Longitudinal librations are periodic changes in the rotation angle of a planet or satellite. Their observation and subsequent interpretation have profoundly increased our understanding of the interior structure of Mercury. Likewise, libration is thought to provide important constraints on the interior structure of icy satellites. Here we study the libration of Mercury and large icy satellites rotating synchronously with their orbital motion and explain how it depends on the interior structure.

The rotation of asymmetric bodies in eccentric Keplerian orbits can be chaotic when there is some overlap of spin-orbit resonances. Here we show that the rotation of two coorbital bodies (two planets orbiting a star or two satellites of a planet) can also be chaotic even for quasi-circular orbits around the central body. When dissipation is present, the rotation period of a body on a nearly circular orbit is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We further show that the rotation becomes chaotic when the natural rotational libration frequency, due to the axial asymmetry, is of the same order of magnitude as the orbital libration frequency.

Representing a post-scriptum supplementary to a previous paper of the authors Brumberg & Ivanova (2011) this note aims to simplify the practical development of the Earth's rotation theory, in the framework of the general planetary theory, avoiding the non–physical secular terms and involving the separation of the fast and slow angular variables, both for planetary–lunar motion and Earth's rotation. In this combined treatment of motion and rotation, the fast angular terms are related to the mean orbital longitudes of the bodies, the diurnal and Euler rotations of the Earth. The slow angular terms are due to the motions of pericenters and nodes, as well as the precession of the Earth. The combined system of the equations of motion for the principal planets and the Moon and the equations of the Earth's rotation is reduced to the autonomous secular system with theoretically possible solution in a trigonometric form. In the above–mentioned paper, the Earth's rotation has been treated in Euler parameters. The trivial change of the Euler parameters to their small declinations from some nominal values may improve the practical efficiency of the normalization of the Earth's rotation equations. This technique may be applied to any three-axial rigid planet. The initial terms of the corresponding expansions are given in the Appendix.

The Saturnian satellite Titan is one of the main targets of the Cassini-Huygens mission, which revealed in particular Titan's shape, gravity field, and rotation state. The shape and gravity field suggest that Titan is not in hydrostatic equilibrium, that it has a global subsurface ocean, and that its ice shell is both rigid (at tidal periods) and of variable thickness. The rotational state of Titan consists of an expected synchronous rotation rate and an unexpectedly high obliquity (0.3○) explained by Baland et al. (2011) to be a resonant behavior. We here combine a realistic model of the ice shell and interior and a 6-degrees of freedom rotational model, in which the librations, obliquity and polar motion of the rigid core and of the shell are modelled, to constrain the structure of Titan from the observations. We consider the gravitational pull of Saturn on the 2 rigid layers, the gravitational coupling between them, and the pressure coupling at the liquid-solid interfaces.

We confirm the influence of the resonance found by Baland et al., that affects between 10 and 13% of the possible Titans. It is due to the 29.5-year periodic annual forcing. The resonant Titans can be obtained in situations in which a mass anomaly at the shell-ocean boundary (bottom loading) is from 80 to 92% compensated. This suggests a 250 to 280 km thick ocean below a 130 to 140 km thick shell, and is consistent with the degree-3 analysis of Hemingway 26 et al. (2013).

Titan longitudinal librations are dependent on the satellite internal structure and the elastic behavior of the surface. The elastic deformation of the surface is related to the perturbing potential through the Love theory. In a previous paper, we described the deformation as a response to the tidal potential exerted by Saturn at orbital frequency. Here we improve the tidal deformation reponse by including the effect of the libration angle and the orbital perturbations. We then provide the libration amplitudes associated with the rotational model of a tidally deformed three-layer Titan evolving on a non-Keplerian orbit.

A moon in synchronous rotation has longitudinal librations because of its non-spherical mass distribution and its elliptical orbit around the planet. We study the librations of Titan with periods of 14.7y and 29.5y and include deformation effects and the existence of a subsurface ocean. We take into account the fact that the orbit is not Keplerian and has other periodicities than the main period of orbital motion around Saturn due to perturbations by the Sun, other planets and moons. An orbital theory is used to compute the orbital perturbations due to these other bodies.

We numerically evaluate the amplitude of the long-period librations for many interior structure models of Titan constrained by the mass, radius and gravity field. Measurements of the librations may give constraints on the interior structure of the icy satellites.

Tidal dissipation is known as one of the main drivers of the secular evolution of planetary systems. It directly results from dissipative mechanisms that occur in planets and stars' interiors and strongly depends on the structure and dynamics of the bodies. This work focuses on the mechanism of viscous friction in stars and planetary layers. A local model is used to study tidal dissipation. It provides general scaling laws that give a qualitative overview of the different possible behaviors of fluid tidal waves. Furthermore, it highlights the sensitivity of dissipation to the tidal frequency and the roles played by the internal parameters of the fluid such as rotation, stratification, viscosity and thermal diffusivity that will impact the spins/orbital architecture in planetary systems.

We simulate the despinning of Mercury, with or without a fluid core, and with a frequency-dependent tidal model employed. The tidal model incorporates the viscoelastic (Maxwell) rebound at low frequencies and a predominantly inelastic (Andrade) creep at higher frequencies. It is combined with a statistically relevant set of histories of Mercury's eccentricity. The tidal model has a dramatic influence on the behaviour of spin histories near spin-orbit resonances. The probabilities of capture into high-order resonances are greatly enhanced. Exploring several scenarios, we conclude that the present 3:2 spin state was achieved by entrapment of an initially prograde cold Mercury when its age was less than 20 Myr, i.e., well before differentiation.

