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| 1. | Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 |
| 2. | School of Electrical Engineering and Computer Science, Washington State University, Pullman, Washington 99164 |
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| 1. | Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 |
| 2. | School of Electrical Engineering and Computer Science, Washington State University, Pullman, Washington 99164 |
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Introduction Asteroid 433 Eros is the second largest member of the near-Earth population and also one of the most elongated. Only the Earth-crosser 1620 Geographos, roughly a tenth the size of Eros, is known to be more elongated (Ostro et al. 1995). Much of our current knowledge of Eros is based on extensive observations made when the asteroid passed within 0.15 AU of Earth in 1975 (Zellner 1975). Initial results from those observations were reported in a special issue of Icarus (Vol. 28, 1976). Eros' visual lightcurves have amplitudes as large as 1.5 magnitudes, with two distinct maxima and two distinct minima, which occur in the order: Max 1, Min 1, Max 2, Min 2, in the convention of Millis et al. (1976). During the 1975 apparition, Max 1 was the brighter, or primary, maximum for solar phase angles less than about 40 degrees. Several lightcurve analyses have yielded estimates of Eros' pole direction and elongation in the context of axisymmetric models (Magnusson 1989 and references therein). Eros' elongation is also evident in 3.5-cm Goldstone radar echoes (Jurgens and Goldstein 1976); as the asteroid rotates, the echo's Doppler bandwidth varies by about a factor of two. A consensus model ellipsoid (Zellner 1976), based primarily on the 1975 lightcurve and radar observations, has dimensions of 36 x 15 x 13 km and a pole direction within a few degrees of 16 degrees ecliptic longitude and +11 degrees latitude. The 1976 consensus ellipsoid provides a reasonable first approximation to Eros' shape, but it is known to be simplistic. Jurgens and Goldstein (1976; hereafter JG76) noted that asymmetries in the echo's frequency distribution at minimum-bandwidth (end-on) rotation phases indicated "a difference in surface reflectivity or surface slopes on opposite sides of the asteroid." A non-axisymmetric shape is also evident from the unequal maxima and unequal minima in Eros' lightcurves.
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Constraints from the Echo Spectral Shape The convex hull estimator considered the rotation phase dependence of the spectral edge frequencies but ignored the spectral shape and therefore ~90% of the available data. Hudson (1993) developed a technique that uses the evolution of the echo's delay/Doppler distribution as the target rotates and moves across the sky to estimate its three-dimensional shape, spin state, and radar scattering properties. This technique has been used to reconstruct the shapes of 4769 Castalia and 4179 Toutatis (Hudson and Ostro 1994, 1995). The Eros radar data obtained in 1975 are not resolved in time delay and therefore depend in a coupled way on size, spin vector, and scattering law. However, armed with external constraints on the asteroid's size and spin vector, we can use the rotational evolution of the spectra to place constraints on Eros' shape. Ephemeris Correction We adopt the 1975 site ephemeris used by Jurgens and Goldstein with the constant Doppler correction determined by ORJ90 from their convex hull analysis. (The variation of the Doppler prediction error during the observations was expected to be much less than the data's 2.73-Hz raw frequency resolution.) We assume that the model's spin axis coincides exactly with the data's (corrected) zero Doppler frequency. Data Filtering We smoothed the spectra by 20 degrees in phase and 30 Hz in frequency, which is the optimal phase/frequency filter for the convex hull analysis. This choice of data filtering effectively limits the model's maximum spatial resolution. Given the data's SNR and our model results (below), it is unlikely that we have "filtered out" any significant shape information contained in the data. The phase smoothing averages data from several days during which Eros' sky coordinates changed by about 4 degrees. This motion results in a synodic spin period of 5.2689 hours (Dunlap 1976) but does not affect the sub-radar latitude. Each of the 18 smoothed spectra is multiplied by a factor to permit the recalibration of its brightness scale. Thus, only the shapes of the spectra are used to constrain the model. Definition of a Reference Ellipsoid
