p7< L /- //V@JL- .i3, 

SPECKLE INTERFEROMETRY OF ASTEROIDS 

NAGW-867 February 1, 1987 - January 31,1988 /y/2 YJ 

FINAL REPORT 

Jack Drummond 

Steward Observatory 

University of Arizona 

Tucson, Arizona 85721 

May 31, 1988 

(NASA-CR-180438) SPECKLE INTERPEROHETRX OF N 88- 228 43 

ASTE8OIDS Final Report, 1 Peb. 1987 - 31 

Jan. 1988 (Arizona Univ.) 24 p CSCL 038 

Unclas 

G3/9Q 0 141 243 

1 
Abstract 

This final report for NASA Contract NAGW867 consists of abstracts of the 

first three papers in a series of four appearing in Icarus that were funded by the 

preceding contract NAG W224: 

1) Speckle Interferometry of Asteroids I. 433 Eros, 

2) Speckle Interferometry of Asteroids II. 532 Herculina, 

3) Speckle Interferometry of Asteroids III. 51 1 Davida and its Photometry; 

the abstract of the fourth acknowledged to NAGW867: 

4) Speckle Interferometry of Asteroids IV. Reconstructed Images of 4 Vests; 

and a review (complete copy enclosed) of the results from the asteroid speckle 

interferometry program at Steward Observatory prepared for the Asteroids II book: 

Two papers on asteroids, indirectly related to speckle interferometry, were written 

in part under NAGW867. One is in press and its abstract is included: 

6) Photometric Geodesy of Main-Belt Asteroids. II. Analysis of Lightcurves 

for Poles, Periods, and Shapes; 

and the other, which has been submitted to Icaru8, is included in full: 

Stellar Occultations. 

5) Speckle Interferometry of Asteroids. 

3) fiiaxid Ellipsoid Dimensiom ab Xctaticnd Pde of 2 Pallas from TRG 

2 
? 

SPECKLE INTERFEROMETRY OF ASTEROIDS 

Jack D. Drummond and E. Keith Hege 

Steward Observatory, University of Arizona 

Abstract 

Assuming that an asteroid can be treated as a smooth, featureless triaxial 

ellipsoid rotating about its shortest axis, we summarize Steward Observatory’s two 

dimensional power spectra analysis for speckle interferometry observations of six 

asteroids. The poles and triaxial ellipsoid dimensions of 4 Vesta, 433 Eros, 511 

Davida, and 532 Herculina have been previously reported. New results for 2 Pallas 

and 29 Amphitrite are given, as well as further results for Vesta. The ultimate goal 

of image reconstruction has been achieved for Vesta and Eros, and images for these 

two are displayed. SI and radiometry diameters are compared, and diameters from 

the two occultations of Pallas are also addressed. 

I. INTRODUCTION 

Speckle interferometry (SI) is a high angular resolution technique first suggested in the form wed here hy Labeyrie (1970). To overcome the generally one 

arcsec limit to resolution imposed by the Earth’s atmwphere, and to approach the 

theoretical resolving power of large telescopes given by the Rayleigh criterion, very 

short exposures (approximately 0.01 sec) of an object are recorded. These short 

exposures reveal many small patches of coherence (‘apertures’), which can then be 

combined (Fourier transformed) to obtain information, as in a multiple-aperture 

interferometer, down to the resolution limit of the telescope. Worden (1979) gives 

a good description and illustrations of the phenomenon. 

Since the angular sizes of all asteroids are less than one arcsec, but many 

are greater than the theoretical resolution limit of larger telescopes, they are ideal 

targets for the application of the technique. The Fourier modulus or power spectrum 

(ps) of the exposures of the ‘speckles’ contains diffraction limited information of 

resolved asteroids. Although this information can be obtained from either the ps or 

its transform, the autocorrelation function (ac), it is easier to calibrate observations 

of extended sources by deconvolving (dividing) with the ps of a point source, which 

then also removes residual seeing effects as well as the effects of the telescope transfer 

function. Furthermore, the impact of individual photons can more accurately be 

1 
gauged in power spectra domain since this photon bias shows up as a simple DC 

background level in the ps of both the object and the point source, rather than 

compacted into one or two pixels in the ac domain. 

