3 Modeling asteroid shapes

A grid of triaxial ellipsoid models is explored as a first step to obtain a preliminary best fit for each single visit, separately for each FGS-axis. In a second step, a model-grid is explored to obtain the best fit solution for the whole set of visits taking into account the asteroid rotation. This iterative two-steps fitting procedure finally provides the pole solution $(\lambda,~\beta)$, the lengths of the axes of the ellipsoid (a,b,c), the separation and the axes (a',b',c') of the secondary in case of a binary, and the rotational phase angle W₀, i.e., the position of the major-axis meridian at a given reference epoch (see Hilton 1992; Seidelmann et al. 2002). In all cases, a fit assuming two components with diameter ratio larger than 0.5 and varying separation has been performed to check the possible binary nature. If the goodness of the fit is poor, the hypothesis of a nearly-contact binary system can be rejected. A complete discussion of this procedure can be found in Paper I.

The values obtained for the physical ephemeris and the ellipsoid sizes are summarized in Table 2. They were computed assuming uniform surface brightness of the projected asteroid shape. The error bars of each listed quantity depends critically on the orientation of the asteroid with respect to the direction of the two FGS axis. For this reason, details are given in the discussion concerning each single object. However, in general the formal error (depending upon the asteroid magnitude) is of the order of a few mas for the best determined axis, and can be as large as 10-20% for the less well constrained length. Moreover, these formal errors refer to the best-fit ellipsoidal model, which may in principle differ from the actual shape of the observed asteroid. Last, as seen in Sect. 2.1, the modeling is based on a limited variation of the apparent projected ellipse during each visit as a consequence of asteroid rotation. In this respect, faster rotations can provide in principle a better reconstructed shape.

Our data analysis procedure yields the best-fit solution for a triaxial ellipsoid assuming a uniform brightness distribution. Introducing a limb-darkening effect of the surface leads to larger resulting sizes with approximately the same goodness of fit. The exact function describing this limb-darkening is generally not known for the asteroids and cannot be retrieved from our data alone. However, in order to quantify the uncertainty in the resulting shape and size determination due to the insufficient knowledge of limb-darkening, we have performed a separate data reduction by assuming a normalized brightness distribution corresponding to a Minnaert's law (Minnaert 1941), $I=\mu_0^k~\mu^{k-1}$ ($\mu_0$ and $\mu$ being the cosine of the incidence and reflection angles, respectively) assuming k=0.6(Hestroffer 1998; Parker et al. 2002), i.e., moderate limb-darkening. We found that, except for (15) Eunomia, the systematic error on size estimate is of the order of 3%. Interestingly, we also found that introducing a limb-darkening effect has no appreciable influence on the resulting ellipsoid flattening. In other words, while the overall size increases slightly for increasing limb-darkening, the change on the resulting a/b and a/c ratios tends to be insignificant.

  

Table 2: Physical ephemeris and shape parameters derived from HST/FGS observations. For (216) Kleopatra the solution with two overlapping ellipsoidal components is given.

$$% latex2html id marker 1236 { \begin{tabular}{lrrrrrccrrcc} \... ...m{0}94 \phantom{0}94 & 2.21 & (2.21) \\ \hline \end{tabular}}$$

(a) Solution number of Magnusson et al. (1994) and coordinates in the ecliptic B1950.
(b) The rotational phase W₀ is computed for the reference epoch given on the second line.
(c) The ellipsoid's flattening coefficient given in parenthesis is determined with lower precision (see text).

  

Table 3: Same as Table 2 for (216) Kleopatra. Here the solution with two "overlapping'' ellipsoidal components is given.

$$\begin{tabular}{ccclccclcc} \hline\hline \noalign{\smallskip ... ...83)} && 114 & 125\\ \noalign{\smallskip } \hline \end{tabular}$$

In the following sections we give, for each asteroid, the results for the duplicity test and the data inversion. The axial ratios of the ellipsoids are compared to the values derived by Magnusson et al. (1994) (and reference therein) on the basis of photometric analyses. The same source provides a set of pole coordinates for each object, as determined by different authors. Coordinate values are normally spread over two intervals of a few degrees, grouped around two independent pole orientations, both compatible with the available photometric data (Taylor 1979). The two orientations differ by about 180 degrees in ecliptic longitude. This is due to the fact that, for both orientations, the same fraction of the asteroid surface is visible from Earth at a given epoch; as a consequence, the integrated disk photometry is the same. However, the orientation of the projected shape of the object on the sky plane is different, therefore high-resolution observations can discriminate between the two solutions. In general, as it will be seen in the detailed discussion about each object, a single pole orientation is consistent with the HST/FGS data. Thus, we are able to eliminate the residual pole ambiguity.