Zero rotation phase is defined to occur in the observatory-based frame on 1975 January 23.34758, an epoch when the convex hull's minimum dimension appeared from the Earth to be normal to the line of sight with the longest (flat) side on the receding limb (ORJ90). (Correcting for light travel time, this orientation occurs about 76 sec earlier in the asteroid's frame.) Since this zero-phase epoch is based on the convex hull analysis, it need not correspond precisely to a time when the ellipsoid's x-axis is along the line of sight. Therefore, we define the parameter phi to be the angle between the x-axis and the plane defined by the z-axis and the line of sight at the zero-phase epoch. We adopt an empirical radar scattering law given by: 
where theta is the angle between the local surface normal and the line of sight, rho is the normal reflectivity, and n is a measure of specularity (see Mitchell et al. 1996). We assume Lambertian scattering (n = 2), which is slightly more diffuse than scattering laws estimated for Castalia (n = 2.8) and Toutatis (n = 2.3) (Hudson and Ostro 1994, 1995), which have circular polarization ratios similar to that of Eros (~0.3; JG76). Since the sub-radar latitude during the 1975 observations was within 10 degrees of the equator, and since the spectra are resolved only in Doppler frequency, there are correlations between the axis dimensions and the scattering law parameters, as noted by JG76. Consequently, we adopt long and intermediate dimensions for the ellipsoid from the convex hull (2a = 34.7; 2b = 17.4 km). With these assumed equatorial dimensions and a Lambertian scattering law, we search for the ellipsoid's polar dimension (2c), the angle phi, and the 18 brightness calibration factors. Using a least-squares approach, we find 2c = 14 km and phi = 6 degrees. The brightness calibration factors are within ~30% of unity. This model constitutes the reference ellipsoid.
In the next two sections, we explore two different approaches to extracting shape information from the Goldstone radar data. A nearly equatorial view, modest SNR, and the lack of delay-resolved echoes preclude the determination of a unique, three-dimensional shape; however, our ability to resolve the echoes in frequency and rotation phase provide sufficient geometric leverage to search for shape attributes "inside" the convex hull of ORJ90. Is there any significance to the fact that the hull's centers of figure and rotation are not aligned? What causes the spectral asymmetry near rotation phases of 180 degrees? |
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Method 1: Perturbations to the Reference Ellipsoid
A vertex of the perturbed shape is given by: B = B(e) + r*D(e) D = D(e) where B(e) and D(e) are a base point and a direction vector of the reference ellipsoid. The rest of the parameter vector consists of 18 brightness calibration factors and the angle phi for a total of 275 free parameters. The parameter vector p is adjusted by minimizing an objective function Q(p) using Powell's method (see Press et al. 1988). The objective function is the sum of chisq (the weighted sum of squared residuals) and penalty functions, which are used to impose dynamical and shape constraints. Dynamical penalties are applied to force the shape to be consistent with a uniform-density, principal-axis rotator. With the reference ellipsoid as the starting point, the dynamical penalty weights are set high enough so that as the model's shape is perturbed away from the ellipsoid, its centroid remains close to the origin of the body-fixed coordinate system and its principal axes of inertia remain nearly aligned with the x, y, and z axes. Eros' rotation period is short enough that the asteroid should be in principal-axis rotation (Harris 1994, Burns and Safranov 1973), so the dynamical penalty functions restrict our attention to only those perturbations that result in dynamically plausible models. Two shape penalty functions are used to limit deviations from the reference ellipsoid and to suppress sharp angles between adjacent facets. The first is a weighting factor (rdev) times the mean square value of r, and the second is a weighting factor (nsmo) times the mean square value of sin(psi/2), where psi is the angle between two adjacent facets. When psi = 0, the adjacent facets are coplanar.
