The Resolved Asteroid Program - Size, shape, and pole of (52) Europa111Based on observations at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
With the adaptive optics (AO) system on the 10 m Keck-II telescope, we acquired a high quality set of 84 images at 14 epochs of asteroid (52) Europa on 2005 January 20, when it was near opposition. The epochs covered its 5.63 h rotation period and, by following its changing shape and orientation on the plane of sky, we obtained its triaxial ellipsoid dimensions and spin pole location. An independent determination from images at three epochs obtained in 2007 is in good agreement with these results. By combining these two data sets, along with a single epoch data set obtained in 2003, we have derived a global fit for (52) Europa of diameters km, yielding a volume-equivalent spherical-diameter of km, and a prograde rotational pole within 7 of [RA; Dec] = [257; +12] in an Equatorial J2000 reference frame (Ecliptic: 255; +35). Using the average of all mass determinations available for (52) Europa, we derive a density of 1.5 0.4 g cm, typical of C-type asteroids. Comparing our images with the shape model of Michalowski et al. (Astron. and Astrophys. 416, p 353, 2004), derived from optical lightcurves, illustrates excellent agreement, although several edge features visible in the images are not rendered by the model. We therefore derived a complete 3-D description of (52) Europa’s shape using the KOALA algorithm by combining our 18 AO imaging epochs with 4 stellar occultations and 49 lightcurves. We use this 3-D shape model to assess these departures from ellipsoidal shape. Flat facets (possible giant craters) appear to be less distinct on (52) Europa than on other C-types that have been imaged in detail, (253) Mathilde and (511) Davida. We show that fewer giant craters, or smaller craters, is consistent with its expected impact history. Overall, asteroid (52) Europa is still well modeled as a smooth triaxial ellipsoid with dimensions constrained by observations obtained over several apparitions.
Direct, accurate measurements of asteroid shapes, sizes, and
pole positions are now possible for larger asteroids
that can be spatially resolved using
the Hubble Space Telescope (HST) or
large ground-based telescopes equipped with adaptive optics (AO).
Physical and statistical
study of asteroids requires accurate knowledge of these parameters. Improved
sizes permit improved estimates of albedo, in turn allowing better interpretation
of surface composition. In those cases where we have an estimate of
the mass, for example from the presence of a satellite, the uncertainty in the
volume of the asteroid is the overwhelming uncertainty in attempts to derive
(Merline et al. 2002). Of course, density is the
single most critical
observable having a bearing on bulk composition, porosity, and
(Merline et al. 2002; Britt et al. 2002, 2006).
With our technique of determining the size of an asteroid by
following its changing apparent size, shape, and orientation, the
volume can now be reduced to the level of the mass uncertainty, vastly
our confidence in the derived asteroid densities.
because we can see the detailed shape, track edge or surface
features during rotation, and often can make an immediate pole
Dedicated study of asteroids now allows directly observable shape profiles, and already has shown that some asteroids show large departures from a reference ellipsoid that may provide clues to the body’s response to large impacts over time (e.g., (4) Vesta, Thomas et al. 1997). For asteroid (511) Davida, we suggested (Conrad et al. 2007) that such features (e.g., large flat facets) may be analogs of the giant craters, seen edge-on, in the images of (253) Mathilde during the NEAR mission (Veverka et al. 1999) flyby. If giant craters are evident on these surfaces, they can be related to the impact history and impact flux over time, and there is some chance they can be associated with asteroid families or clusters that are being identified by numerical back-integration and clustering of orbital elements (e.g., Nesvorný et al. 2002).
As we have demonstrated with asteroid (511) Davida (Conrad et al. 2007), we can derive an asteroid’s triaxial ellipsoid dimensions and rotational pole location in a single night. However, we now have developed the ability to combine sets of similar observations obtained at different viewing aspects to make a global fit to all of the images, drastically reducing dimension uncertainties that might be due to sparse rotational sampling or peculiar observing geometries (Drummond et al., in preparation). The leverage of widely spaced observations and the accompanying range of viewing aspects allows unprecedented accuracy in derived parameters. We can then use these estimates to project the apparent size and shape of an asteroid into the past or future, making the asteroid useful as a reference or calibration object.
