19, 1982, emphasizing extrema.

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FIG. 46. Composite lightcurve of 433 Eros, April 22–24, 1982.

FIG. 44. Single-night lightcurve of Eros, March 24, 1982.

with the long axis of the precession ellipse aligned in the direction of the short equatorial axis of the body. Thus if the moment of inertia difference (C − B) is small, the period of the polar precession becomes very long, and the trajectory of the motion becomes nearly a straight line in the direction perpendicular to the long body axis. If this difference is as small as ∼1%, the period could be of the order of several days. Even if the amplitude of libration is too low to cause a measurable effect on the lightcurve, it may be measurable by NEAR, which would provide a measure of the excitation and damping of the nonprincipal axis rotation and a direct measure of the moment of inertia differences. The images of Eros returned by NEAR in December 1998 (Veverka et al. 1999) reveal that the short equatorial dimension and the polar dimension of Eros are indeed essentially equal. However, the shape is a bit banana-like, so the polar moment of inertia (C) is likely significantly greater than the larger equatorial moment (B). 434 Hungaria. Hungaria was observed extensively in 1979 (Harris and Young 1983), from which a very accurate and unique period of 26.51 h was determined. Those observations were all at large phase angle (26◦ –29◦ ), so it was not possible to derive a meaningful slope parameter G nor an accurate absolute magnitude H . The present observations were clustered at phase angles around 8◦ (September) and 21◦ (October), and fortunately the portions of the lightcurve covered overlapped thoroughly in the 2 months, allowing us to derive much improved values of H and G. The total lightcurve coverage was less than complete, however, requiring a somewhat constrained solution. First, we

FIG. 45. Single-night lightcurve of 433 Eros, April 17, 1982, from Lowell Observatory. Relative photometry only.

adjusted the magnitudes each month to a single phase angle, assuming a slope parameter G = 0.42 as is appropriate for a high albedo (E class) object. The period was constrained to the value found from the 1979 data, 26.51 h. Since our data all overlapped, covering only slightly more than half of the rotation phase, we then performed a Fourier fit with one half the period, or 13.255 h, which in effect ignores any odd harmonics in the full lightcurve. Normally this is not a bad approximation and avoids the large data gap that would exist if a full-period solution were attempted. Figure 48 is the plot of this solution, and the various values of mean magnitude, amplitude of variation, etc., are based on this solution. It is noteworthy that a least squares adjustment of the half period for the present data resulted in a χ 2 minimum exactly on the period value of 13.255 h, confirming the earlier period determination. The value of G computed from the two reference TABLE III Maximum Light Levels of 433 Eros

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FIG. 47. Phase relation of 433 Eros, all data from several apparitions. We plot the average level of maximum light (mean of the two maxima, when different).

FIG. 49. Individual lightcurves of 443 Photographica, in the same format as Fig. 23.

phase angles was 0.43, thus confirming the assumed value used to adjust the magnitudes to a common phase angle each month. The value of H is 0.25 mag fainter than the currently adopted value (e.g., IMPS or EMP), no doubt mostly due to the shallower phase relation than that assumed in reducing the photometric data for those tables. IRAS did not observe Hungaria, although Morrison (1977) reported groundbased radiometry of Hungaria. Curiously, he cites a value of the absolute magnitude, V (1, 0) = 11.76, which is in perfect agreement with our new value of H = 11.46. Thus the diameter we list in Table II is exactly his value, and the albedo we list is simply his value transformed to the (H, G) based magnitude system.

