Convex-Profile Inversion of Asteroid Lightcurves: Theory and Applications

ICARUS 75, 30-63 (1988)

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

ROBERT CONNELLY Department 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 interpretation of an asteroid's lightcurve and then is used to constrain the shapes of selected asteroids. 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 contours parallel to the equatorial plane. C is a unique, rigorously defined, two-dimensional average of the three-dimensional shape and constitutes optimal extraction of shape information from a nonopposition lightcurve. For a lightcurve obtained at opposition, C's odd harmonics are not accessible, but one can estimate the asteroid's symmetrized mean cross section, Cs. To convey visually the physical meaning of the mean cross sections, we show C and Cs calculated numerically for a regular convex shape and for an irregular, nonconvex shape. The ideal conditions for estimating C from a lightcurve are Condition GEO, that the scattering is uniform and geometric; Condition EVIG, that the viewing-illumination geometry is equatorial; Condition CONVEX, that all of the asteroid's surface contours parallel to the equatorial plane are convex; and Condition PHASE, that the solar phase angle 4 * 0. Condition CONVEX is irrelevant for estimation of Cs. Definition of a rotation- and scale-invariant measure of the distance, A2, between two profiles permits quantitative comparison of the profiles' shapes. Useful descriptors of a profile's shape are its noncircularity, 12c, defined as the distance of the profile from a circle, and the ratio ,6* of the maximum and minimum values of the profile's breadth function 13(0), where 0 is rotational phase. C and Cs have identical breadth functions. At opposition, under Conditions EVIG and GEO, (i) the lightcurve is equal 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 physical interpretation. An estimate C of C (or Cs of Cs) can be distorted if the applicable ideal conditions are violated. We present results of simulations designed to calibrate the nature, severity, and predictability of such systematic error. Distortion introduced by violation of Condition EVIG depends on 4, on the asteroid-centered declinations Ss and SE of the 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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shapes. Errors arising from violation of Condition GEO have been studied by generating lightcurves for model asteroids having known mean cross sections and obeying Hapke's photometric function, inverting the lightcurves, and comparing C to C. The distance of C from C depends on k and rarely is negligible, but values of 12, and ,8* for C resemble those of C rather closely for a range of solar phase angles (i - 20° ± 100) generally accessible for most asteroids. Opportunities for reliable estimation of Cs far outnumber those for C. We have examined how lightcurve noise and rotational-phase sampling rate propagate into statistical error in C and offer guidelines for acquisition of lightcurves targeted for convex-profile inversion. Estimates of C and/or Cs are presented for selected asteroids, along with profile shape descriptors and goodness-of-fit statistics for the inverted lightcurves. For 15 Eunomia and 19 Fortuna, we calculate weighted estimates of C from lightcurves taken at different solar phase angles. C 1988 Academic Press, Inc.

1. INTRODUCTION present a comprehensive conceptual foundation for the physical interpretation of One of an asteroid's most fundamental asteroid lightcurves. This foundation en- attributes is its shape, and lightcurves pro- compasses calibration and application of vide the only source of shape information convex-profile inversion as well as estab- for most asteroids. However, the functional lished principles from the oldest lightcurve form of a lightcurve is determined by the literature, so let us begin with a review of viewing/illumination geometry and the as- essential background material in which we teroid's light-scattering properties as well emphasize the progression of logic which as by its shape, and lightcurves offer less has led to convex-profile inversion. powerful shape constraints than do techniques that resolve asteroids spatially. Os- tro and Connelly (1984) introduced a new 11. HISTORICAL BACKGROUND AND REVIEW approach called convex-profile inversion to OF RECENT WORK using a lightcurve to constrain an asteroid's A general statement of the lightcurve in- shape, demonstrating that any lightcurve version problem is "Given that a lightcurve can be inverted to yield a convex profile is determined by asteroid shape, surface that, under certain ideal conditions, repre- scattering properties, and viewing/illumina- sents a certain two-dimensional average of tion geometry, how, if at all, can we isolate the asteroid's three-dimensional shape. The these effects and obtain physically interest- ideal conditions include geometric scattering information about the asteroid?" In a ing and equatorial viewing/illumination ge- classic paper, H. N. Russell (1906) offered ometry and will rarely be satisfied exactly the first analysis of this problem, investigat- for actual lightcurves, so the derived shape ing in fine detail the information content of constraints will usually contain systematic lightcurves taken close to opposition, i.e., error. with the solar phase angle 0 0°. He sum- Therefore, while convex-profile inver- marized his results as follows: If an asteroid sion can help to optimize extraction of has been observed at opposition in all parts shape information, proper application of of its orbit: the technique demands an understanding of (RI) "We can determine by inspection of systematic (as well as statistical) sources of its light-curves whether or not they can be uncertainty. Two objectives of this paper accounted for by its rotation alone, and, if are to explore these effects at some length so, whether the asteroid (a) has an absorb- and then, with a better feel for error ing atmosphere, (b) is not of a convex2 sources, to use convex-profile inversion to constrain the shapes of selected asteroids. 2 An object is convex if the straight line connecting A third, more general, objective is to any two points on the boundary is inside the boundary. 32 OSTRO, CONNELLY, AND DOROGI form, (c) has a spotted surface, or whether the asteroid's shape is not convex. How- these hypotheses are unnecessary." ever, as Russell proved, it is impossible to (R2) "It is always possible (theoretically) separate the effects of albedo variation to determine the position of the asteroid's from effects of surface curvature, so nei- equator, (except that the sign of the inclina- ther the light-scattering properties nor the tion remains unknown)." That is, we can shape can be modeled uniquely. Hence, find the line of equinoxes, of intersection apart from testing the convexity hypothe- between the equatorial and orbital planes, sis, the utility of lightcurves as sources of and the size of the angle between those shape information seemed virtually nil. planes. More than half a century after Russell's (R3) "It is quite impossible to determine treatise, photoelectric photometry and po- the [three-dimensional] shape of the aster- larimetry dramatically changed the empiri- oid. If any continuous convex form is possi- cal setting for lightcurve interpretation, ble, all such forms are possible." demonstrating that with a few notable ex- (R4) "In this case we may assume any ceptions the forms of most broadband opti- such form, and then determine a distribu- cal lightcurves seem less sensitive to sur- tion of brightness on its surface which will face heterogeneity than to gross shape. account for the observed light-curves. This Moreover, laboratory experiments and the- can usually be done in an infinite variety of oretical studies disclosed that geometric ways. " scattering is likely to be a good approxima- (R5) "The consideration of the light- tion for asteroids viewed close to opposi- curve of a planet at phases remote from op- tion (e.g., French and Veverka 1983) but position may aid in determining the mark- that the validity of this approximation dete- ings on its surface, but cannot help us find riorates with increasing solar phase angle. its [three-dimensional] shape." Thus, in the absence of evidence to the con- Despite Russell's skepticism toward trary, the premise that the scattering is uni- prospects for using lightcurves to find an form and geometric has been the starting asteroid's three-dimensional shape, he did point for nearly all efforts to interpret light- describe how to test certain hypotheses curves physically (e.g., Burns and Tedesco about shape, scattering law, and albedo dis- 1979, Harris 1986). tribution. For example, suppose we have Many researchers further constrain the found the asteroid's line of equinoxes (R2) lightcurve inversion problem by assuming a and obtained an "equatorial" opposition simple a priori shape. The usual choice is a lightcurve. Russell showed that such a triaxial ellipsoid, whose shape is com- lightcurve's Fourier series cannot have odd pletely described by the ratios bla and cla, harmonics higher than the first if the scat- with a - b - c the semiaxis lengths. The tering is geometric, 3 regardless of the al- equatorial opposition lightcurve of a geo- bedo distribution; it cannot have any odd metrically scattering ellipsoid (GSE) has harmonics if that distribution is uniform. brightness extrema in the ratio alb = He also showed that if two opposition light- 100 4Am, where Am is the peak-to-valley curves obtained in opposite directions are lightcurve amplitude in magnitudes. A com- different, then either the scattering is not mon practice is to take the maximum ob- geometric or the albedo is not uniform, or served value of Am as a measure of an as- both; if the difference is not a sinusoid, then teroid's elongation. Because GSE lightcurves are easy to calculate, they provide a convenient tool for error-propagation studies (Sections III and I The brightness of a geometrically scattering surface element is proportional to the element's pro- IV below) and for pole-direction determina- jected, visible, illuminated area. tions (e.g., Zappala and Knezevi6 1984, ASTEROID LIGHTCURVE INVERSION 33

