Compact planetary systems perturbed by an inclined companion: II. Stellar spin-orbit evolution
CCompact planetary systems perturbed by an inclined companion:II. Stellar spin-orbit evolution
Gwena¨el Bou´e , and Daniel C. Fabrycky [email protected] ABSTRACT
The stellar spin orientation relative to the orbital planes of multiplanet systems are becomingaccessible to observations. Here, we analyze and classify different types of spin-orbit evolution incompact multiplanet systems perturbed by an inclined outer companion. Our study is based onclassical secular theory, using a vectorial approach developed in a separate paper. When planet-planet perturbations are truncated at the second order in eccentricity and mutual inclination,and the planet-companion perturbations are developed at the quadrupole order, the problembecomes integrable. The motion is composed of a uniform precession of the whole system aroundthe total angular momentum, and in the rotating frame, the evolution is periodic. Here, we focuson the relative motion associated to the oscillations of the inclination between the planet systemand the outer orbit, and of the obliquities of the star with respect to the two orbital planes.The solution is obtained using a powerful geometric method. With this technique, we identifyfour different regimes characterized by the nutation amplitude of the stellar spin-axis relative tothe orbital plane of the planets. In particular, the obliquity of the star reaches its maximumwhen the system is in the Cassini regime where planets have more angular momentum than thestar, and where the precession rate of the star is similar to that of the planets induced by thecompanion. In that case, spin-orbit oscillations exceed twice the inclination between the planetsand the companion. Even if mutual inclination is only (cid:39) ◦ , this resonant case can cause thespin-orbit angle to oscillate between perfectly aligned and retrograde values. Subject headings: methods: analytical — methods: numerical — celestial mechanics — planets andsatellites: dynamical evolution and stability — planets and satellites: general — planet-star interactions
1. Introduction
Hot or eccentric Jupiters only constitute a smallfraction of the exoplanets discovered to date.Among those with short orbital periods, mostare smaller, less massive, and part of compactmultiplanet systems (Howard et al. 2010, 2012;Howard 2013; Petigura et al. 2013). Due to thesmaller planetary radii and larger orbital periods,the efficiency of the standard method to mea- Department of Astronomy and Astrophysics, Univer-sity of Chicago, 5640 South Ellis Avenue, Chicago, IL60637, USA Astronomie et Syst`emes Dynamiques, IMCCE-CNRSUMR 8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France. sure spin-orbit angle in systems with hot Jupiters,based on the Rossiter-McLaughlin effect (Holt1893; Rossiter 1924; McLaughlin 1924), decreasessignificantly in multiplanet systems. Only twomultiplanet systems, called KOI-94 and Kepler-25, have been studied with this technique (Hiranoet al. 2012; Albrecht et al. 2013). Two other meth-ods have been implemented to measure the spin-orbit angle in multiplanet systems, the stellar spotcrossing technique on Kepler-30 (Sanchis-Ojedaet al. 2012), and asteroseismology on Kepler-50and Kepler-65 (Chaplin et al. 2013). The fivesystems prove to be compatible with perfect spin-orbit alignments, a priori suggesting that multi-planet systems are preferentially in the equatorialplane of their star. A sixth system, Kepler-561 a r X i v : . [ a s t r o - ph . E P ] M a y able 1: Notation. variable Ref. description H a m il t o n i a n H tot Eq. (33) secular Hamiltonian of the numerical system H Eqs. (1, 7) secular Hamiltonian of the analytical model K Eq. (8) first integral of the analytical problem α , γ Eq. (35) coefficients of the analytical Hamiltonian H : (cid:80) α k and (cid:80) γ k , respectivelyb ( k ) s Laplace coefficient t i m e s c a l e s ν Eq. (41) precession frequency α/L of s relative to w ν Eq. (41) precession frequency α/G of w relative to s ν Eq. (41) precession frequency γ/G of w relative to w (cid:48) ν Eq. (41) precession frequency γ/G (cid:48) of w (cid:48) relative to w ν a , ν b , ν c , ν d permutation of ν , ν , ν , ν P nut Eq. (20) nutation period P prec Eq. (32) precession period s t e ll a r p a r a m e t e r s m mass R radius J Eq. (36) quadrupole gravitational harmonic k second fluid Love number C moment of inertia along the short axis ω rotation speed P rotation period 2 π/ω (cid:15) obliquity relative to the reference plane ψ precession angle s stellar spin axis L stellar angular momentum Cω s j t hp l a n e t a nd c o m p a n i o n m j m (cid:48) mass a j a (cid:48) semimajor axis b (cid:48) semiminor axis a (cid:48) (1 − e (cid:48) ) / P j P (cid:48) revolution period e j e (cid:48) eccentricity I j I (cid:48) absolute inclination (with respect to the reference plane)Ω j Ω (cid:48) longitude of the ascending node e j eccentricity vector j j dimensionless orbital angular momentum (1 − e j ) / w j w j w (cid:48) unit orbital angular momentum G j G (cid:48) orbital angular momentum o t h e r v a r i a b l e s w unit vector of GG Eq. (38) total angular momentum of the planet system G w = (cid:80) G j w j W Eq. (5) total angular momentum L + G + G (cid:48) x = cos θ x Eq. (6) cosine of the stellar obliquity relative to the planets plane s · w y = cos θ y Eq. (6) cosine of the stellar obliquity relative to the companion’s orbit s · w (cid:48) z = cos θ z Eq. (6) cosine of the mutual inclination between the planets and the companion w · w (cid:48) τ Eq. (17) fictitious time used to parametrize elliptic orbits in ( x, y, z ) g e o m e t r i c o b j e c t s E elliptic orbit in ( x, y, z ) satisfying H ( x, y, z ) = h and K ( x, y, z ) = k for two reals ( h, k ) C , ∂ C cube [ − , × [ − , × [ − ,
1] in ( x, y, z ) and its boundary, respectively B , ∂ B Fig. 1 Cassini Berlingot defined by V ( x, y, z ) ≥ D x , D y , D z Fig. 1 diagonals which are the intersections of ∂ C and ∂ BS Fig. 10 hyperbolic surface equal to the union of all elliptic trajectories intersecting D x V Eqs. (11, 12) oriented volume generated by ( s , w , w (cid:48) ) S Eq. (54) quadric function defining the surface S A x , A z Eq. (10) length scales of the elliptic orbit in ( x, y, z ) G gravitational constant c speed of light ◦ (Liet al. 2014). More generally, in single as well as inmultiplanetary systems, spin-orbit misalignmentmay also result from the magnetic interaction be-tween the protostar and its circumstellar disc (Laiet al. 2011) or from the solid precession of theprotoplanetary disc induced by an inclined com-panion (Batygin 2012; Batygin & Adams 2013;Lai 2014). In multiplanet systems surrounded byan outer stellar companion, apsidal precession fre-quencies are dictated by the companion and bythe planet-planet interactions. As a consequence,even at high inclination, if the planet system is suf-ficiently packed, planet-planet interactions domi-nate the apsidal motion, the evolution is stabilizedwith respect to the Lidov-Kozai mechanism, ec-centricities remain small, and all planets move inconcert (Innanen et al. 1997; Takeda et al. 2008;Saleh & Rasio 2009). These systems are classi-fied as dynamically rigid . Although the Lidov-Kozai evolution is quenched, the planetary meanplane still precesses if it is inclined relative tothe orbit of the companion, and can eventuallylead to spin-orbit misalignment with the centralstar. Kaib et al. (2011) applied this idea to the55 Cancri multiplanet system which has a stel-lar companion, and concluded that the planetsare likely misaligned with respect to the stellarequator. However, the results only hold as longas the stellar spin-axis is weakly coupled to theplanets orbit. We show here that this conditionis not satisfied for the 55 Cancri system unless Note that our definition of dynamically rigid is more strin-gent than that of Takeda et al. (2008) who also include thecase where planet eccentricities increase in concert. the semiminor axis of the perturber is very small,of the order of 180 au (periastron distance (cid:46) ◦ mightbe the signature of an earlier tilt of the planetsystem (Tremaine 1991). Moreover, analyzing asimilar problem where a protoplanetary disk takesthe place of the compact planet system, Batygin(2012) showed that this mechanism is able to tiltforming planetary systems around slow rotatorT Tauri pre-main sequence stars. Here, we re-visit the problem composed of a dynamically rigidsystem perturbed by a stellar or a planetary com-panion on a wide and inclined orbit. The innerplanets are assumed to have low eccentricities andmutual inclinations comparable to or lower thanthose of our own solar system. According to theseassumptions, orbital evolution induced by tidesis expected to be weak and is neglected. Thesehypotheses are motivated by statistical studies ofcompact exoplanet systems detected by Kepler orby radial velocity (e.g., Tremaine & Dong 2012;Figueira et al. 2012; Fabrycky et al. 2012; Wu& Lithwick 2013). However, we allow the overallplane of planets to tilt by an arbitrary angle. A hi-erarchical companion is included, which is allowedto have any eccentricity and inclination. The maingoal of this study is to follow the evolution of theinclination of the planet system with respect to thespin-axis of the parent star. Thus, the interactionbetween the stellar spin-axis and the orbital mo-tion of the inner planets is taken into account. Forthis study, we exploit the results of the so-called“3-vector problem” which has been solved geomet-rically in (Bou´e & Laskar 2006, hereafter BL06)and in (Bou´e & Laskar 2009, hereafter BL09).The 3-vector problem aims to model the secularevolution of three coupled angular motions suchas the lunar problem with the planet spin andthe orbital angular momenta of the satellite andthe star (BL06) or the binary asteroid problemwith two spin-axes and their mutual orbital mo-tion (BL09). Here, the three vectors are the spinof the star, the total orbital angular momentumof the planet system, and that of the companion.3n Section 2, we recall the main results of the 3-vector problem, and we also provide a new integralexpression for the precession frequency. Then, inSection 3 we employ the vectorial formalism of theclassical secular theory that we described in a pre-vious paper (Boue and Fabrycky 2014; BF14) andwe show how the three vector problem emergesfrom this general secular model. The validity ofthe simplification is also discussed. In this work,we thus consider two different models which cor-respond to two levels of approximation. On theone hand, the perturbing function is expanded atthe fourth order in planet eccentricity and mu-tual inclination and at the octupole in the inter-action between each planet and the companion.This model provides accurate results but is non-integrable and has to be solved numerically. Onthe other hand, the system is described by the in-tegrable three vector problem which gives deepergeometrical insight. In the following, we refer tothe former as the numerical model and to the lat-ter as the analytical model. In Section 4, the twomodels are compared in their application to realexoplanet systems. Then, we exploit more deeplythe possibilities of the analytical model to spanthe parameter space and identify four differentregimes of evolution in Section 5. The conclusionsare given in the last section.