Here we report on the physical properties of the asteroid (52) Europa as a part of our on-going Resolved Asteroid Program. We routinely image the apparent disk of asteroids, and search their close vicinity for companions, aiming at setting better constraints on their spin-vector coordinates, 3-D shapes, sizes, and multiplicity. One of our main goals is to derive (or better constrain) their densities. We use two independent methods to determine size, shape, and pole position of the target asteroids. One of these is based on the assumption that the shape is well-described by a smooth triaxial ellipsoid (see Drummond 2000; Drummond et al. 2009a, 2010, for instance). Our other method allows construction of full 3-D shape models by combining our AO images with other data types, when available (e.g., optical lightcurves and stellar occultations, see Carry et al. 2010a, b), in the technique we call KOALA (Knitted Occultation, Adaptive-optics and Lightcurve Analysis, see Carry et al. 2010a; Kaasalainen 2011).
The best angular resolution, approximated by = / (radian), with the wavelength and the diameter of the telescope aperture, of current ground-based optical telescopes is about 0.04 (Keck/NIR). Due to systematics, however, we have found that our ability to accurately measure sizes and details of the apparent shape degrades below about 0.10, based on simulations and observations of the moons of Saturn and simulations (Carry 2009; Drummond et al. 2009b). The sample of observable asteroids (i.e., having angular sizes that get above about 0.10) is therefore limited to about 200.
This limit in angular resolution can be converted to a physical diameter. As can be seen in Fig. 1, we can probe the size distribution of main-belt asteroids down to about 100 km, while Pluto is the only Trans-Neptunian Object (TNO) whose apparent disk can be resolved. At opportune times, we have been able to resolve the disks of Near-Earth Asteroids (NEAs, for example, see Merline et al. 2011, 2012). The next generation of optical facilities will allow an improvement in angular resolution by a factor of 3-4 due to mirror size alone (30 m for TMT and 40 m for E-ELT), allowing the observation of more than 500 asteroids, even if we consider only objects that reach half (or 0.05) of the current size limits (We computed the expected apparent diameter of asteroids for the 2020–2030 period, and counted objects when apparent diameters reach 0.05 within this period.) Second-generation instruments with high-Strehl AO corrections into visible wavelengths are planned for these large ground-based telescopes, providing another factor of 5 improvement due to operation at shorter wavelengths. Together, these two factors should provide more than an order-of-magnitude improvement with respect to current resolution. Almost 7 000 asteroids should then be observable with apparent diameters greater than 0.01. This breakthrough in imaging capabilities will also enable the spatial resolution of apparent disks of TNOs larger than 500 km, larger moons (100 km) of Uranus and Neptune, small moons of Jupiter and Saturn, main-belt asteroids of few tens of kilometers, and NEAs of several hundred meters in favorable conditions (Fig. 1).
2 Disk-resolved imaging observations
For asteroid (52) Europa, our primary data set was taken on
2005 January 20. In addition, we observed (52) Europa at one epoch on
2003 October 12, and at three epochs on 2007 May 28.
In 2005 we obtained adaptive optics images of (52) Europa at H
(1.6 m) and Kp
(2.1 m) bands with NIRC2
(van Dam et al. 2004) on the Keck II 10 m telescope, and give the observing log in
The 2003 and 2005 images were taken using the first generation Keck wave-front
controller; the 2007 images were taken using Keck’s next generation wavefront
(NGWFC, van Dam et al. 2007)
under similar conditions.
Strehl ratios were 30%, 27%, and 40% on average, respectively, for the 2003,
2005, and 2007 epochs.
The latter, higher, value reflects the NGWFC
changes which, in addition to a new detector, include improvements to the
electronics and to the software. The data set consists of 111 images:
9 from 2003, 84 from 2005, and 18 from 2007, that result in 18
composite images (Table 1).
Although less extensive and at a larger distance from Earth, the 2007 data add an important new epoch to our 2005 data. By combining all 3 data sets (2003, 2005, and 2007), our goal was to derive a global fit that spans a wide range of viewing geometries and provide tight constraints on the size, shape, and pole for (52) Europa.
