A&A 601, A114 (2017)
DOI: 10.1051/0004-6361/201629956
c
ESO 2017
Astronomy
&
Astrophysics
Volumes and bulk densities of forty asteroids
from ADAM shape modeling
J. Hanuš
1, 2, 3
, M. Viikinkoski
4
, F. Marchis
5
, J.
ˇ
Durech
3
, M. Kaasalainen
4
, M. Delbo’
2
, D. Herald
6
, E. Frappa
7
,
T. Hayamizu
8
, S. Kerr
9
, S. Preston
9
, B. Timerson
9
, D. Dunham
9
, and J. Talbot
10
1
Centre National d’Études Spatiales, 2 place Maurice Quentin, 75039 Paris Cedex 01, France
e-mail: hanus.home@gmail.com
2
Université Côte d’Azur, OCA, CNRS, Lagrange, France
3
Astronomical Institute, Faculty of Mathematics and Physics, Charles University, V Holešovi
ˇ
ckách 2, 18000 Prague,
Czech Republic
4
Department of Mathematics, Tampere University of Technology, PO Box 553, 33101 Tampere, Finland
5
SETI Institute, Carl Sagan Center, 189 Bernado Avenue, Mountain View, CA 94043, USA
6
RASNZ Occultation Section, 3 Lupin Pl., Murrumbateman, NSW 2582, Australia
7
Euraster, 8 route de Soulomes, 46240 Labastide-Murat, France
8
JOIN/Japan Occultation Infomation Network, 891-0141 Kagoshima, Japan
9
International Occultation Timing Association (IOTA), %SK, SP, BT, DD
10
RASNZ Occultation Section, 3 Hughes Street, Waikanae Beach, 5036 Kapiti Coast, New Zealand
Received 25 October 2016 / Accepted 6 February 2017
ABSTRACT
Context. Disk-integrated photometric data of asteroids do not contain accurate information on shape details or size scale. Additional
data such as disk-resolved images or stellar occultation measurements further constrain asteroid shapes and allow size estimates.
Aims. We aim to use all the available disk-resolved images of approximately forty asteroids obtained by the Near-InfraRed Camera
(Nirc2) mounted on the W.M. Keck II telescope together with the disk-integrated photometry and stellar occultation measurements to
determine their volumes. We can then use the volume, in combination with the known mass, to derive the bulk density.
Methods. We downloaded and processed all the asteroid disk-resolved images obtained by the Nirc2 that are available in the Keck
Observatory Archive (KOA). We combined optical disk-integrated data and stellar occultation profiles with the disk-resolved images
and use the All-Data Asteroid Modeling (ADAM) algorithm for the shape and size modeling. Our approach provides constraints on
the expected uncertainty in the volume and size as well.
Results. We present shape models and volume for 41 asteroids. For 35 of these asteroids, the knowledge of their mass estimates from
the literature allowed us to derive their bulk densities. We see a clear trend of lower bulk densities for primitive objects (C-complex)
and higher bulk densities for S-complex asteroids. The range of densities in the X-complex is large, suggesting various compositions.
We also identified a few objects with rather peculiar bulk densities, which is likely a hint of their poor mass estimates. Asteroid masses
determined from the Gaia astrometric observations should further refine most of the density estimates.
Key words. minor planets, asteroids: general – techniques: photometric – methods: numerical – methods: observational
1. Introduction
Density and internal structure belong to the most important char-
acteristics of asteroids, which are also some of the least con-
strained. Moreover, when compared with the densities of me-
teorites one can deduce the nature of asteroid interiors. These
physical properties of asteroids reflect the accretional and colli-
sional environment of the early solar system. Additionally, be-
cause some asteroids are analogs to the building blocks that
formed the terrestrial planets 4.56 Gyr ago, the density and in-
ternal structures of minor bodies inform us about the formation
conditions and evolution processes of planets and the solar sys-
tem as a whole. To determine the density directly, we need both
the mass and the volume of the object. The current density esti-
mates are mostly governed by the knowledge of these two prop-
erties. On the other hand, indirect density measurements based
on photometric observations of mutual eclipses of small bi-
nary near-Earth asteroids (NEAs; e.g.,
Scheirich & Pravec 2009)
do not require the mass nor the size. However, the achieved
accuracy of such density estimates is usually much lower when
compared with the direct measurements. Additionally, the typi-
cal size range of objects from these methods also differ.