42-in. telescope. Most of the original data and reduction details have been lost. We digitized data from hand-plotted figures in Bowell’s files. Fourier analyses of the data, both with nightly magnitudes unconstrained and with constraints according to a nominal S-class phase relation, G = 0.23, failed to yield unique solutions. Thus we present the individual nightly lightcurves in Fig. 49. In this figure, we have adjusted the light levels according to the nominal phase relation, normalized at α = 10.44◦ (June 17 phase angle). All but the June 16 lightcurve fall within a magnitude range of ±0.15 mag, indicating an amplitude of variation of ∼0.3 magnitudes. The June 16 light level, if correct, would imply a total amplitude of ∼0.55 magnitude. Such a large amplitude of variation is usually characterized by sharper minima compared to broader maxima, a feature which does not appear to be present; thus we suspect that there is a magnitude level error on June 16. Assuming that the full amplitude is ∼0.3 mag, we infer a time from maximum to minimum light of about 4 h, suggesting a period of ∼16 h. This is in fact the period suggested by Bowell shortly after the observations were taken. We are unable to improve upon this estimate.

443 Photographica. This asteroid was observed on five nights in May–June 1982, from Lowell Observatory with the

505 Cava. Our observations of this asteroid were fully reported by Young and Harris (1985), so we do not include figures of the composite lightcurve and phase relation. The data are included in the digital file, and we have reanalyzed the data using the Fourier fitting procedure developed since that earlier publication. The essential results are confirmed. We list slightly revised values of the period, H and G, and error bars in Tables I and II.

FIG. 48.

Composite lightcurve of 434 Hungaria.

519 Sylvania. The only previous lightcurve observations reported in the literature are two nights in August 1991, from which Lagerkvist et al. (1992) suggested “slow rotation.” Our lightcurves confirm this result, yielding a definitive period of

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FIG. 50. Composite lightcurve of 519 Sylvania.

FIG. 51. Composite lightcurve of 699 Hela.

17.962 h. The individual nights’ observations of course cover less than the full rotation cycle and in fact do not overlap very much, so we constrained the magnitude levels each month to a single reference phase angle, using a nominal slope parameter G = 0.23, appropriate for an S-class object. The resulting Fourier fit is unique and quite satisfactory (Fig. 50), and the H–G phase relation fit for the two phase angles yields a G value of 0.23 ± 0.04, right on the nominal value assumed for the subcomposite magnitude adjustments.

that our period corresponds to 7 rotations in 23.773 h, whereas the Pilcher value corresponds to 6.5 rotations in 23.762 h. It appears likely that the Pilcher result corresponds to a one-half cycle error each day from the correct period.

699 Hela. This asteroid was observed on two nights from Lowell Observatory and on one night from McDonald Observatory. The latter observations were reported by Binzel (1987), but are included in our fit for completeness. Those observations were made in the B band, which we convert to V assuming B − V = 0.86. Pilcher (1983), based on timings of extrema from his own visual plus these photoelectric measurements, reported a rotation period of 3.6557 h. Here we present a Fourier fit and composite lightcurve (Fig. 51) based only on the photoelectric observations. We must note, however, that the July 19 observations show a severe monotonic trend, perhaps because the local comparison star was varying, of ∼0.66 mag/day. We have detrended those data using this slope, but note that the absolute magnitude level so derived appears to be off by ∼0.2 mag. This magnitude level has been excluded from our H–G analysis. Using only the July 19 data, we found a best-fit period of 3.40 ± 0.03 h. We then turned to the June 13 and 19 data, which are separated by 6 days (approximately 42 cycles). The uncertainty in rotation phase over that time is ∼1/3 of a rotation, so it is just possible to have a half-cycle ambiguity in connecting these two lightcurves. The best fit value found was 3.398, with the value 3.3585 significantly poorer fit. We then fit the full set of 3 days and obtained a best fit value of 3.3962 ± 0.0002 h. There is some chance of an ambiguity, but it does seem unlikely that the value given by Pilcher is correct. We can note, however,

852 Wladilena. This asteroid was observed by Tedesco (1979) for more than 8 h, obtaining a lightcurve spanning just under two rotation cycles. From this he derived a period of 4.56 h. Our data on 6 nights over 2 months confirms and greatly improves this period determination. Our lightcurve (Fig. 52) is peculiar enough that we performed a Fourier scan of all possible periods from 3 h to 30 h to confirm that our solution, P = 4.6134 ± 0.0003 h, is unique. It can be noted that Tedesco’s lightcurve also exhibits very sharp maxima and relatively broader minima, although his lightcurve showed nearly equal maxima.