Scaltriti et al. 1978). However, for the pur- Convex-Profile Inversion: Averages of pose of extracting information about an as- Asteroid Shapes teroid's shape from its lightcurve, the GSE assumption begs the question. After all, el- Ostro and Connelly (1984) demonstrated lipsoids are a subset of axisymmetric that whereas Russell's paper is valid, and shapes, which in turn constitute a small indeed it is not possible to find an asteroid's subset of convex shapes and an even three-dimensional shape from disk-inte- smaller subset of plausible asteroid shapes. grated measurements, it is possible to ex- The GSE assumption forces one to ignore tract shape constraints that are meaningful much of the lightcurve's information con- and that exploit more of the information in tent, imposing a very particular Fourier the lightcurve. Still, shape constraints from structure on any model lightcurve. As disk-integrated measurements are neces- shown by Russell and noted above, the sarily much less informative than, say, a equatorial, opposition lightcurve of any ob- stellar-occultation profile, which is a two- ject with a geometric scattering law will dimensional projection of the asteroid's have no odd harmonics higher than the three-dimensional shape. Given an aster- first. If the asteroid's surface is symmetric oid's lightcurve, the best constraint we can about the spin axis then the first harmonic obtain is not a projection of the asteroid's will vanish, leaving only even harmonics in three-dimensional shape, but rather a cer- the lightcurve, regardless of viewing as- tain two-dimensional average of that shape. pect. But if, additionally, the asteroid is an Ostro and Connelly (1984) defined an as- ellipsoid, the lightcurve's Fourier structure teroid's mean cross section C as the aver- will be even more restricted: As shown by age of all cross sections C(z) cut by planes a Connelly and Ostro (1984), the square of distance z above the asteroid's equatorial the opposition lightcurve of any geometri- plane. That definition of the C(z) left C un- cally scattering ellipsoid has no harmonics defined unless all the C(z) are convex. except for 0 and 2, regardless of viewing Here, to remedy this situation, we define aspect. the C(z) as the convex envelopes, or hulls, How should we interpret the equatorial on the actual surface contours. With this opposition lightcurve of a uniform, geomet- revision, C is now clearly defined for any rically scattering asteroid which might (by asteroid. the Fourier test just mentioned) be non- C and all the C(z) are convex profiles or, ellipsoidal? More generally, what can any in the language of geometry, convex sets. given lightcurve tell us about an asteroid's We use underlined, uppercase, italic letters shape? Russell wrote that if the scattering to denote such quantities. As discussed in is uniform and geometric, then we might detail by Ostro and Connelly (1984) and re- "seek to account for the light-changes by iterated below, a convex profile can be rep- the form of the surface alone. But this leads resented by a radius-of-curvature function to difficult problems in the theory of sur- or by that function's Fourier series. Dele- faces, and will not be attempted here." He tion of a profile's odd harmonics yields that pointed out that some of these problems profile's symmetrization. For example, an arise because odd harmonics in the expan- asteroid's symmetrized mean cross section sion of a surface's curvature function will Cs has the same even harmonics as C but no be irretrievable from an opposition light- odd harmonics. curve. He also suggested that the odd har- Figure 1 shows C and Cs for two three- monics "may sometimes be determined dimensional, styrofoam models, calculated from nonopposition lightcurves." These by applying formulas in Appendix A of Os- critical issues were not pursued further for tro and Connelly (1984)to measurements of nearly 75 years. photographs of those objects. Several pho- 34 OSTRO, CONNELLY, AND DOROGI

FIG. 1. Values of the mean cross section for (a) an egg-shaped object and (b) an irregular, nonconvex object. tographs of each model are shown to con- (If C contains odd harmonics, then light- vey visually the relation between three-di- curve odd harmonics can be introduced mensional shape and the two-dimensional away from opposition via the rotational- averages accessible from lightcurves. phase dependence of the fraction of the vis- There are four ideal conditions for esti- ible projected area that is illuminated. At mating C from a lightcurve: opposition, no shadows are visible, so that Condition GEO: The scattering is uni- fraction is unity regardless of rotational form and geometric. phase, and the odd harmonics vanish.) Condition EVIG: The viewing-illumina- Although we cannot estimate C from an tion geometry is equatorial, that is, the as- opposition lightcurve, we can estimate C's teroid-centered declinations 8s and 8E of the even harmonics and hence its symmetriza- Sun and Earth are zero. tion, Cs. Estimation of Cs from opposition Condition CONVEX: All of the aster- lightcurves is less "burdened" by ideal oid's surface contours parallel to the equa- conditions than is estimation of C from non- torial plane are convex. opposition lightcurves, for three reasons. Condition PHASE: The solar phase angle First, for estimation of C, Condition 4 is known and does not equal 0° or 1800. CONVEX ensures the absence of visible The last condition states that we cannot shadows between the limb and the termina- find C from an opposition lightcurve and tor, which might distort determination of C. arises for the following reason. Each coeffi- However, at opposition, there are no visi- cient in the Fourier series for C's radius-of- ble shadows, so Condition CONVEX is ir- curvature function is found from the corre- relevant and C, can be estimated reliably as sponding coefficient in the Fourier series long as Conditions GEO and EVIG are sat- for the lightcurve. However, we cannot use isfied. an opposition lightcurve to find C's odd Second, Conditions GEO and EVIG are harmonics because, as Russell showed, an satisfied more easily" for opposition light- opposition lightcurve has no odd harmonics curves than for nonopposition lightcurves. if Conditions GEO and EVIG are satisfied. As noted above, the geometric scattering ASTEROID LIGHTCURVE INVERSION 35

approximation is more valid at opposition nongeometric scattering, and in calibrating than at large solar phase angles. Condition this distortion we have relied heavily on EVIG can be satisfied at large phase angles Hapke's (1981, 1984, 1986) theory for a re- only if the pole lies nearly normal to the alistic description of the optical properties Earth-asteroid-Sun plane, but is satisfied of asteroid surfaces. On the other hand, our at any opposition occurring when the aster- calibrations have not been exhaustive; we oid is at an equinox, i.e., with the pole nor- have not examined coupled effects due to mal to the Earth-asteroid line. Hence, ge- simultaneous violation of two or more ideal ometry dictates that opportunities to conditions, and we have simplified as much estimate Cs far outnumber opportunities to as possible our modeling of three-dimen- estimate C. sional shapes. Nonetheless, our work does Third, to assess how close the viewing/ provide a guide to both the power and the illumination geometry is to equatorial for a limitations of convex-profile inversion. We given nonopposition lightcurve, one must argue that systematic errors in estimation know the asteroid's pole direction; for an of C and Cs are not insurmountable and that opposition lightcurve it is sufficient to know their severity often can be assessed a priori just the location of the asteroid's line of and/or a posteriori. equinoxes. In Section V we briefly examine how Ostro and Connelly (1984) showed that noise, sampling rate, and rotational phase calculation of a mean-cross-section esti- coverage propagate into statistical error in mate C from a lightcurve can be ap- mean-cross-section estimation. Then, hav- proached by weighted-least-squares optimi- ing calibrated various sources of uncer- zation subject to inequality constraints. In tainty in convex-profile inversion, we apply this paper, we use the term "convex-profile it in Section VI to an assortment of asteroid inversion" to denote our theory for the in- lightcurves. Finally we summarize all our formation content of lightcurves as well as results and offer strategies for future efforts the computational technique for estimating in the acquisition and physical interpreta- mean cross sections. tion of asteroid lightcurves. None of the first three ideal conditions are likely to be satisfied exactly for an ac- 111.CONVEX-PROFILE INVERSION tual lightcurve. However, the important is- MATHEMATICS sue is the extent to which their violation The power of convex-profile inversion degrades the accuracy of a mean-cross- rests on the fact that when the ideal condi- section estimate. As noted above, key ob- tions hold, the three-dimensional lightcurve jectives of this paper are to calibrate con- inversion problem, which cannot be solved vex-profile inversion's sensitivity to sys- uniquely, collapses into a two-dimensional tematic and statistical sources of error and problem that can. The geometry of the two- then to use convex-profile inversion to con- dimensional problem is shown in Fig. 2, strain the shapes of selected asteroids. In drawn for 0 = 30°. The asteroid is a convex the next section we briefly review the math- profile whose brightness 1(0) at rotational ematics of convex-profile inversion and in- phase 0 is proportional to the orthogonal troduce techniques for describing, compar- projection toward Earth of the visible, illu- ing, and manipulating profiles. In Section minated portion of the profile. IV we explore the manner in which depar- At opposition, the visible portion of the ture from ideal conditions makes C deviate profile is fully illuminated; the projection of from C. We have tried to maintain at least a this curve is called the profile's width, or modicum of realism in evaluating sources "breadth," in the direction of Earth, and of systematic error. For example, special the lightcurve is proportional to the breadth attention is paid to the distortion caused by function 83(0).Unless this function is con- 36 OSTRO, CONNELLY, AND DOROGI

or as

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

- (an ibJ)/2. Define P(0) - [x(0), y(0)]+ as the point on the curve that would be on the receding limb if the asteroid were at rotational phase 0. In this normal-angle parameterization, the outward normal to the curve at P(0) points in the positive x direction at rotational P(360- -1 i phase 0. Let r(0) = dsIdO be the radius of ...... 0 .... X curvature of the profile at P(0), where I1r(0) is the curvature at P(0) and s = s(0) is arc FIG. 2. Geometry for two-dimensional asteroid light- length defined by curve inversion. The asteroid is a convex profile rotat- at rotational phase 6 = 00. 2 2 2 ing clockwise and is shown (dsldO) = (dx/dO) + (dy/do) . The solar phase angle 4 is indicated, as are the asteroid's illuminated (solid) and unilluminated (dotted) Then portions. The asteroid's scattering law is geometric; i.e., the observed brightness is proportional to the orthogonal projection toward Earth of the visible, illumi- x(0) = x(O) - f r(t) sin t dt nated length 1(6). The point P(O) is on the receding (left-hand) limb and the outward normal in at P(6) and points in the positive x direction when the rotational phase is 0. Reproduction of Fig. I of Ostro and Con- nelly (1984). y(0) = y(O) + r(t) cos t dt.

