ICARUS 84, 334-351 (1990)

The Shape of Eros

S. J. OSTRO, K. D. ROSEMA, AND R. F. JURGENS

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

Received June 26, 1989; revised September 6, 1989

The convex hull of Eros' polar silhouette, estimated from radar echo spectra obtainedin 1975 by R. F. Jurgens and R. M. Goldstein (1976, Icarus 28, 1-15), provides newinformation about this asteroid's shape. Monte Carlo simulations are used to optimize theestimation, to explore the nature and severity of associated errors, and to guide bias-correction procedures. Eros' hull is shaped like a rounded trapezoid, whose long and shortbases faced Earth during epochs of primary and secondary maxima, respectively, inthe January 1975 optical lightcurves. The nonaxisymmetric shape helps to explain oddharmonics in Eros' echo spectral signature as a function of rotation phase, whosepresence cannot be accounted for by homogeneous ellipsoid models. The extreme breadthsof Eros' polar silhouette are within a few kilometers of 35 and 16 km. Additional con-straints on Eros' figure are obtained by inverting an optical lightcurve to estimate theasteroid's "mean cross section," which is a two-dimensional average of the three-dimen-sional shape. Eros' mean cross section and polar silhouette have similar elongations.The hull estimate permits previously reported radar time-delay and Doppler-frequencymeasurements to be referenced directly to Eros' center of mass. e 1990 Academic Press, Inc.

I. INTRODUCTION

433 Eros, discovered by G. Witt in 1898, isdistinguished by being the first known Mars-crosser, the first asteroid to show a light-curve (von Oppolzer 1901), and the first as-teroid to be observed while occulting a star(O'Leary et al. 1976). The discovery ofEros' large orbital eccentricity attractedconsiderable interest and catalyzed a seriesof dynamical studies, including Russell's(1900) doctoral dissertation. Triangulationof the distance to Eros and analyses of per-turbations on Eros' orbit provided the mostreliable values for the solar parallax and theAU (i.e., for the scale of the solar systemexpressed in terrestrial units) until the ad-vent of planetary radar ranging in the early1960s (McGuire et al. 1961).

Eros made its closest approach to Earthin this century (0.15 AU) on January 13,1975, and was favorably placed for tele-scopic investigation within several monthsof that date. A campaign was organized toseize this opportunity using available astro-

0019-1035/90 $3.00Copyright ( 1990 by Academic Press, Inc.All rights of reproduction in any form reserved.

nomical techniques, and the May 1976 issueof Icarus was devoted to papers reportingthe observations. Zellner (1976) noted thatno other solar system body in its size rangehad been so thoroughly observed, and sum-marized the results as follows:

Newly available photometric, polarimetic, spec-troscopic, thermal-radiometric, radar, and occul-tation results are synthesized in order to derive acoherent model for Eros. The geometric albedo is0.19 - 0.01 at the visual wavelength, and the over-all dimensions are approximately 13 x 15 x 36km. The rotation is about the short axis, in thedirect sense, with a sidereal period of 5h16 " 13.4'.The pole of rotation lies within a few degrees ofecliptic coordinates X = 160 and 3 = + 110. Erosis uniformly coated with a particulate surface layerseveral millimeters thick. It has an iron-bearingsilicate composition, similar to that of a minorityof main-belt asteroids, and probably identifiablewith H-type ordinary chondrites.

The constraints on Eros' shape, derivedprimarily from optical lightcurves (e.g.,Dunlap 1976, Millis et al. 1976, Scaltriti andZappala 1976) and 3.5-cm radar observa-tions (Jurgens and Goldstein 1976), were

334

THE SHAPE OF EROS

obtained within the framework of someassumed, axisymmetric model. Such ap-proaches were appropriate first approxima-tions, but their inadequacy was apparentfrom the presence of odd harmonics inthe lightcurves and in the rotation-phasedependence of the radar signatures. Inanalyzing the latter, Jurgens and Goldstein(1976) used a homogeneous, triaxial ellip-soid model (Jurgens 1982), but they warnedof that model's inability to accommodatea strong, predominantly first-harmonicwobble in the echo spectral shape. Thepresence of the fundamental Fourier com-ponent indicated that Eros' figure and/or it's radar scattering properties are notaxisymmetric, but it was not at all clearhow one might obtain information aboutthe nature of the asymmetry.

Recently, Ostro et al. (1988a) introduceda theoretical approach to asteroid echo-spectral analysis that follows naturally fromthe geometric relation between spectraledge frequencies and the shape of a rotatingasteroid. This approach uses the extent ofthe spectra to determine the convex enve-lope, or hull, of the asteroid's polar silhou-ette-a pole-on projection of the asteroidwith concavities "filled in." Ostro et al.(1988a) showed how to estimate the hull,developed error-analysis techniques, andstudied the relation between the accuracyof a hull estimate and the parent data set'ssignal-to-noise ratio.

Here we estimate Eros' hull and use theresults to evaluate, and in a few instancesrefine, the 1976 constraints on the aster-oid's physical properties. In the next sec-tion, we describe the Eros radar data setused in our calculations. In Section III, webriefly review important geometric rela-tions and hull estimation mathematics, andthen describe in considerable detail thelogic and procedures underlying optimiza-tion of our hull estimator. Practical aspectsof hull estimation not confronted by Ostroet al. (1988a) are addressed here. For ex-ample, for a given set of echo spectra, theaccuracy of a hull estimate turns out to

depend on the data set's frequency resolu-tion and rotation-phase resolution. TheEros data are "overresolved" in each do-main, forcing us to smooth the data andto search for the best phase/frequencyfilter. In Section IV, we present our esti-mate of Eros' hull and then, armed withnew information about the asteroid'sshape, revisit several intriguing character-istics of the optical and radar data.