2. Three-vector problem
This section summarizes a few key results as-sociated to the so-called “3-vector problem” de-scribed in (BL06; BL09). In the context of thispaper, the three vectors are the angular momen-tum of the star L = L s , the orbital angular mo-menta of the planet system G = G w , and that ofthe companion G (cid:48) = G (cid:48) w (cid:48) , where s , w , and w (cid:48) are unit vectors. The 3-vector problem assumesthat the evolution is governed by a Hamiltonianof the form H = − α s · w ) − β s · w (cid:48) ) − γ w · w (cid:48) ) , (1)where α , β , and γ are constant parameters repre-senting the coupling between the planetary systemand both the stellar rotation and the binary orbit,respectively. Their expression will be derived inthe subsequent section. Note that in contrast tothe more general 3-vector problem, here we neglectthe direct interaction between the stellar spin and the orbit of the companion, i.e., we set β = 0. Theequations of motion are d s dt = − L s × ∇ s H ; d w dt = − G w × ∇ w H ; d w (cid:48) dt = − G (cid:48) w (cid:48) × ∇ w (cid:48) H , (2)which leads to d s dt = − αL ( s · w ) w × s ; d w dt = − αG ( s · w ) s × w − γG ( w (cid:48) · w ) w (cid:48) × w ; d w (cid:48) dt = − γG (cid:48) ( w (cid:48) · w ) w × w (cid:48) . (3)In prevision of the subsequent analysis, we set ν = α/L ,ν = α/G ,ν = γ/G ,ν = γ/G (cid:48) . (4)These quantities are important as they are thecharacteristic precession frequencies of s around w , of w around s and w (cid:48) , and of w (cid:48) around w ,respectively. The 3-vector problem is integrable (BL06;BL09). Let W = L s + G w + G (cid:48) w (cid:48) (5)be the total angular momentum of the system.The general solution is a uniform rotation of thethree vectors around the total angular momentumcombined with a periodic motion in the rotatingframe (BL06; BL09). The evolution is thus char-acterized by two frequencies or periods. Hereafter,the uniform rotation is referred to as the preces-sion motion with period P prec , and the periodicloops described in the rotating frame are equallyqualified as nutation in reference to the Earth-Moon problem, or simply as the relative motionwith period P nut . The relative motion can besolved elegantly with geometric arguments (BL06;4L09). It is also very important for our study fortwo reasons: it enables 1) to check if any systemcan be misaligned, and 2) to evaluate the timescaleof the secular spin-orbit evolution which can thenbe compared to the lifetime of the system. Next,we recall its solution and main properties as de-rived in (BL06; BL09). Then, we present a newintegral expression of the precession period. In order to get the relative evolution of the sys-tem described by the Hamiltonian (1), we followthe same derivation as in BL06 and BL09. Wedenote x = s · w ,y = s · w (cid:48) ,z = w · w (cid:48) . (6)Sometimes, we will also use the correspondingangles defined by x = cos θ x , y = cos θ y , and z = cos θ z . In this coordinate system, the Hamil-tonian reads as H = − α x − γ z . (7)The conservation of the norm of each angular mo-mentum L , G , and G (cid:48) , as well as the total angularangular momentum of the system W , Eq. (5), leadto the second constant of the motion K = (cid:107) W (cid:107) − L − G − G (cid:48) LGx + LG (cid:48) y + GG (cid:48) z . (8)Each trajectory of the relative motion in the( x, y, z ) frame is at the intersection of a cylin-der defined by H ( x, y, z ) = h and a plane definedby K ( x, y, z ) = k , where h and k are two con-stants given by the initial conditions. Trajectoriesare thus subsets of ellipses defined by the values h and k of the two first integrals of the motion.Hereafter, we denote them as E = { ( x, y, z ) ∈ R | H ( x, y, z ) = h, K ( x, y, z ) = k } . These ellipses canbe parametrized as follows x ( τ ) = A x cos τ ,z ( τ ) = A z sin τ ,y ( τ ) = 1 LG (cid:48) (cid:0) k − LGx ( τ ) − GG (cid:48) z ( τ ) (cid:1) , (9) where A x = (cid:114) − hα , A z = (cid:115) − hγ . (10)The change of time t (cid:55)→ τ leading to theparametrization (9) will be made explicit in sec-tion 2.4. In general, systems do not cover the fullellipses. Indeed, x , y , and z are dot products ofunit vectors and the evolution is restricted insidethe cube C = { ( x, y, z ) ∈ [ − , } . There is also amore stringent additional constraint (BL06). Let V = s · ( w × w (cid:48) ) . (11)V represents the oriented volume of the paral-lelepiped generated by the vectors s , w , and w (cid:48) .In terms of the dot products x , y , z , the square ofthe volume V is given by the Gram determinant V = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x yx zy z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 − x − y − z + 2 xyz . (12)When the three vectors s , w , and w (cid:48) are copla-nar, V ( x, y, z ) = 0. This is the equation ofa cubic surface known as Cayley’s nodal cubic.The restriction of this surface to the cube C isdisplayed in Fig. 1. In BL09, this restrictionis called Cassini Berlingot . The word Berlingotcomes after a french hard candy with a similarshape. The name Cassini has been added in ref-erence to the ‘Cassini states’ characterized by thecoplanarity of the same three vectors as in thisproblem. Thus ‘Cassini states’ are located atthe surface V ( x, y, z ) = 0. Because the cubic(12) represents the square of the volume V , itmust be positive. As a consequence, the evolu-tion of the system is restricted inside the Berlin-got B = { ( x, y, z ) ∈ C | V ( x, y, z ) ≥ } . Here, B denotes the inside of the Berlingot, and ∂ B its surface. In Fig. 1, important diagonals D x , D y , and D z are represented. They all belong tothe intersection ∂ B ∩ ∂ C , in which the three vec-tors ( s , w , w (cid:48) ) lie in the same plane. The point(1 , ,
1) corresponds to the configuration where allthree vectors are aligned and in the same direc-tion. In that case, the system is fully coplanar withonly prograde orbits. At the point (1 , − , − w (cid:48) is pointingin the opposite direction as s and w . In that case,5 x D z D y Fig. 1.— Cassini Berlingot defined by V ( x, y, z ) ≥
0. As V must be greater orequal to zero, the allowed region in the ( x, y, z )space is the interior of the Berlingot shapevolume. The diagonals D x , D y , and D z are alsorepresented. See text for detail.the system is also coplanar, but the outer compan-ion is on a retrograde orbit. Along the diagonal D x joining these two points, the planet system re-mains in the equatorial plane of the star, and thecompanion’s inclination θ y = θ z increases from 0 ◦ at (1 , ,
1) to 180 ◦ at (1 , − , − D y , the angle θ y = cos − ( s · w (cid:48) )is constant and equal to 0, thus the equator ofthe star remains in the plane of the outer body,while the planet system tilts from 0 ◦ at (1 , , ◦ at ( − , , − D z , w · w (cid:48) = cos θ z = 1, the planet system remainsin the plane of the outer perturber and the stel-lar obliquity θ x = θ y increases from 0 ◦ at (1 , , ◦ at ( − , − , special solutions , so called be-cause the three vectors s , w , and w (cid:48) never getcoplanar and remain always almost orthogonal toeach other, which is not usual. In this case, theellipse E is fully contained inside the Berlingot B .The solutions of the second class are called gen-eral solutions . They are such that s , w , and w (cid:48) periodically lie in the same plane, which happens a V ≥ b V ≥ τ τ Fig. 2.— Two types of relative motion. Theshaded area represents the interior of the Berlingot( V ≥
0) and the circles are two different ellipticorbits. (a)
The orbit is fully inside the Berlingot.This configuration is called special solution . (b) The orbit intersects the surface of the Berlingotin τ and τ . This case is named general solution .The dashed section of the orbit (b) is forbiddenbecause V would be negative.when E crosses the surface of the Berlingot ∂ B .In the following, we limit our analysis to systemsbelonging to the second class only. For a detaileddescription of the special solutions, see BL06 andBL09. As stated before, the relative motion ofthe three vectors is integrable. In particular, thetrajectory of any general solution in the ( x, y, z )frame is a continuous piece of ellipse. More pre-cisely, trajectories are connected sections of E ∩ B with extremities at the surface ∂ B of the Berlin-got. These elliptic sections are swept back andforth indefinitely, each round trip defining a nu-tation oscillation, unless the ellipse E is tangentto the Berlingot ∂ B , in which case the point oftangency is a fixed point of the relative motion(BL06). The stability of each relative equilibriumis easily obtained as follows. If the ellipse is tan-gent to the Berlingot ‘from the outside’ (Fig. 3. a ),the trajectory is a singleton and thus the relativeequilibrium is stable. On the other hand, if theellipse is tangent to the Berlingot ‘from the inside’(Fig. 3. b ), the trajectory goes within the Berlin-got, and the relative equilibrium is unstable. Westress again that because the fixed points are lo-cated at the surface ( V ( x, y, z ) = 0) of the Berlin-got, in the equilibrium states the three vectors s , w , and w (cid:48) lie in the same plane, as in the Cassinistates.6 E s V ≥ b E u V ≥ Fig. 3.— Two types of Equilibrium. The shadedareas represent the interior of the Berlingot ( V ≥
