11institutetext:  SETI Institute, Carl Sagan Center, 189 Bernado Avenue, Suite
200, Mountain View CA 94043, USA 22institutetext: Aix Marseille Univ, CNRS,
LAM, Laboratoire d’Astrophysique de Marseille, Marseille, France
33institutetext: Institute of Astronomy, Faculty of Mathematics and Physics,
Charles University, V Holešovičkách 2, 18000 Prague, Czech Republic
44institutetext: IMCCE, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Universités, UPMC Univ Paris 06, Univ. Lille, France 55institutetext:
Department of Earth, Atmospheric and Planetary Sciences, MIT, 77 Massachusetts
Avenue, Cambridge, MA 02139, USA 66institutetext: Mathematics & Statistics,
Tampere University, PO Box 553, 33101, Tampere, Finland 77institutetext: Space
sciences, Technologies and Astrophysics Research Institute, Université de
Liège, Allée du 6 Août 17, 4000 Liège, Belgium 88institutetext: Faculty of
Physics, Astronomical Observatory Institute, Adam Mickiewicz University, ul.
Słoneczna 36, 60-286 Poznań, Poland 99institutetext: Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
1010institutetext: Observatoire du Bois de Bardon, 16110 Taponnat, France
1111institutetext: Astronomical Institute of Romanian Academy, 5, Cutitul de
Argint Street, 040557 Bucharest, Romania 1212institutetext: Thirty-Meter-
Telescope, 100 West Walnut St, Suite 300, Pasadena, CA 91124, USA
1313institutetext: Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA 1414institutetext:
European Space Agency, ESTEC - Scientific Support Office, Keplerlaan 1,
Noordwijk 2200 AG, The Netherlands 1515institutetext: The French Aerospace Lab
BP72, 29 avenue de la Division Leclerc, 92322 Chatillon Cedex, France
1616institutetext: Open University, School of Physical Sciences, The Open
University, MK7 6AA, UK 1717institutetext: Laboratoire Atmosphères, Milieux et
Observations Spatiales, CNRS & Université de Versailles Saint-Quentin-en-
Yvelines, Guyancourt, France 1818institutetext: Sección Física, Departamento
de Ciencias, Pontificia Universidad Católica del Perú, Apartado 1761, Lima,
Perú 1919institutetext: Departamento de Fisica, Ingeniería de Sistemas y
Teoría de la Señal, Universidad de Alicante, Alicante, Spain
2020institutetext: Institut de Ciéncies del Cosmos (ICCUB), Universitat de
Barcelona (IEEC-UB), Martí Franqués 1, E08028 Barcelona, Spain
2121institutetext: European Southern Observatory (ESO), Alonso de Cordova
3107, 1900 Casilla Vitacura, Santiago, Chile

# (216) Kleopatra, a low density critically rotating M-type asteroid ††thanks:
Based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under program 199.C-0074 (PI Vernazza),††thanks: The reduced
images are available at the CDS via anonymous ftp to
http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx

F. Marchis (216) Kleopatra, a low density critically rotating M-type asteroid
††thanks: Based on observations made with ESO Telescopes at the La Silla
Paranal Observatory under program 199.C-0074 (PI Vernazza),††thanks: The
reduced images are available at the CDS via anonymous ftp to
http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx L. Jorda (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx P. Vernazza (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx M. Brož (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx J. Hanuš (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx M. Ferrais (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx F. Vachier (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx N. Rambaux (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx M. Marsset (216) Kleopatra, a low density critically
rotating M-type asteroid ††thanks: Based on observations made with ESO
Telescopes at the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low density critically rotating
M-type asteroid ††thanks: Based on observations made with ESO Telescopes at
the La Silla Paranal Observatory under program 199.C-0074 (PI
Vernazza),††thanks: The reduced images are available at the CDS via anonymous
ftp to http://cdsarc.u-strasbg.fr/ or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/xxx/Axxx M. Viikinkoski (216) Kleopatra, a low density
critically rotating M-type asteroid ††thanks: Based on observations made with
ESO Telescopes at the La Silla Paranal Observatory under program 199.C-0074
(PI Vernazza),††thanks: The reduced images are available at the CDS via
anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx E. Jehin (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx S. Benseguane (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx E. Podlewska-Gaca (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx B. Carry (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx A. Drouard (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx S. Fauvaud (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx M. Birlan (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx J. Berthier (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx P. Bartczak (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx C. Dumas (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx G. Dudziński (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx J. Ďurech (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx J. Castillo-Rogez (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx F. Cipriani (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx F. Colas (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx R. Fetick (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx T. Fusco (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx J. Grice (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx A. Kryszczynska (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx P. Lamy (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx A. Marciniak (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx T. Michalowski (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx P. Michel (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx M. Pajuelo (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx T. Santana-Ros (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx,(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx,(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx P. Tanga (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx A. Vigan (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx O. Witasse (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx B. Yang (216)
Kleopatra, a low density critically rotating M-type asteroid ††thanks: Based
on observations made with ESO Telescopes at the La Silla Paranal Observatory
under program 199.C-0074 (PI Vernazza),††thanks: The reduced images are
available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx(216) Kleopatra, a low
density critically rotating M-type asteroid ††thanks: Based on observations
made with ESO Telescopes at the La Silla Paranal Observatory under program
199.C-0074 (PI Vernazza),††thanks: The reduced images are available at the CDS
via anonymous ftp to http://cdsarc.u-strasbg.fr/ or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/xxx/Axxx

(Received x-x-2020 / Accepted x-x-2019)

###### Abstract

Context. The recent estimates of the 3D shape of the M/Xe-type triple asteroid
system (216) Kleopatra indicated a density of $\sim$5 g.cm-3, which is by far
the highest for a small Solar System body. Such a high density implies a high
metal content as well as a low porosity which is not easy to reconcile with
its peculiar “dumbbell” shape.

Aims. Given the unprecedented angular resolution of the VLT/SPHERE/ZIMPOL
camera, here, we aim to constrain the mass (via the characterization of the
orbits of the moons) and the shape of (216) Kleopatra with high accuracy,
hence its density.

Methods. We combined our new VLT/SPHERE observations of (216) Kleopatra
recorded during two apparitions in 2017 and 2018 with archival data from the
W.M. Keck Observatory, as well as lightcurve, occultation, and delay-Doppler
images, to derive a model of its 3D shape using two different algorithms
(ADAM, MPCD). Furthermore, an N-body dynamical model allowed us to retrieve
the orbital elements of the two moons as explained in the accompanying paper.

Results. The shape of (216) Kleopatra is very close to an equilibrium dumbbell
figure with two lobes and a thick neck. Its volume equivalent diameter
$(118.75\pm 1.40)\,{\rm km}$ and mass $(2.97\pm 0.32)\cdot 10^{18}\,{\rm kg}$
(i.e., 56% lower than previously reported) imply a bulk density of $(3.38\pm
0.50)\,{\rm g}\,{\rm cm}^{-3}$. Such a low density for a supposedly metal-rich
body indicates a substantial porosity within the primary. This porous
structure along with its near equilibrium shape is compatible with a formation
scenario including a giant impact followed by reaccumulation. (216)
Kleopatra’s current rotation period and dumbbell shape imply that it is in a
critically rotating state. The low effective gravity along the equator of the
body, together with the equatorial orbits of the moons and possibly rubble-
pile structure, opens the possibility that the moons formed via mass shedding.

Conclusions.  (216) Kleopatra is a puzzling multiple system due to the unique
characteristics of the primary. This system certainly deserves particular
attention in the future, with the Extremely Large Telescopes and possibly a
dedicated space mission, to decipher its entire formation history.

###### Key Words.:

Minor planets, asteroids: individual: Kleopatra – Methods: observational –
Methods: numerical – Surface modeling

##  1 Introduction

Moons around main-belt asteroids have been known to exist since the discovery
in 1993 of Dactyl, the companion of (243) Ida (Binzel 1995). Since then, using
mostly lightcurve inversion, but also with the use of adaptive optics (AO) on
8-10m class telescopes and the Hubble Space Telescope (HST), $\sim$190
multiple systems have been discovered (Johnston 2021), starting in 1998 with
Petit-Prince, around (45) Eugenia (Merline et al. 1999), followed by the
discovery of the first triple asteroid (87) Sylvia a few years later (Marchis
et al. 2005). Today, about 30 of them have been observed by direct imaging (AO
on 8-10m class telescopes, HST), providing insights into their formation and
evolution (Yang et al. 2016). The images bring indirect information about the
interior based on direct information about mass and volume, and hence density
(Margot et al. 2015; Scheeres et al. 2015).

The arrival of a second generation of AO, such as the Spectro-Polarimetric
High-contrast Exoplanet Research instrument (SPHERE) at the Very Large
Telescope (VLT, Beuzit et al. 2019) and the Gemini Planet Imager (GPI) at
GEMINI-South (Macintosh et al. 2014), offers a great opportunity to constrain,
via direct imaging, the 3D shape and mass of large multiple asteroid systems
where the primary’s diameter exceeds 100 km. In 2017, we started a survey of
about forty large (D$\geq$100km) main-belt asteroids through a European
Southern Observatory (ESO) large programme (id: 199.C-0074, Vernazza et al.
2018), including the following six known multiple systems: (22) Kalliope, (41)
Daphne (Carry et al. 2019), (45) Eugenia, (87) Sylvia (Carry et al. 2021),
(130) Elektra, and (216) Kleopatra. In addition, our programme allowed the
discovery of a new binary asteroid ((31) Euphrosyne, Yang et al. 2020b, a).

Among the six known systems, (216) Kleopatra is of particular interest because
of the various density estimates reported for this object, ranging from
$\sim$3.6 g.cm-3 to $\sim$5 g.cm-3 (Descamps et al. 2011; Hanuš et al. 2017;
Shepard et al. 2018), and for its unique dumbbell shape so far (Ostro et al.
2000; Descamps 2015; Shepard et al. 2018). (216) Kleopatra is a M/Xe-type
(DeMeo et al. 2009; Hardersen et al. 2011) triple asteroid system (Descamps et
al. 2011) with a high radar albedo (Shepard et al. 2018), likely implying the
presence of a substantial fraction of metal at its surface.

Here, we present new AO observations of (216) Kleopatra with VLT/SPHERE/ZIMPOL
(Zurich Imaging Polarimeter), which were obtained as part of our ESO large
programme (Sect. 2). Combining these new observations, with disk-integrated
photometry, stellar occultations, and delay-Doppler images, we derived two
shape models with the ADAM and MPCD reconstruction methods (Viikinkoski et al.
2015a; Capanna et al. 2012) (Sect. 3). The images were further used to
constrain the orbital properties of the two moons and thus constrain the mass
of (216) Kleopatra, and hence its density (see Sect. 4 and accompanying paper
by Brož et al.). In Sect. 5, we perform a thorough analysis of Kleopatra’s
shape and propose a formation scenario of this peculiar triple system in Sect.
6.