II. THE EROS RADAR DATA

We worked with 199 echo spectra fromGoldstone X-band (8495-MHz, 3.53-cm) ob-servations conducted on January 19, 24, 25,and 26, 1975 (Table I of Jurgens andGoldstein 1976). On these dates, receptionwas in the sense of circular polarization or-thogonal to that transmitted. These "OC"echoes, which contain most of the power inback-reflections from smooth componentsof Eros's surface, are approximately threetimes stronger than the SC (same circular)echoes.

The raw frequency resolution was 1.95Hz on January 19 and 2.73 Hz on thelast three dates; we interpolated betweenspectral estimates from the January 19 datato find values at 2.73-Hz intervals. Eros'echo bandwidth was known to range from-300 to -700 Hz, so the fractional spectralresolution was -0.01. The echoes lie nearthe middle of a spectral window -1200 Hzwide.

The integration time per spectrum was- 150 sec, during which Eros rotatedthrough 2.8° of rotation phase. The rota-tion-phase coverage of the 199 spectra isvery thorough and fairly uniform, with nogaps larger than 6°.

The spectral elements have units of stan-dard deviations of the receiver noise, andeach spectrum is tagged with y, the valueof the standard deviation in units of squarekilometers of radar cross section per 2.73-Hz resolution cell. y is constant for anygiven spectrum, but varies between spectrabecause of run-to-run variations in radarsystem sensitivity and target distance.

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OSTRO, ROSEMA, AND JURGENS

An asteroid's instantaneous echo powerspectrum has a bandwidth given by

B = (4rrDIAP) cos 6, (1)

where P is the apparent spin period, 6 isasteroid-centered declination of the radar,and D is the sum of the "support distances"p+ and p- from the plane tfo containing theasteroid's apparent spin vector fl and theline of sight to the surface elements with thegreatest positive (approaching) and negative(receding) radial velocities. In the figure, theplanes qi+ and i- are parallel to q0 and tan-gent to the asteroid's approaching and re-ceding limbs; A+, and j- are at distancesp,and p - from tpo; fo, f<, and f- are the Dop-pler frequencies of echoes from portions ofthe asteroid intersecting qi0, A+,, and qj ; andB = f+ -f- = 4tr(p+ + p -)(XP) -cos 6.

The "support function," p(O) - p,()-p -(0 + 1800), is a periodic function of rota-tion phase 0. Santald (1976) shows that

p(0) + p"(0) = r(0), (2)

where the primes denote differentiation withrespect to 0, and r(0) is the radius of curva-ture of H where A+ touches H. The Carte-sian coordinates of H are given by

FIG. 1. Geometric relationships between an as-teroid's shape and its echo power spectrum. The planeq0 contains the line of sight and the asteroid's spinvector. Echo from any portion of the asteroid intersect-ing qi0 has Doppler frequency fo. The cross-hatchedstrip of power in the spectrum corresponds to echoesfrom the cross-hatched strip on the asteroid. The x, ycoordinate system (dark lines) rotates with the asteroid.Reproduced, with permission, from Ostro et at.(1988a).

III. THEORY AND PRACTICE OFHULL ESTIMATION

A. Geometric Relations and Definitions

Figure 1 sketches the relation between anasteroid's hull and an echo power spectrumobtained at rotation phase 0. One can thinkof the hull, H, as the shape of a rubber bandstretched around the projection of the aster-oid onto its equational plane.

x(0) = p(0) cos 0 - p'(0) sin 0

y(0) = p(0) sin 0 + p'(0) cos 0,(3)

where the x, y coordinate system (dark linesin the figure) rotates with the asteroid.

B. Mathematics of Hull EstimationHenceforth, we assume prior knowledge

of the asteroid's apparent rotation period.For the moment, let us also assume that weknow the Doppler frequencyf, correspond-ing to echoes from the center of mass.

Estimation of H can be thought of as athree-step process:

Step 1. Generate a data vector of supportfunction estimates from the spectra. We de-rive two numbers, p(0) andp(0 + 1800), froma spectrum acquired at rotation phase 0, soL spectra yield a 2L-element data vector.Note: In this section, we use hertz as the

w0:0.

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THE SHAPE OF EROS

units for the support function and the hull,postponing until Section IV the conversionto kilometers, which requires a value of &.

Step 2. Use least squares to fit an M-har-monic Fourier series to the data vector. Thatseries has the form

M

p(O) = A a, cos nO + b0 sin nO. (4)n=O

Let boldface denote matrices and T denotetranspose. We define the data vector of sup-port-function estimates

pT = (pI, P2 , . P2L) (5)

and the vector of Fourier coefficients

XT = (X, X2, . . , X2M+I)

- (a0 , a,, bl, a2, b2 , . . . , am, bM). (6)

If the errors e in the support-function esti-mates p are random variables with zeromean and covariance matrix M = (8ET),then the weighted-least-squares estimate ofx is x = B -IATM- 1p, where B = ATM- 'Aand the i"t row of A is (1, cos Oi, sin 0i, cos20;, sin 20, . . .., cos LO,, sin LO,). Theweighted sum of squared residuals, Q(x) =(p - Ax)TM(p - Ax), defines a quadratichypersurface with minimum Q(i). The co-variance matrix for errors in the parameterestimates is V B 'and the Fourier modelfor the data is p Ax.

Since H is convex, its radius-of-curvaturefunction is nonnegative:

r(0) = p(O) + p"(0) > 0. (7)

In practice, our estimate of H will be an N-sided convex polygon corresponding to amodel support function that satisfies

r(27rkIN) = r(0k) - P(Ok) + P"(Ok) - 0,k = 1, . . ., N. (8)

These N inequality constraints would "au-tomatically" be satisfied by x if the spectrawere continuous, contained no noise, andwere available as a continuous vector func-tion of rotation phase. In practice, echospectra are discrete and noisy, and sample

rotation phase imperfectly, so x might notsatisfy those constraints even though itminimizes the weighted sum of squaredresiduals Q(x), and an additional step isrequired.