Ostro and Connelly (1984) define the stant, it is periodic with a period equal to Fourier series for the radius of curvature 1800 or an integral fraction thereof-the function as lightcurve has no odd harmonics. (This is why most asteroid lightcurves, which are rarely obtained very far from opposition, r(0) = E dnei10 are dominated by even harmonics.) The second harmonic dominates the higher and show that the profile can be found from even harmonics unless the profile possesses the lightcurve because symmetry akin to that of regular convex d0 = C,/vn (1) polygons, in which case the lightcurve's fundamental can be a higher even har- where vn is a known function (Fig. 3) of &. monic. (Asteroids are unlikely to have such The profile will be closed only if d+, = 0 and symmetry, and this is why most asteroid it will be convex only if a set of N linear lightcurves are dominated by the second constraints, r(2F-k/N) > 0 for 1 c k c N, harmonic.) are satisfied. In practice, convex-profile in- The lightcurve's Fourier series can be ex- version finds that profile C which provides pressed either as the least-squares estimator for C by finding that vector x of Fourier coefficients that satisfies the constraints and is as close as I(W) - A (a, cos nO + b, sin nO) n=O possible to the (unconstrained) vector x of ASTEROID LIGHTCURVE INVERSION 37

sense as the asteroid's rotation. We adhere to the semantic convention that 4)satisfies 00 c 141 - 180°, so 4 # 0 is positive if the asteroid rotates through 141from E to S or negative if it rotates through j4) from S to E. For example, in Fig. 2, if the directions of the Earth and Sun were interchanged, 4 would equal -300. Clearly, for any given Sun-Earth-asteroid geometry, the sign of 4 is determined by the sense of rotation. 1 _ 6 7 Thus convex-profile inversion of a nonopposition lightcurve is sensitive to the sense 0 of rotation, and using the wrong sign of 4 can distort C; see Fig. 3 of Ostro and Con- 180° 0° 180° nelly (1984). However, we suspect that if Solar Phase Angle, S prior knowledge of an asteroid's pole direction is adequate to guarantee that the aster- FIG. 3. The function v,(j), discussed in the text, is plotted vs solar phase angle ( for values of the har- oid-centered declinations of the Sun and monic index n from 2 through 7. The following expres- Earth are both close to zero for a given non- sions for v, are valid for n r + 1; see Section II.A of opposition lightcurve, then in many cases Ostro and Connelly (1984). If 0 c

QuantitativeDescription of Profiles Fourier coefficients fit to the data. The We often will wish to quantify relation- lightcurves derived from x and x are, in vector notation, y and 5. ships between profiles to supplement verbal descriptions and subjective comparisons. Truncationof FourierSeries Following Richard and Hemami (1974) and Sato and Honda (1983), we define the dis- As discussed by Ostro and Connelly tance fl between any two profiles (1984), truncation of Fourier series limits PI and P2 as the fidelity with which convex-profile inversion can reproduce profiles with flat sides Q = 10 min ld1 d1o - Oddd2 0 , or sharp corners. We employ Fourier series containing 10 harmonics throughout this where dj is the vector of complex Fourier paper; series this long adequately represent coefficients in the expansion of Pj's radius- all lightcurves discussed herein. We note of-curvature function; djois the constant, or that lightcurves with significant energy in zero-harmonic term, in that expansion; and harmonics higher than 10 are rare, and we the diagonal matrix E rotates d2, by einO. speculate that the mean cross sections of This distance measure is rotation invariant actual asteroids generally lack extremely and scale invariant. flat sides and very sharp corners. The meaning of il can be visualized as follows. If our Fourier series are truncated Sign of the Solar Phase Angle to M harmonics, dj will have 2M + 1 ele- As shown in Fig. 2, the solar phase angle ments, so di and d2 define two points in a is measured from the Earth direction E to (2M + 1)-dimensional space, and it is easy the Sun direction S in the same rotational to calculate the distance between them. 38 OSTRO, CONNELLY, AND DOROGI

However, dj determines P i's circumference TABLE I (i.e., its size) and rotational orientation as GLOSSARYOF SYMBOLS well as its shape. Therefore, we define n as the distance between d, and d2 when PI and C Mean cross section P2 are scaled to the same size and rotated as C, Symmetrized mean cross section much into alignment with each other as C Estimate of C possible. C, Estimate of C, In describing interprofile distances, we j Solar phase angle will use the following scale: Am Lightcurve peak-to-valley amplitude 5E Asteroid-centered declination of Earth Distance (fl) Verbal description 5s Asteroid-centered declination of the Sun j3* Breadth ratio; ratio of a profile's maximum and C2 Nearly identical minimum widths 3-4 Similar but not the same ( Distance between two profiles 5-6 Different Dc Noncircularity; distance of a profile from a 7-8 Very different circle ,!9 Radically different nE Error distance, between C and C (or between C, and C) We also define a profile's "noncircular- f3s Symmetrization distance, between C and C, (or ity" fc as its distance from a circle, and a between C and CQ) profile's breadth ratio ,8* - 1 as the ratio of the profile's maximum breadth to its mini- IV. SENSITIVITY TO DEPARTURE FROM mum breadth. Figure 4 plots fc vs /8* for IDEAL CONDITIONS ellipses and shows the locations of various profiles in this two-parameter space, as well A. Violation of Condition EVIG as some interprofile distances. Table I lists As a first step in studying how departures symbols used frequently in this paper. from ideal conditions can introduce system-

14 l l 13 - 130 7 3 12 A

C: 9 43 10 9.2 I 9 - 9.3/ 6.4 0 7.8 5 0

4. 76

Z 4 z 3i

1.6 1.8 2.0 2.2 2.4 BREADTH 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 in the text, is indicated for selected pairs of profiles. ASTEROID LIGHTCURVE INVERSION 39 atic error in C or C we examine effects of 1984),arises from the constant illumination nonequatorial viewing/illumination geome- and visibility of a polar region. try.,For any given asteroid, the proximity From the viewpoint of convex-profile in- of C to C for a lightcurve obtained when version, nonequatorial viewing/illumina- SE 0 or Ss # 0 will be a function of 6 E, 5S, tion geometry prevents the "clean" col- and t. We have studied this particular lapse of the lightcurve inversion problem source of systematic error by inverting from three dimensions to two dimensions. lightcurves generated analytically for geo- It seems unlikely that many asteroids are as metrically scattering ellipsoids within the long and flat as our most extreme examples, parameter space: and in this respect our figures might exag- gerate the distortion caused by nonequato- 100< < 80° rial viewing/illumination geometry. On the 0.2 < b/a • 0.4 other hand, distortion for nonaxisymmetric and highly irregular shapes might be much 0.05 < c/a c bla. more severe than for ellipsoids. Note that For given values of b/a, c/a, and 4, the in the minimum-distortion region in the fig- field of possible values of NE and Ss is con- ure, the constantly illuminated pole is not veniently displayed in a plot of Ss vs (6 E - visible and vice versa, so neither pola~r re- 5s). Figure 5 shows several contours of con- gion contributes to the lightcurve. C re- stant BE and of constant Ss in this plane. sembles C in this region because C(z) has a Note that the Earth is north of the equator constant shape for an ellipsoid, a situation outside the stippled zone. The figure shows unlikely to hold for most asteroids. Hence, results of our simulations for illustrative even if EVIG were the only violated condi- two-dimensional cuts through the five-di- tion, it would probably be wise to interpret mensional parameter space. the simulations described here as establish- Under Condition EVIG (at thezorigin of ing ,a lower bound on the systematic error the figures, where BE= 5S = 0), C is nearly in C. indistinguishable from the asteroid's mean cross section C, which is an ellipse with B. Violation of Condition GEO axis ratio alb. However, under extremely How much is C distorted if the scattering nonideal viewing/illumination geometry, is nongeometric? Our strategy in answering C is not necessarily elliptical and has breadth this question is to (i) assume that the other extrema not necessarily in the ratio bla. ideal conditions are satisfied; (ii) generate The distortion in each C in Fig. 5 is quan- lightcurves for an asteroid with a known tified by the "error distance" QEbetween shape and a realistic scattering law; (iii) use C and C. It worsens with increasing 0, alb, convex-profile inversion to obtain an esti- or a/c and is most severe when 15EI, mate, C, of the asteroid's mean cross sec- I5sl> 10° and Ss is between 00 and 6 E. The tion C; and (iv) compare C to C. For the 6 distortion is minimal when either 16EI, S < time being, let us also assume that the as- 100 or SE is between 0° and -6s, i.e., in the teroid's three-dimensional shape is a con- triangular region labeled MIN in Fig. 5. vex cylinder, that is, that C(z) is constant Outside those regions, violation of Condi- and congruent with C. These assumptions tion EVIG often makes C more circular reduce the problem's dimensionality to than C. This result, which is consistent with two, letting us treat the asteroid's surface the well-known reduction in lightcurve am- as a convex profile. For convenience we plitude with decreasing aspect angle for op- model an asteroid as a convex polygon with position lightcurves of geometrically scat- 360 sides. The orientation and length of tering ellipsoids and a variety of other each side is known, so it is easy to calculate shapes (Barucci and Fulchignoni 1982, the asteroid's lightcurve for an arbitrary 40 OSTRO, CONNELLY, AND DOROGI

b/a =0.4 c/a =0.2 =20° b/aa 0.2 c/a =0.2 = * 020- I 1 7Q0 - 90100100 O10 70 160 170 170 017 600 100 010010 600 150 160 160 017Q 7 5Q0 -80Q90 90 010010 50 1- 130 150 160 016017 - 4001 4Q0'. 110 130 140 0 15Q017 - - 70 80 90 09 0Qo as 300 - 40 60 80 09 0Qi 30'- 8 Q100 12 0 0 14016- 20°l- 2204 Q8 40 60 07 09 20'k 40 60 80 0 11015- I 0o0-2 20 0 30 05 08 to0 30 20 30 07 013- 0111- 50 20 0 0 00 80 20 o0 0 2 08 I I -20° -Qo0 00 Roe 200 -204 -10 o00 -$ 200 8 - a, 8E- as b/a=0.4 c/a 0.2 0=400