##  2 Observations & data reduction

###  2.1 Disk-resolved data with SPHERE

Asteroid (216) Kleopatra was observed at two different epochs in July–August
2017 and December 2018 – January 2019, using ZIMPOL of SPHERE (Thalmann et al.
2008) in the classical imaging mode with the narrow band filter (N$\\_$R
filter; filter central wavelength = 645.9 nm, width = 56.7 nm). The angular
size of Kleopatra was in the range of 0.09–0.11$\arcsec$. At the time of the
observations, the asteroid was close to an equator-on geometry. Therefore, the
SPHERE images of Kleopatra obtained from seven epochs allowed us to
reconstruct a reliable 3D shape model with well defined dimensions. The
reduced images were further deconvolved with the Mistral algorithm (Fusco et
al. 2003), using a parametric point-spread function (Fétick et al. 2019).
Table 5 lists information about the images, while Figs. 14 and 15 display all
obtained images with SPHERE.

We complemented our dataset with 14 disk-resolved images obtained by the NIRC2
camera mounted on the W. M. Keck II telescope (Table 6 and Fig. 16). These
data were already compiled and used for Kleopatra’s shape modeling in the
study of Hanuš et al. (2017).

The pixel scale of the Zimpol instrument is 3.6 mas, which is almost a factor
of three improvement compared to the Keck’s NIRC2 camera with a pixel scale of
9.942 mas. We also note that the pixel scale of the VLT/NACO instrument, the
decommissioned predecessor of SPHERE, was 13.24 mas.

###  2.2 Disk-integrated optical photometry

We compiled a rich dataset of Kleopatra’s disk-integrated optical photometry
(180 lightcurves from 15 different apparitions). The oldest data were obtained
in 1977 (Scaltriti & Zappala 1978) and the most recent ones in 2015 (Stéphane
Fauvaud). A large fraction of the data spans seven different apparitions in
the 1980s (e.g., Pilcher & Tholen 1982; Weidenschilling et al. 1987).
Additionally, we used data from apparitions in 1994 (Fauvaud et al. 2001) and
2006 (Warner 2006). Finally, the largest dataset covering apparitions in 2008,
2010, 2011, and 2013 comes from the SuperWASP survey (Grice et al. 2017). The
list of lightcurves is summarized in Table LABEL:tab:lcs; a subset is then
shown in Figs. 18, 19. We did not use any sparse data for their redundancy.
Each lightcurve densely sampled the brightness variations for several hours.
We treated the lightcurves as relative only, so we normalized the fluxes to
unity. We corrected the epochs on light-time effect. Because each lightcurve
is only several hours long, it was not necessary to correct for the phase
angle effect.

###  2.3 Stellar occultations

We utilized five stellar occultations of Kleopatra (Table LABEL:tab:occ,
Herald et al. 2020). While three (from 2009, 2015, and 2016) are of a
sufficient quality to be utilized for the shape modeling (i.e., multiple
chords sampling the object’s projection well), the remaining two (from 1980
and 1991) served as validity checks. We note that the occultation in 1980 is
of particular historical interest as two observers independently spotted a
0.9-second star disappearance too far from the primary to be related with it.
At that time, the scientific community was not yet ready to accept the
existence of tiny moons around asteroids. This, however, changed in 1994 when
the Galileo probe sent images of asteroid (243) Ida with its moon Dactyl.
Fortunately, the awareness about these peculiar data persisted. The evident
explanation of the data is the occultation by one of Kleopatra’s two moons. We
list the suspected moon position in Table 2.

###  2.4 Delay-Doppler images

Delay-Doppler images of Kleopatra were obtained in 2008 and 2013 using the
2380 MHz radar at the Arecibo observatory (Shepard et al. 2018). The 2008
observations were almost equatorial, but with a weak signal-to-noise ratio.
Therefore we did not include them in the modeling. Higher quality observations
in 2013 had an aspect angle of –50∘ from the equatorial plane. The nominal
range resolution is 5.25 km in range and 10 Hz in frequency.

##  3 3D shape modeling

###  3.1 ADAM shape model

Figure 1: Comparison between the VLT/SPHERE/ZIMPOL deconvolved images of
Kleopatra and the corresponding projections of our MPCD and ADAM shape models.
The red arrow indicates the orientation of the spin axis. We used a realistic
illumination to highlight the local topography of the model using the OASIS
software (Jorda et al. 2010). The residuals of both models are shown in the
two bottom rows, more specifically those are chi-square pixel residuals in
units of the instrumental noise associated to each pixel (photon and readout
noise).

All-Data Asteroid Modeling (ADAM, Viikinkoski et al. 2015a, and references
therein) is an inversion algorithm commonly used for the reconstruction of
shape models of asteroids from their combined disk-integrated and disk-
resolved data (Viikinkoski et al. 2015b, 2018; Hanuš et al. 2017). The key
elements of ADAM are the a priori knowledge of the rotation state (i.e.,
sidereal rotation period and spin vector orientation) and the existence of
disk-resolved data. The former is usually available as convex shape models
have been derived for the majority of the largest asteroids (see the Database
of Asteroid Models from Inversion Techniques, DAMIT111
https://astro.troja.mff.cuni.cz/projects/damit/, Ďurech et al. 2010). The most
common disk-resolved data are the high-resolution angular images obtained with
the 8–10m class telescopes equipped with AO systems (Keck, VLT, Gemini), but
also the more scarce delay-Doppler images (Shepard et al. 2018) or the ALMA
interferometry (Viikinkoski et al. 2015b). Finally, stellar occultations can
also be considered as disk-resolved data; however, only those with multiple
chords with proper timings, sampling the asteroid’s on-sky projection well,
are useful for constraining the shape.

We applied ADAM to our dataset of 180 optical lightcurves, 14 disk-resolved
images from Keck, 55 disk-resolved images from SPHERE, three stellar
occultations, and 15 delay-Doppler images from the Arecibo Observatory
(Shepard et al. 2018). We used the rotation state (Hanuš et al. 2017) as an
initial value for the ADAM modeling with a low shape model resolution (1152
facets) and the octantoid shape parametrization (Viikinkoski et al. 2015a).
Then, we increased the shape model resolution (2048 facets) and used the low-
resolution model as a starting point for the shape model improvement. We also
increased the relative weight of the SPHERE data with respect to other
datasets. We show the comparison of the shape model projections with the
corresponding SPHERE and Keck/NIRC2 images in Figs. 1 and 16, and with the
stellar occultations in Fig. 17. The fit to a subset of optical lightcurves is
shown in Figs. 18, 19. Our solution is robust against variations in data
weighting and ADAM regularization functions. We generated several models both
with and without delay-Doppler images. The optimization method used for radar
images is described in detail in Viikinkoski et al. (2015a). We used the
cosine scattering law with constant albedo. By increasing the relative weight
of radar data with respect to AO images, the shape solution approaches the
shape presented in Shepard et al. (2018). The choice of weights between
different data sources is always a somewhat subjective matter. However, in
this case, both the coverage and the resolution of AO images is clearly
superior compared to radar images, so it seems prudent that the AO
observations from SPHERE are given predominance. The comparison between radar
data and the shape model is shown in Fig. 20.

Our final ADAM shape model fits all datasets sufficiently well. Specifically,
we have not identified any substantial disagreement between the model and the
data. Considering the superior quality, the resolution, and the coverage of
the SPHERE data, the shape model is already well constrained by them. The
other data (Keck, occultations, radar) are usually fitted naturally and are
mostly complementary.

###  3.2 MPCD shape model

The Multiresolution PhotoClinometry by Deformation (MPCD, Capanna et al. 2012;
Jorda et al. 2016) is a 3D shape reconstruction method that utilizes an
initial shape model to give a better fit to disk-resolved images. Therefore,
our ADAM shape model is further modified by MPCD by fitting solely the high-
resolution SPHERE AO data. The MPCD algorithm minimizes the chi-square pixel-
to-pixel differences between a set of observed images and the synthetic images
built from the shape model for optimization. The reflectance function is
Hapke’s five-parameters function with the parameters listed in Descamps et al.
(2011). The shape is optimized through shifts of the vertices along the local
normal vector. The method goes through several increasing steps of resolutions
of both the observed images and shape before converging toward the final
optimized model. In the case of (216) Kleopatra, we used the sample of 33
SPHERE images obtained during 11 visits. The shape was reconstructed starting
from a decimated ADAM model with only 196 facets and ending with a final model
of 3136 facets, after optimization in three levels. Furthermore, we also
optimized the Euler angles describing the orientation of the spin pole after
noticing unusual systematic residuals between our sets of observed and
synthetic images. The shape was reoptimized with this new pole orientation in
the same manner to produce the final MPCD model of (216) Kleopatra. As
expected, the chi-square between observed and synthetic AO images decreases
from 135 (ADAM model) to 50 (MPCD model). A comparison between synthetic
SPHERE images generated from the ADAM and MPCD models and an observed image
for each visit is shown in Fig. 1. The final MPCD model is shown in Fig. 2
alongside the ADAM model and the radar model of Shepard et al. (2018). The
physical properties of those three models are listed in Tab. 1. The MPCD
method also provides an albedo map calculated together with the slope and
height errors of each facet from their corresponding average residual pixel
values. However, the only significant albedo features are found near limbs and
a careful inspection shows that they likely correspond to faint artifacts
introduced by the deconvolution process (Fétick et al. 2019) at the edges of
the object.