Step 3. Find the value x of x that mini-mizes Q(x) subject to the N inequalityconstraints. x is the closest point to x(in the B metric) that corresponds to anonnegative radius-of-curvature function.Our task, to minimize the quadratic form(x - i)TB(x - i) subject to the constraints,is an example of the strictly convex qua-dratic programming problem. Ostro andConnelly (1984, Section 11C and AppendixB) describe this problem and its solutionvia a recursive projection method. Forall the work described in this paper, wetruncated Fourier series to M = 10 har-monics and applied the inequality con-straints at N = 96 rotation phases.

C. Accuracy of Hull Estimation:Spectral-Frequency Resolution andRotation-Phase Resolution

Whereas execution of steps 2 and 3 isstraightforward, the best way of estimatingthe spectral edge frequencies (step 1) isnot known. For simplicity, we use theclosest zero crossings to fo as the valuesof p(O) and p(0 + 1800) associated with aspectrum taken at rotation phase 0. Notethat this or any other support-function esti-mator might introduce systematic uncer-tainty into the hull estimation. This uncer-tainty will not be known a priori becausethe true shape of the spectral edges isunknown, so it will be prudent to assess theseverity of systematic errors by performingnumerical simulations.

We expect our "zero-crossing estimator"to yield hulls whose accuracy depends onthe frequency resolution, the rotation-phasesampling, and the signal-to-noise ratio of theparent spectra. For the Eros data, the "to-tal" signal-to-noise ratio (SNR) of the opti-mally filtered, weighted sum of all the spec-tra is about 70 standard deviations. Given

337

OSTRO, ROSEMA, AND JURGENS

the analysis of Ostro et al. (1988a), we knewthat with an SNR that low, estimation of Hfrom the raw spectra would be very inaccu-rate. That is, the spectra are so finely re-solved that the echo edges are overwhelmedby noise, and application of the zero-cross-ing estimator would yield a hull with a sizemuch smaller than the true one and presum-ably with a grossly incorrect shape.

To overcome the low SNR of the Erosspectra, we can either (i) smooth the spectrato a coarser frequency resolution, therebyimproving their SNR, and/or (ii) formweighted averages of several spectra takenat similar rotation phases, i.e., increase theSNR by smoothing the spectra to a coarserrotation-phase resolution. Of course, wewould like to do this "frequency and/orphase filtering" in a manner that optimizesthe accuracy of the hull estimate. The fol-lowing strategy was used to evaluate anygiven choice of frequency resolution andphase resolution:

i. Use Jurgens and Goldstein's (1976) tri-axial ellipsoid model to generate 199 noise-free echo spectra having 2.73-Hz resolutionand the same relative phases as the actualEros spectra.

ii. Add Gaussian noise supplied by a ran-dom number generator to each of the modelspectra, scaling the standard deviation ofthe noise to match that in the correspondingspectrum of the actual data.

iii. Smooth the noise-contaminated modelspectra to a frequency resolution Af and aphase resolution AO (Appendix).

iv. Estimate the model's hull according tothe three-step procedure described earlier.Then compare the estimate, H(AO, Af), tothe true hull (a known ellipse), and assessthe accuracy provided by this choice ofphase/frequency smoothing.

In (ii), note that the vector of noise consti-tutes a single realization of the parent ran-dom process. To better assess the varianceof this estimator, we repeated the simula-tion for five independent noise realizations.This entire, five-realization simulation was

done for each choice of phase/frequencysmoothing.

D. Optimization with Respect to AOand AfFigure 2 shows results for all nine combi-

nations of the values: AO = 150, 200, and250, and Af = 25, 30, and 35 Hz, coveringthe region yielding the mpst satisfactory re-sults. The accuracy of H(AO, Af) deterio-rates rapidly outside this region.

It is useful to define a simple measure ofthe "distance" Q between two hulls HI andH2 as the root-sum-square of the distancesbetween those two polygon's correspondingvertices:

NNn 2 = [(X -X 2)2 + (yj - Y2)2 I.k (9)

The figure gives the mean value and rmsdispersions of the error distance betweenH(AO, Af) and the true (elliptical) hull foreach of the nine, five-simulation sets. Ineach set, the five distances' mean valuequantifies the estimator's bias. The squareof the five distances' dispersion about theirmean can be taken as a crude indicator ofthe estimator's variance.

The minimum-bias estimator in the figureis at (AO, Af) = (200, 30 Hz), but the differ-ences among the nine values of (AO, Af)do not seem highly significant. Therefore,although we will refer to E(20°, 30 Hz) asthe optimum estimator, we will apply allnine estimators to the Eros data, and willuse the spread in the results as a gauge ofestimation error.

Our estimator recovers the true hull'smaximum breadth quite accurately but over-estimates the minimum breadth by -20%.This positive bias in the estimated hull's min-imum dimension probably is caused by acombination of the limited SNR and the factthat the minimum-bandwidth orientation isseen very briefly, despite the fairly uniformand thorough phase coverage provided bythe Goldstone data. (Note that phase filter-

338

THE SHAPE OF EROS

Af = 30 Hz Af = 35 Hz

18.1 i 5.1 19.4 ± 5.4

16.7 ± 5.5 18.5 ± 5.5

17.6 + 4.2 17.2 ± 4.9

FIG. 2. Hull estimates for simulations designed to explore the dependence of estimation accuracy onthe data's frequency resolution (Af) and rotation-phase resolution (AO). The dotted profile is the truehull of an ellipsoidal model asteroid and the solid curves are estimates derived from simulated echospectra contaminated with noise. The five profiles at any given value of (Af, AO) correspond to fivedifferent realizations of the noise-generating random process. The difference between a hull estimateand the true ellipse is quantified by an error distance fl defined in Eq. (9), and this figure gives themean value and rms dispersion of fI for each five-simulation set.

ing "works against" our seeing the mini-mum echo bandwidth, but would not keepus from seeing the maximum bandwidth.)