FIG. 52. Composite lightcurve of 852 Wladilena.

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Di Martino and Cacciatori (1984b) observed Wladilena on one night a few days later than our October observations, obtaining a lightcurve that confirms the peculiar shape of our composite. We did not try to combine their results with ours, as the increase in time span would not yield a significant improvement in our period determination, and their phase angle was intermediate between those we observed in September and October, thus no improvement in phase relation could be expected either. Finally, De Angelis and Mottola (1995) present lightcurves from two nights in November 1993, at a similar aspect to ours. Again the strange sharp maximum is confirmed, this time also showing unequal levels of the two maxima. They obtain a period of 4.612 ± 0.001 h, consistent with our value. 951 Gaspra. We observed Gaspra on two nights in 1982, long before it became the target of the first spacecraft flyby of an asteroid by Galileo. Our observations were included in the paper by Wisniewski et al. (1993). We include aspect data in Table II and include the observations in the machine readable data file, but do not reiterate the detailed analysis here. 1204 Renzia. This asteroid was observed on three consecutive nights in September 1982. No other physical observations (lightcurves, radiometry, or colorimetry) appear in the literature. The phase angle range was inadequate to yield a meaningful slope parameter, so a value G = 0.23 consistent with an inner belt asteroid was assumed, and the magnitudes from night-tonight so constrained. Although the data are a bit sparse, the period is uniquely determined by our data (Fig. 53). 1429 Pemba. Three nights of observations in September 1982 failed to yield a period, although it appears that the mean light levels from night-to-night are somewhat different, suggesting a long period. It is possible that this is due to observational errors though, so the result must be assigned a very low reliability. We present the data in a single plot, Fig. 54, in the same

FIG. 53. Composite lightcurve of 1204 Renzia.

FIG. 54. Individual lightcurves of 1429 Pemba, in the same format as Fig. 23.

style as Figs. 23 and 26. Pemba is a Mars-crossing asteroid, so even though it has no taxonomic classification or radiometric albedo, we assumed a slope parameter G = 0.23 appropriate for an inner belt asteroid for night-to-night light level corrections and deriving a value of H . 1862 Apollo. Our observations of Apollo were fully reported by Harris et al. (1987). We include the aspect data in Table II and the digital data in the machine readable file, but do not repeat the analysis here. 1863 Antinous. We observed this asteroid on only one night, March 24, 1982. Binzel (1987) observed it the night before and the night after from McDonald Observatory, and it was observed as a part of the Eight-Color Asteroid Survey (Zellner et al. 1985) on March 23 and 24, 1982. Binzel (1987) presented a composite lightcurve based on a (low reliability) period of 4.02 h. In Fig. 55 we show a combined solution with a period of 4.025 h, with all of the data with the night-to-night magnitude levels constrained by a nominal phase relation, which fits reasonably well, except for the ECAS point of March 24. In this and the next figure, we plot error bars to give an impression of the likely significance of the fits. Binzel (1987) did not provide error bars, so we have assumed uniform error bars of ±0.02 mag. Zellner et al. (1985) only listed time tags to 0.01 day in their tabulation of data, so we have indicated time error bars corresponding to ±0.005 day on those two points. A scan of all possible periods from 2.5 to 10 hours revealed a large number of possible periods. Lightcurves with periods shorter than ∼3.5 hours tended to be highly asymmetrical or only singly periodic; periods longer than ∼5 h tended to yield lightcurves that were triply periodic or more and quite irregular. Only periods in the range near 4 h seem likely. Figure 56 is one such solution, which illustrates the other basic possibility, which is to match up the rise from minimum of our March 24 lightcurve with the start of both of

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levels as constraining the phase relation and adjusted the ECAS magnitudes according to that relation. The magnitude level of the TMO lightcurve was left as a free parameter, since we cannot be sure of the absolute calibrations with respect to the other observatories. Since one of the ECAS points was taken during the interval of the Binzel lightcurve on 23 March, we have a direct check on the magnitude calibration between these observatories, which appears to be good. The zero points on the magnitude scales of the figures appear to be offset from the mean magnitude level of the composite lightcurves by about 0.03 mag, so the levels listed in Table I are adjusted by that amount. Even after all of this analysis, we must conclude that the period determined is still of very low reliability. The diameter and albedo are revised from Veeder et al. (1989) using our value of H .