Radar | ADAM | MPCD
---|---|---
|  |
|  |
Figure 2: Rendered views of the radar shape model of Shepard et al. (2018) (left panels), and ADAM and MPCD models (middle and right panels) as seen from the $Y-$ (top panels) and $Z+$ (bottom panels) body axes. The mass deficit or crater is visible on the small lobe of the MPCD model. Table 1:  Physical properties of (216) Kleopatra based on ADAM and MPCD shape modeling of our VLT/SPHERE images: Sidereal rotation period $P$, spin-axis ecliptic J2000 coordinates $\lambda$ and $\beta$, volume-equivalent diameter $D$, dimensions along the major axis $a$, $b$, $c$, their ratios $a/b$ and $b/c$, mass $M$ from Brož et al. (accompanying paper), volume $V$, and bulk density $\rho$. Uncertainties correspond to 1 $\sigma$ values. The values based on radar data (Shepard et al. 2018) are also reported. The $b$ and $c$ extents reported for the radar model are the maximum extents in those directions. Parameter | S18 | ADAM | MPCD
---|---|---|---
$P$ (h) | 5.385280(1) | 5.385282(1) | 5.385282(1)
$\lambda$ ($\degr$) | 74$\pm$ 2 | 73.5$\pm$0.5 | 74.1$\pm$0.5
$\beta$ ($\degr$) | 20$\pm$ 2 | 20.8$\pm$0.5 | 21.6$\pm$0.5
$D$ (km) | 122$\pm$ 10 | 119.3 $\pm$ 2 | 118.2$\pm$0.8
$a$ (km) | 276$\pm$ 14 | 270$\pm$4 | 267$\pm$6
$b$ (km) | 94$\pm$ 5 | 62$\pm$4 | 61$\pm$6
$c$ (km) | 78$\pm$ 4 | 38$\pm$4 | 48$\pm$6
$a/b$ | 2.9 | 4.35$\pm$0.3 | 4.4$\pm$0.4
$b/c$ | 1.20 | 1.63$\pm$0.2 | 1.3$\pm$0.2
$V$ ($10^{5}$ km3) | 9.56 | 8.90$\pm$0.45 | 8.65$\pm$0.17
$M$ ($10^{18}$ kg) | 4.64$\pm$0.02 | 2.97$\pm$0.32 | 2.97$\pm$0.32
$\rho$ (g cm-3) | 4.9$\pm$0.5 | 3.34$\pm$0.53 | 3.43$\pm$0.38
$\chi^{2}$ | 209 | 135 | 50

##  4 Mass and bulk density

In the first step, each image obtained with SPHERE/ZIMPOL was further
processed to remove the bright halo surrounding Kleopatra, following the
procedure described in detail in Yang et al. (2016) and Pajuelo et al. (2018),
and shown in Fig. 3. The residual structures after the halo removal were
minimized using the processing techniques introduced in Wahhaj et al. (2013),
where the radial structures were removed using a running median in a
$\sim\,$30-pixel box in the radial direction and the images were smoothed by
convolving a Gaussian function with a FWHM of $\sim\,$8 pixels.

In the second step, we measured the relative positions on the plane of the sky
between Kleopatra and its satellites. We used the unmodified images (i.e.,
without halo removal) and we fit both photocenters by a suitable 2D Gaussian
(see Carry et al. 2019). The dispersion of the Gaussian function for the moons
was chosen conservatively as comparable to the residual (AO-corrected) PSF.
The astrometric positions are reported in Table 2. Sometimes, the
identification of the two satellites was ambiguous. Nevertheless, it was
possible to recover the correct identification later (see Brož et al.). We
report the corrected data here.

Furthermore, we estimated the offsets between the photocenter and the center
of mass for Kleopatra. Because the central body is so extended and irregular,
the offset may reach up to a few milli-arcseconds, as reported in Table 3. We
use these offset adjustments for further analysis of orbits because our
dynamical model requires the centers of mass, not the centers of light.
Alternatively, one can use a relative astrometry of the two moons (second with
respect to the first), which is unaffected by these photocenter motions.

Uncertainties in the measurements are approximately 0.01 arcsec, based on
repeated and/or close-in-time measurements. As of now, we do not account for
the orbital motion of the satellites during five consecutive exposures and we
take their average position, although in principle that motion could be
detected. It would however require a fitting by an asymmetric PSF, elongated
along the orbital motion.

The dynamical model required to interpret the motion of the moons is more
complex due to the irregular shape of Kleopatra, and the mutual interactions
of the moons and the solar tides. We thus used an advanced n-body model with
the multipole expansion up to the order of $\ell=10$, which is described in
detail in Brož et al., in order to determine the orbital elements. Our best
solution fits the observed positions with a root mean square (RMS) residuals
of 17 mas. Most importantly, the phase coverage of new VLT/SPHERE observations
allowed us to derive the true periods $P_{1}=(1.822359\pm 0.004156)\,{\rm d}$
and $P_{2}=(2.745820\pm 0.004820)\,{\rm d}$, which results in the revised mass
$m_{1}=(2.97\pm 0.32)\cdot 10^{18}\,{\rm kg}$ for Kleopatra. This is
significantly lower than the previously reported value of $4.64\cdot
10^{18}\,{\rm kg}$ (Descamps et al. 2011). The orbits of both satellites are
circular, prograde, and equatorial, similar to most known satellites around
large main belt asteroids. (e.g., Marchis et al. 2008; Berthier et al. 2014;
Margot et al. 2015; Carry et al. 2019; Yang et al. 2020b).

Taking the average volume of the ADAM and MPCD models (Table 1), the density
of Kleopatra amounts to $(3.38\pm 0.50)\,{\rm g}\,{\rm cm}^{-3}$. A comparison
with the previous estimate (3.6 ${\rm g}\,{\rm cm}^{-3}$, Descamps et al.
2011) is not pertinent given that both the mass and the volume were revised.
This density has important implications for the interpretation of the shape in
Sec. 5.

Figure 3: Processed ZIMPOL images on the left, revealing the presence of the
satellites, CleoSelene and AlexHelios around (216) Kleopatra at two epochs
(bottom: 2017-07-14, top: 2017-08-22). To reveal the moons which are as faint
as the halo (due to imperfect AO correction), we subtracted a rotational
average of the image centered on the primary (right image). The circle points
to the location of the satellites in the images. The other dots in the images
are bad pixels. Table 2: Positions of Kleopatra’s satellites with respect to
its photocenter. Uncertainties are approximately 0.01 arcsec. The position
from 1980 is based on the stellar occultation and those from 2008 are taken
from Descamps et al. (2011). AlexHelios
---
Date | UT |  $u$ [arcsec] |  $v$ [arcsec]
1980-10-10 | 07:00 | $0.2711$ | $0.4564$
2008-09-19 | 11:38:00 | $-0.18\phantom{00}$ | $0.35\phantom{00}$
2008-09-19 | 11:51:00 | $-0.20\phantom{00}$ | $0.36\phantom{00}$
2008-10-05 | 09:13:00 | $-0.27\phantom{00}$ | $0.37\phantom{00}$
2008-10-05 | 09:49:00 | $-0.29\phantom{00}$ | $0.39\phantom{00}$
2008-10-05 | 10:03:00 | $-0.32\phantom{00}$ | $0.39\phantom{00}$
2008-10-09 | 05:46:00 | $-0.32\phantom{00}$ | $0.29\phantom{00}$
2017-07-14 | 05:00:59 | $-0.4158$ | $0.2952$
2017-07-22 | 04:18:07 | $-0.4262$ | $0.2444$
2017-07-22 | 05:00:55 | $-0.4291$ | $0.2614$
2017-08-22 | 01:42:34 | $0.1843$ | $-0.3150$
2018-12-10 | 06:47:17 | $-0.4894$ | $-0.0958$
2018-12-19 | 06:45:02 | $0.1871$ | $-0.2869$
2018-12-22 | 05:58:43 | $-0.2540$ | $-0.3438$
2018-12-26 | 08:14:42 | $-0.4973$ | $-0.0428$
2019-01-14 | 04:57:43 | $-0.5319$ | $-0.1177$
CleoSelene
2008-09-19 | 06:17:00 | $-0.25\phantom{00}$ | $0.50\phantom{00}$
2008-09-19 | 08:44:00 | $-0.34\phantom{00}$ | $0.54\phantom{00}$
2008-09-19 | 11:38:00 | $-0.44\phantom{00}$ | $0.57\phantom{00}$
2008-09-19 | 11:51:00 | $-0.44\phantom{00}$ | $0.57\phantom{00}$
2008-09-19 | 12:02:00 | $-0.44\phantom{00}$ | $0.58\phantom{00}$
2008-10-05 | 09:13:00 | $-0.29\phantom{00}$ | $0.46\phantom{00}$
2008-10-05 | 09:49:00 | $-0.31\phantom{00}$ | $0.47\phantom{00}$
2008-10-05 | 10:03:00 | $-0.32\phantom{00}$ | $0.46\phantom{00}$
2008-10-09 | 09:36:00 | $0.28\phantom{00}$ | $-0.44\phantom{00}$
2017-07-14 | 05:01:00 | $0.3070$ | $-0.2600$
2017-07-22 | 04:18:07 | $-0.2714$ | $0.2941$
2017-07-22 | 05:00:55 | $-0.2621$ | $0.2963$
2017-08-22 | 01:42:34 | $0.4423$ | $-0.2401$
2018-12-10 | 06:47:17 | $0.1949$ | $-0.1505$
2018-12-19 | 06:45:02 | $0.3763$ | $0.0929$
2018-12-22 | 05:58:43 | $0.4555$ | $-0.0119$
2018-12-26 | 08:14:42 | $0.3937$ | $0.1271$
2019-01-14 | 04:57:43 | $-0.1008$ | $-0.3298$
Table 3: Positions of the photocenter minus the center of mass for Kleopatra. JD | UT |  $u$ [mas] |  $v$ [mas]
---|---|---|---
2008-09-19 | 06:17:00 | $0.762$ | $0.789$
2008-09-19 | 08:44:00 | $-0.948$ | $-1.395$
2008-09-19 | 11:38:00 | $1.178$ | $1.069$
2008-09-19 | 11:51:00 | $-1.462$ | $-0.737$
2008-09-19 | 12:02:00 | $-3.723$ | $-1.784$
2008-10-05 | 09:13:00 | $4.586$ | $2.911$
2008-10-05 | 09:49:00 | $4.059$ | $1.363$
2008-10-05 | 10:03:00 | $2.212$ | $-0.395$
2008-10-09 | 05:46:00 | $0.548$ | $-2.190$
2008-10-09 | 09:36:00 | $6.923$ | $5.164$
2017-07-14 | 05:01:00 | $0.223$ | $-1.194$
2017-07-22 | 04:18:07 | $0.609$ | $1.934$
2017-07-22 | 05:00:55 | $0.666$ | $1.929$
2017-08-22 | 01:42:34 | $0.288$ | $0.901$
2018-12-10 | 06:47:17 | $-1.674$ | $-2.402$
2018-12-19 | 06:45:02 | $0.391$ | $-3.132$
2018-12-22 | 05:58:43 | $2.014$ | $-1.430$
2018-12-26 | 08:14:42 | $-0.848$ | $0.791$
2019-01-14 | 04:57:43 | $-0.455$ | $-1.266$

##  5 Shape analysis

###  5.1  Lobes interpretation

A visual inspection of the global shape confirms the presence of two lobes
separated by a neck, at the first order similar to the shape derived in Ostro
et al. (2000). For this reason, we extracted the individual shapes of the two
lobes from the MPCD global shape model in order to characterize their physical
properties. We used an approach similar to that applied to comet 67P/C-G
(Jorda et al. 2016). In this approach, the facets belonging to each lobe are
manually selected in the “Meshlab software” (Cignoni et al. 2008). The best-
fit ellipsoid of each lobe is then computed by fitting the coordinates of the
extracted vertices. Finally, the lobe models are merged with those of the
best-fit ellipsoids to compute their closed shapes and volumes (for details on
the method, see Jorda et al. 2016). This leads to the individual shapes of the
two lobes shown in Fig. 4.