On the other hand, we are not able to provide fully independent improvements of the precise value of pole coordinates, that would require observations over longer time spans, at different aspect angles. Thus, we did not include the pole coordinates in the free parameters of the fit. In fact, our observations (being restricted to a single aspect angle) are mainly sensitive to the projected position of the rotation axis on the sky plane, not its real position in space. Given the large uncertainty (several degrees) affecting the pole coordinates given in Magnusson et al. (1994), we ran several fit solutions, trying to minimize the residuals, for different values of pole coordinates. The values retained for the final fit, listed in Table 2, are always inside the uncertainty interval of the available solutions. Finally, we should note that the different photometrically-derived pole solutions available in the literature are also associated with some corresponding estimates of the axial ratios b/a and c/a, computed from the analysis of the light-curve properties under the general assumption that the objects are triaxial ellipsoids. Our HST observations are able to determine the axial ratios of the objects, given the measured (varying) axial ratios of the projected ellipses in the sky, and assuming different pole solutions. This allows us also to discriminate among different pole solutions in some cases like (43) Ariadne (see below), in which the resulting axial ratios corresponding to one of the pole solutions would be unrealistic (extreme flattening) and totally not compatible with the values derived from the photometry.

A comparison to the asteroid sizes derived by indirect methods, such as radiometric diameters computed on the basis of IRAS-measured thermal IR fluxes, requires one to translate the derived shape parameters into average radii. In the following we discuss our results by computing the radius of a sphere equivalent in volume ($R_{\rm v}$). For a three-axis ellipsoid, we thus have $4/3~\pi~a~b~c=4/3~\pi~R_{\rm v}^3$.

The data for the diameters comparison are summarized in Table 4. The graphs for the final step of our model fit process - providing the ellipsoid parameters - are given in Figs. 1 to 6. The fits of the derived model to the data ("first step'') follow (Figs. 7 to 12).

(15) Eunomia

With an apparent size of approximately 0.26 arcsec, (15) Eunomia is the largest asteroid observed within this program, and hence exhibits the flattest S-curve. The evolution of the S-curve as a function of time (Fig. 1) clearly shows that the size along the X-axis remains approximately constant while the size along the other axis is increasing. This enables us to discriminate between the two possible pole solutions. The one that is retained shows that, consistently with the variation observed along the FGS axis, at the epoch of observation (15) Eunomia was almost equator-on with a sub-earth point (SEP in the following) longitude close to 120$^\circ$. In this viewing conditions, the length of the longest and shortest axes (a and c) are well determined, whereas the length of the b axis is poorly constrained.

The fit with a single-body solution and the pole solution indicated in Table 2 is acceptable for the whole set of visits (see Fig. 1). (15) Eunomia is an almost prolate-spheroid with sizes $361\times 205\times 203$ km. The orientation relatively to the FGS axis is such that the lengths of the a and c axes are well determined. The ellipsoid flattening is larger than the one derived from the photometry (a/b=1.42, a/c=1.6) even if b is not well constrained. The derived volume corresponds to an effective diameter of 248 km, close to the IRAS diameter (255 km).

In contrast with the other asteroids of this program, the large size of (15) Eunomia, combined with the relatively large solar phase angle, implies that introducing a moderate limb-darkening (Minnaert $k\sim 0.6$) provides a slightly better goodness of fit to the data.

A more careful analysis of fit residuals provides further insight into the the shape of this object. In fact, it can be seen that the S-curves in the X-axis are particularly asymmetric in comparison to the model. The fit residual, systematically present especially close to the S-curve maximum (Fig. 7), can reflect a shape or brightness-distribution irregularity (i.e. presence of a spot, non convex or non-symmetric shape, etc.). To test this hypothesis, a dark-spot - as was suggested by Lupishko et al. (1984) on basis of photometric data analysis - was introduced in the fitting grid with varying position, relative albedo, and diameter. The residuals - on a single-scan basis - improved considerably on the X-axis, without affecting the fit on the Y-axis. Nevertheless, the best-fit dark-spot is much too large (about 25% of the visible surface) and/or too dark to be realistic. Such model would thus be in complete disagreement with the observed photometric light-curves of this asteroid.

Such a solution being now discarded, a tentative fit with a binary structure and with diameters ratio (secondary/primary) smaller than 0.6 has been done. The residuals of each independent single visit are, again, considerably improved on the X-axis. Nevertheless, no acceptable solution fitting together all data on both X and Y axis, can be found. In conclusion, the available data show that duplicity is not convincingly suggested by the data at our disposal.