The majority of reported mass estimates are based on or-
bit deflections during close encounters (e.g.,
Michalak 2000,
2001; Pitjeva 2001; Konopliv et al. 2006; Mouret et al. 2009;
Zielenbach 2011) and planetary ephemeris (e.g., Baer & Chesley
2008; Baer et al. 2011; Fienga et al. 2008, 2009, 2011, 2014;
Folkner et al. 2009). These methods give accurate masses for
the largest asteroids (within a few percent), but the accuracy
gets worse very quickly with decreasing size/mass of the objects.
The astrometric observations of the ESA’s Gaia satellite promise
a significant improvement of the poor knowledge of the mass.
More specifically, Gaia will constrain masses for ∼150 asteroids
(∼50 with an accuracy below 10%,
Mouret et al. 2007, 2008) by
the orbit deflection method. The advantage of Gaia masses is
in the uniqueness of the mission, which should result in a com-
prehensive sample with well described biases (e.g., the current
mass estimates are strongly biased toward the inner main belt).
Article published by EDP Sciences A114, page 1 of 41
A&A 601, A114 (2017)
The list of asteroids, for which the masses will be most likely
determined, is already known.
Carry (2012) analyzed available
mass estimates for ∼250 asteroids and concluded that only about
a half of them have a precision better than 20%, although some
values might be still affected by systematic errors. The second
most accurate mass determinations so far (after those determined
by the spacecraft tracking method) are based on the study of
multiple systems (e.g.,
Marchis et al. 2008a,b, 2013; Fang et al.
2012) and reach a typical uncertainty of 10–15%. Masses based
on planetary ephemeris are often inconsistent with those derived
from the satellite orbits, which is the indication that masses from
planetary ephemeris should be treated with caution.
Determining the volume to a similar uncertainty level as the
mass (<20%) is very challenging. The density is proportional
to the mass and inversely proportional to the cube of the as-
teroid size, so one needs a relative size uncertainty three times
smaller than of the mass estimate to contribute with the same
relative uncertainty to the density uncertainty as the mass. The
most frequent method for the size determination is the fitting of
the thermal infrared observations (usually from IRAS, WISE,
Spitzer or AKARI satellites) by simple thermal models such as
the Standard Thermal Model (STM,
Lebofsky et al. 1986) or the
Near-Earth Asteroid Thermal Model (NEATM, Harris 1998) as-
suming a spherical shape model. This size is often called the ra-
diometric diameter and it corresponds to the surface-equivalent
diameter
1
. Because thermal models usually assume a spheri-
cal shape model, the surface equivalent diameter equals the
volume-equivalent diameter
2
. Reported size uncertainties for in-
dividual asteroids are usually very small (within a few percent,
Masiero et al. 2011), however, they are not realistic (Usui et al.
2014). Indeed, the uncertainties are dominated by the model sys-
tematics – the spherical shape assumption is too crude and the
role of the geometry is neglected (e.g., the spin axis orientation).
The sizes determined by thermal models are statistically reliable,
but could easily be inaccurate for individual objects by 10–30%
(Usui et al. 2014). This implies a density uncertainty of 30–90%.
Other size determination methods that assume a sphere or a tri-
axial ellipsoid for the shape model suffer by the same model sys-
tematics. More complex shape models have to be used for the
more accurate size determinations.