When observing at Kp in good seeing conditions, adaptive optics on Keck II delivers diffraction-limited resolution elements of width approximately 50 milli-arcsecond (mas). We used the narrow plate-scale (9.942 0.050 mas/pixel) of the NIRC2 camera, oriented North-up (0.15, Konopacky et al. 2007) for all the observations.
|2003-10-12 - 11:48||3.02||2.07||7.2||10.8||0.25||26||Kp|
|2005-01-20 - 10:39||2.79||1.84||5.5||10.3||0.28||6||Kp|
|2005-01-20 - 10:43||2.79||1.84||5.5||10.3||0.28||9||H|
|2005-01-20 - 11:25||2.79||1.84||5.5||10.3||0.28||55||Kp|
|2005-01-20 - 11:28||2.79||1.84||5.5||10.3||0.28||58||H|
|2005-01-20 - 12:02||2.79||1.84||5.5||10.3||0.28||95||Kp|
|2005-01-20 - 12:04||2.79||1.84||5.5||10.3||0.28||97||H|
|2005-01-20 - 13:01||2.79||1.84||5.5||10.3||0.28||157||Kp|
|2005-01-20 - 13:04||2.79||1.84||5.5||10.3||0.28||160||H|
|2005-01-20 - 13:45||2.79||1.84||5.5||10.3||0.28||204||Kp|
|2005-01-20 - 13:48||2.79||1.84||5.5||10.3||0.28||206||H|
|2005-01-20 - 14:16||2.79||1.84||5.5||10.3||0.28||237||Kp|
|2005-01-20 - 14:18||2.79||1.84||5.5||10.3||0.28||239||H|
|2005-01-20 - 15:02||2.79||1.84||5.5||10.3||0.28||-74||Kp|
|2005-01-20 - 15:05||2.79||1.84||5.5||10.3||0.28||-71||H|
|2007-05-28 - 11:44||3.41||2.69||13.3||11.9||0.19||105||Kp|
|2007-05-28 - 12:54||3.41||2.69||13.3||11.9||0.19||179||Kp|
|2007-05-28 - 13:01||3.41||2.69||13.3||11.9||0.19||186||Kp|
3 Triaxial Ellipsoid (TE) Assumption
3.1 2005 January 20
Each of seven sets of six H-band images and seven sets of
six Kp-band images
of asteroid (52) Europa obtained in 2005 was sky-subtracted, and then fit
in the Fourier plane for the asteroid and Lorentzian PSF, using our
method of Parametric Blind Deconvolution
(PBD, as described by Drummond et al. 1998; Drummond 2000; Conrad et al. 2007).
Asteroid ellipse parameters were computed as weighted means
from each set of six images obtained at each filter and each rotational
phase or epoch. These ellipse parameters
(apparent major axis length ,
minor axis length ,
and an orientation angle PA),
were then used to convert
the series of apparent diameters and
orientations to the full triaxial-ellipsoid diameters and
direction of (52) Europa’s
through a non-linear least squares inversion
(see Drummond 2000, for instance).
The results of the fit are given in Table 2.
|a (km)||377 3||376 3||381 4||378 3|
|b (km)||331 3||332 3||335 4||332 3|
|c (km)||236 9||246 8||249 10||244 8|
|SEP ()||+27 3||+25 3||+25 5||+25 3|
|PA ()||339 1||339 1||338 1||338 1|
|(UT)||10.35 0.03||10.33 0.03||10.28 0.04||10.30 0.03|
|EQJ2000 (, in )||261;+10||260;+11||259;+12||260;+11|
|ECJ2000 (, in )||260;+34||258;+34||257;+35||258;+34|
In addition to the direct PBD methodology, as
cross-checks, we use two additional avenues to get to the
triaxial-ellipsoid solutions. In the first of these, the data
were flat-fielded, shifted, and added at each
rotational epoch (Fig. 2), and
a single deconvolved image was created with the Mistral
algorithm (Mugnier et al. 2004), for each epoch and each filter.
These seven Kp and seven H deconvolved images
(Fig. 3) were again fit in the Fourier
plane for their apparent ellipse parameters, and the series was fit for the full
triaxial solution, also given in Table 2.
