Shape and size of asteroid 4 Vesta: Speckle interferometry and polarimetry

iCARUS92, 342-349 (1991)

Shape and Size of Asteroid 4 Vesta: Speckle Interferometry and Polarimetry V. S. TSVETKOVA, V. N . DUDINOV, S. B. NOVIKOV, 1 YE. A. PLUZHNIK, Y u . G . SHKURATOV, V. G . VAKULIK, AND A . P. ZHELEZNYAK Astronomical Observatory of Kharkov University, Sumskaya sir. 35, Kharkov 310022, USSR Received May 2, 1990; revised April 24, 1991

Vesta's surface. The interpretation of speckle interferoSpeckle interferometric data for Vesta and diffraction-limited metric data has a well-known ambiguity: only autocorrelaimages of Vesta's disk are presented for three measurements near tion of actual light distribution over the object can be the opposition of 1988. It is argued that polarimetric data can be derived from power spectrum measurements. The special utilized to estimate a contribution of albedo features to Vesta's algorithm permitted Drummond et al. (1988) to solve the lightcurve, and thus to derive the geometric component of the "phase problem" for the Vesta speckle series and to refightcurve. The analysis of speckle data and polarimetric data construct the images of Vesta for several moments, distogether with the results of simulation of speckle interferometry tributed over the whole period of its lightcurve. for Vesta enables us to conclude that Vesta must have an almost As is known, a model is required for speckle interferoequilibrium shape, with the axes ratio a/b close to unity. ~ t99t metric measurements of an object's effective size. For AcademicPress, Inc. asteroids, the model of a smooth featureless triaxial ellipsoid is usually adopted, and thus the resolved asteroid must project onto the plane of the sky as an ellipse. A 1. INTRODUCTION discrepancy between the accepted model and the true According to current knowledge based mainly on the appearance of an object may cause a bias in speckle interrecent speckle interferometric results (Drummond and ferometric estimates of its effective size (Drummond and Hege 1989), Vesta has a triaxial ellipsoid shape with axes Hege 1989). Speckle interferometry of asteroids followed ratio a / b = 1.07 -'- 0.04. In comparison to their previous by image reconstruction with the minimum of arbitrary result (Drummond et al. 1988) this value o f a / b is in better suggestions about their structure is thus of particular interagreement with well-known considerations that predict est. Because of the ambiguity and difficulty of interpreting an equilibrium shape for a body with an inferred basaltic speckle data, the application of other methods is very crust (Drake 1979, Drummond and Hege 1989, Cellino et desirable. Here we present our results of Vesta speckle interferometry near the opposition of 1988, and suggest al. 1989). While the early speckle interferometric observations of our view of Vesta's shape and size based on both the Vesta were rather illustrative (Worden et al. 1977, Wor- speckle interferometry and the photopolarimetry perden 1979), now we have the detailed set of Vesta's shape formed by Lupishko et al. (1988) during the opposition of and size determinations made by Drummond and Hege 1986. (1989) for the oppositions of 1983 and 1986. Moreover, the images of Vesta have been reconstructed from speckle 2. O B S E R V A T I O N S series, showing some albedo details on its surface. SugSpeckle series of Vesta and reference stars were obgestion about strong albedo inhomogeneity of Vesta's surtained during January 20 and 21, 1988, with the l-m teleface arose as an attempt to explain its iightcurve and scope of the Institute of Physics of LitSSR (Mount Maidapolarization measurements (Degewij et al. 1979, Gradie et nak, Middle Asia) equipped with a speckle camera al. 1978). While further observations, especially speckle interferometry, confirmed this suggestion, they were not analogous to that described by Dudinov et al. (1982). A able to determine the actual pattern of albedo details on three-stage amplifier and a camera with 35-mm photographic film were used as a detector. Exposures for Vesta t Sternberg State Astronomical Institute, Universitetsky prosp. 13, and reference stars were 20 and 30 ms and the rate was Moscow 119899. eight frames per second, heff = 6800 A,, Ah = 700 A,. 342 0019-1035/91 $3.00 Copyright © 1991 by Academic Press, Inc. All fights of reproduction in any form reserved.

