Noname manuscript No. (will be inserted by the editor) Tidal evolution of hierarchical and inclined systems A.C.M. Correia · J. Laskar · F. Farago · G. Bou´e Received: date / Accepted: date Abstract We investigate the dynamical evolution of Keywords Restricted Problems · Extended Body · hierarchical three-body systems under the effect of tides, Dissipative Forces · Planetary Systems · Rotation when the ratio of the orbital semi-major axes is small and the mutual inclination is relatively large (greater ◦ than 20 ). Using the quadrupolar non-restricted ap- 1 Introduction proximation for the gravitational interactions and the viscous linear model for tides, we derive the averaged At present, about 50 multi-planet systems have been re- equations of motion in a vectorial formalism which is ported, out of which roughly 1/3 possess close-in plan- suitable to model the long-term evolution of a large ets with semi-major axis smaller than 0.1 AU. There are variety of exoplanetary systems in very eccentric and also indications that about half of these stars are likely inclined orbits. In particular, it can be used to derive to have distant companions (Fischer et al, 2001). In constraints for stellar spin-orbit misalignment, capture addition, close binary star systems (separation smaller in Cassini states, tidal-Kozai migration, or damping of than 0.1 AU) are often accompanied by a third star (e.g. the mutual inclination. Because our model is valid for D’Angelo et al, 2006). Therefore, although hierarchical the non-restricted problem, it can be used to study systems are considerably different from our Solar sys- systems of identical mass or for the outer restricted tem, they represent a significant fraction of the already problem, such as the evolution of a planet around a known systems of stars and planets. These systems are binary of stars. Here, we apply our model to three dis- particularly interesting from a dynamical point of view, tinct situations: 1) the HD 80606 planetary system, for as they can be stable for very eccentric and inclined or- which we obtain the probability density function dis- bits and thus present uncommon behaviors. In partic- arXiv:1107.0736v1 [astro-ph.EP] 4 Jul 2011 tribution for the misalignment angle, with two pro- ular, they become interesting when the two innermost ◦ nounced peaks of higher probability around 53 and bodies are sufficiently close to undergo significant tidal ◦ 109 ; 2) the HD98800 binary system, for which we interactions over the age of the system, since the final show that initial prograde orbits inside the observed outcome of the evolution can be in a configuration that disc may become retrograde and vice-versa, only be- is totally different from the initial one. cause of tidal migration within the binary stars; 3) the The origin and evolution of the orbital configura- HD 11964 planetary system, for which we show that tions of multi-body systems can be analyzed with direct tidal dissipation combined with gravitational perturba- numerical integrations of the full equations of motion, tions may lead to a decrease in the mutual inclination, but the understanding of the dynamics often benefits and a fast circularization of the inner orbit. from analytical approximations. Additionally, tidal ef- fects usually act over very long time-scales and there- fore approximate theories also allow to speed-up the nu- A.C.M. Correia merical simulations and to explore the parameter space Department of Physics, I3N, University of Aveiro, Campus much more rapidly. Secular perturbation theories based Universit´ario de Santiago, 3810-193 Aveiro, Portugal on series expansions have been used for hierarchical E-mail: [email protected] triple systems. For low values of the eccentricity, the ex- 2 A.C.M. Correia et al. pansion of the perturbation in series of the eccentricity The same gravitational scattering is simultaneously re- is very efficient (e.g. Wu and Goldreich, 2002), but this sponsible for an increase of the mutual inclination of method is not appropriate for orbits that become very the orbits (e.g. Chatterjee et al, 2008), and the fact eccentric. An expansion in the ratio of the semi-major that inclined systems exchange its inclination with the axis a1/a2 is then preferred, as in this case exact ex- inner’s orbit eccentricity, results that the dissipation pressions can be computed for the secular system (e.g. in the eccentricity can be transmitted to the inclina- Laskar and Bou´e, 2010). tion of the orbits, and vice-versa. The most striking The development to the second order in a1/a2, called example is that the spin and the orbit can be com- the quadrupole approximation, was used by Lidov (1961, pletely misaligned (e.g. Pont et al, 2009; Triaud et al, 1962) and Kozai (1962) for the restricted inner prob- 2010). Previous studies on this subject have been un- lem (the outer orbit