Finally, ellipse parameters were derived from fitting the
edges produced by a Laplacian of Gaussian wavelet transform
(Carry et al. 2008) on the Mistral deconvolved images.
A full triaxial solution can then be found
from these ellipse parameters, and is given in
The adopted triaxial solution for (52) Europa, independently determined
from the 2005 data, is derived from the series of mean ellipse
parameters at each epoch, that is, from the mean of the PBD images,
the Mistral deconvolved images, and
the edge-fitting at each epoch.
This preferred mean fit is
plotted against observations in Fig. 5.
The location of the pole on the Ecliptic
globe is shown in Fig. 6, along with the locations
derived from lightcurves analysis by others.
Our imaging of (511) Davida (Conrad et al. 2007) showed large edge features that could be followed during rotation, even in the raw images. While there may be similar features on (52) Europa, they do not appear as consistently in the edge profiles and are not as easy to track. The features are not as large or prominent as those on Davida, relative to our reference ellipsoid. Later in the paper, we use 3-D shape modeling to try to study these departures from a pure ellipsoid shape.
3.2 2007 May 28
We also acquired AO observations of (52) Europa at Keck in 2007 (Table 1). Following the recipe from the last section, we formed the mean apparent parameters from the three methods already described (PBD, deconvolved images, and outlines from the deconvolved images). Although not expected to yield significant results because the three 2007 observations provide only nine observables to find six unknowns, we nevertheless fit the three observations for a triaxial ellipsoid (Table 3 and Fig. 7), and found that the model is in surprisingly good agreement with the results from the 2005 set in Table 2.
|a (km)||379 1|
|b (km)||330 1|
|c (km)||225 9|
|SEP ()||-41 5|
|PA ()||212 1|
|EQJ2000 (, in )||258;+11|
|ECJ2000 (, in )||256;+34|
3.3 2003 October 12
The single set of AO images of (52) Europa taken in 2003 (Table 1) does not allow an independent fit for a triaxial solution because it only provides three observables for six unknowns. We use these early Keck AO images, however, in a global fit in the next section. Fig. 8 shows the global fit prediction for the 2003 epoch, together with those data.
3.4 A global solution for all epochs
We can tie the 2003, 2005, and 2007 observations of (52) Europa together
into one simultaneous global fit
(Drummond et al., 2012, in preparation), using
the sidereal period of P = 0.2345816 days
(with an uncertainty of 2 in the last digit)
derived by Michałowski et al. (2004).
Along with the global solution for the triaxial
dimensions and pole in Table 4,
we list the three parameters that differ due
to the changing angles for each date.
Statistical uncertainties for the dimensional parameters, as well as those involving angles, such as pole position and longitude of the node, come from the non-linear least-squares fit for the 6 parameters that define the TE model, the 3 diameters and 3 Euler angles. Systematic effects can arise in the process of constructing a 3-D description of an asteroid from information limited to a 2-D plane (images). Therefore, one needs to be particularly vigilant regarding model assumptions, and their appropriateness for a particular situation. While the uncertainties derived for the parameters as fit by the model are straightforward, estimating the systematic effects that are present is not. Deriving realistic (and therefore, directly applicable by other workers) uncertainties for our results, including possible systematics, is the most challenging aspect of our work.
We have carefully calibrated some of these uncertainties by making observations of external sources (e.g., the moons of Saturn) of known size. One of the results of that work has shown that our systematic uncertainties are larger for objects of smaller angular diameter, until we reach a limit (at about 0.09for a 10 m telescope) where we can no longer get reliable sizes. Aspect ratios of projected shapes are still possible, but absolute sizes break down. We have found that our systematics from these tests span about 1–4% per linear dimension. In addition, we have also imaged targets of spacecraft missions prior to flyby (see KOALA section). In the case of (21) Lutetia, despite an angular size of only 0.10, our resulting models were good to 2% in size and 2 km RMS in topography on a 100 km object (see Carry et al. 2012).