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tik-Ro~ U\.,,.f 30: 'N... as -00@000000 *'~:G .E A X. '." 20 -000000~0000 ,b.A, Io0o 00000000o :7 b.NN~::~. 00 @S0C00@@S

-40' -20° 00 20° 40' a -8

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 geometrically scattering ellipsoids, for various values of the asteroid-centered declinations Ss and SE of the Sun and Earth. 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 that coordinate system. Note that the Earth is north of the equator everywhere except within the shaded area. The three arrays of profiles show estimates C of the mean cross section C (shown as the ellipse at the origin) for three different sets of choices for the solar phase angle 4 and the ellipsoid's axis ratios b/a and c/a. The "error distance" OIE, of C from C, is given for each profile. Distortion due to violation of Condition EVIG is minimal within the region labeled "MIN." photometric function. (By virtue of Condi- asteroid's brightness from light reflected tion EVIG, the flat polar surfaces do not from a surface areal element dA as contribute 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 dWto an where the reflectance rR is defined in the ASTEROID LIGHTCURVE INVERSION 41 cited references, 6, and 0e are the angles of value for the 12 remaining objects. These incidence and emergence, and six parame- figures demonstrate that the difference be- ters describe the surface properties. The tween C and C generally depends on the parameter w is the average single-scattering solar phase angle. At opposition, the scat- albedo. The opposition effect is described tering is sufficiently close to geometric that by the parameters h and S(0); h depends on f, is very accurately determined. (In fact, the regolith's porosity and particle size dis- for b less than about 100,one can estimate tribution, and S(0) is the fraction of light the symmetrized mean cross section quite scattered from close to a particle's surface reliably, with QE rarely as large as 5.) How- at 0 = 0. Effects of macroscopic surface ever, for very small phase angles (2° and 50 roughness are parameterized by the aver- in the figures), OE is extremely large for all age slope 0, measured with respect to the profiles except objects No. 8, 14, and 15; mean surface, of surface facets unresolved object No. 12 is an intermediate case. The by the detector. Following Helfenstein and cause of the severe distortion in C at low Veverka (1987), we write the average, sin- solar phase angles for the other 12 profiles gle-particle, angular scattering function as is 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 is P(tk) = 1 + bleg cos d) 2 n a clv,, estimation of odd harmonics + cleg(l.5cos 0 - 0.5). in C becomes increasingly sensitive to lightcurve noise and computational precision as Those authors estimated values of the six b- 0. These simulations and those pre- Hapke parameters for dark, average, and sented in Section V suggest that reliable de- bright terrains on the Moon. In our calcula- termination of odd harmonics is difficult if tions, we adopt their results for average ter- 14.1is less than about 50, and we will not try rain: w = 0.25, h = 0.06, S(O) = 0.78, 0 = to estimate C from lightcurves taken that 20.60, bleg = 0.33, and clg = 0.37. Our close to opposition. Instead, we will use results are essentially insensitive to varia- equatorial lightcurves obtained near oppo- tion in these values over ranges much larger sition to estimate C . than those spanned by their results for the As discussed earlier, Cs has the same three lunar terrain classes. even harmonics as C but no odd harmonics. For a geometric scattering law (rR = con- C, actually tells us quite a bit about C be- stant), convex-profile inversion yields C, at cause the breadth function of any convex opposition and C away from opposition. profile is independent of the profile's odd Figure 6 shows C and Cs for each, of 15 harmonics. That is, Cs and C have the same model asteroids, as well as C or C, esti- breadth function because they have the mated from lightcurves generated under the same even harmonics. It is important to assumption of Hapke scattering at 0 = 00, note that at opposition, under the pertinent 20, 50, 10°, 20°, 300, 50°, and 90°. The figure ideal conditions (EVIG and GEO), an as- also lists for each profile the breadth ratio teroid's lightcurue equals the breadth func- 3*, the error distance flE from C (or Cs), tion of C, (and of C) within a multiplicative and the noncircularity flc. (The 15 model constant. Furthermore, ,3*= 100.4Am,so the asteroids' mean cross sections are plotted lightcurve's amplitude is a trivial function in Fig. 4.) of /8* and thus is a valid gauge of elonga- Figure 7a plots flE for each object. Figure tion. We stress that this elongation or any 7b plots the average value of QE for all 15 other shape-related parameter obtained objects, the average value for the only three from a lightcurve pertains not to the aster- objects (No. 8, 14, and 15) with negligible oid's three-dimensional shape but to a two- energy in odd harmonics, and the average dimensional average of that shape. 42 OSTRO, CONNELLY, AND DOROGI c. X 1 C 2f

0 0e 1 23 2880 , 4 57 5 66 pr I 7 1 07 pr 1 12 1 12

0 0° 2° 50 lo 0io 2° 5o 10o

GE 0 00 6 68 5.12 3 56 E 08 01 6 25 5 58 4 42 Ge 1 23 8 81 6. 26 4.54 n0 4 56 8 23 8 09 7 29 p 107 107 1 07 1 08 r 1 12 1 13 1 13 1 13

0 20' 300 500 90o c 200 300 500 go0

GE 2.10 1 63 14A 113 GE 3 54 3 97 4 74 3.47 0l 3.01 2 56 2609 1.78 DC 5 10 3 90 2 71 4 68 p. 1808 I 08 108 1 07 V 1. 11 1 10 1 08 1 14

sC 3 C5 C 4

Ge 2 77 3 23 DC 4 92 7 99 a. 1.14 1 14 - 1 26 1 26

c/ 0 2° 5° 100 c 0° 20 50 lo

aE 801 6.22 4,95 2.69 GE 0 01 7 45 6 71 289 1E 2 77 7 53 6 49 4 41 Ge 491 7 32 7 28 769 p. 1 14 1 14 1 15 1 15 p. 126 1 26 1 27 1 28

0 200 300 500 90o i 200 300 500 900

nE 184 1 76 2.87 2802 On 4 31 4 78 6 13 6 37 n0 3 12 2 56 I6 2.98 Ge 7.13 6 32 58 0 4 78 pr 1 s5 1 15 1 14 1 14 Jr 1 29 130 1.30 1 26

FIG. 6. Effects of nongeometric scattering on estimation of C and C,. Each box shows results of simulations for a particular model "asteroid." The model's mean cross section and symmetrized mean cross section are shown above the horizontal bar. Below the bar are estimates of those quantities from inversion of lightcurves generated at various solar phase angles q and employing the Hapke photometric function described in the text. (C, is estimated at opposition and C is estimated at the seven nonzero values of s.) Each profile's noncircularity Q, and breadth ratio 13* arc given. {l, is the distance between 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 phase with respect to estimation of C's odd har- angles equal to 100, 200, 300, and 500, the monics, but the scattering becomes dis- average value of QE is 4.1, 3.8, 4.3, and 4.7, ASTEROID LIGHTCURVE INVERSION 43

C-s c 5 C. c 6

'c 3 28 5 64 b 6 89 10 20 P. 1 28 128 p. 129 1 29

¢ 0° 2° 50 10° 00 20 50 100

' 881 0X6 39 6.27 5 26 n 882 7 96 790 6 44 'C 3.28 8 48 8 27 8 19 nE 6 87 8 68 9. s 9 78 p. 1,28 1 29 130 1 31 . 1 28 129 1 30 1 30

P 20° 300 500 900o 200 300 500 90o

' 4.05 3 66 3.14 1.96 nt 5 54 5 97 7 86 48 0 'IE 7.95 7 62 6 68 5.97 'Ic 977 945 8 96 9.21 P- 1.31 1 32 1 31 1 30 1 29 1 27 1 27 1 29

9s C 7 C C 8

flc 4 38 7~64 'Ic 4.59 4 58 jr 1 48 1 48 pr 1 54 1 54

0 00 20 50 10° 00 20 50 100

'I 801 5 97 5 79 5 25 'I 8 01 0 10 0 23 837 'I 4.38 10 27 10 28 10 30 'c 4 59 4 68 4'88 495 P- 1 48 1 49 151 153 Ir 1,54 1 55 1 57 160

S 20° 300 500 900 0 200 300 500 900

DE 4 40 4 14 3 77 2 98 'O 8:64 9 0 87 1 31 0 63 n 974 916 8 62 7 87 'I5 0 5 09 4.95 4 86 rC 155 156 153 1 50 Jr 162 1463 1 61 1.57

FIG. 6-Continued. with an rms dispersion within 10% of 2.0 tain some systematic error. Figurers 6 and 7 throughout. Thus, the systematic error in suggest that the distortion in C can be C from violation of Condition GEO tends to expected to be moderate for most cases, "bottom out" around 200, a fortuitous cir- but negligible for some small fraction of cumstance in view of the inaccessibility of cases and intolerably severe for a compara- main-belt asteroid lightcurves at much ble fraction. On the other hand, Fig. 8 larger solar phase angles. shows that for (Aat least as large as 200 the Our simulations indicate that even for + noncircularity is within about _ I unit of the near 200, an estimate of C is likely to con- asteroid's actual value and that values of 1* 44 OSTRO, CONNELLY, AND DOROGI

C 9 C I01

a 00 0C 7. e7 12 13 1 66 r 1 77 1 77

50 10° 0 0° 2° 50 1 00 ) X( 1 14 6 76 aE 088 88a 7.25 5 40 870 1116 rC 78? 98 6 1l 45 11 69 1.71 1 73 r 1 77 1 88 183 1 88

50° 90, 0 20° 30 0 500 90o

S 14 5 90 nE 4 1244 3 38 14 12 s 29 7 94 81 3 c 12 11 95 1 43 I1 17 1 69 1 65 r 1 92 1 90 18 3 1 82

