Observed tidal evolution of Kleopatra’s outer satellite^★

¹ Institute of Astronomy, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague, Czech Republic
e-mail: mira@sirrah.troja.mff.cuni.cz
² Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
³ IMCCE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Lille, France
⁴ SETI Institute, Carl Sagan Center, 189 Bernado Avenue, Mountain View, CA 94043, USA
⁵ Aix Marseille Univ, CNRS, LAM, Laboratoire d’Astrophysique de Marseille, Marseille, France
⁶ Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstr. 2, 12489 Berlin, Germany
⁷ Geneva Observatory, 1290 Sauverny, Switzerland

Received: 19 August 2021
Accepted: 15 October 2021

Abstract

Aims. The orbit of the outer satellite Alexhelios of (216) Kleopatra is already constrained by adaptive-optics astrometry obtained with the VLT/SPHERE instrument. However, there is also a preceding occultation event in 1980 attributed to this satellite. Here, we try to link all observations, spanning 1980–2018, because the nominal orbit exhibits an unexplained shift by + 60° in the true longitude.

Methods. Using both a periodogram analysis and an ℓ = 10 multipole model suitable for the motion of mutually interacting moons about the irregular body, we confirmed that it is not possible to adjust the respective osculating period P₂. Instead, we were forced to use a model with tidal dissipation (and increasing orbital periods) to explain the shift. We also analysed light curves spanning 1977–2021, and searched for the expected spin deceleration of Kleopatra.

Results. According to our best-fit model, the observed period rate is Ṗ₂ = (1.8 ± 0.1) × 10⁻⁸ d d⁻¹ and the corresponding time-lag Δt₂ = 42 s of tides, for the assumed value of the Love number k₂ = 0.3. This is the first detection of tidal evolution for moons orbiting 100 km asteroids. The corresponding dissipation factor Q is comparable with that of other terrestrial bodies, albeit at a higher loading frequency 2|ω − n|. We also predict a secular evolution of the inner moon, Ṗ₁ = 5.0 × 10⁻⁸, as well as a spin deceleration of Kleopatra, Ṗ₀ = 1.9 × 10⁻¹². In alternative models, with moons captured in the 3:2 mean-motion resonance or more massive moons, the respective values of Δt₂ are a factor of between two and three lower. Future astrometric observations using direct imaging or occultations should allow us to distinguish between these models, which is important for our understanding of the internal structure and mechanical properties of (216) Kleopatra.