We can also compare our TE results with those of KOALA (see below), in cases where we have adequate observations. In particular, we have such comparisons for four asteroids, including (52) Europa. We can look for consistency, not only between the two techniques, but in sub-sets of data to learn how far we fall from the “correct” values. We can also compare the results of data sets from different years. Our upcoming article, mentioned above (Drummond et al., in preparation) will be a stand-alone treatment of the global fitting technique and calibration that will include much detail on uncertainties. For the present results, we have determined that we should add quadratically systematic uncertainties of 4.1%, 2.3%, and 3.8% to the TE-derived fit errors (given in Table 4) for , respectively. The resulting total uncertainty estimates for the diameters are 16 8 10 km, with a 7 degree systematic uncertainty for the orientation of the spin axis. See Fig. 4 for a visualization of the orientation of (52) Europa on the plane of the sky.
|Diameter (km)||Pole||Param||2003 Oct 12||2005 Jan 20||2007 May 28|
|a = 379 2||(,) = 257;+12||SEP ()||+49 1||+23 1||-40 1|
|b = 330 2||radius = 1||PA ()||204 1||339 1||213 1|
|c = 249 3||(,) = 255;+35||(UT)||11.11 0.02||10.31 0.02||9.72 0.02|
4 Comparison of (52) Europa to Lightcurves Inversion Model
From optical lightcurves of (52) Europa,
Michałowski et al. (2004) found a
rotational pole at [, ]=[252, +38],
with a 5 uncertainty in each Ecliptic
coordinate. It is the pole closest to ours in
Fig. 6, about 6 away.
They derived an a/b axial ratio of 1.15, the same as our
and a b/c ratio of 1.3, compared with our
Figure 9 is a side by side comparison of our deconvolved Kp images, from 2005 January 20 (from Fig. 3) and the Michałowski et al. model, using the updated rotational pole for the model at [, ]=[251; +35] from the DAMIT (Ďurech et al. 2010) web site222http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php. Figure 10 shows comparison between our convolved and deconvolved images and the lightcurves inversion model for 2003 and 2007.
The overall agreement between our AO deconvolved images and the model predictions is excellent. A careful examination of Figs 9 and 10, however, will show edge features that are seen in one but not the other, requiring the development of an updated shape model, as discussed in following section. Despite these features, (52) Europa is still well-modeled as a smooth triaxial ellipsoid.
5 KOALA 3-D shape model
We construct a 3-D shape model of (52) Europa to give a better rendering
of the apparent shape visible in the images.
For that, we use our KOALA algorithm
(Carry et al. 2010a; Kaasalainen 2011)
that makes combined use of optical lightcurves, stellar
occultations timings, and profiles from disk-resolved images.
The results of KOALA have been recently validated at (21) Lutetia
by the images taken by the ESA Rosetta mission:
The 3-D shape model and spin orientation
determined before the encounter by combining AO images and
lightcurves (Carry et al. 2010b)
were in complete agreement with images and results from
the flyby (Sierks et al. 2011; Carry et al. 2012).
Axial dimensions from KOALA were determined within
2% of the the actual values and RMS differences in topography
were only 2 km.
We use here the 18 imaging epochs described in Sect. 2, together with 49 lightcurves taken between 1979 and 2011 (we acquired 8 additional lightcurves within the CdR/CdL collaboration with respect to the 41 lightcurves presented by Michałowski et al. 2004), and 4 stellar occultations (timings taken from Dunham et al. 2011). A comparison of the KOALA 3-D shape model with the AO images from 2005 is presented in Fig. 11. The agreement between the 3-D shape model and the data is very good. The typical deviation with the 18 imaging contours is of 0.2 pixel, corresponding to a few km. The 49 lightcurves are rendered at a level of 0.03 mag, i.e., close to the intrinsic level of uncertainty of the data. Finally, the residuals between the occultation chords and the model are 13 km, on average, mainly owing to the lower quality of 1983 occultation timings (residuals of 19 km, compared to 11, 13, and 6 km for the other epochs). Figure 12 shows these chords mapped onto the projections of the 3-D KOALA model for the epochs of the occultations.