Cs O 1 2

1 n' 876 :2. 20 f 8 32 9 57 r' 1 93 3r 2 08 2 08

qb 00 2° 5° lo, 0.0 2°

C52 1, O

8E8.e2 8 72 8.28 759ss n 8 3el2 4 31 l 4.52 4.22 aC 8 75 1 79 1 116 11 64 ac . 9 30 937 9 11 P1 1.93 1 95 1 98 2 02 r 288 2 03 2 07 2 13

t 200 300 500 90° 0 20° 300° 500 90o

DE 5 68 5 85 6843 4.35 4 4 27 4 04 4 72 4.25 DC li 98 11 68 1809 18.76 'I 8 72 a 44 7 99 806 r 2 07 2 08 1.98 1.95 r 219 2 191 2 14 208

FIG. 6-Continued. estimated for C are within 5% of C's value 10° of opposition, the lightcurve amplitude of /3*for all solar phase angles. That is, the provides a reliable estimator for the elonga- salient characteristics of the mean cross tion of C, but overestimates the elongation section are determined fairly accurately, by several tens of percent for 4 as large as even when a realistic scattering law is as- 200, by factors of several at 4 = 500, and by sumed. factors of up to 10 at 4 = 900.It s~eemsclear Figure 8 also plots the ratio 100.4Am/,8*, that a value of /3* derived from C is a gauge where 10041m is the estimate of,8* provided of C's elongation that not only is more reli- by the lightcurve amplitude. Within about able and more physically meaningful than ASTEROID LIGHTCURVE INVERSION 45

C. C 14 0 013U 04 NC10 12 18 C, 12 96 12. 96 P- 2 36 2 36 r 2 47 2 47

0 00 2° 50 10, 0 00° 000'2° 50 1o 00

E88 02 7 97 702 6 77 [E S 03 0 32 0 85 2. 54 n 03 121 2 42 13 17 [I 12 95 13 22 13 19 12 51 r 2 36 2 40 246 2 52 or 2 48 2 53 2.60 2.65

0 200 300 500 90o 0 200 30° 500 90o

D [6 16 6 47 5.18 3.85 nE1 6.408 9 6 7.63 7.90 5.82 DC 12 33 11 36 18. 51 10 96 [c 992 9 la 1 as P- 2 57 2 51 2.39 2 36 r 2.63 2 56 244 2.44

Cs C 1 5

0 0 A

0C9 89 9.89 r 2.58 250

0 0° 20 50 100

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

0 20° 300 500 900

DE 2. 418 3. 44 3.98 2.52 OIci 10 12 9-53 10.30 2.79 276 2.61 2.59

FIG. 6-Continued.

Am, but also compensates for the so-called properties at each point. Our simulations phase-magnitude relation. have used a realistic approach to scattering We have not tried to model lightcurves properties but have ignored components of for three-dimensional objects for which curvature that are not parallel to the equa- C(z) is not constant. For any arbitrary tor. Hence our results are generally valid shape, C can be thought of as the sum of only to the extent that the curvature of sur- potentially distorted versions of the individ- face contours not parallel to the equator is ual C(z), where the distortion depends on irrelevant to distortion in C. In simpler lan- the local surface curvature and scattering guage, we have ignored effects of polar 46 OSTRO, CONNELLY, AND DOROGI

10 -- #1 tering. Partially because asteroidal values a a -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 systematic 4- #13 distortion in C caused by departure from 8- #14 80 i 0-#15 ideal conditions. Since actual lightcurves SOLAR PHASE ANGLE 0 (deg)

.eF #1 a b LL. 5* 4- #3 0 - #4 - #5 4 W -- #6 -J -0- exd. 0- #7 < 3. #8, #14, - 08 #15 << 2 - #9 W 2 4- all fifteen + #10 4- #11 4- just 1- #8, #14. - #12 Us 0- 1 #15 - #13 so - #14 - #15 0 20 40 60 80 1 00 20

SOLAR PHASE ANGLE, 0 (deg) #1 b 4- #2 #3 FIG. 7. (a) Individual values of flE for the 15 model #4 #5 asteroids in the simulations described in the text, and 4-0 -a- #6 (b) average values of fIE for three subsets of the model 2E10 #7 asteroids. #8 #9 0 -6. #10 '0- #11 #12 darkening on the functional form of light- 4- #13 n z. . #14 curves. Nevertheless, we suggest that our 20 40 60 80 1 00 #15 simulations with constant-C(z) shapes may apply to physically realistic convex shapes if Condition EVIG is satisfied, because the w _J weighting of the C(z) is such [Ostro and .00 4A~ P Connelly 1984, Eq. (7) and Appendix A] -*-I/ P. that contributions to the mean cross section CD from equatorial C(z) can easily dominate cc those from polar regions. This situation will be even more pronounced for a Hapke scat- 0 20 40 60 80 terer than for a geometric scatterer. SOLAR PHASE ANGLED, (deg)

C. Violation of Condition CONVEX FIG. 8. Estimated values of breadth ratio /3* and noncircularity ftc for the simulations described in the We have already dealt with this source of text, normalized to the model asteroids' true values. systematic error in a de facto sense, be- (a) and (b) show individual values for the 15 objects, cause Hapke's theory includes effects of while (c) plots average values. f'c and ,8* correspond roughness at scales much larger than the to C. O04A1- is the ratio of the asteroid's maximum observing wavelength. In other words, bias brightness to its minimum brightness, i.e., an estimate of ,1* derived from the lightcurve amplitude Ain. Indi- in Cdue to concavities is indistinguishable, vidual values of /3*/f3*, not shown here, are within 5% either a priori or a posteriori, from bias due of unity for all 15 model asteroids, regardless of solar to nongeometric (or even nonuniform) scat- phase angle. ASTEROID LIGHTCURVE INVERSION 47 are contaminated with measurement errors, trade-offs between noise level and sampling any estimate of C or Cs will also contain rate, and the dependence of flE on solar some statistical error. Whereas actual light- phase angle. Tripling the noise level gener- curve measurement errors may arise from ally increases the error distance more dra- systematic sources (e.g., imperfect removal matically than does tripling the sampling in- of sky background, intermittent cirrus, terval, but there is some coupling between etc.), as well as from photon-counting sto- a- and AO, with the negative effect of in- chastic errors, one can always compute an creasing AOfelt more severely at the high "effective standard deviation" in the over- noise level than at the low noise level. all noise due to all sources. Distortion The accuracy of C increases with in- caused by any given noise level will increasing solar phase angle because noise crease as the rotational-phase sampling in- "masquerading" as odd-harmonic energy terval increases. in a low-+ lightcurve gets "blown up" via We have investigated the effects of noise Eq. (1) into very strong odd harmonics in and sampling rate for five of the model as- C. It should not surprise us that reliable teroids (objects No. 1, 4, 8, 11, and 15) used estimation of C's odd harmonics is intrinsi- in Section IV.B. As shown in Fig. 4, those cally difficult at low solar phase angles; re- five shapes span a large region in -fQc vs 1* call that under the ideal conditions, the space." Here, to focus on statistical manifestation of C's odd harmonics as odd sources of error, we assume that all ideal harmonics in the lightcurve is nil at opposi- conditions are satisfied. Propagation of sta- tion and grows slowly with |4|. (See Fig. 3 tistical error has been modeled by generat- and Section III.) Note that overestimation ing a lightcurve for a given model asteroid, of C's noncircularity appears correlated using a random number generator to pro- with the error distance for the more sym- duce Gaussian noise samples, scaling the metric shapes, but that the noncircularity is standard deviation to some fraction a-of the fairly immune to noise for the shapes (ob- asteroid's mean brightness, adding the jects No. 4 and 11) containing healthy odd scaled noise to the lightcurve, and then esti- harmonics. mating C or Cs. The entire procedure is Estimation of the symmetrized mean repeated to build up an ensemble of five cross section (i.e., of C's even harmonics) profiles, each containing error introduced is accurate even for the higher noise level by a different realization of the same noise- and coarser sampling interval. This situa- generating random process. We repeated tion reflects the intrinsic accessibility of this exercise for two noise levels (o- = 1 and C's even harmonics, regardless of 4. Con- 3%), two sampling intervals (AO 30 and sequently, 13*,whose value depends only 90), and four solar phase angles (¢ = 00, 50, on C's even harmonics, is estimated quite 10°, and 200), for each of the five model reliably for every one of our simulations. asteroids. Our choices of noise level and Since systematic sources of error in C sampling interval span the values corre- can be severe, it is important that statistical sponding to many published asteroid light- sources of error be minimized for lightcurves and let one gauge the severity of sta- curves acquired to constrain asteroid tistical error sources in the inversions to be shapes. A conservative rule of thumb might presented in Section V. be to try to keep o- at or below 1%. At that Figure 9 superposes the five estimates of level, statistical sources of error in C seem C for each choice of the three parameters small compared to distortion introduced by (a-, AO, X) and also lists the five-profile violation of Conditions EVIG and/or GEO. mean values of the error distance flE, the (In the following section, we apply convex- noncircularity Qc, and the breadth ratio ,8*, profile inversion to lightcurves whose noise using the same format as in Fig. 6. Note the levels range from 0.6 to 4% and average 48 OSTRO, CONNELLY, AND DOROGI

Cs C1

1e123 2.00 I 1 07 1 07 560° 5° lo, 20°' 0° 5° lo, 20' a=1% ~00 000 C 0~00

ls 1.46 5 36 3 01 1 26 lb 2 96 6 07 4 55 2 23 nc 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 a=3%

lb 4 27 6 94 5 66 3 60 Q 4 90 7 32 6 63 4 87 nc 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 =303 A= 90