Several methods for reliable size determination that require
lightcurve- or radar-based shape models have been already em-
ployed (the only few exceptions are the largest asteroids that
can be approximated by simple rotational ellipsoids): (i) scal-
ing the asteroid shape projections by disk-resolved images ob-
served by the 8–10 m class telescopes equipped with adaptive
optics systems (e.g., Marchis et al. 2006; Drummond et al. 2009;
Hanuš et al. 2013b); (ii) scaling the asteroid shape projections
to fit the stellar occultation measurements (e.g.,
ˇ
Durech et al.
2011
); or (iii) analyizing thermal infrared measurements by the
means of a thermophysical modeling which allows to scale the
shape from radar or lightcurve inversion to match the size infor-
mation carried by the infra red radiation (e.g.,
Müller et al. 2013;
Alí-Lagoa et al. 2014; Rozitis & Green 2014; Emery et al. 2014;
Hanuš et al. 2015, 2016a). The lightcurve-based shape models
are usually best described as convex (e.g., Kaasalainen et al.
2002a
; Torppa et al. 2003;
ˇ
Durech et al. 2009
; Hanuš et al. 2011,
2016b), the radar models are reconstructed from delay-Doppler
echoes, sometimes in combination with light curve data (e.g.,
1
Surface-equivalent diameter is a diameter of a sphere that has the
same surface as the surface of the shape model.
2
Volume-equivalent diameter is a diameter of a sphere that has the
same volume as the volume of the shape model.
Hudson & Ostro 1999; Busch et al. 2011). Size uncertainties
achieved by these methods are usually below 10%. Recently,
models combining both disk-integrated and disk-resolved data
were developed (e.g., KOALA and ADAM models, Carry 2012;
Viikinkoski et al. 2015a). With these models, both shape and
size are optimized (Merline et al. 2013; Berthier et al. 2014;
Viikinkoski et al. 2015b; Hanuš et al. 2017). For instance, a
KOALA-based shape model of asteroid (21) Lutetia was de-
rived from optical light curves and disk-resolved images by
Carry et al. (2010b). This result was later confirmed by the
ground-truth shape model reconstructed from images obtained
by the camera on board the Rosetta space mission during its
close fly-by (Sierks et al. 2011), which effectively validated the
KOALA shape modeling approach (Carry 2012).
In our work, we use the ADAM algorithm for asteroid shape
modeling from the disk-integrated and disk-resolved data, and
stellar occultation measurements. We describe the optical data
in Sect.
2.1, the disk-resolved data from the Keck II telescope
in Sect. 2.2, and the occultation measurements in Sect. 2.3.
The ADAM shape modeling algorithm is presented in Sect. 3.
We show and discuss derived shape models and corresponding
volume-equivalent sizes and bulk densities in Sect. 4. Finally, we
conclude our work in Sect.
5.
2. Data
2.1. Shape models from disk-integrated photometry
In this work, we mostly focused on asteroids for which the ro-
tation states and shape models had recently been derived or re-
vised. We used rotation state parameters of these asteroids as
initial inputs for the shape and size optimization by ADAM.
The majority of previously published shape models, spin states
and optical data are available in the Database of Asteroid Mod-
els from Inversion Techniques (DAMIT
3
,
ˇ
Durech et al. 2010
),
from where we adopted the disk-integrated light curve datasets
as well. Moreover, we list adopted rotation state parameters and
references to the publications in Table
A.1.
2.2. Keck disk-resolved data
The W.M. Keck II telescope located at Maunakea in Hawaii is
equipped since 2000 with an adaptive optics (AO) system and
the Near-InfraRed Camera (Nirc2). This AO system provides
an angular resolution close to the diffraction limit of the tele-
scope at ∼2.2 µm, so ∼45 mas for bright targets (V < 13.5)
(Wizinowich et al. 2000). The AO system was improved several
times since it was mounted. For example, the correction quality
of the system was improved in 2007 (
van Dam et al. 2004), re-
sulting into reaching an angular resolution of 33 mas at shorter
wavelengths (∼1.6 µm).
