Capture Probability in the 3:1 Mean Motion Resonance with Jupiter

We study the capture and crossing probabilities into the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100%, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (1995) and Quillen (2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations

Orbital evolution of circumbinary planets due to creep tides

Most confirmed circumbinary planets are located very close to their host binary where the tidal forces are expected to play an important role in their dynamics. Here we consider the orbital evolution of a circumbinary planet with arbitrary viscosity, subjected to tides due to both central stars. We adopt the creep tide theory and assume that the planet is the only extended body in the system and that its orbital evolution occurs after acquiring its pseudo-synchronous stationary rotational state. With this aim, we first performed a set of numerical integrations of the tidal equations, using a Kepler-38-type system as a working example. For this case we find that the amount of planetary tidal migration and also, curiously, its direction both depend on the viscosity. However, the effect of tides on its eccentricity and pericenter evolutions is simply a move toward pure gravitational secular solutions. Then we present a secular analytical model for the planetary semimajor axis and eccentricity evolution that reproduces very well the mean behavior of the full tidal equations and provides a simple criterion to determine the migration directions of the circumbinary planets. This criterion predicts that some of the confirmed circumbinary planets are tidally migrating inward, but others are migrating outward. However, the typical timescales are predicted to be very long, and not much orbital tidal evolution is expected to have taken place in these systems. Finally, we revisit the orbital evolution of a circumbinary planet in the framework of the constant time lag model. We find that the results predicted with this formalism are identical to those obtained with creep theory in the limit of gaseous bodies.Fil: Zoppetti, Federico Andrés. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba; Argentina. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba. Grupo de Invest.en Astronomia Teórica y Exptal.; ArgentinaFil: Folonier, H.. Universidade do Sao Paulo. Instituto de Astronomia, Geofísica e Ciências Atmosféricas; BrasilFil: Leiva, Alejandro Martín. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba; ArgentinaFil: Gomes, G.O.. Universidade do Sao Paulo. Instituto de Astronomia, Geofísica e Ciências Atmosféricas; Brasi

Interplay of tidal evolution and stellar wind braking in the rotation of stars hosting massive close-in planets

This paper deals with the application of the creep tide theory (Ferraz-Mello, Cel. Mech. Dyn. Astron. vol. 116, 109, 2013) 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 towards a stationary solution. However, stellar rotation may also be affected by stellar wind braking. Thus, while the rotation of a quiet host star evolves towards a stationary attractor with a frequency ($1+6e^2$) times the orbital mean-motion of the companion, the continuous loss of angular momentum in an active star displaces the stationary solution towards slower values: Active host stars with big close-in companions tend to have rotational periods larger 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 semi-major 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.Comment: 22 pages, 8 figures; Publication in Ap

A self-consistent weak friction model for the tidal evolution of circumbinary planets

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 applied 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 two-body problem. The orbital evolution is more complex. After reaching spin stationarity, the eccentricity was damped in all bodies and for all the parameters analysed 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 the inclusion of 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 two-body case when α → 0 or when one of the masses is taken as equal to zero. This model allows us to find a closed 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

Creep tide model for the three-body problem

We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present a set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin–orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the Solar System planets. The timescales for reaching the equilibrium state are expected to be approximately millions of years for stiff bodies but can be longer than the age of the system for planets with a large gaseous component. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the ratio of the semimajor axes a1∕a2 and low-order expansions of the eccentricities. These are found to very accurately reproduce the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a1∕a2 ~ 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and we find that most of them are probably rotating in a subsynchronous state, although the synchrony shift is much less important than our previous estimations. We present a comparison of our results with those obtained with the Constant Time Lag and find that, as opposed to the assumptions in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N-extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory. In addition to this, the latter formalism considers more realistic rheology that proved to reduce to the Constant Time Lag model in the gaseous limit and also allows several additional relevant physical phenomena to be studied