FIG. 55. All data for 1863 Antinous, composited at the period 4.025 h proposed by Binzel (1987). The ECAS observation of March 24 is badly discordant.

the Binzel curves. Placing our lightcurve on the other minimum (Fig. 55) inevitably places the ECAS data point which was taken on the same night about 0.1 mag above the level measured by the Binzel lightcurves. The P = 4.386 h solution illustrated in Fig. 56 is the best of a few possibilities which avoid this problem, mainly by placing the March 24 ECAS point far from any other measurement. This lightcurve could be fairly regular, with minima at approximately 1.6 and 3.8 h and maxima at 0.8 and 2.6 h. It can be noted that all solutions place the two Binzel lightcurves in coincidence; thus we have taken those two light

FIG. 56. Composite lightcurve of 1863 Antinous for the period 4.386 h, with all the same data as Fig. 50. The previously discordant ECAS data point is compatible with this period, if only by being isolated.

3199 Nefertiti. We observed this asteroid, then known by its provisional designation 1982 RA, on one night in 1982. Our observations on that night and in 1984 are reported in the paper by Pravec et al. (1997). Here we include the aspect information (Table II), and the digital data are included in the machine readable data file. 3288 Seleucus. This asteroid was observed by Debehogne et al. (1983) on the night of March 13, 1982, and on March 21 by M. Carlsson (pers. commun., 1982), both from ESO. We observed it on seven nights in April. A preliminary examination of all the data indicated a long period and large amplitude of variation, but obtaining an exact composite fit proved to be impossible. Since the phase angle changed by only ∼1◦ during the entire span of the April run, and we maintained good magnitude standardization from night to night, we can confidently constrain the magnitude levels from night to night according to a nominal phase relation (G = 0.23, mean value for S class). The solid curve in Fig. 57 is a noise spectrum of the residuals to a fourth-order Fourier fit to the April observations with the magnitude levels constrained. The best period fit is around 75.8 h, but even this fit leaves an RMS residual 3.4 times the a priori estimated value. When we include the March data (dashed line, magnitudes constrained according to G = 0.23), the minimum at ∼75.8 h is split by two possibilities, with periods of 74.55 and 78.4 h, but neither fit is even as good as the Aprilonly fit. In Fig. 58, we show the 74.55-h solution to illustrate the systematic errors which are apparent from any simply periodic solution. Note in particular the systematic misfits of the 3 days’ data that fall near 0–10 h on the plot, from successive rotations. If one forces those successive cycles into a good fit, even larger systematic discrepancies open up in other parts of the composite curve. Thus we conclude that the light variation of 3288 Seleucus is not simply periodic, but is one of the class of “tumbling” asteroids, first noticed in the case of 4179 Toutatis (Hudson and Ostro 1995). Other examples of apparently “tumbling” rotation include 253 Mathilde (Mottola et al. 1995), 3691 1982 FT, and 1997 BR (Pravec et al. 1998). Harris (1994) has estimated the time scale of damping of the rotational wobble to a state of principal axis rotation to be about 5 × 109 years for

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FIG. 57. Noise spectrum of 3288 Seleucus. The solid curve is for the data gathered April 22–30 only, thus has relatively low resolution in the period domain. No period is satisfactory and the various minima are generally not harmonics of one another as is common for objects in principal-axis rotation states. The dashed curve is for all data, March and April. The April data dominates the noise level, but the March data provides a longer time base and hence higher resolution in the period domain. The one best solution at 37.31 h is singly periodic and unacceptable. The double period at 75.8 h is split when the March data are added, with neither value yielding an acceptable composite lightcurve.

