Tidal Evolution of Hierarchical and Inclined Systems 3

arXiv:1107.0736v1 [astro-ph.EP] 4 Jul 2011 ia vlto fheacia n nlndsystems Correia inclined A.C.M. and hierarchical of evolution Tidal eateto hsc,IN nvriyo vio Campus [email protected] Aveiro, E-mail: Portugal of Aveiro, University 3810-193 Santiago, I3N, Universitário de Physics, of Department Correia A.C.M. n atcruaiaino h ne orbit. inner inclination, the mutual of the circularization in fast decrease a a and that to lead show may perturba- we tions gravitational which with combined the for dissipation 3) tidal system, stars; be- binary planetary the only 11964 within HD vice-versa, migration observed tidal and the of retrograde cause inside become orbits may prograde disc initial that show 109 n h uuliciaini eaieylre(greater large small relatively is 20 is axes than inclination semi-major mutual orbital the the and of ratio the tides, of when effect the under systems three-body hierarchical Abstract date Accepted: / date Received: one ek fhge rbblt rud53 pro- around probability two higher with of angle, peaks dis- misalignment nounced function the density for probability tribution the for system, dis- obtain planetary three a we 80606 to HD around which model the planet our 1) apply situations: a restricted we tinct of outer Here, stars. evolution the study of the binary for to as or used such mass be problem, identical can of for it valid systems problem, is model non-restricted our the Because of damping inclination. or mutual migration, the derive tidal-Kozai to states, capture used Cassini misalignment, in be spin-orbit can stellar and it for large constraints eccentric particular, a very In of in orbits. evolution inclined systems exoplanetary is long-term which of the variety formalism model vectorial averaged to a the suitable in derive motion we of tides, equations for the model and linear interactions viscous gravitational the for proximation wl eisre yteeditor) the by inserted be (will No. manuscript Noname ◦ )teH 80 iaysse,frwihwe which for system, binary 98800 HD the 2) ; ◦ .Uigteqarplrnnrsrce ap- non-restricted quadrupolar the Using ). eivsiaetednmcleouinof evolution dynamical the investigate We · .Laskar J. · .Farago F. · ◦ .BouéG. and nsre xasoshv enue o hierarchical ex- for the eccentricity, the used of values been low For have systems. triple expansions series based on theories perturbation space Secular rapidly. parameter more the much explore to nu- and there- the simulations speed-up and merical to time-scales allow also long theories approximate very fore over act ef- benefits usually tidal often fects Additionally, dynamics approximations. the analytical motion, of from of understanding equations the full but the direct of with integrations analyzed be numerical can systems multi-body of tions one. initial that the configuration from a final different in the totally be since is can system, evolution the the of of outcome age tidal significant the undergo over to innermost interactions close two sufficiently the are when partic- bodies interesting In become behaviors. or- they uncommon inclined ular, present and eccentric thus very and for bits stable be view, can of they point are as dynamical systems a These from planets. interesting and particularly already stars the of of systems sys- fraction known significant Solar a our represent from they tem, different considerably hierarchical although are Therefore, systems (e.g. 2006). star third al, a et by In D’Angelo accompanied often smaller 2001). are (separation AU) al, 0.1 systems than et star binary (Fischer close likely companions addition, are stars distant these have of half to about are There that AU. indications 0.1 plan- than also close-in smaller axis possess semi-major 1/3 with roughly ets which of re- out been have ported, systems multi-planet 50 about present, At Introduction 1 isptv Forces Dissipative Keywords h rgnadeouino h ria configura- orbital the of evolution and origin The etitdProblems Restricted · lntr Systems Planetary · xeddBody Extended · Rotation · 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 Boué, 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). Palacián (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 Boué, 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 Bouéand 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 = − . (3) β1µ1 r1 with an external companion with mass m2. Both inner bodies are considered oblate ellipsoids with gravity ﬁeld ai is the semi-major axis (that can also be expressed 3 coeﬃcients given by J20 and J21 , rotating about the axis using the mean motion, ni = µi/ai ), ei is the eccen- of maximal inertia along the directions ˆs0 and ˆs1 (gyro- tricity, and C is the principal moment of inertia. The i p scopic approximation), with rotation rates ω0 and ω1, contributions to the orbits are easily computed from respectively, such that (e.g. Lambeck, 1988) the above potentials as

2 3 ˙ ˙ ωi Ri G1 = r1 × F1 , G2 = r2 × F2 , (4) J2i = k2i , (1) 3Gmi and where G is the gravitational constant, Ri is the radius 1 G1 e˙ = F × + ˙r × G˙ , (5) of each body, and k2i is the second Love number for 1 1 1 1 β1µ1 β1 potential (pertaining to a perfectly fluid body). ! ′ ′ We use Jacobi canonical coordinates, with r1 being where Fi = −∇ri U , with U = U + Gm0m1/r1 + the position of m1 relative to m0, and r2 the position Gm01m2/r2. of m2 relative to the center of mass of m0 and m1. We In Jacobi coordinates, the total orbital angular mo- further assume that |r1| ≪ |r2|, and we shall refer to mentum is equal to G1 + G2 (e.g. Smart, 1953). Since the orbit of m1 relative to m0 as the inner orbit, and the total angular momentum is conserved, the contri- the orbit of m2 relative to the center of mass of m0 and butions to the spin of the bodies can be computed from m1 as the outer orbit. In the following, for any vector the orbital contributions: u, û = u/ kuk is the unit vector. ˙ ˙ ˙ ˙ L0 + L1 = −(G1 + G2) . (6) Because we are only interested in the secular evolution of the system, we further average the equations of 2.1 Conservative motion motion over the mean anomalies of both orbits (see appendix A). The resulting equations are (e.g. Bouéand In the quadrupolar three-body problem approximation, Laskar, 2006; Farago and Laskar, 2010): the potential energy U of the system is given by (e.g. ˙ 2 ˆ ˆ ˆ ˆ Smart, 1953): G1 = −γ(1 − e1)cos I k2 × k1 +5γ(e1 · k2) k2 × e1 ˆ − α1i cos θi ˆsi × k1 , (7) 2 i m0m1 Ri X U = −G 1 − J i P (ˆr · ˆs ) 2 2 1 i 2 r1  r1  ˙ ˆ ˆ ˆ ˆ i=0,1 G2 = −γ(1 − e1)cos I k1 × k2 +5γ(e1 · k2) e1 × k2 X   ˆs kˆ 2 − α2i cos εi i × 2 , (8) m01m2 mi Ri i −G 1 − J2i P2(ˆr2 · ˆsi) X r2  m01 r2  i=0,1 X 2 γ(1 − e1) ˆ ˆ ˆ ˆ ˆ  2  ˙e1 = − cos I k2 × e1 − 2 k1 × e1 − 5(e1 · k2) k2 × k1 β1m2 r1 kG1k −G P2(ˆr2 · ˆr1) , (2) h i r r 2 2 α 1i 1 2 ˆ − cos θi ˆsi × e1 + (1 − 5cos θi) k1 × e1 ,(9) 2 kG k 2 where P2(x)=(3x − 1)/2 is the Legendre polynomial i 1 3 3 X of degree two, and terms in (r1/r2) and (Ri/rj ) have and been neglected (i, j =0, 1). We also have m01 = (m0 + L˙ kˆ ˆs kˆ ˆs m1), β1 = m0m1/m01, β2 = m01m2/(m01 + m2), µ1 = i = −α1i cos θi 1 × i − α2i cos εi 2 × i , (10) Gm01, and µ2 = G(m01 + m2). where The evolution of the spins can be tracked by the 2 3Gm0m1J2i R rotational angular momenta, L ≃ C ω ˆs . In turn, the α = i , (11) i i i i 1i 2a3(1 − e2)3/2 evolution of the orbits can be tracked by the orbital an- 1 1 2 ˆ ˆ 2 gular momenta, Gi = βi µiai(1 − e ) ki (where ki is 3Gm m J i R i α = 2 i 2 i , (12) the unit vector Gˆ ), and the Laplace-Runge-Lenz vec- 2i 3 2 3/2 i p 2a2(1 − e2) 4 A.C.M. Correia et al.