Figure 2 shows the pole-on silhouettes of the shapes that minimize Q(p) for selected values of rdev and nsmo. Large values of rdev force the shape toward the reference ellipsoid, for which chisq/N = 1.15, while large values of nsmo drive the shape toward a sphere and chisq/N > 1.15. When rdev and nsmo are small, the estimator has enough freedom to attain values of chisq/N less than unity. The values of chisq/N given in Fig. 2 sample a chisq surface, which is a smooth, sloping function of rdev and nsmo. A curve on the surface is defined by chisq/N = 1. Starting from a point on this curve and moving in a direction of decreasing rdev and nsmo results in chisq/N < 1, indicating that the estimator is beginning to "fit the noise." Thus, we take models along the chisq/N = 1 curve to bound the region of acceptable deviations from the reference ellipsoid. Although we cannot define a "best-fit" shape, this procedure helps to identify the kinds of perturbations to the reference ellipsoid that can account for the asymmetries observed in the Doppler spectra. Induced Model SymmetriesA more subtle consequence of the penalty functions is their influence on the north/south symmetry of the model as a result of the north/south ambiguity of the data. Since the sub-radar latitude is very near the equator, the north/south ambiguity cannot be resolved with the geometric leverage afforded by the asteroid's rotation. Thus, in attempting to fit a given feature of the Doppler spectra, our estimator is free to perturb northern vertices or southern vertices or both. Since the shape penalty functions depend on higher than linear powers of r, smaller values of Q can generally be attained by distributing a perturbation evenly among northern and southern vertices, thus imposing a degree of north/south symmetry in the model. Symmetries can also be induced by the dynamical penalty functions. As vertices are perturbed on one side of the asteroid to improve the spectral fit near one rotation phase, there must be associated perturbations elsewhere to maintain principal axis rotation about the origin in a manner that does not compromise the goodness of fit at other rotation phases. Such "balancing" perturbations can be quite distributed but can induce a degree of symmetry. The data's low SNR does not provide much leverage in preventing such artifacts; however, the shape penalty functions are effective in suppressing features that do not play a significant role in reducing chisq/N. Selection of a Working Model
Deviations from the reference ellipsoid as a function of rdev and nsmo are readily seen in the model's pole-on silhouette (Fig. 2). As rdev decreases from 2 to 0, concavities begin to form on one of the long sides as the convex envelope of that side flattens. Meanwhile, the opposing long side remains highly convex. As nsmo increases from 0 to 100, concavities are filled in, and the model's ends become more rounded. At nsmo = 100 the model's ends are slightly more rounded than those of the reference ellipsoid. Models with chisq/N = 1 are quite similar for widely ranging combinations of rdev and nsmo, which reflects the influence of the data in defining deviations from the reference ellipsoid subject to the imposed dynamical constraints. Although we cannot define a "best-fit" model, we adopt a working model with (rdev, nsmo) = (0.35, 20) and chisq/N = 1 (Fig. 3). We will henceforth refer to this as Model 1.
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Method 2: Successive Approximations (Vertex Doubling)
Next, we resample the shape that results from minimizing Q(p) for the 16-vertex model with twice the number of vertices, as follows. The new model is initialized with 6 base points placed on the previous model's surface where it intersects the x, y, and z axes. These base points define (2 x 6) - 4 = 8 triangular facets. A new base point is placed at the midpoint of the longest facet side and then moved radially until it lies on the previous model's surface. This procedure is repeated until the desired number of base points have been obtained. The resampled shape is thus defined by more or less equidistant base points, and hence roughly equal-area facets. A triaxial ellipsoid is fit to the new collection of base points, and for each base point we find the point on the ellipsoid's surface such that the base point lies along the surface normal when viewed from that point. The unit direction vector (D) for the base point is then set to be that normal. Then, Q(p) is minimized for the new parameter vector. This process is repeated until the desired accuracy is achieved.