Seleucus; thus a nonprincipal axis spin state is expected for such a small slowly spinning body. The damping time scales for the other asteroids mentioned are similarly long or longer. The total amplitude of variation of Seleucus appears to be ∼0.9 magnitude, and the period is ∼75 h, but since the rotational motion is not simply periodic, exact values cannot be defined. The diam-

FIG. 58. The composite lightcurve of 3288 Seleucus for the period 74.55 h. Note major systematic deviations in the fit, which cannot be reconciled with any simply periodic fit.

FIG. 59. Composite lightcurve of 3757 1982 XB.

eter and albedo are revised from Veeder et al. (1989) using our value of H . 3757 1982 XB. This asteroid was observed within 24 h of its discovery, at Table Mountain and by Tholen at the University of Arizona. It was also observed by Binzel at the McDonald Observatory. We here collect all observations, including those already published by Binzel (1987). Only a period near 9.0 h appears to fit the data. Our composite (Fig. 59) is somewhat noisy and not completely unique; thus we have assigned a reliability of 2 to the period result. We had to eliminate one point from Tholen’s data on Dec. 16 and a string of three points taken by Binzel on Dec. 18. The two lowest points on the composite, taken about half an hour later, are also suspect, so we do not include them in estimating the amplitude of variation. The phase

FIG. 60. Phase relation of 3757 1982 XB.

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relation (Fig. 60) is ordinary for a moderate albedo asteroid. The diameter and albedo are revised from Veeder et al. (1989) using our value of H .

Hainaut-Rouelle, M.-C., O. R. Hainaut, and A. Detal 1995. Lightcurves of selected minor planets. Astron. Astrophys. Suppl. Ser. 112, 125–142.

We thank R. P. Binzel, C. R. Chapman, V. Zappal`a, and M. Di Martino for providing some additional unpublished observations and M. Watt and T. Dockweiler for obtaining some of the observations at Lowell Observatory. This research was supported at the Jet Propulsion Laboratory, Caltech, under contract from NASA, and at Lowell Observatory and the University of Arizona under grants from NASA.

Harris, A. W. 1994. Tumbling asteroids. Icarus 107, 209–211. Harris, A. W. 1998. A thermal model for near-Earth asteroids. Icarus 131, 291–301. Harris, A. W., and A. W. Harris 1997. On the revision of radiometric albedos and diameters of asteroids. Icarus 126, 450–454. Harris, A. W., and D. F. Lupishko 1989. Photometric lightcurve observations and reduction techniques. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 39–53. Univ. of Arizona Press, Tucson. Harris, A. W., and J. W. Young 1980. Asteroid rotation. III. 1978 observations. Icarus 43, 20–32.

REFERENCES

Harris, A. W., and J. W. Young 1983. Asteroid rotation. IV. 1979 observations. Icarus 54, 59–109.

Binzel, R. P. 1987. A photoelectric survey of 130 asteroids. Icarus 72, 135–208.

Harris, A. W., and J. W. Young 1989. Asteroid lightcurve observations from 1979–1981. Icarus 81, 314–364.