0) and the ellipses are two different elliptic orbits.The dashed part of each trajectory is inaccessi-ble. Equilibrium states are located at the pointsof tangency between elliptic trajectories and thesurface of the Berlingot. (a)
When the tangencyis from the outside of the Berlingot, the fixed pointis stable ( E s ). (b) When the tangency is from theinside, the fixed point is unstable ( E u ). The precession motion is a solid rotation of thethree vectors s , w , w (cid:48) leaving their mutual anglesunchanged. Conversely, oscillations of inclinationsand spin-orbit angles constitute the nutation mo-tion. The amplitude of these angles are thus dic-tated by the relative motion taking place in thespace ( x, y, z ). More precisely, the extrema ofthe stellar obliquity θ x with respect to the meanplanet plane, of the stellar obliquity θ y with re-spect to the outer orbit, and of the mutual in-clination θ z between the planet mean plane andthe outer orbit correspond to the extrema of thedot products x, y, z reached during one nutationperiod. In the following, most of the derivationis done for θ x , but the method is equivalent for θ y and θ z . With the parametrization (9), theextrema of x are attained when dx/dτ = 0 orwhen the three vectors lie in the same plane, i.e., V ( τ ) = V ( x ( τ ) , y ( τ ) , z ( τ )) = 0. In the last case,although dx/dτ is not necessarily zero, the tra-jectory on the ellipse E makes a bounce becausethe point M = ( x, y, z ) reaches the surface ∂ B of the Berlingot. Using the parametrization (9),the Gram determinant V ( τ ) becomes a polyno-mial of degree 3 in cos τ and sin τ . The zeros of V ( τ ) can thus be deduced from the roots of apolynomial of degree 6 in cos τ . Given that for non-stationary general solutions, the volume V ( τ )(and the Gram determinant V ( τ )) only cancelsat two different values τ and τ , once the roots of V ( τ ) are known, one has to select τ and τ asthe closest ones surrounding the initial condition τ = atan ( z /A z , x /A x ), where x = x ( t = 0)and z = z ( t = 0). Once the roots ( τ k ) k =1 , areknown, the amplitudes of variation of θ x , θ y , and θ z are straightforward. The angle θ x varies be-tween θ x, = cos − x ( τ ) and θ x, = cos − x ( τ ),idem for θ y , and θ z . Nevertheless, the amplitudeof θ x , θ y , or θ z can be larger than what has justbeen calculated if there exists a value of τ between τ and τ such that dx/dτ = 0, dy/dτ = 0, or dz/dτ = 0, respectively. The derivative dx/dτ cancels at τ ± x = π ± π , (13)the quantity dz/dτ vanishes at τ ± z = 0 ± π , (14)and finally, dy/dτ = 0 for τ ± y = atan ( LA x , − G (cid:48) A z ) ± π . (15)All values θ x associated to τ , τ , and τ ± x areextrema. Thus, if more than two such valuesexist ( τ + x or τ − x falls between τ and τ ), onehas to sort them to get the maximal amplitudeof θ x . Idem for θ y and θ z . For example, inthe systems studied in the following sections, thecondition dy/dτ = 0 is met several times with θ y ( τ ) < θ y ( τ ) < θ y ( τ − y ). In these cases, thevariations of θ y are thus bounded by θ y ( τ ) and θ y ( τ − y ). The equations of motion (3) expressed in termsof the dot products x , y , z (6) are˙ x = ν V z ;˙ y = ν V x − ν V z ;˙ z = − ν V x , (16)where ( ν k ) k =1 ,..., are frequencies defined in (4). Itis then straightforward to check that the changeof time leading to the parametrization (9) satisfies dτ = √ ν ν V dt . (17)7he nutation period P nut corresponds to the os-cillation period of the dot products x ( t ), y ( t ), and z ( t ), which is twice the time needed to reach τ starting from τ , the two roots of V ( τ ) = 0(BL06). Using the definition of τ (17), the resultis P nut = 2 √ ν ν (cid:90) τ τ dτV ( τ ) , (18)where V ( x, y, z ) is given by (12) and the parame-trization x ( τ ), y ( τ ), and z ( τ ) written in (9). Inthe vicinity of the integral boundaries ( τ k ) k =1 , , V ( τ ) evolves like (cid:112) | τ − τ k | , unless the ellipse E is tangent to the Berlingot, a case we discard forthe moment. The integral (18) is thus convergent(BL06). Nevertheless, to avoid the singularitiesat the boundaries, one shall perform the followingchange of variable τ = τ − τ u + τ + τ P nut = τ − τ √ ν ν (cid:90) π/ − π/ cos u duV ( u ) . (20)If the trajectory of ( x, y, z ) is tangent to theBerlingot ‘from the inside’, the point of tangencyis an unstable fixed point. In that case, the nuta-tion period P nut becomes infinite as expected foran evolution to or from an equilibrium. On theother hand, if the trajectory is tangent ‘from theoutside’, the system is in the middle of a librationzone. In that case, the nutation period is derivedfrom a linearization of the equations of motion (3). As stated before, the precession period is moredifficult to derive. We have reached the limitof the geometrical approach. It is now neces-sary to return to a more conventional descriptionbased on action/angle variables that we denote( L z , G z , G (cid:48) z , ψ, Ω , Ω (cid:48) ). The three actions are thethird components of the angular momenta in agiven reference frame while ψ , Ω, and Ω (cid:48) are theprecession angle of the star and the longitudes ofthe ascending nodes of the planet system and ofthe companion, respectively. ( L z , ψ ) are Andoyerconjugate variables; ( G z , Ω) and ( G (cid:48) z , Ω (cid:48) ) are sub-sets of Delaunay variables. The system has thus a priori three degrees of freedom and the Hamilto-nian written in these variables reads H = − α ε sin I cos( ψ − Ω) + cos ε cos I ) − γ I sin I (cid:48) cos(Ω − Ω (cid:48) ) + cos I cos I (cid:48) ) , (21)with cos ε = L z /L , cos I = G z /G , and cos I (cid:48) = G (cid:48) z /G (cid:48) . We recall that ( L, G, G (cid:48) ) are constants ofmotion. After a usual reduction of the node (Ma-lige et al. 2002), the problem reduces to two de-grees of freedom. The associated symplectic trans-formation is ˜ ψ = ψ , ˜Ω = Ω − ψ , ˜Ω (cid:48) = Ω (cid:48) − ψ , ˜ L z = L z + G z + G (cid:48) z , ˜ G z = G z , ˜ G (cid:48) z = G (cid:48) z , (22)and the Hamiltonian expressed in the new vari-ables reads˜ H = − α (cid:16) sin ε sin I cos ˜Ω + cos ε cos I (cid:17) − γ (cid:16) sin I sin I (cid:48) cos( ˜Ω − ˜Ω (cid:48) ) + cos I cos I (cid:48) (cid:17) , (23)with cos ε = ( ˜ L z − ˜ G z − ˜ G (cid:48) z ) /L , cos I = ˜ G z /G , andcos I (cid:48) = ˜ G (cid:48) z /G (cid:48) . The variable ˜ ψ is cyclic, its con-jugated momentum ˜ L z is constant. ˜ L z representsthe third component of the total angular momen-tum of the system. The two remaining degrees offreedom are carried by ( ˜ G z , ˜Ω) and ( ˜ G (cid:48) z , ˜Ω (cid:48) ). Theproblem is further simplified when the referenceplane is chosen to be the invariant plane orthog-onal to the total angular momentum W . Indeed,in that case the first two components of W arezero and thus the two angle variables ˜Ω and ˜Ω (cid:48) are related by L sin ε + G sin I exp(i ˜Ω) + G (cid:48) sin I (cid:48) exp(i ˜Ω (cid:48) ) = 0 . (24)Hence, the relative motion described by the Hamil-tonian (23) has only one degree of freedom whenexpressed in the invariant plane. This agreeswith the fact that the relative motion is charac-terized by a single frequency called the nutationfrequency. But although the Hamiltonian (23)is integrable, the general solution does not seemeasy to derive in action/angle variables. Next, wethus rely on the solution obtained in the variables( x, y, z ). The expression of the precession period8 prec is then obtained by solving the evolution of˜ ψ = ψ . Indeed, consider a 3-vector system in agiven configuration at time t = 0. After a nutationperiod P nut , the system returns to its initial con-figuration (same mutual distances) but its overallorientation is changed by an angle ψ ( P nut ) − ψ (0)around the total angular momentum. Hence, P prec = 2 π ∆ ψ P nut , (25)with ∆ ψ = | ψ ( P nut ) − ψ (0) | is deduced from theequation of motion dψdt = d ˜ ψdt = ∂ ˜ H∂ ˜ L z = 1 L ∂ ˜ H∂ cos (cid:15) . (26)Substituting the expression of the Hamiltonian(23), and remembering that its first parenthesis(in factor of α/