11. AUTOCORRELATION/POWER SPECTRUM ANALYSIS 

A. Background 

For interpretation of the two dimensional autocorrelation (ac) of an asteroid, or 

its Fourier inverse the two dimensional power spectrum (ps), use is made of what has 

evolved into a standard set of assumptions for most techniques, SI and photometry 

in particular. An asteroid is considered to be a smooth (no major topographic 

features such as craters, mountains, etc.), featureless (no albedo variation over 

its surface), uniformly bright (geometric scattering; no limb darkening) triaxial 

ellipsoid (axes diameters u 3 b 3 c) rotating about its shortest axis. Under these 

assumptions the asteroid ellipsoid model will project onto the plane of the Earth’s 

sky a series of ellipses as the body rotates. The observed major and minor axes, and 

position angle, will vary in a unique fashion as a function of rotation. Drammond 

et al. (19858) derive the equations relating these observable parameters back to the 

triauial c!!ipsoid axes dimensiona and to the three Euler angles used to then locate 

the spin axis direction. Thus a non-linear least squares method can be used on two 

or more ellipses (or their ac or ps), to find an asteroid’s true dimensions and pole. 

The assumption of an ellipsoidal shape is a mathematically tractable, first 

order approximation to the actual shape of an asteroid. Since for most purposes 

any body can be described by its principal axes ratios, the ellipsoid assumption is 

a very general and useful abstraction. Departures from the ellipsoid (and perhaps 

other) assumptions are sometimes evident, but generally as secondary effects. The 

assumption of a triaxial ellipsoid rotating about its shortest axis is rather standard, 

and would be a natural stable outcome of a body in gravitational and/or hydrostatic 

equilibrium, such as would be formed by either coalescence or catastrophic collisions 

which result in so-called rubble piles (Davis et al. 1979; Farinella et al. 1981; Catullo 

et al. 1984; Zappala et al. 1984). Rotation about the short axis is the most stable 

configuration, and even precession induced by perturbations and collisions would be 

expected to be damped out over a small fraction of the lifetime of the Solar System 

(Burns and Tedesco 1979). 

For dark atmosphereless bodies observed at low solar phase angles uniform 

2 
brightncss is to be expected for all reasonable scattering laws (Dollfus and Zellner 

1979). Moreover, limb darkening, which may be 5-10% for completely smooth bodies, is shown to be reduced to less than 5% in meteorites by roughness (French and 

Veverka 1983). Deformation of a triaxial ellipsoid shape by the presence of mountains, craters, etc., might be important for smaller bodies, but should be negligible 

for larger asteroids. Even with a random distribution of such deformations, it is 

still possible to treat the object as a triaxial ellipsoid with noise (irregularities of 

outline). And unless the deformation has a different albedo, it has no effect on SI 

measurements until it lies on the limb. Similarly, Fulchignoni and Barucci (1984) 

have shown that even for the largest craters known (in term of the body diameter), 

for Phobos, Mimas, and Thetis, the presence of a crater with the same albedo as 

the surrounding asteroid material cannot be detected in a lightcurve. (See also the 

chapter by Magnusson et al. in this book for further justification of the adopted 

assumptions). Thus we proceed to recount the results of ps analysis of asteroids 

performed at Steward Observatory with the above assumptions. 

B. Results from Power Spectrum Analysis 

2 Pallas 

In an attempt to use asteroids as calibration objects for SI, 2 Pallas was observed in 1979 (Hege et al. 1980a;1980b) and found to be extremely elongated. The 

notion of asteroid satellites was in vogue at the time, and these observations were 

interpreted as indications for a large Pallas satellite. However, this first exploratory 

effort revealed the necessity for much better calibration of the point spread function, the telescope transfer function, and the geometric distortion introduced by 

the detector. Furthermore, it was then realized that in order to properly derive the 

dimensions of an asteroid it needs to be observed throughout its rotation, and in 

fact such a procedure could lead to a pole determination as well. 

