This paper deals with a new formulation of the creep tide theory (Ferraz-Mello, Cel. Mech. Dyn. Astron. 116, 109, 2013 − Paper I) and with the tidal dissipation predicted by the theory in the case of stiff bodies whose rotation is not synchronous but is oscillating around the synchronous state with a period equal to the orbital period. We show that the tidally forced libration influences the amount of energy dissipated in the body and the average perturbation of the orbital elements. This influence depends on the libration amplitude and is generally neglected in the study of planetary satellites. However, they may be responsible for a 27 percent increase in the dissipation of Enceladus. The relaxation factor necessary to explain the observed dissipation of Enceladus (γ = 1.2 − 3.8 × 10 −7 s −1 ) has the expected order of magnitude for planetary satellites and corresponds to the viscosity 0.6 − 1.9 × 10 14 Pa s, which is in reasonable agreement with the value recently estimated by Efroimsky (2018) (0.24 × 10 14 Pa s) and with the value adopted by Roberts and Nimmo (2008) for the viscosity of the ice shell (10 13 − 10 14 Pa s). For comparison purposes, the results are extended also to the case of Mimas and are consistent with the negligible dissipation and the absence of observed tectonic activity. The corrections of some mistakes and typos of paper II (Ferraz-Mello, Cel. Mech. Dyn. Astron. 122, 359, 2015) are included at the end of the paper.

This paper deals with the application of the creep tide theory (Ferraz-Mello) to the study of the rotation of stars hosting massive close-in planets. The stars have nearly the same tidal relaxation factors as gaseous planets and the evolution of their rotation is similar to that of close-in hot Jupiters: they tidally evolve toward a stationary solution. However, stellar rotation may also be affected by stellar wind braking. Thus, while the rotation of a quiet host star evolves toward a stationary attractor with a frequency ( e 1 6 2 + ) times the orbital mean motion of the companion, the continuous loss of angular momentum in an active star displaces the stationary solution toward slower values: active host stars with big close-in companions tend to have rotational periods longer than the orbital periods of their companions. The study of some hypothetical examples shows that, because of tidal evolution, the rules of gyrochronology cannot be used to estimate the age of one system with a large close-in companion, no matter if the star is quiet or active, if the current semimajor axis of the companion is smaller than 0.03-0.04 AU. Details on the evolution of the systems: CoRoT LRc06E21637, CoRoT-27, Kepler-75, CoRoT-2, CoRoT-18, CoRoT-14 and on hypothetical systems with planets of mass 1-4 M Jup in orbit around a star similar to the Sun are given.

We present a self-consistent model for the tidal evolution of circumbinary planets that is easily extensible to any other three-body problem. Based on the weak-friction model, we derive expressions of the resulting forces and torques considering complete tidal interactions between all the bodies of the system. Although the tidal deformation suffered by each extended mass must take into account the combined gravitational effects of the other two bodies, the only tidal forces that have a net effect on the dynamic are those that are applied on the same body that exerts the deformation, as long as no mean-motion resonance exists between the masses. As a working example, we apply the model to the Kepler-38 binary system. The evolution of the spin equations shows that the planet reaches a stationary solution much faster than the stars, and the equilibrium spin frequency is sub-synchronous. The binary components, on the other hand, evolve on a longer timescale, reaching a super-synchronous solution very close to that derived for the 2-body problem. The orbital evolution is more complex. After reaching spin stationarity, the eccentricity is damped in all bodies and for all the parameters analyzed here. A similar effect is noted for the binary separation. The semimajor axis of the planet, on the other hand, may migrate inwards or outwards, depending on the masses and orbital parameters. In some cases the secular evolution of the system may also exhibit an alignment of the pericenters, requiring to include additional terms in the tidal model. Finally, we derived analytical expressions for the variational equations of the orbital evolution and spin rates based on low-order elliptical expansions in the semimajor axis ratio α and the eccentricities. These are found to reduce to the well-known 2-body case when α → 0 or when one of the masses is taken equal to zero. This model allow us to find a close and simple analytical expression for the stationary spin rates of all the bodies, as well as predicting the direction and magnitude of the orbital migration.Key words. planets and satellites: dynamical evolution and stability -planet-disc interactions -planet-star interactions -methods: numerical

We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated nonhomogeneous bodies in non-synchronous rotation (Tisserand, Mécanique Céleste, t.II, Chap. 13 and 14) adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by n homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings ǫ k , µ k and the mean radius R k of each layer, or, equivalently, their semiaxes a k , b k and c k . To first order in the flattenings, the general solution can be written as ǫ k = H k ǫ h and µ k = H k µ h , where H k is a characteristic coefficient for each layer which only depends on the internal structure of the body and ǫ h , µ h are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile, using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: i) a body composed of two homogeneous layers; ii) bodies with simple polynomial density distribution laws and iii) bodies following a polytropic pressure-density law.