Table 4:  Physical parameters (volume and diameters) of the two lobes computed from their reconstructed shape models, as well as geometric parameters (center coordinates, Euler angles, and tilt of the lobes’ Z-axis with respect to that of the object), resulting from the best-fit ellipsoid of the selected facets of the MPCD shape model belonging to each lobe. Parameter | Lobe A | Lobe B
---|---|---
Diameter $a$ (km) | 118 | 126
Diameter $b$ (km) | 94 | 79
Diameter $c$ (km) | 66 | 61
Volume ($10^{5}$ km3) | 4.1 | 3.5
Volume (%) | 47 | 40
Volume-equiv. diameter $D$ (km) | 91.9 | 87.1
Center coordinate $X$ (km) | –69.9 | 79.8
Center coordinate $Y$ (km) | 0.0 | –0.5
Center coordinate $Z$ (km) | 0.2 | –0.9
Euler angle $\psi$($\degr$) | 39.0 | –21.7
Euler angle $\theta$($\degr$) | –5.6 | 5.0
Euler angle $\phi$($\degr$) | –35.2 | 26.3
Z-axis tilt($\degr$) | 5.6 | 5.0
Ellipsoid fit residuals (km) | 2.2 | 1.8
Ellipsoid fit residuals (%) | 3.1 | 2.4

We determined the diameters of the two lobes along the X-, Y-, and Z-axes and
along their principal axes of inertia, as well as their volume (summarized in
Table 4). It appears that the X- and Z- axes have the same diameters within
their error bars. However, the Y-axis is significantly different between the
two lobes, leading to a volume difference of 16 % in favor of lobe A.
Furthermore, both lobes appear highly ellipsoidal, with a deviation between
the lobes and their best-fit ellipsoids of only $\sim\\!2.5-3\,\%$, a value
comparable to those found for large asteroids with equilibrium shapes, such as
10 Hygiea (Vernazza et al. 2020) and 4 Vesta (Ferrais et al. 2020). The size
difference between the two lobes along the Y-axis can possibly be explained by
a depression observed on lobe B (see Fig. 4), possibly formed by an impact.

As a next step, we computed the length and mean radii of the neck as well as
its volume from the parameters of Table 4. Its volume appears to be 13 % of
the total volume of the object, whereas the minimum length of the neck along
the X-axis is $\sim 25$ km.

Figure 4:  Views of the two lobes A (blue) and B (red) extracted from the
shape model as seen from the Y+ (left) and Z+ (right) axes. The initial shape
model is displayed as a wireframe for comparison, allowing one to visualize
the neck between the two lobes.

###  5.2 Dumbbell interpretation

It is striking that in Fig. 2, the shape of Kleopatra resembles the “dumb-
bell” equilibrium shapes studied by Descamps (2015). To test whether Kleopatra
formed at equilibrium, we computed the rms of the deviation between the MPCD
shape model of Kleopatra and several shape models of dumbbell equilibrium
figures corresponding to different values of the normalized angular velocity
$\Omega$ defined by Descamps (2015) as $\omega\sqrt{3/(4\pi G\rho)}$, where
$\rho$ denotes the bulk density. In order to perform this comparison, we
rescaled the equilibrium shapes so that their lengths match that of Kleopatra
and we shifted the center of the figure by $-3.8\,{\rm km}$ along the X-axis.
This shift leads to a displacement of the center of mass toward the larger
lobe (lobe A), which is coherent if we assume that the two lobes have the same
density. The lowest final rms of the distance between the models is equal to
$\sim 3.5$ km for a normalized angular velocity $\Omega=0.32-0.33$, but
figures with values of $\Omega$ in the range from $0.31-0.34$ remain
compatible with the shape of Kleopatra (rms below 4 km), as illustrated in
Fig. 5. The $\Omega$ value calculated from Kleopatra’s current rotation period
$5.385\,{\rm h}$ and density $3.38\pm 0.50\,{\rm g}\,{\rm cm}^{-3}$ is
$0.334$, which is in striking agreement with the values reported above. We
thus confirm that Kleopatra is an equilibrium shape.

Figure 5:  Views of the dumbbell equilibrium shapes for values of the
normalized angular velocity $\Omega$ equal to $0.31$ (top left), $0.32$ (top
right), $0.33$ (middle left), and $0.34$ (middle right). The two bottom panels
show the reconstructed ADAM (left) and MPCD (right) shapes. All shape models
are seen from the $Y-$ axis.

###  5.3 Surface acceleration

To probe the effect of the shape on the internal compaction, we computed the
gravitational plus the centrifugal acceleration at each point of the surface
and compared it with that of a sphere of equivalent mass (Fig. 6). It appears
that the mean acceleration at the surface of Kleopatra amounts to 76% of that
of a sphere of equivalent mass with values below 50% locally. It follows that
Kleopatra’s highly elongated shape does not favor compaction and it supports a
higher macroprosity compared to more spherical or ellipsoidal bodies of an
equivalent mass.

We also computed tangential surface accelerations $|a_{\mathrm{t}}|$, which
indicate possible material motion (Fig. 7). It seems that a convergence is
located on the left lobe, at (x, y, z) = ($-$120, 20, 20) km, or on the neck
($\pm$20, 0, 20) km. On the other hand, a divergence is on the ”hill” at (50,
15, $-$35) km. These locations are common to both ADAM and MPCD models. We do
not report on the locations with possible shape artifacts. The maximum
accelerations reach $\sim$1 cm/s2. Whether this is sufficient to sustain
global motion is uncertain because it depends on local topography, regolith
structure, roughness, friction, impacts, seismicity, etc. Alternatively,
cratering impacts can initiate ballistic transport, with complex near-surface
dynamics, determined by the proximity of the critical L1 equipotential.

Figure 6:  Local acceleration at the surface of Kleopatra normalized by that
of a sphere of an equivalent radius and spin period. View from the $Z+$ (left)
axis and oblique view along the equator (right). We note the very low value at
the edges and higher values around the two lobes.

Figure 7:  Tangential surface accelerations $|a_{\mathrm{t}}|$ computed for
ADAM (top) and MPCD (bottom) models. Views on the left are north pole-on
($+z$), while those on the right are south pole-on ($-z$).

###  5.4 Critical equipotentials

The effective potential $U_{\rm eff}=U_{\rm g}-\frac{1}{2}\omega(x^{2}+y^{2})$
was computed using the same algorithm as in Appendix A. We plotted its
equipotentials together with the four critical points in Fig. 8. The major
result is that the shape extends to a distance that is very close to the
critical L1 equipotential. In fact, the ADAM shape model almost touches it at
one point in the $(x,y)$ plane. At the same time, they are separated by
several (or more) kilometers in the $(x,z)$ plane. An analogous analysis of
the MPCD shape shows a very similar result (see Figs. 9 and 10). A minor
difference is that the shape touches the equipotential at two different points
along its $(x,y)$ and $(x,z)$ circumference. The closest distance is less than
a kilometer.

Our work includes three major differences with respect to Hirabayashi &
Scheeres (2014). (i) We used the currently observed $P=5.385\,{\rm h,}$
together with $\rho=3\,380\,{\rm kg}\,{\rm m}^{-3}$. Consequently, we did not
need any mechanism for a spin-down (e.g., from $2.8\,{\rm h}$) to explain why
the shape is critical. (ii) We did not use any scaling. Instead, the absolute
volume is constrained by the AO observations. (iii) Our L1 point is on the
right ($+x$) and not on the left, implying that the shape model presented here
is different to the previous one used by Hirabayashi & Scheeres (2014) . This
possibly affects near-surface dynamics and surface locations from which
material is more likely to escape.

Figure 8:  Effective potential $U_{\rm eff}$ in the $(x,y)$ plane (gray
lines), critical equipotentials (blue lines), and the ADAM shape model
(orange). The density is $\rho=3.34\,{\rm g}\,{\rm cm}^{-3}$ and the rotation
period $P=5.385\,{\rm h}$. Four critical points are denoted: L1, L2, L3, and
L4. The L1 critical point is on the right. The L1 equipotential touches the
surface of Kleopatra.  Figure 9: Effective potential $U_{\rm eff}$ in the
$(x,y)$ plane (gray lines), critical equipotentials (blue lines), for the MPCD
model (orange) and density $3.43\,{\rm g}\,{\rm cm}^{-3}$.  Figure 10:
Effective potential $U_{\rm eff}$ in the $(x,z)$ plane (gray lines), critical
equipotentials (blue lines), for the MPCD model (orange) and density
$3.43\,{\rm g}\,{\rm cm}^{-3}$.

##  6 Discussion

Our density estimate of $(3.38\pm 0.50)\,{\rm g}\,{\rm cm}^{-3}$ for Kleopatra
is surprising, considering its high radar albedo of 0.43 $\pm$ 0.10 (Shepard
et al. 2018) that implies a high surface bulk density and a large metal
content. In comparison, the density of metal-rich asteroid (16) Psyche is
$(4.2\pm 0.6)\,{\rm g}\,{\rm cm}^{-3}$ (Ferrais et al. 2020). One possible
explanation for such a low density for Kleopatra is the presence of
substantial porosity within the body. Wilson et al. (1999) showed that
gravitationally reaccreted asteroids should have porosities of $\sim$20–40%.
It is of great interest that the highly elongated shape of Kleopatra actually
supports a higher macroporosity than that expected for a spherical or
ellipsoidal body. Accordingly, acknowledging a rubble-pile structure of
Kleopatra from its specific angular momentum, this range of porosities implies
a grain density in the 4.2–5.8 ${\rm g}\,{\rm cm}^{-3}$ range, suggestive of a
mixture of NiFe metal-rich with the inclusion of silicates (Marchis et al.
2003). The presence of silicates is supported by the presence of a 0.9 micron
band in Kleopatra’s spectrum (Hardersen et al. 2011).

This rubble-pile structure along with its near equilibrium shape is compatible
with a formation scenario including a giant impact (Sugiura et al. 2018),
followed by reaccumulation during which Kleopatra behaved as a fluid as
suggested by Descamps (2010). The dumbbell equilibrium shape also explains the
very unusual long neck between the two lobes. The small volume deficit due to
the depression observed on lobe B (see Figures 2 and 4) is coherent with a
later smaller impact.

The critical rotation would be 5.250 h (compared to 5.385 h) if we require the
$L_{1}$ equipotential to be in contact with the surface. Similarly, the
critical density should be 3.2 ${\rm g}\,{\rm cm}^{-3}$. The whole surface
actually does not follow the $L_{1}$ equipotential exactly, since even very
small impacts could eject fragments beyond the equipotential surface.
Moreover, the critical value of $P$ (as well as $\rho$) is sensitive to small
variations in the topography.

