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ICARUS 75, 30-63 (1988)

Convex-Profile Inversion of Asteroid Lightcurves:Theory and Applications

STEVEN J. OSTROJet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

ROBERT CONNELLYDepartment of Mathematics, Cornell Uniuersity, Ithaca, New York 14853

AND

MARK DOROGI'Department of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853

Received October 8, 1987; revised December 8, 1987

Convex-profile inversion, introduced by S. J. Ostro and R. Connelly (1984, Icarus 57,443-463), is reformulated, extended, and calibrated as a theory for physical interpreta-tion of an asteroid's lightcurve and then is used to constrain the shapes of selectedasteroids. Under ideal conditions, one can obtain the asteroid's "mean cross section" C,a convex profile equal to the average of the convex envelopes of all the surface contoursparallel to the equatorial plane. C is a unique, rigorously defined, two-dimensionalaverage of the three-dimensional shape and constitutes optimal extraction of shapeinformation from a nonopposition lightcurve. For a lightcurve obtained at opposition,C's odd harmonics are not accessible, but one can estimate the asteroid's symmetrizedmean cross section, Cs. To convey visually the physical meaning of the mean crosssections, we show C and Cs calculated numerically for a regular convex shape and for anirregular, nonconvex shape. The ideal conditions for estimating C from a lightcurve areCondition GEO, that the scattering is uniform and geometric; Condition EVIG, that theviewing-illumination geometry is equatorial; Condition CONVEX, that all of the aster-oid's surface contours parallel to the equatorial plane are convex; and ConditionPHASE, that the solar phase angle 4 * 0. Condition CONVEX is irrelevant for estima-tion of Cs. Definition of a rotation- and scale-invariant measure of the distance, A2,between two profiles permits quantitative comparison of the profiles' shapes. Usefuldescriptors of a profile's shape are its noncircularity, 12c, defined as the distance of theprofile from a circle, and the ratio ,6* of the maximum and minimum values of theprofile's breadth function 13(0), where 0 is rotational phase. C and Cs have identicalbreadth functions. At opposition, under Conditions EVIG and GEO, (i) the lightcurve isequal to C's breadth function, and (ii) ,8* = 10° m, where Am is the lightcurve peak-to-valley amplitude in magnitudes; this is the only situation where Am has a unique physicalinterpretation. An estimate C of C (or Cs of Cs) can be distorted if the applicable idealconditions are violated. We present results of simulations designed to calibrate the na-ture, severity, and predictability of such systematic error. Distortion introduced by viola-tion of Condition EVIG depends on 4, on the asteroid-centered declinations Ss and SE ofthe Sun and Earth, and on the asteroid's three-dimensional shape. Violations of Condi-tion EVIG on the order of 100 appear to have little effect for convex, axisymmetric

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300019-1035/88 $3.00Copyright e 1988 by Academic Press, Inc.All nghts of reproduction in any form reserved.

ASTEROID LIGHTCURVE INVERSION

shapes. Errors arising from violation of Condition GEO have been studied by generatinglightcurves for model asteroids having known mean cross sections and obeying Hapke'sphotometric function, inverting the lightcurves, and comparing C to C. The distance of Cfrom C depends on k and rarely is negligible, but values of 12, and ,8* for C resemblethose of C rather closely for a range of solar phase angles (i - 20° ± 100) generallyaccessible for most asteroids. Opportunities for reliable estimation of Cs far outnumberthose for C. We have examined how lightcurve noise and rotational-phase sampling ratepropagate into statistical error in C and offer guidelines for acquisition of lightcurvestargeted for convex-profile inversion. Estimates of C and/or Cs are presented for selectedasteroids, along with profile shape descriptors and goodness-of-fit statistics for the in-verted lightcurves. For 15 Eunomia and 19 Fortuna, we calculate weighted estimates of Cfrom lightcurves taken at different solar phase angles. C 1988 Academic Press, Inc.

1. INTRODUCTION

One of an asteroid's most fundamentalattributes is its shape, and lightcurves pro-vide the only source of shape informationfor most asteroids. However, the functionalform of a lightcurve is determined by theviewing/illumination geometry and the as-teroid's light-scattering properties as wellas by its shape, and lightcurves offer lesspowerful shape constraints than do tech-niques that resolve asteroids spatially. Os-tro and Connelly (1984) introduced a newapproach called convex-profile inversion tousing a lightcurve to constrain an asteroid'sshape, demonstrating that any lightcurvecan be inverted to yield a convex profilethat, under certain ideal conditions, repre-sents a certain two-dimensional average ofthe asteroid's three-dimensional shape. Theideal conditions include geometric scatter-ing and equatorial viewing/illumination ge-ometry and will rarely be satisfied exactlyfor actual lightcurves, so the derived shapeconstraints will usually contain systematicerror.

Therefore, while convex-profile inver-sion can help to optimize extraction ofshape information, proper application ofthe technique demands an understanding ofsystematic (as well as statistical) sources ofuncertainty. Two objectives of this paperare to explore these effects at some lengthand then, with a better feel for errorsources, to use convex-profile inversion toconstrain the shapes of selected asteroids.A third, more general, objective is to

present a comprehensive conceptual foun-dation for the physical interpretation ofasteroid lightcurves. This foundation en-compasses calibration and application ofconvex-profile inversion as well as estab-lished principles from the oldest lightcurveliterature, so let us begin with a review ofessential background material in which weemphasize the progression of logic whichhas led to convex-profile inversion.

11. HISTORICAL BACKGROUND AND REVIEWOF RECENT WORK

A general statement of the lightcurve in-version problem is "Given that a lightcurveis determined by asteroid shape, surfacescattering properties, and viewing/illumina-tion geometry, how, if at all, can we isolatethese effects and obtain physically interest-ing information about the asteroid?" In aclassic paper, H. N. Russell (1906) offeredthe first analysis of this problem, investigat-ing in fine detail the information content oflightcurves taken close to opposition, i.e.,with the solar phase angle 0 0°. He sum-marized his results as follows: If an asteroidhas been observed at opposition in all partsof its orbit:

(RI) "We can determine by inspection ofits light-curves whether or not they can beaccounted for by its rotation alone, and, ifso, whether the asteroid (a) has an absorb-ing atmosphere, (b) is not of a convex2

2 An object is convex if the straight line connectingany two points on the boundary is inside the boundary.

3 1

OSTRO, CONNELLY, AND DOROGI

form, (c) has a spotted surface, or whetherthese hypotheses are unnecessary."

(R2) "It is always possible (theoretically)to determine the position of the asteroid'sequator, (except that the sign of the inclina-tion remains unknown)." That is, we canfind the line of equinoxes, of intersectionbetween the equatorial and orbital planes,and the size of the angle between thoseplanes.

(R3) "It is quite impossible to determinethe [three-dimensional] shape of the aster-oid. If any continuous convex form is possi-ble, all such forms are possible."

(R4) "In this case we may assume anysuch form, and then determine a distribu-tion of brightness on its surface which willaccount for the observed light-curves. Thiscan usually be done in an infinite variety ofways. "

(R5) "The consideration of the light-curve of a planet at phases remote from op-position may aid in determining the mark-ings on its surface, but cannot help us findits [three-dimensional] shape."

Despite Russell's skepticism towardprospects for using lightcurves to find anasteroid's three-dimensional shape, he diddescribe how to test certain hypothesesabout shape, scattering law, and albedo dis-tribution. For example, suppose we havefound the asteroid's line of equinoxes (R2)and obtained an "equatorial" oppositionlightcurve. Russell showed that such alightcurve's Fourier series cannot have oddharmonics higher than the first if the scat-tering is geometric, 3 regardless of the al-bedo distribution; it cannot have any oddharmonics if that distribution is uniform.He also showed that if two opposition light-curves obtained in opposite directions aredifferent, then either the scattering is notgeometric or the albedo is not uniform, orboth; if the difference is not a sinusoid, then

I The brightness of a geometrically scattering sur-face element is proportional to the element's pro-jected, visible, illuminated area.

the asteroid's shape is not convex. How-ever, as Russell proved, it is impossible toseparate the effects of albedo variationfrom effects of surface curvature, so nei-ther the light-scattering properties nor theshape can be modeled uniquely. Hence,apart from testing the convexity hypothe-sis, the utility of lightcurves as sources ofshape information seemed virtually nil.

More than half a century after Russell'streatise, photoelectric photometry and po-larimetry dramatically changed the empiri-cal setting for lightcurve interpretation,demonstrating that with a few notable ex-ceptions the forms of most broadband opti-cal lightcurves seem less sensitive to sur-face heterogeneity than to gross shape.Moreover, laboratory experiments and the-oretical studies disclosed that geometricscattering is likely to be a good approxima-tion for asteroids viewed close to opposi-tion (e.g., French and Veverka 1983) butthat the validity of this approximation dete-riorates with increasing solar phase angle.Thus, in the absence of evidence to the con-trary, the premise that the scattering is uni-form and geometric has been the startingpoint for nearly all efforts to interpret light-curves physically (e.g., Burns and Tedesco1979, Harris 1986).

Many researchers further constrain thelightcurve inversion problem by assuming asimple a priori shape. The usual choice is atriaxial ellipsoid, whose shape is com-pletely described by the ratios bla and cla,with a - b - c the semiaxis lengths. Theequatorial opposition lightcurve of a geo-metrically scattering ellipsoid (GSE) hasbrightness extrema in the ratio alb =100 4Am, where Am is the peak-to-valleylightcurve amplitude in magnitudes. A com-mon practice is to take the maximum ob-served value of Am as a measure of an as-teroid's elongation.

Because GSE lightcurves are easy to cal-culate, they provide a convenient tool forerror-propagation studies (Sections III andIV below) and for pole-direction determina-tions (e.g., Zappala and Knezevi6 1984,

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Scaltriti et al. 1978). However, for the pur-pose of extracting information about an as-teroid's shape from its lightcurve, the GSEassumption begs the question. After all, el-lipsoids are a subset of axisymmetricshapes, which in turn constitute a smallsubset of convex shapes and an evensmaller subset of plausible asteroid shapes.The GSE assumption forces one to ignoremuch of the lightcurve's information con-tent, imposing a very particular Fourierstructure on any model lightcurve. Asshown by Russell and noted above, theequatorial, opposition lightcurve of any ob-ject with a geometric scattering law willhave no odd harmonics higher than thefirst. If the asteroid's surface is symmetricabout the spin axis then the first harmonicwill vanish, leaving only even harmonics inthe lightcurve, regardless of viewing as-pect. But if, additionally, the asteroid is anellipsoid, the lightcurve's Fourier structurewill be even more restricted: As shown byConnelly and Ostro (1984), the square ofthe opposition lightcurve of any geometri-cally scattering ellipsoid has no harmonicsexcept for 0 and 2, regardless of viewingaspect.