All data obtained by the Nirc2 extending back to 2001 are
available at the Keck Observatory Archive (KOA). It is possi-
ble to download the raw images with all necessary calibration
and reduction files, and often also images on which the ba-
sic reduction was performed. We downloaded and processed all
disk-resolved images of all observed asteroids. Usually, several
frames were obtained by shift-adding 3–30 frames with an ex-
posure time of fractions of seconds to several seconds depend-
ing on the asteroid’s brightness at the particular epoch. We per-
formed the flat-field correction and we used a bad-pixel sup-
pressing algorithm to improve the quality of the images before
3
http://astro.troja.mff.cuni.cz/projects/asteroids3D
A114, page 2 of 41
J. Hanuš et al.: Bulk densities of asteroids based on ADAM
shift-adding them. Then, we visually checked all images and se-
lected only those where the asteroids were resolved. Typically,
we considered an asteroid as resolved if its maximum size on the
image was at least approximately ten pixels. Also, we rejected
fuzzy and saturated images, and images with various artifacts.
We obtained about 500 individual images of about 80 asteroids.
Finally, we deconvolved each resolved image by the AIDA al-
gorithm (
Hom et al. 2007) to improve its sharpness.
Many images had already been used independently in previ-
ous shape studies (
Marchis et al. 2006; Drummond et al. 2009;
Descamps et al. 2009; Merline et al. 2013; Hanuš et al. 2013b;
Berthier et al. 2014). In Table A.4, we list all used disk-resolved
images for each studied asteroid and the name of the principal
investigator of the scientific project within which the data were
obtained.
2.3. Occultation data
Stellar occultations are publicly available in the OCCULT soft-
ware
4
maintained by David Herald. In Table A.5, we list all ob-
servers that participated in each stellar occultation measurement
we used for the shape modeling. To achieve a better conver-
gence of the shape modeling, we visually examined each occul-
tation measurement and removed chords with large uncertain-
ties in their timings (mostly visual observations) and chords that
were clearly inconsistent with the remaining ones (mostly due
to the incorrect timing). The chord removal was a relatively safe
procedure, because the offset of the incorrect chord with respect
to several close-by chords was always obvious. Moreover, such
cases were quite rare. We also rejected occultation events that
had less then three reliable chords.
2.4. Asteroid masses
The most accurate mass estimates are based on space probe fly-
by measurements or the satellite’s orbits in the multiple sys-
tems. We adopted these estimates from the corresponding stud-
ies. Densities based on these masses should be the most reliable
ones.
Masses derived from astrometric observations (close en-
counters or planetary ephemeris methods) are available for most
asteroids in our sample. Moreover, multiple determinations for
individual asteroids are common. However, these determinations
are often inconsistent or result in an unrealistic density determi-
nation. To select the most reliable mass estimates, we decided
to use values from the work of
Carry (2012), who investigated
available mass estimates for ∼250 asteroids and present a single
value for each of them. The author also provides bulk density
estimates and ranks their quality. A low rank is usually a sugges-
tion that the mass estimate is not reliable. Recently, Fienga et al.
(2014) computed masses for tens of asteroids from INPOP plan-
etary ephemerides. However, several masses of multiple aster-
oids are inconsistent with masses from
Carry (2012). It is not
obvious which values should be the better ones. For example,
masses for the (45) Eugenia and (107) Camilla multiple sys-
tems are clearly wrong in Fienga et al. (2014), because their re-
liable mass estimates based on the satellite’s orbits are too dif-
ferent. On the other hand, the mass of the (41) Daphne sys-
tem is consistent. Moreover, masses for several asteroids from
Fienga et al. (2014) lead to more realistic bulk densities than
those from
Carry (2012). Beacuse of this, we decided to use
masses from Fienga et al. (2014) only in cases where the density
4
http://www.lunar-occultations.com/iota/occult4.htm
would be unrealistic otherwise. All these cases are individu-
ally commented in Sect. 4.2. Additionally, we also comment the
cases where the masses are inconsistent within each other.