2) is precisely x = s · w , we get dψdt = − αL x (cid:16) − cos (cid:15) sin (cid:15) sin I cos ˜Ω + cos I (cid:17) . (27)This expression is further simplified after express-ing cos ˜Ω in terms of x , (cid:15) , and I as dψdt = − αL x cos I − x cos (cid:15) − cos (cid:15) . (28)Moreover, cos (cid:15) and cos I are functions of ( x, y, z )given bycos (cid:15) = W · s W = L + Gx + G (cid:48) yW , cos I = W · w W = Lx + G + G (cid:48) zW , (29)where W = (cid:107) W (cid:107) is the norm of the total angularmomentum of the system and is a constant of themotion. Thus, dψdt = − αL W x G (1 − x ) + G (cid:48) ( z − xy ) W − ( L + Gx + G (cid:48) y ) . (30)Finally, using the change of time t (cid:55)→ τ (17) andthe fact that the oriented volume V ( τ ) changes itssign at a rebounce, we get∆ ψ = 1 √ ν ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) τ τ dψdt dτV ( τ ) − (cid:90) τ τ dψdt dτV ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) , = 2 √ ν ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) τ τ dψdt dτV ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) . (31) The expression of the precession period is thus P prec = π √ ν ν P nut (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) τ τ dψdt dτV ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) − , (32)with dψ/dt given by (30). N -body problem3.1. Numerical model The generic system we wish to study is com-posed of p packed planets orbiting an oblate starwith a companion on a wide eccentric inclined or-bit. The planet system is supposed to be relativelycoplanar with low eccentricities. Furthermore, weassume that mean-motion resonances are not dom-inating the secular motion. In that case, using theformalism developed in (BF14), the secular Hamil-tonian of the system can be approximated by H tot = (cid:88) ≤ j 1, with thenotation of (BF14), reads d e p dt = G m p − m p a p G p (cid:16) c w × e p + c w × e p − (cid:17) (43)where G p = m m p / ( m + m p ) (cid:112) G ( m + m p ) a p and c and c are two functions of ( a p − /a p ). Wethen set τ pp as the inverse of the coefficient of w × e p in (43). At the lowest order in ( a p − /a p ),the result is τ pp P p ≈ π m m p − (cid:18) a p a p − (cid:19) , (44)with P p the orbital period of the (outermost)planet p . On the other hand, the Lidov-Kozaitimescale (Lidov & Ziglin 1976), written in canon-ical astrocentric elements is τ Koz P p = 23 π m m (cid:48) (cid:18) b (cid:48) a p (cid:19) , (45)where b (cid:48) = a (cid:48) √ − e (cid:48) is the semiminor axis ofthe companion. The criterion of validity of theanalytical model is thus τ pp (cid:28) τ Koz , (46)where τ pp and τ Koz are given by (44) and (45),respectively. We have checked this criterion on11 ′ /a p (cid:16) m ′ m p − (cid:17) / (cid:16) a p − a p (cid:17) − / m ′ / m p − = m ′ / m p − = m ′ / m p − = large increase of eccentricity e p dynamically rigid a p − / a p b ′ / a p Fig. 4.— Determination of the validity of the an-alytical model. Top: results of the 2 400 simula-tions done with I (cid:48) = 89 ◦ . Dots, crosses, and opencircles represent solutions with max( e p ) greaterthan 0.8, lower than 0.01, and intermediate, re-spectively. The solid line is the threshold (47).Bottom: shaded areas are regions of high eccen-tricity increase as delimited by Eq. (47). The cri-terion is given for different m (cid:48) /m p − (mass of thecompanion / mass of the penultimate planet) interms of the two outermost planet semimajor axisratio ( a p − /a p ) versus the ratio b (cid:48) /a p (companionsemiminor axis / outer planet semimajor axis).4-body problems through 4 800 integrations of thenumerical model. The simulated system is com-posed of two planets with mass 10 − M (cid:12) orbit-ing a 1 M (cid:12) star. We chose three different massesfor the companion: m (cid:48) = 10 − , 10 − , 10 − M (cid:12) ,two eccentricities e (cid:48) = 0, 0 . 5, and two inclina-tions relative to the initial planet plane I (cid:48) = 50,89 ◦ . For each combination ( m (cid:48) , e (cid:48) , I (cid:48) ), the semi-major axis of the inner planet and the semiminoraxis of the companion take 20 values uniformlydistributed in the ranges 0 . ≤ a p − /a p ≤ . ≤ b (cid:48) /a p ≤ I (cid:48) = 89 ◦ , we founda sharp transition between low (max( e p ) < . e p ) > . 8) eccentricity regimes at τ Koz ∼ . τ pp leading to the criterion b (cid:48) a p (cid:38) . (cid:18) m (cid:48) m p − (cid:19) / (cid:18) a p − a p (cid:19) − / (47)represented in Fig. 4. For I (cid:48) = 50 ◦ , the ampli-tude of max( e p ) (not shown) is smaller than inthe I (cid:48) = 89 ◦ case. Nevertheless, a sharp transi-tion is still observed although at lower values of b (cid:48) (the limits obtained with the two inclinationsdiffer by 25%). The constraint given by Eq. (47)is thus more stringent. Finally, we would like tostress that the criterion (47) displayed in Fig. 4 hasonly been verified in limited configurations. Forinstance, we have only considered inner systemscomposed of two planets with equal mass. We ex-pect the criterion to remain informative in othercases, but close to its threshold we recommend in-tegrations of the numerical model to certify thevalidity of the analytical one. Moreover, the cri-terion has to be adapted if the system containsseveral well separated compact groups of planets.It is indeed necessary to check whether they allbehave coherently or not. 4. Numerical tests and applications In the previous paper BF14, we tested the nu-merical model against N-body integrations, find-ing them to hold well in for a system like the solarsystem, with low eccentricities and inclinations,and hierarchical system which is highly inclinedand undergoing Kozai-Lidov oscillations. In thissection, we assume the numerical model can wellrepresent the motion of planetary systems, and weuse it to test the analytic model in a few cases in-volving two observed exoplanet systems. The firstone, 55 Cancri, is composed of five planets orbitingone member of a wide binary system. This systemis actually the prototype for which the formalismof the previous section has been developed. Thesecond system, HD 20794, contains three super-Earths, but no wide perturber detected to date.We have thus added an arbitrary planetary com-panion to it. This latter system has been chosenbecause of the low angular momentum in the plan-12able 2: Masses and orbital elements of the planetsof the 55 Cnc system.Planet m a e (cid:36) I Ω( M J ) (au) ( ◦ ) ( ◦ ) ( ◦ )e 0.025 0.015 0.00 0 0.70 0b 0.788 0.113 0.01 147 1.00 0c 0.164 0.237 0.06 99 0.30 0f 0.143 0.771 0.13 180 0.00 0d 3.660 5.700 0.03 189 0.03 180 notes. Data are those of Dawson & Fabrycky (2010),except semimajor axes which have been computed fora central mass m = 0 . M (cid:12) instead of 0 . M (cid:12) ,and initial inclinations which are introduced to avoidcoplanar evolutions. ets orbit in comparison to the 55 Cancri system.The two systems allow to explore relatively differ-ent regions of the parameter space. The system 55 Cnc is our first example becauseit is actually a binary system and five planets or-bit the most massive component 55 Cnc A. More-over, the evolution of the inclination of the planetsystem has already been studied in Kaib et al.(2011) (hereafter noted K11) using another ap-proach. The masses and the orbital parameters ofthe five planets are summarized in Tab. 2. Plan-ets b and c are close to the 3:1 mean-motion res-onance (Fischer et al. 2008), but we assume thatthis proximity does not affect significantly the sec-ular motion derived from the numerical model. Asin K11, we introduce small initial inclinations toavoid purely coplanar evolutions of the planet sys-tem. The central star 55 Cnc A has a mass m = 0 . M (cid:12) , and a radius R = 0 . R (cid:12) (von Braun et al. 2011). Its rotation period deter-mined photometrically is P = 42 . k = 0 . 028 and C = 0 . m R , have been takenfrom stellar models (Landin et al. 2009, Tab. 1a).The stellar companion is at a projected separa-tion of 1065 au and has a mass m (cid:48) = 0 . M (cid:12) (Mugrauer et al. 2006). In all the following stud-ies, the initial obliquity of the central star rela-tive to the planet plane θ x ( t = 0) is set to zero.As a first test, we reproduce the evolution pre-sented in K11’s Fig. 1. The orbital parameters of time (Myr) i n c li n a t i o n ( d e g ) Fig. 5.— Evolution of the inclination of all theplanets of the 55 Cancri system vs. time obtainedwith the numerical model. Inclination is mea-sured with respect to the initial planetary orbitalplane. The inset plot resolves the evolution of eachplanet’s inclination.the companion are a (cid:48) = 1250 au, e (cid:48) = 0 . 93, and I (cid:48) = 115 deg. For this simulation, we use the nu-merical Hamiltonian expressed in vectorial form(BF14). Thus planet-planet interactions are mod-eled at the fourth order in eccentricity inclination,the interaction with the companion is modeled atthe third order in semi-major axis ratio. Relativis-tic precession is included, but the central star isconsidered as a point mass, we thus neglect the in-teractions with the stellar figure. The result of thesimulation is displayed in Fig. 5. The numericalevolution agrees very well with the n -body inte-gration performed in K11. All the planet orbitalinclinations follow the same track. The planetaryorbital plane tilts periodically like a rigid body.The main difference between our work and K11’sresult is a slight shift in the precession frequency.In K11’s figure, there are twelve full precession os-cillations within 1 Gyr, while in ours the twelfthone is not finished. Now, we focus our attentionon the evolution of the spin axis of the centralstar. Figure 5, showing the solid rotation of theplanetary system, justifies the analytical modelof the previous section: the planetary system re-main almost coplanar. The numerical values ofthe precession frequencies, Eq. (41), are ν ≈ ν ≈ . 49 deg/Myr, ν ≈ . ν ≈ . 030 deg/Myr. The equations of motion13re thus d s dt = − 323 cos θ x w × s ; d w dt = − . 49 cos θ x s × w − . θ z w (cid:48) × w ; d w (cid:48) dt = − . 