How should we interpret the equatorialopposition lightcurve of a uniform, geomet-rically scattering asteroid which might (bythe Fourier test just mentioned) be non-ellipsoidal? More generally, what can anygiven lightcurve tell us about an asteroid'sshape? Russell wrote that if the scatteringis uniform and geometric, then we might"seek to account for the light-changes bythe form of the surface alone. But this leadsto difficult problems in the theory of sur-faces, and will not be attempted here." Hepointed out that some of these problemsarise because odd harmonics in the expan-sion of a surface's curvature function willbe irretrievable from an opposition light-curve. He also suggested that the odd har-monics "may sometimes be determinedfrom nonopposition lightcurves." Thesecritical issues were not pursued further fornearly 75 years.

Convex-Profile Inversion: Averages ofAsteroid Shapes

Ostro and Connelly (1984) demonstratedthat whereas Russell's paper is valid, andindeed it is not possible to find an asteroid'sthree-dimensional shape from disk-inte-grated measurements, it is possible to ex-tract shape constraints that are meaningfuland that exploit more of the information inthe lightcurve. Still, shape constraints fromdisk-integrated measurements are neces-sarily much less informative than, say, astellar-occultation profile, which is a two-dimensional projection of the asteroid'sthree-dimensional shape. Given an aster-oid's lightcurve, the best constraint we canobtain is not a projection of the asteroid'sthree-dimensional shape, but rather a cer-tain two-dimensional average of that shape.

Ostro and Connelly (1984) defined an as-teroid's mean cross section C as the aver-age of all cross sections C(z) cut by planes adistance z above the asteroid's equatorialplane. That definition of the C(z) left C un-defined unless all the C(z) are convex.Here, to remedy this situation, we definethe C(z) as the convex envelopes, or hulls,on the actual surface contours. With thisrevision, C is now clearly defined for anyasteroid.

C and all the C(z) are convex profiles or,in the language of geometry, convex sets.We use underlined, uppercase, italic lettersto denote such quantities. As discussed indetail by Ostro and Connelly (1984) and re-iterated below, a convex profile can be rep-resented by a radius-of-curvature functionor by that function's Fourier series. Dele-tion of a profile's odd harmonics yields thatprofile's symmetrization. For example, anasteroid's symmetrized mean cross sectionCs has the same even harmonics as C but noodd harmonics.

Figure 1 shows C and Cs for two three-dimensional, styrofoam models, calculatedby applying formulas in Appendix A of Os-tro and Connelly (1984) to measurements ofphotographs of those objects. Several pho-

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FIG. 1. Values of the mean cross section for (a) an egg-shaped object and (b) an irregular, nonconvexobject.

tographs of each model are shown to con-vey visually the relation between three-di-mensional shape and the two-dimensionalaverages accessible from lightcurves.

There are four ideal conditions for esti-mating C from a lightcurve:

Condition GEO: The scattering is uni-form and geometric.

Condition EVIG: The viewing-illumina-tion geometry is equatorial, that is, the as-teroid-centered declinations 8s and 8E of theSun and Earth are zero.

Condition CONVEX: All of the aster-oid's surface contours parallel to the equa-torial plane are convex.

Condition PHASE: The solar phase angle4 is known and does not equal 0° or 1800.

The last condition states that we cannotfind C from an opposition lightcurve andarises for the following reason. Each coeffi-cient in the Fourier series for C's radius-of-curvature function is found from the corre-sponding coefficient in the Fourier seriesfor the lightcurve. However, we cannot usean opposition lightcurve to find C's oddharmonics because, as Russell showed, anopposition lightcurve has no odd harmonicsif Conditions GEO and EVIG are satisfied.

(If C contains odd harmonics, then light-curve odd harmonics can be introducedaway from opposition via the rotational-phase dependence of the fraction of the vis-ible projected area that is illuminated. Atopposition, no shadows are visible, so thatfraction is unity regardless of rotationalphase, and the odd harmonics vanish.)

Although we cannot estimate C from anopposition lightcurve, we can estimate C'seven harmonics and hence its symmetriza-tion, Cs. Estimation of Cs from oppositionlightcurves is less "burdened" by idealconditions than is estimation of C from non-opposition lightcurves, for three reasons.First, for estimation of C, ConditionCONVEX ensures the absence of visibleshadows between the limb and the termina-tor, which might distort determination of C.However, at opposition, there are no visi-ble shadows, so Condition CONVEX is ir-relevant and C, can be estimated reliably aslong as Conditions GEO and EVIG are sat-isfied.

Second, Conditions GEO and EVIG aresatisfied more easily" for opposition light-

curves than for nonopposition lightcurves.As noted above, the geometric scattering

34


approximation is more valid at oppositionthan at large solar phase angles. ConditionEVIG can be satisfied at large phase anglesonly if the pole lies nearly normal to theEarth-asteroid-Sun plane, but is satisfiedat any opposition occurring when the aster-oid is at an equinox, i.e., with the pole nor-mal to the Earth-asteroid line. Hence, ge-ometry dictates that opportunities toestimate Cs far outnumber opportunities toestimate C.

Third, to assess how close the viewing/illumination geometry is to equatorial for agiven nonopposition lightcurve, one mustknow the asteroid's pole direction; for anopposition lightcurve it is sufficient to knowjust the location of the asteroid's line ofequinoxes.

Ostro and Connelly (1984) showed thatcalculation of a mean-cross-section esti-mate C from a lightcurve can be ap-proached by weighted-least-squares optimi-zation subject to inequality constraints. Inthis paper, we use the term "convex-profileinversion" to denote our theory for the in-formation content of lightcurves as well asthe computational technique for estimatingmean cross sections.

None of the first three ideal conditionsare likely to be satisfied exactly for an ac-tual lightcurve. However, the important is-sue is the extent to which their violationdegrades the accuracy of a mean-cross-section estimate. As noted above, key ob-jectives of this paper are to calibrate con-vex-profile inversion's sensitivity to sys-tematic and statistical sources of error andthen to use convex-profile inversion to con-strain the shapes of selected asteroids. Inthe next section we briefly review the math-ematics of convex-profile inversion and in-troduce techniques for describing, compar-ing, and manipulating profiles. In SectionIV we explore the manner in which depar-ture from ideal conditions makes C deviatefrom C. We have tried to maintain at least amodicum of realism in evaluating sourcesof systematic error. For example, specialattention is paid to the distortion caused by

nongeometric scattering, and in calibratingthis distortion we have relied heavily onHapke's (1981, 1984, 1986) theory for a re-alistic description of the optical propertiesof asteroid surfaces. On the other hand, ourcalibrations have not been exhaustive; wehave not examined coupled effects due tosimultaneous violation of two or more idealconditions, and we have simplified as muchas possible our modeling of three-dimen-sional shapes. Nonetheless, our work doesprovide a guide to both the power and thelimitations of convex-profile inversion. Weargue that systematic errors in estimationof C and Cs are not insurmountable and thattheir severity often can be assessed a prioriand/or a posteriori.

In Section V we briefly examine hownoise, sampling rate, and rotational phasecoverage propagate into statistical error inmean-cross-section estimation. Then, hav-ing calibrated various sources of uncer-tainty in convex-profile inversion, we applyit in Section VI to an assortment of asteroidlightcurves. Finally we summarize all ourresults and offer strategies for future effortsin the acquisition and physical interpreta-tion of asteroid lightcurves.

111. CONVEX-PROFILE INVERSIONMATHEMATICS

The power of convex-profile inversionrests on the fact that when the ideal condi-tions hold, the three-dimensional lightcurveinversion problem, which cannot be solveduniquely, collapses into a two-dimensionalproblem that can. The geometry of the two-dimensional problem is shown in Fig. 2,drawn for 0 = 30°. The asteroid is a convexprofile whose brightness 1(0) at rotationalphase 0 is proportional to the orthogonalprojection toward Earth of the visible, illu-minated portion of the profile.

At opposition, the visible portion of theprofile is fully illuminated; the projection ofthis curve is called the profile's width, or"breadth," in the direction of Earth, andthe lightcurve is proportional to the breadthfunction 83(0). Unless this function is con-

35


P(360- -1 i

......... 0 .... X

FIG. 2. Geometry for two-dimensional asteroid light-curve inversion. The asteroid is a convex profile rotat-ing clockwise and is shown at rotational phase 6 = 00.The solar phase angle 4 is indicated, as are the aster-oid's illuminated (solid) and unilluminated (dotted)portions. The asteroid's scattering law is geometric;i.e., the observed brightness is proportional to the or-thogonal projection toward Earth of the visible, illumi-nated length 1(6). The point P(O) is on the receding(left-hand) limb and the outward normal in at P(6)points in the positive x direction when the rotationalphase is 0. Reproduction of Fig. I of Ostro and Con-nelly (1984).

or as

A(0) = E Ce inl,n = -

where- (an + ibJ)/2.

Define P(0) - [x(0), y(0)] as the point on thecurve that would be on the receding limb ifthe asteroid were at rotational phase 0. Inthis normal-angle parameterization, theoutward normal to the curve at P(0) pointsin the positive x direction at rotationalphase 0. Let r(0) = dsIdO be the radius ofcurvature of the profile at P(0), where I1r(0)is the curvature at P(0) and s = s(0) is arclength defined by

(dsldO)2 = (dx/dO)2 + (dy/do)2 .

Then

and

x(0) = x(O) - f r(t) sin t dt

y(0) = y(O) + r(t) cos t dt.

stant, it is periodic with a period equal to1800 or an integral fraction thereof-thelightcurve has no odd harmonics. (This iswhy most asteroid lightcurves, which arerarely obtained very far from opposition,are dominated by even harmonics.) Thesecond harmonic dominates the highereven harmonics unless the profile possessessymmetry akin to that of regular convexpolygons, in which case the lightcurve'sfundamental can be a higher even har-monic. (Asteroids are unlikely to have suchsymmetry, and this is why most asteroidlightcurves are dominated by the secondharmonic.)