Finally, the low gravity at the edges and along the equator of the body,
together with its rubble-pile structure and the equatorial orbits of the
moons, opens the possibility that the latter formed via mass shedding. The
large offset between the L1 equipotential and the surface at the edges of the
asteroid supports this interpretation.

##  7 Summary & conclusion

New AO observations of the triple system (216) Kleopatra with VLT/SPHERE
allowed us to constrain the 3D shape and mass of the primary with high
accuracy (see Brož et al., companion paper for the mass estimate). Both mass
and volume estimates of (216) Kleopatra imply a low density of $(3.38\pm
0.50)\,{\rm g}\,{\rm cm}^{-3}$. Such low density for a metallic asteroid
suggests the presence of substantial porosity within the metal-rich primary or
a significant content of silicate in the composition of the asteroid.

This rubble-pile structure along with its near equilibrium shape is compatible
with a formation scenario including a giant impact followed by reaccumulation.
(216) Kleopatra’s rotation period and dumbbell shape imply that it is in a
critically rotating state. The low gravity at the edges and along the equator
of the body, together with its rubble-pile structure and the equatorial orbits
of the moons, opens the possibility that the latter formed via mass shedding
as suggested by Descamps et al. (2008).

Future observations of (216) Kleopatra with current AO systems such as
SPHERE/ZIMPOL could reveal long-term perturbations in the moon orbits related
to the shape of the primary. Similar observations of the primary, but at low
phase angles (less than 5 deg), could also provide more accurate contours and
thus help refine its shape.

We can speculate that high resolution images of Kleopatra’s surface could help
to truly understand the origin of the moons by revealing the presence of
surface heterogeneities (e.g., albedo variegations), or anomalies such as
concavities, that would help to link the moons directly to their parent body.
Future observations with high angular resolution imaging data provided by the
next generation of extremely large telescopes could help marginally by
providing color and spectroscopic constraints on the moons and refining the
shape model of the primary. A future space mission to (216) Kleopatra and its
two moons CleoSelene and AlexHelios would definitively shed light on the
origin and current dynamics of this complex system. In situ measurements
could, for instance, reveal the ejection of particles from Kleopatra similar
to what was recently seen on the near-Earth asteroid (101955) Bennu (Lauretta
et al. 2019).

###### Acknowledgements.

This material is partially based upon work supported by the National Science
Foundation under Grant No. 1743015. This work has been supported by the Czech
Science Foundation through grant 20-08218S (J. Hanuš, J. Ďurech), 21-11058S
(M. Brož) and by the Charles University Research program No. UNCE/SCI/023. P.
Vernazza, A. Drouard, M. Ferrais and B. Carry were supported by CNRS/INSU/PNP.
M.M. was supported by the National Aeronautics and Space Administration under
grant No. 80NSSC18K0849 issued through the Planetary Astronomy Program. The
work of TSR was carried out through grant APOSTD/2019/046 by Generalitat
Valenciana (Spain). This work was supported by the MINECO (Spanish Ministry of
Economy) through grant RTI2018-095076-B-C21 (MINECO/FEDER, UE). The research
leading to these results has received funding from the ARC grant for Concerted
Research Actions, financed by the Wallonia-Brussels Federation. TRAPPIST is a
project funded by the Belgian Fonds (National) de la Recherche Scientifique
(F.R.S.-FNRS) under grant FRFC 2.5.594.09.F. TRAPPIST-North is a project
funded by the University of Liège, and performed in collaboration with Cadi
Ayyad University of Marrakesh. E. Jehin is a FNRS Senior Research Associate.
The data presented herein were obtained partially 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. The
authors wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Maunakea 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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##  Appendix A Surface slopes

Another useful characteristics is the statistics of surface slopes. We
computed the slope $\alpha$ as the angle between the local normal and the
total acceleration in the corotating frame. To evaluate the respective
volumetric integrals, we triangulated the volume via the Tetgen program (Si
2006), with 24099 elements, and the brute-force algorithm. The result is shown
in Fig. 11. Most of the slopes are between $2^{\circ}$ and $15^{\circ}$.
Steeper slopes (up to $20^{\circ}$) are located on the lobes, at the largest
distances from the $z$-axis, and some of them are on the neck. A few outliers
(over $25^{\circ}$) are related to individual facets, which might be elongated
or degenerate.

For comparison, we also show the slopes computed on the basis of the MPCD
model (Fig. 12). This shape model has a lower number of faces which are more
regular. Interestingly, the steeper slopes are located elsewhere, mostly on
the neck. The slopes on the $+x$ lobe are shallower (around $10^{\circ}$).
Given the nature of the shape models (ADAM versus MPCD), we think that local
normals are better constrained by the MPCD model. The overall statistics (Fig.
13) from both models is comparable, though. A comparison to Shepard et al.
(2018) shows that our mean slope is significantly lower ($9^{\circ}$ versus
$12^{\circ}$). Let us recall, however, that this difference might be partly
”enforced” by our/their regularization techniques.

Figure 11:  Surface slopes $\alpha$ (shown as color) for the ADAM shape model.
They were computed for centers of faces and gravitational plus centrifugal
accelerations. The density is $\rho=3\,340\,{\rm kg}\,{\rm m}^{-3}$ and the
rotation period $P=5.385\,{\rm h}$. Higher slopes are present on the lobes.
Figure 12:  Same as Fig. 11, but for the MPCD model and density
$\rho=3\,430\,{\rm kg}\,{\rm m}^{-3}$. Higher slopes are on the neck.  Figure
13:  Histograms of slopes $\alpha$ for the ADAM and MPCD models. They are most
commonly between $2^{\circ}$ and $15^{\circ}$. The distribution is a bit wider
for the ADAM model, with a few outliers. For comparison, we also plotted a
histogram from Shepard et al. (2018).

##  Appendix B Additional figures and tables

Figure 14: Full set of VLT/SPHERE/ZIMPOL images of (216) Kleopatra from the
apparition in 2017. The images were deconvolved by the Mistral algorithm. The
pixel scale is 3.6 mas. Additional information about the data can be found in
Table 5.

Figure 15: Full set of VLT/SPHERE/ZIMPOL images of (216) Kleopatra from the
apparition in 2018/2019. The images were deconvolved by the Mistral algorithm.
The pixel scale is 3.6 mas. Additional information about the data can be found
in Table 5.

Figure 16: Comparison between the disk-resolved images from the Keck/NIRC2
camera and the corresponding projections of our ADAM shape model. The red
arrow indicates the orientation of the spin axis. The ordering of the images
follows the chronological order in Table 6. Given the lower resolution of the
Keck data (pixel scale 9.942 mas) compared to those of the SPHERE (pixel scale
3.6 mas), the agreement between the images and the model projections is quite
reasonable and sufficient.

Figure 17: Comparison between the stellar occultations of Kleopatra (Herald et
al. 2020) and the corresponding projections of our ADAM shape model. Dashed
lines are the negative observations. Occultations are ordered chronologically;
the first two were not included in the ADAM shape modeling and are shown here
for consistency reasons.

Figure 18: Comparison between a subset of optical lightcurves of Kleopatra and
the corresponding modeled lightcurves based on our ADAM shape model. The fit
to all lightcurves from our dataset is available in the DAMIT database.

Figure 19: Comparison between a subset of optical lightcurves of Kleopatra and
the corresponding modeled lightcurves based on our ADAM shape model. The fit
to all lightcurves from our dataset is available in the DAMIT database.

Figure 20: Comparison between the delay Doppler images (right) and the
synthetic images (middle) based on the best-fitting MPCD shape solution. Three
images at slightly different observation times ($0.6\%$ of full rotation)
overlap to simulate the effect of integration and to improve the contrast. The
third column shows the plane of the sky view of the asteroid and the direction
of the rotation axis.