343

SHAPE AND SIZE OF ASTEROID 4 VESTA TABLE I Aspect D a t a for Speckle Observations

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"

- 1

.

--0.8

.

.

-0.6

.

2or •

-0.2

0

0.2

0.4

0.6

NORMALIZEDSPATIALFREQUENCY

0.8

1.0

FIG. 1. Radial scan of Vesta's power spectrum which illustrates the signal (dots) and the noise (line) components of the power spectrum.

Power spectrum measurements have been carried out with the coherent optical processor of the Kharkov Astronomical Observatory (Dudinov et al. 1982). Speckle series as long as 180-200 frames are necessary to provide significant information over the noise at spatial frequencies up t o f = f0, where f0 is the diffraction-limited frequency, fo = D / X F (D is the diameter of a telescope, F its focal distance). In Fig. 1 the radial scan of Vesta's power spectrum illustrates the character of the signal (dots) and the noise (line) in the power spectrum space. In Fig. 2 the time of observations is plotted at Vesta's lightcurve. Aspect data for the observations are given in Table I. Reference stars for determining the modulation transfer function (MTF) and several binaries for scale calibration were recorded for each of Vesta's speckle runs. To make the scale determination more precise the binary 3' Vir was observed by us and at the same time by our colleagues at the 1.3-m telescope in Abastumani (Astrophysical Observatory of Georgian Academy of Science) with a device for precise scale calibration (Bakhtin e t a / . 1988). A relative error of scale determination of 4% was derived from

1

2

3

Ore2 TIME (MRS) FIG. 2.

RA 8hi6TI

Dec. 23°!2!7

Solar phase angle 2.°25

Distance from Earth 1.514 AU

Long. 8h03.~8

Lat. 3°33~!

Distance from Sun 2.495 AU

, , "~,

.

-0.4

Date Jan. 21,875, 1988

Times of observations projected on the lightcurve of Vesta.

different binaries observations with y Vir as a reference binary. 3. P O W E R S P E C T R U M A N A L Y S I S

Coherent optical techniques for processing the speckle data are appropriate when a photographic film is used as a detector. A question about accuracy usually arises when coherent optical processors are discussed. Numerous optical surfaces are the main sources of their intrinsic noise, which noticeably increases the errors of power spectrum measurements and decreases the dynamic range of processors. The processor of Kharkov University has an extremely low intrinsic noise level due to its optical design, which reduces the number of refracting surfaces. The main sources of errors for power spectrum measurements include: I. Statistical error, connected with the limited number of frames in the speckle run. When this number is too low there may be a bias in the power spectrum estimates. 2. Random noise, connected with the irregularities of the amplifier's cathode and the photon nature of light. The effect of this source of errors is clear from Fig. I. In a logarithmic scale the noise component is parabolic (solid), and the parameters of a parabola can be determined by a least-squares routine. If the model of additive noise is accepted, the noise component should be subtracted to obtain the correct values of the object's power spectrum. 3. As the time interval for each speckle series is short, the long-exposure (i.e., low frequency) part of the MTF retains its random character. For objects with Vesta's size the most informative part of the power spectrum lies in the transitional zone between the long-exposure and the diffraction-limited part of the MTF, and thus the estimates of an object's power spectrum are affected by seeing mismatches. 4. The spatial frequency range, where the reliable power spectrum estimates are possible, is always limited from both the low and the high frequencies. For Vesta, power spectrum measurements were possible from 0.15f0 to 0.Tf0 (see Fig. 1), while the deconvolved power spectrum of Vesta was fit for ellipse parameters in the region of 0.15f0 to 0.4f0. At frequencies less than 0.15f0 the effects of seeing and seeing mismatches become considerable while

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T S V E T K O V A ET AL.

ments with a coherent processor is due to photographic nonlinearity. To remove photographic nonlinearity, the input signal must be presented as a positive with the fulfillment of a well-known condition: ')/neg -- ')/pos ~-

