Rotation properties of four L5 Trojan asteroids from CCD photometry

ICARUS 72, 3 2 5 - 3 4 1

(1987)

Rotation Properties of Four L5 Trojan Asteroids from CCD Photometry L I N D A M. F R E N C H 1 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received March 30, 1987; revised June 22, 1987 CCD observations made with the Cerro Tololo 0.9-m telescope of four L5 Trojan asteroids during June and July of 1986 are presented. Partial lightcurves for 1208 Troilus and 1867 Deiphobus suggest long periods (P > 24 hr). Complete lightcurves were obtained for two asteroids, 1173 Anchises (P = 11.6095 ± 0.0036 hr, Am --- 0.57 -- 0.01 mag) and 2674 Pandarus (P = 8.4803 ± 0.0019, Am = 0.58 ± 0.01). The large iightcurve amplitudes are unique: from a sample of 182 main-belt asteroids, A. W. Harris and J. A. Burns (1979, Icarus 40, 115-144) found none in a comparable size range (50-100 km) with amplitudes greater than 0.5 mag. We suggest that the Trojans may represent a primordial asteroid population, with shapes not significantly altered since the formation of the solar system. Sufficient phase angle coverage was attained for 1173 Anchises to determine a near-opposition phase coefficient. No opposition effect was observed. The two-parameter function of E. Bowell, A. W. Harris, and K. Lumme (1985, preprint) and a linear phase coefficient both fit the data well. However, the multiple scattering parameter derived from the Bowell et al. model implies a geometric albedo of near unity, in conflict with the low radiometric albedos observed for Anchises and other Trojans (D. P. Cruikshank, 1977, Icarus 30, 224-230). The lack of an opposition effect implies that the surface of Anchises may be significantly less rough than other asteroid sufaces. © 1987 Academic Press, Inc.

INTRODUCTION O v e r the past 10 years, the n u m b e r of main-belt asteroids with k n o w n rotation properties has risen dramatically (see, for example, Burns and T e d e s c o 1979, Binzel 1984). This w o r k has made possible searches for trends in rotation period and amplitude with size, composition, heliocentric distance, and family m e m b e r s h i p a m o n g the main-belt asteroids; b e c a u s e of various observational selection effects m a n y of the a t t e m p t e d surveys have c o m e to conflicting and contradictory results (Harris and Burns 1979, T e d e s c o and ZappalS. 1980, D e r m o t t et al. 1984, Binzel 1984). N o n e t h e l e s s , for the main-belt asterVisiting Astronomer, Cerro Tololo Interamerican Observatory, La Serena, Chile. 325

oids a sufficient data base exists to pose, and to answer, basic physical questions about the rotation of asteroids. Because of their great heliocentric distance and their corresponding faintness, little is known abut the rotation properties of the Trojan asteroids. The objects themselves, however, are of considerable interest, because of recent discoveries about both their composition and the possible evolution of their orbits. In an a t t e m p t to explain the dark, reddish surfaces of D-type asteroids, which m a k e up m o r e than half of the Trojans, Gradie and V e v e r k a (1980) suggested that one could r e p r o d u c e the spectra of D material b y a mixture of silicates with c a r b o n a c e o u s c o m p o u n d s e v e n m o r e primitive than those found in the c a r b o n a c e o u s chondrites. This is in accord with current condensation the0019-1035/87 $3.00 Copyright© 1987by AcademicPress,Inc. All rightsofreproductionin anyformreserved.

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LINDA M. FRENCH

ories about the formation of the solar sys- two nearly equally sized spherical asteroids tem, and is supported by the CCD spectral (Hartmann and Cruikshank 1978), and a bistudies of Vilas and Smith (1985), who ob- nary system of two Darwin ellipsoids served an increasing reddening of asteroid nearly in contact (Weidenschilling 1980, spectra with heliocentric distance among Farinella et al. 1981). Many binary models the Cybele, Hilda, and Trojan groups of as- involve a collisional origin; to evaluate the teroids (found at 3.4, 4.0 and 5.2 AU, re- likelihood of such models one must conspectively). Eight-color photometry of the sider the frequency and relative velocity of outer Jovian satellites, at the same helio- collisions in the Trojan regions. Hartmann centric distance as the Trojans, however, (1979), for example, finds the dumbbell shows that these objects are probably shape to be a likely end product of a low mostly C type (Tholen and Zellner 1984). relative velocity collision (v < 1 km/sec) This "mixing" of C and D types--D's in and postulates that such low-velocity collithe Trojan groups and C's in the Jovian sys- sions are more common among the Trojans t e m - p o s e s a complication for the standard than among the main-belt asteroids, beformation model of direct correlation be- cause of their "clustering" near the Jovian tween asteroid composition and heliocen- L4 and L5 points. If such is the case, one tric distance. Tholen and Zellner raised the might expect a higher fraction of elongated possibility that objects composed of D ma- objects among the Trojans than in the main terial formed at a greater heliocentric dis- belt. The most direct way to find out tance than Jupiter's; there is evidence for whether elongated objects are more or less this scenario in the identification of D mate- common among the Trojans, of course, is rial in the Saturnian satellite system and through observation of their lightcurves also in the similarities between the contin- over time. Davis and Weidenschilling uum spectra of some old comets to those of (1981) show that the apparent lower volume D objects (Hartmann et al. 1987). The com- density of Trojans compared to main-belt positional evidence, therefore, suggests asteroids should have produced few, if any, that the Trojans may not have formed at catastrophic collisions among Trojans since their present location, but further out, and the formation of the solar system; the could be related to comets. present-day rotation properties of the TroPrior to this work, a well-deterimined jan asteroids may therefore be indicative of lightcurve existed for only one Trojan, 624 asteroid rotation states during the formaHektor (Dunlap and Gehrels 1969). The tion of the solar system. lightcurve has a maximum amplitude of In summary, both compositional studies 1.09 mag; if due to projected area variations and dynamical models suggest that the Troalone, this implies an axial ratio of almost jan asteroids may have a history signifi3: 1. Hektor's extreme brightness varia- cantly different from that of main-belt astion, its large size (the dimensions are esti- teroids. Recent lightcurve studies of the mated to be approximately 150 × 300 km by larger Hirayama families by Binzel (1986) Hartmann and Cruikshank 1978), and its are persuasive that, from a statistically sigrelatively short rotation period of 6.92 hr nificant number of lightcurves, we can learn have led many to suspect that the asteroid about the collisional history of families and is not a single elongated body held together groups of asteroids. Such a survey of the by internal material strength, but either a Trojans, with enough evidence to draw figure of hydrostatic equilibrium or a binary conclusions about their origin, is the goal of asteroid. The models proposed include a re- this ongoing study. assembled brecciated ellipsoid (Cook 1971, The present paper presents the results Poutanen et al. 1981), a dumbbell-shaped from the first observations of L5 Trojans; object formed by the partial coalescence of details of the observing strategy are also

TROJAN ASTEROID LIGHTCURVES included and the method of period determination is described in detail.

