We revisit the tidal stability of extrasolar systems harboring a transiting planet and demonstrate that, independently of any tidal model, none but one (HAT-P-2b) of these planets has a tidal equilibrium state, which implies ultimately a collision of these objects with their host star. Consequently, conventional circularization and synchronization timescales cannot be defined because the corresponding states do not represent the endpoint of the tidal evolution. Using numerical simulations of the coupled tidal equations for the spin and orbital parameters of each transiting planetary system, we confirm these predictions and show that the orbital eccentricity and the stellar obliquity do not follow the usually assumed exponential relaxation but instead decrease significantly, reaching eventually a zero value, only during the final runaway merging of the planet with the star. The only characteristic evolution timescale of {\it all} rotational and orbital parameters is the lifetime of the system, which crucially depends on the magnitude of tidal dissipation within the star. These results imply that the nearly circular orbits of transiting planets and the alignment between the stellar spin axis and the planetary orbit are unlikely to be due to tidal dissipation. Other dissipative mechanisms, for instance interactions with the protoplanetary disk, must be invoked to explain these properties.
Deep Dive into Falling Transiting Extrasolar Giant Planets.
We revisit the tidal stability of extrasolar systems harboring a transiting planet and demonstrate that, independently of any tidal model, none but one (HAT-P-2b) of these planets has a tidal equilibrium state, which implies ultimately a collision of these objects with their host star. Consequently, conventional circularization and synchronization timescales cannot be defined because the corresponding states do not represent the endpoint of the tidal evolution. Using numerical simulations of the coupled tidal equations for the spin and orbital parameters of each transiting planetary system, we confirm these predictions and show that the orbital eccentricity and the stellar obliquity do not follow the usually assumed exponential relaxation but instead decrease significantly, reaching eventually a zero value, only during the final runaway merging of the planet with the star. The only characteristic evolution timescale of {\it all} rotational and orbital parameters is the lifetime of the
Jupiter-like extra-solar planets have been detected transiting their parent star at an unexpectedly small distance of less than 0.1 AU (e.g. Pont 2008). Most of these systems have nearly circular orbits (e.g. Pont 2008) and first measurements of the sky-projected angle λ between the stellar rotation axis and the planetary orbital axis through the Rossiter-MacLaughlin effect (for HD 209458, HD 149026, HD 189733, TrES-1, XO-1, HD 17156) indicate a nearly perfect spin-orbit alignment (Winn et al. 2005(Winn et al. , 2007;;Gaudi & Winn 2007;Narita et al. 2007;Loeillet et al. 2008;Cochran et al. 2008). These observational properties are commonly interpreted as an outcome of tidal dissipation between the host star and the planet and these same effects are also believed to lead to synchronization of the planetary and stellar rotation with the orbital motion. As a consequence, corresponding timescales associated to these processes are usually evaluated by assuming an exponential relaxation towards equilibrium parameters as obtained from any evolution perturbation calculation near an equilibrium state (e.g. Hut 1981). This leads to timescale estimates of spin-orbit alignment, synchronization and circularization which differ by several orders of magnitude, ranging typically from ∼ 10 5 yrs to a Hubble time (e.g. Rasio et al. 1996;Sasselov 2003;Dobbs-Dixon et al. 2004;Ogilvie & Lin 2008;Mazeh 2008). All these conclusions, however, implicitly assume the existence of such tidal equilibrium states.
It has already been suggested that short-period planets could be unstable to tidal dissipation but these calculations were based on the assumption of the existence of (unstable) tidal equilibrium states (Rasio et al. 1996;
1 also at IMCCE-CNRS UMR 8028, 77 Avenue Denfert-Rochereau, 75014, Paris. blevrard@ens-lyon.fr Dobbs-Dixon et al. 2004), leading to an erroneous application of the tidal stability criterion derived by Hut (1980). More recently, numerical simulations of the orbits of some transiting planets from the OGLE survey indicated a possible collapse with the host star but the effect of tides raised by the star within the planet was ignored (Pätzold et al. 2004;Carone & Pätzold 2007). Jackson et al. (2008) noticed the importance of considering both tides raised by the star and the planet as well as the non-linear coupled evolution of the eccentricity and the orbital distance, but the global stability of the system and the additional coupling with the rotational evolution were not investigated.
In this Letter, we reconsider the stability of transiting extra-solar planets to tidal dissipation through theoretical and numerical considerations and show that none but one of the transiting planets has a tidal equilibrium state. We investigate the consequences for the tidal evolution timescales of orbital (semi-major axis, eccentricity) and rotational (stellar obliquity, stellar and planetary rotational velocities) parameters, taking both tides raised by the planet and the star into account.
A binary star-planet system that conserves the total angular momentum L tot but dissipates its energy is known to dynamically evolve towards only two possible solutions (Counselman 1973;Hut 1980). On one hand, if L tot < L c , where L c is the critical angular momentum defined by
where M p , C p and M ⋆ , C ⋆ denote the masses and polar moments of inertia of the planet and the star, respec-tively, and G is the gravitational constant, no equilibrium state exists and the system ultimately merges, independently of any tidal model. On the other hand, if L tot > L c , two equilibrium states exist that are characterized by the coincidence between equatorial and orbital planes, circularity of the orbit and synchronization between rotational and orbital periods when no further dissipation occurs (Hut 1980). The furthest equilibrium orbital distance a 1 is stable while the closest a 2 is unstable, with a 1 > a c > a 2 , where a c is the marginal equilibrium orbital distance for L tot = L c , with a c ≃ 3 C ⋆ /M p ∼ 0.064 (M Jup /M p ) 1/2 AU for a planet orbiting a Sun-like star. Therefore, exploring whether a system is stable or not, and deriving characteristic timescales, first requires to find out whether the condition L tot < L c is satisfied or not. Neglecting the spin of the planet, the total angular momentum of the system is given by the sum of the orbital angular momentum and the spin of the star:
assuming that the stellar obliquity is zero to maximize L tot , where e is the eccentricity, ω ⋆ is the rotational velocity of the star and
where k is set to the typical value 0.06 for centrally condensed stars with dominantly radiative interiors, characteristic of Sun-like stars (Claret & Gimenez 1989). In order to investigate the fate of observed transiting planetary systems, we have computed the ratio L tot /L c for each of them. All the quantities relevant to our study and their uncertainties have been collected from up-to-date published
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