2,

COHERENT LIGHT

FIG. 3. Optical scheme of coherent spectrum analyzer. The film with the initial speckle images is inside the Fourier transforming lens C in immersion liquid.

the signal-to-noise ratio decreases rapidly toward the high frequencies inhibiting the reliable power spectrum measurements. 5. The most important error of speckle interferometric measurements is due to scale calibration. Usually the scale is extracted from the observations of binaries with reliable orbits. Speckle interferometry demands, however, much more precision. That is why we used the speckle interferometric results for the binary 7 Vir, which were obtained both by us at the 1-m telescope and simultaneously by our collegues at the 1.3-m telescope in Abastumani with the device of precise scale calibration (Bakhtin et al. 1988). The errors for values of major and minor axes in Table II are due to the spread of the scale determination data, obtained for binaries, while the formalistic error is much smaller. In addition to the peculiarities listed above, which are inherent in any method of speckle data processing, there are some specific difficulties of coherent optical processing of speckle series. Here we shall briefly describe the main steps of coherent optical processing for speckle data. The film with speckle interferograms (positives with the proper value of contrast) is put into the immersion cell C which is also the Fourier transforming optical element (Fig. 3). In this picture the coherent optical processor of Kharkov University is shown in the spectrum analyzer mode. The input window is illuminated by a coherent spherical wave from a point source S situated at the double focal distance from C, and thus the light field distribution in the plane P differs from the genuine Fourier transform only by a phase factor, which is of no importance. The way the number of refracting surfaces (i.e., reducing the processor's intrinsic noise) is considerably reduced can be easily understood from Fig. 3. In addition, simple low-power spherical lenses are used as the Fourier transforming elements instead of complex high-power objectives with aspherical surfaces, as is usual in coherent processors. The most significant error of power spectrum measure-

where Yncgand Yposare the contrasts of the initial negative and of its positivecopy, respectively. The range of linearity of approximately 15 dB is easily achieved by such a two-step photographic procedure. The output signal is accumulated in the plane P and also must be standardized. The best way to do it in coherent light is to use a diffraction pattern with a known energy distribution, for example, diffraction by a rectangular or a circular aperture. As the background of positives with speckle interferograms cannot be made zero, the lobes of the input window are the most important features of an output signal at low spatial frequencies in a coherent spectrum analyzer. These lobes may cause a significant bias of power spectrum estimates. That is why a special input window with side lobes, which can be neglected at spatial frequencies larger than 0.15 f0 was used. This is the low-frequency limit for power spectrum measurements in our case, and this is approximately the high-frequency boundary of the long-exposure part of the MTF. We think that this rather archaic processing technique is justified for objects with Vesta's magnitude when speckle interferometry in analogous regimes is preferable. 4. SPECKLE INTERFEROMETRIC ESTIMATES OF VESTA'S PROJECTED ELLIPSE PARAMETERS The power spectra of Vesta and reference stars were digitized with the automatic microdensitometer, so that matrices of 100 × 100 pixels for the frequency range up to 0.7f0 were the initial data. The first step was to account for the noise component of the power spectra. As was mentioned above, for the range from 0.4f0 to f0 (where Vesta's spectrum is negligible), the noise component is well approximated by a Gaussian function (parabola in logarithmic scale). The low-frequency trend of the noise component is unknown, but we anticipate that it will not cause a noticeable error of the final result. For reference stars the noise component of a power spectrum cannot be separated from the MTF, which is significant up to the diffraction-limited frequency. This requires use of the above approximation derived for Vesta. The remaining problem is to determine the weight of the noise component with the other parameters (dispersion in the case of Gaussian function) obtained from Vesta's power spectrum measurements. The weight was fit so that after subtracting noise the resulting MTF was positive up to its diffraction limit. After subtracting noise

345

SHAPE AND SIZE OF ASTEROID 4 VESTA

TABLE 1I Speckle Interferometric Data for Vesta for Jan. 21, 1988 UT

M ~ o r a x i s (km)

M i n o r a x i s (km)

