Observed tidal evolution of Kleopatra’s outer satellite

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.

Discovery of an asteroid family linked to (22) Kalliope and its moon Linus

“…Unfortunately, it is unconstrained by observations. According to our tests, a model with tides (Mignard 1979;Brož et al 2022) is statistically equal to a model without tides. Nevertheless, we can assume tides to be either strong (Q = 40, k = 0.305 as for (216) Kleopatra; Brož et al 2022) or weak (Q = 280 as for the Earth, k = 0.024 as for Moon).…”

Discovery of an asteroid family linked to (22) Kalliope and its moon Linus

“…Unfortunately, it is unconstrained by observations. According to our tests, a model with tides (Mignard 1979;Brož et al 2022) is statistically equal to a model without tides. Nevertheless, we can assume tides either strong (Q = 40, k = 0.305 as for (216) Kleopatra; Brož et al 2022), or weak (Q = 280 as for the Earth, k = 0.024 as for Moon).…”

M-type (22) Kalliope: A tiny Mercury

“…where M denotes the mass; R the reference radius; r, θ, φ the body-frozen spherical coordinates; P m the Legendre polynomials; and ρ the density. The derivatives of U were presented in Brož et al (2021). Moreover, our model includes the external tides of the Sun.…”

“…We used the Genoid algorithm (Vachier et al 2012) to explore the dynamics of Linus, as we did for several other targets of our Large Program (Euphrosyne, Daphne, Sylvia, see Yang et al 2020;Carry et al 2019Carry et al , 2021 As illustrated in the case of (216) Kleopatra (Brož et al 2021), the topology of the parametric space is highly nontrivial to explore, with numerous local minima for the figure of merit (χ 2 ) in a high-dimensional space. Whereas gradient descent minimization is unlikely to find the absolute minimum, brute-force grids like Monte Carlo would be extremely CPU expensive.…”

“…To describe the orbital motion of the moon more precisely, we also use the multipole numerical model of Brož et al (2021) 4 , previously used for ( 216) Kleopatra. This model accounts for the irregular shape of the central body, using its multipole expansion up to the order = 10.…”