030 cos θ z w × w (cid:48) (48)in deg/Myr. The precession motion of the spinaxis s of the central star relative to the pole w ofthe planet mid-plane is about eighty times fasterthan any of the other secular motions (when tak-ing into account cos θ z ≈ − . s can thus be computedassuming all the other vectors as constant. As aresult, the obliquity θ x of the star with respectto the orbital plane of the planets should remainconstant and equal to its initial value. If it be-gins aligned, it will remain aligned. A numericalintegration similar to that of Fig. 5, but includ-ing the evolution of the orientation of the centralstar (represented by the vector s ), confirms thisanalytical conclusion. The trajectory of the unitvectors s , w , and w (cid:48) are plotted in Fig. 6 a , c aswell as w to check whether the innermost planetof the system follows the evolution of the outerplanets. These trajectories are a combination ofa solid rotation (precession motion) around thetotal angular momentum and quasi-periodic nuta-tion motions (BL06; BL09). In Fig. 6 b , d - f , thetrajectories, displayed in the frame rotating withthe main precession frequency, show that the stel-lar spin axis s never moves away from w and w by more than about 2 ◦ . For comparison, the an-alytical model (section 2.3) predicts a spin-orbitamplitude of 1 . ◦ . The star is thus pulled alongwith the planetary orbits as their inclinations os-cillate. As a consequence, the sole presence of aninclined stellar companion in the 55 Cnc systemis not enough to generate a spin-orbit misalign-ment. The above conclusions have been derivedfor a specific set of physical and orbital parame-ters. Is there any other choices producing effectivemisalignment ? In this system, the star follows thetilt of the orbits of the planets because ν (cid:29) ν .To cancel this effect, one should decrease ν or in-crease ν . Except for the planet e observed in tran-sit, only the minimum masses m k sin i k , where theinclinations i k are measured with respect to theplane of the sky, are known. Nevertheless, the fre- quency ν depends linearly on m k while ν is notaffected by a rescaling of the planet masses. Thus,increasing the planet masses would only cause thestar to be more strongly coupled to the planets. Inthe expression of ν (41), the next less known pa-rameter is the Love number k of the star. Basedon the results of the internal structure model ofLandin et al. (2009), we estimate the error on k tobe of the order of 20%. But even if k is wrong bya factor two, it is not enough to compensate for thelarge ratio ν /ν ∼ 33. On the contrary, the pre-cession frequency ν is much more undetermined.Indeed, ν scales as m (cid:48) / [ a (cid:48) (1 − e (cid:48) ) / ]. The mass m (cid:48) = 0 . M (cid:12) has been derived by Mugrauer et al.(2006) from the absolute infrared magnitude usingthe evolutionary models of Baraffe et al. (1998),whereas the semimajor axis a (cid:48) = 1025 au andthe eccentricity e (cid:48) = 0 . 93 are only constrained bythe projected separation d = 1065 au. There isthus a full range of semimajor axes and eccen-tricities compatible with the observations. Thecloser and the more eccentric the perturber’s or-bit is, the larger is the frequency ν . Fixing thedistance of the pericenter at q = 35 au, and as-suming that the projected separation correspondsto the distance of the apocenter Q = 1065 au, weget a (cid:48) = 550 au, and e (cid:48) = 0 . ν is proportional to cos θ z , where θ z is the or-bital inclination of the stellar companion relativeto the planetary mid-plane, we chose a lower ini-tial inclination I (cid:48) ( t = 0) = θ z ( t = 0) = 30 ◦ only(we recall that I (cid:48) is measured with respect to thereference plane, which is also the planet plane at t = 0, while θ z is the mutual inclination betweenthe orbits of the companion and of the planets).With this set of parameters, | ν cos θ z | increasesfrom 4.2 deg/Myr to 114 deg/Myr. This quantityis still less that ν but it is of the same order ofmagnitude. The trajectories of the unit vectorscomputed with this new set of initial conditionsare plotted in Fig. 7. We indeed observe that thestellar axis and the planets orbital pole gets peri-odically misaligned from 0 ◦ to about 50 ◦ . In thiscase, the amplitude of 32 ◦ given by the analyti-cal model is underestimated by about 36%. Thisdiscrepancy is mainly due to the relatively weakcoupling between 55 Cnc d and the four inner-most planets. The semimajor axis ratio betweenthe two outermost planets is about 7.4. Thereis a better agreement on the nutation period de-14 b f × − e × − -2.803-2.805 × − d x -coordinate 10.50-0.5-10.10-0.1 s , w , w w ′ c y - c oo r d i n a t e Fig. 6.— Trajectory of the stellar spin axis s (blue), and of the unit orbital angular momentum of theinnermost planet w (red), of all the planets w (green), and of the stellar companion w (cid:48) (magenta) inthe 55 Cancri system. (a) Representation in a fixed reference frame on the unit sphere whose north polecoincides with the direction of the total angular momentum of the system. (b) Representation in a framerotating with the main precession frequency. (c) Projection of the trajectory on the x - y plane in the fixedreference frame. (d) Projection on the x - y plane in the rotating frame. (e) Zoom on the trajectory of w (cid:48) inthe x - y plane. (f ) Zoom on the trajectory of s , w , and w (large green point) in the x - y plane. The initialconditions are the same as in Fig. 5. 15 b ww sw ′ d x -coordinate y - c oo r d i n a t e c y - c oo r d i n a t e Fig. 7.— Same as Fig. 6 but with the stellar companion 55 Cnc B on a closer and less inclined orbit: a (cid:48) = 550au, e (cid:48) = 0 . I (cid:48) = 30 ◦ , initially. In panel d , w , w , and w follow w , while w ∼ w .rived from the numerical integration 2.3 Myr, andanalytically, 1.8 Myr. We have shown on thisexample (55 Cancri) that the central star can bestrongly linked to the motion of the orbital planeof its planets. This is mainly due to the fact thatmost of the angular momentum is in the orbit ofthe planets rather than in the rotation of the star.Nevertheless, if the interaction with the outer com-panion is strong enough, the evolution of the stel-lar spin axis can be decoupled from the motion ofits planet. A detailed analysis of the statisticaldistribution of the spin-orbit misalignment in the55 Cnc system is beyond of the scope of the presentpaper. Nevertheless, a global analysis of the max-imal spin-orbit misalignment of compact systems perturbed by a wide companion is performed insection 5. The system HD 20794 is composed of threesuper-Earths orbiting a G8V type star with mass m = 0 . M (cid:12) and radius R = 0 . R (cid:12) (Pepe et al.2011; Bernkopf et al. 2012). In comparison to 55Cancri, this system is not known to harbor anyouter companion. We nevertheless choose this sys-tem because the angular momentum is more bal-anced between the star and the planets than inthe 55 Cancri system, and this difference allowsnew kinds of spin-orbit angle evolutions. The ro-tation period of the star is P = 33 . k = 0 . C = 0 . m R . The parameters of the threesuper-Earths are summarized in Tab. 3. Becauseno stellar companion has been detected in this sys-tem until now, we arbitrarily add a perturber: agiant planet with mass m (cid:48) = 1 M J at a (cid:48) = 20au with an eccentricity e (cid:48) = 0 . ◦ with respect to the plane ofthe planets. This planet would have a 97 yr or-bital period and a radial velocity semi-amplitude K = 7 . 25 m s − . For an observation time span of 7years (Pepe et al. 2011), the amplitude of the driftinduced by the new planet would range between0.19 m s − and 3.3 m s − , depending on the or-bital phase. Although the upper limit is well abovethe 0.82 m s − rms of the residuals in Pepe et al.(2011), the lower limit is significantly below andsuch a planet could have been missed. In this sys-tem, the central star and the planet system havevery similar angular momenta. Thus, the frequen-cies ν and ν are almost equal. The equations ofmotion are d s dt = − . w × s ; d w dt = − . s × w − . w (cid:48) × w ; d w (cid:48) dt = − . w × w (cid:48) (49)in deg/Myr. From (49), we deduce that the outergiant planet possesses most of the angular mo-mentum of the system, its orbit remains close tothe invariant plane of the system. The equations(49) also tell us that the angular momentum ofTable 3: Masses and orbital elements of the planetsof the HD 20794 system.Planet m a e (cid:36) I Ω( M ⊕ ) (au) ( ◦ ) ( ◦ ) ( ◦ )b 3.28 0.1288 0.00 0 0.10 0c 2.91 0.2172 0.01 90 0.30 0d 5.83 0.3733 0.03 270 0.15 180 notes. Data are derived from Pepe et al. (2011),masses and semimajor axis have computed for acentral mass m = 0 . M (cid:12) instead of 0 . M (cid:12) . Theeccentricity and the inclination are introduced solelyto avoid circular and coplanar evolutions. sw , ww ′ HD 20794 x -coordinate y - c oo r d i n a t e Fig. 8.— Trajectory of the unit angular momentaof the system HD 20794 to which a giant planetwith mass m (cid:48) = 1 M J , semimajor axis a (cid:48) = 20 au,eccentricity e (cid:48) = 0 . 1, and inclination I (cid:48) = 30 ◦ hasbeen added. The colors are the same as in Fig. 6.The frame is rotating at the stellar precession fre-quency.the planet system w precesses around w (cid:48) about26 times faster than the star s around w (withcos θ z ≈ . (cid:104) w (cid:105) whichis collinear to w (cid:48) , but at a much slower frequencythan w . Figure 8 shows the trajectories of s , w , w , and w (cid:48) in the frame rotating at the precessionfrequency of the stellar axis s around the totalangular momentum. Since the precession motionof the planets pole is faster than that of the star,the planets mid-plane is still precessing around thetotal angular momentum of the system in the ro-tating frame. As a consequence, the spin-orbit an-gle oscillates periodically between 0 ◦ and 60 ◦ , thelatter being equal to twice the initial inclination I (cid:48) ( t = 0). This amplitude is in perfect agreementwith the analytical model, as well as the nutationperiod which is equal to 67.4 Myr. In a last ex-periment, we reduce the mass m (cid:48) of the perturbingcompanion by a factor 25, such that the frequen-cies ν becomes of the same order of magnitude as ν and ν . As a result, w is expected to describe anutation motion around a center located half-waybetween s and w (cid:48) . Moreover, the amplitude of the17 w , ww ′ HD 20794 x -coordinate y - c oo r d i n a t e Fig. 9.