Positional angle

20h50 m 21h38 m 22h21 m

534 ± 16 646 ± 19 561 ± 17

436 ± 13 534 ± 16 445 ± 13

-- 12° ~ 1° --27 ° ± i ° 4* ± 1°

Vesta's power spectra were divided by the power spectra of the reference stars, and the deconvoived power spectra were fit for ellipse parameters by the least-squares routine, assuming that the asteroid can be treated as a uniformly bright triaxial ellipsoid, and thus the observed disk is a two-dimensional ellipse. An annular frequency zone from 0.15f0 to 0.4f0 was used for fitting. The reasons for this were discussed above, and the additional argument for fitting inside this frequency zone is the considerable growth of formalistic errors when the broader zone is used, while the estimates do not vary noticeably. The major and minor axes of Vesta's elliptical disk and positional angle of the semiminor axis, measured eastward from north, are presented in Table II. 5. I M A G E

RECONSTRUCTION

In addition to speckle interferometric estimates of effective size an attempt was made to synthesize the diffraction-limited images of Vesta with minimal prior information. The images were reconstructed from speckle interferograms by the technique analogous to the "shiftand-add" method (Bates 1977). This method is based on the assumption that each regular and isolated bright spot in a speckle image can be regarded as the diffractionlimited image of an object. Only nonoverlapping zeroorder interference ("white") speckles may form such images, while the rest of the speckles produce a random background. The number of such white speckles depends on the seeing conditions, and particularly on the depth of the atmospheric phase modulation of a wave front. In nonmonochromatic light they must prevail over the speckles of the higher orders of interference and thus be easily identified. The coordinates of these speckles can be used to produce an artificial image of a point source, and then the problem reduces to speckle holography. The "shift-and-add" method is restricted to the case of rather small objects, i.e., objects which do not exceed the average distance between the neighboring white speckles in dimension. For larger objects only far, not neighboring, white speckles may produce isolated diffraction images, so the number of frames which can be used falls as the object's size increases. We therefore selected only 25% of the whole speckle run to obtain Vesta's images.

We applied a very simple analogous technique to synthesize Vesta's diffraction images. Summing of zero-order speckles, which carry the diffraction disk of Vesta ( - 5 0 for each speckle run) was fulfilled with the device of optical superposition, which operates on the principle of blinking. The synthesized images of Vesta are shown in Fig. 4 (top row), together with the diffraction image of a reference star, obtained with the same method. North is up and east is to the right; the effective resolution is 15 km on Vesta's disk (approximately the FWHM of the diffraction point spread function for the l-m telescope at hcff -- 6800 ,~, i.e., 0.13'~. Compensation of the diffraction-limited MTF was not undertaken as it may only increase the noise. Some peculiarities can be noted on the images; however, they hardly can be regarded as real because of the low signal-to-noise ratio. With the same extent of uncertainty they can be interpreted both as features on the disk and as irregularities of the edge. All the images are elliptical with the eccentricity and orientation changing noticeably for the time interval 40-50 min. In the second row of Fig. 4 the comparison of the reconstructed images of Vesta (contours) with the speckle interferometric estimates (ellipses) are presented. The contour with the maximal value of gradient is accepted as an edge. The step for the contours is approximately 0.2 in a logarithmic scale. For technical reasons calibration was not possible, so these pictures are rather illustrative. 6. S I M U L A T I O N

OF SPECKLE

INTERFEROMETRY

FOR ASTEROIDS

The previous section had clearly demonstrated that agreement between our speckle interferometric measurements and the synthesized images could hardly be regarded as satisfactory. Unfortunately our speckle data do not permit us to obtain the solution for Vesta's body, as we have estimates of projected ellipse parameters only for three time periods. As was mentioned above, any discrepancy between the accepted model and the genuine appearance of an object must cause a bias of speckle interferometric estimates. To illustrate this, a simulation accounting for possible albedo spots as well as irregularities of shape has been undertaken. In Fig. 5 the results of the simulation are shown. The models are a round disk without features, an irregular disk with an effective area equal to that of the round disk, a round disk with two slight edge irregularities, and a round disk with six albedo features, similar to those which were reconstructed by Drummond et al. (1988) (top row). The second row of Fig. 5 shows corresponding power spectra. The last power spectrum is that of a uniform circular disk. While slight irregularities of the edge (a) result in an