327

an important one for efficient lightcurve observing.

Instrumentation OBSERVATIONS AND DATA ANALYSIS

Observing Strategy In June and July of 1986, many of the L5 Trojans were located in the general direction of the galactic center. In an effort to avoid overcrowded star fields, the asteroid paths were studied on the Palomar Sky Survey while the observations were planned. An asteroid was put on the observing program if it had a relatively clear path through the stars for at least 3 nights, with enough stars of comparable brightness on one CCD frame to make differential photometry possible. Differential photometry between the asteroid and comparison stars within a single frame (typically a few minutes of arc on a side) enables one to ignore extinction corrections and possible transparency variations in a quick search for a period of variation. In addition, UBVRI standard stars from the published lists of Landolt (1983) and Graham (1982) were selected to determine the absolute V magnitude of the asteroid and comparison stars on each night, so that several nights' observations could be tied together on the absolute system. When looking for time variability of objects whose period is initially unknown, one must always trade off coverage of a large number of objects against high time resolution on the lightcurve of any one. In this pilot observing run, brightness variations were monitored throughout a night of observing by means of on-line photometry software; a plot of each night's data was studied before planning the next night's observing. In this way the observing plan could be modified to give relatively shortperiod asteroids the attention needed for complete phase coverage. Thus, the goal at the beginning of the run was to have raw magnitudes for each lightcurve at the end of each night of observing. By the end of the run, that goal was in large part attained; it is

All observations were made using the 0.9-m Cassegrain reflector of the Cerro To1olo Inter-American Observatory (CTIO) and that observatory's Cassegrain Focus Charge-Coupled Device (CFCCD) detector with a GEC chip. The chip was preflashed before each exposure to a level of approximately 200 electrons (full well corresponds to aproximately 33,000 electrons) to improve charge transfer. Each data frame was corrected for the bias level by subtracting the average value for the overscan region on that frame. Flat fields were obtained by observing a white screen, illuminated by color-balanced lamps, through each of the two filters used for the observations. Between 24 and 32 flat frames for each filter were corrected for the bias level, averaged, and median filtered to remove the effects of readout noise. The dark current of the CFCCD is negligible because the amplifier is turned off during exposures, making dark frames unnecessary. The telescope was tracked at the sidereal rate, leaving the asteroid images slightly trailed over the typical exposure times of 300-500 sec. However, because of the low readout noise of the GEC chip (approximately 12 electrons per pixel), the scatter introduced was minimal, and slightly larger aperture sizes allowed all the asteroid light to be measured.

Photometric Reduction After the data frames were corrected for bias and flat field effects, the KPNO Mountain Photometry Code program was used for aperture photometry of the asteroid and between one and four comparison stars on the frame. Typically, aperture sizes of 1318 pixels were used, corresponding to 3.9 to 5.4 arcsec at the plate scale of the 0.9-m telescope. Internal errors are computed based on the photon noise from the star and sky signals and on the uncertainty in the

328

LINDA M. FRENCH

sky level determination. Typical internal uncertainties are -<0.005 mag for objects of V = 15-17. F o r the final reduction, a leastsquares fitting routine was used to determine the extinction, transformation, and zero-point coefficients for each night. The transformation coefficients for photometric nights were averaged, and the mean was used in final reductions of the asteroid data. For nights which were only marginally photometric, or for which observing a shortperiod asteroid made a large number of standard star observations impossible, the mean extinction and zero-point coefficients were used. The photometric coefficients in V and R are given in Table I. The final error bars shown on the data points (typically, -+0.02-0.03 mag) include the internal uncertainty as well as uncertainties in the photometric coefficients propagated in quadrature; uncertainties in the extinction and zero-point corrections dominate the error bars for the absolute photometry. Period Determination

After the photometric corrections were applied to each set of asteroid data, the preliminary plots made at CTIO were used to estimate a trial rotation period, and the data were folded together based on that period. Visually one can determine the best fit quite well for a short-period, large-amplitude asteroid with data from several nights. To search for periods in less ideal cases, a modified version of the phase-dispersion minimization technique (hereafter, PDM) first described by Stellingwerf (1978) was developed. In Stellingwerf's approach, the data are

TABLE

I

PHOTOMETRIC COEFFICIENTS

binned according to the rotational phase based on a trial period, and the average variance of the binned subsets of the observations is compared to the overall variance of the full set of the observations. When the " t r u e " period is selected, the variance within any given bin is small, since all the data within the bin are from the same region of the true mean curve and hence have similar mean intensity. On the other hand, when an erroneous period is selected, a given bin could well contain data from both the maxima and the minima of the true curve, and the scatter would typically be larger. Thus, the best period is that for which the ratio of the average variance within the bins to the overall variance is minimized. A characteristic of Stellingwerf's implementation is that, even when the data are noise free and the period is known exactly, the variance with respect to the m e a n within any bin is nonzero because, in general, the lightcurve is not constant over the width of a bin. In practice, if the mean lightcurve has a large slope within a given bin, the variance with respect to the mean could be much larger than the variance associated with the observational error of the data themselves. We have modified Stellingwerf's method by defining the " m e a n lightcurve" within each bin to be the bestfitting straight line to the data within the bin, rather than the mean of the data within the bin. This is a more realistic approximation; we have found that the revised method is better at discriminating between the true period and nearby aliases. Following Stellingwerf's notation, we represent the set of N observations as {Xi, ti}, where xi is the ith magnitude observed at time ti, and we define the variance of the ensemble as N

Filter

V R

Extinction coefficient

Transformation coefficient

0 . 1 4 -+ 0.01 0.11 -+ 0 . 0 2

0.08 -+ 0.01 0 . 0 0 -+ 0 . 0 2

o-2 _ i = l N _

1

'

(1)

where £ is the mean of the entire set of observations. For a trial period, II, we de-

TROJAN ASTEROID LIGHTCURVES fine the rotational phase as ~bi = ti mod(II), and bin the data into M samples by requiring that all members of the j t h sample have phase ~bi within a specified range. We approximate the variance of each sample about the mean lightcurve as Ix, - fj(xk)] 2 k=l

sj- =

nj - 2

'

(2)

where nj is the number of members of the j t h sample and the best-fitting straight line to the points in the sample is defined as fj(xk) = ajxk + bj,