— Same as Fig. 8 but with a lighterperturbing planet m (cid:48) = 0 . M J comparable toUranus.nutation of s should be similar to that of w . Thisis confirmed by the Figure 9 showing the trajecto-ries of the unit angular momenta in the frame ro-tating at the precession frequency. Because of theinitial condition θ x ( t = 0) = 0 ◦ , the nutation loopsof w and s are tangent. In that case, the spin-orbitangle θ x between the stellar axis and the planetsorbit pole oscillate between 0 ◦ and 36 ◦ at a muchlonger nutation period of 1.07 Gyr. The analyti-cal model reproduces exactly these two quantities(amplitude and period). 5. Global analysis In the previous section, the application of thenumerical and the analytical models on a few sys-tems revealed different types of evolution. Here,we exploit more deeply the geometric structure ofthe analytical model to derive very general results.In particular, we abandon studies of individualmotions in favor of more global analysis of groupof trajectories. In all cases, we assume that theinitial spin-orbit angle θ x ( t = 0) is nil. In the analytical approximation, section 2, sys-tems composed of packed planets surrounded by aperturber on a wide orbit have fixed physical and orbital parameters except inclinations and obli-quities. The constant orbital parameters are thesemimajor axes of the planets and the semimi-nor axis of the perturber b (cid:48) = a (cid:48) √ − e (cid:48) . It isthus natural to keep masses, semimajor axes andeccentricities at given values, and study the ef-fect of initial inclinations on the behavior of thosesystems. Furthermore, considering the hypothe-sis where the initial stellar obliquity θ x ( t = 0) isnil, i.e., x = 1 and y = z ( D x in Fig. 1), evolu-tions are only characterized by the initial inclina-tion of the perturber I (cid:48) = I (cid:48) (0) = θ y (0) = θ z (0)with respect to the inner system. Thus, in the fol-lowing we consider the surface resulting from theunion of the trajectories in the ( x, y, z ) frame with0 ≤ I (cid:48) ≤ ◦ , or equivalently, − ≤ z ≤ 1. Moregenerally, we define S as the set of all ( x, y, z ) ∈ R such that there exists a z ∈ R verifying H ( x, y, z ) = H (1 , z , z ) K ( x, y, z ) = K (1 , z , z ) (50)where H (7) and K (8) are the two first integrals ofthe motion. The second equation of (50) providesthe expression of z as a function of ( x, y, z ) z = p ( x − 1) + qy + rz , (51)with p = LG ( L + G ) G (cid:48) ,q = LL + G ,r = GL + G . (52)The substitution into the first equation of (50)gives S = { ( x, y, z ) ∈ R | S ( x, y, z ) = 0 } , (53)with S = αγ ( x − 1) + z − (cid:0) p ( x − 1) + qy + rz (cid:1) . (54) S being a quadratic function of ( x, y, z ), S is aquadric surface. Moreover, the diagonal D x be-longs to S . Therefore, S is an hyperboloid ofone sheet (neither ellipsoid nor hyperboloid of twosheets contain a straight line). For some peculiarvalues of the parameters, S can also be a cylinder18r a cone, but this set of parameters is negligiblein the sense that its measure is zero. In the ap-proximation G (cid:48) (cid:29) ( L, G ) (or p (cid:28) ( q, r )), the casein which the outer body dominates the system’sangular momentum, S simplifies into S ≈ (cid:0) x y z (cid:1) [ S ] xyz − , (55)where[ S ] = − q γ/α − qrγ/α − qrγ/α (1 − r ) γ/α . (56)The matrix [ S ] has at least one positive eigenvalue(equal to 1), and its determinant − ( γq/α ) is neg-ative. There is thus exactly two positive and onenegative eigenvalues, and S is indeed an hyper-boloid of one sheet. Such an hyperboloid is plot-ted in Fig. 10 with the parameters of the system55 Cancri and where b (cid:48) = 190 au. The figure alsoshows the cube C = [ − , as well as a sectionof an elliptic trajectory in red whose initial con-ditions are x (0) = 1 and y (0) = z (0) = cos − ◦ meaning that the companion is initially inclinedby 45 ◦ with respect to the equator of the star andthe planet system mean plane. As expected, thediagonal D x representing the locus of all the initialconditions (see Fig. 1) belongs to the hyperboloid S (54), and is represented by the thick black line inFig. 10. Within the approximation (55), this hy-perboloid is centered on the origin, and given that r = 1 − q with 0 ≤ q ≤ 1, it can be shown that theaxes of symmetry are deduced from ( Ox, Oy, Oz )by a rotation of an angle 0 ≤ χ ≤ π/ x -axis. The limit χ = π/ q → 0, i.e., when planets have more angular mo-mentum than the star ( G (cid:29) L ), which is the casein Fig. 10, while the limit χ = 0 corresponds to q → L (cid:29) G ). We stress that the approxi-mate equation (55) defining S has been providedin order to get an idea of the shape and of theorientation of the hyperboloid. In Fig. 10 and inthe following, we always use the exact expression(54). Now that the quadric S is determined, we ex-amine its intersections with the surface of the Fig. 10.— Quadric S (red) defined as the unionof all elliptic trajectories starting at x (0) = 1 and y (0) = z (0) (thick black diagonal). The red curveis an example of elliptic trajectory with initial con-dition y (0) = z (0) = cos − ◦ . The cube C isrepresented in gray.Berlingot ∂ B . The maximal amplitudes of thespin-orbit angle θ x are then deduced from theseintersections. To make the approach more con-crete, we use again the two examples of section 4.The two surfaces S and ∂ B are plotted in figure11 with parameters of 55 Cancri for b (cid:48) =190, 182,180, and 170 au. As in Fig. 10, the initial con-ditions are the thick black diagonal D x . In thecase b (cid:48) = 190 au (panel a ), and for prograde or-bits like the red curve, elliptic trajectories evolvethrough the Berlingot, toward decreasing valuesof x and y (increasing spin-orbit angles betweenthe star and both the planet system and the com-panion). Trajectories exit the Berlingot after arelatively short distance and reenter the Berlingotat negative values of x and y (retrograde rotationof the star) before exiting again. Because the evo-lution of the system is restricted to the interior ofthe Berlingot, the maximal spin-orbit angles cor-respond to the first exit from the Berlingot. Thistopology is equivalent to the case studied by Kaibet al. (2011). As the distance of the companiondecreases (panel b , c , and d ), the hyperboloidshrinks so that the first exit from the Berlingotoccurs later and later. As a consequence the max-imal obliquity increases more and more. Further-19 bc d Fig. 11.— Modification of the topology of the spin-orbit evolution in the 55 Cancri system as the companion’sorbit shrinks. Parameters are those of Fig. 6 except the companion semiminor axis which takes the values b (cid:48) =190, 182, 180, and 170 in the panels a , b , c , and d , respectively. As b (cid:48) decreases from panel a to d , thequadric surface (violet) also shrinks. In panels c and d , a gap is open allowing some trajectories (blue) toreach low values of x – and thus high spin-orbit misalignment – while remaining inside the Berlingot (green).The elliptic orbit of Fig. 10 is still represented in red.20 b ′ = 190 au f initial inclination I ′ (deg) b ′ = 182 au e b ′ = 180 au d b ′ = 170 au c n u t a t i o n p e r i o d P n ( M y r ) b ′ =190 au b ′ =182 au b ′ =180 au b ′ =170 au b initial inclination I ′ (deg) a s p i n - o r b i t a n g l e θ x ( d e g ) Fig. 12.— a Intersections S ∩ B seen in the plane ( I (cid:48) , θ x ). b Range of accessible spin-orbit angles θ x withinitial condition θ x (0) = 0 ( x -axis). c-f Nutation periods computed from Eq. (20). Parameters are those ofthe 55 Cancri system (same as in Fig. 11), colors from dark to light correspond to b (cid:48) =190, 182, 180, and170 au, respectively. Points in the panels b and c are the results of numerical integrations with b (cid:48) = 170 au.more, from panel b to c , we see a modification ofthe topology of the intersection S ∩ ∂ B . If we imag-ine B as an ocean and S as a continent, panel c is astrait which gets larger as the companion semimi-nor axis decreases (panel d ). Once the strait isopen, trajectories passing through it reach the ret-rograde side of the Berlingot and produce highspin-orbit misalignment. The intersections be-tween the two surfaces for each distance of the per-turber have been computed with Maple. Then, foreach point ( x, y, z ) of the intersections, the initialcondition I (cid:48) = cos − z are computed from (51),and the spin-orbit angle from θ x = cos − x . Theresults are displayed in figure 12 for 0 ≤ θ z ≤ ◦ and 0 ≤ I (cid:48) ≤ ◦ . In Fig. 12. a , the dashed curvesrepresent the intersections S ∩ ∂ B associated to the different values of the semiminor axis b (cid:48) , andthe shaded areas show the regions S ∩ B where thehyperboloid is inside the Berlingot. The darkestshaded areas, associated to the largest distancesof the perturber, are disconnected. In these con-ditions, systems starting with zero spin-orbit an-gle must stay in the lowest regions and cannotreach misalignment larger than about 50 ◦ . Onthe contrary, when the strait is open, shaded ar-eas become connected and at the vertical of thestrait, systems reach high obliquities of the orderof 100 ◦ . Nevertheless, in the ( I (cid:48) , θ x ) plane, evolu-tions are vertical, thus at both sides of the straitsystems conserve relatively low obliquities. Fig-ure 12. b shows the maximal reachable spin-orbitmisalignments deduced from Fig. 12. a . Interest-21 b ′ = g initial inclination I ′ (deg) b ′ = f b ′ = e b ′ = d b ′ = c n u t a t i o n p e r i o d P n ( G y r ) b ′ =70.0 au b ′ =60.0 au b ′ =58.6 au b ′ =55.0 au b ′ =50.0 au b initial inclination I ′ (deg) a s p i n - o r b i t a n g l e θ x ( d e g ) Fig. 13.