ICARUS 75, 30-63 (1988) Convex-Profile Inversion of Asteroid Lightcurves: Theory and Applications 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 interpreta- tion 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 aster- oid'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 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. 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 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 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 Current address: Philips ECG, 50 Johnston St., Seneca Falls, NY 13148. 30 0019-1035/88 $3.00 Copyrighte 1988by Academic Press, Inc. All nghts of reproduction in any form reserved. ASTEROID LIGHTCURVE INVERSION 3 1 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 in- verted 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 foun- dation 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 tech- niques 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 scatter- ing 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.