C 4

lb 4 92 7 99 ¢ 1 26 1 26

5 °10100 20' 0° 5° 10 ° 200

0=1%

1 5E 174 5 78 2 37 1.09 0k 2 48 6 72 4 11 1 87 c 4 74 7 76 7 37 7 71 I 4 95 7 56 6 3 7.47 1 26 1 26 1 25 1 25 ' 211 2 6 1 25 1 25

0° 5° 100 200 10lo 0O 200

a= 3%

f4 32 6 82 5 88 2 79 a, 4 74 8 00 733 4 57 n1 5 45 0 35 7 66 7 37 Sc 6 61 8 59 7 68 7 23 P- 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 as Fig. 6. Each box shows results of simulations for a particular model. The model's mean cross section and symmetrized mean cross section are shown above the horizontal bar. Below the bar are estimates of those quantities from inversion of noisy lightcurves generated under ideal conditions at solar phase angle k, noise level o-, and rotational-phase sampling interval AO. ((, is estimated at opposition and C is estimated at the three nonzero values of q>.)Five cross section estimates, corresponding to five different 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. ASTEROID LIGHTCURVE INVERSION 49

c8C

0, 4 59 4 58 1 54 15

0. 5° 10° 20° 0° 5° lo, 200 a= 1%

11E 1 93 6 84 3.26 1 47 DA, 3 e4 6 62 4 73 2 52 c 4 98 826 565. 4 81 fl, 558 855 8659 5.24 154 1 54 1054 1254 55 51 1 55

0 0000 20 00 20 a =3%

, 4. 46 a 18 6 79 4 5 556s s 5E 7 44 7 09 s 55 fl 6.43 9 48 8 22 612 4 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 28 1 93 1 93

!0 5° lo, 20' 9 a 5o l, 200

a= 1%

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 94 p1 93 1 93 1 92 1 92 1 94 1 93 193 1 92

5° 100lo 20o 50 lo10 200

SE 2. 59 5 28 3~ 38 It 94 11E 3 73 6~ 46 5~ 19 3~ 31 Se 8 88 11, 38 11 38 11 82 De 9 23 18 79 11 25 11 65 c 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 for I distance unit or less. The bias in an esti- such noise levels, statistical error in an estimate of ,B*will usually be negligible, but the mate of the breadth ratio will not exceed a bias in an estimate of C's fC obtained at few percent of 83*,while that in an estimate low solar phase angle can be several dis- of the noncircularity will typically be about tance units at the higher noise levels.) 50 OSTRO, CONNELLY, AND DOROGI

C 15

11 9 89 9 89 2 50 250 5 °lo100 200 0o 50 0o, 200 a = 1%

n I 55 537 2 46 1 36 6O 2 00 6 64 365 1 93 110 I8 Be 11 33 18. 21 18 81 ~ 0 9. 92 1 1 79 18 39 9 98 O- 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 .-3%

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

AO= 3° AO 90

FIG. 9-Continued.

The sampling interval is less critical, es- this section includes values for several pecially at low noise levels and large solar goodness-of-fit statistics. If the ideal condi- phase angles. However, simulations not tions were satisfied perfectly, then the (un- presented here indicate that gaps in rota- constrained) Fourier model 5 for the light- tional-phase coverage as large as a few tens curve data y would be the same as the of degrees can increase [IE by several tens (constrained) Fourier model § correspond- of percent. For this reason, it seems pru- ing to C. (See Section III.) Ostro and Con- dent to interpret the parameter AO in the nelly (1984) describe a test of the null hy- Fig. 9 simulations as the minimum interval pothesis, Ho, that the vector of Fourier between any two data points. coefficients for § is a statistically acceptable estimate of the Fourier vector for the "true" lightcurve. This "variance ratio" VI. APPLICATIONS test compares an estimate of c-2 obtained In this section, we present results of ap- from the residuals between the data and y plying convex-profile inversion to selected to an estimate obtained from the residuals asteroids, beginning with lightcurves for between the data and 5. Let M be the num- which prior knowledge of the asteroid's ber of Fourier harmonics and L the number pole direction indicates viewing-illumina- of data points, and let us assume that the tion geometry not far from equatorial. Note errors in y are independent, identically dis- that for all lightcurves shown below, we tributed, zero-mean, Gaussian random plot the brightness in linear units scaled so variables. Then the following goodness-of- that the average brightness is near unity; fit statistic, w1, is distributed as accordingly, we express Am in linear units under the null hypothesis (Plackett 1960, instead of magnitudes. p. 52 ff.): To help guide interpretation of our results, each of the figures referred to in w, = (R - OWV2 ASTEROID LIGHTCURVE INVERSION 51 where fied perfectly and the lightcurves are not noiseless, but the similarity of the profiles - §)(y - y)T(y - R = (y - §)T(y obtained from the two lightcurves lends v, = 2M + 1 confidence to their validity and is consistent with the net effect of the violations of V2 = L-vj . the ideal conditions being fairly small. Also note that the fractional variation in the esti- We can reject Ho at the 100(1 - a)% confi- mates of 13*(which pertains to the aster- dence level whenever w1 > FV1 V2.^AFor ex- oid's mean cross section) is half that in the ample, Ho can be rejected at the 99% confi- estimates of Am (which pertains to the dence level if w, > F, 1, 2,001 . For all lightcurves); see the discussion in Section inversions in this paper, M = 10, L > 45, IV.B. and F 1,,12 ,0. 01 < 3. The fact that the values of w, and w2 ex- In the figures below, we give up to three ceed 3 is interpretable as very strong evi- goodness-of-fit statistics for each inversion. dence for violation of ideal conditions. The first, wI, tests Ho for y, as just de- However, caution is advised even when scribed; if w, exceeds 3, we can safely re- those statistics are near unity, since viola- ject Ho. The second, w2, tests the same hy- tions of ideal conditions might conspire to pothesis, but for the lightcurve model Y's produce a lightcurve yielding a good match corresponding to The third, W , tests the C,. 3 between § and the lightcurve data, but a C same hypothesis as w2, but assumes that the that nevertheless is inaccurate. By the correct lightcurve model is y, instead of y as same token, it is conceivable that a mean for w, and w2. If the third w statistic is >3, cross section estimate providing a poor fit then ywecan safely reject the hypothesis to the data might be physically valid. Given that C, provides as good a fit to the light- these various possibilities, it clearly is de- curve data as does C. Whenever we esti- sirable to estimate an asteroid's mean cross mate C, we also give the "symmetrization section at a variety of solar phase angles. distance" fQsbetween C and C,. Figure 10 also shows the weighted average of the two estimates of Eunomia's A. Objects with Published Pole Directions mean cross section, calculated by summing 15 Eunomiac. Figure 10 shows C (solid the statistically weighted vector of Fourier profile) and C, (dotted profile) estimated coefficients for the radius-of-curvature from lightcurves at solar phase angles equal functions for the individual profiles, with to -12° and -21°. The signs of / assume the profiles' relative rotational phase cho- retrograde rotation, as deduced by authors sen to maximize the cross correlation of the cited in the figure caption. Using Magnus- radius-of-curvature functions. This exam- son's (1986)values for the ecliptic longitude ple demonstrates how convex-profile inver- (106° + 50) and latitude (-730 + 10°) of sion can combine the shape information Eunomia's north pole, we calculate (OE,5S) contained in any number of independent equal to (19°, 180) and (80, -2°) for the two lightcurves, perhaps taken at a variety of Eunomia curves. solar phase angles and/or during different The figure shows that the values of the apparitions. elongation /3*and the noncircularity fQcfor 19 Fortuna. Figure II shows C for two the two estimates of C vary by less than lightcurves obtained by Weidenschilling et 10%. If the ideal conditions held perfectly al. (1987) for 19 Fortuna, and their and if the lightcurves were free of noise, weighted average. This asteroid's light- then C and those two shape descriptors curves have peak-to-valley amplitudes would be independent of solar phase angle. close to 0.25 mag, independent of ecliptic The ideal conditions certainly are not satis- longitude, suggesting that the asteroid-cen- 52 OSTRO, CONNELLY, AND DOROGI

15 EUNOMIII 5 - 120

. s Am 1 49 1 41 QC 8 40 w 9 4 22 15

E s Qs 7 46

AVERAGE

9* 1 43 QC 8 16

Qs 7 15

15 EUNOMIA -210 IY Am 1 62 1 48 I-c 7 65 w 11 33 26 Qs 6 26 180 ROTATIONAL PHASE

FIG. 10. Estimates of C (the solid profile) and C. (the dotted profile) for lightcurves obtained for asteroid 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 are plotted on a linear scale, with the average brightness set to unity. In the lightcurve plots, the large symbols are the data, the solid lightcurve 9 was derived from the unconstrained vector of Fourier coefficients 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 breadth ratio (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 us test hypotheses described in the text. Briefly, since the first statistic (wl) exceeds 3, we can safely state that the lightcurve corresponding to the mean cross section estimate is not a good fit to the data. Since the second statistic (W2) exceeds 3, we can make the same statement about the symmetrized mean cross section. Since the third statistic (W3) exceeds 3, we can state that the difference between the lightcurves corresponding to C and C. is significant, i.e., that thze odd harmonics in y are statistically significant. (In some of the subsequent figures we just estimate C., and in those we do not give values for f1s, wl, or W3.) In the middle of this figure are the weighted average of the two individual estimates of C, its symmetrization, and relevant shape descriptors. Note the distances between the cross section estimates. 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- ASTEROID LIGHTCURVE INVERSION 53