The lightcurve's Fourier series can be ex-pressed either as

I(W) - A (a, cos nO + b, sin nO)n=O

Ostro and Connelly (1984) define theFourier series for the radius of curvaturefunction as

r(0) = E dnei10

and show that the profile can be found fromthe lightcurve because

d0 = C,/vn (1)

where vn is a known function (Fig. 3) of &.The profile will be closed only if d+, = 0 andit will be convex only if a set of N linearconstraints, r(2F-k/N) > 0 for 1 c k c N,are satisfied. In practice, convex-profile in-version finds that profile C which providesthe least-squares estimator for C by findingthat vector x of Fourier coefficients thatsatisfies the constraints and is as close aspossible to the (unconstrained) vector x of

36


1 _ 6 7

0

180° 0° 180°Solar Phase Angle, S

FIG. 3. The function v,(j), discussed in the text, isplotted vs solar phase angle ( for values of the har-monic index n from 2 through 7. The following expres-sions for v, are valid for n r + 1; see Section II.A ofOstro and Connelly (1984). If 0 c <P < 7r, then 2uv -ei(- 4)('-°/(n - I) - ei(e+ )(-0'/(n + 1) -2/(n 2 - 1). If-ar < f) c 0, then 2(n2 - 1)v. = (n - l)e i('+l)0 - (n +1)e i(- 1)° + 2(-I)e+l.

Fourier coefficients fit to the data. Thelightcurves derived from x and x are, in vec-tor notation, y and 5.Truncation of Fourier Series

As discussed by Ostro and Connelly(1984), truncation of Fourier series limitsthe fidelity with which convex-profile inver-sion can reproduce profiles with flat sidesor sharp corners. We employ Fourier seriescontaining 10 harmonics throughout thispaper; series this long adequately representall lightcurves discussed herein. We notethat lightcurves with significant energy inharmonics higher than 10 are rare, and wespeculate that the mean cross sections ofactual asteroids generally lack extremelyflat sides and very sharp corners.

Sign of the Solar Phase AngleAs shown in Fig. 2, the solar phase angle

is measured from the Earth direction E tothe Sun direction S in the same rotational

sense as the asteroid's rotation. We adhereto the semantic convention that 4) satisfies00 c 141 - 180°, so 4 # 0 is positive if theasteroid rotates through 141 from E to S ornegative if it rotates through j4) from S to E.For example, in Fig. 2, if the directions ofthe Earth and Sun were interchanged, 4would equal -300. Clearly, for any givenSun-Earth-asteroid geometry, the sign of4 is determined by the sense of rotation.Thus convex-profile inversion of a nonop-position lightcurve is sensitive to the senseof rotation, and using the wrong sign of 4can distort C; see Fig. 3 of Ostro and Con-nelly (1984). However, we suspect that ifprior knowledge of an asteroid's pole direc-tion is adequate to guarantee that the aster-oid-centered declinations of the Sun andEarth are both close to zero for a given non-opposition lightcurve, then in many casesthe data (lightcurve, radar, or whatever)used to constrain the pole direction will beadequate to permit reliable determinationof rotation sense, too.

Quantitative Description of ProfilesWe often will wish to quantify relation-

ships between profiles to supplement verbaldescriptions and subjective comparisons.Following Richard and Hemami (1974) andSato and Honda (1983), we define the dis-tance fl between any two profiles PI and P2as

Q = 10 min ld1 d1o - Odd 2d 0 ,

where dj is the vector of complex Fouriercoefficients in the expansion of Pj's radius-of-curvature function; djo is the constant, orzero-harmonic term, in that expansion; andthe diagonal matrix E rotates d2, by einO.This distance measure is rotation invariantand scale invariant.

The meaning of il can be visualized asfollows. If our Fourier series are truncatedto M harmonics, dj will have 2M + 1 ele-ments, so di and d2 define two points in a(2M + 1)-dimensional space, and it is easyto calculate the distance between them.

37


However, dj determines P i's circumference(i.e., its size) and rotational orientation aswell as its shape. Therefore, we define n asthe distance between d, and d2 when PI andP2 are scaled to the same size and rotated asmuch into alignment with each other aspossible.

In describing interprofile distances, wewill use the following scale:Distance (fl)

C23-45-67-8,!9

Verbal description

Nearly identicalSimilar but not the sameDifferentVery differentRadically different

We also define a profile's "noncircular-ity" fc as its distance from a circle, and aprofile's breadth ratio ,8* - 1 as the ratio ofthe profile's maximum breadth to its mini-mum breadth. Figure 4 plots fc vs /8* forellipses and shows the locations of variousprofiles in this two-parameter space, as wellas some interprofile distances. Table I listssymbols used frequently in this paper.

14 l l13 - 7 313012 A

C: 9 4310 9.2

I 9 - 9.3/ 6.4

7.8 50 0 764.

Z 4z 3i

TABLE I

GLOSSARY OF SYMBOLS

C Mean cross sectionC, Symmetrized mean cross section

C Estimate of C

C, Estimate of C,j Solar phase angleAm Lightcurve peak-to-valley amplitude5E Asteroid-centered declination of Earth5s Asteroid-centered declination of the Sunj3* Breadth ratio; ratio of a profile's maximum and

minimum widths( Distance between two profilesDc Noncircularity; distance of a profile from a

circlenE Error distance, between C and C (or between

C, and C)f3s Symmetrization distance, between C and C, (or

between C and CQ)

IV. SENSITIVITY TO DEPARTURE FROMIDEAL CONDITIONS

A. Violation of Condition EVIGAs a first step in studying how departures

from ideal conditions can introduce system-

1.6 1.8 2.0 2.2 2.4BREADTH RATIO, d*

FIG. 4. Shape descriptors for convex profiles. Each profile's noncircularity S1c and breadth ratio /*is indicated by the profile's position. Note the locus of ellipses. The interprofile distance Q, defined inthe text, is indicated for selected pairs of profiles.

38


atic error in C or C we examine effects ofnonequatorial viewing/illumination geome-try.,For any given asteroid, the proximityof C to C for a lightcurve obtained whenSE 0 or Ss # 0 will be a function of 6 E, 5S,and t. We have studied this particularsource of systematic error by invertinglightcurves generated analytically for geo-metrically scattering ellipsoids within theparameter space:

100 < < 80°0.2 < b/a • 0.4

0.05 < c/a c bla.For given values of b/a, c/a, and 4, thefield of possible values of NE and Ss is con-veniently displayed in a plot of Ss vs (6 E -5s). Figure 5 shows several contours of con-stant BE and of constant Ss in this plane.Note that the Earth is north of the equatoroutside the stippled zone. The figure showsresults of our simulations for illustrativetwo-dimensional cuts through the five-di-mensional parameter space.

Under Condition EVIG (at thez origin ofthe figures, where BE = 5S = 0), C is nearlyindistinguishable from the asteroid's meancross section C, which is an ellipse withaxis ratio alb. However, under extremelynonideal viewing/illumination geometry,C is not necessarily elliptical and has breadthextrema not necessarily in the ratio bla.The distortion in each C in Fig. 5 is quan-tified by the "error distance" QE betweenC and C. It worsens with increasing 0, alb,or a/c and is most severe when 15EI,I5sl > 10° and Ss is between 00 and 6 E. Thedistortion is minimal when either 16EI, 6S <100 or SE is between 0° and -6s, i.e., in thetriangular region labeled MIN in Fig. 5.Outside those regions, violation of Condi-tion EVIG often makes C more circularthan C. This result, which is consistent withthe well-known reduction in lightcurve am-plitude with decreasing aspect angle for op-position lightcurves of geometrically scat-tering ellipsoids and a variety of othershapes (Barucci and Fulchignoni 1982,

1984), arises from the constant illuminationand visibility of a polar region.

From the viewpoint of convex-profile in-version, nonequatorial viewing/illumina-tion geometry prevents the "clean" col-lapse of the lightcurve inversion problemfrom three dimensions to two dimensions.It seems unlikely that many asteroids are aslong and flat as our most extreme examples,and in this respect our figures might exag-gerate the distortion caused by nonequato-rial viewing/illumination geometry. On theother hand, distortion for nonaxisymmetricand highly irregular shapes might be muchmore severe than for ellipsoids. Note thatin the minimum-distortion region in the fig-ure, the constantly illuminated pole is notvisible and vice versa, so neither pola~r re-gion contributes to the lightcurve. C re-sembles C in this region because C(z) has aconstant shape for an ellipsoid, a situationunlikely to hold for most asteroids. Hence,even if EVIG were the only violated condi-tion, it would probably be wise to interpretthe simulations described here as establish-ing ,a lower bound on the systematic errorin C.

B. Violation of Condition GEOHow much is C distorted if the scattering

is nongeometric? Our strategy in answeringthis question is to (i) assume that the otherideal conditions are satisfied; (ii) generatelightcurves for an asteroid with a knownshape and a realistic scattering law; (iii) useconvex-profile inversion to obtain an esti-mate, C, of the asteroid's mean cross sec-tion C; and (iv) compare C to C. For thetime being, let us also assume that the as-teroid's three-dimensional shape is a con-vex cylinder, that is, that C(z) is constantand congruent with C. These assumptionsreduce the problem's dimensionality totwo, letting us treat the asteroid's surfaceas a convex profile. For convenience wemodel an asteroid as a convex polygon with360 sides. The orientation and length ofeach side is known, so it is easy to calculatethe asteroid's lightcurve for an arbitrary

39


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Io0o

00

b/a = 0.4 c/a =0.2

- 90100-80Q90- 70 80- 4 0 60

2 204 40-2 20 0- 5 0 2 0- Q8

=20° b/aa 0.2

O1001001001001009 0Qo09 0Qi07 0905 08

I I

10010090908060300 0

-20° -Qo0 00 Roe 2008 - a,

b/a=0.4 c/a 0.2 0=400

-40' -20° 00a -8

70

600

as4Q0'.

30'-

20'k

to0

00

c/a =0.2

160 170 170150 160 16013 0 150 160110 130 140

8 Q100 12 0

40 60 8030 20 3080 20 o0

* 020-I 1

017017Q 7016017 -0 15Q017 -

0 14016 -0 11015 -07 013-0 2 08

-204 -10 o 00 -$ 2008E- as

'\ ** \ .*\l

tik-Ro~ U\., ,.f

*'~:G A .E

0 ,b.A,:7 b.NN~::~.

'N...