(3)

where at and bj are determined from a fit to the data within the j t h bin. (In contrast, Stellingwerf uses Eq. (1) to approximate the sample variance about the mean lightcurve.) The overall variance for all the samples is given by M

E

(ni-

M

(4) -

l)

j-I

The measure of goodness of the fit is the ratio S2 =

ventionally expressed in magnitudes per degree. Equation (6) is frequently used because of its simple form, but it is devoid of physical content. The phase function first developed by Lumme and Bowell (1981) is more directly associated with the scattering properties and physical makeup of the surface and offers some prospect of relating the observed brightness variation with phase to actual conditions on the surface of the asteroid. The form of the relation used here, from Bowell et al. (1985), hereafter referred to as BHL, has two free parameters: G, a term describing the importance of multiple scattering for the surface material, and H, the magnitude at zero-phase angle with the opposition effect included. The functional form of the two-parameter relation is V(1, c0 = H - 2.5 log10 [(1 - G)qbl(a ) + Gqb2(a)], (7)

1)s 2

$2 ~- j = l

0

329

--

0-2,

(5)

where 0 reaches a local minimum near the correct period, II. For each data set, we determined 0 for a wide range of periods, and used Stellingwerf's (1978) recommended binning scheme to help reduce aliasing. Phase Functions

To describe the brightness variations with solar phase angle, two relations were used. A linear phase relation relates the zero-phase magnitude at unit heliocentric and geocentric distance, V(1, 0), to the magnitude at phase angle a, as V(1, ~) = V(1, 0) + a/3,

(6)

where/3 is the linear phase coefficient, con-

where ~ ( a ) is an empirical function fitted to observations of low-albedo asteroids and • 2(a) is fitted to observations of brighter asteroids and satellites, for which multiple scattering is more important. The simple form of the BHL relation is intended to give a measure of the albedo of the surface material for surfaces with textures similar to those of the asteroids which were used to derive ~l(a) and ~2(a). Major differences in surface roughness or porosity will give values of G which might not give accurate values for the albedo of the astero•d surface material. When deriving phase relations from observations of individual asteroids, it is common practice to use only the observed magnitude at one maximum of each rotational cycle; thus most of the data in the lightcurve are not used in the fit. It is desirable to maintain the information in the present large data set, yet to do so it is necessary to remove the effects of the large-amplitude rotational lightcurve. To do this, a Fourier component fitting routine has been used to model the shape of the lighteurve and sub-

LINDA M. FRENCH

330

TABLE II

ASTEROIDPHYSICALPARAMETERS Asteroid

Taxonomic class

D (km)

TRIAD V(1, 0)

Geometric albedo

D (km)

IRAS V(1, 0)

Geometric albedo

1173 Anchises 1208 Troilus 1867 Deiphobus 2674 Pandarus

P FCU D D

92 103 127 --

9.08 9.12 8.68 --

0.047 ----

135 111 131 102

8.91 9.00 8.60 9.05

0.026 0.036 0.037 0.041

Note: Taxonomic classifications are from Tholen (1984). Diameters, magnitudes, and geometric albedos are from the TRIAD file (Zellner 1979) and the IRAS asteroid catalog (Tedesco 1986).

tract its contribution to the overall brightness variation. The lightcurve as a function of time and p h a s e angle is represented as V(A, r, c~, t) = V(1, c~) + 5 log10 (Ar) - ~ Al sin(~bt(t) - ~b0j), (8)

l-1

where A is the distance from the Earth to the asteroid in A U , r is the heliocentric asteroid distance in A U , V(1, a) is the phase function, given by Eq. (6) or (7), and the rotational c o m p o n e n t of the lightcurve is represented by m Fourier c o m p o n e n t s of amplitude At and phase offset ~b01. The rotational p h a s e of a given observation is

vary with p h a s e angle (see, for example, Zappal~t et al. 1983), any variation should be small for the c o m p a r a t i v e l y narrow range of phase angles c o v e r e d by our observations. In practice, four Fourier components were used in the fits (m = 4). These produced a good m a t c h to the m e a n curve without introducing artificial wiggles in the model rotation curve. This method is best suited for well-sampled lightcurves; when the c o v e r a g e in rotational phase is incomplete, the n u m b e r of Fourier c o m p o n e n t s should be reduced. RESULTS 1173 A n c h i s e s

~l(t) = 277"mod [l(' ~l-i/ref)],

(9)

where tref is a reference time chosen for c o n v e n i e n c e to be just prior to the onset of observations, and II is the a s s u m e d rotation period. The least-squares fit solves simultaneously for the best-fitting phase function, the rotational c o m p o n e n t of the lightcurve (expressed in terms of the Fourier amplitudes and phases), and the rotation period. The Fourier model to the rotational lightcurve is not m e a n t to imply any knowledge of the true shape or rotation state of the asteroid; it is simply an empirical way of parameterizing the o b s e r v e d rotational variation of the asteroid so that this c o m p o n e n t of the light variation m a y be removed. Although asteroid lightcurves are known to

This asteroid was o b s e r v e d on 5 nights at low-phase angle (0.3 ° -< o~ -< 2.0°). The physical properties of Anchises and the other p r o g r a m asteroids determined by the I R A S satellite and for the T R I A D data file are given in Table II. We note especially the very low ratiometrically determined geometric albedo, which is typical of P- and D-type asteroids. The observational data are presented in Table III, and the lightcurve corrected for atmospheric extinction and transformation but without any correction for differing phase angle or distance is shown in Fig. 1. The period is shown by both the P D M analysis (Figs. 2 and 3) and direct overlay of the lightcurves to be 11.61 hr; note that the deep minimum in the PDM curve at 5.8 hr c o r r e s p o n d s to one-half the

TROJAN ASTEROID LIGHTCURVES

331

T A B L E III OBSERVATIONS OF 1173 ANCHISES Date 3 July 3 July 3 July 3 July 3 July 3 July 3 July 3 July 3 July 3 July 3 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 4 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 5 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July

1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

Time (UT) 1:36 1:41 3:55 4:02 4:10 4:18 4:26 4:33 4:41 4:49 4:56 2:30 2:40 4:31 4:39 4:47 4:55 5:30 5:37 5:56 6:04 6:19 6:27 7:48 7:56 8:04 8:12 1:31 1:40 1:48 1:55 2:03 2:10 2:18 5:26 5:34 5:42 5:49 5:57 7:41 7:49 7:57 8:04 8:13 8:21 1:12 1:19 1:35 1:43 2:45 3:04 3:12 4:02