E. Center Frequency as a Free ParameterSo far, we have assumed prior knowledge

of f0, the Doppler frequency corresponding

to echoes from Eros' center of mass. In real-ity, that frequency is a function of time,t. During the Eros radar observations, thereceiver was continuously tuned to theecho using a prediction ephemeristhat later proved to be accurate to withinseveral tens of hertz (Fig. 2 of Jurgens

Af = 25 Hz

AO = 250

AO = 200

17.0 ± 4.1

17.0 + 4.4

AO = 15°

17.6 ± 2.5

339

OSTRO, ROSEMA, AND JURGENS

-60 0 60

IbaaA!k~ ~I-~5

-60 0 60

I0

.1 . d.I_

i .

-60 0 60

-60 0 60

/

* 1�A

A'.-"C.-.-

-60 0 60

I /A

I.

-60 0 60

-60 0 60

-7

-60 0 60

/A

-60 0 60

Center Frequency, Hz

FIG. 3. Sensitivity of the hull estimators to center frequency for the model-ellipsoid simulationsdescribed in Section IIIE. The nine blocks of curves correspond to the nine drawings in Fig. 2. Withineach block, the five curves correspond to five different noise realizations, as in Fig. 2. RWMS, definedin Eq. (10), is the root-weighted-mean-square residual between support-function data p and the support-function values p corresponding to a hull estimate. Here, RWMS is plotted on arbitrary linear scalesversus the center frequency f0 used in the estimation, measured with respect to the true value.

and Goldstein 1976). The variation of theDoppler prediction error during the weekof observations was expected to be muchless than the data's frequency resolution(2.73 Hz).

Henceforth, let f0 represent Eros' echocenter frequency relative to the observingephemeris. We wish to treat this offset as afree parameter, so we explored each estima-tor's sensitivity to fo by repeating each ofthe 45 simulations in Fig. 2 six more times,offsettingfo from its true value by ±5, ± 10,

and + 15 frequency resolution cells. Foreach simulation, we calculated the root-weighted-mean-square postfit residualRWMS, defined by

RWMS2

(p - ' ') S- 2 1 S2 (10)

and plotted the five-simulation-averageRWMS versus offset frequency in Fig. 3.

3I

340

THE SHAPE OF EROS

Af = 25 Hz

AO = 25°

AO = 200

Af = 30 Hz

I

Af = 35 Hz

397 Hz

1 X

AO = 150

FIG. 4. "Raw" estimates of Eros' hull at nine values of phase/frequency resolution. The maximumand minimum breadths for the optimal estimate, H(200 , 30 Hz), are indicated. All profiles are plottedat the same scale. For each profile, the center of rotation is denoted by a tiny dot and the centroid isdenoted by a large symbol. Dotted replicas of the eight profiles on the figure's periphery are superposedon H(20', 30 Hz). The simulations described in Section IIID indicate that these estimations are biased,with the hull's minimum dimension being overestimated by -20%. A bias-corrected version of thecentral drawing is shown in Fig. 6. See text.

On average, RWMS's minimum appears tobe within -10 Hz of the true value of fo,although biases as large as -20 Hz are evi-dent in some of the individual curves.

IV. EROS' SHAPE

A. The Hull Estimate

Figure 4 shows results of applying eachof the nine best estimators (Fig. 2) to the

Eros data. The center drawing superposesIt(20', 30 Hz) and dotted replicas of the eightother estimates to permit more direct com-parison of the profiles' sizes and shapes.Table I gives corresponding values of thehull's maximum and minimum widths, theirratio, and the distance of each estimate ofEros' hull from H(200, 30 Hz). Table I alsolists the goodness-of-fit statistic RWMS.

341

OSTRO, ROSEMA, AND JURGENS

TABLE I

"RAW" EROS HULL ESTIMATES'

Af = 25 Hz Af = 30 Hz Af = 35 Hz

Distance (Hz) from H(200 , 30 Hz)AO = 250 26.0 25.0 28.0AB = 200 37.3 0 12.5AO = 150 56.7 42.8 31.8

Support-function RWMS (Hz)AO = 250 42.8 45.5 51.6A9 = 200 41.0 42.6 44.5A 0= 150 48.5 51.1 48.0

"Raw" maximum dimension (Hz)A = 25° 638 663 702AA = 200 638 651 668A 0 = 150 630 647 666

"Raw" minimum dimension (Hz)AO = 250 368 397 414A = 200 343 397 406AO = 15° 296 356 373

"Raw" ElongationAO = 250 1.74 1.67 1.70aO = 20' 1.86 1.64 1.65AO = 150 2.13 1.82 1.79

a Results for each of the nine estimators E(AO, Af), where AO and Af are the rotation phase and spectralfrequency resolutions. Table II shows results of applying bias corrections described in the text to the three lower3 x 3 subtables.

The minimum in RWMS at (AO, Af) = (200,

25 Hz) is not a global minimum; another,nearly identical, local minimum occurs at(AO, Af) = (200, 15 Hz).

In Fig. 4, the Earth is toward the bottomof the page and the hull rotates clockwiseabout its center of mass (the small dot). Wehave rotated H(200, 30 Hz) to bring its mini-mum dimension normal to the line of sightwith the longest side on the receding limb.We define this orientation to have rotationphase 6 = 00. Equivalent "minimum-breadth" orientations of the eight other esti-mates occur at rotation phases within 2.50of 00. This is excellent agreement in view ofthe 2.80 phase resolution of the individualspectra and the angular quantization (3600/N = 3.750) used in our estimations. Conse-

quently, we have rotated all nine hull esti-mates in Fig. 4 by the same amount.

Additional results associated with Eros'hull estimation tare shown in Fig. 5 for thecalculation of H(20', 30 Hz). The left boxshows p and p, and the right box shows theradius-of-curvature functions r and r. Thehull's bandwidth function,

B(O) = p(- ) + pJ(O + 1800), (11)

is the uppermost curve in the top left plot;note that it contains only even harmonics.The corresponding "middle frequencyfunction,"

f id(-) [=(0) - p3(O + 1800)1/2, (12)

is the lowest curve in that plot; note that itcontains only odd harmonics.