ACKNOWLEDGMENTS

Black, G. J., P. D. Nicholson, W. F. Bottke, J. A. Burns, and A. W. Harris 1999. On a possible rotation state of (433) Eros. Icarus 140, 239–242. Bowell, E., B. Hapke, D. Domingue, K. Lumme, J. Peltoniemi, and A. W. Harris 1989. Application of photometric models to asteroids. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 524–556. Univ. of Arizona Press, Tucson. De Angelis, G. 1993. Three asteroid pole determinations. Planet. Space Sci. 41, 285–290. De Angelis, G. 1995. Asteroid spin, pole and shape determinations. Planet. Space Sci. 43, 649–682. De Angelis, G., and S. Mottola 1995. Lightcurves and pole determinations for the asteroids 69 Heperia, 79 Eurynome and 852 Wladilena. Planet. Space Sci. 43, 1013–1017. Debehogne, H., and V. Zappal`a 1980. Photoelectric lightcurves of the asteroids 139 Juewa and 161 Athor, obtained with the 50 cm photometric telescope at ESO, La Silla. Astron. Astrophys. Suppl. Ser. 42, 85–89. Debehogne, H., G. De Sanctis, and V. Zappal`a 1983. Photoelectric photometry of asteroids 45, 120, 776, 804, 814, and 1982 DV. Icarus 55, 236–244. Debehogne, H., C.-I. Lagerkvist, and V. Zappal`a 1982. Physical studies of asteroids VIII. Photoelectric photometry of the asteroids 42, 48, 93, 105, 145, and 245. Astron. Astrophys. Suppl. Ser. 50, 277–281. Denchev, P., P. Magnusson, and Z. Donchev 1998. Lightcurves of nine asteroids, with pole and sense of rotation of 42 Isis. Planet. Space Sci. 46, 673–682. De Young, J. A., and R. E. Schmidt 1996. The lightcurve and period of the C-type minor planet 38 Leda. Minor Planet Bull. 23, 44. Di Martino, M., and S. Cacciatori 1984a. Rotation periods and light curves of the large asteroids 409 Aspasia and 423 Diotima. Astron. Astrophys. 130, 206–207. Di Martino, M., and S. Cacciatori 1984b. Photoelectric photometry of 14 asteroids. Icarus 60, 75–82. Di Martino, M., C. Blanco, D. Riccioli, and G. De Sanctis 1994. Lightcurves and rotational periods of nine main belt asteroids. Icarus 107, 269–275. Di Martino, M., E. Dotto, A. Celino, M. A. Barucci, and M. Fulchignoni 1995. Intermediate size asteroids: Photoelectric photometry of 8 objects. Astron. Astrophys. Suppl. Ser. 112, 1–7. Dotto, E., G. De Angelis, M. Di Martino, M. A. Barucci, M. Fulchignoni, G. De Sanctis, and R. Burchi 1995. Pole orientation and shape of 12 asteroids. Icarus 117, 313–327. Dunlap, J. L. 1976. Lightcurves and the axis of rotation of 433 Eros. Icarus 28, 69–78. Gil Hutton, R. 1990. Rotation period and light curve of asteroid 423 Diotima. Bol. Asoc. Argent. Astron. 36, 193–199. Goguen, J., J. Veverka, J. L. Elliot, and C. Church 1976. The lightcurve and rotation period of asteroid 139 Juewa. Icarus 29, 137–142.