2 3Gm2β1a1 compare the theoretical results with the observations, γ = 3 2 3/2 , (13) 4a2(1 − e2) as the effect of tides are very small and can only be de- and tected efficiently after long periods of time. The qual- itative conclusions are more or less unaffected, so, for cos θ = ˆs · kˆ , cos ε = ˆs · kˆ , cos I = kˆ · kˆ ,(14) i i 1 i i 2 1 2 simplicity, we adopt here a model with constant ∆ti, are the direction cosines of the spins and orbits: θi is the which can be made linear (Mignard, 1979; Néron de obliquity to the orbital plane of the inner orbit, εi is the Surgy and Laskar, 1997): obliquity to the orbital plane of the outer companion, ′ r1 ≃ r1 + ∆ti (ωiˆsi × r1 − ˙r1) . (19) and I is the inclination between orbital planes. They can also be expressed as As for the conservative motion, we can obtain the equations of motion directly from equations (4), (5) and cos ε = cos I cos θ + sin I sin θ cos ϕ , (15) ′ i i i i (6) using UT instead of U (see appendix A), that is, where ϕi is a precession angle. ˙ ˙ ˙ ˙ G2 =0 , G1 = −L0 − L1 , (20)

5 2.2 General relativity correction 15 m(1−i) Ri ˆ e˙ 1 = k2i n1 f4(e1) k1 × e1 i 2 mi ! a1 ! We may add to Newton’s equations the dominant con- X tribution from general relativistic effects. These effects Ki ωi ˆ − 2 f4(e1) (e1 · ˆsi) k1 are mainly felt by eccentric orbits on close encounters β a 2n1 i 1 1 between the central bodies and contribute to the gravi- X 11 ωi tational force with a small correction (e.g. Schutz, 1985) − f4(e1)cos θi − 9f5(e1) e1 , (21) 2 n1 3µ kG k2 F 1 1 ˆr gr = − 2 4 1 , (16) and β1c r1 ωi 2 L˙ = K n f (e ) 1 − e2 (ˆs − cos θ kˆ ) (22) where kG1k = β1 µ1a1(1 − e ), and c is the speed of i i 1 4 1 1 i i 1 1 2n1 light. To this order, the dominant relativistic secular q p ω (e · ˆs )(6 + e2) ω contribution is on the precession of the periapsis, leav- −f (e ) i ˆs + f (e )kˆ + 1 i 1 i e , 1 1 n i 2 1 1 4(1 − e2)9/2 n 1 ing eccentricity, orientation and semi-major axis of the 1 1 1 orbit unaffected (Eq. 5): where, 2 5 3µ1n1 ˆ 3k2i Gm(1−i)Ri e˙ 1 = k1 × e1 . (17) K = ∆t , (23) c2a (1 − e2) i i 6 1 1 a1 and 2.3 Tidal effects 1+3e2 +3e4/8 f (e)= , (24) 1 (1 − e2)9/2 Neglecting the tidal interactions with the external body 2 4 6 m , the tidal potential for the inner pair writes (e.g. 1+15e /2+45e /8+5e /16 2 f2(e)= , (25) Kaula, 1964): (1 − e2)6 5 2 4 6 8 G 2 Ri ′ 1+31e /2+255e /8+185e /16+25e /64 U = − k i m P (ˆr · ˆr ) , (18) f (e)= ,(26) T r3 2 (1−i) r′3 2 1 1 3 (1 − e2)15/2 1 i=0,1 1 X ′ 1+3e2/2+ e4/8 where r1 is the position of the interacting body at a f4(e)= 2 5 , (27) time delayed of ∆ti. The dissipation of the mechani- (1 − e ) cal energy of tides in the body’s interior is responsi- 1+15e2/4+15e4/8+5e6/64 ble for this delay between the initial perturbation and f (e)= . (28) 5 (1 − e2)13/2 the maximal deformation. As the rheology of stars and planets is badly known, the exact dependence of ∆ti on The first term in expression (21) corresponds to a the tidal frequency is unknown. Many different authors permanent tidal deformation, while the second term have studied the problem and several models have been corresponds to the dissipative contribution. The pre- ˆ developed so far, from the simplest ones to the more cession rate of e1 about k1 is usually much faster than complex (for a review see Correia et al, 2003; Efroim- the evolution time-scale for the dissipative tidal effects. sky and Williams, 2009). The huge problem in validat- As a consequence, when the eccentricity is constant over ing one model better than others is the difficulty to a precession cycle, we can average expression (23) over Tidal evolution of hierarchical and inclined systems 5 the argument of the periapsis and get (Correia, 2009, are thus combination of three eigenmodes. More pre- appendix A): cisely, these solutions are composed of a uniform rotation of all the vectors around the total angular momentum, and in the rotating frame, vectors describe quasi- ˆs kˆ ˙ i + cos θi 1 ωi ˆ Li = −Ki n1 f1(e1) − f2(e1)k1 (29), periodic motions with only 2 proper frequencies (Boué 2 n1 ! and Laskar, 2006, 2009). The bodies are in an equilibrium configuration, usually called Cassini state, when the amplitudes of the quasi-periodic motion vanish. Al- 3 Secular evolution though analytical approximations of the solutions can be obtained (Bouéand Laskar, 2006, 2009), we assume We have presented the equations that rule the tidal evo- that α ≪ α ≪ γ, and also kG k≪kG k. Then, lution of a hierarchical system of three bodies in terms 2i 1i 1 2 equation (33) simplifies of a vectorial formalism. However, the spin and orbital dkˆ γ 3 quantities are better represented by the rotation angles 1 ≈− 1+ e2 cos I kˆ × kˆ . (34) dt kG k 2 1 2 1 and elliptical elements. The direction cosines (Eq.14) 1 are obtained from the angular momenta vectors, since The evolution of G1 is thus independent of Li, and