X. '."

20° 40'

FIG. 5. Systematic errors due to nonequatorial viewing/illumination geometry (violation of Condi-tion EVIG). Convex profile inversion has been applied to lightcurves generated for geometricallyscattering ellipsoids, for various values of the asteroid-centered declinations Ss and SE of the Sun andEarth. The field of viewing/illumination geometries is displayed in plots with Ss on the ordinate and(SE 6s) on the abscissa. The bottom right-hand drawing identifies several regions and lines in thatcoordinate system. Note that the Earth is north of the equator everywhere except within the shadedarea. The three arrays of profiles show estimates C of the mean cross section C (shown as the ellipse atthe origin) for three different sets of choices for the solar phase angle 4 and the ellipsoid's axis ratiosb/a and c/a. The "error distance" OIE, of C from C, is given for each profile. Distortion due to violationof Condition EVIG is minimal within the region labeled "MIN."

photometric function. (By virtue of Condi- asteroid's brightness from light reflectedtion EVIG, the flat polar surfaces do not from a surface areal element dA ascontribute to the lightcurve.)

Hapke's (1981, 1984, 1986) theory of dif- d rR[O;, Oe, its

fuse scattering from a particulate medium w, h, S(0), 0, b1 g, clog] dA cosO0

lets one express the contribution dW to an where the reflectance rR is defined in the

* @0000000-00 @000000-000000~0000000 0000o@S0C00@@S

=

40

50 1-

11

:


cited references, 6, and 0e are the angles ofincidence and emergence, and six parame-ters describe the surface properties. Theparameter w is the average single-scatteringalbedo. The opposition effect is describedby the parameters h and S(0); h depends onthe regolith's porosity and particle size dis-tribution, and S(0) is the fraction of lightscattered from close to a particle's surfaceat 0 = 0. Effects of macroscopic surfaceroughness are parameterized by the aver-age slope 0, measured with respect to themean surface, of surface facets unresolvedby the detector. Following Helfenstein andVeverka (1987), we write the average, sin-gle-particle, angular scattering function as

P(tk) = 1 + bleg cos d)+ cleg(l.5 cos2 0 - 0.5).

Those authors estimated values of the sixHapke parameters for dark, average, andbright terrains on the Moon. In our calcula-tions, we adopt their results for average ter-rain: w = 0.25, h = 0.06, S(O) = 0.78, 0 =20.60, bleg = 0.33, and clg = 0.37. Ourresults are essentially insensitive to varia-tion in these values over ranges much largerthan those spanned by their results for thethree lunar terrain classes.

For a geometric scattering law (rR = con-stant), convex-profile inversion yields C, atopposition and C away from opposition.Figure 6 shows C and Cs for each, of 15model asteroids, as well as C or C, esti-mated from lightcurves generated under theassumption of Hapke scattering at 0 = 00,20, 50, 10°, 20°, 300, 50°, and 90°. The figurealso lists for each profile the breadth ratio3*, the error distance flE from C (or Cs),and the noncircularity flc. (The 15 modelasteroids' mean cross sections are plottedin Fig. 4.)

Figure 7a plots flE for each object. Figure7b plots the average value of QE for all 15objects, the average value for the only threeobjects (No. 8, 14, and 15) with negligibleenergy in odd harmonics, and the average

value for the 12 remaining objects. Thesefigures demonstrate that the difference be-tween C and C generally depends on thesolar phase angle. At opposition, the scat-tering is sufficiently close to geometric thatf, is very accurately determined. (In fact,for b less than about 100, one can estimatethe symmetrized mean cross section quitereliably, with QE rarely as large as 5.) How-ever, for very small phase angles (2° and 50in the figures), OE is extremely large for allprofiles except objects No. 8, 14, and 15;object No. 12 is an intermediate case. Thecause of the severe distortion in C at lowsolar phase angles for the other 12 profilesis evident from Fig. 3, which plots v(o,n)and shows that for odd n, v- 0 as q- 0.Since the basic Eq. (1) for calculating C isn a clv,, estimation of odd harmonics

in C becomes increasingly sensitive to light-curve noise and computational precision asb- 0. These simulations and those pre-sented in Section V suggest that reliable de-termination of odd harmonics is difficult if14.1 is less than about 50, and we will not tryto estimate C from lightcurves taken thatclose to opposition. Instead, we will useequatorial lightcurves obtained near oppo-sition to estimate C .

As discussed earlier, Cs has the sameeven harmonics as C but no odd harmonics.C, actually tells us quite a bit about C be-cause the breadth function of any convexprofile is independent of the profile's oddharmonics. That is, Cs and C have the samebreadth function because they have thesame even harmonics. It is important tonote that at opposition, under the pertinentideal conditions (EVIG and GEO), an as-teroid's lightcurue equals the breadth func-tion of C, (and of C) within a multiplicativeconstant. Furthermore, ,3* = 100.4Am, so thelightcurve's amplitude is a trivial functionof /8* and thus is a valid gauge of elonga-tion. We stress that this elongation or anyother shape-related parameter obtainedfrom a lightcurve pertains not to the aster-oid's three-dimensional shape but to a two-dimensional average of that shape.

41

42 OSTRO, CONNELLY, AND DOROGI

FIG. 6. Effects of nongeometric scattering on estimation of C and C,. Each box shows results ofsimulations for a particular model "asteroid." The model's mean cross section and symmetrized meancross section are shown above the horizontal bar. Below the bar are estimates of those quantities frominversion of lightcurves generated at various solar phase angles q and employing the Hapke photomet-ric function described in the text. (C, is estimated at opposition and C is estimated at the seven nonzerovalues of s.) Each profile's noncircularity Q, and breadth ratio 13* arc given. {l, is the distancebetween C and C or between C, and C,.

For phase angles larger than 50, convex- tinctly nongeometric as IcI, increases,profile inversion is increasingly "stable" thereby causing distortion in C. For phasewith respect to estimation of C's odd har- angles equal to 100, 200, 300, and 500, themonics, but the scattering becomes dis- average value of QE is 4.1, 3.8, 4.3, and 4.7,

c. X 1

0e 1 23 2880pr I 7 1 07

0 0° 2° 50 lo

GE 0 00 6 68 5.12 3 56Ge 1 23 8 81 6. 26 4.54p 107 107 1 07 1 08

0 20' 300 500 90o

GE 2.10 1 63 14A 1130l 3.01 2 56 2609 1.78p. 1808 I 08 108 1 07

C 2f

0, 4 57 5 66

pr 1 12 1 12

0i o 2° 5o 10o

E 08 01 6 25 5 58 4 42n0 4 56 8 23 8 09 7 29r 1 12 1 13 1 13 1 13

c 200 300 500 go0

GE 3 54 3 97 4 74 3.47DC 5 10 3 90 2 71 4 68V 1. 11 1 10 1 08 1 14

sC 3

Ge 2 77 3 23a. 1.14 1 14

c/ 0 2° 5° 100

aE 801 6.22 4,95 2.691E 2 77 7 53 6 49 4 41

p. 1 14 1 14 1 15 1 15

0 200 300 500 90o

nE 184 1 76 2.87 2802n0 3 12 2 56 I6 2.98pr 1 s5 1 15 1 14 1 14

C5 C 4

DC 4 92 7 99- 1 26 1 26

c 0° 20 50 lo

GE 0 01 7 45 6 71 289Ge 491 7 32 7 28 769p. 126 1 26 1 27 1 28

i 200 300 500 900

On 4 31 4 78 6 13 6 37Ge 7.13 6 32 58 0 4 78Jr 1 29 130 1.30 1 26

C-s

'c 3 28P. 1 28

¢ 0° 2°

0X' 881 6 39'C 3.28 8 48p. 1,28 1 29

P 20° 300

' 4.05 3 66'IE 7.95 7 62P- 1.31 1 32

with an rms dispersion within 10% of 2.0throughout. Thus, the systematic error inC from violation of Condition GEO tends to"bottom out" around 200, a fortuitous cir-cumstance in view of the inaccessibility ofmain-belt asteroid lightcurves at muchlarger solar phase angles.

Our simulations indicate that even for +near 200, an estimate of C is likely to con-

43ASTEROID LIGHTCURVE INVERSION

c 5 C.

5 64 b 6 89128 p. 129

50 10° 00 20

6.27 5 26 n 882 7 968 27 8 19 nE 6 87 8 68130 1 31 . 1 28 129

500 900 o 200 300

3.14 1.96 nt 5 54 5 976 68 5.97 'Ic 977 9451 31 1 30 1 29 1 27

C 7 C

7~64 'Ic 4.591 48 pr 1 54

50 10° 00 20

5 79 5 25 'I 8 01 0 1010 28 10 30 'c 4 59 4 68

151 153 Ir 1,54 1 55

500 900 0 200 300

3 77 2 98 'O 8:64 0 878 62 7 87 ' I5 0

9 5 09

153 1 50 Jr 162 1463

FIG. 6-Continued.

c 6

10 201 29

50 100

790 6 449. s 9 781 30 1 30

500 90o

7 86 48 08 96 9.211 27 1 29

C 8

4 581 54

50 100

0 23 8374'88 4951 57 160

500 900

1 31 0 634.95 4 861 61 1.57

9s

flc 4 38jr 1 48

0 00 20

'I 801 5 97'I 4.38 10 27

P- 1 48 1 49

S 20° 300

DE 4 40 4 14n 974 916rC 155 156

tain some systematic error. Figurers 6 and 7suggest that the distortion in C can beexpected to be moderate for most cases,but negligible for some small fraction ofcases and intolerably severe for a compara-ble fraction. On the other hand, Fig. 8shows that for (A at least as large as 200 thenoncircularity is within about _ I unit of theasteroid's actual value and that values of 1*

44

estimated for C are within 5% of C's valueof /3* for all solar phase angles. That is, thesalient characteristics of the mean crosssection are determined fairly accurately,even when a realistic scattering law is as-sumed.

Figure 8 also plots the ratio 100.4Am/,8*,where 10041m is the estimate of,8* providedby the lightcurve amplitude. Within about

10° of opposition, the lightcurve amplitudeprovides a reliable estimator for the elonga-tion of C, but overestimates the elongationby several tens of percent for 4 as large as200, by factors of several at 4 = 500, and byfactors of up to 10 at 4 = 900. It s~eems clearthat a value of /3* derived from C is a gaugeof C's elongation that not only is more reli-able and more physically meaningful than


C 9 C

a 00 0C 7. e71 66 r 1 77

50 10° 0 0° 2°

1 14 6 76 aE 088 88a870 1116 rC 78? 98 61.71 1 73 r 1 77 1 88

50° 90, 0 20° 30

S 14 5 90 nE 4 24 3 387 94 81 3 c 12

1 4 11 95

1 69 1 65 r 1 92 1 90

2. 20 f 8 323r 2 08

5° lo, 0.0 2°

8.28 759 ss n el 4 31116 11 64 ac

8.