On April 9 and 10, 1982, nine observations were made at Steward Observatory’s 2.3m telescope with the same equipment as used for the other asteroids since 

reported in the literature. The results from these Pallas observations are given (for 

the first time) in Table I. Substantial, systematic departures from a triaxial ellipsoid fit are evident, which we have come to realize as characteristic of the impact 

of albedo features on our ps analysis. 

Pallas is the only asteroid with two well observed stellar occultation outlines, 

3 
which are free from albedo effects and which give apparently very reliable direct 

measurements of its size and shape at two epochs. In considering these occultations, 

and the two models of the triaxial ellipsoid shape and rotational pole derived from 

them by Wasserman et al. (1979) and Magnusson (1986), it can be shown that 

neither model yields the occultation outlines as observed. Drummond and Cocke 

(1988) show that with the ellipsoid equations at the heart of the SI ps analysis 

derived in Drummond et al. (1985a), two occultations give a narrowly defined pole 

(to within a two-fold sense of rotation ambiguity) and triaxial dimensions. Taking 

the major and minor axes dimensions and the position angle for the two events as 

given by Wasserman et al. (1979) and as reported by Dunham in a private communication to Magnusson (1986), the c axis is found to be no greater than 120 km. 

Since this is unrealistically small, and since the first occultation had much larger 

uncertainties on the measured parameters than the second, a range in solutions is 

found by varying the first occultation parameters by their uncertainties. It then 

becomes obvious that the size of the outline on the first occasion was (only slightly) 

overestimated, since only smaller dimensions yield better (larger c) solutions. Taking the average and standard deviations for the various possible realistic (c 2 300 

km) solutions results in a pole at (70;--15) or (250;tE) with a 9" radius error circle, 

and dimensions of 596( f19)x528 (fl)x407( f55). 

The weighted averages between the direct techniques of SI and occultations are 

given for the axial dimensions in Table 11. Because of the geometries of the observations, SI determines the c axis better than the occultations, but the occultations 

define the b axis more precisely. Lebofsky et al. (1987) have recently revised the 

radiometric scale with Ceres and Pallas occultation results and derive an average 

diameter of 532km for Pallas. The IRAS diameter (Matson et d. 1986) is 523f20. 

The mean diameter (u~c)'/~ of the model of Pallas from SI and the occultations 

given in Table I1 is 528km, with a range of 504 for the minimum cross section seen 

at equatorial aspects to 552 for the maximum cross section seen at polar aspects. 

The pole in Table I1 is the average between the speckle pole at (100;-22), the occultation pole from the mean of the possible solutions at (70;-15), and the pole at 

(54;-6) from lightcuwe analyses by Magnusson (1986). 

4 Vesta 

Departing from our traditional intensified photon counting video raster detec4 
tor in favor of Harvard’s PAPA and Stanford’s MAMA two-dimensional photon 

counting arrays, we have obtained excellent observations of 4 Vesta. As reported 

by Drummond et al. (1988a), ten observations were made with the PAPA detector 

over two nights in November, 1983. Triaxial ellipsoid dimensions and the usual two 

poles were obtained from ps analysis. The superior characteristics and geometric 

fidelity of the detector allowed successful image reconstructions, discussed further 

in the next section. Three years later, over three nights in October, 1986, we obtained some 65 observations with the MAMA detector, which fully corroborated the 

results from the earlier run. In fact the two-fold ambiguity inherent to SI, which 

leads to a two-fold ambiguity in the location of the rotational pole and is normally 

resolved by appealing to lightcurves, is fully resolved with SI alone because of the 

differing geometries for the two runs. The two possible poles from the PAPA run 

were (336:+55) and (209;-50), while the two from the MAMA run were around 

(324;+52) and (83;-64); the correct region of the pole is obvious. Treating the 

results from each of the three nights of the MAMA run independently, the weighted 

average of four pole determinations involving 10,17,23, and 25 observations from the 

two runs is given in Table I, and since it can be regarded as definitive, is repeated 

in Tabie II. 