— Same as Fig. 12 with the parameters of HD 20794, m (cid:48) = 1 M J , and b (cid:48) =70, 60, 58.6, 55, and 50au from dark to light gray, respectively. A series of numerical simulations with b (cid:48) = 58 . b and e , showing that the analytic model performs excellently for this system. The dash-dotted linein panel b corresponds to θ x = 2 I (cid:48) .ingly, we see that when the companion’s semimi-nor axis is of the order of 170 au (lightest grayarea), even an inclination as small as 15 degreesis enough to generate a spin-orbit misalignmentof about 100 degrees. It is worth noting that,in the limit where most of the angular momen-tum is in the orbit of the companion, the hyper-boloid S and the Berlingot B are invariant by thetransformation ( x, y, z ) → ( x, − y, − z ) accordingto (55) and (12). The spin-orbit angle θ x is thusthe same whether the orbit of the outer companionis prograde or retrograde. Hence, Fig. 12 stays un-changed if the abscissa I (cid:48) is replaced by 180 ◦ − I (cid:48) .Once the intersections between the Berlingot andthe hyperboloid are known, the nutation periods P nut associated to the oscillation of θ x are ob- tained numerically from Eq. (20). These periodsare plotted in Fig. 12. c-f . When the perturber isclose ( b (cid:48) = 170 , 180 au, Fig. 12. c,d ), P nut showsspikes at the exact position of the borders of thestrait. As the perturber semiminor axis increases(Fig. 12. e,f ), the strait disappears as well as thespikes. The spikes are actually associated to tra-jectories along separatrices where the period is in-finite. Indeed, these trajectories are tangent to theintersection S ∩ ∂ B and thus, they are tangent tothe surface of the Berlingot ∂ B . Moreover, fromFig. 11. d , it is manifest that the tangency is ‘fromthe inside’, as in Fig. 3. b , so these trajectories ac-tually pass through an hyperbolic fixed point. Be-sides those unstable fixed points, two stable fixedpoints exist in all panels of Fig. 12. They are at22 (cid:48) = 0 and 90 ◦ . In Fig. 11, these initial conditionsare located at the extremity (1 , , , , D x , respectively.In contrast to the previous fixed points, these onesbelong to orbits which are tangent to the surfaceof the Berlingot ‘from the outside’, as in Fig. 3. a .They are thus stable, and indeed, the amplitudeof nutation at these two points (Fig. 12. b ) are nil.To check the analytical results, we performed nu-merical integrations with a (cid:48) = 170 au and e (cid:48) = 0starting at 1 ◦ ≤ I (cid:48) ≤ ◦ by step of 1 ◦ . Theresults are shown in Fig. 12. b,c . The strait isactually present in the numerical simulations, al-though its width derived analytically was under-estimated by about 10 degrees. The nutation pe-riod obtained numerically has the expected shapewith two spikes at the border of the strait exceptthat a small discrepancy exists inside the straitwhere the analytical method slightly overestimatesthe period. Nevertheless, the qualitative behavioris well recovered, and the quantitative differencesmust be due to the weak coupling between theinner planet and the outer ones as observed inFig. 7. The same exercise has been carried outon the system HD 20794, with m (cid:48) = 1 M J and b (cid:48) ∈ { , , . , , } au. Results are plottedin Fig. 13. The two systems present a few quali-tative similarities: 1) for large semiminor axis ofthe perturber, S ∩ B is disconnected, the innersystem is quite decoupled from the outer compan-ion and the stellar obliquity θ x remains low what-ever is the inclination of the perturber; 2) at smallsemiminor axis, a strait forms, spikes appear in thenutation period and some initial conditions leadto very high spin-orbit angle amplitudes. Never-theless, behaviors are quantitatively different. Inthe case of the system 55 Cnc, the strait opens atlow inclination I (cid:48) ∼ ◦ and high values of I (cid:48) donot produce any strong spin-orbit misalignments,while for HD 20794, the strait is created at largerinclination I (cid:48) ∼ ◦ and a high initial inclinationis required to produce high stellar obliquity. More-over, in contrary to 55 Cancri, the maximal obliq-uity θ x of HD 20794 never exceeds twice the initialinclination I (cid:48) (dash-dotted line in Fig.13. b ). Theorigin of these differences will be discussed in thefollowing. For this system, numerical simulationsdone at b (cid:48) = 58 . b,e ) are in perfectagreement with the analytical results because theplanets are well coupled. The relative evolution described by the analyti-cal model (section 2) depends on four parameters:the frequencies ν , ν , ν , ν (see Eq. (16)). If wediscard the temporal evolution, we can normalizeall these frequencies by ν , for example, and reducethe dimension of the parameter space to three.This dimension is still too large to be spanned en-tirely. Moreover, for each set of frequencies, wewish to analyze the curve representing the maxi-mal obliquity of the star as a function of the initialinclination which adds one more dimension. Tosolve this issue, we limit the study to asymptoticcases described as follows. Let ( ν a , ν b , ν c , ν d ) bea permutation of ( ν , ν , ν , ν ). We distinguish8 classes A, B, C, D, E, F, G, H of parametersdefined by class A : ν a (cid:28) ν b (cid:28) ν c (cid:28) ν d , class B : ( ν a ∼ ν b ) (cid:28) ν c (cid:28) ν d , class C : ν a (cid:28) ( ν b ∼ ν c ) (cid:28) ν d , class D : ν a (cid:28) ν b (cid:28) ( ν c ∼ ν d ) , class E : ( ν a ∼ ν b ∼ ν c ) (cid:28) ν d , class F : ( ν a ∼ ν b ) (cid:28) ( ν c ∼ ν d ) , class G : ν a (cid:28) ( ν b ∼ ν c ∼ ν d ) , class H : ν a ∼ ν b ∼ ν c ∼ ν d .The number of distinct permutations in class A is n (A) = 4! = 24. In class B, there are half as manydistinct permutations because switching ν a and ν b does not change the order of the frequencies.Thus, n (B) = 12. Similarly, n (C) = n (D) = 12.Using simple enumerative combinatorics, one gets n (E) = 4, n (F) = 6, n (G) = 4, and n (H) = 1.Adding these cases all together only leads to 75different configurations. We analyze them all. Foreach pair of frequencies satisfying ν a (cid:28) ν b , we take ν b = 10 ν a , and the cases ν a ∼ ν b are replaced byexact equalities ν a = ν b . As a result, we identifyfour different regimes. We call systems in the Cassini regime thosesatisfying ( ν , ν ) (cid:28) ( ν ∼ ν ). This hierarchyof frequencies implies that 1) the planet system23 k ν ∼ ν ∼ ν ∼ ν j initial inclination (deg) ν ≪ ( ν ∼ ν ∼ ν ) i ν ≪ ( ν ∼ ν ∼ ν ) h ( ν ∼ ν ) ≪ ( ν ∼ ν ) g ν ≪ ν ≪ ( ν ∼ ν ) f ν ≪ ( ν ∼ ν ∼ ν ) e s p i n - o r b i t a n g l e ( d e g ) ( ν , ν ) ≪ ( ν ∼ ν ) d ( ν , ν ) ≪ ( ν ∼ ν ) c ( ν , ν , ν ) ≪ ν b ( ν , ν ) ≪ ( ν ∼ ν ) a Fig. 14.— Nutation amplitude θ x, max as a function of the initial inclination I (cid:48) of the perturber in differentregimes. Similar behaviors are grouped together in same panels. a Cassini regime. b-c Pure orbital regime. d-e Laplace regime. f-j Hybrid regime. k All remaining configurations leading to low nutation amplitudes.The dashed line and the dotted line represent θ x, max = 2 I (cid:48) and θ x, max = I (cid:48) , respectively.is not affected by the orientation of the centralstar ( ν is small); 2) the plane of the outer or-bit is fixed ( ν is small); and 3) the precessionfrequency of the star matches that of the planetsystem ( ν ∼ ν ). These hypothesis are thoseleading to the well-known Cassini problem (e.g.,Colombo 1966; Peale 1969; Henrard & Murigande1987; Ward & Hamilton 2004). Among the 75different configurations, three of them fulfill thecondition ( ν , ν ) (cid:28) ( ν ∼ ν ), with ν (cid:28) ν , ν ∼ ν , or ν (cid:28) ν . They all present similarnutation amplitudes θ x, max as a function of theinclination I (cid:48) (Fig. 14. a ). Interestingly, the max-imal stellar obliquity relative to the planet plane exceeds twice the initial inclination I (cid:48) between theplanet plane and the outer orbit. As an exam-ple, the system 55 Cnc where the perturber is puton a rather close-in orbit b (cid:48) (cid:46) 170 au belongs tothe Cassini regime with ν (cid:28) ν (cid:28) ( ν ∼ ν ).The dynamics of this type of system is usuallyanalyzed in a frame rotating at the precession fre-quency (as in Figs. 6. d , 7. d , 8, and 9), by plot-ting the projected trajectories of the spin-axis onthe invariant plane for different initial obliquities.Numerical integrations of the 55 Cnc system with b (cid:48) = 170 au and I (cid:48) = 15 ◦ are displayed in Fig. 15.To make the comparison easier, the axes are thesame as in Ward & Hamilton (2004). Trajecto-24 x -coordinate y - c oo r d i n a t e Fig. 15.