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تسجيل مستخدم جديد(216) Kleopatra, a low density critically rotating M-type asteroid
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Context. The recent estimates of the 3D shape of the M/Xe-type triple asteroid system (216) Kleopatra indicated a density of 5 g.cm$^{-3}$. Such a high density implies a high metal content and 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, we aim to constrain the mass and the shape of Kleopatra with high accuracy, hence its density. Methods. We combined our new VLT/SPHERE observations of Kleopatra recorded in 2017 and 2018 with archival data, as well as lightcurve, occultation, and delay-Doppler images, to derive its 3D shape model 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 Kleopatra is very close to an equilibrium dumbbell figure with two lobes and a thick neck. Its volume equivalent diameter (118.75$pm$1.40) km and mass (2.97$pm$0.32) 10$^{18}$ kg imply a bulk density of (3.38$pm$0.50) g cm$^{-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. Kleopatras 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. Kleopatra is a puzzling multiple system due to the unique characteristics of the primary. It deserves particular attention in the future, with the Extremely Large Telescopes and possibly a dedicated space mission.
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To interpret adaptive-optics observations of (216) Kleopatra, we need to describe an evolution of multiple moons, orbiting an extremely irregular body and including their mutual interactions. Such orbits are generally non-Keplerian and orbital elemen
ts are not constants. Consequently, we use a modified $N$-body integrator, which was significantly extended to include the multipole expansion of the gravitational field up to the order $ell = 10$. Its convergence was verified against the `brute-force algorithm. We computed the coefficients $C_{ell m},S_{!ell m}$ for Kleopatras shape, assuming a~constant bulk density. For solar-system applications, it was also necessary to implement a variable distance and geometry of observations. Our $chi^2$ metric then accounts for the absolute astrometry, the relative astrometry (2nd moon with respect to 1st), angular velocities, and also silhouettes, constraining the pole orientation. This allowed us to derive the orbital elements of Kleopatras two moons. Using both archival astrometric data and new VLT/SPHERE observations (ESO LP 199.C-0074), we were able to identify the true periods of the moons, $P_1 = (1.822359pm0.004156),{rm d}$, $P_2 = (2.745820pm0.004820),{rm d}$. They orbit very close to the 3:2 mean-motion resonance, but their osculating eccentricities are too small compared to other perturbations (multipole, mutual), so that regular librations of the critical argument are not present. The resulting mass of Kleopatra, $m_1 = (1.49pm0.16)cdot10^{-12},M_odot$ or $2.97cdot10^{18},{rm kg}$, is significantly lower than previously thought. An implication explained in the accompanying paper (Marchis et al.) is that (216) Kleopatra is a critically rotating body.
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