it ˆ ˆsi = Li/ kLik and ki = Gi/ kGik, as well as the ro- has a uniform precession motion at frequency g around tation rate ωi = Li · ˆsi/Ci. The eccentricity and the the total orbital angular momentum, where semi-major axis can be obtained from e1 = ke1k and γ 3 2 2 2 2 g ≈− 1+ e cos I. (35) a1 = kG1k /(β1 µ1(1 − e1)), respectively. kG k 2 1 1 The evolution of the spin-axes is then simpler in the kˆ kˆ 3.1 Conservative motion rotating frame where 1 and 2 are constant. Since α1i ≫ α2i, we have for each body 3.1.1 Cassini states dˆsi α1i ˆ ˆ = − cos θi k1 × ˆsi − g k2 × ˆsi , (36) dt kLik The obliquities are functions of G1, G2, and Li (Eq. 14). which leads to Their evolution can be obtained from (Eqs. 7, 8, 10) as θ˙i = g sin I sin ϕi , (37) d cos θ G˙ · (ˆs − cos θ kˆ ) L˙ · (kˆ − cos θ ˆs ) i = 1 i i 1 + i 1 i i ,(30) and dt kG1k kLik α cos ϕ ϕ˙ = − 1i cos θ + g cos I + g sin I i . (38) i kL k i tan θ for the obliquity to the orbital plane of the inner orbit, i i θ . An identical expression could be obtained for the ˆ ˆ i We used the fact that k1 and k2 are coplanar. Equation obliquity to the outer orbit, εi, replacing G1 by G2. To (37) shows that the obliquity is oscillating around an simplify, we may average the equations over the argu- equilibrium value given by ϕi =0 or π. Stable configu- ment of the periapsis (appendix A), and the norms of rations for the spin can be found whenever the vectors kG k, kG k, and kL k become constant. Thus, ˆ ˆ 1 2 i (ˆsi, k1, k2) are coplanar and precess at the same rate d cos θ dkˆ dˆs g (e.g. Colombo, 1966; Peale, 1969). The equilibrium i = 1 ·(ˆs −cos θ kˆ )+ i ·(kˆ −cos θ ˆs ) ,(31) dt dt i i 1 dt 1 i i obliquities can be found settingϕ ˙ i = 0 (Eq. 38), which provides a single relationship (e.g. Ward and Hamilton, where 2004): ˆ dk1 γ 3 2 = − 1+ e cos I kˆ × kˆ (32) λi cos θ sin θ + sin(θ − I)=0 , (39) dt kG k 2 1 2 1 i i i 1 α where λi = α1i/(kLik g) is a dimensionless parameter. − 1i cos θ ˆs × kˆ , kG k i i 1 The above equation has two or four real roots for θi, i 1 X which are known by Cassini states. For nearly coplanar and orbits, we have I ∼ 0, and these solutions are approxi- dˆsi α1i ˆ α2i ˆ mately given by: = − cos θi k1 × ˆsi − cos εi k2 × ˆsi . (33) dt kLik kLik sin I cos I tan−1 , ± cos−1 − . (40) cos I ± λ λ This system has a priori four degrees of freedom asso- i i ciated to the four precession angles (one for each orbit, For a generic value of I, when |λi| ≪ 1 the first expres- and one per solid body). However, because the total sion gives the only two real roots of equation (39). On angular momentum is conserved, there is only three de- the other hand, when |λi| ≫ 1, we have four real roots grees of freedom. Regular solutions of (Eqs. 31, 33, 33) approximately given by expressions (40). 6 A.C.M. Correia et al.

3.1.2 Lidov-Kozai cycles The variations in the inclination and eccentricity (Eqs. 43, 43) become zero when sin2̟1 = 0, that is, The variations in the mutual inclination between the for cos2̟1 = ±1. When cos2̟1 = −1 (̟1 = ±π/2) it inner orbit and the orbit of the external companion is possible to show that̟ ˙ = 0 has a solution if h1 ≤ can be obtained from the direction cosine (Eq.14) in a 3/5 (Lidov, 1962; Kozai, 1962). Thus, when the mu- similar way as the obliquity (Eq. 30): tual inclination is greater than arccos 3/5 ≃ 39.23◦ p ˙ ˆ ˆ ˙ ˆ ˆ we have a libration regime for the periapsis about ̟1 = d cos I G1 · (k2 − cos Ik1) G2 · (k1 − cos Ik2) p = + .(41) ±π/2. In this regime, one can observe large variations dt kG1k kG2k in both I and e1, known by Lidov-Kozai cycles. If the Since kG1k≪kG2k, we obtain from expression (7): inner orbit is initially circular, the maximum eccentric- 2 ity achieved is given by e = 1 − (5/3)cos2 I. d cos I 5 γe1 2 1 = cos I sin I sin 2̟1 (42) Lidov-Kozai cycles persist as long as the perturba- dt 2 kG1k p tion from the outer body is the dominant cause of pre- α1i cos θi ˆ ˆ − (ˆsi × k1) · k2 , cession in the inner orbit. However, additional sources kG1k i X of precession, such as general relativity or tides, can where ̟1 is the argument of the periapsis of the inner compensate the libration mechanism and suppress the orbit, that is, the angle between the line of nodes of the large eccentricity/inclination oscillations (e.g. Migaszewski two orbits and the periapsis of the inner orbit. and Go´zdziewski, 2009). For h1 > 3/5 the periapsis On the other hand, the variations in the eccentricity of the inner orbit is always in a circularization regime, p are easily obtained from the Laplace-Runge-Lenz vector so there is only small variations in the eccentricity and (Eq.9): inclination. 2 ˙e1 · e1 5 γ(1 − e1)e1 2 e˙1 = = sin I sin 2̟1 . (43) e1 2 kG1k 3.2 Tidal evolution Thus, combining expressions (43) and (43) and neglecting the contributions from the rotational ﬂattening 3.2.1 Spin evolution (terms in α1i), we get