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m

0.13"

@ FIG. 4. Images of Vesta, synthesized from the speckle interferograms by the "shift-and-add" method. North is up, and east is to the right; the effective resolution is 0.13" (150 km at Vesta's disk). The bottom row is the result of a comparison of the reconstructed images (contours) and the speckle interferometric projected ellipses (see Table 1I).

FIG. 5. Simulation of asteroid speckle interferometry. Top row, the models: bottom row, the corresponding power spectra. Spectrum (d) is that of a uniformly bright circular disk of the same diameter as in (a) and (c).

SHAPE AND SIZE OF ASTEROID 4 VESTA eilipticity of the zero order of the spectrum, case (b) demonstrates distortions of the first and the second diffraction orders, while the central spot remains almost undisturbed. The prominent changes of power spectrum, especially of the central maximum, are visible for the round disk with albedo features (c). It should be noted that in a real experiment only the central spot of a power spectrum can be measured with a sufficient signal-to-noise ratio. Thus for a heterogeneous albedo the model of a uniformly bright disk will cause the wrong solution for speckle interferometric estimates. For example, in case (c) the solution will be an elliptical disk with a noticeable eccentricity and a cross section approximately 25-30% larger than that of the actual disk. The simulation shows that speckle data for asteroids must be treated with great precaution. Albedo features that may exist on asteroid surfaces cause wrong solutions for the projected disk parameters and thus for those of the asteroid's body if the model of a uniformly bright disk is used. In the next section the results o f Vesta's polarimetry in opposition of 1986 (Lupishko et al. 1988) are used as an additional approach to the problem of Vesta's shape.

7. POLARIMETRY OF VESTA: A NEW APPROACH TO

THE SHAPE ESTIMATES Polarimetry of Vesta shows noticeable variations of polarization near its minimum Pmi. with the rotational phase (Lupishko et al. 1988). Polarimetry of samples in the laboratory has demonstrated that variations of Pmi. may be due to albedo variations in some cases (Shkuratov 1988). For the totality of asteroids a correlation between Pmi, and albedo A has been stated by Zellner (1977), with the equation o f regression

JPm~,IA = 0.16,

(1)

where Pminand A are percent. A similar correlation was recently discovered for features of the Moon (Opanasenko et al. 1990). For high aibedos (0.15 -< A -< 0.23) the regression line is

A = Cl{emin{ + C 2,

(2)

where C1 = - 15.3 and C2 = 30.6 (A and Pmi, are percent again). Expressions (1) and (2) can be utilized to reconstruct the " a l b e d o " part of a lightcurve from polarimetric measurements. The results of such reconstruction are pre-

347

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2

4

5 TIME (HRS)

6

7

FIG. 6. Reconstruction of the "albedo" component of Vesta's lightcurve from polarimetric measurements. The crosses are the observed lightcurve; the dots are results of reconstruction using the asteroid type of correlation between Pminand albedo; the circles are results of reconstruction with the lunar type of correlation.