Masses based on astrometric observations of the ESA’s Gaia
satellite should be available in 2019. After that, our volume esti-
mates of several asteroids studied here could be used for future
bulk density refinements.
3. All-Data Asteroid Modeling (ADAM) algorithm
Reconstructing a 3D shape model of an asteroid from various ob-
servations is a typical ill-posed problem, since noise-corrupted
observations contain only low-frequency information. To miti-
gate effects of ill-posedness, we use parametric shape represen-
tations combined with several regularization methods.
While the reconstruction can be made well-behaved in the
sense that the optimization process converges to a shape model,
there is also the problem of uniqueness: often it is not obvious
whether features present in the shape model are supported by
data or if they are artifacts caused by the parameterization and
regularization methods. The chance that these features are spu-
rious can be alleviated by the use of several different parameter-
izations and regularization methods: it is conceivable that all the
representations should produce similar shapes if the solution is
well constrained by the data. Therefore, in this article, we de-
rive shape models for asteroids using two different parametric
shape representations – subdivision surfaces and octanoids (see,
Viikinkoski et al. 2015a). If the resulting shape models for the
asteroid are significantly different, we conclude that the available
data are not sufficient for reliable reconstruction and discard the
model.
The procedure used in this article for shape reconstruction is
called ADAM (Viikinkoski et al. 2015a). It is an universal inver-
sion technique for various disk-resolved data types. ADAM fa-
cilitates the usage of adaptive optics images directly, without re-
quiring deconvolution or boundary extraction. The software used
in this article is freely available on the internet
5
.
Utilizing the Levenberg-Marquardt optimization algorithm,
ADAM minimizes an objective function
χ
2
:= χ
2
LC
+ λ
AO
χ
2
AO
+ λ
OC
χ
2
OC
+
X
i
λ
i
γ
2
i
, (1)
where terms χ
2
LC
, χ
2
AO
and χ
2
OC
are, respectively, model fit to light
curves, adaptive optics images, and stellar occultation chords.
The final sum corresponds to regularization functions measuring
the smoothness and complexity of the mesh.
The formulation of terms χ
2
LC
and χ
2
AO
is covered in
(
Viikinkoski et al. 2015a), and the theoretical foundations of
stellar occultations relating to the shape reconstruction of aster-
oids are well established in
ˇ
Durech et al.
(2011), so we describe
here how the goodness-of-fit measure χ
2
OC
for occultation chords
is implemented in ADAM.
As an asteroid occults a star, its shadow travels on the sur-
face of the Earth. The positions of the observers, together with
the disappearance and reappearance times of the star, determine
a chord on the fundamental plane, which is the plane perpendic-
ular to the line determined by the asteroid and the star. Given the
fundamental plane determined by the occultation, we project the
shape model represented by a triangular mesh M into the plane
by using an orthogonal projection P. To form the goodness-of-fit
measure χ
2
OC
, we must first define a reasonable distance function
d(C, PM).
5
https://github.com/matvii/adam
A114, page 3 of 41
A&A 601, A114 (2017)
Let C be the occultation chord with the endpoints p
1
and
p
2
on the plane, and L the line determined by the chord. We
consider the case where the line L intersects the boundary of the
projected shape model PM at two points q
1
and q
2
. Assuming
the points are ordered so that the vectors p
1
− p
2
and q
1
− q
2
are
parallel, we set
d(C, PM) = kq
1
− p
1
k
2
+ kq
2
− p
2
k
2
. (2)
If the line does not intersect the projected shape, let δ be the
perpendicular distance from the line L to the closest vertex in
PM. We define
d(C, PM) = 2kp
2
− p
1
k
2
L(δ), (3)
where
L(x) =
1
1 + e
−kx
, (4)
is the logistic function with the parameter k.