342

THE SHAPE OF EROS

N

4-

I-

co

C.,0:5a

90 180 270Rotational Phase, deg

90 180 270Rotational Phase, deg

FIG. 5. Quantities associated with Eros' hull estimation, shown here for the calculation of H(20', 30Hz), which is the solid profile in the center of Fig. 4. In the left-hand box, the vertical bars representthe support-function data pi obtained from the echo spectra; each bar extends from pi - Ei to pi + si,where the errors si were determined a posteriori, as described in the Appendix. The solid curve throughthe support-function data is p, the unconstrained Fourier model. The dotted curve through the data isP, the closest Fourier model to p yielding a nonnegative radius-ofcurvature function; p correspondsto the hull estimate via Eq. (3). The hull's bandwidth function B(O) and middle-frequency functionfmnd(

6) are defined by Eqs. (11) and (12). The radius-of-curvature functions r and i in the right-hand

figure are related to p and p via Eq. (2).

The support function data are shown asvertical bars extending from pi - ej to pi +e;, where the values of the errors e; weredetermined a posteriori, as described in theAppendix. Note that the fractional precisionof individual support-function estimate~s isvery low; the leverage in determining H isfurnished by the geometrical constraints inEq. (7), i.e., by the fact that we are estimat-ing a convex quantity.

How can we gauge the uncertainty in ourestimate of Eros' hull? Recall from the dis-cussion of the ellipsoid simulations that thesystematic uncertainty in E(AO, Af) is quan-tified by the mean error distance El associ-ated with any particular estimator, while thestatistical uncertainty in E(AO, Af) is quan-tified by fl's rms dispersion (Fig. 2). ForE(20', 30 Hz), II's mean and rms values are16.7 ± 5.5 Hz, so the systematic uncertaintyis several times more severe than the statis-tical uncertainty. The sum of these two errorcomponents is 22.2 Hz, which is 72% aslarge as the average distance (31 Hz) of the

solid profile [H(AO, Af)] from the dottedprofiles in Fig. 4 and 40% as large as themaximum distance (56.7 Hz) of the solidprofile from a dotted profile. Thus, the "col-lective smear" of the profiles in the centerdrawing of Fig. 4 overstates the magnitudeof the total error in our optimum estimator.

B. Bias Correction

As discussed earlier, the ellipsoid simula-tions yielded hull estimates whose longestdimension was estimated quite accuratelybut whose shortest dimension was overesti-mated by up to -20%. Table II gives correc-tion factors which, when multiplied by thefive-realization-average values of the ex-treme breadths of the estimated hulls, resultin the extreme breadths of the ellipsoid's ac-tual, elliptical hull. For example, that ellipsehas minimum and maximum breadths thatare 82 and 100% of those of the five-realiza-tion-average value for the optimal estimatorE(20', 30 Hz). Thus, if we express Eros' hullestimate in a Cartesian coordinate system

N

0

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OSTRO, ROSEMA, AND JURGENS

433 EROS

Min 2

174 km

Time Delay

Max 1 ,Max 2

Minf

Mn 1{ |Occultation

FiG. 6. Bias-corrected estimate of the convex hullof Eros' polar silhouette. Each profile in the centraldrawing of Fig. 4 has been scaled by the horizontal andvertical correction factors in Table 11. The solid profileis our final hull estimate, corresponding to a phase/frequency filter (AO, Af) = (200, 30 Hz), and the dottedprofiles correspond to the other eight choices of (AO,Af) in Fig. 4. As discussed in the text, the arrowsindicate orientations at epochs of lightcurve extrema(Millis et al. 1976), the radar time delay measurementby Campbell et al. (1976), and the stellar occultationobservations by O'Leary et al. (1976).

with x and y axes parallel to the ellipse's mi-nor and major axes, respectively, and thenmultiply the hull'sx andy coordinates by 0.82and 1.00, we can compensate forthe most se-rious bias in our estimator.

We have applied bias corrections to thenine estimates of Eros' hull in Fig. 4. Figure6 shows a bias-corrected version of the cen-tral drawing from Fig. 4, and Table II liststhe corresponding hull dimensions. Sincewe wish to ensure that errors assigned toEros' hull estimate and derived quantitiesare not underestimated, we will quote un-certainties based on the total range of valuesspanned by the nine, bias-corrected estima-tions, and whenever practicable will expressthe results as an interval estimate (Freundand Walpole 1980).

C. Size and Shape of Eros'Polar SilhouetteFor the bandwidth equivalents of the

maximum and minimum dimensions ofEros' hull, our bias-corrected estimationsyield

638 ' Bmax ' 681 Hz

263 c Bm_ < 329 Hz.(13)

Eros' synodic spin period during theradar observations was 0.21954 + 0.00001day = 5.2689 ± 0.0002 hr (Dunlap 1976,Table IV). Substituting that value into Eq.(1) yields a conversion factor of (0.05327/cos 5) km Hz- '. Constraints on Eros' poledirection (Dunlap 1976, Millis et al. 1976,Scaltriti and ZappalA 1976, Drummond et al.1985, Drummond and Hege 1989) establish

TABLE 11

BIAS-CORRECTED EROS HULL ESTIMATES0

Af = 25 Af = 30 Af = 35Hz Hz Hz

Al) = 250A = 20°AO = 150

AO = 250AO = 200A = 150

AO = 250AO= 200AO = 150

AO=AO=AO=

250200150

Bias-correction factors fromellipsoid simulations

(x, y) (x, y) (x, y)0.84,1.00 0.82,0.98 0.79,0.970.86,1.02 0.82,1.00 0.81,0.980.89,1.05 0.85,1.01 0.84,0.99

Maximum dimension (Hz)638 650 681651 651 655662 653 659

Minimum dimension (Hz)309 326 327295 326 329263 303 313

2.062.212.52

Elongation1.992.002.16

2.081.992.11

a The three lower 3 x 3 subtables show results ofcorrecting corresponding values in Table I for bias inthe hull estimators, by multiplying the maximum andminimum dimensions by the x and y correction factorsgiven in the top subtable.

