Context. The secular dynamics of small planetesimals in tight binary systems play a fundamental role in establishing the possibility of accretional collisions in such extreme cases. The most important secular parameters are the forced eccentricity and secular frequency, which depend on the initial conditions of the particles, as well as on the mass and orbital parameters of the secondary star. Aims. We construct a second-order theory (with respect to the masses) for the planar secular motion of small planetasimals and deduce new expressions for the forced eccentricity and secular frequency. We also reanalyze the radial velocity data available for γ-Cephei and present a series of orbital solutions leading to residuals compatible with the best fits. Finally, we discuss how different orbital configurations for γ-Cephei may affect the dynamics of small bodies in circumstellar motion. Methods. The secular theory is constructed using a Lie series perturbation scheme restricted to second order in the small parameter. The orbital fits were analyzed with a minimization code that employs a genetic algorithm for a preliminary solution plus a simulated annealing for the fine tuning. Results. For γ-Cephei, we find that the classical first-order expressions for the secular frequency and forced eccentricity lead to large inaccuracies ∼50% for semimajor axes larger than one tenth the orbital separation between the stellar components. Low eccentricities and/or masses reduce the importance of the second-order terms. The dynamics of small planetesimals only show a weak dependence with the orbital fits of the stellar components, and the same result is found including the effects of a nonlinear gas drag. Thus, the possibility of planetary formation in this binary system largely appears insensitive to the orbital fits adopted for the stellar components, and any future alterations in the system parameters (due to new observations) should not change this picture. Finally, we show that planetesimals migrating because of gas drag may be trapped in mean-motion resonances with the binary, even though the migration is divergent.

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