R A (1950.0) 19 24 19 24 19 24 19 24 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 23 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 22 19 20 19 20 19 20 19 20 19 20 19 19 19 19 19 19

03 03 00 00 59 59 59 59 59 58 58 28 27 25 25 24 24 23 23 23 23 22 22 20 20 20 20 55 55 54 54 54 54 54 49 49 49 49 49 46 46 46 46 45 45 02 02 01 01 00 59 59 58

Dec (1950.0) -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21

22 22 22 22 22 22 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23

52 52 53 53 53 53 53 53 53 54 54 02 02 03 ~33 03 03 03 03 03 03 03 03 04 04 04 04 11 11 11 11 11 11 11 12 12 13 13 13 13 13 13 13 13 14 57 57 57 58 58 58 58 58

a

A

r

Vob~

V(1, 0) ~

1.991 1.991 1.969 1.968 1.967 1.965 1.964 1.963 1.962 1.960 1.959 1.750 1.749 1.731 1.729 1.728 1.727 1.721 1.720 1.717 1.716 1.713 1.712 1.699 1.697 1.696 1.695 1.527 1.525 1.524 1.523 1.522 1.520 1.519 1.489 1.487 1.486 1.485 1.484 1.467 1.466 1.464 1.463 1.462 1.460 0.380 0.379 0.377 0.376 0.367 0.364 0.363 0.356

3.663 3.663 3.663 3.663 3.663 3.663 3.663 3.662 3.662 3.662 3.662 3.660 3.660 3.660 3.660 3.660 3.660 3.660 3.660 3.660 3.660 3.660 3.659 3.659 3.659 3.659 3.659 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.657 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649

4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.669 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.668 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665

15.47 15.45 15.46 15.49 15.53 15.57 15.60 15.66 15.61 15.75 15.77 15.30 15.35 15.81 15.81 15.81 15.81 15.76 15.77 15.76 15.68 15.56 15.51 15.34 15.35 15.34 15.35 15.34 15.35 15.38 15.39 15.39 15.39 15.45 15.64 15.57 15.50 15.43 15.44 15.37 15.39 15.39 15.43 15.44 15.47 15.73 15.70 15.53 15.49 15.29 15.31 15.30 15.36

9.26 9.24 9.25 9.28 9.32 9.36 9.39 9.45 9.40 9.54 9.56 9.10 9.15 9.60 9.61 9.61 9.61 9.56 9.57 9.56 9.48 9.36 9.31 9.14 9.15 9.14 9.15 9.14 9.16 9.19 9.19 9.19 9.19 9.26 9.44 9.37 9.31 9.23 9.25 9.17 9.19 9.20 9.23 9.25 9.28 9.57 9.53 9.36 9.33 9.12 9.15 9.13 9.20

332

L I N D A M. FRENCH T A B L E Ill--Continued

Date 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July 10 July

Time (UT)

1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

R A (1950.0)

4:12 4:20 4:27 4:38 4:48 4:55 5:04 5:12 6:09 6:17 6:59 7:10 7:18 7:25 7:33 7:40 7:48 7:55 8:03 8:11 8:18 8:26 8:33 8:41 8:48 8:56 9:03

19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

Dec (1950.0)

58 57 57 57 57 57 56 56 55 55 54 53 53 53 53 53 52 52 52 52 52 51 51 51 51 51 51

-21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 -21 21 -21 -21 21 -21 21 -21 -21 -21 -21 -21 -21

23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24

58 59 59 59 59 59 59 59 59 59 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

c~

z~

r

Vob s

V(l, 0) a

0.354 0.353 0.352 0.351 0.349 0.348 0.347 0.346 0.337 0.336 0.330 0.329 0.328 0.327 0.326 0.325 0.323 0.323 0.321 0.320 0.319 0.318 0.317 0.316 0.315 0.314 0.313

3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649 3.649

4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665 4.665

15.40 15.42 15.40 15.49 15.52 15.56 15.60 15.64 15.77 15.71 15.69 15.63 15.58 15.50 15.48 15.42 15.38 15.34 15.31 15.32 15.27 t5.25 15.24 15.23 15.23 15.24 15.24

9.23 9.26 9.23 9.33 9.36 9.39 9.44 9.48 9.60 9.55 9.53 9.47 9.42 9.34 9.32 9.26 9.21 9.18 9.15 9.15 9.11 9.08 9.08 9.07 9.07 9.08 9.08

Zero-phase angle magnitude calculated using a linear phase coefficient of 0.023 -+ 0.008 mag/deg. See text for details.

15.2 , 15.3

~

15.4

1.0

" 15.5

0.8

¢

~ v >

15.6 ra

q}_

15.7

0.6

I--0.4

15.8 15.9 0.0

i

t

0.2

i

i

i

0.4 Rotational

i

0.6

L

i

0.8

0.2

i

1.0

Phase

FIG. t. O b s e r v a t i o n s of 1173 A n c h i s e s , uncorrected for distance and solar-phase angle. The different symbols denote observations from different nights. Squares denote 2 - 3 July data, triangles 3 - 4 July, c r o s s e s 4 - 5 July, and circles 9 - 1 0 July. The observational error bars are d o m i n a t e d by uncertainties in the extinction and zero-point coefficients.

o o.o

5.0

lO.O

15.o

2o.o

25.o

30.0

Period (hrs)

FIG. 2. P h a s e dispersion minimization (PDM) plot for 1173 A n c h i s e s . T h e t a is a m e a s u r e of the goodness of the fit for a trial period (see text for details). The deepest m i n i m u m , at ~5.8 hr, c o r r e s p o n d s to one-half of the rotational period.

TROJAN

ASTEROID

LIGHTCURVES

333

9.£

1.0 9.1 9.;' 9.-7:

0.6 E-

>

9.4

0.4 9.5 0.2 9.e 0 8.0

i

i 9.0

i 10.0

11.0

12.0

9.7 0.0

13.0

0.2

0.4

P e r i o d (hrs)

FIG. 3. High-resolution P D M scan for A n c h i s e s , showing the rotational period near 11.6 hr.

