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The rotation of a star and the revolutions of its planets are not necessarily aligned. This article reviews the measurement techniques, key findings, and theoretical interpretations related to the obliquities (spin–orbit angles) of planet-hosting stars. The best measurements are for stars with short-period giant planets, which have been found on prograde, polar, and retrograde orbits. It seems likely that dynamical processes such as planet–planet scattering and secular perturbations are responsible for tilting the orbits of close-in giant planets, just as those processes are implicated in exciting orbital eccentricities. The observed dependence of the obliquity on orbital separation, planet mass, and stellar structure suggests that in some cases, tidal dissipation damps a star’s obliquity within its main-sequence lifetime. The situation is not as clear for stars with smaller or wider-orbiting planets. Although the earliest measurements of such systems tended to find low obliquities, some glaring exceptions are now known in which the star’s rotation is misaligned with respect to the coplanar orbits of multiple planets. In addition, statistical analyses based on projected rotation velocities and photometric variability have found a broad range of obliquities for F-type stars hosting compact multiple-planet systems. The results suggest it is unsafe to assume that stars and their protoplanetary disks are aligned. Primordial misalignments might be produced by neighboring stars or more complex events that occur during the epoch of planet formation.

Despite numerous attempts, no exomoon has firmly been confirmed to date. New missions like CHEOPS aim to characterize previously detected exoplanets and potentially discover exomoons. In order to optimize search strategies, we need to determine those planets which are the most likely to host moons. We investigate the tidal evolution of hypothetical moon orbits in systems consisting of a star, one planet, and one test moon. We study a few specific cases with ten billion years integration time where the evolution of moon orbits follows one of these three scenarios: (1) “locking,” in which the moon has a stable orbit on a long timescale (≳10 9 yr); (2) “escape scenario” where the moon leaves the planet’s gravitational domain; and (3) “disruption scenario,” in which the moon migrates inwards until it reaches the Roche lobe and becomes disrupted by strong tidal forces. Applying the model to real cases from an exoplanet catalog, we study the long-term stability of moon orbits around known exoplanets. We calculate the survival rate which is the fraction of the investigated cases when the moon survived around the planet for the full integration time (which is the age of the star, or if not known, then the age of the Sun). The most important factor determining the long-term survival of an exomoon is the orbital period of the planet. For the majority of the close-in planets (<10 days orbital periods) there is no stable orbit for moons. Between 10 and 300 days we find a transition in survival rate from about zero to 70%. Our results give a possible explanation for the lack of successful exomoon discoveries for close-in planets. Tidal instability causes moons to escape or being tidally disrupted around close-in planets which are mostly favored by current detection techniques.

WASP-12b stands out among the planets of its class of hot Jupiters because of the observed fast orbital decay attributed to tidal dissipation. The measured rate of the orbital period change is P⋅orb=−29±3ms/yr= −(9.2±1.0)×10−10s/s. In the literature heretofore, all attempts to explain this high rate were based on the assumption that the orbital evolution is dominated by the tides in the star. Since the modified tidal quality factor in yellow dwarfs is insufficient to warrant such a decay rate, a hypothesis was put forward that the star may actually be a subgiant. Using the latest data from the Gaia mission, we deduce that WASP-12 at 1.36M☼ is an evolving dwarf at an early stage of post-turn-off evolution that has not yet depleted hydrogen in its core. Its unremarkable position in the color-magnitude diagram and the existence of close planets orbiting red giants of similar mass contradict the hypothesis of an abrupt boost of tidal quality due to structural internal changes. On the other hand, the previous research neglected the tidal dissipation in the planet, assuming it to be negligible due to the likely synchronisation of its rotation and a presumed high quality factor. We critically reassess this assumption in the light of recent astrometric results for Jupiter and Saturn. Assuming that the structure of WASP-12b is similar to that of our Jupiter and Saturn, we find that the observed orbital decay is well explained by the tides in the planet. The estimated value of the planet’s modified quality factor coincides almost precisely with that of our Jupiter.

Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

The distribution of period ratios for 580 known two-planet systems is apparently nonuniform, with several sharp peaks and troughs. In particular, the vicinity of the 2:1 commensurability seems to have a deficit of systems. Using Monte Carlo simulations and an empirically inferred population distribution of period ratios, we prove that this apparent dearth of near-resonant systems is not statistically significant. The excess of systems with period ratios in the wider vicinity of the 2:1 resonance is significant, however. Long-term WHFast integrations of a synthetic two-planet system on a grid period ratios from 1.87 through 2.12 reveal that the eccentricity and inclination exchange mechanism between non-resonant planets represents the orbital evolution very well in all cases, except at the exact 2:1 mean motion resonance. This resonance destroys the orderly exchange of eccentricity, while the exchange of inclination still takes place. Additional simulations of the Kepler-113 system on a grid of initial inclinations show that the secular periods of eccentricity and inclination variations are well fitted by a simple hyperbolic cosine function of the initial mutual inclination. We further investigate the six known two-planet systems with period ratios within 2% of the exact 2:1 resonance (TOI-216, KIC 5437945, Kepler-384, HD 82943, HD 73526, HD 155358) on a grid of initial inclinations and for two different initial periastron longitudes corresponding to the aligned and anti-aligned states. All these systems are found to be long-term stable except HD 73526, which is likely a false positive. The periodic orbital momentum exchange is still at work in some of these systems, albeit with much shorter cycling periods of a few years.