The 3-D shape derived with KOALA is close to an ellipsoid, validating (52) Europa as a Standard Triaxial Ellipsoid Asteroid (STEA, Drummond et al. 2008). Fitting the KOALA model as a triaxial ellipsoid yields diameters of 368 327 255 km, in excellent agreement with the diameters and total uncertainties in Table 4. The volume-equivalent spherical-diameter of the KOALA 3-D shape model, derived by summing volume cells, is 312 6 km, in excellent agreement with the TE analysis presented above. The KOALA model yields a spin pole within 3of [, ]=[254; +37] or [, ]=[257; +15], also close to the TE result. The shape model can be downloaded from the DAMIT web page.
6 Occurrence of large facets on C-type asteroids
The 3-D shape model presents two broad shallow depressions, probably best noted in the lower right of Fig. 12. They can also be seen on the tops and bottoms of the asteroid images in column 1, panel 3, and column 3, panel 2. The departures from an ellipsoid, however, are not nearly as significant as the apparent giant facets seen in our analysis of (511) Davida (Conrad et al. 2007), nor as prominent, relative to body size, as the giant craters seen on (253) Mathilde (Veverka et al. 1997). We chose Mathilde as a prototypical object displaying giant features seen in profile (craters/facets), although Mathilde was a much smaller asteroid than Davida. But it turns out that (52) Europa is almost a twin of Davida in many respects: both are C-type asteroids of very nearly the same size, they have similar spin periods, and they have similar orbital properties, so they have likely seen the same impact environment (although Davida does have a bit larger eccenticity and inclination). In the Davida paper, we went to some length to demonstrate that Davida could have encountered impacts of the size necessary to produce the giant facets seen, without having broken up the body. So given the similarities between Davida and (52) Europa, one might now wonder how likely it is that (52) Europa would not show such facets (or at least not show facets that are quite as prominent).
Returning to our analysis in the Davida paper, we estimated that Davida should have had about 2.5 impacts large enough to make such a giant crater during its lifetime. This led to the conclusion that if the facets seen were indeed craters, seen edge-on, as on Mathilde, they would not be unexpected. The same statistics should hold true for (52) Europa. But with an expected total of only 2.5 impacts of this size during its lifetime, the chances are also reasonable that it did not encounter any such impacts. We therefore conclude that not seeing such prominent features on a twin such as (52) Europa could also be expected. Of course, the flux of smaller impactors would be higher, and these would be responsible for the perhaps less prominent edge features that we do see. Given that the viewing geometry has to be just right to see these types of facets, it is possible that observational circumstances have conspired such that we missed some giant feature, or such that those facets we do see are less pronounced or are particularly hard to follow with rotation. We have a fairly wide range of latitudes and longitudes in our data set, however, so the chances of missing something as prominent as a Davida-style facet are diminished, and we assert that Europa appears qualitatively different than Davida.
7 Density of (52) Europa
There are 17 estimates of the mass of (52) Europa
available in the literature, derived either from the analysis of the
orbit’s deflection during close approaches of minor planets to
(52) Europa (e.g., Michalak 2001),
or from a general adjustment of the parameters used to generate
the ephemeris of the planets and
asteroids in the Solar System
(e.g., Fienga et al. 2009).
We adopt here the weighted mean of these determinations
(following the selection discussed in Carry 2012):
In general, the differences in volume between the triaxial and the KOALA models are small. Here, that difference is less than 1%, which would lead to a volume difference of less than 3%. When assigning uncertainties to our sizes (from either method), we not only assess the derivable statistical uncertainties, but we must also provide an estimate of systematic effects, of which this difference is an example. The uncertainties used already include potential differences between the models. Because of the added topographic detail provided by the KOALA model, we choose, in this case, to use the KOALA-derived volume of 0.10) km, giving a density of 1.5 0.4 g cm. This bulk density falls within the observed range of densities for C-type asteroids. Here, the uncertainty is mainly due to the uncertainty on the mass determination (24%) rather than the volume uncertainty of 6%. Thus, we are at the point in the study of the density of asteroids where the uncertainty on the volume is no longer the limiting factor (volume determination remains generally the limiting factor when the mass is estimated from a spacecraft encounter or a satellite, see the review by Carry 2012).