rotational period. The raw lightcurve has a peak-to-peak amplitude of about 0.6 mag. It is somewhat surprising that the lightcurves from different nights superpose so well without any correction for differing phase angles, because for most dark asteroids and satellites the opposition effect--a strong, nonlinear surge in brightness below tx 6°--produces a brightening of up to 0.15 mag/deg at such low-phase angles (see, for example, the observations of the Uranian satellites by Brown and Cruikshank (1983) and those of Franz and Millis (1975) for three Saturnian satellites). The lightcurve, corrected for distance and solar phase angle, is shown in Fig. 4. The values derived for the linear phase coefficient are quite small for an object as dark as Anchises. The geometric albedo predicted by the simple two-parameter BHL model is nearly unity, based on the observed value of G. Cruikshank (1977) and the IRAS satellite (Tedesco 1986) observed the albedo of Anchises to be only a few percent, ruling out such a high albedo. We conclude that its surface is different from that of the asteroids used by Bowell et al. (1985) to define their phase relation. The moderate opposition effect could be produced by an unusually smooth surface, either at the macroscopic or microscopic scale. The typical BHL values of G for C asteroids, which are of approximately the

0.6

0.8

1.0

Rotational Phase FIG. 4. A n c h i s e s lightcurve, corrected for distance and for solar-phase angle with the best-fit linear phase coefficient derived from this data set. The m e a n rotation curve obtained from the f o u r - c o m p o n e n t Fourier fit is also s h o w n .

same albedo, is 0.15; the curve corresponding to this value is shown with the bestfitting models superposed on the data in Fig. 5. Although some scatter remains in the magnitudes because the effects of shape have not been completely removed, the Casteroid curve is clearly a poor fit. The BHL and linear phase function fits are virtually identical and match the data much better. Although the range of phase angles covered by these observations is small, the 9.0 "E

9.1

°°

9.2

8E

9.3

E

_.e

13

~

9.4 r't" 9.5

....

Linear Phase Coefficienl BowelI-Harris-Lumme Model

o

N

9.6 9.'~ 0.0

Typical C Object i

i

i

i

r

I

i

1.0

l

i

2.0

Phase Angle (deg) FIG. 5. O b s e r v e d m a g n i t u d e s m i n u s the four-component Fourier model fit for A n c h i s e s as a function of solar-phase angle. S u p e r i m p o s e d on the data are the best-fit straight line (the best-fit B H L curve is virtually identical to the straight line) and the B H L curve appropriate for m o s t dark C objects, for which G = 0.15 (E. Bowell, personal communication). T h e C-object curve is clearly a poor match.

334

LINDA M. FRENCH

2674 Pandarus

T A B L E IV FITTED

ROTATION

Parameter Period (hr) Am (max - min) /3 (mag/deg) V(I, 0) G H (V mag)

AND

PHOTOMETRIC

1173 Anchises 11.6095 0.57 0.023 9.35 1.05 9.35

-+ 0.0036 +- 0.01 -+ 0.008 -+ 0.01 -+ 0.09 -+ 0.01

PROPERTIES

2674 Pandarus 8.4803 -+ 0.0019 0.58 -+ 0.0l (0.023)" 9.70 +- 0.01 (1.05)" 9.70 -+ 0.01

Insufficient phase angle coverage to determine phase coefficient. Adopted values shown in parentheses.

goodness of the fit implies that the phase curve has been adequately sampled to determine reliable values of the fitted parameters. It is of interest to compare the absolute magnitude derived from these observations (Table IV) with the values from the earlier IRAS and TRIAD surveys. Both earlier studies obtained a brighter zero-phase angle magnitude; this is undoubtedly because the near-opposition brightness "surge" of Anchises is practically nonexistent. In particular, we wish to stress that the albedos and diameters derived from the IRAS data set in most cases are based on assumed values of the BHL multiple scattering parameter G. For dark outer-belt objects, a value of GB = 0.15 was assumed (Tedesco 1986). Our data show that such a low value of G is inappropriate for Anchises; thus the diameter and geometric albedo derived from the IRAS data are suspect. Others who make use of the IRAS data set should take care not to substitute physical parameters derived on the basis of assumed scattering functions for parameters determined directly. As this survey continues, photometric data will be obtained for Anchises at a wider range of phase angles. This work represents the first published phase relation for a P- or D-type asteroid; further phase-curve studies of these objects will ascertain whether Anchises is unique or typical of the outer-belt classes of asteroids.

The observational data for Pandarus are given in Table V, and a composite lightcurve uncorrected for phase angle and distance is shown in Fig. 6, based on a period of 8.48 hr. The initial PDM period search showed this period as well as two other candidate periods, 8.18 and 8.78 hr; as Pandarus was extensively observed on only 2 nights, these aliases represent shifting of the second lightcurve relative to the first by one-half period in each direction. The ambiguity was resolved by visual inspection of the folded lightcurves for each period and by inclusion of data from 6-7 July, a night which was photometric in the beginning (as evidenced by standard star measurements), but which became cloudy after 5 hr FT. The night's data were not used in the original PDM fits (shown in Figs. 7 and 8), but the first six points, standardized with mean photometric coefficients, fall nicely on the 8.48-hr curve. Both aliases were ruled out by the inclusion of the 6-7 July data. The amplitude of the lightcurve is quite large, >0.5 mag.

15.9

i

i

i

i

i

~

i

o

16.0

"

oo

16.1

g g

16.2

16.3 16.4

o

r

16.5

++

16.6 16.7 0.0

l

0.2

i

+ i

0.4

i

i

0.6

I

i 0.8

L

1.0

Rotational Phase

FIG. 6. Observations of 2674 Pandarus, uncorrected for distance and solar-phase angle. The different symbols denote observations from different nights. Circles denote 5-6 July, crosses 6-7 July, and triangles 10-11 July. As with Anchises, extinction is the major source of observational uncertainty. Points from 6-7 July, which was photometric for only a few hours, were not used in the initial PDM scans or in the fit for lightcurve parameters, but were used to eliminate possible aliases (see Figs. 7 and 8).

TABLE V OBSERVATIONSOF2674PANDARUS Date 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 6 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July 11 July

1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 •986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

Time (UT)

RA (1950.0)

1:12 1:23 1:33 2:06 2:17 2:27 2:39 2:49 3:12 3:27 3:38 3:48 3:59 4:42 4:54 5:05 5:33 5:44 5:55 6:06 6:16 6:27 7:37 7:47 7:58 8:08 8:19 8:29 9:25 9:36 9:46 9:57 1:23 1:34 1:44 1:55 2:06 2:18 2:29 2:39 2:50 3:00 3:11 5:00 5:11 6:07 6:17 7:12 7:23 7:34 7:44 7:55 8:47 8:58 9:09 9:19