Super-Earth b and sub-Neptunes c and d are orbiting about the M3.0V dwarf TOI-270 in that order from the star. Their global resonant chain (3:5, 1:1, 2:1) is extremely surprising because planet d appears to be the only known planet occupying the 2:1 resonant orbit without participating in a Laplace resonance (LR) or another planet intervening between the 1:1 and 2:1 orbits as in HD 110067. We do not believe that TOI-270 d is an exception to the empirical rule calling for 2:1 vacancy except in 1:2:4 LRs and Laplace-like 2:3:4 chains. Instead, a LR might exist in this system, and we searched (to no avail) the TESS light curves of TOI-270 for hints of an outer planet that would complete the LR chain. Alternative explanations would be an unknown planet more massive than planet c (Mc=6.20M⊕) establishing the actual 1:1 orbit, or planet b residing in the 1:2 Laplace orbit with a period shorter by 0.53 days. However, these possibilities are ruled out by current data. This leaves only one other option to explore: the observed orbits could be in a stable 35:1:2 resonant chain. Preliminary calculations do not preclude this possibility that should be investigated further by numerical orbit integrations. To this end, we determine two potentially resonant angles, φ and φ^, related via the Laplace phase φL by φ^=φL+2φ. In contrast, HD 110067 is shown to have planets d-e-f in a Laplace-like 1:32:2 resonance with phase φ=2φL precisely.

Close-in co-orbital planets (in a 1:1 mean-motion resonance) can experience strong tidal interactions with the central star. Here, we develop an analytical model adapted to the study of the tidal evolution of those systems. We use a Hamiltonian version of the constant time-lag tidal model, which extends the Hamiltonian formalism developed for the point-mass case. We show that co-orbital systems undergoing tidal dissipation favour either the Lagrange or the anti-Lagrange configurations, depending on the system parameters. However, for all range of parameters and initial conditions, both configurations become unstable, although the timescale for the destruction of the system can be larger than the lifetime of the star. We provide an easy-to-use criterion to determine whether an already known close-in exoplanet may have an undetected co-orbital companion.

We describe a method of determining three-body and four-body Laplace-like phase angles with the potential to librate about a mean value in multiplanet extrasolar systems. Unlike in past searches of N-body results, this method relies on global mean-motion resonances (MMRs) and takes into consideration the location of the most massive planet that defines the 1:1 global MMR in each (sub)system. We compiled lists of potentially librating phase angles and prevalent MMRs in 35 real multibody systems, and we discuss their properties in conjunction with recent investigations of librations discovered in sophisticated N-body simulations. We hope that our results will facilitate systematic libration searches in dynamical models of compact systems with three or more orbiting bodies.

We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and at least one of them is characterised by a high value of the forced eccentricity for the ‘first planet’: here, the paradigmatic case is the exo-planetary system GJ-876, in which the combination of libration centres admits triple conjunctions otherwise not possible in the Galilean case. The normal form obtained by averaging with respect to the free eccentricity oscillations describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance. The agreement of the analytic predictions with the numerical integration of the toy models is very good.

We study orbital evolution of multi-planet systems that form a resonant chain, with nearest neighbours close to first order commensurabilities, incorporating orbital circularisation produced by tidal interaction with the central star. We develop a semi-analytic model applicable when the relative proximities to commensurability, though small, are large compared to ϵ2/3, with ϵ being a measure of the characteristic planet to central star mass ratio. This enables determination of forced eccentricities as well as which resonant angles enter libration. When there are no active linked three body Laplace resonances, the rate of evolution of the semi-major axes may also be determined. We perform numerical simulations of the HD 158259 and EPIC 245950175 systems finding that the semi-analytic approach works well in the former case but not so well in the latter case on account of the effects of three active three body Laplace resonances which persist during the evolution. For both systems we estimate that if the tidal parameter, Q′, significantly exceeds 1000, tidal effects are unlikely to have influenced period ratios significantly since formation. On the other hand if Q′<∼100 tidal effects may have produced significant changes including the formation of three body Laplace resonances in the case of the EPIC 245950175 system.