In a similar fashion, the weighted average of the four SI triaxial ellipsoid dimension determinations are given in both Tables I and II. The mean diameter of 

Vesta would vary between extremes of 498 and 549 km, with (u~c)’/~ = 520. This 

is much closer to the radiometrically determined 530 in the TRIAD file than the 

579 from polarimetry. (Note that these diameters were inadvertently attributed to 

the opposite techniques by Drummond et al. 1988a). Brown et al. (1982) have suggested that the radiometric diameters in the TRIAD file should be reduced by 5% 

on the average, which would make Vesta’s radiometric diameter 504km. The IRAS 

diameter is listed as 501f24. It appears that most of the radiometric observations 

were made near Vesta’s equatorial plane, where the average diameter would range 

between 498 and 514 km. 

29 Amphitrite 

Previously unpublished results from ps analysis of five observations of Amphitrite on July 1, 1985, are given in Table I. It is well known from its irregular 

photoelectric lightcurves that Amphitrite must possess substantial albedo varia5 
tions over its surface, making conclusions and interpretations somewhat dangerous. 

Nevertheless, with the large number of lightcurves available and the converging 

consensus of axial ratios from photometric analysis, this asteroid is suggested as a 

calibration standard for S asteroids in the chapter on poles and shapes by Magnusson et al. in this book. 

From the photometric determinations of the rotational axis listed in Part VI 

of this book, two ambiguous poles emerge, at (139;-33) and (328;-41), both with 

about 15" errors. Considering the two possible poles from SI listed in Table I, the 

pole ambiguity is resolved in favor of the first listed pole; the average from the six 

photometric and one SI determination is listed in Table 11. 

The consensus axial ratios for Amphitrite given in the chapter by Magnusson 

et al. includes the ratios of a/b = 1.22 f .17 and b/c = 1.06 f .13 contributed by 

SI. In light of the good agreement between the photometric and SI axial ratios, it 

is somewhat surprising to find that the mean TRIAD polarimetric and radiometric 

diameters of 197 and 200km, respectively, and the IRAS value of 219f5 do not agree 

that well with the mean SI diameter of 160, with an extreme range of 148 to 168 

corresponding to (bc) V2 and (ab) V2 , respectively. The reason for the discrepancy 

undoubtedly lies in the surface albedo structure inferred from the lightcurves, but 

it is not clear whether this influences SI or radiometry more. Adjusting the mean 

diameter to 190km, midway between the SI and the IRAS values, and adopting the 

consensus axial ratios of 1.21~1.08~1.00, the best estimate of the dimensions are 

given in Table 11, where the uncertainties are the vector sum of the uncertainties in 

Table I and the differences between the dimensions in the two tables. 

433 Eros 

Key to the development of the present form of speckle interferometry is the 

small Earth-approaching asteroid Eros. Spurred by observations separated by a 

month, but for substantially different geometries at high solar phase angles, equations expressing the projection of triaxial ellipsoids to ellipses were derived and 

converted into a non-linear method to relate the observed major and minor axes 

and position angles back to the triaxial ellipsoid dimensions and rotational pole 

(Drummond et al. 1985a). Figure 1 shows the seven 2-D power spectra from two 

nights in December 1981, arranged in order of rotational phase. The lower half of 

each frame is the actual power spectra, and the upper half is the ellipse fit to the ps. 