19 12 15 19 12 15 19 12 15 19 12 14 19 12 14 19 12 14 19 12 13 19 12 13 19 12 13 19 12 12 19 12 12 19 12 12 19 12 12 19 12 11 19 12 10 19 12 10 19 12 09 19 12 09 19 12 09 19 12 09 19 12 08 19 12 08 19 12 07 19 12 06 19 12 06 19 12 06 19 12 06 19 12 05 19 12 04 19 12 04 19 12 04 19 12 03 19 09 30 19 09 29 19 09 29 19 09 29 19 09 29 19 09 28 19 09 28 19 09 28 19 09 28 19 09 27 19 09 27 19 09 25 19 09 24 19 09 23 19 09 23 19 09 22 19 09 21 19 09 21 19 09 21 19 09 21 19 09 19 19 09 19 19 09 19 19 09 19

Dec (1950,0) -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20 -20

09 51 09 52 09 52 09 53 09 54 09 54 09 55 09 55 09 56 09 57 09 57 09 58 09 58 10 O0 10 O0 10 Ol 10 02 10 03 10 03 10 03 lO 04 10 04 10 07 10 08 10 08 10 08 10 09 10 09 10 12 10 12 10 13 10 13 14 54 14 54 14 55 14 55 14 55 14 56 14 56 14 57 14 57 14 58 14 58 15 03 15 03 15 06 15 06 15 08 15 09 15 09 15 10 15 10 15 12 15 13 15 13 15 14

a

A

r

gob s

V(1, 0)"

0.878 0.876 0.875 0.871 0.869 0.868 0.866 0.865 0.862 0.860 0.859 0.857 0.856 0.850 0.849 0.848 0.844 0.843 0.841 0.840 0.838 0.837 0.828 0.827 0.825 0.824 0.823 0.821 0.814 0.813 0.812 0.810 0.561 0.562 0.563 0.564 0.565 0.567 0.568 0.568 0.569 0.570 0.571 0.582 0.583 0.588 0.589 0.595 0.596 0.597 0.598 0.599 0.605 0.606 0.607 0.608

4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4,023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.023 4.024 4.024 4.024 4.024 4.024 4.024 4.024 4.024 4.024 4,024 4.024 4.024 4.024 4.024 4.024 4,025 4.025 4.025 4.025 4.025 4.025 4.025 4.025 4.025

5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.038 5.040 5.040 5.040 5.040 5.040 5.040 5.040 5.040 5.040 5.040 5.040 5,040 5.040 5.040 5.040 5.040 5.040 5.040 5,040 5.040 5.040 5.040 5.040 5.040

16.03 16.09 16.08 16.26 16.28 16.34 16.40 16.41 16.55 16.56 16.53 16.41 16.29 16.07 16.01 16.01 15.97 16.05 16.10 16.14 16.22 16.25 16.52 16.52 16.42 16.33 16.30 16.22 16.03 16.04 16.05 16.10 16.36 16.43 16.52 16.57 16.58 16.50 16.40 16.34 16.25 16.18 16.15 16.18 16.25 16.59 16.59 16.18 16.11 16.09 16.07 16.03 16.09 16.12 16.20 16.25

9,48 9,53 9,53 9,70 9,73 9,79 9,85 9.86 9,99 10,00 9.98 9,85 9.74 9.52 9.46 9.46 9.41 9.50 9.54 9.59 9.66 9.70 9.97 9.96 9.86 9.78 9.74 9.66 9.48 9.49 9.50 9.54 9.81 9.88 9.97 10.02 10.03 9.95 9.85 9.79 9.70 9.63 9.60 9.63 9.71 10.05 10.04 9.63 9.57 9.55 9.52 9.48 9.54 9.57 9.65 9.70

a Zero-phase angle magnitude calculated using a linear phase coefficient of 0.023 mag/deg. See text for details. 335

336

LINDA M. FRENCH 9.3 1.0 9.4 9.5

0.8

9.6 0.6 ~ >

I--0.4

9.7 9.8 9.9

0.2 10.0 0 0.0

5.0

10,0

15.0

20.0

25.0

10.1 0.0

30.0

0.2

0.4

FIG. 7. Coarse PDM scan for Pandarus. The true half period, near 4.24 hr, and an alias at 4.09 hr are apparent, along with several overtones.

The Pandarus observations span too small a range of phase angles (0.6 ° -< a -< 0.9 °) for the phase coefficient to be reliably determined. However, as for Anchises, inspection of the raw lightcurve suggests that an opposition effect as large as 0.I mag/deg cannot be present. The linear coefficient derived for Anchises has been used to determine zero-phase magnitudes for Pandarus; these are the values given in the last column of Table V. The derived lightcurve parameters for Pandarus are given in Table IV. The lightcurve, corrected for distance and solar phase angle, is shown in Fig. 9.

i

0.6

0.8

1.0

RotationalPhase

Period (hrs)

FIG. 9. Pandarus observations reduced for distance effects and solar-phase angle, using the linear phase coefficient derived for Anchises, 0.023 mag/deg. The small range of phase angles for which Pandarus was observed made the uncertainties in the directly determined phase coefficient large; the Anchises coefficient is consistent with the value derived for Pandarus.

1208 Troilus and 1867 Deiphobus Incomplete lightcurves were obtained for these asteroids; the observational data are summarized in Tables VI and VII. Figure 10 shows the PDM plot for Troilus; the sharp minimum near 13 hr suggests that the true period is a multiple of this number. The drop in minimum at periods beyond ~25 hr suggests that the time baseline is not sufficiently long to determine the period; folding

i

1,0

1.0If 0.8

0.8

0.6

0.6 I"--

0.4

0.4

0.2

0.2

8.0

8.2

8.4

8.6

8.8

9.0

Period(hrs) FIG. 8. High-resolution PDM scan for Pandarus, using only data from 5-6 and 10-11 July. Severe aliasing problems at 8.18 and 8.78 hr, corresponding to shifting one peak forward and backward with respect to the other, are obvious. Data from 6-7 July were used to eliminate the aliased periods; see text for details.

0 0.0

i

i

i

i

i

i

6.0

12.0

18.0

24.0

30.0

36.0

Period(hrs) FIG. 10. PDM scan for 1208 Troilus. The steep dip at ~13 hr, by analogy with Anchises and Pandarus, should indicate a rotational period of ~26 hr. The data give unacceptably large scatter when folded to that period, however. The true period is probably a higher multiple of 13 hr.