3 2 9 30

1 98 2 02 r 288 2 03

500 90° 0 20° 30

6843 4.35 4 4 27 4 04809 18.76 'I 8 72 a 441.98 1.95 r 219 2 19

FIG. 6-Continued.

1 I0

12 131 77

50 1 00

) X(7.25 5 40

1l 45 11 69183 1 88

0 500 90o

14 12 s 291 43 I1 1718 3 1 82

Cs

n'1

876 :r' 1 93

qb 00 2°

8E 8.e2 8 72aC 8 75 1 79 1P1 1.93 1 95

t 200 300

DE 5 68 5 85DC li 98 11 68 1r 2 07 2 08

O 1 2

9 572 08

C 52 1, O

l 4.52 4.22937 9 112 07 2 13

0° 500 90o

4 72 4.257 99 806

1 2 14 208


Cs C 1 5

0 0 A0C 9 89 9.89r 2.58 250

0 0° 20 50 100

882 023 853 892nC 9 89 18.11 18 34 18.50Ir 2.51 256 2.63 2.72

0 20° 300 500 900

DE 2. 1 3. 44 3.98 2.52OIc i 4 8

10 12 9-53 10.302.79 276 2.61 2.59

FIG. 6-Continued.

Am, but also compensates for the so-calledphase-magnitude relation.

We have not tried to model lightcurvesfor three-dimensional objects for whichC(z) is not constant. For any arbitraryshape, C can be thought of as the sum ofpotentially distorted versions of the individ-ual C(z), where the distortion depends onthe local surface curvature and scattering

properties at each point. Our simulationshave used a realistic approach to scatteringproperties but have ignored components ofcurvature that are not parallel to the equa-tor. Hence our results are generally validonly to the extent that the curvature of sur-face contours not parallel to the equator isirrelevant to distortion in C. In simpler lan-guage, we have ignored effects of polar

45

0 013UNC 10 04 12 18P- 2 36 2 36

0 00 2° 50 10,

E88 02 7 97 702 6 77n 03 121 2 42 13 17r 2 36 2 40 246 2 52

0 200 300 500 90o

D [6 16 6 47 5.18 3.85DC 12 33 11 36 18. 51 10 96P- 2 57 2 51 2.39 2 36

C. C 14

C, 12 96 12. 96r 2 47 2 47

0 0° 2° 50 1 000 000' o[E S 03 0 32 0 85 2. 54[I 12 95 13 22 13 19 12 51or 2 48 2 53 2.60 2.65

0 200 30° 500 90o

nE 6.40 7.63 7.90 5.82[c

1 8 9 6 992 9 la 1 as

r 2.63 2 56 244 2.44

46

10a

SOLAR PHASE ANGLE 0 (deg)

0

b

20 40 60 80 1 00

SOLAR PHASE ANGLE, 0 (deg)

-0- exd.#8, #14,#154- all fifteen4- just#8, #14.#15

FIG. 7. (a) Individual values of flE for the 15 modelasteroids in the simulations described in the text, and(b) average values of fIE for three subsets of the modelasteroids.

darkening on the functional form of light-curves. Nevertheless, we suggest that oursimulations with constant-C(z) shapes mayapply to physically realistic convex shapesif Condition EVIG is satisfied, because theweighting of the C(z) is such [Ostro andConnelly 1984, Eq. (7) and Appendix A]that contributions to the mean cross sectionfrom equatorial C(z) can easily dominatethose from polar regions. This situation willbe even more pronounced for a Hapke scat-terer than for a geometric scatterer.

C. Violation of Condition CONVEXWe have already dealt with this source of

systematic error in a de facto sense, be-cause Hapke's theory includes effects ofroughness at scales much larger than theobserving wavelength. In other words, biasin C due to concavities is indistinguishable,either a priori or a posteriori, from bias dueto nongeometric (or even nonuniform) scat-

<< 2

1-

20

2E 100

w_J

CD

cc

20 40 60 80 1 00

0 20 40 60 80

SOLAR PHASE ANGLED, (deg)

.eF #1

4- #3- #4- #5-- #60- #7- 08- #9+ #10

4- #11- #12

- #13- #14

- #15

#1#2#3#4#5#6#7#8#9#10#11#12#13#14#15

.00 4A~ P

-*-I/ P.

FIG. 8. Estimated values of breadth ratio /3* andnoncircularity ftc for the simulations described in thetext, normalized to the model asteroids' true values.(a) and (b) show individual values for the 15 objects,while (c) plots average values. f'c and ,8* correspondto C. O04A1- is the ratio of the asteroid's maximumbrightness to its minimum brightness, i.e., an estimateof ,1* derived from the lightcurve amplitude Ain. Indi-vidual values of /3*/f3*, not shown here, are within 5%of unity for all 15 model asteroids, regardless of solarphase angle.

4-

4-0-a-

-6.'0-

4-


-- #1 tering. Partially because asteroidal valuesa -0-8#2

-- #3 of 0 are unavailable, we have not explored-0 #4- #5 the dependence of flE on 0, but such a study-a- # might be worth pursuing in the future.

4-M #9 V. PROPAGATION OF LIGHTCURVE ERROR-- #10

#12 So far, we thave focused on systematic4- #13 distortion in C caused by departure from8- #14

80 i 0-#15 ideal conditions. Since actual lightcurves

aLL. 5 *0W 4

-J< 3.

W 2

0-Us 1

so

b

. .n z


are contaminated with measurement errors,any estimate of C or Cs will also containsome statistical error. Whereas actual light-curve measurement errors may arise fromsystematic sources (e.g., imperfect removalof sky background, intermittent cirrus,etc.), as well as from photon-counting sto-chastic errors, one can always compute an"effective standard deviation" in the over-all noise due to all sources. Distortioncaused by any given noise level will in-crease as the rotational-phase sampling in-terval increases.

We have investigated the effects of noiseand sampling rate for five of the model as-teroids (objects No. 1, 4, 8, 11, and 15) usedin Section IV.B. As shown in Fig. 4, thosefive shapes span a large region in -fQc vs 1*space." Here, to focus on statisticalsources of error, we assume that all idealconditions are satisfied. Propagation of sta-tistical error has been modeled by generat-ing a lightcurve for a given model asteroid,using a random number generator to pro-duce Gaussian noise samples, scaling thestandard deviation to some fraction a- of theasteroid's mean brightness, adding thescaled noise to the lightcurve, and then esti-mating C or Cs. The entire procedure isrepeated to build up an ensemble of fiveprofiles, each containing error introducedby a different realization of the same noise-generating random process. We repeatedthis exercise for two noise levels (o- = 1 and3%), two sampling intervals (AO 30 and90), and four solar phase angles (¢ = 00, 50,10°, and 200), for each of the five modelasteroids. Our choices of noise level andsampling interval span the values corre-sponding to many published asteroid light-curves and let one gauge the severity of sta-tistical error sources in the inversions to bepresented in Section V.

Figure 9 superposes the five estimates ofC for each choice of the three parameters(a-, AO, X) and also lists the five-profilemean values of the error distance flE, thenoncircularity Qc, and the breadth ratio ,8*,using the same format as in Fig. 6. Note the

trade-offs between noise level and samplingrate, and the dependence of flE on solarphase angle. Tripling the noise level gener-ally increases the error distance more dra-matically than does tripling the sampling in-terval, but there is some coupling betweena- and AO, with the negative effect of in-creasing AO felt more severely at the highnoise level than at the low noise level.

The accuracy of C increases with in-creasing solar phase angle because noise"masquerading" as odd-harmonic energyin a low-+ lightcurve gets "blown up" viaEq. (1) into very strong odd harmonics inC. It should not surprise us that reliableestimation of C's odd harmonics is intrinsi-cally difficult at low solar phase angles; re-call that under the ideal conditions, themanifestation of C's odd harmonics as oddharmonics in the lightcurve is nil at opposi-tion and grows slowly with |4|. (See Fig. 3and Section III.) Note that overestimationof C's noncircularity appears correlatedwith the error distance for the more sym-metric shapes, but that the noncircularity isfairly immune to noise for the shapes (ob-jects No. 4 and 11) containing healthy oddharmonics.

Estimation of the symmetrized meancross section (i.e., of C's even harmonics)is accurate even for the higher noise leveland coarser sampling interval. This situa-tion reflects the intrinsic accessibility ofC's even harmonics, regardless of 4. Con-sequently, 13*, whose value depends onlyon C's even harmonics, is estimated quitereliably for every one of our simulations.

Since systematic sources of error in Ccan be severe, it is important that statisticalsources of error be minimized for light-curves acquired to constrain asteroidshapes. A conservative rule of thumb mightbe to try to keep o- at or below 1%. At thatlevel, statistical sources of error in C seemsmall compared to distortion introduced byviolation of Conditions EVIG and/or GEO.(In the following section, we apply convex-profile inversion to lightcurves whose noiselevels range from 0.6 to 4% and average

47


a=1%

a=3%

0=1%

a= 3%

Cs C1

1e 123 2.00I 1 07 1 07

56 0° 5° lo, 20° ' 0° 5° lo, 20'~00 0 00 C 0~ 00ls 1.46 5 36 3 01 1 26 lb 2 96 6 07 4 55 2 23nc 2 30 6 39 4 00 2 57 4b 3 57 6 95 5 48 3 30

1 07 1 07 1 07 1 07 En0 1 0 1 08 0 1 08

00 50 100 200 'lo 200

lb 4 27 6 94 5 66 3 60 Q 4 90 7 32 6 63 4 87nc 5 09 a 07 6 69 4 73 1c 5 49 8 22 7 56 5 81

i 00 1 08 1 00 1 080 1 0 1 09 1 09 1 09

A 3 =30 A= 90

C 4

lb 4 92 7 99¢ 1 26 1 26

100 5 ° 10 20' 0° 5° 10 ° 200

E1 174 5 78 2 37 1.09 k 2 48 6 72 4 11 1 875c 4 74 7 76 7 37 7 71

0I 4 95 7 56 6 3 7.47

1 26 1 26 1 25 1 25 ' 21 1 2 6 1 25 1 25

0° 5° 100 200 10 0O lo 200

f4 32 6 82 5 88 2 79 a, 4 74 8 00 733 4 57n1 5 45 0 35 7 66 7 37 Sc 6 61 8 59 7 68 7 23P- 1 27 1 26 1 26 1 26 A- 1 26 1 26 1 26 1 25

AO = 30 AO 90

FIG. 9. Propagation of statistical error for five model asteroids. This figure is in the same format asFig. 6. Each box shows results of simulations for a particular model. The model's mean cross sectionand symmetrized mean cross section are shown above the horizontal bar. Below the bar are estimatesof those quantities from inversion of noisy lightcurves generated under ideal conditions at solar phaseangle k, noise level o-, and rotational-phase sampling interval AO. ((, is estimated at opposition and Cis estimated at the three nonzero values of q>.) Five cross section estimates, corresponding to fivedifferent realizations of the noise-generating random process, are superposed for each choice of X, C-,AO. Five-profile mean values of noncircularity fic, breadth ratio A*, and error distance "E are listed.See text.