6 
Since ps are mirror symmetric, it is sufficient to fit and display a half-plane. If Eros 

were a smooth trixial ellipsoid these elliptical power spectra would have the same 

shape as the ellipses projected onto the plane of the sky. At the very least, Fig. 1 

clearly shows an elongated object spinning in space when the sub-Earth point was 

close to the rotational pole. 

The results from the two runs are summarized in Table I. The pole is resolved 

from SI alone because of the differing geometries of the two runs, and is consistent 

with the same resolution known from photometric methods. Table 11 gives the 

average of the eight most recent pole determinations (including the SI pole) listed 

in Part VI of this book. The dimensions in Table I1 are the averages and standard 

deviations from SI, radiometry (Lebofsky and Reike 1979), radar (Jurgens and 

Goldstein 1976), and the consensus model of Zellner (1976), all listed by Drrlmmond 

et al. (1985a). The biggest discrepancy between the SI and other results appears to 

be in the pole, but given the small target, the large (40 and 52O) solar phase angles 

involved in the SI observations and the distinctly non-ellipsoid shape shown in the 

reconstructed image in the next section, the agreement is actually quite good. 

51 1 Davida 

From only five observations on one night covering only a quarter of a rotation, 

the results given in Table I were derived from ps analysis (Drummond et al. 1986). 

Although it was suggested that the difficulty in finding a triaxial ellipsoid solution 

was perhaps due to albedo features, subsequent photometric analysis strongly euggests the contrary, that Davida is very smooth and uniform. In fact, Magnusson 

et al in this book offer it as the best calibration standard available. Despite the 

large error associated with the SI poles, it is still easy to resolve the ambiguity in 

photometric pole determinations. Part VI in this book shows that the two possible 

poles are located around (95;+30) and (300;+35), but considering the two possible 

poles from SI in Table I it is easy to identify the correct one. 

Because of the small number of observations the SI diameters have large associated errors. Therefore, it is not surprising that the mean SI diameter of 350km, 

with a range of (bc)'i2 = 304 to (ab)'/' = 408, is larger than the TRIAD radiometric diameter of 323km or the downward revised 307km, or the IR.AS diameter 

of 337f5. For our best estimate of the diameters in Table I1 we use the well determined axial ratios from the Magnusson et al. chapter and a mean diameter midway 

7 
between the IRAS and SI values. The preferred uncertainties are then simply the 

difference between the diameters in Tables I and 11. The errors from SI are not used 

because they are unrealistic. The diameters used in Table I1 are certainly within 

the quoted uncertainties of the true values without considering the SI errors. 

532 Herculina 

Herculina presents a peculiar case. Its sometimes single max/min and sometimes double max/min lightcurves have proven difficult to explain. The first attempt to develop a model was given by Drummond et al. (1985b) to explain the 

SI observations. The pole and dimensions from ps analysis are given in Table I. 

In order to account for peculiar SI measurements at certain rotational phases, and 

to account for the lightcurve history, a bright spot was postulated. However, with 

a photometric astrometry method, Taylor et al. (1987) found an entirely different 

pole, and offered a spherical model with two dark spots. But this model was in 

turn rejected because Lebofsky et al. (1988) showed from thermal lightcurves that 

the visible lightcurve amplitude was caused by changing cross sectional area and 

therefore Herculina could not be a sphere. Current efforts (Drummond et al. 1988b) 

are concentrating on a triaxial ellipsoid model with major albedo or topographic 

(crater) features. 

It would be premature to list parameters with any degree of confidence in Table 

11. While the original TRIAD radiometric diameter of 220km is in good agreement 

with the SI mean diameter of 231km, the revised diameter of 209 falls outside the 

range inferred from SI of 216 to 239km. However, the IRAS diameter of 231f4 

agrees exactly with the SI mean diameter. Further observations with all techniques 

are needed before anything firm can be said about the shape of Herculina, but 

its average size seems well-determined and the pole from Taylor et d. (1987) at 

(96;-1) derived from the timings of lightcurve features appears firm. 