sented in Fig. 6. The crosses are the observed lightcurve (Lupishko et al. 1988), the clots are the lightcurve reconstructed from polarimetric measurements for the same night (Lupishko et al. 1988) with expression (I), and the circles, with expression (2). The two reconstructed lightcurves should be regarded as lightcurves of a rotational ellipsoid with albedo features responsible for variations of brightness. So the contribution of Vesta's shape in the lightcurve can be obtained from a comparison of the observed and.the reconstructed lightcurves, if Eq. (1) or (2) is right for Vesta. Here we suppose that polarimetric variations connected with the rotation are due only to the distribution of the albedo. One can object that variations in microstructure characteristics should change polarization too. But, as a first approximation, we suggest that these microstructure characteristics have been unified by impact processes during Vesta's lifetime. Subtracting the observed lightcurve from the reconstructed ones we obtained curve 2 in Fig. 7 for the " l u n a r " case (2) and curve 1 for Eq. (1). For both cases there are only one minimum and one maximum over a 5-hr 20.5-min period for the geometric component of Vesta's lightcurve. This fact is inconsistent however with the 5-hr 20.5-rain rotational period. It is obvious that no choice of coefficients in Eqs. (1) and (2) affords two maxima o v e r the period for the geometric component of Vesta's lightcurve. The only thing which can be achieved is to make the geometric component o f the lightcurve constant (points (3) in Fig. 7). In this case the coefficients C l and C z are - 2 4 and 36, respectively, with the average o v e r a period albedo of 0.25, if the "lun a r " type of correlation is accepted for Vesta. At present we have no opportunity to choose between the two types o f correlation. H o w e v e r , there are certain similarities between the lunar and the Vesta spectra that suggest the lunar variant rather than that for asteroids, in particular, the existence of mafic mineral absorptions, and

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a l m o s t the same values o f visual albedo. In any case the contribution o f albedo features to V e s t a ' s lightcurve is significant. T o illustrate the contribution o f the albedo features to a lightcurve and a polarization c u r v e o f an a t m o s p h e r e l e s s b o d y , the integral lightcurve o f the M o o n was calculated (Fig. 8) using its a l b e d o m a p ( S h e v c h e n k o 1980). A range o f brightness variations as great as 0.3 m was obtained, three times larger than that for Vesta. With e x p r e s s i o n (2) the variations o f IPminl with the rotational p h a s e w e r e calculated, giving c u r v e 2 in Fig. 8. F o r the M o o n the amplitude o f IP~,.] variations is twice as large as that for Vesta. So, either the pattern o f albedo features for V e s t a is m o r e s y m m e t r i c a l than that for the M o o n , or the contrast o f V e s t a ' s features is lower.

1.1

10 o

~ 0.9

FIG. 8. The integral lightcurve of the Moon, synthesized with the full albedo chart of the Moon (a). The integral polarization of the Moon, reconstructed from its lightcurve (b).

8. CONCLUSIONS H e r e we did not discuss the p r o b l e m o f V e s t a ' s rotational period, and a s s u m e d that there are important reasons for a c c e p t i n g a 5-hr 20.5-rain period rather than one o f 10 hr 41 min. But in this case one should explain w h y there are o n l y o n e m i n i m u m and o n e m a x i m u m o v e r a 5-hr 20.5-min period at o u r g e o m e t r i c c o m p o n e n t o f V e s t a ' s lightcurve, w h i c h was r e c o n s t r u c t e d f r o m polarimetric data. T h e r e m a y be t w o alternative explanations o f this fact: either V e s t a has a 10-hr 41-rain rotational period, or it has a l m o s t a rotational ellipsoid shape. In the latter case we must fit the coefficients o f Eq. (2) so that the resulting g e o m e t r i c c o m p o n e n t o f a lightcurve should bec o m e constant. Figure 7 (curve 3) s h o w s that this p r o c e dure is feasible, with a quite r e a s o n a b l e a v e r a g e a l b e d o o f 0.25. H e r e we p r e s e n t e d the results o f o u r three best speckle runs o f Vesta. T h e r e are four m o r e speckle runs w h i c h are not so g o o d , but are w o r t h y o f being p r o c e s s e d . T h e s e

eo° o oo ~o eoo~Qo IPOqND~OI OQOO000e ~OOQO0OtOi 3

runs are u n d e r p r o c e s s i n g n o w , and w e e x p e c t to obtain additional speckle i n t e r f e r o m e t r i c estimates as well as m o r e images o f V e s t a ' s disk. W e think that V e s t a ' s s h a p e is far f r o m being c o m pletely u n d e r s t o o d . S p e c k l e i n t e r f e r o m e t r y f o l l o w e d by successful image r e c o n s t r u c t i o n is o f great i m p o r t a n c e , but o t h e r m e t h o d s are n e c e s s a r y , s u c h as p h o t o m e t r y and polarimetry, c o v e r i n g several rotational p e r i o d s o f this interesting asteroid. ACKNOWLEDGMENTS We thank F. Velitshko for projecting our observations at Vesta's lightcurve. REFERENCES BAKHTIN, V. D., V. G. VAKULIK, A. P. ZHELEZNYAK, V. V. KONiCHEK,