I 19 FORTUNA 120 - Am 27 23 (Qc 8 06 In W 6 1 10 2 3 Qs 7 05

AVERAGE

p. 1 22 5.59 (IC 6 73

5 81

3.47

19 FORTUNA

190

I Am 1 28 10 - p. I - .. Il 1 22 a, =T C - - (Ic 5 72 W 0 5 1 2 1 7

I IB I 36 Qs 4 49 0 180 3PA ROTATIONAL PHASE

FIG. 11. Similar to Fig. 10, but for lightcurves of asteroid 19 Fortuna obtained by Weidenschilling et al. (1987). teroids. For each object, Magnusson's dence to their mutual validity. The poor (1986) pole position estimates are consis- goodness of fit for the inversion of the op- tent with the asteroid-centered declinations position lightcurve is disconcerting, but of the Sun and Earth being within a few note that the residuals are spread out fairly tens of degrees of zero during the observa- uniformly over 360° of rotational phase. tions. For Antigone, the two estimates of Cs are If an estimate of C from a nonopposition 6.20 distance units apart-they are defi- lightcurve were accurate, then its symme- nitely different-and the residuals are se- trization would provide an estimate of C. vere, so the cross section estimates are that would also be accurate and hence likely to contain significant systematic er- would resemble an estimate of C. from an rors. The presence of a strong first har- opposition lightcurve. We expect C to be monic in the lightcurve taken only 70 from more reliably estimated from an opposition opposition and, if Magnusson's pole direc- lightcurve because the scattering is likely to tion is correct, at (8 E, 5s) equal to (17°, 12°) be closer to geometric at small (P. is interpretable as evidence for a nonuni- For Massalia, the two estimates of CSare form albedo distribution. nearly identical (fl = 2.33), lending confi- 624 Hektor. Figure 14 shows Cs esti- 54 OSTRO, CONNELLY, AND DOROGI

N 20 MASSALIA

130 Am 1 16 1 14 9* w 5 85 W 2 8 4 9 3 4 nS 5 03 180 360 ROTATIONAL PHASE

S -

j 20 MASSALIA

m 0 0 2° Am 1 17 1 15 0, 2 82 w 13

180 360 ROTATIONAL PHASE

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

129 ANTIGONE I 0 -160 Am 1 34 1 23 4 03 W 18 20 1 7 Q5 2 90 IB1 ROTATIONAL PHASE

6.20

129 ANTIGONE

Am 1 31 IQ- CS 1 26 5 03 W 4 3

08 180 360 ROTATIONAL 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. ASTEROID LIGHTCURVE INVERSION 55

624 HEKTOR a- 40

- - Am 2 52 a: 2 45 dc 12 64 w 1 4 0

ISe 36e ROTATIONAL PHASE

44 NYSA

el Q- I Am 1 44 1 41 8 06 w 2 7 4 2 3 1 1 . Qs 6 38 0 18e 36e ROTATIONAL PHASE

Q -s I

Ir 1685 TORO 330

ZU Am 2 02 I 67 Ia 5 92 9* 10 21 0. w 17? 2 14 18 3GR ROTATIONAL PHASE

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 symmetrized mean cross section from a lightcurve obtained by di Martino et al. (1987). Bottom: Estimates of C and C, for 1685 Toro, from inversion of a lightcurve obtained by Dunlap et al. (1973). See text and Fig. 10 caption. mated for 624 Hektor from a lightcurve ob- of C Note that the profile has ,3* = 2.5 and tained at 0 = 4° by Dunlap and Gehrels is distinctly nonelliptical. Since the mean (1969). The pole directions estimated for cross section of an ellipsoid rotating about a this Trojan asteroid by those authors (and principal axis is an ellipse, our results sug- more recently by ZappalA and Knezevi6 gest that neither the asteroid nor its convex 1984, Pospieszalska-Surdej and Surdej hull are ellipsoids. On the other hand, our C 1985, Magnusson 1986)indicate that 6 E and is quite consistent with many other models 8s have absolute values less than 10°. The for Hektor's three-dimensional shape, in- constancy of Hektor's color indices with cluding a cylinder with rounded ends rotational phase (Dunlap and Gehrels (Dunlap and Gehrels 1969), a dumbbell 1969), the absence of odd harmonics in the (Hartmann and Cruikshank 1978),and vari- lightcurve, and the low value of W2 concur ous binary configurations (Weidenschilling in supporting the reliability of our estimate 1980). Of course, in making these state- 56 OSTRO, CONNELLY, AND DOROGI ments, we ignore effects of possible small harmonic structure of a lightcurve and that violations of the ideal conditions. of a mean-cross-section estimate. Unless 44 Nysa. As noted by di Martino et al. otherwise noted, all estimates of C assume (1987) and Magnusson (1986), this aster- that 4)'s sign is positive; the profile obtained oid's pole is relatively well known. The in- for a negative sign generally is similar, with verted lightcurve (middle row of Fig. 14), nearly the same /3* and fic as the profile from di Martino et al. (1987), was taken at a shown. solar phase angle of 8.50, under nearly 36 Atalante. As di Martino et al. (1987) equatorial viewing/illumination geometry. point out, even though this lightcurve's am- The pole estimate yields 5E = 90 and 5s = plitude is similar to that in lightcurves ob- 30, so the longitudinal separation of the sub- tained 700 of longitude away, the viewing/ Earth and sub-Sun points is only about 50, illumination geometry might not neces- and our "leverage" in estimating C's odd sarily be equatorial. The fit is awful (w, is harmonics is marginal at best. On the other huge), with strong, positive residuals for hand, our estimate of Cs probably is reli- half a cycle but strong negative residuals able. It is interesting that our estimate, for the other half. This situation is consis- /3* = 1.41, for the elongation of Nysa's tent with a hemispheric albedo asymmetry, mean cross section is within several per- so perhaps this object should be targeted cent of values derived for a model ellip- for polarimetry or multicolor photometry. soid's equatorial axis ratio from analysis 60 Echo. The fit of the model lightcurve of a multi-apparition lightcurve data set corresponding to C, to Zeigler and Flor- (Pospieszalska-Surdej and Surdej 1985, ence's (1985) data is okay, with no major Magnusson 1986). residuals except at the primary maximum. 1685 Toro. We assumed the pole direc- 61 Danae. To the best of our knowledge, tion (200°, 550) quoted by Dunlap et al. this object has not been observed photo- (1973) in inverting their July 1972 lightcurve metrically since Wood and Kuiper (1963) data (bottom row of Fig. 14), and the nega- acquired this lightcurve. At such a small tive sign of 4 corresponds to direct rota- solar phase angle a mean cross section esti- tion. The fit is excellent. Note that C is mate is probably unreliable, and we show C much less elongated than one might infer primarily as an example of a shape with a from the lightcurve's large amplitude. Also large noncircularity. note that whereas fQ = 2, so Ce is not far 69 Hesperia. The residuals are large, sug- from C, w2 and W3 are each much greater gesting violation of ConditionAEVIG and/or than 3. That is, the lightcurve's odd har- Condition GEO. Compare Cs to Danae's, monics are prominent while C's are not. which has similar values of the shape de- There is no discrepancy here; odd harmon- scriptors /* and Qc. ics in shape generate increasingly large odd 118 Peitho. This Stanzel and Schober harmonics in a lightcurve as the solar phase (1980) opposition lightcurve has an unusu- angle increases. [See (I) and Fig. 3.] ally strong fourth harmonic, which mani- fests itself in the rectangular appearance of B. Objects with Unknown Pole Directions Cs. The large residuals could be due to al- Figure 15 shows C and/or Ct for objects bedo heterogeneities and/or to non-equato- for which pole-direction estimates are lack- rial viewing/illumination geometry. ing. We present these profiles because the 164 Eva. The large value of w, is a bit goodness-of-fit statistics are small, or in misleading here. The number of points plot- some cases because the profiles are inter- ted in the Schober (1982) lightcurve is huge, esting, or simply to illustrate the mapping and our digitization of it yielded more than from lightcurve space to profile space, e.g., 700. With this many data points the F-ratio the relation [Eq. (1) and Fig. 3] between the test becomes extremely sensitive. The ASTEROID LIGHTCURVE INVERSION 57

- 36 ATALtNTE 0 140 Am 1 18 1 11 p. - - 8 50 w 57 75 6 0 (Is 7 93 0 6TA 360 RO TATI ONAL PHASE

-ED 60 ECHO 0 60 Am 1 20 A - -Cs I 1 16 (I 2 52 S w 2 8 m C m. 0 186 36 ;D ROTATIONAL PHASE

An< 61 DANCE D W 0 Am 1 30 SC p. 1 27 8 11 W 2 5 3 1 1 0 2 m nCS 6 13 e 1I60 360 ROTATIONAL PHASE

FIG. 15. Estimates of C and/or C. for asteroids lacking published estimates of pole direction. See text and Fig. 10 caption. Sources of the lightcurves are 36 Atalante (di Martino et al. 1987, Fig. 4), 60 Echo (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), 704 Interamnia (Lustig and Hahn 1976, Fig. 11), 739 Mandeville (ZappalA et al. 1983, Fig. 28), 1173 Anchises (French 1987), 1627Ivar (Hahn et al. 1987), 2156 Kate (Binzel and Mulholland 1983, Fig. 18).