It should be stressed that this example clearly illustrates the need to have more than two single baselines for the interferometer, and that observations with a different scanning geometry (i.e. different "roll'' position angle) would be valuable for the shape reconstruction. As shown in Paper I, an egg-shaped convex profile (Gaffey & Ostro 1987) or an octants-shape model (Cellino et al. 1989) would provide features similar to those observed on the present S-curves (see Fig. 13). In summary, (15) Eunomia is hardly a binary system nor a regular ellipsoid, but this work confirms that probably it has an egg-like shape that could be accurately modeled with more HST/FGS data.

Figure 13: Octants shape model for (15) Eunomia on the first visit (top), and interferogram (bottom).

(43) Ariadne

In contrast to the other asteroids of this program, two different published poles yield a possible solution for the triaxial ellipsoid that adequately fit the data. Nevertheless, the shape corresponding to one of the two solutions would be unrealistic, being characterized by $(a\!:\!b\!:\!c)=(1\!:\!1\!:\!0.26)$. As we will discuss in more detail in a future work, such a flattened body is not compatible with photometric observations. Our results suggest that (43) Ariadne should be a prolate-spheroid with axial lengths $90\times 53\times 53$ km. At the epoch of observation, however, (43) Ariadne had an intermediate aspect angle (with SEP latitude 44$^\circ$), so that the length of the (c) axis is not well constrained. The ellipsoid flattening is fairly in agreement with that derived from photometry (a/b=1.6, a/c=1.8) taking into account the uncertainty on c. The derived volume corresponds to an effective diameter of 63 km, close to the IRAS diameter (66 km).

Figure 14: "Binary'' shape model for (43) Ariadne on the last visit (top), and interferogram (bottom).

Although the single-ellipsoid model provides an acceptable goodness of fit, the S-curves in the Y-axis exhibit a tendency to a slight, increasing deviation from the model at the end of the observing run (see Fig. 8). In fact, a better goodness of fit is obtained with a binary model in contact where the components diameters are in the ratio 0.35 (see Fig. 14). Such a binary model, however, would not adequately reproduce the observed light-curves, and it is not clear from the available data if the components are actually separated or if (43) Ariadne could be a single object with non-ellipsoidal shape. In particular, the actual shape of the "secondary'' (a sphere here) is not well constrained. More data are certainly needed to refine the overall shape of (43) Ariadne, but for the moment we have strong indication that the asteroid is a bi-lobated or bifurcated non-convex body. In any case, our data reliably reject the tentative model of Cellino et al. (1985) (based on photometric analysis) that predicts a slightly separated binary system having nearly equal components (diameter ratio 0.93).

(44) Nysa

The fit with a single-body solution and the indicated pole solution is good for the whole set of visits (see Figs. 3 and 9). Any ambiguity on the pole coordinates is thus removed. Nysa is well modeled by a prolate-spheroid of size $119\times 69\times 69$ km. At the epoch of observation this asteroid was almost equator-on with a SEP longitude close to 220$^\circ$, so that the major and minor axis (a and c) are well determined while the intermediate one  b) is poorly constrained. The ellipsoid flattening is coherent with the one derived from the photometry analysis (a/b=1.44, a/c=1.6-2.3) taking into account the inherent uncertainty on b. The derived volume corresponds to an effective diameter of 83 km, larger by 16% than the IRAS diameter (71 km).

Kaasalainen et al. (2002), from analysis of photometric data, derive a cone-like shape for (44) Nysa, that they suggest to be the signature of a "compound asteroid" consisting of two components of unequal size. Although there is no strong indication of such a contact-binary structure in the available HST/FGS data for this object, we tested this hypothesis by fitting the data with a binary model. The best-fit solution, given in Fig. 15, is obtained for a contact structure with a 0.6 diameter ratio. It is stressed that the goodness of fit is neither considerably improved nor is it degraded in this case, and that hence the FGS data alone cannot confirm or rule out such a solution. A more careful analysis, combining both photometric and interferometric data, should help to constrain a possible non-convex model and reveal the actual shape of (44) Nysa.

Figure 15: "Binary'' shape model for (44) Nysa on the last visit (top), and interferogram (bottom).