TROJAN ASTEROID LIGHTCURVES the data together with periods near 26, 39, and 51 hr gives, in each case, an unsatisfactory light " c u r v e . " We conclude that the period is probably longer than 26 hr. The data show slightly more internal scatter than for the other asteroids; because of its large inclination (i = 33°) Troilus had a significant component of motion in both right ascension and declination, unlike the other program asteroids. It is possible that the images are slightly more trailed than those of other asteroids. This object remains high on the observing priority list because of the need not to introduce bias against long periods into the rotation data set for Trojan asteroids. The observed amplitude--a lower limit to the true amount since a full period was not sampled--is ---0.2 mag. 1867 Deiphobus was observed over a significant fraction of a night for only 1 night, 1-2 July (see Table VII). A decrease in brightness of ~0.1 mag was seen over 6 hr; if the "classical" two-maxima-per-period assumption applies for Deiphobus, this would imply a period of ---24 hr. The V magnitudes corrected for phase and distance from 30 June-1 July are in agreement with those from 24 hr later; at present we can only say that the period is long. In the only other published lightcurve study of Deiphobus, Degewij and van Houten (1979) reported a lower limit to the amplitude of the lightcurve of 0.28 mag. Over the ~ 10 years since the Degewij and van Houten work, the amplitude of the lightcurve may have changed significantly if the asteroid's rotational pole is significantly inclined with respect to the pole of the ecliptic. Again, a relatively large-amplitude lightcurve is suggested, and again, long-term observations are called for. DISCUSSION The rotation and phase parameters for 1173 Anchises and 2674 Pandarus are summarized in Table IV. The most striking property of these asteroids is the large peak-to-peak amplitude of the two complete lightcurves--nearly 0.6 mag for each.

337

If the asteroid rotation axes are perpendicular to the line of sight, this corresponds to B* ~ 0.6, where B* is the ratio of the minimum to maximum breadths of the asteroid's mean cross section as defined by Ostro and Connelly (1984). While the coverage of the lightcurve for Troilus is not complete, the amplitude exceeds 0.2 mag. By comparison, the average lightcurve amplitude of main-belt asteroids in the 50-100 km diameter range is approximately 0.2 mag (Burns and Tedesco 1979, Binzel 1984). It should be noted that care has been taken to eliminate as much as possible selection effects favoring large-amplitude lightcurves, within the realistic limits of telescope scheduling. Of course, the rotation of irregularly shaped objects causes the flux received at all wavelengths to vary with time. Because of the large amplitudes of the lightcurves for Anchises and Pandarus, we have not attempted to derive new geometric albedos and diameters based only on visible-band observations. In order to eliminate systematic errors in the radiometric method of albedo-diameter determination, one must either observe simultaneously in the infrared and visible wavelength regions or have sufficiently precise knowledge of the lightcurve to remove its effects on data sets taken at different times. With their unusual rotation and scattering properties, such multiband observations of the Trojan asteroids should have high priority. Given the pronounced asphericity of at least three more Trojan asteroids, what can we say about the likelihood of another--or several more--Hektors? The lightcurve amplitudes observed here are only half that observed for Hektor, and the rotation frequencies are lower. Thus, the implied shapes are less elongated; the high centrifugal accelerations which would severely stress primitive material of low internal strength are not present. If the observed amplitudes are near the maximum values for Anchises and Pandarus, the exotic models invoked to explain Hektor are not needed; the lightcurves of both Anchises

LINDA M. FRENCH

338

T A B L E VI OBSERVATIONS OF 1208 TROILUS Date 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29

June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June June

Time (UT) 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 •986

3:39 3:48 4:01 4:12 5:10 5:27 5:39 5:53 6:05 6:17 6:29 6:42 6:54 7:06 7:19 8:47 9:00 9:15 9:27 1:21 1:41 1:53 2:05 2:17 2:29 2:42 3:02 3:17 3:31 3:44 3:57 4:09 4:29 5:35 5:47 3:18 3:33 4:39 5:02 1:40 2:04 2:20 2:37 2:54 3:18 3:33 4:39 5:02 5:16 6:28 6:42 7:10 7:23

RA (1950.0) 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 30 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 29 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28 19 28

25 24 24 24 21 21 20 20 20 19 19 18 18 17 17 14 13 13 12 38 37 37 36 36 36 35 34 34 33 33 32 32 31 29 28 41 41 38 37 45 44 43 43 42 41 41 38 37 37 34 34 33 32

Dec (1950.0) -57 57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57

30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 34 34 34 34 34 34 35 35 35 35 35 35 35 35 35 35 39 39 40 40 39 39 39 39 39 39 39 40 40 40 40 40 40 40

22 24 27 29 41 44 47 50 52 55 57 00 02 05 07 25 28 31 33 45 49 51 54 56 59 01 05 08 11 14 16 19 23 36 38 50 53 06 10 31 36 39 42 46 50 53 06 10 13 27 29 35 37

a

A

r

gobs

V(1, 0)"

7.081 7.081 7.080 7.080 7.079 7.078 7.078 7.078 7.077 7.077 7.077 7.077 7.076 7.076 7.076 7.074 7.073 7.073 7.073 7.052 7.052 7.051 7.051 7.051 7.051 7.050 7.050 7.050 7.049 7.049 7.049 7.049 7.048 7.047 7.047 7,022 7,022 7.021 7.021 7.024 7.024 7.023 7.023 7.023 7.022 7.022 7.021 7.021 7.020 7.019 7.019 7.018 7.018

4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.039 4.038 4.038 4.038 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.037 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.035 4.034 4.034 4.034 4.034

4.895 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.894 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893

15.97 15.94 15.94 15.97 15.99 15.99 15.94 15.93 15.95 15.94 15.94 15.94 15.94 15.95 15.96 15.91 15.96 15.94 15.94 16.15 16.17 16.14 16.12 16.10 16.12 16.12 16.09 16.09 16.09 16.10 16.08 16.07 16.03 16.05 16.00 16.09 16.08 16.07 16.09 16.05 16.11 16.13 16.16 16.09 16.09 16.08 16.07 16.09 16.08 16.09 16.10 16.07 16.05

9.49 9.46 9.46 9.49 9.51 9.51 9.46 9.45 9.47 9.46 9.46 9.46 9.46 9.47 9.48 9.44 9.48 9.46 9.47 9.67 9.70 9.66 9.64 9.62 9.64 9.64 9.61 9.61 9.61 9.62 9.60 9.59 9.55 9.57 9.53 9.61 9.61 9.60 9.61 9.57 9.63 9.65 9.68 9.62 9.61 9.61 9.60 9.61 9,61 9.62 9.62 9.59 9.58

TROJAN

ASTEROID

339

LIGHTCURVES

T A B L E VI--Continued Date 29 29 29 29 29 29 30 30 30 30 30 30 30 30

June June June June June June June June June June June June June June

T i m e (UT) 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

7:39 7:59 8:13 8:42 9:09 9:25 2:32 2:44 2:54 3:07 3:30 5:07 5:29 5:39

RA (1950.0) 19 28 19 28 19 28 19 28 19 28 19 28 19 27 19 27 19 27 19 27 19 27 19 27 19 27 19 27

32 31 30 29 28 28 50 49 49 48 48 44 43 43

Dec (1950.0) -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57 -57

40 40 40 40 40 41 44 44 44 44 44 44 44 44

40 44 47 53 58 01 15 17 19 21 25 44 47 49

a

A

r

Vob~

V(1, 0) "

7.018 7.018 7.017 7.017 7.016 7.016 7.000 7.000 7.000 7.000 6.999 6.998 6.998 6.997

4.034 4.034 4.034 4.034 4.034 4.034 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033

4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893 4.893

16.06 16.01 16.00 15.99 15.99 16.02 16.04 16.06 16.07 16.06 16.08 16.09 16.07 16.06

9.59 9.54 9.52 9.51 9.51 9.54 9.56 9.59 9.59 9.58 9.60 9.61 9.60 9.59

a Phase angle coverage was insufficient to c o m p u t e a phase coefficient; these values have been corrected for r and A only.