For the negative chords (i.e., chords along which no occulta-
tion is observed), we use a slightly different approach. We set
d(C, PM) = γ · (1 − L(δ)), (5)
where γ is a constant weight, and δ is defined as follows: if the
chord C intersects PM, let δ
1
be the distance to the farthest ver-
tex on the positive side of the line L, and similarly let δ
2
be the
distance to the farthest vertex on the negative side of the line. We
set
δ = − min{δ
1
, δ
2
}. (6)
If the chord does not intersect PM, let δ be the perpendicular
distance from the line to the closest vertex on PM. The idea here
is that if the negative chord intersects the projected shape, the
distance function attains its maximum value γ. The weight γ is
chosen large enough to ensure that an optimization step causing
an intersection is rejected. The logistic function is used instead
of the step function to make the distance function differentiable.
Given an occultation event consisting of n chords C
i
, we
define
χ
2
OC
:=
X
i
d(C
i
, PM + (O
x
, O
y
)), (7)
where (O
x
, O
y
) is the offset from the projection origin, to be de-
termined during the optimization.
4. Results and discussions
Here we present shape models of asteroids based on the ADAM
shape modeling algorithm. All derived shape models as well
as all their optical disk-integrated and disk-resolved data, and
occultation measurements are available online in the DAMIT
database. Our observation datasets always contain all three types
of these data. The uncertainties in the spin vector determinations
were estimated from the differences between the solutions based
on the usage of the two different shape supports in ADAM (i.e.,
subdivission surfaces and octanoids, see Sect.
3), and the usage
of raw and deconvolved versions of the disk-resolved data. These
uncertainties are usually between two and five degrees.
To estimate the size, first, we computed the volume from the
scaled shape model and estimated its uncertainty from the differ-
ences between the solutions based on the usage of different shape
supports in ADAM the same way as for the pole solution. Then,
we computed the corresponding volume-equivalent diameter and
its possible range from the volume. We only report the volume-
equivalent size with its 1σ uncertainty in Table A.2, however,
the volume can be easily accessed based on this size. The bulk
density in Table A.2 is then the ratio between the mass and the
volume, and its uncertainty was computed from the propagation
of the volume and mass uncertainties.
4.1. Shape models of primaries in multiple systems
Several main-belt binaries that consist of a large primary
(&100 km) and a few-kilometer sized secondary (or even two
satellites) were discovered in the images obtained by the 8–
10 m class telescopes equipped with AO systems during the last
decade. Usually, tens of large asteroids were surveyed during a
few-year campaign and their close proximity was searched for
a potential presence of a satellite. Once a satellite was detected,
the system was then imaged in other epochs, so the satellite’s or-
bit could be constrained. Fortunately, some of the primaries were
large enough to be resolved, often during multiple distant epochs
(apparitions). On the other hand, single objects were usually ob-
served only once or twice, so the observations available for them
are mostly single- or double-apparition.
4.1.1. Comparison with previously modeled primaries
Asteroids with multiple disk-resolved images were natural can-
didates for shape modeling, so several shape models have
already been published for those. All these shape models
are based on methods that use the 2D contours extracted
from the disk-resolved images. Such approach is sensitive
to the boundary condition applied when extracting the con-
tour. Shape models of asteroids (22) Kalliope, (87) Sylvia,
(93) Minerva, and (216) Kleopatra have been previously derived
(
Descamps et al. 2008; Berthier et al. 2014; Marchis et al. 2013;
Kaasalainen & Viikinkoski 2012). As the first step, we decided
to validate our modeling approach on these asteroids – we have
similar or even larger optical, disk-resolved and stellar occul-
tation datasets for them. We present shape models for all four
asteroids and reproduce well the previous results.
Before listing the asteroids one by one, we note that there
are plenty of previous shape and spin pole studies for each
asteroid in our work, including single-epoch methods assum-
ing triaxial ellipsoids as well as a more general modeling by
the lightcurve inversion method. Below, we mainly comment
on the shape modeling results based on the lightcurve inver-
sion of optical photometry and neglect most other studies for
the sake of simplicity. Moreover, we most often used spin states
based on lightcurve inversion as necessary initial inputs for the
shape modeling by ADAM, because the one-apparition ellip-
soidal shape and spin solutions lack the necessary precision in
the sidereal rotation period.