Comparison of SI to other Diameters 

Fig. 2 shows the mean diameters (~bc)'/~ from Table I plotted against the 

radiometric diameters from the IRAS file (Matson et d. 1986). The vertical lines 

on the mean diameters are not error bars, but represent the extreme possible range 

of cross section, from (bc)'j2 in the equatorial plane to (ub)'I2 at polar aspects 

according to the SI values in Table 1. Observations should statistically tend to 

occur near the bottom of this range since an asteroid's orbit guarantees that the 

8 
Earth will cross the equatorial plane every sidereal period, but only in exceptional 

circumstances (the rotational pole lying in the orbital plane) could a polar view be 

provided. The horizontal lines are error bars from the IRAS list. For comparison, 

the diameters from the TRIAD file (Morrison and Zellner 1979), reduced by the 5% 

suggested by Brown et ul. (1982), are shown as open circles. The Pallas diameter, 

however, is from Lebofsky et al. (1986), where a refined thermal model was used. 

Also note that Eros does not have an IRAS measurement, so only a reduced TRIAD 

diameter is used. 

Except for 29 Amphitrite, the agreement between the SI and radiometric diameters is quite good. Pallas' radiometric diameter has been considerably reduced 

from the original TRIAD diameter of 589 km. For two asteroids (511 and 532) 

the agreement between the SI and IRAS diameters are much better (moved to the 

right in Fig. 2) than the TRIAD diameters with the indiscriminate 5% reduction. 

Perhaps some of the discrepancies in Fig. 2 can be explained by albedo features 

since such features can have an impact on diameters derived from SI ps analysis, 

and could move the SI diameters up or down in the figure. Or perhaps individual treatment of asteroids with non-standard thermal modeling would reduce the 

discrepancies. One of the strong pints =bout the SI resrlltn ic! that all three axes 

diameters are derived at once, so that the vertical lines can be drawn in the first 

place. With SI each asteroid is treated independently and completely, requiring no 

assumptions or modelling with indeterminant parameters. 

111. IMAGE RECONSTRUCTION 

A. Background 

The image of an object, O(r), can be reconstructed from measurements of the 

object's complex visibility function by inverse Fourier transform: 

O(r) = FT-'(V(f)), where r and f represent image ordinary space (x,y) and 

image frequency space (u,v), respectively. The complex visibility V (f), measured 

in image frequency space, is conveniently represented, after Euler, in terms of its 

amplitude and phases: V(f) = A(f)e'"('). The image amplitudes A(f) are just the 

square root of the image power spectra calibrated as required for the ps/ac analysis 

as described above. 

The image phase measurements in the Knox-Thompson approximation are obtained from the same complex Fourier transform, &(f) = FT(ik(r)) of the observed 

9 
specklegrams ik(r). The ps is accumulated as the time averaged square modulus of 

the set k = 1,2,3, ..., m, of Fourier transformed specklegrams, 

PS(f) =< I;(f)Ik(f) >m where < ... >m denotes the average. Knox and 

Thompson (1974) showed that phase differences, and hence by numerical integration 

the phases, can be accumulated from two quite similar complex accumulations in 

the two dimensional f = (u, u) plane: 

CS,(f) =< I;(u,u)I(u + 1,~) >m and CS,(f) =< I;(U,U)~(U,U + 1) >m. 

Just as there is an image plane analog of the ps (the ac), there are equivalently 

image plane analogs to the two phase difference cross spectra, the two corresponding 

complex cross correlation functions. Hence the complex visibilty measurements can 

be made in either data domain. The phases and amplitudes can be combined to 

retrieve the image of the object (see Drummond et al., 1988, for details of the 

procedures BS applied to Vesta). 