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Comparison of Models 1 and 2 The two approaches outlined above yield distinct shapes. Model 1 exhibits several features (concavities and ridges) along one side, whereas Model 2 has a single large concavity along the same side and a smoother, more rounded appearance. What can we learn about Eros' shape from these two models? Their dissimilarity is one measure of our limited ability to constrain Eros' shape with the available radar data. Nevertheless, both models share some attributes. By a careful consideration of how each model attempts to improve the spectral fit, we can identify the types of features that Eros is likely to possess. We begin this section be comparing the pole-on silhouettes of Models 1 and 2 with the convex hull of ORJ90. There is reasonable agreement between these very different approaches to modeling the radar data. Following this is a detailed comparison of Models 1 and 2, where we show that each model accounts for the observed spectral asymmetries in basically the same way. Pole-on SilhouetteRotational variation of the echo bandwidth influences the convex hull and the pole-on silhouette of the three-dimensional shape in a similar manner. Both approaches seek to confine Eros' extent orthogonal to the spin axis such that the echo power becomes much less than the noise level near the Doppler frequencies of the apparent spectral edges. (Our ability to detect echoes from the target's limbs depends on the scattering law and the data's SNR.) The convex hull approach does this directly through the use of a spectral edge estimator, whereas three-dimensional shape reconstruction does this indirectly as Q(p) is minimized.
Imperfect agreement between the silhouettes and the hull is not unexpected, given the data's noise level. Model 1 shows somewhat better agreement with the dotted hull profiles, which provide an estimate of the nominal hull's uncertainty (ORJ90). Model 2 tends to have larger radii of curvature than the nominal hull, especially at the ends. These discrepancies probably result from our use of penalty functions, which suppress sharp angles between facets for both models and departures from the (smooth) reference ellipsoid for Model 1. The trade-off between producing a smooth model and accurately fitting the spectral edges may account for some of psi's offset from zero in both models. In addition, psi might differ from zero since the principal axes of inertia need not be aligned with the maximum and minimum radii of either model's pole-on silhouette. |
Concavities, Tapering, and Bends
The high degree of north/south symmetry is an artifact of our approach to modeling the north/south ambiguous data, as described above. There are also symmetries about the x-z plane. The subdued ridge at 60 degrees opposes the concavity at 300 degrees, while the subdued trough at 90 degrees opposes the ridge at 270 degrees. This ridge/trough pair on the M2 side is barely noticeable in the pole-on silhouette (Fig. 2) and does not play an obvious role in improving the spectral fit. Thus, we cannot rule out the possibility that some of the subdued structure on the M2 side arises from "balancing" perturbations, which are an expected consequence of the dynamical penalty functions. The lower half of Fig. 7 shows a map of deviation scalars (r) for Model 2. The deviations are relative to the ideal reference ellipsoid along normals to that ellipsoid's surface. There are fewer and larger deviations in Model 2; however, they show some correspondence to the deviations in Model 1. For example, the large concavity from 240 to 320 degrees longitude in Model 2 is in roughly the same location as the trough-ridge-trough pattern centered at 270 degrees in Model 1. In addition, both models have a rounded m2 end and a bend in the m1 end, which is especially prominent in Model 2. Symmetries in Model 2 are much less pronounced than those in Model 1. The pattern of alternating ridges and troughs is virtually absent, and the north/south symmetry is also greatly reduced, although deviations remain centered roughly about the model's "equator." The comparative weakness of symmetries in Model 2 probably results from the successive approximation approach, which places less weight on the reference ellipsoid as an initial condition, and the fact that rdev is set to zero, thereby eliminating one of the causes of north/south symmetries.