Dedicated observing programs and theoretical work are now needed to derive more accurate masses of large main-belt asteroids. The advent of the Gaia mission (expected launch 2013) should contribute a large number of new, improved masses (see Mouret et al. 2007, for instance). With these more reliable volumes and masses, we can derive improved densities and porosities, which in turn will allow us to better understand how density and porosity may be related to taxonomic class, absolute diameters, or location (e.g., inner vs. outer main belt). And this highlights the importance of continuing to push for more AO observations of asteroids for size/shape determination, from the best facilities, and the continued development of techniques, such as KOALA, that combine multiple data types (hopefully, eventually to include thermal radiometry and radar echoes).
At this point, (52) Europa can be considered for
membership as a Standard Triaxial Ellipsoid Asteroid
(STEAs, see Drummond et al. 2008)
is so well modeled as an ellipsoid
(like asteroid (511) Davida, see Conrad et al. 2007).
The ellipses projected by these standard ellipsoids
can be predicted well into the future or past, and therefore, can be used as
calibration objects for other techniques used in studying asteroids.
Conrad et al. (2007) and Drummond et al. (in preparation)
detail the equations necessary to
predict the asterocentric latitudes and longitudes, and
show how to derive the projected ellipse parameters from the asterocentric
latitudes and longitudes. For example, (52) Europa’s asterocentric latitude can
be predicted to within the error of its rotational pole,
and its asterocentric
longitude to within 0.5/yr since the date of the
most recent epoch reported here (2007 May 28).
The longitude uncertainty arises from the formal uncertainty in the
sidereal period, but in fact, judging by the good agreement shown between
the images and lightcurves inversion model projected forward from 1983,
longitudes should be predictable to a much higher
accuracy than these values indicate.
The projected major or minor axis dimensions can
be predicted to within approximately the uncertainty found here of
and the orientation of the apparent ellipse to within
We are fortunate to have both the triaxial ellipsoid (TE, Drummond et al. 2009a) and the KOALA (Carry et al. 2010a) techniques available for our analysis of AO images of asteroids. Each has its own strengths. TE requires relatively few images, can return shape/size/pole information amazingly quickly, is generally insenstive to changes in the PSF, and is usually adequate to get the basic asteroid parameters. For more detailed 3-D shape information we can rely on KOALA. Unlike lightcurve inversion alone, KOALA can obtain absolute sizes, and is sensitive to concavities. The methods can be used to validate each other, as we found exceedingly useful during our analysis of the Lutetia data, prior to the Rosetta flyby (Drummond et al. 2010; Carry et al. 2010b). And while a detailed 3-D shape model might be seen to supercede the triaxial assumption of TE, that is not necessarily the case. As an example, our AO imaging of the close flyby of Near-Earth Asteroid 2005 YU55 from Keck, in November 2011, resulted in almost immediate size and shape information from TE (Merline et al. 2011). In futher analysis, we had hoped to use numerous lightcurves, taken near the time of the flyby, to help refine the size/shape with KOALA. But despite our efforts, the lightcurve information on 2005 YU55 so-far is insufficient (mostly due to a very slow spin period) to allow KOALA to improve significantly on TE. This demonstrates the importance of having both methods available for analysis of our asteroid data.
New imaging, lightcurve, and occultation data will be added to our overall analysis for (52) Europa as they become available. These may allow us to distinguish whether any of the somewhat-flattened edges seen on (52) Europa in our existing data sets are indeed facets or craters of the type seen on Davida and Mathilde, and to better evaluate the extent and morphology of any departure from a pure ellipsoid. The techniques we are developing here (both observational and in data analysis) will allow us to make immediate and substantial advances once data from new, larger telescopes can be acquired.
This work presented here was supported by grants from NASA’s Planetary
Astronomy Program and the U.S. National Science Foundation, Planetary
Astronomy Program. We are grateful for telescope time made available to us
by the NASA TAC, and also for the support of our collaborators on Team Keck,
the Keck science staff.
The work of J. Ď was supported by grant P209/10/0537
of the Czech Science Foundation and by the Research Program
MSM0021620860 of the Ministry of Education.
This research has made use of NASA’s Astrophysics Data System. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
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