A comparison of several known dynamical indicators of chaos based on the numerical integration of differential variational equations is performed. The comparison is implemented on the examples of studying dynamics in the planar circular restricted three-body problem.

A method of analyses of dynamical system is applied to the planetary restricted three-body problem (RTBP). It is well known, that equations of motion of restricted 3-body problem in rotating rectangular frame may be reduced to the second order differential equation with periodic coefficients (Hills equation). Here Hills equation in cylindrical coordinate frame is derived. It gives the possibility to estimate width and position of the unstable zones. The dependence of the position of unstable zones on orbital eccentricity of the test particle is derived. Some followings of this simple linear model are noted.

The theoretical three body problem, with three given non infinitesimal point masses, has two types of oscillating motions. In the first type at least two mutual distances are unbounded, but their inferior limit is bounded: there are an infinite number of larger and larger ejections, but without escape. In the second type, it is the velocities that are unbounded: there are an infinite number of nearer and nearer quasi-collisions, without exact collisions.

The first type has only a theoretical interest: its measure in phase space is zero. But the second type has a positive measure in phase space and a physical interest: it governs most of the collisions of stars.

In this paper we consider the special case of the planar circular restricted three-body problem by the example of the problem of the Earth, the Moon and a point mass, where the gravitational potentials of the Earth and the Moon are given as the Kislik potential. The Kislik potential takes into account the flattening of a celestial body on the poles. We find the relative equilibria solutions for a point mass and analyze their stability. We describe the difference between the obtained points and the classical solution of the three-body problem.

HD 82943 hosts a mysterious multi-planet system in the 2:1 mean-motion resonance that puzzles astronomers for more than a decade. We describe our new analysis of all radial velocity data currently available for this star, including both the most recent Keck data and the older but more numerous CORALIE measurements.

Here we pay a major attention to the task of optimal scheduling of the future observation of this system. Applying several optimality criteria, we demonstrate that in the forthcoming observational season of HD 82943 (the winter 2014/2015) rather promising time ranges can be found. Observations of the near future may give rather remarkable improvement of the orbital fit, but only if we choose their time carefully.

Determining planetary habitability is a complex matter, as the interplay between a planet's physical and atmospheric properties with stellar insolation has to be studied in a self consistent manner. Standardized atmospheric models for Earth-like planets exist and are commonly accepted as a reference for estimates of Habitable Zones. In order to define Habitable Zone boundaries, circular orbital configurations around main sequence stars are generally assumed. In gravitationally interacting multibody systems, such as double stars, however, planetary orbits are forcibly becoming non circular with time. Especially in binary star systems even relatively small changes in a planet's orbit can have a large impact on habitability. Hence, we argue that a minimum model for calculating Habitable Zones in binary star systems has to include dynamical interactions.

A large number of observed exoplanets are part of multiple planet systems. Most of these systems are sufficiently close-in to be tidally evolving. In such systems, there is a competition between the excitation caused by planet-planet interactions and tidal damping. Using as an example two multiple planet systems, which host planets in the surface liquid water habitable zone (HZ): Kepler-62 and Kepler-186, we show the importance and effect of both planetary and stellar tides on the dynamical evolution of planets and on the climate of the HZ planets.

The stellar spin orientation relative to the orbital planes of multiplanet systems are becoming accessible to observations. For example, 55 Cancri is a system composed of 5 planets orbiting a member of a stellar binary for which a projected obliquity of 72±12 ° relative to the orbit of the innermost planet has been reported (Bourrier & Hébrard 2014). This large obliquity has been attributed to the perturbation induced by the binary. Here we describe the secular evolution of similar systems and we discuss the case of the 55 Cancri system more deeply. We provide two different orbital configurations compatible with the currently available observations.

Inside-Out Planet Formation (IOPF; Chatterjee & Tan 2014, hereafter CT14) is a scenario for sequential in situ planet formation at the pressure traps of retreating dead zone inner boundaries (DZIBs) motivated to explain the many systems with tightly packed inner planets (STIPs) discovered by Kepler. The scenario involves build-up of a pebble-dominated protoplanetary ring, supplied by radial drift of pebbles from the outer disk. It may also involve further build-up of planetary masses to gap-opening scales via continued pebble accretion. Here we study radial drift & growth of pebbles delivered to the DZIB in fiducial IOPF disk models.

We consider the dynamics of a small trojan companion of a hypothetical giant exoplanet under the secular perturbations of additional planets. By a suitable choice of action-angle variables, the problem is amenable to the study of the slow modulation, induced by secular perturbations, to the dynamics of an otherwise called ‘basic’ Hamiltonian model of two degrees of freedom (planar case). We present this Hamiltonian decomposition, which implies that the slow chaotic diffusion at resonances is best described by the paradigm of modulational diffusion.

We study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

Fomalhaut harbours a moderately eccentric dust belt with a planet candidate (Fom b) imaged near its inner edge. MCMC-based orbital determination of Fom b shows that the orbit is highly eccentric (e≃0.9), nearly apsidally aligned with the belt. We study the secular interaction between the planet and the dust ring. We show that only if it is a small mass object, Fom b can perturb the belt without destroying it. But Fom b's perturbing action inevitably drives the belt to high eccentricity and apsidal misalignment. This behaviour is due to the planet's high eccentricity.

This dynamical outcome contradicts both observations and orbital determination. We conclude that another, more massive and less eccentric planet (Fom c) is required to dynamically control the belt. We show that Fom b is likely to have been formerly trapped in mean-motion resonance with Fom c and that subsequent eccentricity increase caused it to cross Fom c's orbit and to jump on its present day orbit via a scattering event.