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null (Ed.)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 modified N -body integrator, which was significantly extended to include the multipole expansion of the gravitational field up to the order ℓ = 10. Its convergence was verified against the ‘brute-force’ algorithm. We computed the coefficients C ℓm , S ℓm for 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. Our χ 2 metric 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, P 1 = (1.822359 ± 0.004156) d, P 2 = (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, m 1 = (1.49 ± 0.16) × 10 −12 M ⊙ or 2.97 × 10 18 kg, is significantly lower than previously thought. An implication explained in the accompanying paper is that (216) Kleopatra is a critically rotating body.more » « less
High-eccentricity tidal migration is a potential formation channel for hot Jupiters. During this process, the planetary f-mode may experience a phase of diffusive growth, allowing its energy to quickly build up to large values. In Yu et al., we demonstrated that nonlinear mode interactions between a parent f-mode and daughter f- and p-modes expand the parameter space over which the diffusive growth of the parent is triggered. We extend that study by incorporating (1) the angular momentum transfer between the orbit and the mode, and consequently the evolution of the pericenter distance; (2) a prescription to regulate the nonlinear frequency shift at high parent mode energies; and (3) dissipation of the parent’s energy due to both turbulent convective damping of the daughter modes and strongly nonlinear wave-breaking events. The new ingredients allow us to follow the coupled evolution of the mode and orbit over ≳104yr, covering the diffusive evolution from its onset to its termination. We find that the semimajor axis shrinks by a factor of nearly 10 over 104yr, corresponding to a tidal quality factor
. The f-mode’s diffusive growth terminates while the eccentricity is still high, at around e= 0.8–0.95. Using these results, we revisit the eccentricity distribution of proto-hot Jupiters. We estimate that less than 1 proto-HJ with eccentricity >0.9 should be expected in Kepler's data once the diffusive regime is accounted for, explaining the observed paucity of this population.
A planet’s orbital alignment places important constraints on how a planet formed and consequently evolved. The dominant formation pathway of ultra-short-period planets (P < 1 day) is particularly mysterious as such planets most likely formed further out, and it is not well understood what drove their migration inwards to their current positions. Measuring the orbital alignment is difficult for smaller super-Earth/sub-Neptune planets, which give rise to smaller amplitude signals. Here we present radial velocities across two transits of 55 Cancri (Cnc) e, an ultra-short-period super-Earth, observed with the Extreme Precision Spectrograph. Using the classical Rossiter–McLaughlin method, we measure 55 Cnc e’s sky-projected stellar spin–orbit alignment (that is, the projected angle between the The star 55 Cancri (Cnc) A hosts five known exoplanets with minimum mass estimates ranging from approximately 8M⊕ to 3MJup and periods less than one day to nearly 20 years1–4. Of particular interest has been 55 Cnc e, one of the most massive known ultra-short-period planets (USPs) and the only planet around 55 Cnc found to transit5,6. It has an star’s spin axis and the planet’s orbit normal—will shed light on the formation and evolution of USPs, especially in the case of compact, multiplanet systems. It has been shown that USPs form a statistically distinct popula- tion of planets9 that tend to be misaligned with other planetary orbits in their system10. This suggests that USPs experience a unique migra- tion pathway that brings them close in to their host stars. This inward migration is most likely driven by dissipation due to star–planet tidal interactions that result from either non-zero eccentricities11,12 or plan- etary spin-axis tilts13. orbital period of 0.7365474 +1.3 × 10−6 days, a mass of 7.99 ± 0.33M −1.4 × 10−6 ⊕ and a radius of 1.853 +0.026 R⊕ (refs. 7,8). A precise measure of the −0.027 stellar spin–orbit alignment of 55 Cnc e—the angle between the host planet’s orbital axis and its host star’s spin axis) to be λ = 10 +17∘ with an +14∘ −20∘ unprojected angle of ψ = 23 −12∘. The best-fit Rossiter–McLaughlin model to the Extreme Precision Spectrograph data has a radial velocity semi- amplitude of just 0.41 +0.09 m s−1. The spin–orbit alignment of 55 Cnc e −0.10 favours dynamically gentle migration theories for ultra-short-period planets, namely tidal dissipation through low-eccentricity planet–planet interactions and/or planetary obliquity tides.more » « less
Each of the giant planets within the Solar system has large moons but none of these moons have their own moons (which we call submoons). By analogy with studies of moons around short-period exoplanets, we investigate the tidal-dynamical stability of submoons. We find that 10 km-scale submoons can only survive around large (1000 km-scale) moons on wide-separation orbits. Tidal dissipation destabilizes the orbits of submoons around moons that are small or too close to their host planet; this is the case for most of the Solar system’s moons. A handful of known moons are, however, capable of hosting long-lived submoons: Saturn’s moons Titan and Iapetus, Jupiter’s moon Callisto, and Earth’s Moon. Based on its inferred mass and orbital separation, the newly discovered exomoon candidate Kepler-1625b-I can in principle host a large submoon, although its stability depends on a number of unknown parameters. We discuss the possible habitability of submoons and the potential for subsubmoons. The existence, or lack thereof, of submoons may yield important constraints on satellite formation and evolution in planetary systems.
Large eddy simulations are employed to investigate the role of tidal modulation strength on wake vortices and dissipation in flow past three‐dimensional topography, specifically a conical abyssal hill. The barotropic current is of the form
U c+ U tsin(Ω t t), where U cand U tare the mean and oscillatory components, respectively, and Ω tis the tidal frequency. A regime with strong stratification and weak rotation is considered. The velocity ratio R= U t/ U cis varied from 0 to 1. Simulation results show that the frequency of wake vortices reduces gradually with increasing Rfrom its natural shedding frequency at R= 0 to Ω t/2 when R≥ 0.2. The ratio of Rand the excursion number, denoted as , controls the shift in the vortex frequency. When , vortices are trapped in the wake during tidal deceleration, extending the vortex shedding cycle to two tidal cycles. Elevated dissipation rates in the obstacle lee are observed in the lateral shear layer, hydraulic jet, and the near wake. The regions of strong dissipation are spatially intermittent, with values exceeding during the maximum‐velocity phase, where Dis the base diameter of the hill. The maximum dissipation rate during the tidal cycle increases monotonically with Rin the downstream wake. Additionally, the normalized area‐integrated dissipation rate in the hydraulic response region scales with Ras (1 + R)4. Results show that the wake dissipation energetically dominates the internal wave flux in this class of low‐Froude number geophysical flows.