ANDI. YE. SINELNIKOV1988. Precise scale calibration for astronomical investigations. Kinematika i Fis. Neb. Tel. 4(2), 93-96. BATESR. H. T. 1977. Recovery of fringe visibility from recorded speckle images quantized to two levels. Mon. Not. R. Astron. Soc. 181(2), 365-374. BROGLIA,P., ANDA. MANARA1989. Rotational variations in the optical polarization of 4 Vesta. Astron. Astrophys. 214(1-2), 389-390. CELLINO, A., M. D. DI MARTINO, J. DRUMMOND, P. FARINELLA, P.

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FIG. 7. Reconstruction of the geometric component of Vesta's lightcurve: 1, with Eq. (1) and 2, with Eq. (2). Note that there are only one maximum and one minimum over a 5-hr 20.5-min period; this is inconsistent with the short rotational period of Vesta. The dots (3) are the result of fitting the coefficients in Eq. (2) to make the geometric component of Vesta's lightcurve constant.

PAOLICCHI,AND V. ZAPPALA1989. Vesta's shape, density and albedo features. Astron. Astrophys. 219(1-2), 320-321. DEGEWU, J., E. F. TEDESCO, AND B. ZELLNER 1979. A l b e d o and co lo u r

contrasts on asteroid surfaces. Icarus 40, 364-374. DRAKE, M. J. 1979. In Asteroids (T. Gehrels, Ed.), pp. 765-782. Univ. of Arizona Press, Tucson. DRUMMOND,J. D., A. ECKART,ANDE. K. HEGE 1988. Speckle interferometry of Asteroids. IV. Reconstructed images of 4 Vesta. Icarus 73, 1-14. DRUMMOND,J. D., AND E. K. HEGE 1989. In Asteroids II (T. Gehrels and M. S. Mattews, Eds.), pp. 171-190. Univ. of Arizona Press, Tucson. DUDINOV, V. N., V. W. KONICHEK, S. G. KUZ'MENKOV, V. S. TSVETKOVA, V. S. RYLOV, L. V. GUAVGUANEN, AND V. N. EROKHIN 1982.

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In Instrumentation f o r Large Optical Telescopes, pp. 191-198. Proc. IAU Colloq. N68, Dordrecht.

Photometric and polarimetric studies of Lunar regions at small angles. Kinematika i Fis. Neb. Tel 6(1), 3-9.

GRADtE, J., E. TEDESCO, AND B. ZELLNER 1978. Rotational variants in the optical polarization and reflection spectrum of Vesta. Bull. Am. Astron. Soc. 10~ 595. LUPISHKO, D. F., I. N. BELSKAYA,O. I. KVARATSKHELIA, N. N. KISELYOV, A. V. MOROZHENKO, AND N. M. SHAKHOVSKOY 1988. Polarimetry of Vesta in opposition of 1986. Astron. Vestnik 2, 142146. OPANASENKO,N. V., Yr. G. SHKURATOV,AND V. A. KUCHEROV 1990.

SHEVCrlEr~KO V. V. 1980. Modern Selenography. Nauka, Moscow. SHKURATOV, YU. G. 1988. On the nature of polarimetric inhomogeneity of asteroid 4 Vesta surface. Astron. Vesmik, 2, 152-158. WOROEN, S. P., M. K. STEm, G. D. SHMIDT, ANO J. R. P. ANOEL 1977. The angular diameter of Vesta from speckle interferometry. Icarus 32, 450-457. WORDErq, S. P. 1979. In Asteroids (T. Gehrels, Ed.), pp. 119-13 I. Univ. of Arizona Press, Tucson.