(63) Ausonia

The fit of the observations by means of a single triaxial ellipsoid model is the best we could find in our sample. In fact, residuals are very small for the whole set of visits (see Figs. 4 and 10). (63) Ausonia is not a binary asteroid but a regular prolate spheroid with sizes $151\times 66\times 66$ km. At the epoch of observation the SEP latitude for (63) Ausonia was large (57$^\circ$), so that the length of the smallest axis (c) is poorly constrained. The ellipsoid flattening is in good agreement with the one derived from the photometry analysis (a/b=2.2, a/c=2.2), and in particular with the resulting shape of Zappalà & Knezevic (1984). The derived volume corresponds to an effective diameter of 87 km, smaller by 16% than the IRAS diameter (103 km).

(216) Kleopatra

Unlike the other asteroids of our sample, the observed S-curves of (216) Kleopatra are not consistent with a single triaxial ellipsoid model, but are best explained by a double-lobed shape model with the pole indicated in Table 2 (Fig. 5). The fit procedure is based on assuming an ellipsoidal companion with varying size, flattening, and separation. The best-fit model is obtained by using two similarly sized elongated bodies overlapping each other. Consistently to their volume, we call them "primary'' and "secondary''. The details about this solution are given in (Tanga et al. 2001). This bi-lobated model is coherent with the radar observations of Ostro et al. (2000) and adaptive optics observations of Marchis et al. 1999.

At the epoch of observation the SEP latitude for (216) Kleopatra was large (-43$^\circ$), so that the length of the smallest axis (c) is poorly constrained. No direct comparison should be done with the ellipsoid flattening derived from photometry. Nevertheless the resulting a/b ratio for the "primary'' and the "secondary'' is of the order of 2.1, which is still a somewhat high value for such a large asteroid. Due to the peculiar shape, different from the simple models usually employed to derive sizes from thermal data, a comparison with the IRAS diameter can only be tentative. Taking into account that the two ellipsoids of the model are overlapping by approximately 10%, the derived volume corresponds to an effective diameter of 95 km, smaller than the IRAS diameter (135 km).

The fit of the S-curves - while satisfactory - is not as good as for other asteroids in this program (see Fig. 11), and reflects non-modeled shape and/or brightness anomalies. Nevertheless, it is shown in Hestroffer et al. (2002c) that such a simple model of overlapping ellipsoids better reproduces the presently observed S-curves than would the topographic nominal model obtained by Ostro et al. (2000) from inversion of radar data.

(624) Hektor

(624) Hektor, a member of the Jupiter Trojans, is the only non main-belt asteroid in this program. It is also the faintest object observed, close to the limit of acceptable S/N ratio for the FGS#3. At the epoch of observation the best-fitting pole solution suggested a SEP latitude relatively large (-33$^\circ$), so that the length of the smallest axis (c) is poorly constrained. The fit with a single-body solution is satisfactory for the whole set of visits (Figs. 6 and 12).

The ellipsoid flattening is smaller than the one derived from the photometric analysis (a/b=2.4, a/c=3.1), but it should be taken into account that c is not well constrained. The derived volume would correspond to an effective diameter of 245 km. No IRAS diameter is available for (624) Hektor, but on the other hand the size $416\times 188$ km found here is consistent with the size estimate ( $370\times 195$ km) given by Storrs et al. (1999) from a deconvolution of HST/WFPC data.

Fit residuals indicate that there is no strong evidence of a binary structure. However, to test the binary equal-sized double as hypothesized by Hartmann & Cruikshank (1978), a fit with a binary model, with either overlapping or separated components, has also been done. The varying parameters are the diameters of the primary and of the secondary, and their separation.

This model does not improve fit residuals in comparison to the single ellipsoid. Due to the geometry of our observations, the data on the FGS X-axis are not very sensitive to a binary structure (see Paper I). The best fit is obtained for a "binary'' with two overlapping components and with a relatively large diameter ratio (0.9), thus for a shape that, given the resolution of the instrument, is not significantly different from that of a single ellipsoid (see Fig. 16). Thus, our HST/FGS data do not conclusively reject the hypothesis of a dumbbell-shape made of two large and similarly sized bodies. The data S/N ratio together with the limited (u,v) plane coverage are not high enough to separate those two shape models. Nevertheless our analysis suggests that a single-ellipsoid model better matches the data. Observations with the recently installed astrometer  FGS#1, providing higher S/N ratio, would be helpful for a better reconstruction of the shape of (624) Hektor.

 

 

Table 4: Comparison to IRAS diameters (unavailable for (624) Hektor). All values are in km.

Name	IRAS	Ellipsoid $2\times R_{\rm v}$
(15) Eunomia	255	248
(43) Ariadne	66	63
(44) Nysa	71	83
(63) Ausonia	103	87
(216) Kleopatra	135	95
(624) Hektor	-	245

Figure 16: "Binary'' shape model for (624) Hektor on the last visit (top), and interferogram (bottom).