Table 5: VLT/SPHERE disk-resolved images obtained in the N_R filter by the ZIMPOL camera within the ESO’s large program. For each observation, we provide the epoch, the exposure time, the airmass, the distance to the Earth $\Delta$ and the Sun $r$, the phase angle $\alpha$, and the angular diameter $D_{\mathrm{a}}$ of Kleopatra. 241 $s$ (176 $s$) total exposure time corresponds to 1.24% (0.91%) of the rotation period. Date | UT | Exp | Airmass | $\Delta$ | $r$ | $\alpha$ | $D_{\mathrm{a}}$
---|---|---|---|---|---|---|---
|  | (s) |  | (AU) | (AU) | (°) | (″)
2017-07-14 | 5:00:59 | 241 | 1.11 | 1.73 | 2.69 | 8.9 | 0.095
2017-07-14 | 5:05:09 | 241 | 1.11 | 1.73 | 2.69 | 8.9 | 0.095
2017-07-14 | 5:09:17 | 241 | 1.11 | 1.73 | 2.69 | 8.9 | 0.095
2017-07-14 | 5:13:26 | 241 | 1.11 | 1.73 | 2.69 | 8.9 | 0.095
2017-07-14 | 5:17:36 | 241 | 1.11 | 1.73 | 2.69 | 8.9 | 0.095
2017-07-22 | 4:18:07 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 4:22:16 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 4:26:27 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 4:30:37 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 4:34:46 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 5:00:54 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 5:05:06 | 241 | 1.11 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 5:09:17 | 241 | 1.12 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 5:13:25 | 241 | 1.12 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-22 | 5:17:33 | 241 | 1.12 | 1.69 | 2.66 | 7.9 | 0.097
2017-07-27 | 4:11:03 | 241 | 1.11 | 1.68 | 2.65 | 7.9 | 0.098
2017-07-27 | 4:15:12 | 241 | 1.11 | 1.68 | 2.65 | 7.9 | 0.098
2017-07-27 | 4:19:23 | 241 | 1.11 | 1.68 | 2.65 | 7.9 | 0.098
2017-07-27 | 4:23:33 | 241 | 1.11 | 1.68 | 2.65 | 7.9 | 0.098
2017-07-27 | 4:27:42 | 241 | 1.11 | 1.68 | 2.65 | 7.9 | 0.098
2017-08-10 | 5:05:17 | 241 | 1.25 | 1.67 | 2.61 | 10.5 | 0.098
2017-08-10 | 5:09:29 | 241 | 1.26 | 1.67 | 2.61 | 10.5 | 0.098
2017-08-10 | 5:13:40 | 241 | 1.28 | 1.67 | 2.61 | 10.5 | 0.098
2017-08-10 | 5:17:48 | 241 | 1.29 | 1.67 | 2.61 | 10.5 | 0.098
2017-08-10 | 5:21:58 | 241 | 1.31 | 1.67 | 2.61 | 10.5 | 0.098
2017-08-22 | 1:42:34 | 241 | 1.10 | 1.71 | 2.58 | 13.9 | 0.096
2017-08-22 | 1:46:44 | 241 | 1.10 | 1.71 | 2.58 | 13.9 | 0.096
2017-08-22 | 1:50:54 | 241 | 1.10 | 1.71 | 2.58 | 13.9 | 0.096
2017-08-22 | 1:55:03 | 241 | 1.09 | 1.71 | 2.58 | 13.9 | 0.096
2017-08-22 | 1:59:11 | 241 | 1.09 | 1.71 | 2.58 | 13.9 | 0.096
2018-12-10 | 6:41:03 | 176 | 1.13 | 1.54 | 2.37 | 15.9 | 0.107
2018-12-10 | 6:44:11 | 176 | 1.13 | 1.54 | 2.37 | 15.9 | 0.107
2018-12-10 | 6:47:17 | 176 | 1.13 | 1.54 | 2.37 | 15.9 | 0.107
2018-12-10 | 6:50:22 | 176 | 1.13 | 1.54 | 2.37 | 15.9 | 0.107
2018-12-10 | 6:53:28 | 176 | 1.13 | 1.54 | 2.37 | 15.9 | 0.107
2018-12-19 | 6:45:02 | 176 | 1.12 | 1.51 | 2.40 | 12.9 | 0.109
2018-12-19 | 6:48:07 | 176 | 1.13 | 1.51 | 2.40 | 12.9 | 0.109
2018-12-19 | 6:51:13 | 176 | 1.13 | 1.51 | 2.40 | 12.9 | 0.109
2018-12-19 | 6:54:19 | 176 | 1.13 | 1.51 | 2.40 | 12.9 | 0.109
2018-12-19 | 6:57:24 | 176 | 1.13 | 1.51 | 2.40 | 12.9 | 0.109
2018-12-22 | 5:46:19 | 176 | 1.12 | 1.50 | 2.40 | 12.0 | 0.109
2018-12-22 | 5:49:27 | 176 | 1.12 | 1.50 | 2.40 | 12.0 | 0.109
2018-12-22 | 5:52:32 | 176 | 1.12 | 1.50 | 2.40 | 12.0 | 0.109
2018-12-22 | 5:55:38 | 176 | 1.12 | 1.50 | 2.40 | 12.0 | 0.109
2018-12-22 | 5:58:43 | 176 | 1.12 | 1.50 | 2.40 | 12.0 | 0.109
2018-12-26 | 8:08:27 | 176 | 1.37 | 1.50 | 2.41 | 10.7 | 0.109
2018-12-26 | 8:11:34 | 176 | 1.38 | 1.50 | 2.41 | 10.7 | 0.109
2018-12-26 | 8:14:41 | 176 | 1.40 | 1.50 | 2.41 | 10.7 | 0.109
2018-12-26 | 8:17:45 | 176 | 1.41 | 1.50 | 2.41 | 10.7 | 0.109
2018-12-26 | 8:20:51 | 176 | 1.43 | 1.50 | 2.41 | 10.7 | 0.109
2019-01-14 | 4:57:42 | 176 | 1.13 | 1.52 | 2.46 | 8.4 | 0.108
2019-01-14 | 5:00:49 | 176 | 1.14 | 1.52 | 2.46 | 8.4 | 0.108
2019-01-14 | 5:03:55 | 176 | 1.14 | 1.52 | 2.46 | 8.4 | 0.108
2019-01-14 | 5:07:00 | 176 | 1.14 | 1.52 | 2.46 | 8.4 | 0.108
2019-01-14 | 5:10:04 | 176 | 1.15 | 1.52 | 2.46 | 8.4 | 0.108
Table 6: Disk-resolved images obtained by the NIRC2 camera mounted on the Keck II telescope. For each observation, we provide the epoch, the filter, the exposure time, the airmass, the distance to the Earth $\Delta$ and the Sun $r$, the phase angle $\alpha$, the angular diameter $D_{\mathrm{a}}$ of Kleopatra and the program PI. Date | UT | Filter | Exp | Airmass | $\Delta$ | $r$ | $\alpha$ | $D_{\mathrm{a}}$ | Program PI
---|---|---|---|---|---|---|---|---|---
|  |  | (s) |  | (AU) | (AU) | (°) | (″) |
2002-05-07 | 10:54:36 | H | 60 | 1.20 | 2.45 | 3.46 | 1.8 | 0.067 | J.L. Margot
2003-07-14 | 11:11:35 | Kp | 10 | 1.05 | 1.70 | 2.65 | 9.8 | 0.097 | W. Merline
2008-09-19 | 06:16:59 | PK50$\\_$1.5 | 60 | 1.47 | 1.24 | 2.22 | 7.5 | 0.132 | F. Marchis
2008-09-19 | 06:28:57 | PK50_1.5 | 60 | 1.40 | 1.24 | 2.22 | 7.5 | 0.132 | F. Marchis
2008-09-19 | 11:38:20 | PK50_1.5 | 60 | 1.18 | 1.24 | 2.22 | 7.5 | 0.132 | F. Marchis
2008-10-05 | 09:12:57 | PK50_1.5 | 60 | 1.05 | 1.26 | 2.19 | 12.3 | 0.130 | F. Marchis
2008-10-05 | 09:48:54 | PK50_1.5 | 60 | 1.10 | 1.26 | 2.19 | 12.3 | 0.130 | F. Marchis
2008-10-05 | 10:03:40 | H | 30 | 1.13 | 1.26 | 2.19 | 12.3 | 0.130 | F. Marchis
2008-10-06 | 07:18:06 | PK50_1.5 | 60 | 1.05 | 1.26 | 2.19 | 12.6 | 0.130 | F. Marchis
2008-10-06 | 09:49:50 | PK50_1.5 | 60 | 1.11 | 1.26 | 2.19 | 12.7 | 0.130 | F. Marchis
2008-10-09 | 05:45:41 | PK50_1.5 | 60 | 1.23 | 1.27 | 2.18 | 13.8 | 0.129 | F. Marchis
2008-10-09 | 09:35:21 | PK50_1.5 | 60 | 1.11 | 1.27 | 2.18 | 13.8 | 0.129 | F. Marchis
2012-08-10 | 06:14:05 | Kp | 0.8 | 1.14 | 2.23 | 2.92 | 16.6 | 0.074 | W. Merline
2013-08-26 | 15:38:25 | Kp | 0.2 | 1.00 | 1.69 | 2.11 | 28.2 | 0.097 | W. Merline
Table 7: Optical disk-integrated lightcurves of Kleopatra used for ADAM shape modeling . For each lightcurve, the table gives the epoch, the number of individual measurements $N_{p}$, asteroid’s distances to the Earth $\Delta$ and the Sun $r$, phase angle $\varphi$, photometric filter and reference. N | Epoch | $N_{p}$ | $\Delta$ | $r$ | $\varphi$ | Filter | Reference
---|---|---|---|---|---|---|---
|  |  | (AU) | (AU) | (°) |  |
1 | 1977-01-15.9 | 277 | 1.57 | 2.51 | 8.1 | V | Scaltriti & Zappala (1978)
2 | 1977-01-16.9 | 245 | 1.57 | 2.51 | 8.1 | V | Scaltriti & Zappala (1978)
3 | 1980-09-06.3 | 23 | 1.24 | 2.20 | 10.3 | C | Pilcher & Tholen (1982)
4 | 1980-09-07.3 | 27 | 1.23 | 2.20 | 9.9 | C | Pilcher & Tholen (1982)
5 | 1980-09-09.3 | 20 | 1.23 | 2.20 | 9.2 | C | Pilcher & Tholen (1982)
6 | 1980-09-11.1 | 11 | 1.22 | 2.19 | 8.6 | C | Pilcher & Tholen (1982)
7 | 1980-09-14.3 | 25 | 1.21 | 2.19 | 7.8 | C | Pilcher & Tholen (1982)
8 | 1980-09-15.6 | 41 | 1.20 | 2.19 | 7.5 | V | Kennedy & Tholen (1982)
9 | 1980-09-18.2 | 18 | 1.20 | 2.18 | 7.1 | C | Pilcher & Tholen (1982)
10 | 1980-09-20.3 | 14 | 1.19 | 2.18 | 7.0 | V | Harris & Young (1989)
11 | 1980-09-20.3 | 18 | 1.19 | 2.18 | 7.0 | C | Pilcher & Tholen (1982)
12 | 1980-09-21.3 | 10 | 1.19 | 2.18 | 7.0 | C | Pilcher & Tholen (1982)
13 | 1980-09-27.1 | 11 | 1.19 | 2.17 | 7.7 | C | Pilcher & Tholen (1982)
14 | 1980-09-29.2 | 21 | 1.19 | 2.16 | 8.2 | C | Pilcher & Tholen (1982)