d cos I e1e˙1 The variation in the body’s rotation rate can be com- = cos I, (44) ˙ 2 puted from equation (29) asω ˙ i = Li · ˆsi/Ci, giving dt (1 − e1) (Correia and Laskar, 2010a): which can be integrated to give 2 2 Ki n1 1+cos θi ωi 1 − e1 cos I = h1 = Cte , (45) ω˙ i = − f1(e1) − f2(e1)cos θi .(47) Ci 2 n q 1 ! where h1 is constant in absence of tides. The above re- lation shows that an increase in the eccentricity of the For a given obliquity and eccentricity, the equilib- inner orbit must be accompanied by a decrease in the rium rotation rate, obtained whenω ˙ i = 0, is attained mutual inclination and vice-versa. It corresponds to a for: happy coincidence, as called by Lidov and Ziglin (1976), ωi f2(e1) 2cos θi = 2 , (48) which results from the fact that in the quadrupolar ap- n1 f1(e1) 1+cos θi proximation the potential energy (Eq. 2) does not de- Notice, however, that the above expression is only valid pend on argument of the periapsis of the external body for constant eccentricity (and also constant semi-major ̟ (e.g. Farago and Laskar, 2010). As a consequence, 2 axis), or at least if tidal effects modify the eccentricity the conjugate Delaunay variable kG k = β µ a (1 − e2) 2 2 2 2 2 faster than other effects. Indeed, when the eccentric- is constant, and, as the semi-major axis are constant in p ity is forced by the gravitational perturbations from a the secular problem, e is also constant. Indeed, when 2 companion body, the limit solution of expression (47) we consider only the orbital contributions, the total an- is no longer given by equation (48), but more generally gular momentum is (Correia and Laskar, 2004, 2009): G 2 G 2 G G k 1k + k 2k +2 k 1kk 2k cos I = Cte . (46) t ωi(t) Ki Since kG2k is constant and kG1k≪kG2k, it remains = f2(e1(τ))cos θi(τ)gi(τ) dτ , (49) n1 Cigi(t) 0 kG1k cos I = Cte, and we retrieve the result given by Z equation (45). This is no longer true for the octopole with or higher order approximations (e.g. Laskar and Boué, K t g (t) = exp i f (e (t))(1 + cos2 θ (τ)) dτ .(50) 2010). i 2C 1 1 i i Z0 Tidal evolution of hierarchical and inclined systems 7

It corresponds to a solution that pursues the instan- 2Ki ωi = 2 f2(e1)cos θi − f3(e1) . (56) taneous equilibrium rotation for a given eccentricity β1a n1 i 1 (Eq.48), but delayed and with a smaller amplitude, de- X The inner orbit can either expand or contract, depending on the relative strength of tidal effects. The pending on the initial spin of the bodies. Considering stronger these effects are, the shorter is the time delay for simplicity dissipation in only one component, for and closer are the amplitudes to expression (48). fast initial rotating rates (ω ≫ n ), the semi-major In turn, the dissipative obliquity variations are com- i 1 axis and the eccentricity usually increase, except for puted by substituting equation (29) in (30) with kL k ≪ i retrograde spins (θ > π/2). The ratio between or- kG k, giving: i 1 bital and spin evolution time-scales is roughly given Kin1 ωi 2 θ˙ ≃ sin θ f (e )cos θ − f (e ) . (51) by Ci/(m1a1) ≪ 1, meaning that the spin achieves an i C ω i 1 1 i 2n 2 1 i i 1 equilibrium position much faster than the orbit. Thus, Because of the factor n1/ωi in the magnitude of the as the rotation rate decreases, the increasing tendency obliquity variations, for an initial fast rotating body, in the orbital parameters is reversed when d kG1k /dt = the time-scale for the obliquity evolution is longer than 0 (Eq. 54), the time-scale for the rotation rate evolution (Eq.47). ωi f2(e1) As a consequence, it is expected that the rotation rate cos θi = , (57) n1 f1(e1) approaches its equilibrium value (Eq.48) earlier than the obliquity. Replacing equation (48) in (51), we have that is, when the rotation rate is close to its equilib- for constant eccentricity: rium value (Eq. 48). After an initial increase in the semi- major axis and in the eccentricity, we can then always K n sin θ θ˙ ≃− i 1 f (e ) i . (52) expect a contraction of the inner orbit until it becomes i C ω 2 1 1+cos2 θ i i i completely circularized. The final evolution of the sys- We then conclude that, although the initial behavior of tem is then achieved when e1 = 0, ωi = n1 (Eq.48) and the obliquity depends on the initial rotation rate, tidal θ ≃− sin I/λ (Eq.53). ˙ i i effects always end by decreasing the obliquity, since θi ≤ The variations in the mutual inclination can be ob- 0. Thus, the final obliquity tends to be captured in a ˙ ˙ tained using expression (41) with G1 = − i Li (Eq. 29). small obliquity Cassini state (Eq.40), that is, We have then: P sin I Ciωiγ 3 2 d cos I Kiωi θi ≃− = 1+ e1 cos I sin I. (53) = f1(e1)(cos εi − cos I cos θi) , (58) λi α1i kG1k 2 dt 2 kG1k i X or, making use of expression (15), 3.3 Orbital evolution dI Kiωi = − f1(e1) sin θi cos ϕi . (59) dt 2 kG1k The variations in the norm of the orbital angular mo- i mentum can be computed directly from expression (29), X ˙ ˙ ˙ Just after the spin of one body is trapped in a small since G1 = −L0 − L1: obliquity Cassini state, ϕi = 0 and sin θi is constant d kG k = − L˙ · k and given by expression (53). Thus, dt 1 i 1 i dI Kiωi X ≃ f1(e1) sin I. (60) ωi dt 2λi kG1k = Kin1 f1(e1)cos θi − f2(e1) . (54) i n1 X i X The variations in the eccentricity are easily obtained 4 Application to exoplanets from the Laplace-Runge-Lenz vector (Eq.21):