— Evolution of s obtained by numeri-cal integration of the 55 Cnc system with m (cid:48) =0 . M (cid:12) , b (cid:48) = 170 au, and I (cid:48) = 15 ◦ . Trajectoriesare plotted in the invariant plane in a frame rotat-ing with the precession period of the system. TheCassini states are labeled 1, 2, and 4. The point la-beled w represents the position of w which evolvesvery slightly in this frame.ries in Fig. 15 present small amplitude oscillationswhich are due to the individual motion of the orbitof each planet. Nevertheless, Cassini states 1, 2,and 4 are easily identifiable, and the large increaseof the obliquity observed in Figs. 11 and 14. a isdue to the wide oscillation of s around the Cassinistate 2 along the trajectory passing through thepoint labeled w in Fig. 15 (the initial condition θ x (0) = 0 implies s (0) = w (0)). When the frequencies associated to the spin-orbit coupling ν and ν are much smaller thanthose of the planet-companion interaction ν and ν , the evolution of the system becomes purelyorbital. The stellar orientation remains fixedand does not perturbed the motion of the orbitalplanes. According to the distribution of angularmomentum between the planets and the compan-ion, this regime leads to different spin-axis evolu-tions displayed in Fig. 14. b,c,k . If the companioncontains most of the angular momentum, its or-bital plane is almost invariant and the planet sys- tem precesses around it at constant inclination.In this case, ( ν , ν , ν ) (cid:28) ν and the spin-orbitangle oscillates between 0 and min( I (cid:48) , ◦ − I (cid:48) )(Fig. 14. b ). The period associated to this mo-tion is simply P nut = 2 π/ν . In this regime, wecan cite in particular Kepler-56, the only multi-planet system with an observed spin-orbit mis-alignment (Huber et al. 2013). Also, the five plan-ets of the 55 Cancri system, with the outer oneplaying the role of the perturber and the otherscoupled together, lies in this regime. In that case,the inclination of the innermost planet with re-spect to the plane of the sky is known by transitto be 82 . ± . ◦ (Gillon et al. 2012) while theoutermost is measured at 53 ± ◦ by astrometry(McArthur et al. 2004). According to these ob-servations, the orbits should be mutually inclinedby at least 20 ◦ which would give rise to a pe-riodic misalignment of (cid:38) ◦ . When the angu-lar momentum is equally distributed between theplanets and the companion, ν ∼ ν and both or-bital planes precess around the total orbital an-gular momentum G + G (cid:48) . As a consequence, ifthe initial obliquity θ x (0) is nil, the two angles θ x and θ y oscillate in antiphase between 0 and I (cid:48) (Fig.14. c ). The corresponding nutation period is P nut = π/ ( ν cos I (cid:48) cos( I (cid:48) / ν (cid:28) ν . As a result, the planetsplane does not move while the outer body pre-cesses around it at the frequency ν . Once thelatter motion is averaged out, the remaining inter-action in the system is between the stellar equa-torial bulge and the planets, but the angular mo-menta are aligned, so this configuration does notproduce any spin-orbit misalignment (Fig. 14. k ).Such would be the case if a hot Jupiter were per-turbed by an Earth analogue. More generally,within the pure orbital regime and considering allorbital angular momentum distributions, the am-plitude of the spin-orbit angle θ x varies from 0 to2 I (cid:48) as G (cid:48) /G increases from zero to infinity, themutual inclination θ z remains constant equal to I (cid:48) , and the associated nutation period is P nut = 2 πG (cid:48) Gγ cos θ z √ G (cid:48) + G + 2 G (cid:48) G cos θ z . (57)25 .3.3. Laplace regime We now consider the case where the frequency ν is one of the largest frequencies in the system,but ν remains much smaller. This occurs onlyif the star has more angular momentum than theplanets ( L (cid:29) G ). With these parameters, thestellar orientation does not evolve and the mo-tion of the planet plane is dictated by the incli-nation of the Laplace surface with respect to theequator of the star. The Laplace surface, com-monly used in satellite problems, is such that in-clinations measured relative to it remain roughlyconstant (BL06; Tremaine et al. 2009). For close-in planets with ν (cid:29) ν , the Laplace surface isaligned with the stellar equator and the spin-orbitangle remains constant (Fig. 14. k ). In systemswith longer period planets satisfying ν (cid:29) ( ν , ν ),the Laplace surface is that of the companion’s or-bit. The latter configuration falls into the abovementioned pure orbital regime characterized by θ x, max = I (cid:48) . In between the two extremes ( ν ∼ ν , Fig. 14. d,e ), the Laplace surface is situatedhalf way between the stellar equator and the outerorbit. In particular, when ( ν , ν ) (cid:28) ( ν ∼ ν )(Fig. 14. d ), θ x and θ z oscillate in antiphase be-tween 0 and min( I (cid:48) , ◦ − I (cid:48) ). The evolution ismore complicated when all the three frequencies ν , ν , and ν are similar (Fig. 14. e ). The hybrid regime gather systems in which thestellar precession frequency ν is larger than orof the same order as the dominant orbital one(s).Thus, both the equator of the star and the or-bits have a nutation motion. The Cassini regime( ν , ν ) (cid:28) ( ν ∼ ν ) fulfills this criterion, butgiven its importance for spin-orbit misalignment,it has been studied separately. Other configu-rations are ν (cid:28) ν (cid:28) ( ν ∼ ν ) (Fig. 14. f ),( ν ∼ ν ) (cid:28) ( ν ∼ ν ) (Fig. 14. g ), ν (cid:28) ( ν ∼ ν ∼ ν ) (Fig. 14. h ), ν (cid:28) ( ν ∼ ν ∼ ν )(Fig. 14. i ), and ν ∼ ν ∼ ν ∼ ν (Fig. 14. j ).These cases lead to intermediate spin-orbit angleamplitudes. All the others do not produce anysignificant misalignment (Fig. 14. k ). For exam-ple, using the parameters of Kaib et al. (2011),we obtained ν (cid:29) ( ν , ν , ν ) which is a particularcase of the hybrid regime that does not produceany increase of the stellar obliquity. 6. Conclusion The dynamics of compact multiplanet systemsperturbed by an outer companion reveals itselfvery rich. These systems can be represented bythree unit vectors s , w , and w (cid:48) along the angu-lar momentum of the star, of the planet system,and of the companion’s orbit, respectively. Theirdynamics is mainly the combination of a uniformrotation of the three vectors around the total an-gular momentum and of a periodic motion in therotating frame. These two evolutions are calledprecession and nutation in reference to the Earth-Moon problem perturbed by the Sun. The rela-tive motion described in the frame ( x, y, z ), with x = s · w , y = s · w (cid:48) , and z = w · w (cid:48) , is inte-grable whatever are the four precession constants ν , ν , ν , ν (see Eqs. (41)) involved in the prob-lem. The solutions are derived geometrically as inBL06 and BL09. In this work, we went even fur-ther in the geometrical approach. In particular,gathering all evolutions starting with a spin-orbitalignment and an arbitrary inclination of the com-panion, nutation amplitudes (and in particular theamplitude of the spin-orbit angle) are obtained alltogether without numerical simulations from theintersection of a Berlingot shaped surface definedin BL06 with an hyperboloid of one sheet equal tothe union of all trajectories in the ( x, y, z ) frame.The parameter space of the problem controlled bythe precession constants is three-dimensional. It isthus not possible to explore the whole dynamics.Nevertheless, using simple enumerative combina-torics, we identified 75 different asymptotic config-urations. We computed the nutation amplitudeson all these cases and distinguished four differentregimes:1. the Cassini regime where the stellar spin evo-lution is well described by the Cassini ap-proximation and the obliquity can exceedtwice the initial inclination of the perturber.This occurs when the star has much less an-gular momentum than the planets and thecompanion;2. the pure orbital regime where the stellar ori-entation remains fixed and does not perturbthe orbital evolution. In that case, due tothe motion of the planet system, the stellarnutation amplitude varies between zero and26wice the initial inclination of the perturberas the orbital angular momentum ratio ofthe latter over that of the planet system in-creases from zero to infinity;3. the Laplace regime where the stellar spin-axis is fixed and planets precess around aLaplace plane between the equator of thestar and the orbit of the companion. In thisregime, the amplitude of the spin-orbit an-gle varies between zero and the initial incli-nation of the perturber. The maximum isreached when the equatorial bulge and theouter body produce similar torques on theplanets;4. the hybrid regime where the precession rateof the stellar equator is larger than or similarto those of the orbit planes. Systems in theCassini regime are excluded since they havealready been studied. The hybrid regime ischaracterized by intermediate nutation am-plitudes ranging from zero to almost twicethe initial inclination of the outer body.This formalism applied to the 55 Cancri systemshowed that the central star is linked to the planetsunless the outer companion is on a very high eccen-tric orbit ( e (cid:48) (cid:38) . REFERENCES Albrecht, S., Winn, J. N., Marcy, G. W., et al.2013, ApJ, 771, 11Baraffe, I., Chabrier, G., Allard, F., & Hauschildt,P. H. 1998, A&A, 337, 403Batygin, K. 2012, Nature, 491, 418Batygin, K., & Adams, F. 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