15 | 1980-09-30.5 | 27 | 1.19 | 2.16 | 8.6 | V | Kennedy & Tholen (1982)
16 | 1980-10-01.5 | 37 | 1.19 | 2.16 | 8.9 | V | Kennedy & Tholen (1982)
17 | 1980-10-02.0 | 63 | 1.19 | 2.16 | 9.0 | V | Zappala et al. (1983)
18 | 1980-10-02.1 | 44 | 1.19 | 2.16 | 9.1 | V | Zappala et al. (1983)
19 | 1980-10-02.6 | 9 | 1.19 | 2.16 | 9.3 | V | Kennedy & Tholen (1982)
20 | 1980-10-06.2 | 37 | 1.20 | 2.15 | 10.6 | C | Pilcher & Tholen (1982)
21 | 1980-10-09.2 | 30 | 1.21 | 2.15 | 11.7 | C | Pilcher & Tholen (1982)
22 | 1980-11-06.2 | 5 | 1.36 | 2.11 | 22.0 | V | Harris & Young (1989)
23 | 1980-11-07.8 | 199 | 1.38 | 2.11 | 22.4 | V | Grossman et al. (1981)
24 | 1980-11-10.2 | 24 | 1.40 | 2.11 | 23.0 | C | Pilcher & Tholen (1982)
25 | 1981-12-01.2 | 19 | 2.48 | 2.72 | 21.2 | V | Weidenschilling et al. (1987)
26 | 1981-12-02.4 | 17 | 2.47 | 2.72 | 21.2 | V | Weidenschilling et al. (1987)
27 | 1982-02-13.9 | 74 | 1.96 | 2.91 | 6.4 | V | Zappala et al. (1983)
28 | 1982-02-14.0 | 36 | 1.96 | 2.91 | 6.4 | V | Zappala et al. (1983)
29 | 1982-02-20.3 | 30 | 1.97 | 2.93 | 6.0 | V | Weidenschilling et al. (1987)
30 | 1982-03-23.2 | 162 | 2.18 | 3.00 | 12.6 | V | Carlsson & Lagerkvist (1983)
31 | 1982-03-26.9 | 126 | 2.22 | 3.01 | 13.5 | V | Zappala et al. (1983)
32 | 1983-02-20.3 | 18 | 3.01 | 3.48 | 15.4 | V | Weidenschilling et al. (1987)
33 | 1983-02-21.3 | 8 | 2.99 | 3.48 | 15.3 | V | Weidenschilling et al. (1987)
34 | 1983-02-22.4 | 5 | 2.98 | 3.48 | 15.2 | V | Weidenschilling et al. (1987)
35 | 1983-03-28.3 | 26 | 2.60 | 3.49 | 8.5 | V | Weidenschilling et al. (1987)
36 | 1983-03-29.3 | 12 | 2.59 | 3.49 | 8.2 | V | Weidenschilling et al. (1987)
37 | 1983-05-21.3 | 15 | 2.60 | 3.50 | 8.8 | V | Weidenschilling et al. (1987)
38 | 1983-05-22.3 | 18 | 2.61 | 3.50 | 9.1 | V | Weidenschilling et al. (1987)
39 | 1983-05-23.2 | 6 | 2.61 | 3.50 | 9.4 | V | Weidenschilling et al. (1987)
40 | 1984-05-08.4 | 10 | 2.40 | 3.10 | 15.3 | V | Weidenschilling et al. (1987)
41 | 1984-05-09.4 | 18 | 2.39 | 3.10 | 15.2 | V | Weidenschilling et al. (1987)
42 | 1984-05-10.2 | 7 | 2.38 | 3.10 | 15.0 | V | Weidenschilling et al. (1987)
43 | 1984-07-05.3 | 24 | 2.00 | 2.97 | 6.9 | V | Weidenschilling et al. (1987)
44 | 1985-10-21.3 | 35 | 1.25 | 2.11 | 17.5 | V | Weidenschilling et al. (1987)
45 | 1985-11-09.9 | 63 | 1.17 | 2.13 | 8.8 | V | Lupishko & Velichko (1987)
46 | 1985-12-02.9 | 117 | 1.19 | 2.16 | 7.3 | V | Dotto et al. (1992)
47 | 1985-12-08.0 | 84 | 1.22 | 2.16 | 9.4 | V | Dotto et al. (1992)
48 | 1986-01-17.2 | 34 | 1.58 | 2.23 | 22.7 | V | Weidenschilling et al. (1987)
49 | 1987-02-03.4 | 33 | 2.41 | 3.15 | 13.6 | V | Weidenschilling et al. (1990)
50 | 1987-03-06.0 | 35 | 2.25 | 3.21 | 5.2 | C | Frank & Frevert (1988)
51 | 1988-04-23.4 | 18 | 2.51 | 3.46 | 6.0 | V | Weidenschilling et al. (1990)
52 | 1988-04-25.2 | 21 | 2.49 | 3.46 | 5.5 | V | Weidenschilling et al. (1990)
53 | 1989-07-05.2 | 48 | 1.77 | 2.68 | 12.1 | VB | Hutton (1990)
54 | 1994-09-05.1 | 16 | 1.26 | 2.23 | 9.7 | R | Fauvaud et al. (2001)
55 | 1994-09-06.1 | 15 | 1.25 | 2.22 | 9.4 | R | Fauvaud et al. (2001)
56 | 1994-09-07.2 | 50 | 1.25 | 2.22 | 9.1 | R | Fauvaud et al. (2001)
57 | 1994-09-08.1 | 163 | 1.25 | 2.22 | 8.8 | R | Fauvaud et al. (2001)
58 | 2006-04-02.3 | 134 | 2.42 | 3.42 | 2.1 | R | Warner (2006)
59 | 2006-04-03.2 | 149 | 2.42 | 3.42 | 2.1 | R | Warner (2006)
60 | 2008-07-26 | 56 | 1.59 | 2.33 | 21.0 | C | Grice et al. (2017)
61 | 2008-07-27 | 61 | 1.58 | 2.33 | 20.8 | C | Grice et al. (2017)
62 | 2008-07-28 | 65 | 1.57 | 2.32 | 20.6 | C | Grice et al. (2017)
63 | 2008-07-29 | 53 | 1.56 | 2.32 | 20.4 | C | Grice et al. (2017)
64 | 2008-07-30 | 61 | 1.55 | 2.32 | 20.1 | C | Grice et al. (2017)
65 | 2008-08-01 | 83 | 1.53 | 2.31 | 19.7 | C | Grice et al. (2017)
66 | 2008-08-02 | 37 | 1.52 | 2.31 | 19.4 | C | Grice et al. (2017)
67 | 2008-08-04 | 41 | 1.50 | 2.31 | 18.9 | C | Grice et al. (2017)
68 | 2008-08-05 | 49 | 1.49 | 2.31 | 18.7 | C | Grice et al. (2017)
69 | 2008-08-06 | 59 | 1.48 | 2.30 | 18.4 | C | Grice et al. (2017)
70 | 2008-08-07 | 38 | 1.47 | 2.30 | 18.1 | C | Grice et al. (2017)
71 | 2008-08-09 | 69 | 1.45 | 2.30 | 17.6 | C | Grice et al. (2017)
72 | 2008-08-10 | 61 | 1.44 | 2.30 | 17.3 | C | Grice et al. (2017)
73 | 2008-08-11 | 65 | 1.43 | 2.29 | 17.0 | C | Grice et al. (2017)
74 | 2008-09-23 | 39 | 1.24 | 2.21 | 8.4 | C | Grice et al. (2017)
75 | 2008-09-23 | 40 | 1.24 | 2.21 | 8.4 | C | Grice et al. (2017)
76 | 2008-10-03 | 67 | 1.25 | 2.19 | 11.7 | C | Grice et al. (2017)
77 | 2008-10-04 | 70 | 1.25 | 2.19 | 12.1 | C | Grice et al. (2017)
78 | 2008-10-05 | 37 | 1.26 | 2.19 | 12.5 | C | Grice et al. (2017)
79 | 2008-10-06 | 48 | 1.26 | 2.19 | 12.9 | C | Grice et al. (2017)
80 | 2008-10-07 | 64 | 1.26 | 2.19 | 13.2 | C | Grice et al. (2017)
81 | 2008-10-08 | 64 | 1.27 | 2.19 | 13.6 | C | Grice et al. (2017)
82 | 2008-10-14 | 47 | 1.29 | 2.18 | 15.9 | C | Grice et al. (2017)
83 | 2008-10-14 | 124 | 1.29 | 2.18 | 15.9 | C | Grice et al. (2017)
84 | 2008-10-15 | 38 | 1.30 | 2.17 | 16.3 | C | Grice et al. (2017)
85 | 2008-10-16 | 96 | 1.30 | 2.17 | 16.7 | C | Grice et al. (2017)
86 | 2008-10-17 | 41 | 1.31 | 2.17 | 17.0 | C | Grice et al. (2017)
87 | 2008-10-18 | 34 | 1.31 | 2.17 | 17.4 | C | Grice et al. (2017)
88 | 2008-10-20 | 92 | 1.33 | 2.17 | 18.1 | C | Grice et al. (2017)
89 | 2008-10-23 | 59 | 1.35 | 2.16 | 19.1 | C | Grice et al. (2017)
90 | 2008-10-24 | 28 | 1.35 | 2.16 | 19.4 | C | Grice et al. (2017)
91 | 2008-10-24 | 76 | 1.35 | 2.16 | 19.4 | C | Grice et al. (2017)
92 | 2008-10-25 | 30 | 1.36 | 2.16 | 19.8 | C | Grice et al. (2017)
93 | 2008-10-26 | 44 | 1.37 | 2.16 | 20.1 | C | Grice et al. (2017)
94 | 2008-10-28 | 29 | 1.38 | 2.16 | 20.7 | C | Grice et al. (2017)
95 | 2008-10-31 | 35 | 1.40 | 2.15 | 21.6 | C | Grice et al. (2017)
96 | 2008-11-01 | 40 | 1.41 | 2.15 | 21.8 | C | Grice et al. (2017)
97 | 2008-11-03 | 37 | 1.43 | 2.15 | 22.4 | C | Grice et al. (2017)
98 | 2008-11-04 | 28 | 1.44 | 2.15 | 22.6 | C | Grice et al. (2017)
99 | 2008-11-12 | 55 | 1.51 | 2.14 | 24.5 | C | Grice et al. (2017)
100 | 2008-11-12 | 78 | 1.51 | 2.14 | 24.5 | C | Grice et al. (2017)
101 | 2008-11-14 | 38 | 1.52 | 2.14 | 24.8 | C | Grice et al. (2017)
102 | 2009-12-30 | 71 | 2.10 | 2.76 | 17.4 | C | Grice et al. (2017)
103 | 2010-01-05 | 80 | 2.05 | 2.78 | 16.0 | C | Grice et al. (2017)
104 | 2010-01-07 | 31 | 2.04 | 2.78 | 15.5 | C | Grice et al. (2017)
105 | 2010-01-10 | 25 | 2.02 | 2.79 | 14.7 | C | Grice et al. (2017)
106 | 2010-01-11 | 60 | 2.01 | 2.79 | 14.5 | C | Grice et al. (2017)
107 | 2010-01-12 | 27 | 2.00 | 2.79 | 14.2 | C | Grice et al. (2017)
108 | 2010-01-13 | 41 | 2.00 | 2.80 | 13.9 | C | Grice et al. (2017)
109 | 2010-01-14 | 26 | 1.99 | 2.80 | 13.7 | C | Grice et al. (2017)
110 | 2010-01-15 | 47 | 1.99 | 2.80 | 13.4 | C | Grice et al. (2017)
111 | 2010-01-28 | 60 | 1.93 | 2.83 | 9.6 | C | Grice et al. (2017)
112 | 2010-02-02 | 122 | 1.92 | 2.85 | 8.3 | C | Grice et al. (2017)
113 | 2010-02-03 | 55 | 1.92 | 2.85 | 8.0 | C | Grice et al. (2017)
114 | 2010-02-04 | 49 | 1.92 | 2.85 | 7.7 | C | Grice et al. (2017)
115 | 2010-02-04 | 78 | 1.92 | 2.85 | 7.8 | C | Grice et al. (2017)