22 Kalliope. Reliable size and bulk density of Kalliope have
already been derived from observations of mutual events by
Descamps et al. (2008). Our shape model and size based on
102 light curves, 23 disk-resolved images and one occultation
is in an agreement with the previous results (e.g., the difference
in the pole orientation is only one degree). Our size (161 ± 6 km)
is slightly smaller (compared to 166 ± 3 km), but still within
the small uncertainties. So, we derived a similar bulk density of
(3.7 ± 0.4) g cm
−3
that is consistent with the M-type taxonomic
classification of Kalliope.
A114, page 4 of
41
J. Hanuš et al.: Bulk densities of asteroids based on ADAM
87 Sylvia. As Sylvia is a multiple system, a large number of
22 disk-resolved images could be used for the shape model-
ing. Together with 55 optical light curves and two occultation
measurements, the ADAM modeling resulted in a reliable shape
model and size that is in a perfect agreement with an indepen-
dent shape model derived by the KOALA algorithm from a sim-
ilar dataset by
Berthier et al. (2014). The size of (273 ± 5) km
combined with the mass estimate gave us a bulk density of
(1.39 ± 0.08) g cm
−3
. Sylvia is the only P-type asteroid in our
sample and its bulk density is one of the most precise so far es-
timated for an asteroid. Most of the C-complex asteroids have
similar bulk densities as Sylvia.
93 Minerva. Size and bulk density of the C-type aster-
oid Minerva based on optical light curves, disk-resolved im-
ages and occultation data have already been determined by
Marchis et al. (2013). We used a similar optical dataset and a
subset of disk-resolved data and derived a shape model and
size (159 ± 3 km) that are consistent with those of Marchis et al.
(2013, 154 ± 6 km). The difference in the pole orientation is only
four degrees. Our size estimate is slightly higher, which resulted
in a smaller though more consistent with respect to its taxonomic
type (i.e., C-complex), bulk density of (1.59 ± 0.27) g cm
−3
.
216 Kleopatra. So far, the disk-resolved images of Kleopa-
tra have not been sufficient for a proper shape model
Kaasalainen & Viikinkoski (2012). Using all available data, we
derived a model with ADAM. Only one pole solution of (λ,
β) ∼ (73, 21)
◦
is consistent with the AO data. Our ADAM model
is based on 55 optical light curves, 14 AO images and three oc-
cultations. All occultations consist of multiple chords that sam-
ple most of the shape projection. The issue with the shape model
is that there are no AO nor occultation data that were obtained at
a view close to the pole. Closest is the 2016 occultation, ∼70 de-
grees above equator. Similarly there is one AO image 70 degrees
below the equator, but it is too fuzzy to be useful. We obtained a
pole solution of ∼(74, 20)
◦
, that is in a perfect agreement with the
one of
Kaasalainen & Viikinkoski (2012). Our size (121 ± 5) km
and adopted mass of Descamps et al. (2011) lead to a bulk den-
sity of (5.0 ± 0.7) g cm
−3
, an unusually large value within the M-
type asteroids, however, with a larger uncertainty. Additionally,
Ostro et al. (2000) derived a shape model of Kleopatra from the
delay-Doppler observations obtained by Arecibo. The spin state
is similar to ours, though the shape models differ. The model of
Ostro et al. (2000) has a dumbbell appearance with a handle that
is substantially narrower than the two lobes. In our shape model,
the handle is of about the same thickness as the lobes.