The fact that both models account for the spectral asymmetry at rotation phases from ~190-210 degrees by placing one or more concavities on the asteroid's M1 side suggests that this is a real characteristic of Eros' shape. This interpretation is compatible with the "difference in surface reflectivity or surface slopes on opposite sides of the asteroid" noted by Jurgens and Goldstein (JG76) as well as the misalignment of the convex hull's centers of figure and rotation (ORJ90). Comparing Models 1 and 2, we cannot say whether a single large concavity or several smaller ones are responsible for the observed spectral asymmetry. Furthermore, the data's north/south ambiguity combined with an equatorial view precludes the determination of the three-dimensional shape of any of these concavities, which would help to clarify whether they are craters, troughs, or bends in Eros' overall shape. In any event, the topography of the M1 side is apparently much more dramatic than that of the M2 side. The pole-on silhouette of Model 2 looks like a kidney bean, which resembles a nearly pole-on optical image derived from speckle interferometry (Drummond and Hege 1989). The speckle image is based on data obtained in December, 1981, when the target-centered declination of the Earth was probably between -55 and -75 degrees. The similarity between the speckle image and Model 2 is suggestive, given that the two experimental techniques are completely independent. However, the uncertainties are sufficiently large that we cannot rule out a shape more similar to Model 1. |
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Shape and Mineralogy Visual and infrared observations establish Eros as an S-class asteroid (Chapman et al. 1975, Tholen 1989). Recently, Murchie and Pieters (1996) reanalyzed visual/infrared spectra that were acquired in January, 1975, at nearly the same time of the Goldstone radar observations. They found that Eros' average spectrum falls within the S-IV sub-group in the classification system of Gaffey et al. (1993) but that the spectrum varies with rotation phase. Murchie and Pieters identified two sides of Eros, each spanning ~180 degrees of rotation phase. Both sides fall within the S-IV group, but one side has a higher olivine/pyroxene ratio than the opposite side. According to Murchie and Pieters, such mineralogical variations could be interpreted in different ways. Eros could be an intact fragment from a (highly or slightly) differentiated parent body, in which case mineralogical variations in that body might be preserved in Eros. Alternatively, Eros could be a "rubble pile" consisting of fragments from one or more parent bodies, and the relative locations of mineralogically distinct units could be random. Table I shows the correspondence between the mineralogical variations and our shape models. Eros' px-rich side is marked by one or more prominent concavities, while the ol-rich side lacks such features. We do not have sufficient information to determine the exact shapes of the concavities, much less their origin. The concavities on the M1 side of Model 1 could plausibly be impact features; however, the large M1 concavity of Model 2 could just as plausibly define a bend in Eros' shape. Thus, specific geologic interpretations for Eros' shape and mineralogy await NEAR's rendezvous in 1999.
Acknowledgments We are grateful for thoughtful reviews by S. Murchie and an anonymous referee. The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by Washington State University through an agreement with the National Aeronautics and Space Administration. References Asphaug, E., J. M. Moore, D. Morrison, W. Benz, M. C. Nolan, and R. J. Sullivan 1996. Mechanical and Geological Effects of Impact Cratering on Ida. Icarus 120, 158-184. Burns, J. A., and V. S. Safranov 1973. Asteroid nutation angles. Mon. Not. R. Astron. Soc. 165, 403-411. Chapman, C. R., D. Morrison, and B. Zellner 1975. Surface properties of asteroids: A synthesis of polarimetry, radiometry, and spectrophotometry. Icarus 25, 104-130. Drummond, J. D., and E. K. Hege 1989. Speckle interferometry of asteroids. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 171-191. Univ. of Arizona Press, Tucson. Dunlap, J. L. 1976. Lightcurves and the axis of rotation of 433 Eros. Icarus 28, 69-78. Gaffey, M., J. Bell, R. Brown, T. Burbine, J. Piatek, K. Reed, and D. Chaky 1993. Mineralogic variations within the S-type asteroid class. Icarus 106, 573-602. Geissler, P., J.