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THE SHAPE OF EROS

that 181 was no more than -100 from zeroduring the radar observations, i.e., that cosS was within a few percent of unity. Thus,converting units establishes a length scalefor the extreme dimensions of Eros' hull:

34 c Dmax c 37 km

14 s Dmin c 18 km.

These intervals are in decent agreementwith the extreme dimensions of various axi-symmetric models of Eros reported in theliterature, including Zellner's (1976) consen-sus dimensions (16 x 37 km) and Lebofskyand Rieke's (1979) values for a thermalmodel based on a cylinder with hemispheri-cal ends (16.1 ± 0.8 km x 39.3 + 2.0 km).

Eros' hull is shaped like a trapezoidwhose bases have different lengths andwhose sides are distinctly nonparallel. Thepolar silhouette is clearly not axisymmetric,suggesting that some of the odd-harmoniccharacter of the 1975-1976 optical light-curves and radar echoes can be attributedto the asteroid's shape, as discussed furtherbelow.

D. The Hull's Centroid

In Figs. 4 and 6, the hull's center of rota-tion, i.e., Eros' center of mass, is indicatedby a dot, while the hull's centroid is markedby a large symbol. The offset of the hull'scentroid from Eros' center of mass, between1 and 2 km, constrains Eros' mass distribu-tion, albeit very loosely. A modest concav-ity along the silhouette's lower left edge of-fers a simple explanation for the offset, butthere obviously are a myriad of other possi-bilities.

E. Eros' Echo Power Spectra RevisitedFigure 7 plots weighted sums of the Eros

echo spectra within 12 30'-wide windowscentered on 0 = 00, 300, . . ., 330°. Aboveeach spectrum is a replica of the bias-cor-rected estimate of Eros' hull (the solid pro-file in Fig. 6), drawn at the same linear scaleas the Doppler axis and at the indicated rota-tion phase (cf. Fig. 1).

Note how the spectral bandwidth tracksthe horizontal breadth of the hull. Thereader might wish to follow Eros through afull rotation, exploring the relation betweenspectral shape and the inclination of thehull's boundary to the line of sight. In con-trast to the situation at optical wavelengths,radar backscattering laws for virtually allplanetary surfaces studied so far exhibitlimb darkening. To the extent that the hullreveals the orientation of surface elementsresponsible for the radar echoes, we expectsome correlation between a spectrum's am-plitude at any given Doppler frequency andthe angle of incidence at the correspondingpoint on the hull. Some correlation of thissort is indeed present at severalphases-more normally oriented portionsof the hull often appear to return more echopower.

The 12 spectra comprise six pairs of spec-tra taken 180° apart. The differences be-tween "opposing" spectra constitute oddharmonics in Eros' spectral signature,which were noted by Jurgens and Goldstein(1976). These differences, plus the facet thatthe hull's middle-frequency function imid(o)

wobbles around zero (Fig. 5), demonstratesthat the asymmetry in the shape of Eros'polar silhouette explains that characteristicof the echoes.

F. Rotation Phase of Eros' Hull: Relationto Lightcurve Extrema

A UTC epoch near the middle of the weekof Goldstone observations and correspond-ing to 0 = 00 is 1975 January 23.34758. Milliset al. (1976) report epochs of lightcurve ex-trema, which they identify as Max 1, Min 1,Max 2, and Min 2. Those authors translateepochs to the asteroid's frame, while weretain an observatory-based frame, whichis "uncorrected for light time." Taking thedifference between these two conventionsinto account, we find that the Min 1 configu-ration occurred 0.00063 day, or 54 sec, or1.00 of rotation phase, before the minimum-breadth configuration.

345

OSTRO, ROSEMA, AND JURGENS

433 EROS

e80 0 aDoppler Frwqjuny. Hz

FIG. 7. Eros echo spectra and hull at 12 rotation phases. Weighted sums of spectra within 30'-widewindows have been filtered to a frequency resolution Af = 40 Hz and plotted on identical, linear scalesof echo power versus Doppler frequency. A horizontal line is drawn at zero echo power. A verticalbar at the origin indicates + 1 standard deviation of the receiver noise. A replica of Eros hull estimate,the solid profile in Fig. 6, is drawn above each spectrum at the same scale as the Doppler frequencyaxis and at the indicated, weighted-mean rotational phase. The phases are rarely integral multiples of30° because the data's phase coverage is not perfectly uniform.

During January 19-26, Max I was the pri-mary maximum (-0.1 mag brighter thanMax 2), and Min 2 was the primary minimum(-0.1 dimmer than Min 1). Thus, assketched in Fig. 6, the primary maximumoccurred when Eros' longest side was facingEarth. At the secondary maximum, the, op-posite side of the asteroid faced the viewerand the hull's projected length was the sameas at primary maximum, but less of it wasnormally oriented at secondary maximumthan at primary maximum.

If we were dealing with a two-dimensionalasteroid, then these differences in the distri-bution of incidence angle, coupled with theassumption that the asteroid is limb-dark-ened, would furnish a simple explanation forthe difference between the lightcurve max-ima; the same logic would explain the differ-

ences between the lightcurve minima. How-ever, the solar phase angle (i.e., theSun-Eros-Earth angle) was within 1° of 90during the radar observations, and very littlelimb darkening is expected this close to op-position (e.g., French and Veverka 1983,Hapke 1986, Lumme and Bowell 1981).

In the absence of limb darkening (i.e., forpurely geometric scattering), odd harmonicsin the lightcurve can arise from many differ-ent kinds of shape effects and albedo varia-tions. However, the differences betweenmatched extrema constitute roughly 10% ofthe lightcurve amplitude while optical polar-ization and color variations are only at the-1% level (Zellner and Gradie 1976, Larson

et al. 1976, Pieters et al. 1976, Miner andYoung 1976, Tedesco 1976, Veeder et al.1976), so it seems likely that the lightcurve

346

THE SHAPE OF EROS

characteristics are caused more by Eros'shape than by photometric heterogeneity.