B. Images of 433 Eros and 4 Vesta 

In collaboration with the speckle group at Steward Observatory, R. H. Bates of 

the University of Canterbury in New Zealand has produced a reconstructed image 

of the second observation of Eros on December 18, 1981, which corresponds to the 

third ps from the right in the top row in Fig. 1. (In Drummond et al. 1985a, the 

observation is the sixth point in Figs. 2 and 3, the third frame in Fig. 9, and is the 

sixth drawing in Fig. 10.) Line drawings of this projected image are given in Fig. 3a, 

and intensity profiles are presented in two different perspectives in 3b and 3c. (The 

intensities between the dotted and solid line in the projected image and exterior 

to the dotted line in the intermediate perspective are rather uncertain because of 

instrumental artifacts along these rasters.) The solar phase angle at the time was 

40" and the latitude of the sub-Earth point was -74" according to the pole in Table 

I, or -62O with the pole from Table 11. The projected image reveals that Eros is 

shaped more like a peanut than a strict triaxial ellipsoid and the intensity profiles 

suggest that the brightness distribution across the small asteroid is not uniform at 

40° solar phase angles, both of which would contribute to the difference between 

the poles in the two tables. Does this picture of Eros correspond to the idea that it 

may be a 'chip' off a larger body or does it look more like the nucleus of a comet? 

Eight images of Vesta were reconstructed and discussed by Drtlmmond et al. 

(1988a). The data were taken with Harvard's PAPA detector, and with the close 

10 
cooperation of C. Papalilios, P. Nisenson, and S. Ebstein. The images are due to A. 

Eckart. Although there was a problem with multiple detection of photons, which 

have since been corrected, the images were sufficient to reveal dark and bright 

areas on the surface. A spot model was developed based on the images which 

when combined with the triaxial ellipsoid shape derived from the pa analysis could 

reproduce all low solar phase angle lightcurves ever taken, down to the rotational 

phase and amplitude. 

From theoretical considerations of its inferred basaltic crust, it could be argued 

that the derived triaxial shape (Drummond et al. 1988a), where u/b = 1.10 f .04 

from ten observations, cannot be correct, that a/b must be 1.0, and that Veta is an 

equilibrium figure (Cellino et al. 1987). It should be pointed out that the SI results 

are observational, with the model being driven by the images, and not vice versa. 

However, m the observations stand, the possibilty certainly exists that Vesta’s u/b 

ratio may be closer to unity, with the observed spots dominating the lightcurves 

more than the unequal axes. In fact, with some 65 later observations with Stanford’s 

MAMA detector, provided through the courtesy of J. Morgan and J. G. Timothy, 

the u/b ratio is now considered to be 1.06f.04, as reported in Table I. The question 

then hecomes how clwe to unity mi& the ratio be to satisfy theoretical objections. 

Perhaps analysis of images reconstructed from the newer data set will settle the 

issue. One from the set of 65 images currently being constructed is shown in Fig. 

4. 

11 
Acknowledgments 

The results presented in this chapter were obtained under NASA grants NAGW-224 

and NAGW-867. The chapter was written with funds from the latter. 

12 
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13 
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Labeyrie, A. 1970. Attainment of diffraction limited resolution in large telescopes 

by Fourier analyzing speckle patterns in star images. Astron. and Aatrophya. 

6: 85-87. 

Lebofsky, L. A., and Reike, G. H. 1979. Thermal properties of 433 Eros. Icarua 40: 

Lebofsky, L. A.;Sykes, M. V.;Tedesco, E. F.;Veeder, G. J.;Matson, D. L.;Brown, R. 

H.;Gradie, J. C.;Feierberg, M. A.;and Rudy, R. J. 1987. A refined ‘standard’ 

thermal model for asteroids based on observations of 1 Ceres and 2 Pallas. 

Icarua 68: 239-251. 

Icarus 54: 38-47. 

297-308. 

Lebofsky, L. A.;Greenberg, R.;Tedesco, E. F.;and Veeder, G. J. 1988. Infrared 

lightcurvews of asteroids 532 Herculina and 45 Eugenia: Proof of the absence 

of significant albedo markings. Icarua accepted for publication: . Magnusson, P. 1986. Distribution of spin axes and senses of rotation for 20 large 

asteroids. Icarus 68: 1-39. 