T A B L E VII OBSERVATIONS OF 1867 DEIPHOBUS Date 1 July 1 July 1 July 1 July 1 July 1 July 1 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July 2 July

1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986

Time (UT) 6:53 7:04 7:13 7:21 8:01 8:09 8:17 0:53 1:06 1:15 1:26 2:09 4:35 4:44 4:54 5:05 6:32 6:42 6:52 7:23 7:34 7:43 7:53 8:15

R A (1950.0) 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40 17 40

59 59 58 58 57 57 57 30 30 30 30 28 25 24 24 24 22 21 21 20 20 20 19 19

Dec (1950.0) -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32 -32

21 38 21 36 21 34 21 33 21 27 21 26 21 25 18 49 18 47 18 46 18 44 18 37 18 14 18 13 18 11 18 09 17 56 17 54 17 53 17 48 17 46 17 45 17 43 17 40

et

A

r

Vob~

V(I, 0) 0

3.156 3.157 3.158 3.159 3.164 3.164 3.165 3.284 3.286 3.287 3.288 3.293 3.311 3.312 3.313 3.314 3.325 3.326 3.327 3.331 3.332 3.333 3.335 3.337

3.957 3.957 3.957 3.957 3.957 3.957 3.957 3.959 3.959 3.959 3.959 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960 3.960

4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944 4.944

15.49 15.49 15.49 15.48 15.51 15.51 15.53 15.40 15.41 15.42 15.43 15.41 15.46 15.51 15.49 15.48 15.51 15.48 15.48 15.51 15.51 15.48 15.48 15.47

9.03 9.03 9.03 9.03 9.05 9.05 9.07 8.94 8.96 8.96 8.97 8.95 9.00 9.05 9.03 9.02 9.05 9.02 9.02 9.05 9.05 9.02 9.02 9.01

a P h a s e angle coverage was insufficient to c o m p u t e a phase coefficient; these values have been corrected for r and A only.

340

LINDA M. FRENCH

and Pandarus would be better explained as the main-belt asteroids? A positive answer collision fragments (Farinella et al. 1981). to this question would be consistent with Of course, the present work has set only the calculation of Davis and Weidenschilllower limits to the asphericity; further mon- ing (1981) that Trojans in both Lagrangian itoring will determine the shapes of these regions should be stable against cataasteroids more precisely. strophic collisions for longer than the age of It is quite remarkable that no main-belt the solar system. asteroids in the 50-100 km size range have CONCLUSIONS been observed with mean lightcurve amplitudes as large as ~0.6 mag, and yet we have Two of the L5 Trojan asteroids observed, observed two 100-km scale objects (Table 1173 Anchises and 2674 Pandarus, have exII) in the L5 Trojan group with such ceptionally large amplitudes for their size large amplitudes. The dichotomy would be range, 0.57 and 0.58 mag, respectively. The lessened if the radiometric albedos were lightcurve amplitude for 1208 Troilus exshown to be systematically low; the aster- ceeds 0.2 mag. The periods of Anchises and oids would then be smaller, and Binzel Pandarus are typical for asteroids in their (1984) has shown that the scatter in light- size range, while those of Troilus and Deicurve amplitudes tend to increase with de- phobus probably exceed 24 hr. Anchises creasing diameter for main-belt asteroids. shows no opposition effect over the range Higher albedos would also be more consis- of phase angles 0.3 ° -< a - 2.0 °, suggesting tent with the observed phase relation for that its surface texture may be much smoother than that of most asteroids and Anchises. The observations presented here suggest satellites. In upcoming observing runs, monitoring that the Trojans may, on the average, be more elongated than main-belt asteroids of of these objects will continue and rotation equal size. What this is telling us about the properties for other Trojan asteroids will be collision history of the Trojans, or about determined. The lightcurves of P and D obthe effects of collisions on asteroid shape, jects in the main asteroid belt are also being is not yet clear. For example, at present studied to identify any systematic effect of there is no general agreement about how composition on shape and rotation freasteroid shapes evolve through collisions. quency for primitive asteroids. Some believe an asteroid is equally diminACKNOWLEDGMENTS ished in all dimensions, thus increasing the It is a pleasure to thank the staff of CTIO for their nonsphericity with time. Others believe capable and gracious assistance. Special thanks are that collisions "round off" the edges--the due to Mauricio Fern~indez, Ram6n Gfilvez, Ricardo "pebble in a stream" model of Degewij Gonz~ilez, Gabriel Martin, Lorenzo Martfnez-Conde, (1978). Only the Earth-crossing asteroids, Mauricio Navarrete, Dan Smith, Hugo Vargas, and as a group, have as pronounced a deviation Ricardo Venegas. I am grateful to Ted Bowell and from nonspherical shapes, and these aster- Richard French for informative and stimulating discussions. David Kramer assisted with the preparation of oids are typically much smaller than the finder charts and with data reduction; his work was Trojans studied here. For this reason, a di- supported in part by the MIT Undergraduate Research rect comparison of Earth crossers and Tro- Opportunities Program. jans is inappropriate. However, it is interREFERENCES esting that the two groups--both out of the main asteroid " s t r e a m " - - a p p e a r to be BINZEL, R. P. 1984. The rotation of small asteroids. Icarus 57, 294-306. more irregularly shaped than the main-belt BINZEL, R. P. 1986. Collisional Evolution o f the Asterobjects. Are the Trojans primordial collioid Belt: An Observational and Numerical Study. sion fragments, with shapes not as evolved Ph.D. thesis, University of Texas, Austin. by many subsequent collisions as those of BOWELL, E., A. W. HARRIS, AND K. LUMME 1985. A

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