48

ASTEROID LIGHTCURVE INVERSION 49

C c8

0, 4 59 4 581 54 15

0. 5° 10° 20° 0° 5° lo, 200

11E 1 93 6 84 3.26 1 47 DA, 3 e4 6 62 4 73 2 52c 4 98 826 .565 4 81 fl, 558 8 55 8659 5.24

154 1 54 1054 1254 51 55 1 55

0000 20 0 00 20

, 4. 46 a 18 6 79 4 556 5E s s 7 44 7 09 s 55fl 6.43 9 48 8 22 612

54 7 21 8 72 8 43 7 19

155 155 155 14156 156 1 56 1 56

A =30 A =90

CsC

fl 8 76 12 281 93 1 93

!0 5° lo, 20' 9 a 5o l, 200

11, 1 26 2 75 1.56 884 fl, 1 64 3 53 2 55 1 38

no 8 78 11 25 175 12 85 fl 8 79 16 13 11 56 14 94p1 93 1 93 1 92 1 92 1 94 1 93 193 1 92

100 5° lo 20o 10 50 lo 200

SE 2. 59 5 28 3~ 38 It 94 11E 3 73 6~ 46 5~ 19 3~ 31Se 8 88 11, 38 11 38 11 82 De 9 23 18 79 11 25 11 65c 1 93 1 93 1 93 1 91 P- 1 95 195 194 192

AO = 30 AO = 90

FIG. 9-Continued.

-1.5%. Our simulations suggest that forsuch noise levels, statistical error in an esti-mate of the breadth ratio will not exceed afew percent of 83*, while that in an estimateof the noncircularity will typically be about

I distance unit or less. The bias in an esti-mate of ,B* will usually be negligible, but thebias in an estimate of C's fC obtained atlow solar phase angle can be several dis-tance units at the higher noise levels.)

a= 1%

a =3%

a= 1%


AO = 3° AO 90

FIG. 9-Continued.

The sampling interval is less critical, es-pecially at low noise levels and large solarphase angles. However, simulations notpresented here indicate that gaps in rota-tional-phase coverage as large as a few tensof degrees can increase [IE by several tensof percent. For this reason, it seems pru-dent to interpret the parameter AO in theFig. 9 simulations as the minimum intervalbetween any two data points.

VI. APPLICATIONS

In this section, we present results of ap-plying convex-profile inversion to selectedasteroids, beginning with lightcurves forwhich prior knowledge of the asteroid'spole direction indicates viewing-illumina-tion geometry not far from equatorial. Notethat for all lightcurves shown below, weplot the brightness in linear units scaled sothat the average brightness is near unity;accordingly, we express Am in linear unitsinstead of magnitudes.

To help guide interpretation of ourresults, each of the figures referred to in

this section includes values for severalgoodness-of-fit statistics. If the ideal condi-tions were satisfied perfectly, then the (un-constrained) Fourier model 5 for the light-curve data y would be the same as the(constrained) Fourier model § correspond-ing to C. (See Section III.) Ostro and Con-nelly (1984) describe a test of the null hy-pothesis, Ho, that the vector of Fouriercoefficients for § is a statistically acceptableestimate of the Fourier vector for the"true" lightcurve. This "variance ratio"test compares an estimate of c-2 obtainedfrom the residuals between the data and yto an estimate obtained from the residualsbetween the data and 5. Let M be the num-ber of Fourier harmonics and L the numberof data points, and let us assume that theerrors in y are independent, identically dis-tributed, zero-mean, Gaussian randomvariables. Then the following goodness-of-fit statistic, w1, is distributed asunder the null hypothesis (Plackett 1960,p. 52 ff.):

w, = (R - OWV2

a = 1%

.-3%

C 15

11 9 89 9 892 50 250

100 5 ° lo 200 0o 50 0o, 200

n I 55 537 2 46 1 36 6O 2 00 6 64 365 1 93110 I8 Be 11 33 18. 21 18 81 ~ 0 9. 92 1 1 79 18 39 9 98O- 2 58 2 5 2 50 2 51 P 2 49 2 49 2 49 2 49

o O. 50 l00 200 ¢ 0o 5- l00 200

f)O2 95 6B8 507 2 70 lb 3 55 7 51 G 28 3 88n, 10 34 12 12 115 19 10 36 110 1007 12 08 11 36 10 33P*2 5I 2 51 2 51 2. 52 r2. 47 2 47 2 47 2 48

50


whereR = (y - §)T(y - §)(y - y)T(y -v, = 2M + 1V2 = L-vj .

We can reject Ho at the 100(1 - a)% confi-dence level whenever w1 > FV1 V2.^A For ex-ample, Ho can be rejected at the 99% confi-dence level if w, > F, 1, 2,001 . For allinversions in this paper, M = 10, L > 45,and F 1,,12 ,0. 01 < 3.

In the figures below, we give up to threegoodness-of-fit statistics for each inversion.The first, wI, tests Ho for y, as just de-scribed; if w, exceeds 3, we can safely re-ject Ho. The second, w2, tests the same hy-pothesis, but for the lightcurve model Y'scorresponding to C,. The third, W3, tests thesame hypothesis as w2, but assumes that thecorrect lightcurve model is y, instead of y asfor w, and w2. If the third w statistic is >3,then ywe can safely reject the hypothesisthat C, provides as good a fit to the light-curve data as does C. Whenever we esti-mate C, we also give the "symmetrizationdistance" fQs between C and C,.

A. Objects with Published Pole Directions15 Eunomiac. Figure 10 shows C (solid

profile) and C, (dotted profile) estimatedfrom lightcurves at solar phase angles equalto -12° and -21°. The signs of / assumeretrograde rotation, as deduced by authorscited in the figure caption. Using Magnus-son's (1986) values for the ecliptic longitude(106° + 50) and latitude (-730 + 10°) ofEunomia's north pole, we calculate (OE, 5S)equal to (19°, 180) and (80, -2°) for the twoEunomia curves.

The figure shows that the values of theelongation /3* and the noncircularity fQc forthe two estimates of C vary by less than10%. If the ideal conditions held perfectlyand if the lightcurves were free of noise,then C and those two shape descriptorswould be independent of solar phase angle.The ideal conditions certainly are not satis-

fied perfectly and the lightcurves are notnoiseless, but the similarity of the profilesobtained from the two lightcurves lendsconfidence to their validity and is consis-tent with the net effect of the violations ofthe ideal conditions being fairly small. Alsonote that the fractional variation in the esti-mates of 13* (which pertains to the aster-oid's mean cross section) is half that in theestimates of Am (which pertains to thelightcurves); see the discussion in SectionIV.B.

The fact that the values of w, and w2 ex-ceed 3 is interpretable as very strong evi-dence for violation of ideal conditions.However, caution is advised even whenthose statistics are near unity, since viola-tions of ideal conditions might conspire toproduce a lightcurve yielding a good matchbetween § and the lightcurve data, but a Cthat nevertheless is inaccurate. By thesame token, it is conceivable that a meancross section estimate providing a poor fitto the data might be physically valid. Giventhese various possibilities, it clearly is de-sirable to estimate an asteroid's mean crosssection at a variety of solar phase angles.

Figure 10 also shows the weighted aver-age of the two estimates of Eunomia'smean cross section, calculated by summingthe statistically weighted vector of Fouriercoefficients for the radius-of-curvaturefunctions for the individual profiles, withthe profiles' relative rotational phase cho-sen to maximize the cross correlation of theradius-of-curvature functions. This exam-ple demonstrates how convex-profile inver-sion can combine the shape informationcontained in any number of independentlightcurves, perhaps taken at a variety ofsolar phase angles and/or during differentapparitions.

19 Fortuna. Figure II shows C for twolightcurves obtained by Weidenschilling etal. (1987) for 19 Fortuna, and theirweighted average. This asteroid's light-curves have peak-to-valley amplitudesclose to 0.25 mag, independent of eclipticlongitude, suggesting that the asteroid-cen-

51


. s

E s

Qs

IY

180ROTATIONAL PHASE

15

Am

I-cwQs

EUNOMIA-210

1 621 487 65

11 33

7 15

266 26

FIG. 10. Estimates of C (the solid profile) and C. (the dotted profile) for lightcurves obtained forasteroid 15 Eunomia at solar phase angle S = -12° (Fig. 8 of Groeneveld and Kuiper 1954) and at b =-210 (Fig. 14 of van Houten-Groeneveld and van Houten 1958). All the lightcurves in this paper areplotted on a linear scale, with the average brightness set to unity. In the lightcurve plots, the largesymbols are the data, the solid lightcurve 9 was derived from the unconstrained vector of Fouriercoefficients i, andzthe dotted lightcurve y was derived from the constrained Fourier coefficients (x)corresponding to C.On the right, Am is the ratio of y's maximum to its minimum; f3* is the breadthratio (the same for C and Ci); flc is the noncircularity of the profile drawn with a solid curve, here C;and fs is the distance between C and C.. To the right of "w" are values for three statistics that let ustest hypotheses described in the text. Briefly, since the first statistic (wl) exceeds 3, we can safely statethat the lightcurve corresponding to the mean cross section estimate is not a good fit to the data. Sincethe second statistic (W2) exceeds 3, we can make the same statement about the symmetrized meancross section. Since the third statistic (W3) exceeds 3, we can state that the difference between thelightcurves corresponding to C and C. is significant, i.e., that thze odd harmonics in y are statisticallysignificant. (In some of the subsequent figures we just estimate C., and in those we do not give valuesfor f1s, wl, or W3.) In the middle of this figure are the weighted average of the two individual estimatesof C, its symmetrization, and relevant shape descriptors. Note the distances between the cross sectionestimates.

tered declinations of Earth and the Sun are 12 and 13 show results of inverting an oppo-within a few tens of degrees of zero. sition lightcurve and a nonopposition light-

20 Massalia and 129 Antigone. Figures curve for each of these large, main-belt as-

52

15 EUNOMIII5 - 120

Am 1 491 41

QC 8 40w 9 4 22 15Qs 7 46

AVERAGE

9* 1 43QC 8 16


19 FORTUNA

Am

(QcWQs

86 1

12027230610

I

-

In

AVERAGE

p.(IC

5.591 226 73

5 81

3.47

19

Amp.