is unperturbed). In this case, the dertaken by Eggleton and Kiseleva-Eggleton (2001) for conservation of the normal component of the angular binary stars and by Wu and Murray (2003) and Fab- momentum enables the inner orbit to periodically ex- rycky and Tremaine (2007) for a planet in a wide bi- change its eccentricity with inclination (the so-called nary. Despite the success obtained by these works in ex- Lidov-Kozai mechanism). There is, however, another plaining the observations, they all used the same set of limit case to the massive problem, which is the outer re- equations, derived by Eggleton and Kiseleva-Eggleton stricted problem (the inner orbit is unperturbed). Palaci´an (2001), which is not easy to implement and has been ob- et al (2006) have studied this case and discussed the ex- tained in the frame of the inner restricted quadrupolar istence and stability of equilibria in the non-averaged approximation. As a consequence, their model cannot system. Farago and Laskar (2010) derived a simple model be applied to a large number of situations, where the of the outer restricted case and described the possible outer orbit cannot be held constant, such as regular motions of the bodies. They also looked at the quadrupo- planetary systems or planets around close binaries. lar problem of three masses and show how the inner and outer restricted cases are related to the general case. In this paper we intend to go deeper into the study of hierarchical three-body systems, where the inner- For planar problems, the series expansions in a /a 1 2 most bodies undergo tidal interactions. We do not make should be conducted to the octopole order (e.g. Mar- any restrictions on the masses of these bodies, and use chal, 1990; Ford et al, 2000; Laskar and Bou´e, 2010), the quadrupolar approximation for gravitational inter- as the quadrupole approximation fails to reproduce the actions with general relativity corrections. Our study eccentricity oscillations (e.g. Lee and Peale, 2003). How- is then suitable for binary star systems, planetary sys- ever, the inclinations of the already known hierarchical tems, and also for planet-satellite systems. We also systems have not been yet determined, and it can be as- consider in our model the full effect on the spins of sumed that high values for the eccentricities may also the two closest bodies, including the rotational flatten- indicate that their mutual inclinations are large as well ing of their figures. This allows us to correctly describe (e.g. Laskar, 1997; Chatterjee et al, 2008). the precession of the spin axis and subsequent capture As for Mercury, Venus and the majority of the nat- in Cassini states. We adopt a viscous linear model for ural satellites in the Solar system, close-in bodies un- tides (Singer, 1968; Mignard, 1979), as it provides sim- dergo significant tidal interactions, resulting that their ple expressions for the tidal torques for any eccentricity spins and orbits are slowly modified. The ultimate stage value. Since we are interested in the secular behavior, for tidal evolution is the synchronization of the spin we average the motion equations over the mean anoma- and the circularization of the orbit. Indeed, the ob- lies of the orbits and express them using the vectorial served mean eccentricity for planets and binary stars methods developed by Bou´eand Laskar (2006), Correia with a1 < 0.1 AU is close to zero within the obser- (2009), and Tremaine et al (2009). vational limitations (e.g. Pont et al, 2011). Although tidal effects modify the spin in a much shorter time- In Section2 we derive the averaged equations of mo- scale than they modify the orbit, synchronous rotation tion that we use to evolve hierarchical systems by tidal can only occur when the eccentricity is very close to effect. In Section 3 we obtain the secular evolution of the zero: the rotation rate tends to be locked with the or- spin and orbital quantities in terms of reference angles bital speed at the periapsis, because tidal effects are and elliptical elements, that are useful and more intu- stronger when the two bodies are closer to each other. itive to understand the outcomes of the numerical sim- During the formation process, the orbital eccentric- ulations. In Section 4 we apply our model to three dis- ity can increase due to gravitational scattering, so that tinct situations of extra-solar systems: HD 80606, HD98800, the inner bodies become close enough at periapsis for and HD 11964. Finally, last section is devoted to the tidal interactions to occur (e.g. Nagasawa et al, 2008). conclusions. Tidal evolution of hierarchical and inclined systems 3 2 The model tor, which points along the major axis in the direction of periapsis with magnitude e1: We consider here a hierarchical system of bodies com- ˙r1 × G1 r1 posed of a central pair with masses m0 and m1, together e1 = − .