-M. Petit, D. D. Durda, R. Greenberg, W. F. Bottke, and M. C. Nolan 1996. Erosion and Ejecta Reaccretion on 243 Ida and its Moon. Icarus 120, 140-157. Greenberg, R., M. C. Nolan, W. F. Bottke, Jr., R. Kolvoord, and J. Veverka 1994. Collisional History of Gaspra. Icarus 107, 84-97. Harris, A. W. 1994. Tumbling asteroids. Icarus 107, 209-211. Hudson, R. S. 1993. Three-dimensional reconstruction of asteroids from radar observations. Remote Sensing Reviews 8, 195-203. Hudson, R. S., and S. J. Ostro 1994. Shape of asteroid 4769 Castalia (1989 PB) from inversion of radar images. Science 263, 940-943. Hudson, R. S., and S. J. Ostro 1995. Shape and non-principal axis spin state of asteroid 4179 Toutatis. Science 270, 84-86. Jurgens, R. F., and R. M. Goldstein 1976. Radar observations at 3.5 and 12.6 cm wavelength of asteroid 433 Eros. Icarus 28, 1-15. Magnusson, P. 1989. Pole determinations of asteroids. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1180-1190. Univ. of Arizona Press, Tucson. Millis, R. L., E. Bowell, and D. T. Thompson 1976. UBV photometry of asteroid 433 Eros. Icarus 28, 53-67. Mitchell, D. L., S. J. Ostro, R. S. Hudson, K. D. Rosema, D. B. Campbell, R. Veléz, J. F. Chandler, I. I. Shapiro, J. D. Giorgini, and D. K. Yeomans 1996. Radar observations of asteroids 1 Ceres, 2 Pallas, and 4 Vesta. Icarus, in press. Murchie, S. L., and C. M. Pieters 1996. Spectral properties and rotational spectral heterogeneity of 433 Eros. J. Geophys. Res. 101 (No. E1), 2201-2214. Ostro, S. J., K. D. Rosema, and R. F. Jurgens 1990. The shape of Eros. Icarus 84, 334-351. Ostro, S. J., K. D. Rosema, R. S. Hudson, R. F. Jurgens, J. D. Giorgini, R. Winkler, D. K. Yeomans, D. Choate, R. Rose, M. A. Slade, S. D. Howard, and D. L. Mitchell 1995. Extreme elongation of asteroid 1620 Geographos from radar images. Nature 375, 474-477. Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling 1988. Numerical Recipes in C, Chap. 10, pp. 309-317. Cambridge Univ. Press, Cambridge. Santo, A. G., S. C. Lee, and R. E. Gold 1995. NEAR spacecraft and instrumentation. Journal of the Astronomical Sciences 43 (No. 4), 373-397. Scheeres, D. J., S. J. Ostro, R. S. Hudson, and R. A. Werner 1996. Orbits close to asteroid 4769 Castalia. Icarus 121, 67-87. Taylor, R. C. 1985. The pole orientation of asteroid 433 Eros determined by photometric astrometry. Icarus 61, 490-496. Tholen, D. J., 1989. Asteroid taxonomic classifications. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1139-1150. Univ. of Arizona Press, Tucson. Zellner, B. 1975. New findings about Eros. Sky and Telescope (December 1975), pp. 376-379. Zellner, B. 1976. Physical properties of asteroid 433 Eros. Icarus 28, 149-153.
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A uniform-density, triaxial ellipsoid is a useful first
approximation. We adopt a body-fixed coordinate system, with origin at
the ellipsoid's center of mass and axes (x, y, z)
that correspond to the principal axes of smallest, intermediate, and
largest moment of inertia. The ellipsoid rotates about the
z-axis, which points in a direction with ecliptic coordinates
of 22 degrees longitude and +9 degrees latitude (Taylor 1985). All of
the post-1975 pole estimates predict that the target-centered
declination of the radar during the 1975 Goldstone observations was
within ~10 degrees of zero, so our choice of pole direction is not
critical. 
In this approach, we search for the minimum deviations from the
reference ellipsoid that can reduce chisq/N from 1.15 to unity.
We begin by approximating the reference ellipsoid's surface with 508
triangular facets defined by 256 vertices. Each vertex is defined by
a base point (B) and a unit direction vector (D).
Initially, all the base points lie on the surface of the reference
ellipsoid, and the direction vector at each base point is the local
surface normal. Deviations from the reference ellipsoid are
parameterized by 256 scalars (r), which form the first part of
a parameter vector p. 
In this approach, we begin with a polyhedral approximation to the
reference ellipsoid consisting of 28 triangular facets defined by 16
vertices. Dynamical and shape penalty functions are used as before,
except that rdev is fixed at zero - that is, there is no
penalty associated with deviations from the reference ellipsoid. Even
after Q(p) has been minimized for this 16-vertex model,
chisq/N is larger than that of the ideal reference
ellipsoid. Evidently, 16 shape parameters (as opposed to only three
for the reference ellipsoid) are insufficient to produce a polyhedral
model that is superior to the reference ellipsoid, even though some
gross features of the shape are beginning to emerge. 