(IcWQs

- .. Il

I I B I 360 180 3PAROTATIONAL PHASE

FORTUNA190

1 281 225 72

0 5 1 2 1 74 49

FIG. 11. Similar to Fig. 10, but for lightcurves of asteroid 19 Fortuna obtained by Weidenschilling etal. (1987).

teroids. For each object, Magnusson's(1986) pole position estimates are consis-tent with the asteroid-centered declinationsof the Sun and Earth being within a fewtens of degrees of zero during the observa-tions.

If an estimate of C from a nonoppositionlightcurve were accurate, then its symme-trization would provide an estimate of C.that would also be accurate and hencewould resemble an estimate of C. from anopposition lightcurve. We expect C to bemore reliably estimated from an oppositionlightcurve because the scattering is likely tobe closer to geometric at small (P.

For Massalia, the two estimates of CS arenearly identical (fl = 2.33), lending confi-

dence to their mutual validity. The poorgoodness of fit for the inversion of the op-position lightcurve is disconcerting, butnote that the residuals are spread out fairlyuniformly over 360° of rotational phase.

For Antigone, the two estimates of Cs are6.20 distance units apart-they are defi-nitely different-and the residuals are se-vere, so the cross section estimates arelikely to contain significant systematic er-rors. The presence of a strong first har-monic in the lightcurve taken only 70 fromopposition and, if Magnusson's pole direc-tion is correct, at (8 E, 5s) equal to (17°, 12°)is interpretable as evidence for a nonuni-form albedo distribution.

624 Hektor. Figure 14 shows Cs esti-

2 37 05

= a, -

10 -II

T C -

53


20 MASSALIAN

S -

j

m 0

20 MASSALIA0 2°Am 1 17

1 150, 2 82w 13

180 360ROTATIONAL PHASE

FIG. 12. Inversion of lightcurves obtained for 20 Massalia by Gehrels (1956, composite of his Figs. 4and 5) at ) = 13° and by Barucci et al. (1985) at 4 = 20. Note the resemblance between the symmetriza-tion (dotted profile) of our estimate of C from the nonopposition lightcurve and the estimate of C,from the opposition lightcurve. See Fig. 10 caption.

IB1ROTATIONAL PHASE

6.20

CS

129 ANTIGONE

129 ANTIGONE

IQ-

0 8 180 360ROTATIONAL PHASE

FIG. 13. Convex-profile inversion of lightcurves of asteroid 129 Antigone obtained by Barucci et al.(1985) at 4 = 70 and by Scaltriti and Zappala (1977) at 4) -160. See Fig. 10 caption, Fig. 12, and text.


Am

w9*

WnS

130

1 161 145 85

2 8 4 9 3 45 03

0Am

WQ5

-1601 341 234 03

18 20 1 72 90

Am

W

1 311 265 03

4 3

54

I


624

Am

dcw

HEKTOR40

2 522 4512 64

1 4

a-

- -a:

0

44 NYSA

Am

wQs

0 18e 36eROTATIONAL PHASE

el

1 441 418 06

2 7 4 2 3 16 38

1685 TORO

Am

w9*

2

517?

33002679210 21

2 14

Q-I

1 .

Q -sI

Ir

ZUI

Ia

0.

FIG. 14. Top: Estimate of the symmetrized mean cross section of asteroid 624 Hektor for a near-opposition lightcurve obtained by Dunlap and Gehrels (1969). Middle: Estimation of 44 Nysa's symme-trized mean cross section from a lightcurve obtained by di Martino et al. (1987). Bottom: Estimates ofC and C, for 1685 Toro, from inversion of a lightcurve obtained by Dunlap et al. (1973). See text andFig. 10 caption.

mated for 624 Hektor from a lightcurve ob-tained at 0 = 4° by Dunlap and Gehrels(1969). The pole directions estimated forthis Trojan asteroid by those authors (andmore recently by ZappalA and Knezevi61984, Pospieszalska-Surdej and Surdej1985, Magnusson 1986) indicate that 6 E and8s have absolute values less than 10°. Theconstancy of Hektor's color indices withrotational phase (Dunlap and Gehrels1969), the absence of odd harmonics in thelightcurve, and the low value of W2 concurin supporting the reliability of our estimate

of C Note that the profile has ,3* = 2.5 andis distinctly nonelliptical. Since the meancross section of an ellipsoid rotating about aprincipal axis is an ellipse, our results sug-gest that neither the asteroid nor its convexhull are ellipsoids. On the other hand, our Cis quite consistent with many other modelsfor Hektor's three-dimensional shape, in-cluding a cylinder with rounded ends(Dunlap and Gehrels 1969), a dumbbell(Hartmann and Cruikshank 1978), and vari-ous binary configurations (Weidenschilling1980). Of course, in making these state-

36eISeROTATIONAL PHASE

18 3GRROTATIONAL PHASE

55


ments, we ignore effects of possible smallviolations of the ideal conditions.

44 Nysa. As noted by di Martino et al.(1987) and Magnusson (1986), this aster-oid's pole is relatively well known. The in-verted lightcurve (middle row of Fig. 14),from di Martino et al. (1987), was taken at asolar phase angle of 8.50, under nearlyequatorial viewing/illumination geometry.The pole estimate yields 5E = 90 and 5s =30, so the longitudinal separation of the sub-Earth and sub-Sun points is only about 50,and our "leverage" in estimating C's oddharmonics is marginal at best. On the otherhand, our estimate of Cs probably is reli-able. It is interesting that our estimate,/3* = 1.41, for the elongation of Nysa'smean cross section is within several per-cent of values derived for a model ellip-soid's equatorial axis ratio from analysisof a multi-apparition lightcurve data set(Pospieszalska-Surdej and Surdej 1985,Magnusson 1986).

1685 Toro. We assumed the pole direc-tion (200°, 550) quoted by Dunlap et al.(1973) in inverting their July 1972 lightcurvedata (bottom row of Fig. 14), and the nega-tive sign of 4 corresponds to direct rota-tion. The fit is excellent. Note that C ismuch less elongated than one might inferfrom the lightcurve's large amplitude. Alsonote that whereas fQ = 2, so Ce is not farfrom C, w2 and W3 are each much greaterthan 3. That is, the lightcurve's odd har-monics are prominent while C's are not.There is no discrepancy here; odd harmon-ics in shape generate increasingly large oddharmonics in a lightcurve as the solar phaseangle increases. [See (I) and Fig. 3.]

B. Objects with Unknown Pole DirectionsFigure 15 shows C and/or Ct for objects

for which pole-direction estimates are lack-ing. We present these profiles because thegoodness-of-fit statistics are small, or insome cases because the profiles are inter-esting, or simply to illustrate the mappingfrom lightcurve space to profile space, e.g.,the relation [Eq. (1) and Fig. 3] between the

harmonic structure of a lightcurve and thatof a mean-cross-section estimate. Unlessotherwise noted, all estimates of C assumethat 4)'s sign is positive; the profile obtainedfor a negative sign generally is similar, withnearly the same /3* and fic as the profileshown.

36 Atalante. As di Martino et al. (1987)point out, even though this lightcurve's am-plitude is similar to that in lightcurves ob-tained 700 of longitude away, the viewing/illumination geometry might not neces-sarily be equatorial. The fit is awful (w, ishuge), with strong, positive residuals forhalf a cycle but strong negative residualsfor the other half. This situation is consis-tent with a hemispheric albedo asymmetry,so perhaps this object should be targetedfor polarimetry or multicolor photometry.

60 Echo. The fit of the model lightcurvecorresponding to C, to Zeigler and Flor-ence's (1985) data is okay, with no majorresiduals except at the primary maximum.

61 Danae. To the best of our knowledge,this object has not been observed photo-metrically since Wood and Kuiper (1963)acquired this lightcurve. At such a smallsolar phase angle a mean cross section esti-mate is probably unreliable, and we show Cprimarily as an example of a shape with alarge noncircularity.

69 Hesperia. The residuals are large, sug-gesting violation of ConditionA EVIG and/orCondition GEO. Compare Cs to Danae's,which has similar values of the shape de-scriptors /* and Qc.

118 Peitho. This Stanzel and Schober(1980) opposition lightcurve has an unusu-ally strong fourth harmonic, which mani-fests itself in the rectangular appearance ofCs. The large residuals could be due to al-bedo heterogeneities and/or to non-equato-rial viewing/illumination geometry.

164 Eva. The large value of w, is a bitmisleading here. The number of points plot-ted in the Schober (1982) lightcurve is huge,and our digitization of it yielded more than700. With this many data points the F-ratiotest becomes extremely sensitive. The

56


36 ATALtNTE0Am

p.

w(Is

- -

-ED

A -

I

S

1401 181 118 50

57 75

m C

0 186 36ROTATIONAL PHASE

60 ECHO

-Cs0Am

(Iw

60

1 201 162 52

2 8

61 DANCEAn<SC

0Amp.

WnCS

D W

2 m1 e I60 360

ROTATIONAL PHASE

1 301 278 11

2 5 3 1 1 06 13

FIG. 15. Estimates of C and/or C. for asteroids lacking published estimates of pole direction. Seetext and Fig. 10 caption. Sources of the lightcurves are 36 Atalante (di Martino et al. 1987, Fig. 4), 60Echo (Zeigler and Florence 1985, Fig. 4), 61 Danae (Wood and Kuiper 1963, Fig. 7), 69 Hesperia(Poutanen et al. 1985, Fig. 4), 118 Peitho (Stanzel and Schober 1980, Fig. 1), 164 Eva (Schober 1982,Fig. 3), 246 Asporina (Harris and Young 1983, Fig. 29), 270 Anahita (Harris and Young 1980, Fig. 18),434 Hungaria (Harris and Young 1983, Fig. 38), 505 Cava (Harris and Young 1985, Fig. 2), 704Interamnia (Lustig and Hahn 1976, Fig. 11), 739 Mandeville (ZappalA et al. 1983, Fig. 28), 1173Anchises (French 1987), 1627 Ivar (Hahn et al. 1987), 2156 Kate (Binzel and Mulholland 1983, Fig. 18).

model lightcurve y corresponding to ourmean cross section estimate actually is veryslightly lower than the unconstrained modellightcurve 5 at each of the extrema exceptthe deepest minimum, where it is slightlyhigher than 5.