model lightcurve y corresponding to our ference between y and 5. Contrast with the mean cross section estimate actually is very situation for 164 Eva. slightly lower than the unconstrained model 270 Anahita. Differences between the lightcurve 5 at each of the extrema except shapes of the two peaks (i.e., odd harmon- the deepest minimum, where it is slightly ics in the lightcurve) cause the large value higher than 5. of W2. 246 Asporina. The Harris and Young 434 Hungaria. Despite the large residuals (1983) lightcurve has only 65 points, so the over rotational phases from 2500 to 3600, F-ratio test is not very sensitive to the dif- this inversion demonstrates that, at large 58 OSTRO, CONNELLY, AND DOROGI

69 HESPERIA 0 100 Am 1 25 C d- 1 19 UN - 0, 7 89 . I I I w 4 0 6 6 4 6 Os 6 94 ; 0 180 360 ROTATIONAL PHASE

118 PEITHO

0 40 Am 1 34 a* 1 29 M. (Ql 6 91 w 14 = 0

N 8 108 368 ROTATIONAL PHASE

164 EVA

I' 260 ~I I 0 Am 1 40 C 1 28 Qc 5 64 w 22 306 474

I N 4 58 : 0 180 360 ROTATIONAL PHASE

FIG. 15-Continued. solar phase angles, the breadth ratio /3*is depths: The three pairs of extrema arise be- superior to Am as an elongation measure. cause C resembles an equilateral triangle, 505 Cava. The lightcurve from Harris while differences between the minima arise and Young (1985) is composited from data because the triangle's vertices have differ- taken at phase angles ranging from 100 to ent "sharpnesses." 25°, and we use an average value of 150 for 739 Mandeville. The slightly different the inversion. The fit of to the data is good shapes of the minima in this Zappala et al. except near 4 2700; readers might care to (1983) fightcurve are the source of the odd speculate about the source of this "fea- harmonics in C, which provides a decent fit ture." to the data and, in view of the values of fis 704 Interamnia. The fit of y to the light- and W3, a significantly better one than does curve obtained by Lustig and Hahn (1976) Cs. is not good, but the three pairs of extrema 1173 Anchises. This opposition light- are replicated, and we show this inversion curve obtained by French (1987) yields an because C offers a simple explanation for estimate of Cs which is notable for its elon- the existence of three minima with different gation (/3* = 1.6) and its interesting shape, ASTEROID LIGHTCURVE INVERSION 59

L - 246 ASPORINA a I 0 6° Am 1 18 I 91* 1 16 oc 3 08 en w 1 1 I I mI

270 ANAHITA I - 0 (S Am 1 32 Cs 1 30 4 94 w 9 4 - m

S e 180 36 ROTATIONAL PHASE

434 HUNGARIA 0 27° C Am 2 02 ,0 X p. 1 62 8 16 E m w 10 16 5 2 4 81 m 0 180 360 ROTATIONAL PHASE

FIG. 15-Continued. which has a weak sixth harmonic. The that the aspect during September was lightcurve has a significant first harmonic closer to equatorial than during the closest (note the poor fit of y to the lightcurve max- approach to Earth in July, but that Ivar is ima), which could arise from a nonequato- actually more elongated than one would in- rial view of a nonaxisymmetric shape or, if fer from any of the lightcurve data or from the shape is axisymmetric and the aspect is our estimate of C. equatorial, from a heterogeneous albedo 2156 Kate. Our estimate of this small distribution. main-belt asteroid's mean cross section is 1627 Ivar. This lightcurve from Hahn et elongated and highly noncircular. The al. (1987) is a composite of data taken dur- worst residuals are confined to the minima. ing September 1985 at solar phase angles between 180 and 330; a value of 250 was VIL. CONCLUSIONS used in the inversion. Radar data (Ostro et This paper has been an attempt to formu- al. 1986) and the progression of lightcurve late and apply a comprehensive framework amplitude during the 1985 apparition (A. for using a lightcurve to constrain an aster- W. Harris, private communication) suggest oid's shape. We have seen that convex-pro- 60 OSTRO, CONNELLY, AND DOROGI

- I 505 CAVA

I 150 Am 1 24

T~ 1 19 - --- C, Oc 6 16 w 3 5 4 8 2 4 m I, I Qs 4 0' 0 199 360 ROTATIONAL PHASE

N - 704 INTERAMNIA 120 Am 1 10 91~ 1 07 .ll zIn m Oc 6 41 w 20 42 17 Qs 6 15 X 0 99 360 ROTATIONAL PHASE

= " 739 MANDEVILLE I 10° 1 13 I . Am \C 1 11 oc 5 84 w 2 8 7 4 10 I; m Cs 5 45 9 .A80 360 ROTATIONAL PHASE

FIG. 15-Continued. file inversion builds on the canonical work ever, it is encouraging that our simulations of Russell by (i) identifying the mean cross suggest that salient characteristics of C re- section C as the optimal shape constraint semble those of C rather closely for equato- available from a lightcurve, (ii) defining the rial lightcurves obtained at solar phase an- differences between the potential informa- gles near 20°, at least for the simplistic tion content of opposition lightcurves and shapes we examined. When the viewing/il- that of nonopposition lightcurves, and (iii) lumination geometry is as much as a few specifying the conditions that determine the tens of degrees from equatorial, the validity accessibility of that information. of estimates of C or Cs should be consid- The methodology for estimating C and Cs ered suspect, and simulations like those in was introduced by Ostro and Connelly Section IV.A can be used to gauge the se- (1984), and here we have tried to assess verity of distortion. (On the other hand, principal sources of systematic and statisti- even when C is apparently biased, it re- cal error. Perhaps the most severe obstacle mains a visual representation of the light- to estimating the mean cross section stems curve in a language befitting the light- from violation of Condition GEO. How- curve's intrinsic information content.) ASTEROID LIGHTCURVE INVERSION 61

- 1173 ANCHISES

Am 1 64 I- - 1 59 Oc 9 34 w 0 2

0 t80 360 ROTATIONAL PHASE

1627 I VAR 25°

T - Am 1 72 -i - C I. 1 51 5 34 I) w 2 0 3 1 2 6 Qs 2 28 0 180 360 ROTATIONAL PHASE

tI ,s 2156 KATE 6O Am 1 75 w* 1 67 7 69 to w, 7 5 C t 180 360 ROTATIONAL PHASE

FIG. 15-Continued.

We cannot hope to escape the reality that example, one could apply convex-profile a lightcurve is determined by the viewing/ inversion to analysis of an opposition light- illumination geometry and the surface's curve to constrain the average longitudinal light-scattering behavior as well as by the dependence of an asteroid's albedo distri- three-dimensional shape. Hopefully, we bution, as follows. If Condition EVIG is have demonstrated the value of convex- satisfied close to opposition, where Condi- profile inversion as a tool for assessing the tion GEO is most likely to be satisfied, we relative roles of each of these factors. can use Russell's Fourier tests to seek evi- Since shape effects are inextricably in- dence for a nonuniform albedo distribution. volved in the interpretation of any disk- If the opposition lightcurve has a first har- integrated observations, including those monic but no higher odd harmonics, we meant to probe physical properties other could estimate Cs with confidence and at- than body shape, we conjecture that contribute residuals between the lightcurve and vex-profile inversion will prove useful in Cs's breadth function to albedo variegation. "removing" shape effects in studies de- Then one might try to account for those signed to constrain those properties. For residuals in terms of a nonuniform, two-di- 62 OSTRO, CONNELLY, AND DOROGI mensional model asteroid, i.e., by varying inversion is not readily applicable to data the albedos of each side of the polygonal sets consisting of many lightcurves taken estimate of Csobtained from convex-profile under a variety of (nonequatorial) viewing/ inversion. This exercise would collapse illumination geometries. To accommodate both the three-dimensional shape and the this most common kind of lightcurve data albedo distribution on the three-dimen- set, one must parameterize aspects of the sional surface into two-dimensional aver- asteroid's shape not described by the mean ages, and as such could help to convey the cross section. Satisfactory accomplishment range of possible configurations for the as- of this task could greatly increase the num- teroid. Of course, from R3 and R4, we can- ber of asteroids for which mean cross sec- not expect lightcurves to reveal which of tions can be accurately estimated and might the infinite number of admissible configura- enhance the value of multi-lightcurve con- tions is correct. Unique models of either straints on asteroid pole directions. shape or of the distribution of surface properties simply cannot be obtained from disk- ACKNOWLEDGMENTS integrated measurements. We thank B. Hapke and J. Goguen for discussions If a reliable estimate of an asteroid's of the light-scattering properties of particulate sur- mean cross section were available, one faces; A. Harris, M. Gaffey, J. Surdej, K. Lumme, could substitute C for Cs in the above exer- and P. Magnusson for critiques of an earlier version of this paper; and L. Belkora, D. Casey, D. Long, and A. cise. It might also be interesting to incorpo- Zaruba for technical assistance. Part of this research rate Hapke's scattering formalism into this was conducted at the Jet Propulsion Laboratory, Cali- "two-dimensional modeling," as in Section fornia Institute of Technology, under contract with the IV.B, to explore the range of possible val- National Aeronautics and Space Administration, and ues for Hapke parameters and perhaps to part (RC) was supported by NSF Grant MCS77-01740. model effects of surface heterogeneities. At this level of modeling, however, even two- REFERENCES dimensional simplifications cannot be ex- BARUCCI, M. A.. AND M. FULCHIGNONI 1982. The dependence of asteroid lightcurves on the orienta- pected to yield unique solutions. Still, these tion parameters and the shapes of asteroids. Moon approaches might help to separate shape Planets 27, 47-57. from compositional variations in analysis of BARUCCI,M. A., AND M. FULCHIGNONI1984. On the multispectral lightcurves, from regolith inversion of asteroidal lightcurve functions. 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