G. Lightcurve Inversion: Estimation ofEros' Mean Cross Section

If Eros' lightcurves are attributable pri-marily to the asteroid's shape, what can thelightcurves tell us about that shape? Ostroet al. (1988b) show that under certain idealconditions, one can use "convex-profile in-version" of a lightcurve to estimate a profilewhich, unlike the hull, is a two-dimensionalaverage of the asteroid's shape. That profileis called the mean cross section, C, and isdefined as the average of the envelopes onall the surface contours parallel to the equa-tor. The ideal conditions for estimating Cinclude Condition GEO, that the scatteringis uniform and geometric; Condition EVIG,that the viewing-illumination geometry isequatorial; and Condition PHASE, that thesolar phase angle 4 is known and nonzero.The logic behind these conditions is thatthey collapse the three-dimensional light-curve inversion problem, which cannot besolved uniquely, into a two-dimensionalproblem that can.

During late January 1975, the sub-Sun andsub-Earth points were not very far fromEros' equator and, given the 90 solar phaseangle, the scattering was probably almostgeometric. Condition EVIG, that the sub-Earth and sub-Sun points lie on the equator,and Condition PHASE imply that the solarphase angle's "equatorial component" 4 )

eq

equals 4 itself; this component of 4 createsthe mapping between odd harmonics in Cand those in the lightcurve. If 4)

eq = 0, thenan equatorial lightcurve of a uniform, geo-metrically scattering asteroid will contain noodd harmonics. For this reason, inversion ofan opposition lightcurve furnishes an even-harmonic-only version of C called the aster-oid's "symmetrized mean cross section,"C,. The accuracy of an estimate of C canbe degraded if the value of 4) used in theinversion is very different from eq. ForEros, available pole-direction estimates(Dunlap 1976, Millis et al. 1976, Scaltriti

and Zappala 1976, Drummond et al. 1985,Drummond and Hege 1989) have uncertain-ties on the order of 15°, so it is hard to saywhether O., is closer to zero or 4.

A subtlety of Condition PHASE is that thesign of 4 used in "convex-profile inversion"corresponds to the asteroid's rotation sense.The sign is positive if the asteroid rotatesthrough 4 from the Earth direction to theSun direction, a configuration that pertainsto Eros in late January 1975 if, as deducedby Dunlap (1976) and Morrison (1976), theasteroid's rotation is direct.

Convex-profile inversion of an Eros light-curve obtained by Tedesco (1976) on Jan 20at 4) 9° yields the estimates of C and C,shown in Fig. 8. These averages of Eros'shape rotate clockwise and are shown at4 = 0°, as in the hull figures. The Earthdirection is toward the bottom of the figureand the Sun direction is 90 clockwise fromthere.

Our estimate of Eros' mean cross sectionis "tapered," and some support for the va-lidity of this result is offered by Lebofskyand Rieke (1979), who argue that a taperedshape can help to explain aspects of the as-teroid's thermal-infrared signature (i.e., theIR lightcurve as a function of wavelengthbetween 1.2 and 22 /im) observed by thoseauthors and by Morrison (1976). Brown(1985) has used radiometric models basedon ellipsoids to demonstrate that diur-nal-thermal lightcurves depend dramati-cally on asteroid shape; however, axisym-metric models might prove inadequate forhighly irregular asteroids. In this context,we suggest that our estimate of Eros' meancross section reveals shape characteristicsresponsible for odd-harmonic componentsof the asteroid's thermal-IR "emission"lightcurve as well as phase differences be-tween the emission and "reflected-light"curves.

For the circumstances at hand, we expectthe estimate of the mean cross section'sbreadth ratio, /3* = 2.36, to be accurate toseveral percent. We stress that this elonga-tion pertains not to Eros' three-dimensional

347

OSTRO, ROSEMA, AND JURGENS

^ ED-J

U) -

U)Liz

M W0 180 360

ROTATIONAL PHASE

FIG. 8. Eros' mean cross section C, a two-dimen-sional average of the asteroid's shape. The solid profileis the estimate of C and the dotted profile is its symme-trization Q,, as described in the text. The profiles werederived from Tedesco's (1976) lightcurve via convex-profile inversion (Ostro et al. 1988b). The profiles rotateclockwise and are shown at rotational phase 6 = 0°.Earth's direction is toward the bottom of the figure andthe Sun's direction is 9° clockwise from there. At left,the large symbols are the lightcurve data, the solidcurve is a Fourier model fit to the data, and the dottedcurve is the model lightcurve corresponding to the esti-mate of C.

shape but to a two-dimensional average ofthat shape. Nevertheless, since the "con-tour averaging" implicit in the definition ofC and C, is done in radius-of-curvaturespace (Appendix A of Ostro and Connelly1984), the largest contours are weightedmost heavily, so we should not be surprisedthat f3* is similar to the hull's elongation,Dma/ID in, which apparently lies between1.9 and 2.6 (Table II).

Our estimates of Eros' hull and meancross section have their longest sides on thereceding limb at zero rotational phase. Per-haps that face of Eros lacks prominent posi-tive relief. For the other "sides," we specu-late that the appar~ent differences betweenthe curvatures of C and H arise from inter-esting topographic structure off Eros'equator.

H. Astrometric Results Referenced toEros' Center of MassEcho Doppler frequency. Recall from

Section IIME that one of our goals was to

estimate f0 , the Doppler frequency corre-sponding to echoes from Eros' center ofmass, measured with respect to the predic-tion ephemeris. Figure 4 is the culminationof a search for the value offi that minimizesthe postfit RWMS, defined in (10). The opti-mum estimator gives fo = - 17 Hz and theeight surrounding estimators give values be-tween -24 and - 13 Hz. Correcting the ob-serving ephemeris by -17 + 15 Hz, we ob-tain the Doppler frequency estimate in TableIII.