˙e1 · e1 In this section we apply the model described in Sect. 2 e˙1 = e1 to three distinct situations of exoplanetary systems: 9K 11 ω HD 80606 (inner restricted problem), HD 98800 (outer = i f (e )cos θ i − f (e ) e , (55) β a2 18 4 1 i n 5 1 1 restricted problem), and HD 11964 (intermediate non- i 1 1 1 X restricted problem). We numerically integrate the set while the semi-major axis variations are obtained from of equations (7), (8), (9), (10) and (17) for the conser- the eccentricity and the norm of the orbital angular vative motion, together with equations (20), (21) and momentum: (23) for tidal effects. ˙ a˙ 1 2e ˙1e1 2 G1 · G1 There exist many systems containing a “hot Jupiter” = 2 + 2 a1 (1 − e1) kG1k in a wide binary for which one could apply the present 8 A.C.M. Correia et al. model to illustrate the tidal migration combined with Table 1 Initial parameters for the HD 80606 system (Naef Lidov-Kozai cycles. However, we prefer to reproduce the et al, 2001; Eggenberger et al, 2004; Wu and Murray, 2003). results for HD80606 in order to compare our results HD 80606 (obtained with a non-restricted model) with previous parameter body 0 body 1 body 2 studies by Wu and Murray (2003) and Fabrycky and m (M⊙) 1.1 0.0037 1.1 Prot (day) 20 0.5 − Tremaine (2007) (obtained with a restricted model). θ (deg) 10 35 − The stability of a disc around the multi-binary HD 98800 ϕ (deg) 0 0 − system has been recently studied by Verrier and Evans R ( 106m) 695 75 × 2 − C/mR 0.08 0.25 (2009) and Farago and Laskar (2010), which have iso- − k2 0.028 0.51 − lated two different regimes for the trajectories of the ∆t (s) 0.1 40 − disc. We then apply our model to an hypothetical planet parameter orbit 1 orbit 2 around the binary stars and show how transitions be- a (AU) 5.0 1000 tween the two regimes are possible, and how particles e 0.1 0.5 ̟ (deg) 0.0 in initial prograde orbits may end in retrograde orbits. − I (deg) 85.6 We finally use our model to study the HD 11964 system, which is a hierarchical planetary system composed of two planets, the inner one in the Neptune-mass regime and the outer one similar to Jupiter. There is dergoing tidal evolution, and is thus a perfect example no determination of the mutual inclination between the to test our model. two planets, but the system is stable for very large val- In order to compare our results with the previous ues, so we analyze its behavior for different situations. studies of Wu and Murray (2003) and Fabrycky and Tremaine (2007) we use the same initial conditions for the planet and the stellar companions: the planet is 4.1 HD 80606 initially set in a Jupiter-like orbit with a1 = 5 AU, ◦ e1 = 0.1, and I = 85.6 , while the stellar companion In current theories of planetary formation, the region is supposed to be a Sun-like star at a2 = 1000 AU, within 0.1 AU of a protostar is too hot and rarefied and e2 = 0.5 (Table 1). In Figure 1 we plot an ex- for a Jupiter-mass planet to form, so “hot Jupiters” ample of combined tidal-Kozai migration of the planet likely form further away and then migrate inward. A HD 80606 b. significant fraction of “hot Jupiters” has been found in Prominent eccentricity oscillations are seen from the systems of binary stars (e.g. Eggenberger et al, 2004), very beginning and the energy in the planet’s spin is suggesting that the stellar companion may play an im- transferred to the orbit increasing the semi-major axis portant role in the shrinkage of the planetary orbits. for the first 0.1 Gyr (Eq. 47). As the equilibrium ro- In addition, close binary stars (separation compara- tation is approached around 0.3 Gyr (Eq. 49) the tidal ble to the stellar radius) are also often accompanied evolution is essentially controlled by equations (55) and by a third star. For instance, Tokovinin et al (2006) (56), whose contributions are enhanced when the eccen- found that 96% of a sample of spectroscopic binaries tricity reaches high values. The semi-major axis evo- with periods less than 3 days has a tertiary compo- lution is executed by apparent “discontinuous” transi- nent. Indeed, in some circumstances the distant com- tions precisely because the tidal dissipation is only effi- panion induces tidal interactions in the inner binary cient during periods of high eccentricity. As dissipation by means of the Lidov-Kozai mechanism (Sect. 3.1.2), reduces the semi-major axis, periapsis precession be- causing the binary semi-major axis to shrink to the comes gradually dominated by general relativity rather currently observed values (e.g. Eggleton and Kiseleva- than by the third body, and the periapsis starts circu- Eggleton, 2001). The same mechanism has been subse- lating as the eccentricity approaches to 0 near 0.5 Gyr. quently proposed to be at the origin of “hot Jupiters” Tidal evolution stops when the orbit is completely cir- as an alternative to migration in a disk (e.g. Wu and cularized. The final semi-major axis is estimated to 2 Murray, 2003). about af = a1(1 − e1) ≃ 0.07 AU, which corresponds The HD80606 system is composed of two Sun-like to a regular “hot Jupiter” (Correia and Laskar, 2010b). stars in a very wide orbit (a2 ∼ 1000AU) (Eggenberger The angle between the spin axis of the planet and et al, 2004), and a short-period planet in a very eccen- its orbit (denoted by θ1) is quickly brought by tides tric orbit (a1 = 0.45 AU and e1 = 0.92) (Naef et al, to a small obliquity Cassini state (Eq.53). On the other 2001). Because at periapsis the distance to the main hand, the angle between the spin axis of the star and the star is only 0.036 AU, the orbit of the planet is still un- orbit of the planet (denoted by θ0) is not tidally evolved, Tidal evolution of hierarchical and inclined systems 9