4.1.2. New shape models of primaries
41 Daphne. All recent shape model studies of the C/Ch-
type asteroid Daphne reported a consistent pole solution of
∼(200, −30)
◦
(
Kaasalainen et al. 2002a;
ˇ
Durech et al. 2011
;
Matter et al. 2011; Hanuš et al. 2016b), which we used as an ini-
tial input for the shape modeling with ADAM. Because Daphne
is a binary asteroid (Conrad et al. 2008b), the number of disk-
resolved images is rather high due to the attraction to the satel-
lite’s position (Conrad et al. 2008a). We also have occultation
observations from two distant epochs. Our shape model fits all
the AO, light curve and occultation data well, so the size is
reliably constrained to be (188 ± 5) km. Moreover, this size is
compatible with the size estimate of
Matter et al. (2011) based
on interpretation of interferometric data by a thermophysical
model and the use of a shape model with local topography. Our
size and the precise mass estimate from
Carry (2012) lead to
a bulk density of (1.81 ± 0.15) g cm
−3
, which belongs to higher
values within the C-complex asteroids.
45 Eugenia. The small moonlet “Petit-Prince” of Eugenia
was discovered by
Merline et al. (1999), which made Euge-
nia a target for several AO campaigns studying orbit of the
moon (e.g., Marchis et al. 2010). As a consequence, 23 disk-
resolved images were obtained by the Nirc2 during six dif-
ferent apparitions. Hanuš et al. (2013b) rejected one of the
mirror solutions and our shape modeling with ADAM con-
firmed that conclusion. The shape model nicely fits all the
disk-resolved images as well as two occultation measurements,
which lead to a precise size estimate of (186 ± 4) km. The cor-
responding density of (1.69 ± 0.11) g cm
−3
is consistent with
typical densities of C-type asteroids. The reliable mass esti-
mate of (5.69 ± 0.12) × 10
18
kg is based on the moon’s orbit
(
Marchis et al. 2008a).
107 Camilla. The single pole solution of ∼(72, 51)
◦
(Torppa
et al. 2003;
ˇ
Durech et al. 2011
; Hanuš et al. 2013b, 2016b) is
well established and we used it as an initial input for the shape
modeling by ADAM. Camilla is another binary (actually triple)
asteroid that was often observed by the NICR2 at Keck. So, we
gathered 21 disk-resolved images obtained at seven different ap-
paritions. As a result, our shape and size solution that explains
all the observations is well constrained. The size of (254 ± 6) km
combined with the mass from Marchis et al. (2008a) resulted in
a typical C-type asteroid bulk density of (1.31 ± 0.10) g cm
−3
.
4.2. Shape models of single asteroids
Only few single asteroids were observed during multiple epochs
by the Keck AO system, namely mostly the largest ones (e.g.,
2 Pallas, 52 Europa) and the space mission targets (e.g., 1 Ceres,
4 Vesta, 21 Lutetia), because most AO surveys at the Keck tele-
scope were dedicated to the discovery of satellites and then the
system follow-up. Shape models based on disk-resolved data
were previously independently derived for asteroids (2) Pallas,
(16) Psyche and (52) Europa (
Carry et al. 2010a; Shepard et al.
2017; Merline et al. 2013), so we provide our ADAM solutions
for comparison and as a reliability test. For the remaining as-
teroids, we present their first shape model solutions from disk-
resolved data and sometimes their first non-radiometric size esti-
mates. For the majority of the asteroids studied, we used adopted
mass estimates and derived their bulk densities.
2 Pallas. Our ADAM modeling started with the single pole so-
lution from
Carry et al. (2010a) as an initial input and converged
to a solution that fit nicely all optical light curves, 18 disk-
resolved images and two occultations. We note that the occul-
tation from the year 1983 (Dunham et al. 1990) is of an excep-
tional quality, the projection consists of 131 chords (there are
also 117 observations outside the path of Pallas shadow) and
sample almost the whole projected disk. Our shape and size
of (523 ± 10) km is consistent with the solution of Carry et al.
(2010a) of (512 ± 6) km. Pallas is one of the three B-type aster-
oids in our sample and has a bulk density of (2.72 ± 0.17) g cm
−3
.
Clearly, such density is exceptionally high among the primitive
C-complex asteroids, it is even higher than the bulk density of
A114, page 5 of 41