Harris, A. W., J. W. Young, E. Bowell, L. J. Martin, R. L. Millis, M. Poutanen, F. Scaltriti, V. Zappal`a, H. J. Schober, H. Debehogne, and K. W. Zeigler 1989a. Photoelectric observations of asteroids 3, 24, 60, 261, and 863. Icarus 77, 171–186. Harris, A. W., J. W. Young, L. Contreiras, T. Dockweiler, L. Belkora, H. Salo, W. D. Harris, E. Bowell, M. Poutanen, R. P. Binzel, D. J. Tholen, and S. Wang 1989b. Phase relations of high albedo asteroids: The unusual opposition brightening of 44 Nysa and 64 Angelina. Icarus 81, 365–374. Harris, A. W., J. W. Young, T. Dockweiler, J. Gibson, M. Poutanen, and E. Bowell 1992. Asteroid lightcurve observations from 1981. Icarus 95, 115– 147. Harris, A. W., J. W. Young, J. Goguen, H. B. Hammel, G. Hahn, E. F. Tedesco, and D. J. Tholen 1987. Photoelectric lightcurves of the asteroid 1862 Apollo. Icarus 70, 246–256. Harris, A. W., J. W. Young, F. Scaltriti, and V. Zappal`a 1984. Lightcurves and phase relations of the asteroids 82 Alkmene and 444 Gyptis. Icarus 57, 251– 258. Holliday, B. 1996a. Photometric observations of minor planet 50 Virginia. Minor Planet Bull. 23, 13. Holliday, B. 1996b. Photometric observations of minor planet 212 Medea. Minor Planet Bull. 23, 43. Hudson, R. S., and S. J. Ostro 1995. Shape and non-principal axis spin state of asteroid 4179 Toutatis. Science 270, 84–86. Lagerkvist, C.-I., M. A. Barucci, M. T. Capria, M. Fulchignoni, L. Guerriero, E. Perozzi, and V. Zappal`a 1987. Asteroid Photometric Catalog. CNR, Rome. Lagerkvist, C.-I., P. Magnusson, H. Debehogne, M. Hoffmann, A. Erikson, A. De Campos, and G. Cutispoto 1992. Physical studies of asteroids. XXV. Photoelectric photometry of asteroids obtained at ESO and Hoher List Observatory. Astron. Astrophys. Suppl. Ser. 95, 461–470. Lebofsky, L. A., and G. H. Rieke 1979. Thermal properties of 433 Eros. Icarus 40, 297–308. Magnusson, P. 1986. Distribution of spin axes and senses of rotation for 20 large asteroids. Icarus 68, 1–39. Millis, R. L., E. Bowell, and D. T. Thompson 1976. UBV photometry of Asteroid 433 Eros. Icarus 28, 53–67. Millis, R. L., L. H. Wasserman, E. Bowell, O. G. Franz, A. Klemola, and D. W. Dunham 1984. The diameter of 375 Ursula from its occultation of AG + 39◦ 303. Astron. J. 89, 592–596. Millis, R. L., L. H. Wasserman, E. Bowell, O. G. Franz, R. Nye, W. Osborn, and A. Klemola 1985. The occultation of AG + 39◦ 303 by 93 Minerva. Icarus 61, 124–131. Morrison, D. 1977. Asteroid sizes and albedos. Icarus 31, 185–220. Mottola, S., W. D. Sears, A. Erikson, A. W. Harris, J. W. Young, G. Hahn, M. Dahlgren, B. E. A. Mueller, B. Owen, R. Gil-Hutton, J. Licandro, M. A.

ASTEROID LIGHTCURVES Barucci, C. Angeli, G. Neukum, C.-I. Lagerkvist, and J. F. Lahulla 1995. The slow rotation of 253 Mathilde. Planet. Space Sci. 43, 1609–1613. Pilcher, F. 1983. Visual lightcurves of 321 Florentina, 699 Hela and 1192 Prisma. Minor Planet Bull. 10, 18–20. ˇ Pravec, P., M. Wolf, L. Sarounov´ a, S. Mottola, A. Erikson, G. Hahn, A. W. Harris, A. W. Harris, and J. W. Young 1997. The near-Earth objects followup program. II. Results for 8 asteroids from 1982 to 1995. Icarus 130, 275– 286. ˇ Pravec, P., M. Wolf, L. Sarounov´ a 1998. Lightcurves of 26 near-Earth asteroids. Icarus 136, 124–153. Scaltriti, F., and V. Zappal`a 1976. Photometric lightcurves and pole determination of 433 Eros. Icarus 28, 29–35. Scaltriti, F., V. Zappal`a, and A. W. Harris 1981. Photometric lightcurves and rotation periods of the asteroids 46 Hestia and 115 Thyra. Icarus 46, 275– 280. Schober, H. J. 1983. The large C-type asteroid 423 Diotima: Rotation period, lightcurve and implications for a possible satellite. Astron. Astrophys. 127, 301–303. Schober, H. J. 1987. Rotation and variability of the large C-type asteroid 375 Ursula. Astron. Astrophys. 183, 151–155. Shevchenko, V. G., I. N. Belskaya, V. G. Chiorny, J. Piironen, A. Erikson, G. Neukum, and R. Mohamed 1997. Asteroid observations at low phase angles. I. 50 Virginia, 91 Aegina and 102 Miriam. Planet. Space Sci. 45, 1615–1623. Spencer, J. R., and 48 colleagues 1995. The lightcurve of 4179 Toutatis: Evidence for complex rotation. Icarus 117, 71–89. Taylor, R. C., P. V. Birch, A. Pospieszalska-Surdej, and J. Surdej 1988. Asteroid 45 Eugenia: Lightcurves and the pole orientation. Icarus 73, 314–323. Tedesco, E. F. 1976. UBV lightcurves of Asteroid 433 Eros. Icarus 28, 21– 28. Tedesco, E. F. 1979. A Photometric Investigation of the Colors, Shapes and Spin Rates of Hirayama Family Asteroids. Ph.D. dissertation, New Mexico State Univ. Tedesco, E. F. 1989. Asteroid magnitudes, UBV colors, and IRAS albedos and diameters. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1090–1138. Univ. of Arizona Press, Tucson. Tedesco, E. F., D. J. Tholen, and B. Zellner 1982. The eight-color asteroid survey: Standard stars. Astron. J. 87, 1585–1592.