◦ Fig. 1 Possible evolution of the planet HD 80606 b initially in an orbit with a1 = 5 AU, e1 = 0.1, and I = 85.6 (Table 1). We show the evolution of the eccentricity e1(a), semi-major axis a1, and periapsis a1(1 e1) (b), mutual inclination I, obliquity of − the star θ0, and obliquity of the planet θ1 (c), and orbital period Porb, rotation period of the star Prot,0, and rotation period of the planet Prot,1 (d). Results plot here are identical to those in Wu and Murray (2003) and Fabrycky and Tremaine (2007), obtained with a restricted model. since the dissipation within the star is much smaller As the semi-major axis decreases, the precession than the dissipation within the planet (k20 ∆t0 ≪ k21 ∆t1, of the planet’s orbit becomes progressively dominated Table 1). As a consequence, capture of the spin of the by the equatorial bulge of the star, so that α10 ∼ γ star in a Cassini state does not occur during the time around 2.8 Gyr (Eq. 33). From that point, the orbit of length of the simulations, but one can observe large os- the planet precesses around the spin of the star and θ0 cillations corresponding to the variations in the orienta- becomes constant, retaining a value between 10◦ and tion of the orbital plane of the planet. Indeed, as long as 160◦. One then expects to observe a misalignment be- α10 ≪ γ, the angle between the spin of the star and the tween the spin of the star and the orbit of the planet. In ◦ outer companion (denoted by ε0) is approximately con- the simulation shown in Figure 1, we obtain θ0 ≈ 60 , stant (Eq.36), and thus |ε0 − I|≤ θ0 ≤ ε0 + I (Eq. 15). but this value depends on the initial conﬁguration of ◦ ◦ ◦ In our simulation ε0 = 85.6 −10 = 75.6 (Table 1), so the system. we approximately observe that 10◦ ≤ θ ≤ 161◦ (Fig- 0 In order to get a statistical distribution of the ﬁ- ure 1c). nal misalignment we have integrated a series of 40 000 10 A.C.M. Correia et al.

Table 2 Initial parameters for the HD 98800 B system (Tor- res et al, 1995; Tokovinin, 1999). HD 98800 B parameter body 0 body 1 body 2 m (M⊙) 0.699 0.582 0.001 Prot (day) 20 0.5 − θ (deg) 10 35 − ϕ (deg) 0 0 − R ( 106m) 758 591 × 2 − C/mR 0.08 0.08 − k2 0.028 0.028 − Fig. 2 Histogram of the final distribution of the misalign- ∆t (s) 100 100 − ment angle θ0. We have integrated a series of 40 000 systems parameter orbit 1 orbit 2 with the same initial conditions from Table 1 except for the a (AU) 0.983 5.2 ◦ ◦ obliquity θ0 = 0 , and inclination I, which range from 84.3 e 0.785 0.5 ◦ ± to 90 . We observe two pronounced peaks of higher probabil- ̟ (deg) 82.0 ◦ ◦ − ity around θ0 53 and θ0 109 , which is consistent with I (deg) 20.0 the observations≈ of the Rossiter-McLaughlin≈ anomaly for the HD 80606 system (Pont et al, 2009). systems with the same initial conditions from Table 1 1995; Kastner et al, 1997). Moreover, it has a large in- ◦ frared excess attributed to a circumbinary disc around except for the obliquity of the star θ0 =0 , and for the mutual inclination I, which ranged from ±84.3◦ to 90◦ the B pair (Koerner et al, 2000; Furlan et al, 2007). (on an evenly spaced grid of −0.1 ≤ cos I ≤ +0.1). A Substantial extinction towards this pair suggests that histogram with the results of these simulations is given it is observed through some of this material (Tokovinin, in Figure 2, with a bin width of 1◦. Our results do not 1999). An apparent absence of CO molecular gas in the differ much from the histogram obtained by Fabrycky disc combined with infrared spectrum modeling indi- and Tremaine (2007), except that we observe two pro- cate that this is a T-Tauri transition disc that is just ◦ reaching the debris disc stage, with a collisional cascade nounced peaks of higher probability around θ0 ≈ 53 ◦ having been recently initiated (e.g. Furlan et al, 2007). and θ0 ≈ 109 . They performed 1 000 simulations dis- tributed in bins widths of 10◦, which partially explains The orbits of the stars are all highly eccentric and in- this difference, but the main reason is the fact that they clined, creating a dynamical environment unlike almost all other known debris discs (Tokovinin, 1999). The dust adopted for the rotation of the star Prot,0 = 10 h, while we used a value closer to the Sun’s rotation period. Be- disc is generally agreed to be an annulus around the cause we executed so much simulations our histogram B binary, but the exact structure varies from model to is close to the probability density function distribution model. Koerner et al (2000) estimate a coplanar nar- for the misalignment. Observations of the Rossiter- row ring outwards of about ∼ 5 AU from the two stars. McLaughlin anomaly for the HD80606 star reinforced However, Boden et al (2005) argue that the line of sight the hypothesis of spin-orbit misalignment in this sys- extinction means that the disc cannot be coplanar un- tem (alignment excluded at > 95% level), with a pos- less it is very flared. itive median projected angle of 50◦ (Pont et al, 2009). Verrier and Evans (2009) investigated numerically These observations are in perfect agreement with our the stability of a family of particles at large inclinations simulations. around the B binary, which remain stable even under Although we used the full quadrupolar problem, the perturbation of an outer third stellar companion. while previous studies (Wu and Murray, 2003; Fabrycky The results show that the Lidov-Kozai mechanism of and Tremaine, 2007) used the inner restricted problem the outer star is disrupted by a nodal libration induced (where the outer orbit is considered constant), we re- by the inner binary pair on a shorter time-scale. An trieve similar results and then verify that their approx- analytical study by Farago and Laskar (2010) confirmed imation is appropriate for this specific situation. that equilibria at large inclination circumbinary orbits exist for the outer restricted quadrupolar problem. This raises the possibility that planets and asteroids with 4.2 HD 98800 large inclination may survive in multi-stellar systems. In this section we test the evolution of a Jupiter- HD98800 is an interesting and unusual system of four like mass planet around the B binary in an orbit that 10Myr old post T-Tauri K stars: two spectroscopic bi- is presently occupied by the dust disc. Since we aver- naries A and B in orbit about one another (Torres et al, age the orbit of the planet over the mean longitude, the Tidal evolution of hierarchical and inclined systems 11

Fig. 3 Energy levels in the (I sin ̟, I cos ̟) plane for diﬀerent values of the eccentricity of the B binary HD 98800, e1 = 0.5 (a), e1 = 0.785 (b), and e1 = 0.9 (c) (Farago and Laskar, 2010). The dot marks the initial position of the outer orbit (Table 2).