Morrison, D., and Zellner, B. 1979. Polarimetry and radiometry of the asteroids. 

In Aster~& ed. T. Gehrels, pp. 1090.1097.: . Taylor, R. C; Birch, P. V.;Drummond, J.;Pospieszalska-Surdej, A.; and Surdej, J. 

1987. Asteroid 532 Herculina: Lightcurves, pole orientation, and a model. 

IC~~US 69: 354-369. 

14 
Wasserman, L. H.;Millis, R. L.;Franz, 0. G.;Bowell, E.;White, N. M.;Giclas, H. 

L.;Martin, L. J.;Elliott, J. L.;Dunham, E.;Mink, D.;Baron, R.;Honeycutt , R. 

K.;Hendon, A. A.;Kephart, J. E.;A’Hearn, M. F.; Reitsema, H. J.;Radic, 

R.;and Taylor, G. E. 1979. The diameter of Pallas from its occultation of 

SA0 85009. Astron. J. 84: 259-268. 

Worden, S. P. 1979. Interferometric determinations of asteroid diameters. In Asteroids ed. T. Gehrels, pp. 119-131.: . Zappala, V.;Farinella, P.;Knezevic,Z.;and Paolicchi, P. 1984. Collisional origin of 

the asteroid families: Mass and velocity distributions. Icarus 59: 261-285. 

Zellner, B. 1976. Physical properties of asteroid 433 Eros. Icczrus 28: 149-153. 

15 
Figure Captions 

Fig. 1. Two dimensional power spectra of 433 Eros from December 17,18 1981. 

The first figure contains rings, in outward order, of .3, .l, and .06 arcsec in power 

spectra domain used for scaling illustrations. The upper part of each frame is the 

elliptical fit to the lower half. 

Fig. 2. SI vs IRAS radiometric diameters. The range in SI diameters, from 

(bc)'l2 to corresponds to the minimum and maximum possible values according to Table I. The dots and their horizontal error bars are from the IRAS file 

(Matson et d. 1986), and the open circles are TRIAD diameters reduced by 5%, 

except for 2 Pallas which comes from Lebofsky et uf. (1987). 

Fig. 3. Intensity profiles from a reconstructed image of 433 Eros. 3a is the 

outline of the asteroid's reconstructed image, 3b and 3c are intensity profiles from 

different perspectives. 

Fig. 4. Reconstructed image of 4 Vesta. To be provided. 
. 

Table I 

Diameters and Poles from SI 

Asteroid a b C Ecliptic Pole (1950) 

2 Pallas 537 f 29 488 f 11 485 f 11 100 -22 f 8 

295 +16 f 8 

4 Vesta 566 f 15 532 f 15 466 f 15 327 +55 f 4 

29 Amphitrite 186 f 18 152 f 15 144 f 10 134 -36 f 12 

303 +35 f 12 

433 Erog 41 f 3 15 f 2 14 f 2 23 +37 f 14 

511 Davida 465 f 90 358 f 58 258 f 356 196 -12 f 29 

291 +37 f 29 

532 Herculina 263 f 14 218 f 12 215 f 12 132 -59 f 7 

128 +?4 f ? 

Table I1 

Diameters and Poles from Combining SI and Others 

Asteroid a b C Ecliptic Pole (1950) 

~~ 

2 Pallas 578 f 27 528 f 4 482 f 15 74 -15 f 24 

4 Vesta 566 f 15 532 f 15 466 f 15 327 +55 f 4 

29 Amphitrite 210 f 30 188 f 39 174 f 19 138 -33 f 16 

433 Erog 38 f 2 15 f 1 14 f 1 17 +16 f 11 

511 Davida 417 f 48 333 f 25 292 f 34 299 +34 f 9 

532 Herculina ?*? ?f? ? f? ? f? 
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