Echo time delay. During the week of theGoldstone experiment, Eros radar observa-tions were also conducted at Arecibo Obser-vatory, at 70-cm wavelength (Campbell etal. 1976). On January 22, a "ranging" wave-form was used in attempts to measure theecho's round-trip time delay. The single runyielding detection of an echo was at an ep-och (1976 Jan 22, 0 4h30j 0 0s UTC, referredto the instant of reception) close to primarylightcurve maximum (Fig. 6).

Campbell et al. report a time delay(150,885,295 ± 5 UTC /isec) measured "tothe weighted center of the surface scatteringfunction, which is closer to Earth than, butnot precisely located with respect to, theobject's center of mass. A best guess of theoffset is that it equals the radius of Eros inthe line-of-sight at the time of the observa-tion: about 8 km (or 50 usec in round-tripdelay). The quoted error, of course, doesnot reflect this uncertainty."

Since the subradar latitude was close tozero, it seems reasonable to use our hullestimate to determine that offset. Thequoted epoch corresponds to a rotationphase of 2580, at which the hull's center ofmass is 9.7 ± 1.5 km (or 65 + 10 /usec)further from the radar than the subradarpoint. Adding this offset to the measurementreported by Campbell et al. yields an esti-mate of the time delay corresponding to ech-oes from Eros' center of mass (Table III).

V. CONCLUSION

Our calculations have elucidated the non-axisymmetric, quasi-trapezoidal shape of

348

THE SHAPE OF EROS

TABLE III

RADAR ASTROMETRIC RESULTS FOR EROS'

Date UTC Observatory Transmitter Doppler Time delay(1975) (hh:mm:ss) frequency frequency (UTC ,usec)

(MHz) (Hz)

Jan 22 04:30:00 Arecibo 430 150,885,360 + 15Jan 23 07:00:00 Goldstone 8495 670 + 15

a Measurements correspond to echoes from the asteroid's center of mass. Epochs are referred to the instantof reception. For the delay entry, our hull estimate has been used to refine a measurement reported by Campbellet al. (1976). See text.

Eros' polar silhouette. This result helps toexplain the rotation-phase variations in theradar echo spectra and, to a lesser extent,the presence of odd harmonics in the opticallightcurves. Our estimate of Eros' meancross section C is a "weaker" shape con-straint than H; however, visual and thermal-IR lightcurves are disc-integrated mappingsof precisely those average shape character-istics conveyed by the mean cross section.

Unambiguous information about suchshape attributes as concavities, nonpolarprojections, and nonequatorial curvaturecomponents must be sought via stellar-oc-cultation timings, radar delay-Doppler im-aging, and speckle interferometry. The soleobservations of a stellar occultation by Eros(O'Leary et al. 1976) suffer from incompleteground-track coverage and the lack of pho-toelectric records. On the other hand, initialefforts to reconstruct images of Eros fromspeckle interferometric visibility functionsare promising (Drummond and Hege 1989).Their speckle image, taken at a solar phaseangle -40° and with the sub-Earth point.20° from the south pole, has a "peanut-

shaped" outline consistent with our hull es-timate, and also reveals concavities andbrightness variations.

Unlike speckle images and occultationprofiles, a radar-derived hull has an a prioriorientation with respect to the spin vector;it is conveniently perpendicular, so the spinvector's projection is a point whose locationinside the hull can be determined. Conse-

quently, we have been able to tie the 1975radar time-delay and Doppler-frequencymeasurements to Eros' center of mass, set-ting a precedent for small-body radar as-trometry. The resultant pair of refinedastrometric measurements can be used toimprove the accuracy of prediction ephem-erides for future ground-based and space-craft studies of Eros (Yeomans et al. 1987).

APPENDIX: PHASE SMOOTHING DETAILS,

In the phase-smoothing operation, all theNSPEC spectra whose phases Oi fall withina specified window are combined into aweighted mean; the weights are wi = y7 2where yi is the noise level in the ith spectrum(Section II). The noise level Y/ in theweighted-mean spectrum satisfies y-2 =

Eli 2

Let the zero crossings in the weighted-mean spectrum be p+ and p-. These twosupport-function values are associated withrotation phases 0+ and 0, + 180°, and itmight seem proper to set 0+ equal to theweighted mean (0w) of the 0i. However, wewish to ensure that the least-squares Fourierfit (step 2 in Section IIIB) "be aware" of thephases Oi and noise levels yi of the NSPECparent spectra. Therefore, we let the win-dow's contribution to the support-function"data vector" p be NSPEC identical pairsof zero crossings (p +, p -) whose associatedphases (0, and 0, + 180°, 02 and 02 +s 180,*. , ONSPEc and ONSPEC + 180°) and noiselevels are those of the parent spectra. In

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OSTRO, ROSEMA, AND JURGENS

different words, we repeat the pair (p+,p-) NSPEC times in p, and the pair's ithappearance is associated with the rotationphase and noise level of the ith parentspectrum.

The error assigned to the ith pair can bewritten &i= Fy, where F is a proportionalityfactor whose value is assumed to be con-stant for the entire Eros data set but is notknown a priori. This is all right, because Fmerely scales the data-covariance matrix Mand drops out of the least-squares calcula-tion of x. If the errors £ are normally distrib-uted and our estimator is unbiased, wewould expect the postfit residuals to be x2

distributed (Jenkins and Watts 1968, Appen-dix A4. 1). Under this assumption, the valueof F satisfying

N(2L - K) F2 = 1 I(p- p)1y2 (A1)

i-I

with K = 2M + I and 2L equal to the lengthof p, let us calculate the effective error inthe support-function data a posteriori. Thisprocedure was followed in assigning theerror bars to the support-function data inFig. 5.

In exploring phase-averaging approaches,we experimented with independent win-dows as well as overlapping windows (i.e.,running boxes), for a variety of windowsizes. The results led us to settle on overlap-ping windows, offset from each other bymultiples of 5°.

ACKNOWLEDGMENTS

We thank P. Ford for a constructive critique of thispaper. This research was conducted at the Jet Propul-sion Laboratory, California Institute of Technology,under contract with the National Aeronautics andSpace Administration.

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