orbital evolution of this planet can also be regarded as grade orbit), while in the second case the inclination the orbital evolution of the particles in the disc. The oscillates around ±90◦. ◦ HD98800 system is still very young, and the B binary In our simulation, we use ̟1 = 82 , as this value stars are very close to each other, so one can expect places the initial orbit of the planet at the edge between that the two components will undergo significant tidal the two different regimes of behavior (Fig. 3b). Initially, evolution throughout its life. In order to speed-up the k2 is precessing around k1 and the mutual inclination simulations we assume that both stars experience in- oscillates between 20◦ and nearly 90◦. As the system tense tidal dissipation, with ∆t = 102 s (Table 2). Be- evolves, the norm of the orbital angular momentum of cause the gravitational effects of the A binary do not the planet remains constant, but the norm of the or- destabilize the system, we do not consider its presence bital angular momentum of the binary is modified by in the simulations. exchanges with the rotational angular momentum of The inclination of the B binary with respect to the the stars. Since we have considered ω1 ≫ ω0 (Table 2) line of sight is estimated to be about 23◦ (Boden et al, in our example the initial exchanges are mainly between 2005). The inclination of the disc with respect to the the binary’s orbit and the spin of the star 1. B binary is unknown, but one can contest its copla- During the first stages of the evolution ω1 ≫ n1 and narity, since substantial extinction of the light implies the norm of the orbital angular momentum of the bi- that the system is observed through some of this ma- nary increases (Eq.54), as well as the eccentricity and terial (Tokovinin, 1999). Therefore, we assume that the the semi-major axis (Eqs. 55, 56). As a consequence, the inclination is about 20◦ (Table 2). The argument of the separatrix between the two regimes contracts (Fig. 3c). periapsis and the longitude of the node were also de- Since the norm of the orbital angular momentum of termined with respect to the plane of the sky (Boden the planet remains constant (the planet is too far to et al, 2005), but again, we miss the value of these two undergo dissipation), the orbit of the planet crosses quantities measured with respect to the plane of the the separatrix and switches to the regime of preces- disc. As a consequence, ̟1 is a free parameter for the sion around the direction of the periapsis of the binary planet/disc in our simulation. The choice of this pa- e1. In our simulation this transition occurs shortly after rameter is critical, as it places the initial system in a 80 Myr since we placed the initial system very close to different regime of libration, and it also determines the the separatrix (Fig. 4a). libration amplitude (Fig. 3b). Indeed, we can differen- According to expression (51) the obliquity of the tiate two kinds of regimes (Farago and Laskar, 2010): star θ1 also increases, but the rotation rate decreases closed trajectories where the orbital angular moment (Eq.47). Around 2Gyr, the obliquity begins to decrease of the planet k2 precesses around the orbital angular and the orbit of the binary initiates its contraction. In momentum of the binary k1 (or its opposite −k1); or general, this contraction is much slower than the pre- closed trajectories where the orbital angular momen- vious expansion because the spin is already near an tum of the planet precesses around the direction of the equilibrium position and the major source of dissipa- periapsis of the binary e1 (or its opposite −e1). In the tion becomes the orbital energy. As the eccentricity is first situation the inclination is strictly inferior to 90◦ damped, the separatrix between the two regimes ex- (or strictly superior to 90◦, corresponding to a retro- pands and catches again the planet’s orbit (Fig. 3b). 12 A.C.M. Correia et al.

−3 Fig. 4 Possible evolutions for a m2 10 M⊙ planet or disc at a2 = 5.2 AU and e2 = 0.5, initially in a prograde orbit around the binary HD 98800 B (Table 2). During≈ the ﬁrst evolutionary stages the orbit of the binary expands and the planet changes of regime after about 80 Myr (a). As the binary orbit shrinks, the planet changes of regime again (somewhere between 25 and 30 Gyr), but depending when the planet crosses the separatrix between regimes, its orbit may become retrograde (b) or prograde (c).

After the second separatrix transition, the orbital an- c) and m2 =0.62 MJup (planet b), at a1 =0.229 AU and gular momentum of the planet precesses again around a2 = 3.16 AU, respectively (Butler et al, 2006; Wright the orbital angular momentum of the binary or its op- et al, 2009). Veras and Ford (2010) performed extensive posite ±k1, depending on the ̟1 value at the time the n-body simulations for this system and concluded that transition occurs. We have performed two different sim- it is always stable for mutual inclinations I < 60◦ (or ulations, with a 10−2 difference in the initial obliquity I > 120◦), stable up to 85% for I < 75◦ (or I > 105◦), ◦ ◦ of the star θ0, and each one led to the a different final and stable up to 25% for 75

Table 3 Initial parameters for the HD 11964 system (Butler et al, 2006; Wright et al, 2009). HD 11964 body 0 body 1 body 2 parameter (star) (c) (b) m (MJup) 1178 0.0788 0.622 Prot (day) 20 0.5 − θ (deg) 10 35 − ϕ (deg) 0 0 − R ( 106m) 695 75 × 2 − C/mR 0.08 0.25 − k2 0.028 0.50 − ∆t (s) 0.1 100 − parameter orbit 1 (c) orbit 2 (b) a (AU) 0.229 3.16 e 0.300 0.041 ̟ (deg) 0.0 − I (deg) 30.0

