Abstract
Aims.To interpret adaptive-optics observations of (216) Kleopatra, we need to describe an evolution of multiple moons orbiting an extremely irregular body and include their mutual interactions. Such orbits are generally non-Keplerian and orbital elements are not constants.Methods.Consequently, we used a modifiedN-body integrator, which was significantly extended to include the multipole expansion of the gravitational field up to the orderℓ= 10. Its convergence was verified against the ‘brute-force’ algorithm. We computed the coefficientsCℓm,Sℓmfor Kleopatra’s shape, assuming a constant bulk density. For Solar System applications, it was also necessary to implement a variable distance and geometry of observations. Ourχ2metric then accounts for the absolute astrometry, the relative astrometry (second moon with respect to the first), angular velocities, and silhouettes, constraining the pole orientation. This allowed us to derive the orbital elements of Kleopatra’s two moons.Results.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,P1= (1.822359 ± 0.004156) d,P2= (2.745820 ± 0.004820) 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), meaning that regular librations of the critical argument are not present. The resulting mass of Kleopatra,m1= (1.49 ± 0.16) × 10−12M⊙or 2.97 × 1018kg, is significantly lower than previously thought. An implication explained in the accompanying paper is that (216) Kleopatra is a critically rotating body.
Highlights
In the interest of emphasis, the orbital elements are not constants in our dynamical model; we demonstrate this in the accompanying Fig. 11
We plan to use our multipole model for analyses of other triple systems observed by the VLT/SPHERE (e.g. (45) Eugenia, (130) Elektra)
Even if the velocity magnitude is incorrect because of residual seeing and an under-corrected point-spread function (PSF), it is sufficient to measure its direction (‘sign’), which would prevent some of the ambiguities
Summary
Them to 10−16 M , be approximately m2 = 2 × 10−16 M and which together with diameters Kamndaρn3d=D136=008.k9gkmm−c3.orTrhesepseonadretosodmeneswithieast lower than the value for Kleopatra, but the 1σ uncertainties are still too large (50%) for any robust conclusions to be made. A hypothetical possibility of ‘zero-mass’ moons, with χ2 = 214 versus 182 after a manual adjustment of P1 and P2, cannot be excluded. If we believe that Ds > 0, we should believe that ms > 0.
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