246 Asporina. The Harris and Young(1983) lightcurve has only 65 points, so theF-ratio test is not very sensitive to the dif-

ference between y and 5. Contrast with thesituation for 164 Eva.

270 Anahita. Differences between theshapes of the two peaks (i.e., odd harmon-ics in the lightcurve) cause the large valueof W2.

434 Hungaria. Despite the large residualsover rotational phases from 2500 to 3600,this inversion demonstrates that, at large

0 6TA 360RO TA TI ON AL PHASE

6 07 93

m.

57

;D

-


UN -

. I I I

; 0 180 360ROTATIONAL PHASE

M.

= 0

N 8 108 368ROTATIONAL PHASE

I' ~I I

I N

: 0 180 360ROTATIONAL PHASE

C

69 HESPERIA0 100Am 1 25d- 1 190, 7 89w 4 0 6 6 4 6Os 6 94

118 PEITHO0 40

Am 1 34a* 1 29(Ql 6 91w 14

1640Am

Qcw

C

EVA260

1 401 285 64

22 306 4744 58

FIG. 15-Continued.

solar phase angles, the breadth ratio /3* issuperior to Am as an elongation measure.

505 Cava. The lightcurve from Harrisand Young (1985) is composited from datataken at phase angles ranging from 100 to25°, and we use an average value of 150 forthe inversion. The fit of to the data is goodexcept near 4 2700; readers might care tospeculate about the source of this "fea-ture."

704 Interamnia. The fit of y to the light-curve obtained by Lustig and Hahn (1976)is not good, but the three pairs of extremaare replicated, and we show this inversionbecause C offers a simple explanation forthe existence of three minima with different

depths: The three pairs of extrema arise be-cause C resembles an equilateral triangle,while differences between the minima arisebecause the triangle's vertices have differ-ent "sharpnesses."

739 Mandeville. The slightly differentshapes of the minima in this Zappala et al.(1983) fightcurve are the source of the oddharmonics in C, which provides a decent fitto the data and, in view of the values of fisand W3, a significantly better one than doesCs.

1173 Anchises. This opposition light-curve obtained by French (1987) yields anestimate of Cs which is notable for its elon-gation (/3* = 1.6) and its interesting shape,

58


2460Am91*ocw

Cs

2700Am

w

ASPORINA6°

1 181 163 08

1 1

ANAHITA(S

1 321 304 94

9 4

C

4340Amp.

w

, 0 XE m

m 0 180 360ROTATIONAL PHASE

HUNGARIA27°

2 021 628 16

10 16 5 24 81

FIG. 15-Continued.

which has a weak sixth harmonic. Thelightcurve has a significant first harmonic(note the poor fit of y to the lightcurve max-ima), which could arise from a nonequato-rial view of a nonaxisymmetric shape or, ifthe shape is axisymmetric and the aspect isequatorial, from a heterogeneous albedodistribution.

1627 Ivar. This lightcurve from Hahn etal. (1987) is a composite of data taken dur-ing September 1985 at solar phase anglesbetween 180 and 330; a value of 250 wasused in the inversion. Radar data (Ostro etal. 1986) and the progression of lightcurveamplitude during the 1985 apparition (A.W. Harris, private communication) suggest

that the aspect during September wascloser to equatorial than during the closestapproach to Earth in July, but that Ivar isactually more elongated than one would in-fer from any of the lightcurve data or fromour estimate of C.

2156 Kate. Our estimate of this smallmain-belt asteroid's mean cross section iselongated and highly noncircular. Theworst residuals are confined to the minima.

VIL. CONCLUSIONS

This paper has been an attempt to formu-late and apply a comprehensive frameworkfor using a lightcurve to constrain an aster-oid's shape. We have seen that convex-pro-

I -

- m

S e 180 36ROTATIONAL PHASE

59

L -aI

Ien

I

ImI


I, I0 199 360

ROTATIONAL PHASE

N -

In.ll

z m

X 0 99 360ROTAT IONAL PHASE

= "I

I .

I; m9 .A80 360

ROTATIONAL PHASE

704 INTERAMNIA

Am91~OcwQs

1201 101 076 41

20 42 176 15

739 MANDEVILLE

\C

FIG. 15-Continued.

Am

ocwCs

10°1 131 115 84

2 8 7 4 105 45

file inversion builds on the canonical workof Russell by (i) identifying the mean crosssection C as the optimal shape constraintavailable from a lightcurve, (ii) defining thedifferences between the potential informa-tion content of opposition lightcurves andthat of nonopposition lightcurves, and (iii)specifying the conditions that determine theaccessibility of that information.

The methodology for estimating C and Cswas introduced by Ostro and Connelly(1984), and here we have tried to assessprincipal sources of systematic and statisti-cal error. Perhaps the most severe obstacleto estimating the mean cross section stemsfrom violation of Condition GEO. How-

ever, it is encouraging that our simulationssuggest that salient characteristics of C re-semble those of C rather closely for equato-rial lightcurves obtained at solar phase an-gles near 20°, at least for the simplisticshapes we examined. When the viewing/il-lumination geometry is as much as a fewtens of degrees from equatorial, the validityof estimates of C or Cs should be consid-ered suspect, and simulations like those inSection IV.A can be used to gauge the se-verity of distortion. (On the other hand,even when C is apparently biased, it re-mains a visual representation of the light-curve in a language befitting the light-curve's intrinsic information content.)

I

T~

505 CAVA

C,

1501 241 196 16

3 5 4 8

Am

OcwQs

2 44 0'

60

- I

- ---

m


1173

Am

Ocw

C

1627

AmI.w 2Qs

2156

Amw*

w,

ANCHISES

1 641 599 34

0 2

I VAR

25°1 721 515 34

0 3 1 2 62 28

KATE6O

1 751 677 69

7 5


FIG. 15-Continued.

We cannot hope to escape the reality thata lightcurve is determined by the viewing/illumination geometry and the surface'slight-scattering behavior as well as by thethree-dimensional shape. Hopefully, wehave demonstrated the value of convex-profile inversion as a tool for assessing therelative roles of each of these factors.

Since shape effects are inextricably in-volved in the interpretation of any disk-integrated observations, including thosemeant to probe physical properties otherthan body shape, we conjecture that con-vex-profile inversion will prove useful in"removing" shape effects in studies de-signed to constrain those properties. For

example, one could apply convex-profileinversion to analysis of an opposition light-curve to constrain the average longitudinaldependence of an asteroid's albedo distri-bution, as follows. If Condition EVIG issatisfied close to opposition, where Condi-tion GEO is most likely to be satisfied, wecan use Russell's Fourier tests to seek evi-dence for a nonuniform albedo distribution.If the opposition lightcurve has a first har-monic but no higher odd harmonics, wecould estimate Cs with confidence and at-tribute residuals between the lightcurve andCs's breadth function to albedo variegation.Then one might try to account for thoseresiduals in terms of a nonuniform, two-di-

I- -

0 t80 360ROTATIONAL PHASE

T -

-i -

I)

tI ,s

to

C t

0 180 360ROTATIONAL PHASE

61

-


mensional model asteroid, i.e., by varyingthe albedos of each side of the polygonalestimate of Cs obtained from convex-profileinversion. This exercise would collapseboth the three-dimensional shape and thealbedo distribution on the three-dimen-sional surface into two-dimensional aver-ages, and as such could help to convey therange of possible configurations for the as-teroid. Of course, from R3 and R4, we can-not expect lightcurves to reveal which ofthe infinite number of admissible configura-tions is correct. Unique models of eithershape or of the distribution of surface prop-erties simply cannot be obtained from disk-integrated measurements.

If a reliable estimate of an asteroid'smean cross section were available, onecould substitute C for Cs in the above exer-cise. It might also be interesting to incorpo-rate Hapke's scattering formalism into this"two-dimensional modeling," as in SectionIV.B, to explore the range of possible val-ues for Hapke parameters and perhaps tomodel effects of surface heterogeneities. Atthis level of modeling, however, even two-dimensional simplifications cannot be ex-pected to yield unique solutions. Still, theseapproaches might help to separate shapefrom compositional variations in analysis ofmultispectral lightcurves, from regoliththermal characteristics in analysis of ther-mal-infrared lightcurves, or from surfacemicrostructure in analysis of asteroid"phase functions." The last possibility in-volves the observation that brightness gen-erally increases as zero phase angle isapproached-the opposition effect. Inprinciple, one can use Hapke's scatteringtheory or any other applicable formalism(e.g., Lumme and Bowell 1981, Goguen1981) to model phase curves measured forasteroids and then use values of the rele-vant free parameters to constrain such sur-face characteristics as porosity. However,any such model necessarily makes some as-sumption about asteroid shape, and con-vex-profile inversion offers a powerful toolfor dealing with shape effects.

In its current incarnation, convex-profile

inversion is not readily applicable to datasets consisting of many lightcurves takenunder a variety of (nonequatorial) viewing/illumination geometries. To accommodatethis most common kind of lightcurve dataset, one must parameterize aspects of theasteroid's shape not described by the meancross section. Satisfactory accomplishmentof this task could greatly increase the num-ber of asteroids for which mean cross sec-tions can be accurately estimated and mightenhance the value of multi-lightcurve con-straints on asteroid pole directions.

ACKNOWLEDGMENTSWe thank B. Hapke and J. Goguen for discussions

of the light-scattering properties of particulate sur-faces; A. Harris, M. Gaffey, J. Surdej, K. Lumme,and P. Magnusson for critiques of an earlier version ofthis paper; and L. Belkora, D. Casey, D. Long, and A.Zaruba for technical assistance. Part of this researchwas conducted at the Jet Propulsion Laboratory, Cali-fornia Institute of Technology, under contract with theNational Aeronautics and Space Administration, andpart (RC) was supported by NSF Grant MCS77-01740.

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