Fig. 5 Evolution of the HD 11964 c inclination (top) and eccentricity (bottom) during 1 Myr, starting with the orbital solution from Wright et al (2009) and I = 30◦ (Table 3). The but the general long-term behavior of the system re- solid lines curves are the values obtained with the quadrupo- mains essentially the same. lar secular model, while the dashed green lines are the com- Though the present age of the star is estimated to plete solutions obtained with a n-body numerical model. be about 10Gyr (Saffe et al, 2005), we start the system as if it has been recently formed. We assume the we follow the system over Gyr time-scales one can ob- present orbits as the original ones, and an initial rota- serve those variations occurring (Figure 6b). The most tion period of the inner planet of about 10 h (Table 3). striking effect is an initial increase in the eccentricity Because tidal effects on the inner planet are strong, the of the inner planet, that is accompanied by a signifi- choice of the initial spin is irrelevant, as it evolves to cant reduction in the mutual inclination between the an equilibrium configuration in about one million years orbits of the two planets. This behavior results from a (Figure 6a). Nevertheless, this procedure allows us to combination of tidal effects, rotational flattening, and understand simultaneously the initial behavior of the the gravitational perturbations from the outer planet, system and also its future evolution. In addition, we 2 ◦ which tend to conserve the quantity h1 = 1 − e1 cos I arbitrarily assume a mutual inclination I = 30 . This (Eq.45). Tidal effects alone also decrease the eccentric- p inclination value sets the system well outside a coplanar ity (Eq. 55), so when its contribution becomes dominat- configuration, but yet below the critical value of about ◦ ing the eccentricity is progressively damped to zero. 39 that allows Lidov-Kozai cycles. The exchanges between eccentricity and inclination During the first million years (Figure 6a), the ro- (Fig. 5) are more significant for large values of the ini- tation rate of the inner planet and its obliquity are tial mutual inclination. The maximum value for the ec- brought to their equilibrium positions, where one be- centricity oscillations is then higher and therefore the lieve that the spin of the planet can be found today. The planet experiences stronger tidal effects. As a result, rotation rate is slowed down until it reaches an equilib- the orbit of the inner planet evolves in a shorter time- rium near ω1/n1 ≈ f2(e1)/f1(e1), that is, Prot,1 ≈ 25 d scale. In Figure 7 we plot the simultaneous evolution (Eq. 49), while the obliquity, after an initial increase of the inclination and eccentricity of the inner orbit (since the initial rotation rate is fast (Eq.51)), quickly for different initial mutual inclinations, I. For I < 30◦ drops until it is captured in the small obliquity Cassini there is almost no reduction in the inclination, and the ◦ state θ1 ≈ sin I/λ1 ≈ 0.018 (Eq.53). Because the ec- eccentricity damping is very slow (Figure 7a). On the centricity and the inclination are oscillating, so does the other hand, for I > 40◦ we observe a reduction of more equilibrium rotation rate (Eq. 48) and the equilibrium than 20◦ in the inclination, which is accompanied by an obliquity (Eq. 53). initial increase in the eccentricity, followed by a rapid During the first two million years (Figure 6a), the damping to zero (Figure 7c,d). eccentricity and the inclination of the inner orbit do not For large values of the initial inclination, the system experience any substantial secular modification (apart evolves much faster to its final configuration, so the or- the periodical ones), since tidal effects are much less ef- bit becomes circular in a shorter time-scale. Since the ficient over the orbit than over the spin. However, when present orbit of the inner planet still presents an ec- 14 A.C.M. Correia et al.

Fig. 6 Long-term evolution of the planet HD 11964 c for an initial mutual inclination of 30◦ (Table 3). During the ﬁrst million years, the rotation rate is slowed down into its equilibrium position (Eq. 48) and the obliquity is trapped in a small obliquity Cassini state (Eq. 53). As the system evolves, tidal eﬀects decrease the mutual inclination of the planetary orbits, while the eccentricity increases until around 2 Gyr, time after which it is damped to zero. The rotation rate closely follows the eccentricity, and the obliquity also decreases, as the equilibrium in a Cassini state depends on the inclination of the perturber.

Fig. 7 Long-term evolution of the inclination (top) and eccentricity (bottom) of the planet HD 11964 c for diﬀerent values of the initial mutual inclination: 15◦ (a), 30◦ (b), 40◦ (c), and 60◦ (d) (Table 3). As the initial inclination increases, so does the amplitude of the eccentricity oscillations, resulting that tidal eﬀects are enhanced, and the evolution time-scale shorter. Tidal evolution of hierarchical and inclined systems 15

Fig. 8 Time needed to circularize the orbit of the planet HD 11964 c (e1 < 0.01) as a function of the initial mutual inclination and for different values of the initial eccentricity (e1 = 0.1, 0.2, 0.3, and 0.4). The system is 10 Gyr old (Saffe et al, 2005), so we conclude that initial inclinations between 40◦ and 140◦ were not possible, as the present eccentricity is still 0.3. centricity close to 0.3, one may assume that the initial to the star and undergoes tidal evolution. Here we have inclination could not have been excessively large. In or- studied this particular sub-group of multi-planet sys- der to estimate an upper limit for the initial inclination, tems. Using a very general and simplified averaged vec- we can run some simulations and check which of those torial formalism, we have shown that inclined hierar- can evolve to the present configuration within 10 Gyr, chical planetary systems undergoing tidal dissipation the estimated age for the system. can evolve in many different and sometimes unexpected However, the choice of the initial orbit is not easy, ways. since many different initial configurations can bring the We re-analyzed the case of HD80606, a system where planet to the present orbit, depending on the tidal damp- the planet migrated inwards by a combined tidal-Kozai ing coefficients (k2 and ∆t) and also on the initial eccen- mechanism, confirming previous results. We looked at tricity, inclination, and semi-major axis. It is out of the the behavior of a planet (or disc) around the binary scope of this study to explore exhaustively the initial stars HD98800B and showed that initial prograde or- configurations of the HD 11964 system, but rather to il- bits may become retrograde and vice-versa, only be- lustrate some interesting evolutionary scenarios. As we cause of tidal migration within the binary stars. Fi- just have seen, the present eccentricity combined with nally, we studied the regular 2-planet HD 11964 system a large inclination can later evolve to an identical ec- and showed that tidal dissipation combined with grav- centricity but with smaller inclination. The semi-major itational perturbations may lead to a decrease in the axis will be slightly smaller than today, but this pa- mutual inclination, and a fast circularization of the in- rameter essentially plays over the evolution time-scale. ner orbit. Thus, the present system can be a representation of the We have chosen the above three examples, as they initial system and we adopt it for simplicity. In Fig- are representative of the diversity of behaviors among ure 8 we used the initial conditions listed in Table 3, inclined hierarchical systems. Many other systems are but modified the initial inclinations and eccentricities. awaiting to be studied. The fact that we use average We observe that inclinations in the interval [40◦, 140◦] equations for both tidal and gravitational effects, makes circularize the orbit (e1 < 0.01) in less than 10Gyr, so our method suitable to be applied in long-term studies. they can be discarded. It allows to run many simulations for different initial conditions in order to explore the entire phase space and evolutionary scenarios. In particular, it can be very 5 Conclusion useful to put constraints on the inclinations and dissipation ratios of hierarchical planetary systems. Our study Many multi-planet systems have been reported in hier- can also be extended to systems of binary stars, and to archical configurations. For the most part, their mutual planet-satellite systems. inclinations are unknown, but the fact that they exhibit significant values in the eccentricities led to think that Acknowledgements We acknowledge support from PNP-CNRS, the inclinations can also be large. In addition, very of- France, and from Funda¸cão para a Ciência e a Tecnologia, ten the innermost planet in these systems is very close Portugal (grant PTDC/CTE-AST/098528/2008). 16 A.C.M. Correia et al.

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