201

Tedesco, E. F., G. V. Veeder, J. W. Fowler, and J. R. Chillemi 1992. The IRAS Minor Planet Survey, Tech. Rep. PL-TR-92-2049. Phillips Laboratory, Hanscom AFB, MA. Tedesco, E. F., J. G. Williams, D. L. Matson, G. J. Veeder, J. C. Gradie, and L. A. Lebofsky 1989. Three-parameter asteroid taxonomy classifications. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1151– 1161. Univ. of Arizona Press, Tucson. Tempesti, P., and R. Burchi 1969. A photometric research on the minor planet 12 Victoria. Mem. Soc. Astron. Ital. 40, 415–432. Tholen, D. J. 1989. Asteroid taxonomic classifications. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1139–1150. Univ. of Arizona Press, Tucson. Veeder, G. J., M. S. Hanner, D. L. Matson, E. F. Tedesco, L. A. Lebofsky, and A. T. Tokunaga 1989. Radiometry of near-Earth asteroids. Astron. J. 97, 1211–1219. Veverka, J., P. C. Thomas, J. F. Bell III, M. Bell, B. Carcich, B. Clark, A. Harch, J. Joseph, P. Martin, M. Robinson, S. Murchie, N. Izenberg, E. Hawkins, J. Warren, R. Farquhar, A. Cheng, D. Dunham, C. Chapman, W. J. Merline, L. McFadden, D. Wellnitz, M. Malin, W. M. Owen Jr., J. K. Miller, B. G. Williams, and D. K. Yeomans 1999. Imaging of Asteroid 433 Eros during NEAR’s flyby reconnaissance. Science 285, 562–564. Weidenschilling, S. J., C. R. Chapman, D. R. Davis, R. Greenberg, D. H. Levy, R. P. Binzel, S. M. Vail, M. Magee, and D. Spaute 1990. Photometric geodesy of main-belt asteroids. III. Additional lightcurves. Icarus 86, 402–447. Wisniewski, W. Z., M. A. Barucci, M. Fulchignoni, C. De Sanctis, E. Dotto, A. Rotundi, R. P. Binzel, C. D. Madras, S. F. Green, M. L. Kelly, P. J. Newman, A. W. Harris, J. W. Young, C. Blanco, M. Di Martino, W. Ferreri, M. Gonano-Beurer, S. Mottola, D. J. Tholen, J. D. Goldader, M. Coradini, and P. Magnusson 1993. Ground-based photometry of Asteroid 951 Gaspra. Icarus 101, 213–222. Wisniewski, W. Z., T. M. Michalowski, A. W. Harris, and R. S. McMillan 1997. Photometric observations of 125 asteroids. Icarus 126, 395–449. Young, J. W., and A. W. Harris 1985. Photoelecric lightcurve and phase relation of the asteroid 505 Cava. Icarus 64, 528–530. Zappal`a, V., M. Di Martino, A. Hanslmeier, and H. J. Schober 1985. New cases of ambiguity among large asteroids’ spin rates. Astron. Astrophys. 147, 35–38. Zellner, B., D. J. Tholen, and E. F. Tedesco 1985. The eight-color asteroid survey: Results for 589 minor planets. Icarus 61, 355–416.