116 | 2010-02-05 | 45 | 1.92 | 2.86 | 7.5 | C | Grice et al. (2017)
117 | 2010-02-05 | 46 | 1.92 | 2.85 | 7.5 | C | Grice et al. (2017)
118 | 2010-02-06 | 29 | 1.92 | 2.86 | 7.3 | C | Grice et al. (2017)
119 | 2010-02-06 | 34 | 1.92 | 2.86 | 7.3 | C | Grice et al. (2017)
120 | 2010-02-07 | 35 | 1.92 | 2.86 | 7.1 | C | Grice et al. (2017)
121 | 2010-02-08 | 22 | 1.92 | 2.86 | 6.9 | C | Grice et al. (2017)
122 | 2010-02-08 | 25 | 1.92 | 2.86 | 6.9 | C | Grice et al. (2017)
123 | 2010-02-09 | 48 | 1.92 | 2.87 | 6.7 | C | Grice et al. (2017)
124 | 2010-02-10 | 28 | 1.92 | 2.87 | 6.6 | C | Grice et al. (2017)
125 | 2010-02-11 | 55 | 1.92 | 2.87 | 6.5 | C | Grice et al. (2017)
126 | 2010-02-12 | 58 | 1.92 | 2.87 | 6.4 | C | Grice et al. (2017)
127 | 2010-02-14 | 58 | 1.92 | 2.88 | 6.3 | C | Grice et al. (2017)
128 | 2010-02-15 | 37 | 1.93 | 2.88 | 6.2 | C | Grice et al. (2017)
129 | 2010-02-15 | 38 | 1.93 | 2.88 | 6.2 | C | Grice et al. (2017)
130 | 2010-02-18 | 41 | 1.93 | 2.89 | 6.3 | C | Grice et al. (2017)
131 | 2010-02-19 | 67 | 1.94 | 2.89 | 6.4 | C | Grice et al. (2017)
132 | 2010-02-20 | 30 | 1.94 | 2.89 | 6.5 | C | Grice et al. (2017)
133 | 2010-02-23 | 28 | 1.95 | 2.90 | 6.9 | C | Grice et al. (2017)
134 | 2010-02-23 | 34 | 1.95 | 2.90 | 6.9 | C | Grice et al. (2017)
135 | 2010-02-24 | 89 | 1.96 | 2.90 | 7.1 | C | Grice et al. (2017)
136 | 2010-02-24 | 90 | 1.96 | 2.90 | 7.1 | C | Grice et al. (2017)
137 | 2010-03-01 | 32 | 1.98 | 2.91 | 8.1 | C | Grice et al. (2017)
138 | 2010-03-02 | 17 | 1.99 | 2.92 | 8.4 | C | Grice et al. (2017)
139 | 2010-03-02 | 23 | 1.99 | 2.92 | 8.4 | C | Grice et al. (2017)
140 | 2010-03-02 | 23 | 1.99 | 2.92 | 8.4 | C | Grice et al. (2017)
141 | 2010-03-03 | 44 | 2.00 | 2.92 | 8.6 | C | Grice et al. (2017)
142 | 2010-03-03 | 46 | 2.00 | 2.92 | 8.6 | C | Grice et al. (2017)
143 | 2010-03-03 | 53 | 2.00 | 2.92 | 8.6 | C | Grice et al. (2017)
144 | 2010-03-10 | 45 | 2.05 | 2.94 | 10.5 | C | Grice et al. (2017)
145 | 2010-03-13 | 36 | 2.07 | 2.94 | 11.2 | C | Grice et al. (2017)
146 | 2010-03-13 | 48 | 2.07 | 2.94 | 11.3 | C | Grice et al. (2017)
147 | 2010-03-14 | 54 | 2.08 | 2.94 | 11.5 | C | Grice et al. (2017)
148 | 2010-03-14 | 71 | 2.08 | 2.94 | 11.5 | C | Grice et al. (2017)
149 | 2010-03-15 | 32 | 2.09 | 2.95 | 11.7 | C | Grice et al. (2017)
150 | 2010-03-16 | 35 | 2.10 | 2.95 | 12.0 | C | Grice et al. (2017)
151 | 2010-03-17 | 53 | 2.11 | 2.95 | 12.3 | C | Grice et al. (2017)
152 | 2010-03-18 | 24 | 2.12 | 2.95 | 12.5 | C | Grice et al. (2017)
153 | 2010-03-18 | 43 | 2.12 | 2.95 | 12.5 | C | Grice et al. (2017)
154 | 2010-03-19 | 15 | 2.13 | 2.96 | 12.8 | C | Grice et al. (2017)
155 | 2010-03-20 | 50 | 2.14 | 2.96 | 13.0 | C | Grice et al. (2017)
156 | 2010-03-20 | 96 | 2.14 | 2.96 | 13.0 | C | Grice et al. (2017)
157 | 2010-03-21 | 33 | 2.15 | 2.96 | 13.2 | C | Grice et al. (2017)
158 | 2010-03-21 | 36 | 2.15 | 2.96 | 13.3 | C | Grice et al. (2017)
159 | 2010-03-22 | 23 | 2.16 | 2.96 | 13.5 | C | Grice et al. (2017)
160 | 2010-03-22 | 40 | 2.16 | 2.96 | 13.5 | C | Grice et al. (2017)
161 | 2010-03-23 | 64 | 2.18 | 2.97 | 13.7 | C | Grice et al. (2017)
162 | 2010-03-24 | 26 | 2.19 | 2.97 | 13.9 | C | Grice et al. (2017)
163 | 2010-03-24 | 62 | 2.19 | 2.97 | 14.0 | C | Grice et al. (2017)
164 | 2010-03-29 | 55 | 2.25 | 2.98 | 15.0 | C | Grice et al. (2017)
165 | 2010-03-29 | 116 | 2.24 | 2.98 | 15.0 | C | Grice et al. (2017)
166 | 2010-03-30 | 50 | 2.26 | 2.98 | 15.2 | C | Grice et al. (2017)
167 | 2010-03-30 | 70 | 2.26 | 2.98 | 15.2 | C | Grice et al. (2017)
168 | 2010-03-31 | 61 | 2.27 | 2.98 | 15.4 | C | Grice et al. (2017)
169 | 2010-04-01 | 67 | 2.28 | 2.99 | 15.6 | C | Grice et al. (2017)
170 | 2010-04-04 | 41 | 2.32 | 2.99 | 16.1 | C | Grice et al. (2017)
171 | 2010-04-06 | 24 | 2.35 | 3.00 | 16.5 | C | Grice et al. (2017)
172 | 2010-04-07 | 40 | 2.36 | 3.00 | 16.6 | C | Grice et al. (2017)
173 | 2010-04-08 | 37 | 2.37 | 3.00 | 16.8 | C | Grice et al. (2017)
174 | 2010-04-10 | 39 | 2.40 | 3.01 | 17.1 | C | Grice et al. (2017)
175 | 2010-04-12 | 35 | 2.43 | 3.01 | 17.4 | C | Grice et al. (2017)
176 | 2011-02-20 | 9 | 2.96 | 3.47 | 15.2 | C | Grice et al. (2017)
177 | 2013-10-23 | 200 | 1.18 | 2.11 | 13.1 | C | Grice et al. (2017)
178 | 2013-10-28 | 174 | 1.16 | 2.11 | 10.7 | C | Grice et al. (2017)
179 | 2013-10-30 | 115 | 1.15 | 2.11 | 9.7 | C | Grice et al. (2017)
180 | 2015-04-15.0 | 245 | 2.41 | 3.25 | 11.4 | R | This work
Table 7: continued. Table 8: Individual observers that participated in the
stellar occultation campaigns targeting asteroid Kleopatra (Herald et al.
2020). Observer
---
1980-10-10
S. Krysko, Linbrook, Alberta
G. Stokes, Richland, WA
Beals/Belcher/Loehde, Cooking Lake
D. Scarlett, Kaslo, B.C.
Jones/Bowen, Castlegar, B.C.
D. Hube, Devon, U. of Alberta, AL
G. Fouts, Goldendale, WA
Mitchell, Ellensburg, WA
E. Mannery, Ellensburg, WA
1991-01-19
D. Dunham/W. Warren, Maryland, NJ
Jeff Guerber, Greenbelt, MD
C. Aikman/M. Fletcher, Victoria, BC
Robert J. Mordic, Richmond Hts, OH
Samuel Storch, Jones Beach, NY
Jim Pyral, North Bend, WA
Edwin Lurcott, West Chester, PA
M. Henry, Canton, OH
James H. Fox, Afton, MN
Robert Bolster, Alexandria, VA
Dan Grieser, Columbus, OH
2009-12-24
D. Dunham et al, Piedra, AZ
J. Ray, Glendale, AZ
R. Peterson, Phoenix, AZ
R. Peterson, Scottsdale, AZ
P. Maley, Sun Lakes, AZ
G. Rattley, Gilbert, AZ
P. Maley, Santan, AZ
S. Degenhardt, Quijotoa, AZ
L. Martinez, Casa Grande, AZ
S. Degenhardt, Gu Oldak, AZ
S. Degenhardt, Sells, AZ
S. Degenhardt, Ali Chukson, AZ
S. Degenhardt, Schuchk, AZ
S. Degenhardt, Three Points, AZ
J. Stamm, Oro Valley, AZ
2015-03-12
Henk Bulder, NL
Friedhelm Dorst, DE
Oliver Kloes, DE
Jan-Maarten Winkel, NL
Otto Farago, DE
Vasilis Metallinos, GR
Harrie Rutten, NL
Henk De Groot, NL
Bernd Gaehrken, DE
Hans Kostense, NL
D. Fischer, H.G. Purucker, R. Stoyan, DE
Eberhard Bredner, FR
Andre Mueller, DE
Lex Blommers, NL
K. Moddemeijer, P. Bastiaansen, W. Nobel, NL
Christof Sauter, CH
Maxime Devogele, BE
Mike Kohl, CH
Jose De Queiroz, CH
Karl-Ludwig Bath, DE
Martin Federspiel, DE
Fernand Emering, LU
F. Van Den Abbeel, BE
Rene Bourtembourg, BE
Jonas Schenker, CH
J. Lecacheux, E. Meza, FR
Stefano Sposetti, CH
Roberto Di Luca, IT
T. Pauwels, P. De Cat, BE
C. Demeautis. D. Matter, FR
Stefano Sposetti, CH
Andrea Manna, CH
Alberto Ossola, CH
Carlo Gualdoni, IT
Roland Decellier, BE
Fabrizio Ciabattari, IT
Mauro Bachini, IT
Giancarlo Bonatti, IT
Alex Pratt, UK
Gilles Sautot, FR
Roland Boninsegna, BE
Fausto Delucchi, CH
Martin Federspiel, DE
G. Sautot, E. Vauthrin, FR
Olivier Dechambre, FR
Jerome Berthier, FR
Frederic Vachier, FR
B. Carry, M. Pajuelo, FR
Joan Rovira, ES
2016-04-05
B. Dunford, Naperville, IL
A. Olsen, Urbana, IL
D. Dunham/J. Dunham, Yemasse, SC
D. Dunham/J. Dunham, Coosawhatchie, SC
D. Dunham/J. Dunham, Hardeesville, SC
D. Dunham/J. Dunham, Savannah, GA
D. Dunham/J. Dunham, Midway, GA
D. Dunham/J. Dunham, South Newport, GA
D. Dunham/J. Dunham, Darien, GA
N. Smith, Trenton, GA
R. Venable, Yonkers, GA
R. Venable, Hawkinsville, GA
S. Messner, Moravia, IA
S. Messner, Iconium, IA
R. Venable, Oakfield, GA
R. Venable, Newton, GA
Table 8: continued.